Control of switching constrained systems

We consider a particular class of hybrid systems characterized by a finite state machine and a set of discretetime linear dynamical systems, each corresponding to a state of the machine. The hybrid control problem addressed consists of maintaining the output of the systems in a given subset of the output space, independently of the state of the FSM. Necessary and sufficient conditions are given for the existence of a solution. Introduction We consider a hybrid system (see e.g. (Morse 1997)) consisting of a finite state machine (FSM) and a set dynamical systems, each corresponding to a state of the FSM. We assume that, at each location of the FSM, the actual configuration of the dynamical system is known. The time at which a transition occurs between two different states of the FSM (also called switching time) not known a priori but is determined by an external uncontrollable event. The simpler case where the time of occurrence of the transitions is not completely unknown but can be predicted is also considered. The control problem we solve in this paper consists in maintaining the output of the systems in a given subset of the output space, independently of the state of the FSM. We work in an infinite time horizon framework. This model can be used to represent a number of control problems of practical interest. In our case, the motivation to study this formulation comes from the engine control problem in automotive applications. Our research group has been involved with the formulation of the engine control problem as a hybrid system control problem and has developed guaranteed performance efficient algorithms for its solution (Balluchi et al. 1997, 1998). The plant to be controlled consists of the engine, modeled as a FSM, and of the powertrain, modeled as a linear dynamical system. A goal of engine control is to ensure comfort during the entire operation of the automobile. Acceleration oscillations caused by torque variations on the powertrain are source of discomfort for the driver. A possible control objective is to maintain such oscillations, which can be expressed as a linear combination of state variables of the powertrain, below a given threshold. We assumed in (Balluchi et al. 1997, 1998) that there is no gear change during the operation of the automobile and, hence, the powertrain model is the same for all the regions of operation of the engine. A gear change has the effect of changing the "structure" of the model of the powertrain dynamics, in the sense that the matrices describing the linear system change. A first step to solve the more general automobile control problem when gear change is considered is to "relax" the hybrid control problem as follows: the plant is the powertrain equipped with a gear change mechanism and the control input is the torque generated by the engine. The plant can then be represented by a hybrid model with an FSM part whose states correspond to a particular gear, and the dynamical system part corresponding to the appropriate powertrain dynamics for that gear. Then, the hybrid control problem described above is an adequate representation for the relaxed control problem. Switching systems have been considered e.g. in (Marro and Piazzi 1993), (d’Alessandro and De Santis 1996) in the case where the switching instants are known a priori. In particular, in (d’Alessandro and De Santis 1996), an optimal solution is derived for systems with linear state constraints and linear cost functional. In (Marro and Piazzi 1993), the problem of robust regulation without error transients is solved. Some interesting control problems are solved in (Sontag 1996) when the transitions between two different states of the FSM are enabled by some guard conditions that may depend on the input and/or on the state of the dynamical system. In this paper, we derive necessary and sufficient conditions for the existence of a controller which maintains the output of the switching system in a given set, in the case where the switching instants are unknown a priori. Similar conditions can be obtained using the general results of (Tomlin, Lygeros and Sastry 1998). Our results are less general since they apply to a particular class of hybrid systems but have the advantage of exploiting the structure of the FSM. This allows to simplify the procedure for the determination of a solution and to derive convergence conditions which are unknown in the general case. The paper is organized as follows, in Section 2, the problem of controlling a switching system is formulated 15 From: AAAI Technical Report SS-99-05. Compilation copyright © 1999, AAAI (www.aaai.org). All rights reserved. and a solution is given in the case of unknown switching times. In Section 3, the problem is relaxed to the case where the switching times can be predicted. In Section 4, possible extensions of our results are described. Conclusions are offered in Section 5. Switching constrained systems The transition structure of an FSM determines the reachability of its states. A connected FSM is such that, for all state bi-partitions, there is always at least a transition from one set of the bi-partition to the other. Without loss of generality, we assume that the FSMs considered in this paper are connected. Consider a connected FSM F with state set S = {Si,i = 1,... ,N}. To each state Si of the FSM we associate a discrete time dynamical system (also called for simplicity configuration Si) described by x(t + 1) = Aix(t) + Bin(t) (1) y(t) = C~x(t) + D~u(t) where t E N, x E ]~, u E ]~m, y E ]~P, and A~, Bi, C~, Di (i = 1,... ,N) are matrices of suitable dimensions. The state evolution of the FSM in time is described by the function s : 1~ --~ S, so that s(t) denotes the state of the FSM at time t. Let to E N be the initial time. Only one switching is allowed at any t E N and the switching times are supposed to be not known a priori. The system described by the FSM F, its evolution in time s(t) and the dynamical systems (1) is called switching system. Our goal is to find under which conditions it is possible to maintain the output y(t) in a given set ~t (where is a region of ]~v), for all t _> to, i.e.