Lyapunov functions a hierarchical nonlinear control strat- egy is developed that stabilizes a given nonlinear system by stabilizing a collection of nonlinear ...
TP11 16:20
Proceedings of the 37th IEEE Conference on Decision & Control Tampa, Florida USA. December 1998
Nonlinear System Stabilization via Stability-Based Switching Alexander Leonessa, Wassim M. Haddad, and VijaySekhar Chellaboina School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150 Abstract In this paper a nonlinear control-system design framework predicated on a stability-based switching controller architecture parameterized over a set of system equilibria is developed. Specifically, using equilibria-dependent Lyapunov functions a hierarchical nonlinear control strategy is developed that stabilizes a given nonlinear system by stabilizing a collection of nonlinear controlled subsystems. The switching nonlinear controller architecture is designed based on a generalized lower semicontinuous Lyapunov function obtained by minimizing a potential function over a given switching set induced by the parameterized system equilibria. The proposed framework provides a rigi.. .us alternative to designing gain scheduled feedback controllers and guarantees local and global closed-loop system stability for general nonlinear systems.
1. Introduction Since all physical systems are inherently nonlinear with system nonlinearities arising from numerous sources including, for example, friction (eg., Coulomb, hysteresis), gyroscopic effects (eg., rotational motion), kinematic effects (eg., backlash), input constraints (eg., saturation, deadband), and geometric constraints, plant nonlinearities must be accounted for in the control-system design process. However, since nonlinear systems can exhibit multiple equilibria, limit cycles, bifurcations, jump resonance phenomena, and chaos, general nonlinear system stabilization is notoriously hard and remains an open problem. Control system designers have usually resorted to Lyapunov methods [l-31 in order to obtain stabilizing controllers for nonlinear systems. In particular, for smooth feedback, Lyapunov-based methods were inspired by Jurdjevic and Quinn [4]who give sufficient conditions for smooth stabilization based on the ability of constructing a Lyapunov function for the closed-loop system [5]. Unfortunately, however, there does not exist a unified procedure for finding a control Lyapunov function candidate that will stabilize the closed-loop system for general nonlinear systems. Recent work involving differential geometric methods [6-111 has made the design of controllers for certain classes of nonlinear systems more methodical. Such frameworks include the concepts of zero dynamics and feedback linearization and require that the system zero dynamics are asymptotically stable assuring the existence of globally defined diffeomorphisms to transform the nonlinear system
into a normal form [7]. For this class of systems, feedback linearization techniques usually rely on cancelling out system nonlinearities using feedback and may therefore lead to inefficient designs since feedback linearizing controllers may generate unnecessarily large control effort t o cancel beneficial system nonlinearities [12]. Backstepping control has also recently received a great deal of attention in the nonlinear control literature [13-151. The popularity of this control methodology can be explained in a large part due to the fact that it provides a systematic procedure for finding a control Lyapunov function for nonlinear closed-loop systems. Furthermore, the controller is obtained in such a way that the nonlinearities of the dynamical system, which may be useful in reaching performance objectives, need not t o be canceled, as in state or output feedback linearization techniques. However, once again this approach is limited t o strict-feedback systems [15,16]. To address optimality issues within nonlinear controlsystem design, one may consider an optimal control problem in which a performance functional is minimized along the closed-loop system trajectories. This problem leads to the maximum principle [17,18] which usually provides open-loop optimal control laws characterized via nonlinear two-point boundary-value problems requiring iterative solution schemes. Alternatively, to obtain feedback optimal controllers, one may formulate the optimal control problem as a dynamic programming problem [19] by considering a value function [20],or return function [17], that minimizes a cost functional among all possible system trajectories. In this case, the value function is given by the solution to the Hamilton-Jacobi-Bellman equation [17,18]. Computational methods for solving the Hamilton- JacobiBellman partial differential equation have not been developed to a level comparable to solving Riccati equations arising in linear optimal control problems and in general is very difficult to solve. In fact, solutions may not even exist unless one allows a generalized notion of viscosity solutions [21]. In order to avoid the complexity in solving the Hamilton-Jacobi-Bellman equation one may consider an inverse optimal control problem [12,16,22] where one does not attempt to minimize a given cost functional, but rather, minimizes a derived cost functional. The basic underlying idea of inverse optimality is the fact that the steady-state solution to the Hamilton-Jacobi-Bellman equation is a Lyapunov function for the nonlinear controlled system [23,24]. However, in this case the complexity of solving the Hamilton-Jacobi-Bellman equation is shifted to finding a Lyapunov function for the closedloop system limiting this approach to strict-feedback systems. Furthermore, the performance of inverse optimal controllers can be arbitrarily poor when compared to the optimal performance as measured by a designer specified
This research was supported in part by NSF under Grant ECS-9496249, AFOSR under Grant F49620-96-1-0125,and ARO under Grant DAAH04-96-1-0008. 0-7803-4394-8198 $10.00 0 1998 IEEE 2983
cost functional. If the operating range of the control system is small and if the system nonlinearities are smooth, then the control system can be locally approximated by a linearized system about a given operating condition and linear multivariable control theory can be used to maintain local stability and performance. However, in high performance engineering applications such as advanced tactical fighter aircraft and variable-cycle gas turbine aeroengines, the locally approximated linearized system does not cover the operating range of the system dynamics. In this case, gain scheduled controllers can be designed over several fixed operating points covering the system operating range and controller gains interpolated over this range [25]. However, due to approximation linearization errors and neglected operating point transitions, the resulting gain scheduled system does not have any guarantees of performance or stability. Even though stability properties of gain scheduled controllers are analyzed in [26,27] and stability guarantees are provided for plant output scheduling, a design framework for gain scheduling control guaranteeing system stability over an operating range of the nonlinear plant dynamics has not been addressed in the literature. In an attempt t o develop a design framework for gain scheduling control, linear parameter-varying system theory has been developed [28-311. Since gain scheduling control involves a linear parameter-dependent plant, linear parameter-varying methods for gain scheduling seem natural. However, even though this is indeed the case for linear dynamical systems involving exogenous parameters, this is not the case for nonlinear dynamical systems. This is due t o the fact that a nonlinear system cannot be represented as a true linear parameter-varying system since the varying system parameters are endogenous, that is, functions of the system state. Hence, stability and performance guarantees of linear parameter-varying systems do n o t extend to the nonlinear system. Of course, in the case where the magnitude and rate of the endogenous parameters are constrained such that the linear parametervarying system hopefully behaves closely to the actual nonlinear system, then a posteriori stable controllers can be designed. However, in the case of unexpectedly large amplitude uncertain exogenous disturbances and/or system parametric uncertainty, a priori assumptions on magnitude and rate constraints on endogenous parameters are unverifiable. For nonlinear controlled systems, in general, null controllability does not imply continuous or smooth stabilizability [32,33]. Variable structure control is perhaps the quintessential discontinuous nonlinear stabilization approach for controlling nonlinear systems [34,35]. In particular, variable structure control utilizes a high-speed switching control law to drive the system state trajectories onto an invariant sliding manifold. In order to establish the existence of a sliding mode, the controller is constructed such that the system state velocity vector is directed towards the switching surface. In an ideal case, this results in infinitely fast switching. However, in practical implementation, infinitely fast switching leads t o high frequency chattering which may excite unmodeled high
frequency system dynamics leading t o system instabilities. In an attempt to overcome this problem, boundary layer controllers which continuously approximate the discontinuous control action in a neighborhood of the switching surface have been proposed [35]. However, in this case the resulting controller does not guarantee asymptotic stability but rather uniform ultimate boundedness [35]. Finally, variable structure controllers are generally limited to nonlinear affine systems. In this paper a nonlinear control design framework predicated on a stability-based switching controller architecture parameterized over a set of system equilibria is developed. Specifically, using equilibria-dependent Lyapunov functions or, instantaneous (with respect to a given parameterized equilibrium) Lyapunov functions, a hierarchical nonlinear control strategy is developed that stabilizes a given nonlinear system by stabilizing a collection of nonlinear controlled subsystems. Each subcontroller can be nonlinear and thus local set point designs can be nonlinear. Furthermore, for each parameterized equilibrium the collection of subcontrollers provide guaranteed domains of attraction with nonempty intersections that cover the region of operation of the nonlinear system in the state space. A switching nonlinear controller architecture is developed based on a generalized lower semicontinuous Lyapunov function obtained by minimizing a potential function, associated with each domain of attraction, over a given switching set induced by the parameterized system equilibria. The switching set specifies the subcontroller t o be activated at the point of switching, which occurs within the intersections of the domains of attraction. The hierarchical switching nonlinear controller guarantees that the generalized Lyapunov function is nonincreasing along the closed-loop system trajectories with strictly decreasing values at the switching points, establishing asymptotic stability. In the case where one of the parameterized system equilibrium points is globally asymptotically stable, the proposed nonlinear stabilization framework guarantees global asymptotic stability of any given system equilibrium on the parameterized system equilibrium branch. Furthermore, since the proposed switching nonlinear control strategy is predicted on a generalized Lyapunov function framework with strictly decreasing values at the switching points, the possibility of a sliding mode is precluded. Hence, the proposed nonlinear stabilization framework avoids the undesirable effects of high-speed switching onto an invariant sliding manifold which is one of the main limitations of variable structure controllers. Limited to system analysis, related but different approaches t o the proposed stability-based switching control design framework are given in [36-391. Specifically, analysis of switched linear systems in the plane (R2)are given in [36,37]. More recently, asymptotic stability analysis of m-linear systems using Lyapunov-like functions is given in [38]. Stability of a multi-controller switched system is. analyzed using Lyapunov functions and sliding surfaces in [39]. Finally, we note that the concept of equilibriadependent Lyapunov functions was first introduced by the authors in [40] where a globally stabilizing control design
2984
framework for controlling rotating stall and surge in axial flow compressors was developed. In parallel research to 1401 a related but different approach was introduced in [41] wherein control Lyapunov functions for nonlinear systems linearized about a finite number of "trim" points t o guarantee stability of a range of operating conditions were given. The contents of the paper are as follows. In Section 2 we establish definitions, notation, and several key results used later in the paper. Then in Section 3, in order to address stability of closed-loop switching systems, we develop generalized Lyapunov and invariant set theorems for nonlinear closed-loop feedback dynamical systems wherein all regularity assumptions on the Lyapunov function and the closed-loop system dynamics are removed. In particular, local and global stability theorems are presented using generalized Lyapunov functions that are lower semicontinuous. Furthermore, generalized invariant set theorems are derived wherein closed-loop system trajectories converge to a union of largest invariant sets contained on the boundary of the intersections over finite intervals of the closure of generalized Lyapunov level surfaces. In Section 4 we concentrate on nonlinear stabilization of local set points over a set of parameterized equilibria of the nonlinear system. In Section 5 a nonlinear stabilization framework predicated on a hierarchical stability-based switching controller architecture is developed. Extensions to continuous switching using a fixed-order dynamic compensator architecture is presented in Section 6. Finally, we draw some conclusions in Section 7.
2.
Mathematical Preliminaries
In this section we establish definitions, notation, and several key results used later in the paper. Let E% denote the set of real numbers, let Rn denote the set of n x 1 real column vectors, let RnX" denote the set of real n x m matrices, let (.)T denote transpose, and let 1, or 1 denote the n x n identity matrix. Furthermore, we write 11 . 11 for the Euclidean vector norm, V'(x)for the Frechet derivative of V(.) at Z, and A 2 0 (resp., A > 0) to denote the fact that the Hermitian matrix A is nonnegative (resp., positive) definite. For a subset S c R", we write as,
input, U is the set of all admissible controls such that U(.) is a measurable function with 0 E U ,and F : V x U -+ Rn is continuous on V x U.
Definition 2.1. The point 3 E D is an equilibrium point of (1)if there exists fi E U such that F(Z,ii) = 0. In this paper we assume that given an equilibrium point Z E V corresponding to ii E U and a mapping p : V x A - + U , A ~ R ~ , O E A , s u c h t h a t c p ( ~ ,=G,there O) exist neighborhoods V, c V of ?tand A, c A of 0, and a continuous function $ : A, -+ V, such that Z = $(O), and, for every X E A,, xx = $(A) is an equilibrium point, that is, F($(A),p($(A),X)) = 0, X E A,. Note that the connected set A C E%q corresponds to a parameterization set with the function $(.) parameterizing the system equilibria. In the special case where q = m and p ( ~A), = A, it follows that the parameterized system equilibria are given by the constant control u ( t )E A. A parameterization that provides a local characterization of an equilibrium surface, including in neighborhoods of bifurcation points, is given in [42]. Alternatively, the well known Implicit Function Theorem provides suficient conditions for guaranteeing the existence of such a parameterization under the more restrictive condition of continuous differentiability of the mapping $(.I.
Theorem 2.1 [43]. Assume the function I?(z,A)
F(x,p(z, A)), x E D,X E A, is C' at each point ( 2 ,A) E V x A. Suppose p(Z,0) = 0 for (3,O) E V x A and the Jacobian matrix g ( 3 , O ) is full rank. Then there exist open n_eighborhoods V, C V of Z and A, c A of 0 such that F ( z , X ) = 0, X E A,, has a unique solution zx E 23,. In particular, there exists a unique C1 mapping $ : A, -+ V, such that xx = $(A), X E A,, and 3 = $ ( O ) . Next, we consider feedback controlled nonlinear dynamical systems. A measurable mapping 4 : D - + U satisfying $(Z) = fi is called a control law. Furthermore, if u ( t ) = $(x(t)), where 4(.)is a control law and x(t), t E R, satisfies (l),then U ( . ) is called a feedback control law. Here, we consider closed-loop nonlinear dynamical systems of the form
0
S , for the boundary, the interior, and the closure of 0 0 S, respectively. Recall that, as = S \ S , S = S \ as, S S. A set S 5 Rn is connectedif = S u d s , and there does not exist open sets 01 and 0 2 in E%" such that S c 01u02,SnO1 # 0 ,Sn02 # 0, and SnO1nO2 = 0. Recall that S is a connected subset of R if and only if S is either an interval or a single point. Finally, let CO denote the set of continuous functions and C" denote the set of
s s
functions with n-continuous derivatives. In this paper we consider controlled nonlinear dynamical systems of the form k ( t ) = F ( x ( t ) ,u ( t ) ) ,
Z(0) = 20,
t E Iw,
(1)
where ~ ( tE )V C R",t E R, is the system state vector, V is an open set, 0 E V ,u(t) E U 5 Iwm, t E R,is the control
A function x : Z -+ V is said to be a solution to (2) on the interval Z C E% if ~ ( tsatisfies ) (2) for all t E Z. Note that we do not assume any regularity condition on the function 4(.).However, we do assume that F ( . , $ ( . ) ) is such that the solution x(t), t E Iw, to (2) is well-defined on Z = E%. That is, we assume that for every y E V there exists a unique solution xi.) of (2) defined on E% satisfying x(0)= y. Furthermore, we assume that all the solutions x ( t ) ,t E R,to (2) are continuous functions of the system initial conditions zoE V. Remark 2.1. If F ( . , +(.)) is Lipschitz continuous on V then there exists a unique solution to (2). In this case, the translation property s ( t T,2 0 ) = s ( t , S(T, zO)), t , T E R, and the continuity of s ( t , . ) on V , t E R, hold, where
2985
+
s(., 2 0 ) denotes the solution of the controlled feedback dynamical system (2). Alternatively, uniqueness of solutions ensure that in time along with the continuity of F(., the solutions to (2) satisfy the translation property and are continuous functions of the initial condition 20 E V even when F ( - ,q5(-)) is not Lipschitz continuous on V (see [44, Theorem 4.3, p. 591). More generally, F ( - ,q5(.)) need not to be continuous. In particular, if F(., $(.)) is discontinuous but bounded and x(.) is the unique solution to (2) in the sense of Filippov [45], then the translation property along with the continuous dependence of solutions on initial conditions hold [45]. + ( e ) )
Definition 2.6. Let 730 c 2) be a compact positively invariant set for the nonlinear dynamical system (2). DO is Lyapunov stable if for every open neighborhood 01 23 of 270, there exists an open neighborhood 0 2 01 of V O such that x ( t ) E 01, t 2 0, for all 20 E 0 2 . DOis attractive if there exists an open neighborhood 0 3 C V of V O such Vo for all 20 E 0 3 . V O is asymptotically that P L stable if it is Lyapunov stable and attractive. V Ois globally asymptotically stable if it is Lyapunov stable and P A E V O for all z o E Rn . Finally, DOis unstable if it is not Lyapunov stable.
Next, we introduce several definitions and key results that are necessary for the main results of this paper.
Next, we give a set theoretic definition involving the domain, or region, of attraction of the compact positively invariant set 23, of (2).
Definition 2.2. Let V, V and let V : V, + R {x E D, : V(z) = a } For a E R, the set V-'(a) is called the a-level set. For a , @E R, a 5 p, the set V-'([a,p])2 {z E V, : a 5 V ( x ) 5 p } is called the [a,,f3]-sublevel set.
Definition 2.7. Suppose the compact positively invariant set V, c V of (2) is attractive. Then the domain of attraction V Aof V, is defined as
Definition 2.3. The trajectory z ( t ) E 2) Rn, t E R, of (2) denotes the solution to (2) corresponding to the initial condition z(0) = zo evaluated at time t. The trajectory x ( t ) , t E R, is bounded on Z 2 R if there exists y > 0 such that Ilx(t)II < y, t E 2. Definition 2.4. A set M + 2 V Rn (resp., M - ) is a positively (resp., negatively) invariant set for the nonlinear dynamical system (2) if xo E M + (resp., M - ) implies that x ( t ) E M + (resp., M - ) for all t 2 0 (resp., t 5 0). A set M E V Rn is an invariant set for the nonlinear dynamical system (2) if zo E M implies that x ( t ) E M for all t E R.
V, {xo E V : P& c 2%). (3) Recall that V A is an open, connected invariant set [3]. Next, we present a key theorem due to Weierstrass involving lower semicontinuous functions on compact sets. For the statement of the result the following definition is needed.
Definition 2.8. Let V, c V. A function V : V, -+ R is lower semicontinuous on Vc if for every sequence { x , } ~ = ,CV, such that lim z, =x, V ( x )Sliminf V ( x n ) . n-im
n+m
Definition 2.5. p E 'zi C IW" is a positive limit point of the trajectory z ( t ) , t E R, if there exists a sequence {tn}F=o,with t , + 0;) as n + co,such that x(tn) -+ p as n + 0;). The set of all positive limit points of x ( t ) , t E R, is the positive limit set P A of z ( t ) ,t E R.
Note that a bounded function V : D, -+ R is lower semicontinuous at x E D, if and only if for each e > 0 there exists 6 > 0 such that 112 - yIJ < 6 , y E a,, implies V ( x )- V(y) 5 E . Furthermore, if V, V is compact and V : V, + R is lower semicontinuous at x E V,,then for each a E R the set {x E V, : V ( x )5 a } is compact.
The following result on positive limit sets is fundamental and forms the basis for all the generalized stability and invariant set theorems developed in Section 3.
Theorem 2.2 [46]. Suppose V, c V is compact and V : V, + IR is lower semicontinuous. Then there exists x* E V, such that V ( x * )5 V ( x ) ,x E V,.
Lemma 2.1 [3]. Suppose the forward solution x ( t ) ,t 2 0, to (2) corresponding to an initial condition x ( 0 ) = xo is bounded. Then the positive limit set P A of ~ ( t t) E, R, is a nonempty, compact, connected invariant set. Furthermore, x ( t ) + P& as t + m. Remark 2.2. It is important to note that Lemma 2.1 holds for time-invariant nonlinear feedback controlled systems (2) possessing unique solutions with solutions being continuous functions of the system initial conditions. More generally, Lemma 2.1 holds if s ( t T , X O ) = s ( t , S(T,zo)), t , T E R,and s(., 20) is a continuous function of 20 E V.
+
The following definition introduces three types of stability as well attraction of (2) with respect to a compact positively invariant set.
3.
Generalized Stability Theorems for Nonlinear Feedback Systems
Most Lyapunov stability and invariant set theorems presented in the literature require that the Lyapunov function for a nonlinear system be a C1 function with a negativedefinite derivative (see [l,3,47-501 and the numerous references therein). However, even though in the case of discontinuous system dynamics with continuous motions standard Lyapunov theory is applicable, it might be simpler to construct discontinuous LLLyapunov"functions to establish closed-loop system stability. In particular, as mentioned in the introduction, the key step in obtaining a general nonlinear stabilization framework is to use several different controllers designed over several fixed operating points covering the system's operating range in the
2986
state space, and to switch between them over this range. Even though for each operating range one can construct a C1 Lyapunov function, t o show closed-loop system stability over the whole system operating envelope for a given switching control strategy, a generalized Lyapunov function obtained by minimizing a potential function associated with domains of attraction for each operating range is constructed. As will be shown in Section 5 the generalized Lyapunov function is non-smooth and non-continuous. Hence, in this section we develop generalized Lyapunov and invariant set theorems for nonlinear feedback systems wherein all regularity assumptions on the Lyapunov function and the closed-loop system dynamics are removed. The following result generalizes the Barbashin-KrasovskiiLaSalle invariant set theorems [3,49,51-531 to the case where the Lyapunov function is lower semicontinuous.
Theorem 3.1. Consider the nonlinear controlled dynamical system (2), let z ( t ) , t E R, denote the solution t o (2), and let V, c 2) be a compact positively invariant set with ..espect t o (2). Assume that there exists a lower semicontinuous function V : D, + R such that V(z(t)) IV(Z(T)), 0 5 T 5 t , for all 20 E D,. Let R, n V-l([y,c]), y E R, and let M , be the C>Y
largest invariant set contained in R,. If z ( t ) - + M A U M , as t + C O .
20
E
y E R. Hence,
k, 4
where
n k,,c, is such that V(z) < y,z E 2,. C>,
Remark 3.2. Note that if V : D, + R is a lower semicontinuous function such that all the conditions of Theorem 3.1 are satisfied, then for every zo E V, there exists yzo5 V(z0) such that P A MY,, C M . Remark 3.3. It is important to note that as in standard Lyapunov and invariant set theorems involving C1 functions, Theorem 3.1 allows one to characterize the invariant set M without knowledge of the closed-loop system trajectories z ( t ) ,t E R. Similar remarks hold for the remainder of the theorems in this section. Next, we sharpen the results of Theorem 3.1 by providing a refined construction of the invariant set M . In particular, we show that the closed-loop system trajectories converge to a union of largest invariant sets contained on the boundary of the intersections over finite intervals of the closure of generalized Lyapunov level surfaces.
D,,then
-YEW
Proof. Let z ( t ) , t E R, be the solution to (2) with E D,. Since V(.)is lower semicontinuous on the compact set D,,there exists b E R such that V(z) 2 b, z E D,. Hence, since V(z(t)), t 2 0, is nonincreasing, yzo hlV(z(t)), 20 E D,,exists. Now, for all p E P& there exists an increasing unbounded sequence to = 0, such that lim z(tn) = p . Next, since n-iw V(z(tn)), with n 2 0, is nonincreasing it follows that for all n L 0, ~ $ 0IV(z(tn)) IV(z(tN)), n 2 N , or, equivalently, since D, is positively invariant, z ( t n ) E V-l([y,,, V ( z ( t ~ ) )n] )2, N . Now, since mlim z ( t n )= p -420
Theorem 3.2. Consider the nonlinear controlled dynamical system (2), let D, and 'DO be compact positively invariant sets with respect to (2) such that DO C V, C D, and let z ( t ) ,t E R, denote the solution to (2) corresponding to z o E D,.Assume that there exists a lower semicontinuous function V : D, -+ R such that V(z) = 0, z E Do, V(z) > 0, z E D,, z @Do, 0I ,T L t. V(z:(t)) i V ( 4 4 ) ,
{tn}r!o,
Remark 3.1. Note that since V-'([y,c]) = {z E D, : V(z) 2 r}n{z E D, : V(z) 5 c} and {z E Dc: V(z) I c} is a closed set, it follows that %!,,c C {z E 27, : V(z) < y}, where fi,,c A V-l([y,c]) \ V-'([y, c]), c > y,for a fixed
(6)
Furthermore, assume that for all 20 E D,,z o $! DO,there exists an increasing unbounded sequence {tn}F=o, with to = 0, such that
." .--
it follows that p E V-l([yz0,V(z(tn))]),n 2 0. Furthermore, since lim V(z(tn)) = yzoit follows that for every n-+w c > yzo, there exists n 2 0 such that yzo5 V(z(tn)) 5 c which implies that for every c > yzo,p E V-l([y,,,c]). Hence, p -E R,zo which implies that- P& C R,,;. Now, since D, is compact and positively invariant it follows that the forward solution z ( t ) ,t 2 0, t o (2) is bounded for all z o E Dc and hence it follows from Lemma 2.1 that P A is a nonempty compact connected invariant set which further implies that P& is a subset of the largest invariant . for all set contained in R,,,, that is, P& C M T Z oHence, 50 E Dc, P& C M . Finally, since z ( t ) -+ P& as t -+ 00 it follows that z ( t ) + M as t + CO. 13
(4) (5)
V(z(tn+l)) < V(z(tn)), Define
R,
71
= 0 , L . ...
(7)
n V-l([y,c]), y 2 0, and let M , be the C>Y
largest invariant set contained in R,. Then, M Y C 2, R,\V-l(y), y > 0. Furthermore, if 20 E D,,then z ( t ) -+ M A U M , as t + CO, where G A {y 2 0 : E , nDo # Y€G
0 ) . If, in addition, Do c Dc and V(.) is continuous on DO then Do is locally asymptotically stable and V, is a subset of t,he domain of attraction.
Proof. Since 23, is a compact positively invariant set, it follows that for all 20 E V,,the forward solution z ( t ) , t 2 0, to (2) is bounded. Hence, it follows from Lemma 2.1 that, for all z o E D,,PA is a nonempty, compact, connected invariant set. Next, it follows from Theorem 3.1, Remark 3.2, and the fact that V(.) is positive-definite that for every zo E D, there (with respect to V c \DO), exists yzo2 0 such that PA M,,, C R,,,.Now, given
2987
ditions (4)-(6) in Theorem 3.2 specialize t o the standard z o E V - ' ( y z 0 ) ,yzo > 0 , ( 7 ) implies that there exists tl > 0 such that V(x(t1))< yzo and x(t1) $ V - ' ( y z 0 ) . Lyapunov stability conditions [3,47,49].
Hence, V - ' ( y z 0 )c R,,, does not contain any invariant set which implies that M,,, c 7?,,=,, yzo > 0. Now, ad absurdum, suppose DOf l P.& = 0. Since V ( . ) is lower semicontinuous it follows from Theorem 2.2 that there exists P E P&, such that a = V ( P ) V ( z ) ,z E P&. Now, with x(0) = 2 $ DO it follows from (7) that there exists an increasing unbounded sequence {tn}p!.o,with t o = 0, such that V(z(t,+l))< V ( z ( t , ) ) ,n = 0 , 1 , . . ., which implies that there exists t > 0 such that V ( z ( t ) < ) a which further implies that z ( t ) $Z P A contradicting the fact that P A is an invariant set. Hence, there exists q E DO such that q E P A C R,, which implies that R,,,n DO # 8. Thus, yzo E G for afl zo E Dc which further implies that M . NOW, since x ( t ) -+ C_ M as t + 00 it follows that z ( t )-+ M as t -+ CO.
0, z E R,, 2 $Do, V ( z ( t > > V(X(T)), 0 57- t , V ( z ) -+ 00 as 11x11 -+ 00.
PA
0, x E D,,E DO. Next, using the facts that V ( z )= 0, x E DO, and V ( . )is continuous on DO,it follows that the set 0 2 g {z E 0 1 : V ( z ) < a } is not empty. Now, it follows froin ( 6 ) that for all z(0) E 0 2 ,
Define
R, 4 C n V - l ( [ y , c ] )y, 2 0 , and let M Y be >7
the largest invariant set contained in R,. Then for all 20 E Rn, z ( t ) -+ M A U M,, as t + 00. If, in addi720
tion, for all z o E R", z o $ DO,there exists a n increasing unbounded sequence {tn}p=o, with to = 0, such that
V(4tn+l)< ) V(z(tn)),
n = 0 , L . . .,
(12)
then M , C 'li, 4 R, \ V - ' ( y ) , y > 0, and x ( t ) -+ M 4 U M , as t + ca, where G {y 2 0 : R, n DO # ,E0
V ( z ( t ) )L V ( z ( 0 ) )< Q,
t 2 0,
which, since V ( x ) 2 a , z E 801, implies that x(t) $ 801, t 2 0. Hence, for every open neighborhood 01 Vc of V O ,there exists an open neighborhood 0 2 C 01 of DO such that, if z(0) E 0 2 , then z ( t ) E 0 1 , t 2 0 , which proves Lyapunov stability of the compact positively invariant set DO of (2). Finally, from the continuity of V ( - )on DO and the fact that V ( z )= 0 for all z E DO,it follows that G = (0) and M O .Hence, P A DOfor all 20 E D, establishing local asymptotic stability of the compact positively invariant set Do of (2) with a subset 0 of the domain of attraction given by V c .
.a
Remark 3.4. If in Theorem 3.2 M C DO,then the compact positively invariant set Do of (2) is attractive. If, in addition, V ( . )is continuous on DO c D, then the compact positively invariant set Do of (2) is locally asymptotically stable. In both cases, Dc is a subset of the domain of attraction .
8 ) . Finally, if V ( . )is continuous on DOthen the compact positively invariant set DOof (2) is globally asymptotically stable. Proof. Note that since V ( z )-+ 00 as llzll -+ CO it follows that for every p > 0 there exists r > 0 such that V ( z ) > p, 1 1 z l 1 > T , or, equivalently, V-'([O,P])C { x : llzll r } which implies that V-'([0,p]) is bounded for all p > 0. Hence, for all 20 E Rn, V-l([O,bzo]) is bounded where pz, A V(z0). Furthermore, since V ( . )is a positivedefinite lower semicontinuous function, it follows that V-'([O,p,,]) is closed and since V ( z ( t ) ) t, 2 0 , is nonincreasing it follows that V-' ([0,,&,]) is positively invariant. Hence, for every 20 E R*, V-'([O,p,,]) is a compact positively invariant set. Now, with Vc = V-'([O,,Ozo]) it follows from Theorem 3.1 and Remark 3.2 tha: there exists 0 yzo ,&, such that P A C M,,, C R,,, which implies that z ( t ) -+ M as t -+ CO. If, in addition, for all zo E R", 20 $Z DO, there exists an increasing unbounded sequence with t o = 0, such that (12) holds then it follows from Theorem 3.2 that z ( t ) + M as t -+ 00. Finally, if V ( . )is continuous on DOthen Lyapunov stability follows as in the proof of Theorem 3.2. Furthermore, in this case, 6 = {0} which implies that M = MO. Hence, P A DO establishing global asymptotic stability of the compact positively invariant set DO of (2). 0
0 : V;'([O,@]) C XA}.
Proof. Sufficiency is a direct consequence of Theorem 3.2 and Remark 3.5. Alternatively, the result is a restatement of Lyapunov's direct theorem as applied to the closed-loop system (14). Necessity follows from converse Lyapunov theory. Finally, the domain of attraction estimate (19) is given by the largest closed Lyapunov sublevel set contained in XA. 0 Remark 4.1. It follows from Theorem 4.1 that for all E V A ,lim V ~ ( z ( t ) = ) 0 or, equivalently, for each 6 > 0 t+cc there exists a finite time T > 0 such that Vx(x(t)) 5 6, t 2 T . Hence, given the initial condition zo E V X ,it follows that for every 6 > 0 there exists a finite time T > 0 such that ~ ( tE)Vc'([O,6]), t 2 T . 20
E CO, E As),
A, 5 A,, (13)
X E A,, the closed-loop feedback
4 t ) = FA(4t)) 4 F ( z ( t ) , 4A(4t))), 5(0) = 5 0 , t E R,
(14)
has an asymptotically stable equilibrium point ZA E V, C: D. Here, we assume that for each X E A,, the linear or (.) are given. In particnonlinear feedback controllers ular, these controllers correspond t o local set point designs and can be obtained using any appropriate standard
We stress that the aim of Theorem 4.1 is not to make direct comparisons with existing methods for estimating domains of attraction, but rather in aiding t o provide a streamlined presentation of the main results of Section 5 requiring estimates of domains of attraction for local set point designs. Since V A given in Theorem 4.1 gives an estimate of the domain of attraction using closed Lyapunov sublevel sets, it may be conservative. To reduce
2989
conservatism in estimating a subset of the domain of attraction several alternative methods can be used. For example, maximal Lyapunov functions [57], Zubov's method [47, 581, ellipsoidal estimate mappings [59], Carlemann linearizations [60], computer generated Lyapunov functions [61], iterative Lyapunov function constructions [62], trajectory-reversing methods [3], and open Lyapunov sublevel sets [63], can be used to construct less conservative estimates of the domain of attraction. 5.
Nonlinear System Stabilization via StabilityBased Switching
In this section we develop a nonlinear stabilization framework predicated on a stability-based switching controller architecture parameterized over a set of system equilibria. Specifically, using equilibria-dependent Lyapunou functions or, instantaneous (with respect to a given parameterized equilibrium) Lyapunov functions, a hierarchical nonlinear control strategy is developed that stabilizes a given nonlinear system by stabilizing a collection of nonlinear controlled subsystems while providing an explicit expression for a guaranteed domain of attraction. A switching nonlinear controller architecture is developed based on a generalized lower semicontinuous Lyapunov function obtained by minimizing a potential function, associated with each domain of attraction, over a given switching set induced by the parameterized system equilibria. In the case where one of the parameterized equilibrium points is globally asymptotically stable with a given subcontroller, the proposed nonlinear stabilization framework guarantees global asymptotic stability of any given system equilibrium on the parameterized equilibrium branch. To state the main results of this section several definitions and a key assumption are needed. Recall that the set A, C Ao, 0 E A,, is such that for every X E A, there exists a feedback control law q5x(.) E such that the equilibrium point xx E Do 2) of (14) is asymptotically stable with an estimate of the domain of attraction given by VA.Since xx, X E A,, is an asymptotically stable equilibrium point of (14), it follows from Theorem 4.1 that there exists a Lyapunov function Vx(.)satisfying (16)-( 18), and hence, without loss of generality, we can take Dx,X E A,, given by (19). Furthermore, we assume that the set-valued map Q : A, U 2'O, where 2'O denotes the collection of all subsets of D,is such that Dx = *(A), X E A,, is continuous. In particular, since Dx, X E A,, is given by (19), the continuity of the set-valued map Q(.) is guaranteed provided that Vx(x),x E Dx,and C A are continuous functions of the parameter X E A,. Next, let S A,, 0 E S, denote a switching set such that the following key assumption is satisfied.
Assumption 5.1. The switching set S C A, is such that the following properties hold:
i) There exists a continuous positive-definite function p : S + R such that for all X E S, X # 0, there exists 0 A1 E S such that p(X1) < p(X), xx E Dx,.
ii) If A, A1 E S, X # XI, is such that p(X) = XI), then D x nvxl= 0. Note that Assumption 5.1 assumes the existence of a positive-definite potential function p(X), for all X in the switching set S, associated with each domain of attraction Dx. Furthermore, Assumption 5.1 guarantees that domains of attraction corresponding to different local set point designs intersect only if their corresponding potentials are different. In particular, given Dx,X E S \ {0}, it is always possible to find at least one intersecting domain of attraction Dx,, A1 E S , such that the potential function decreases and contains xx as an internal point. This guarantees that if a forward trajectory x ( t ) ,t 0 , approaches xx,then there exists a finite time T > 0 such that the trajectory enters Vxl. Finally, it is important to note that the switching set S is arbitrary. In particular, we do not assume that S is countable or countably infinite. For example, the switching set S can have a hybrid topological structure involving isolated points and connected closed sets homeomorphic to intervals on the real line. Next, we show that Assumption 5.1 implies that every level set of the potential function p ( . ) is either empty or consists of only isolated points. Furthermore, in a neighborhood of the origin every level set of p ( . ) consists of at most one isolated point. For the statement of this result, let Bx(r), X E 2.40, r > 0, denote the open ball centered at xx with radius r , that is, B x ( r ) {x E 2, : 11x-xxII < r } .
>
Lemma 5.1. Let S A, be such that Assumption 5.1 holds. Then for every a > 0, p - l ( a ) is either empty or consists of only isolated points. Furthermore, there exists /3 > 0 such that for every a < 0,p - ' ( a ) consists of at most one isolated point.
Proof. Suppose, ad absurdum, that there exists E p - ' ( a ) , a > 0, such that is not an isolated point in p - ' ( a ) . Now, let fi C p - ' ( a ) be a neighborhood of and note that, by continuity of $(.) and the fact that 1 E p-' ( a )is not an isolated point, there exist XI, A2 E such that llxxl - xxzII < E , E > 0, and p(X1) = ~ ( X Z = ) a.
fl
0
However, since zx, E Vxl, it follows that there exists Dxl. Now, choosing E < r r > 0 such that B,,(r) yields z ~ ,E Bx1(r) g 'DA,and zxz E 'DA~ which implies that Vxl n Dx2 # 0 contradicting ai) of Assumption 5.1. Hence, if p - ' ( a ) , a > 0, is non-empty, it must consist of only isolated points. Next, suppose, ad absurdum, that for all S > 0 there { A E S : llXll < exist two isolated points X I , & E #d S} such that p(X1) = p(X2). Now, repeating the above arguments leads to a contradiction. Hence, there exists 8 > 0 such that if X 1 E #A, then f l n ~ p-'(p(X1)) = {XI}. NOW,since p ( . ) is continuous and positive definite, it follows that p ( . ) is locally convex in a neighborhood of the origin. Hence, there exists P > 0 such that p - l ( a ) Nh, a < P, consists of at most one isolated point. 0 Note that Lemma 5.1 implies that there exists a finite distance between isolated points contained in the
2990
same level set of p ( - ) . Furthermore, if p-l(a), a > 0 , is bounded, p-l ( a )consists of at most a finite number of isolated points. Finally, since in a neighborhood of the origin every level set of p ( - ) consists of at most one isolated point, a particular topology for S, in a neighborhood of the origin, is homeomorphic t o the interval [O,a],a > 0 , with 0 E S corresponding t o 0 E R. Now, for every x E D, A U V A ,define the viable XES
switching set Vs(x)A {A E S : z E D,},which contains all X E S such that x E Dx.Note that if we consider a sequence c Vs(x),that is, x E Vx,,, such that lim An = A, it follows from the continuity of the setn+w valued map g(-)that x E Vx. Thus, E Vs(x) which implies that Us(x)is a non-empty closed set since it contains all of its accumulation points. Next, we introduce the switching function Xs(x), x E D,,such that the following definition holds
iXn}z?l
V ( z )A p ( ~ s ( x=min{p(X) )) : X E V,(x)},
x
E
V,.(20)
The following :!ma shows that “min” in (20) is attained and hence V ( x )is well defined. 16.:
Lemma 5.2. Let S C A, and let p : S + R be a continuous positive-definite function such that Assumption 5.1 holds. Then, for all 2 E V,, there exists a unique Xs(x) such that p ( X ~ ( z )=) min{p(X) : X E Vs(x)}. Proof. Existence follows from the fact that p ( . ) is lower bounded and V,(x), x E D,,is a non-empty closed set. Now, to prove uniqueness suppose, ad absurdum, Xs(x) is not unique. In this case, there exist XI, A2 E S , XI # X2, such that p ( X l ) = p ( X 2 ) and x E Vxl n Dx2 # 0 which 0 contradicts ii) in Assumption 5.1. The next result shows that V ( . )given by (20) is a generalized Lyapunov function candidate, that is, V ( - )is lower semicontinuous in V,.
c
Theorem 5.1. Let S A, be such that Assumption 5.1 holds. Then the function V ( x ) = min{p(X) : X E V s ( z ) } ,x E D,,is lower semicontinuous on V, and con0
tinuous on Dxs(,).
Since fi is closed, it follows that P E that there exist nl 2 n o such that 2 E contradiction. 0 To show that V ( z )is continuous on
fi which
implies which is
it need only 0
be shown that V ( 2 )is upper semicontinuous on D A ~ ( ~ ) , or, equivalently, V ( 2 ) 2 p ( i ) . Since lim x n = P and n+w
0
2 E ’ D x s ( i ) , there exists a positive integer 722 such that x, E Dxs(g),n 2 n2. Hence, AS(?) E Vs(xn) and V ( 2 )2 0 V ( z n )n , 2 122, which implies that V ( 2 )2 p ( i ) . Next, we show that with the hierarchical nonlinear feedback control strategy u = d ~ ~ ( , ) ( x )E, D,,V ( . )given by (20) is a generalized Lyapunov function for the nonlinear closed-loop dynamical system (2). The controller notation 4Xs(,) (x) denotes a switching nonlinear feedback ’ , is controller where the switching function XS(Z), x E D such that definition (20) of the generalized Lyapunov function V ( x ) ,x E D,,holds for a given potential function p ( . ) and switching set S satisfying Assumption 5.1. Furthermore, note that since 4xs(,)(x) is defined for x E V,,it follows that the solution x(.) to (2) with xo E V, and U = #As (x)is defined for all values of t E Iw such that x ( t ) E D,,that is, solutions are defined for t E I,, C R. Now, since 23, is a positively invariant set [0, +m) I,,, while if xo E V c is such that x ( t ) , t < 0, is always contained in V,,then Z,,= R.
c
Theorem 5.2. Consider the controlled nonlinear system (1) with F(0,O) = 0 and assume there exists a continuous function : A. + Do, 0 E A,, parameterizing an equilibrium branch of (l),such that xx = +(A), X E Ao. Furthermore, assume that there exists a CO feedback control law &,(.), X E A, C A. with 0 E A,, that locally stabilizes x~ with a domain of attraction estimate D Aand A,, 0 E S , be such that Assumption 5.1 holds. let S If Xs(x), x E D,,is such that V ( z ) ,x E V,,given by (20) holds and x ( t ) , t E Z,,, is the solution to (1) with x ( 0 ) = zo E D, and feedback control law
+
U
= 4Xs(z)(4,
x E DC,
(21)
then V ( z ( t ) )t, 2 0 , is nonincreasing. Furthermore, for all t l , t 2 2 0 , V ( x ( t ) )= V ( x ( t l ) )t, E [tl,t z ] , if and only if Xs(z(t)) = Xs(z(tl)),t E [tl,t2].Finally, for all t E I,,, such that X s ( x ( t ) ) # 0 there exists a finite time T > 0 such that V ( x ( t T ) )< V ( x ( t ) ) .
Proof. Let the sequence (x,}r==, C D, be such that lim zn = P and define A liminf Xs(z,). Here we asn-+w n+w sume without loss of generality that { X S ( ~ ~ ) } ~con=~ if this is not the case, it is always possible to verges t o i; Proof. Let z ( t ) , x(O) = 20, t E I,,, satisfy (1) construct a subsequence having this property. Since p(.) is with u(t) = q5xt(x(t)), where At A X s ( x ( t ) ) , and let, continuous (and hencep( lim X5(xn)) = lim p ( X s ( z n ) ) ) , for an arbitrary time tl 2 0, the feedback control law n+w n+w ~ asymptotically ~ ~ stabilize the equilibrium point it suffices to show that V(P)5 p ( i ) . Suppose, ad absur- U = q $ (x) dum, that V ( 2 ) > p ( i ) . In this case, there exists a pos- zxtl of (1) with domain of attraction Dxtl. Since x(t1) E itive integer no such that V ( 2 )> V(z,), n 2 no. Now, Dxt1, it follows from Theorem 4.1 that there exists a (z(t1))= since by definition Xs(2) minimizes p(X) for X E V s ( 2 ) , C1 Lyapunov function V A ,(.)~ such that it follows that V(P) 5 p(X), X E V,(P). Hence, since %(x(tl))Fxtl ( x ( t 1 ) ) < 0. Next, since Fxtl is a V ( 2 ) > V ( z n ) = p ( X s ( z n ) ) , n 2 no, it follows that CO function, it follows that there exists 6 > 0 such Xs(zn) Vs(P) and P 9 DAs(,,), n 2 no. Now, define that Vxtl ( x ( t ) )< 0 , t E [tl,tl 61, which implies that 00 the closed set 9 A U Dxs(,,,) such that { X ~ } : = ~ , C V . Vxt1( z ( t ) ) < VXt1( S ( t l ) ) t, E [tl,tl Hence, d t ) E 7L=ll.,
+
(e)
+
2991
+ 4.
Vx,,,t E [tl,tl + a], which implies that At, E V s ( z ( t ) ) , t E [ t l , t l +a], and therefore V(z(t)) 5 V(z(t1)) = p ( X t , ) , t E [tl,t1+6]. Now, since tl > 0 is arbitrary, it follows that V(z(t)), t 2 0, is a nonincreasing function along the forward trajectories z ( t ) ,t > 0, of (1) with u(t)= 4x, ( z ( t ) ) .
Next, assume that z ( t ) E Vxt, and V(z(t)) = V(z(tl)), t E [tl,tl + 61. Now, suppose, ad absurdum, that At, t E [tl,tl+6], is not constant, that is, there exists t 2 E (tl, tl + 61 such that At, # At,. In this case z(t2) E V A nDxt, ~, and p(At,) = p(At,), which contradicts ii) of Assumption 5.1. Hence, it follows that if V(z(t)) = V(z(t2)), t E [tl,tl+b], then At = At,, t E [tl, tl + 61. Conversely, if for tl, t 2 > 0, At = A t , , t E [tl,t2], then v ( z ( t ) ) = V(z(tl)), t E [tl,t21, is immediate. Finally, for an arbitrary tl 2 0, suppose, ad absurdum, that V(z(t)) = V(z(t1)) # 0, t 2 t l , or, equivalently, At = At, E S \ {0), t 2 t l . Then the feedback control law 4xl (.) = bx, (.) stabilizes the equilibrium point zx,, . In this case, it follows from Assumption 5.1 that there exists 0
A1 # At, such that p(A1) < V(z(t1)) and zxtl E VxI, which implies that there exists a > 0 such that zx,, E V,;ll([O,a]) C Vxl. Hence, it follows from Remark 4.1 that z ( t ) approaches the level set V,,'(a) in a finite time T > 0 so that V(z(t1 T ) ) 5 p(A1) < V(z(t,)), which contradicts the original supposition. 0
+
It follows from Theorem 5.2 that the hierarchical switching nonlinear controller (21) guarantees that the generalized Lyapunov function (20) is nonincreasing along the closed-loop system trajectories with strictly decreasing values only at the switching points which occur on the intersections of the domains of attractio? over a finite time interval. Specifically, if z(t2) E V x t z ,where At, 4 As(z(t2)) and t2 > 0, then from continuity of the closed-loop system trajectories z(.) it follows that there exists tl < t2 such that z(t1) E Vxtz, which implies that At, E V s ( z ( t l ) )and V(z(t1)) 5 V(z(t2)). Since V(z(t)), t 2 0, is a nonincreasing function of time, it follows that V(z(t)) = V(z(t,)), t E [tl,t2]. Alternatively, if z(t2) E dVxt2,then, for all tl < t2, z(t1) $! VX,,. If there existed t l < t 2 such that z(t1) E Dxt2, then At = At,, t E [ t l , h I , and, since Vxt, (21, 2 E %,, attains its maximum at z(t2) E dVxt,, it follows that Vxt2( z ( t ) )5 Vx,, (z(t2)), t E [tl,t2], which contradicts the fact that Vxtz( z ( t ) ) t, 2 0, is a decreasing function of time.
Remark 5.1. Note that for z o E V,,the assumptions of Theorem 5.2 guarantee that V(z(t)), t 2 0, is nonincreasing along the solution z ( t ) ,t E Zzo,t o (1) with feedback control law (21). However, since V(z) is only defined for z E V,,it follows that for all 20 E V,,z ( t ) E V,,t 2 0, and hence D, is a compact positively invariant set. Finally, we present the main result of this section. Specifically, we show that the hierarchical switching nonlinear controller given by (21) guarantees that the closedloop system trajectories converge to a union of largest
invariant sets contained on the boundary of intersections over finite intervals of the closure of the generalized Lyapunov level surfaces. In addition, if the scheduling set S is homeomorphic t o an interval on the real line and/or consists of only isolated points, then the hierarchical switching nonlinear controller establishes asymptotic stability of the origin.
Theorem 5.3. Consider the controlled nonlinear dynamical system (1) with F(0,O) = 0 and assume there exists a continuous function $ : A, + Do, 0 E A,, parameterizing an equilibrium branch of (1), such that zx = $(A), A E A,. Furthermore, assume that there exists a CO feedback control law &,(-), X E A, C A, with 0 E A,, that locally stabilizes z x with a domain of attraction estimate V A and let S C A,, 0 E S , be such that Assumption 5.1 holds. In addition, assume As(z), z E De, is such that V(z), z E D,,given by (20) holds, and, for zo E V,,z ( t ) , t E I,,,is the solution t o (2) with the feedback control law
Finally, define R, 4 n V-l([y,c]), y 2 0, and let M , C>-Y
be the largest invariant set contained in R,. If zo E De, then z ( t ) -+ M 4 U M-, as t -+ CO, where G {y 0 : ,€G
>
R,nVo # 0). If, in addition, SO4 i {A E S : DxnDo # 0 ) is homeomorphic t o [0,a], a > 0, with 0 E So corresponding to 0 E E, or SOconsists of only isolated points, then the zero solution z ( t ) E 0 t o (2) is locally asymptotically stable with an estimate of domain of attraction given by = En and there exists E S such that the feedback control law 4i(.) globally stabilizes z i , then the above results are global.
D,. Finally, if V
Proof. The result follows from Theorems 3.2, 5.1, and 5.2. Specifically, Theorem 5.2 implies that if z ( f ) E DO for an arbitrary t^ 2 0, then V(z(t)) = V(z(i!)) = 0, t 2 t^. Hence, As(z(t)) = 0, t 2 t^, and the feedback control law U = 40(z) asymptotically stabilizes the origin with an estimate of the domain of attraction given by DO.In this case, DO and (see Remark 5.1) D ' , are compact positively invariant sets of (2) with the feedback control law (22). Next, it follows from Theorems 5.1 and 5.2 that V(.) is a generalized Lyapunov function defined on V,. Now, it follows from Theorem 3.2 tha.t, for all 50 E V,,z ( t ) -+ M ast-kco. Next, if SOis homeomorphic t o [0,a], a > 0, with 0 E SO corresponding t o 0 E R, so that Assumption 5.1 is satisfied, it follows from the continuity of the set-valued map Q ( . ) restricted t o SOthat V(-) is continuous on DO. Now, it follows from Theorem 3.2 and the fact that the origin is the largest invariant set contained in M O that M M O3 (0). Hence, z ( t ) -+ 0 as t -+ CO establishing local asymptotic stability with an estimate of domain of attraction given by V,. Alternatively, if So consists of only isolated points with finite pairwise distance, it follows that 6 consists of the isolated values of V(-) evaluated on the elements of SO. Hence, since fi, R, \ V-'(y),
2992
y E GI is bounded, V ( . )can only assume a finite number of distinct values on y E 6 , including the zero value. Now, it follows from Theorem 3.2 that M, C fi, and from Theorem 5.2 that for all 20 E V,,20 # DO,there , exists an increasing unbounded sequence { t n } ~ = owith t o = 0, such that V ( x ( t n + l ) )< V ( x ( t ) ) ,n = O , l , . ...
e,,
Thus, no forward trajectories can be entirely contained in R,, y E 6 \ (0). Hence, M , = 0, y E 6 \ {0}, and &t E M O 5 (0) establishing local asymptotic stability with an estimate of domain of attraction given by D,. Finally, let 2) = Rn and assume S, 4 {A E S : D X 3 Rn} is not empty. In particular, if fi E S,, then the globally stabilizes xi. Now, feedback control law $A(.) E S,, AI # i z , Assumption 5.1 implies that if then p(A1) # p(X2). Furthermore, since Di = R n I Vs(x),x E Rn, and hence E S,, it follows that S, V ( x ) 5 min{p(i) : iE S g } ,x E Rn. Next, note that for all iE S,, only the (unique) value that minimizes p ( . ) on S, is assumed by the switching function Xs(x), x E Rn, so that all the other elements of S, can be discarded from S. Hence, assume, without loss of generality, that there exists a unique 1 E S such that 4i(.) globally {A E S : p(A) < p ( i ) } stabilizes xi. Now, define and 9,2 U D A which is a compact positively invariant
AI,&
XES
Figure 1: Stability-based switching control strategy
is constructed such that the switching function As(x), x E V,, assures that V ( x ) ,x E Dc, defined by (20) is a generalized Lyapunov function with strictly decreasing values at the switching points, the possibility of a sliding mode is precluded with the proposed control scheme. In particular, Theorem 5.2 guarantees that the closed-loop state trajectories cross the boundary of adjacent regions of attraction in the state space in a inward direction. Thus, the closed-loop state trajectories enter the lower potentialvalued domain of attraction before subsequent switching can occur. Hence, the proposed nonlinear stabilization framework avoids the undesirable effects of high-speed switching onto an invariant sliding manifold.
set. Hence, if xo E V,, it follows from the first part of the theorem that M is a local attractor and, if SOis homeomorphic to an interval on R or consists of only isolated points, the origin is asymptotically stable, with (in both cases) an estimate of the domain of attraction given by $,. Now, global attraction to M as well as global asymptotic stability of the origin is immediate by noting that if 6. Extension to Nonlinear Dynamic Compensation xo # V,, then the forward trajectory of (2) approaches 8, in a finite time. If, in fact, x # fi,, then As(%)= 1 which In this section we provide an online procedure for comimplies that for all x # 5, the feedback control law (22) puting the switching function AS(%), x E D,,such that 0 stabilizes xi and, by Assumption 5.1, xi E D,. In this (20) holds, by constructing an initial value problem for AS (x) having a fixed-order dynamic compensator struccase, it follows from Remark 4.1 that for all zo # 5, there ture. Specifically, we consider a scheduling set S diffeoexists a finite time T > 0 such that x(T)E V,. Hence, morphic to an interval on the real line R and assume that global attraction as well as global stability of the origin is the switching function As(x), z E D,,is C1. To present 0 this result, consider the nonlinear plant given by (1) and established for the respective cases.
Remark 5.2. Note that the switching set S is quite general in the sense that it can have a hybrid topological structure involving isolated points and connected closed sets homeomorphic to intervals on the real line. In the special case where the switching set S consists of only isolated points the stability-based switching control strategy given by (22) is piecewise continuous (see Figure 1). Alternatively, in the special case where the switching set S is homeomorphic to an interval on R, the stability-based switching control strategy given by (22) is not necessarily continuous. The continuous control case will be discussed in Section 6 .
consider a nonlinear feedback dynamic controller of the form
& ( t ) = Fc(z(t),xc(t)), ~ ( 0=)G O ,
tE
(23) (24) where x c ( t ) E C Rnc, t E R,is the controller state vector,Cisanopenset,F, : V x C + R n c , a n d $ , : D x C + U . Note that we do not assume any regularity condition on .) and cjc(.,.). However, we do assume the mappings F,(., that the nonlinear closed-loop dynamical system given by ( l ) ,(23), and (24) is such that the solutions of the closedloop system in R are unique and continuously dependent on the closed-loop system initial conditions. To construct dynamic controllers of the form (23), (24) Finally, it is important to note that since the hierarchi- we assume that the scheduling set S is such that there cal switching nonlinear controller U = ~ X ~ ( ~ ) ( Xx)E, D,, exists a closed interval [O,a], a > 0, and a diffeomor-
2993
4t) =
$C(34t),G(t)),
Theorem 6.1. Assume D : [O,a] + S is a diffeomorphism D : [0, a] + As, such that D ( S ) E S , s E [0, a ] , and ~ ( 0= ) 0. Furthermore, we assume that VA(.)and CA are phism, XS : V, -+ S is a C' function, and VA(-)and CA C' functions of X E S . Next, recall that Theorem 5.3 guar- are C1 functions of X E S . Then, the solutions z ( t ) and antees that the feedback control law U(.) = ~A~(,)(x), X ( t ) , t E I,,,of the closed-loop system z E V,, where Xs(z>, z E V,, is given by (20), lo(29) cally asymptotically stabilizes the origin of the nonlin- k ( t ) = F(a:(t),4qt,(zc(t)>),4 0 ) = zo, t E I,, = Q A ( t ) ( 4 t ) ) F ( z ( th) ,( t )( z ( t ) ) ) , Xo = Xs(zo), (30) ear closed-loop dynamical system (2), and V, is a subset of the domain of attraction. Furthermore, note that if are such that X ( t ) = X,(z(t)), t E Z,,, or, equivalently, 20 E V, \ VO it follows by continuity of the closed-loop VA(t)(z(t)) = CA(t), t E L o . trajectories x(.) that there exists a finite time T,, > 0 Proof. The result follows by differentiating both sides such that z(T,,) E dV0 and Xs(z(T,,)) = 0. Now, define of (25) with respect to time and noting that At = wA,s(t). Xt Xs(z(t)) and note that since S is a connected set U and the set-valued map $E(.) is continuous, it follows that Note that the switching function dynamics characterR, = V-'(y), y 2 0, which implies that V(z(t)), t 2 0, can be constant on the interval [ t l ,t 2 ] C [0, T,,] only if ized by (30) defines a fixed-order dynamic compensator z ( t ) E ~ V A t, E, [tl,t z ] , which contradicts the fact that of the form given by (23), (24). Specifically, defining the V ~ , ( z ( t< ) ) 0. Hence, it follows that V(z(t)), t E [O,T,,], compensator state as z c ( t ) 4 X ( t ) so that n, = q, the is a strictly decreasing function and Theorem 5.2 further dynamic compensator structure is given by = ,V < l ( c ~ t ) t, E [O,T,,]. Thus, implies that z ( t ) E ~ V A kc ( t )= Qz, ( t )(z ( t )1F ( z ( t )7 4 z c( t )(E( t )1) 7 (31) 4 t ) = 4 z , ( t ) ( z ( t ) ) , 4 0 ) = Xs(zo), t E L o . (32) VA,(z(t)) = CA,, t E [O,%O], (25) relates the state trajectories z(.) to the scheduling func- Now, it follows from Theorems 5.3 and 6.1 that for all tion Xs(z(.)). For the statement of the main result of zo E V, the closed-loop dynamical system given by ( l ) , this section the following definitions and proposition are (31), and (32), drives z,(t) to 0 E S in a finite time T,, > 0. Note that this result does not violate the assumpneeded. For ( 2 ,A) E V, x S define tion of uniqueness of solutions of (31) since x(Tz0) E dVo at the finite time T,, > 0 does not correspond to a system equilibrium point. Next, since X ( t ) = 0, t 2 T,,, z ( t ) reaches 0 E V asymptotically which guarantees asymptotic stability of the origin with an estimate of the do--!% which, as shown = in Propo- main of attraction given by V,. Similar arguments hold and &A(.) VA(X)WA sition 6.1, is well defined for all z E V,\Vo and X = Xs(z). for global asymptotic stability in the case where V,E Rn. Finally, we note that the compensator dynamics (30) In the case where z E V O ,define &A(.) E 0. characterize the fastest admissible rate of change of the Proposition 6.1. Assume (T : [O,a] -+ S is a diffeo- switching function for which the feedback control (32) morphism, X s : V, -+ S is a C1 function, and VA(.) maintains stability of the closed-loop system. It is not and CA are C' functions of X E S . Then .A(.) and surprising to note that this rate is an explicit function of W A , (.,A) E V, x S , defined in (26), are such that the gradient of the equilibria-dependent Lyapunov func~ A s ( ~ ) ( ~ ) w A s#( z 0 )for all z E V c \Do. tions and the gradient of the domains of attraction esProof. The result follows by differentiating both sides timates with respect to the parameterized surface. Furof (25) and noting that for all zo E V,, VA,(ZO)< 0, thermore, the rate of change of the switching function also XO = Xs(z0). Specifically, differentiating both sides of depends on the gradient of the diffeomorphism evaluated along the scheduling set S ; such a dependence can be used (25) yields to enforce desirable structural properties of the scheduling set. In particular, introducing a C1 positive-definite extended potential function pe : A, + R and defining the scheduling set S as the projection on A, of the steepest Next, let s ( t ) , t E [O,T,,], be such that At = a ( s ( t ) ) , descent curve on the graph of pe(-)containing the point t E [O,T,,], and note that (27) evaluated at t = 0 yields (Xo,pe(Xo)), Xo = Xs(zo), it follows that WA = -pL(X). In this case, the scheduling set S is such that the Euclidean norm of the gradient of the generalized Lyapunov V A O ( ~ O ) W A O . ~ ( O )= avAo(Zo)k(0) ,& = V A ~ ( X
