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STABILIZATION OF DISCRETE-TIME SWITCHED SYSTEMS
Preprints of the 5th IFAC Symposium on Robust Control Design
ROCOND'06, Toulouse, France, July 5-7, 2006
STABILIZATION OF DISCRETE-TIME SWITCHED SYSTEMS Jos´ e C. Geromel# and Patrizio Colaneri⋆
# DSCE / School of Electrical and Computer Engineering UNICAMP, CP 6101, 13083 - 970 Campinas, SP, Brazil, [email protected] ⋆
Politecnico di Milano Dipartimento di Elettronica e Informazione Piazza Leonardo da Vinci 32, 20133 Milano, Italy [email protected]
Abstract: This paper addresses two strategies for stabilization of discrete time linear switched systems. The first one, is of open loop nature and is based on the determination of an upper bound of the minimum dwell time by means of a family of quadratic Lyapunov functions. The relevant point on dwell time calculation is that the proposed stability condition does not require the Lyapunov function be uniformly decreasing at every switching time. The second one, is of closed loop nature and is designed from the solution of what we call Lyapunov-Metzler c inequalities from which the stability condition is expressed. Copyright ° 2005 IFAC. Keywords: Switched systems, Discrete-time systems, Linear matrix inequality
1. INTRODUCTION
This paper aims at providing procedures for stabilizing control synthesis for a discrete time switched linear system of the following general form x(k + 1) = Aσ(k) x(k) , x(0) = x0
(1)
n
defined for all k ∈ N where x(k) ∈ R is the state, σ(k) is the switching rule, x0 is the initial condition and N is the set of natural numbers. Considering a set of matrices Ai ∈ Rn×n , i = 1, · · · , N be given, the switching rule σ(k), for each k ∈ N, is such that Aσ(k) ∈ {A1 , · · · , AN }
(2)
Clearly, (2) imposes that Aσ(k) is constrained to jump among the N vertices of the matrix polytope {A1 , · · · , AN }.
Stability analysis of continuous time switched linear systems have been addressed in (Branicky, 1998), (Hespanha, 2004), (Hockerman et al., 1998), (Johansson et al., 1998), (Ye et al., 1998) and (Liberzon, 2003), among others. For discretetime switched systems, see (Daafouz et al., 2001), (Xie et al., 2003) and (Zhai, 2001). References (DeCarlo et al., 2000) and (Liberzon et al., 1999) provide excellent overviews on switched and hybrid systems. In this paper, the determination of an upper bound of the minimum dwell time preserving global stability is accomplished from the determination of a family of quadratic Lyapunov functions. Each of them is associated to the corresponding matrix of the set {A1 , · · · , AN } such that at every switching time lq the sequence v(x(lq )), for q = 0, · · · , ∞, converges uniformly to zero. In some instances, the determination of
Preprints of the 5th IFAC Symposium on Robust Control Design
ROCOND'06, Toulouse, France, July 5-7, 2006
the minimum dwell time, based on a quadratic guaranteed cost, is related to the results of (Xu et al., 2004) assuming further that the switching rule is not a priori given but can be taken arbitrarily, among the feasible ones. For switched systems with σ(·) being state dependent, the stability condition is expressed in terms of a set of inequalities that we call LyapunovMetzler inequalities because the variables involved are a set of symmetric and positive matrices {P1 , · · · , PN } and a Metzler matrix Π. The point to be noticed is that our asymptotical stability condition contains as special cases the quadratic stability condition and the results of (Daafouz et al., 2001) and (Zhai, 2001). The price to be payed, however, is the non-convex nature of the the Lyapunov-Metzler inequalities being thus difficult to solve numerically. For this reason, a more conservative but easier to solve asymptotical stability condition is proposed. This paper, is the discretetime counterpart of (Geromel et al., 2005).
which enables us to conclude that there exist scalars α ∈ (0, 1) and β > 0 such that kx(k)k22 ≤ βα(k−lq ) v(x(lq )) , ∀k ∈ [lq , lq+1 ) (7) On the other hand, using the inequalities (5) we have v(x(lq+1 )) = x(lq+1 )′ Pj x(lq+1 ) ′∆q
= x(lq )′ Ai
∆
Pj Ai q x(lq )
′(∆q −∆)
< x(lq )′ Ai
(∆q −∆)
Pi Ai
x(lq )
′
< x(lq ) Pi x(lq ) < v(x(lq ))
(8)
where the second inequality holds from the fact that for every τ = ∆q − ∆ ∈ N it is true that τ A′τ i Pi Ai ≤ Pi . The consequence is that there exists µ ∈ (0 , 1) such that v(x(lq )) ≤ µq v(x0 ) , ∀q ∈ N
(9)
which together with (7) implies that the equilibrium solution x = 0 of (1) is globally asymptotically stable. 2
2. TIME SWITCHING CONTROL Assuming that the matrices {A1 , · · · , AN } are asymptotically stable, the problem under consideration can be stated as follows : Determine a minimum dwell time ∆∗ ∈ N such that the equilibrium point x = 0 of the system (1) is globally asymptotically stable with the time switching control σ(k) = i ∈ {1, · · · , N } , k ∈ [lq , lq+1 )
(3)
where lq and lq+1 are successive switching times satisfying lq+1 − lq ≥ ∆∗ ≥ 1 for all q ∈ N. It is interesting to observe that the index i ∈ {1, · · · , N } selected at each instant of time k ∈ N is arbitrary. Theorem 1. Assume that, for some ∆ ≥ 1, there exists a collection of positive definite matrices {P1 , · · · , PN } of compatible dimensions such that A′i Pi Ai − Pi < 0 , ∀ i = 1, · · · , N ∆ A′∆ i Pj Ai − Pi < 0 , ∀ i 6= j = 1, · · · , N
(4) (5)
The time switching control (3) with lq+1 − lq ≥ ∆ makes the equilibrium solution x = 0 of (1) globally asymptotically stable. Proof: Consider, in accordance to (3), that σ(k) = i ∈ {1, · · · , N } for all k ∈ [lq , lq+1 ) where lq+1 = lq + ∆q with ∆q ≥ ∆ ≥ 1 and that at k = lq+1 the time switching control jumps to σ(k) = j ∈ {1, · · · , N }, otherwise the result trivially follows. From (4), it is seen that, for all k ∈ [lq , lq+1 ), the Lyapunov function v(x(k)) = x(k)′ Pσ(k) x(k), along an arbitrary trajectory of (1) satisfies v(x(k + 1)) < v(x(k))
(6)
This result deserves some comments. First, since it is assumed that all matrices A1 , · · · , AN are asymptotically stable then the constraints (4) are always feasible and the constraints (5) are satisfied whenever ∆ ∈ N is taken large enough. Second, assume that matrices A1 , · · · , AN are quadratically stable, which is the same to say that they share a positive definite matrix P such that A′i P Ai − P < 0 for all i = 1, · · · , N . In this case, the inequality (5) is satisfied for P1 = · · · = PN = P for all ∆ ≥ 1 meaning that the switching policy (3) may jump from i to j at each instant of time k ∈ N, preserving once again asymptotical stability. Hence, Theorem 1 contains, as a particular case, the quadratic stability condition and the stability results of (Daafouz et al., 2001) which are less conservative than quadratic stability. Finally, with ∆ ≥ 1, it is always possible to define a time switching control strategy (3) such that Aσ(k) is periodic. As a consequence, a necessary condition for the feasibility of those constraints is ¯ ÃN !¯ ¯ ¯ Y ¯ ∆ ¯ Bp ¯ < 1 θ(∆) := max ¯λq (10) q=1,··· ,n ¯ ¯ p=1
where λq (·) denotes a generic eigenvalue of (·) and {B1 , · · · , BN } are matrices corresponding to any permutation among those of the set {A1 , · · · , AN }.
In this setting, an upper bound for the minimum dwell time ∆∗ can be computed by taking the minimum value of ∆ satisfying the constraints of Theorem 1. Hence, it can be calculated with no big difficulty from the optimal solution of the
Preprints of the 5th IFAC Symposium on Robust Control Design
ROCOND'06, Toulouse, France, July 5-7, 2006
optimization problem 1 {∆ : (4) − (5)}
which, for ∆ ≥ 1 fixed, reduces to a convex programming feasibility problem with linear matrix inequalities constraints that can be handled by any LMI solver available in the literature to date, see (Boyd et al., 1994) for an important study on systems and LMIs. Finally, we generalize the result of Theorem 1 to define a guaranteed cost to go from any initial condition to the origin. The next result is related to the time switching strategy (3) where it is further assumed that ∆ ≥ lq+1 − lq ≥ ∆ holds for all q ∈ N for some ∆ ≥ ∆ ≥ 1. From Theorem 1 asymptotical stability is preserved whenever ∆ ≥ ∆∗ . Theorem 2. Let Q ≥ 0 ∈ Rn×n and ∆ ≥ ∆ ≥ 1 be given. Define the set of positive semidefinite matrices Ri :=
∆−1 X
100
50
0
0
, i = 1, · · · , N
(12)
Assume that there exists a collection of positive definite matrices {P1 , · · · , PN } of compatible dimensions such that A′i Pi Ai − Pi + Q < 0 , ∀ i = 1, · · · , N ∆ A′∆ i Pj Ai −Pi +Ri
(13)
< 0 , ∀ i 6= j = 1, · · · , N (14)
The time switching control (3) with ∆ ≥ lq+1 − lq ≥ ∆ makes the equilibrium solution x = 0 of (1) globally asymptotically stable and x(k)′ Qx(k) < x′0 Pσ(0) x0
(15)
k=0
For illustration purpose let us consider the following example characterized by N = 2, matrices A1 = eB1 T and A2 = eB2 T where · ¸ · ¸ 0 1 0 1 B1 = , B2 = (16) −10 −1 −0.1 −0.5 and T = 0.5 which corresponds to get samples from two continuous-time systems at each T seconds. First, from (11), we have calculated the upper bound of the minimum dwell time as being ∆∗ ≤ 6. To give an idea of its conservativeness we have calculated the value ∆per = 6 corresponding to the necessary condition for stability (10), arising from linear periodic systems. Both being equal indicates, for this simple example, that the minimum dwell time is ∆∗ = 6.
5
10
15
20
This problem should be stated with inf instead of min. All feasible sets of problems expressed in terms of LMIs must be considered closed from the interior within a precision defined by the user.
30
Fig. 1. The Lyapunov function Figure 1 has been constructed by simulation of system (1) with the time switching control (3), lq+1 −lq = 6, initial condition x0 = [1 1]′ , σ(0) = 2 and Q = I. The family od Lyapunov functions has been determined from the optimal solution of the convex programming problem max {x′0 Pi x0 : (13)−(14)} (17)
P1 >0,··· ,PN >0 i=1,··· ,N
which puts in evidence that, from the result of Theorem 2 a guaranteed cost can be determined for the worst case as far as the initial value of σ(0) is concerned. For ∆ = ∆ = 6 we have obtained the associated minimum guaranteed cost equal to δ ∗ = 152.17, valid for any logical initial state. As commented before, the Lyapunov function v(x(k)) = x(k)′ Pσ(k) x(k) goes to zero as k goes to infinity however, it is not uniformly decreasing with respect to time. In Figure 1, due to the stability conditions of Theorem 2, the points marked with ”o”, defines a globally convergent sequence v(x(lq )), for all q ∈ N. Solving again problem (17) but for ∆ = +∞ and ∆ = 6 the minimum cost increases to δ ∗ = 229.01 as a consequence of allowing a more flexible switching rule (3) with lq+1 − lq ≥ 6. This assures the global convergence of the sequence marked in dashed line.
3. STATE SWITCHING CONTROL In this section we consider again the system (1) where the switching rule satisfies (2). The main difference from the previous section is that, presently, it is assumed that the state vector x(k) is available for feedback for all k ∈ N and our goal is to determine the function u(·) : Rn → {1, · · · , N }, such that σ(k) = u(x(k))
1
25
k
min
k A′k i QAi
k=0
∞ X
150
(11)
v(x(k))
min
∆≥1,P1 >0,··· ,PN >0
(18)
makes the equilibrium point x = 0 of (1) globally asymptotically stable. To this end, let us define the simplex
Preprints of the 5th IFAC Symposium on Robust Control Design
Λ :=
(
λ∈R
N
N X
:
ROCOND'06, Toulouse, France, July 5-7, 2006
)
λi = 1 , λi ≥ 0
(19)
i=1
which together with the set of positive definite matrices {P1 , · · · , PN } enables us to introduce the following non-quadratic Lyapunov function v(x) :=
min x′ Pi x
i=1,··· ,N
(20)
Before proceed, let us recall the class of Metzler matrices denoted by M and constituted by all matrices Π ∈ RN ×N with elements πij , such PN that πij ≥ 0 and i=1 πij = 1. It is clear that any Π ∈ M is such that Π presents an unitary eigenvalue since c′ Π = c′ where c′ = [1 · · · 1]. In addition, it is well known that the eigenvector associated to the unitary eigenvalue of Π is nonnegative yielding the conclusion that there always exists λ∞ ∈ Λ such that Πλ∞ = λ∞ . Theorem 3. Assume that there exist Π ∈ M and a set of positive definite matrices {P1 , · · · , PN } satisfying the Lyapunov-Metzler inequalities N X A′i πji Pj Ai − Pi < 0 , i = 1, · · · , N (21) j=1
The state switching control (18) with u(x(k)) = arg
min x(k)′ Pi x(k)
i=1,··· ,N
(22)
makes the equilibrium solution x = 0 of (1) globally asymptotically stable. Proof: It follows from the Lyapunov function (20). Assume, in accordance to (22), that at an arbitrary k ∈ N, the state switching control is given by σ(k) = u(x(k)) = i for some i ∈ {1, · · · , N }. Hence, v(x(k)) = x(k)′ Pi x(k) and from the system dynamic equation (1) we have N X v(x(k + 1)) = min x(k)′ A′i λj Pj Ai x(k) λ∈Λ
j=1
≤ x(k)′ A′i
N X j=1
πji Pj Ai x(k) (23)
where the inequality holds from the fact that each column of Π belongs to Λ. Finally, using the Lyapunov-Metzler inequalities (21) one gets v(x(k + 1)) < x(k)′ Pi x(t) = v(x(k))
(24)
which proves the proposed theorem since the Lyapunov function v(x(k)) defined in (20) is radially unbounded. 2 With Π ∈ M, a necessary condition for the Lyapunov-Metzler inequalities to be feasible with 1/2 respect to {P1 , · · · , PN } is matrices πii Ai for
all i = 1, · · · , N be asymptotically stable. Since 0 ≤ πii ≤ 1 this condition does not imply on the asymptotical stability of Ai . However, an interesting case occurs when all matrices {A1 , · · · , AN } are asymptotically stable for which the choice Π = I is possible and the state switching strategy proposed preserves stability. Furthermore, if the set {A1 , · · · , AN } is quadratically stable then the Lyapunov-Metzler inequalities admit a solution P1 = · · · = PN = P . In this classical but particular case, at any k ∈ N, the control law u(x(k)) = i ∈ {1, · · · , N } is arbitrary and asymptotical stability is once again guaranteed. Hence, Theorem 3, contains as a particular case the quadratic stability condition. In the literature, the Lyapunov-Metzler inequalities with Π ∈ M fixed, have been introduced in order to study the Mean-Square (MS) stability of Markov Jump Linear Systems (MJLS). In that context, the Metzler matrix Π = Π0 ∈ M is given and Π′0 represents the transition matrix of a Markov chain σ(k) governing the dynamical system (1). In this respect, each component of the vector λ(k) ∈ Λ is the probability of the Markov chain to be on the i − th logical state and obeys the linear equation λ(k + 1) = Π0 λ(k) with λ(0) = λ0 ∈ Λ, where the eigenvector λ∞ ∈ Λ associated to the unitary eigenvalue of Π0 represents the stationary probability vector. It has been shown (see e.g. (Costa et al., 1993)) that the system is MS-stable if and only if there exists a set of positive definite matrices {P1 , · · · , PN } satisfying the Lyapunov-Metzler inequalities (21) for Π = Π0 . Numerically speaking, this particular case of the Lyapunov-Metzler inequalities is simpler to solve, since (21) reduces to a set of linear matrix inequalities. Lemma 1. Suppose that there exist λ ∈ Λ and P > 0 such that N X
λi (A′i P Ai − P ) < 0
(25)
i=1
then the Lyapunov-Metzler inequalities (21) admit a solution. Invoking the fact that P > 0 and λ ∈ Λ, the condition (25) whenever satisfied, implies that PN Aλ = stable, since i=1 λi Ai is asymptotically PN using (25) we have A′λ P Aλ ≤ i=1 λi A′i P Ai < P . Unfortunately, the converse does not hold in general, that is, the existence of a stable convex combination does not suffice for (25) to be true. However, we conjecture that the existence of a stable combination suffices to guarantee the existence of a solution to the Lyapunov-Metzler inequalities. Although this point will not be further addressed our claim is based on the proof of
Preprints of the 5th IFAC Symposium on Robust Control Design
ROCOND'06, Toulouse, France, July 5-7, 2006
Lemma 1 where only a Metzler matrix with a very particular structure has been used to construct a feasible solution. Furthermore, notice that if the Lyapunov-Metzler inequalities (21) admit a solution with Π = [λ · · · λ] ∈ M for some λ ∈ Λ PN λi Ai is asymptotically stable then Aλ = Pi=1 N since Pλ = i=1 λi Pi is positive definite and multiplying (21) by λi ≥ 0 and summing up we get PN Pλ > i=1 λi A′i Pλ Ai ≥ A′λ Pλ Aλ . The existence of this particular solution to the inequalities (21) meets exactly the stability condition provided in (Zhai, 2001). To see this, in our present context, let us assume that there exist λ = λ∞ ∈ Λ and P > 0 such that (25) holds. Hence, the switching rule (18) with u(x(k)) = arg
min x(k)′ A′i P Ai x(k)
i=1,··· ,N
(26)
makes the equilibrium point x = 0 of the switched system (1) globally asymptotically stable. Indeed, considering the Lyapunov function v(x(k)) = x(k)′ P x(k) we have v(x(k + 1)) = x(k)′ A′σ(k) P Aσ(k) x(k) ÃN ! X = min x(k)′ λi A′i P Ai x(k) λ∈Λ
≤ x(k)′
i=1
Ã
N X
λ∞i A′i P Ai
!
x(k)
∞ X
x(k)′ Qx(k) <
k=0
min x′0 Pi x0
i=1,··· ,N
(29)
The numerical determination, if any, of a solution of the Lyapunov-Metzler inequalities with respect to the variables (Π, {P1 , · · · , PN }) is not a simple task and certainly deserves additional attention. The main source of difficulty stems from its nonconvex nature due to the products of variables and so LMI solvers do not apply. The main idea is to get a simpler, although certainly more conservative stability condition that can be expressed by means of LMIs being thus solvable by the machinery available in the literature to date. Theorem 4. Let Q ≥ 0 be given. Assume that there exist a set of positive definite matrices {P1 , · · · , PN } and a scalar 0 ≤ γ < 1 satisfying the modified Lyapunov-Metzler inequalities for all j 6= i = 1, · · · , N A′i (γPi + (1 − γ)Pj )Ai − Pi + Q < 0
(30)
The state switching control (18) with u(x(k)) given by (22) makes the equilibrium solution x = 0 of (1) globally asymptotically stable and ∞ X
x(k)′ Qx(k) <
N X
x′0 Pi x0
(31)
i=1
k=0
i=1
< v(x(k))
(27)
It is important to keep in mind that if the set of matrices {A1 , · · · , AN } does not admit Aλ asymptotically stable for some λ ∈ Λ then the above stabilizing switching rule can not be determined. Even if it is known that there exists λ ∈ Λ such that Aλ is asymptotically stable, this information is of little use since, as commented before, the condition (25) of Lemma 1 may still be infeasible. Furthermore, the numerical determination, if any, of λ ∈ Λ and P > 0 satisfying (25) is not a simple task due to its nonlinear nature. The LyapunovMetzler inequalities introduced in Theorem 3 suffer the same difficulty but fortunately a simple numerical procedure based on line search can be settled to determine its solution. Lemma 2. Let Q ≥ 0 be given. Assume that there exist a set of positive definite matrices {P1 , · · · , PN } and Π ∈ M satisfying the Lyapunov-Metzler inequalities N X πji Pj Ai − Pi + Q < 0 (28) A′i j=1
for i = 1, · · · , N . The state switching control (18) with u(x(k)) given by (22) makes the equilibrium solution x = 0 of (1) globally asymptotically stable and
Proof: The proof follows from the choice of Π ∈ M such that πii = γ and the remaining elements satisfying −1
(1 − γ)
N X
πji = 1
(32)
j6=i=1
for all i = 1, · · · , N . Taking into account that πji ≥ 0 for all j 6= i = 1, · · · , N , multiplying (30) by πji , summing up for all j 6= i = 1, · · · N and finally multiplying the result by (1 − γ)−1 > 0 we get the Lyapunov-Metzler inequalities (28). From Lemma 2, the upper bound (29) holds which trivially implies that (31) is verified. The proposed theorem is thus proved. 2 The basic theoretical features of Theorem 3 and Lemma 2 are still present in Theorem 4. The most important is that the asymptotic stability of the set of matrices {A1 , · · · , AN } still is not required. In addition, notice that the guaranteed cost (31) is clearly worse than the one provide d by Lemma 2 but the former being convex makes possible to solve the problem (N ) X min x′0 Pi x0 : (30) (33) 0<γ<1,P1 >0,··· ,PN >0
i=1
by LMI solvers and line search.
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REFERENCES 1.5
1
x(k)
0.5
0
−0.5
−1
−1.5
0
10
20
30
40
50
60
70
80
90
100
k
Fig. 2. Time simulation The next example illustrates some aspects of the theoretical results obtained so far. Consider the system (1) with N = 2 and matrices A1 = eB1 T , A2 = eB2 T where · ¸ · ¸ 0 1 0 1 B1 = , B2 = (34) 2 −9 −2 8 and T = 0.1. As it can be verified matrices A1 and A2 are both unstable but for λ = [0.69 0.31]′ the matrix Aλ obtained by convex combination is stable. For these matrices it has also been verified numerically that the condition of Lemma 1 does not hold. Even though, in this particular case, the modified version of the Lyapunov-Metzler inequalities (30) have a solution which supports our claim on this point discussed before. Considering Q = I and the initial condition x0 = [1 1]′ , problem (33) has been solved by line search fixing γ and minimizing its objective function, denoted by δ(γ), with respect to the remaining variables. We have determined its minimum value δ ∗ = 4.81 × 103 , corresponding to γ ∗ = 6 × 10−3 . Figure 2 shows the trajectories of the state variable x(k) ∈ R2 versus time for the system controlled by the state switching rule σ(k) = u(x(k)) given by (22) with the positive definite matrices P1 and P2 obtained from the optimal solution of problem (33). As it can be seen, the proposed control strategy is effective to stabilize the system under consideration.
4. CONCLUSION In this paper we have introduced stability conditions for switched discrete-time linear systems. They have been used for control synthesis of state independent (open loop) and state dependent (closed loop) switching rules. In both cases, the determination of a guaranteed cost associated to each control strategy has been addressed. Special attention has been devoted towards the numerical solvability of the design problems by means of methods based on linear matrix inequalities.
S. P. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan (1994). Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994. M. S. Branicky (1998). Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automat. Contr., vol. 43, pp. 475–482. O. L. V. Costa and M. D. Fragoso (1993). Stability results for discrete-time linear systems with Markovian jumping parameters, Journal of Mathematical Analysis and Applications, vol. 179, pp. 154–178. R. A. DeCarlo, M. S. Branicky, S. Pettersson and B. Lennartson (2000). Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE, vol. 88, pp. 1069–1082. J. Daafouz and J. Bernussou (2001). Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties, Systems & Control Letters, vol. 43, pp. 355–359. J. C. Geromel and P. Colaneri (2005). Stability and stabilization of continuous time switched systems, submitted for publication. J. P. Hespanha (2004). Uniform stability of switched linear systems : extensions of LaSalle’s principle, IEEE Trans. Automat. Contr., vol. 49, pp. 470–482. J. Hockerman-Frommer, S. R. Kulkarni and P. J. Ramadge (1998). Controller switching based on output predictions errors, IEEE Trans. Automat. Contr., vol. 43, pp. 596–607. M. Johansson and A. Rantzer (1998). Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Trans. Automat. Contr., vol. 43, pp. 555–559. D. Liberzon (2003). Switching in Systems and Control, Boston, Birkhauser, 2003. D. Liberzon and A. S. Morse (1999). Basic problems in stability and design of switched systems, IEEE Control Systems Magazine, vol. 19, pp. 59–70. X. Xu and P. J. Antsaklis (2004). Optimal control of switched systems based on parameterization of the switching instants, IEEE Trans. Automat. Contr., vol. 49, pp. 2–16. G. M. Xie and L. Wang (2003). Reachability realization and stabilizability of switched linear discrete-time systems, Journal of Math. Analysis and Applications, vol. 280, pp. 209–220. H. Ye, A. N. Michel and L. Hou (1998). Stability theory for hybrid dynamical systems, IEEE Trans. Automat. Contr., vol. 43, pp. 461–474. G. Zhai (2001). Quadratic stability of discretetime switched systems via state and output feedback, Proceedings of the 40th IEEE CDC, Orlando, pp. 2165–2166.
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STABILIZATION OF DISCRETE-TIME SWITCHED SYSTEMS Jos´ e C. Geromel# and Patrizio Colaneri⋆
# DSCE / School of Electrical and Computer Engineering UNICAMP, CP 6101, 13083 - 970 Campinas, SP, Brazil, [email protected] ⋆
Politecnico di Milano Dipartimento di Elettronica e Informazione Piazza Leonardo da Vinci 32, 20133 Milano, Italy [email protected]
Abstract: This paper addresses two strategies for stabilization of discrete time linear switched systems. The first one, is of open loop nature and is based on the determination of an upper bound of the minimum dwell time by means of a family of quadratic Lyapunov functions. The relevant point on dwell time calculation is that the proposed stability condition does not require the Lyapunov function be uniformly decreasing at every switching time. The second one, is of closed loop nature and is designed from the solution of what we call Lyapunov-Metzler c inequalities from which the stability condition is expressed. Copyright ° 2005 IFAC. Keywords: Switched systems, Discrete-time systems, Linear matrix inequality
1. INTRODUCTION
This paper aims at providing procedures for stabilizing control synthesis for a discrete time switched linear system of the following general form x(k + 1) = Aσ(k) x(k) , x(0) = x0
(1)
n
defined for all k ∈ N where x(k) ∈ R is the state, σ(k) is the switching rule, x0 is the initial condition and N is the set of natural numbers. Considering a set of matrices Ai ∈ Rn×n , i = 1, · · · , N be given, the switching rule σ(k), for each k ∈ N, is such that Aσ(k) ∈ {A1 , · · · , AN }
(2)
Clearly, (2) imposes that Aσ(k) is constrained to jump among the N vertices of the matrix polytope {A1 , · · · , AN }.
Stability analysis of continuous time switched linear systems have been addressed in (Branicky, 1998), (Hespanha, 2004), (Hockerman et al., 1998), (Johansson et al., 1998), (Ye et al., 1998) and (Liberzon, 2003), among others. For discretetime switched systems, see (Daafouz et al., 2001), (Xie et al., 2003) and (Zhai, 2001). References (DeCarlo et al., 2000) and (Liberzon et al., 1999) provide excellent overviews on switched and hybrid systems. In this paper, the determination of an upper bound of the minimum dwell time preserving global stability is accomplished from the determination of a family of quadratic Lyapunov functions. Each of them is associated to the corresponding matrix of the set {A1 , · · · , AN } such that at every switching time lq the sequence v(x(lq )), for q = 0, · · · , ∞, converges uniformly to zero. In some instances, the determination of
Preprints of the 5th IFAC Symposium on Robust Control Design
ROCOND'06, Toulouse, France, July 5-7, 2006
the minimum dwell time, based on a quadratic guaranteed cost, is related to the results of (Xu et al., 2004) assuming further that the switching rule is not a priori given but can be taken arbitrarily, among the feasible ones. For switched systems with σ(·) being state dependent, the stability condition is expressed in terms of a set of inequalities that we call LyapunovMetzler inequalities because the variables involved are a set of symmetric and positive matrices {P1 , · · · , PN } and a Metzler matrix Π. The point to be noticed is that our asymptotical stability condition contains as special cases the quadratic stability condition and the results of (Daafouz et al., 2001) and (Zhai, 2001). The price to be payed, however, is the non-convex nature of the the Lyapunov-Metzler inequalities being thus difficult to solve numerically. For this reason, a more conservative but easier to solve asymptotical stability condition is proposed. This paper, is the discretetime counterpart of (Geromel et al., 2005).
which enables us to conclude that there exist scalars α ∈ (0, 1) and β > 0 such that kx(k)k22 ≤ βα(k−lq ) v(x(lq )) , ∀k ∈ [lq , lq+1 ) (7) On the other hand, using the inequalities (5) we have v(x(lq+1 )) = x(lq+1 )′ Pj x(lq+1 ) ′∆q
= x(lq )′ Ai
∆
Pj Ai q x(lq )
′(∆q −∆)
< x(lq )′ Ai
(∆q −∆)
Pi Ai
x(lq )
′
< x(lq ) Pi x(lq ) < v(x(lq ))
(8)
where the second inequality holds from the fact that for every τ = ∆q − ∆ ∈ N it is true that τ A′τ i Pi Ai ≤ Pi . The consequence is that there exists µ ∈ (0 , 1) such that v(x(lq )) ≤ µq v(x0 ) , ∀q ∈ N
(9)
which together with (7) implies that the equilibrium solution x = 0 of (1) is globally asymptotically stable. 2
2. TIME SWITCHING CONTROL Assuming that the matrices {A1 , · · · , AN } are asymptotically stable, the problem under consideration can be stated as follows : Determine a minimum dwell time ∆∗ ∈ N such that the equilibrium point x = 0 of the system (1) is globally asymptotically stable with the time switching control σ(k) = i ∈ {1, · · · , N } , k ∈ [lq , lq+1 )
(3)
where lq and lq+1 are successive switching times satisfying lq+1 − lq ≥ ∆∗ ≥ 1 for all q ∈ N. It is interesting to observe that the index i ∈ {1, · · · , N } selected at each instant of time k ∈ N is arbitrary. Theorem 1. Assume that, for some ∆ ≥ 1, there exists a collection of positive definite matrices {P1 , · · · , PN } of compatible dimensions such that A′i Pi Ai − Pi < 0 , ∀ i = 1, · · · , N ∆ A′∆ i Pj Ai − Pi < 0 , ∀ i 6= j = 1, · · · , N
(4) (5)
The time switching control (3) with lq+1 − lq ≥ ∆ makes the equilibrium solution x = 0 of (1) globally asymptotically stable. Proof: Consider, in accordance to (3), that σ(k) = i ∈ {1, · · · , N } for all k ∈ [lq , lq+1 ) where lq+1 = lq + ∆q with ∆q ≥ ∆ ≥ 1 and that at k = lq+1 the time switching control jumps to σ(k) = j ∈ {1, · · · , N }, otherwise the result trivially follows. From (4), it is seen that, for all k ∈ [lq , lq+1 ), the Lyapunov function v(x(k)) = x(k)′ Pσ(k) x(k), along an arbitrary trajectory of (1) satisfies v(x(k + 1)) < v(x(k))
(6)
This result deserves some comments. First, since it is assumed that all matrices A1 , · · · , AN are asymptotically stable then the constraints (4) are always feasible and the constraints (5) are satisfied whenever ∆ ∈ N is taken large enough. Second, assume that matrices A1 , · · · , AN are quadratically stable, which is the same to say that they share a positive definite matrix P such that A′i P Ai − P < 0 for all i = 1, · · · , N . In this case, the inequality (5) is satisfied for P1 = · · · = PN = P for all ∆ ≥ 1 meaning that the switching policy (3) may jump from i to j at each instant of time k ∈ N, preserving once again asymptotical stability. Hence, Theorem 1 contains, as a particular case, the quadratic stability condition and the stability results of (Daafouz et al., 2001) which are less conservative than quadratic stability. Finally, with ∆ ≥ 1, it is always possible to define a time switching control strategy (3) such that Aσ(k) is periodic. As a consequence, a necessary condition for the feasibility of those constraints is ¯ ÃN !¯ ¯ ¯ Y ¯ ∆ ¯ Bp ¯ < 1 θ(∆) := max ¯λq (10) q=1,··· ,n ¯ ¯ p=1
where λq (·) denotes a generic eigenvalue of (·) and {B1 , · · · , BN } are matrices corresponding to any permutation among those of the set {A1 , · · · , AN }.
In this setting, an upper bound for the minimum dwell time ∆∗ can be computed by taking the minimum value of ∆ satisfying the constraints of Theorem 1. Hence, it can be calculated with no big difficulty from the optimal solution of the
Preprints of the 5th IFAC Symposium on Robust Control Design
ROCOND'06, Toulouse, France, July 5-7, 2006
optimization problem 1 {∆ : (4) − (5)}
which, for ∆ ≥ 1 fixed, reduces to a convex programming feasibility problem with linear matrix inequalities constraints that can be handled by any LMI solver available in the literature to date, see (Boyd et al., 1994) for an important study on systems and LMIs. Finally, we generalize the result of Theorem 1 to define a guaranteed cost to go from any initial condition to the origin. The next result is related to the time switching strategy (3) where it is further assumed that ∆ ≥ lq+1 − lq ≥ ∆ holds for all q ∈ N for some ∆ ≥ ∆ ≥ 1. From Theorem 1 asymptotical stability is preserved whenever ∆ ≥ ∆∗ . Theorem 2. Let Q ≥ 0 ∈ Rn×n and ∆ ≥ ∆ ≥ 1 be given. Define the set of positive semidefinite matrices Ri :=
∆−1 X
100
50
0
0
, i = 1, · · · , N
(12)
Assume that there exists a collection of positive definite matrices {P1 , · · · , PN } of compatible dimensions such that A′i Pi Ai − Pi + Q < 0 , ∀ i = 1, · · · , N ∆ A′∆ i Pj Ai −Pi +Ri
(13)
< 0 , ∀ i 6= j = 1, · · · , N (14)
The time switching control (3) with ∆ ≥ lq+1 − lq ≥ ∆ makes the equilibrium solution x = 0 of (1) globally asymptotically stable and x(k)′ Qx(k) < x′0 Pσ(0) x0
(15)
k=0
For illustration purpose let us consider the following example characterized by N = 2, matrices A1 = eB1 T and A2 = eB2 T where · ¸ · ¸ 0 1 0 1 B1 = , B2 = (16) −10 −1 −0.1 −0.5 and T = 0.5 which corresponds to get samples from two continuous-time systems at each T seconds. First, from (11), we have calculated the upper bound of the minimum dwell time as being ∆∗ ≤ 6. To give an idea of its conservativeness we have calculated the value ∆per = 6 corresponding to the necessary condition for stability (10), arising from linear periodic systems. Both being equal indicates, for this simple example, that the minimum dwell time is ∆∗ = 6.
5
10
15
20
This problem should be stated with inf instead of min. All feasible sets of problems expressed in terms of LMIs must be considered closed from the interior within a precision defined by the user.
30
Fig. 1. The Lyapunov function Figure 1 has been constructed by simulation of system (1) with the time switching control (3), lq+1 −lq = 6, initial condition x0 = [1 1]′ , σ(0) = 2 and Q = I. The family od Lyapunov functions has been determined from the optimal solution of the convex programming problem max {x′0 Pi x0 : (13)−(14)} (17)
P1 >0,··· ,PN >0 i=1,··· ,N
which puts in evidence that, from the result of Theorem 2 a guaranteed cost can be determined for the worst case as far as the initial value of σ(0) is concerned. For ∆ = ∆ = 6 we have obtained the associated minimum guaranteed cost equal to δ ∗ = 152.17, valid for any logical initial state. As commented before, the Lyapunov function v(x(k)) = x(k)′ Pσ(k) x(k) goes to zero as k goes to infinity however, it is not uniformly decreasing with respect to time. In Figure 1, due to the stability conditions of Theorem 2, the points marked with ”o”, defines a globally convergent sequence v(x(lq )), for all q ∈ N. Solving again problem (17) but for ∆ = +∞ and ∆ = 6 the minimum cost increases to δ ∗ = 229.01 as a consequence of allowing a more flexible switching rule (3) with lq+1 − lq ≥ 6. This assures the global convergence of the sequence marked in dashed line.
3. STATE SWITCHING CONTROL In this section we consider again the system (1) where the switching rule satisfies (2). The main difference from the previous section is that, presently, it is assumed that the state vector x(k) is available for feedback for all k ∈ N and our goal is to determine the function u(·) : Rn → {1, · · · , N }, such that σ(k) = u(x(k))
1
25
k
min
k A′k i QAi
k=0
∞ X
150
(11)
v(x(k))
min
∆≥1,P1 >0,··· ,PN >0
(18)
makes the equilibrium point x = 0 of (1) globally asymptotically stable. To this end, let us define the simplex
Preprints of the 5th IFAC Symposium on Robust Control Design
Λ :=
(
λ∈R
N
N X
:
ROCOND'06, Toulouse, France, July 5-7, 2006
)
λi = 1 , λi ≥ 0
(19)
i=1
which together with the set of positive definite matrices {P1 , · · · , PN } enables us to introduce the following non-quadratic Lyapunov function v(x) :=
min x′ Pi x
i=1,··· ,N
(20)
Before proceed, let us recall the class of Metzler matrices denoted by M and constituted by all matrices Π ∈ RN ×N with elements πij , such PN that πij ≥ 0 and i=1 πij = 1. It is clear that any Π ∈ M is such that Π presents an unitary eigenvalue since c′ Π = c′ where c′ = [1 · · · 1]. In addition, it is well known that the eigenvector associated to the unitary eigenvalue of Π is nonnegative yielding the conclusion that there always exists λ∞ ∈ Λ such that Πλ∞ = λ∞ . Theorem 3. Assume that there exist Π ∈ M and a set of positive definite matrices {P1 , · · · , PN } satisfying the Lyapunov-Metzler inequalities N X A′i πji Pj Ai − Pi < 0 , i = 1, · · · , N (21) j=1
The state switching control (18) with u(x(k)) = arg
min x(k)′ Pi x(k)
i=1,··· ,N
(22)
makes the equilibrium solution x = 0 of (1) globally asymptotically stable. Proof: It follows from the Lyapunov function (20). Assume, in accordance to (22), that at an arbitrary k ∈ N, the state switching control is given by σ(k) = u(x(k)) = i for some i ∈ {1, · · · , N }. Hence, v(x(k)) = x(k)′ Pi x(k) and from the system dynamic equation (1) we have N X v(x(k + 1)) = min x(k)′ A′i λj Pj Ai x(k) λ∈Λ
j=1
≤ x(k)′ A′i
N X j=1
πji Pj Ai x(k) (23)
where the inequality holds from the fact that each column of Π belongs to Λ. Finally, using the Lyapunov-Metzler inequalities (21) one gets v(x(k + 1)) < x(k)′ Pi x(t) = v(x(k))
(24)
which proves the proposed theorem since the Lyapunov function v(x(k)) defined in (20) is radially unbounded. 2 With Π ∈ M, a necessary condition for the Lyapunov-Metzler inequalities to be feasible with 1/2 respect to {P1 , · · · , PN } is matrices πii Ai for
all i = 1, · · · , N be asymptotically stable. Since 0 ≤ πii ≤ 1 this condition does not imply on the asymptotical stability of Ai . However, an interesting case occurs when all matrices {A1 , · · · , AN } are asymptotically stable for which the choice Π = I is possible and the state switching strategy proposed preserves stability. Furthermore, if the set {A1 , · · · , AN } is quadratically stable then the Lyapunov-Metzler inequalities admit a solution P1 = · · · = PN = P . In this classical but particular case, at any k ∈ N, the control law u(x(k)) = i ∈ {1, · · · , N } is arbitrary and asymptotical stability is once again guaranteed. Hence, Theorem 3, contains as a particular case the quadratic stability condition. In the literature, the Lyapunov-Metzler inequalities with Π ∈ M fixed, have been introduced in order to study the Mean-Square (MS) stability of Markov Jump Linear Systems (MJLS). In that context, the Metzler matrix Π = Π0 ∈ M is given and Π′0 represents the transition matrix of a Markov chain σ(k) governing the dynamical system (1). In this respect, each component of the vector λ(k) ∈ Λ is the probability of the Markov chain to be on the i − th logical state and obeys the linear equation λ(k + 1) = Π0 λ(k) with λ(0) = λ0 ∈ Λ, where the eigenvector λ∞ ∈ Λ associated to the unitary eigenvalue of Π0 represents the stationary probability vector. It has been shown (see e.g. (Costa et al., 1993)) that the system is MS-stable if and only if there exists a set of positive definite matrices {P1 , · · · , PN } satisfying the Lyapunov-Metzler inequalities (21) for Π = Π0 . Numerically speaking, this particular case of the Lyapunov-Metzler inequalities is simpler to solve, since (21) reduces to a set of linear matrix inequalities. Lemma 1. Suppose that there exist λ ∈ Λ and P > 0 such that N X
λi (A′i P Ai − P ) < 0
(25)
i=1
then the Lyapunov-Metzler inequalities (21) admit a solution. Invoking the fact that P > 0 and λ ∈ Λ, the condition (25) whenever satisfied, implies that PN Aλ = stable, since i=1 λi Ai is asymptotically PN using (25) we have A′λ P Aλ ≤ i=1 λi A′i P Ai < P . Unfortunately, the converse does not hold in general, that is, the existence of a stable convex combination does not suffice for (25) to be true. However, we conjecture that the existence of a stable combination suffices to guarantee the existence of a solution to the Lyapunov-Metzler inequalities. Although this point will not be further addressed our claim is based on the proof of
Preprints of the 5th IFAC Symposium on Robust Control Design
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Lemma 1 where only a Metzler matrix with a very particular structure has been used to construct a feasible solution. Furthermore, notice that if the Lyapunov-Metzler inequalities (21) admit a solution with Π = [λ · · · λ] ∈ M for some λ ∈ Λ PN λi Ai is asymptotically stable then Aλ = Pi=1 N since Pλ = i=1 λi Pi is positive definite and multiplying (21) by λi ≥ 0 and summing up we get PN Pλ > i=1 λi A′i Pλ Ai ≥ A′λ Pλ Aλ . The existence of this particular solution to the inequalities (21) meets exactly the stability condition provided in (Zhai, 2001). To see this, in our present context, let us assume that there exist λ = λ∞ ∈ Λ and P > 0 such that (25) holds. Hence, the switching rule (18) with u(x(k)) = arg
min x(k)′ A′i P Ai x(k)
i=1,··· ,N
(26)
makes the equilibrium point x = 0 of the switched system (1) globally asymptotically stable. Indeed, considering the Lyapunov function v(x(k)) = x(k)′ P x(k) we have v(x(k + 1)) = x(k)′ A′σ(k) P Aσ(k) x(k) ÃN ! X = min x(k)′ λi A′i P Ai x(k) λ∈Λ
≤ x(k)′
i=1
Ã
N X
λ∞i A′i P Ai
!
x(k)
∞ X
x(k)′ Qx(k) <
k=0
min x′0 Pi x0
i=1,··· ,N
(29)
The numerical determination, if any, of a solution of the Lyapunov-Metzler inequalities with respect to the variables (Π, {P1 , · · · , PN }) is not a simple task and certainly deserves additional attention. The main source of difficulty stems from its nonconvex nature due to the products of variables and so LMI solvers do not apply. The main idea is to get a simpler, although certainly more conservative stability condition that can be expressed by means of LMIs being thus solvable by the machinery available in the literature to date. Theorem 4. Let Q ≥ 0 be given. Assume that there exist a set of positive definite matrices {P1 , · · · , PN } and a scalar 0 ≤ γ < 1 satisfying the modified Lyapunov-Metzler inequalities for all j 6= i = 1, · · · , N A′i (γPi + (1 − γ)Pj )Ai − Pi + Q < 0
(30)
The state switching control (18) with u(x(k)) given by (22) makes the equilibrium solution x = 0 of (1) globally asymptotically stable and ∞ X
x(k)′ Qx(k) <
N X
x′0 Pi x0
(31)
i=1
k=0
i=1
< v(x(k))
(27)
It is important to keep in mind that if the set of matrices {A1 , · · · , AN } does not admit Aλ asymptotically stable for some λ ∈ Λ then the above stabilizing switching rule can not be determined. Even if it is known that there exists λ ∈ Λ such that Aλ is asymptotically stable, this information is of little use since, as commented before, the condition (25) of Lemma 1 may still be infeasible. Furthermore, the numerical determination, if any, of λ ∈ Λ and P > 0 satisfying (25) is not a simple task due to its nonlinear nature. The LyapunovMetzler inequalities introduced in Theorem 3 suffer the same difficulty but fortunately a simple numerical procedure based on line search can be settled to determine its solution. Lemma 2. Let Q ≥ 0 be given. Assume that there exist a set of positive definite matrices {P1 , · · · , PN } and Π ∈ M satisfying the Lyapunov-Metzler inequalities N X πji Pj Ai − Pi + Q < 0 (28) A′i j=1
for i = 1, · · · , N . The state switching control (18) with u(x(k)) given by (22) makes the equilibrium solution x = 0 of (1) globally asymptotically stable and
Proof: The proof follows from the choice of Π ∈ M such that πii = γ and the remaining elements satisfying −1
(1 − γ)
N X
πji = 1
(32)
j6=i=1
for all i = 1, · · · , N . Taking into account that πji ≥ 0 for all j 6= i = 1, · · · , N , multiplying (30) by πji , summing up for all j 6= i = 1, · · · N and finally multiplying the result by (1 − γ)−1 > 0 we get the Lyapunov-Metzler inequalities (28). From Lemma 2, the upper bound (29) holds which trivially implies that (31) is verified. The proposed theorem is thus proved. 2 The basic theoretical features of Theorem 3 and Lemma 2 are still present in Theorem 4. The most important is that the asymptotic stability of the set of matrices {A1 , · · · , AN } still is not required. In addition, notice that the guaranteed cost (31) is clearly worse than the one provide d by Lemma 2 but the former being convex makes possible to solve the problem (N ) X min x′0 Pi x0 : (30) (33) 0<γ<1,P1 >0,··· ,PN >0
i=1
by LMI solvers and line search.
Preprints of the 5th IFAC Symposium on Robust Control Design
ROCOND'06, Toulouse, France, July 5-7, 2006
REFERENCES 1.5
1
x(k)
0.5
0
−0.5
−1
−1.5
0
10
20
30
40
50
60
70
80
90
100
k
Fig. 2. Time simulation The next example illustrates some aspects of the theoretical results obtained so far. Consider the system (1) with N = 2 and matrices A1 = eB1 T , A2 = eB2 T where · ¸ · ¸ 0 1 0 1 B1 = , B2 = (34) 2 −9 −2 8 and T = 0.1. As it can be verified matrices A1 and A2 are both unstable but for λ = [0.69 0.31]′ the matrix Aλ obtained by convex combination is stable. For these matrices it has also been verified numerically that the condition of Lemma 1 does not hold. Even though, in this particular case, the modified version of the Lyapunov-Metzler inequalities (30) have a solution which supports our claim on this point discussed before. Considering Q = I and the initial condition x0 = [1 1]′ , problem (33) has been solved by line search fixing γ and minimizing its objective function, denoted by δ(γ), with respect to the remaining variables. We have determined its minimum value δ ∗ = 4.81 × 103 , corresponding to γ ∗ = 6 × 10−3 . Figure 2 shows the trajectories of the state variable x(k) ∈ R2 versus time for the system controlled by the state switching rule σ(k) = u(x(k)) given by (22) with the positive definite matrices P1 and P2 obtained from the optimal solution of problem (33). As it can be seen, the proposed control strategy is effective to stabilize the system under consideration.
4. CONCLUSION In this paper we have introduced stability conditions for switched discrete-time linear systems. They have been used for control synthesis of state independent (open loop) and state dependent (closed loop) switching rules. In both cases, the determination of a guaranteed cost associated to each control strategy has been addressed. Special attention has been devoted towards the numerical solvability of the design problems by means of methods based on linear matrix inequalities.
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