On the Hamilton–Jacobi–Bellman approach to the relaxed optimal feedback control problems
Article, Journal of the Franklin Institute, 2025, DOI Link
View abstract ⏷
This paper deals with a novel approximation technique for constrained optimal feedback control problems. We consider a conventional optimal control processes governed by the closed-loop dynamics and apply the β-relaxation approach. We next establish some useful convergence properties of the resulting relaxations for the extended and for the originally given Feedback Optimal Control Problems (FOCPs). The approximative approach involving the β-relaxation reduces the nonlinear closed-loop dynamic system to a control-affine model. This system reduction makes it possible to apply the generic Hamilton–Jacobi–Bellman (HJB) solution methodology and determine an exact optimal feedback in the relaxed FOCP. The resulting explicit solution scheme is finally applied to the originally given constrained FOCP. The obtained theoretical results provide a rigorous mathematical basis for the possible numerical approaches to the optimal policy iterations.
A Formal Approach to Optimally Configure a Fully Connected Multilayer Hybrid Neural Network
Chakraborty G., Azhmyakov V., Guzman Trujillo L.A.
Article, Mathematics, 2025, DOI Link
View abstract ⏷
This paper is devoted to a novel formal analysis, optimizing the learning models for feedforward multilayer neural networks with hybrid structures. The proposed mathematical description replicates a specific switched-type optimal control problem (OCP). We have developed an equivalent, optimal control-based formulation of the given problem of training a hybrid feedforward multilayer neural network, to train the target mapping function constrained by the training samples. This novel formal approach makes it possible to apply some well-established optimal control techniques to design a versatile type of full connection neural networks. We next discuss the irrelevance of the necessity of Pontryagin-type optimality conditions for the construction of the obtained switched-type OCP. This fact motivated us to consider the so-called direct-solution approaches to the switched OCPs, which can be associated with the learning of hybrid neural networks. Concretely, we consider the generalized reduced-gradient algorithm in the framework of the auxiliary switched OCP.
Advanced Statistical Analysis of the Predicted Volatility Levels in Crypto Markets
Azhmyakov V., Shirokov I., Guzman Trujillo L.A.
Article, Journal of Risk and Financial Management, 2024, DOI Link
View abstract ⏷
Our paper deals with an advanced statistical tool for the volatility prediction problem in financial (crypto) markets. First, we consider the conventional GARCH-based volatility models. Next, we extend the corresponding GARCH-based forecasting and calculate a specific probability associated with the predicted volatility levels. As the probability evaluation is based on a stochastic model, we develop an advanced data-driven estimation of this probability. The novel statistical estimation we propose uses real market data. The obtained analytical results for the statistical probability of the levels are also discussed in the framework of the integrated volatility concept. The possible application of the established probability estimation approach to the volatility clustering problem is also mentioned. Our paper includes a concrete implementation of the proposed volatility prediction tool and considers a novel trading and volatility estimation module for crypto markets recently developed by the 1ex Trading Board group in collaboration with GoldenGate Venture. We also briefly discuss the possible application of a model combined with the data-driven volatility prediction methodology to financial risk management.
On the Optimal Control Approach to the Hybrid Deep Learning
Azhmyakov V., Trujillo L.A.G., Shirokov I.
Conference paper, 10th 2024 International Conference on Control, Decision and Information Technologies, CoDIT 2024, 2024, DOI Link
View abstract ⏷
This paper proposes a novel interpretation of the hybrid Deep Learning (DL) models in the form of an auxiliary semiclassical Optimal Control Problem (OCP) with a switched structure. The proposed equivalent reformulation of the problem of training a hybrid deep neural network allows to apply a widely developed solution methodology for OCPs to the trainable design of the network under consideration. In particular, we consider the reduced gradient method for a concrete numerical solution of the obtained OCP. We also establish the non-effectiveness of the generic necessary optimality conditions for a possible numerical treatment of the auxiliary OCP in the hybrid DL framework.
On a Data-Driven Optimization Approach to the PID-Based Algorithmic Trading
Azhmyakov V., Shirokov I., Dernov Y., Guzman Trujillo L.A.
Article, Journal of Risk and Financial Management, 2023, DOI Link
View abstract ⏷
This paper proposes an optimal trading algorithm based on a novel application of conventional control engineering (CE). We consider a fundamental CE concept, namely, the feedback control, and apply it to algorithmic trading (AT). The concrete feedback control strategy is designed in a form of the celebrated proportional–integral–derivative (PID) model. The highly fluctuating nature of the modern financial markets has led to the adoption of a model-free realization of the generic PID framework. The control theoretical methodology we propose is combined with the advanced statistics for the historical market data. We obtain a specific log-normal probability distribution function (pdf) associated with the specific quantities associated with the available stock data. The empirical log-normal pdf mentioned above enables the necessary PID gains optimization. For this aim, we apply the data-driven optimization approaches and consider the corresponding Monte Carlo solution procedure. The optimized PID trading algorithm we propose is also studied in the Fourier analysis framework. This equivalent frequency domain representation involves a new concept in financial engineering, namely, the “stock market energy” concept. For the evaluation, we implement the proposed PID optimal trading algorithm and develop a Python-based prototype software. We finally apply the corresponding prototype software to a data set from the Binance BTC/USDT (Bitcoin/Tether) stock market. The experimental result illustrates the implementability of the proposed optimal PID trading scheme and also shows the effectiveness of the proposed CE methods in the modern AT.
On the Approximate Optimal Feedback Control for Dynamical Systems Modeled by Ordinary Differential Equations
Azhmyakov V., Trujillo L.A.G., Salazar F.S., Shirokov I.
Conference paper, 9th 2023 International Conference on Control, Decision and Information Technologies, CoDIT 2023, 2023, DOI Link
View abstract ⏷
This paper is devoted to a novel approximation technique for optimal feedback control. We study a constrained Optimal Control Problem (OCP) in the context of a closed-loop dynamic system described by Ordinary Differential Equations (ODEs). The approximate optimal feedback control design we develop involves a specific β-relaxation technique (see [1]). We present here an initial research related to explicit approximations of the optimal control processes governed by the closed-loop dynamic systems. The obtained mathematical result has a potential to be extended to some concrete numerical solution algorithms for optimal feedback control problems.
Application of a Switched PIDD Type Control Strategy to the Model-Free Algorithmic Trading
Azhmyakov V., Shirokov I., Trujillo L.A.G.
Conference paper, IFAC-PapersOnLine, 2022, DOI Link
View abstract ⏷
Our paper is devoted to an applications of the feedback control methodology to the modern Algorithmic Trading (AT). The controller we propose performs a specific four term control design consisting of proportional, integral, derivative, and second order derivative terms (PIDD). The designed PIDD based trading strategy also incorporates a switched (dynamic) structure. We develop a specific switching mechanism that involves a backtesting driven profit maximization procedure on the historical data. We also study here the resulting model-free PIDD trading strategy in the frequency domain. Additionally, we give a rigorous formal estimation of the win rate for the trading algorithm under consideration. We finally apply the obtained AT strategy to a real-world stock market, namely, to the Binance Bitcoin/USD price dynamics.
Non-parametric identification of upper bound covariance matrices for min-sup Robust Kalman Filter: Application to the AR case
Castano N., Fernandez-Gutierrez J.P., Azhmyakov V., Graczyk P.
Conference paper, IFAC-PapersOnLine, 2022, DOI Link
View abstract ⏷
The min-sup type Robust Kalman Filter (RKF) introduced in Azhmyakov [2002] guarantees a robust estimate in uncertain linear dynamic systems under relatively weak assumptions related to the state and observation noises. In particular, it is supposed that the system and observation noises have some unknown probability distribution functions from some classes of centered distributions with bounded covariances with the known upper bound matrices. In our paper, we address the identification problem for upper-bound matrices in RKF, in the case of scalar observations. We use a novel Penalized Uncoverage (PU) function and an advanced optimization technique for this purpose. The novel PU-RKF methodology we develop in this paper is applied to robust state estimation in the stationary autoregressive model. We finally compare computationally our new PU-RKF algorithm with a classical approach involving a combination of the maximum-likelihood estimation and Kalman Filter (ML-KF) for Gaussian and some non-Gaussian noises.
Optimal control methodology for the Counter-Terrorism strategies: The relaxation based approach
Azhmyakov V., Verriest E.I., Bonilla M., Pickl S.
Article, Journal of the Franklin Institute, 2022, DOI Link
View abstract ⏷
This paper deals with a novel application of the advanced optimal control techniques to a Counter-Terrorism optimal decision-making problem. The recently developed model of the Counter-Terrorism security policies (see [1]) naturally incorporates a competing interest group dynamics. The optimal dynamics can be found in this case by considering a specific Optimal Control Problem. In our work we propose an essential formal improvement of the above applied Optimal Control Problem and its numerical treatment. We formulate a suitable phase-constrained Optimal Control Problem for the initial Counter-Terrorism model and study a specific relaxation of this model. The proposed β-relaxation approach makes it possible to eliminate some model inconsistences and to design an optimal Counter-Terrorism strategy. We develop a new numeric algorithm for the improved dynamic optimization problem. We next study some analytic properties of this algorithm. The resulting computational technique involves a gradient based method in combination with the proposed relaxation scheme (see [2]). The obtained numerical approach is finally applied to an illustrative example.
Stabilization of a class of switched dynamic systems: the Riccati-equation-based Approach
Bonilla M., Aguillon N.A., Ortiz Castillo M.A., Jacques Loiseau J., Malabre M., Azhmyakov V., Salazar S.
Article, IMA Journal of Mathematical Control and Information, 2022, DOI Link
View abstract ⏷
Our paper deals with the stabilization of a class of time-dependent linear autonomous complex systems with a switched structure. The initially given switched dynamic system is assumed to be controlled by a specific state feedback strategy associated with the linear quadratic regulator (LQR) type control. The proposed control design guarantees stabilization of the closed-loop system for all of the possible location transitions. In the solution procedure of the Algebraic Riccati Equation related to the LQR control strategy, only the knowledge of the algebraic structure related to the switched system are needed. We prove that the proposed optimal LQR type state feedback control design stabilizes the closed-loop switched system for every possible active location. The theoretical approach proposed in this paper is finally applied to a model of the Single Wing Quadrotor Aircraft, when changing from its Quadrotor Flight Envelope to its Airplane Flight Envelope.
Robust Stabilization for a Class of Descriptor Systems under Input Delays: The Attractive Invariant Ellipsoid Approach
Toro R.J.D., Azhmyakov V., Corrales Astorga M.A., Salcido C.B.
Conference paper, 8th International Conference on Engineering and Emerging Technologies, ICEET 2022, 2022, DOI Link
View abstract ⏷
The paper deals with the robust control design for descriptor (implicit) linear systems governed by a semi-explicit Differential Algebraic Equation (DAE) under two conditions: The unknown time-varying input delay and the unbounded exogenous perturbation presence. In order to provide numeric stabilizing conditions of the descriptor system solution, a linear feedback control and a Luenberguer-like observer are designed by the application of the conventional Attractive Invariant Ellipsoid Method (AIEM). The AIEM ensures both practical stability and robustness properties of the descriptor system solution. Practical stability is associated with the implicit system solution convergence into a specific geometrical region delimited by an Ellipsoid. Robustness properties are described in the sense of the exogenous perturbation rejection. The AIEM is an effective and well-known approach to ensure stability and robustness of dynamical systems governed by ordinary differential equations (ODE) but not for implicit systems governed by DAE. The main contribution on this work is the use of AIEM under unknown time-varying input delay presence. From the principles of AIEM for time-delay systems we propose an adequate linear feedback control and a Lyapunov-Krasovskii functional associated to a convergence zone in form of an ellipsoid. Minimization of the attractive ellipsoid is numerically described by the solution of a Bilinear Matrix Inequiality (BMI). Finally, an academic example supports the theoretical results.
Robust State Estimations in Controlled ARMA Processes with the Non-Gaussian Noises: Applications to the Delayed Dynamics
Azhmyakov V., Arango J.P., Bonilla M., del Toro R.J., Pickl S.
Conference paper, IFAC-PapersOnLine, 2021, DOI Link
View abstract ⏷
This paper deals with a novel theoretic approach to the robust state estimations in discrete-time dynamic systems with the non-Gaussian correlated stochastic noises. The methodology we develop is based on the so-called”worst case” robust Kalman Filter (KF) approach proposed in [2,3]. We are interested in the robust state estimation for the controlled ARMA models under assumption of the colored noises. Since an ARMA model involves the correlated noises in the equivalent Linear Model (LM) representation, the resulting dynamic system also includes the correlated stochastic variables. These two crucial properties of the ARMA models under consideration imply the impossibility of application of the classic KF-type state estimations. We use the modified”instrumental variable” method and derive an auxiliary LM with the uncorrelated noises. Application of the robust KF to this auxiliary LM makes it possible to derive a guaranteed state estimation in the initially given ARMA model. The proposed non-standard KF based state estimations are finally applied to the linear stochastic dynamic systems with the time delays.
A SEPARATION BASED OPTIMIZATION APPROACH TO DYNAMIC MAXIMAL COVERING LOCATION PROBLEMS WITH SWITCHED STRUCTURE
Azhmyakov V., Fernandez-Gutierrez J.P., Verriest E.I., Pickl S.W.
Article, Journal of Industrial and Management Optimization, 2021, DOI Link
View abstract ⏷
This paper extends a newly developed computational optimization approach to a specific class of Maximal Covering Location Problems (MCLPs) with a switched dynamic structure. Most of the results obtained for the conventional MCLP address the “static” case where an optimal decision is determined on a fixed time-period. In our contribution we consider a dynamic MCLP based optimal decision making and propose an effective computational method for the numerical treatment of the switched-type Dynamic Maximal Covering Location Problem (DMCLP). A generic geometrical structure of the constraints under consideration makes it possible to separate the originally given dynamic optimization problem and reduce it to a specific family of relative simple auxiliary problems. The generalized Separation Method (SM) for the DMCLP with a switched structure finally leads to a computational solution scheme. The resulting numerical algorithm also includes the classic Lagrange relaxation. We present a rigorous formal analysis of the DMCLP optimization methodology and also discuss computational aspects. The proposed SM based algorithm is finally applied to a practically oriented example, namely, to an optimal design of a (dynamic) mobile network configuration.
Robust tracking scheme for an experimental quadrotor
Blas L.A., Bonilla M., Salazar S., Malabre M., Azhmyakov V.
Conference paper, 2021 European Control Conference, ECC 2021, 2021, DOI Link
View abstract ⏷
In this paper, we present a robust tracking scheme for an experimental quadrotor, with outdoor real-time implementation. This control scheme is based on the robust structural feedback linearization scheme, presented in [2]. This control scheme has the advantage of combining the classical linear control techniques with the sophisticated robust control techniques. This control scheme is specially ad hoc for unmanned aerial vehicles, where it is important not only to reject the actual nonlinearities and the unexpected changes of the structure, but also to look for the simplicity and effectiveness of the control scheme.
On the Take-Off of a Single-Wing Quadrotor
Aguillon N.A., Bonilla M., Salazar S., Malabre M., Azhmyakov V.
Conference paper, 2021 European Control Conference, ECC 2021, 2021, DOI Link
View abstract ⏷
This paper focusses on differential thrust quad-copters, whose flight envelope can be divided in two, namely: the quadrotor flight envelope, and the airplane flight envelope. We define a transition maneuver as a controlled path followed by an aircraft in order to move between the two flight envelopes. We propose a control law that is robust with respect to wind disturbances and unmodeled aerodynamic effects. This approach ensures that the aircraft does not fall down during the transition maneuver.
A Consistent Numerical Approach to a Class of Optimal Control Processes Governed by Volterra Integro-Differential Equations
Azhmyakov V., Verriest E.I., Londono C., Del Toro R.J.
Conference paper, Proceedings of the IEEE Conference on Decision and Control, 2020, DOI Link
View abstract ⏷
This paper is devoted to the numerical analysis of the recently obtained theoretic results for a class of optimal control processes governed by Volterra integro-differential equations (see [5]). Using the specific structure of the delayed Volterra integro-differential equations under consideration, we reduce the initially given Optimal Control Problem (OCP) to a numerically tractable separate convex-concave program. This reduction involves optimization techniques in real Hilbert spaces and makes it possible to apply the first-order solution algorithms to the sophisticated Volterra type OCPs. We next consider the celebrated Armijo gradient method for the purpose of a concrete computation and establish the numerical consistency of the resulting algorithm. Finally, we consider an illustrative example.
An Implicit Class of Robust Control System Characterized by a Convex-Type Matrix Constraints
Del Toro R.J., Azhmyakov V., Hernandez M.M., Salas Perez F.G.
Conference paper, 2020 17th International Conference on Electrical Engineering, Computing Science and Automatic Control, CCE 2020, 2020, DOI Link
View abstract ⏷
Considering the use of the so-called Composite Quadratic Lyapunov Function (CQLF) based on descriptor techniques from the stability theory of continuous systems, this paper designs a linear feedback control to deal with bounded uncertainties in a class of nonlinear dynamical system described by semi-explicit Differential Algebraic Equations (DAE) by application of an extended version of the conventional Attractive Invariant Ellipsoid Method (AEM) the resulting closed-loop system converges to a minimal size set generated by the convex hull of some ellipsoids the use of CQLF in the modified AEM allows obtaining an appropriate feedback control to ensure robustness and stability properties of the system. An academic example is presented to illustrate the applicability of the contribution.
An implicit class of continuous dynamical system with data-sample outputs: A robust approach
Juarez R., Azhmyakov V., Espinoza A.T., Salas F.G.
Article, IMA Journal of Mathematical Control and Information, 2020, DOI Link
View abstract ⏷
This paper addresses the problem of robust control for a class of nonlinear dynamical systems in the continuous time domain. We deal with nonlinear models described by differential-algebraic equations (DAEs) in the presence of bounded uncertainties. The full model of the control system under consideration is completed by linear sampling-type outputs. The linear feedback control design proposed in this manuscript is created by application of an extended version of the conventional invariant ellipsoid method. Moreover, we also apply some specific Lyapunov-based descriptor techniques from the stability theory of continuous systems. The above combination of the modified invariant ellipsoid approach and descriptor method makes it possible to obtain the robustness of the designed control and to establish some well-known stability properties of dynamical systems under consideration. Finally, the applicability of the proposed method is illustrated by a computational example. A brief discussion on the main implementation issue is also included.
Robust structural feedback linearization based on the nonlinearities rejection
Bonilla M., Blas L.A., Azhmyakov V., Malabre M., Salazar S.
Article, Journal of the Franklin Institute, 2020, DOI Link
View abstract ⏷
In this paper, we consider a class of affine control systems and propose a new structural feedback linearization technique. This relatively simple approach involves a generic linear-type control scheme and follows the classic failure detection methodology. The robust linearization idea proposed in this contribution makes it possible an effective rejection of nonlinearities that belong to a specific class of functions. The nonlinearities under consideration are interpreted here as specific signals that affect the initially given systems dynamics. The implementability and efficiency of the proposed robust control methodology is illustrated via the attitude control of a Planar Vertical Take Off Landing (PVTOL) system.
Application of the nested convex programming to the optimal power flow in MT-HVDC grids
Conference paper, IFAC-PapersOnLine, 2020, DOI Link
View abstract ⏷
This paper deals with an application of the nested convex programming to the optimal power flow (OPF) in multi-terminal high-voltage direct-current grids (MT-HVDC). The real-world optimization problem under consideration is non-convex. This fact implies some possible inconsistencies of the conventional numerical minimization algorithms (such as interior point method). Moreover, the constructive numerical treatment of this problem is usually based on some approximative approaches, namely, on the suitable linearizations and problem relaxations. The resulting convex programming model constitutes an approximated model and can naturally involve the significant (approximation) errors. In difference to the strongly approximate computational approaches mentioned above, the numerical scheme we propose takes into account the specific bi-linear structure of the problem and operates with the originally given non-convex formulation of the problem. We implement the proposed nested optimization approach and study the numerical consistency of the resulting optimal design. The Python based numerical experiments demonstrate the imlementability of the proposed methodology. Optimization problem of the modified version of the CIGRE MT-HVDC is next used as a benchmark test for the approach we developed.
Advances of implicit description techniques in modelling and control of switched systems
Bonilla Estrada M., Malabre M., Azhmyakov V.
Book chapter, Lecture Notes in Control and Information Sciences, 2020, DOI Link
View abstract ⏷
Our contribution is devoted to a constructive overview of the implicit system approach in modern control of switched dynamic models. We study a class of non-stationary autonomous switched systems and formally establish the existence of solution. We next incorporate the implicit systems approach into our consideration. At the beginning of the contribution, we also develop a specific system example that is used for illustrations of various system aspects that we consider. Our research involves among others a deep examination of the reachability property in the framework of the implicit system framework that we propose. Based on this methodology, we finally propose a resulting robust control design for the switched systems under consideration and the proposed control strategy is implemented in the context of the illustrative example.
On the LQ based stabilization for a class of switched dynamic systems
Castillo M.A.O., Bonilla M., Loiseau J.J., Malabre M., Azhmyakov V.
Conference paper, IFAC-PapersOnLine, 2020, DOI Link
View abstract ⏷
This paper deals with the stabilization of a class of time-dependent linear autonomous systems with a switched structure. For this aim, the switched dynamic system is modeled by means of an implicit representation combined with a Linear-Quadratic (LQ) type control design. The proposed control design stabilizes the resulting system for all of the possible realizations of its locations. In order to solve the Algebraic Riccati Equation (ARE) associated with the LQ control strategy one only needs the knowledge of the algebraic structure related to the switched system. We finally prove that the proposed optimal LQ type state feedback stabilizes the closed-loop switched system no matter which location is active. The proposed theoretical approaches are illustrated by a numerical example.
On the Optimal Control of Volterra Integro-Differential Equations
Azhmyakov V., Egerstedt M., Verriest E.I.
Conference paper, Proceedings of the IEEE Conference on Decision and Control, 2019, DOI Link
View abstract ⏷
This contribution constitute an theoretic work devoted to the class of optimal control problems (OCPs) involving a specific dynamics described by Volterra integro-differential equations. We study OCPs associated with the Volterra integro-differential systems and establish the solvability property of this class of problems. A special structure of the abstract dynamic optimization problem under consideration makes it possible to interpret the initially given sophisticated OCP as a separate convex optimization problem in a suitable Hilbert space. This fact implies effective splitting type solution schemes in combination with the first-order computational methods for the numerical treatment of the initially given OCPs. We concretely consider the celebrated Armijo-type gradient method for this purpose. Finally, we give a mathematically rigorous proof of the numerical consistence of splitting algorithm applied to the main OCP.
Consistent approximations of the control – Affine systems with bounded uncertainties: Application to the controlled lotka – Volterra dynamics
Azhmyakov V., Londono C., Osorio P., Rojas A., Puerta M.E., Trujillo L.A.G.
Conference paper, 4th IEEE Colombian Conference on Automatic Control: Automatic Control as Key Support of Industrial Productivity, CCAC 2019 - Proceedings, 2019, DOI Link
View abstract ⏷
Our contribution is devoted to constructive approximations of a wide class of the modern control systems, namely, of the general control-affine dynamic models. We consider the generic tracking control problem associated with the control-affine systems in the presence of bounded uncertainties. The conventional linearization based solution approach to the tracking problem under consideration is finally refined and extended by a novel analytic errors estimation technique. Theoretic results developed in our contribution are applied to a practically motivated example of the controlled Lotka-Volterra system.
Synthesis of a robust linear structural feedback linearization scheme for an experimental quadrotor
Blas L.A., Bonilla M., Salazar S., Malabre M., Azhmyakov V.
Conference paper, 2019 18th European Control Conference, ECC 2019, 2019, DOI Link
View abstract ⏷
In this paper, we show in detail a synthesis procedure of the control scheme recently proposed in [3]. This control scheme has the advantage of combining the classical linear control techniques with the sophisticated robust control techniques. This control scheme is specially ad hoc for unmanned aircraft vehicles, where it is important not only to reject the actual nonlinearities and the unexpected changes of the structure, but also to look for the simplicity and effectiveness of the control scheme.
Advances in attractive ellipsoid method for robust control design
Article, International Journal of Robust and Nonlinear Control, 2019, DOI Link
View abstract ⏷
Our contribution is devoted to a further theoretic development of the attractive ellipsoid method (AEM). We consider dynamic models given by nonlinear ordinary differential equations in the presence of bounded disturbances. The resulting robustness analysis of the closed-loop system incorporates the celebrated Clarke invariancy concept (an analytic extension of the celebrated Lyapunov methodology). We finally obtain a new general geometric characterization of the AEM-based approach to the robust systems design. Moreover, we also discuss the corresponding numerical aspects of the proposed theoretical extensions of the method. The theoretic results obtained in this contribution are finally illustrated by a practically oriented computational example.
A Relaxation-Based Approach to Optimal Control of Hybrid and Switched Systems: A Practical Guide for Engineers
Book, A Relaxation-Based Approach to Optimal Control of Hybrid and Switched Systems: A Practical Guide for Engineers, 2019, DOI Link
View abstract ⏷
A Relaxation Based Approach to Optimal Control of Hybrid and Switched Systems proposes a unified approach to effective and numerically tractable relaxation schemes for optimal control problems of hybrid and switched systems. The book gives an overview of the existing (conventional and newly developed) relaxation techniques associated with the conventional systems described by ordinary differential equations. Next, it constructs a self-contained relaxation theory for optimal control processes governed by various types (sub-classes) of general hybrid and switched systems. It contains all mathematical tools necessary for an adequate understanding and using of the sophisticated relaxation techniques. In addition, readers will find many practically oriented optimal control problems related to the new class of dynamic systems. All in all, the book follows engineering and numerical concepts. However, it can also be considered as a mathematical compendium that contains the necessary formal results and important algorithms related to the modern relaxation theory.
On the Optimal Robust Time-Delay Robot Dynamics
Azhmyakov V., Trujillo L.A.G., Vargas M.G.F.
Conference paper, 2018 IEEE 2nd Colombian Conference on Robotics and Automation, CCRA 2018, 2018, DOI Link
View abstract ⏷
Our paper is devoted to a specific class of Optimal Control Problems (OCPs) in theoretical mechanics. We consider a minimax-Type optimal control processes governed by dynamic systems with randomly varying time delays. In particular we deals with the minimax-Type OCPs associated with a family of delayed Lagrange differential equations for the robot dynamics. The mathematical abstractions under consideration provide an adequate approach to many real-world robotic systems. Moreover, the proposed minimax dynamic optimization approach has a fundamental interpretation as a system robustness with respect to the unavoidable delays in robot control. The obtained convex structure of a linearized robot dynamics makes it possible to reduce the originally given delayed OCP to an auxiliary convex program in a suitable Euclidean space. The equivalent transformation we propose involves the wide range of effective algorithms for an effective computational treatment of the resulting convex OCP. We finally propose a concrete gradient based computational approach for the optimal control design of the controlled Lagrange-Type robot dynamics.
On the Optimal Control of Multidimensional Dynamic Systems Evolving with State Suprema
Azhmyakov V., Verriest E.I., Guzman Trujillo L.A., Pickl S.W.
Conference paper, Proceedings of the IEEE Conference on Decision and Control, 2018, DOI Link
View abstract ⏷
This paper constitutes a further generalization of the numerical solution approaches to Optimal Control Problems (OCPs) of systems evolving with state suprema. We study multidimensional control systems described by differential equations with the sup-operator in the right hand sides. A specific state-observer model and the linear type feedback control design under consideration imply a resulting closed-loop system that can formally be characterized as a multidimensional Functional Differential Equation (FDE) with delays. We study OCPs associated with the obtained FDEs and establish some fundamental solution properties of this class of problems. A particular structure of the resulting dynamic optimization problem makes it possible to consider the originally given sophisticated OCP in the framework of the nonlinear separate programming in some Euclidean spaces. This fact makes it possible to apply effective and relative simple splitting type computational algorithms to the initially given sophisticated OCPs for systems evolving with state suprema.
Robust Optimal Control of Linear-Type Dynamic Systems with Random Delays
Azhmyakov V., Verriest E.I., Guzman Trujillo L.A., Lahaye S., Delanoue N.
Conference paper, IFAC-PapersOnLine, 2018, DOI Link
View abstract ⏷
Our contribution deals with a class of Optimal Control Problems (OCPs) of dynamic systems with randomly varying time delays. We study the minimax-type OCPs associated with a family of delayed differential equations. The presented minimax dynamic optimization has a natural interpretation as a robustness (in optimization) with respect to the possible delays in control system under consideration. A specific structure of a delayed model makes it possible to reduce the originally given sophisticated OCP to an equivalent convex program in an Euclidean space. This analytic transformation implies a possibility to derive the necessary and sufficient optimality conditions for the original OCP. Moreover, it also allows consideration of the wide range of effective numerical procedures for the constructive treatment of the obtained convex-like OCP. The concrete computational methodology we follow in this paper involves a gradient projected algorithm. We give a rigorous formal analysis of the proposed solution approach and establish the necessary numerical consistence properties of the resulting robust optimization algorithm.
A first-order numerical approach to switched-mode systems optimization
Article, Nonlinear Analysis: Hybrid Systems, 2017, DOI Link
View abstract ⏷
This paper studies optimal control processes governed by switched-mode systems. We consider Optimal Control Problems (OCPs) with smooth cost functionals and apply a newly elaborated abstraction for the system dynamics under consideration. The control design we finally obtain includes an optimal switching times selection (”timing”) as well as an optimal modes sequence scheduling (”sequencing”). For purpose of numerical treatment of the initially given OCP we use a newly elaborated relaxation concept and analyse the resulting ”weakly relaxed” optimization problems. In contrast to the conventional relaxations our approach is based on the infimal prox convolution technique and does not use the celebrated Chattering Lemma. This fact causes a lower relaxation gap. Our aim is to propose a gradient-based computational algorithms for the OCPs with switched-mode dynamics. In particular, we deal with the celebrated Armijo-type gradient methods and establish the corresponding convergence properties. The numerical consistency (numerical stability) analysis makes it possible to apply a class of relative simple first-order numerical procedures to a sophisticated initial OCP involved in specific switched-mode dynamics.
Optimization of affine dynamic systems evolving with state suprema: New perspectives in maximum power point tracking control
Azhmyakov V., Vemest E.I., Trujillo L.A.G., Valenzuela P.A.
Conference paper, 2017 IEEE 3rd Colombian Conference on Automatic Control, CCAC 2017 - Conference Proceedings, 2017, DOI Link
View abstract ⏷
This paper studies optimization of dynamic systems described by affine Functional Differential Equations (FDEs) involving a sup-operator. We deal with a class of FDEs-featured Optimal Control Problems (OCPs) in the presence of some additional control constraints. Our aim is to derive the first-order optimality conditions and propose an effective solution algorithm. Moreover, we are interested in applications of the resulting optimal design techniques to the Maximum Power Point Tracking (MPPT) control of solar energy plants. We develop a conceptual computational approach to the specific class of OCPs under consideration and also point possible applications of this new methodology in MPPT control.
On the minimax robust Kalman Filter: A bounded estimation resources approach
Azhmyakov V., Castano N., Arango J.P., Graczyk P., Murillo F.H.S.
Conference paper, 2017 IEEE 3rd Colombian Conference on Automatic Control, CCAC 2017 - Conference Proceedings, 2017, DOI Link
View abstract ⏷
This paper is devoted to a generalization of the non-standard Kalman Filter (KF) introduced in [4]. We deal with some restrictions of the technical resources in the context of a state estimation problem and study a constrained convex program. Moreover, we replace two main concepts of the conventional KF, namely, the fundamental Normality Hypothesis (NH) and the unconstrained optimization approach. The minimax methodology we propose make it possible to develop an effective quasi-explicit solution method for the practically motivated generalization of the Kalman-type filter. We present a rigorous formal analysis of the obtained algorithm. The resulting non-linear filter possesses a strong optimality properties.
A separation method for maximal covering location problems with fuzzy parameters
Azhmyakov V., Fernandez-Gutierrez J.P., Pickl S.
Article, Italian Journal of Pure and Applied Mathematics, 2017,
View abstract ⏷
Our paper discusses a novel computational approach to the extended Maximal Covering Location Problem (MCLP). We consider a fuzzy-type formulation of the generic MCLP and develop the necessary theoretical and numerical aspects of the proposed Separation Method (SM). A speciffic structure of the originally given MCLP makes it possible to reduce it to two auxiliary Knapsack-type problems. The equivalent separation we propose reduces essentially the complexity of the resulting computational algorithms. This algorithm also incorporates a conventional relaxation technique and the scalarizing method applied to an auxiliary multiobjective optimization problem. The proposed solution methodology is next applied to Supply Chain optimization in the presence of incomplete information. We study two illustrative examples and give a rigorous analysis of the obtained results.
Linearization by means of Linear Implicit Rectangular Descriptions
Conference paper, IFAC-PapersOnLine, 2017, DOI Link
View abstract ⏷
This paper discusses a novel implementable approach to an exact linearization procedure based on the implicit systems techniques. The formal procedure we propose includes a specific “splitting” of the nonlinear state representation in two parts that involve a basic rectangular representation and an auxiliary nonlinear algebraic equation. The proposed linear implicit systems description makes it possible to apply the conventional linear control techniques to an initially given sophisticated nonlinear dynamic model.
Structural feedback linearization based on nonlinearities rejection
Blas L.A., Bonilla M., Malabre M., Azhmyakov V., Salazar S.
Conference paper, IFAC-PapersOnLine, 2017, DOI Link
View abstract ⏷
In this paper, a structural feedback linearization technique is proposed. This is a quite simple and effective linear control scheme based on failure detection techniques. Our proposed linear control approach is intended to reject the nonlinearities, which are treated as failure signals affecting the systems dynamics. The proposed control methodology is illustrated via the attitude control of a quadrotor in hover flying.
Advances in optimal control of differential systems with the state suprema
Conference paper, 2017 IEEE 56th Annual Conference on Decision and Control, CDC 2017, 2017, DOI Link
View abstract ⏷
This paper deals with a further development of analytic techniques for Optimal Control Problems (OCPs) involving differential systems with the state suprema. Differential equations evolving with state suprema (maxima) provide a useful modelling framework for various real-world applications, namely, in electrical engineering and in biology. The corresponding dynamic models lead to Functional Differential Equations (FDEs) in the presence of state-dependent delays. We study some particular (but important) cases of optimal control processes governed by systems with sup-operator in the right hand sides of the differential equations and obtain constructive characterizations of optimal solutions. The constrained OCPs we examine are formulated assuming the (linear) feedback-type control law. The case study presented in this article constitutes a formal extension of the concept of positive dynamic systems to differential systems with the state suprema.
A modified pseudospectral method for solving trajectory optimization problems with singular arc
Foroozandeh Z., Shamsi M., Azhmyakov V., Shafiee M.
Article, Mathematical Methods in the Applied Sciences, 2017, DOI Link
View abstract ⏷
This paper presents a direct method based on Legendre–Radau pseudospectral method for efficient and accurate solution of a class of singular optimal control problems. In this scheme, based on a priori knowledge of control, the problem is transformed to a multidomain formulation, in which the switching points appear as unknown parameters. Then, by utilizing Legendre-Radau pseudospectral method, a nonlinear programming problem is derived which can be solved by the well-developed parameter optimization algorithms. The main advantages of the present method are its superior accuracy and ability to capture the switching times. Accuracy and performance of the proposed method are examined by means of some numerical experiments. Copyright © 2016 John Wiley & Sons, Ltd.
On the geometry of the attractive ellipsoids method: Applications to the robust control design of switched systems
Book chapter, Robust Control: Systems, Theory and Analysis, 2017,
View abstract ⏷
This contribution deals with a robust control design for general switched affine control systems. Dynamical models under consideration are described by ordinary differential equations involving a switching mechanism and in the presence of bounded uncertainties. The design procedure we analyse is based on the newly elaborated attractive ellipsoids method ([32]). The stability and robustness of the resulting closed-loop systeminvolves an abstract Clarke stability theoremand a theoretic extension of the celebrated Lyapunov-typemethodology. A short discussion on the obtained analytic results and possible applications and extensions is also included.
The singular optimal control of switched systems
Book chapter, Advances in Communications and Media Research, 2017,
View abstract ⏷
This chapter studies a singular case of Optimal Control Problems(OCPs) governed by a class of switched control systems. We proposea new mathematical formalism for this type of switched dynamic systemsand study OCPs with a quadratic cost functionals. The original sophisticatedoptimization problem is next replaced by an auxiliary "weaklyrelaxed" OCP. Our main result includes a formal proof of the local convexityproperty of the obtained auxiliary OCP. The convex structure ofthe OCP implies a possibility to apply a variety of powerful and relativelysimple optimization schemes to the sophisticated singular OCP involvingswitched dynamics. The conceptual numerical approach we finallydevelop includes an optimal switching times selection ("timing") and asimultaneous optimal switched modes sequence scheduling ("sequencing").
On the linear quadratic dynamic optimization problems with fixed-levels control functions
Article, Italian Journal of Pure and Applied Mathematics, 2017,
View abstract ⏷
This paper deals with a constrained LQ-type optimal control problem (OCP) in the presence of fixed levels input restrictions. We consider control processes governed by linear differential equations with a priori known control switching structure. The set of admissible inputs reflects some important natural engineering applications and moreover, can also be interpreted as a result of a quantization procedure applied to the original dynamic system. We propose a novel implementable algorithm that makes it possible to calculate a (numerically consistent) approximative solution to the constrained LQ-type OCPs under consideration. Our contribution mainly discusses theoretic aspects of the proposed solution scheme and contains an illustrative numerical example.
On the optimal control of systems evolving with state suprema
Azhmyakov V., Ahmed A., Verriest E.I.
Conference paper, 2016 IEEE 55th Conference on Decision and Control, CDC 2016, 2016, DOI Link
View abstract ⏷
This paper studies optimal control processes governed by a specific family of systems described by functional differential equations (FDEs) involving the sup-operator. Systems evolving with the state suprema constitute a useful abstraction for various models of technological and biological processes. The specific theoretic framework incorporates state suprema in the right hand side of the initially given differential equation and finally leads to a FDE with the state-dependent delays. We study a class of nonlinear FDE-featured optimal control problems (OCPs) in the presence of some additional control constraints. Our aim is to develop implementable first-order optimality conditions for the retarded OCPs under consideration. We use the celebrated Lagrange approach and prove a variant of the Pontryagin-like Minimum Principle for the given OCPs. Moreover, we discuss a computational approach to the main dynamic optimization problems and also consider a possible application of the developed methodology to the Maximum Power Point Tracking (MPPT) control of solar energy plants.
Attractive ellipsoids based robust control design of switched systems: A geometrical approach
Azhmyakov V., Juarez R., Gadi S.K., Guzman Trujillo L.A.
Conference paper, 2016 13th International Conference on Electrical Engineering,Computing Science and Automatic Control, CCE 2016, 2016, DOI Link
View abstract ⏷
This contribution deals with a robust control design for general switched affine control systems. Dynamical models under consideration are described by ordinary differential equations involving a switching mechanism and in the presence of bounded uncertainties. The design procedure we analyse is based on the newly elaborated attractive ellipsoids method ([33]). The stability and robustness of the resulting closed-loop system involves an abstract Clarke stability theorem. A short discussion on the obtained analytic results and possible applications and extensions is also included.
Robust control for a class of continuous dynamical system governed by semi-explicit DAE with data-sample outputs
Juarez R., Azhmyakov V., Gadi S.K., Salas F.G.
Conference paper, 2016 13th International Conference on Electrical Engineering,Computing Science and Automatic Control, CCE 2016, 2016, DOI Link
View abstract ⏷
This paper addresses the problem of robust control for a class of nonlinear dynamical systems in the continuous time domain. We deal with nonlinear models described by differential algebraic equations (DAEs) in the presence of bounded uncertainties. The full model of the control system under consideration is completed by linear sampling-Type outputs. Main contribution of present manuscript is the proposal of a linear state feedback control design for this type of uncertain nonlinear system obtaining the required stability and robustness. Due to the need to the state feedback there is a need of the implementation of a Luenberguer observer. Finally, the applicability of the proposed method is illustrated by a computational example. A brief discussion on the main implementation issue is also included.
Optimal fixed-levels control for nonlinear systems with quadratic cost-functionals
Azhmyakov V., Martinez J.C., Poznyak A.
Article, Optimal Control Applications and Methods, 2016, DOI Link
View abstract ⏷
This paper is devoted to general optimal control problems (OCPs) associated with a family of nonlinear continuous-time switched systems in the presence of some specific control constraints. The stepwise (fixed-level type) control restrictions we consider constitute a common class of admissible controls in many real-world engineering systems. Moreover, these control restrictions can also be interpreted as a result of a quantization procedure appglied to the inputs of a conventional dynamic system. We study control systems with a priori given time-driven switching mechanism in the presence of a quadratic cost functional. Our aim is to develop a practically implementable control algorithm that makes it possible to calculate approximating solutions for the class of OCPs under consideration. The paper presents a newly elaborated linear quadratic-type optimal control scheme and also contains illustrative numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.
Constructive approximations of the Zeno dynamics in affine switched systems: The projection based approach
Azhmyakov V., Bonilla M., Pickl S., Trujillo L.A.G.
Conference paper, Proceedings of the American Control Conference, 2016, DOI Link
View abstract ⏷
This paper deals with a new theoretic approach to a specific interaction of continuous and discrete dynamics in switched control systems known as a Zeno dynamics. We study executions of switched control systems with affine structure that admit infinitely many discrete transitions on a finite time interval. Although the real-world processes do not present the corresponding behavior, mathematical models of some interconnected engineering systems may be Zeno due to the corresponding formal abstraction. We propose a useful approximative approach to Zeno dynamics, namely, a projection based characterization of this phenomena. A resulting trajectory associated with the Zeno dynamics can finally be described as a result of a specific dynamic projection procedure applied to the original model. We use here the projected dynamic systems methodology. The obtained formal description provides an effective theoretic basis for a constructive treatment of the Zeno behaviour. We also discuss an application of the proposed technique to the conventional sliding mode control processes.
Consistent approximations of the zeno behaviour in affine-type switched dynamic systems
Article, Abstract and Applied Analysis, 2016, DOI Link
View abstract ⏷
This paper proposes a new theoretic approach to a specific interaction of continuous and discrete dynamics in switched control systems known as a Zeno behaviour. We study executions of switched control systems with affine structure that admit infinitely many discrete transitions on a finite time interval. Although the real world processes do not present the corresponding behaviour, mathematical models of many engineering systems may be Zeno due to the used formal abstraction. We propose two useful approximative approaches to the Zeno dynamics, namely, an analytic technique and a variational description of this phenomenon. A generic trajectory associated with the Zeno dynamics can finally be characterized as a result of a specific projection or/and an optimization procedure applied to the original dynamic model. The obtained analytic and variational techniques provide an effective methodology for constructive approximations of the general Zeno-type behaviour.We also discuss shortly some possible applications of the proposed approximation schemes.
Application of the attractive ellipsoid methodology to robust control design of a class of switched systems
Article, Lecture Notes in Mechanical Engineering, 2016, DOI Link
View abstract ⏷
Our contribution is devoted to an application of the newly elaborated robust feedback-type control methodology to a class of industrial robotic systems. We consider a formal prototype of an automated Continuous Stirred Tank Reactor (CSTR) in the presence of bounded (operating) uncertainties and external disturbances. The nonlinear model of the CSTR has a switched nature and implies a sophisticated dynamical behaviour. Moreover, the resulting control design is supposed to be the defined only by the given system output. The robustness property of the closed-loop automated system is determined here in the sense of a “practical stability” concept and is based on the Attractive Ellipsoid (AE) approach. The implementable control design scheme we propose involves the Bilinear Matrix Inequalities (BMIs) techniques in combination with the Multiple Lyapunov functions analysis.
A Novel Numerical Approach to the MCLP Based Resilent Supply Chain Optimization
Azhmyakov V., Fernandez-Gutierrez J.P., Gadi S.K., Pickl S.
Conference paper, IFAC-PapersOnLine, 2016, DOI Link
View abstract ⏷
This paper deals with the Maximal Covering Location Problem (MCLP) for Supply Chain optimization in the presence of incomplete information. A specific linear-integer structure of a generic mathematical model for Resilient Supply Chain Management System (RSCMS) makes it possible to reduce the originally given MCLP to two auxiliary optimization Knapsack-type problems. The equivalent transformation (separation) we propose provides a useful tool for an effective numerical treatment of the original MCLP and reduces the complexity of algorithms. The computational methodology we follow involves a specific Lagrange relaxation procedure. We give a rigorous formal analysis of the resulting algorithm and apply it to a practically oriented example of an optimal RSCMS design.
Practical stabilization of a class of switched systems: Dwell-time approach
Article, IMA Journal of Mathematical Control and Information, 2015, DOI Link
View abstract ⏷
This paper addresses a problem of robust stabilization for a class of switched systems in the presence of bounded perturbations. We consider non-linear dynamic models under arbitrary switching mechanisms. The practical stabilization method we propose is carried out by a linear-type feedback switching control strategy subject to an average dwell-time scheme. We apply the newly elaborated (extended) version of the conventional invariant ellipsoid method for this purpose. The numerically implementable sufficient conditions for the practical stability of systems are derived using bilinear matrix inequalities. The obtained results are illustrated by the example of a continuous stirred tank reactor.
Decoupling of internal variable structure for a class of switched systems
Conference paper, 2015 European Control Conference, ECC 2015, 2015, DOI Link
View abstract ⏷
We consider particular linear switched systems (LSS) with autonomous transitions, that we model by rectangular implicit representations. We design proper approximations for proportional and derivative descriptor variable feedback laws that were previously proposed for making unobservable (decoupling) the effect of the variable structure, and guarantee BIBO-stability for the switched controlled system. An example is used for illustration.
An implicit systems characterization of a class of impulsive linear switched control processes. Part 2: Control
Article, Nonlinear Analysis: Hybrid Systems, 2015, DOI Link
View abstract ⏷
Our paper focuses on a fundamental structural property associated with a family of linear switched control systems in the presence of impulsive dynamics. We consider dynamic processes governed by piecewise linear ODEs with controlled location transitions. Using the newly developed implicit systems characterization technique, we rewrite the initially given impulsive switched system as an implicit dynamic model. This auxiliary representation of the original control system makes it possible to hide the switched phenomenon and to apply the conventional approach to the resulting implicit dynamic model. The proposed algebraic-based modeling framework follows the celebrated behavioral approach (see Polderman and Willems, 1998) and makes it possible to apply to the switched dynamics some classic techniques from the well-established time-invariant implicit systems theory. The analytic results of our paper constitute a formal theoretical extension of switched control systems methodology and can be used (as an auxiliary step) in a concrete control design procedure. The second part of our manuscript is devoted to control aspects.
On the local convexity of singular optimal control problems associated with the switched-mode dynamic systems
Conference paper, IFAC-PapersOnLine, 2015, DOI Link
View abstract ⏷
This paper studies a class of singular Optimal Control Problems (OCPs) governed by general switched-mode control processes. We develop a new constructive formalism for this class of switched dynamic systems and study OCPs with quadratic cost functionals. The initially given optimization problem is replaced by an auxiliary "weakly relaxed" OCP. The main aim of our contribution is with the formal proof of the local convexity property of the obtained auxiliary OCP. This fundamental convexity property makes it possible to apply powerful and relatively simple numerical optimization schemes to the initially given sophisticated singular OCP. The conceptual optimal control design framework we propose includes an optimal switching times selection ("optimal timing") as well as optimal modes scheduling ("optimal sequencing").
Optimization of a class of nonlinear switched systems with fixed-levels control inputs
Azhmyakov V., Martinez J.C., Poznyak A., Serrezuela R.R.
Conference paper, Proceedings of the American Control Conference, 2015, DOI Link
View abstract ⏷
This paper is devoted to a consistent numerical method for Optimal Control Problems (OCPs) associated with a specific class of nonlinear systems. We study dynamic models equipped with the piecewise constant (fixed-level) control inputs. The presented formalism also includes some switched dynamic models. We develop an implementable control algorithm for the above-mentioned classes of switched-type OCPs, study convexity characteristics and discuss some key properties of the proposed numeric scheme. We also consider a computational example. The computational scheme we propose is based on the Armijo-type projected gradient methodology.
A robust pseudo-spectral method for a class of optimal control problems
Azhmyakov V., Shamsi M., Trujillo L.A.G., Rojas J.H.C.
Conference paper, IFAC-PapersOnLine, 2015, DOI Link
View abstract ⏷
Our paper presents a numerically robust control design algorithm for a class of Optimal Control Problem (OCPs). The computational approach we develop constitutes an extension of the celebrated Gauss pseudo-spectral method. The last one makes it possible to reduce the Pontryagin Maximum Principle related Hamiltonian boundary value problem to an auxiliary algebraic system. The implementable scheme we propose is numerically consistent (by implementation) and implies an effective robust control procedure for a relative small discretization grids. The pseudo-spectral methodology is used in combination with a classic differential continuation approach. The presented solution procedure can be extremely useful when a generic shooting-type solution method fails (the case of a high sensitivity or stiffness). We finally consider an illustrative example and discuss the obtained results.
An implicit systems characterization of a class of impulsive linear switched control processes. Part 1: Modeling
Article, Nonlinear Analysis: Hybrid Systems, 2015, DOI Link
View abstract ⏷
Our paper focuses on a fundamental structural property associated with a family of linear switched control systems in the presence of impulsive dynamics. We consider dynamic processes governed by piecewise linear ODEs with controlled location transitions and describe the resulting system in a constructive form of an implicit dynamic system. The proposed algebraic-based modeling framework follows the celebrated behavioral approach (see Polderman and Willems (1998)) and makes it possible to apply to the switched dynamics some conventional techniques from the well-established time-invariant implicit systems theory. The analytic results of our paper constitute a formal theoretical extension of switched control systems methodology and can be used (as an auxiliary step) in a concrete control design procedure. In this first part, we consider modeling aspects.
On the Projected Gradient Methods for Switched – Mode Systems Optimization
Conference paper, IFAC-PapersOnLine, 2015, DOI Link
View abstract ⏷
This paper is devoted to optimal control problems (OCPs) associated with a class of switched - mode systems. We propose a new formalism for the autonomous variant of these dynamic systems and study an OCP with a smooth cost functional. The control design we finally propose includes an optimal switching times selection ("timing") as well as an optimal modes sequence scheduling ("sequencing"). The initial nonlinear optimization problem is replaced by a suitable relaxed OCP with some specific control restrictions. We consider a fully convexified OCP as well as a "weakly" relaxed optimization problems. Our aim is to analyse a family of numerically tractable control algorithms, namely, the projected gradient methods involved Armijo line search and to establish the corresponding convergence properties. The abstract consistency analysis we implement makes it possible to apply relative simple first - order numerical procedures to a sophisticated initial OCP involved an autonomous switched - mode dynamics.
An efficient solution of hamiltonian boundary value problems by combined gauss pseudospectral method with differential continuation approach
Mehrpouya M.A., Shamsi M., Azhmyakov V.
Article, Journal of the Franklin Institute, 2014, DOI Link
View abstract ⏷
Our paper deals with an effective application of the pseudospectral method to solution of Hamiltonian boundary value problems in optimal control theory. The developed numerical methodology is based on the celebrated Gauss pseudospectral approach. The last one makes it possible to reduce the conventional Hamiltonian boundary value problem to an auxiliary algebraic system. The implementable algorithm we propose is computationally consistent and moreover, involves numerically tractable results for a relative small discretization grids. However, the solution of the obtained algebraic equations system may has a low convergence radius. We next use the differential continuation approach in order to weaken the necessity of the well-defined initial conditions for the above algebraic system. The presented solution procedure can be extremely useful when the generic shooting-type methods fail because of sensitivity or stiffness. We discuss some numerical results and establish the efficiency of the proposed methodology.
Internal stability of a class of switched systems designed by implicit control techniques
Bonilla M., Alvarez N., Malabre M., Azhmyakov V.
Conference paper, 2014 European Control Conference, ECC 2014, 2014, DOI Link
View abstract ⏷
This paper is devoted to the Lyapunov - type stability analysis for a class of dynamic systems in the presence of a specific time - driven switchings mechanism. We consider dynamic stability problem for the whole set of implicit global representations associated with a class of linear switched systems. The resulting closed - loop dynamic model is determined by a PD - like state feedback control. For the corresponding systemanalysis associated with the resulting system we propose a common Lyapunov function that makes it possible to establish some basic stability properties. The obtained results have a constructive nature and can be applied to a wide class of switched systems.
Approximations based optimal control design for a class of switched dynamic systems
Azhmyakov V., Serrezuela R.R., Trujillo L.A.G.
Conference paper, IECON Proceedings (Industrial Electronics Conference), 2014, DOI Link
View abstract ⏷
This contribution deals with a new computational approach to optimal control design for a class of switched systems. The control strategy we develop is based on the proximal point method applied to a specific optimal control problem (OCP) with switched dynamics. The class of OCPs under consideration is widely applicable in optimization of electronic systems. We create constructive approximations for the initial sophisticated OCP, establish numerical stability of the resulting computational algorithm and develop an optimal control strategy. We finally discuss some numerical issues, study an illustrative example and also point possible generalizations of the elaborated control design in the context of hybrid dynamic systems.
Preface
Poznyak A., Polyakov A., Azhmyakov V.
Editorial, Systems and Control: Foundations and Applications, 2014,
Introduction
Poznyak A., Polyakov A., Azhmyakov V.
Book chapter, Systems and Control: Foundations and Applications, 2014, DOI Link
View abstract ⏷
This introductory chapter briefly reviews the evolution of optimal control design. First, it considers the classical control principles of optimal design for ideal completely known systems. Then, the case of incomplete information is studied. The ideas of robust control design and related optimization issues are discussed. The general principles of ellipsoid-based control design are introduced.
Robust state feedback control
Poznyak A., Polyakov A., Azhmyakov V.
Book chapter, Systems and Control: Foundations and Applications, 2014, DOI Link
View abstract ⏷
In this chapter, a particular family of nonlinear affine control systems with a sufficiently general type of uncertainties is considered. Nonlinear uncertain systems, considered here, are governed by vector ordinary differential equations with so-called quasi-Lipschitz right-hand sides admitting a wide class of external and internal uncertainties (including discontinuous nonlinearities such as relay and hysteresis elements, time-delay blocks, and so on). Here, the simplest class of linear state-feedback controllers is analyzed. Sufficient conditions guaranteeing the boundedness of all possible trajectories of controlled systems are presented. Since bounded dynamics can always be imposed on an ellipsoid, it is suggested that the “robust-optimal” gain matrix of the designated linear feedback be selected in such a way that the “size” of this attractive ellipsoid will be minimal. Several numerical and experimental illustrative examples are considered.
Attractive ellipsoids in robust control
Poznyak A., Polyakov A., Azhmyakov V.
Book, Systems and Control: Foundations and Applications, 2014,
Sample data and quantifying output control
Poznyak A., Polyakov A., Azhmyakov V.
Book chapter, Systems and Control: Foundations and Applications, 2014, DOI Link
View abstract ⏷
In this chapter, we consider the analysis and design of an output feedback controller for a perturbed nonlinear system in which the output is sampled and quantized. Using the attractive ellipsoid method, which is based on Lyapunov analysis techniques, together with the relaxation of a nonlinear optimization problem, sufficient conditions for the design of a robust control law are obtained. Since the original conditions result in nonlinear matrix inequalities, a numerical algorithm to obtain the solution is presented. The obtained control ensures that the trajectories of the closed-loop system will converge to a minimal (in a sense to be made specific) ellipsoidal region. Finally, numerical examples are presented to illustrate the applicability of the proposed design method.
Optimal LQ-type switched control design for a class of linear systems with piecewise constant inputs
Azhmyakov V., Basin M., Reincke-Collon C.
Conference paper, IFAC Proceedings Volumes (IFAC-PapersOnline), 2014, DOI Link
View abstract ⏷
This paper is devoted to a specific LQ-type optimal control problem (OCP) in the presence of additional control constraints. We consider control processes governed by linear differential equations with a priori known control switchings. The piecewise constant structure of the admissible controls under consideration is motivated by variety of concrete engineering applications and moreover, can be interpreted as a result of a quantization procedure applied to the original dynamics. We propose an new implementable numeric algorithm that makes it possible to calculate a consistent approximating solution to the initial constrained LQ-type OCPs. The contribution discusses some theoretic aspects of the obtained computational scheme and also contains a numerical example.
Control with sample-data measurements
Poznyak A., Polyakov A., Azhmyakov V.
Book chapter, Systems and Control: Foundations and Applications, 2014, DOI Link
View abstract ⏷
In this chapter, we formulate our main problem and discuss some necessary mathematical concepts related to feedback control design for nonlinear systems under sample-data output measurements. Then we present a theoretical analysis of an extended version of the invariant ellipsoid method. Then two feedbacks are analyzed: a linear feedback proportional to the current state estimate obtained by a Luenberger-type estimator; anda full-order linear dynamic controller governed by a linear ordinary differential equation with available sample data as input. Then we construct a minimal attractive ellipsoid that guarantees stability of the system in a practical sense by varying all parameters of the suggested feedbacks. The associated numerical techniques are also presented. An implementable algorithm for the constructive treatment of the robust control design problem is proposed.
Attractive ellipsoid method with adaptation
Poznyak A., Polyakov A., Azhmyakov V.
Book chapter, Systems and Control: Foundations and Applications, 2014, DOI Link
View abstract ⏷
This chapter deals with the development of a state estimator and adaptive controller based on the attractive ellipsoid method (AEM) for a class of uncertain nonlinear systems having “quasi-Lipschitz” nonlinearities as well as external perturbations. The set of stabilizing feedback matrices is given by a specific matrix inequality including the characteristic matrix of the attractive ellipsoid that contains all possible bounded trajectories around the origin. Here we present two modifications of the AEM that allow us to use online information obtained during the process and to adjust matrix parameters participating in constraints that characterize the class of adaptive stabilizing feedbacks. The proposed method guarantees that under a specific persistent excitation condition, the controlled system trajectories converge to an ellipsoid of “minimal size” having a minimal trace of the corresponding inverse ellipsoidal matrix.
Bounded robust control
Poznyak A., Polyakov A., Azhmyakov V.
Book chapter, Systems and Control: Foundations and Applications, 2014, DOI Link
View abstract ⏷
This chapter deals with a bounded control design for a class of nonlinear systems whose mathematical model may not be explicitly given. This class of uncertain nonlinear systems is governed by a system of ordinary differential equations with quasi-Lipschitz right-hand sides and contains external perturbations as well. The attractive ellipsoid method (AEM) allows us to describe the class of nonlinear feedbacks (containing a nonlinear projection operator, a linear state estimator, and a feedback matrix gain) guaranteeing the boundedness of all possible trajectories around the origin. To satisfy this property, some modifications of the AEM are introduced: basically, some sort of sample-time corrections of the feedback parameters are required. The optimization of feedback within this class of controllers is associated with the selection of the feedback parameters such that the trajectory converges within an ellipsoid of “minimal size.” The effectiveness of the suggested approach is illustrated by its application to a flexible arm system.
Consistent approximations and variational description of some classes of sliding mode control processes
Azhmyakov V., Polyakov A., Poznyak A.
Article, Journal of the Franklin Institute, 2014, DOI Link
View abstract ⏷
This paper is devoted to constructive approximations and an alternative theoretic characterization of some classes of sliding mode control processes. We construct the consistent approximations of the differential inclusions associated with the first-order variable structures dynamics and also propose a variational description of the sliding mode control in the framework of an auxiliary Hamiltonian based formalism. A trajectory of the closed-loop systems can be then constructively specified as a result of a particular system optimization procedure applied to the original model. The presented approximations and variational description of the sliding mode-type control design can provide a new analytic basis for constructive numerical schemes and implementable control algorithms. The mathematical tool elaborated in our contribution constitutes a formal extension of the classic Filippovïs results. © 2013 The Franklin Institute.
Robust control of switched systems
Poznyak A., Polyakov A., Azhmyakov V.
Book chapter, Systems and Control: Foundations and Applications, 2014, DOI Link
View abstract ⏷
This chapter deals with the problem of robust feedback design for a class of switched systems in the presence of bounded model uncertainties as well as external perturbations. Only the output of the system is supposed to be available for a designer. We consider nonlinear dynamic models under arbitrary switching mechanisms assuming that sample-switching times are known. Online state estimates are obtained by the use of a Luenberger-like observer using only current inputs and general information on the class of model uncertainties. The stabilization issue is solved in the sense of practical stability, and it is carried out by a linear (with respect to a current state estimate) feedback switching controller subject to an average dwell time scheme. We apply the newly elaborated (extended) version of the conventional attractive ellipsoid method for this purpose. Numerically implementable sufficient conditions for the practical stability of systems are derived using bilinear matrix inequalities. The effectiveness of the proposed method is illustrated by an example of a continuous stirred tank reactor in which only the temperature (not the concentration) is available during the process.
Optimal control processes associated with a class of discontinuous control systems: Applications to sliding mode dynamics
Azhmyakov V., Basin M.V., Gil Garcia A.E.
Article, Kybernetika, 2014, DOI Link
View abstract ⏷
This paper presents a theoretical approach to optimal control problems (OCPs) governed by a class of control systems with discontinuous right-hand sides. A possible application of the framework developed in this paper is constituted by the conventional sliding mode dynamic processes. The general theory of constrained OCPs is used as an analytic background for designing numerically tractable schemes and computational methods for their solutions. The proposed analytic method guarantees consistency of the resulting approximations related to the original infinite-dimensional optimization problem and leads to specific implementable algorithms.
Robust control of implicit systems
Poznyak A., Polyakov A., Azhmyakov V.
Book chapter, Systems and Control: Foundations and Applications, 2014, DOI Link
View abstract ⏷
This chapter deals with a new approach to robust control design for a class of nonlinearly affine control systems. The dynamic models under consideration are described by implicit differential equations in the presence of additive bounded uncertainties. The proposed robust feedback design procedure is based on an extended version of the classical invariant ellipsoid technique. In this book, this extension is called the attractive ellipsoid method. The stability/robustness analysis of the resulting closed-loop system involves a modified descriptor approach associated with the usual Lyapunov-type methodology. The theoretical schemes elaborated in our contribution are finally illustrated by a simple computational example.
Robust stabilization of time-delay systems
Poznyak A., Polyakov A., Azhmyakov V.
Book chapter, Systems and Control: Foundations and Applications, 2014, DOI Link
View abstract ⏷
In this chapter, we consider the class of uncertain time-delay affine-controlled systems in which a delay is admitted in state variables, and we show that the attractive ellipsoid method allows us to create a feedback that provides the convergence of any state trajectory of the controlled system from a given class to an ellipsoid whose “size” depends on the parameters of the applied feedback. Finally, we present a method for numerical calculation of these parameters that provides the “smallest” zone convergence for controlled trajectories.
Mathematical background
Poznyak A., Polyakov A., Azhmyakov V.
Book chapter, Systems and Control: Foundations and Applications, 2014, DOI Link
View abstract ⏷
This chapter is of a tutorial character. The aim here is to introduce some main concepts and auxiliary facts related to the class of control systems and problems under consideration. Moreover, we also give a first abstract problem formulation in the framework of robust and practically stable control design. Some conventional and advanced results related to ordinary differential equations in the framework of the examined class of dynamical systems are also included. Moreover, we take a quick look at the basic computational tool for our problems, namely, at linear matrix inequality techniques. We focus our attention primarily on motivations of the proposed “attractive ellipsoid” method and illustrate it in some simple situations.
Attractive ellipsoids in sliding mode control
Poznyak A., Polyakov A., Azhmyakov V.
Book chapter, Systems and Control: Foundations and Applications, 2014, DOI Link
View abstract ⏷
In this chapter, a new sliding mode control design algorithm for a linear and a class of nonlinear quasi-Lipschitz disturbed systems is presented. It is based on the appropriate selection of a sliding surface via the invariant ellipsoid method. The designed control guarantees minimization of unmatched disturbance effects to system motions in a sliding mode. The theoretical results are verified by numerical simulations. Additionally, a methodology for the design of sliding mode controllers for linear systems subjected to matched and unmatched perturbations is proposed. It is considered that the control signal is applied through a first-order low-pass filter. The technique is based on the existence of an attracting (invariant) ellipsoid such that the convergence to a quasiminimal region of the origin using the suboptimal control signal is guaranteed. The design procedure is given in terms of the solution of a set of matrix inequalities. Benchmark examples illustrating the design are given.
Robust output feedback control
Poznyak A., Polyakov A., Azhmyakov V.
Book chapter, Systems and Control: Foundations and Applications, 2014, DOI Link
View abstract ⏷
In this chapter, we consider three types of possible linear feedbacks using only the current output information: static feedback proportional to the output measurable signal, observer-based feedback proportional to the state estimation vector, and full-order linear dynamic controllers. For each type of possible linear feedback, we suggest that one characterize the set of all stabilizing gain-feedback matrices by a system of the corresponding linear matrix inequalities, providing the boundedness of all possible trajectories of every controlled plant from the considered class of uncertain systems. We also suggest selecting the optimal feedback gain matrix from the described class of stabilizing feedbacks as the one that minimizes the “size” of the attractive ellipsoid containing all possible bounded dynamic trajectories. The corresponding numerical procedures for designing the best feedback gain matrices are introduced and discussed for each type of considered feedback. Several illustrative examples clearly show the effectiveness of the suggested technique.
An approximations based approach to optimal control of switched dynamic systems
Azhmyakov V., Serrezuela R.R., Gallardo A.M.R., Vargas W.G.
Article, Mathematical Problems in Engineering, 2014, DOI Link
View abstract ⏷
Our paper deals with a new computational approach to optimal control design for a class of switched systems. The control strategy we propose is based on the conventional proximal point method applied to a specific optimal control problem (OCP) with switched dynamics. The class of OCPs under consideration is widely applicable in optimization of real-world electronic systems. We create constructive approximations for the initially given sophisticated OCP, establish numerical stability (consistency) of the resulting algorithm, and develop an optimal control strategy. We finally discuss some implementability issues, study an illustrative example, and also point to possible generalizations of the elaborated control design in the context of nonlinear hybrid systems.
Optimal switched-type control design for a class of nonlinear systems
Conference paper, IEEE International Conference on Automation Science and Engineering, 2013, DOI Link
View abstract ⏷
This paper is devoted to a general optimal control problem (OCP) associated with a family of nonlinear continuous-time switched systems in the presence of additional control constraints. The step wise control restrictions under consideration can usually be interpreted as a result of specific quantization procedures applied to the conventional systems inputs. We consider dynamic systems with a priori known switching strategies and a quadratic costs functional. Our aim is to elaborate a constructive implementable algorithm that makes it possible to calculate approximating solutions for the above-mentioned class of switched OCPs. The paper discuss such a computational scheme and also contains a numerical example. © 2013 IEEE.
On the practical stability of control processes governed by implicit differential equations: The invariant ellipsoid based approach
Article, Journal of the Franklin Institute, 2013, DOI Link
View abstract ⏷
This paper deals with a new approach to the robust control design for a class of nonlinearly affine control systems. The dynamical models under consideration are described by implicit differential equations in the presence of additive bounded uncertainties. The proposed robust feedback design procedure is based on an extended version of the classical invariant ellipsoid technique. We call this extension the Attractive Ellipsoid (AE) method. The stability/robustness analysis of the resulting closed-loop system involves a modified descriptor approach associated with the usual Lyapunov-type methodology. The theoretic schemes elaborated in our contribution are finally illustrated by a simple computational example. © 2013 The Franklin Institute.
Practical stability of control processes governed by semiexplicit DAEs
Article, Mathematical Problems in Engineering, 2013, DOI Link
View abstract ⏷
This paper deals with a new approach to robust control design for a class of nonlinearly affine control systems. The dynamical models under consideration are described by a special class of structured implicit differential equations called semi-explicit differential-algebraic equations (of index one), in the presence of additive bounded uncertainties. The proposed robust feedback design procedure is based on an extended version of the classical invariant ellipsoid technique that we call the Attractive Ellipsoid (AE) method. The theoretic schemes elaborated in our contribution are illustrated by a simple computational example. © 2013 R. Juárez et al.
On the robust control design for a class of nonlinearly affine control systems: The attractive ellipsoid approach
Azhmyakov V., Poznyak A., Gonzalez O.
Article, Journal of Industrial and Management Optimization, 2013, DOI Link
View abstract ⏷
This paper is devoted to a problem of robust control design for a class of continuous-time dynamic systems with bounded uncertainties. We study a family of nonlinearly affine control systems and develop a computational extension of the conventional invariant ellipsoid techniques. The obtained method can be considered as a powerful numerical approach that makes it possible to design a concrete stabilizing control strategies for the resulting closed-loop systems. The design procedure for this feedback-type control is based on the classic Lyapunov-type stability analysis of invariant sets for the given dynamic system. We study the necessary theoretic basis and propose a computational algorithm that guarantee some minimality properties of the stability/attractivity regions for dynamic systems under consideration. The complete solution procedure contains an auxiliary LMI-constrained optimization problem. The effectiveness of the proposed robust control design is illustrated by a numerical example.
On the practical stability for a class of switched system
Conference paper, CCE 2012 - 2012 9th International Conference on Electrical Engineering, Computing Science and Automatic Control, 2012, DOI Link
View abstract ⏷
This paper addresses a problem of robust control for a class of switched systems with time-delays. We deal with controllable nonlinear models that satisfy a quasi-Lipschitz condition and include bounded perturbations. The stabilization issue is carried out by two linear feedback controls: the static and the switching one. We also apply the extended version of the invariant ellipsoid method in combination with the Lyapunov-Krasovskii stability approach. Sufficient conditions of practical stability are derived in terms of Linear Matrix Inequalities (LMIs). The result is illustrated by a numerical example. © 2012 IEEE.
A general approach to optimal control processes associated with a class of discontinuous control systems: Applications to the sliding mode dynamics
Azhmyakov V., Basin M., Gil Garcia A.E.
Conference paper, Proceedings of the IEEE International Conference on Control Applications, 2012, DOI Link
View abstract ⏷
This paper extends a theoretical approach to optimal control problems (OCPs) governed by a class of control systems with discontinuous right hand sides. A possible application of the framework developed in this paper is constituted by the conventional sliding mode dynamic processes. The general theory of the general constrained OCPs is finally used as an analytic basis for some conceptual numerically tractable schemes from a wide family of computational methods for OCPs. The proposed analytic method guarantees consistency of the resulting approximations related to the sophisticated initial infinite-dimensional optimization problem and can provide a fundament for some concrete implementable algorithms. © 2012 IEEE.
Optimization of hybrid mechanical systems
Garcia A.E.G., Azhmyakov V., Basin M.V.
Conference paper, CCE 2012 - 2012 9th International Conference on Electrical Engineering, Computing Science and Automatic Control, 2012, DOI Link
View abstract ⏷
This paper deals with a general variational approach to optimal control design for a class of hybrid mechanical systems. We study a special structure of the optimal control problems (OCPs) in the classic mechanics and apply the multiobjective optimization techniques in the solution procedure. The main dynamic model under consideration is given in the form of a nonlinear hybrid control system and incorporates the conventional two points boundary-value problems for the Euler-Lagrange or Hamilton equations. We consider the first order optimality conditions for OCPs governed by hybrid mechanical systems and also discuss some conceptual computational algorithms. © 2012 IEEE.
Practical stability of control processes governed by semi-explicit DAEs
Conference paper, CCE 2012 - 2012 9th International Conference on Electrical Engineering, Computing Science and Automatic Control, 2012, DOI Link
View abstract ⏷
This paper deals with a new approach to robust control design for a class of nonlinearly affine control systems. The dynamical models under consideration are described by a special class of structured implicit differential equations called semi-explicit differential-algebraic equations (of index one), in the presence of additive bounded uncertainties. The proposed robust feedback design procedure is based on an extended version of the classical invariant ellipsoid technique that we call the Attractive Ellipsoid (AE) method. The theoretic schemes elaborated in our contribution are illustrated by a simple computational example. © 2012 IEEE.
A robust dynamic controller for a class of nonlinear systems with sample-data outputs
Mera M., Poznyak A., Azhmyakov V., Polyakov A.
Conference paper, CCE 2012 - 2012 9th International Conference on Electrical Engineering, Computing Science and Automatic Control, 2012, DOI Link
View abstract ⏷
In this paper we study a class of continues nonlinear control systems associated with the specific sampled outputs. The dynamical model under consideration is described by ordinary differential equations in the presence of additive bounded uncertainties. We propose a linear-type control design procedure that guarantee robustness property of the closed-loop realization. The proposed design method incorporates a concept of the dynamic feedback controller and a newly elaborated extended version of the classic invariant ellipsoid method. The applicability of the proposed control design scheme is illustrated by a computational example. A brief discussion on the numerical issues is also included. © 2012 IEEE.
On the set-valued approach to optimal control of sliding mode processes
Article, Journal of the Franklin Institute, 2012, DOI Link
View abstract ⏷
This paper is devoted to a theoretic framework for a general optimal control problem (OCP) associated with the classic sliding mode process. The sliding dynamic behavior is interpreted here as a special kind of additional constraints related to the main optimization problem. We are specially interested in the development of some adequate constructive approximations of the original OCPs. The mathematical approach based on the set-valued analysis allows to study the discontinuity of sliding mode dynamics in the abstract setting. Moreover, we also establish some sensitivity properties of the optimal solutions. The obtained results provide an universal analytical tool for the corresponding conceptual approximation schemes related to the original OCPs. The constructive approximations proposed in this paper are numerically stable and can be applied to various classes of optimal control processes governed by the affine control systems. © 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
A proximal point based approach to optimal control of affine switched systems
Article, Discrete Event Dynamic Systems: Theory and Applications, 2012, DOI Link
View abstract ⏷
This paper focuses on the proximal point regularization technique for a class of optimal control processes governed by affine switched systems. We consider switched control systems described by nonlinear ordinary differential equations which are affine in the input. The affine structure of the dynamical models under consideration makes it possible to establish some continuity/approximability properties and to specify these models as convex control systems. We show that, for some classes of cost functionals, the associated optimal control problem (OCP) corresponds to a conventional convex optimization problem in a suitable Hilbert space. The latter can be reliably solved using standard first-order optimization algorithms and consistent regularization schemes. In particular, we propose a conceptual numerical approach based on the gradient-type method and classic proximal point techniques. © 2011 Springer-Verlag.
Approximability and variational description of the Zeno behavior in affine switched systems
Conference paper, IFAC Proceedings Volumes (IFAC-PapersOnline), 2012, DOI Link
View abstract ⏷
This paper deals with a new theoretic description of a specific interaction of continuous and discrete dynamics in switched control systems known as a Zeno dynamics. We consider solutions to the closed loop realizations of control systems with affine structure that admit infinitely many discrete transitions on a finite time interval. Although the real-world processes do not present the corresponding behavior, mathematical models of some interconnected engineering systems may be Zeno due to the formal abstraction. In this paper we give a variational interpretation of the Zeno-like dynamics. A generic trajectory associated with this dynamics is finally specified as a result of a particular optimization procedure applied to the original model. The variational description of the Zeno effect provides an analytic basis for the constructive approximations techniques. We also study an intersection of a class of switched systems and the conventional sliding mode control processes. © 2012 IFAC.
A variational characterization of the sliding mode control processes
Conference paper, Proceedings of the American Control Conference, 2012, DOI Link
View abstract ⏷
This paper deals with a new theoretic description of some classes of the conventional sliding mode control processes. We use a variational characterization of the first-order variable structures dynamics and extend the corresponding state space formalism. A trajectory of the original systems can finally be specified as a result of a particular system optimization procedure applied to the original model. The presented variational theory of some sliding mode-type control systems is motivated by an alternative approach to the control design procedure. The mathematical tool of the sliding mode systems theory presented in our paper constitutes a formal extension of the classic Fillipov results. © 2012 AACC American Automatic Control Council).
On applications of attractive ellipsoid method to dynamic processes governed by implicit differential equations
Juarez R., Poznyak A.S., Azhmyakov V.
Conference paper, CCE 2011 - 2011 8th International Conference on Electrical Engineering, Computing Science and Automatic Control, Program and Abstract Book, 2011, DOI Link
View abstract ⏷
This paper deals with the application of the attractive (invariant) ellipsoid method for stabilization of class of the, so-called, implicit systems whose dynamics cannot be represented in the standard Cauchy form given by some ODE resolved with respect to the states of derivates. This class of dynamics systems includes, as a particular case, the models whose part of state-components is given in ODE-format while the rest of them represent only some algebraic nonlinear relations of states. To design a stabilizer as a linear state-feedback we suggest to apply the descriptive method with vector Lagrange multipliers in the Lyapunov stability analysis. The suggested technique leads to the sufficient conditions of the global practical stability which are shown to be expressed in BMI (bilinear matrix inequality) form. The last, after some coordinate transformation, can be converted to LMI (linear matrix inequalities) under fixed scalar parameters arising during the Lyapunov function construction. Results of numerical simulation realized by the standard MATLAB packages application illustrates the effectiveness of the suggested approach. © 2011 IEEE.
Applications of the strong approximability property to a class of affine switched systems and to relaxed differential equations with affine structure
Article, International Journal of Systems Science, 2011, DOI Link
View abstract ⏷
This article deals with applications of the newly elaborated fundamental continuity property to a class of affine switched systems and to some differential equations with discontinuous right-hand sides. We study conventional, switched and relaxed controllable dynamical models described by nonlinear ordinary differential equations that are affine in the input. The special structure of these systems makes it possible to prove the strong approximability property that can provide a useful tool for some specific robustness results. The mathematical approach based on the nonlinear and set-valued analysis, allows to consider the controllable dynamics in an abstract setting and to obtain some general theoretical results. The latter can be effectively applied to a wide classes of switched control systems and to differential inclusions with affine structure. © 2011 Taylor & Francis.
A gradient-type algorithm for a class of optimal control processes governed by hybrid dynamical systems
Article, IMA Journal of Mathematical Control and Information, 2011, DOI Link
View abstract ⏷
Advanced optimization techniques play a key role in various problems from the modern control engineering. Our contribution deals with a class of hybrid control systems and the associated optimal control problems. We study a specific type of non-stationary hybrid control processes with autonomous location transitions and propose a constructive gradient-based optimization method. In this paper, we use the celebrated Lagrange method, the technique of the reduced gradient and obtain the first-order necessary optimality conditions for the above class of hybrid optimal control problems. These optimality conditions are closely related to the corresponding variant of the hybrid maximum principle and can be used effectively as a part of optimization algorithms. Finally, we consider some numerical aspects of the developed theory and propose an implementable computational scheme. © 2011 The author.
Practical output feedback stabilisation for a class of continuous-time dynamic systems under sample-data outputs
Article, International Journal of Control, 2011, DOI Link
View abstract ⏷
This article deals with a class of continuous nonlinear control systems in the presence of sampled outputs. The dynamical models under consideration are described by ordinary differential equations with additive bounded uncertainties. The linear feedback control design proposed in this article is based on an extended version of the classical invariant ellipsoid method. The stability/robustness analysis of the resulting closed-loop system involves the celebrated 'descriptor techniques' from the extended Lyapunov methodology. Finally, the implementability of the proposed control design scheme is illustrated by a computational example. A brief discussion on the principal numerical issues is also included. © 2011 Taylor & Francis.
Approximability of nonlinear affine control systems
Azhmyakov V., Egerstedt M., Fridman L., Poznyak A.
Article, Nonlinear Analysis: Hybrid Systems, 2011, DOI Link
View abstract ⏷
This paper focuses on a strong approximability property for nonlinear affine control systems. We consider control processes governed by ordinary differential equations (ODEs) and study an initial system and the associated generalized system. Our theoretical approach makes it possible to prove a strong approximability result for the above dynamical systems. The latter can be effectively applied to some classes of variable structure and hybrid control systems. In particular, this paper deals with applications of the strong approximability property obtained to the conventional sliding mode processes and to hybrid control systems with autonomous location transitions. We also take into consideration some optimal control problems for the above class of hybrid systems. © 2010 Elsevier Ltd.
On the geometric aspects of the invariant ellipsoid method: Application to the robust control design
Conference paper, Proceedings of the IEEE Conference on Decision and Control, 2011, DOI Link
View abstract ⏷
This paper deals with a robust control design for a class of nonlinear affine control systems. The dynamical models under consideration are described by ordinary differential equations in the presence of some additive bounded uncertainties. The design procedure for the robust linear feedback control associated with the linearized dynamic model is based on an extended version of the classical invariant ellipsoid method. The stability/robustness analysis of the resulting closed-loop system involves the celebrated Clarke stability theorem that represents a theoretic extension of the celebrated Lyapunov-type methodology. The obtained analytic results are illustrated by a simple computational example. © 2011 IEEE.
The proximal point approach to optimal control of affine switched systems
Conference paper, IFAC Proceedings Volumes (IFAC-PapersOnline), 2011, DOI Link
View abstract ⏷
This paper deals with the optimal control techniques for a class of affine switched systems. The affine structure of the state equations makes it possible to establish a continuity property (with respect to the control parameter) for the above systems. Under some additional assumptions, the optimal control problem (OCP) associated with the affine dynamic models corresponds to a standard convex optimization problem. The latter can be reliably solved using some standard numerical algorithms and consistent regularization schemes. © 2011 IFAC.
Optimization of multiagent systems with increasing state dimensions: Hybrid LQ approach
Galvan-Guerra R., Azhmyakov V., Egerstedt M.
Conference paper, Proceedings of the American Control Conference, 2011, DOI Link
View abstract ⏷
In this paper, we study a specific optimal control problem associated with a multiagent dynamic system. The interest is placed on minimization of the tracking error in the multiagent leader-follower model. We replace this problem by a specific hybrid optimal control problem. In particular, we consider control systems with monotonically increasing dimensions of the state vector. The change of the state dimension has the character of a jump and is modeled by an impulsive hybrid system. The paper proposes an effective computational procedure for the above optimal tracking control problem in a multiagent setting. The theoretical and numerical approaches obtained in this contribution are tested on a practically motivated example. © 2011 AACC American Automatic Control Council.
On the robust control design for a class of continuous-time dynamical systems with a sample-data output
Conference paper, IFAC Proceedings Volumes (IFAC-PapersOnline), 2011, DOI Link
View abstract ⏷
This paper deals with a class of continuous nonlinear control systems in the presence of sampled outputs. The dynamical models under consideration are described by ordinary differential equations with additive bounded uncertainties. The linear feedback control design proposed in the manuscript is based on an extended version of the classical invariant ellipsoid method. The stability/robustness analysis of the resulting closed-loop system involves the celebrated "descriptor techniques" from the extended Lyapunov methodology. Finally, the implementability of the proposed control design scheme is illustrated by a computational example. A brief discussion on the principal numerical issues is also included. © 2011 IFAC.
Explicit switching times and locations detection for linear hybrid systems
Galvan-Guerra R., Velazquez-Velazquez J.-E., Azhmyakov V.
Conference paper, Program and Abstract Book - 2010 7th International Conference on Electrical Engineering, Computing Science and Automatic Control, CCE 2010, 2010, DOI Link
View abstract ⏷
This paper is concerned with the exact calculus of the switching times for linear time invariant hybrid systems. We study the system's behavior that incorporates dynamical switches from a current location to another and propose an explicit procedure for a constructive treatment of the switching times. An implementable computational method elaborated in this paper makes it possible to solve the multidimensional problem of switching times detections in an algorithmic form. In this sense, the above numerical procedure also determines the sequence of locations of the hybrid system. Finally, we apply the proposed constructive approach to two illustrative examples. © 2010 IEEE.
On a variational approach to optimization of hybrid mechanical systems
Article, Mathematical Problems in Engineering, 2010, DOI Link
View abstract ⏷
This paper deals with multiobjective optimization techniques for a class of hybrid optimal control problems in mechanical systems. We deal with general nonlinear hybrid control systems described by boundary-value problems associated with hybrid-type Euler-Lagrange or Hamilton equations. The variational structure of the corresponding solutions makes it possible to reduce the original "mechanical" problem to an auxiliary multiobjective programming reformulation. This approach motivates possible applications of theoretical and computational results from multiobjective optimization related to the original dynamical optimization problem. We consider first order optimality conditions for optimal control problems governed by hybrid mechanical systems and also discuss some conceptual algorithms. © 2010 Vadim Azhmyakov and Ruben Velazquez.
Optimal control of sliding mode processes: A general approach
Conference paper, Proceedings of the 2010 11th International Workshop on Variable Structure Systems, VSS 2010, 2010, DOI Link
View abstract ⏷
This paper addresses a general theoretical framework of optimal control problems (OCPs) associated with the conventional sliding mode dynamics. We deal with a class of constrained OCPs governed by nonlinear affine control systems and propose some numerically stable approximations to the sophisticated dynamical optimization problem. The above-mentioned structure of the state equations makes it also possible to consider some sensitivity properties of the optimal solutions. The mathematical approach based on the set-valued analysis allows to study the usual discontinuous sliding mode-type dynamics in the abstract setting and to obtain some general analytical results. These facts can be effectively applied to wide classes of OCPs governed by sliding mode processes and variable structure systems (VSSs). © 2010 IEEE.
On the hybrid LQ-based control design for linear networked systems
Azhmyakov V., Galvan-Guerra R., Poznyak A.
Article, Journal of the Franklin Institute, 2010, DOI Link
View abstract ⏷
This paper addresses a problem of optimal control design associated with the linear networked control systems (NCSs). We study a class of the conventional networked systems in the presence of time delays and propose a hybrid LQ-based theoretical and computational approach to the above NCSs. In particular, we develop an explicit theoretical representation of the networked control processes by an adequate auxiliary hybrid systems. For the constructive feedback control design procedure we derive the necessary hybrid Riccati-formalism and propose an implementable solution procedure. © 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Hybrid LQ optimization of linear network based systems
Conference paper, Proceedings of the IEEE International Conference on Industrial Technology, 2010, DOI Link
View abstract ⏷
This paper deals with an optimal design of linear network based control systems. We study a class of delayed dynamical systems in the continuous time-domain and propose an optimization approach based on the newly elaborated hybrid LQ-type techniques. Moreover, we develop an explicit connection between the given network based control processes and auxiliary hybrid models. The optimal feedback control strategy for the network based control systems is obtained as a consequence of the Riccati-formalism for the associated hybrid LQ problems. We also discuss some alternative optimization techniques and stability properties of the optimal trajectories. ©2010 IEEE.
Optimization of hybrid mechanical systems
Azhmyakov V., Garrido Moctezuma R.A., Gil Garcia A.E.
Conference paper, Proceedings of the IEEE International Conference on Industrial Technology, 2010, DOI Link
View abstract ⏷
This paper deals with multiobjective optimization techniques for a class of hybrid optimal control problems in mechanical systems. We deal with a general nonlinear hybrid control systems described by boundary-value problems associated with hybrid-type Euler-Lagrange or Hamilton equations. The variational structure of the corresponding solutions makes it possible to reduce the original "mechanical" problem to an auxiliary multiobjective programming reformulation. This approach motivates possible applications of theoretical and computational results from multiobjective optimization related to the original dynamical optimization problem.We consider first order optimality conditions for optimal control problems governed by hybrid mechanical systems and also discuss some conceptual algorithms. ©2010 IEEE.
An application of the proximal point algorithm to optimal control of affine switched systems
Azhmyakov V., Raisch J., Velazquez R.
Conference paper, Proceedings of the IEEE International Conference on Industrial Technology, 2010, DOI Link
View abstract ⏷
This paper focuses on the proximal point regularization technique for a class of optimal control processes governed by affine switched systems. We consider some controllable dynamical systems described by nonlinear ordinary differential equations which are affine in the input. The affine structure of the control systems under consideration makes it possible to establish some continuity/approximability properties of the dynamical models and to characterize these dynamical models as convex control systems. We show that, for some classes of cost functionals, the associated optimal control problems (OCPs) correspond to the standard convex optimization problems in a suitable Hilbert space. The latter can be reliably solved using standard first-order optimization algorithms and consistent regularization schemes. In particular, we propose a conceptual numerical approach based on gradient-type methods and proximal point techniques. ©2010 IEEE.
On an optimization problem for a class of impulsive hybrid systems
Article, Discrete Event Dynamic Systems: Theory and Applications, 2010, DOI Link
View abstract ⏷
This contribution addresses the problem of optimal control for a class of hybrid systems, where discrete transitions are accompanied by instantaneous changes in the continuous state variables, and where these changes can be considered as control variables. Based on a variational approach, necessary conditions of optimality are first established. The problem is then cast as a parametric optimization problem for which gradient information is derived. Finally, we discuss assumptions that guarantee convergence of a conceptual algorithm to a stationary solution. A brief discussion on the main implementation issues is also included. © The Author(s) 2009.
Proximal point method for optimal control processes governed by ordinary differential equations
Article, Asian Journal of Control, 2010, DOI Link
View abstract ⏷
This paper is concerned with the proximal-based approach to linear and finite-difference approximations of constrained convex optimal control problems. We consider control systems governed by ordinary differential equations in the presence of additional terminal/state inequalities and propose a numerical method derived from the proximal point algorithm. The aim of the paper is to study the convergence properties of the obtained conceptual algorithm and to show that it can be used to compute approximate optimal controls. © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society.
Discussion on: “A decentralized sliding control approach for distributed simulation of differential-algebraic equation systems”
Azhmyakov V., Galvan-Guerra R., Velazquez R.
Article, European Journal of Control, 2010, DOI Link
On the LQ-based optimization techniques for impulsive hybrid control systems
Azhmyakov V., Galvan-Guerra R., Egerstedt M.
Conference paper, Proceedings of the 2010 American Control Conference, ACC 2010, 2010, DOI Link
View abstract ⏷
This paper deals with a quadratic optimization problem for linear impulsive hybrid systems. We study a class of LQ-type impulsive hybrid optimal control problems (OCPs) and consider the application of the hybrid Maximum Principle (MP). Our aim is to investigate the natural relationship between the Pontryagin-type MP and the Bellman Dynamic Programming (DP) approach. As next we develop the "hybrid" Riccati formalism and discuss some related computational aspects. © 2010 AACC.
Numerically stable approximations of optimal control processes associated with a class of switched systems
Azhmyakov V., Velazquez R., Galvan-Guerra R.
Conference paper, IFAC Proceedings Volumes (IFAC-PapersOnline), 2010, DOI Link
View abstract ⏷
In this contribution, we propose a numerical approach to optimal control processes governed by some specific switched systems. Our principal computational scheme can be characterized as an extension of the conventional proximal point method to optimal control problems (OCPs) with switched dynamical models. We study proximal based constructive approximations of the initial OCPs and establish the numerical stability (numerical consistency) of the resulting algorithm. We also discuss the practical implementability of the elaborated method and point some possible generalizations of our idea in the context of hybrid control systems.
An application of the proximal point algorithm to optimal control problems with constraints
Conference paper, CERMA 2009 - Electronics Robotics and Automotive Mechanics Conference, 2009, DOI Link
View abstract ⏷
This paper is concerned with the proximal-based approach to linear and finite-difference approximations of constrained convex optimal control problems (OCPs). We consider control systems governed by ordinary differential equations in the presence of additional terminal/state inequalities and propose a numerical method derived from the proximal point algorithm. The aim of the paper is to study the convergence properties of the obtained conceptual algorithm and to show that it can be used to compute approximate optimal controls. © 2009 IEEE.
On the robust control design for a class of nonlinear affine control systems: The invariant ellipsoid approach
Gonzalez O., Poznyak A., Azhmyakov V.
Conference paper, 2009 6th International Conference on Electrical Engineering, Computing Science and Automatic Control, CCE 2009, 2009, DOI Link
View abstract ⏷
This paper is devoted to the problem of robust design for a class of continuous-time control systems with bounded uncertainties. We study a family of nonlinearly affine dynamical system and apply a modified invariant ellipsoid technique. This makes it possible to obtain practically stable closed-loop controllable models. The design of the stabilizing feedback control strategy is based on the conventional Lyapunov-like approach to invariant sets of dynamical systems. We propose a computational scheme for a constructive treatment a feedback control law such that the region of the practical stability of the resulting system is minimized. The corresponding solution procedure contains an auxiliary LMI-constrained optimization problem. The effectiveness of the elaborated invariant ellipsoid based method is illustrated by a numerical example.
Robust control for a class of continuous-time dynamical systems with sample-data outputs
Mera M., Poznyak A., Azhmyakov V., Fridman E.
Conference paper, 2009 6th International Conference on Electrical Engineering, Computing Science and Automatic Control, CCE 2009, 2009, DOI Link
View abstract ⏷
This paper addresses the problem of robust control for a class of nonlinear dynamical systems in the discretecontinuous time domain. We deal with nonlinear controllable models described by ordinary differential equations in the presence of bounded uncertainties. The full model of the control system under consideration is completed by linear samplingtype outputs. The linear feedback control design proposed in this manuscript is created by application of an extended version of the conventional invariant ellipsoid method. Moreover, we also apply some specific Lyapunov-based "descriptor techniques" from the stability theory of delayed systems. The above combination of the modified invariant ellipsoid approach and descriptor method make it possible to obtain the robustness of the designed control and to establish some well known stability properties of dynamical systems under consideration. Finally, the applicability of the proposed method is illustrated by a computational example. A brief discussion on the main implementation issue is also included.
An approach to optimization of linear networked systems based on the hybrid LQ methodology
Galvan-Guerra R., Azhmyakov V., Velazquez-Velazquez J.E., Poznyak A.
Conference paper, 2009 6th International Conference on Electrical Engineering, Computing Science and Automatic Control, CCE 2009, 2009, DOI Link
View abstract ⏷
This paper deals with an optimal design of linear networked control systems. We study a class of delayed dynamical systems in the continuous time-domain and propose an optimization approach based on the newly elaborated hybrid LQ-type techniques. Moreover, we develop an explicit connection between the given networked control processes and auxiliary hybrid models. The optimal feedback control strategy for the networked systems is obtained as a consequence of the Riccati-formalism for the associated hybrid LQ problems. We also consider some alternative optimization techniques and discuss stability properties of the optimal trajectories.
Convexity of the set of fixed points generated by some control systems
Article, Journal of Applied Mathematics, 2009, DOI Link
View abstract ⏷
We deal with an application of the fixed point theorem for nonexpansive mappings to a class of control systems. We study closed-loop and open-loop controllable dynamical systems governed by ordinary differential equations (ODEs) and establish convexity of the set of trajectories. Solutions to the above ODEs are considered as fixed points of the associated system-operator. If convexity of the set of trajectories is established, this can be used to estimate and approximate the reachable set of dynamical systems under consideration. The estimations/approximations of the above type are important in various engineering applications as, for example, the verification of safety properties. Copyright © 2009 Vadim Azhmyakov.
Hybrid LQ-optimization using dynamic programming
Azhmyakov V., Galvan-Guerra R., Egerstedt M.
Conference paper, Proceedings of the American Control Conference, 2009, DOI Link
View abstract ⏷
In this paper we study the linear quadratic optimal control problem for linear hybrid systems in which transitions between different discrete locations occur autonomously when the continuous state intersects given switching surfaces. In particular, we make an explicit connection between the newly developed, Pontryagin-type Hybrid Maximum Principle and the Bellman Dynamic Programming approach. As a consequence, we extend the classic Riccati-formalism, derive the associated Riccati-type equations, and prove the discontinuity of the full "hybrid" Riccati matrix. Finally, we discuss some computational aspects of the obtained theoretical results and propose a numerical algorithm in the framework of an optimal feedback control law. © 2009 AACC.
The dynamic programming approach to multi-model robust optimization
Azhmyakov V., Boltyanski V., Poznyak A.
Article, Nonlinear Analysis, Theory, Methods and Applications, 2009, DOI Link
View abstract ⏷
The aim of this paper is to extend the dynamic programming (DP) approach to multi-model optimal control problems (OCPs). We deal with robust optimization of multi-model control systems and are particularly interested in the Hamilton-Jacobi-Bellman (HJB) equation for the above class of problems. In this paper, we study a variant of the HJB for multi-model OCPs and examine the natural relationship between the Bellman DP techniques and the Robust Maximum Principle (MP). Moreover, we describe how to carry out the practical calculations in the context of multi-model LQ-problems and derive the associated Riccati-type equation. © 2009 Elsevier Ltd. All rights reserved.
On the method of dynamic programming for linear-quadratic problems of optimal control in hybrid systems
Azmyakov V.V., Galvan-Guerra R., Polyakov A.E.
Article, Automation and Remote Control, 2009, DOI Link
View abstract ⏷
For a sufficiently wide class of the linear hybrid systems, an algorithm of optimal feedback control was proposed. Consideration was given to the hybrid control systems with autonomous switching, as well as the corresponding problems of the hybrid linear-quadratic optimal control based on the recently suggested principle of maximum. Interrelations between the hybrid principle of maximum and the method of dynamic programming for the systems of this class were discussed. The classical formalism was extended, the corresponding Riccati equations were obtained, and discontinuity of the "hybrid" Riccati matrix was proved. The computational aspects of the established theoretical results were considered. © 2009 Pleiades Publishing, Ltd.
On the optimal design of linear networked systems
Azhmyakov V., Galvan-Guerra R., Velazquez Cuevas R., Poznyak A.S.
Conference paper, IFAC Proceedings Volumes (IFAC-PapersOnline), 2009, DOI Link
View abstract ⏷
This paper addresses the problem of optimal design for linear networked control systems (NCSs). We deal with the delayed NCSs in the continuous time-domain and propose a computational approach to optimization problems based on the hybrid LQ-type techniques. In particular, we develop an explicit connection between the networked control processes and the corresponding hybrid systems. For the constructive feedback control design procedure we derive the necessary Riccati-formalism and propose an implementable solution algorithm. © 2009 IFAC.
Continuity properties of nonlinear affine control systems: Applications to hybrid and sliding mode dynamics
Azhmyakov V., Egerstedt M., Fridman L., Poznyak A.
Conference paper, IFAC Proceedings Volumes (IFAC-PapersOnline), 2009, DOI Link
View abstract ⏷
This paper focuses on the continuity and approximability properties for nonlinear affine control systems. We consider dynamical systems governed by ordinary differential equations and establish the continuity properties of the given and relaxed (in the sense of Filippov) systems with respect to controls and initial state variables. The approach based on the set-valued analysis makes it possible to study discontinuous models in the abstract setting and to obtain general theoretical results. The latter can be effectively applied to wide classes of variable structure control systems. In particular, this paper deals with applications of the above-mentioned continuity and approximability to some hybrid control systems and to the classical sliding mode control processes.
Relationship between dynamic programming and the maximum principle for impulsive hybrid LQ optimal control problems
Conference paper, 2008 5th International Conference on Electrical Engineering, Computing Science and Automatic Control, CCE 2008, 2008, DOI Link
View abstract ⏷
This paper deals with optimization techniques for linear impulsive hybrid systems (LIHSs). We study the LQ (linear quadratic) impulsive hybrid optimal control problem (OCP) and apply the corresponding Pontryagin-type Maximum Principle (MP) (see [11], [14], [4]). Our aim is to investigate the natural relationship between the above MP and the Bellman Dynamic Programming (DP) approach to the above hybrid OCP under consideration. We derive the associated Riccatitype formalism and discuss some related numerical schemes. © 2008 IEEE.
Optimal control of impulsive hybrid systems
Azhmyakov V., Boltyanski V.G., Poznyak A.
Article, Nonlinear Analysis: Hybrid Systems, 2008, DOI Link
View abstract ⏷
In this paper, we deal with optimization techniques for a class of hybrid systems that comprise continuous controllable dynamics and impulses (jumps) in the state. Using the mathematical techniques of distributional derivatives and impulse differential equations, we rewrite the original hybrid control system as a system with autonomous location transitions. For the obtained auxiliary dynamical system and the corresponding optimal control problem (OCP), we apply the Lagrange approach and derive the reduced gradient formulas. Moreover, we formulate necessary optimality conditions for the above hybrid OCPs, and discuss the newly elaborated Pontryagin-type Maximum Principle for impulsive OCPs. As in the case of the conventional OCPs, the proposed first order optimization techniques provide a basis for constructive computational algorithms. © 2008 Elsevier Ltd. All rights reserved.
On the dynamic programming approach to multi-model robust optimal control problems
Azhmyakov V., Boltyanski V.G., Poznyak A.S.
Conference paper, Proceedings of the American Control Conference, 2008, DOI Link
View abstract ⏷
The aim of this paper is to extend the Dynamic Programming (DP) approach to multi-model optimal control problems (OCPs). We deal with robust optimization of multimodel control systems and are particularly interested in the Hamilton-Jacobi-Bellman (HJB) equation for the above class of problems. In this paper, we study a variant of the HJB for multi-model OCPs and examine the natural relationship between the Bellman DP techniques and the Robust Maximum Principle (MP) [7], [8], [9], [18]. Moreover, we describe a concept for practical calculations in the context of multi-model LQ-problems and derive the associated Riccati-type equation. ©2008 AACC.
First order optimization techniques for impulsive hybrid dynamical systems
Azhmyakov V., Boltyanski V.G., Poznyak A.
Conference paper, IEEE 10th International Workshop on Variable Structure Systems, VSS'08, 2008, DOI Link
View abstract ⏷
In this paper, we deal with optimization techniques for a class of hybrid systems that comprise continuous controllable dynamics and impulses (jumps) in the state. Using the mathematical techniques of distributional derivatives and impulse differential equations, we rewrite the original hybrid control system as a system with autonomous location transitions. For the obtained auxiliary dynamical system and the corresponding optimal control problem (OCP) we apply the Lagrange approach and derive the reduced gradient formulas. Moreover, we formulate necessary optimally conditions for the above hybrid OCPs. As in the case of a classic problem, the proposed optimization techniques also provided a basis for constructive computational algorithms. ©2008 IEEE.
Convex control systems and convex optimal control problems with constraints
Article, IEEE Transactions on Automatic Control, 2008, DOI Link
View abstract ⏷
This note discusses the concepts of convex control systems and convex optimal control problems. We study control systems governed by ordinary differential equations in the presence of state and target constraints. Our note is devoted to the following main question: under which additional assumptions is a "sophisticated" constrained optimal control problem equivalent to a "simple" convex minimization problem in a related Hilbert space. We determine some classes of convex control systems and show that, for suitable cost functionals and constraints, optimal control problems for these classes of systems correspond to convex optimization problems. The latter can be reliably solved using standard numerical algorithms and effective regularization schemes. In particular, we propose a conceptual computational approach based on gradient-type methods and proximal point techniques. © 2008 IEEE.
An approach to controlled mechanical systems based on the multiobjective optimization technique
Article, Journal of Industrial and Management Optimization, 2008, DOI Link
View abstract ⏷
This paper discusses the multiobjective optimization techniques for a class of optimal control problems in mechanics. We deal with constrained nonlinear control systems described by the Euler-Lagrange or Hamilton equations and study the variational structure of the solution of the corresponding boundary-value problems. We also reduce the original "mechanical" problem to an auxiliary multiobjective optimization problem. This approach makes it possible to apply the effective theoretical and computational results from multiobjective programming to the original problem. We consider first order computational schemes for optimal control problems governed by mechanical systems and examine some illustrative examples.
On the maximum principle for impulsive hybrid systems
Conference paper, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2008, DOI Link
View abstract ⏷
In this contribution, we consider a class of hybrid systems with continuous dynamics andjumps in the continuous state (impulsive hybrid systems). By using a newly elaborated version ofthe Pontryagin-type Maximum Principle (MP) for optimal control processes governed by hybriddynamics with autonomous location transitions, we extend the necessary optimality conditions to aclass of Impulsive Hybrid Optimal Control Problems (IHOCPs). For these problems, we obtain aconcise characterization of the Impulsive Hybrid MP (IHMP), namely, the correspondingboundary-value problem and some additional relations. As in the classical case, the proposed IHMPprovides a basis for diverse computational algorithms for the treatment of IHOCPs. © 2008 Springer-Verlag Berlin Heidelberg.
A computational approach to optimization of controlled mechanical systems
Conference paper, 2007 Mediterranean Conference on Control and Automation, MED, 2007, DOI Link
View abstract ⏷
In the present work we consider a class of nonlinear optimal control problems, which can be called "optimal control problems in mechanics". We deal with control systems whose dynamics can be described by a system of Euler-Lagrange or Hamilton equations. Using the variational structure of the solution of the corresponding boundary-value problems, we reduce the initial optimal control problem to an auxiliary problem of multiobjective programming. This technique makes it possible to apply some consistent numerical approximations of a multiobjective optimization problem to the initial optimal control problem. For solving the auxiliary problem we propose an implementable numerical algorithm. ©2007 IEEE.
State jump optimization for a class of hybrid autonomous systems
Conference paper, Proceedings of the IEEE International Conference on Control Applications, 2007, DOI Link
View abstract ⏷
In this contribution, optimization of state jumps for a class of hybrid systems is considered. Basically, the control variables to be determined are the amounts of jump in the continuous states such that a corresponding cost functional is minimized. Based on a variational approach, necessary conditions of optimality are first established. The problem is then cast as a parametric optimization problem where the gradient information is derived. Finally and under some assumptions, convergence to the optimal solution of a conceptual algorithm is established. A brief discussion on the main implementation issues is also included. © 2007 IEEE.
Optimal control of mechanical systems
Article, Differential Equations and Nonlinear Mechanics, 2007, DOI Link
View abstract ⏷
In the present work, we consider a class of nonlinear optimal control problems, which can be called "optimal control problems in mechanics." We deal with control systems whose dynamics can be described by a system of Euler-Lagrange or Hamilton equations. Using the variational structure of the solution of the corresponding boundary-value problems, we reduce the initial optimal control problem to an auxiliary problem of multiobjective programming. This technique makes it possible to apply some consistent numerical approximations of a multiobjective optimization problem to the initial optimal control problem. For solving the auxiliary problem, we propose an implementable numerical algorithm.£.
On convexity of reachable sets for nonlinear control systems
Azhmyakov V., Flockerzi D., Raisch J.
Conference paper, 2007 European Control Conference, ECC 2007, 2007, DOI Link
View abstract ⏷
In this paper, we investigate reachable sets for some classes of open- and closed-loop control systems governed by ordinary differential equations. We are particularly interested in convexity properties of both reachable sets and sets of trajectories. If such properties exist they can be used to estimate the reachable set under investigation. This is important in applications as, e.g., the verification of safety properties. Convexity of the set of trajectories is also important for a constructive reformulation of some constrained dynamical optimization problems in the form of a convex minimization problem.
Necessary optimality conditions for a class of hybrid optimal control problems
Azhmyakov V., Attia S.A., Gromov D., Raisch J.
Conference paper, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2007, DOI Link
View abstract ⏷
In this paper we study a class of Mayer-type hybrid optimal control problems. Using Lagrange techniques, we formulate a version of the Hybrid Maximum Principle for optimal control problems governed by hybrid systems with autonomous location transitions in the presence of additional target constraints. © Springer-Verlag Berlin Heidelberg 2007.
A fixed point theorem for a class of differentiable stable operators in Banach spaces
Article, Fixed Point Theory and Applications, 2006, DOI Link
View abstract ⏷
We study Fréchet differentiable stable operators in real Banachspaces. We present the theory of linear and nonlinear stableoperators in a systematic way and prove solvability theorems for operator equations with differentiable expanding operators. In addition, some relations to the theory of monotone operators in Hilbert spaces are discussed. Using the obtained solvability results, we formulate the corresponding fixed point theorem for aclass of nonlinear expanding operators. Copyright © 2006 Vadim Azhmyakov.
Stability of differential inclusions: A computational approach
Article, Mathematical Problems in Engineering, 2006, DOI Link
View abstract ⏷
We present a technique for analysis of asymptotic stability for a class of differential inclusions. This technique is based on the Lyapunov-type theorems. The construction of the Lyapunov functions for differential inclusions is reduced to an auxiliary problem of mathematical programming, namely, to the problem of searching saddle points of a suitable function. The computational approach to the auxiliary problem contains a gradient-type algorithm for saddle-point problems. We also extend our main results to systems described by difference inclusions. The obtained numerical schemes are applied to some illustrative examples.
A gradient-based approach to a class of hybrid optimal control problems
Conference paper, IFAC Proceedings Volumes (IFAC-PapersOnline), 2006, DOI Link
View abstract ⏷
We investigate optimal control problems for a class of non-stationary hybrid systems with autonomous location transitions. Using the Lagrange approach and the technique of the reduced gradient, we derive necessary optimality conditions for the considered class of problems. These optimality conditions are closely related to a variant of the Hybrid Maximum Principle and can be used for constructive optimization algorithms. Copyright © 2006 IFAC.
Approximations of relaxed optimal control problems
Article, Journal of Optimization Theory and Applications, 2006, DOI Link
View abstract ⏷
In the present paper, we investigate an approximation technique for relaxed optimal control problems. We study control processes governed by ordinary differential equations in the presence of state, target, and integral constraints. A variety of approximation schemes have been recognized as powerful tools for the theoretical studying and practical solving of Infinite-dimensional optimization problems. On the other hand, theoretical approaches to the relaxed optimal control problem with constraints are not sufficiently advanced to yield numerically tractable schemes. The explicit approximation of the compact control set makes it possible to reduce the sophisticated relaxed problem to an auxiliary optimization problem. A given trajectory of the relaxed problem can be approximated by trajectories of the auxiliary problem. An optimal solution of the introduced optimization problem provides a basis for the construction of minimizing sequences for the original optimal control problem. We describe how to carry out the numerical calculations in the context of nonlinear programming and establish the convergence properties of the obtained approximations. © 2006 Springer Science+Business Media, Inc.
A gradient-based approach to a class of hybrid optimal control problems
Book chapter, Analysis and Design of Hybrid Systems 2006, 2006, DOI Link
View abstract ⏷
This chapter develops a new approach to a class of hybrid optimal control problems of the Mayer type. Hybrid control systems are mathematical modes of heterogeneous systems consisting of a continuous part, a finite number of continuous controllers and a discrete supervisor. For a hybrid optimal control problem, the main tool toward the construction of optimal trajectories is the Hybrid Maximum Principle. This approach is based on explicit formulae for the reduced gradient of the cost functional of the given hybrid optimal control problem. The corresponding relations make it possible to formulate first-order necessary optimality conditions for the considered hybrid optimal control problems and provide a basis for effective computational algorithms. The idea of reduced gradients can also be used for some linearization procedures of the initial optimal control problem, the main tool toward the construction of optimal trajectories is the Hybrid Maximum Principle. © 2006 Elsevier Ltd All rights reserved.
On the optimal design of elastic beams
Article, Structural and Multidisciplinary Optimization, 2004, DOI Link
View abstract ⏷
In this paper we investigate the optimal dynamics of simply supported nonlinearly elastic beams with rectangular cross-sections. We consider the elastic beam under the assumption of time-dependent intensive transverse loading. The state of the beam is described by a system of partial differential equations of the fourth order. We deal with the problem of choosing the optimal shape for the beam. The optimal shape is determined in such a way that the deflection of the nonlinearly elastic beam for any given time is minimal. The problem of choosing the optimal shape is formulated as an optimal control problem. To solve the obtained problem effectively, we use the optimality principle of Bellman (Bellman and Dreyfus 1962; Bryson and Ho 1975) and the penalty function method (Polyak 1987). We present a constructive algorithm for the optimal design of non-linearly elastic beams. Some simple examples of the implementation of the proposed numerical algorithm are given.
Strong convergence of a proximal-based method for convex optimization
Article, Mathematical Methods of Operations Research, 2003, DOI Link
View abstract ⏷
In this work we study a proximal-like method for the problem of convex minimization in Hubert spaces. Using the classical proximal mapping, we construct a new stable iterative procedure. The strong convergence of obtained sequences to the normal solution of the optimization problem is proved. Some results of this paper are extended for uniformly convex Banach spaces. © Springer-Verlag 2003.
Optimal Control of a Well-Stirred Bioreactor in the Presence of Stochastic Perturbations
Article, Informatica, 2002,
View abstract ⏷
We study the stochastic model for bioremediation in a bioreactor with ideal mixing. The dynamics of the examined system is described by stochastic differential equations. We consider an optimal control problem with quadratic costs functional for the linearized model of a well-stirred bioreactor. The optimal control is based on the optimal robust state estimates. The approximate optimal solution is obtained as a linear feedback.
Exponential stabilization and synthesis of discrete stochastic systems
Article, Avtomatika i Telemekhanika, 1994,
View abstract ⏷
There are given necessary and sufficient conditions of exponential stabilization that are obtained using the direct Lyapunov method. For constructive solution of the stabilization problem generalization of the stabilizing stochastic pair method for stochastic system is used. A numerical method of stabilizing pair construction in some region of phase space is substantiated. The criteria of exponential stabilization of the controlled Markov networks are formulated. The effective algorithms of stabilizing control synthesis are presented.
Nonlocal synthesis of stabilizing systems of discrete stochastic control processes
Article, Avtomatika i Telemekhanika, 1994,
View abstract ⏷
Criteria are formulated for nonlocal stabilizability of discrete systems with random parameters. Algorithms are presented for the constructive solution of a stabilization problem of discrete stochastic systems.
