On Stability Analysis of Switched Circulant Systems

On Stability Analysis of Switched Circulant Systems N. Rakoto-Ravalontsalama ∗ ∗

Ecole des Mines de Nantes and IRCCyN Institute, 44307 Nantes cedex 03, France, (e-mail: [email protected]).

Abstract: In this paper we study the properties of a class of switched systems. Circulant systems are systems defined by circulant matrices that exhibit special structural properties. We consider switched circulant systems where each subsystem is a defined by a circulant matrix. Necessary and sufficient conditions for stability of such switched systems are given. Necessary conditions for switching stabilization are also given. Both results are based on matrix pencil approaches. Keywords: Stability analysis; switched systems; circulant systems. 1. INTRODUCTION There has been a particular attention for circulant systems during the past years. Indeed, these systems exhibit particular mathematical structures that can be found in some engineering applications Brockett (1974), Hovd and Skogestad (1994), Motee et al. (2008). On the other hand there has been a growing interest in the stability analysis of switched and hybrid systems. See e.g., DeCarlo et al. (2000); Liberzon (2003); Lin and Antsaklis (2009) and the references therein. In this paper we consider a particular class of switched systems, that is a switched linear system where each subsystem is a circulant system. We study then the stability analysis of switched circulant systems. To our knowledge, the only work that addresses the stability of switched circulant systems is the work of Li et al. (2004). The work in this paper is different from the one in Li et al. (2004) since we derive some necessary and sufficient conditions for the stability of switched circulant systems, particularly by using the matrix pencil calculation. Moreover, in this paper, we also give necessary and sufficient conditions for the stabilizability of switched circulant systems when one of the subsystem is not stable. The remaining of this paper is structured as follows. Some definitions and preliminaries are given in the next section. Then the main results are given and discussed in Section III. Finally, some conclusions are given.

2. DEFINITIONS AND PRELIMINARIES Definition 1. Davis (1994) A n×n matrix A is said to be circulant if it is specified by one vector v = [a1 a2 . . . an ]T , which is rotated one element to the right for the next row vectors, as follows:



a1  an a n−1 A=  .  .. a2

a2 a1 an .. . a3

a3 a2 a1 .. . a4

 . . . an . . . an−1  . . . an−2   ..  .. . .  . . . a1

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In this case, matrix A is denoted A = circ{a1 , a2 , . . . , an }. Definition 2. The linear system x(t) ˙ = Ax(t), x(t0 ) = x0

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is said to be a circulant system if A = circ{a1 , a2 , . . . , an }, where x(t) = [x1 x2 . . . xn ]T ∈ Rn is the vector of state variables. Definition 3. The linear system x(t) ˙ = Aσ x(t) = Ai x(t), t ∈ R+ , i ∈ I.

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is called switched system if the switching signal σ takes value on the index set I and {Ai : i ∈ I} is a parametrized compact family of x × n matrices. Definition 4. The switched system x(t) ˙ = Ai x(t), t ∈ R+ , i ∈ I.

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is called switched circulant system if (3) holds and for all i ∈ I, Ai is a circulant matrix. Proposition 1. Davis (1994) Let A and B two n × n circulant matrices. Then the following properties are true: (1) The product AB is also a circulant matrix, and AB = BA. (2) The sum A + B is also a circulant matrix; (3) The transpose AT is also a circulant matrix; (4) The inverse inv(A) is also a circulant matrix. Proposition 2. Davis (1994) The eigenvectors of a n×n circulant matrix A are given by vj = (1, ωj , ωj2 , . . . , ωjn−1 )T ,

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where ωj = exp(2iπj/n); j = 1 . . . n. For a given n, n ∈ N, the constants ω0 , ω1 , . . . ωn−1 satisfy the equation ωjn = 1, thus they constitute the nth roots of unity. The corresponding eigenvalues are given by λj = a1 +a2 ωj +a3 ωj2 +· · ·+an ωjn−1 , j = 0 . . . n−1. (6) Theorem 1. A necessary and sufficient condition for asymptotic stability of the circulant system (2) is that the circulant elements a1 , a2 , . . . an satisfy Re(a1 +a2 ωj +a3 ωj2 +· · ·+an ωjn−1 ) < 0, j = 1, . . . , n (7) Proof. The proof is straightforward, since the circulant matrix is diagonalizable i.e., there exists an invertible matrix T such that T −1 AT = diag(λ1 , λ2 , . . . , λn ), where λi are the eigenvalues of the matrix A. Corollary 1. Following Theorem 1, the necessary and sufficient conditions for the stability of the circulant system (2) are given in the following equations: If n = 2, ( If n = 3,

If n = 4,

 a1 + a2 < 0 a1 − a2 < 0

a1 + a2 + a3 < 0 1 1 a1 − a2 − a3 < 0 2 2

  a1 + a2 + a3 + a4 < 0 a − a3 < 0  1 a1 − a2 + a3 − a4 < 0

Proof. The results are obtained by direct calculation of ωj = exp(2iπj/n); j = 1 . . . n. in (7).

Example 1. The system defined by x(t) ˙ = A1 x(t), x(t0 ) = x0 where A1 is given by " # −1 2 −3 A1 = 3 −1 2 2 3 −1 is stable since A1 = circ{a1 , a2 , a3 } = circ{−1, 2, −3} and the elements of this circulant matrix verify the conditions of stability for the case n = 3: (a1 + a2 + a3 < 0) and (a1 − 12 a2 − 21 a3 < 0). 3. STABILITY ANALYSIS 3.1 Common Quadratic Lyapunov Functions

Finding a common quadratic Lyapunov function is not straightforward. Some results dealing with the existence of a CQLF use the notion of matrix pencil. The matrix pencil is defined as follows. Given two matrices A1 and A2 , the matrix pencil γα (A1 , A2 ) is defined as the one-parameter family of matrices γα (A1 , A2 ) = αA1 +(1−α)A2 , α ∈ [0, 1]. The matrix pencil γα (A1 , A2 ) is said to be Hurwitz if its eigenvalues are in the open left half plane for all 0 ≤ α ≤ 1. The following theorem gives some equivalent conditions for the stability of a switched system composed of two Hurwitz matrices of size (2 × 2). Theorem 2. Shorten and Narendra (1999, 2002); Shorten et al. (2003) Let A1 , A2 be two Hurwitz matrices in R2×2 . The following conditions are equivalent: (1) There exists a CQLF for the switch system (3) with A1 , A2 as the 2 subsystems; (2) The matrix pencils γα (A1 , A2 ) and γα (A1 , A−1 2 ) are equivalent; (3) The matrices A1 A2 and A1 A−1 do not have any 2 negative real eigenvalues. Now consider the special case of Theorem 2 where the matrices A1 , A2 are two Hurwitz matrices in R2×2 which are both circulant. Then we derive the following new theorem. Theorem 3. Let A1 , A2 be two Hurwitz circulant matrices in R2×2 . Then there always exists a CQLF for the switched systems (3) with A1 , A2 as the 2 subsystems; Proof. We will prove that Statement 3) of Theorem 2 is always true in the case of Hurwitz circulant matrices. Indeed, since A1 and A2 are both circulant matrices, then inv(A2 ) is also circulant. Without loss of generality, assume that matrices A1and A  2 are simultaneously   diagonalizable such λ1 0 λ2 0 that A1 = and A2 = . Then the inverse of 0 λ1 0 λ2   1 0  λ2  A2 is given by inv(A2 ) = A−1 2 = 1 . 0 λ2   λ λ 0 Then consider now the product A1 A2 = 1 2 and the 0 λ1 λ2   λ1 0  λ2  product A1 A−1 =  2 λ1 . 0 λ2

One tool to study the stability of a switched system is to find a common quadratic Lyapunov function (CQLF). The conditions for the existence of a CQLF is given as linear matrix inequalities (LMIs) Boyd et al. (1994).

Since A1 and A2 are both Hurwitz, then their eigenvalues are strictly negative, respectively, λ1 < 0 and λ2 < 0. Then λ1 λ2 > 0 and λλ12 > 0. Hence the matrices A1 A2 and A1 A−1 do not have any negative real eigenvalues. This 2 completes the proof.

There exists a positive definite symmetric matrix P , P ∈ Rn×n , such that P Ai + ATi P < 0, ∀ i ∈ I. (8) P is called the common quadratic Lyapunov function (CQLF).

The following remark can now be made. This theorem says that if a switched system is made of 2 Hurwitz circulant matrices of (2 × 2) size, then this switched system is

always stable, whatever the switching law between the subsystems. Example 2. The switched system defined by x(t) ˙ = Ai x(t), i ∈ I where A1 and A2 are given by     −2 0 −5 0 A1 = , A2 = 0 −2 0 −5 is always stable and it does not depend on the switching law between the 2 subsystems.

Until now, we have assumed that all the subsystems are Hurwitz. We consider now a particular class of switch circulant systems, where not all the subsystem matrices Ai , i ∈ I are Hurwitz. In that case we study the stabilizability of such class of switched systems. 3.2 Quadratic Switching Stabilization

Consider now a system composed of a pair of n-th order of LTI systems, with n > 2. An extension of Theorem 2 is proposed for a pair of n-th order of LTI systems. A necessary condition for the existence of a CQLF was derived in Shorten et al. (2004, 2003).

Quadratic stabilization is one way of bringing possibly unstable systems to become stable. A switched system is called quadratically stabilizable when there exist switching signals which stabilize the switched system along a quadratic Lyapunov function, V (x) = xT P x. The following theorem has been proposed by Feron (1996) and Wicks et al. (1998).

Theorem 4. Shorten et al. (2004, 2003). Let A1 , A2 be two Hurwitz matrices in Rn×n . A necessary condition for the existence of a CQLF is that the matrix products A1 [αA1 + (1 − α)A2 ] and A1 [αA1 + (1 − α)A2 ]−1 do not have any negative real eigenvalues for all 0 ≤ α ≤ 1.

Theorem 6. Feron (1996); Wicks et al. (1998) A switched system that contains two linear time-invariant (LTI) subsystems, x(t) ˙ = Ai x(t), i = 1, 2, is quadratically stabilizable if and only if the matrix pencil γ(A1 , A2 ) contains a stable matrix. 2

2

Consider now the special case of switch circulant systems, of (2 × 2) size. The following new theorem is derived.

Consider now the specific case where each subsystem is defined by a circulant matrix, respectively. The following theorem can be derived. We provide the theorem for the case n = 3 but it can be easily extended to higher order cases n > 3. Theorem 5. Let A1 , A2 be two Hurwitz matrices in R3×3 . Then there always exists a CQLF for the switched system (3). Proof. We will prove that the matrix products A1 [αA1 + (1 − α)A2 ] and A1 [αA1 + (1 − α)A2 ]−1 do not have any negative real eigenvalues for all 0 ≤ α ≤ 1, and by Theorem 4, there always exists a CQLF. Indeed, since A1 and A2 are both 3 × 3 Hurwitz circulant matrices, then without loss of generality, assume that matrices A1 and A2 are simultaneously diagonalizable such that " # " # λ1 0 0 λ2 0 0 A1 = 0 λ1 0 , A2 = 0 λ2 0 0 0 λ1 0 0 λ2 It is easy to see that the eigenvalues of the product of 1 matrices are λ1 [αλ1 + (1 − α)λ2 ] and λ1 [ αλ1 +(1−α)λ ], 2 respectively. Since A1 and A2 are both Hurwitz, then λ1 < 0 and λ2 < 0. Hence λ1 [αλ1 + (1 − α)λ2 ] > 0 and 1 ] > 0. Thus these matrix products do not λ1 [ αλ1 +(1−α)λ 2 have any negative real eigenvalues for all 0 ≤ α ≤ 1. This completes the proof.

Theorem 7. Consider a switched circulant system (2) that contains two circulant subsystems, x(t) ˙ = Ai x(t), i = 1, 2. If matrices A1 and A2 are both unstable, then the switched circulant system is not quadratically stabilizable. Proof. We will prove that there exists all the eigenvalues of the matrix pencil αA1 + (1 − α)A2 have positive real part. Indeed matrices A1 and A2 are both unstable. Then there exists at least an eigenvalue λi of A1 such that Re(λi > 0). In the same way, there exists also an eigenvalue γj of A2 such that Re(γj > 0). Without loss of generality, we can choose i = j by an appropriate change of coordinates. It follows that Re(αλi + (1 − α)γi ) > 0 thus, by Theorem 6, the matrix pencil αA1 +(1−α)A2 does not contain a stable matrix. Thus the switched system is not quadratically stabilizable. This completes the proof. This theorem deserves a remark. This result says that in order to be stabilizable, a switched circulant subsystem must have at least one Hurwitz subsystem. A generalization to more than two LTI subsystems was suggested in Pettersson and Lennartson (2001) by using a ”min-projection strategy”, i.e., σ(t) = arg min x(t)T P Ai x(t). i∈I

Example 3. The switched system defined by x(t) ˙ = Ai x(t), i ∈ I where A1 and A2 are given by " # " # −3 0 0 −7 0 0 A1 = 0 −3 0 , A2 = 0 −7 0 0 0 −3 0 0 −7 is always stable and it does not depend on the switching law between the 2 subsystems.

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Theorem 8. Pettersson and Lennartson (2001) If there P exist constants αi ∈ [0, 1], and i∈I αi = 1 such that X αi Ai Aα = i∈I

is stable, then the min-projection strategy (9) quadratically stabilizes the switched system.

However, the existence of a stable convex combination matrix Aα is only sufficient for switched LTI systems with more than two modes.

Brockett, R.W., and Willems, J.L. (1974). Discretized partial differential equations: Examples of control systems defined on modules, Automatica, vol. 10, no 4, pp. 507– 515. The following necessary and sufficient condition for quadratic Davis, P.J. (1994). ”Circulant Matrices”, New York: John Wiley and Sons. stabilizability is due to Skafidas et al. (1999). DeCarlo, R.A., Branicky, M.S., Pettersson, S., and Lennartson, B. (2000). P.J. Antsaklis Ed., ”Perspectives Theorem 9. Skafidas et al. (1999) The switched system and results on the stability and stabilizability of hybrid is quadratically stabilizable if and only if there exists a posT systems”, Proc. IEEE: Special Issue Hybrid Systems, itive definite real symmetric matrix P = P > 0 such that T vol. 88, pp. 1069-1082. the set of matrices {Ai P + P Ai } is strictly complete, i.e., Feron, E. (1996). ”Quadratic stabilizability of switched T for any x ∈ R/{0}, there exists i ∈ I such that x (Ai P + systems via state output feedback”, MIT, Cambridge, T P Ai )x < 0. In addition a stabilizing switching signal can MA, Tech Rep. CICS-P-468. T T be selected as σ(t) = mini {x (t)(Ai P + P Ai )x(t)}. 2 Hespanha, J.P. (2001) Extending LaSalle’s invariance principle to switched linear systems, Proc. of the 40rd IEEE Conference on Decision and Control, pp. 2968The following remarks can be made about Theorem 8 and 2973. Theorem 9. These above two theorems are quite powerful Hovd, M., and Skogestad, S. (1994). Control of symmetrisince they generalize the case to more than two modes. cally interconnected plants, Automatica, vol. 30, no 6, However, no specific properties can be derived for the pp. 957–973. particular cases of switched circulant matrices. In the Khalil, H.K. (1996) ”Nonlinear Systems”, New York: next subsection, we will consider switched state feedback Prentice-Hall. control. Li, J.H., Zhao, S.Z., Zhao, J., and Li, Y.P. (2004) Stability analysis for circulant systems and switched circulant 3.3 Switched State Feedback systems, Proc. of the 43rd IEEE Conference on Decision and Control, pp. 2805-2809. We consider now the case when all the matrices Ai , i ∈ I only differ by a state feedback matrix gain, i.e., all the Liberzon, D. (2003) ”Switching in Systems and Control”, Boston: Birkhauser. Ai , i ∈ I are of the form Lin, H. and Antsaklis, P.J. (2009) Stability and stabilizAi = A + BFi , (10) ability of switched linear systems: A Survey of recent where A and B are given matrices and {Fi , i ∈ I} is a results, Trans. of Automatic Control, vol. 54, N.2, pp. compact set of state feedback matrix gains. This type of 308-322. structure arises, e.g., when a fixed time-invariant process is Motee, N., Jadbabaie, A., and Bamieh, B. (2008) On decontrolled using a switched state feedback gain Hespanha centralized optimal control and information structures, (2001). Proc. of American Control Conf., pp. 4985-4990. Pettersson, S., and Lennartson, B. (2001) Stabilization of Theorem 10. Suppose that Ai = A + BFi , i ∈ I, and hybrid systems using a min-projection strategy, Proc. there exists a compact family {Pi : i ∈ I} of symmetric of American Control Conf., pp. 223-228. positive definite n × n matrices such that Shorten, R., Mason, O., Cairbre, F.O., and Curran, P. ATi Pi + Pi Ai ≤ 0 i ∈ I (11) (2004) A unifying framework for the SISO circle criT terion and other quadratic stability criteria, Int. J. and that z Pi1 ≤ z T Pi2 , at every point z in the state Control, vol. 77, no 1, pp. 1–8. space, at which the switching signal σ can switch from i1 + Shorten, R., and Narendra, K. (1999). Necessary and to i2 . Then x(t) ˙ = Aσ x(t) = Ai x(t), t ∈ R , i ∈ I, is sufficient conditions for the existence of a common uniformly stable. quadratic Lyapunov function for two stable second order 2. linear time invariant systems, Proc. of American Control Conf., pp. 1410-1414. 4. CONCLUSIONS Shorten, R., and Narendra, K. (2002) Necessary and sufficient conditions for the existence of a CQLF for a In this paper we have studied some properties of a class finite number of stable LTI systems, Int. J. Adaptive of switched systems: the circulant switched systems. A Control Signal Processing, vol. 16, no 10, pp. 709–728. switched circulant system is composed of subsystems Shorten, R., Narendra, K., and Mason, O. (2003) A result which are a specified by a circulant matrix. Necessary and on common quadratic Lyapunov function, IEEE Trans. sufficient conditions for stability of such switched systems of Automatic Control, vol. 48, no 1, pp. 110–113. are given. 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