Stabilizability of Switched Linear Systems Does Not Imply the ...
Proceedings of the 45th IEEE Conference on Decision & Control Manchester Grand Hyatt Hotel San Diego, CA, USA, December 13-15, 2006
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Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions Franco Blanchini and Carlo Savorgnan Abstract— Counterexamples are given which show that a linear switched system (with controlled switching) that can be stabilized by means of a suitable switching law does not necessarily admit a convex Lyapunov function. Both continuous and discrete–time cases are considered. This fact contributes in focusing the difficulties encountered so far in the theory of stabilization of switched systems. In particular the result is in contrast with the case of uncontrolled switching in which it is known that if a system is stable under arbitrary switching then admits a polyhedral norm as a Lyapunov function.
I. INTRODUCTION Switched systems are a class of known practical significance and they are currently receiving a deep attention by the control community. They are commonly viewed as a special case of hybrid systems. Notwithstanding their special nature, they still present fundamental unsolved problems and thus they are currently attracting a significant investigation effort as it is documented in recent surveys [8], [13], special issues [1], [16] and books [17], [12]. We are referring to the class of systems of the form x(t) ˙ = fi(·) (x(t)), t ∈ R+ x(k + 1) = fi(·) (x(k)), k ∈ Z+ where x(t) ∈ Rn is the state vector, R+ is the set of non– negative real numbers and Z+ is the set of non–negative integer numbers. i(·) ∈ I indicates the discrete signal which instantaneously determines the dynamics. Throughout the paper, we will use the notation i = i∗ to indicate that the i∗ dynamics is active x˙ = fi∗ (x). For fixed i, function fi has standard regularity properties, so that the corresponding dynamical system admits a solution in the usual sense. It is admitted that instantaneous commutations among different values in I may occur. Special restrictions are often imposed to the system such as dwell time [17], [12] or constraints (typically a transition graph) on the discrete sequence i. We do not explicitly consider these aspects since the presented results, in view of their negative nature, apply also when such constraints are present. A major distinction of our interest is whether the switch is controlled or not. For the sake of brevity we will call • Switching: a system in which the commutation is not controlled but is determined by exogenous agents; F. Blanchini is with Dipartimento di Matematica e Informatica, Universit`a degli Studi di Udine, Via delle Scienze 208, 33100 Udine - Italy
[email protected] C. Savorgnan is with Dipartimento di Ingegneria Elettrica, Gestionale e Meccanica, Universit`a degli Studi di Udine, Via delle Scienze 208, 33100 Udine - Italy [email protected]
1-4244-0171-2/06/$20.00 ©2006 IEEE.
•
Switched: a system in which the commutation is controlled, that means it can be decided by a control law.
In the second case, it is fundamental to specify which is the control law, that is typically of the form i = i(i− , x(t), t), where i− represents the last value assumed by the discrete variable. In this paper we are considering the problem of stabilizability of switched systems. This problem is most often treated by means of a Lyapunov approach [6]. In particular, a well known result [8], [12] is that, for continuous– time systems, as long there exists a convex combination of the functions fi (x) which is stable and admits a smooth Lyapunov function then the system can be stabilized via a switching law. This means that, in the linear case, the existence of a single stable matrix in the convex hull of the system matrices implies switched quadratic stabilizability [10]. Quite naturally, in the case of linear systems, Lyapunov functions of convex nature are considered. This preference is mainly detected by technical reasons. Perhaps a possible motivation is suggested by the complementary problem of stability of switching system. Indeed it is well known that if a linear switching system is stable (for any possible switching sequence) then it necessarily admits a norm (actually a polytopic norm) as a Lyapunov function [15] [2]. It is also known that the class of quadratic Lyapunov functions does not lead to necessary and sufficient conditions for the stabilization of uncertain (including switching) systems. Therefore several contributions of the past dealt with a more general class of function, the polyhedral norms [7] [3] [4] [5] which allow for non–conservative results for the stability and the stabilization of uncertain systems. Here we show that the situation for switching and switched systems is quite different. Indeed, in the case of switched systems, not even the class of convex positive definite functions (including norms and quadratic functions) are suitable as candidate Lyapunov functions to establish the stabilizability. In brief we show that the existence of a convex Lyapunov function is sufficient but not necessary for the stabilizability of a switched system. In particural, this means that classes of Lyapunov functions that are possibly nonconvex should be preferred to face the problem. An example is given by the piecewise quadratic functions (see for instance [11] [9]) although they do not appear to be useful in the counterexamples presented in this paper.
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II. PROBLEM FORMULATION AND PRELIMINARIES
By the definition of Lyapunov function we can write Ψ(x) ≤ α(¯ µ, T ) < µ ¯ ∀x ∈ C(T )
We deal with switched linear systems of the form x(t) ˙ = Ai(·) x(t)
t ∈ R+ ,
Since Ψ(x) is convex, we have
i ∈ I = {1, · · · , M }
Ψ(x) ≤ α(¯ µ, T ) < µ ¯ ∀x ∈ conv{C(T )}
for the continuous time case and x(k + 1) = Ai(·) x(k)
k ∈ Z+ ,
If we would assume (1) is not true and R ⊆ conv{C(T )} then we would have
i ∈ I = {1, · · · , M }
for the discrete time case. We consider the case in which i(·) can be used to stabilize the system. In particular, state– dependent switching rules i(x) are taken into account. Next we provide an extremely general definition of a Lyapunov function without introducing any specific request of smoothness. Definition 2.1: Consider a system of the form x(t) ˙ = fi(·) (x(t)) (we include both switching and switched cases) for which x = 0 is an equilibrium point. A continuous function which is bounded as follows κ1 (x) ≤ Ψ(x) ≤ κ2 (x) where κ1 (·) κ2 (·) are κ–functions 1 is said to be a Lyapunov function if Ψ(x(t)) ≤ α(Ψ(x(0)), t)
max Ψ(x) ≤ x∈R
sup
x∈conv{C(T )}
Ψ(x) ≤ α(¯ µ, T ) < µ ¯,
in contrast with the definition of µ ¯. The Lemma obviously holds for both discrete and continuous–time case. III. MAIN RESULT We present now two cases (continuous and discrete time respectively) of stabilizable linear switched systems such that, whatever switching rule it is chosen, the confinement set from a considered set R violates the condition of Lemma 2.1, thus proving the non–existence of a convex Lyapunov function. A. CONTINUOUS-TIME CASE Consider the system
∀t,
where α(ψ, t) is continuous and defined for ψ, t ≥ 0, it is strictly increasing w.r.t. ψ, strictly decreasing w.r.t. t and it is such that α(ψ, 0) = ψ. Note that in case of a Lyapunov function assuring exponential stability we just have α(Ψ(x(0)), t) = Ψ(x(0))e−σt , for some σ > 0. An equivalent definition can be given for the discrete time case. We introduce now, for technical reasons, the concept of confinement set in time t ≥ T . Definition 2.2: Given a set R and a switching control law i = i(i− , x(t), t), define as C(T ) the confinement set in time greater or equal to T , as the set of all possible solutions x(t) evaluated at time t ≥ T with initial condition x(0) ∈ R. 2 Clearly for stable systems set C(T ) collapses to 0 as T → ∞. Note that in general C(0) = R unless for special cases. Given a set S we denote by conv{S} its convex hull, namely the smallest convex set containing S (in the case of a discrete set of points xk , we simply write conv{x1 , x2 , . . . }). Now we are able to introduce the next preliminary and fundamental lemma. Lemma 2.1: Consider a system of the form x(t) ˙ = fi(·) (x(t)). Let R be a compact set. If the system admits a convex Lyapunov function, then ∀T > 0, R conv{C(T )}. (1) Proof: Consider the convex Lyapunov function Ψ(x) and let µ ¯ = max Ψ(x). x∈R
1 a κ–function β(·) is a continuous strictly increasing function defined on R+ such that β(0) = 0 2 this definition holds for the switching case if we consider all the possible solutions x(t) achieved by some i(·)
x(t) ˙ = Ai(x(t)) x(t)
i ∈ I = {1, 2}
where A1 and A2 are unstable matrices given by 1 0 γ −1 A1 = A2 = 0 −1 1 γ
(2)
with γ > 0. The system trajectories given the initial condition x(0) are et 0 x(t) = x(0) 0 e−t when i(x(t)) = 1, and eγt cos(t) −eγt sin(t) x(t) = γt x(0) e sin(t) eγt cos(t) when i(x(t)) = 2 (see Fig. 1). The following facts should be noticed. • When the state is of the form (0, x2 ) and the dynamics i = 1 is active, x(t) converges to 0. • When the dynamics i = 2 is active, the phase of x(t) increases with angular speed ω = 1 and its Euclidean norm increases as x(t) = x(0)eγt . Therefore, as a first result of the previous considerations, we can state the following. Proposition 3.1: The state of the system can be driven to 0 by switching, no matter how γ > 0 is taken. Proof: To prove the proposition we can simply notice that the following switching law stabilizes the system: if the state is of the form (0, x2 ) apply the dynamics i = 1, otherwise apply the dynamics i = 2 (from any point, the state will reach the x2 axis in time no greater than π).
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xb
b1
C
−xc
−xg
−˜ x
x˜
xg
xc
B
x1
D
g
x1 −xb b3
A
b4
Fig. 2. The sector D, the extremal trajectory originating from ±˜ x and the points ±xc ,±xb ,±xg
g Q4
Fig. 1. Possible trajectories for dynamics i = 1 (dashed) and dynamics i = 2 (plain).
The previous proposition shows how to construct a trajectory converging to 0 for any initial state. Actually activating the i = 1 only on the x2 axis is not realistic. Better stabilizing switching laws will be proposed later. Before giving a formal proof to show that this is a counterexample, we provide a pictorial explanation. Denote by Qi with i = 1, 2, 3, 4 the four quadrants in counterclockwise direction. Consider as initial state set R the unit circle and in particular choose the initial condition √ √ x ˜ = [1/ 2 − 1/ 2]T ∈ Q4 . We can identify the ”most internal” trajectory (Fig. 3) converging to zero. This extremal trajectory crosses the x1 – axis in a point xc , it crosses the bisector line b1 of Q1 in a point xb and eventually crosses f in a point xf . Any other trajectory from x ˜, which converges to 0 must necessarily cross all these axes in points of greater norm. The argument holds, symmetrically, for the opposite point −˜ x. Then it turns out that, for γ large enough, the norms xc and xb (andxf ) can be made arbitrarily large and that the convex hull conv{xc , −xc , xb , −xb } includes the circle. To provide a formal proof, consider the system trajectories as shown in Figure 1. The lines g and f represent the points where the trajectories of the two subsystems are tangent (i.e the two vector fields are aligned). The equations for these lines can be derived by the alignment condition A1 x = λA2 x. By eliminating λ we derive the angular coefficients . line g: xx21 = − √1 2 = −τ γ+γ +1 line f: xx21 = γ + γ 2 + 1 = 1/τ
Perhaps, the simplest way to show that condition (1) of the lemma is violated is to consider the convex sector D defined by x1 ≥ |x2 | and therefore delimited by the bisectors of the first and the fourth quadrants denoted respectively by b 1 and b4 (see Fig. 2). We can state the following Claim 1 : any trajectory converging to 0 starting form a non–zero initial condition in D must abandon D. This is true because, if we consider the square Euclidean norm as Lyapunov–like function V (x1 , x2 ) = (x21 + x22 )/2, then the derivatives associated with the two dynamics are V˙ i=1 = x21 − x22 ≥ 0, V˙ i=2 = γx21 + γx22 ≥ 0 as long as x(t) ∈ D. As a second step we show the following Claim 2 : any trajectory converging to 0 with initial condition inside D (which leaves D) must cross the bisector of the first quadrant. To prove this claim, we consider the phase of the vector which is well defined for x ∈ D as x2 θ = atan x1 whose derivative is x˙ 2 x1 − x˙ 1 x2 . θ˙ = x21 + x22 For i = 2 we have θ˙ = 1 while for i = 1 we have θ˙ = 2 2 −2x2 x1 /(x1 +x2 ), thus in the lower part of D (D Q4 ) the rotation is positive (counterclockwise), unless x2 = 0. Thus the bisector b4 cannot be crossed clockwise. Conversely we can reach the bisector b1 from any point in D (by setting i = 2). The third step consists in considering the unit disk as the initial set R required by the lemma. In particular we consider the point √ √ x(0) = x ˜ = [1/ 2 − 1/ 2]T ∈ Q4
(the two lines are orthogonal). Observe that τ → 0 as γ → ∞.
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Claim 3 : Any trajectory converging to 0 with initial condition x ˜ crosses the x1 axis in a point xc and
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f
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xf
To prove this claim note that in sector Q4 we have x˙ 1 > 0, no matter which dynamics is active. Any trajectory can be therefore represented by the function x2 (x1 ) whose derivative is
x2 − x1 if i = 1 dx2 = x1 +γx2 if i = 2 dx1 γx1 −x2 The curves that maximizes such a derivative is achieved by taking i = 1 as long as x(t) ∈ D Q4 is below the line g (thus x2 ≤ −τ x1 ) and by activating i = 2 when x(t) ∈ D Q4 is above the line g (i.e. x2 > −τ x1 )3 . The curve achieved by means of this strategy is the extremal one depicted in Fig. 2. Since such a curve maximizes the derivative, any other curve is below it. If we denote by xc the intersection of this curve with the x1 axis, then any other trajectory converging to 0 intersects the line x2 = 0 in a point which is necessarily equalor to the right of xc . To prove that xc ≥ 1/(2τ ), denote by xg the intersection of the extremal curve with the line g (x2 = −τ x1 ). Both x ˜ and xg lie on a trajectory achieved by the first dynamics i = 1 whose analytic expression is, as we have ˜2 e−t . By multiplying the seen, x1 (t) = x˜1 et , x2 (t) = x two expressions and denoting by tg the intersection time for which x(tg ) = xg , we have x1 (tg )x2 (tg ) = x ˜1 x ˜2 = −1/2.
x2
Q2
the bisector b1 of Q1 in a point xb such that 1 xb ≥ xc ≥ 2τ
b1 xb
−˜ x −xc xc
g
−xb
−xf
Q3
Now we have to remind that V (x1 , x2 ) = (x21 + x22 )/2, the square of the norm, is non-decreasing in D. This means that 1 xc ≥ xg = 2τ
Q4
Fig. 3. Trajectories of the stabilized continuous time system starting from the initial states x ˜ and −˜ x. A little hysteresis has been applied on the switching surface s.
On the other hand, for τ > 0 small enough, the unit circle is in the parallelogram 4
(3)
On the other hand, the intersection is on line g, so that x2 (tg ) = −τ x1 (tg ) hence 1 xg ≥ x1 (tg ) = 2τ
x1
x˜
R ⊂ conv{−xc , −xb , xc , xb } Therefore R ⊂ convC(T ) and, in view of the Lemma, we can conclude the following. Proposition 3.2: There do not exist convex Lyapunov functions for this system, no matter which strategy is used. We conclude this section by showing that a sector dependent stabilizing rule exists for this system. Define by θg = atan(−τ ) and θf = atan(1/τ ) the angles of the lines g and f . The following switching surfaces are chosen: g(x) = − sin(θg )x1 + cos(θg )x2 = 0
Since for any trajectory the crossing point xc is on the right of that achieved by the extremal trajectory, this inequality holds for any solution. Let us denote by xb the first intersection of any converging trajectory with the bisector b 1 of Q1 . Again, since V (x1 , x2 ) is non-decreasing in D, we have xb ≥ xc , so that Claim 3 is proved. Now we have just to repeat exactly the symmetric argument for the initial condition −˜ x to identify the symmetric and −x (having obviously norms greater or points −x c b equal to 1/(2τ )). For T > 0 small enough (smaller than the minimum time to reach the x1 axis from x ˜) we have that C(T ) includes the points {−xc , −xb , xc , xb }, so that
(any value if g(x)s(x) = 0). By means of this strategy, we always reach the line s (from one side or the other depending on where we start). If we consider a sufficiently small sector including this line, the following property holds
conv{−xc , −xb , xc , xb } ⊂ convC(T )
s(x)s(x) ˙ 0
4 easy
the line the two derivatives are equal
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computations show that τ < sin2 (π/8)/2
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on both sides which means that s(x) = 0 is a sliding surface. To prove this consider the quadrant Q1 . Then for s(x) < 0 the first dynamics is activated, so we have
i=1
s(x) ˙ = sin(θf + δθ )x1 + cos(θf + δθ )x2 > 0 Conversely for s(x) > 0 the second dynamics is activated, so we have
−˜ x
i=2
s(x) ˙ = sin(θf + δθ )(γx1 − x2 ) − cos(θf + δθ )(x1 + γx2 ) < 0 x1
To show that the sliding mode is converging one have just to consider the theory of the equivalent derivative [18]. For each point on the line s one must find a convex combination of A1 x and A2 x aligned with this line.
i=2
x˜
x˙ eq = (1 − λ)A1 x + λA2 x = µ[cos(θf + δθ ) sin(θf + δθ )]T for some 0 ≤ λ ≤ 1. Tedious computation show that µ < 0, so the sliding mode is convergent. Remark 3.1: A simple way to drive the state to 0 without chattering is to introduce an hysteresis in a sector including line s (see Fig. 3). This basically shows that our negative result holds also in the presence of dwell time. Remark 3.2: In the proof of Proposition 3.1, the proposed switching law applies the dynamics i = 1 only on the x2 –axis which is a 0–measure manifold. This is source of practical trouble. In particular arbitrarily small switching delays would cause instability. To avoid this problem, we can enlarge the set on which the dynamics i = 1 is applied. For example a better switching law is given by 1 if (x1 + x2 )x1 ≤ 0; i(x) = 2 otherwise (i.e. dymamics 1 is applied in the two sectors included by the x2 –axis and the dashed line in Figure 4). Figure 4 pictorially shows the trajectories of the state starting from the initial conditions x ˜ and −˜ x, where we allowed for a small switching delay on the x2 –axis. Also in this case the dynamics i = 1 is instantaneosly activated on the x2 –axis and, from an ideal point of view, the state eventually approaches the origin with a vertical motion. In case of such a small switching delay τ the trajectory still converges, provided that τ is small enough. Indeed the trajectory after such a switching is an hiperbola which gets closer to the x2 axis as τ → 0. It quite simple to show that this switching law is stabilizing, so we omit a formal proof. Besides, this new control shows that the system is stabilizable also by a switching law which does not produce sliding motions. B. DISCRETE-TIME CASE Consider the unstable matrices cos(θ) − sin(θ) A1 = ν sin(θ) cos(θ)
A2 =
µ 0 0 0
where θ = 5π 4 , ν and µ > 0 are sufficiently large numbers (ν = 10 and µ = 100 is enough). The matrix A1 rotates the
i=1
Fig. 4. Trajectories of the continuous time system when the switching law proposed in Remark 3.2 is used. A small switching delay is applied when crossing the x2 –axis.
state anti-clockwise and increases its norm while the matrix A2 has the property that A2 x¯ = 0, if x ¯ = (0, x2 )T Define the set X
=
{x = γ(cos(φ), sin(φ)), γ ∈ R, i5π , i = 0, 1, 2, 3} φ= 4 Note that the second dynamics i = 2 drives the state to the x1 –axis that belongs to X . Furthermore, for each vector in this set, in a finite number of steps the state is brought to the x2 –axis (which is not included in X ). From this axis A2 drives the state to 0. The following stabilizing control strategy can be adopted. • When x ∈ / X we activate A2 (to obtain a new state of the form x = (x1 , 0) ∈ X ) ; • When x ∈ X we activate A1 (this rotates the system state until it is of the form x = (0, x2 )); The strategy can be formally written as 2 if x ∈ X i(x) = 1 otherwise In figure 5 the trajectories obtained from the chosen switching rule are shown for the initial states x ˜ = (1, 0) and −˜ x. Take as initial set R = {x : x = (x1 , 0) f or x1 ∈ [−1; 1]}. The counterexample is based on a simple observation. Consider x(0) = [1 0]T and its opposite. If we apply the proposed strategy, the state is driven to 0 by applying A1 , repeatedly until we reach the x2 –axis, where in one shot we reach zero by activating A2 . In applying this strategy
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x2
−˜ x
x˜
x1
Fig. 5. Trajectories of the stabilized discrete time system starting from the initial states x ˜ and −˜ x.
[9] J.C. Geromel and P. Colaneri. Stability and stabilization of discrete– time switched systems. Int. Journal of Control, to appear, 2006. [10] B. Hu, G. Zhai, and A. N. Michel. Common quadratic Lyapunov-like functions with associated switching regions for two unstable secondorder lti systems. Int. Journal of Control, 75(14):1127–1135, 2002. [11] M. Johansson and A. Rantzer. Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Automat. Contr., 43(4):555–559, 1998. [12] D. Liberzon. Switching in System and Control. Systems & and Control: Foundations & Applications. Birkhauser, Boston, 2003. [13] D. Liberzon and A. S. Morse. Basic problems in stability and design of switched systems. IEEE Contr. Syst. Mag., 19(5):59–70, 1999. [14] Y. Lin, E. D. Sontag, and Y. Wang. A smooth converse Lyapunov theorem for robust stability. SIAM J. on Contr. and Optim., 34(1):124– 160, 1996. [15] A. P. Molchanov and E. S. Pyatnitskii. Lyapunov functions that define necessary and sufficient conditions for absolute stability of nonlinear nonstationary control systems. IIIIII. Automat. Remote Control, 47:I:344–354, II: 443–451, III:620–630, 1986. [16] A. S. Morse, C.C. Pantelides, S. S. Sastry, and J.M. Schumacher. Special issue on hybrid systems. Automatica, 35(3), 1989. [17] Z. Sun and S. Ge. Switched Linear Systems Control and Design. Communications and Control Engineering. Springer-Verlag, London, 2005. [18] V.I. Utkin. Sliding modes and their application in variable structure systems. MIR, Moskow, 1978.
the norm of x(k) increases at each step (see Fig 5) so the resulting C(1) is the union of rotated segments of increasing length, then its convex hull includes R. Conversely, if we change strategy and we apply A2 before reaching the vertical axis, as a result we send back x to the x1 –axis but with an norm greater than 1 thus the condition (1) is violated anyway. IV. CONCLUSIONS By means of counterexamples, in this paper it has been shown that, even in the linear case, a system which can be stabilized via switching control, does not necessarily admit a convex Lyapunov function. Clearly, in view of known converse results [14], it is expected that stabilizability implies the existence of a Lyapunov functions. However, these should be sought in more general classes than the convex ones which are currently the preferred ones. R EFERENCES [1] P.J. Antsaklis and A. Nerode. Special issue on hybrid control systems. IEEE Trans. Automat. Contr., 43(4), 1988. [2] N. E. Barabanov. Lyapunov indicator of discrete inclusion. part I-II-III. Automat. Remote Control, 49:I: 152–157, II: 283–287, III: 558–566, 1988. [3] A. Bhaya and C. Mota. Equivalence of stability concepts for discrete time-varying system. Int. Journal on Robust and Nonlinear Control, 4(6):725–740, 1994. [4] F. Blanchini. Non-quadratic Lyapunov function for robust control. Automatica, 31(3):451–461, 1995. [5] F. Blanchini and S. Miani. On the transient estimate for linear systems with time-varying uncertain parameters. IEEE Trans. on Circuit and Systems, 43(7):000–000, 1996. [6] M. S. Branicky. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Automat. Contr., 43(4):475–482, 1998. [7] R. K. Brayton and C. H. Tong. Constructive stability and asymptotic stability of dynamical systems. IEEE Trans. on Circuit and Systems, 27(11):1121–1130, April 1980. [8] R. A. Decarlo, M. S. Branicky, S. Pettersson, and B. Lennatson. Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of the IEEE, 88(7):1069–1082, 2000.
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Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions Franco Blanchini and Carlo Savorgnan Abstract— Counterexamples are given which show that a linear switched system (with controlled switching) that can be stabilized by means of a suitable switching law does not necessarily admit a convex Lyapunov function. Both continuous and discrete–time cases are considered. This fact contributes in focusing the difficulties encountered so far in the theory of stabilization of switched systems. In particular the result is in contrast with the case of uncontrolled switching in which it is known that if a system is stable under arbitrary switching then admits a polyhedral norm as a Lyapunov function.
I. INTRODUCTION Switched systems are a class of known practical significance and they are currently receiving a deep attention by the control community. They are commonly viewed as a special case of hybrid systems. Notwithstanding their special nature, they still present fundamental unsolved problems and thus they are currently attracting a significant investigation effort as it is documented in recent surveys [8], [13], special issues [1], [16] and books [17], [12]. We are referring to the class of systems of the form x(t) ˙ = fi(·) (x(t)), t ∈ R+ x(k + 1) = fi(·) (x(k)), k ∈ Z+ where x(t) ∈ Rn is the state vector, R+ is the set of non– negative real numbers and Z+ is the set of non–negative integer numbers. i(·) ∈ I indicates the discrete signal which instantaneously determines the dynamics. Throughout the paper, we will use the notation i = i∗ to indicate that the i∗ dynamics is active x˙ = fi∗ (x). For fixed i, function fi has standard regularity properties, so that the corresponding dynamical system admits a solution in the usual sense. It is admitted that instantaneous commutations among different values in I may occur. Special restrictions are often imposed to the system such as dwell time [17], [12] or constraints (typically a transition graph) on the discrete sequence i. We do not explicitly consider these aspects since the presented results, in view of their negative nature, apply also when such constraints are present. A major distinction of our interest is whether the switch is controlled or not. For the sake of brevity we will call • Switching: a system in which the commutation is not controlled but is determined by exogenous agents; F. Blanchini is with Dipartimento di Matematica e Informatica, Universit`a degli Studi di Udine, Via delle Scienze 208, 33100 Udine - Italy
[email protected] C. Savorgnan is with Dipartimento di Ingegneria Elettrica, Gestionale e Meccanica, Universit`a degli Studi di Udine, Via delle Scienze 208, 33100 Udine - Italy [email protected]
1-4244-0171-2/06/$20.00 ©2006 IEEE.
•
Switched: a system in which the commutation is controlled, that means it can be decided by a control law.
In the second case, it is fundamental to specify which is the control law, that is typically of the form i = i(i− , x(t), t), where i− represents the last value assumed by the discrete variable. In this paper we are considering the problem of stabilizability of switched systems. This problem is most often treated by means of a Lyapunov approach [6]. In particular, a well known result [8], [12] is that, for continuous– time systems, as long there exists a convex combination of the functions fi (x) which is stable and admits a smooth Lyapunov function then the system can be stabilized via a switching law. This means that, in the linear case, the existence of a single stable matrix in the convex hull of the system matrices implies switched quadratic stabilizability [10]. Quite naturally, in the case of linear systems, Lyapunov functions of convex nature are considered. This preference is mainly detected by technical reasons. Perhaps a possible motivation is suggested by the complementary problem of stability of switching system. Indeed it is well known that if a linear switching system is stable (for any possible switching sequence) then it necessarily admits a norm (actually a polytopic norm) as a Lyapunov function [15] [2]. It is also known that the class of quadratic Lyapunov functions does not lead to necessary and sufficient conditions for the stabilization of uncertain (including switching) systems. Therefore several contributions of the past dealt with a more general class of function, the polyhedral norms [7] [3] [4] [5] which allow for non–conservative results for the stability and the stabilization of uncertain systems. Here we show that the situation for switching and switched systems is quite different. Indeed, in the case of switched systems, not even the class of convex positive definite functions (including norms and quadratic functions) are suitable as candidate Lyapunov functions to establish the stabilizability. In brief we show that the existence of a convex Lyapunov function is sufficient but not necessary for the stabilizability of a switched system. In particural, this means that classes of Lyapunov functions that are possibly nonconvex should be preferred to face the problem. An example is given by the piecewise quadratic functions (see for instance [11] [9]) although they do not appear to be useful in the counterexamples presented in this paper.
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II. PROBLEM FORMULATION AND PRELIMINARIES
By the definition of Lyapunov function we can write Ψ(x) ≤ α(¯ µ, T ) < µ ¯ ∀x ∈ C(T )
We deal with switched linear systems of the form x(t) ˙ = Ai(·) x(t)
t ∈ R+ ,
Since Ψ(x) is convex, we have
i ∈ I = {1, · · · , M }
Ψ(x) ≤ α(¯ µ, T ) < µ ¯ ∀x ∈ conv{C(T )}
for the continuous time case and x(k + 1) = Ai(·) x(k)
k ∈ Z+ ,
If we would assume (1) is not true and R ⊆ conv{C(T )} then we would have
i ∈ I = {1, · · · , M }
for the discrete time case. We consider the case in which i(·) can be used to stabilize the system. In particular, state– dependent switching rules i(x) are taken into account. Next we provide an extremely general definition of a Lyapunov function without introducing any specific request of smoothness. Definition 2.1: Consider a system of the form x(t) ˙ = fi(·) (x(t)) (we include both switching and switched cases) for which x = 0 is an equilibrium point. A continuous function which is bounded as follows κ1 (x) ≤ Ψ(x) ≤ κ2 (x) where κ1 (·) κ2 (·) are κ–functions 1 is said to be a Lyapunov function if Ψ(x(t)) ≤ α(Ψ(x(0)), t)
max Ψ(x) ≤ x∈R
sup
x∈conv{C(T )}
Ψ(x) ≤ α(¯ µ, T ) < µ ¯,
in contrast with the definition of µ ¯. The Lemma obviously holds for both discrete and continuous–time case. III. MAIN RESULT We present now two cases (continuous and discrete time respectively) of stabilizable linear switched systems such that, whatever switching rule it is chosen, the confinement set from a considered set R violates the condition of Lemma 2.1, thus proving the non–existence of a convex Lyapunov function. A. CONTINUOUS-TIME CASE Consider the system
∀t,
where α(ψ, t) is continuous and defined for ψ, t ≥ 0, it is strictly increasing w.r.t. ψ, strictly decreasing w.r.t. t and it is such that α(ψ, 0) = ψ. Note that in case of a Lyapunov function assuring exponential stability we just have α(Ψ(x(0)), t) = Ψ(x(0))e−σt , for some σ > 0. An equivalent definition can be given for the discrete time case. We introduce now, for technical reasons, the concept of confinement set in time t ≥ T . Definition 2.2: Given a set R and a switching control law i = i(i− , x(t), t), define as C(T ) the confinement set in time greater or equal to T , as the set of all possible solutions x(t) evaluated at time t ≥ T with initial condition x(0) ∈ R. 2 Clearly for stable systems set C(T ) collapses to 0 as T → ∞. Note that in general C(0) = R unless for special cases. Given a set S we denote by conv{S} its convex hull, namely the smallest convex set containing S (in the case of a discrete set of points xk , we simply write conv{x1 , x2 , . . . }). Now we are able to introduce the next preliminary and fundamental lemma. Lemma 2.1: Consider a system of the form x(t) ˙ = fi(·) (x(t)). Let R be a compact set. If the system admits a convex Lyapunov function, then ∀T > 0, R conv{C(T )}. (1) Proof: Consider the convex Lyapunov function Ψ(x) and let µ ¯ = max Ψ(x). x∈R
1 a κ–function β(·) is a continuous strictly increasing function defined on R+ such that β(0) = 0 2 this definition holds for the switching case if we consider all the possible solutions x(t) achieved by some i(·)
x(t) ˙ = Ai(x(t)) x(t)
i ∈ I = {1, 2}
where A1 and A2 are unstable matrices given by 1 0 γ −1 A1 = A2 = 0 −1 1 γ
(2)
with γ > 0. The system trajectories given the initial condition x(0) are et 0 x(t) = x(0) 0 e−t when i(x(t)) = 1, and eγt cos(t) −eγt sin(t) x(t) = γt x(0) e sin(t) eγt cos(t) when i(x(t)) = 2 (see Fig. 1). The following facts should be noticed. • When the state is of the form (0, x2 ) and the dynamics i = 1 is active, x(t) converges to 0. • When the dynamics i = 2 is active, the phase of x(t) increases with angular speed ω = 1 and its Euclidean norm increases as x(t) = x(0)eγt . Therefore, as a first result of the previous considerations, we can state the following. Proposition 3.1: The state of the system can be driven to 0 by switching, no matter how γ > 0 is taken. Proof: To prove the proposition we can simply notice that the following switching law stabilizes the system: if the state is of the form (0, x2 ) apply the dynamics i = 1, otherwise apply the dynamics i = 2 (from any point, the state will reach the x2 axis in time no greater than π).
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b2
Q1
xb
b1
C
−xc
−xg
−˜ x
x˜
xg
xc
B
x1
D
g
x1 −xb b3
A
b4
Fig. 2. The sector D, the extremal trajectory originating from ±˜ x and the points ±xc ,±xb ,±xg
g Q4
Fig. 1. Possible trajectories for dynamics i = 1 (dashed) and dynamics i = 2 (plain).
The previous proposition shows how to construct a trajectory converging to 0 for any initial state. Actually activating the i = 1 only on the x2 axis is not realistic. Better stabilizing switching laws will be proposed later. Before giving a formal proof to show that this is a counterexample, we provide a pictorial explanation. Denote by Qi with i = 1, 2, 3, 4 the four quadrants in counterclockwise direction. Consider as initial state set R the unit circle and in particular choose the initial condition √ √ x ˜ = [1/ 2 − 1/ 2]T ∈ Q4 . We can identify the ”most internal” trajectory (Fig. 3) converging to zero. This extremal trajectory crosses the x1 – axis in a point xc , it crosses the bisector line b1 of Q1 in a point xb and eventually crosses f in a point xf . Any other trajectory from x ˜, which converges to 0 must necessarily cross all these axes in points of greater norm. The argument holds, symmetrically, for the opposite point −˜ x. Then it turns out that, for γ large enough, the norms xc and xb (andxf ) can be made arbitrarily large and that the convex hull conv{xc , −xc , xb , −xb } includes the circle. To provide a formal proof, consider the system trajectories as shown in Figure 1. The lines g and f represent the points where the trajectories of the two subsystems are tangent (i.e the two vector fields are aligned). The equations for these lines can be derived by the alignment condition A1 x = λA2 x. By eliminating λ we derive the angular coefficients . line g: xx21 = − √1 2 = −τ γ+γ +1 line f: xx21 = γ + γ 2 + 1 = 1/τ
Perhaps, the simplest way to show that condition (1) of the lemma is violated is to consider the convex sector D defined by x1 ≥ |x2 | and therefore delimited by the bisectors of the first and the fourth quadrants denoted respectively by b 1 and b4 (see Fig. 2). We can state the following Claim 1 : any trajectory converging to 0 starting form a non–zero initial condition in D must abandon D. This is true because, if we consider the square Euclidean norm as Lyapunov–like function V (x1 , x2 ) = (x21 + x22 )/2, then the derivatives associated with the two dynamics are V˙ i=1 = x21 − x22 ≥ 0, V˙ i=2 = γx21 + γx22 ≥ 0 as long as x(t) ∈ D. As a second step we show the following Claim 2 : any trajectory converging to 0 with initial condition inside D (which leaves D) must cross the bisector of the first quadrant. To prove this claim, we consider the phase of the vector which is well defined for x ∈ D as x2 θ = atan x1 whose derivative is x˙ 2 x1 − x˙ 1 x2 . θ˙ = x21 + x22 For i = 2 we have θ˙ = 1 while for i = 1 we have θ˙ = 2 2 −2x2 x1 /(x1 +x2 ), thus in the lower part of D (D Q4 ) the rotation is positive (counterclockwise), unless x2 = 0. Thus the bisector b4 cannot be crossed clockwise. Conversely we can reach the bisector b1 from any point in D (by setting i = 2). The third step consists in considering the unit disk as the initial set R required by the lemma. In particular we consider the point √ √ x(0) = x ˜ = [1/ 2 − 1/ 2]T ∈ Q4
(the two lines are orthogonal). Observe that τ → 0 as γ → ∞.
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Claim 3 : Any trajectory converging to 0 with initial condition x ˜ crosses the x1 axis in a point xc and
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xf
To prove this claim note that in sector Q4 we have x˙ 1 > 0, no matter which dynamics is active. Any trajectory can be therefore represented by the function x2 (x1 ) whose derivative is
x2 − x1 if i = 1 dx2 = x1 +γx2 if i = 2 dx1 γx1 −x2 The curves that maximizes such a derivative is achieved by taking i = 1 as long as x(t) ∈ D Q4 is below the line g (thus x2 ≤ −τ x1 ) and by activating i = 2 when x(t) ∈ D Q4 is above the line g (i.e. x2 > −τ x1 )3 . The curve achieved by means of this strategy is the extremal one depicted in Fig. 2. Since such a curve maximizes the derivative, any other curve is below it. If we denote by xc the intersection of this curve with the x1 axis, then any other trajectory converging to 0 intersects the line x2 = 0 in a point which is necessarily equalor to the right of xc . To prove that xc ≥ 1/(2τ ), denote by xg the intersection of the extremal curve with the line g (x2 = −τ x1 ). Both x ˜ and xg lie on a trajectory achieved by the first dynamics i = 1 whose analytic expression is, as we have ˜2 e−t . By multiplying the seen, x1 (t) = x˜1 et , x2 (t) = x two expressions and denoting by tg the intersection time for which x(tg ) = xg , we have x1 (tg )x2 (tg ) = x ˜1 x ˜2 = −1/2.
x2
Q2
the bisector b1 of Q1 in a point xb such that 1 xb ≥ xc ≥ 2τ
b1 xb
−˜ x −xc xc
g
−xb
−xf
Q3
Now we have to remind that V (x1 , x2 ) = (x21 + x22 )/2, the square of the norm, is non-decreasing in D. This means that 1 xc ≥ xg = 2τ
Q4
Fig. 3. Trajectories of the stabilized continuous time system starting from the initial states x ˜ and −˜ x. A little hysteresis has been applied on the switching surface s.
On the other hand, for τ > 0 small enough, the unit circle is in the parallelogram 4
(3)
On the other hand, the intersection is on line g, so that x2 (tg ) = −τ x1 (tg ) hence 1 xg ≥ x1 (tg ) = 2τ
x1
x˜
R ⊂ conv{−xc , −xb , xc , xb } Therefore R ⊂ convC(T ) and, in view of the Lemma, we can conclude the following. Proposition 3.2: There do not exist convex Lyapunov functions for this system, no matter which strategy is used. We conclude this section by showing that a sector dependent stabilizing rule exists for this system. Define by θg = atan(−τ ) and θf = atan(1/τ ) the angles of the lines g and f . The following switching surfaces are chosen: g(x) = − sin(θg )x1 + cos(θg )x2 = 0
Since for any trajectory the crossing point xc is on the right of that achieved by the extremal trajectory, this inequality holds for any solution. Let us denote by xb the first intersection of any converging trajectory with the bisector b 1 of Q1 . Again, since V (x1 , x2 ) is non-decreasing in D, we have xb ≥ xc , so that Claim 3 is proved. Now we have just to repeat exactly the symmetric argument for the initial condition −˜ x to identify the symmetric and −x (having obviously norms greater or points −x c b equal to 1/(2τ )). For T > 0 small enough (smaller than the minimum time to reach the x1 axis from x ˜) we have that C(T ) includes the points {−xc , −xb , xc , xb }, so that
(any value if g(x)s(x) = 0). By means of this strategy, we always reach the line s (from one side or the other depending on where we start). If we consider a sufficiently small sector including this line, the following property holds
conv{−xc , −xb , xc , xb } ⊂ convC(T )
s(x)s(x) ˙ 0
4 easy
the line the two derivatives are equal
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computations show that τ < sin2 (π/8)/2
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on both sides which means that s(x) = 0 is a sliding surface. To prove this consider the quadrant Q1 . Then for s(x) < 0 the first dynamics is activated, so we have
i=1
s(x) ˙ = sin(θf + δθ )x1 + cos(θf + δθ )x2 > 0 Conversely for s(x) > 0 the second dynamics is activated, so we have
−˜ x
i=2
s(x) ˙ = sin(θf + δθ )(γx1 − x2 ) − cos(θf + δθ )(x1 + γx2 ) < 0 x1
To show that the sliding mode is converging one have just to consider the theory of the equivalent derivative [18]. For each point on the line s one must find a convex combination of A1 x and A2 x aligned with this line.
i=2
x˜
x˙ eq = (1 − λ)A1 x + λA2 x = µ[cos(θf + δθ ) sin(θf + δθ )]T for some 0 ≤ λ ≤ 1. Tedious computation show that µ < 0, so the sliding mode is convergent. Remark 3.1: A simple way to drive the state to 0 without chattering is to introduce an hysteresis in a sector including line s (see Fig. 3). This basically shows that our negative result holds also in the presence of dwell time. Remark 3.2: In the proof of Proposition 3.1, the proposed switching law applies the dynamics i = 1 only on the x2 –axis which is a 0–measure manifold. This is source of practical trouble. In particular arbitrarily small switching delays would cause instability. To avoid this problem, we can enlarge the set on which the dynamics i = 1 is applied. For example a better switching law is given by 1 if (x1 + x2 )x1 ≤ 0; i(x) = 2 otherwise (i.e. dymamics 1 is applied in the two sectors included by the x2 –axis and the dashed line in Figure 4). Figure 4 pictorially shows the trajectories of the state starting from the initial conditions x ˜ and −˜ x, where we allowed for a small switching delay on the x2 –axis. Also in this case the dynamics i = 1 is instantaneosly activated on the x2 –axis and, from an ideal point of view, the state eventually approaches the origin with a vertical motion. In case of such a small switching delay τ the trajectory still converges, provided that τ is small enough. Indeed the trajectory after such a switching is an hiperbola which gets closer to the x2 axis as τ → 0. It quite simple to show that this switching law is stabilizing, so we omit a formal proof. Besides, this new control shows that the system is stabilizable also by a switching law which does not produce sliding motions. B. DISCRETE-TIME CASE Consider the unstable matrices cos(θ) − sin(θ) A1 = ν sin(θ) cos(θ)
A2 =
µ 0 0 0
where θ = 5π 4 , ν and µ > 0 are sufficiently large numbers (ν = 10 and µ = 100 is enough). The matrix A1 rotates the
i=1
Fig. 4. Trajectories of the continuous time system when the switching law proposed in Remark 3.2 is used. A small switching delay is applied when crossing the x2 –axis.
state anti-clockwise and increases its norm while the matrix A2 has the property that A2 x¯ = 0, if x ¯ = (0, x2 )T Define the set X
=
{x = γ(cos(φ), sin(φ)), γ ∈ R, i5π , i = 0, 1, 2, 3} φ= 4 Note that the second dynamics i = 2 drives the state to the x1 –axis that belongs to X . Furthermore, for each vector in this set, in a finite number of steps the state is brought to the x2 –axis (which is not included in X ). From this axis A2 drives the state to 0. The following stabilizing control strategy can be adopted. • When x ∈ / X we activate A2 (to obtain a new state of the form x = (x1 , 0) ∈ X ) ; • When x ∈ X we activate A1 (this rotates the system state until it is of the form x = (0, x2 )); The strategy can be formally written as 2 if x ∈ X i(x) = 1 otherwise In figure 5 the trajectories obtained from the chosen switching rule are shown for the initial states x ˜ = (1, 0) and −˜ x. Take as initial set R = {x : x = (x1 , 0) f or x1 ∈ [−1; 1]}. The counterexample is based on a simple observation. Consider x(0) = [1 0]T and its opposite. If we apply the proposed strategy, the state is driven to 0 by applying A1 , repeatedly until we reach the x2 –axis, where in one shot we reach zero by activating A2 . In applying this strategy
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x2
−˜ x
x˜
x1
Fig. 5. Trajectories of the stabilized discrete time system starting from the initial states x ˜ and −˜ x.
[9] J.C. Geromel and P. Colaneri. Stability and stabilization of discrete– time switched systems. Int. Journal of Control, to appear, 2006. [10] B. Hu, G. Zhai, and A. N. Michel. Common quadratic Lyapunov-like functions with associated switching regions for two unstable secondorder lti systems. Int. Journal of Control, 75(14):1127–1135, 2002. [11] M. Johansson and A. Rantzer. Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Automat. Contr., 43(4):555–559, 1998. [12] D. Liberzon. Switching in System and Control. Systems & and Control: Foundations & Applications. Birkhauser, Boston, 2003. [13] D. Liberzon and A. S. Morse. Basic problems in stability and design of switched systems. IEEE Contr. Syst. Mag., 19(5):59–70, 1999. [14] Y. Lin, E. D. Sontag, and Y. Wang. A smooth converse Lyapunov theorem for robust stability. SIAM J. on Contr. and Optim., 34(1):124– 160, 1996. [15] A. P. Molchanov and E. S. Pyatnitskii. Lyapunov functions that define necessary and sufficient conditions for absolute stability of nonlinear nonstationary control systems. IIIIII. Automat. Remote Control, 47:I:344–354, II: 443–451, III:620–630, 1986. [16] A. S. Morse, C.C. Pantelides, S. S. Sastry, and J.M. Schumacher. Special issue on hybrid systems. Automatica, 35(3), 1989. [17] Z. Sun and S. Ge. Switched Linear Systems Control and Design. Communications and Control Engineering. Springer-Verlag, London, 2005. [18] V.I. Utkin. Sliding modes and their application in variable structure systems. MIR, Moskow, 1978.
the norm of x(k) increases at each step (see Fig 5) so the resulting C(1) is the union of rotated segments of increasing length, then its convex hull includes R. Conversely, if we change strategy and we apply A2 before reaching the vertical axis, as a result we send back x to the x1 –axis but with an norm greater than 1 thus the condition (1) is violated anyway. IV. CONCLUSIONS By means of counterexamples, in this paper it has been shown that, even in the linear case, a system which can be stabilized via switching control, does not necessarily admit a convex Lyapunov function. Clearly, in view of known converse results [14], it is expected that stabilizability implies the existence of a Lyapunov functions. However, these should be sought in more general classes than the convex ones which are currently the preferred ones. R EFERENCES [1] P.J. Antsaklis and A. Nerode. Special issue on hybrid control systems. IEEE Trans. Automat. Contr., 43(4), 1988. [2] N. E. Barabanov. Lyapunov indicator of discrete inclusion. part I-II-III. Automat. Remote Control, 49:I: 152–157, II: 283–287, III: 558–566, 1988. [3] A. Bhaya and C. Mota. Equivalence of stability concepts for discrete time-varying system. Int. Journal on Robust and Nonlinear Control, 4(6):725–740, 1994. [4] F. Blanchini. Non-quadratic Lyapunov function for robust control. Automatica, 31(3):451–461, 1995. [5] F. Blanchini and S. Miani. On the transient estimate for linear systems with time-varying uncertain parameters. IEEE Trans. on Circuit and Systems, 43(7):000–000, 1996. [6] M. S. Branicky. Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Trans. Automat. Contr., 43(4):475–482, 1998. [7] R. K. Brayton and C. H. Tong. Constructive stability and asymptotic stability of dynamical systems. IEEE Trans. on Circuit and Systems, 27(11):1121–1130, April 1980. [8] R. A. Decarlo, M. S. Branicky, S. Pettersson, and B. Lennatson. Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of the IEEE, 88(7):1069–1082, 2000.
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