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H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function
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H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function Panshuo Li, Pengxu Li, Yun Liu, Hong Bao, Renquan Lu∗ School of Automation and Guangdong Province Key Laboratory of Intelligent Decision and Cooperative Control, Guangdong University of Technology, Guangzhou 510006, China Received 26 September 2018; received in revised form 30 May 2019; accepted 13 June 2019 Available online xxx
Abstract In this paper, the H∞ control problem of periodic piecewise systems with polynomial time-varying subsystems is addressed. Based on a periodic Lyapunov function with a continuous time-dependent Lyapunov matrix polynomial, the H∞ performance is studied. The result can be easily reduced to the conditions for periodic piecewise systems with constant subsystems or linear time-varying systems based on a common Lyapunov function or a linear time-varying Lyapunov matrix. Moreover, an H∞ controller with time-varying polynomial controller gain is proposed as well, which could be directly solved with the linear matrix inequalities. A numerical example is presented to demonstrate the effectiveness of the proposed method. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
1. Introduction The periodic time-varying system is a class of time-varying systems with periodic properties. It may come from the system inherent dynamic characteristics such as rotor-blade systems [1], spacecraft attitude adjustment [2] or from the artificial operation such as multirate sampling [3,4]. Control problems for continuous-time periodic systems are more challenging compared with discrete-time periodic systems since the discrete-time periodic systems could ∗
Corresponding author. E-mail address: [email protected] (R. Lu).
https://doi.org/10.1016/j.jfranklin.2019.06.008 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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be transformed into linear time-invariant systems with lifting techniques [5]. Efforts of solving the control problems for continuous-time periodic systems can be found in [6–8], where the L∞ and L2 semi-global stabilization problems with bounded controls are studied in [6], the state feedback stabilizing controller is designed in [7] based on the solutions of parametric periodic Lyapunov differential equations and pointwise frequency-domain approach is used to develop the stability analysis in [8]. The periodic piecewise system is a special class of continuous-time periodic system. It has several subsystems in one period, the switching sequence and dwell time of each subsystem are fixed. Such kind of systems could be regarded as the approximation system of continuoustime periodic systems [9,10]. Studies on periodic piecewise systems can help the research of continuous-time periodic systems [8]. Moreover, the periodic piecewise system also has many prototypes in engineering applications, such as vibration systems [11], power converters [12]. Because of its value both in studying continuous-time periodic systems and engineering applications, lots of efforts have been put into this area. The stability and L2 -gain performance are studied in [13] and [14] for the periodic piecewise system with non-Hurwitz subsystems based on continuous time-varying Lyapunov function and discontinuous time-varying Lyapunov function, respectively. Apart from analysis problems, the controller design investigation could be found in [11,15–17]. The saturated H∞ controller with piecewise constant controller gains is designed in [11] for periodic piecewise mechanical systems to attenuate the system vibration, a corresponding algorithm is proposed as well. A finite-time H∞ controller is synthesized for periodic piecewise system in [15]. The H∞ control problem of periodic piecewise system with time-delay is studied in [16] based on a piecewise Lyapunov function with a discontinuous time-varying matrix function with two global terms. Guaranteed cost control of periodic piecewise time-delay systems is investigated in [17] based on a novel Lyapunov function with relaxes constraints on positive definiteness of the involved symmetric matrices. In the above results, on one hand, one may observe that the considered periodic piecewise systems are composed with constant subsystems. However, it could be seen that the constant subsystems may loss dynamic characteristics of its approximated continuous-time periodic systems. From this perspective, the periodic piecewise system with time-varying subsystems might be more accurate. The results on periodic piecewise time-varying systems are not so profound as the results on periodic piecewise constant systems. The analysis and synthesis problems of periodic piecewise time-varying systems are addressed in [18], where the subsystems are linear time-varying. On the other hand, continuous and multiple Lyapunov functions are employed with a time-varying Lyapunov matrix. The Lyapunov matrix is given as a linear interpolation of time in each subsystem [11,14] or each segment [15]. To improve the results, a Lyapunov matrix given in matrix polynomial form is employed in [19] for periodic piecewise linear systems, the modified stability and L2 -gain results are reported. However, the obtained results could not cover the condition based on a linear time-varying Lyapunov matrix or a common Lyapunov function. More techniques for system with time-varying properties can be found in [20,21], where passivity-based asynchronous sliding mode control [20], fault detection filtering [21] are used. From the above discussion, one may notice that neither the study on periodic piecewise polynomial systems nor the results based on Lyapunov polynomial matrix is abundant, not to mention results of periodic piecewise polynomial systems based on a polynomial Lyapunov function. Hence, motivated by the above-mentioned works and the broad applications of periodic piecewise systems, a periodic piecewise system with time-varying polynomial subsystems Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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is considered in this paper. A continuous Lyapunov function with a time-varying Lyapunov matrix polynomial is adopted. The contributions of this paper lie in the H∞ performance analysis and the corresponding controller design for periodic piecewise polynomial time-varying systems. Conditions of the negativeness and positiveness of a matrix polynomial are introduced to obtain the results, which can be reduced to the results for the periodic piecewise systems with constant or linear time-varying systems based on common Lyapunov function or linear time-varying Lyapunov function. Since the continuous-time periodic system is a special case of the periodic piecewise polynomial time-varying systems, the results obtained in this work may further help the study in control of continuous-time periodic systems. This paper is organized as follows. The system is described and preliminaries are given in Section 2. The main results are obtained in Section 3. Simulation results are given in Section 4. Conclusion is provided in Section 5. 2. System description and preliminaries Notation: Rr denotes the r-dimensional Euclidean space, N + denotes the set of the positive integers. · stands for the Euclidean vector norm, the superscript refers to matrix transposition, sym(P) stands for P + P and P > 0 means that P is a real symmetric and positive definite matrix, max(·) or min(·) stand for the maximum or minimum value of (·). To facilitate the description, a non-negative integer is used to represent the nominal number of fundamental periods, that is, = 0, 1, . . .. Consider a T-periodic piecewise time-varying system given as, for t ∈ [T + ti−1 , T + ti ], i ∈ N , N = {1, 2, . . . , S}, x˙(t ) = Ai (t )x(t ) + Bi (t )u(t ) + Bwi (t )w(t ), z(t ) = Ci x(t )
(1)
where ti is the switching instant in the first period from the ith subsystem to the (i + 1)th subsystem and t0 = 0, S is the number of subsystems in one period, and one has Ai (t ) = Ai (t + T ), Bi (t ) = Bi (t + T ), Bwi (t ) = Bwi (t + T ). x(t ) ∈ Rr , u(t ) ∈ Rd , z(t ) ∈ Rk are the state vector, control input, measurement output, respectively. Define the dwell time of the ith subsystem is Ti , that is, the time between two successive discrete events, then one has Ti = ti − ti−1 [22]. Each subsystem is formulated in time-varying polynomial form, where Ai (t ), Bi (t ), Bwi (t ) are given by n t − T − ti−1 (t − T − ti−1 )n (t − T − ti−1 ) j Ai (t ) = Ai,0 + Ai,1 + · · · + Ai,n = Ai, j , Ti Ti n Ti j j=0 n n (t − T − ti−1 ) j (t − T − ti−1 ) j Bi (t ) = Bi, j , Bwi (t ) = Bwi, j , (2) Ti j Ti j j=0 j=0 where Ai, j , Bi, j , Bwi, j are constant matrices. It can be seen that A(t ), B(t ), Bwi (t ) are continuous time-varying in each subsystems, and discontinuous at switching instants. However, A(t ), B(t ), Bwi (t ) could becontinuous at switching instantsunder the condition that Ai,0 = nj=0 Ai−1, j , Bi,0 = nj=0 Bi−1, j , Bwi,0 = nj=0 Bwi−1, j , Si=1 nj=1 Ai, j = 0, Si=1 nj=1 Bi, j = 0, Si=1 nj=1 Bwi, j = 0, then system (1) will become a general continuous-time periodic system. It should be noticed that the output matrix Ci is settled Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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as a constant matrix here to facilitate the the system state observation, one could also choose a time-varying Ci (t). The definition of the exponential stability of a periodic piecewise system is given as follows. Definition 1 [14]. The periodic piecewise system (1) with u(t ) = 0, w(t ) = 0 is said to be λ∗ -exponentially stable if the solution of the system from x(0) satisfies x(t ) ≤ ∗ κe−λ t x(0), ∀t > 0 for some constants κ ≥ 1, λ∗ > 0. To facilitate the development, the Dini-derivative [23] is introduced for a continuous function P(t) as P (t + h) − P (t ) D + P (t ) = lim+ sup , t ∈ [T + ti−1 , T + ti ) (3) h→0 h and a general condition concerning the exponential stability of periodic piecewise system is given in Lemma 1. Lemma 1 [19]. Consider a periodic piecewise time-varying system (1) with u(t ) = 0, w(t ) = 0, let λ∗ > 0, λi , i ∈ N , be given constants. If there exist λi , i ∈ N , and real symmetric Tperiodic, continuous and Dini-differentiable periodic matrix function P(t) defined on t ∈ [0, ∞) such that, for i ∈ N , t ∈ [T + ti−1 , T + ti ), P (t ) = Pi (t ) > 0, satisfies Ai (t ) Pi (t ) + Pi (t )Ai (t ) + D + Pi (t ) + λi Pi (t ) < 0, 2λ∗ T −
S
(4)
λi Ti ≤ 0
(5)
i=1
then system (1) is λ∗ -exponentially stable. A lemma concerning the negative definiteness of a matrix polynomial is given below. Lemma 2 [18]. Consider a bounded matrix polynomial f (τ1 , τ2 , . . . , τn ) given as n f (τ1 , τ2 , . . . , τn ) = ϒ0 + τ1 ϒ1 + τ1 τ2 ϒ2 + · · · + τk ϒn ,
(6)
k=1
where n ∈ N + , ϒ j , j = 0, 1, . . . , n are real symmetric matrices, τk , k = 1, 2, . . . n are variables and τ k ∈ [0, 1]. If d
ϒk < 0, d = 0, 1, . . . , n,
k=0
then the matrix polynomial f (τ1 , τ2 , . . . , τn ) < 0. Similarly, one can obtain Lemma 3 concerning the positive definiteness of a matrix polynomial, the proof can be easily extended from the proof of Lemma 2. Lemma 3. Consider a bounded matrix polynomial f (τ1 , τ2 , . . . , τn ) given as in Eq. (6), where n ∈ N + , ϒ j , j = 0, 1, . . . , n are real symmetric matrix, τk , k = 1, 2, . . . n are variables and τ k ∈ [0, 1]. If d
ϒk > 0, d = 0, 1, . . . , n,
(7)
k=0
then the matrix polynomial f (τ1 , τ2 , . . . , τn ) > 0. Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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3. Main results In this Section, a general condition of H∞ performance for periodic piecewise time-varying system based on a periodic continuous Lyapunov function is obtained at first. Then, based on a time-varying Lyapunov matrix polynomial and the time-interval information, a sufficient condition of H∞ performance is proposed. The H∞ performance condition based on a linear time-varying Lyapunov matrix is given as well. Finally, under the state-feedback control strategy, a controller which guarantees the H∞ performance of periodic piecewise polynomial systems is designed. 3.1. H∞ performance analysis For periodic piecewise polynomial system (1), consider a Lyapunov function V (t ) = x (t )P (t )x(t ), where P(t) > 0 is continuous and bounded, and it is periodic with period T, that is, P (t ) = P (t + T ). For t ∈ [T + ti−1 , T + ti ), i ∈ N , rewrite P (t ) = Pi (t ), then V(t) can be given as V (t ) = Vi (t ) = x (t )Pi (t )x(t ).
(8)
Then, based on the above Lyapunov function, the general H∞ performance index for periodic piecewise polynomial system can be established as in Theorem 1. Theorem 1. Consider a periodic piecewise time-varying system (1) with u(t ) = 0, given γ > 0, λ∗ > 0. If there exist λi , i ∈ N and continuous, Dini differentiable matrix function P(t) defined on t ∈ [0, ∞) such that, for t ∈ [T + ti−1 , T + ti ), i ∈ N , P (t ) = Pi (t ) > 0, satisfies ⎡ ⎤ Ai (t )Pi (t ) + Pi (t )Ai (t ) + D + Pi (t ) + λi Pi (t ) Pi (t )Bwi (t ) Ci ⎣ ∗ −γ 2 I 0 ⎦ < 0, (9) ∗ ∗ −I 2λ∗ T −
S
λi Ti ≤ 0,
(10)
i=1
then system (1) is λ∗ -exponentially stable and satisfies ∞ ∞ z (τ )z(τ )dτ ≤ γ 2 w (τ )w(τ )dτ, 0
where
γ = γe
(11)
0
T max (2λ∗ −λmin ,0)
λmax , λmax = max (λi ), λmin = min (λi ). i i 2λ∗
Proof. See the Appendix. Remark 1. One may observe that the constraints on λi are independent with the stability of each subsystem. It is not necessary to allocate a positive λi for a Hurwitz subsystem, and a negative λi for a non-Hurwitz subsystem as in previous works. It relaxes the condition, and facilitates the theory development since determining the stability of a time-varying system is not an easy work. The proof details could be found in [19]. Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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It should be noticed that H∞ index measures the output energy for the worst case input energy, it is a well-known performance index to describe a system in engineering application. In the following, we will study the H∞ performance condition based on a continuous timevarying Lyapunov matrix polynomial. Consider a continuous Lyapunov matrix P(t) given in a time-varying matrix polynomial form as, for t ∈ [T + ti−1 , T + ti ), i ∈ N P (t ) = Pi (t ) = Pi,0 + Pi,0 =
n
t − T − ti−1 (t − T − ti−1 )n Pi,1 + · · · + Pi,n Ti Ti n
Pi−1, j , P1,0 =
j=0
n
PS, j ,
(12)
(13)
j=0
where Pi,j are constant matrices and n ≥ 0. It can be seen that P(t) is continuous both in subsystems and switching instants. One could also obtain P2,0 = P1,0 +
n (t − T ) j
T1j
j=1
P2, j = P1,0 +
P1, j , P3,0 = P2,0 +
j=1
2 n (t − T − t1 ) j
T2j
i=1 j=1
n (t − T − t1 ) j
T2j
P1, j ,
··· P1,0 +
S n (t − T − ti−1 ) j i=1 j=1
hence, Pi,0 = Pi,0 = P1,0 +
Ti j
n j=0
Pi−1, j , P1,0 =
i−1 n k
Pi, j = P1,0 , n j=0
PS, j , then Eq. (13) could also be written as
Pi, j , i = 2, 3, . . . , S, and
j=1
S n
Pi, j = 0.
i=1 j=1
Remark 2. It should be noticed that the polynomial degree of Pi (t) in different subsystems could be different, which can be denoted as ni . The degrees of the subsystem matrices and Lyapunov matrix are independent as well. In this work, to facilitate the development, the degree of all polynomial functions are chosen to be an uniform n. It is easy to extend the result to general cases with different degrees. Based on the above Lyapunov matrix polynomial, the H∞ performance condition could be established as in Theorem 2. Theorem 2. Consider a periodic piecewise time-varying system (1) with u(t ) = 0, given γ > 0, λ∗ > 0. If there exist λi and matrices Pi, j , i ∈ N , j = 0, 1, . . . , n, satisfying k
Pi, j > 0, k = 0, 1, . . . , n,
(14)
j=0
Pi,0 =
n
Pi−1, j , i = 2, 3, . . . , S,
(15)
j=0
S i=1
n
Pi, j = 0,
(16)
j=1
Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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i, j < 0, k = 0, 1, . . . , n,
7
(17)
j=0 S
2λ∗ T −
λi Ti ≤ 0,
(18)
i=1
where i,0
i, j
⎡
Ai,0 Pi,0 + Pi,0 Ai,0 + T1i Pi,1 + λi Pi,0 =⎣ ∗ ∗ ⎡ ⎤ φi, j ϕi, j 0 ⎣ 0 0 ⎦, = ϕi, j 0 0 0
φi, j =
j
(Ai,q Pi, j−q + Pi, j−q Ai,q ) +
q=0
φi,n =
n
Pi,0 Bwi,0 −γ 2 I ∗
⎤ Ci 0 ⎦, −I
j+1 Pi, j+1 + λi Pi, j , Ti
j = 1, . . . , n − 1,
(Ai,q Pi, j−q + Pi, j−q Ai,q ) + λi Pi, j ,
q=0
φi, j =
n
(Ai,q Pi, j−q + Pi, j−q Ai,q ),
j = n + 1, . . . , 2n,
q= j−n
ϕi, j =
j
Pi,q Bi, j−q ,
j = 1, . . . , 2n,
(19)
q=0
then system (1) is λ∗ -exponentially stable and satisfies Eq. (11). Proof. See the Appendix. Remark 3. It is worth mentioning that the sum of squares (SOS) [24] is an alternative method to tackle the matrix polynomial problem, it has been successfully applied in the stability analysis of periodic piecewise linear systems [19]. However, apart from the polynomial degree limitation, it is not an easy work to use the SOS method to develop the H∞ controller. On contrast, the method used in this work is more convenient to develop the H∞ controller with no constraint on the matrix polynomial degree. Moreover, the condition based on SOS method would be valid independent of time for periodic piecewise system [19], while the time-interval information is considered in this work which may result in less conservative result. If the degree of P(t) equals 1, then a corollary could be established as below. Corollary 1. Consider a periodic piecewise time-varying system (1) with u(t ) = 0, given γ > 0, λ∗ > 0. If there exist λi and matrices Pi,0 , Pi,1 , satisfying Pi,0 > 0, Pi,0 + Pi,1 > 0,
(20)
Pi,0 = Pi−1,0 + Pi−1,1 , i = 2, 3, . . . , S,
(21)
S
Pi,1 = 0,
(22)
i=1
Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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k
i, j < 0, k = 0, 1, . . . , n + 1,
(23)
j=0
2λ∗ T −
S
λi Ti ≤ 0,
(24)
i=1
where i,0
i, j
⎡
Ai,0 Pi,0 + Pi,0 Ai,0 + T1i Pi,1 + λi Pi,0 ⎣ = ∗ ∗ ⎡ ⎤ φi, j ϕi, j 0 0 0 ⎦, = ⎣ϕi, j 0 0 0
Pi,0 Bwi,0 −γ 2 I ∗
⎤ Ci 0 ⎦, −I
φi,1 = sym (Ai,0 Pi,1 + Ai,1 Pi,0 ) + λi Pi,1 ,
φi, j = sym Ai, j Pi,0 + Ai, j−1 Pi,1 , j = 2, . . . , n, φi,n+1 = Ai,n Pi,1 + Pi,1 Ai,n , ϕi, j = Pi,0 Bwi, j + Pi,1 Bwi, j−1 ,
j = 1, . . . , n,
(25)
then system (1) is λ∗ -exponentially stable and satisfies Eq. (11). The proof can be easily obtained from the proof of Theorem 2. It is worth mentioning that Corollary 1 is the condition based on the Lyapunov function with a linear time-varying Lyapunov matrix. The condition based on a common Lyapunov function could be obtained from the Theorem 2 as well, which is omitted here. 3.2. Time-varying H∞ controller synthesis Consider a state-feedback controller u(t ) = K (t )x(t ),
(26)
where K(t) is periodic with period T, that is, K (t ) = K (t + T ). Under this controller, the closed-loop system can be given as x˙(t ) = Aci (t )x(t ) + Bwi (t ), z(t ) = Ci x(t ),
(27)
where Aci (t ) = Ai (t ) + Bi (t )Ki (t ). Then an H∞ control scheme condition could be obtained as in Theorem 3. Theorem 3. Consider a periodic piecewise time-varying system (1), given γ > 0, λ∗ > 0. If there exist λi and matrices Wi, j , Qi, j , i ∈ N , j = 0, 1, . . . , n, such that k
Wi, j > 0, k = 0, 1, . . . , n
(28)
j=0
Wi,0 =
n
Wi−1, j , i = 2, 3, . . . , S
(29)
j=0
Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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Wi, j = 0,
9
(30)
i=1 j=1 k
i, j < 0, k = 0, 1, . . . , n,
(31)
j=0
2λ∗ T −
S
λi Ti ≤ 0,
(32)
i=1
where i,0
i, j
⎡ Wi,0 Ai,0 + Ai,0Wi,0 + Bi,0 Qi,0 + Qi, 0 Bi, 0 − T1i Wi,1 + λiWi,0 =⎣ ∗ ∗ ⎡ ⎤ i, j ϒi, j i, j 0 0 ⎦, = ⎣ ϒi, j
i, j 0 0
i, j =
j
Bwi,0 −γ 2 I ∗
⎤ Wi,0Ci 0 ⎦, −I
(Wi, j−q Ai,q + Ai,qWi, j−q + Qi, j−q Bi,q + Bi,q Qi, j−q )
q=0
j+1 Wi, j+1 + λiWi, j , j = 1, . . . , n − 1, Ti n = (Wi,n−q Ai,q + Ai,qWi,n−q + Qi,n−q Bi,q + Bi,q Qi,n−q ) + λiWi,n , −
i,n
q=0 n
i, j =
(Wi, j−q Ai,q + Ai,qWi, j−q + Qi, j−q Bi,q + Bi,q Qi, j−q ),
j = n + 1, . . . , 2n,
q= j−n
ϒi, j = Bwi, j , i, j = Wi, j Ci , j = 1, 2, . . . , n, ϒi, j = 0, i, j = 0, j = n + 1, . . . , 2n.
(33)
then the periodic piecewise polynomial system (1) with state-feedback controller u(t ) = Ki (t )x(t ) is λ∗ -exponentially stable and satisfies Eq. (11). And the controller gain can be given as Ki (t ) = Qi (t )Wi−1 (t ) with Qi (t ) =
n j=0
Wi (t ) =
n j=0
(34)
(t − T − ti−1 ) j Ti j
(t − T − ti−1 ) j Ti j
Qi, j ,
(35)
Wi, j .
(36)
Proof. See the Appendix. Moreover, an H∞ controller design based on a linear time-varying Lyapunov function could also be established with Corollary 2. Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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Corollary 2. Consider a periodic piecewise time-varying system (1), given γ > 0, λ∗ > 0. If there exist λi and matrices Wi, j , Qi, j , i ∈ N , j = 1, 2, . . . , n, such that Wi,0 > 0, Wi,0 + Wi,1 > 0, Wi,0 = Wi−1,0 + Wi−1,1 , S
i = 1, 2, . . . , S
(37)
i = 2, 3, . . . , S,
(38)
Wi,1 = 0,
(39)
i=1 k
i, j < 0,
i = 1, 2, . . . , S, k = 0, 1, . . . , n + 1,
(40)
j=0
2λ∗ T −
S
λi Ti ≤ 0,
(41)
i=1
where ⎡
i,0
i, j
Ai,0Wi,0 + Wi,0 Ai,0 + Bi,0 Qi,0 + Qi, 0 Bi, 0 − T1i Wi,1 + λiWi,0 =⎣ ∗ ∗ ⎡ ⎤ φi, j ϕi, j i, j 0 0 ⎦, = ⎣ ϕi, j i, j 0 0
Bwi,0 −γ 2 I ∗
⎤ Wi,0Ci 0 ⎦, −I
φi,1 = sym (Ai,0Wi,1 + Ai,1Wi,0 + Bi,0 Qi,1 + Bi,1 Qi,0 ) + λiWi,1 ,
φi, j = sym Ai, j Wi,0 + Ai, j−1Wi,1 + Bi, j Qi,0 + Bi, j−1 Qi,1 , j = 2, . . . , n, φi,n+1 = Ai,nWi,1 + Wi,1 Ai,n + Bi,n Qi,1 + Qi, 1 Bi,n , ϕi, j = Bwi, j , i, j = Wi, j Ci , j = 1, . . . , n,
ϕi,n+1 = 0, i,n+1 = 0,
(42)
then the periodic piecewise polynomial system (1) with state-feedback controller u(t ) = Ki (t )x(t ) is λ∗ -exponentially stable and satisfies Eq. (11). And the controller gain can be given as Ki (t ) = Qi (t )Wi−1 (t ) with t − T − ti−1 Qi,1 , Ti t − T − ti−1 Wi (t ) = Wi,0 + Wi,1 . Ti
Qi (t ) = Qi,0 +
The proof can be easily obtained from the proof of Theorem 3 with let the degrees of W(t), Q(t) equal 1. Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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4. Simulation In this section, a numerical example is used to demonstrate the effectiveness of the proposed controller. The simulation is carried within Matlab, where the LMI Toolbox is adopted to solve the condition. Consider a periodic piecewise polynomial system with T = 2, t1 = 0.4, t2 = 1, t3 = 2 and 1 −3 −3 −2.5 −2 −3 + 2.5(t − T ) + 6.25(t − T )2 , A1 (t ) = 2 −2.8 −3 −4.5 −2 3.5 5(t − T − 0.4) −1 −0.5 25(t − T − 0.4)2 −4 −2.4 −5 −3 + + , A2 (t ) = 0 −4 0 −2 −1 −4 3 9 1 3 −2 −3 −4 −2.6 + (t − T − 1) + (t − T − 1)2 , A3 (t ) = 2 −3.2 3 −2 2 −4 −5 −8 −16 + 2.5(t − ) + 6.25(t − ) , B1 (t ) = 14 15 15 5(t − T − 0.4) −3 25(t − T − 0.4) −11 −13.6 + + , B2 (t ) = 13 10 4 3 9 −13 −10 −8 + (t − T − 1) + (t − T − 1)2 , B3 (t ) = 8 12 2 −6 −8 −6 + 2.5(t − ) + 6.25(t − ) , Bw1 (t ) = 3 11 3 5(t − T − 0.4) −5 25(t − T − 0.4) −15 −3 + + , Bw2 (t ) = 8 15 5 3 9 −15 −4 −11 + (t − T − 1) + (t − T − 1)2 , Bw3 (t ) = 5 10 8 1 0 C1 = C2 = C3 = 0 1 Time histories of A(t ), B(t ), Bw (t ) over a period are shown in Figs. 1–3. Given λ1 = −0.5, λ2 = 1, λ3 = 1, and the disturbance is w(t ) = e−0.1t . An H∞ controller with degree 2 is designed for the above system according to Theorem 3 with λ∗ = (λ1 T1 + λ2 T2 + λ3 T3 )/(2T ) = 0.425. The obtained r is 1.3512, it is related to the H∞ index under the worst case of this systems. The obtained W(t), Q1 (t), Q2 (t), Q3 (t) are shown in Figs. 4 and 5. It can be seen that W(t) is continuous in the whole time span, and Qi (t) is continuous in each subsystem and discontinuous at switching instant. The corresponding controller gain is shown in Fig. 6, which is discontinuous at the switching instant, because of the discontinuous Qi (t). It should be noticed that a continuous time-varying controller could be obtained with Corollary 2. Fig. 7 compares the system outputs of the open-loop and closed-loop systems. It can be seen that the system is exponentially stable, and the outputs, which are the same with the system state, are obviously attenuated under the designed H∞ controller. It implies that the closed-loop system displays good disturbance attenuation performance. From the above numerical example, one can conclude that for a periodic piecewise polynomial time-varying system, one could design a periodic time-varying controller proposed in this work to attenuate the system output under the disturbance excitation. Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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8
11 (
) ) 21( ) 22 ( ) 12 (
6 4
magnitude
2 0 -2 -4 -6 -8 -10 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t Fig. 1. Time history of A(t) over a period.
50
11 ( 12 (
) )
40 30
magnitude
20 10 0 -10 -20 -30 -40 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t Fig. 2. Time history of B(t) over a period.
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13
30
11 ( 12 (
) )
20
magnitude
10
0
-10
-20
-30 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.8
2
t Fig. 3. Time history of Bw (t ) over a period.
14
11 (
) ) 22 ( ) 12 (
12 10
magnitude
8 6 4 2 0 -2 -4 -6 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
t Fig. 4. Time history of W(t) over a period. Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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1 1( 1 2(
) )
2 1( 2 2(
) )
3 1( 3 2(
) )
60
magnitude
40 20 0 -20 -40 -60 -80 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t Fig. 5. Time history of Q1 (t), Q2 (t), Q3 (t) over a period.
8
1 1( 1 2(
6
) )
2 1( 2 2(
) )
3 1( 3 2(
) )
4
magnitude
2 0 -2 -4 -6 -8 -10 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t Fig. 6. Time history of controller gain over a period.
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3
1(
) ) 1( ) 2( ) 2(
2
15
without control without control with control with control
magnitude
1
0
-1
-2
-3 0
5
10
15
20
25
30
35
40
45
50
t Fig. 7. System state comparison of the open-loop and closed-loop system.
5. Conclusion The H∞ control problem of periodic piecewise polynomial time-varying systems is addressed in this paper. A continuous Lyapunov function which relaxes the constraints on the exponential orders of subsystems is adopted with time-dependent Lyapunov matrix polynomial. The H∞ performance index and the corresponding H∞ controller design are studied. The H∞ performance condition for the periodic piecewise constant system or periodic piecewise linear time-varying systems with a common Lyapunov function or a continuous Lyapunov function with linear time-varying Lyapunov matrix could be easily obtained from the results in this work. Since the continuous-time periodic system in polynomial form is a special case of the periodic piecewise system with polynomial subsystems, the result in the work may help the controller design for general continuous-time periodic systems. The future work of this paper may extend to the periodic piecewise system with time-delay or with high-order uncertain nonlinearities, techniques in [25–27] may help the future studies. Acknowledgment The work was supported by National Natural Science Foundation of China under Grant (61703111), the Innovative Research Team Program of Guangdong Province Science Foundation (2018B0330312006). Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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Appendix A A1. Proof of Theorem 1 Proof. For t ∈ [T + ti−1 , T + ti ), i ∈ N , construct a Lyapunov function as in Eq. (8). With Eqs. (9) and (10), according to Lemma 1, one could conclude that system (1) is λ∗ -exponential stability. On one hand, define F (t ) = z (t )z(t ) − γ 2 w (t )w(t ), then one has D +Vi (t ) + λiVi (t ) + F (t ) = (x (t )Ai (t ) + w (t )Bwi (t ))Pi (t )x(t ) + x (t )P (t )(Ai (t )x(t ) + Bwi (t )w(t )) + x (t )D + Pi (t )x(t )
+ λi x Pi (t )x(t ) + z (t )z(t ) − γ w (t )w(t ) = x (t )(Ai (t )Pi (t ) + Pi (t )Ai (t ) + D + Pi (t ) + λi Pi (t ) + Ci (t )Ci (t ))x(t ) + w (t )(Bwi (t )Pi (t ))x(t ) 2 + x (t )(Pi (t )Bwi (t ))w(t ) + w (t )(−γ I )w(t ) x(t ) x(t ) = i (t ) w(t ) w(t )
where
A (t )Pi (t ) + Pi (t )Ai (t ) + D + Pi (t ) + λi Pi (t ) i (t ) = i Bwi (t )Pi (t )
Pi (t )Bwi (t ) C + i Ci −γ 2 I 0
0 .
(43)
On the other hand, applying Schur complement equivalence to Eq. (9), one could obtain that i (t) < 0, which indicates D +Vi (t ) + λiVi (t ) + F < 0 if x = 0 or w = 0. Then integrate it within t ∈ [T + ti−1 , T + ti ), with zero initial condition, one could obtain t1 S S i−1 V (t ) < − e−λ1 (t1 −τ )− i=2 λ j Ti −(−1) i=1 λi Ti − i=1 λi Ti −λi (t−(T +ti−1 ))F (τ )dτ 0 t2 S S i−1 e−λ2 (t2 −τ )− i=3 λi Ti −(−1) i=1 λi Ti − i=1 λi Ti −λi (t−(T +ti−1 ))F (τ )dτ − · · · − t1
− − −
T
e−λS (T −τ )−(−1)
tS−1 T +t1 T t T +ti−1
S i=1
λi Ti − i−1 i=1 λi Ti −λi (t−(T +ti−1 ))
i−1
e−λ1 (T +t1 −τ )−
k=2
λk Tk −λi (t−(T +ti−1 ))
F (τ )dτ − · · ·
F (τ )dτ − · · ·
e−λi (t−τ ) F (τ )dτ.
(44)
With V(t) > 0 and Eq. (44), one has S k=1 j=1
+
i−1 j=1
(k−1)T +t j
S
e−λ j ((k−1)T +t j −τ )−
l= j+1
λl Tl −(−k)
S l=1
λl Tl − i−1 l=1 λl Tl −λi (t−(T +ti−1 ))
(k−1)T +t j−1 T +t j T +t j−1
i−1
e−λ j (T +t j −τ )−
l= j+1
z (τ )z(τ )dτ
λl Tl −λi (t−(T +ti−1 ))
z (τ )z(τ )dτ
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+
T +ti−1
≤ γ2
+
t
e−λi (t−τ ) z (τ )z(τ )dτ
⎧ S ⎨ ⎩
k=1 j=1
i−1 j=1
17
(k−1)T +t j
S
e−λ j ((k−1)T +t j −τ )−
l= j+1
λl Tl −(−k)
S l=1
λl Tl − i−1 l=1 λl Tl −λi (t−(T +ti−1 ))
w (τ )w(τ )dτ
(k−1)T +t j−1
T +t j
e
−λ j (T +t j −τ )− i−1 l= j+1 λl Tl −λi (t−(T +ti−1 ))
T +t j−1
w (τ )w(τ )dτ +
t
e
−λi (t−τ )
T +ti−1
w (τ )w(τ )dτ T
(45) Then, following the similar arguments in [18], one has − λ j ((k − 1)T + t j − τ ) −
S
λl Tl − ( − k)
l= j+1
−
i−1
S
λl Tl
l=1
λl Tl − λi (t − (T + ti−1 )) > −λmax (t − τ ),
l=1
− λ j (T + t j − τ ) −
i−1
λl Tl − λi (t − (T + ti−1 )) > −λmax (t − τ ),
l= j+1
− λi (t − τ ) > −λmax (t − τ ), S S i−1 − λ j ((k − 1)T + t j − τ ) − λl Tl − ( − k) λl Tl − λl Tl − λi (t − (T + ti−1 )) l= j+1
l=1
l=1
< −2λ∗ (t − τ ) + max (2λ∗ − λmin , 0)2T , i−1 − λ j (T + t j − τ ) − λl Tl − λi (t − (T + ti−1 )) ∗
l= j+1 ∗
< −2λ (t − τ ) + max (2λ − λmin , 0)2T , − λi (t − τ ) < −2λ∗ (t − τ ) + max (2λ∗ − λmin , 0)2T then Eq. (45) can be rewritten as t t ∗ ∗ −λmax (t−τ ) 2 e z (τ )z(τ )dτ ≤ γ e2T (2λ −λmin )−2λ (t−τ ) w (τ )w(τ )dτ 0
(46)
0
Then, integrating t from 0 to ∞, one obtains Eq. (11). The proof is complete.
A2. Proof of Theorem 2 Proof. Construct a Lyapunov function (8) with matrix (12), for t ∈ [T + ti−1 , T + ti ), one i−1 has 0 ≤ t−TT−t ≤ 1, then with Eqs. (14)–(16) and according to Lemma 3, one has Pi (t) > 0. i Moreover, one also has
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Ai (t )Pi (t ) + Pi (t )Ai (t ) + D+ P + λi Pi (t ) ⎛ n ⎞ n j j (t − T − ti−1 ) (t − T − ti−1 ) = sym⎝ Ai, j Pi, j ⎠ j Ti Ti j j=0 j=0 n n j(t − T − ti−1 ) j−1 (t − T − ti−1 ) j + Pi, j−1 + λi Pi, j Ti j Ti j j=1 j=0 = Ai,0 Pi,0 + Pi,0 Ai,0 +
1 Pi,1 + λi Pi,0 Ti
t − T − ti−1 2 Ai,1 Pi,0 + Pi,0 Ai,1 + Ai,0 Pi,1 + Pi,1 Ai,0 + Pi,2 + λi Pi,1 + Ti Ti (t − T − ti−1 )2 Ai,0 Pi,2 + Pi,2 Ai,0 + Ai,1 Pi,1 + Pi,1 Ai,1 + Ai,2 Pi,0 + Pi,2 Ai,0 Ti 2 3 + Pi,3 + λi Pi,2 + · · · Ti ⎛ ⎞ n−1 (t − T − ti−1 )n−1 ⎝ n + (Ai, j Pi,n−1− j + Pi,n−1− j Ai, j ) + Pi,n + λi Pi,n−1 ⎠ Ti Ti n−1 j=0 +
+
n (t − T − ti−1 )n Ai,q Pi,n−q + Pi,n−q Ai,q + · · · n Ti q=0
+
(t − T − ti−1 )2n Ai,n Pi,n + Pi,n Ai,n . Ti 2n t − T − ti−1 Pi,0 Bwi,1 + Pi,1 Bwi,0 Ti (t − T − ti−1 )2 + Pi,0 Bwi,2 + Pi,1 Bwi,1 + Pi,2 Bwi,0 + · · · 2 Ti (t − T − ti )2n Pi,n Bwi,n + 2n Ti
Pi Bwi (t ) = Pi,0 Bwi,0 +
Then, one has ⎡ Ai (t )Pi (t ) + Pi (t )Ai (t ) + D + Pi (t ) + λi Pi (t ) ⎣ ∗ i (t ) = ∗ = i,0 +
Pi (t )Bwi (t ) −γ 2 I ∗
t − T − ti−1 (t − T − ti )2n i,1 + · · · + i,2n . Ti Ti 2n
(47)
⎤ Ci 0⎦ −I (48)
i−1 Since 0 ≤ t−TT−t ≤ 1, with Eq. (17) and according to Lemma 2, one obtains i (t) < 0. i Then, combining with Eq. (18), according to Theorem 1, one could conclude that system (1) is λ∗ -exponentially stable and satisfies Eq. (11). The proof is complete.
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A3. Proof of Theorem 3 Proof. Construct a Lyapunov function V (t ) = x (t )W −1 (t )x(t ) = x (t )Z (t )x(t ) and define Z (t ) = Zi (t ) for t ∈ [T + ti−1 , T + ti ), with the formulation of W(t) given as in Eqs. (29), (30) and (36), one has W(t) is continuous. With Eq. (28), and according to Lemma 3, one has that W(t) > 0, then one could conclude Z(t) > 0 and it is continuous. Similar to the proof of Theorem 2, for t ∈ [t − T − ti−1 , t − T − ti ), one has Wi (t )Ai (t ) + Ai (t )Wi (t ) + Qi (t )Bi (t ) + Bi (t )Qi (t ) − D +Wi (t ) + λiWi (t ) 1 = Wi,0 Ai,0 + Ai,0Wi,0 + Bi,0 Qi,0 + Qi, 0 Bi, 0 − Wi,1 + λiWi,0 + · · · Ti t − T − ti−1 + Wi,0 Ai,1 + Ai,1Wi,0 + Wi,1 Ai,0 + Ai,0Wi,1 + Qi, 0 Bi, 1 + Bi,1 Qi,0 Ti 2 + Qi,1 Bi,0 + Bi,0 Qi,1 − Wi,2 + λiWi,1 + · · · Ti 2n (t − T − ti−1 ) + Wi,n Ai,n + Ai,nWi,n + Qi,n Bi,n + Bi,n Qi,n . 2n Ti and (t − T − ti−1 ) (t − T − ti−1 )n Bwi,1 + · · · + Bwi,n . Ti Ti n (t − T − ti−1 ) (t − T − ti−1 )n CiWi (t ) = CiWi,0 + CiWi,1 + · · · + CiWi,n . Ti Ti n Bwi (t ) = Bwi,0 + +
Substitute them in the following matrix, then one could obtain that ⎡ Wi (t )Ai (t ) + Ai (t )Wi (t ) + Qi (t )Bi (t ) + Bi (t )Qi (t ) − D +Wi (t ) + λiWi (t ) ⎣ ∗ ∗ = i,0 +
Since 0 ≤
t−T −ti−1 Ti
(t−T −ti−1 ) i,1 Ti
+ ··· +
(49)
Bwi (t ) −γ 2 I ∗
(t−T −ti−1 )2n i,2n Ti 2n
⎤ Wi (t )Ci 0 ⎦ −I
(50)
≤ 1, with Eq. (31) and according to Lemma 2, one has
⎡ Wi (t )Ai (t ) + Ai (t )Wi (t ) + Qi (t )Bi (t ) + Bi (t )Qi (t ) − D +Wi (t ) + λiWi (t ) ⎣ ∗ ∗
Bwi (t ) −γ 2 I ∗
⎤ Wi (t )Ci 0 ⎦ < 0, −I
(51) then left-multiply and right-multiply diag{Zi (t), I, I} of Eq. (51), one has ⎡
Aci (t )Wi−1 (t ) + Wi−1 (t )Aci (t ) − Wi−1 (t )D +Wi (t )Wi−1 (t ) + λiWi−1 (t ) ⎣ ∗ ∗
Wi−1 (t )Bwi (t ) −γ 2 I ∗
⎤ Ci 0 ⎦ < 0, −I
(52)
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Since D +Wi−1 (t ) = −W −1 (t )D +Wi (t )Wi−1 (t ), then Eq. (52) equals to ⎡ ⎤ Aci (t )Zi (t ) + Zi (t )Aci (t ) + D + Zi (t ) + λi Zi (t ) Zi (t )Bwi (t ) Ci ⎣ ∗ −γ 2 I 0 ⎦ < 0, ∗ ∗ −I
(53)
Combining with Eq. (32) and according to Theorem 1, one could conclude that the closedloop system achieves the H∞ performance index (11). This completes the proof. References [1] D. Ju, Q. Sun, Modeling of wind turbine rotor blade system, J. Vib. Acoust. Trans. ASME 139 (5) (2017) 051013. [2] B. Zhou, Global stabilization of periodic linear systems by bounded controls with applications to spacecraft magnetic attitude control, Automatica 60 (2015) 145–154. [3] S. Xue, X. Yang, Z. Li, H. Gao, An approach to fault detection for multirate sampled-data systems with frequency specifications, IEEE Trans. Syst. Man Cybern. Syst. 48 (7) (2018) 1155–1165. [4] W. Ma, X. Jia, D. Zhang, Networked continuous-time filtering for quadratically inner-bounded time-delay systems with multirate sampling, J. Frankl. Inst. Eng. Appl. Math. 354 (17) (2017) 7946–7967. [5] J. Zhou, Classification and characteristics of Floquet factorisations in linear continuous-time periodic systems, Int. J. Control 81 (11) (2008) 1682–1698. [6] B. Zhou, M. Hou, G. Duan, L∞ and L2 semi-global stabilisation of continuous-time periodic linear systems with bounded controls, Int. J. Control 86 (4) (2013) 709–720. [7] B. Zhou, G. Duan, Periodic Lyapunov equation based approaches to the stabilization of continuous-time periodic linear systems, IEEE Trans. Autom. Control 57 (8) (2012) 2139–2146. [8] J. Zhou, H.M. Qian, Pointwise frequency responses framework for stability analysis in periodically time-varying systems, Int. J. Syst. Sci. 48 (4) (2017) 715–728. [9] K. Farhang, A. Midha, Steady-state response of periodic time-varying linear systems, with application to an elastic mechanism, J. Mech. Des. 117 (1995) 633–639. [10] T.J. Selstad, K. Farhang, On efficient computation of the steady-state response of linear systems with periodic coefficients, J. Vib. Acoust. 118 (3) (1996) 522–526. [11] P. Li, J. Lam, K.C. Cheung, H∞ control of periodic piecewise vibration systems with actuator saturation, J. Vib. Control 23 (20) (2017) 3377–3391. [12] R.Z. Scapini, L.V. Bellinaso, L. Michels, Stability analysis of AC-DC full-bridge converters with reduced DC-link capacitance, IEEE Trans. Power Electron. 33 (1) (2018) 899–908. [13] P. Li, J. Lam, Y. Chen, K.C. Cheung, Y. Niu, Stability and L2 -gain analysis of periodic piecewise linear systems, in: Proceedings of the American Control Conference, Chicago, 2015, pp. 3509–3514. July 1–3 [14] P. Li, J. Lam, K.C. Cheung, Stability, stabilization and L2 -gain analysis of periodic piecewise linear systems, Automatica 61 (2015) 218–226. [15] X. Xie, J. Lam, P. Li, Finite-time H∞ control of linear periodic piecewise linear systems, Int. J. Syst. Sci. 48 (11) (2017) 2333–2344. [16] X. Xie, J. Lam, P. Li, H∞ control problem of linear periodic piecewise time-delay systems, Int. J. Syst. Sci. 49 (5) (2018) 997–1011. [17] X. Xie, J. Lam, Guaranteed cost control of periodic piecewise linear time-delay systems, Automatica 94 (2018) 274–282. [18] P. Li, J. Lam, R. Lu, K.W. Kwok, Stability and L2 synthesis of a class of periodic piecewise time-varying systems, IEEE Trans. Autom. Control (2018), doi:10.1109/TAC.2018.2880678. [19] P. Li, J. Lam, K.W. Kwok, R. Lu, Stability and stabilization of periodic piecewise linear systems: a matrix polynomial approach, Automatica 94 (2018) 1–8. [20] F. Li, C. Du, C. Yang, W. Gui, Passivity-based asynchronous sliding mode control for delay singular Markovian jump systems, IEEE Trans. Autom. Control 63 (8) (2018) 2715–2721. [21] T. Wu, F. Li, C. Yang, W. Gui, Event-based fault detection filtering for complex networked jump systems, IEEE/ASME Trans. Mechatron. 23 (2) (2018) 497–505. [22] C. Briat, Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints, Automatica 49 (11) (2013) 3449–3457. Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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[23] K.M. Garg, Theory of Differentiation: A Unified Theory of Differentiation via New Derivate Theorems and New Derivatives, Wiley-Interscience, New York, 1998. [24] G. Chesi, A. Garulli, A. Tesi, A. Vicino, Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems, Springer-Verlag, London, 2009. [25] Z. Feng, W. Zheng, Improved stability condition for Takagi–Sugeno fuzzy systems with time-varying delay, IEEE Trans. Cybern. 47 (3) (2017) 661–670. [26] S. Tong, D. Qian, J. Fang, Joint estimation of parameters, states and time-delay based on singular pencil model, Int. J. Innov. Comput. Inf. Control 12 (1) (2016) 225–242. [27] W. Sun, S. Su, Y. Wu, J. Xia, V. Nguyen, Adaptive fuzzy control with high-order barrier Lyapunov functions for high-order uncertain nonlinear systems with full-state constraints, IEEE Trans. Cybern. (2018), doi:10.1109/ TCYB.2018.2890256.
Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function Panshuo Li, Pengxu Li, Yun Liu, Hong Bao, Renquan Lu∗ School of Automation and Guangdong Province Key Laboratory of Intelligent Decision and Cooperative Control, Guangdong University of Technology, Guangzhou 510006, China Received 26 September 2018; received in revised form 30 May 2019; accepted 13 June 2019 Available online xxx
Abstract In this paper, the H∞ control problem of periodic piecewise systems with polynomial time-varying subsystems is addressed. Based on a periodic Lyapunov function with a continuous time-dependent Lyapunov matrix polynomial, the H∞ performance is studied. The result can be easily reduced to the conditions for periodic piecewise systems with constant subsystems or linear time-varying systems based on a common Lyapunov function or a linear time-varying Lyapunov matrix. Moreover, an H∞ controller with time-varying polynomial controller gain is proposed as well, which could be directly solved with the linear matrix inequalities. A numerical example is presented to demonstrate the effectiveness of the proposed method. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
1. Introduction The periodic time-varying system is a class of time-varying systems with periodic properties. It may come from the system inherent dynamic characteristics such as rotor-blade systems [1], spacecraft attitude adjustment [2] or from the artificial operation such as multirate sampling [3,4]. Control problems for continuous-time periodic systems are more challenging compared with discrete-time periodic systems since the discrete-time periodic systems could ∗
Corresponding author. E-mail address: [email protected] (R. Lu).
https://doi.org/10.1016/j.jfranklin.2019.06.008 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.
Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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be transformed into linear time-invariant systems with lifting techniques [5]. Efforts of solving the control problems for continuous-time periodic systems can be found in [6–8], where the L∞ and L2 semi-global stabilization problems with bounded controls are studied in [6], the state feedback stabilizing controller is designed in [7] based on the solutions of parametric periodic Lyapunov differential equations and pointwise frequency-domain approach is used to develop the stability analysis in [8]. The periodic piecewise system is a special class of continuous-time periodic system. It has several subsystems in one period, the switching sequence and dwell time of each subsystem are fixed. Such kind of systems could be regarded as the approximation system of continuoustime periodic systems [9,10]. Studies on periodic piecewise systems can help the research of continuous-time periodic systems [8]. Moreover, the periodic piecewise system also has many prototypes in engineering applications, such as vibration systems [11], power converters [12]. Because of its value both in studying continuous-time periodic systems and engineering applications, lots of efforts have been put into this area. The stability and L2 -gain performance are studied in [13] and [14] for the periodic piecewise system with non-Hurwitz subsystems based on continuous time-varying Lyapunov function and discontinuous time-varying Lyapunov function, respectively. Apart from analysis problems, the controller design investigation could be found in [11,15–17]. The saturated H∞ controller with piecewise constant controller gains is designed in [11] for periodic piecewise mechanical systems to attenuate the system vibration, a corresponding algorithm is proposed as well. A finite-time H∞ controller is synthesized for periodic piecewise system in [15]. The H∞ control problem of periodic piecewise system with time-delay is studied in [16] based on a piecewise Lyapunov function with a discontinuous time-varying matrix function with two global terms. Guaranteed cost control of periodic piecewise time-delay systems is investigated in [17] based on a novel Lyapunov function with relaxes constraints on positive definiteness of the involved symmetric matrices. In the above results, on one hand, one may observe that the considered periodic piecewise systems are composed with constant subsystems. However, it could be seen that the constant subsystems may loss dynamic characteristics of its approximated continuous-time periodic systems. From this perspective, the periodic piecewise system with time-varying subsystems might be more accurate. The results on periodic piecewise time-varying systems are not so profound as the results on periodic piecewise constant systems. The analysis and synthesis problems of periodic piecewise time-varying systems are addressed in [18], where the subsystems are linear time-varying. On the other hand, continuous and multiple Lyapunov functions are employed with a time-varying Lyapunov matrix. The Lyapunov matrix is given as a linear interpolation of time in each subsystem [11,14] or each segment [15]. To improve the results, a Lyapunov matrix given in matrix polynomial form is employed in [19] for periodic piecewise linear systems, the modified stability and L2 -gain results are reported. However, the obtained results could not cover the condition based on a linear time-varying Lyapunov matrix or a common Lyapunov function. More techniques for system with time-varying properties can be found in [20,21], where passivity-based asynchronous sliding mode control [20], fault detection filtering [21] are used. From the above discussion, one may notice that neither the study on periodic piecewise polynomial systems nor the results based on Lyapunov polynomial matrix is abundant, not to mention results of periodic piecewise polynomial systems based on a polynomial Lyapunov function. Hence, motivated by the above-mentioned works and the broad applications of periodic piecewise systems, a periodic piecewise system with time-varying polynomial subsystems Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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is considered in this paper. A continuous Lyapunov function with a time-varying Lyapunov matrix polynomial is adopted. The contributions of this paper lie in the H∞ performance analysis and the corresponding controller design for periodic piecewise polynomial time-varying systems. Conditions of the negativeness and positiveness of a matrix polynomial are introduced to obtain the results, which can be reduced to the results for the periodic piecewise systems with constant or linear time-varying systems based on common Lyapunov function or linear time-varying Lyapunov function. Since the continuous-time periodic system is a special case of the periodic piecewise polynomial time-varying systems, the results obtained in this work may further help the study in control of continuous-time periodic systems. This paper is organized as follows. The system is described and preliminaries are given in Section 2. The main results are obtained in Section 3. Simulation results are given in Section 4. Conclusion is provided in Section 5. 2. System description and preliminaries Notation: Rr denotes the r-dimensional Euclidean space, N + denotes the set of the positive integers. · stands for the Euclidean vector norm, the superscript refers to matrix transposition, sym(P) stands for P + P and P > 0 means that P is a real symmetric and positive definite matrix, max(·) or min(·) stand for the maximum or minimum value of (·). To facilitate the description, a non-negative integer is used to represent the nominal number of fundamental periods, that is, = 0, 1, . . .. Consider a T-periodic piecewise time-varying system given as, for t ∈ [T + ti−1 , T + ti ], i ∈ N , N = {1, 2, . . . , S}, x˙(t ) = Ai (t )x(t ) + Bi (t )u(t ) + Bwi (t )w(t ), z(t ) = Ci x(t )
(1)
where ti is the switching instant in the first period from the ith subsystem to the (i + 1)th subsystem and t0 = 0, S is the number of subsystems in one period, and one has Ai (t ) = Ai (t + T ), Bi (t ) = Bi (t + T ), Bwi (t ) = Bwi (t + T ). x(t ) ∈ Rr , u(t ) ∈ Rd , z(t ) ∈ Rk are the state vector, control input, measurement output, respectively. Define the dwell time of the ith subsystem is Ti , that is, the time between two successive discrete events, then one has Ti = ti − ti−1 [22]. Each subsystem is formulated in time-varying polynomial form, where Ai (t ), Bi (t ), Bwi (t ) are given by n t − T − ti−1 (t − T − ti−1 )n (t − T − ti−1 ) j Ai (t ) = Ai,0 + Ai,1 + · · · + Ai,n = Ai, j , Ti Ti n Ti j j=0 n n (t − T − ti−1 ) j (t − T − ti−1 ) j Bi (t ) = Bi, j , Bwi (t ) = Bwi, j , (2) Ti j Ti j j=0 j=0 where Ai, j , Bi, j , Bwi, j are constant matrices. It can be seen that A(t ), B(t ), Bwi (t ) are continuous time-varying in each subsystems, and discontinuous at switching instants. However, A(t ), B(t ), Bwi (t ) could becontinuous at switching instantsunder the condition that Ai,0 = nj=0 Ai−1, j , Bi,0 = nj=0 Bi−1, j , Bwi,0 = nj=0 Bwi−1, j , Si=1 nj=1 Ai, j = 0, Si=1 nj=1 Bi, j = 0, Si=1 nj=1 Bwi, j = 0, then system (1) will become a general continuous-time periodic system. It should be noticed that the output matrix Ci is settled Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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as a constant matrix here to facilitate the the system state observation, one could also choose a time-varying Ci (t). The definition of the exponential stability of a periodic piecewise system is given as follows. Definition 1 [14]. The periodic piecewise system (1) with u(t ) = 0, w(t ) = 0 is said to be λ∗ -exponentially stable if the solution of the system from x(0) satisfies x(t ) ≤ ∗ κe−λ t x(0), ∀t > 0 for some constants κ ≥ 1, λ∗ > 0. To facilitate the development, the Dini-derivative [23] is introduced for a continuous function P(t) as P (t + h) − P (t ) D + P (t ) = lim+ sup , t ∈ [T + ti−1 , T + ti ) (3) h→0 h and a general condition concerning the exponential stability of periodic piecewise system is given in Lemma 1. Lemma 1 [19]. Consider a periodic piecewise time-varying system (1) with u(t ) = 0, w(t ) = 0, let λ∗ > 0, λi , i ∈ N , be given constants. If there exist λi , i ∈ N , and real symmetric Tperiodic, continuous and Dini-differentiable periodic matrix function P(t) defined on t ∈ [0, ∞) such that, for i ∈ N , t ∈ [T + ti−1 , T + ti ), P (t ) = Pi (t ) > 0, satisfies Ai (t ) Pi (t ) + Pi (t )Ai (t ) + D + Pi (t ) + λi Pi (t ) < 0, 2λ∗ T −
S
(4)
λi Ti ≤ 0
(5)
i=1
then system (1) is λ∗ -exponentially stable. A lemma concerning the negative definiteness of a matrix polynomial is given below. Lemma 2 [18]. Consider a bounded matrix polynomial f (τ1 , τ2 , . . . , τn ) given as n f (τ1 , τ2 , . . . , τn ) = ϒ0 + τ1 ϒ1 + τ1 τ2 ϒ2 + · · · + τk ϒn ,
(6)
k=1
where n ∈ N + , ϒ j , j = 0, 1, . . . , n are real symmetric matrices, τk , k = 1, 2, . . . n are variables and τ k ∈ [0, 1]. If d
ϒk < 0, d = 0, 1, . . . , n,
k=0
then the matrix polynomial f (τ1 , τ2 , . . . , τn ) < 0. Similarly, one can obtain Lemma 3 concerning the positive definiteness of a matrix polynomial, the proof can be easily extended from the proof of Lemma 2. Lemma 3. Consider a bounded matrix polynomial f (τ1 , τ2 , . . . , τn ) given as in Eq. (6), where n ∈ N + , ϒ j , j = 0, 1, . . . , n are real symmetric matrix, τk , k = 1, 2, . . . n are variables and τ k ∈ [0, 1]. If d
ϒk > 0, d = 0, 1, . . . , n,
(7)
k=0
then the matrix polynomial f (τ1 , τ2 , . . . , τn ) > 0. Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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3. Main results In this Section, a general condition of H∞ performance for periodic piecewise time-varying system based on a periodic continuous Lyapunov function is obtained at first. Then, based on a time-varying Lyapunov matrix polynomial and the time-interval information, a sufficient condition of H∞ performance is proposed. The H∞ performance condition based on a linear time-varying Lyapunov matrix is given as well. Finally, under the state-feedback control strategy, a controller which guarantees the H∞ performance of periodic piecewise polynomial systems is designed. 3.1. H∞ performance analysis For periodic piecewise polynomial system (1), consider a Lyapunov function V (t ) = x (t )P (t )x(t ), where P(t) > 0 is continuous and bounded, and it is periodic with period T, that is, P (t ) = P (t + T ). For t ∈ [T + ti−1 , T + ti ), i ∈ N , rewrite P (t ) = Pi (t ), then V(t) can be given as V (t ) = Vi (t ) = x (t )Pi (t )x(t ).
(8)
Then, based on the above Lyapunov function, the general H∞ performance index for periodic piecewise polynomial system can be established as in Theorem 1. Theorem 1. Consider a periodic piecewise time-varying system (1) with u(t ) = 0, given γ > 0, λ∗ > 0. If there exist λi , i ∈ N and continuous, Dini differentiable matrix function P(t) defined on t ∈ [0, ∞) such that, for t ∈ [T + ti−1 , T + ti ), i ∈ N , P (t ) = Pi (t ) > 0, satisfies ⎡ ⎤ Ai (t )Pi (t ) + Pi (t )Ai (t ) + D + Pi (t ) + λi Pi (t ) Pi (t )Bwi (t ) Ci ⎣ ∗ −γ 2 I 0 ⎦ < 0, (9) ∗ ∗ −I 2λ∗ T −
S
λi Ti ≤ 0,
(10)
i=1
then system (1) is λ∗ -exponentially stable and satisfies ∞ ∞ z (τ )z(τ )dτ ≤ γ 2 w (τ )w(τ )dτ, 0
where
γ = γe
(11)
0
T max (2λ∗ −λmin ,0)
λmax , λmax = max (λi ), λmin = min (λi ). i i 2λ∗
Proof. See the Appendix. Remark 1. One may observe that the constraints on λi are independent with the stability of each subsystem. It is not necessary to allocate a positive λi for a Hurwitz subsystem, and a negative λi for a non-Hurwitz subsystem as in previous works. It relaxes the condition, and facilitates the theory development since determining the stability of a time-varying system is not an easy work. The proof details could be found in [19]. Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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It should be noticed that H∞ index measures the output energy for the worst case input energy, it is a well-known performance index to describe a system in engineering application. In the following, we will study the H∞ performance condition based on a continuous timevarying Lyapunov matrix polynomial. Consider a continuous Lyapunov matrix P(t) given in a time-varying matrix polynomial form as, for t ∈ [T + ti−1 , T + ti ), i ∈ N P (t ) = Pi (t ) = Pi,0 + Pi,0 =
n
t − T − ti−1 (t − T − ti−1 )n Pi,1 + · · · + Pi,n Ti Ti n
Pi−1, j , P1,0 =
j=0
n
PS, j ,
(12)
(13)
j=0
where Pi,j are constant matrices and n ≥ 0. It can be seen that P(t) is continuous both in subsystems and switching instants. One could also obtain P2,0 = P1,0 +
n (t − T ) j
T1j
j=1
P2, j = P1,0 +
P1, j , P3,0 = P2,0 +
j=1
2 n (t − T − t1 ) j
T2j
i=1 j=1
n (t − T − t1 ) j
T2j
P1, j ,
··· P1,0 +
S n (t − T − ti−1 ) j i=1 j=1
hence, Pi,0 = Pi,0 = P1,0 +
Ti j
n j=0
Pi−1, j , P1,0 =
i−1 n k
Pi, j = P1,0 , n j=0
PS, j , then Eq. (13) could also be written as
Pi, j , i = 2, 3, . . . , S, and
j=1
S n
Pi, j = 0.
i=1 j=1
Remark 2. It should be noticed that the polynomial degree of Pi (t) in different subsystems could be different, which can be denoted as ni . The degrees of the subsystem matrices and Lyapunov matrix are independent as well. In this work, to facilitate the development, the degree of all polynomial functions are chosen to be an uniform n. It is easy to extend the result to general cases with different degrees. Based on the above Lyapunov matrix polynomial, the H∞ performance condition could be established as in Theorem 2. Theorem 2. Consider a periodic piecewise time-varying system (1) with u(t ) = 0, given γ > 0, λ∗ > 0. If there exist λi and matrices Pi, j , i ∈ N , j = 0, 1, . . . , n, satisfying k
Pi, j > 0, k = 0, 1, . . . , n,
(14)
j=0
Pi,0 =
n
Pi−1, j , i = 2, 3, . . . , S,
(15)
j=0
S i=1
n
Pi, j = 0,
(16)
j=1
Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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i, j < 0, k = 0, 1, . . . , n,
7
(17)
j=0 S
2λ∗ T −
λi Ti ≤ 0,
(18)
i=1
where i,0
i, j
⎡
Ai,0 Pi,0 + Pi,0 Ai,0 + T1i Pi,1 + λi Pi,0 =⎣ ∗ ∗ ⎡ ⎤ φi, j ϕi, j 0 ⎣ 0 0 ⎦, = ϕi, j 0 0 0
φi, j =
j
(Ai,q Pi, j−q + Pi, j−q Ai,q ) +
q=0
φi,n =
n
Pi,0 Bwi,0 −γ 2 I ∗
⎤ Ci 0 ⎦, −I
j+1 Pi, j+1 + λi Pi, j , Ti
j = 1, . . . , n − 1,
(Ai,q Pi, j−q + Pi, j−q Ai,q ) + λi Pi, j ,
q=0
φi, j =
n
(Ai,q Pi, j−q + Pi, j−q Ai,q ),
j = n + 1, . . . , 2n,
q= j−n
ϕi, j =
j
Pi,q Bi, j−q ,
j = 1, . . . , 2n,
(19)
q=0
then system (1) is λ∗ -exponentially stable and satisfies Eq. (11). Proof. See the Appendix. Remark 3. It is worth mentioning that the sum of squares (SOS) [24] is an alternative method to tackle the matrix polynomial problem, it has been successfully applied in the stability analysis of periodic piecewise linear systems [19]. However, apart from the polynomial degree limitation, it is not an easy work to use the SOS method to develop the H∞ controller. On contrast, the method used in this work is more convenient to develop the H∞ controller with no constraint on the matrix polynomial degree. Moreover, the condition based on SOS method would be valid independent of time for periodic piecewise system [19], while the time-interval information is considered in this work which may result in less conservative result. If the degree of P(t) equals 1, then a corollary could be established as below. Corollary 1. Consider a periodic piecewise time-varying system (1) with u(t ) = 0, given γ > 0, λ∗ > 0. If there exist λi and matrices Pi,0 , Pi,1 , satisfying Pi,0 > 0, Pi,0 + Pi,1 > 0,
(20)
Pi,0 = Pi−1,0 + Pi−1,1 , i = 2, 3, . . . , S,
(21)
S
Pi,1 = 0,
(22)
i=1
Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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k
i, j < 0, k = 0, 1, . . . , n + 1,
(23)
j=0
2λ∗ T −
S
λi Ti ≤ 0,
(24)
i=1
where i,0
i, j
⎡
Ai,0 Pi,0 + Pi,0 Ai,0 + T1i Pi,1 + λi Pi,0 ⎣ = ∗ ∗ ⎡ ⎤ φi, j ϕi, j 0 0 0 ⎦, = ⎣ϕi, j 0 0 0
Pi,0 Bwi,0 −γ 2 I ∗
⎤ Ci 0 ⎦, −I
φi,1 = sym (Ai,0 Pi,1 + Ai,1 Pi,0 ) + λi Pi,1 ,
φi, j = sym Ai, j Pi,0 + Ai, j−1 Pi,1 , j = 2, . . . , n, φi,n+1 = Ai,n Pi,1 + Pi,1 Ai,n , ϕi, j = Pi,0 Bwi, j + Pi,1 Bwi, j−1 ,
j = 1, . . . , n,
(25)
then system (1) is λ∗ -exponentially stable and satisfies Eq. (11). The proof can be easily obtained from the proof of Theorem 2. It is worth mentioning that Corollary 1 is the condition based on the Lyapunov function with a linear time-varying Lyapunov matrix. The condition based on a common Lyapunov function could be obtained from the Theorem 2 as well, which is omitted here. 3.2. Time-varying H∞ controller synthesis Consider a state-feedback controller u(t ) = K (t )x(t ),
(26)
where K(t) is periodic with period T, that is, K (t ) = K (t + T ). Under this controller, the closed-loop system can be given as x˙(t ) = Aci (t )x(t ) + Bwi (t ), z(t ) = Ci x(t ),
(27)
where Aci (t ) = Ai (t ) + Bi (t )Ki (t ). Then an H∞ control scheme condition could be obtained as in Theorem 3. Theorem 3. Consider a periodic piecewise time-varying system (1), given γ > 0, λ∗ > 0. If there exist λi and matrices Wi, j , Qi, j , i ∈ N , j = 0, 1, . . . , n, such that k
Wi, j > 0, k = 0, 1, . . . , n
(28)
j=0
Wi,0 =
n
Wi−1, j , i = 2, 3, . . . , S
(29)
j=0
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Wi, j = 0,
9
(30)
i=1 j=1 k
i, j < 0, k = 0, 1, . . . , n,
(31)
j=0
2λ∗ T −
S
λi Ti ≤ 0,
(32)
i=1
where i,0
i, j
⎡ Wi,0 Ai,0 + Ai,0Wi,0 + Bi,0 Qi,0 + Qi, 0 Bi, 0 − T1i Wi,1 + λiWi,0 =⎣ ∗ ∗ ⎡ ⎤ i, j ϒi, j i, j 0 0 ⎦, = ⎣ ϒi, j
i, j 0 0
i, j =
j
Bwi,0 −γ 2 I ∗
⎤ Wi,0Ci 0 ⎦, −I
(Wi, j−q Ai,q + Ai,qWi, j−q + Qi, j−q Bi,q + Bi,q Qi, j−q )
q=0
j+1 Wi, j+1 + λiWi, j , j = 1, . . . , n − 1, Ti n = (Wi,n−q Ai,q + Ai,qWi,n−q + Qi,n−q Bi,q + Bi,q Qi,n−q ) + λiWi,n , −
i,n
q=0 n
i, j =
(Wi, j−q Ai,q + Ai,qWi, j−q + Qi, j−q Bi,q + Bi,q Qi, j−q ),
j = n + 1, . . . , 2n,
q= j−n
ϒi, j = Bwi, j , i, j = Wi, j Ci , j = 1, 2, . . . , n, ϒi, j = 0, i, j = 0, j = n + 1, . . . , 2n.
(33)
then the periodic piecewise polynomial system (1) with state-feedback controller u(t ) = Ki (t )x(t ) is λ∗ -exponentially stable and satisfies Eq. (11). And the controller gain can be given as Ki (t ) = Qi (t )Wi−1 (t ) with Qi (t ) =
n j=0
Wi (t ) =
n j=0
(34)
(t − T − ti−1 ) j Ti j
(t − T − ti−1 ) j Ti j
Qi, j ,
(35)
Wi, j .
(36)
Proof. See the Appendix. Moreover, an H∞ controller design based on a linear time-varying Lyapunov function could also be established with Corollary 2. Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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Corollary 2. Consider a periodic piecewise time-varying system (1), given γ > 0, λ∗ > 0. If there exist λi and matrices Wi, j , Qi, j , i ∈ N , j = 1, 2, . . . , n, such that Wi,0 > 0, Wi,0 + Wi,1 > 0, Wi,0 = Wi−1,0 + Wi−1,1 , S
i = 1, 2, . . . , S
(37)
i = 2, 3, . . . , S,
(38)
Wi,1 = 0,
(39)
i=1 k
i, j < 0,
i = 1, 2, . . . , S, k = 0, 1, . . . , n + 1,
(40)
j=0
2λ∗ T −
S
λi Ti ≤ 0,
(41)
i=1
where ⎡
i,0
i, j
Ai,0Wi,0 + Wi,0 Ai,0 + Bi,0 Qi,0 + Qi, 0 Bi, 0 − T1i Wi,1 + λiWi,0 =⎣ ∗ ∗ ⎡ ⎤ φi, j ϕi, j i, j 0 0 ⎦, = ⎣ ϕi, j i, j 0 0
Bwi,0 −γ 2 I ∗
⎤ Wi,0Ci 0 ⎦, −I
φi,1 = sym (Ai,0Wi,1 + Ai,1Wi,0 + Bi,0 Qi,1 + Bi,1 Qi,0 ) + λiWi,1 ,
φi, j = sym Ai, j Wi,0 + Ai, j−1Wi,1 + Bi, j Qi,0 + Bi, j−1 Qi,1 , j = 2, . . . , n, φi,n+1 = Ai,nWi,1 + Wi,1 Ai,n + Bi,n Qi,1 + Qi, 1 Bi,n , ϕi, j = Bwi, j , i, j = Wi, j Ci , j = 1, . . . , n,
ϕi,n+1 = 0, i,n+1 = 0,
(42)
then the periodic piecewise polynomial system (1) with state-feedback controller u(t ) = Ki (t )x(t ) is λ∗ -exponentially stable and satisfies Eq. (11). And the controller gain can be given as Ki (t ) = Qi (t )Wi−1 (t ) with t − T − ti−1 Qi,1 , Ti t − T − ti−1 Wi (t ) = Wi,0 + Wi,1 . Ti
Qi (t ) = Qi,0 +
The proof can be easily obtained from the proof of Theorem 3 with let the degrees of W(t), Q(t) equal 1. Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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11
4. Simulation In this section, a numerical example is used to demonstrate the effectiveness of the proposed controller. The simulation is carried within Matlab, where the LMI Toolbox is adopted to solve the condition. Consider a periodic piecewise polynomial system with T = 2, t1 = 0.4, t2 = 1, t3 = 2 and 1 −3 −3 −2.5 −2 −3 + 2.5(t − T ) + 6.25(t − T )2 , A1 (t ) = 2 −2.8 −3 −4.5 −2 3.5 5(t − T − 0.4) −1 −0.5 25(t − T − 0.4)2 −4 −2.4 −5 −3 + + , A2 (t ) = 0 −4 0 −2 −1 −4 3 9 1 3 −2 −3 −4 −2.6 + (t − T − 1) + (t − T − 1)2 , A3 (t ) = 2 −3.2 3 −2 2 −4 −5 −8 −16 + 2.5(t − ) + 6.25(t − ) , B1 (t ) = 14 15 15 5(t − T − 0.4) −3 25(t − T − 0.4) −11 −13.6 + + , B2 (t ) = 13 10 4 3 9 −13 −10 −8 + (t − T − 1) + (t − T − 1)2 , B3 (t ) = 8 12 2 −6 −8 −6 + 2.5(t − ) + 6.25(t − ) , Bw1 (t ) = 3 11 3 5(t − T − 0.4) −5 25(t − T − 0.4) −15 −3 + + , Bw2 (t ) = 8 15 5 3 9 −15 −4 −11 + (t − T − 1) + (t − T − 1)2 , Bw3 (t ) = 5 10 8 1 0 C1 = C2 = C3 = 0 1 Time histories of A(t ), B(t ), Bw (t ) over a period are shown in Figs. 1–3. Given λ1 = −0.5, λ2 = 1, λ3 = 1, and the disturbance is w(t ) = e−0.1t . An H∞ controller with degree 2 is designed for the above system according to Theorem 3 with λ∗ = (λ1 T1 + λ2 T2 + λ3 T3 )/(2T ) = 0.425. The obtained r is 1.3512, it is related to the H∞ index under the worst case of this systems. The obtained W(t), Q1 (t), Q2 (t), Q3 (t) are shown in Figs. 4 and 5. It can be seen that W(t) is continuous in the whole time span, and Qi (t) is continuous in each subsystem and discontinuous at switching instant. The corresponding controller gain is shown in Fig. 6, which is discontinuous at the switching instant, because of the discontinuous Qi (t). It should be noticed that a continuous time-varying controller could be obtained with Corollary 2. Fig. 7 compares the system outputs of the open-loop and closed-loop systems. It can be seen that the system is exponentially stable, and the outputs, which are the same with the system state, are obviously attenuated under the designed H∞ controller. It implies that the closed-loop system displays good disturbance attenuation performance. From the above numerical example, one can conclude that for a periodic piecewise polynomial time-varying system, one could design a periodic time-varying controller proposed in this work to attenuate the system output under the disturbance excitation. Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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8
11 (
) ) 21( ) 22 ( ) 12 (
6 4
magnitude
2 0 -2 -4 -6 -8 -10 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t Fig. 1. Time history of A(t) over a period.
50
11 ( 12 (
) )
40 30
magnitude
20 10 0 -10 -20 -30 -40 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t Fig. 2. Time history of B(t) over a period.
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13
30
11 ( 12 (
) )
20
magnitude
10
0
-10
-20
-30 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.8
2
t Fig. 3. Time history of Bw (t ) over a period.
14
11 (
) ) 22 ( ) 12 (
12 10
magnitude
8 6 4 2 0 -2 -4 -6 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
t Fig. 4. Time history of W(t) over a period. Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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1 1( 1 2(
) )
2 1( 2 2(
) )
3 1( 3 2(
) )
60
magnitude
40 20 0 -20 -40 -60 -80 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t Fig. 5. Time history of Q1 (t), Q2 (t), Q3 (t) over a period.
8
1 1( 1 2(
6
) )
2 1( 2 2(
) )
3 1( 3 2(
) )
4
magnitude
2 0 -2 -4 -6 -8 -10 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t Fig. 6. Time history of controller gain over a period.
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3
1(
) ) 1( ) 2( ) 2(
2
15
without control without control with control with control
magnitude
1
0
-1
-2
-3 0
5
10
15
20
25
30
35
40
45
50
t Fig. 7. System state comparison of the open-loop and closed-loop system.
5. Conclusion The H∞ control problem of periodic piecewise polynomial time-varying systems is addressed in this paper. A continuous Lyapunov function which relaxes the constraints on the exponential orders of subsystems is adopted with time-dependent Lyapunov matrix polynomial. The H∞ performance index and the corresponding H∞ controller design are studied. The H∞ performance condition for the periodic piecewise constant system or periodic piecewise linear time-varying systems with a common Lyapunov function or a continuous Lyapunov function with linear time-varying Lyapunov matrix could be easily obtained from the results in this work. Since the continuous-time periodic system in polynomial form is a special case of the periodic piecewise system with polynomial subsystems, the result in the work may help the controller design for general continuous-time periodic systems. The future work of this paper may extend to the periodic piecewise system with time-delay or with high-order uncertain nonlinearities, techniques in [25–27] may help the future studies. Acknowledgment The work was supported by National Natural Science Foundation of China under Grant (61703111), the Innovative Research Team Program of Guangdong Province Science Foundation (2018B0330312006). Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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Appendix A A1. Proof of Theorem 1 Proof. For t ∈ [T + ti−1 , T + ti ), i ∈ N , construct a Lyapunov function as in Eq. (8). With Eqs. (9) and (10), according to Lemma 1, one could conclude that system (1) is λ∗ -exponential stability. On one hand, define F (t ) = z (t )z(t ) − γ 2 w (t )w(t ), then one has D +Vi (t ) + λiVi (t ) + F (t ) = (x (t )Ai (t ) + w (t )Bwi (t ))Pi (t )x(t ) + x (t )P (t )(Ai (t )x(t ) + Bwi (t )w(t )) + x (t )D + Pi (t )x(t )
+ λi x Pi (t )x(t ) + z (t )z(t ) − γ w (t )w(t ) = x (t )(Ai (t )Pi (t ) + Pi (t )Ai (t ) + D + Pi (t ) + λi Pi (t ) + Ci (t )Ci (t ))x(t ) + w (t )(Bwi (t )Pi (t ))x(t ) 2 + x (t )(Pi (t )Bwi (t ))w(t ) + w (t )(−γ I )w(t ) x(t ) x(t ) = i (t ) w(t ) w(t )
where
A (t )Pi (t ) + Pi (t )Ai (t ) + D + Pi (t ) + λi Pi (t ) i (t ) = i Bwi (t )Pi (t )
Pi (t )Bwi (t ) C + i Ci −γ 2 I 0
0 .
(43)
On the other hand, applying Schur complement equivalence to Eq. (9), one could obtain that i (t) < 0, which indicates D +Vi (t ) + λiVi (t ) + F < 0 if x = 0 or w = 0. Then integrate it within t ∈ [T + ti−1 , T + ti ), with zero initial condition, one could obtain t1 S S i−1 V (t ) < − e−λ1 (t1 −τ )− i=2 λ j Ti −(−1) i=1 λi Ti − i=1 λi Ti −λi (t−(T +ti−1 ))F (τ )dτ 0 t2 S S i−1 e−λ2 (t2 −τ )− i=3 λi Ti −(−1) i=1 λi Ti − i=1 λi Ti −λi (t−(T +ti−1 ))F (τ )dτ − · · · − t1
− − −
T
e−λS (T −τ )−(−1)
tS−1 T +t1 T t T +ti−1
S i=1
λi Ti − i−1 i=1 λi Ti −λi (t−(T +ti−1 ))
i−1
e−λ1 (T +t1 −τ )−
k=2
λk Tk −λi (t−(T +ti−1 ))
F (τ )dτ − · · ·
F (τ )dτ − · · ·
e−λi (t−τ ) F (τ )dτ.
(44)
With V(t) > 0 and Eq. (44), one has S k=1 j=1
+
i−1 j=1
(k−1)T +t j
S
e−λ j ((k−1)T +t j −τ )−
l= j+1
λl Tl −(−k)
S l=1
λl Tl − i−1 l=1 λl Tl −λi (t−(T +ti−1 ))
(k−1)T +t j−1 T +t j T +t j−1
i−1
e−λ j (T +t j −τ )−
l= j+1
z (τ )z(τ )dτ
λl Tl −λi (t−(T +ti−1 ))
z (τ )z(τ )dτ
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+
T +ti−1
≤ γ2
+
t
e−λi (t−τ ) z (τ )z(τ )dτ
⎧ S ⎨ ⎩
k=1 j=1
i−1 j=1
17
(k−1)T +t j
S
e−λ j ((k−1)T +t j −τ )−
l= j+1
λl Tl −(−k)
S l=1
λl Tl − i−1 l=1 λl Tl −λi (t−(T +ti−1 ))
w (τ )w(τ )dτ
(k−1)T +t j−1
T +t j
e
−λ j (T +t j −τ )− i−1 l= j+1 λl Tl −λi (t−(T +ti−1 ))
T +t j−1
w (τ )w(τ )dτ +
t
e
−λi (t−τ )
T +ti−1
w (τ )w(τ )dτ T
(45) Then, following the similar arguments in [18], one has − λ j ((k − 1)T + t j − τ ) −
S
λl Tl − ( − k)
l= j+1
−
i−1
S
λl Tl
l=1
λl Tl − λi (t − (T + ti−1 )) > −λmax (t − τ ),
l=1
− λ j (T + t j − τ ) −
i−1
λl Tl − λi (t − (T + ti−1 )) > −λmax (t − τ ),
l= j+1
− λi (t − τ ) > −λmax (t − τ ), S S i−1 − λ j ((k − 1)T + t j − τ ) − λl Tl − ( − k) λl Tl − λl Tl − λi (t − (T + ti−1 )) l= j+1
l=1
l=1
< −2λ∗ (t − τ ) + max (2λ∗ − λmin , 0)2T , i−1 − λ j (T + t j − τ ) − λl Tl − λi (t − (T + ti−1 )) ∗
l= j+1 ∗
< −2λ (t − τ ) + max (2λ − λmin , 0)2T , − λi (t − τ ) < −2λ∗ (t − τ ) + max (2λ∗ − λmin , 0)2T then Eq. (45) can be rewritten as t t ∗ ∗ −λmax (t−τ ) 2 e z (τ )z(τ )dτ ≤ γ e2T (2λ −λmin )−2λ (t−τ ) w (τ )w(τ )dτ 0
(46)
0
Then, integrating t from 0 to ∞, one obtains Eq. (11). The proof is complete.
A2. Proof of Theorem 2 Proof. Construct a Lyapunov function (8) with matrix (12), for t ∈ [T + ti−1 , T + ti ), one i−1 has 0 ≤ t−TT−t ≤ 1, then with Eqs. (14)–(16) and according to Lemma 3, one has Pi (t) > 0. i Moreover, one also has
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Ai (t )Pi (t ) + Pi (t )Ai (t ) + D+ P + λi Pi (t ) ⎛ n ⎞ n j j (t − T − ti−1 ) (t − T − ti−1 ) = sym⎝ Ai, j Pi, j ⎠ j Ti Ti j j=0 j=0 n n j(t − T − ti−1 ) j−1 (t − T − ti−1 ) j + Pi, j−1 + λi Pi, j Ti j Ti j j=1 j=0 = Ai,0 Pi,0 + Pi,0 Ai,0 +
1 Pi,1 + λi Pi,0 Ti
t − T − ti−1 2 Ai,1 Pi,0 + Pi,0 Ai,1 + Ai,0 Pi,1 + Pi,1 Ai,0 + Pi,2 + λi Pi,1 + Ti Ti (t − T − ti−1 )2 Ai,0 Pi,2 + Pi,2 Ai,0 + Ai,1 Pi,1 + Pi,1 Ai,1 + Ai,2 Pi,0 + Pi,2 Ai,0 Ti 2 3 + Pi,3 + λi Pi,2 + · · · Ti ⎛ ⎞ n−1 (t − T − ti−1 )n−1 ⎝ n + (Ai, j Pi,n−1− j + Pi,n−1− j Ai, j ) + Pi,n + λi Pi,n−1 ⎠ Ti Ti n−1 j=0 +
+
n (t − T − ti−1 )n Ai,q Pi,n−q + Pi,n−q Ai,q + · · · n Ti q=0
+
(t − T − ti−1 )2n Ai,n Pi,n + Pi,n Ai,n . Ti 2n t − T − ti−1 Pi,0 Bwi,1 + Pi,1 Bwi,0 Ti (t − T − ti−1 )2 + Pi,0 Bwi,2 + Pi,1 Bwi,1 + Pi,2 Bwi,0 + · · · 2 Ti (t − T − ti )2n Pi,n Bwi,n + 2n Ti
Pi Bwi (t ) = Pi,0 Bwi,0 +
Then, one has ⎡ Ai (t )Pi (t ) + Pi (t )Ai (t ) + D + Pi (t ) + λi Pi (t ) ⎣ ∗ i (t ) = ∗ = i,0 +
Pi (t )Bwi (t ) −γ 2 I ∗
t − T − ti−1 (t − T − ti )2n i,1 + · · · + i,2n . Ti Ti 2n
(47)
⎤ Ci 0⎦ −I (48)
i−1 Since 0 ≤ t−TT−t ≤ 1, with Eq. (17) and according to Lemma 2, one obtains i (t) < 0. i Then, combining with Eq. (18), according to Theorem 1, one could conclude that system (1) is λ∗ -exponentially stable and satisfies Eq. (11). The proof is complete.
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A3. Proof of Theorem 3 Proof. Construct a Lyapunov function V (t ) = x (t )W −1 (t )x(t ) = x (t )Z (t )x(t ) and define Z (t ) = Zi (t ) for t ∈ [T + ti−1 , T + ti ), with the formulation of W(t) given as in Eqs. (29), (30) and (36), one has W(t) is continuous. With Eq. (28), and according to Lemma 3, one has that W(t) > 0, then one could conclude Z(t) > 0 and it is continuous. Similar to the proof of Theorem 2, for t ∈ [t − T − ti−1 , t − T − ti ), one has Wi (t )Ai (t ) + Ai (t )Wi (t ) + Qi (t )Bi (t ) + Bi (t )Qi (t ) − D +Wi (t ) + λiWi (t ) 1 = Wi,0 Ai,0 + Ai,0Wi,0 + Bi,0 Qi,0 + Qi, 0 Bi, 0 − Wi,1 + λiWi,0 + · · · Ti t − T − ti−1 + Wi,0 Ai,1 + Ai,1Wi,0 + Wi,1 Ai,0 + Ai,0Wi,1 + Qi, 0 Bi, 1 + Bi,1 Qi,0 Ti 2 + Qi,1 Bi,0 + Bi,0 Qi,1 − Wi,2 + λiWi,1 + · · · Ti 2n (t − T − ti−1 ) + Wi,n Ai,n + Ai,nWi,n + Qi,n Bi,n + Bi,n Qi,n . 2n Ti and (t − T − ti−1 ) (t − T − ti−1 )n Bwi,1 + · · · + Bwi,n . Ti Ti n (t − T − ti−1 ) (t − T − ti−1 )n CiWi (t ) = CiWi,0 + CiWi,1 + · · · + CiWi,n . Ti Ti n Bwi (t ) = Bwi,0 + +
Substitute them in the following matrix, then one could obtain that ⎡ Wi (t )Ai (t ) + Ai (t )Wi (t ) + Qi (t )Bi (t ) + Bi (t )Qi (t ) − D +Wi (t ) + λiWi (t ) ⎣ ∗ ∗ = i,0 +
Since 0 ≤
t−T −ti−1 Ti
(t−T −ti−1 ) i,1 Ti
+ ··· +
(49)
Bwi (t ) −γ 2 I ∗
(t−T −ti−1 )2n i,2n Ti 2n
⎤ Wi (t )Ci 0 ⎦ −I
(50)
≤ 1, with Eq. (31) and according to Lemma 2, one has
⎡ Wi (t )Ai (t ) + Ai (t )Wi (t ) + Qi (t )Bi (t ) + Bi (t )Qi (t ) − D +Wi (t ) + λiWi (t ) ⎣ ∗ ∗
Bwi (t ) −γ 2 I ∗
⎤ Wi (t )Ci 0 ⎦ < 0, −I
(51) then left-multiply and right-multiply diag{Zi (t), I, I} of Eq. (51), one has ⎡
Aci (t )Wi−1 (t ) + Wi−1 (t )Aci (t ) − Wi−1 (t )D +Wi (t )Wi−1 (t ) + λiWi−1 (t ) ⎣ ∗ ∗
Wi−1 (t )Bwi (t ) −γ 2 I ∗
⎤ Ci 0 ⎦ < 0, −I
(52)
Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008
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Since D +Wi−1 (t ) = −W −1 (t )D +Wi (t )Wi−1 (t ), then Eq. (52) equals to ⎡ ⎤ Aci (t )Zi (t ) + Zi (t )Aci (t ) + D + Zi (t ) + λi Zi (t ) Zi (t )Bwi (t ) Ci ⎣ ∗ −γ 2 I 0 ⎦ < 0, ∗ ∗ −I
(53)
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Please cite this article as: P. Li, P. Li and Y. Liu et al., H∞ control of periodic piecewise polynomial time-varying systems with polynomial Lyapunov function, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin. 2019.06.008