Non-fragile observer design for nonlinear switched time delay systems ...
UKACC International Conference on Control 2012 Cardiff, UK, 3-5 September 2012
Non-fragile observer design for nonlinear switched time delay systems using delta operator Ronghao Wang, Jianchun Xing, IEEE member
Ping Wang, Qiliang Yang
PLA University of Science and Technology Nanjing, China [email protected]
PLA University of Science and Technology Nanjing, China [email protected] been developed[12,13]. The problem of system instability in fast sampling can be solved by using delta operator model [13].
Abstract—This paper considers the non-fragile observer design method for nonlinear switched time delay systems using the delta operator. Based on multiple Lyapunov function method and delta operator theory, an asymptotic stability criterion for delta operator switched system with time delay and Lipschitz nonlinearity is presented. By using the key technical lemma, a new sampling period and delay dependent design approach to the non-fragile observer is addressed. The proposed non-fragile observer can guarantee the estimated state error dynamics of delta operator time delay switched system can be asymptotically convergent for observer gain perturbations. The solution to the observer is formulated in the form of a set of linear matrix inequalities. A numerical example is employed to verify the proposed method. Keywords-delta operator; non-fragile observer; nonlinear; switched systems; time delay
I.
Recently, delta operator approach is used to investigate robust control for a class of uncertain switched systems, and the stabilization conditions of the delta operator switched systems are formulated in terms of a set of linear matrix inequalities (LMIs) [14]. The developed method is valid for the system. However, in actual operation, the states of the systems are not all measurable. It is necessary to design state observers for systems of this type. Several design procedures have been proposed to design state observers for switched systems[15, 16]. It is considered that the state observer gain variations could not be avoided in several applications, a kind of non-fragile observer is proposed and the design method is proved to be effective[17]. However, to date and to the best of our knowledge, the problem of the non-fragile observer design for time delay delta operator switched nonlinear systems has not been investigated, which motivated us for this study.
Lipschitz
INTRODUCTION
Switched systems have attracted the interest of several scientists in the last several years. Switched systems are a class of hybrid systems consisting of subsystems and a switching law, which define a specific subsystem being activated during a certain interval of time. Switched systems exist widely in engineering and social systems, such as mechanical systems, automotive industry, aircraft and air traffic control and many other fields[1-3]. A lot of research in this direction have appeared recently[4,5]. Many important progress and remarkable achievements have been made on issues about stability and stabilization for the system. Based on common quadratic Lyapunov functions (CQLFs), a series of methods and conditions have been given for analyzing the stability of switched systems for arbitrary switching law[6, 7]. As effective tools, multiple Lyapunov function (MLF), switched Lyapunov function(SLF) and the average dwell-time approaches have been proposed to analyze the stability of switched systems, and many valuable results have been obtained for switched systems[8-10].
In this paper, we deal with the problem of the non-fragile observer design for time delay switched nonlinear systems using a delta operator, where the observer gain perturbations are assumed to be time-varying and unknown, but are normbounded. The aim is to design the non-fragile observer such that the estimated state error dynamics can be asymptotically convergent for observer gain perturbations. The desired nonfragile observer can be constructed by solving a set of LMIs. The remainder of the paper is organized as follows. In Section II, problem formulation and some necessary lemmas are given. In Section III, based on the MLF approach and delta operator theory, the stability analysis for a delta operator time delay switched nonlinear system is considered, and the result is dependent of time delay and will be employed to develop a non-fragile observer. A numerical example is given to illustrate the feasibility and effectiveness of the developed technique in Section IV, and concluding remarks are given in Section V. II. PROBLEM FORMULATION AND PRELIMINARY
It is recognized that most discrete-time signals and systems are the results of sampling continuous-time signals and systems. When sampling is fast, all resulting signals and systems tend to become ill conditioned and thus difficult to deal with using the conventional algorithms. The delta operator-based algorithms are numerically better behaved under finite precision implementations for fast sampling[11]. A class of uncertain systems in delta domain has been studied and several results about robust stability for the system have
Consider the following delta operator switched nonlinear system with time delay: (1) δ xk = Aσ ( k ) xk + Adσ ( k ) xk −n + Bσ ( k )uk + fσ ( k ) ( xk , k )
yk = Cσ ( k ) xk
(3) xk = φk , ∀k ∈ {−n, −n + 1," , 0} where xk ∈ R is the state vector at the k th instant, yk ∈ R m p
This work is supported by the Pre-Research Foundation of PLA University of Science and Technology under Grant No. 20110316.
978-1-4673-1558-6/12/$31.00 ©2012 IEEE
(2)
387
is the measurement output vector. φk is a vector-valued initial function, f i (⋅, ⋅) : R p × R → R p is an unknown nonlinear
Assumption 2 The gain perturbations ΔLi are of the normbounded form: (15) ΔLi = H i Fik Ei where H i , Ei which denote the structure of the uncertainties are known real constant matrices with proper dimensions, and Fik are unknown time-varying matrices which satisfy:
function, n is the state delay of the system. σ (k ) : Z + → N = {1, 2," , N } is a switching signal. Moreover, σ (k ) = i means that the i th subsystem is activated. uk ∈ R l is the control input of the i th subsystem at the k th instants. Ai ∈ R p× p , Adi ∈ R p× p , Bi ∈ R p×l , Ci ∈ R m× p for i ∈ N are known real-valued matrices with appropriate dimensions, δ denotes the delta operator, the definition can be seen in [11], ie, δ xk = ( xk +1 − xk ) / T , where T is a sample period.
FikT Fik ≤ I The unknown matrices Fik contain the uncertain parameters in the linear part of the subsystem and the matrices H i , Ei
specify how the unknown parameters in Fik affect the elements of the nominal matrices Li .
We construct the following discrete-time switched system using delta operator to estimate the state of system (1)-(3):
δ xˆk = Aσ ( k ) xˆk + Adσ ( k ) xˆk − n + Bσ ( k )uk + fσ ( k ) ( xˆk , k )
The parameter uncertainty structure in equation (15) has been widely used and can represent parameter uncertainty in many physical cases. Equation (15) is called additive gain variations. Lemma 1 [18] Let U, V, W and X be real matrices of appropriate dimensions with X satisfying X = X T , then for all V TV ≤ I X + UVW + W T V T U T < 0 if and only if there exists a scalar ε > 0 such that X + ε UU T + ε −1W T W < 0 [19] Consider the following system Lemma 2 (16) xk +1 = f k ( xk )
(4)
+ Lσ ( k ) ( yk − yˆ k ) yˆ k = Cσ ( k ) xˆk
(5)
(6) xˆk = 0, ∀k ∈ {− n, − n + 1," , 0} where xˆk ∈ R p is the estimated state vector of xk , yˆ k ∈ R m is the observer output vector, Li ∈ R p×m for i ∈ N is the observer gain. Let xk = xk − xˆk be the estimated state error. By the definition of the delta operator, we have δ xk = δ xk − δ xˆk , then we can obtain the following error system from (1)-(6): δ xk = ( Aσ ( k ) − Lσ ( k )Cσ ( k ) ) xk + Adσ ( k ) xk − n + fσ ( k ) ( xk , k ) (7)
If there is a function V : Z + × R n → R such that: (1) V is a positive-definite function, decreasing and radially unbounded. (2) ΔV (k , xk ) = V (k + 1, xk +1 ) − V (k , xk ) < 0 is negative definite along the solution of (16). then system (16) is asymptotically stable. Lemma 3[20] For any constant positive semi-definite symmetric matrix W , two positive integers r and r0 satisfying r ≥ r0 ≥ 1 , the following inequality holds
− fσ ( k ) ( xˆk , k )
(8) xk = φk , ∀k ∈ {−n, −n + 1," , 0} If the state observer gain variations could not be avoided, a kind of non-fragile state observer will be designed as follows: (9) δ xˆk = Aσ ( k ) xˆk + Adσ ( k ) xˆk − n + Bσ ( k ) uk + fσ ( k ) ( xˆk , k ) + ( Lσ ( k ) + ΔLσ ( k ) )( yk − yˆ k ) yˆ k = Cσ ( k ) xˆk
T
r ⎛ r ⎞ ⎛ r ⎞ T ⎜ ∑ x(i ) ⎟ W ⎜ ∑ x(i ) ⎟ ≤ ρ ∑ x (i )Wx(i ) = = = i r i r i r 0 ⎝ 0 ⎠ ⎝ 0 ⎠ where ρ = r − r0 + 1 . The objective of this paper is to design non-fragile observer gain Li for delta operator time delay switched nonlinear systems (1)-(3) such that the estimated state error dynamics is asymptotically convergent.
(10)
(11) xˆk = 0, ∀k ∈ {−n, −n + 1," , 0} are uncertain real-valued matrix functions
where ΔLi ∈ R representing norm-bounded parameter uncertainties. According to system (1)-(3) and (9)-(11), the dynamic equations of error switched system for non-fragile observer can be prescribed: δ xk = [ Aσ ( k ) − ( Lσ ( k ) + ΔLσ ( k ) )Cσ ( k ) ]xk + Adσ ( k ) xk − n (12) + fσ ( k ) ( xk , k ) − fσ ( k ) ( xˆk , k ) p× m
III.
A. Stability analysis In this subsection, we investigate the stability of the following delta operator time delay switched nonlinear system (17) δ xk = Aσ ( k ) xk + Adσ ( k ) xk −n + fσ ( k ) ( xk , k )
(13) xk = φk , ∀k ∈ {−n, −n + 1," , 0} Without loss of generality, we make the following assumptions. Assumption 1 f i ( xk , k ) for i ∈ N are nonlinear functions
where
satisfying: f i ( xk , k ) − fi ( xˆk , k ) ≤ U i ( xk − xˆk ) where U i are known real constant matrices.
MAIN RESULTS
(14)
f i ( xk , k ) ≤ U i xk
for i ∈ N . Define the indicator
function:
ξ (k ) = (ξ1 (k ), ξ 2 (k )," , ξ N (k ))T
388
(18)
with i ∈ {1, 2," , N } , where ⎧1 when the ith subsystem is activated , ξi (k ) = ⎨ ⎩0 others then system (17) can be written as
xkT+1 P (ξ (k + 1)) xk +1 − xkT P (ξ (k )) xk 2T ( xk + T δ xk )T P(ξ (k + 1))( xk + T δ xk ) − xkT P (ξ (k )) xk = 2T 1 1 T T T ≤ xk [ Ai Pj + Pj Ai + TAi Pj Ai + ε iTU iTU i + ( Pj − Pi )]xk T 2 T T T T +2 xk [(TAi + I ) Pj Adi ]xk − n + xk − n (TAdi Pj Adi ) xk − n =
N
δ xk = ∑ ξi (k )( Ai xk + Adi xk −n + fσ ( k ) ( xk , k )) i =1
∀i ∈ {1, 2," , N }
> 0 and scalar ε i > 0 , i ∈ {1, 2," , N } , r = 1, 2," , s − 1 , s = 1, 2," , n − 1 , such that (19) Pi < ε i I
(21)
⎡ AiT Pj + Pj Ai + TAiT Pj Ai + ε i (T + 1)U iT U i ⎢ ⎢ + 1 ( P − 1 P ) − 1 R ( s ) + Q (1) j ⎢ T j 2 i n i ⎢ * ⎢ ⎢ * ⎢ ⎢ ⎢⎣ 1 (TAiT + I ) Pj Adi + Ri( s ) n Pi Adi
s =1
By (21), we can obtain that T (n) δ V2 (k , xk ) < xkT Q (1) j xk − xk − n Qi xk − n 1 T
A Pi − Pi + nT 2 R (1) j →
(22)
s
( r +1) j k −r
e
n s−1
n
s =2 r =1
s=1
− ∑∑ekT−r Ri(r )ek −r − ∑ekT−s Ri( s)ek −s
≤ xkT [ AiT Pj + Pj Ai + TAiT Pj Ai + εiTUiTUi 1 1 + (Pj − Pi )]xk + xkT−n (TAdiT Pj Adi ) xk −n T 2 2 T (1) +2xkT [(TAiT + I )Pj Adi ]xk −n + xkT Q(1) j xk + nT δ xk Rj δ xk
(27)
1 −xkT−nQi( n) xk −n − ( xk −n − xk )T Ri( s) ( xk −n − xk ) n Notice that (28) 0 = −2δ xkT Pi (δ xk − Ai xk − Adi xk − n − f i ( xk , k )) Combining (27) and (28) leads to ⎡ AiT Pj + Pj Ai + TAiT Pj Ai + εi (T +1)UiTUi ⎢ AiT Pi ⎢+ 1 (P − 1 P ) − 1 R( s) + Q(1) j ⎢ T j 2 i n i T ⎢ −Pi + nT 2 R(1) δV (k, xk ) ≤ η ⎢ * j → ⎢ * * ⎢ ⎢ ⎢⎣ 1 ⎤ (TAiT + I ) Pj Adi + Ri( s ) ⎥ n ⎥ Pi Adi ⎥η ⎥ 1 TAdiT Pj Adi − Qi( n ) − Ri( s ) ⎥ n ⎦
N
V3 (k , xk ) = T ∑∑ ekT− r R ( r ) (ξ (k ))ek − r = T ∑∑∑ ekT− rξi (k ) Ri( r ) ek − r s =1 r =1
T k −r
(24), (25) and (26), we have δV (k, xk ) = δV1 (k, xk ) + δV2 (k, xk ) + δV3 (k, xk )
N
n
s=1 r =1
n s −1
follows that this has to hold for special configuration ξi (k ) = 1 , ξ h ≠i (k ) = 0 , ξ j (k + 1) = 1 , ξ g ≠ j (k + 1) = 0 and for all xk ∈ R p . By
s =1 i =1
s
(1) j k
s=2 r =1
V2 (k , xk ) = T ∑ xkT− s Q ( s ) (ξ (k )) xk − s = T ∑∑ xkT− sξi (k )Qi( s ) xk − s n
(25)
s
By (20), we can obtain that n 1 n (∑ ek − s )T Ri( s ) (∑ ek − s ) δ V3 (k , xk ) ≤ nekT R (1) j ek − (26) n s =1 s =1 1 = − ( xk − n − xk )T Ri( s ) ( xk − n − xk ) + nT 2δ xkT R (1) j δ xk n If δ V (k , xk ) < 0 holds under arbitrary switching signal, it
Proof Consider the following switched LyapunovKrasovskii functional: (23) V (k , xk ) = V1 (k , xk ) + V2 (k , xk ) + V3 (k , xk ) with 1 1 ⎛ N ⎞ V1 (k , xk ) = xkT P (ξ (k )) xk = xkT ⎜ ∑ ξi (k ) Pi ⎟ xk 2 2 ⎝ i =1 ⎠ s =1
n
= ne R e + ∑∑e R
under arbitrary switching.
n
s
s =1 r =1
T k
⎤ ⎥ ⎥ ⎥ 0 , Ri( r ) > 0 , Ri( s ) > 0 , Qi( s ) > 0 ,
R (j r +1) < Ri( r ) , r = 1, 2," , s − 1
n
1 T
(24)
s =1 r =1 i =1
with Pi , Qi( s ) , Ri( r ) , i ∈ {1, 2," , N } being symmetric positivedefinite matrices and ek = xk − xk +1 . Then switched Lyapunov-Krasovskii functional in delta domain has the following form: V (k + 1, xk +1 ) − V1 (k , xk ) δ V1 (k , xk ) = 1 T
389
B. Observer Design Now we consider system (7)-(8). The following theorem presents sufficient conditions for the existence of asymptotic stability of system (7)-(8). Theorem 2 Consider system (7)-(8), if there exist a set of matrices X i > 0 , Vi ( s ) > 0 , Vi ( n ) > 0 , Z i( r ) > 0 , Z i( s ) > 0 , Vij( s ) > 0 , Vij( n ) > 0 , Z ij( r ) > 0 , Z ij( s ) > 0 , ε i > 0 , i ∈ {1, 2," , N } ,
T
where η = ⎡ xkT δ xkT xkT−n ⎤ . From (22), we have δ V (k , xk ) < 0 . ⎣ ⎦ By Lemma 2, the system (17) is asymptotically stable. The proof is completed. Remark 1 When time delay n ≡ 0 and fσ ( k ) ( xk , k ) ≡ 0 , system (17) becomes as follows:
δ xk = Aσ ( k ) xk where Aσ ( k ) = Aσ ( k ) + Adσ ( k ) . From (22), we know that the matrix inequality imply that the following inequality holds: 1 AiT Pj + Pj Ai + TAiT Pj Ai + ( Pj − Pi ) < 0 T The above inequality is just a sufficient condition of stability for the delta operator switched system without time delay, which can be expressed in [14].
r = 1, 2," , s − 1 , s = 1, 2," , n − 1 , and Wi such that the following LMIs have feasible solutions (35) X i > ε i−1 I ( r +1) (r) (36) Z ij < Z i , r = 1, 2," , s − 1 Vij( s +1) < Vi ( s ) , s = 1, 2," , n − 1 1 (s) ⎡ 1 (1) ⎢ − 2T X i + Vij − n Z i ⎢ * ⎢ ⎢ * ⎢ ⎢ ⎢ * ⎢ * ⎣
Remark 2 When sample period T = 0 and fσ ( k ) ( xk , k ) ≡ 0 , system (17) becomes a continuous-time switched linear system as follows: x (t ) = Aσ ( t ) x(t ) + Adσ ( t ) x(t − d )
(29)
We can obtain sufficient condition of stability for system (29) by Theorem 1. Corollary 1 Consider system (29). If there exist symmetric positive definite matrices P > 0 , R > 0 , Q > 0 , such that 1 1 ⎤ ⎡ T T ⎢ Ai P + PAi − d R + Q Ai P PAdi + d R ⎥ (30) ⎢ ⎥ −2 P PAdi ⎥ < 0 * ⎢ ⎢ 1 ⎥ −Q − R ⎥ * * ⎢ d ⎦ ⎣ ∀i ∈ {1, 2," , N } , then system (29) is asymptotically stable under arbitrary switching. Remark 3 Let Aσ ( k ) = Aσ ( k ) + I . When sample period T = 1
−2 Pi + nR (1) j *
−Vi ( n ) −
*
*
*
*
1 (s) → Zi n
(38)
T ( Ai X i − Wi ) + X i
X iU
0
0
T i
T T ⎡( Ai − LC i i ) Pj + Pj ( Ai − LC i i ) +T ( Ai − LC i i ) Pj ( Ai − LC i i) T ⎢ (Ai − LC i i ) Pi ⎢+ε (T +1)UTU + 1 (P − 1 P) − 1 R(s) + Q(1) i i j i i j ⎢ i 2 T n ⎢ * Mij = ⎢ −Pi + nT 2R(1) j → ⎢ * * ⎢ ⎢ ⎢⎣ 1 (s ) ⎤ T [T(Ai − LC i i ) + I ]Pj Adi + Ri ⎥ n ⎥ PA ⎥ i di ⎥ (n) 1 ( s) T TAdi Pj Adi −Qi − Ri ⎥ n ⎦
Qi( n ) > 0 , i ∈ {1, 2," , N } , r = 1, 2," , s − 1 , s = 1, 2," , n − 1 , such that (32) R (j r +1) < Ri( r ) , r = 1, 2," , s − 1
AiT Pi
*
can guarantees system (7)-(8) is asymptotically stable under arbitrary switching. Proof Denote
Corollary 2 Consider system (31). If there exist symmetric positive definite matrices Pi > 0 , Ri( r ) > 0 , Ri( s ) > 0 , Qi( s ) > 0 ,
⎡ AiT Pj + Pj Ai + AiT Pj Ai ⎢ ⎢ − P − P − 1 R ( s ) + Q(1) j ⎢ j i n i ⎢ * ⎢ ⎢ * ⎢ ⎢ ⎢⎣
− X i + nT 2 Z ij(1)
1 (s) Zi n Adi X i
⎤ ⎥ ⎥ ⎥ 0 TX i AdiT ⎥ ε i−1 I (43) Z ij( r +1) < Z i( r ) , r = 1, 2," , s − 1
(41)
⎤ TX i ( Ai − Li Ci )T + X i ⎥ ⎥ 0 ⎥ ⎥ 0 , Vi ( s ) > 0 , Vi ( n ) > 0 , Z i( r ) > 0 , Z i( s ) > 0 , (s) ij
V
>0 ,V
(n) ij
>0, Z
(r ) ij
>0 , Z
(s) ij
T ( Ai X i − Wi )T + X i μ iTH i H iT TX i AdiT −TX j + μiT 2 H i H iT
> 0 , ε i > 0 , i ∈ {1, 2," , N } ,
r = 1, 2," , s − 1 , s = 1, 2," , n − 1 and Wi , LMIs (35)-(38) have feasible solutions, then go to Step 2. Step 2. If the equations Li Ci = Wi X i−1 have feasible solutions for unknown matrices Li , then the observer can be constructed
* *
X iU iT 0 0 0 1 − I ε i (T + 1) *
(44) 1 (s) Zi n Adi X i 1 (n) −Vi − Zi( s) → n * * *
− X i CiT EiT ⎤ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥
Non-fragile observer design for nonlinear switched time delay systems using delta operator Ronghao Wang, Jianchun Xing, IEEE member
Ping Wang, Qiliang Yang
PLA University of Science and Technology Nanjing, China [email protected]
PLA University of Science and Technology Nanjing, China [email protected] been developed[12,13]. The problem of system instability in fast sampling can be solved by using delta operator model [13].
Abstract—This paper considers the non-fragile observer design method for nonlinear switched time delay systems using the delta operator. Based on multiple Lyapunov function method and delta operator theory, an asymptotic stability criterion for delta operator switched system with time delay and Lipschitz nonlinearity is presented. By using the key technical lemma, a new sampling period and delay dependent design approach to the non-fragile observer is addressed. The proposed non-fragile observer can guarantee the estimated state error dynamics of delta operator time delay switched system can be asymptotically convergent for observer gain perturbations. The solution to the observer is formulated in the form of a set of linear matrix inequalities. A numerical example is employed to verify the proposed method. Keywords-delta operator; non-fragile observer; nonlinear; switched systems; time delay
I.
Recently, delta operator approach is used to investigate robust control for a class of uncertain switched systems, and the stabilization conditions of the delta operator switched systems are formulated in terms of a set of linear matrix inequalities (LMIs) [14]. The developed method is valid for the system. However, in actual operation, the states of the systems are not all measurable. It is necessary to design state observers for systems of this type. Several design procedures have been proposed to design state observers for switched systems[15, 16]. It is considered that the state observer gain variations could not be avoided in several applications, a kind of non-fragile observer is proposed and the design method is proved to be effective[17]. However, to date and to the best of our knowledge, the problem of the non-fragile observer design for time delay delta operator switched nonlinear systems has not been investigated, which motivated us for this study.
Lipschitz
INTRODUCTION
Switched systems have attracted the interest of several scientists in the last several years. Switched systems are a class of hybrid systems consisting of subsystems and a switching law, which define a specific subsystem being activated during a certain interval of time. Switched systems exist widely in engineering and social systems, such as mechanical systems, automotive industry, aircraft and air traffic control and many other fields[1-3]. A lot of research in this direction have appeared recently[4,5]. Many important progress and remarkable achievements have been made on issues about stability and stabilization for the system. Based on common quadratic Lyapunov functions (CQLFs), a series of methods and conditions have been given for analyzing the stability of switched systems for arbitrary switching law[6, 7]. As effective tools, multiple Lyapunov function (MLF), switched Lyapunov function(SLF) and the average dwell-time approaches have been proposed to analyze the stability of switched systems, and many valuable results have been obtained for switched systems[8-10].
In this paper, we deal with the problem of the non-fragile observer design for time delay switched nonlinear systems using a delta operator, where the observer gain perturbations are assumed to be time-varying and unknown, but are normbounded. The aim is to design the non-fragile observer such that the estimated state error dynamics can be asymptotically convergent for observer gain perturbations. The desired nonfragile observer can be constructed by solving a set of LMIs. The remainder of the paper is organized as follows. In Section II, problem formulation and some necessary lemmas are given. In Section III, based on the MLF approach and delta operator theory, the stability analysis for a delta operator time delay switched nonlinear system is considered, and the result is dependent of time delay and will be employed to develop a non-fragile observer. A numerical example is given to illustrate the feasibility and effectiveness of the developed technique in Section IV, and concluding remarks are given in Section V. II. PROBLEM FORMULATION AND PRELIMINARY
It is recognized that most discrete-time signals and systems are the results of sampling continuous-time signals and systems. When sampling is fast, all resulting signals and systems tend to become ill conditioned and thus difficult to deal with using the conventional algorithms. The delta operator-based algorithms are numerically better behaved under finite precision implementations for fast sampling[11]. A class of uncertain systems in delta domain has been studied and several results about robust stability for the system have
Consider the following delta operator switched nonlinear system with time delay: (1) δ xk = Aσ ( k ) xk + Adσ ( k ) xk −n + Bσ ( k )uk + fσ ( k ) ( xk , k )
yk = Cσ ( k ) xk
(3) xk = φk , ∀k ∈ {−n, −n + 1," , 0} where xk ∈ R is the state vector at the k th instant, yk ∈ R m p
This work is supported by the Pre-Research Foundation of PLA University of Science and Technology under Grant No. 20110316.
978-1-4673-1558-6/12/$31.00 ©2012 IEEE
(2)
387
is the measurement output vector. φk is a vector-valued initial function, f i (⋅, ⋅) : R p × R → R p is an unknown nonlinear
Assumption 2 The gain perturbations ΔLi are of the normbounded form: (15) ΔLi = H i Fik Ei where H i , Ei which denote the structure of the uncertainties are known real constant matrices with proper dimensions, and Fik are unknown time-varying matrices which satisfy:
function, n is the state delay of the system. σ (k ) : Z + → N = {1, 2," , N } is a switching signal. Moreover, σ (k ) = i means that the i th subsystem is activated. uk ∈ R l is the control input of the i th subsystem at the k th instants. Ai ∈ R p× p , Adi ∈ R p× p , Bi ∈ R p×l , Ci ∈ R m× p for i ∈ N are known real-valued matrices with appropriate dimensions, δ denotes the delta operator, the definition can be seen in [11], ie, δ xk = ( xk +1 − xk ) / T , where T is a sample period.
FikT Fik ≤ I The unknown matrices Fik contain the uncertain parameters in the linear part of the subsystem and the matrices H i , Ei
specify how the unknown parameters in Fik affect the elements of the nominal matrices Li .
We construct the following discrete-time switched system using delta operator to estimate the state of system (1)-(3):
δ xˆk = Aσ ( k ) xˆk + Adσ ( k ) xˆk − n + Bσ ( k )uk + fσ ( k ) ( xˆk , k )
The parameter uncertainty structure in equation (15) has been widely used and can represent parameter uncertainty in many physical cases. Equation (15) is called additive gain variations. Lemma 1 [18] Let U, V, W and X be real matrices of appropriate dimensions with X satisfying X = X T , then for all V TV ≤ I X + UVW + W T V T U T < 0 if and only if there exists a scalar ε > 0 such that X + ε UU T + ε −1W T W < 0 [19] Consider the following system Lemma 2 (16) xk +1 = f k ( xk )
(4)
+ Lσ ( k ) ( yk − yˆ k ) yˆ k = Cσ ( k ) xˆk
(5)
(6) xˆk = 0, ∀k ∈ {− n, − n + 1," , 0} where xˆk ∈ R p is the estimated state vector of xk , yˆ k ∈ R m is the observer output vector, Li ∈ R p×m for i ∈ N is the observer gain. Let xk = xk − xˆk be the estimated state error. By the definition of the delta operator, we have δ xk = δ xk − δ xˆk , then we can obtain the following error system from (1)-(6): δ xk = ( Aσ ( k ) − Lσ ( k )Cσ ( k ) ) xk + Adσ ( k ) xk − n + fσ ( k ) ( xk , k ) (7)
If there is a function V : Z + × R n → R such that: (1) V is a positive-definite function, decreasing and radially unbounded. (2) ΔV (k , xk ) = V (k + 1, xk +1 ) − V (k , xk ) < 0 is negative definite along the solution of (16). then system (16) is asymptotically stable. Lemma 3[20] For any constant positive semi-definite symmetric matrix W , two positive integers r and r0 satisfying r ≥ r0 ≥ 1 , the following inequality holds
− fσ ( k ) ( xˆk , k )
(8) xk = φk , ∀k ∈ {−n, −n + 1," , 0} If the state observer gain variations could not be avoided, a kind of non-fragile state observer will be designed as follows: (9) δ xˆk = Aσ ( k ) xˆk + Adσ ( k ) xˆk − n + Bσ ( k ) uk + fσ ( k ) ( xˆk , k ) + ( Lσ ( k ) + ΔLσ ( k ) )( yk − yˆ k ) yˆ k = Cσ ( k ) xˆk
T
r ⎛ r ⎞ ⎛ r ⎞ T ⎜ ∑ x(i ) ⎟ W ⎜ ∑ x(i ) ⎟ ≤ ρ ∑ x (i )Wx(i ) = = = i r i r i r 0 ⎝ 0 ⎠ ⎝ 0 ⎠ where ρ = r − r0 + 1 . The objective of this paper is to design non-fragile observer gain Li for delta operator time delay switched nonlinear systems (1)-(3) such that the estimated state error dynamics is asymptotically convergent.
(10)
(11) xˆk = 0, ∀k ∈ {−n, −n + 1," , 0} are uncertain real-valued matrix functions
where ΔLi ∈ R representing norm-bounded parameter uncertainties. According to system (1)-(3) and (9)-(11), the dynamic equations of error switched system for non-fragile observer can be prescribed: δ xk = [ Aσ ( k ) − ( Lσ ( k ) + ΔLσ ( k ) )Cσ ( k ) ]xk + Adσ ( k ) xk − n (12) + fσ ( k ) ( xk , k ) − fσ ( k ) ( xˆk , k ) p× m
III.
A. Stability analysis In this subsection, we investigate the stability of the following delta operator time delay switched nonlinear system (17) δ xk = Aσ ( k ) xk + Adσ ( k ) xk −n + fσ ( k ) ( xk , k )
(13) xk = φk , ∀k ∈ {−n, −n + 1," , 0} Without loss of generality, we make the following assumptions. Assumption 1 f i ( xk , k ) for i ∈ N are nonlinear functions
where
satisfying: f i ( xk , k ) − fi ( xˆk , k ) ≤ U i ( xk − xˆk ) where U i are known real constant matrices.
MAIN RESULTS
(14)
f i ( xk , k ) ≤ U i xk
for i ∈ N . Define the indicator
function:
ξ (k ) = (ξ1 (k ), ξ 2 (k )," , ξ N (k ))T
388
(18)
with i ∈ {1, 2," , N } , where ⎧1 when the ith subsystem is activated , ξi (k ) = ⎨ ⎩0 others then system (17) can be written as
xkT+1 P (ξ (k + 1)) xk +1 − xkT P (ξ (k )) xk 2T ( xk + T δ xk )T P(ξ (k + 1))( xk + T δ xk ) − xkT P (ξ (k )) xk = 2T 1 1 T T T ≤ xk [ Ai Pj + Pj Ai + TAi Pj Ai + ε iTU iTU i + ( Pj − Pi )]xk T 2 T T T T +2 xk [(TAi + I ) Pj Adi ]xk − n + xk − n (TAdi Pj Adi ) xk − n =
N
δ xk = ∑ ξi (k )( Ai xk + Adi xk −n + fσ ( k ) ( xk , k )) i =1
∀i ∈ {1, 2," , N }
> 0 and scalar ε i > 0 , i ∈ {1, 2," , N } , r = 1, 2," , s − 1 , s = 1, 2," , n − 1 , such that (19) Pi < ε i I
(21)
⎡ AiT Pj + Pj Ai + TAiT Pj Ai + ε i (T + 1)U iT U i ⎢ ⎢ + 1 ( P − 1 P ) − 1 R ( s ) + Q (1) j ⎢ T j 2 i n i ⎢ * ⎢ ⎢ * ⎢ ⎢ ⎢⎣ 1 (TAiT + I ) Pj Adi + Ri( s ) n Pi Adi
s =1
By (21), we can obtain that T (n) δ V2 (k , xk ) < xkT Q (1) j xk − xk − n Qi xk − n 1 T
A Pi − Pi + nT 2 R (1) j →
(22)
s
( r +1) j k −r
e
n s−1
n
s =2 r =1
s=1
− ∑∑ekT−r Ri(r )ek −r − ∑ekT−s Ri( s)ek −s
≤ xkT [ AiT Pj + Pj Ai + TAiT Pj Ai + εiTUiTUi 1 1 + (Pj − Pi )]xk + xkT−n (TAdiT Pj Adi ) xk −n T 2 2 T (1) +2xkT [(TAiT + I )Pj Adi ]xk −n + xkT Q(1) j xk + nT δ xk Rj δ xk
(27)
1 −xkT−nQi( n) xk −n − ( xk −n − xk )T Ri( s) ( xk −n − xk ) n Notice that (28) 0 = −2δ xkT Pi (δ xk − Ai xk − Adi xk − n − f i ( xk , k )) Combining (27) and (28) leads to ⎡ AiT Pj + Pj Ai + TAiT Pj Ai + εi (T +1)UiTUi ⎢ AiT Pi ⎢+ 1 (P − 1 P ) − 1 R( s) + Q(1) j ⎢ T j 2 i n i T ⎢ −Pi + nT 2 R(1) δV (k, xk ) ≤ η ⎢ * j → ⎢ * * ⎢ ⎢ ⎢⎣ 1 ⎤ (TAiT + I ) Pj Adi + Ri( s ) ⎥ n ⎥ Pi Adi ⎥η ⎥ 1 TAdiT Pj Adi − Qi( n ) − Ri( s ) ⎥ n ⎦
N
V3 (k , xk ) = T ∑∑ ekT− r R ( r ) (ξ (k ))ek − r = T ∑∑∑ ekT− rξi (k ) Ri( r ) ek − r s =1 r =1
T k −r
(24), (25) and (26), we have δV (k, xk ) = δV1 (k, xk ) + δV2 (k, xk ) + δV3 (k, xk )
N
n
s=1 r =1
n s −1
follows that this has to hold for special configuration ξi (k ) = 1 , ξ h ≠i (k ) = 0 , ξ j (k + 1) = 1 , ξ g ≠ j (k + 1) = 0 and for all xk ∈ R p . By
s =1 i =1
s
(1) j k
s=2 r =1
V2 (k , xk ) = T ∑ xkT− s Q ( s ) (ξ (k )) xk − s = T ∑∑ xkT− sξi (k )Qi( s ) xk − s n
(25)
s
By (20), we can obtain that n 1 n (∑ ek − s )T Ri( s ) (∑ ek − s ) δ V3 (k , xk ) ≤ nekT R (1) j ek − (26) n s =1 s =1 1 = − ( xk − n − xk )T Ri( s ) ( xk − n − xk ) + nT 2δ xkT R (1) j δ xk n If δ V (k , xk ) < 0 holds under arbitrary switching signal, it
Proof Consider the following switched LyapunovKrasovskii functional: (23) V (k , xk ) = V1 (k , xk ) + V2 (k , xk ) + V3 (k , xk ) with 1 1 ⎛ N ⎞ V1 (k , xk ) = xkT P (ξ (k )) xk = xkT ⎜ ∑ ξi (k ) Pi ⎟ xk 2 2 ⎝ i =1 ⎠ s =1
n
= ne R e + ∑∑e R
under arbitrary switching.
n
s
s =1 r =1
T k
⎤ ⎥ ⎥ ⎥ 0 , Ri( r ) > 0 , Ri( s ) > 0 , Qi( s ) > 0 ,
R (j r +1) < Ri( r ) , r = 1, 2," , s − 1
n
1 T
(24)
s =1 r =1 i =1
with Pi , Qi( s ) , Ri( r ) , i ∈ {1, 2," , N } being symmetric positivedefinite matrices and ek = xk − xk +1 . Then switched Lyapunov-Krasovskii functional in delta domain has the following form: V (k + 1, xk +1 ) − V1 (k , xk ) δ V1 (k , xk ) = 1 T
389
B. Observer Design Now we consider system (7)-(8). The following theorem presents sufficient conditions for the existence of asymptotic stability of system (7)-(8). Theorem 2 Consider system (7)-(8), if there exist a set of matrices X i > 0 , Vi ( s ) > 0 , Vi ( n ) > 0 , Z i( r ) > 0 , Z i( s ) > 0 , Vij( s ) > 0 , Vij( n ) > 0 , Z ij( r ) > 0 , Z ij( s ) > 0 , ε i > 0 , i ∈ {1, 2," , N } ,
T
where η = ⎡ xkT δ xkT xkT−n ⎤ . From (22), we have δ V (k , xk ) < 0 . ⎣ ⎦ By Lemma 2, the system (17) is asymptotically stable. The proof is completed. Remark 1 When time delay n ≡ 0 and fσ ( k ) ( xk , k ) ≡ 0 , system (17) becomes as follows:
δ xk = Aσ ( k ) xk where Aσ ( k ) = Aσ ( k ) + Adσ ( k ) . From (22), we know that the matrix inequality imply that the following inequality holds: 1 AiT Pj + Pj Ai + TAiT Pj Ai + ( Pj − Pi ) < 0 T The above inequality is just a sufficient condition of stability for the delta operator switched system without time delay, which can be expressed in [14].
r = 1, 2," , s − 1 , s = 1, 2," , n − 1 , and Wi such that the following LMIs have feasible solutions (35) X i > ε i−1 I ( r +1) (r) (36) Z ij < Z i , r = 1, 2," , s − 1 Vij( s +1) < Vi ( s ) , s = 1, 2," , n − 1 1 (s) ⎡ 1 (1) ⎢ − 2T X i + Vij − n Z i ⎢ * ⎢ ⎢ * ⎢ ⎢ ⎢ * ⎢ * ⎣
Remark 2 When sample period T = 0 and fσ ( k ) ( xk , k ) ≡ 0 , system (17) becomes a continuous-time switched linear system as follows: x (t ) = Aσ ( t ) x(t ) + Adσ ( t ) x(t − d )
(29)
We can obtain sufficient condition of stability for system (29) by Theorem 1. Corollary 1 Consider system (29). If there exist symmetric positive definite matrices P > 0 , R > 0 , Q > 0 , such that 1 1 ⎤ ⎡ T T ⎢ Ai P + PAi − d R + Q Ai P PAdi + d R ⎥ (30) ⎢ ⎥ −2 P PAdi ⎥ < 0 * ⎢ ⎢ 1 ⎥ −Q − R ⎥ * * ⎢ d ⎦ ⎣ ∀i ∈ {1, 2," , N } , then system (29) is asymptotically stable under arbitrary switching. Remark 3 Let Aσ ( k ) = Aσ ( k ) + I . When sample period T = 1
−2 Pi + nR (1) j *
−Vi ( n ) −
*
*
*
*
1 (s) → Zi n
(38)
T ( Ai X i − Wi ) + X i
X iU
0
0
T i
T T ⎡( Ai − LC i i ) Pj + Pj ( Ai − LC i i ) +T ( Ai − LC i i ) Pj ( Ai − LC i i) T ⎢ (Ai − LC i i ) Pi ⎢+ε (T +1)UTU + 1 (P − 1 P) − 1 R(s) + Q(1) i i j i i j ⎢ i 2 T n ⎢ * Mij = ⎢ −Pi + nT 2R(1) j → ⎢ * * ⎢ ⎢ ⎢⎣ 1 (s ) ⎤ T [T(Ai − LC i i ) + I ]Pj Adi + Ri ⎥ n ⎥ PA ⎥ i di ⎥ (n) 1 ( s) T TAdi Pj Adi −Qi − Ri ⎥ n ⎦
Qi( n ) > 0 , i ∈ {1, 2," , N } , r = 1, 2," , s − 1 , s = 1, 2," , n − 1 , such that (32) R (j r +1) < Ri( r ) , r = 1, 2," , s − 1
AiT Pi
*
can guarantees system (7)-(8) is asymptotically stable under arbitrary switching. Proof Denote
Corollary 2 Consider system (31). If there exist symmetric positive definite matrices Pi > 0 , Ri( r ) > 0 , Ri( s ) > 0 , Qi( s ) > 0 ,
⎡ AiT Pj + Pj Ai + AiT Pj Ai ⎢ ⎢ − P − P − 1 R ( s ) + Q(1) j ⎢ j i n i ⎢ * ⎢ ⎢ * ⎢ ⎢ ⎢⎣
− X i + nT 2 Z ij(1)
1 (s) Zi n Adi X i
⎤ ⎥ ⎥ ⎥ 0 TX i AdiT ⎥ ε i−1 I (43) Z ij( r +1) < Z i( r ) , r = 1, 2," , s − 1
(41)
⎤ TX i ( Ai − Li Ci )T + X i ⎥ ⎥ 0 ⎥ ⎥ 0 , Vi ( s ) > 0 , Vi ( n ) > 0 , Z i( r ) > 0 , Z i( s ) > 0 , (s) ij
V
>0 ,V
(n) ij
>0, Z
(r ) ij
>0 , Z
(s) ij
T ( Ai X i − Wi )T + X i μ iTH i H iT TX i AdiT −TX j + μiT 2 H i H iT
> 0 , ε i > 0 , i ∈ {1, 2," , N } ,
r = 1, 2," , s − 1 , s = 1, 2," , n − 1 and Wi , LMIs (35)-(38) have feasible solutions, then go to Step 2. Step 2. If the equations Li Ci = Wi X i−1 have feasible solutions for unknown matrices Li , then the observer can be constructed
* *
X iU iT 0 0 0 1 − I ε i (T + 1) *
(44) 1 (s) Zi n Adi X i 1 (n) −Vi − Zi( s) → n * * *
− X i CiT EiT ⎤ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥
