Observer Design for a Class of Nonlinear ... - Semantic Scholar
Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
FrC07.5
Observer design for a class of nonlinear descriptor systems Chunyu Yang, Qingling Zhang and Tianyou Chai
Abstract— This paper considers the problem of observer design for a class of nonlinear descriptor systems whose nonlinear terms are time-varying and satisfy a quadratic inequality. The error system is firstly represented by a Lur’e descriptor system (LDS). Then, motivated by the basic idea of absolute stability theory, both types of full-order and reduced-order observers are constructed by a unified approach. The proposed observers guarantee exponential convergence of the state estimation error to zero. Finally, the design methods are reduced to a linear matrix inequality problem and illustrated by an example.
I. I NTRODUCTION Observer design for descriptor systems has been an active field of research in the past several decades. Many approaches have been developed for linear descriptor systems (see [1], [2], [3], [4], [5], [6], [7] and the references therein). Nonlinear descriptor systems are also considered by many researchers and some recent progresses are reported in [8], [9], [10], [11], [6], [12], [13]. In [8], a local asymptotic observer is obtained for nonlinear descriptor systems by means of coordinate transformation and a reduced-order observer design approach is developed by using a generalized Sylvester equation. In [9], a full-order observer is constructed for a class of nonlinear descriptor systems subject to unknown inputs and faults by dividing the system into dynamic system and static system. [10] considers a class of nonlinear descriptor systems in quasi-linear form and presents a full-order observer design method. The approach is based on rewriting the descriptor system as an equivalent system of (explicit) differential equations on a restricted manifold. In [11], the observer design problem is studied for semi-explicit nonlinear descriptor systems of index one. The proposed observer is formulated as a differential-algebraic equation of index one and the observer error dynamics are ensure to be locally stable. In [6], [12], descriptor systems subject to nonlinear uncertainty and external disturbances are considered. The nonlinear terms are assumed to be globally Lipschitz and the observers are constructed to estimate the states and/or the disturbances. [13] addresses the issues of full-order and reduced-order observer designs for a class of descriptor systems with global Lipschitz constraint. The design of both types of observers is formulated as a unified LMI conditions. This work is partially supported by the National Fundamental Research Program of China (No. 2009CB320601) and the Funds for Creative Research Groups of China (60521003) Chunyu Yang and Tianyou Chai are with the Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang, 110004, P.R. China. [email protected],
[email protected] Qingling Zhang is with Institute of Systems Science, Northeastern University, Shenyang, 110004, P.R. China. [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
In this paper, we consider the observer design problem for a class of nonlinear descriptor systems. We remove the globally Lipschitz restriction and only assume that the nonlinearities satisfy a given quadratic inequality. Under our assumption, the observer error system is represented as the interconnection of a linear time-invariant descriptor system and a time-varying nonlinearity in the unbounded sector, that is, an LDS. Thus the convergence of the observer error is equivalent to the stability of the obtained LDS. Though some stability results for LDS have been given by [14] and [15], they can not be used directly in this paper since [14] and [15] require the nonlinearities to be bounded sector constrained and time-invariant, respectively. Motivated by the basic idea of [14] and [15], we present a unified approach to design both types of full-order and reduced-order observers. The error systems are guaranteed to be globally exponentially convergent. It is shown that the obtained results are more general than the existing results for standard state-space systems given in [16] and [17]. Finally, we reduce the obtained design methods to an LMI problem and present an illustrative example. II. P RELIMINARIES Consider a linear time-invariant descriptor system E x˙ = Ax + Bu,
(1)
where x ∈ Rn is the state variable, u ∈ Rm is the input variable, the matrices A, E ∈ Rn×n , B ∈ Rn×m are constant and rank(E) = s ≤ n. The following lemmas will be used in the sequel. L EMMA 2.1: [19] Assume that (E, A) is in the form of · ¸ · ¸ Is 0 A11 A12 E= ,A = , A21 A22 0 0 then (E, A) is regular and impulsive-free if and only if A22 is nonsingular. L EMMA 2.2: [21] The pair (E, A) is admissible if and only if there exists a matrix X ∈ Rn×n satisfying E T X = X T E ≥ 0, AT X + X T A < 0. III. M AIN RESULTS Consider the following descriptor system E x˙ = Ax + Bγ(t, Hx) + ψ(t, y, u), y = Cx,
(2)
where x ∈ Rn , u ∈ Rm , y ∈ Rp are the system state, input and output, respectively. The system matrices A, E ∈ Rn×n , B ∈ Rn×r , C ∈ Rp×n , H ∈ Rr×n are constant and
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FrC07.5 rank(E) = s ≤ n. The functions γ(·, ·) and ψ(·, ·) are locally Lipschitz. Through out this paper, the nonlinear term γ(t, v) is assumed to have the following decoupled structure γ1 (t, v1 ) v1 γ2 (t, v2 ) v2 v = . , γ(t, v) = , .. .. . γk (t, vk ) vk
Θ = diag{Θ1 , Θ2 , · · · , Θk } Pk ri ×ri where vi ∈ R , ,i = i=1 ri = r, and Θi ∈ R 1, 2, · · · , k are given and satisfy Θi > 0. Furthermore, γ(t, v) is assumed to satisfy the quadratic inequality ri
(a − b)T Θ(γ(t, a) − γ(t, b)) ≥ 0, ∀a, b ∈ Rr , t ≥ 0.
(4)
where the matrices L ∈ Rn×p and K ∈ Rr×p are to be designed. Defining the state estimation error by e = x − x ˆ, from (2) and (4), the dynamics of the observer error e = x − x ˆ are governed by E e˙ = (A + LC)e + B[γ(t, v)) − γ(t, w)],
(5)
where v = Hx, w = H x ˆ + K(C x ˆ − y). We begin the observer design by representing the observer error system (5) as the feedback interconnection of a linear system and a multivariable sector nonlinearity. To this end, we view γ(t, v))−γ(t, w) as a function of t and z , v−w = (H + KC)e, that is, a time-varying nonlinearity in z φ(t, z) , γ(t, v) − γ(t, w). Then, we can rewrite the observer error system (5) as E e˙ = (A + LC)e + Bφ(t, z), z = (H + KC)e.
E T P = P T E ≥ 0,
(3)
R EMARK 3.1: A nonlinearity satisfying inequality (3) may not satisfy the globally Lipschitz condition, for example, a3 . Thus, the observer design methods given by [6], [12], [13] are not valid for system (2) with nonlinearities satisfying inequality (3). In [16] and [17], observer design problems for system (2) with E = In are considered. One can see that the admissible nonlinearities of this paper are more general than those of [16] and [17]. For system (2), we construct a full-order observer Ex ˆ˙ = Aˆ x + L(C x ˆ − y) + Bγ(t, H x ˆ + K(C x ˆ − y)) +ψ(t, y, u),
sector (7). Thus, the convergence of the observer error is equivalent to the stability of the LDS (6). When E = In , using the absolute stability theory for standard statespace Lur’e systems, the observer design problem can be reduced to the feasibility problem of a set of LMIs [16], [17]. However, for general E which may be singular, the preliminary stability results of LDS (6) in [14] and [15] are not suitable for the observer design problem because they require the nonlinearities to be bounded sector constrained or time-invariant. Now we are ready to give the full-order observer design method. T HEOREM 3.1: If there exist matrices Λ as in (8), P ∈ Rn×n , L ∈ Rn×p , K ∈ Rr×p and scalar τ > 0 such that
(6)
and ·
(7)
⋆ 0
¸
≤ 0, (10)
hold, then there exists a full-order observer in the form of (4) for system (2) and the resulting observer gains L, K ensure that the estimation error is globally exponentially convergent. Proof: Denote A = A + LC, C = H + KC. Using Lemma 2.2, inequalities (9) and (10) imply that (E, A) is admissible. Then there exist two nonsingular matrices M, N ∈ Rn×n , such that · ¸ · ¸ Is 0 A1 0 M EN = , M AN = , 0 0 0 In−s where A1 ∈ Rs×s . Correspondingly, let · ¸ £ ¤ B1 MB = , CN = C1 C2 , B2 · ¸ P11 P12 −T M PN = . P21 P22
Then pre- and post-multiplying (9) by N T and N , reT spectively, we have P11 = P11 ≥ 0, P12 = 0. Further, it follows from (10) that P11 is nonsingular which shows T P11 = P11 > 0. ¸ · e1 −1 , and pre-multiplying (6) by M , Letting N e = e2 we have e˙ 1 = A1 e1 + B1 φ(t, z), e2 = −B2 φ(t, z), (11) z = C1 e1 − C2 B2 φ(t, z). For any scalar µ > 0, pre- and post-multiplying (10) by # " T In µC 0 Ir
Taking into account the structure of γ and inequality (3), we have that φ(t, z) satisfies z T ΛΘφ(t, z) ≥ 0, ∀z ∈ Rr , t ≥ 0,
(A + LC)T P + P T (A + LC) + τ I B T P + ΘΛ(H + KC)
(9)
and its transpose, respectively, show that (A + µBC)T P + P T (A + µBC) < 0.
where Λ = diag{λ1 Ir1 , λ2 Ir2 , · · · , λk Irk }, λi > 0.
(8)
The error system (6) is an LDS and the interconnected nonlinearity φ(t, z) is time-varying and in the unbounded
Then, using Lemma 2.2, we have that (E, A + µBC) is admissible. In addition, we note that · ¸ Is 0 M EN = , 0 0
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·
A1 + µB1 C1 µB2 C1
µB1 C2 In−s + µB2 C2
¸
where
,
which indicates, by Lemma 2.1, det(In−s + µB2 C2 ) 6= 0.
λmax (P11 ) . λmin (P11 ) From (11), we have z = C1 e1 − C2 B2 φ. Then taking into account (7) and (23), there exists δ > 0, such that ζ=
(12)
0 ≤ 2φT ΘΛz ≤ 2φT ΘΛC1 e1 − δφT φ.
It follows from inequality (12) that det(Ir + µC2 B2 ) 6= 0.
(13)
Since µ > 0 is arbitrary, inequality (13) shows 1 det( Ir + C2 B2 ) 6= 0 µ
(14)
det(C2 B2 ) 6= 0
(15)
which implies that
φT φ ≤ ρ2 eT1 e1 ,
by letting µ → +∞ in (14). Then, we have that B2 is full collum rank. Pre- and post-multiplying (10) by ¸ · T N 0 0 I and its transpose, respectively, we have T A1 P11 + P11 A1 + τ I ⋆ T P P + P 21 22 22 + τ I T ΘΛC1 + B1 P11 +B2T P21 ΘΛC2 + B2T P22 Pre- and post-multiplying (16) I 0 0 −B2T 0 I
where ρ2 =
ke(t)k2
⋆ ⋆ ≤ 0. 0 (16)
0 I 0
and its transpose, respectively, give (20) shown on the top of the next page. And (20) implies (21) shown on the top of the next page. Then −ΘΛC2 B2 − B2T C2T ΛΘ + τ B2T B2 ≤ 0.
(22)
Since B2 is full collum rank, we have (23)
Let V (e1 ) := eT1 P11 e1 . Calculating the derivative of V (e1 ) along the trajectories of system (11) and using inequalities (7) and (21), we have V˙ (e1 )(11)
= e˙ T1 P11 e1 + eT1 P11 e˙ 1 ¸ · £ T ¤ e1 T e1 φ = S φ ≤ −τ eT1 e1 .
1 η kΘΛC1 k
δ−η
(28)
(29)
> 0.
Then, we have
by
ΘΛC2 B2 + B2T C2T ΛΘ > 0.
For arbitrary scalar η > 0, it holds that 1 2φT ΘΛC1 e1 ≤ ηφT φ + kΘΛC1 keT1 e1 . η Then, we can choose η < δ, such that 1 (δ − η)φT φ ≤ kΘΛC1 keT1 e1 , η which shows that
(27)
(24)
Then, there exists a scalar α > 0, such that V˙ (e1 ) ≤ −αV (e1 ),
(25)
eT1 (t)e1 (t) ≤ ζe−αt eT1 (0)e1 (0),
(26)
which indicates
≤ ke1 (t)k2 + ke2 (t)k2 ≤ (1 + ρ2 kB2 k2 )ζe−αt ke(0)k2 ,
(30)
that is, ke(t)k2 ≤ µe−αt ke(0)k2 , 2
2
(31)
where µ = (1 + ρ kB2 k k)ζ. Then, the estimation error is globally exponentially convergent. R EMARK 3.2: When E = In , Theorem 3.1 reduces to Theorem 1 of [17]. But the admissible nonlinearities in this paper are more general than those in [17]. In fact, a standard state-space system can be realized by a descriptor system by introducing an additional descriptor variable (see, for example, [22]). In this case, a direct application of Theorem 1 in [17] to the system is not good enough since it has less design variables in the design procedure than our Theorem 3.1, which will be illustrated by Example 1 in the next section. R EMARK 3.3: Theorem 3.1 also describes a stability criterion for LDS (6) whose interconnected nonlinearities are admitted to be time-varying and in unbounded sector. Thus, this criterion is complementary to the existing stability results given in [14] and [15]. In applications, it may be more convenient to employ a reduced-order observer, which generates estimates only for the unmeasured states. To this end, we make the following common assumption. £ ¤ 3.1: rank E T C T = n and C = £ A SSUMPTION ¤ Ip 0 . In the following discussion, · ¸ Assumption 3.1 always holds. 0 Let M1 , E . It is easy to show that In−p rank(M1 ) = n − p which guarantees that we can find a matrix M0 ∈ Rn×p such that £ ¤−1 M , M 0 M1 (32)
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AT1 P11 + P11 A1 + τ I ΘΛC1 + B1T P11 P21 S,
·
⋆ −ΘΛC2 B2 − B2T C2T ΛΘ + τ B2T B2 −P22 B2 − τ B2 + C2T ΛΘ
AT1 P11 + P11 A1 ΘΛC1 + B1T P11
⋆ −ΘΛC2 B2 − B2T C2T ΛΘ
exists. Then, we have that M E and M A are of the following structure · ¸ · ¸ E1 0 A11 A12 ME = , MA = , (33) E2 In−p A21 A22 where E1 ∈ Rp×p , E2 ∈ R(n−p)×p , A11 ∈ Rp×p , A12 ∈ Rp×(n−p) , A21 ∈ R(n−p)×p , A22 ∈ R(n−p)×(n−p) . We introduce the following new state £ ¤ ω = 0 In−p x + (Lr E1 + E2 )y,
where Lr ∈ R(n−p)×p is the observer gain to be design. For convenience, let · ¸ Ip 0 Γ1 = , −E2 − Lr E1 In−p
then we have x = Γ1 and ω=
£
Lr
·
y ω
In−p
¸ ¤
M Ex.
(34)
(35)
It can be seen from (34) that x can be easily estimated if we can obtain the estimator of ω. The dynamics of state ω can be represented by £ ¤ Lr In−p M E x˙ ω˙ = ¤ £ Lr In−p M (Ax + Bγ(t, Hx) = +ψ(t, y, u)). (36) Then, we have
¤ £ ω˙ = (A22 + Lr A12 )ω + Lr In−p M B e y, u), ×γ(t, (H1 − H2 (E2 + Lr E1 ))y + H2 ω) + ψ(t, (37) £ ¤ e y, u) = Lr In−p M ψ(t, y, u) + [A21 + where ψ(t, r×p Lr A11 − (A22 + Lr A12 )(E2 +£Lr E1 )]y and ¤ H1 ∈ R , r×(n−p) H2 ∈ R such that H = H1 H2 . For plant (37), the observer is given by £ ¤ ω ˆ˙ = (A22 + Lr A12 )ˆ ω + Lr In−p M B e y, u). ×γ(t, (H1 − H2 (E2 + Lr E1 ))y + H2 ω ˆ ) + ψ(t, (38) Denoting δ = ω − ω ˆ as the error state, we have £ ¤ δ˙ = (A22 + Lr A12 )δ + Lr In−p M Bφ(t, z), (39)
¸
⋆ ≤0 ⋆ T + τI P22 + P22 ≤
·
−τ I 0
0 0
¸
(20)
(21)
Based on the above discussion, we describe a reduced order observer design method in the following theorem whose proof is partially motivated by [13]. T HEOREM 3.2: Under Assumption 3.1, if the conditions of Theorem 3.1 hold, there exists a reduced-order observer in the form of (38) for system (2) and the estimation error is globally exponentially convergent. Proof: Assume that the conditions of Theorem 3.1 hold, that is, inequalities (9) and (10) are feasible. Let ¸ · P1 P2 , M −T P = P3 P4 where M is defined as (32). Based on the new decomposition of matrices of E, A in (33), inequality (9) shows · T ¸ E1 P1 + E2T P3 E1T P2 + E2T P4 P3 P4 · T ¸ T T P1 E1 + P3 E2 P3 = ≥ 0, (40) P2T E1 + P4T E2 P4T which implies that · T E1 P1 + E2T P3 P3
P3T P4
¸
≥ 0.
(41)
By inequality £ (10), we ¤have that P is nonsingular, which indicates that P3 P4 is full row rank. Then, it follows from (41) that P4 > 0. Substituting (33) into (10) ¸ (42) shown on the top of · yields £ ¤ Ip LT P +P T L Ip 0 + the next page, where Γ2 = 0 τ In . Inequality (42) implies that T A12 P2 + AT22 P4 + P2T A12 +P4·A22 +¸τ In−p ⋆ ≤ 0. (43) P 2 T T B M + ΘΛH2 0 P4
where φ(t, z) = γ(t, (H1 − H2 (E2 + Lr E1 ))y + H2 ω) − γ(t, (H1 − H2 (E2 + Lr E1 ))y + H2 ω ˆ ) and z = H2 δ. It can be shown that φ(t, z) satisfies inequality (7).
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Choosing Lr = P4−1 P2T , inequality (43) is equivalent to P4 (A22 + Lr A12 ) +(A22 + Lr A12 )T P4 + τ In−p ⋆ ≤ 0. ¸ · (44) T T LTr B M P4 + ΘΛH2 0 In−p Defining Lyapunov function
V (δ) = δ T P4 δ,
(45)
FrC07.5 ·
A11 A21
¸T ·
¸ · ¸T · ¸ P1 P2 P1 P2 A11 A12 + + Γ2 A21 A22 ·P3 P4 ¸ P3 P4 £ ¤ P1 P2 BT M T + ΘΛ H1 + K H2 P3 P4
A12 A22
we have V˙ |(39)
= 2δ T P4 (A22 + Lr A12 )δ £ ¤ +2δ T P4 Lr In−p M Bφ
(42)
IV. N UMERICAL EXAMPLES
By inequality (44), we have V˙ |(39) ≤ −τ δ T δ which implies that the error system (39) is globally exponentially stable. R EMARK 3.4: When E = In , Q = Ir and k = r, the matrix inequality conditions in Theorem 3.2 are equivalent to those given by [16]. Thus, our design methods are generalizations of the nonlinear observer design methods for standard state-space systems given in [16]. Using Theorem 3.1 and 3.2, both types of full-order and reduced-order nonlinear observers are constructed by a unified approach which can be reduced to the LMI problem in the following theorem. T HEOREM 3.3: If there exist a matrix Λ in the form of (8), a nonsingular matrix P ∈ Rn×n , matrices Q ∈ Rn×p , F ∈ Rr×p and scalar τ > 0, such that the LMIs ET P = P T E ≥ 0 AT P + P T A + C T QT + QC + τ I B T P + ΘΛH + ΘF C
⋆ ≤0 0
quadratic inequality. We will consider this problem in our future work.
≤ 2δ T P4 (A22 + Lr A12 )δ £ ¤ +2δ T P4 Lr In−p M Bφ + 2δ T H2T ΛΘφ. (46)
and ·
(47) ⋆ 0
¸
≤0
(48)
hold, then there exist a full-order observer of the form (4) with L = P −T Q, K = Λ−1 F , and a Q reduced-order Q observer of the form (38) with Lr = ( 1 )−1 ( 2 )T , £ ¤ −T £ ¤T Q Q and 2 = where 1 = 0 In−p M P 0 In−p £ ¤ £ ¤T Ip 0 M −T P 0 In−p . The proof of Theorem 3.3 is direct according to Theorems 3.1 and 3.2. R EMARK 3.5: It is known that LMIs can be solved by highly efficient numerical algorithms in polynomial time [23], [24]. Some of these algorithms have been incorporated into different computer tools for the resolution of LMI problems [25], [26]. R EMARK 3.6: In [27], the observer design problem is considered for standard state-space systems whose nonlinear terms satisfy an incremental quadratic constraint. This kind of nonlinearities include many commonly encountered nonlinearities, such as, globally Lipschitz ones, monotonic ones and also encompass the nonlinearities satisfying (3). However, because of the complex nature of descriptor systems, it is more complex to design observers for descriptor systems with nonlinearities satisfying the so-called incremental
Example 1: Descriptor systems arise naturally as dynamic models of a wide range of engineering applications. This example considers the circuit displayed in Fig. 1, where a dc source with voltage µ is connected in series to a linear resistor, a linear inductor and a nonlinear capacitor with q−v characteristic q = z(v) = (v − v0 )1/3 + q0 . Similar nonlinear capacitors are considered in [28] and [29]. This circuit may be easily shown to admit the charge-flux description q˙ = φ/L, (49) φ˙ = −φR/L − v + µ, 0 = v − v0 − (q − q0 )3 . where φ is the magnetic flux in the inductor. T £ Choose the state ¤ £and input vector as ¤follows: x = x1 x2 x3 = q − q0 φ v − v0 , u = µ − v0 . Assume that x1 and x3 can be measured. Then system (49) can be expressed by system (2) with 1 0 0 0 1/L 0 E = 0 1 0 , A = 0 −R/L −1 , 0 0 0 0 0 1 0 £ ¤ £ ¤ B = 0 ,H = 1 0 0 ,C = 1 0 1 −1 £ ¤ and γ(·) = (·)3 , ψ T = 0 u 0 . Let R = 1, L = 0.5, µ = 2, v0 = 1, q0 = 1. It can be seen that γ(Hx) does not satisfy the globally Lipschitz condition, thus the results given by [6], [12] and [13] are not valid for this example. Letting Θ = 1, we have that γ(Hx) satisfies (3), then, our observer design method can be used. Using Theorem 3.3 with the aid of YALMIP [26], the full-observer gains are given by −2.9131 L = 0.8290 , K = −0.4598. 0.0000
The simulation results are given in Fig. 2. On the other hand, descriptor system (49) can be rewritten as the following standard state-space system q˙ = φ/L, φ˙ = −φR/L − (q − q0 )3 − v0 + µ.
(50)
We found that the observer design methods given by [16], [17] are not feasible.
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FrC07.5 R
L
P
v q z(v)
Fig. 1.
A nonlinear circuit
3
3 x
x
1
2
estimate of x
2
estimate of x
1
2
2
1 0
1
−1 0
−2 0
5 sec
10
10
0
5 sec
10
3 x
8
norm of estimate error
3
estimate of x
3
2
6 4
1
2 0
0 0
5 sec
Fig. 2.
10
0
5 sec
10
States and the estimation performance.
This example shows that the circuit system shown in Fig. 1 can be modeled as the descriptor system (49) and also the standard state-space system (50). Theorem 3.1 can design a full-order observer, but the methods proposed by [16], [17] do not work. The reason is that Theorem 3.1 admit more free parameters in the design procedure. V. C ONCLUSIONS In this paper, we have studied the observer design problem for a class of nonlinear descriptor systems. The nonlinear term may be not globally Lipschitz and is only assumed to satisfy a quadratic inequality. The observer error system is represented by an LDS. Both types of full-order and reducedorder nonlinear observers are constructed by a unified LMI approach, by which the observer error system is guaranteed to be globally exponentially convergent. The design example has illustrated the obtained methods. R EFERENCES [1] M. M. Fahmy and J. O’Reilly, “Observers for descriptor systems,” Int. J. Control, vol. 49, no. 6, pp. 2013–2028, 1989. [2] M. Darouach and M. Boutayeb, “Design of observers for descriptor systems,” IEEE Trans. Autom. Control, vol. 40, no. 7, pp. 1323–1327, 1995. [3] M. Darouach, M. Zasadzinski, and M. Hayar, “Reduced-order observer design for descriptor systems with unknown inputs,” IEEE Trans. Autom. Control, vol. 41, no. 7, pp. 1068–1072, 1996. [4] M. Hou and P. C. Muller, “Observer design for descriptor systems,” IEEE Trans. Autom. Control, vol. 44, no. 1, pp. 164–169, 1999.
[5] D. Koenig and S. Mammar, “Design of proportional-integral observer for unknown input descriptor systems,” IEEE Trans. Autom. Control, vol. 47, no. 12, pp. 2057–2062, 2002. [6] Z. Gao and D. W. C. Ho, “State/noise estimator for descriptor systems with application to sensor fault diagnosis,” IEEE Trans. Signal Processing, vol. 54, no. 4, pp. 1316–1326, 2006. [7] H. Trinh, S. Nahavandi and T. D. Tran, “Algorithms for Designing Reduced-order Functional Observers of Linear Systems,” Int. J. of Innovative Computing, Information and Control, vol. 4, no. 2, pp. 321-334, 2008. [8] M. Boutayeb and M. Darouach, “Observers design for nonlinear descriptor systems,” In Proc. IEEE Conf. Decision and Control, New Orleans, LA, 1995, pp. 2369–2374. [9] N. D. Shields, “Observer design and detection for nonlinear descriptor systems,” Int. J. Control, vol. 67, no. 2, pp. 153–168, 1997. [10] G. Zimmera and J. Meierb, “On observing nonlinear descriptor systems,” Syst. Control Lett., vol. 32, no. 1, pp. 43–48, 1997. [11] J. Aslund and E. Frisk, “An observer for non-linear differentialalgebraic systems,” Automatica, vol. 42, no. 6, pp. 959–965, 2006. [12] D. Koenig, “Observer design for unknown input nonlinear descriptor systems via convex optimization,” IEEE Trans. Autom. Control, vol. 51, no. 6, pp. 1047–1052, 2006. [13] G. Lu and D. W. C. Ho, “Full-order and reduced-order observers for Lipschitz descriptor systems: the unified LMI approach,” IEEE Trans. Circiuts and Syst. II: Express Briefs, vol. 53, no. 7, pp. 563–567, 2006. [14] C. Y. Yang, Q. L. Zhang, Y. P. Lin, and L. N. Zhou, “Positive realness and absolute stability problem of descriptor systems,” IEEE Trans. Circuits and Syst. I: Regular papers, vol. 54, no. 5, pp. 1142–1149, 2007. [15] C. Y. Yang, Q. L. Zhang, and L.N. Zhou, “Strongly absolute stability of Lur’e type differential-algebraic systems,” J. Math. Anal. Appl., vol. 336, no. 1, pp. 188–204, 2007. [16] M. Arcak and P. Kokotovic, “Nonlinear observers: a circle criterion design and robustness analysis,” Automatica, vol. 37, no. 12, pp. 1923– 1930, 2001. [17] X. Fan and M. Arcak, “Observer design for systems with multivariable monotone nonlinearities,” Syst. Control Lett., vol. 50, no. 4, pp. 319– 330, 2003. [18] S. L. Campbell, Singular Systems of Differential Equations. Pitman, London, 1980. [19] L. Dai, Singular Control Systems. Springer, Berlin, Germany, 1989. [20] E. L. Lewis, “A survey of linear singular systems,” Circuits, Systems and Signal Processing, vol. 5, no. 1, pp. 3–36, 1986. [21] I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda, “H∞ control for descriptor systems: a matrix inequalities approach” Automatica, vol. 33, no. 4, pp. 669–673, 1997. [22] Y. Cao and Z. Lin, “A descriptor system approach to robust stability analysis and controller synthesis,” IEEE Trans. Autom. Control, vol. 49, no. 11, pp. 2081–2084, 2004. [23] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [24] L. El Ghaoui and S. Niculescu, Advances in Linear Matrix Inequality Methods in Control. Philadelphia, PA: SIAM, 2000. [25] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox. Technical report, The Mathworks Inc., 1995. [26] J. L¨ofberg. YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference, Taipei, Taiwan, 2004. [27] A. B. Acikmese and M. Corless, “Observers for systems with nonlinearities satisfying an incremental quadratic inequality,” In Proc. Amer. Control Conf., Portland, OR, USA, 2005, pp. 3622–3629. [28] V. Venkatasubramanian, A. Saberi, and Z. Lin, “On the validity of solutions and equilibrium points in a nonlinear network,” IEEE Trans. Circuits Syst. I:Fundam. Theory Appl., vol. 43, no. 4, pp. 233–235, 1996. [29] R. Riaza, “Double SIB points in differential-algebraic systems,” IEEE Trans. Autom. Control, vol. 48, no. 9, pp. 1625–1629, 2003.
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Observer design for a class of nonlinear descriptor systems Chunyu Yang, Qingling Zhang and Tianyou Chai
Abstract— This paper considers the problem of observer design for a class of nonlinear descriptor systems whose nonlinear terms are time-varying and satisfy a quadratic inequality. The error system is firstly represented by a Lur’e descriptor system (LDS). Then, motivated by the basic idea of absolute stability theory, both types of full-order and reduced-order observers are constructed by a unified approach. The proposed observers guarantee exponential convergence of the state estimation error to zero. Finally, the design methods are reduced to a linear matrix inequality problem and illustrated by an example.
I. I NTRODUCTION Observer design for descriptor systems has been an active field of research in the past several decades. Many approaches have been developed for linear descriptor systems (see [1], [2], [3], [4], [5], [6], [7] and the references therein). Nonlinear descriptor systems are also considered by many researchers and some recent progresses are reported in [8], [9], [10], [11], [6], [12], [13]. In [8], a local asymptotic observer is obtained for nonlinear descriptor systems by means of coordinate transformation and a reduced-order observer design approach is developed by using a generalized Sylvester equation. In [9], a full-order observer is constructed for a class of nonlinear descriptor systems subject to unknown inputs and faults by dividing the system into dynamic system and static system. [10] considers a class of nonlinear descriptor systems in quasi-linear form and presents a full-order observer design method. The approach is based on rewriting the descriptor system as an equivalent system of (explicit) differential equations on a restricted manifold. In [11], the observer design problem is studied for semi-explicit nonlinear descriptor systems of index one. The proposed observer is formulated as a differential-algebraic equation of index one and the observer error dynamics are ensure to be locally stable. In [6], [12], descriptor systems subject to nonlinear uncertainty and external disturbances are considered. The nonlinear terms are assumed to be globally Lipschitz and the observers are constructed to estimate the states and/or the disturbances. [13] addresses the issues of full-order and reduced-order observer designs for a class of descriptor systems with global Lipschitz constraint. The design of both types of observers is formulated as a unified LMI conditions. This work is partially supported by the National Fundamental Research Program of China (No. 2009CB320601) and the Funds for Creative Research Groups of China (60521003) Chunyu Yang and Tianyou Chai are with the Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang, 110004, P.R. China. [email protected],
[email protected] Qingling Zhang is with Institute of Systems Science, Northeastern University, Shenyang, 110004, P.R. China. [email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
In this paper, we consider the observer design problem for a class of nonlinear descriptor systems. We remove the globally Lipschitz restriction and only assume that the nonlinearities satisfy a given quadratic inequality. Under our assumption, the observer error system is represented as the interconnection of a linear time-invariant descriptor system and a time-varying nonlinearity in the unbounded sector, that is, an LDS. Thus the convergence of the observer error is equivalent to the stability of the obtained LDS. Though some stability results for LDS have been given by [14] and [15], they can not be used directly in this paper since [14] and [15] require the nonlinearities to be bounded sector constrained and time-invariant, respectively. Motivated by the basic idea of [14] and [15], we present a unified approach to design both types of full-order and reduced-order observers. The error systems are guaranteed to be globally exponentially convergent. It is shown that the obtained results are more general than the existing results for standard state-space systems given in [16] and [17]. Finally, we reduce the obtained design methods to an LMI problem and present an illustrative example. II. P RELIMINARIES Consider a linear time-invariant descriptor system E x˙ = Ax + Bu,
(1)
where x ∈ Rn is the state variable, u ∈ Rm is the input variable, the matrices A, E ∈ Rn×n , B ∈ Rn×m are constant and rank(E) = s ≤ n. The following lemmas will be used in the sequel. L EMMA 2.1: [19] Assume that (E, A) is in the form of · ¸ · ¸ Is 0 A11 A12 E= ,A = , A21 A22 0 0 then (E, A) is regular and impulsive-free if and only if A22 is nonsingular. L EMMA 2.2: [21] The pair (E, A) is admissible if and only if there exists a matrix X ∈ Rn×n satisfying E T X = X T E ≥ 0, AT X + X T A < 0. III. M AIN RESULTS Consider the following descriptor system E x˙ = Ax + Bγ(t, Hx) + ψ(t, y, u), y = Cx,
(2)
where x ∈ Rn , u ∈ Rm , y ∈ Rp are the system state, input and output, respectively. The system matrices A, E ∈ Rn×n , B ∈ Rn×r , C ∈ Rp×n , H ∈ Rr×n are constant and
8232
FrC07.5 rank(E) = s ≤ n. The functions γ(·, ·) and ψ(·, ·) are locally Lipschitz. Through out this paper, the nonlinear term γ(t, v) is assumed to have the following decoupled structure γ1 (t, v1 ) v1 γ2 (t, v2 ) v2 v = . , γ(t, v) = , .. .. . γk (t, vk ) vk
Θ = diag{Θ1 , Θ2 , · · · , Θk } Pk ri ×ri where vi ∈ R , ,i = i=1 ri = r, and Θi ∈ R 1, 2, · · · , k are given and satisfy Θi > 0. Furthermore, γ(t, v) is assumed to satisfy the quadratic inequality ri
(a − b)T Θ(γ(t, a) − γ(t, b)) ≥ 0, ∀a, b ∈ Rr , t ≥ 0.
(4)
where the matrices L ∈ Rn×p and K ∈ Rr×p are to be designed. Defining the state estimation error by e = x − x ˆ, from (2) and (4), the dynamics of the observer error e = x − x ˆ are governed by E e˙ = (A + LC)e + B[γ(t, v)) − γ(t, w)],
(5)
where v = Hx, w = H x ˆ + K(C x ˆ − y). We begin the observer design by representing the observer error system (5) as the feedback interconnection of a linear system and a multivariable sector nonlinearity. To this end, we view γ(t, v))−γ(t, w) as a function of t and z , v−w = (H + KC)e, that is, a time-varying nonlinearity in z φ(t, z) , γ(t, v) − γ(t, w). Then, we can rewrite the observer error system (5) as E e˙ = (A + LC)e + Bφ(t, z), z = (H + KC)e.
E T P = P T E ≥ 0,
(3)
R EMARK 3.1: A nonlinearity satisfying inequality (3) may not satisfy the globally Lipschitz condition, for example, a3 . Thus, the observer design methods given by [6], [12], [13] are not valid for system (2) with nonlinearities satisfying inequality (3). In [16] and [17], observer design problems for system (2) with E = In are considered. One can see that the admissible nonlinearities of this paper are more general than those of [16] and [17]. For system (2), we construct a full-order observer Ex ˆ˙ = Aˆ x + L(C x ˆ − y) + Bγ(t, H x ˆ + K(C x ˆ − y)) +ψ(t, y, u),
sector (7). Thus, the convergence of the observer error is equivalent to the stability of the LDS (6). When E = In , using the absolute stability theory for standard statespace Lur’e systems, the observer design problem can be reduced to the feasibility problem of a set of LMIs [16], [17]. However, for general E which may be singular, the preliminary stability results of LDS (6) in [14] and [15] are not suitable for the observer design problem because they require the nonlinearities to be bounded sector constrained or time-invariant. Now we are ready to give the full-order observer design method. T HEOREM 3.1: If there exist matrices Λ as in (8), P ∈ Rn×n , L ∈ Rn×p , K ∈ Rr×p and scalar τ > 0 such that
(6)
and ·
(7)
⋆ 0
¸
≤ 0, (10)
hold, then there exists a full-order observer in the form of (4) for system (2) and the resulting observer gains L, K ensure that the estimation error is globally exponentially convergent. Proof: Denote A = A + LC, C = H + KC. Using Lemma 2.2, inequalities (9) and (10) imply that (E, A) is admissible. Then there exist two nonsingular matrices M, N ∈ Rn×n , such that · ¸ · ¸ Is 0 A1 0 M EN = , M AN = , 0 0 0 In−s where A1 ∈ Rs×s . Correspondingly, let · ¸ £ ¤ B1 MB = , CN = C1 C2 , B2 · ¸ P11 P12 −T M PN = . P21 P22
Then pre- and post-multiplying (9) by N T and N , reT spectively, we have P11 = P11 ≥ 0, P12 = 0. Further, it follows from (10) that P11 is nonsingular which shows T P11 = P11 > 0. ¸ · e1 −1 , and pre-multiplying (6) by M , Letting N e = e2 we have e˙ 1 = A1 e1 + B1 φ(t, z), e2 = −B2 φ(t, z), (11) z = C1 e1 − C2 B2 φ(t, z). For any scalar µ > 0, pre- and post-multiplying (10) by # " T In µC 0 Ir
Taking into account the structure of γ and inequality (3), we have that φ(t, z) satisfies z T ΛΘφ(t, z) ≥ 0, ∀z ∈ Rr , t ≥ 0,
(A + LC)T P + P T (A + LC) + τ I B T P + ΘΛ(H + KC)
(9)
and its transpose, respectively, show that (A + µBC)T P + P T (A + µBC) < 0.
where Λ = diag{λ1 Ir1 , λ2 Ir2 , · · · , λk Irk }, λi > 0.
(8)
The error system (6) is an LDS and the interconnected nonlinearity φ(t, z) is time-varying and in the unbounded
Then, using Lemma 2.2, we have that (E, A + µBC) is admissible. In addition, we note that · ¸ Is 0 M EN = , 0 0
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FrC07.5 M (A + µBC)N =
·
A1 + µB1 C1 µB2 C1
µB1 C2 In−s + µB2 C2
¸
where
,
which indicates, by Lemma 2.1, det(In−s + µB2 C2 ) 6= 0.
λmax (P11 ) . λmin (P11 ) From (11), we have z = C1 e1 − C2 B2 φ. Then taking into account (7) and (23), there exists δ > 0, such that ζ=
(12)
0 ≤ 2φT ΘΛz ≤ 2φT ΘΛC1 e1 − δφT φ.
It follows from inequality (12) that det(Ir + µC2 B2 ) 6= 0.
(13)
Since µ > 0 is arbitrary, inequality (13) shows 1 det( Ir + C2 B2 ) 6= 0 µ
(14)
det(C2 B2 ) 6= 0
(15)
which implies that
φT φ ≤ ρ2 eT1 e1 ,
by letting µ → +∞ in (14). Then, we have that B2 is full collum rank. Pre- and post-multiplying (10) by ¸ · T N 0 0 I and its transpose, respectively, we have T A1 P11 + P11 A1 + τ I ⋆ T P P + P 21 22 22 + τ I T ΘΛC1 + B1 P11 +B2T P21 ΘΛC2 + B2T P22 Pre- and post-multiplying (16) I 0 0 −B2T 0 I
where ρ2 =
ke(t)k2
⋆ ⋆ ≤ 0. 0 (16)
0 I 0
and its transpose, respectively, give (20) shown on the top of the next page. And (20) implies (21) shown on the top of the next page. Then −ΘΛC2 B2 − B2T C2T ΛΘ + τ B2T B2 ≤ 0.
(22)
Since B2 is full collum rank, we have (23)
Let V (e1 ) := eT1 P11 e1 . Calculating the derivative of V (e1 ) along the trajectories of system (11) and using inequalities (7) and (21), we have V˙ (e1 )(11)
= e˙ T1 P11 e1 + eT1 P11 e˙ 1 ¸ · £ T ¤ e1 T e1 φ = S φ ≤ −τ eT1 e1 .
1 η kΘΛC1 k
δ−η
(28)
(29)
> 0.
Then, we have
by
ΘΛC2 B2 + B2T C2T ΛΘ > 0.
For arbitrary scalar η > 0, it holds that 1 2φT ΘΛC1 e1 ≤ ηφT φ + kΘΛC1 keT1 e1 . η Then, we can choose η < δ, such that 1 (δ − η)φT φ ≤ kΘΛC1 keT1 e1 , η which shows that
(27)
(24)
Then, there exists a scalar α > 0, such that V˙ (e1 ) ≤ −αV (e1 ),
(25)
eT1 (t)e1 (t) ≤ ζe−αt eT1 (0)e1 (0),
(26)
which indicates
≤ ke1 (t)k2 + ke2 (t)k2 ≤ (1 + ρ2 kB2 k2 )ζe−αt ke(0)k2 ,
(30)
that is, ke(t)k2 ≤ µe−αt ke(0)k2 , 2
2
(31)
where µ = (1 + ρ kB2 k k)ζ. Then, the estimation error is globally exponentially convergent. R EMARK 3.2: When E = In , Theorem 3.1 reduces to Theorem 1 of [17]. But the admissible nonlinearities in this paper are more general than those in [17]. In fact, a standard state-space system can be realized by a descriptor system by introducing an additional descriptor variable (see, for example, [22]). In this case, a direct application of Theorem 1 in [17] to the system is not good enough since it has less design variables in the design procedure than our Theorem 3.1, which will be illustrated by Example 1 in the next section. R EMARK 3.3: Theorem 3.1 also describes a stability criterion for LDS (6) whose interconnected nonlinearities are admitted to be time-varying and in unbounded sector. Thus, this criterion is complementary to the existing stability results given in [14] and [15]. In applications, it may be more convenient to employ a reduced-order observer, which generates estimates only for the unmeasured states. To this end, we make the following common assumption. £ ¤ 3.1: rank E T C T = n and C = £ A SSUMPTION ¤ Ip 0 . In the following discussion, · ¸ Assumption 3.1 always holds. 0 Let M1 , E . It is easy to show that In−p rank(M1 ) = n − p which guarantees that we can find a matrix M0 ∈ Rn×p such that £ ¤−1 M , M 0 M1 (32)
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FrC07.5
AT1 P11 + P11 A1 + τ I ΘΛC1 + B1T P11 P21 S,
·
⋆ −ΘΛC2 B2 − B2T C2T ΛΘ + τ B2T B2 −P22 B2 − τ B2 + C2T ΛΘ
AT1 P11 + P11 A1 ΘΛC1 + B1T P11
⋆ −ΘΛC2 B2 − B2T C2T ΛΘ
exists. Then, we have that M E and M A are of the following structure · ¸ · ¸ E1 0 A11 A12 ME = , MA = , (33) E2 In−p A21 A22 where E1 ∈ Rp×p , E2 ∈ R(n−p)×p , A11 ∈ Rp×p , A12 ∈ Rp×(n−p) , A21 ∈ R(n−p)×p , A22 ∈ R(n−p)×(n−p) . We introduce the following new state £ ¤ ω = 0 In−p x + (Lr E1 + E2 )y,
where Lr ∈ R(n−p)×p is the observer gain to be design. For convenience, let · ¸ Ip 0 Γ1 = , −E2 − Lr E1 In−p
then we have x = Γ1 and ω=
£
Lr
·
y ω
In−p
¸ ¤
M Ex.
(34)
(35)
It can be seen from (34) that x can be easily estimated if we can obtain the estimator of ω. The dynamics of state ω can be represented by £ ¤ Lr In−p M E x˙ ω˙ = ¤ £ Lr In−p M (Ax + Bγ(t, Hx) = +ψ(t, y, u)). (36) Then, we have
¤ £ ω˙ = (A22 + Lr A12 )ω + Lr In−p M B e y, u), ×γ(t, (H1 − H2 (E2 + Lr E1 ))y + H2 ω) + ψ(t, (37) £ ¤ e y, u) = Lr In−p M ψ(t, y, u) + [A21 + where ψ(t, r×p Lr A11 − (A22 + Lr A12 )(E2 +£Lr E1 )]y and ¤ H1 ∈ R , r×(n−p) H2 ∈ R such that H = H1 H2 . For plant (37), the observer is given by £ ¤ ω ˆ˙ = (A22 + Lr A12 )ˆ ω + Lr In−p M B e y, u). ×γ(t, (H1 − H2 (E2 + Lr E1 ))y + H2 ω ˆ ) + ψ(t, (38) Denoting δ = ω − ω ˆ as the error state, we have £ ¤ δ˙ = (A22 + Lr A12 )δ + Lr In−p M Bφ(t, z), (39)
¸
⋆ ≤0 ⋆ T + τI P22 + P22 ≤
·
−τ I 0
0 0
¸
(20)
(21)
Based on the above discussion, we describe a reduced order observer design method in the following theorem whose proof is partially motivated by [13]. T HEOREM 3.2: Under Assumption 3.1, if the conditions of Theorem 3.1 hold, there exists a reduced-order observer in the form of (38) for system (2) and the estimation error is globally exponentially convergent. Proof: Assume that the conditions of Theorem 3.1 hold, that is, inequalities (9) and (10) are feasible. Let ¸ · P1 P2 , M −T P = P3 P4 where M is defined as (32). Based on the new decomposition of matrices of E, A in (33), inequality (9) shows · T ¸ E1 P1 + E2T P3 E1T P2 + E2T P4 P3 P4 · T ¸ T T P1 E1 + P3 E2 P3 = ≥ 0, (40) P2T E1 + P4T E2 P4T which implies that · T E1 P1 + E2T P3 P3
P3T P4
¸
≥ 0.
(41)
By inequality £ (10), we ¤have that P is nonsingular, which indicates that P3 P4 is full row rank. Then, it follows from (41) that P4 > 0. Substituting (33) into (10) ¸ (42) shown on the top of · yields £ ¤ Ip LT P +P T L Ip 0 + the next page, where Γ2 = 0 τ In . Inequality (42) implies that T A12 P2 + AT22 P4 + P2T A12 +P4·A22 +¸τ In−p ⋆ ≤ 0. (43) P 2 T T B M + ΘΛH2 0 P4
where φ(t, z) = γ(t, (H1 − H2 (E2 + Lr E1 ))y + H2 ω) − γ(t, (H1 − H2 (E2 + Lr E1 ))y + H2 ω ˆ ) and z = H2 δ. It can be shown that φ(t, z) satisfies inequality (7).
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Choosing Lr = P4−1 P2T , inequality (43) is equivalent to P4 (A22 + Lr A12 ) +(A22 + Lr A12 )T P4 + τ In−p ⋆ ≤ 0. ¸ · (44) T T LTr B M P4 + ΘΛH2 0 In−p Defining Lyapunov function
V (δ) = δ T P4 δ,
(45)
FrC07.5 ·
A11 A21
¸T ·
¸ · ¸T · ¸ P1 P2 P1 P2 A11 A12 + + Γ2 A21 A22 ·P3 P4 ¸ P3 P4 £ ¤ P1 P2 BT M T + ΘΛ H1 + K H2 P3 P4
A12 A22
we have V˙ |(39)
= 2δ T P4 (A22 + Lr A12 )δ £ ¤ +2δ T P4 Lr In−p M Bφ
(42)
IV. N UMERICAL EXAMPLES
By inequality (44), we have V˙ |(39) ≤ −τ δ T δ which implies that the error system (39) is globally exponentially stable. R EMARK 3.4: When E = In , Q = Ir and k = r, the matrix inequality conditions in Theorem 3.2 are equivalent to those given by [16]. Thus, our design methods are generalizations of the nonlinear observer design methods for standard state-space systems given in [16]. Using Theorem 3.1 and 3.2, both types of full-order and reduced-order nonlinear observers are constructed by a unified approach which can be reduced to the LMI problem in the following theorem. T HEOREM 3.3: If there exist a matrix Λ in the form of (8), a nonsingular matrix P ∈ Rn×n , matrices Q ∈ Rn×p , F ∈ Rr×p and scalar τ > 0, such that the LMIs ET P = P T E ≥ 0 AT P + P T A + C T QT + QC + τ I B T P + ΘΛH + ΘF C
⋆ ≤0 0
quadratic inequality. We will consider this problem in our future work.
≤ 2δ T P4 (A22 + Lr A12 )δ £ ¤ +2δ T P4 Lr In−p M Bφ + 2δ T H2T ΛΘφ. (46)
and ·
(47) ⋆ 0
¸
≤0
(48)
hold, then there exist a full-order observer of the form (4) with L = P −T Q, K = Λ−1 F , and a Q reduced-order Q observer of the form (38) with Lr = ( 1 )−1 ( 2 )T , £ ¤ −T £ ¤T Q Q and 2 = where 1 = 0 In−p M P 0 In−p £ ¤ £ ¤T Ip 0 M −T P 0 In−p . The proof of Theorem 3.3 is direct according to Theorems 3.1 and 3.2. R EMARK 3.5: It is known that LMIs can be solved by highly efficient numerical algorithms in polynomial time [23], [24]. Some of these algorithms have been incorporated into different computer tools for the resolution of LMI problems [25], [26]. R EMARK 3.6: In [27], the observer design problem is considered for standard state-space systems whose nonlinear terms satisfy an incremental quadratic constraint. This kind of nonlinearities include many commonly encountered nonlinearities, such as, globally Lipschitz ones, monotonic ones and also encompass the nonlinearities satisfying (3). However, because of the complex nature of descriptor systems, it is more complex to design observers for descriptor systems with nonlinearities satisfying the so-called incremental
Example 1: Descriptor systems arise naturally as dynamic models of a wide range of engineering applications. This example considers the circuit displayed in Fig. 1, where a dc source with voltage µ is connected in series to a linear resistor, a linear inductor and a nonlinear capacitor with q−v characteristic q = z(v) = (v − v0 )1/3 + q0 . Similar nonlinear capacitors are considered in [28] and [29]. This circuit may be easily shown to admit the charge-flux description q˙ = φ/L, (49) φ˙ = −φR/L − v + µ, 0 = v − v0 − (q − q0 )3 . where φ is the magnetic flux in the inductor. T £ Choose the state ¤ £and input vector as ¤follows: x = x1 x2 x3 = q − q0 φ v − v0 , u = µ − v0 . Assume that x1 and x3 can be measured. Then system (49) can be expressed by system (2) with 1 0 0 0 1/L 0 E = 0 1 0 , A = 0 −R/L −1 , 0 0 0 0 0 1 0 £ ¤ £ ¤ B = 0 ,H = 1 0 0 ,C = 1 0 1 −1 £ ¤ and γ(·) = (·)3 , ψ T = 0 u 0 . Let R = 1, L = 0.5, µ = 2, v0 = 1, q0 = 1. It can be seen that γ(Hx) does not satisfy the globally Lipschitz condition, thus the results given by [6], [12] and [13] are not valid for this example. Letting Θ = 1, we have that γ(Hx) satisfies (3), then, our observer design method can be used. Using Theorem 3.3 with the aid of YALMIP [26], the full-observer gains are given by −2.9131 L = 0.8290 , K = −0.4598. 0.0000
The simulation results are given in Fig. 2. On the other hand, descriptor system (49) can be rewritten as the following standard state-space system q˙ = φ/L, φ˙ = −φR/L − (q − q0 )3 − v0 + µ.
(50)
We found that the observer design methods given by [16], [17] are not feasible.
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This example shows that the circuit system shown in Fig. 1 can be modeled as the descriptor system (49) and also the standard state-space system (50). Theorem 3.1 can design a full-order observer, but the methods proposed by [16], [17] do not work. The reason is that Theorem 3.1 admit more free parameters in the design procedure. V. C ONCLUSIONS In this paper, we have studied the observer design problem for a class of nonlinear descriptor systems. The nonlinear term may be not globally Lipschitz and is only assumed to satisfy a quadratic inequality. The observer error system is represented by an LDS. Both types of full-order and reducedorder nonlinear observers are constructed by a unified LMI approach, by which the observer error system is guaranteed to be globally exponentially convergent. The design example has illustrated the obtained methods. R EFERENCES [1] M. M. Fahmy and J. O’Reilly, “Observers for descriptor systems,” Int. J. Control, vol. 49, no. 6, pp. 2013–2028, 1989. [2] M. Darouach and M. Boutayeb, “Design of observers for descriptor systems,” IEEE Trans. Autom. Control, vol. 40, no. 7, pp. 1323–1327, 1995. [3] M. Darouach, M. Zasadzinski, and M. Hayar, “Reduced-order observer design for descriptor systems with unknown inputs,” IEEE Trans. Autom. Control, vol. 41, no. 7, pp. 1068–1072, 1996. [4] M. Hou and P. C. Muller, “Observer design for descriptor systems,” IEEE Trans. Autom. Control, vol. 44, no. 1, pp. 164–169, 1999.
[5] D. Koenig and S. Mammar, “Design of proportional-integral observer for unknown input descriptor systems,” IEEE Trans. Autom. Control, vol. 47, no. 12, pp. 2057–2062, 2002. [6] Z. Gao and D. W. C. Ho, “State/noise estimator for descriptor systems with application to sensor fault diagnosis,” IEEE Trans. Signal Processing, vol. 54, no. 4, pp. 1316–1326, 2006. [7] H. Trinh, S. Nahavandi and T. D. Tran, “Algorithms for Designing Reduced-order Functional Observers of Linear Systems,” Int. J. of Innovative Computing, Information and Control, vol. 4, no. 2, pp. 321-334, 2008. [8] M. Boutayeb and M. Darouach, “Observers design for nonlinear descriptor systems,” In Proc. IEEE Conf. Decision and Control, New Orleans, LA, 1995, pp. 2369–2374. [9] N. D. Shields, “Observer design and detection for nonlinear descriptor systems,” Int. J. Control, vol. 67, no. 2, pp. 153–168, 1997. [10] G. Zimmera and J. Meierb, “On observing nonlinear descriptor systems,” Syst. Control Lett., vol. 32, no. 1, pp. 43–48, 1997. [11] J. Aslund and E. Frisk, “An observer for non-linear differentialalgebraic systems,” Automatica, vol. 42, no. 6, pp. 959–965, 2006. [12] D. Koenig, “Observer design for unknown input nonlinear descriptor systems via convex optimization,” IEEE Trans. Autom. Control, vol. 51, no. 6, pp. 1047–1052, 2006. [13] G. Lu and D. W. C. Ho, “Full-order and reduced-order observers for Lipschitz descriptor systems: the unified LMI approach,” IEEE Trans. Circiuts and Syst. II: Express Briefs, vol. 53, no. 7, pp. 563–567, 2006. [14] C. Y. Yang, Q. L. Zhang, Y. P. Lin, and L. N. Zhou, “Positive realness and absolute stability problem of descriptor systems,” IEEE Trans. Circuits and Syst. I: Regular papers, vol. 54, no. 5, pp. 1142–1149, 2007. [15] C. Y. Yang, Q. L. Zhang, and L.N. Zhou, “Strongly absolute stability of Lur’e type differential-algebraic systems,” J. Math. Anal. Appl., vol. 336, no. 1, pp. 188–204, 2007. [16] M. Arcak and P. Kokotovic, “Nonlinear observers: a circle criterion design and robustness analysis,” Automatica, vol. 37, no. 12, pp. 1923– 1930, 2001. [17] X. Fan and M. Arcak, “Observer design for systems with multivariable monotone nonlinearities,” Syst. Control Lett., vol. 50, no. 4, pp. 319– 330, 2003. [18] S. L. Campbell, Singular Systems of Differential Equations. Pitman, London, 1980. [19] L. Dai, Singular Control Systems. Springer, Berlin, Germany, 1989. [20] E. L. Lewis, “A survey of linear singular systems,” Circuits, Systems and Signal Processing, vol. 5, no. 1, pp. 3–36, 1986. [21] I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda, “H∞ control for descriptor systems: a matrix inequalities approach” Automatica, vol. 33, no. 4, pp. 669–673, 1997. [22] Y. Cao and Z. Lin, “A descriptor system approach to robust stability analysis and controller synthesis,” IEEE Trans. Autom. Control, vol. 49, no. 11, pp. 2081–2084, 2004. [23] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM, 1994. [24] L. El Ghaoui and S. Niculescu, Advances in Linear Matrix Inequality Methods in Control. Philadelphia, PA: SIAM, 2000. [25] P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali, LMI Control Toolbox. Technical report, The Mathworks Inc., 1995. [26] J. L¨ofberg. YALMIP: A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference, Taipei, Taiwan, 2004. [27] A. B. Acikmese and M. Corless, “Observers for systems with nonlinearities satisfying an incremental quadratic inequality,” In Proc. Amer. Control Conf., Portland, OR, USA, 2005, pp. 3622–3629. [28] V. Venkatasubramanian, A. Saberi, and Z. Lin, “On the validity of solutions and equilibrium points in a nonlinear network,” IEEE Trans. Circuits Syst. I:Fundam. Theory Appl., vol. 43, no. 4, pp. 233–235, 1996. [29] R. Riaza, “Double SIB points in differential-algebraic systems,” IEEE Trans. Autom. Control, vol. 48, no. 9, pp. 1625–1629, 2003.
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