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Finite-time H∞ control for nonlinear discrete Hamiltonian descriptor systems
Accepted Manuscript
Finite-time H control for nonlinear discrete Hamiltonian descriptor systems Xiaodong Lu, Xianfu Zhang, Liying Sun PII: DOI: Reference:
S0016-0032(17)30327-7 10.1016/j.jfranklin.2017.07.013 FI 3052
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
15 December 2016 3 May 2017 2 July 2017
Please cite this article as: Xiaodong Lu, Xianfu Zhang, Liying Sun, Finite-time H control for nonlinear discrete Hamiltonian descriptor systems, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.07.013
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Finite-time H∞ control for nonlinear discrete Hamiltonian descriptor systems✩ Xiaodong Lua , Xianfu Zhanga,∗, Liying Sunb School of Control Science and Engineering, Shandong University, Jinan 250061, P.R. China b School of Mathematical Sciences, University of Jinan, Jinan 250022, P.R. China
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a
Abstract
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This paper studies the finite-time H∞ control problem for the nonlinear discrete Hamiltonian descriptor system (NDHDS). Firstly, based on a proper state-feedback controller, a strict dissipative system, which is equivalent to the NDHDS, is obtained. Then, two sufficient conditions are derived for finite-time boundedness and finite-time H∞ boundedness of the strict dissipative system, and the obtained results are applied to finite-time H∞ control problem for the NDHDS. Finally, a simulation example is given to illustrate the effectiveness of our results.
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Keywords: Nonlinear descriptor system, Discrete Hamiltonian system, Finite-time boundedness, H∞ control. 1. Introduction
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Due to the wide applications in some practical fields, many special nonlinear systems have been studied in the past decades, such as descriptor systems [1, 2, 3, 4, 5], Hamiltonian systems [6, 7] and fuzzy systems [8, 9, 10]. Descriptor systems appear in many areas, such as engineering systems, social economic systems, network systems, and so on [1]. In the past decades, the analysis and control design for discrete descriptor systems have received considerable attention, and a lot of significant results have been obtained. A novel fault estimation filter design method without the constant fault assumption was proposed for linear discrete descriptor systems in [2]. Feng [3] presented a less conservative and numerically tractable solution to the positive real control problem for affinely parameter dependent discrete descriptor systems. The problem of common input/output triangular decoupling for multi model descriptor linear systems was solved in [4] with the state-feedback control method. A robust state and fault observer was designed in [5] for discrete switched nonlinear descriptor systems. ∗
Corresponding author Email addresses: [email protected] (Xiaodong Lu), [email protected] (Xianfu Zhang), [email protected] (Liying Sun) Preprint submitted to Journal of The Franklin Institute
July 14, 2017
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In recent years, the problem of finite-time stability for nonlinear systems have received considerable attention, and a lot of results have been obtained [12, 11, 13]. The concept of finite-time boundedness, which can be viewed as an extension of finite-time stability for systems with external constant disturbances, was introduced in [14]. Since then, a great deal of significant results about finite-time boundedness have been derived. A new condition for finite-time boundedness of linear systems was presented in [15]. In [16], the authors investigated finite-time boundedness and dissipativity analysis for a class of networked cascade control systems. In [17], finite-time boundedness and finite-time stability of switched systems with sector bounded nonlinearity and constant time delay were investigated.
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Although there have been some results about discrete descriptor systems, to the best of our knowledge, few results are on finite-time boundedness and finite-time H∞ boundedness for the NDHDS. As is mentioned in [18], any nonlinear system has an orthogonal decomposition Hamiltonian realization. Different from the results about continuous descriptor Hamiltonian system in [6], in this paper, we take advantage of this strict dissipative Hamiltonian realization and make use of the energy-based approach to study finite-time boundedness and finite-time H∞ boundedness for the NDHDS, which have never been studied in the existing literature. Compared with the results in [6], the singular matrix here is decomposed into a more usual form, which leads to the equivalent strict dissipative form of the NDHDS being more general.
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The paper is organized as follows. In Section 2, we recall some preliminary results and present an equivalent strict dissipative discrete Hamiltonian descriptor form of the NDHDS. Two sufficient conditions are given for finite-time boundedness and finite-time H∞ boundedness of the strict dissipative system and the NDHDS in Section 3. In Section 4, we give a simulation result, which is followed by a brief conclusion in Section 5. Notations. In the sequel, the following notations will be used: xk = x(k), uk = u(k), ωk = ω(k), Ak = A(x(k)), Bk = B(x(k)), Ck = C(x(k)); in symmetric block matrices, we use an asterisk (?) to represent a term that is induced by symmetry; In denotes the n × n dimension identity matrix; C represents the set of all complex numbers; N+ represents the set of all positive integer numbers; rank(A) represents the rank of the matrix A; λmin (A) and λmax (A) represent the minimum and maximum eigenvalues of the matrix A. 2. Preliminaries First, we recall some definitions and lemmas. Definition 2.1. ([7]) Let H(y) be a differential scalar function of y. ∇H(yk+1 , yk ) is a discrete gradient of H if it is continuous in y and T
∇ H(yk+1 , yk )[yk+1 − yk ] = H(yk+1 ) − H(yk ). 2
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For convenience, we use ∇Hk instead of ∇H(yk+1 , yk ) in the sequel. Definition 2.2. The following nonlinear discrete descriptor system Eyk+1 = fk + pk ωk
(2.1)
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is said to be finite-time bounded with respect to (δy , δω , ε, N, R) if for any k ∈ {1, . . . , N }, the following condition holds: y0T E T REy0 ≤ δy2 , =⇒ ykT E T REyk < ε2 , ω0T ω0 ≤ δω2 , where fk = f (y(k)), pk = p(y(k)), matrix E may be singular with rank(E) = r ≤ n, R is a positive definite matrix, 0 ≤ δy < ε, δω ≥ 0 and N ∈ N+ . Definition 2.3. The following descriptor system
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Eyk+1 = fk + pk ωk , zk = gk
(2.2a) (2.2b)
is said to be finite-time H∞ bounded with respect to (δy , δω , ε, N, R, γ) if the following two conditions hold: 1. The system (2.2a) is finite-time bounded with respect to (δy , δω , ε, N, R). 2. The system (2.2) has an L2 gain less than or equal to γ from ωk to zk , i.e.
k=0
N X
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N X
k zk k2 ≤ γ 2
k=0
k ωk k2 ,
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where zk is the penalty signal, gk = g(y(k)) and γ > 0 is a disturbance attenuation level.
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Remark 2.4. When E = I, from the definitions 2.2 and 2.3, one can get the definitions of finite-time boundedness and finite-time H∞ boundedness for the standard nondescriptor systems.
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Definition 2.5. ([19]) Consider the following system Eyk+1 = f (yk ) + g(yk )ωk .
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The system is causal (impulsive-free) at each y0 if deg(det(zE − Ξ)) = rank(E), ∀z ∈ C, ∂f where Ξ = (y0 ). ∂y According to the definition of index 1 of the differential-algebraic equations in [20], we give the definition of index 1 of the difference-algebraic equations as follows: Definition 2.6. Consider the following difference-algebraic equation: ∆yk1 = f (yk1 , yk2 , ωk ), 0 = g(yk1 , yk2 , ωk ), if
∂g is nonsingular, then we say that the system is of index 1,where ωk is the disturbance. ∂yk2 3
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Lemma 2.7. (Bounded real lemma [21, 22]) Suppose that a nonlinear discrete system of the form yk+1 = f (yk , ωk ), zk = h(yk , ωk )
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has y = 0 as a locally asymptotically stable equilibrium for ωk = 0. Then the system has an L2 gain less than or equal to γ (> 0) if there exists a nonnegative storage function V , 1 such that V (yk+1 ) − V (yk ) ≤ (γ 2 k ωk k2 − k zk k2 ). 2 Lemma 2.8. ([18]) If J(y) ∈ Rn×n is a skew-symmetric matrix and R(y) ∈ Rn×n is a positive (or negative) definite matrix, then J(y) − R(y) is invertible.
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Next, we study an equivalent strict dissipative transformation of the following NDHDS. Consider the following NDHDS ( E∆xk = Ak ∇Hk + Bk uk + Ck ωk , ωk+1 = F ωk ,
(2.3)
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where xk ∈ Rn is the system state; uk ∈ Rm is the control input; ωk ∈ Rs is the external disturbance; rank(E) = r ≤ n; Ak ∈ Rn×n , Bk ∈ Rn×m , Ck ∈ Rn×s ; Hk = H(xk ) is the k discrete Hamiltonian function with x = 0 as its minimum point, ∇Hk = ∂H , F ∈ Rs×s . ∂xk
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Let M0 ∈ Rn×n , S ∈ Rn×n be two nonsingular matrices, such that ! Ir 0 E = M0 S. 0 0
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Then we can obtain an equivalent form of (2.3) as following: ! Ir 0 S∆xk = M Ak ∇Hk + M Bk uk + M Ck ωk , 0 0 where M = M0−1 .
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! x e1k e xk ) = H e k = Hk , where x Denote x ek = := Sxk , H(e e1k ∈ Rr , x e2k ∈ Rn−r . Then x e2k ek ∂Hk ∂H e k , and ∇Hk = =S = S∇H ∂xk ∂e xk ! ∆e x1k e k + M Bk uk + M Ck ωk . (2.4) = M Ak S∇H 0 In order to obtain an equivalent strict dissipative transformation of the system (2.3), we need to design a state-feedback controller. We have the following result. 4
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Proposition 2.9. Suppose there exists a symmetric matrix L ∈ Rm×m such that M Bk LBkT M T −
1 M Ak S + S T ATk M T > 0, 2
under the state-feedback controller
e k + vk , uk = −LBkT M T ∇H
(2.6)
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then the system (2.3) can be transformed into the following strict dissipative form 1 xk ∆e e k + M Bk vk + M Ck ωk , = (Jk − Rk )∇H 0 ωk+1 = F ωk ,
(2.5)
(2.7)
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where Jk ∈ Rn×n is a skew-symmetric matrix, Rk ∈ Rn×n is a positive definite matrix, and vk ∈ Rm is a new reference input. Proof: Substituting (2.7) into (2.4), we can derive (2.6), where
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1 Rk = M Bk LBkT M T − [M Ak S + S T ATk M T ], 2 1 Jk = [M Ak S − S T ATk M T ]. 2
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Obviously, Jk is skew-symmetric and Rk is positive definite. ! ! ! Mr×n J11 J12 R11 R12 Denote M = , Jk = , Rk = . Then J11 , J22 are T T M(n−r)×n −J12 J22 R12 R22 skew-symmetric matrices, and R11 , R22 are positive definite matrices. By Lemma 2.8, we know that J22 − R22 is invertible. Then the system (2.6) can be transformed into the following form: ek ek ∂H ∂H 1 ∆e x = (J + (J − R ) + Mr×n Bk vk + Mr×n Ck ωk , 11 − R11 ) 12 12 k ∂e x1k ∂e x2k e ek ∂H (2.8) T ∂ Hk 0 = −(J + R ) + (J − R ) + M(n−r)×n Bk vk + M(n−r)×n Ck ωk , 12 12 22 22 1 2 ∂e xk ∂e xk ωk+1 = F ωk . Since J22 − R22 is invertible, we have e 1 ek − R ek ) ∂ Hk + B ek vk + C ek ωk , ∆e x = ( J k 1 ∂e x k e ek ∂H (2.9) T ∂ Hk 0 = −(J + R ) + (J − R ) + M(n−r)×n Bk vk + M(n−r)×n Ck ωk , 12 12 22 22 1 2 ∂e xk ∂e xk ωk+1 = F ωk ,
where
ek = J11 − R11 + (J12 − R12 )(J22 − R22 )−1 (J12 + R12 )T , Jek − R 5
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ek = Mr×n Bk − (J12 − R12 )(J22 − R22 )−1 M(n−r)×n Bk , B ek = Mr×n Ck − (J12 − R12 )(J22 − R22 )−1 M(n−r)×n Ck . C
ek is positive definite. Similar to the description in [6], Jek is skew-symmetric and R
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Remark 2.10. From (2.6), (2.8) and (2.9) we know that system (2.3) is transformed into a strict dissipative Hamiltonian form, thus we can take the advantages of the Hamiltonian function being a good Lyapunov function candidate and the strict dissipative Hamiltonian realization to study finite-time H∞ problems of system (2.3). In order to study finite-time boundedness and finite-time H∞ boundedness of the system (2.9), the following assumption is needed.
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Assumption 2.11. For any xk = S −1 x ek , x e1k 6= 0, the following condition holds: n ∂ T Hk −T −1 S M ES M ES −1 + M Ak S − S T ATk M T − M Bk LBkT M T ∂xk o−1 1 + (M Ak S + S T ATk M T ) (I − M ES −1 ) M Bk 6= 0. 2
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According to the Assumption 2.11, we have the following lemma.
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Lemma 2.12. Consider the system (2.3) and its equivalent form (2.9). Assume that the system (2.3) is of index 1 and causal. If the Assumption 2.11 holds, then there exist a reference input vk and a constant α, such that
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T e ek ∂ H ek ek ∂T H ∂T H 2 ek vk + ∂ Hk ∆e B x + α ≤ 0. k 1 2 1 ∂e xk ∂e xk ∂e xk ∂e x1k
Proof: From the Assumption 2.11, we have
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n ∂ T Hk −T S M ES −1 M ES −1 + M Ak S − S T ATk M T − M Bk LBkT M T ∂xk o−1 1 + (M Ak S + S T ATk M T ) (I − M ES −1 ) M Bk 2 o e k Ir 0 n Ir 0 −1 ∂T H 0 0 Mr×n Bk = + [Jk − Rk ] 00 00 0 In−r M(n−r)×n Bk ∂e xk T −1 ek Ir J12 − R12 Mr×n Bk ∂ H = 0 1 0 J22 − R22 M(n−r)×n Bk ∂e xk =
ek ∂T H ek 6= 0. B ∂e x1k
The rest of proof is similar to the Lemma 3 in [6], so it is omitted here.
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(2.10)
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3. Main results In this section, according to the equivalent transformation in Section 2, we study finite-time boundedness and finite-time H∞ boundedness of the system (2.9). 3.1. Finite-time boundedness of the NDHDS
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In the following theorem, we give a sufficient condition for finite-time boundedness of the system (2.9).
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Theorem 3.1. Assume that the system (2.3) is of index 1 and causal, and the Assumption 2.11 holds. If there exist two constants β (> 0), α1 and two positive definite matrices P , Q, such that following conditions hold: ek ek + 1 C ΛP Ω1 ΛP C 2 (3.1) ? eT P < 0, Ω2 C k ? ? − βP and
i h e 0 + λmax (Pe)δx2 + λmax (Q)δω2 (β + 1)N H
M
λmin (Pe)
< ε2 ,
(3.2)
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then the system (2.9) is finite-time boundedness with respect to (δx , δω , ε, N, R) under the state-feedback controller e ekT ∂ Hk , vk = −B (3.3) ∂e x1k ek − α1 I, Ω1 = ΛP ΛT + Jek − R ekT P C ek + F T QF − (β + 1)Q, Ω2 = C
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where
e k )T − B ek B ekT , Λ = (Jek − R
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1 1 and Pe = R− 2 P R− 2 , R is a positive definite matrix.
Proof: Let
e k + (e Vk = H x1k )T P (e x1k ) + ωkT Qωk .
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Then, for all k = 1, 2, . . . , N , we have, T e k+1 − H e k + (e Qωk+1 − ωkT Qωk x1k ) + ωk+1 x1k )T P (e x1k+1 ) − (e Vk+1 − Vk = H x1k+1 )T P (e T e T e T e ek e ∂T H ek − R ek ) ∂ Hk + ∂ Hk B ek vk + ∂ Hk C ek ωk + ∂ Hk ∆e ( J x2k ∂e x1k ∂e x1k ∂e x1k ∂e x1k ∂e x2k T e e ek ∂T H ek − R ek )T P (Jek − R ek ) ∂ Hk + ∂ Hk (Jek − R ek )T P B ek vk ( J + ∂e x1k ∂e x1k ∂e x1k T e ek ∂T H ek − R ek )T P C ek ωk + ∂ Hk (Jek − R ek )T P x + ( J e1k ∂e x1k ∂e x1k e T eT eT e e T P (Jek − R ek ) ∂ Hk + v T B e +vkT B k k P Bk vk + vk Bk P Ck ωk k ∂e x1k e ekT P B ek vk eT P x eT P (Jek − R ek ) ∂ Hk + ωkT C +vkT B e1k + ωkT C k k ∂e x1k e ek ) ∂ Hk eT e1 + (e ek ωk + ω T C eT P C x1k )T P (Jek − R +ωkT C k k k Px k ∂e x1k ek vk + (e ek ωk + ω T (F T QF − Q)ωk . +(e x1 )T P B x1 ) T P C k
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=
k
k
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By Lemma 2.12, we know that for the reference input (3.3), there exists a constant α1 , such that T e ek ek ∂ H ek ∂T H ∂T H 2 ek vk + ∂ Hk ∆e B x + α ≤ 0. 1 k ∂e x1k ∂e x2k ∂e x1k ∂e x1k
Then, by the controller (3.3), we have,
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i e T e ek h ∂T H ek − R ek )T P (Jek − R ek ) + (Jek − R ek ) − α1 I ∂ Hk + ∂ Hk C ek ωk ( J ∂e x1k ∂e x1k ∂e x1k i T e h ek e ∂T H ek − R ek )T P B ek B ekT ∂ Hk + 2 ∂ Hk (Jek − R ek )T − B ek B ekT P C ek ωk −2 ( J ∂e x1k ∂e x1k ∂e x1k i ek h ∂T H ek − R ek )T − B ek B eT P x eT P C ek + F T QF − Q)ωk +2 ( J e1k + ωkT (C k k ∂e x1k ek e ∂T H 1 eT P x ek B eT P B ek B e T ∂ Hk . +2ωkT C e + B k k k k ∂e x1k ∂e x1k
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Vk+1 − Vk ≤
By (3.1), we have, and hence,
Vk+1 − Vk ≤ βωkT Qωk + β(e x1k )T P x e1k < βVk , Vk < (β + 1)Vk−1 =⇒ Vk < (β + 1)N V0 .
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x10 ) ≤ δx2 , and ω0T ω0 ≤ δω2 , then we have, If (e x10 )T R(e
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λmin (Pe)(e x1k )T R(e x1k ) ≤ (e x1k )T P x e1k < Vk < (β + 1)N V0 i h e 0 + (e e10 + ω0T Qω0 = (β + 1)N H x10 )T P x h i e 0 + λmax (Pe)(e ≤ (β + 1)N H x10 )T R(e x10 ) + ω0T Qω0 h i N e 2 2 e ≤ (β + 1) H0 + λmax (P )δx + λmax (Q)δω .
By (3.2), we know, (e x1k )T Re x1k < ε2 , which completes the proof. 3.2. Finite-time H∞ boundedness of the NDHDS
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In this subsection, we consider the following NDHDS: E∆xk = Ak ∇Hk + Bk uk + Ck ωk , ωk+1 = F ωk , zk = ΓM ES −2 ∇Hk ,
(3.4)
where E, Ak , Bk , Ck , F , M , S are the same as those in the system (2.3); zk ∈ Rs is the penalty signal, Γ = CkT ΣT 0s×(n−r) ,
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Σ = Mr×n − (J12 − R12 )(J22 − R22 )−1 M(n−r)×n .
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Similar to the analysis in Section 2, the system (3.4) can be transformed into the e k + vk , following form under the controller uk = −LBkT M T ∇H e 1 ek − R ek ) ∂ Hk + B ek vk + C ek ωk , = ( J ∆e x k ∂e x1k ωk+1 = F ωk , (3.5) e eT ∂ Hk . zk = C k ∂e x1k
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ek = ΣCk . It’s easy to show that C Next, a sufficient condition is derived to guarantee that the system (3.5) is finite-time H∞ bounded in the following theorem.
Theorem 3.2. Suppose the Assumption 2.11 holds. Assume that the system (2.3) is of index 1 and causal, and has xk = 0 as a locally asymptotically stable equilibrium for ωk = 0. If there exist two constants β (> 0), α2 and two positive definite matrices P , Q, such that (3.2) and following conditions hold:
ek + 1 C ek Ω3 ΛP C 2 (a) : ? Ω2 ? ? 9
ΛP eT P < 0; C k − βP
(3.6)
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ek − Jek + α2 I − 1 C ek C ekT is invertible, R 2 −1 2 1 eT e 1 e eT T e ek − Q < γ I, F QF + Ck Rk − Jk + α2 I − Ck Ck C 4 2 2
(b) : and
(3.7)
e e T ∂ Hk , vk = −B k ∂e x1k
where
ek − α2 I, Ω3 = ΛP ΛT + Jek − R
and Λ, Ω2 are the same as those in Theorem 3.1.
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then the system (3.5) is finite-time H∞ boundedness with respect to (δx , δω , ε, N, R, γ) under the state-feedback controller (3.8)
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Proof: First, we prove that the system (3.5) has an L2 gain less than or equal to γ from ωk to zk . By Lemma 2.12, we know that for the reference input (3.8), there exists a constant α2 , such that T e ek ek ∂ H ek ∂T H ∂T H 2 ek vk + ∂ Hk ∆e B x + α ≤ 0. 2 k ∂e x1k ∂e x2k ∂e x1k ∂e x1k
(3.9)
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e k + ω T Qωk . Then, for all k = 1, 2, . . . , N , we have, Let Vk1 = H k 1 e k+1 − H e k + ω T Qωk+1 − ω T Qωk Vk+1 − Vk1 = H k+1 k
T e T e T e ek e ∂T H ek ωk + ∂ Hk B ek vk + ∂ Hk ∆e ek − R ek ) ∂ Hk + ∂ Hk C ( J x2k 1 1 1 1 2 ∂e xk ∂e xk ∂e xk ∂e xk ∂e xk
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=
+ωkT (F T QF − Q)ωk
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T e e ek ∂T H ek − R ek − α2 I) ∂ Hk + ∂ Hk C ek ωk + ωkT (F T QF − Q)ωk ( J ∂e x1k ∂e x1k ∂e x1k ek e 1 1 ∂T H ek C eT ∂ Hk + 1 γ 2 k ωk k2 − 1 k zk k2 − γ 2 ωkT ωk + C k 1 2 2 ∂e xk ∂e x1k 2 2
e 1 e eT 1 ∂ Hk
e e =− R k − Jk − Ck Ck + α2 I ] 2
2 ∂e x1k
2 − 21
1 e 1 ek C e T + α2 I ek ωk − [Rk − Jek − C C
k
2 2 ) ( −1 1 1 1 ek − Q − γ 2 I ωk eT R ek − Jek + α2 I − C ek C ekT +ωkT F T QF + C C 4 k 2 2
≤
1 1 + γ 2 k ωk k2 − k zk k2 . 2 2
By (3.7), we have,
1 1 1 Vk+1 − Vk1 < γ 2 k ωk k2 − k zk k2 . 2 2 10
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From Lemma 2.7, we know that the system (3.5) has an L2 gain less than or equal to γ from ωk to zk . Next, we prove that the system (3.5) is finite-time bounded. The proof is similar to that in the Theorem 3.1, so it is omitted here.
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Remark 3.3. From the foregoing analysis, we know that if (2.5) holds, then the system (2.3) is equivalent to the system (2.9) under the state-feedback controller (2.7) , and it is also true for the systems (3.4) and (3.5). Therefore, Theorem 3.1 and Theorem 3.2 can also guarantee that the systems (2.3) and (3.4) achieve finite-time boundedness and finite-time H∞ boundedness. In fact, for a given ε, there always exists a positive constant δ > 0 such that T RSr×n xk < ε2 ⇐⇒ (xk )T R∗ xk < ε2 , (e x1k )T Re x1k < ε2 ⇐⇒ (xk )T Sr×n
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T where R∗ = Sr×n RSr×n + δI > 0.
In the sequel, from Theorem 3.2, we can derive a finite-time H∞ boundedness criteria for the system (3.4) as follows:
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Corollary 3.4. Suppose the Assumption 2.11 holds. Assume that the system (2.3) is of index 1 and causal, and has xk = 0 as a locally asymptotically stable equilibrium for ωk = 0. If there exist two constants β (> 0), α2 , two positive definite matrices P , Q, and a symmetric matrix L ∈ Rm×m such that the following conditions hold:
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Ω4 Θ1
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(a0 ) : ?
where
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Ω4 = Λ5 P
∗
ΛT5
?
Θ2 ?
+ Λ2 − α2 I,
∗ Λ5 P Ir 0 ΛT3 P∗ < 0, 00 − βP ∗
Ir 0r×(n−r) P
∗
Ir 0(n−r)×r
= P,
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1 Ir 0 ∗ Θ1 = Λ5 P + I Λ3 , 00 2 Ir 0 I 0 Θ2 = ΛT3 P∗ r Λ3 + F T QF − (β + 1)Q, 00 00 −1 0r×(n−r) 0r×(n−r) 0(n−r)×r In−r (Jk − Rk ) × 0(n−r)×r In−r , Λ1 = In−r In−r Λ2 = Jk − Rk − (Jk − Rk )Λ1 (Jk − Rk ), Λ4 = M Bk − (Jk − Rk )Λ1 M Bk ,
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Λ3 = M Ck − (Jk − Rk )Λ1 M Ck , Ir 0 T T Λ5 = Λ2 − Λ4 Λ4 ; 00
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(b0 ): 1 F QF + ΛT3 4 T
n
Ir
o −1 1 Ir T Ir 0r×(n−r) α2 I − Λ2 − Λ3 Λ3 0(n−r)×r 2
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0(n−r)×r γ2 × Ir 0r×(n−r) Λ3 − Q < I; 2 n h io 1 (c0 ) : (β + 1)N H0 + λmax (Pe∗ )δx2 + λmax (Q)δω2 < ε2 , ∗ e λmin (P )
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then the system (3.4) is finite-time H∞ boundedness with respect to (δx , δω , ε, N, R, γ) under the state-feedback controller Ir 0 T T −1 T −1 uk = − LBk M S + Λ4 S ∇Hk , 00 o n ∗ =diag Pe, λmin (Pe)In−r , Pe is the same as that in Theorem 3.1. where Pen×n 4. An example
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∆xk = Ak ∇Hk + Bk uk + Ck ωk , ωk+1 = F ωk ,
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∇Hk ,
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(4.1a) (4.1b) (4.1c)
1 1 as x(0) = (1, 1, 0)T and ω(0) = ( , )T , where n = 3, s = m = r = 2 2 1 1 2 1 2 2 (x ) + (xk ) + (x3k )2 ; 4 k 4
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with initial conditions 1 xk 2, xk = x2k , Hk = x3k
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In this section, we give a simulation example to illustrate the results obtained in this paper. Consider the following NDHDS:
− sin x1k − 1 0 0 , 0 − cos x2k − 1 0 Ak = 3 2 0 0 −(xk ) − 1 1 0 1 0 0 , Bk = 0 1 , Ck = 0 1 2 0 0 xk − xk x3k
1 F = diag(1, ), M = diag(1, 1, 1), S = diag(1, 1, 2). 2 The corresponding parameters and matrices are given as follows: 12
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where
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3 N = 5, α2 = β = γ = δ = 1, δx = δω = , ε = 13, 4 1 1 1 1 ∗ e P = diag( , 1, ), Q = diag(1, 1), R = diag( , ). By some simple calculation, we can 3 3 3 9 verify that (b0 ) and (c0 ) hold. And (a0 ) can be transformed into the following form: Φ1 Φ2 Φ3 ? Φ4 Φ5 < 0, ? ? Φ6
1 2 1 1 1 0 0 9 sin xk − 3 sin xk − 2 1 1 Φ1 = , 2 2 2 0 cos x − cos x − 2 0 k k 9 3 0 0 −1 1 1 1 1 1 1 − 9 sin xk + 6 0 − 9 sin xk − 3 0 0 0 Φ2 = , 1 (x1k − x2k ) 0 0 2 8 1 0 0 0 −9 9 1 7 1 2 Φ3 = − cos xk − 0 , Φ4 = ? − 0 , 9 3 4 1 0 0 ? ? 9 1 Φ5 = (0)3×2 , Φ6 = diag(− , −1). 9 1 xk By designing the controller uk = − , (4.1a) can be transformed into Exk+1 = x2k 1 0 −0.5x1k (sin x1k + 1) (g1 )2×2 0 . = 0 f (xk ) + g(xk )ωk , where f = −0.5x2k (cos x2k + 1) , g := (g2 )1×2 3 2 2 1 2 −2xk ((xk ) + 1)) xk − xk x3k ∂[g2 wk − 2x3k ((x2k ) + 1)] It is easy to check that deg(det(zE − Ξ)) = 2 = rank(E) and is ∂x3k nonsingular, so the system (4.1a) is of index 1 and causal. From Corollary 3.4, we know that the system (4.1) can achieve finite-time H∞ boundedness. Figures 1 and 2 show the simulation results of the closed-loop system of (4.1) and the response of the control signal, respectively. 1 1 By some simple calculations, we have R∗ = diag( , , 1). From figure 1 we can 3 9 9 T ∗ and see that xk R xk < 169 (k = 1, 2, 3, 4, 5) under the initial conditions xT0 R∗ x0 ≤ 16 9 ω0T ω0 ≤ . 16
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Sample k
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Closed-loop system of (4.1) to be finite-time H∞ boundedness with respect to 3 3 ( , , 13, 5, R, 1). 4 4
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Figure. 1
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xTk R*xk
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Control u
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Figure. 2
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5. Conclusion
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In this paper, by an appropriate state-feedback controller, we have shown that the nonlinear discrete Hamiltonian descriptor system can be transformed into an equivalent strict dissipative system. Then, the problems of finite-time boundedness and finite-time H∞ boundedness for this strict dissipative system have been studied. Finally, the results obtained have been extended to the finite-time control problem of the NDHDS. Future works will focus on stochastic descriptor systems, stochastic Hamiltonian systems and so on. 6. Acknowledgments This work is supported by the National Natural Science Foundations of China (61374065, 61573215, 6150021593, 61374002) and the Natural Science Fund for Distinguished Young 14
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Scholars of Shandong Province under grant JQ201613. References [1] L. Dai, Singular control systems. Berlin, Germany: Springer-Verlag, 1989.
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[2] Z. Wang, M. Rodrigues, D. Theilliol, Fault estimation filter design for discrete descriptor systems, IET Control Theory and Applications 9 (10) (2015) 1587-1594. [3] Y. Feng, Positive real control for discrete descriptor systems with affine parameter dependence, Journal of the Franklin Institute 352 (3) (2015) 882-896. [4] F. N. Koumboulis, Common I/O triangular decoupling for multi model descriptor systems, Systems and Control Letters 97 (2016) 108-114.
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[5] D. Koenig, B. Marx, S. Varrier, Filtering and fault estimation of descriptor switched systems, Automatica 63 (2016) 116-121. [6] L. Sun, G. Feng, Y. Wang, Finite-time stabilization and H∞ control for a class of nonlinear Hamiltonian descriptor systems with application to affine nonlinear descriptor systems, Automatica 50 (8) (2014) 2090-2097.
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[7] Y. Yaprak, L. G¨ oren-S¨ umer, A. Astolfi, Some results on disturbance attenuation for Hamiltonian systems via direct discrete design, International Journal of Robust and Nonlinear Control 25 (13) (2015) 1927-1940.
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[8] Y. Wei, J. Qiu, H. Lam, L. Wu, Approaches to TS fuzzy-affine-model-based reliable output feedback control for nonlinear Ito stochastic systems, IEEE Transactions on Fuzzy Systems, http://dx.doi.org/10.1109/TFUZZ.2016.2566810.
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[9] Y. Wei, J. Qiu, P. Shi, M. Chadli, Fixed-order memory piecewise affine output feedback controller for fuzzy-affine-model-based nonlinear systems with time-varying delay, IEEE Transactions on Circuits and Systems I: Regular Papers, doi: 10.1109/TCSI.2016.2632718.
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[10] Y. Wei, J. Qiu, H. R. Karimi, Reliable output feedback control of discrete-time fuzzy affine systems with actuator faults, IEEE Transactions on Circuits and Systems I: Regular Papers, 64 (1) (2017) 170-181. [11] X. Zhang, G. Feng, Y. Sun, Finite-time stabilization by state-feedback control for a class of time-varying nonlinear systems, Automatica 48 (3) (2012) 499-504. [12] X. Zhang, G. Feng, Global finite-time stabilisation of a class of feedforward non-linear systems, IET control theory and applications 5 (12) (2011) 1450-1457. [13] S. Huang, Z. Xiang, Finite-time stabilization of switched stochastic nonlinear systems with mixed odd and even powers, Automatica 73 (2016) 130-137. [14] F. Amato, M. Ariola, P. Dorato, Finite-time control of linear systems subject to parametric uncertainties and disturbances, Automatica 37 (9) (2001) 1459-1463.
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[15] H. T. M. Kussaba, R. A. Borges, & T. Y. Ishihara, A new condition for finite time boundedness analysis, Journal of the Franklin Institute 352 (12) (2015) 5514-5528. [16] K. Mathiyalagan, J. H. Park, & R. Sakthivel, Finite-time boundedness and dissipativity analysis of networked cascade control systems, Nonlinear Dynamics (2016) 1-12.
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[17] X. Lin, X. Li, S. Li, et al, Finite-time boundedness for switched systems with sector bounded nonlinearity and constant time delay, Applied Mathematics and Computation, 274 (2016) 25-40. [18] Y. Wang, C. Li, D. Cheng, Generalized Hamiltonian realization of time-invariant nonlinear systems, Automatica 39 (8) (2003) 1437-1443.
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[19] M. D. S. Aliyu, M. Perrier, H∞ filtering for discrete-time affine non-linear descriptor systems, IET Control Theory and Applications 6 (6) (2012) 787-795. [20] K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical solution of initial-value problems in differential-algebraic equations (Classics in Applied Mathematics Series). Philadelphia, PA: SIAM, 1996. [21] W. Lin, C. I. Byrnes, Discrete nonlinear H∞ control with measurement feedback, Automatica 31 (3) (1995) 419-434.
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[22] W. Lin, C. I. Byrnes, H∞ -control of discrete nonlinear systems, IEEE Transactions on Automatic Control 41 (4) (1996) 494-510.
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Finite-time H control for nonlinear discrete Hamiltonian descriptor systems Xiaodong Lu, Xianfu Zhang, Liying Sun PII: DOI: Reference:
S0016-0032(17)30327-7 10.1016/j.jfranklin.2017.07.013 FI 3052
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
15 December 2016 3 May 2017 2 July 2017
Please cite this article as: Xiaodong Lu, Xianfu Zhang, Liying Sun, Finite-time H control for nonlinear discrete Hamiltonian descriptor systems, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.07.013
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Finite-time H∞ control for nonlinear discrete Hamiltonian descriptor systems✩ Xiaodong Lua , Xianfu Zhanga,∗, Liying Sunb School of Control Science and Engineering, Shandong University, Jinan 250061, P.R. China b School of Mathematical Sciences, University of Jinan, Jinan 250022, P.R. China
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a
Abstract
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This paper studies the finite-time H∞ control problem for the nonlinear discrete Hamiltonian descriptor system (NDHDS). Firstly, based on a proper state-feedback controller, a strict dissipative system, which is equivalent to the NDHDS, is obtained. Then, two sufficient conditions are derived for finite-time boundedness and finite-time H∞ boundedness of the strict dissipative system, and the obtained results are applied to finite-time H∞ control problem for the NDHDS. Finally, a simulation example is given to illustrate the effectiveness of our results.
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Keywords: Nonlinear descriptor system, Discrete Hamiltonian system, Finite-time boundedness, H∞ control. 1. Introduction
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Due to the wide applications in some practical fields, many special nonlinear systems have been studied in the past decades, such as descriptor systems [1, 2, 3, 4, 5], Hamiltonian systems [6, 7] and fuzzy systems [8, 9, 10]. Descriptor systems appear in many areas, such as engineering systems, social economic systems, network systems, and so on [1]. In the past decades, the analysis and control design for discrete descriptor systems have received considerable attention, and a lot of significant results have been obtained. A novel fault estimation filter design method without the constant fault assumption was proposed for linear discrete descriptor systems in [2]. Feng [3] presented a less conservative and numerically tractable solution to the positive real control problem for affinely parameter dependent discrete descriptor systems. The problem of common input/output triangular decoupling for multi model descriptor linear systems was solved in [4] with the state-feedback control method. A robust state and fault observer was designed in [5] for discrete switched nonlinear descriptor systems. ∗
Corresponding author Email addresses: [email protected] (Xiaodong Lu), [email protected] (Xianfu Zhang), [email protected] (Liying Sun) Preprint submitted to Journal of The Franklin Institute
July 14, 2017
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In recent years, the problem of finite-time stability for nonlinear systems have received considerable attention, and a lot of results have been obtained [12, 11, 13]. The concept of finite-time boundedness, which can be viewed as an extension of finite-time stability for systems with external constant disturbances, was introduced in [14]. Since then, a great deal of significant results about finite-time boundedness have been derived. A new condition for finite-time boundedness of linear systems was presented in [15]. In [16], the authors investigated finite-time boundedness and dissipativity analysis for a class of networked cascade control systems. In [17], finite-time boundedness and finite-time stability of switched systems with sector bounded nonlinearity and constant time delay were investigated.
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Although there have been some results about discrete descriptor systems, to the best of our knowledge, few results are on finite-time boundedness and finite-time H∞ boundedness for the NDHDS. As is mentioned in [18], any nonlinear system has an orthogonal decomposition Hamiltonian realization. Different from the results about continuous descriptor Hamiltonian system in [6], in this paper, we take advantage of this strict dissipative Hamiltonian realization and make use of the energy-based approach to study finite-time boundedness and finite-time H∞ boundedness for the NDHDS, which have never been studied in the existing literature. Compared with the results in [6], the singular matrix here is decomposed into a more usual form, which leads to the equivalent strict dissipative form of the NDHDS being more general.
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The paper is organized as follows. In Section 2, we recall some preliminary results and present an equivalent strict dissipative discrete Hamiltonian descriptor form of the NDHDS. Two sufficient conditions are given for finite-time boundedness and finite-time H∞ boundedness of the strict dissipative system and the NDHDS in Section 3. In Section 4, we give a simulation result, which is followed by a brief conclusion in Section 5. Notations. In the sequel, the following notations will be used: xk = x(k), uk = u(k), ωk = ω(k), Ak = A(x(k)), Bk = B(x(k)), Ck = C(x(k)); in symmetric block matrices, we use an asterisk (?) to represent a term that is induced by symmetry; In denotes the n × n dimension identity matrix; C represents the set of all complex numbers; N+ represents the set of all positive integer numbers; rank(A) represents the rank of the matrix A; λmin (A) and λmax (A) represent the minimum and maximum eigenvalues of the matrix A. 2. Preliminaries First, we recall some definitions and lemmas. Definition 2.1. ([7]) Let H(y) be a differential scalar function of y. ∇H(yk+1 , yk ) is a discrete gradient of H if it is continuous in y and T
∇ H(yk+1 , yk )[yk+1 − yk ] = H(yk+1 ) − H(yk ). 2
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For convenience, we use ∇Hk instead of ∇H(yk+1 , yk ) in the sequel. Definition 2.2. The following nonlinear discrete descriptor system Eyk+1 = fk + pk ωk
(2.1)
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is said to be finite-time bounded with respect to (δy , δω , ε, N, R) if for any k ∈ {1, . . . , N }, the following condition holds: y0T E T REy0 ≤ δy2 , =⇒ ykT E T REyk < ε2 , ω0T ω0 ≤ δω2 , where fk = f (y(k)), pk = p(y(k)), matrix E may be singular with rank(E) = r ≤ n, R is a positive definite matrix, 0 ≤ δy < ε, δω ≥ 0 and N ∈ N+ . Definition 2.3. The following descriptor system
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Eyk+1 = fk + pk ωk , zk = gk
(2.2a) (2.2b)
is said to be finite-time H∞ bounded with respect to (δy , δω , ε, N, R, γ) if the following two conditions hold: 1. The system (2.2a) is finite-time bounded with respect to (δy , δω , ε, N, R). 2. The system (2.2) has an L2 gain less than or equal to γ from ωk to zk , i.e.
k=0
N X
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N X
k zk k2 ≤ γ 2
k=0
k ωk k2 ,
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where zk is the penalty signal, gk = g(y(k)) and γ > 0 is a disturbance attenuation level.
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Remark 2.4. When E = I, from the definitions 2.2 and 2.3, one can get the definitions of finite-time boundedness and finite-time H∞ boundedness for the standard nondescriptor systems.
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Definition 2.5. ([19]) Consider the following system Eyk+1 = f (yk ) + g(yk )ωk .
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The system is causal (impulsive-free) at each y0 if deg(det(zE − Ξ)) = rank(E), ∀z ∈ C, ∂f where Ξ = (y0 ). ∂y According to the definition of index 1 of the differential-algebraic equations in [20], we give the definition of index 1 of the difference-algebraic equations as follows: Definition 2.6. Consider the following difference-algebraic equation: ∆yk1 = f (yk1 , yk2 , ωk ), 0 = g(yk1 , yk2 , ωk ), if
∂g is nonsingular, then we say that the system is of index 1,where ωk is the disturbance. ∂yk2 3
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Lemma 2.7. (Bounded real lemma [21, 22]) Suppose that a nonlinear discrete system of the form yk+1 = f (yk , ωk ), zk = h(yk , ωk )
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has y = 0 as a locally asymptotically stable equilibrium for ωk = 0. Then the system has an L2 gain less than or equal to γ (> 0) if there exists a nonnegative storage function V , 1 such that V (yk+1 ) − V (yk ) ≤ (γ 2 k ωk k2 − k zk k2 ). 2 Lemma 2.8. ([18]) If J(y) ∈ Rn×n is a skew-symmetric matrix and R(y) ∈ Rn×n is a positive (or negative) definite matrix, then J(y) − R(y) is invertible.
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Next, we study an equivalent strict dissipative transformation of the following NDHDS. Consider the following NDHDS ( E∆xk = Ak ∇Hk + Bk uk + Ck ωk , ωk+1 = F ωk ,
(2.3)
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where xk ∈ Rn is the system state; uk ∈ Rm is the control input; ωk ∈ Rs is the external disturbance; rank(E) = r ≤ n; Ak ∈ Rn×n , Bk ∈ Rn×m , Ck ∈ Rn×s ; Hk = H(xk ) is the k discrete Hamiltonian function with x = 0 as its minimum point, ∇Hk = ∂H , F ∈ Rs×s . ∂xk
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Let M0 ∈ Rn×n , S ∈ Rn×n be two nonsingular matrices, such that ! Ir 0 E = M0 S. 0 0
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Then we can obtain an equivalent form of (2.3) as following: ! Ir 0 S∆xk = M Ak ∇Hk + M Bk uk + M Ck ωk , 0 0 where M = M0−1 .
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! x e1k e xk ) = H e k = Hk , where x Denote x ek = := Sxk , H(e e1k ∈ Rr , x e2k ∈ Rn−r . Then x e2k ek ∂Hk ∂H e k , and ∇Hk = =S = S∇H ∂xk ∂e xk ! ∆e x1k e k + M Bk uk + M Ck ωk . (2.4) = M Ak S∇H 0 In order to obtain an equivalent strict dissipative transformation of the system (2.3), we need to design a state-feedback controller. We have the following result. 4
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Proposition 2.9. Suppose there exists a symmetric matrix L ∈ Rm×m such that M Bk LBkT M T −
1 M Ak S + S T ATk M T > 0, 2
under the state-feedback controller
e k + vk , uk = −LBkT M T ∇H
(2.6)
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then the system (2.3) can be transformed into the following strict dissipative form 1 xk ∆e e k + M Bk vk + M Ck ωk , = (Jk − Rk )∇H 0 ωk+1 = F ωk ,
(2.5)
(2.7)
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where Jk ∈ Rn×n is a skew-symmetric matrix, Rk ∈ Rn×n is a positive definite matrix, and vk ∈ Rm is a new reference input. Proof: Substituting (2.7) into (2.4), we can derive (2.6), where
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1 Rk = M Bk LBkT M T − [M Ak S + S T ATk M T ], 2 1 Jk = [M Ak S − S T ATk M T ]. 2
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Obviously, Jk is skew-symmetric and Rk is positive definite. ! ! ! Mr×n J11 J12 R11 R12 Denote M = , Jk = , Rk = . Then J11 , J22 are T T M(n−r)×n −J12 J22 R12 R22 skew-symmetric matrices, and R11 , R22 are positive definite matrices. By Lemma 2.8, we know that J22 − R22 is invertible. Then the system (2.6) can be transformed into the following form: ek ek ∂H ∂H 1 ∆e x = (J + (J − R ) + Mr×n Bk vk + Mr×n Ck ωk , 11 − R11 ) 12 12 k ∂e x1k ∂e x2k e ek ∂H (2.8) T ∂ Hk 0 = −(J + R ) + (J − R ) + M(n−r)×n Bk vk + M(n−r)×n Ck ωk , 12 12 22 22 1 2 ∂e xk ∂e xk ωk+1 = F ωk . Since J22 − R22 is invertible, we have e 1 ek − R ek ) ∂ Hk + B ek vk + C ek ωk , ∆e x = ( J k 1 ∂e x k e ek ∂H (2.9) T ∂ Hk 0 = −(J + R ) + (J − R ) + M(n−r)×n Bk vk + M(n−r)×n Ck ωk , 12 12 22 22 1 2 ∂e xk ∂e xk ωk+1 = F ωk ,
where
ek = J11 − R11 + (J12 − R12 )(J22 − R22 )−1 (J12 + R12 )T , Jek − R 5
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ek = Mr×n Bk − (J12 − R12 )(J22 − R22 )−1 M(n−r)×n Bk , B ek = Mr×n Ck − (J12 − R12 )(J22 − R22 )−1 M(n−r)×n Ck . C
ek is positive definite. Similar to the description in [6], Jek is skew-symmetric and R
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Remark 2.10. From (2.6), (2.8) and (2.9) we know that system (2.3) is transformed into a strict dissipative Hamiltonian form, thus we can take the advantages of the Hamiltonian function being a good Lyapunov function candidate and the strict dissipative Hamiltonian realization to study finite-time H∞ problems of system (2.3). In order to study finite-time boundedness and finite-time H∞ boundedness of the system (2.9), the following assumption is needed.
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Assumption 2.11. For any xk = S −1 x ek , x e1k 6= 0, the following condition holds: n ∂ T Hk −T −1 S M ES M ES −1 + M Ak S − S T ATk M T − M Bk LBkT M T ∂xk o−1 1 + (M Ak S + S T ATk M T ) (I − M ES −1 ) M Bk 6= 0. 2
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Lemma 2.12. Consider the system (2.3) and its equivalent form (2.9). Assume that the system (2.3) is of index 1 and causal. If the Assumption 2.11 holds, then there exist a reference input vk and a constant α, such that
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T e ek ∂ H ek ek ∂T H ∂T H 2 ek vk + ∂ Hk ∆e B x + α ≤ 0. k 1 2 1 ∂e xk ∂e xk ∂e xk ∂e x1k
Proof: From the Assumption 2.11, we have
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n ∂ T Hk −T S M ES −1 M ES −1 + M Ak S − S T ATk M T − M Bk LBkT M T ∂xk o−1 1 + (M Ak S + S T ATk M T ) (I − M ES −1 ) M Bk 2 o e k Ir 0 n Ir 0 −1 ∂T H 0 0 Mr×n Bk = + [Jk − Rk ] 00 00 0 In−r M(n−r)×n Bk ∂e xk T −1 ek Ir J12 − R12 Mr×n Bk ∂ H = 0 1 0 J22 − R22 M(n−r)×n Bk ∂e xk =
ek ∂T H ek 6= 0. B ∂e x1k
The rest of proof is similar to the Lemma 3 in [6], so it is omitted here.
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3. Main results In this section, according to the equivalent transformation in Section 2, we study finite-time boundedness and finite-time H∞ boundedness of the system (2.9). 3.1. Finite-time boundedness of the NDHDS
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In the following theorem, we give a sufficient condition for finite-time boundedness of the system (2.9).
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Theorem 3.1. Assume that the system (2.3) is of index 1 and causal, and the Assumption 2.11 holds. If there exist two constants β (> 0), α1 and two positive definite matrices P , Q, such that following conditions hold: ek ek + 1 C ΛP Ω1 ΛP C 2 (3.1) ? eT P < 0, Ω2 C k ? ? − βP and
i h e 0 + λmax (Pe)δx2 + λmax (Q)δω2 (β + 1)N H
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< ε2 ,
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then the system (2.9) is finite-time boundedness with respect to (δx , δω , ε, N, R) under the state-feedback controller e ekT ∂ Hk , vk = −B (3.3) ∂e x1k ek − α1 I, Ω1 = ΛP ΛT + Jek − R ekT P C ek + F T QF − (β + 1)Q, Ω2 = C
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where
e k )T − B ek B ekT , Λ = (Jek − R
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1 1 and Pe = R− 2 P R− 2 , R is a positive definite matrix.
Proof: Let
e k + (e Vk = H x1k )T P (e x1k ) + ωkT Qωk .
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Then, for all k = 1, 2, . . . , N , we have, T e k+1 − H e k + (e Qωk+1 − ωkT Qωk x1k ) + ωk+1 x1k )T P (e x1k+1 ) − (e Vk+1 − Vk = H x1k+1 )T P (e T e T e T e ek e ∂T H ek − R ek ) ∂ Hk + ∂ Hk B ek vk + ∂ Hk C ek ωk + ∂ Hk ∆e ( J x2k ∂e x1k ∂e x1k ∂e x1k ∂e x1k ∂e x2k T e e ek ∂T H ek − R ek )T P (Jek − R ek ) ∂ Hk + ∂ Hk (Jek − R ek )T P B ek vk ( J + ∂e x1k ∂e x1k ∂e x1k T e ek ∂T H ek − R ek )T P C ek ωk + ∂ Hk (Jek − R ek )T P x + ( J e1k ∂e x1k ∂e x1k e T eT eT e e T P (Jek − R ek ) ∂ Hk + v T B e +vkT B k k P Bk vk + vk Bk P Ck ωk k ∂e x1k e ekT P B ek vk eT P x eT P (Jek − R ek ) ∂ Hk + ωkT C +vkT B e1k + ωkT C k k ∂e x1k e ek ) ∂ Hk eT e1 + (e ek ωk + ω T C eT P C x1k )T P (Jek − R +ωkT C k k k Px k ∂e x1k ek vk + (e ek ωk + ω T (F T QF − Q)ωk . +(e x1 )T P B x1 ) T P C k
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By Lemma 2.12, we know that for the reference input (3.3), there exists a constant α1 , such that T e ek ek ∂ H ek ∂T H ∂T H 2 ek vk + ∂ Hk ∆e B x + α ≤ 0. 1 k ∂e x1k ∂e x2k ∂e x1k ∂e x1k
Then, by the controller (3.3), we have,
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i e T e ek h ∂T H ek − R ek )T P (Jek − R ek ) + (Jek − R ek ) − α1 I ∂ Hk + ∂ Hk C ek ωk ( J ∂e x1k ∂e x1k ∂e x1k i T e h ek e ∂T H ek − R ek )T P B ek B ekT ∂ Hk + 2 ∂ Hk (Jek − R ek )T − B ek B ekT P C ek ωk −2 ( J ∂e x1k ∂e x1k ∂e x1k i ek h ∂T H ek − R ek )T − B ek B eT P x eT P C ek + F T QF − Q)ωk +2 ( J e1k + ωkT (C k k ∂e x1k ek e ∂T H 1 eT P x ek B eT P B ek B e T ∂ Hk . +2ωkT C e + B k k k k ∂e x1k ∂e x1k
AC
CE
Vk+1 − Vk ≤
By (3.1), we have, and hence,
Vk+1 − Vk ≤ βωkT Qωk + β(e x1k )T P x e1k < βVk , Vk < (β + 1)Vk−1 =⇒ Vk < (β + 1)N V0 .
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x10 ) ≤ δx2 , and ω0T ω0 ≤ δω2 , then we have, If (e x10 )T R(e
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λmin (Pe)(e x1k )T R(e x1k ) ≤ (e x1k )T P x e1k < Vk < (β + 1)N V0 i h e 0 + (e e10 + ω0T Qω0 = (β + 1)N H x10 )T P x h i e 0 + λmax (Pe)(e ≤ (β + 1)N H x10 )T R(e x10 ) + ω0T Qω0 h i N e 2 2 e ≤ (β + 1) H0 + λmax (P )δx + λmax (Q)δω .
By (3.2), we know, (e x1k )T Re x1k < ε2 , which completes the proof. 3.2. Finite-time H∞ boundedness of the NDHDS
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In this subsection, we consider the following NDHDS: E∆xk = Ak ∇Hk + Bk uk + Ck ωk , ωk+1 = F ωk , zk = ΓM ES −2 ∇Hk ,
(3.4)
where E, Ak , Bk , Ck , F , M , S are the same as those in the system (2.3); zk ∈ Rs is the penalty signal, Γ = CkT ΣT 0s×(n−r) ,
M
Σ = Mr×n − (J12 − R12 )(J22 − R22 )−1 M(n−r)×n .
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PT
ED
Similar to the analysis in Section 2, the system (3.4) can be transformed into the e k + vk , following form under the controller uk = −LBkT M T ∇H e 1 ek − R ek ) ∂ Hk + B ek vk + C ek ωk , = ( J ∆e x k ∂e x1k ωk+1 = F ωk , (3.5) e eT ∂ Hk . zk = C k ∂e x1k
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ek = ΣCk . It’s easy to show that C Next, a sufficient condition is derived to guarantee that the system (3.5) is finite-time H∞ bounded in the following theorem.
Theorem 3.2. Suppose the Assumption 2.11 holds. Assume that the system (2.3) is of index 1 and causal, and has xk = 0 as a locally asymptotically stable equilibrium for ωk = 0. If there exist two constants β (> 0), α2 and two positive definite matrices P , Q, such that (3.2) and following conditions hold:
ek + 1 C ek Ω3 ΛP C 2 (a) : ? Ω2 ? ? 9
ΛP eT P < 0; C k − βP
(3.6)
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ek − Jek + α2 I − 1 C ek C ekT is invertible, R 2 −1 2 1 eT e 1 e eT T e ek − Q < γ I, F QF + Ck Rk − Jk + α2 I − Ck Ck C 4 2 2
(b) : and
(3.7)
e e T ∂ Hk , vk = −B k ∂e x1k
where
ek − α2 I, Ω3 = ΛP ΛT + Jek − R
and Λ, Ω2 are the same as those in Theorem 3.1.
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then the system (3.5) is finite-time H∞ boundedness with respect to (δx , δω , ε, N, R, γ) under the state-feedback controller (3.8)
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Proof: First, we prove that the system (3.5) has an L2 gain less than or equal to γ from ωk to zk . By Lemma 2.12, we know that for the reference input (3.8), there exists a constant α2 , such that T e ek ek ∂ H ek ∂T H ∂T H 2 ek vk + ∂ Hk ∆e B x + α ≤ 0. 2 k ∂e x1k ∂e x2k ∂e x1k ∂e x1k
(3.9)
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e k + ω T Qωk . Then, for all k = 1, 2, . . . , N , we have, Let Vk1 = H k 1 e k+1 − H e k + ω T Qωk+1 − ω T Qωk Vk+1 − Vk1 = H k+1 k
T e T e T e ek e ∂T H ek ωk + ∂ Hk B ek vk + ∂ Hk ∆e ek − R ek ) ∂ Hk + ∂ Hk C ( J x2k 1 1 1 1 2 ∂e xk ∂e xk ∂e xk ∂e xk ∂e xk
ED
=
+ωkT (F T QF − Q)ωk
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CE
PT
T e e ek ∂T H ek − R ek − α2 I) ∂ Hk + ∂ Hk C ek ωk + ωkT (F T QF − Q)ωk ( J ∂e x1k ∂e x1k ∂e x1k ek e 1 1 ∂T H ek C eT ∂ Hk + 1 γ 2 k ωk k2 − 1 k zk k2 − γ 2 ωkT ωk + C k 1 2 2 ∂e xk ∂e x1k 2 2
e 1 e eT 1 ∂ Hk
e e =− R k − Jk − Ck Ck + α2 I ] 2
2 ∂e x1k
2 − 21
1 e 1 ek C e T + α2 I ek ωk − [Rk − Jek − C C
k
2 2 ) ( −1 1 1 1 ek − Q − γ 2 I ωk eT R ek − Jek + α2 I − C ek C ekT +ωkT F T QF + C C 4 k 2 2
≤
1 1 + γ 2 k ωk k2 − k zk k2 . 2 2
By (3.7), we have,
1 1 1 Vk+1 − Vk1 < γ 2 k ωk k2 − k zk k2 . 2 2 10
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From Lemma 2.7, we know that the system (3.5) has an L2 gain less than or equal to γ from ωk to zk . Next, we prove that the system (3.5) is finite-time bounded. The proof is similar to that in the Theorem 3.1, so it is omitted here.
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Remark 3.3. From the foregoing analysis, we know that if (2.5) holds, then the system (2.3) is equivalent to the system (2.9) under the state-feedback controller (2.7) , and it is also true for the systems (3.4) and (3.5). Therefore, Theorem 3.1 and Theorem 3.2 can also guarantee that the systems (2.3) and (3.4) achieve finite-time boundedness and finite-time H∞ boundedness. In fact, for a given ε, there always exists a positive constant δ > 0 such that T RSr×n xk < ε2 ⇐⇒ (xk )T R∗ xk < ε2 , (e x1k )T Re x1k < ε2 ⇐⇒ (xk )T Sr×n
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T where R∗ = Sr×n RSr×n + δI > 0.
In the sequel, from Theorem 3.2, we can derive a finite-time H∞ boundedness criteria for the system (3.4) as follows:
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Corollary 3.4. Suppose the Assumption 2.11 holds. Assume that the system (2.3) is of index 1 and causal, and has xk = 0 as a locally asymptotically stable equilibrium for ωk = 0. If there exist two constants β (> 0), α2 , two positive definite matrices P , Q, and a symmetric matrix L ∈ Rm×m such that the following conditions hold:
ED
Ω4 Θ1
PT
(a0 ) : ?
where
CE
Ω4 = Λ5 P
∗
ΛT5
?
Θ2 ?
+ Λ2 − α2 I,
∗ Λ5 P Ir 0 ΛT3 P∗ < 0, 00 − βP ∗
Ir 0r×(n−r) P
∗
Ir 0(n−r)×r
= P,
AC
1 Ir 0 ∗ Θ1 = Λ5 P + I Λ3 , 00 2 Ir 0 I 0 Θ2 = ΛT3 P∗ r Λ3 + F T QF − (β + 1)Q, 00 00 −1 0r×(n−r) 0r×(n−r) 0(n−r)×r In−r (Jk − Rk ) × 0(n−r)×r In−r , Λ1 = In−r In−r Λ2 = Jk − Rk − (Jk − Rk )Λ1 (Jk − Rk ), Λ4 = M Bk − (Jk − Rk )Λ1 M Bk ,
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Λ3 = M Ck − (Jk − Rk )Λ1 M Ck , Ir 0 T T Λ5 = Λ2 − Λ4 Λ4 ; 00
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(b0 ): 1 F QF + ΛT3 4 T
n
Ir
o −1 1 Ir T Ir 0r×(n−r) α2 I − Λ2 − Λ3 Λ3 0(n−r)×r 2
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0(n−r)×r γ2 × Ir 0r×(n−r) Λ3 − Q < I; 2 n h io 1 (c0 ) : (β + 1)N H0 + λmax (Pe∗ )δx2 + λmax (Q)δω2 < ε2 , ∗ e λmin (P )
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then the system (3.4) is finite-time H∞ boundedness with respect to (δx , δω , ε, N, R, γ) under the state-feedback controller Ir 0 T T −1 T −1 uk = − LBk M S + Λ4 S ∇Hk , 00 o n ∗ =diag Pe, λmin (Pe)In−r , Pe is the same as that in Theorem 3.1. where Pen×n 4. An example
1
0
PT
1
∆xk = Ak ∇Hk + Bk uk + Ck ωk , ωk+1 = F ωk ,
zk = ΓM ES
−2
∇Hk ,
AC
(4.1a) (4.1b) (4.1c)
1 1 as x(0) = (1, 1, 0)T and ω(0) = ( , )T , where n = 3, s = m = r = 2 2 1 1 2 1 2 2 (x ) + (xk ) + (x3k )2 ; 4 k 4
CE
with initial conditions 1 xk 2, xk = x2k , Hk = x3k
ED
M
In this section, we give a simulation example to illustrate the results obtained in this paper. Consider the following NDHDS:
− sin x1k − 1 0 0 , 0 − cos x2k − 1 0 Ak = 3 2 0 0 −(xk ) − 1 1 0 1 0 0 , Bk = 0 1 , Ck = 0 1 2 0 0 xk − xk x3k
1 F = diag(1, ), M = diag(1, 1, 1), S = diag(1, 1, 2). 2 The corresponding parameters and matrices are given as follows: 12
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where
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3 N = 5, α2 = β = γ = δ = 1, δx = δω = , ε = 13, 4 1 1 1 1 ∗ e P = diag( , 1, ), Q = diag(1, 1), R = diag( , ). By some simple calculation, we can 3 3 3 9 verify that (b0 ) and (c0 ) hold. And (a0 ) can be transformed into the following form: Φ1 Φ2 Φ3 ? Φ4 Φ5 < 0, ? ? Φ6
1 2 1 1 1 0 0 9 sin xk − 3 sin xk − 2 1 1 Φ1 = , 2 2 2 0 cos x − cos x − 2 0 k k 9 3 0 0 −1 1 1 1 1 1 1 − 9 sin xk + 6 0 − 9 sin xk − 3 0 0 0 Φ2 = , 1 (x1k − x2k ) 0 0 2 8 1 0 0 0 −9 9 1 7 1 2 Φ3 = − cos xk − 0 , Φ4 = ? − 0 , 9 3 4 1 0 0 ? ? 9 1 Φ5 = (0)3×2 , Φ6 = diag(− , −1). 9 1 xk By designing the controller uk = − , (4.1a) can be transformed into Exk+1 = x2k 1 0 −0.5x1k (sin x1k + 1) (g1 )2×2 0 . = 0 f (xk ) + g(xk )ωk , where f = −0.5x2k (cos x2k + 1) , g := (g2 )1×2 3 2 2 1 2 −2xk ((xk ) + 1)) xk − xk x3k ∂[g2 wk − 2x3k ((x2k ) + 1)] It is easy to check that deg(det(zE − Ξ)) = 2 = rank(E) and is ∂x3k nonsingular, so the system (4.1a) is of index 1 and causal. From Corollary 3.4, we know that the system (4.1) can achieve finite-time H∞ boundedness. Figures 1 and 2 show the simulation results of the closed-loop system of (4.1) and the response of the control signal, respectively. 1 1 By some simple calculations, we have R∗ = diag( , , 1). From figure 1 we can 3 9 9 T ∗ and see that xk R xk < 169 (k = 1, 2, 3, 4, 5) under the initial conditions xT0 R∗ x0 ≤ 16 9 ω0T ω0 ≤ . 16
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PT
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25
20
10
5
0
0
1
2
3
4
Sample k
5
Closed-loop system of (4.1) to be finite-time H∞ boundedness with respect to 3 3 ( , , 13, 5, R, 1). 4 4
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Figure. 1
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xTk R*xk
15
1
u1 u2
M
0
−0.5
−1
PT
−1.5
ED
Control u
0.5
0
1
3
4
5
Sample k
The control signal uk .
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Figure. 2
2
5. Conclusion
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In this paper, by an appropriate state-feedback controller, we have shown that the nonlinear discrete Hamiltonian descriptor system can be transformed into an equivalent strict dissipative system. Then, the problems of finite-time boundedness and finite-time H∞ boundedness for this strict dissipative system have been studied. Finally, the results obtained have been extended to the finite-time control problem of the NDHDS. Future works will focus on stochastic descriptor systems, stochastic Hamiltonian systems and so on. 6. Acknowledgments This work is supported by the National Natural Science Foundations of China (61374065, 61573215, 6150021593, 61374002) and the Natural Science Fund for Distinguished Young 14
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Scholars of Shandong Province under grant JQ201613. References [1] L. Dai, Singular control systems. Berlin, Germany: Springer-Verlag, 1989.
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[2] Z. Wang, M. Rodrigues, D. Theilliol, Fault estimation filter design for discrete descriptor systems, IET Control Theory and Applications 9 (10) (2015) 1587-1594. [3] Y. Feng, Positive real control for discrete descriptor systems with affine parameter dependence, Journal of the Franklin Institute 352 (3) (2015) 882-896. [4] F. N. Koumboulis, Common I/O triangular decoupling for multi model descriptor systems, Systems and Control Letters 97 (2016) 108-114.
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[5] D. Koenig, B. Marx, S. Varrier, Filtering and fault estimation of descriptor switched systems, Automatica 63 (2016) 116-121. [6] L. Sun, G. Feng, Y. Wang, Finite-time stabilization and H∞ control for a class of nonlinear Hamiltonian descriptor systems with application to affine nonlinear descriptor systems, Automatica 50 (8) (2014) 2090-2097.
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[8] Y. Wei, J. Qiu, H. Lam, L. Wu, Approaches to TS fuzzy-affine-model-based reliable output feedback control for nonlinear Ito stochastic systems, IEEE Transactions on Fuzzy Systems, http://dx.doi.org/10.1109/TFUZZ.2016.2566810.
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[9] Y. Wei, J. Qiu, P. Shi, M. Chadli, Fixed-order memory piecewise affine output feedback controller for fuzzy-affine-model-based nonlinear systems with time-varying delay, IEEE Transactions on Circuits and Systems I: Regular Papers, doi: 10.1109/TCSI.2016.2632718.
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[15] H. T. M. Kussaba, R. A. Borges, & T. Y. Ishihara, A new condition for finite time boundedness analysis, Journal of the Franklin Institute 352 (12) (2015) 5514-5528. [16] K. Mathiyalagan, J. H. Park, & R. Sakthivel, Finite-time boundedness and dissipativity analysis of networked cascade control systems, Nonlinear Dynamics (2016) 1-12.
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[22] W. Lin, C. I. Byrnes, H∞ -control of discrete nonlinear systems, IEEE Transactions on Automatic Control 41 (4) (1996) 494-510.
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