Issues in the design of switched linear control systems: A benchmark study

In this paper we present a tutorial overview of some of the issues that arise in the design of switched linear control systems. Particular emphasis is given to issues relating to stability and control system realisation. A benchmark regulation problem is then presented. This problem is most naturally solved by means of a switched control design. The challenge to the community is to design a control system that meets the required performance specifications and permits the application of rigorous analysis techniques. A simple design solution is presented and the limitations of currently available analysis techniques are illustrated with reference to this example. 1. Introductory remarks Recent years have witnessed an enormous growth of interest in dynamic systems that are characterised by a mixture of both continuous and discrete dynamics. Such systems are commonly found in engineering practice and are referred to as hybrid or switching systems. The widespread application of such systems is motivated by ever increasing performance requirements, and by the fact that high performance control systems can be realised by switching between relatively simple LTI systems. However, the potential gain of switched systems is offset by the fact that the switching action introduces behaviour in the overall system that is not present in any of the composite subsystems. For example, it can be easily shown that switching between stable sub-systems may lead to instability or chaotic behaviour of the overall system, or that switching between unstable sub-systems may result in a stable overall system. In this paper we present a tutorial introduction to the design of switched linear systems. We begin by discussing how switching arises naturally in many situations. Examples include: the design of control systems for plants that are themselves characterised by switching action (i.e. plants with gears); the design of reconfigurable (fault tolerant) control systems; a switched controller that combines the advantages of several LTI controllers; and using switching to improve the transient response of adaptive control systems. We then discuss the issues in the design of such systems. Of primary practical importance are the issues of asymptotic stability, and issues concerning the realisation of switched linear controllers (and the associated transient response). Each of these issues is illustrated by means of simple illustrative examples. The final part of the paper presents a wind turbine regulation problem. This problem is can be solved using a switched linear controller; the challenge to the community is to design such a controller, while providing theoretical guarantees concerning the issues raised in the paper. A control design is presented which is characterised by a number of switches, and exhibits control performance that is superior to single LTI and non-linear control design. Unfortunately, while this control system works well, both in simulation, and in practice, no guarantees can be given regarding performance and stability. In particular, the benchmark solution provides a basis for evaluating analysis techniques for analysing the stability of switched systems. In this context, a fundamental contribution of this paper is to document the limitations of these techniques and to motivate the study of theoretical issues that arise in the context of real industrial examples. 2. The Need for Switching Switching between a number of control strategies has long been a valuable tool in the design of automatic control systems. Figure 1: Schematic of a switching system. ‘S’ denotes a supervisory algorithm that controls the switching between the various controllers. The need for (supervisory) switching arises for many reasons, some of which are listed below (Narendra et al., 1996, Goodwin, 2001): (i) Plant dynamics: Many physical systems can be represented by switching or interpolating between locally valid models. Controllers that encompass switching are often a natural method for dealing with such systems. (ii) Performance: Switching between a number of control structures automatically results in control systems that are no longer constrained by the limitations of linear design. It is therefore not surprising that switching based control strategies can result in algorithms that offer significant performance improvements over traditional linear control. For example, different controllers may be encoded within a single structure, resulting in a control system with enhanced functionality by exploiting the benefits of each of the constituent controllers. (iii) Robustness: An important motivation for designing switching control strategies is to ensure robust control performance in the presence of component failure. For example, if an operating condition changes (a sensor failure, a change in sampling rate, or even a controller failure), then a more appropriate control action may be initiated by the supervisor. In extreme cases, switching to a new controller, or even continuous switching between a number of controllers, may be required to maintain closed loop stability. (iv) Adaptive control: Much of the recent interest in switching systems has been motivated by developments in supervisory adaptive control. It is generally accepted that multiple-models, and multiple controllers are required in reconfigurable systems to detect and respond to changes in plant parameters and structure. The paradigm, multiple-models, switching and tuning, is based upon this assumption; namely, that by switching between several adaptive controllers, each initialised to different states, a rapidly converging adaptive control system may be constructed that is capable of coping with both unknown and time-varying parametric uncertainty (Narendra and Balakrishnan, 1994b, Narendra and Chen, 2002). (v) Decentralised design: Many complex engineering systems are designed in a decentralised manner. Each component sub-system is usually designed in relative isolation, and the overall system is constructed by combining each of the sub-systems by means of some appropriate supervisory logic. Often, this approach leads to switched linear control systems. (vi) Control system constraints: Constraints are a common feature of practical control systems. Switching between several controllers is often a natural way of satisfying such constraints. Examples in this context are given in (Goodwin, 2002). Despite the prevalence of switching based control algorithms in engineering design, such systems have only recently, in the context of the more general hybrid systems, attracted the interest of the academic community. Typically, switched linear systems are modelled by vector differential equations of the following form, 0 0 , x ) t ( x , u ) t ( B x ) t ( A x = + = where n R ) t ( u ), t ( x ∈ and where matrices A(t), B(t), are constructed by switching between a set of matrices. Formally we define a switched linear system, referred in the sequel to as the switching system, and its fundamental properties as follows (Shorten et. al., 2002). The switching system: Consider the time-varying system x ) x(t , B(t)u(t) A(t)x (t) x : 0 0 = + = (1) where p n R R ∈ ∈ u(t) , x(t) and where matrices A(t), B(t), are constructed by switching between a set of stable matrices, , R B }, A ,...., A , A { ) t ( A n n i m × ∈ ∈ 2 1 p n i m R B }, B ,...., B , B { ) t ( B × ∈ ∈ 2 1 and where for each time t the matrices A(t), B(t) equals one and only one matrix Ai , Bi in the above sets. Typically the matrix pairs (Ai,Bi) are chosen such that equation (1) is bounded-input bounded-output stable for all fixed values of t (Khalil, 1992). This corresponds to switching between a number of stable systems. Further, we assume that once the matrices A(t), B(t) assume the values Ai , Bi they assume these values for an interval of time τ where > ≥ τ τ where the constant τ is arbitrarily small and independent of i. For instance, suppose that the dynamics in (1) are given by u B x A x i i + = over the finite time interval ) t , t [ 1 + . At time + γ the system switches and the dynamics in the following interval ) t , t [ 2 1 + + are given by u B x A x j j + = . We assume that the state vector x(t) does not jump discontinuously at + γ , and hence the initial state at time + γ for u B x A x j j + = is the terminal state of u B x A x i i + = . If we further assume that u = Kix then the following convenient representation of (1) is obtained, 0 0 , x ) t ( x x ) t ( A x : = = (2) where } A ,...., A , A { ) t ( A m 2 1 ∈ , and i i i i B K A A + = . We refer to systems (1) and (2) interchangeably as the switching system. Associated with the switching system (1) we also define the i constituent system, the switching sequence SW and a switching signal ρ The i constituent system: Consider the linear time-invariant system u B x A x : i i + = i (3) with the switching system defined above. Then Σ is referred to as the i constituent system of (1) . The switching sequence SW: In the spirit of (Branicky, 1994) one can associate the following switching sequence with (1) SW = (i0, t0),(i1,t1),.....,(iN,tN),....., where the SW sequence may or may not be infinite. The th switching interval is the th element of this sequence and defines the evolution of (1) as follows. The switching system evolves according to u B x A : i