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Robust Reduced-order Output-feedback H∞ Control
Proceedings of the 6th IFAC Symposium on Robust Control Design Haifa, Israel, June 16-18, 2009
Robust Reduced-order Output-feedback H∞ Control ⋆ I. Yaesh ∗ U. Shaked ∗∗ Advanced Systems Division, Control Dept., Israel Military Industries, Israel. (Tel: +972-3640-6351; e-mail: [email protected]). ∗∗ School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel (e-mail: [email protected]) ∗
Abstract: The problem of robust H∞ reduced-order control design for linear systems in presence of polytopic type uncertainties is addressed. The controller is obtained by transforming the problem into a robust static output-feedback control design problem. Applying a recent method for solving the latter by means of linear matrix inequalities, a sufficient condition is obtained for the existence of a solution to the problem. A simple method for deriving the reduced-order controller is given in the case where the latter condition is satisfied. Keywords: Reduced-order control, static output-feedback, H-infinity control 1. INTRODUCTION The problem of controlling linear systems applying a reduced order controller has attracted a lot of interest in the past. In the case where the parameters of the system were all known, some iterative methods [1]-[7] have been suggested that achieve, if they converge, a controller of a prescribed order that satisfy prescribed demands on the performance of the closed -loop system. The solutions achieved by these methods, even if they converge to the global minimum of the performance index cannot guarantee the performance in the case where the parameters of the system are not certain and are known to reside in a given polytope. Even if the case where a full order controller is allowed (its order will be the one of the given plant) there exists no solution that finds a solution to the problem that will satisfy the required specifications over the entire uncertainty polytope. In the present paper we introduce a method by which the robust control problem can be solved. By adopting the H∞ setting, the idea is to transfer the robust reducedorder output-feedback control problem into an equivalent problem of seeking a static output-feedback that satisfies the H∞ performance requirements over the entire uncertainty polytope [9]. It is well known, however, that the problem of the static-output-feedback problem is bilinear in the decision variables and thus, even in the case where the system is perfectly known some iterative methods should be applied that will produce a local optimum result solution to the problem. Most of the methods that have been suggested in the past for solving the problem have addressed a single plant with known parameters and they cannot be applied to plants with polytopic type uncertainties. ⋆ This work was supported in part by C&M Maus Chair at Tel Aviv University.
978-3-902661-45-6/09/$20.00 © 2009 IFAC
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In [8] a Linear Matrix Inequality (LMI) based method has been presented which yields a sufficient condition for the existence of an output-feedback gain matrix that satisfies the H∞ performance requirements for polytopic uncertainties. The controller that is obtained, if the condition is satisfied, has been shown to be very effective and it entails only little overdesign [8]. It is thus the purpose of the present paper to apply a modified version of this method to the solution of the corresponding robust output-feedback control problem. 2. PROBLEM FORMULATION We consider the following linear time-invariant system. x˙ = Ax + B2 u + B1 w, y = C2 x + D21 w z = C1 x + D12 u
x(0) = 0 (1)
where x ∈ Rn is the state vector, y ∈ Rp is the measured output, u ∈ Rm is the controlled input, and z ∈ Rq is the objective vector. We assume that the system parameters lie within the following polytope ∆
Ω = [ A B1 C1 C2 D21 ] which is described by its vertices. That is, for i h ∆ (i) Ωi = A(i) B1(i) C1(i) C2(i) D21
(2)
(3)
we have Ω = Co{Ω1 , Ω2 , ..., ΩN } where N is the number of vertices. In other words: N N X X fi = 1 , fi ≥ 0. Ωi f i , Ω= i=1
(4)
(5)
i=1
10.3182/20090616-3-IL-2002.0051
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
We note that we have assumed no uncertainty in B2 and D12 in order to ensure that the LMIs we will obtain will be affine in the uncertain matrices. Uncertainty in these matrices also can be considered using the lowpass augmentation of Remark 8 below. We seek the following controller of order r ≤ n. x˙ c = Ac xc + Bc y, u = Cc xc + Dc y
xc (0) = 0
(6)
where xc ∈ Rr such that the following H∞ criterion is satisfied over the entire uncertainty polytope Ω, for a prescribed attenuation level γ > 0. J = kzk22 − γ 2 kwk22 < 0, ∀w 6= 0 ∈ L2
(7)
Applying the controller (6) to (1), the closed-loop system is described by: ξ˙ = Aξ + Bw, z = Cξ + Dw
ξ(0) = 0
(8)
∆
where ξ = col{x, xc } and: A 0 | B1 0 B2 A|B ∆ Ac Bc 0 Ir | 0 − − − = 0 0 | 0 + Ir 0 C |D
− − −− C1 0 | 0
−− 0 D12
Cc Dc
(9)
C2 0 | D21
Denoting: ∆ Ac Bc K= Cc Dc
(10)
the system (8) can be described as one that results by applying the ’static’ control law u¯ = K y¯ to the following system ¯ +B ¯2 u ¯1 w x¯˙ = Ax ¯+B ¯ ¯ y¯ = C2 x¯ + D21 w (11) ¯ ¯ z = C1 x ¯ + D12 u ¯ where x ¯ ∈ Rn+r , y¯ ∈ Rp+r , and u ¯ ∈ Rm+r and where: A 0 ¯1 = B1 , B ¯2 = 0 B2 , C¯1 = [C1 0] , , B A¯ = 0 0 0 I 0 (12) 0 I ¯ 21 = 0 ¯ 12 = [0 D12] , and D , D C¯2 = D21 C2 0 Clearly, i Ω describe the vertices h the N vertices of polytope ∆ (i) (i) (i) (i) (i) ¯ ¯ C¯ C¯ D Ωi = A¯ B 21 2 1 1 ¯ The robust output feedof the corresponding polytope Ω. back control design is thus reduced to the problem of finding the static output feedback gain K so that by applying u ¯ = K y¯ to the system (11) the requirement (7) ¯ is satisfied over the entire polytope Ω. 3. THE EQUIVALENT STATIC-FEEDBACK GAIN Given the system (11) whose parameters reside in the ¯ a gain matrix K is sought such that by polytope Ω applying u¯ = K y¯, (7) is satisfied over the entire polytope.
156
∆
Defining ξ¯ = col{¯ x, y¯} we obtain from (11) the following descriptor representation for the closed-loop system. ¯2 K ¯ A¯ B B E ξ˙¯ = ¯ ξ¯ + ¯1 w C2 −I D21 (13) ¯ 12 K ξ¯ z = C¯1 D h i ∆ where E = I0 00 . Consider the following Lyapunov function: 0 ∆ P ¯ P¯ = V = ξ¯T E P¯ ξ, −C¯2 P α−1 Pˆ
(14)
where P ∈ R(n+r)×(n+r) is a positive definite matrix, Pˆ is a matrix in R(r+p)×(r+p) and α is chosen for simplicity to be a scalar. It is readily found that V ≥ 0 and that the following holds. V˙ = 2¯ xT P x ¯˙ = 2ξ¯T P¯ Tcol{x ¯˙, 0} ¯ ¯ ¯ (15) A B2 K ¯ B ξ + ¯1 w) = 2ξ¯T P¯ T ( ¯ D21 C2 −I and stability for a single plant in Ω is assured if there exist P¯ and K that satisfy the following inequality ¯2 K T ¯2 K A¯ B A¯ B + ¯ P¯ < 0 P¯ T ¯ C2 −I C2 −I To satisfy (7) we require: V˙ − γ 2 wT w + z T z < 0
(16)
and readily obtain the following inequality. h ¯ i ¯ T T hA¯ B¯ Ki A¯T ¯T C1 ¯ ξ¯ P¯ T ¯ 2 + ¯ T T C2 P¯ P¯ T B¯1 ¯T ξ D21 K TD C2 −I K B2 −I 12 w w <0 ∗ −γ 2 I 0 z z ∗
∗
−I
(17) The latter inequality is not affine in the decision variable P¯ and K. Thus, denoting: Q 0 −1 ¯ ¯ Q=P = ˆ¯ (18) ˆ QC2 αQ where Q > 0, we multiply the matrix in (17) by ¯ T , I} and diag{Q, ¯ I}, from the left and the right, diag{Q respectively, and the following requirement is obtained. T ¯T ¯ ¯ ¯2 Y +QC¯ T − C¯ TQ ˆT AQ+Q A +B2 Y C¯2 + C¯2T Y TB2T αB 2 2 T ˆ ˆ ∆ ∗ −α(Q + Q ) Γ1 = ∗ ∗ ∗ ∗ ¯T ¯1 QC¯ T + C¯ T Y T D B 12 1 2 T ¯T ¯ D21 αY D12 < 0, Q > 0 (19) 2 −γ I 0 ∗ −I ˆ Since the second block on the diagonal of where Y = K Q. ˆ−Q ˆ T ), Q ˆ that solves (19) is not singular for Γ1 is −α(Q α > 0 and thus K is recovered from Y by ˆ −1 . K = YQ (20) The latter leads to the following theorem.
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
Theorem 1. Consider the system (11). There exists a static output-feedback that for a prescribed H∞ -norm bound 0 < γ satisfies (7) for a single plant in the uncertainty ¯ if there exist Q ∈ R(n+r)×(n+r) , Q ˆ polytope Ω ∈ (p+r)×(p+r) (r+m)×(r+p) R and Y ∈ R that, for some tuning scalar parameter 0 < α satisfy (19) and Q > 0. If a solution to the latter set of LMIs exists, the static outputfeedback gain that stabilizes the system (11) and achieves the required performance is given by (20). It is noted here that the latter result is an improvement to the one obtained in [8]. In the present result there is no need to add an artificial lowpass element with very large bandwidth to the system and the additional semi positive element that appears in [8] in the second diagonal block in Γ1 is now missing. The above provides a sufficient condition for the existence of the gain matrix K that guarantees the required per¯ As formance for a single plant (11) in the polytope Ω. such, this gain matrix provides by (10) the reduced order controller for this single plant that satisfies (7). Since the elements are affine in the system matrices the condition of Theorem 1 can be extended to cope with all ¯ the systems in Ω. Denoting the matrices of (11) that correspond to the i¯ (and thus also to the vertices of Ω) by: th vertex of Ω (i) ¯ (i) ¯ (i) ¯ (i) ¯ (i) , i = 1, ..., N , we obtain the ¯ A , B1 , C1 , C2 and D 21 following result. Corollary 2. There exists a reduced-order controller (6) that, for a prescribed γ > 0 satisfies, when applied to (1), the requirement (7) over the entire polytope Ω, if there ˆ ∈ R(p+r)×(p+r) and exist matrices: Q ∈ R(n+r)×(n+r) , Q (r+m)×(r+p) Y ∈R that, for some tuning scalar parameter 0 < α satisfy the following set of LMIs. ˆT ¯2 Y+QC¯ (i)T− C¯ (i)TQ Σ1 αB 2 2 ∗ ˆ+Q ˆT ) −α(Q ∗ ∗ ∗ ∗ (i) (i)T (i)T T ¯ ¯ 12 B QC¯1 + C¯2 Y TD 1 (i) T T ¯ ¯ 12 D αY D < 0, i = 1,...,N Q > 0 (21) 21 2 −γ I 0 ∗ −I where ¯2 Y C¯ (i)+ C¯ (i)TY T . Σ1 = A¯(i)Q+QTA¯(i)T+ B 2 2 If a solution to the latter set of LMIs exists, the matrices of the controller (6) that stabilizes the system (1) and achieves the required performance over the polytope Ω is given by ∆ Ac Bc ˆ −1 . K= =YQ (22) Cc Dc ¯ that will satisfy the The above corollary seeks a single Q N + 1 LMIs of (21). As such it provides what is called the quadratic stabilizing solution [2]. This solution which may be sometimes quite conservative can be replaced by another, less conservative solution as follows. It turns out, ¯ may sometimes however, that the special structure of Q lead to zero Bc and Cc , thus resulting in suboptimal values
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of γ and will lead to static, rather than dynamic, outputfeedback controllers. To circumvent this difficulty, the structure of the augmented system of (9) can be slightly adjusted to include an extra degree of freedom which we denote by C3 . We write: " # A 0 | B1 0 B2 A|B ∆ Ir 0 Ac Bc −1 0 Ir | 0 − − − = 0 0 | 0 +− T T − Cc Dc C2 0 | D21 − − −− C |D C1 0 | 0 0 D12 I 0 . Absorbing T −1 in K we readily where T = −C3 I obtain Ac + Bc C3 Bc K= (23) Cc + Dc C3 Dc and the following model matrices which include the new degree of freedom: A 0 ¯1 = B1 , B ¯2 = 0 B2 , C¯1 = [C1 0] , , B A¯ = I 0 0 0 (24) 0 0 I ¯ 21 = 0 ¯ 12 = [0 D12 ] , and D C¯2 = , D C2 −C3 D21 The latter allows the application of Corollary 2 above, while performing a search on C3 in C¯2 so as to minimize γ. The controller is then derived by applying the modification (23) on the controller (22) of Corollary 2. Remark 3. To allow a more efficient optimization of the controller which avoids the search for C3 , one may alternatively restrict the structure of the controller to one of the following structures where F is a design matrix of the appropriate dimensions: • Fixed Cc and Dc controller with Cc = B2T F where F ∈ Rn×r and arbitrary Dc ∈ Rm×p but otherwise free. The matrices Ac and Bc are then sought which lead to the required performance. In this case, the closed-loop system is described by: ξ˙ = Aξ + Bw (25) z = Cξ + Dw ∆
where ξ = col{x, xc } and where B1 +B2 Dc D21 A+B2 Dc C2 B2 B2T F , , B= A= Bc D21 Bc C2 Ac C = C1 +D12 Dc C2 D12 B2T F and D = D12 Dc D21 . ∆
Defining K = [ Ac Bc ] we obtain the following. ¯ B ¯2 K C¯2 , B = B ¯1 + K D ¯ 21 A = A+ ¯ 12 K C¯2 , D = D ¯ 11 + D ¯ 12 K D ¯ 21 C = C¯1 + D where B1 + B2 Dc D21 A+B2 Dc C2 B2 B2T F ¯ ¯ , B1 = A= , 0 0 0 ¯2 = 0 , C¯1 = C1 +D12 Dc C2 D12 B T F , B 2 I 0 0 I ¯ ¯ 11 = D12 Dc D21 , D ¯ 12 = 0. ¯ , D21 = ,D C2 = D21 C2 0 We may then choose Dc = 0 and an arbitrary F ∈ Rn×r . Obviously, this case is suitable for the case where B2 is without uncertainty.
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
• When B2 involves uncertainty and C2 is perfectly known, we may fix Bc and Dc with Bc = F C2T and seek Ac and Cc that will satisfy the system requirement. In this case, the closed-loop system is described by: ξ˙ = Aξ + Bw z = Cξ + Dw
where: ∆ Aˆ =
This condition is, in fact, a BRL condition for the system: ˆ + Bw, ˆ ˆ ζ˙ = Aζ z = Cζ.
(26)
Thus, applying Finsler’s lemma along the lines of [4], we find the following result: ¯ with the structure of Lemma 5. There exists matrices Q, ¯ (18), and K that satisfy (28) iff there exist matrices Q, ¯ and H ¯ that satisfy the with the above structure, K, G following LMI. T ¯ −G ¯ T + AˆT H ¯ +Q ¯T G ¯T B ˆ Cˆ T ¯ Aˆ + AˆT G G ˆ 0 ¯ −H ¯T ¯TB H ∗ −H < 0 (29) ∗ ∗ −γ 2 I 0 ∗ ∗ ∗ −I
∆
where ξ = col{x, xc } and where: A+B2 Dc C2 B2 Cc B1 +B2 Dc D21 , B = , F C2T C2 Ac F C2T D21 A= C = [C1 +D12 Dc C2 D12 Cc ] and D = D12 Dc D21 . Ac ∆ Defining K = we obtain: Cc ¯ B ¯2 K C¯2 , B = B ¯1 +K D ¯ 21, A = A+ ¯ 12 K C¯2 , D = D ¯ 11 + D ¯ 12 K D ¯ 21 C = C¯1 + D
Proof : We first note that (28) is, in fact, equivalent to ¯ V˙ − γ 2 wT w + z T z < 0 where V = ζ T Qζ. ∆ ˙ ¯ of the We then denote η = col{ζ, ζ, w, z} and we seek Q, structure of (18), such that the following will be satisfied.
where A + B2 Dc C2 0 ¯1 = B1 + BT2 Dc D21 , , B A¯ = T F C2 C2 0 F C2 D21 ¯2 = 0 B2 , C¯1 = [C1 +D12 Dc C2 0], C¯2 = [0 I ], B I 0 ¯ ¯ 11 = D12 Dc D21 , D ¯ 12 = [0 D12 ] . D21 = 0, D
η T Ψη < 0, ∀ζ satisfying Ση = 0 where
We may choose, then, Dc = 0 and an arbitrary F ∈ Rr×n .
4. PARAMETER DEPENDENT LYAPUNOV APPROACH ¯ T , I} and Multiplying the matrix in (17) by diag{Q ¯ I}, from the left and the right, respectively, we diag{Q, obtain the following LMI condition for the existence of (16). ˆ Cˆ T ¯ Q ¯T B ¯ T Aˆ + AˆT Q Q (28) ∗ −γ 2 I 0 < 0 ∗ ∗ −I
¯T 0 Q 0 0 0 −γ 2 I 0 0
(30)
0 ˆ ˆ 0 , Σ = A −I B 0 . 0 Cˆ 0 0 −I I
Applying Finsler’s lemma [9] we find that a sufficient condition for the performance requirement of (7) to be ¯ is the existence of X and satisfied, for a single plant in Ω, ¯ such that Q Ψ + XΣ + ΣT X T < 0 ¯ ¯ GH 0 0 we readily obtain that the suffiTaking X = 0 0 0 I ¯ H ¯ and Q ¯ that satisfy cient condition is the existence of G, (29). 2 ¯ ¯ We next choose the following structures for G and H: 0 H 0 ¯= G ¯ G ˆ and H = α3 G ˆ , ˆ C¯2 α1 G ˆ C¯2 α2 G G
(27)
¯ that will satisfy the N + 1 Corollary 2 seeks a single Q LMIs of (21). As such, it provides what is called the quadratic stabilizing solution [2]. This solution which may be sometimes quite conservative can be replaced by another, less conservative solution as follows.
0 Q ¯ Ψ= 0 0
In the case where there exists, via Theorem 1, a static output-feedback solution (r = 0) to the problem, for a given γ, the question arises whether this solution can be recovered by solving (19) for r = n applying a special ˆ and Y . The following result is readily structure to Q, Q, obtained by substitution. Corollary 4. Consider the output-feedback control problem of Theorem 1 for r = n. The static output-feedback solution (r = 0) is obtained by solving the inequality ˆ and Y that possess the following structure Γ1 ≤ 0 for Q, Q for any positive scalar β. ˜ˆ ˜ βIn }, Q ˆ = diag{βIn , Q}, Q = diag{Q, and Y = diag{0n×n, Y˜ }.
C¯1T A¯T C¯2T ∆ ˆ ˆ ∆ ¯T ¯ T = , B 1 T T , C = B1 D21 . T ¯T K D12 K B2 −I
where αi , i = 1, 2, 3 are tuning scalar parameters. Denotˆ substituting for A, ˆ B, ˆ C, ˆ and allowing for ing Y = K G, ¯ vertex dependent Q the following result is thus obtained. Theorem 6. There exists a reduced-order controller (6) that, for a prescribed γ > 0 satisfies, when applied to (1), the requirement (7) over the entire polytope Ω, if there exist matrices: Q(i) > 0, i = 1, ..., N , G, H ∈ ˆ (i) > 0, i = 1, ..., N , G ˆ ∈ R(p+r)×(p+r) R(n+r)×(n+r) , Q and Y ∈ R(r+m)×(r+p) that, for some positive tuning scalar parameters α1 , α2 and α3 satisfy the following set of LMIs. # " ¯ (i)Y ¯ (i) Y C¯ (i) α2 B ¯(i)H +α3 B Σ2 Σ3 A (i)T T 2 2 2 ¯ ¯ ∗−α1(G+ ˆ G ˆ T ) Q − G + C¯ (i)H −α G ˆ ˆ ¯ (i) α2 G 3 C2 2 ¯ −H ¯T ∗ −H ∗ ∗ ∗ ∗
158
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
# " ¯ (i) B 1 ¯ (i) D 21 # # " " (i)T (i)T ¯ (i) < 0, i = 1, ..., N (31) ¯ (i)T B H T C¯1 + α3 C¯2 Y T D 1 12 ¯ (i) ¯ (i)T D α2 Y T D 21 12 −γ 2 I 0 ∗ −I # " (i)T (i)T ¯ (i)T GT C¯1 + C¯2 Y T D 12 ¯ (i)T α1 Y T D 12
where (i)T ¯ (i)T+B ¯ (i) Y C¯ (i) Σ2 = GTA¯(i)T+A¯(i) G+C¯2 Y TB 2 2 2 and (i)T (i)T ˆ T ¯ (i) Y. +α1B Σ3 = GTC¯2 −C¯2 G 2
Operating point Mach number Altitude (ft)
1 .5 5000
2 .9 35000
3 .85 5000
4 1.5 35000
a11 a12 a13 a21 a22 a23 b1
-.9896 17.41 96.15 .2648 -.8512 -11.39 -97.78
-.6607 18.11 84.34 .08201 -.6587 -10.81 -272.2
-1.702 50.72 263.5 .2201 -1.418 -31.99 -85.09
-.5162 29.96 178.9 -.6896 -1.225 -30.38 -175.6
Table 1: The parameters of the four operating points.
If a solution to the latter set of LMIs exists, the matrices of the controller that stabilizes the system (1) and achieves the required performance over the polytope Ω are given by Ac Bc ˆ −1 . =YG Cc Dc Remark 7. The results of the above two theorems can be readily applied to the case where a robust controller of full order is sought. By taking r = n the solution of the relevant LMIs, if exist, will provide the matrices of the state space model of the controller. Remark 8. The condition of (19) can be applied to reduced order gain scheduling control in cases where uncertainty is simultaneously encountered in B2 , C2 and D12 . The latter can be achieved by introducing lowpass elements in the control input and the measurement output of the system [8]. Since, under the latter assumption, the LMI (19) is convex in Y over the polytope Ω, the matrix Y can be made vertex dependent while Q is kept constant over the polytope. The resulting K and, therefore, the resulting controller will by vertex dependent, thus allowing the scheduling. 5. EXAMPLE Consider a modified version of the pitch control [10] of F4E described by Nz Nz 0 a11 a12 a12 b1 1 0 0 d q a21 a22 a23 0 q 0 0 1 0 = 0 0 −30 30 δe + 0 u+ 0 0 1w dt δe 0 0 0 104 0 0 0 −104 δ˜e δ˜e # " # " 0 10 0 0 z = 0 1 0 0 x+ 0 u 1 00 1 0 1 0 0 0 x y= 0 1 0 0 The state-vector consists of the aerodynamic components which are the load-factor Nz , the pitch-rate q and the servo components which are the elevon angle δe , and an additional state δ˜e . The state δe relates to the elevon command u via a first-order model of the mechanical lag of the servo of a bandwidth of 30rad/sec, whereas δ˜e is the electrical lag of the servo. The parameters ai,j , i = 1, 2; j = 1, 2, 3 and b1 are given in [10] in the following four operating points which are assumed here as vertices of a polytopic plant
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We first designed a static output-feedback controller which achieved a minimal γ = 7.54 with the feedback gain matrix K = [ 0.1093 1.0461 ]. The closed-loop singular values and the resulting feedback gains are depicted versus frequency in Fig. 1a and Fig. 1b., respectively. In Fig. 1a blue solid line, red dash-dotted line, green dashed line and magenta dotted lines respectively corresponds to operating points 1 − 4. In Fig. 1b, the blue solid line depicts K1 whereas red dash-dotted depicts K2 . We next aim at reducing γ by adding dynamics to the controller. To this end, we applied the design approach of Remark 1, with a fixed Cc = B2T F . For each order r ∈ {1, 2, 3, 4} we have chosen F ∈ Rn×r from 30 random trials. The results are described in Table 2. order r γ
0 7.52
1 3.98
2 4.05
3 3.96
4 3.95
SF 3.86
Table 2: The minimum achievable γ for different orders in comparison with the state-feedback. For r = 1 we have obtained the most significant improvement. The resulting closed-loop singular values for r = 1 and the corresponding Bode gains of the controller are depicted in Fig. 2a and Fig. 2b, respectively. In Fig. 2a blue solid line, red dash-dotted line, green dashed line and magenta dotted lines respectively corresponds to operating points 1 − 4. In Fig. 2b, the blue solid line depicts K1 whereas red dash-dotted depicts K2 . We note that the actual closed-loop singular values for r = 1 are somewhat smaller than those obtained for the static-output feedback controller. We also note that the low frequency gains of the controller obtained for r = 1 are higher than those found for r = 0. These gains roll off, however, at about 83 rad/sec. 6. CONCLUSIONS A Linear Matrix Inequalities based approach for designing static and dynamic output feedback controllers of reduced order for a linear time-invariant system has been considered. Both quadratically stable and parameter-dependent Lyapunov approaches have been considered where various design simplification schemes have been proposed, at the cost of some conservatism. The new design method’s effectiveness and controller order effect have been illustrated via a robust flight control example. REFERENCES [1] T. Iwasaki and R. E. Skelton, “The XY-centering algorithm for the dual LMI problem. A new approach
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
[5] [6]
[7]
[8] [9] [10]
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10 a)
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1.4 1.2 1 |K(ω)|
[4]
3
0.8 0.6 0.4 0.2 0 −1 10
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10 ω [rad/sec]
2
10 b)
3
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Fig. 1. Static controller, a) Closed-loop singular values, b) Gain Bode of the controller Dynamic output feedback r = 1 , γ = 3.98 3.5 3 2.5 2 σ
[3]
Static output feedback r = 0 , γ = 7.54 4
1.5 1 0.5 0 −1 10
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to fixed order control designs,” Int. J. Contr., vol. 62, pp. 1257-1272, 1995. I.R. Petersen, “Quadratic stabilizability of uncertain linear systems containing both constant and time varying uncertain parameters”, J. of Optimization Theory and Applications, vol. 57(3), pp. 439-461, 1988. I. Yaesh and U. Shaked, “Minimum entropy static feedback control with an H∞ -norm performance bound”, IEEE Trans. on Automat. Contr., AC-42, pp. 853-858, 1997. M. C. de Oliveira and R. E. Skelton, “Stability test for constrained linear systems”, in Perspectives in Robust Control , S. O. Reza Moheimani Ed., Lecture Notes in Control and Information Sciences 268, Springer, London 2001. T. Iwasaki, “The dual iteration for fixed-order control,” IEEE Trans. on Automat. Contr., AC-44, pp. 783-788, 1999. P. L. D. Peres , J. C. Geromel and S. R. Souza, “H∞ control design by static output-feedback,” Proc. of the IFAC Symposium on Robust Control Design, Rio de Janeiro, Brasil, 1994, pp. 243-248. F. Leibfritz, “An LMI-based algorithm for designing suboptimal static H2 /H∞ output feedback controllers”,SIAM J. of Control and Optimization, vol. 39, pp.1711-1735, 2001. U. Shaked, “An LPD approach to robust H2 and H∞ static output-feedback design,” IEEE Trans. on Automat. Contr., AC-48, pp. 866-872, 2003. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequality in Systems and Control Theory, SIAM Frontier Series, April 1994. J. Ackermann, “Longitudinal control of fighter aircraft F4E with additional canards”. In K. P. Sondergeld (compiler), A collection of plant Models and Design Specifications for Robust Control.DFVLR, Oberpfaffenhoffen, Germany.
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10 b)
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Fig. 2. 1st order controller, a) Closed-loop singular values, b) Gain Bode of the controller
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Robust Reduced-order Output-feedback H∞ Control ⋆ I. Yaesh ∗ U. Shaked ∗∗ Advanced Systems Division, Control Dept., Israel Military Industries, Israel. (Tel: +972-3640-6351; e-mail: [email protected]). ∗∗ School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel (e-mail: [email protected]) ∗
Abstract: The problem of robust H∞ reduced-order control design for linear systems in presence of polytopic type uncertainties is addressed. The controller is obtained by transforming the problem into a robust static output-feedback control design problem. Applying a recent method for solving the latter by means of linear matrix inequalities, a sufficient condition is obtained for the existence of a solution to the problem. A simple method for deriving the reduced-order controller is given in the case where the latter condition is satisfied. Keywords: Reduced-order control, static output-feedback, H-infinity control 1. INTRODUCTION The problem of controlling linear systems applying a reduced order controller has attracted a lot of interest in the past. In the case where the parameters of the system were all known, some iterative methods [1]-[7] have been suggested that achieve, if they converge, a controller of a prescribed order that satisfy prescribed demands on the performance of the closed -loop system. The solutions achieved by these methods, even if they converge to the global minimum of the performance index cannot guarantee the performance in the case where the parameters of the system are not certain and are known to reside in a given polytope. Even if the case where a full order controller is allowed (its order will be the one of the given plant) there exists no solution that finds a solution to the problem that will satisfy the required specifications over the entire uncertainty polytope. In the present paper we introduce a method by which the robust control problem can be solved. By adopting the H∞ setting, the idea is to transfer the robust reducedorder output-feedback control problem into an equivalent problem of seeking a static output-feedback that satisfies the H∞ performance requirements over the entire uncertainty polytope [9]. It is well known, however, that the problem of the static-output-feedback problem is bilinear in the decision variables and thus, even in the case where the system is perfectly known some iterative methods should be applied that will produce a local optimum result solution to the problem. Most of the methods that have been suggested in the past for solving the problem have addressed a single plant with known parameters and they cannot be applied to plants with polytopic type uncertainties. ⋆ This work was supported in part by C&M Maus Chair at Tel Aviv University.
978-3-902661-45-6/09/$20.00 © 2009 IFAC
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In [8] a Linear Matrix Inequality (LMI) based method has been presented which yields a sufficient condition for the existence of an output-feedback gain matrix that satisfies the H∞ performance requirements for polytopic uncertainties. The controller that is obtained, if the condition is satisfied, has been shown to be very effective and it entails only little overdesign [8]. It is thus the purpose of the present paper to apply a modified version of this method to the solution of the corresponding robust output-feedback control problem. 2. PROBLEM FORMULATION We consider the following linear time-invariant system. x˙ = Ax + B2 u + B1 w, y = C2 x + D21 w z = C1 x + D12 u
x(0) = 0 (1)
where x ∈ Rn is the state vector, y ∈ Rp is the measured output, u ∈ Rm is the controlled input, and z ∈ Rq is the objective vector. We assume that the system parameters lie within the following polytope ∆
Ω = [ A B1 C1 C2 D21 ] which is described by its vertices. That is, for i h ∆ (i) Ωi = A(i) B1(i) C1(i) C2(i) D21
(2)
(3)
we have Ω = Co{Ω1 , Ω2 , ..., ΩN } where N is the number of vertices. In other words: N N X X fi = 1 , fi ≥ 0. Ωi f i , Ω= i=1
(4)
(5)
i=1
10.3182/20090616-3-IL-2002.0051
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
We note that we have assumed no uncertainty in B2 and D12 in order to ensure that the LMIs we will obtain will be affine in the uncertain matrices. Uncertainty in these matrices also can be considered using the lowpass augmentation of Remark 8 below. We seek the following controller of order r ≤ n. x˙ c = Ac xc + Bc y, u = Cc xc + Dc y
xc (0) = 0
(6)
where xc ∈ Rr such that the following H∞ criterion is satisfied over the entire uncertainty polytope Ω, for a prescribed attenuation level γ > 0. J = kzk22 − γ 2 kwk22 < 0, ∀w 6= 0 ∈ L2
(7)
Applying the controller (6) to (1), the closed-loop system is described by: ξ˙ = Aξ + Bw, z = Cξ + Dw
ξ(0) = 0
(8)
∆
where ξ = col{x, xc } and: A 0 | B1 0 B2 A|B ∆ Ac Bc 0 Ir | 0 − − − = 0 0 | 0 + Ir 0 C |D
− − −− C1 0 | 0
−− 0 D12
Cc Dc
(9)
C2 0 | D21
Denoting: ∆ Ac Bc K= Cc Dc
(10)
the system (8) can be described as one that results by applying the ’static’ control law u¯ = K y¯ to the following system ¯ +B ¯2 u ¯1 w x¯˙ = Ax ¯+B ¯ ¯ y¯ = C2 x¯ + D21 w (11) ¯ ¯ z = C1 x ¯ + D12 u ¯ where x ¯ ∈ Rn+r , y¯ ∈ Rp+r , and u ¯ ∈ Rm+r and where: A 0 ¯1 = B1 , B ¯2 = 0 B2 , C¯1 = [C1 0] , , B A¯ = 0 0 0 I 0 (12) 0 I ¯ 21 = 0 ¯ 12 = [0 D12] , and D , D C¯2 = D21 C2 0 Clearly, i Ω describe the vertices h the N vertices of polytope ∆ (i) (i) (i) (i) (i) ¯ ¯ C¯ C¯ D Ωi = A¯ B 21 2 1 1 ¯ The robust output feedof the corresponding polytope Ω. back control design is thus reduced to the problem of finding the static output feedback gain K so that by applying u ¯ = K y¯ to the system (11) the requirement (7) ¯ is satisfied over the entire polytope Ω. 3. THE EQUIVALENT STATIC-FEEDBACK GAIN Given the system (11) whose parameters reside in the ¯ a gain matrix K is sought such that by polytope Ω applying u¯ = K y¯, (7) is satisfied over the entire polytope.
156
∆
Defining ξ¯ = col{¯ x, y¯} we obtain from (11) the following descriptor representation for the closed-loop system. ¯2 K ¯ A¯ B B E ξ˙¯ = ¯ ξ¯ + ¯1 w C2 −I D21 (13) ¯ 12 K ξ¯ z = C¯1 D h i ∆ where E = I0 00 . Consider the following Lyapunov function: 0 ∆ P ¯ P¯ = V = ξ¯T E P¯ ξ, −C¯2 P α−1 Pˆ
(14)
where P ∈ R(n+r)×(n+r) is a positive definite matrix, Pˆ is a matrix in R(r+p)×(r+p) and α is chosen for simplicity to be a scalar. It is readily found that V ≥ 0 and that the following holds. V˙ = 2¯ xT P x ¯˙ = 2ξ¯T P¯ Tcol{x ¯˙, 0} ¯ ¯ ¯ (15) A B2 K ¯ B ξ + ¯1 w) = 2ξ¯T P¯ T ( ¯ D21 C2 −I and stability for a single plant in Ω is assured if there exist P¯ and K that satisfy the following inequality ¯2 K T ¯2 K A¯ B A¯ B + ¯ P¯ < 0 P¯ T ¯ C2 −I C2 −I To satisfy (7) we require: V˙ − γ 2 wT w + z T z < 0
(16)
and readily obtain the following inequality. h ¯ i ¯ T T hA¯ B¯ Ki A¯T ¯T C1 ¯ ξ¯ P¯ T ¯ 2 + ¯ T T C2 P¯ P¯ T B¯1 ¯T ξ D21 K TD C2 −I K B2 −I 12 w w <0 ∗ −γ 2 I 0 z z ∗
∗
−I
(17) The latter inequality is not affine in the decision variable P¯ and K. Thus, denoting: Q 0 −1 ¯ ¯ Q=P = ˆ¯ (18) ˆ QC2 αQ where Q > 0, we multiply the matrix in (17) by ¯ T , I} and diag{Q, ¯ I}, from the left and the right, diag{Q respectively, and the following requirement is obtained. T ¯T ¯ ¯ ¯2 Y +QC¯ T − C¯ TQ ˆT AQ+Q A +B2 Y C¯2 + C¯2T Y TB2T αB 2 2 T ˆ ˆ ∆ ∗ −α(Q + Q ) Γ1 = ∗ ∗ ∗ ∗ ¯T ¯1 QC¯ T + C¯ T Y T D B 12 1 2 T ¯T ¯ D21 αY D12 < 0, Q > 0 (19) 2 −γ I 0 ∗ −I ˆ Since the second block on the diagonal of where Y = K Q. ˆ−Q ˆ T ), Q ˆ that solves (19) is not singular for Γ1 is −α(Q α > 0 and thus K is recovered from Y by ˆ −1 . K = YQ (20) The latter leads to the following theorem.
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
Theorem 1. Consider the system (11). There exists a static output-feedback that for a prescribed H∞ -norm bound 0 < γ satisfies (7) for a single plant in the uncertainty ¯ if there exist Q ∈ R(n+r)×(n+r) , Q ˆ polytope Ω ∈ (p+r)×(p+r) (r+m)×(r+p) R and Y ∈ R that, for some tuning scalar parameter 0 < α satisfy (19) and Q > 0. If a solution to the latter set of LMIs exists, the static outputfeedback gain that stabilizes the system (11) and achieves the required performance is given by (20). It is noted here that the latter result is an improvement to the one obtained in [8]. In the present result there is no need to add an artificial lowpass element with very large bandwidth to the system and the additional semi positive element that appears in [8] in the second diagonal block in Γ1 is now missing. The above provides a sufficient condition for the existence of the gain matrix K that guarantees the required per¯ As formance for a single plant (11) in the polytope Ω. such, this gain matrix provides by (10) the reduced order controller for this single plant that satisfies (7). Since the elements are affine in the system matrices the condition of Theorem 1 can be extended to cope with all ¯ the systems in Ω. Denoting the matrices of (11) that correspond to the i¯ (and thus also to the vertices of Ω) by: th vertex of Ω (i) ¯ (i) ¯ (i) ¯ (i) ¯ (i) , i = 1, ..., N , we obtain the ¯ A , B1 , C1 , C2 and D 21 following result. Corollary 2. There exists a reduced-order controller (6) that, for a prescribed γ > 0 satisfies, when applied to (1), the requirement (7) over the entire polytope Ω, if there ˆ ∈ R(p+r)×(p+r) and exist matrices: Q ∈ R(n+r)×(n+r) , Q (r+m)×(r+p) Y ∈R that, for some tuning scalar parameter 0 < α satisfy the following set of LMIs. ˆT ¯2 Y+QC¯ (i)T− C¯ (i)TQ Σ1 αB 2 2 ∗ ˆ+Q ˆT ) −α(Q ∗ ∗ ∗ ∗ (i) (i)T (i)T T ¯ ¯ 12 B QC¯1 + C¯2 Y TD 1 (i) T T ¯ ¯ 12 D αY D < 0, i = 1,...,N Q > 0 (21) 21 2 −γ I 0 ∗ −I where ¯2 Y C¯ (i)+ C¯ (i)TY T . Σ1 = A¯(i)Q+QTA¯(i)T+ B 2 2 If a solution to the latter set of LMIs exists, the matrices of the controller (6) that stabilizes the system (1) and achieves the required performance over the polytope Ω is given by ∆ Ac Bc ˆ −1 . K= =YQ (22) Cc Dc ¯ that will satisfy the The above corollary seeks a single Q N + 1 LMIs of (21). As such it provides what is called the quadratic stabilizing solution [2]. This solution which may be sometimes quite conservative can be replaced by another, less conservative solution as follows. It turns out, ¯ may sometimes however, that the special structure of Q lead to zero Bc and Cc , thus resulting in suboptimal values
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of γ and will lead to static, rather than dynamic, outputfeedback controllers. To circumvent this difficulty, the structure of the augmented system of (9) can be slightly adjusted to include an extra degree of freedom which we denote by C3 . We write: " # A 0 | B1 0 B2 A|B ∆ Ir 0 Ac Bc −1 0 Ir | 0 − − − = 0 0 | 0 +− T T − Cc Dc C2 0 | D21 − − −− C |D C1 0 | 0 0 D12 I 0 . Absorbing T −1 in K we readily where T = −C3 I obtain Ac + Bc C3 Bc K= (23) Cc + Dc C3 Dc and the following model matrices which include the new degree of freedom: A 0 ¯1 = B1 , B ¯2 = 0 B2 , C¯1 = [C1 0] , , B A¯ = I 0 0 0 (24) 0 0 I ¯ 21 = 0 ¯ 12 = [0 D12 ] , and D C¯2 = , D C2 −C3 D21 The latter allows the application of Corollary 2 above, while performing a search on C3 in C¯2 so as to minimize γ. The controller is then derived by applying the modification (23) on the controller (22) of Corollary 2. Remark 3. To allow a more efficient optimization of the controller which avoids the search for C3 , one may alternatively restrict the structure of the controller to one of the following structures where F is a design matrix of the appropriate dimensions: • Fixed Cc and Dc controller with Cc = B2T F where F ∈ Rn×r and arbitrary Dc ∈ Rm×p but otherwise free. The matrices Ac and Bc are then sought which lead to the required performance. In this case, the closed-loop system is described by: ξ˙ = Aξ + Bw (25) z = Cξ + Dw ∆
where ξ = col{x, xc } and where B1 +B2 Dc D21 A+B2 Dc C2 B2 B2T F , , B= A= Bc D21 Bc C2 Ac C = C1 +D12 Dc C2 D12 B2T F and D = D12 Dc D21 . ∆
Defining K = [ Ac Bc ] we obtain the following. ¯ B ¯2 K C¯2 , B = B ¯1 + K D ¯ 21 A = A+ ¯ 12 K C¯2 , D = D ¯ 11 + D ¯ 12 K D ¯ 21 C = C¯1 + D where B1 + B2 Dc D21 A+B2 Dc C2 B2 B2T F ¯ ¯ , B1 = A= , 0 0 0 ¯2 = 0 , C¯1 = C1 +D12 Dc C2 D12 B T F , B 2 I 0 0 I ¯ ¯ 11 = D12 Dc D21 , D ¯ 12 = 0. ¯ , D21 = ,D C2 = D21 C2 0 We may then choose Dc = 0 and an arbitrary F ∈ Rn×r . Obviously, this case is suitable for the case where B2 is without uncertainty.
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
• When B2 involves uncertainty and C2 is perfectly known, we may fix Bc and Dc with Bc = F C2T and seek Ac and Cc that will satisfy the system requirement. In this case, the closed-loop system is described by: ξ˙ = Aξ + Bw z = Cξ + Dw
where: ∆ Aˆ =
This condition is, in fact, a BRL condition for the system: ˆ + Bw, ˆ ˆ ζ˙ = Aζ z = Cζ.
(26)
Thus, applying Finsler’s lemma along the lines of [4], we find the following result: ¯ with the structure of Lemma 5. There exists matrices Q, ¯ (18), and K that satisfy (28) iff there exist matrices Q, ¯ and H ¯ that satisfy the with the above structure, K, G following LMI. T ¯ −G ¯ T + AˆT H ¯ +Q ¯T G ¯T B ˆ Cˆ T ¯ Aˆ + AˆT G G ˆ 0 ¯ −H ¯T ¯TB H ∗ −H < 0 (29) ∗ ∗ −γ 2 I 0 ∗ ∗ ∗ −I
∆
where ξ = col{x, xc } and where: A+B2 Dc C2 B2 Cc B1 +B2 Dc D21 , B = , F C2T C2 Ac F C2T D21 A= C = [C1 +D12 Dc C2 D12 Cc ] and D = D12 Dc D21 . Ac ∆ Defining K = we obtain: Cc ¯ B ¯2 K C¯2 , B = B ¯1 +K D ¯ 21, A = A+ ¯ 12 K C¯2 , D = D ¯ 11 + D ¯ 12 K D ¯ 21 C = C¯1 + D
Proof : We first note that (28) is, in fact, equivalent to ¯ V˙ − γ 2 wT w + z T z < 0 where V = ζ T Qζ. ∆ ˙ ¯ of the We then denote η = col{ζ, ζ, w, z} and we seek Q, structure of (18), such that the following will be satisfied.
where A + B2 Dc C2 0 ¯1 = B1 + BT2 Dc D21 , , B A¯ = T F C2 C2 0 F C2 D21 ¯2 = 0 B2 , C¯1 = [C1 +D12 Dc C2 0], C¯2 = [0 I ], B I 0 ¯ ¯ 11 = D12 Dc D21 , D ¯ 12 = [0 D12 ] . D21 = 0, D
η T Ψη < 0, ∀ζ satisfying Ση = 0 where
We may choose, then, Dc = 0 and an arbitrary F ∈ Rr×n .
4. PARAMETER DEPENDENT LYAPUNOV APPROACH ¯ T , I} and Multiplying the matrix in (17) by diag{Q ¯ I}, from the left and the right, respectively, we diag{Q, obtain the following LMI condition for the existence of (16). ˆ Cˆ T ¯ Q ¯T B ¯ T Aˆ + AˆT Q Q (28) ∗ −γ 2 I 0 < 0 ∗ ∗ −I
¯T 0 Q 0 0 0 −γ 2 I 0 0
(30)
0 ˆ ˆ 0 , Σ = A −I B 0 . 0 Cˆ 0 0 −I I
Applying Finsler’s lemma [9] we find that a sufficient condition for the performance requirement of (7) to be ¯ is the existence of X and satisfied, for a single plant in Ω, ¯ such that Q Ψ + XΣ + ΣT X T < 0 ¯ ¯ GH 0 0 we readily obtain that the suffiTaking X = 0 0 0 I ¯ H ¯ and Q ¯ that satisfy cient condition is the existence of G, (29). 2 ¯ ¯ We next choose the following structures for G and H: 0 H 0 ¯= G ¯ G ˆ and H = α3 G ˆ , ˆ C¯2 α1 G ˆ C¯2 α2 G G
(27)
¯ that will satisfy the N + 1 Corollary 2 seeks a single Q LMIs of (21). As such, it provides what is called the quadratic stabilizing solution [2]. This solution which may be sometimes quite conservative can be replaced by another, less conservative solution as follows.
0 Q ¯ Ψ= 0 0
In the case where there exists, via Theorem 1, a static output-feedback solution (r = 0) to the problem, for a given γ, the question arises whether this solution can be recovered by solving (19) for r = n applying a special ˆ and Y . The following result is readily structure to Q, Q, obtained by substitution. Corollary 4. Consider the output-feedback control problem of Theorem 1 for r = n. The static output-feedback solution (r = 0) is obtained by solving the inequality ˆ and Y that possess the following structure Γ1 ≤ 0 for Q, Q for any positive scalar β. ˜ˆ ˜ βIn }, Q ˆ = diag{βIn , Q}, Q = diag{Q, and Y = diag{0n×n, Y˜ }.
C¯1T A¯T C¯2T ∆ ˆ ˆ ∆ ¯T ¯ T = , B 1 T T , C = B1 D21 . T ¯T K D12 K B2 −I
where αi , i = 1, 2, 3 are tuning scalar parameters. Denotˆ substituting for A, ˆ B, ˆ C, ˆ and allowing for ing Y = K G, ¯ vertex dependent Q the following result is thus obtained. Theorem 6. There exists a reduced-order controller (6) that, for a prescribed γ > 0 satisfies, when applied to (1), the requirement (7) over the entire polytope Ω, if there exist matrices: Q(i) > 0, i = 1, ..., N , G, H ∈ ˆ (i) > 0, i = 1, ..., N , G ˆ ∈ R(p+r)×(p+r) R(n+r)×(n+r) , Q and Y ∈ R(r+m)×(r+p) that, for some positive tuning scalar parameters α1 , α2 and α3 satisfy the following set of LMIs. # " ¯ (i)Y ¯ (i) Y C¯ (i) α2 B ¯(i)H +α3 B Σ2 Σ3 A (i)T T 2 2 2 ¯ ¯ ∗−α1(G+ ˆ G ˆ T ) Q − G + C¯ (i)H −α G ˆ ˆ ¯ (i) α2 G 3 C2 2 ¯ −H ¯T ∗ −H ∗ ∗ ∗ ∗
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6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
# " ¯ (i) B 1 ¯ (i) D 21 # # " " (i)T (i)T ¯ (i) < 0, i = 1, ..., N (31) ¯ (i)T B H T C¯1 + α3 C¯2 Y T D 1 12 ¯ (i) ¯ (i)T D α2 Y T D 21 12 −γ 2 I 0 ∗ −I # " (i)T (i)T ¯ (i)T GT C¯1 + C¯2 Y T D 12 ¯ (i)T α1 Y T D 12
where (i)T ¯ (i)T+B ¯ (i) Y C¯ (i) Σ2 = GTA¯(i)T+A¯(i) G+C¯2 Y TB 2 2 2 and (i)T (i)T ˆ T ¯ (i) Y. +α1B Σ3 = GTC¯2 −C¯2 G 2
Operating point Mach number Altitude (ft)
1 .5 5000
2 .9 35000
3 .85 5000
4 1.5 35000
a11 a12 a13 a21 a22 a23 b1
-.9896 17.41 96.15 .2648 -.8512 -11.39 -97.78
-.6607 18.11 84.34 .08201 -.6587 -10.81 -272.2
-1.702 50.72 263.5 .2201 -1.418 -31.99 -85.09
-.5162 29.96 178.9 -.6896 -1.225 -30.38 -175.6
Table 1: The parameters of the four operating points.
If a solution to the latter set of LMIs exists, the matrices of the controller that stabilizes the system (1) and achieves the required performance over the polytope Ω are given by Ac Bc ˆ −1 . =YG Cc Dc Remark 7. The results of the above two theorems can be readily applied to the case where a robust controller of full order is sought. By taking r = n the solution of the relevant LMIs, if exist, will provide the matrices of the state space model of the controller. Remark 8. The condition of (19) can be applied to reduced order gain scheduling control in cases where uncertainty is simultaneously encountered in B2 , C2 and D12 . The latter can be achieved by introducing lowpass elements in the control input and the measurement output of the system [8]. Since, under the latter assumption, the LMI (19) is convex in Y over the polytope Ω, the matrix Y can be made vertex dependent while Q is kept constant over the polytope. The resulting K and, therefore, the resulting controller will by vertex dependent, thus allowing the scheduling. 5. EXAMPLE Consider a modified version of the pitch control [10] of F4E described by Nz Nz 0 a11 a12 a12 b1 1 0 0 d q a21 a22 a23 0 q 0 0 1 0 = 0 0 −30 30 δe + 0 u+ 0 0 1w dt δe 0 0 0 104 0 0 0 −104 δ˜e δ˜e # " # " 0 10 0 0 z = 0 1 0 0 x+ 0 u 1 00 1 0 1 0 0 0 x y= 0 1 0 0 The state-vector consists of the aerodynamic components which are the load-factor Nz , the pitch-rate q and the servo components which are the elevon angle δe , and an additional state δ˜e . The state δe relates to the elevon command u via a first-order model of the mechanical lag of the servo of a bandwidth of 30rad/sec, whereas δ˜e is the electrical lag of the servo. The parameters ai,j , i = 1, 2; j = 1, 2, 3 and b1 are given in [10] in the following four operating points which are assumed here as vertices of a polytopic plant
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We first designed a static output-feedback controller which achieved a minimal γ = 7.54 with the feedback gain matrix K = [ 0.1093 1.0461 ]. The closed-loop singular values and the resulting feedback gains are depicted versus frequency in Fig. 1a and Fig. 1b., respectively. In Fig. 1a blue solid line, red dash-dotted line, green dashed line and magenta dotted lines respectively corresponds to operating points 1 − 4. In Fig. 1b, the blue solid line depicts K1 whereas red dash-dotted depicts K2 . We next aim at reducing γ by adding dynamics to the controller. To this end, we applied the design approach of Remark 1, with a fixed Cc = B2T F . For each order r ∈ {1, 2, 3, 4} we have chosen F ∈ Rn×r from 30 random trials. The results are described in Table 2. order r γ
0 7.52
1 3.98
2 4.05
3 3.96
4 3.95
SF 3.86
Table 2: The minimum achievable γ for different orders in comparison with the state-feedback. For r = 1 we have obtained the most significant improvement. The resulting closed-loop singular values for r = 1 and the corresponding Bode gains of the controller are depicted in Fig. 2a and Fig. 2b, respectively. In Fig. 2a blue solid line, red dash-dotted line, green dashed line and magenta dotted lines respectively corresponds to operating points 1 − 4. In Fig. 2b, the blue solid line depicts K1 whereas red dash-dotted depicts K2 . We note that the actual closed-loop singular values for r = 1 are somewhat smaller than those obtained for the static-output feedback controller. We also note that the low frequency gains of the controller obtained for r = 1 are higher than those found for r = 0. These gains roll off, however, at about 83 rad/sec. 6. CONCLUSIONS A Linear Matrix Inequalities based approach for designing static and dynamic output feedback controllers of reduced order for a linear time-invariant system has been considered. Both quadratically stable and parameter-dependent Lyapunov approaches have been considered where various design simplification schemes have been proposed, at the cost of some conservatism. The new design method’s effectiveness and controller order effect have been illustrated via a robust flight control example. REFERENCES [1] T. Iwasaki and R. E. Skelton, “The XY-centering algorithm for the dual LMI problem. A new approach
6th IFAC ROCOND (ROCOND'09) Haifa, Israel, June 16-18, 2009
[5] [6]
[7]
[8] [9] [10]
σ
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0 −1 10
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10
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10 ω [rad/sec]
2
10 a)
3
10
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1.4 1.2 1 |K(ω)|
[4]
3
0.8 0.6 0.4 0.2 0 −1 10
0
10
1
10 ω [rad/sec]
2
10 b)
3
10
4
10
Fig. 1. Static controller, a) Closed-loop singular values, b) Gain Bode of the controller Dynamic output feedback r = 1 , γ = 3.98 3.5 3 2.5 2 σ
[3]
Static output feedback r = 0 , γ = 7.54 4
1.5 1 0.5 0 −1 10
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to fixed order control designs,” Int. J. Contr., vol. 62, pp. 1257-1272, 1995. I.R. Petersen, “Quadratic stabilizability of uncertain linear systems containing both constant and time varying uncertain parameters”, J. of Optimization Theory and Applications, vol. 57(3), pp. 439-461, 1988. I. Yaesh and U. Shaked, “Minimum entropy static feedback control with an H∞ -norm performance bound”, IEEE Trans. on Automat. Contr., AC-42, pp. 853-858, 1997. M. C. de Oliveira and R. E. Skelton, “Stability test for constrained linear systems”, in Perspectives in Robust Control , S. O. Reza Moheimani Ed., Lecture Notes in Control and Information Sciences 268, Springer, London 2001. T. Iwasaki, “The dual iteration for fixed-order control,” IEEE Trans. on Automat. Contr., AC-44, pp. 783-788, 1999. P. L. D. Peres , J. C. Geromel and S. R. Souza, “H∞ control design by static output-feedback,” Proc. of the IFAC Symposium on Robust Control Design, Rio de Janeiro, Brasil, 1994, pp. 243-248. F. Leibfritz, “An LMI-based algorithm for designing suboptimal static H2 /H∞ output feedback controllers”,SIAM J. of Control and Optimization, vol. 39, pp.1711-1735, 2001. U. Shaked, “An LPD approach to robust H2 and H∞ static output-feedback design,” IEEE Trans. on Automat. Contr., AC-48, pp. 866-872, 2003. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequality in Systems and Control Theory, SIAM Frontier Series, April 1994. J. Ackermann, “Longitudinal control of fighter aircraft F4E with additional canards”. In K. P. Sondergeld (compiler), A collection of plant Models and Design Specifications for Robust Control.DFVLR, Oberpfaffenhoffen, Germany.
4
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0 −1 10
0
10
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10 ω [rad/sec]
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10 b)
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Fig. 2. 1st order controller, a) Closed-loop singular values, b) Gain Bode of the controller
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