Vadim Azhmyakov

On the Maximum Principle for Impulsive Hybrid Systems

In this contribution, we consider a class of hybrid systems with continuous dynamics and jumps in the continuous state (impulsive hybrid systems). By using a newly elaborated version of the Pontryagin-type Maximum Principle (MP) for optimal control processes governed by hybrid dynamics with autonomous location transitions, we extend the necessary optimality conditions to a class of Impulsive Hybrid Optimal Control Problems (IHOCPs). For these problems, we obtain a concise characterization of the Impulsive Hybrid MP (IHMP), namely, the corresponding boundary-value problem and some additional relations. As in the classical case, the proposed IHMP provides a basis for diverse computational algorithms for the treatment of IHOCPs.

Robust control for a class of continuous-time dynamical systems with sample-data outputs

This paper addresses the problem of robust control for a class of nonlinear dynamical systems in the discrete-continuous 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.

On the hybrid LQ-based control design for linear networked systems

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.

Practical stability of control processes governed by semi-explicit DAEs

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.

On the robust control design for a class of nonlinear affine control systems: The invariant ellipsoid approach

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.

On applications of Attractive Ellipsoid Method to dynamic processes governed by implicit differential equations

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.

Optimal control of sliding mode processes: A general approach

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).

Attractive Ellipsoid Method with Adaptation

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.

Sample Data and Quantifying Output Control

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.

Practical stabilization of a class of switched systems: dwell-time approach

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.