A Model Matching Solution of Robust Observer Design for Time-Delay Systems ? Anas Fattouh1 and Olivier Sename2 Automatic Laboratory of Aleppo, Faculty of Electrical and Electronic Engineering, University of Aleppo, Aleppo, Syria
[email protected] LAG, ENSIEG-BP 46, 38402 Saint Martin d’H`eres Cedex, France
[email protected]
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Summary. Uncertainties are unavoidable in practical situations and they have to be taken into consideration in control system design. In this paper, a method for designing a robust observer for linear time-delay systems is proposed. Under the assumption that the considered time-delay system is spectrally controllable and spectrally observable, a double B´ezout factorization of its nominal transfer matrix is obtained. Next, based on this factorization, all stable observers for the nominal system are parameterized. By applying those observers on the real system, the parameterization transfer matrix has to be found such that the error between the real estimation and the nominal one is minimized. This problem is rewritten as an infinite dimensional model matching problem for different types of uncertainty. In order to solve this infinite dimensional model matching problem, it is transformed into a finite dimensional one, and therefore a suboptimal solution can be obtained using existing algorithms.
Key words: Time-delay systems, Robust observer, Factorization polynomial, model matching.
1 Introduction Time-delay systems appear naturally in many engineering applications and, in fact, in any situation in which transmission delays cannot be ignored. The control of such systems using state feedback has been thoroughly studied (see [2, 19, 13] and the references therein). One of the major difficulties in implementing such control laws is that all the state variables of the system (at least some state variables) are required for the controller synthesis. However, in most practical situations, this condition is rarely satisfied and an observer has to be built up in order to estimate the state variables from the measured output and the controlled input. Many observer schemes have been proposed in the literature under the assumption that there are no model uncertainties [10, 20, 7, 16]. However in practice, model ?
This work has been done during the post doctoral stay of Dr Anas Fattouh at LAG, INPG, France and it is supported by AUF (Agence Universitaire de la Francophonie).
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Anas Fattouh and Olivier Sename
uncertainties have to be taken into consideration when designing observers. These uncertainties have been considered in the literature as additive disturbances [21, 3, 4] or as an unknown input [5]. In this paper, unstructured uncertainties (additive uncertainty and input multiplicative uncertainty) are considered when designing observers. A set of all stable observers for the nominal system is firstly parameterized based on a double B´ezout factorization of its transfer matrix. Then, the problem of finding the parameterization transfer matrix such that the resulting observer satisfies a specified robustness property is written as an infinite dimensional model matching problem. A suboptimal solution of this infinite dimensional problem has been proposed by the authors in [6, 17] in terms of multiple finite dimensional model matching problems. However the solution of those multiple finite dimensional model matching problems is not easy to obtain. Therefore, another solution is proposed in this paper by transforming the infinite dimensional model matching problem into only one modified finite dimensional model matching problem. Then by applying an existing optimization algorithm on the modified finite dimensional problem a solution can be obtained. The paper is organized as follows. Section 2 gives some basic definitions concerning time-delay systems. Section 3 shows how to get a double B´ezout factorization of a given delayed transfer matrix and based on this factorization how to parameterize the set of all stable observers. Section 4 describes how to find the parameterization transfer matrix for different types of system uncertainties. Section 5 provides a suboptimal solution of the infinite dimensional problem described in the previous section. Illustrative examples are given in Section 6 and some concluding remarks are given in Section 7. Throughout this paper the following notations will be used: is the field of real numbers is the set of integer numbers = {a ∈ F : 0 < a < +∞}, F denotes R or N is the ring of polynomials in • with coefficients in R denotes variable, z = e−sh and h ∈ R+ fixed Pm the Laplace k ={ k=0 ak (z)s : ak (z) ∈ R[z], m ∈ N+ } is the field of rational functions in s and z with coefficients in R = {p(s, z) = b(s,z) a(s) ∈ R(s, z) : b(s, z) ∈ R[z][s], a(s) ∈ R[s], degs (a(s)) > degs (b(s, z)) and p(s, z) is entire} Θ[z] is the ring of polynomials in z with coefficients in Θ F = {p(s, z) = b(s,z) a(s) ∈ Θ[z] : a(s) is monic and stable} M(•) denotes a transfer matrix of appropriate dimension with entries in • In denotes the (n × n) identity matrix 0i×j denotes the (i × j) zero matrix XT denotes the transpose of a matrix X k.k∞ is the H∞ -norm defined by: kX(s)k∞ = σmax (X(jω)); σmax (X) denotes the maximum singular value of X, j is the imaginary number, ω denotes the frequency C[a, b] is the set of continuous functions [a, b] → Rn R N F+ R[•] s R[z][s] R(s, z) Θ
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The elements of Θ can be viewed as transfer matrices of distributed time delays while the elements of Θ[z] can be viewed as transfer matrices of finite interconnections of point and distributed time delays [9]. In order to simplify the notation, X will denote either X(s, z) or X(s) when there is no confusion.
2 Preliminaries In this section, some basic definitions concerning time-delay systems are recalled. Consider the following model of a time-delay system ˙ = A(∇)x(t) + B(∇)uo (t) x(t) yo (t) = C(∇)x(t) (1) x(t) = φ(t); t ∈ [−mh, 0]
where • x(t) ∈ Rn , uo (t) ∈ Rr and yo (t) ∈ Rp are the state, the control input and the measured output vectors respectively. • A(∇) ∈ R[∇]n×n , B(∇) ∈ R[∇]n×r , C(∇) ∈ R[∇]p×n and φ(t) ∈ C[−mh, 0] is the functional initial condition of (1). • ∇ is the delay operator ((∇x)(t) = x(t − h), (∇2 x)(t) = x(t − 2h), ...), h ∈ R+ is the fixed known delay duration and m is a positive integer such that mh represents the maximal delay in the system. Definition 1. [9] Consider the time-delay system (1). Let z be a complex variable, the pair (A(z), B(z)) is
i) R[z]-controllable if and only if rank[sIn − A(z), B(z)] = n ∀ (s, z) ∈ C2 . ii) spectrally controllable if and only if rank[sIn −A(e−sh ), B(e−sh )] = n ∀ s ∈ C. iii) R(z)-controllable if and only if rank[sIn − A(z), B(z)] = n for all but finitely many pairs (s, z) ∈ C2 . The pair (C(z), A(z)) is (i) R[z]-observable (resp. (ii) spectrally observable or (iii) R(z)-observable) if and only if the pair (AT (z), C T (z)) is (i) R[z]-controllable (resp. (ii) spectrally controllable or (iii) R(z)-controllable). Moreover, the realization (1) is i) canonical if and only if the pair (A(z), B(z)) is R[z]-controllable and the pair (C(z), A(z)) is R(z)-observable. ii) co-canonical if and only if the pair (A(z), B(z)) is R(z)-controllable and the pair (C(z), A(z)) is R[z]-observable. iii) spectrally canonical if and only if the pair (A(z), B(z)) is spectrally controllable and the pair (C(z), A(z)) is spectrally observable. ¤ It should be noted that if the pair (A(z), B(z)) is R[z]-controllable then there exists a polynomial matrix K(z) ∈ M(R[z]) (i.e. K(z) is the Laplace transformation of point time delays) such that det[sIn − A(z) − B(z)K(z)] = δ1 (s)
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where δ1 (s) ∈ R[s] is a stable polynomial. In return, if the pair (A(z), B(z)) is spectrally controllable then there exists a matrix K(s, z) ∈ M(Θ[z]) (i.e. K(s, z) is the Laplace transformation of point and distributed time delays) such that det[sIn − A(z) − B(z)K(s, z)] = δ2 (s) where δ2 (s) ∈ R[s] is a stable polynomial. Remark 1. When system (1) is spectrally canonical then there exist feedback matrices F1 (s, z), F2 (s, z) ∈ M(Θ[z]) and an observer gain matrix K(s, z) ∈ M(Θ[z]) such that · ¸ sIn − A(z) −B(z) det = α(s) −F1 (s, z) Ir − F2 (s, z) (2) det[sIn − A(z) − K(s, z)C(z)] = β(s) where α(s), β(s) ∈ R[s] and are stable polynomials.
¤
The transfer matrix of system (1) is given by [18] (3)
G(s, z) = C(z)(sIn − A(z))−1 B(z) b (s,z)
Note that G(s, z) is a (p × r) matrix with i, j entry Gij (s, z) = aij where ij (s,z) bij (s, z), aij (s, z) ∈ R[z][s], aij (s, z) is monic and degs (aij (s, z)) ≥ degs (bij (s, z)). It has been shown in [9] that any transfer matrix G(s, z) as defined above has a canonical realization and a co-canonical realization. In the sequel it is assumed that the co-canonical realization (1) of G(s, z) is spectrally controllable which will be noted as spectrally co-canonical realization. Note that this assumption is not restrictive as G(s, z) has a spectrally canonical realization if and only if the canonical realization of G(s, z) is spectrally observable which is equivalent to requiring that the co-canonical realization of G(s, z) be spectrally controllable (see [9] for details).
3 Parametrization of all stable observers In this section, the set of all stable observers for system (1) are parameterized based on a polynomial factorization of its transfer matrix (3). The following lemma provides us a polynomial factorization of the transfer matrix (3) associated with a spectrally co-canonical realization (1). Lemma 1. [14] Consider the transfer matrix (3) associated with a spectrally cocanonical realization (1). The transfer matrix (3) can be factorized as follows ¯ −1 (s, z)N ¯ (s, z) G(s, z) = N (s, z)M −1 (s, z) = M ¯,N ¯ satisfy the following double B´ezout equation where N, M, M · ¸· ¸ · ¸ ¯ Y X M −X Ir 0r×p = ¯ M ¯ 0p×r Ip −N N Y¯
(4)
(5)
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The eight matrices in (5) belong to M(F) and they can be obtained using the following equations M N Y X
= Ir + Fe (sIe − Ao )−1 Be = Ce (sIe − Ao )−1 Be = Ir − Fe (sIe − A¯o )−1 Be = Fe (sIe − A¯o )−1 Ke
¯ M ¯ N Y¯ ¯ X
= Ip + Ce (sIe − A¯o )−1 Ke = Ce (sIe − A¯o )−1 Be = Ip − Ce (sIe − Ao )−1 Ke = Fe (sIe − Ao )−1 Ke
(6)
where · ¸ A(z) B(z) Ao (s, z) = Ae + Be Fe , A¯o (s, z) = Ae + Ke Ce , Ae (z) = 0r×n −Ir · ¸ £ ¤ £ ¤ 0 Be (z) = n×r , Ce (z) = C(z) 0p×r , Fe (s, z) = F1 (s, z) F2 (s, z) Ir · ¸ · ¸ K(z) In 0n×r Ke (z) = , Ie = 0r×p 0r×n 0r×r and F1 (s, z), F2 (s, z) ∈ M(Θ[z]), K(z) ∈ M(R[z]) are chosen such that (2) is satisfied for some stable polynomials α(s), β(s) ∈ R[s]. ¤ Now, let r(t) = E(∇)x(t) ∈ Rk×1 be the vector to be estimated. An asymptotic observer of r(t) is defined as a dynamic system of the following form rˆ(s) = U (s, z)uo (s) + V (s, z)yo (s)
(7)
where U (s, z), V (s, z) ∈ M(Θ[z]), such that for all control input limt→∞ (r(t) − rˆ(t)) = 0 is satisfied. A parameterization of all stable observers (7) based on the factorization (4) and (5) is given in the following lemma. Lemma 2. [21] Consider systems (1) and (7). The set of M(Θ[z])-matrices U (s, z), V (s, z) such that system (7) is an observer of r(s) = E(z)x(s) is given by ½ ¯ (s, z) U (s, z) = P (s, z)Y (s, z) − Q(s, z)N (8) ¯ (s, z) V (s, z) = P (s, z)X(s, z) + Q(s, z)M where P (s, z) ∈ M(Θ[z]) is given by P (s, z) = Ee (z)(sIe − Ao (s, z))−1 Be (z) Ee (z) = [E(z) 0k×r ]
(9)
¯ , X, M ¯ are given by the double B´ezout factorization (6) and Q(s, z) is any maY, N trix belonging to M(Θ[z]). ¤ It should be noted that Q(s, z) can be properly chosen in order to achieve certain performance specifications as it will be shown in the next sections.
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4 Model matching problem for different types of uncertainties ˜ z) as a real time-delay system and its corresponding model is Let us consider G(s, given by (3). Let (7)-(8) be the set of all stable observers for the nominal model (3) associated with a spectrally co-canonical realization (1). Applying the set of observers (7)-(8) on the real system, the parameterized transfer matrix Q(s, z) has to be found such that the difference between the real estimation and the nominal one is minimized as shown in Fig. 1. This problem is formulated in the following proposition as an infinite dimensional model matching problem for different types of uncertainties.
Fig. 1. Error due to system uncertainty.
˜ z) with a nominal model G(s, z) Proposition 1. Given a real time-delay system G(s, associated with a spectrally co-canonical realization (1). Let (7)-(8) be the set of all ˜ z) is given by (7)-(8) where stable observers for nominal model. An observer of G(s, Q(s, z) is the solution of the following model matching problem min kT1 (s, z) − Q(s, z)T2 (s, z)k∞
Q∈M(F)
(10)
where T1 (s, z) and T2 (s, z) ∈ M([Θ[z]) are defined according to the considered model uncertainty. ¯ −1 (s, z)N ¯ (s, z) be the nominal Proof. Let G(s, z) = N (s, z)M −1 (s, z) = M model with a spectrally co-canonical state-space realization (1). Let (7)-(8) be a set of all stable observers of realization (1). Let W (s) be a fixed stable transfer matrix and ∆(s, z) be variable stable transfer matrix with k∆(s, z)k∞ ≤ 1. Applying (7)-(8) on the real system, the real estimation rˆ(s) can be calculated for different model uncertainties as follows (see Fig. 1 and Fig. 2):
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Fig. 2. Different types of uncertainty.
1. Additive uncertainty: ˜ = G + W ∆ and y(s) = yo (s) + W ∆uo (s). The real estimation is In this case G given by rˆ(s) = U (s, z)uo (s) + V (s, z)y(s) = (U + V W ∆)uo (s) + V yo (s) which implies that ¯ W )∆uo (s) e(s) = rˆ(s) − rˆo (s) = V W ∆uo (s) = (P XW + QM Now, as W and ∆ are stable, then e(s) is stable. Moreover, as u0 (s) is bounded then e(s) is bounded. From the above equation, we can write ¯ W k∞ k∆(s, z)k∞ ku0 (s)k2 ke(s)k2 ≤ kP XW + QM As k∆(s, z)k∞ ≤ 1, then in order to minimize the effects of uncertainty on the real estimation, the optimization problem minQ kT1 − QT2 k∞ has to be solved for ½ T1 (s, z) = P (s, z)X(s, z)W (s) (11) ¯ (s, z)W (s) T2 (s, z) = −M 2. Multiplicative input uncertainty: Following the same steps applied on the case of additive uncertainty, we get ¯ W )∆u(s) e(s) = −U W ∆u(s) = −(P Y W − QN Now, as W and ∆ are stable, then e(s) is stable. Moreover, as u(s) is bounded, then e(s) is bounded. In order to minimize the effects of uncertainty on the real estimation, the optimization problem minQ kT1 − QT2 k∞ has to be solved for ½ T1 (s, z) = P (s, z)Y (s, z)W (s) (12) ¯ (s, z)W (s) T2 (s, z) = N
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From the above cases one can conclude that in order to minimize the effect of the model uncertainty on the estimated states one has to solve the model matching problem (10) for T1 (s, z) and T2 (s, z) are given by (11) or (12) according to the model uncertainty type. ¤ Remark 2. The result of Proposition 1 can be interpreted as follows: the estimation rˆ(s) depends on the input and output information signals. When there are uncertainties in those information in some frequency intervals, this dependence has to be wasted away in these intervals by a proper choice of transfer matrix Q(s, z). ¤ A suboptimal solution to (10) can be obtained by finding a transfer matrix Q(s, z) ∈ M(F) solution to (13)
kT1 (s, z) − Q(s, z)T2 (s, z)k∞ ≤ γ
for some positive scalar γ. A solution for this optimization problem is given in the next section.
5 Suboptimal solution In this section, the infinite dimensional optimization problem (13) is transformed into a finite dimensional one. Proposition 2. Given two transfer matrices T1 (s, z), T2 (s, z) ∈ M(F). Suppose that degz (T1 (s, z)) ≥ degz (T2 (s, z)), then there exists a matrix Q(s, z) ∈ M(F) with degz (Q(s, z)) = degz (T1 (s, z)) − degz (T2 (s, z)) such that kT1 (s, z) − Q(s, z) T2 (s, z)k∞ ≤ γ where γ is some positive scalar. Proof. The proof provides a method to construct Q(s, z) ∈ M(F) solution to (13). First, note that kT1 (s, z) − Q(s, z)T2 (s, z)k∞ = kT1T (s, z) − T2T (s, z)QT (s, z)k∞
(14)
= PmNote alsoithat any matrix T (s, z) belonging to M(F) can be written as T (s, z) + T (s)z where T (s) are proper stable rational transfer matrices and m ∈ N . i i i=0 Suppose that T1T (s, z) =
m1 X
T1i (s)z i and T2T (s, z) =
i=0
m2 X
T2j (s)z j
(15)
j=0
where T1i (s) and T2j (s) are proper stable transfer matrices and m1 , m2 ∈ N+ . Since degz (T1 (s, z)) ≥ degz (T2 (s, z)), then m1 ≥ m2 . Let the parameterized matrix QT (s, z) be of the form (m1 −m2 ) T
Q (s, z) :=
X
k=0
Qk (s)z k
(16)
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Then, using (14), (15) and (16), the optimization problem (13) can be rewritten as follows £ ¤ kT1 (s, z) − Q(s, z)T2 (s, z)k∞ = k 1 z . . . z m1 (T1 (s) − T2 (s)Q(s))k∞
(17)
≤ kT1 (s) − T2 (s)Q(s)k∞ ≤ γ
where
T10 (s) T11 (s) . T1 (s) = , . . T1m (s) 1
T (s) 0 ... 0 20 . . T21 (s) T20 (s) ... . . . . . . . 0 T20 (s) . T2 (s) = , . T2m (s) T2m −1 (s) . 2 2 . . . 0 T2m (s) 2 . . . . . . 0 0 ... 0 T2m (s) 2 {z } |
Q(s) =
Q0 (s) Q1 (s) . . . Qm −m (s) 1 2
(m1 +1)×(m1 −m2 +1)
Now, using an algorithm from [8] or [1], a solution Q(s) to the finite dimensional optimization problem (17) can be found, which is a solution to the original infinite dimensional optimization problem (13). ¤ Remark 3. The problem is solved here as an optimization problem of the H∞ -norm of some transfer matrix. The analysis of the structure at infinity of T1 (s, z) and T2 (s, z) may allow to know whether there exists an exact solution for the model matching problem (13) or not (see [15, 11] for more details). ¤
6 Illustrative examples In this section the proposed method is applied on two examples. In the first one, the system has delay on its input but the value of this delay is uncertain. In the second one, the system has internal delay and the system parameters are uncertain. 6.1 Example 1 Consider the following transfer function [1] G(s, z) =
kz 1 + τs
(18)
The parameters of the system are k = 2.5, τ = 2.5 and the delay is presumed to be in the interval [2, 3], i.e. 2 ≤ h ≤ 3 (note that z = e−sh ). The nominal value of the delay is h = 2.5, thus the nominal transfer function is Gn (s, z) =
z s + 0.4
(19)
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A spectrally co-canonical state space representation of (19) is given by ½ x(t) ˙ = −0.4x(t) + ∇u(t) y(t) = x(t)
(20)
A double B´ezout factorization for the transfer matrix (19) associated with the state space representation (20) can be obtained using (4) and (6) as follows ¯ = M =M Y = Y¯ =
s+0.4 s+1.4
s2 +2.8s+1.96−e−1 z (s+0.4)(s+1.4)
¯ = N =N
z s+1.4
¯= X=X
e−1 s+1.4
Based on this factorization, the set of all stable observers for r(t) = x(t) is given by (7) and (8) with z P (s) = s + 1.4 and Q(s, z) is any matrix belonging to M(Θ[z]). Now, the changes in the delay h has to be specified in frequency domain by a weighting matrix W (s) representing the uncertainties on the nominal transfer matrix (19). In this example, we have modelled these parameter variations as an additive uncertainty. Therefore, a transfer function W (s) has to be found such that the following inequality is satisfied ¯ ¯ ¯ ¯ −sh −sh ¯ ¯ (21) ) − Gnominal (s, e )¯ ≤ |W (s)| max ¯ G(s, e 2≤h≤3
By plotting the left hand side of the above inequality, we can choose the following weighting function (see Fig. 3) W (s) =
0.1s2 + 2.8s + 0.15 s2 + s + 0.3
(22)
Now the model matching problem (13) has to be solved for T1 (s, z) and T2 (s, z) given by (12). For this example they equal to T1 (s, z) =
(0.1s2 +2.8s+0.15)e−1 z (s+1.4)2 (s2 +s+0.3)
2
+2.8s+0.15)(s+0.4) T2 (s, z) = − (0.1s (s+1.4)2 (s2 +s+0.3)
The model matching problem (13) can be rewritten in the form (17) with 0 T1 (s) = −1 2 0.1e (s +28s+1.5) (s+1.4)2 (s2 +s+0.3)
T2 (s) =
2
+2.8s+0.15)(s+0.4) − (0.1s (s+1.4)2 (s2 +s+0.3)
0
0
2 +2.8s+0.15)(s+0.4) − (0.1s (s+1.4)2 (s2 +s+0.3)
£ Q(s) = Q0 (s)
Q1 (s)
¤T
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20
W(s)=(0.1s2+2.8s+0.15)/(s2+s+0.3) 10
Singular values of G−Gnominal (dB)
0
−10
−20
−30
−40
−50
−60
−70 −3 10
−2
−1
10
0
10
1
10
2
10
3
10
10
Frequency (rad/sec)
Fig. 3. Left hand side of (21) and the corresponding weighting function W (s). 1 1 0.8 0.8 0.6
0.6
0.4
0.4
0.2 r(t)−re(t)
0.2 input u(t)
h=3
0 −0.2
0 −0.2
−0.4
h=2.5
−0.4
−0.6
h=2
−0.6
−0.8 −0.8 −1 0
10
20 30 time [sec.]
40
50
−1
0
10
20 30 time [sec.]
40
50
Fig. 4. Estimated error for for different values of h.
Applying the algorithm of Francis [8] on (17), we found that Q0 (s) ≡ 0 and Q1 (s) = −0.3679/(s2 + 1.8s + 0.56) with γ = 0, i.e. the robust observer is an exact observer. The robust observer of r(t) = x(t) is given by (7) with U (s, z) =
z , s + 0.4
V (s, z) ≡ 0
Fig. 4 shows the estimated error for different values of h. From this figure, we can state that estimated error converges to zero for all admissible values of h but the convergence rate is different for each value.
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6.2 Example 2 In this example, a wind tunnel system is considered which has the following state space model [12] ½ x(t) ˙ = A(∇)x(t) + B(∇)u(t) (23) y(t) = C(∇)x(t) with 0 −a ka∇ 0 £ ¤ 1 , B(∇) = 0 , C(∇) = 1 0 0 A(∇) = 0 0 ω2 0 −ω 2 −2ζω
where a = τ1 and τ , k, ζ, ω are parameters depending on the operating point and they lie in the following intervals 0.739 ≤ τ ≤ 2.58, −0.0144 ≤ k ≤ −0.0029. Their nominal values are τ = 1.964, k = −0.0117. Other parameters are presumed constant ζ = 0.8, ω = 6. The components of the state vector x(t) are the change in Mach number, the guide vane angle and the guide vane angle velocity respectively. Applying the test conditions of Definition 1 on system (23), we can find that system (23) is spectrally canonical, i.e., it is spectrally controllable and spectrally observable. A state feedback control law is calculated in [12]. Based on this control law, an observer-based state feedback for system (23) can be obtained as follows (note that a constant gain for the observer is sufficient here) (
R0 u(t) = F1 (∇)x(t) + F2 −h eaθ x(t + θ)dθ x ˆ˙ (t) = A(∇)ˆ x(t) + B(∇)u(t) + K (y(t) − C(∇)ˆ x(t))
(24)
where £ ¤ F1 (∇) = 661.5612 1.2399 + 0.0432∇ 0.1003 × 10−3 £ ¤ F2 = 0 −4.6881 0.0510 × 10−3 £ ¤T K = 0.5 0 0
The transfer matrix of system (23) is given by G(s, z) =
kaω 2 z (s + a)(s2 + 2ζωs + ω 2 )
(25)
Based on the above information, a double B´ezout factorization for the transfer matrix (25) for nominal values of its parameters associated with the state space representation (23) can be obtained using (4) and (6) as follows
Robust Observer Design for Time-Delay Systems 3
2
¯ = M
N=
s +10.11s +40.89s+18.33 s3 +10.11s2 +40.84s+18.48 −0.2145 s3 +10.11s2 +40.84s+18.48 z
¯ = N
s+0.5092 s+0.009165 −0.2145 s3 +9.609s2 +36.09s+0.3299 z
Y =
s3 +10.11s2 +40.84s+18.48 s3 +10.11s2 +40.89s+18.33
Y¯ =
s+0.009165 s+0.5092
M=
2
2
−0.001582s+0.07092 + −3.168e−006s s4 +10.12s3 +40.98s2 +18.7s+0.168 z
X=
13
−0.001582s+0.07092 + −3.168e−006s s4 +10.61s3 +45.99s2 +39.27s+9.407 z
¯= X
0.3308 s+0.009165
0.3308s2 +3.175s+11.91 s3 +10.11s2 +40.84s+18.48
The set of all stable observers for r(t) = [0 1 0]x(t) is given by (7) and (8) with P (s) =
36s + 18.33 s3 + 10.11s2 + 40.84s + 18.48
and Q(s, z) is any matrix belonging to M(Θ[z]). The matrix Q(s, z) has to be found such that the effects of the τ and k parameter changes on the estimation are minimized. To this end, changes in the parameters τ and k have to be specified in frequency domain by a weighting matrix W (s) representing the uncertainties on the nominal transfer matrix (25). In this example, we have modelled these parameter variations as direct input multiplicative uncertainty. Therefore, a transfer function W (s) has to be found such that the following inequality is satisfied ¯ ¯ ¯ ¯ G(s, e−sh ) ¯ ¯ max − 1 (26) ¯ ¯ ≤ |W (s)| −sh ) 0.739≤τ ≤2.58 ¯ ¯ G (s, e nominal −0.0144≤k≤−0.0029 30
W(s)=(2.5s+1)/(s+1.5)
20
Singular values of G/Gnominal−1 (dB)
10
0
−10
−20
−30
−40
−50
−60 −3
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
10
Frequency (rad/sec)
Fig. 5. Left hand side of (26) and the corresponding weighting function W (s).
By plotting the left hand side of the above inequality, we can choose the following weighting function (see Fig. 5)
Anas Fattouh and Olivier Sename 1
0.5
0.5 input u(t)
1
input u(t)
14
0 −0.5 −1
−1 0
100
200
300
400
500
0
different values for τ
1
200
300
400
500
400
500
different values for k
0.5 r(t)− re(t)
r(t)− re(t)
100
1
0.5 0 −0.5 −1
0 −0.5
0 −0.5
0
100
200 300 time [sec.]
400
−1
500
0
100
200 300 time [sec.]
Fig. 6. Estimated error for for different values of τ and k.
W (s) =
2.5s + 1 s + 1.5
(27)
Now the model matching problem (13) has to be solved for T1 (s, z) and T2 (s, z) given by (12). For this example, they equal to T1 (s, z) =
90s+36 s3 +11.1s2 +50.4s+54 3
2
−0.1425s +6.326s+2.553 + s7 +21.21s6−0.0002851s +203.6s5 +1037s4 +2819s3 +3162s2 +1026s+9.144 z
T2 (s, z) =
−0.5362s−0.2145 s4 +11.11s3 +50.5s2 +54.46s+0.4949 z
The model matching problem (13) can be rewritten in the form (17) with # " 90s+36 s3 +11.1s2 +50.4s+54
T1 (s) =
T2 (s) =
−0.0002851s3 −0.1425s2 +6.326s+2.553 s7 +21.21s6 +203.6s5 +1037s4 +2819s3 +3162s2 +1026s+9.144
"
0 −0.5362s−0.2145 s4 +11.11s3 +50.5s2 +54.46s+0.4949
#
Q(s) = Q0 (s) Applying the algorithm of Francis [8] on (17), we found that Q0 (s) ≡ 0 with γ = 1.26 which means that the nominal (central) observer is robust. This observer is given by (7) with U = V =
s2
36 −0.0001141s2 − 0.05693s + 2.553 + 6 z 5 + 9.6s + 36 s + 19.71s + 174s4 + 775.9s3 + 1655s2 + 680.2s + 6.096
11.91s + 6.063 s4 + 10.11s3 + 40.93s2 + 18.85s + 0.1693
Robust Observer Design for Time-Delay Systems
15
Fig. 6 shows the estimated error for different values of τ and k. From those figures we can state that estimated error converges to zero for all admissible values of τ and k but the convergence rate is different for each value.
7 Concluding remarks A robust observer has been designed for time-delay systems. The time-delay system is assumed to be spectrally controllable and spectrally observable and the uncertainties are assumed to be defined by a weighting matrix in the frequency domain. The design method is based on a polynomial factorization of the nominal transfer matrix and on the parameterization of the set of all stable observers. The parameterization term is then found as a solution of an optimization problem depending on the uncertainty type. The assigned eigenvalues by the observer-based state feedback in the factorization stage influence the solution of the final optimization problem. This relationship between the closed-loop eigenvalues and the optimization problem will be considered in the future work.
References 1. Doyle, J. C., B. A. Francis and A. R. Tannenbaum (1992). Feedback control theory. Macmillan Publishing Company. 2. Dugard, L. and E. I. Verriest (1998). Stability and control of time-delay systems. Springer. 3. Fattouh, A., O. Sename and J.-M. Dion (1998). H∞ observer design for time-delay systems. In ‘Proc. 37th IEEE Confer. on Decision & Control’. Tampa, Florida, USA. pp. 4545–4546. 4. Fattouh, A., O. Sename and J.-M. Dion (1999a). ‘Robust observer design for time-delay systems: A Riccati equation approach’. Kybernetika 35(6), 753–764. 5. Fattouh, A., O. Sename and J.-M. Dion (1999b). An unknown input observer design for linear time-delay systems. In ‘Proc. 38th IEEE Confer. on Decision & Control’. Phoenix, Arizona, USA. pp. 4222–4227. 6. Fattouh, A., O. Sename and J.-M. Dion (2000). Robust observer design for linear uncertain time-delay systems: A factorization approach. In ‘14th Int. Symp. on Mathematical Theory of Networks and Systems’. Perpignan, France, June, 19-23. 7. Fiagbedzi, Y. A. and A. E. Pearson (1990). ‘Exponential state observer for time-lag systems’. Int. Journal of Control 51(1), 189–204. 8. Francis, B. (1987). A course in H∞ control theory. Springer-Verlag. 9. Kamen, E. W., P. P. Khargonekar and A. Tannenbaum (1986). ‘Proper stable bezout factorizations and feedback control of linear time-delay systems’. Int. Journal of Control 43(3), 837–857. 10. Lee, E. B. and A. Olbrot (1981). ‘Observability and related structural results for linear hereditary systems’. Int. Journal of Control 34(6), 1061–1078. 11. Malabre, M. and R. Rabah (1993). ‘Structure at infinity, model matching and disturbance rejection for linear systems with delays’. Kybernetika 29(5), 485–498.
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12. Manitius, A. Z. (1984). ‘Feedback controllers for a wind tunnel model involving a delay: Analytical design and numerical simulation’. IEEE Trans. on Automatic Control 29(12), 1058–1068. 13. Niculescu, S.-I. (2001). Delay effects on stability. A robust control approach. Vol. 269. Springer-Verlag: Heidelbeg. 14. Nobuyama, E. and T. Kitamori (1990). Spectrum assignment and parameterization of all stabilizing compensators for time-delay systems. In ‘Proc. 29th Confer. on Decision and Control’. Honolulu, Hawaii. pp. 3629–3634. 15. Picard, P., J. F. Lafay and V. Ku˘cera (1998). ‘Model matching for linear systems with delays and 2D systems’. Automatica 34(2), 183–191. 16. Ramos, J. L. and A. E. Pearson (1995). ‘An asymptotic modal observer for linear autonomous time lag systems’. IEEE Trans. on Automatic Control 40(7), 1291–1294. 17. Sename, O. (2001). ‘New trends in design of observers for time-delay systems’. Kybernetika 37(4), 427–458. 18. Sontag, E. D. (1976). ‘Linear systems over commutative rings: A survey’. Ricerche di Automatica 7(1), 1–34. 19. Wang, Q.-G., T. H. Lee and K. K. Tan (1999). Finite spectrum assignment for time-delay systems. Springer-Verlag. 20. Watanabe, K. and T. Ouchi (1985). ‘An observer of systems with delays in state variables’. Int. Journal of Control 41(1), 217–229. 21. Yao, Y. X., Y. M. Zhang and R. Kovacevic (1996). ‘Parameterization of observers for time delay systems and its application in observer design’. IEE Proc.-Control Theory Appl. 143(3), 225–232.
