H∞ Observer Design for Time-Delay Systems A. FATTOUH, O. SENAME and J.-M. DION Laboratoire d’Automatique de Grenoble, INPG - CNRS UMR 5528 ENSIEG-BP 46, 38402 Saint Martin d’H`eres Cedex, FRANCE
[email protected] Abstract A method for H∞ observer design for linear time-delay systems based on the algebraic Riccati equation is proposed. A ”weak” sufficient condition for the existence of such an observer is given.
1 Introduction Since the work of Lee et al. [1], few works have considered the problem of an asymptotic observer design for time-delay systems. In this paper, an asymptotic observer for linear time-delay systems is proposed. A delay-dependent method based on the algebraic Riccati equation is developed in order to stabilize the observer and to minimize the H∞ norm between the disturbance and the estimated error.
where A¯ 0 := A0 − LC0 and A¯ 1 := A1 − LC1 The purpose of this paper is to design a constant gain L for the observer (2) with the following properties: 1. 2.
lim e(t) → 0 for w(t) ≡ 0,
(4)
kek2 ≤ γkwk2 for e(t) ≡ 0 ∀t ≤ 0 and for any non zero w(t) ∈ L2
(5)
t→∞
where L2 is the space of square integrable function on [0, ∞), k.k2 is the L2 -norm, γ > 0 is the disturbance attenuation level to be minimized.
3 The Observer Gain Design In order to solve the above problem, equation (3) can be rewritten as follows: e(t) ˙ =
2 Problem Statement Consider the following linear time-delay system: ½ x(t) ˙ = A0 x(t) + A1 x(t − d) + Ew(t) y(t) = C0 x(t) +C1 x(t − d) + Fw(t) x(t) = φ(t);
(1)
t ∈ [−d, 0]
A0 x(t) ˆ + A1 x(t ˆ − d) −L(C0 x(t) ˆ +C1 x(t ˆ − d) − y(t))
(2)
where x(t) ˆ ∈ Rn is the estimated state vector of x(t) and n×p L∈R is the observer gain matrix to be designed. The estimated error, e(t) := x(t) − x(t), ˆ is given by: e(t) ˙ = A¯ 0 e(t) + A¯ 1 e(t − d) + (E − LF)w(t)
(6)
where w(t) ¯ := [wT (t) γ¯ wT (t)]T , γ¯ is some positive constant and D¯ := [E − 1γ¯ LF]. Consider now the following output:
where x(t) ∈ Rn is the state vector, y(t) ∈ R p is the measured output vector, w(t) ∈ Rq is the square-integrable disturbance vector, φ(t) ∈ C [−d, 0] is the continuous initialvalue function vector, A0 , A1 , C0 , C1 , E and F are real matrices with appropriate dimensions and 0 ≤ d < +∞ is the constant known time-delay duration. It should be noted that matrices E and F may include parameter uncertainties and/or modeling errors. An asymptotic observer for system (1) can be given by the following dynamical system: ˙ˆ x(t) =
A¯ 0 e(t) + A¯ 1 e(t − d) + D¯ w(t) ¯
(3)
z(t) := Pe(t)
(7)
where P = PT > 0 is an n × n matrix to be determined. After some algebraic manipulations, one can p easily find that [2]: if kzk2 ≤ kwk ¯ 2 , then kek2 ≤ kP−1 k 1 + γ¯ 2 kwk2 . Let us now consider the following Lemma. Lemma 1 [2] Consider the system: ½ x(t) ˙ = Ax(t) + Ad x(t − d) + Dw(t) z(t) = Cx(t) x(t) = φ(t);
(8)
t ∈ [−d, 0]
where z(t) ∈ R p is the measured output vector, A, Ad , D and C = CT > 0 are real matrices with appropriate dimension. Given a scalar δ > 0, system (8) is asymptotically stable and kzk2 ≤ δkwk2 for any constant time-delay 0 ≤ d < d¯ if there exist two symmetric positive definite matrices P and Q satisfying the following algebraic Riccati equation: (A + Ad )T P + P(A + Ad ) +CT C +
1 PDDT P + Q = 0 δ2
where d¯ is given by:
x(t) = φ(t);
t ∈ [−0.3, 0]
1 d¯ = kQ− 2 [β1 PAd AP−1 AT ATd P + β2 Ad Ad P−1
Note that the above system is weakly observable but neither strongly observable nor hyper observable. Note also that the pair [(C0 +C1 ), (A0 + A1 )] is detectable. Applying the algorithm given in [2] on the above system gives: for ε = 2.5 and γ¯ = 1.5, the ARE (9) has the solution
1
ATd ATd P + (β1 −1 + β2 −1 )P]Q− 2 k−1 for any positive real numbers β1 and β2 . Applying the above Lemma on the modified system (6)-(7) for δ = 1 leads to the following Theorem. Theorem 1 [2] Consider the time-delay system (1) and the observer (2). If the following algebraic Riccati equation (A0 + A1 )T P + P(A0 + A1 ) + P(EE T + (1 + ε)In )P 1 −(C0 +C1 )T {2I p − 2 FF T }(C0 +C1 ) = 0 γ¯
(9)
has a symmetric positive definite solution P for some posi¯ tive constants γ¯ and ε, then, for all 0 ≤ d < d: 1- Observer (2) is asymptotically stable, 2- kek2 ≤ γ kwk2 . where:
P=
·
4.3629 −1.6193
−1.6193 6.7804
¸
, L=
·
0.9122 1.8402
¸
d¯ = 0.3565sec (for β1 = 0.06 and β2 = 1) and γmin = 0.5. The estimated errors are shown in figure (1) for d = 0.3sec. The disturbance applied at t = 2sec. is of magnitude 1. For the simulation an initial value at time t = −1sec. is used to generate an initial value function on t ∈ [−0.3, 0]. In figure (2), the maximum singular value of Tew ( jω) is traced as a function of the frequency. It shows that kTew ( jω)k∞ ≤ γmin ∀ω ∈ R.
1.5
d¯ = γ
− 21
k(εPP) [β1 PA¯ 1 A¯ 0 P−1 A¯ T0 A¯ T1 P + β2 PA¯ 1 A¯ 1 1 P−1 A¯ T1 A¯ T1 P + (β1 −1 + β2 −1 )P](εPP)− 2 k−1 −1
= kP k
q
1 + γ¯ 2
disturbance
1
0.5
PSfrag replacements
e2 (t)
0
e1 (t)
-0.5
In this case the required observer gain is given by: −1
L = P (C0 +C1 )
-1
T
-1.5 -1
-0.5
0
0.5
1
1.5 time (sec.)
2
2.5
3
3.5
4
Figure 1: Estimated errors for d=0.3 sec. Note that the proofs of Lemma 1 and Theorem 1 given in [2] are based on some results of [3, 4, 5, 6]. In [2], an algorithm is proposed in order to minimize γ and the following sufficient condition is provide. PSfrag replacements
0.05
0.045 0.04
Proposition 1 [2] For a given time-delay system of the form (1), if the pair [(C0 +C1 ), (A0 + A1 )] is detectable, then there exists an H∞ observer of the form (2) with certain disturbance attenuation level.
0.035
disturbance e1 (t) e2 (t)
0.03
0.025 0.02
0.015 0.01 0.005 0 0 10
1
10
2
frequency
10
3
10
It should be noted that the detectability condition given above is weaker than the strong and the hyper observability (see [1] for the definitions).
Figure 2: Maximum SV of Tew ( jω), γmin = 0.5.
4 Illustrative example
5 Concluding Remarks
Consider the following system: ·
−10 0 £ 0 y(t) =
x(t) ˙ =
1 10 x(t) + 1 ¤ £ 1 10 x(t) + 1 ¸
·
1 1 x(t − 0.3) + 1 1 ¤ 1 x(t − 0.3) + w(t) ¸
·
¸
w(t)
In this paper, a method for H∞ asymptotic observer design for linear time-delay systems has been developed. The proposed method can be generalized to the case of multiple delays in state and output equations.
Note that the algorithm proposed in [2] to minimize γ is very simple and it allows to find a reasonable solution in many cases. Few is known about the geometry of the criterion to be minimized and further sophisticated optimization methods could be used. Note also that the maximum delay d¯ obtained in the proposed example is conservative and the simulation results point out that the observer retains its properties for some values of d larger than d.¯
References [1] E. B. Lee and A. Olbrot, “Observability and related structural results for linear hereditary systems,” Int. Journal of Control, vol. 34, no. 6, pp. 1061–1078, 1981. [2] A. Fattouh, O. Sename, and J.-M. Dion, “h∞ observer design for time-delay systems,” in Proc. 37th IEEE Confer. on Decision & Control, (Tampa, Florida, USA), pp. 4545– 4546, 1998. [3] J. Hale, Theory of functional differential equations. Springer-Verlag, 1977. [4] J.-H. Su, “Further results on the robust stability of linear systems with a single time delay,” Systems & Control Letters, vol. 23, pp. 375–379, 1994. [5] S. I. Niculescu, A. Trofino-Neto, J.-M. Dion, and L. Dugard, “Delay-dependent stability of linear systems with delayed state: An l.m.i. approach,” in Proc. 34th Confer. on Decision & Control, (New Orleans, Louisiana), pp. 1495–1496, 1995. [6] X. Li and C. E. de Souza, “Robust stabilization and h∞ control of uncertain linear time-delay systems,” in IFAC Symp. on System Structure and Control, (San Francisco, USA), pp. 113–118, 1996.
