Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
WeCIn1.11
Dynamic Output-Feedback H∞ Control for Polytopic Delta Operator Systems Ying Zhang, Rui Zhang and Guangren Duan Abstract— The problem of designing H∞ dynamic outputfeedback controllers for polytopic Delta operator systems is considered. Given a transfer function matrix of a system with polytopic uncertainty, an appropriate, not necessarily minimal, state-space model of the system is described which permits reconstruction of all its states. To this model, a new polynomial parameter-dependent approach to state-feedback H∞ control is then used to design robust output-feedback controllers, which is different from conventional design framework. These controllers ensure the stability and guarantee a prescribed H∞ performance level. A numerical example is employed to demonstrate the feasibility and advantage of the proposed design.
I. INTRODUCTION The robust H∞ control of linear systems subject to polytopic uncertainties has attracted considerable attention in robust control literature for many years ([1, 2]). Applying the method of convex programming, relatively simple solutions have been obtained to the problems of state-feedback stabilizing and disturbance attenuation. But it has soon been realized that the corresponding output-feedback control problems are not convex and NP-hard, this means that any algorithm which is guaranteed to find the global optimum cannot be expected to have a polynomial time complexity. The lack of a general robust H∞ output-feedback controller design method has intrigued many researchers, and various iterative methods have been suggested to derive the required controllers ([3]). These methods are known to converge locally and they do not necessarily achieve a global minimum for the disturbance attenuation level. Only recently, an alternative method has been proposed for the design of robust H∞ output-feedback control for polytopic systems ([4, 5]). The idea there is to represent the system in an augmented state-space model for the nonretarded system. Since in that representation all the “states” are accessible, the robust state-feedback controller design method has been used to design dynamic outputfeedback controllers. This work was supported by National Nature Science Foundation of China (No. 60772046), Program for Changjiang Scholars and Innovative Research Team in University, Natural Science Foundation of Guangdong Province (No. 7301716, 9451805707002681), and Project Supported by Basic Research Plan in Shenzhen City (No. JC200903120195A). Ying Zhang is with Institute of Information and Control, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055 and Dept. of Automation, Harbin University of Science and Technology, Harbin 150080,
[email protected] Rui Zhang is with Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518054, zhangruiemail
@yahoo.com.cn
Guangren Duan is with Institute of Information and Control, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055,
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
On the other hand, Delta operator has been introduced as a novel formulation of discrete-time systems by Goodwin in [6]. The Delta operator creates the rapprochement between continuous- and discrete-time systems, and establishes the natural framework to investigate the behavior of discrete-time systems in the fast sampling limit, i.e. as the sampling period tends to zero ([7]). Over the years, engineers have recognized that the application of the Delta operator leads to reliable and robust numerical algorithms for computer control. So far, a great amount of effort has been devoted to Delta operator systems, and many fundamental results have been reported ([8-11]). However, few robust H∞ output-feedback control results for polytopic Delta operator systems have been reported, systemic and less conservative design method still remains challenging, which motivates the present study. In this paper, we consider the problem of robust H∞ output-feedback control for polytopic Delta operator systems. More specifically, an augmented state-space model is achieved to which a state-feedback controller corresponds to a dynamic output-feedback controller that is applied to the original system. On the resulting state-space models, the new polynomial parameter-dependent idea ([12, 13]) is introduced to solve the “state-feedback” control problem, which is different from the quadratic framework that entails fixed matrices for the entire uncertainty domain, or the linear parameter-dependent framework that uses linear convex combinations of n matrices. The obtained controller is then applied to obtain the required dynamic output-feedback controller. II. PROBLEM FORMULATION Consider the following linear, time-invariant, multi-input multi-output Delta operator system ([4]): G(γ) = M −1 (γ)N (γ)
(1)
Ps Ps−1 where M (γ) = k=0 Mk γ k , N (γ) = k=0 Ns−1−k γ k are m × m and m × r polynomial matrices, respectively, m is the dimension of output vector y and r is the dimension of the control input u to the plant. Assume P that M (γ) is m row reduced, namely that deg det{M (γ)} = i=1 si = n, where si is the degree of ith row of M (γ) and n is the minimal order of M (γ). Denoting by Mh the coefficient matrix of the highest order term in each row of M (γ). To simplify derivation in the sequel, we assume that Mh does not incorporate uncertain parameters. Thus, we can assume, without loss of generality, that Mh = Im .
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WeCIn1.11 The system parameters are not completely known. Assume that they lie within a polytope Ωλ = Co{Ω1 , Ω2 , . . . , Ωn } where
· Ωi =
N0,i M0,i
... ...
Ns−1,i Ms−1,i
where
−Ms−1 Im 0 · 0 Aλ = 0 0 0 · 0
(2)
¸ , i = 1, . . . , n
are the vertices of the polytope. We seek a dynamic output-feedback controller whose transfer function matrix is given by the following left matrix fractional description: (3)
Bλ =
of an appropriate order. In the development below we consider an additional finite energy disturbance vector w and objective vector signal z defined as v v u∞ u∞ uX uX T t w (k)w(k)h, kzkL2 = t z T (k)z(k)h kwkL2 =
K=
H(γ) = A−1 (γ)B(γ)
k=0
k=0
where h stands for the sampling period. We address the following problem Robust H∞ control problem: For a prescribed scalar α>0, find a controller that asymptotically stabilizes the system and satisfies kzkL2 sup < α2 (4) kwkL2 over the entire uncertainty polytope. We note that the initial condition of the system is assumed to be zero. III. AUGMENTED STATE-SPACE MODEL It follows from (1) and the stated model assumptions that à ! Ãs−1 ! s−1 X X s k k Im γ + Mk γ Y (γ) = Ns−1−k γ U (γ) k=0
k=0
(5) Note, that the last j rows in Mi and Ns−1−i ,0 ≤ i ≤ s − sm−j+1 − 1,j = 1, . . . , m − 1, are identically zero. Choosing in (3) A(γ) = Im γ s−1 +
s−2 X
Ak γ k , B(γ) =
k=0
we obtain à Im γ
s−1
+
s−2 X k=0
Ak γ
Bs−1−k γ k (6)
k=0
! k
s−1 X
U (γ) =
Ãs−1 X
B0
0 0 B1
B2
· −M0 · 0 · 0 · · Im 0 · 0 · 0 · 0 · · · 0
. . . 0 Ir ...
Bs−1
N1 0 0 · 0 0 Ir 0 · 0 0
· Ns−1 · 0 · 0 · · · 0 · 0 · 0 · 0 · · Ir 0 (10) ¤ 0 ... 0 (11)
· · · · · · 0 Ir · 0
· · · · · · · · · ·
−As−2
. . . −A0
−As−3
¤
(12)
Remark 1. The representation above is appropriate for s > 1. In the case where s = 1, the input u does not appear in the above vector ε. The resulting controller will then be a static output-feedback controller so that (7) is replaced by U (γ) = B0 Y (γ). Remark 2. The obtained representation is nonminimal, but it has the merit that the original output-feedback problem for the uncertain plant has been transformed into a statefeedback problem where the matrices Aλ and Bλ lie in the polytope defined by (2) and any state-feedback control which is designed for this representation can be translated into a dynamic output-feedback controller. Note that the resulting controller H(γ) is proper but not necessarily strictly proper. If the state-feedback control law stabilizes the augmented system (8), the corresponding controller of (3) stabilizes the original plant. We consider next the problem of achieving a prescribed bound on the H∞ performance level. The vector sequence w is the disturbance input in L2 that acts through the input matrix Bwλ , and the objective vector is z which is characterized by the output matrix C1λ and the disturbance coupling matrix D11λ . The corresponding state-space model of the system is then given by ( δε = Aλ ε + Bλ u + Bwλ w (13) z = C1λ ε + D11λ w + D12λ u where
k
Ωλ = {Aλ , Bλ , Bwλ , C1λ , D11λ , D12λ } ∈ R ( ¯ ) n ¯ X ¯ R = Ωλ ¯ Ωλ = λ i Ωi ; λ ∈ Γ ¯
Y (γ)
k=0
(7) Define the following state vector ε = col{δ (s−1) y, . . . , y, δ (s−2) u, . . . , u}. Hence, we have the following state-space realization for the system of (1)-(3). δε = Aλ ε + Bλ u u = Kε
£
N0T
· · · · · · · · · ·
−As−4
! Bs−1−k γ
£
· 0 Im · · · · · · ·
(8) (9)
(14)
i=1
with Ωi = (Ai , Bi , Bwi , C1i , D11i , D12i ) denoting the vertices of the polytope, and ( ) n X Γ = (λ1 , λ2 , . . . , λn ) : λi = 1, λi ≥ 0
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i=1
WeCIn1.11 The entries of Bwλ , C1λ , and D11λ are determined according to the type of the problem solved. For example, the complementary sensitivity problem is achieved by choosing T
Bwλ = [Im , 0, . . . , 0] C1λ = [0, . . . , 0, N1 , . . . , Ns−1 ] + N0 K D11λ = 0
(15) (16) (17)
and the sensitivity problem can be achieved by using Bwλ and C1λ in (15), (16) and by taking D11λ = Im . The closedloop system can be given by ( δε = Aλ ε + B λ w (18) z = C λ ε + Dλ w where Aλ = Aλ + Bλ K, B λ = Bwλ
(19)
C λ = C1λ + D12λ K, Dλ = D11λ IV. ROBUST H∞ CONTROL
We consider system (13) with the performance index (4). A robust state-feedback controller is sought that achieves a minimum upper-bound on the disturbance attenuation level α over the entire uncertainty polytope. The basic tool in H∞ design is the bounded real lemma. Lemma 1.([7]) The closed-loop system (18) is asymptotically stable and satisfies (4) if and only if there exists a matrix function Pλ > 0 such that T √ T T Aλ Pλ + Pλ Aλ Pλ B λ C λ hAλ Pλ √ T T ∗ −αI Dλ hB λ Pλ < 0 (20) ∗ ∗ −αI 0 ∗ ∗ ∗ −Pλ where asterisk (∗) represents the term that is induced by symmetry. Theorem 1. The closed-loop system (18) is asymptotically stable and satisfies (4) if and only if there exist matrix functions Pλ > 0, Fλ and Gλ such that √ T T ∆1 ∆2 FλT B λ C λ hAλ Fλ 0 0 ∗ −GTλ − Gλ GTλ B λ √ T T ∗ 0 for j = 1, . . . , J(g). For each set K(g), define also that set Y (g) with elements Yj (g) given by subsets of i, i ∈ {1, 2, . . . , n}, associated to the n-tuples Kj (g) whose ki ’s are nonzero. For each i, i = 1, . . . , n, define the n-tuples Kji (g) as being equal to Kj (g) but with ki > 0 replaced by ki − 1. Note that the n-tuples Kji (g) are defined only in the cases where the corresponding ki is positive. Note also that, when applied to the elements of K(g + 1), the n-tuples Kli (g + 1) define subscripts k1 k2 . . . kn of matrices Pk1 k2 ...kn associated to a homogeneous polynomial parameter-dependent matrix of degree g. Finally, define the scalar constant coefficients ηji (g+1) = g!/(k1 !k2 ! . . . kn ), with k1 k2 . . . kn ∈ Kji (g+1). As an example, we consider a polytope with n = 2 vertices and g = 2. Then, J(2) = 3, K(2) = {02, 11, 20} and Pλ = λ22 P02 + λ1 λ2 P11 + λ21 P20 Moreover, Y (2) = {{2}, {1, 2}, {1}} and K12 (2) = 01, K21 (2) = 01, K22 (2) = 10, K31 (2) = 10 are the only possible triples Kji (3), j = 1, 2, 3, associated to K(2). In what follows, sufficient conditions for existence of Pλ given by (27) satisfying Theorem 2 is given. Theorem 3. If there exist matrices PKj (g) > 0, Kj (g) ∈ K(g), j = 1, . . . , J(g), R, K, and a scalar ρ such that the following inequalities hold for all Kl (g + 1) ∈ K(g + 1), l = 1, . . . , J(g + 1) Φ1 Φ2 ηji (g + 1)Bwi Φ3 Φ4 i X ∗ Φ5 ηj (g + 1)ρBwi 0 0 i Θλ = ∗ ∗ −ηj (g + 1)αI Φ6 Φ7 < 0 i∈Yl (g+1) ∗ ∗ ∗ Φ8 0 ∗ ∗ ∗ ∗ Φ9 (28) where
J(g)
Pλ =
X
λk11 λk22 . . . λknn PKj (g) , k1 k2 . . . kn = Kj (g) (27)
j=1
where PKj (g), j = 1, . . . , J(g) are constant symmetric matrices to be determined. The notation in the above is explained as follows. Define K(g) as the set of n-tuples obtained as all possible combination of k1 k2 . . . kn , with ki being nonnegative integers, such that k1 + k2 + . . . + kn = g. Kj (g) is the j-th n-tuples of K(g) which is lexically ordered, j = 1, . . . , J(g). Since the number of vertices in the polytope
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T
Φ1 = ηji (g + 1)(Ai R + RT ATi + Bi K + K BiT ) T
Φ2 = PKli (g+1) + ηji (g + 1)(ρRT ATi + ρK BiT − R) T
T T Φ3 = ηji (g + 1)(RT C1i + K D12i ) √ √ T T T i Φ4 = ηj (g + 1)( hR Ai + hK BiT )
Φ5 = −ηji (g + 1)(ρRT + ρR) T Φ6 = ηji (g + 1)D11i √ T Φ7 = ηji (g + 1) hBwi
Φ8 = −ηji (g + 1)αI Φ9 = PKli (g+1) − ηji (g + 1)(RT + R)
WeCIn1.11 Then, the homogeneous polynomial parameter-dependent Lyapunov function given by (27) assures (25), and the controller matrix can be given by (26). Moreover, if the LMIs of (28) are fulfilled for a given degree g, then the LMIs corresponding to any degree g > g are also satisfied. Proof. (Proof of the first part) Since PKj (g) > 0, Kj (g) ∈ K(g), j = 1, . . . , J(g), we know that Pλ defined in (27) is positive definite matrix. Now, note that Pλ given by (27) is a homogeneous polynomial matrix equation of degree g + 1, then Θλ in (25) can be written as J(g+1)
Θλ =
X
£ ¤ λk11 λk22 . . . λknn Θλ , k1 k2 . . . kn = Kl (g + 1).
Corollary 1. For system (13) and the uncertainties (14), there exists an H∞ controller such that the closed-loop system is asymptotically stable and the H∞ -norm level is smaller than some given α if there exist Pi > 0, R, K, Θij and a scalar ρ satisfying Ξij + Ξji − Θij − ΘTij ≤ 0, 1 ≤ i < j ≤ n Ξ11 Θ12 · · · Θ1n ∗ Ξ22 · · · Θ2n Θ= .. < 0 . . ∗ . ∗ . ∗ ∗ ∗ Ξnn where
l=1
(29) Condition (28) imposed for all l,l = 1, . . . , J(g + 1), assures that Θλ < 0 for λ ∈ Γ, and thus the first part is proved. (Proof of the second part) Suppose that inequalities (28) are fulfilled for a certain g, that is, there exist J(g), matrices PKj (g) > 0, j = 1, . . . , J(g), such that Pλ defined in (27) is a homogeneous polynomial parameter-dependent Lyapunov matrix assuring Θλ < 0. Then, the terms of the polynomial matrix P λ = (λ1 + λ2 + . . . + λn )Pλ also satisfy (28) corresponding to the degree g + 1, which can be obtained in this case by linear combination of (28) for g. ¤ Remark 4. As Theorem 3 is based on the Delta operator, a close correspondence is shown between the continuousand discrete-time system robust H∞ control. Remark 5. The matrices composing the homogeneous polynomial parameter-dependent Lyapunov function Pλ as well as the inequalities (28) can be generated from sets K(g) and Y (g), which can be constructed from simple routines using, for instance, a recursive code. As the degree g of the polynomial increases, the conditions become less conservative since new free variables are added to the LMIs. Although the number of LMIs is also increased, each LMI becomes easier to be fulfilled due to the extra degrees of freedom provided by the new free variables and smaller values of H∞ guaranteed costs can be obtained. Remark 6. The optimal α can be readily found by solving the following optimization problem
Ξij =
e1 Ξ ∗ ∗ ∗ ∗
e2 Ξ e5 Ξ ∗ ∗ ∗
e3 Ξ 0 T D11i −αI ∗
Bwi λBwi −αI ∗ ∗
e4 Ξ √ 0 T hBwi 0 Pj − RT − R
(30)
(31)
e 1 = Ai R + RT ATi + Bi K + K T BiT Ξ e 2 = Pj − R + ρRT ATi + ρK T BiT Ξ T T T e 3 = RT C1i Ξ + K D12i √ √ e 4 = hRT ATi + hK T BiT Ξ e 5 = −λR − λRT Ξ
Furthermore, if the above LMIs are feasible, the matrix K can be given by (26), then the designed dynamic outputfeedback controller can be constructed by (3) and (12). Proof. Select the following matrices (linearly dependent on the parameter λ) Pλ =
n X
λi Pi
i=1
Then, with the above equation, it is not difficult to rewrite Θλ in (25) as Θλ =
n X n X
λi λj Ξij
i=1 j=1
=
n X
λ2i Ξii +
i=1
min α s.t. (28)
n−1 X
n X
λi λj (Ξij + Ξji )
(32)
i=1 j=i+1
On the other side, (30) is equivalent to
over PKj (g) > 0, Kj (g) ∈ K(g), j = 1, . . . , J(g), R, K, ρ. Remark 7. It is worth mentioning that, when g = 0, we have Pλ = P0 which will lead to the result in P the quadratic n framework. In addition, when g = 1, Pλ = i=1 λi Pi is linearly dependent on the parameter λ, that is the linear parameter-dependent approach. Then, we have the following corollary. Remark 8. Note that for the given ρ, (28) is linear with respect to PKj (g) > 0, Kj (g) ∈ K(g), j = 1, . . . , J(g), R, and K. The problem is then how to find the optimal values of ρ. This can be readily solved by using standard numerical software.
Ξij + Ξji ≤ Θij + ΘTij , 1 ≤ i ≤ j ≤ n
(33)
Then, from (32) and (33), we have Θλ ≤
n X
λ2i Ξii +
i=1
n−1 X
n X
λi λj (Θij +Θji ) = θT Θθ (34)
i=1 j=i+1
where Θ is defined as (31) and θ = [λ1 I, λ2 I, · · · , λn I]T From λi ≥ 0 and (31) indicates that
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n P
(35)
λi = 1, one has θ 6= 0. Inequality
i=1 Θλ
