A Relaxation-Based Approach to Optimal Control of Hybrid and Switched Systems: A Practical Guide for Engineers

By Vadim Azhmyakov

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.

• Illustrates the use of the relaxation approaches in engineering optimization
• Presents application of the relaxation methods in computational schemes for a numerical treatment of the sophisticated hybrid/switched optimal control problems
• Offers a rigorous and self-contained mathematical tool for an adequate understanding and practical use of the relaxation techniques
• Presents an extension of the relaxation methodology to the new class of applied dynamic systems, namely, to hybrid and switched control systems

• Language: English
• Publisher: Elsevier Science
• Release date: Feb 14, 2019
• ISBN: 9780128147894

Vadim Azhmyakov

Vadim Azhmyakov graduated in 1989 from the Department of Applied Mathematics of the Technical University of Moscow. He gained a Ph.D. in Applied Mathematics in 1994, and a Postdoc in Mathematics in 2006 of the EMA University of Greifswald, Greifswald, Germany. He has experience in Applied Mathematics: optimal control, optimization, numerical methods nonlinear analysis, convex analysis, differential equations and differential inclusions, engineering mathematics; and Control Engineering: hybrid and switched dynamic systems, systems optimization, robust control, control over networks, multiagent systems, robot control, Lagrange mechanics, stochastic dynamics, smart grids, energy management systems.

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Preface

This book is written as an application-oriented extension of the conventional relaxation theory to hybrid and switched optimal control problems. Modern optimal control theory is concerned with the analysis and design of sophisticated dynamical systems, where the aim is at steering such a system from a given configuration to some desired target by minimizing a suitable performance. Nowadays this theory constitutes a powerful design methodology for the computer-oriented development of several types of high-performance controllers. General optimal control problems associated with various types of advanced control systems have been comprehensively studied due to their natural engineering applications. The material discussed in this research monograph is the result of the author's work at the Ernst-Moritz-Arndt University of Greifswald (Greifswald, Germany), the Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico Nacional (Mexico City, Mexico), and the University of Medellin (Medellin, Colombia). The main purpose of this book is to propose a unified approach to effective and numerically tractable relaxation schemes for optimal control of hybrid and switched systems. We study several generic classes of hybrid and switched dynamic models and corresponding optimal control problems in the context of suitable relaxation schemes.

Relaxing the initial problem has various meanings in applied mathematics, depending on the areas where it is defined, depending also on what one relaxes (a functional, the underlying space, etc.). In the context of an optimal control problem, when dealing with the minimization of an objective functional, the most common way of looking at relaxation is to consider the lower semicontinuous hull of this functional determined on a convexification of the set of admissible controls. The concept of relaxed controls was introduced by L.C. Young in 1937 under the name of generalized curves and surfaces. It has been used extensively in the professional literature for the study of diverse optimal control problems. It is common knowledge that a real-world optimal control problem does not always have a (mathematical) solution. On the other hand, the corresponding relaxed problem has an optimal solution under some mild assumptions. In practice, this solution can be considered as a suitable approximation for the sophisticated initial problem. In the absence of the so-called relaxation gap, the generalized problem is of prime interest for the initial optimal control problem. In this case, the minimal value of the objective functional in the initial problem coincides with the minimum of the objective functional in the relaxed problem. Therefore, in that situation, an adequately relaxed problem can be used as a theoretical fundament for adequate numerical solution algorithms for the initial problem. When solving optimal control problems with ordinary differential equations, we deal with functions and systems which, except in very special cases, are to be replaced by numerically tractable approximations. In contrast to the conventional optimal control problems an effective implementation of adequate computational schemes for hybrid/switched system optimization is predominantly based on the relaxed controls. Therefore, our aim is to consider the relaxations of the hybrid and switched optimal control problems in a close methodological relationship to the corresponding numerical methods and possible engineering applications.

Recall that various types of hybrid and switched control systems and the related optimal control problems have been comprehensively studied in the past several years due to their important engineering applications. Let us mention here some real-world applications from the mobile robot technology, intelligent automotive control, modern telecommunications, process control, and data science. We first give an extensive overview of the existing (conventional and newly developed) relaxation techniques associated with the conventional systems described by ordinary differential equations. Next we construct a self-contained relaxation theory for optimal control processes governed by various types (subclasses) of general hybrid and switched systems. Note that due to the extreme complexity of hybrid/switched dynamic systems this construction is a challenging analytic and computational problem and cannot be considered as a simple theory/fact transfer from the conventional optimal control to hybrid and switched cases. Let us also note that the book we propose contains all mathematical tools that are necessary for an adequate understanding and use of the sophisticated relaxation techniques. All in all, this manuscript follows the engineering and numerical concepts. However, it can also be considered as a mathematical compendium that contains all the necessary formal results and some important algorithms related to the modern relaxation theory. This fact makes it possible to use this book in systems engineering (specifically in electrical, aerospace, and financial engineering) and in practical systems optimization.

The target audience of this book includes but is not restricted to academic researchers and PhD candidates from Electrical Engineering and/or Applied Mathematics faculties (technical and regular universities, academic research centers) and R&D departments of electrical/electronic companies (research engineers, developers). This book can also be useful for economic schools and mathematically oriented economists. Parts of this book can also be used for advanced courses such as Dynamic Optimization, Optimal Control of Modern Dynamic Systems, and Advanced Mathematics for Engineers. The book can also be included into PhD qualification programs in control engineering, applied mathematics, advanced computer science, and mathematical economics. We expect that this monograph will be useful to the interested graduate students and some undergraduate students with sufficient knowledge of functional analysis, mathematical optimization and dynamic systems. The book can also be considered as a complementary text to graduate courses in applied mathematics. We assume that the reader has knowledge of analysis and linear algebra, while a little more is presupposed from nonlinear analysis, optimization theory, and optimal control. We decided to use a style with detailed and often transparent proofs of the significant results. Of course, the book only claims to present an (extended) introduction to the relaxation theory for hybrid/switched optimal control problems and possible applications. We have made an attempt to unify, simplify, and relate many scattered results in the literature. Some of the topics discussed here are new, others are not. Therefore the book is not a collection of research papers, but it is a monograph to present recent developments of the theory that could be the foundations for further developments.

Many people have influenced the contents and final presentation of this book, and I am grateful to all of them. I would like to thank Professor W.H. Schmidt (Ernst-Moritz-Arndt University of Greifswald, Greifswald, Germany), Professor M.V. Basin (Universidad Autonoma de Nuevo Leon, Monterrey, Mexico), and Professor A. Poznyak (Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico Nacional, Mexico City, Mexico). I also would like to express my warm gratitude to Professor M. Egerstedt, Professor Y. Wardi, and Professor E. Verriest (Georgia Institute of Technology, Atlanta, USA). Many methodological aspects of the book are finally improved due to several professional discussions with my Master and PhD students from Germany, Mexico, and Colombia.

I wish to thank Professor M.V. Basin (Universidad Autonoma de Nuevo Leon, Monterrey, Mexico), Professor St. Pickl (University of Bundeswehr Munich, Munich, Germany), and Professor M. Shamsi (Amirkabir University of Technology, Tehran, Iran) for reading the first version of this monograph.

Finally, I wish to express my appreciation to Elsevier for their excellent and accomplished handling of the manuscript, their understanding, and their patience.

Chapter 1

Introduction and Motivation

Abstract

The increasing complexity associated with many formal models from the real-world engineering applications, including flight control, autonomous robot and vehicle guidance, automation of complex technological processes, power electronics, process control in sensor-rich environments, and control of biological and social systems, has far reaching implications for modern systems design. As an example, switched, hybrid, and in general complex interconnected dynamical systems, interacting among themselves and with remote users over control and communication networks, introduce a whole new set of system-level challenges. In this new situation the main classic objectives of the mathematical systems theory, such as stability, optimality, performance, identification, and robustness, are being complemented with a number of the conceptually new theoretical and technical modifications and extensions. These necessary theoretic extensions also cause the necessity of a diversification of the classic relaxation theory for optimal control. This chapter is devoted to the informal and formal concepts of the general hybrid systems (HSs) and switched systems (SS). Moreover, we give an initial formulation of a hybrid optimal control problem (HOCP)/switched optimal control problem (SOCP) and briefly discuss the related relaxation concepts. Some historical remarks close this chapter.

Keywords

hybrid and switched control systems; relaxation approach; general systems theory

1.1 Optimal Control of Hybrid and Switched Dynamic Systems

Modern optimal control theory (OCT) is concerned with the analysis and design of sophisticated dynamical systems, aiming at steering such a system from a given configuration to some desired target by minimizing a suitable performance. Nowadays this theory constitutes a powerful design methodology for the computer-oriented development of several types of high-performance controllers. General optimal control problems (OCPs) associated with various types of advanced control systems have been comprehensively studied due to their natural engineering applications.

Optimization of sophisticated constrained dynamic models (including hybrid systems [HSs] and switched systems [SSs]) is, nowadays, not only a mathematical theory buy also a mature and relative simple design methodology for the practical development of several types of modern controllers. Consideration of the general OCPs usually incorporates two conceptual sophisticated aspects: the presence of various additional constraints and the existence of an optimal solution of an OCP under consideration. These two basic aspects determine the basic facets in optimization of the dynamics systems described by ordinary differential equations (ODEs). It is easy to understand that in the context of a hybrid optimal control problem (HOCP)/switched optimal control problem (SOCP) the same theoretic points, namely, constraints and existence of an optimal solution, obtain an increasing complexity. This fact is a simple consequence of the following observation: the general HSs and SSs constitute a formal generalization of the ODEs involved in control systems. The same is evidently true for the corresponding OCPs.

It is readily appreciated that the real-world control systems have a corresponding set of constraints; for example, inputs always have maximum and minimum values, and states are usually required to lie within certain ranges. The second problem mentioned above, namely, the existence of an optimal solution, can in fact be denoted as a millennium problem from the classic OCT. A constructive consideration of this sophisticated problem was finally realized in the framework of the relaxation theory (RT). Of course, in a concrete OCP one could proceed by ignoring the theoretically complicated existence question, ignore various state-control constraints, and hope that no serious consequences result from this approach. This simple procedure may be sufficient at times. It is generally true that optimal levels of a suitable performance are associated with operating on, or near, constraint boundaries. Thus, a control engineer really cannot ignore constraints without incurring a performance penalty. Since a relaxed OCP usually possesses some newly determined admissible sets of constraints, the above observation is also true for relaxed OCPs.

For the classic as well as hybrid OCPs the suitable relaxation procedures also imply new numerical methodologies. Relaxed OCPs involve specific computational treatment and the corresponding practical algorithms. Recently, the problem of effective numerical methods for constrained systems optimization has attracted a lot of attention, thus both theoretical results and applications were developed. The handling constraints in an original or relaxed form in practical systems design are an important issue in most, if not all, real-world applications. Our book deals with some specific classes of (finite) hybrid and switched control systems and with the corresponding OCPs. We give in fact a reformulation of the conventional and extended RT for the case of these OCPs. Taking into consideration the wide use of the relaxation schemes in numerical treatment of the classic OCPs, we are also interested to elaborate consistent computational algorithms for HOCPs/SOCPs (OCP) in the presence of constraints. We study classes of OCPs with convex (or convexified) costs functionals. The structure of the admissible control functions we consider in this book is mainly motivated by some important practical control applications as well as by the widely applicable modern quantization procedure associated with the original system dynamics. We refer to the interesting self-closed results on the convex-like and generalized dynamics. For example, the formal treatment of linear quadratic OCPs was based on the backward solutions of the Riccati differential equations, and the optimum had to be recomputed for each new final state. Computation of nonlinear gains using the Hamilton–Jacobi–Bellman (HJB) equation and the convex optimization techniques has also been performed.

On the other hand, the existing optimization approaches to the relaxed dynamics are not sufficiently advanced for the hybrid and switched types of OCPs. In our book, we propose analytic as well as numerical methods based on a combination of the convex-like relaxation schemes and the first-order projection approach. Also, it should be noted already at this point that a computational algorithm we propose can be effectively used in a concrete control synthesis phase associated with a practical engineering design of hybrid and switched dynamic systems. Recall that the general HSs and SSs constitute a class of mathematical abstractions where two types of dynamics are present, i.e., continuous and discrete event dynamic behavior. In order to understand how these systems can be operated efficiently, both aspects of the actual dynamics have to be taken into account during the optimization and the corresponding control design procedure. The nonlinear systems we study can be interpreted as a particular family of the general systems with the state-driven (HSs) and time-driven (SSs) location transitions.

Dynamic processes described by HSs/SSs constitute an adequate and more detailed modeling approach to many important real-world problems in engineering, social science, and bioscience. Variety of suitable mathematical abstraction for control and measurements (engineering instrumentation) can be characterized by this novel modeling framework. These include, for example, the cost of hardware implementation, measured for example not only by computational requirements such as speed and memory, but also by communication requirements of complex and costly systems. In general, a sophisticated interplay of complex nonlinear dynamic objects, such as autonomous robots/vehicles, airplanes, and satellites, in the presence of the associated communication networks is a prevailing attribute of many modern applicable control systems. Let us also note that the switched discrete-continuous structure of the generic HSs and SSs makes it possible to use these dynamic models in the (optimal) decision science. Note that the necessary analysis, optimization, and adequate design procedures for these systems have been recognized as major problems in modern control engineering.

Let us give a qualitative illustration (the structural scheme) of a generic HS (Fig. 1.1).

Figure 1.1 A general structure of an HS.

(a natural number) controllers and actuators that are triggered by a supervisor. We later give a formal definition of a wide class of HSs under consideration.

Recently a vast body of research on mathematical models for hybrid and switched control systems has been produced, drawing its motivation from the fact that many modern application domains involve complex systems, in which subsystem interconnections, mode transitions, and heterogeneous computational devices are presented. A common example of an interconnected (hybrid) dynamic system is given by the following qualitative model of a batch reactor (see Fig. 1.2).

Figure 1.2 Structural scheme of a batch reactor.

An adequate detailed model of the abovementioned complexly interconnected technical objects is usually characterized by switched or hybrid dynamics and, moreover, includes an associated (local or global) communication network. Note that in the case of the interconnected control-communication systems the real discrete transitions of the state vectors (switches) can be triggered not only by simple discrete control commands but also autonomously (for example, by an autonomous location transition mechanism). An illustration of an HS that contains a communication network is given in Fig. 1.3.

Figure 1.3 An HS with the communication network.

Evidently, an optimal control synthesis for systems presented in Figs. 1.1–1.3 is a very sophisticated task. However, we are optimistic as regards the use of the relaxation bases control design schemes. This optimism is due to the solid mathematical and computational foundations of the classic optimization methodologies in complex (real Banach) spaces and also to the clear engineering interpretation of the switched and hybrid dynamical models, their reliability, and the existence of well-established examples and simulation results. Evidently, a possible generalization of the relaxation approach for HOCPs and SOCPs is an extremely interesting theoretic and numeric problem which has a high potential in engineering applications. The aim of our book is to extend the theoretic and numeric aspects of the classic RT to the powerful and effective computational schemes associated with the abovementioned classes of optimal control systems.

Let us now pay attention to some formal concepts. This brief issue illustrates some conceptual aspects mentioned above. We start by introducing a variant of a useful definition of an HS under consideration.

Definition 1.1

An HS is a 7-tuple , where

is a finite set of discrete states (called locations);

is a family of smooth manifolds, indexed by ;

is a set of admissible control input values (called control set);

, is a family of maps

where is the tangent bundle of ;

is the set of all admissible control functions;

is a family of adjoint subintervals of such that

is a subset of Ξ, where

An HS from . Note that in contrast to the general definition of an HS, the control set U is also independent of a location. We generally assume that U is a compact set. Let us assume here that

:

from F are differentiable;

•  

such that

from F are Lipschitz on

one can also define the switching set

from location q , indicate the lengths of time intervals on which the system can stay in location q. We say that a location switching from q occurs at a switching time .

, where

is not defined a priori. A hybrid control system remains in location, i.e.,

.

Definition 1.2

Let be an admissible control for an HS. Then a continuous trajectory of HSs is an absolutely continuous function

such that

and

•  

for almost all and all ;

•  the switching condition

holds if .

The vector

is called a discrete trajectory of the hybrid control system.

) are defined for

Under the above assumptions for the given family of vector fields Ffor an HS are also uniquely defined. Therefore, it is reasonable to introduce the following concept.

Definition 1.3

Let an HS be defined as above. For an admissible control , the triplet

where τ is the set of the corresponding switching times and and are the corresponding continuous and discrete trajectories, respectively, is called a hybrid trajectory of the HS.

from Definition 1.2. Finally we are ready to formulate the following Mayer-type HOCP:

(1.1)

Note that the main dynamic optimization problem, namely, problem (1.1), constitutes a useful abstract framework for the concrete engineering systems optimization.

Analogously to Definition 1.1 one can introduce a concept of an SS. In this section we only give a definition of an affine SS (ASS). The general concept will be studied in Chapter 6.

Definition 1.4

An ASS is a 7-tuple , where

is a finite set of indices;

•  

is a family of state spaces such that ;

is a set of admissible control input values (called control set);

•  

are families of uniformly bounded on an open set Carathéodory functions

is the set of admissible control functions introduced above;

•  Ψ is a subset of Ξ, where

An admissible trajectory associated with an ASS is an absolutely continuous function such that

•  

is an absolutely continuous function on continuously prolongable to ;

•  

for almost all times , where is a restriction of the chosen control function on the time interval .

Using Definition 1.4 we can formulate an OCP involving an SS.

(1.2)

In spite of a visual identity of the OCPs (1.1) and (1.2), these two problems are conceptually different. In Section 1.2 we will discuss the conceptual difference between problems (1.1) and (1.2).

The generic HOCP , namely, in the case of a conventional OCP, the existence of an optimal solution is a very complicated question. As mentioned above, this existence problem is in fact solved (for the conventional case) in the framework of the classic RT. Additionally to this abstract result (existence), the celebrated RT provides various constructive numerical approaches to the initial and generalized OCPs. Summarizing, we can say that the possible theoretic extension of the conventional RT to the new classes of systems, namely, to hybrid and switched dynamic models, can be useful in the practical (optimal) control design. As a main result of the presented book, one can indicate the creation of the analytic basis for practically relaxed optimal control strategies to be associated with the wide classes of modern hybrid/switched dynamical systems.

1.2 Questions Relaxation Theory Can Answer

Consider an HS from . This fact makes it clear that the switching times

is the continuous trajectory of the HS introduced in . For example, the formal proof of the main optimality tool in OCT of HSs, namely, the celebrated hybrid Pontryagin maximum principle (HPMP), is technically more complex compared with the classic case. The same is also true with respect to possible generalizations of the usual techniques of the RT in the framework of HSs.

On a conceptual level the switching times τ introduced in , we next define the characteristic function

Using these introduced characteristic functions, we can rewrite the differential equations from Definition 1.2 in the following compact form:

(1.3)

. Under the basic technical assumptions for the family of vector fields F (see , i.e.,

Evidently, this dependence is highly nonlinear and all the usual first-order techniques from the classic control/systems theory are affected by this fact. For example, a simple linearization technique is being converted to a mathematically nontrivial procedure. Let us use the following compact notation:

Following the main constitutes an additional state of the HS under consideration.

On the other side the SS from Definition 1.4 does not possess any additional state. An SS can formally also be presented in the form (1.3). However, the characteristic function

and needs to be interpreted as additional systems input. Summarizing we can conclude that the conceptual difference between HSs and SSs consists in a formal determination of the corresponding switching mechanism. In (1.3) this switching mechanism (determined by the set of switching times τrepresents a posteriori information, namely, the system state, and in the case of the SSs this vector is an additional control input (a priori information). As mentioned in Section 1.1, HSs and SSs have a similarity from the point of view of the visibility of the formal representation (for example, by (1.3)) but have strong conceptual differences.

Recall that the classic RT for OCPs is usually used for two principal tasks. The first is establishment of the existence of an optimal solution to the given (usually sophisticated) OCP. This use of the RT is more known for the experts in optimization theory as well as for research control engineers, computer scientists, and researchers in mathematical economy. The second benefit which the conventional RT provides is related to the constructive computational schemes for OCPs. Note that this second application possibility of the RT is far less known for the experts and researchers from control engineering and practical optimization.

Questions RT can answer in the context of HSs/SSs and the corresponding OCPs are in fact the same. We will discuss in this book the generic existence questions for hybrid and switched OCPs as well as generalize the relaxation-based numerical approaches. This knowledge transfer from classic OCT to hybrid and switched cases cannot be considered as a simple formal transfer. The conceptually new dynamic aspects of HSs and SSs in comparison to the conventional ODEs involving control systems imply some mathematical challenges and the necessity of additional theoretical development and effort for a successful knowledge transfer mentioned above.

1.3 A Short Historical Remark

It is common knowledge that an OCP does not always have a solution (see, e.g., [159,321,154]). On the other hand, the corresponding relaxed problem has, under mild assumptions, an optimal solution [127,167,279]. This solution can be considered, in practice, for constructing an approximating solution for the initial problem. In the absence of the so-called relaxation gap (see, e.g., [246,247,121]) the relaxed problem is of primary interest for the initial OCP. In this case the minimal value of the objective functional in the initial OCP coincides with the minimum of the objective functional in the relaxed problem. Therefore, in this situation a solution of the relaxed problem can be used as a basis for constructing a minimizing sequence for the initial problem [302,154].

Extensions in problems of variational calculus, beginning with the idea of Hilbert, were realized many times for various purposes. Let us consider the following abstract regular variational problem:

in case we generalize the notion of solution in an appropriate sense?" Today we know the answer is positive. Under some natural assumption (see, e.g., [328]) for a bounded region

from the Sobolev space

The first constructive investigation on this subject was a work of N.N. Bogoljubov [94]. The concept of relaxed controls was introduced by L.C. Young in 1937 under the name of generalized curves and surfaces [326]. It has been used extensively in the literature for the study of diverse OCPs [321,127,173,192,154,279]. As Clarke points out in [130], a relaxed problem is, in general, the only one for which existence theorems can be proved and, for this reason, there are many who deem it the only reasonable problem to consider in practice. However, though relaxation of a problem is important in order to prove existence theorems, one is also interested in proving convergence of numerical approximation