itle Delay-dependent L-L-infinity model reduction for polytopic systems with time-varying delay Authors) Wang, Q; Lam, J Citation he 8 IEEE International Conference on Automation and Logistics ICAL 8), Qingdao, China, 1-3 September 8. In Conference Proceedings, 8, p. 5-55 Issued Date 8 URL http://hdl.handle.net/17/6 Rights IEEE International Conference on Automation and Logistics. Copyright IEEE.; 8 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.; his work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4. International License.
Proceedings of the IEEE International Conference on Automation and Logistics Qingdao, China September 8 Delay-dependent L -L Model Reduction for Polytopic Systems with ime-varying Delay Qing Wang School of information Science and echnology Sun Yat-sen University Guangzhou, 5175, China wangq5@mail.sysu.edu.cn James Lam Department of Mechanical Engineering he University of Hong Kong Pokfulam Road, Hong Kong james.lam@hku.hku.hk Abstract his paper considers the L -L model reduction problems for polytopic system with time-varying delay. In terms of the solution of linear matrix inequalities LMIs) and inverse constraints, sufficient conditions are presented to construct the reduced order models such that the L -L gain of the error system between the full order model and the reduced order one is less than a given scalar. Index erms model reduction problems, polytopic systems with time-varying delay, L -L norm I. INRODUCION he L -L gain of a system Σ is defined as Σ = sup y L u L 1 with zero initial state, where yt) is the output of system Σ and ut) is the input of system Σ with u L = 1/ u t) dt) and u L = sup t u t), u t) = u t) u t) 1 and 16. he L -L gain of error system, which is also referred as the energy-to-peak gain in 14 is a natural criterion for model reduction. he L -L model reduction problem for continuous-time and discretetime systems without delay is studied in 11. Necessary and sufficient conditions are given in terms of LMIs and a rank constraint, which can be solved by the alternating projection algorithm 4, Chap.13. he model reduction problem has been extensively studied in the literature, as can be seen from 13 and the references listed therein. In many physical, industrial and engineering systems, control systems cannot be described accurately without the introduction of delay element. Delays can lead to poor performance and instability of control systems. Considerable research has been carried out on systems with time-varying delay in recent years. he criteria for asymptotic stability of such systems can be classified as delay-independent 18 or delay-dependent 3, 6, 9, 1, which is less conservative than delay-independent stability criteria in general. For his work was supported by CRCG 611159157, Sun Yat-sen University Young eachers Foundation under grant 3171911 and Young eachers Foundation of School of information Science and echnology under grant 3577. polytopic systems with constant delay, the robust stability and stabilization are studied in 17. he extension to timevarying delay is investigated in 1 with stability criteria less conservative through the introduction of slack variables in the LMIs. he polytopic system has been extensively studied in the literature, as can be seen from 8, 19 and the references listed therein. o the authors knowledge, the L -L model reduction problems for polytopic systems with time-varying delay are a challenging topic worthwhile to tackle. II. L -L MODEL REDUCION A polytopic system Σ with time-varying delay is given by Σ:ẋ t) = At)x t)+a h t)xt ht)) + Bt)u t) y t) = Ct)x t)+c h t)xt ht)) x t) =, t h, where x t) R n is the state, u t) R m is the control input which belongs to L, ) and y t) R p is the controlled output. At), A h t), Bt), Ct) and C h t) are appropriately dimensioned continuous functions of time t, and satisfy the real convex polytopic model A t) A h t) B t)) = α i t)a i A hi B i ), C t) C h t)) = α i t)c i C hi ) q α i t), α it) =1, where α i t), i =1,,...,q, are time-varying functions of t, C i,c hi,b i,a i and A hi, i =1,,...,q,are constant matrices with appropriate dimensions. ht) is the time-varying delay satisfying <ht) h <, ḣt) μ<1 with h > and μ>. System Σ, which is to be reduced, is assumed to be quadratically stable. Definition.1: If there exists an ˆnth-order quadratically stable system ˆΣ ˆΣ :.ˆx t) = Ât)ˆx t)+âht)ˆxt ht)) + ˆBt)u t) ŷ t) = Ĉt)ˆx t)+ĉht)xt ht)) ˆx t) =, t h, 978-1-444-53-7/8/$. 8 IEEE 5
where ˆx t) Rˆn, ŷ t) R p, and ˆn <n,such that Σ ˆΣ = sup y ŷ L <γ u L 1 then we say the L -L model reduction problem for system Σ is solvable. he L -L gain of retarded system Σ R Σ R : ẋ t) = Ax t)+a h xt ht)) + But) y t) = Cxt)+C h xt ht)) can be characterized by the following algebraic conditions, where A, A h,b, C and C h are contant matrices. Lemma.: 15 If there exist matrices P >, Q>, X>, Z>and Y, and a scalar <α<1 such that A P + PA+ hx + Y +Y + Q PA h Y PB τa Z A h P Y 1 μ)q τa h Z B P I m τb Z τza τza h τzb τz αp C 1 α)p Ch C C h γ I p X Y Y Z < < then system Σ R is asymptotically stable and Σ R < γ, where τ = h 1 u. In the case of ht) =h, if there exists matrices P>, Q>, and a scalar <α<1 such that A P + PA+ hx + Y + PA h Y PB Y + Q A h P Y 1 μ)q < B P I m αp C 1 α)p Ch < C C h γ I p he following theorem gives the solution of L -L model reduction problem for system Σ. heorem.3: With ñ = n+ˆn, if there exist matrices P>, P >, X>, X >, Q>, Q >, M>, M >, N>, Ñ>, Z>, Z > and Y, and a scalar <α<1 such that for i =1,,...,q, X11 X 1 X1 < 1) X Y11 Y 1 Y1 < ) Y αi 1P I1 Ci 1 α)i 1 P I1 Chi < 3) γ I p X Y Z N Y M 4) 5) P P = Iñ, Q Q = Iñ, X X = I 6) Z Z = Iñ, MM = Iñ, ÑN = Iñ 7) where denotes the symmetric terms in a symmetric matrix and Λ i11 Λ i1 B i Λ i14 Λ i15 X 11 = Λ Λ i1 I m Bi Λ i44 Λ i45 Λ i55 I 1 P I1 P I1 P I1 P X 1 = Iñ X = I P I P I P I P h 1 X Q M Ñ Y 11 = A i I 1P I1 + I 1 P I ) 1 A i hx + Y +I 1 +Y I + Q 1 Y 1 = I 1 P I1 A hi I 1 Y I1 τa i I 1Z 1 μ)i1 QI1 Y = τa hi I 1Z τz and Λ i11 = I 1 P I 1 A i + A i I 1 P I 1, Λ i1 = A hi I 1 8) Λ i14 = I 1 P I 1 A i, Λ i15 = A i I 1 P I 9) Λ = 1 μ)q, Λ i44 = τ 1 I 1 ZI 1 1) Λ i55 = τ 1 I ZI 11) Λ i45 = τ 1 I 1 ZI A i I 1 P I 1) then there exists a quadratically stable system ˆΣ such that the L -L model reduction problem is sovable with Σ ˆΣ <γ. In this case, a desired reduced system corresponding to a feasible solution P, P, X, X, Q, Q, M, M, N, Ñ, Z, Z, Y, α) to 1) 7) is given by ˆB t) Â t) Âh t) ) = Ĉ t) Ĉh t) ) = α i t)ḡi1 13) α i t)ḡi 14) 51
where Ḡ i1 = ˆBi  i  hi = U 1 Ω i1 1 V i1 Λ i1 Λ1 V i1 Λ 1 +U 1 i1 W 1 ) 1 i1 L i1 Λ1 V i1 Λ 1 15) Ḡ i = Ĉ i Ĉ hi = U 1 Ω i V i Λ Λ V i Λ +U 1 i W 1 ) 1 i L i Λ V i Λ 16) and for i = 1,,...,q, L i1 and L i are any matrices satisfying L i1 < 1 and L i < 1, and U i1 > and U i > such that V i1 = Ω1 U 1 Ω i1 1 Φ i1 > V i = Ω U 1 Ω i Φ > W i1 = U i1 Ω 1 W i = U i Ω Φ i1 = Φ i = Ω 1 = Λ 1 = Λ = ) V i1 V i1 Λ 1 Ω 1 Λ1 V i1 Λ 1 Λ1 V i1 ) V V i Λ Ω Λ V i Λ Λ V i Ā i P + P hx Āi+ P + Y + Āhi Y P B i τā i Z Y + Q 1 μ)q τā hi Z I m τ B i Z τz 17) αp C i 1 α)p C hi 18) γ I p P Iˆn, Ω = 19) I p τz Iˆn I m Iˆn ) Iˆn Iˆn 1) Iˆn Ā i = B i = Ai Bi Ahi, Ā hi = ), Ci = C i 3) C hi = C hi 4) I 1 = I n, I = Iˆn 5) Proof. We first consider the error system Σ ˆΣ given by x t) = à t) x t)+ãh t) xt ht)) + B t) u t) ỹ t) = C t) x t)+ C h t) xt ht)) where x t) = x t) ˆx t), ỹ t) =y t) ŷ t), = B t) à t) Ãh t) = Bi + F Ḡi1 N α i t) Ā i + F Ḡi1 H Āhi + F Ḡi1 M 6) C t) Ch t) α i t) Ci + Ḡi J Chi + Ḡi K 7) with Ḡi1 and Ḡi, i =1,,...,q, defined in 15) and 16), B i, Ā i, Ā hi, Ci and C hi given in ) and 3), and F = M = J =, Iˆn H =, Iˆn Iˆn Iˆn N = I m 8) 9), K = Iˆn 3) From Lemma., if there exist matrices Ḡ i1 and Ḡi satisfying 4) and Āi + F Ḡi1 H ) P +P ) Ā i + F Ḡi1 H P ) Ā hi + F Ḡi1 M + hx + Y + Y Y + Q 1 μ)q P ) Bi + F Ḡi1 N τ Ā i + F Ḡi1 H ) Z τ Ā hi + F Ḡi1 M ) Z I m τ Bi + F Ḡi1 N ) Z τz < 31) αp Ci + Ḡi J ) 1 α)p Chi + Ḡi K ) γ I p < 3) 5
It follows from 31) and 3) that < Ã t) P + P Ã t) + hx + Y +Y + Q ) F Ḡi1 N τã t) Z τãh t) Z I m τ B t) Z τz P Bi + P Ãh t) Y 1 μ)q αp C t) 1 α)p Ch t) < γ I p hen we have Σ ˆΣ <γ.it is easy to show that matrix inequalities 31) and 3) can be rewritten as Φ i1 + Ω 1 Ḡ i1 Λ1 + Ω1 ) Ḡ i1 Λ1 < 33) Φ i + Ω Ḡ i Λ + Ω ) Ḡ i Λ < 34) Ω 1 Φ i1 Ω 1 <, equals to where Λ i11 Λ i1 B i 1 μ)q I m Λ i14 Λ i15 I 1 P I1 P Λ i1 Bi Λ i44 Λ i45 Λ i55 I P I P h 1 X Q +M 1 Y N 1 +M 1 Y N 1 ) < 37) M 1 = P I 1 Iñ P I N 1 = P I 1 P I where Φ ij, Ω j, and Λ j,,,...,q, j =1,, are defined in 17) 1). LMIs 33) and 34) have solutions Ḡi1 and Ḡi, if and only if Ω j Ω Φ 1 Ω i1 1 <, Λ 1 Φi1 Λ 1 < 35) Ω Φ Ω i <, Λ Φi Λ < 36) and Λ j,j=1,, can be chosen as follows: Ω 1 = Λ 1 = Ω = I 1 P 1 Iñ I m τ 1 I 1 Z 1 I P 1 τ 1 I Z 1 I 1 I 1 Iñ Iñ Iñ Λ = I 1 I 1 I p where I 1 and I are defined in 5). hen, by Schur complement and the conditions in 6) and 7), we obtain that and Λ i11, Λ i1, Λ i14, Λ i15, Λ i44 and Λ i45,,,...,q, are defined in 8) 1). Since for any matrices M, N,Y and M>and N>, MY N +MY N ) MMM + N NN, holds if YM 1 Y N. LMI 37) is true if 1) and 5) hold. Λ 1 Φi1 Λ 1 <, is equivalent to ). For the inequalities in 36), Ω Φ αp Ω i = 1 α)p < and Λ Φ i Λ < is equivalent to 3). From LMIs 1) 5), if there exist matrices P, P, X, X, Q, Q, M, M, N, Ñ, Z, Z and Y satisfying 35) and 36), there exist matrices Ḡi1 and Ḡi such that LMIs 33) and 34) hold. herefore, we have Σ ˆΣ <γ.all the parameters of the reduced order models satisfying 33) and 34) can be constructed by the parametrization method of Gahinet and Apkarian 7. his completes the proof. In the case of ht) =h, a delay-independent sufficient condition is given for the L -L model reduction problem from Lemma. and the proof of heorem.3. Corollary.4: If there exist matrices P>, P >, Q> and Q >, and a scalar <α<1such that for i = 53
1,,...,q, I 1 P I 1 A i + A i I 1 P I 1 A hi I 1 B i I 1 P Q I m < Q A i I 1P I1 + I 1 P I1 A i + I 1 QI1 I 1 P I1 A hi < I 1 QI1 αi 1 P I1 Ci 1 α)i 1 P I1 Chi < γ I p P P = Iñ, Q Q = Iñ there exists a quadratically stable system ˆΣ with ht) = h solving the L -L model reduction problem with Σ ˆΣ <γ. Remark.5: If P, P = P 1 and Q have the following special form ˆX X P = 1 X1, P 1 Ŷ 1 Y = 1 X Y1 Y Ẑ X Q = 1 X1 38) X Corollary.4 can be expressed by LMIs with rank constraint: if there exist matrices ˆX >, Ŷ > and Ẑ>, and a scalar <α<1 such that for i =1,,...,q, A i Ŷ + ŶA i+ Ẑ + Ŷ ˆX ŶA hi ŶB i Ẑ + Ŷ ˆX < I m A i ˆX + ˆXAi + Ẑ ˆXA hi < Ẑ α ˆX Ci 1 α) ˆX Chi < γ I p rankŷ ˆX) ˆn there exists a quadratically stable system ˆΣ with ht) = h solving the L -L model reduction problem with Σ ˆΣ <γ. If ht) =,C h =and A h =,q=1, system Σ reduces to a continuous time-invariant system without delay Σ C : ẋ t) = A 1 x t)+b 1 u t) y t) = C 1 x t) Corollary.6: If there exist matrices P > and P > such that I 1 P I 1 A 1 + A 1 I 1 P I 1 + B 1 B1 < 39) A 1 I 1 P I1 + I 1 P I1 A 1 < 4) I 1 P I1 + γ C1 C 1 < 41) P P = I n+ˆn 4) there exists an asymptotically stable continuous system ˆΣ C ˆΣ C :. ˆx t) = Â1ˆx t)+ ˆB 1 u t) ŷ t) = Ĉ1ˆx t) that solves the L -L model reduction problem with Σ C ˆΣ C < γ. If P > has special form 38), conditions 39) 4) are equivalent to the existence of matrices ˆX > and Ŷ > such that A 1 Ŷ + ŶA 1 + B 1 B1 <,A ˆX 1 + ˆXA 1 <, ˆX + γ C1 C 1 < and rankŷ ˆX) ˆn, which recovers the result in 11. In this particular case, the model reduction condition is both necessary and sufficient. III. MODEL REDUCION ALGORIHM In order to use the cone complementarity linearization CCL) algorithm 5 to solve the L -L model reduction problem with fixed < α < 1, we define a convex set C := {X X satisfies LMIs 1) 5)} and a nonconvex set = {X X satisfies the inverse constraints in 6-7)} where X = P>, P >, X>, X >, Q>, Q >, Z >, Z >, M >, M >, N>, Ñ>, Y) he solvability of the L -L model reduction problem can be translated into the following nonconvex feasibility problem: Find X C subject to X 43) through the sufficient condition in heorem.3. hen, by the CCL approach, the above nonconvex problem 43) has a solution if and only if the minimization problem min X C Cin { tracep P + X X + Q Q+ Z Z + M M + NÑ) where P I X I,, I P I X Q I Z I C in =,, I Q I Z M I N I, I M I Ñ } 44) 54
achieves a minimum, 1ñ, that is, an optimal solution of problem 44) satisfying tracep P ) = tracex X) =traceq Q) =tracez Z) = tracem M) =tracenñ) =ñ Otherwise 43) is infeasible. herefore, in order to solve the L -L model reduction problem for systems with polytopic uncertainties and time-varying delay, we transform it to a global solution of the minimization problem 44). he CCL algorithm solves problem like 44) effectively although it is still a nonconvex optimization problem. Interested readers may refer to 5 for the details of the CCL algorithm. An algorithm is presented to solve the L -L model reduction problem based on the above analysis. L -L CCL Model Reduction Algorithm: Step 1 Given system Σ, the order of the reduced order model ˆn, prescribed model reduction error γ>and δ > is a sufficiently small prescribed a scalar to control the convergence accuracy. Step Set k =and for a fixed α, choose any initial guess X = P, P,X, X,Q, Q,Z, Z, M, M,N, Ñ, Y ) C Step 3 Define f k P, P,X, X,Q, Q, Z, Z,M, M,N,Ñ) = tracep k P + Pk P + X k X + X k X + Q k Q + Qk Q + Z k Z + Zk Z +M k M + Mk M + N k Ñ + ÑkN) Solve the following convex minimization problem: { } fk P, P,X, X,Q, Q, min X C Cin Z, Z,M, M,N,Ñ) and denote the minimizer Xk and compute the minimum value fk = f kxk ). Step 4 If fk 1ñ <δ, then construct a reduced order model based on 13) and 14); otherwise set k = k +1, and assign Xk 1 = P k, Pk,X k, Xk,Q k, Q k,z k, Z k,m k, M k,n k, Ñk,Y k ) and go to step 3. It can be seen that Step is a simple LMI feasibility problem, and Step 3 is a convex programming with LMI constraints. From the explanation in, 5, {fk } is decreasing and bounded below by 1ñ. Once it converges, then 43) is feasible, which implies that the L -L model reduction problem is solvable for given γ>. IV. 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