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1 Available online at ScienceDirect IFAC PapersOnLine Inversion of Separable Kernel Operators in Coupled Differential-Functional Equations and Application to Controller Synthesis Guoying Miao Matthew M. Peet Keqin Gu School of Information and Control, Nanjing University of Information Science and Technology, Nanjing 2144R China. School of Matter, Transport and Energy, Arizona State University, Tempe, AZ 85287, USA. Department of Mechanical and Industrial Engineering, Southern Illinois University, Edwardsville, IL 6226, USA. Abstract: This article presents the inverse of the kernel operator associated with the complete quadratic Lyapunov-Krasovskii functional for coupled differential-functional equations when the kernel operator is separable. Similar to the case of time-delay systems of retarded type, the inverse operator is instrumental in control synthesis. Unlike the power series epansion approach used in the previous literature, a direct algebraic method is used here. It is shown that the domain of definition of the infinitesimal generator is an invariant subspace of the inverse operator if it is an invariant subspace of the kernel operator. The process of control synthesis using the inverse operator is described, and a numerical eample is presented using the sum-of-square formulation. 217, IFAC International Federation of Automatic Control Hosting by Elsevier Ltd. All rights reserved. Keywords: Lyapunov-Krasovskii functional, linear operator, time delay, sum-of-squares. 1. INTRODUCTION It is known that an accurate stability analysis using the Lyapunov approach requires a complete quadratic Lyapunov-Krasovskii functional. Such an approach was first implemented in the form of the discretized Lyapunov- Krasovskii functional method in Gu 1997, and a refined version was presented in Gu 21. In this method, the kernel of the Lyapunov-Krasovskii functional is piecewise linear. An alternative approach is the Sum-Of-Squares SOS method presented in Peet et al. 29. In the SOS method, the kernel is polynomial. In both approaches, the stability problem is reduced to a semi-definite programming problem, or more specifically, a linear matri inequality problem. For many practical systems, the number of state variables with delays is very small compared with the total number of state variables. For such systems, a special form of the coupled differential-difference equation formulation, or its generalized counterpart, the coupled differential-functional equation formulation proves to be much more efficient in numerical computation. The differential-difference formulations can also model systems of neutral type. The discretized Lyapunov-Krasovskii functional approach to stability of differential-difference equations is documented This work was supported by National Natural Science Foundation of PR China under Grant , , the Natural Science Foundation of Jiangsu Province under Grant BK This work was also supported by NSF Grants , 13166, in Gu and Liu 29, Gu 21 and Li 212, and the SOS formulation can be found in Zhang et al Control synthesis based on complete quadratic Lyapunov- Krasovskii functional stability conditions is still rare. An early eample is Fridman 22, in which a more limited class of Lyapunov-Krasovskii functional is used, and some parameter constraints are imposed. Recently, a synthesis based on the inverse of kernel operator associated with the Lyapunov-Krasovskii functional for time-delay systems of retarded type in the SOS formulation was developed in Peet and Papachristodoulou 29b and Peet 213. This paper etends the method by Peet et al. to coupled differential-functional equations. The inverse operator is derived using direct algebraic approach rather than the series epansion approach. The basic idea of such synthesis is outlined as follows. Consider the coupled differential functional equations ẋt=atb r Hθ θdθ, 1 =CtD r, 2 A R n n, B R n m, C R m n, Hθ R n m, D R m n, and r> is time delay, R n and R m n denote the set of real vectors and matrices with m n dimensions, respectively. The initial conditions are defined as = R n y = PCr, m, , IFAC International Federation of Automatic Control Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control /j.ifacol

2 6514 Guoying Miao et al. / IFAC PapersOnLine y τ represents a shift and restriction of defined by y τ θ =yτ θ, θ Cr, m represents the set of piecewise continuous functions from, to R m. Let Z := R n PCr, m. 3 The solutions to the system described by 1 and 2 may be represented by a strongly continuous semigroupc - semigroup S : Z Z, zt =St τzτ. 4 System 1-2 may be written as an abstract differential equation on Z, ż = Az, 5 A is the infinitesimal generator of the C -semigroup S. Then the stability of the system can be investigated using a quadratic Lyapunov-Krasovskii functional V z =< z,pz >, 6 P is a self-adjoint operator, and, represents inner product. The system is stable if V z is positive definite in some sense, and its derivative along the system trajectory V z =< z,pa A Pz > is negative definite in some sense, A is the adjoint operator of A. Thus testing the stability of the system can be accomplished by searching for a P which satisfies the above conditions, and the problem can be reduced to a semi-definite programming problem when the kernel of the integral operator which defines P is restricted to be either piecewise linear or polynomial because the operator P appears linearly in V z and V z. However, the situation is quite different for control synthesis. Consider a system with input described by the abstract differential equation ż = Az Fu. 7 If we want to design a linear feedback control in the form of u = K, 8 so that the closed-loop system is stable, and use the Lyapunov-Krasovskii functional given in 6, then the derivative becomes V z =< z,pa A P PFK PFK z>. 9 Because we need to determine the feedback gain K in addition to the operator P, V z becomes a bilinear function of the parameters. Determining the eistence of parameters to make V z positive definite and V z negative definite poses a formidable numerical problem, for which there is not yet any established reliable method to implement. One solution to this difficulty is to make a variable transformation ẑ = Pz, 1 and use new parameters Q = P 1, 11 ˆK = KP 1 12 instead of P and K to epress V and V. It can be easily obtained that V z=< ẑ,qẑ >, 13 V z=< ẑ,aq QA F ˆK ˆK F ẑ >, 14 which are linear with respect to the new parameters. Once Q and ˆK have been determined, the original parameters P and K may be obtained by solving 11 and 12, at least symbolically. Critical to implementing the above idea is the inversion of the linear operator P. Unfortunately, such an inversion is not easy in general. It turns out that a relatively simple epression for P 1 is possible when P is separable, as is utilized in Peet and Papachristodoulou, 29b to carry out control synthesis for time-delay systems of retarded type. In this paper, we present the inversion of P for coupled differential-functional equations. Unlike the series epansion method used in Peet, 213, a direct algebraic approach is used here. Control synthesis is also described, and a numerical eample is presented to illustrate the method. 2. PRELIMINARIES Consider the coupled differential functional equations given in 1 and 2. Stability of such a system may be verified using a complete Lyapunov-Krasovskii functional of the following form, V, =r T P2r T Qηηdη T ξrξ,ηηdξdη T ηsηηdη, 15 P = P T R n n, 16 Qη R n m, 17 Rξ,η=R T η, ξ R m m, 18 Sη=S T η S n, 19 and S n represents the set of symmetric matrices, the superscript T denotes the transpose of a matri or vector. Lemma 1. Gu and Liu, 29; Li, 212 System 1-2 with ρd < 1 is eponentially stable if there eists a quadratic Lyapunov-Krasovskii functional from 15-19, such that ǫ 2 V,, for some ǫ >, and its derivative along the system trajectory V, V t, t,,,y t, V, lim sup t t satisfies V, ǫ 2,, denotes 2-norm.

3 Guoying Miao et al. / IFAC PapersOnLine Define inner product on Z given in 3, 1 1, 2 2 = r1 T 2 T 1 s 2sds. For matri P and matri functions Q,R,S that satisfy 16-19, we define the linear operator P : Z Z P = P Qsds, 2 Π Π=rQ T s Rs, θθdθ Ss. Obviously is a bounded and self-adjoint linear operator in view of 16-19, and the Lyapunov-Krasovskii functional may be epressed as V, =. As mentioned in Section 1, system 1-2 define a strongly C -semigroup S : Z Z that satisfies 4. System 1-2 may also be written as an abstract differential equation 5 on Z. Let the domain of definition of A be X. Then, { } X := Z C, = C D, C represents the set of continuous functions. It is of interest in some cases to restrict P so that X is invariant subspace of P, PX X. 21 The specific conditions for such a P to satisfy is given in the following. Lemma 2. P satisfies 21 if and only if the following conditions are satisfied, rq T SC = CP rdq T, 22 R, s=cqsdr, s, s, 23 DS =SD. 24 Proof. See Miao et al Obviously, the above is generalization of Theorem 3 in Peet INVERSE OPERATOR In this section, we will present an analical epression for the inverse of the operator P when it is separable. Similar to Peet 213, such an analic epression for the inverse operator can be used to epedite the construction of the stabilizing controller in the controller synthesis problem. Definition 1: An operator P defined in 2 is said to be separable if Rs, θ =Z T sγzθ, 25 Qs =HZs, 26 for some constant matrices Γ = Γ T and H, and column vector function Zs. Theorem 3. Assume P in 2 is separable. Then, provided that all the inverse matrices below are well defined, its inverse may be epressed as P 1 s ˆP ˆQθθdθ = r ˆQ T s Ŝs, 27 ˆRs, θθdθ ˆRs, θ =ẐT sˆγẑθ, 28 ˆQθ =ĤẐθ, 29 Ŝs =S 1 s, 3 Ẑs =ZsS 1 s, 31 Ĥ = P 1 HT, 32 T =I KΓ rkh T P 1 H 1, 33 ˆP =I rp 1 HTKH T P 1, 34 ˆΓ =rt T H T P 1 H ΓI KΓ 1, 35 K = ZsS 1 sz T sds, 36 and I denotes the identity matri with appropriate dimension. Proof. Let the operator defined by the right hand side of 27 be denoted as ˆP, then Λ1 ˆPP s =, Λ 2 Λ 1 = ˆPQθ ˆQθSθ θdθ ˆPP Λ 2 = r ˆQ T sp ŜsQT s ˆQξRξ,θdξ r ˆQθQ T θdθ, ˆRs, θq T θdθ ŜsSs T r ˆQ sqθŝsrs, θ ˆRs, θsθ ˆRs, ξrξ,θdξ θdθ. Using and 28-36, we obtain ˆPP r ˆQθQ T θdθ = ˆPP rĥkht =I rp 1 HI KΓ rkh T P 1 H 1 KH T P 1 HTKH T = I, ˆPQθ ˆQθSθ ˆQξRξ,θdξ = ˆPH Ĥ ĤKΓZθ = I rp 1 HTKH T P 1 H P 1 HT P 1 HTKΓ Zθ = P 1 H P 1 HTrKH T P 1 H I KΓ Zθ = P 1 H P 1 H Zθ =,

4 6516 Guoying Miao et al. / IFAC PapersOnLine ˆQ T sp ŜsQT s ˆRs, θq T θdθ = ẐT sĥt P H T ˆΓKH T = ẐT s { T T H T P 1 P H T rt T H T P 1 H ΓI KΓ 1 KH T } = ẐT s { I T T I rh T P 1 HI KΓ 1 K ΓI KΓ 1 K } H T = ẐT s { I T T I rh T P 1 HKI ΓK 1 ΓKI ΓK 1} H T = ẐT s { I T T I ΓK rh T P 1 HKI ΓK 1 ΓKI ΓK 1} H T = ẐT si I ΓK 1 ΓKI ΓK 1 H T =, r ˆQ T sqθŝsrs, θ ˆRs, θsθ ˆRs, ξrξ,θdξ = ẐT srĥt H ΓˆΓˆΓKΓZθ = ẐT s { T T H T P 1 H Γ rt T H T P 1 H ΓI KΓ 1 I KΓ } Zθ = ẐT s T T H T P 1 H ΓrT T H T P 1 H Γ Zθ =. Thus, we have shown ˆPP =, 37 for all Z. Similarly, we can show P ˆP =. 38 From 37 and 38, we conclude that ˆP = P 1. Theorem 4. If the separable operator P satisfies PX X, Then 1 X X holds. Proof. See Miao et al CONTROLLER SYNTHESIS In this section, we consider a control system as follows ẋ = A B rfut, 39 =CtD τ. 4 Define the infinitesimal generator A as follows. A B r A s = d ds y. ts Likewise, we define the input operator F : R q X as Fu Fus :=. We define the controller synthesis problem as the search for matrices K,K 1 and matri-valued function K 2 s such that the System of Equations 39-4 is stable if ut =K, 41 we define K : X R q as K s = K tk 1 r K 2 s sds. 42 Before we give the main result of the section, we briefly address SOS methods for enforcing joint positivity of coupled multiplier and integral operators using positive matrices. These methods have been developed in a series of papers, a summary of which can be found in the survey paper Peet 214. Specifically, for matri-valued functions Ms, Ns, θ, we say that {M,N} Ξ if M and N satisfy the conditions of Theorem 8 in Peet 214. The constraint {M,N} Ξ can be cast as an LMI using SOSTOOLS as described in Peet 214 and this constraint ensures that the operator P, defined as P = M 11 rm21s T M 12 sds Ns, θθdθ M 22 s is positive on X. Furthermore, we note that {M,N} Ξ implies that P is separable and P = M 11 sds and S = M 22 are invertible. We now state the main result. Proposition 1: Let Z : R R q n be an arbitrary continuously differentiable function. Suppose there eist matrices M, M 1 = P T, matri-valued functions M 2 s, Qs, Rs, θ, Ss = S T s S n, and scalar ǫ> such that are satisfied and the following conditions hold {T,R} Ξ, { U, V } Ξ, P rqs T s = rq T s Ss ǫi, 43 ΓǫI BSFM 1 1 r CT SD Υ Us = 1 r S DT SD, Ṡs 44 V s, θ = d ds Rs, θ d Rs, θ, 45 dθ and denotes the corresponding symmetric part, Γ=AP PA T rbq T QB T 1 r CT SC FM M T F, Υ=r QsBR, saqsfm 2 s. Then System 39-4 is stabilizable with a controller of the form 41. Furthermore, let ˆP, ˆQ, ˆR and Ŝ be as defined in Theorem 3. Then if ut =K tk 1 r K 2 s sds,

5 Guoying Miao et al. / IFAC PapersOnLine K = M ˆP rm1 ˆQT r M 2 s ˆQ T sds, 46 K 1 = M 1 Ŝ, 47 K 2 s =M ˆQsM1 ˆR, sm2 sŝs 48 M 2 θ ˆRθ, sdθ, 49 then the System 39-4 is stable. Proof. Define P = P rq T s Qsds. Rs, θθdθ Ss Then as per Lemma 2 = P : X X and P ǫi. Furthermore is bounded and as per Theorem 3, the inverse P 1 is defined as in 27 and is likewise bounded and coercive with P 1 ǫ I. Furthermore, from Theorem 4 1 : X X and P 1 = P. Now define the Lyapunov functional for V = 1 X. Since P 1 = P, ǫ 1 A A 1 = P 1, APP 1 APP 1 1. Net, we note that if we define K as and M as K = K tk 1 r K 2 s sds, M = M tm 1 r M 2 s sds, then K := MP 1. We construct the controller 2 MP 1 y = M ˆP ˆQθyθdθ M 1 r ˆQ T Ŝy M 2 s r ˆQ T s Ŝsys ˆRs, θyθdθ ds = M ˆP rm1 ˆQT r ˆR, θyθdθ M 2 s ˆQ T sds M 1 Ŝy M ˆQsM1 ˆR, s M 2 sŝs M 2 θ ˆRθ, sdθ ysds = K. y ˆ Now we define a new state = P ˆ 1 X. Continuing, if u = K = KPP 1 = y t MP, 1 then the closed-loop system is stable if y V < t. Applying conditions 44-45, after calculation, we have ˆ ˆ V =, AP ˆ ˆ ˆ ˆ AP, ˆ ˆ ˆ ˆ FM, ˆ ˆ ˆ ˆ, FM. ˆ ˆ <. Therefore, the closed-loop System 39-4 is stable. Remark 5. When F = in System 39-4, we recover the standard delay-differential framework studied in Peet 216 and Peet 213: ẋt=a t l A i t r i t=t. The primary computational advantage of the differentialdifference framework over control of System 39-4 is that we can replace A i R n n with BC B R n m and C R m n and m is typically strictly less than n. Because the dimension of the decision variables in the optimization problem defined in this paper scale as n2m as opposed to 3n using the framework in Peet 216 and Peet 213, the compleity of the resulting algorithm is significantly reduced. i=1

6 6518 Guoying Miao et al. / IFAC PapersOnLine Remark 6. The feedback controller in 41 does not include delay in the input. However, the case of delay in the input can also be treated using a different form of F. This is left for future work. In the following, we present a numerical eample to illustrate the controller obtained from the condition in Proposition 1. We consider the following system with a feedback controller as follows ẋt= = t.5 1 r ut, t, 51 1 r = 1.6s. By using Proposition 1, together with the tools of MuPad, Matlab, SOSTOOLS and polynomials with degree 2, we obtain the controller ut= T t T.239 r.343 K 2 sds, s.122s K 2 = 2.12s 3.32s 4 T s.7s 2.37s 3.1s 4., Using Controller 52 coupled with System 5-51 we simulate the closed-loop system, which is illustrated in Fig.1. t t/s Fig. 1. States of System 5-51 coupled with stabilizing Controller from Prop CONCLUSIONS In this paper, we have obtained an analic formulation for the inverse of jointly positive multiplier and integral operators as defined in Peet 216. This formulation has the advantage that it eliminates the need for either individual positivity of the multiplier and integral operators or the need to use a series epansion to find the inverse. This inversion formula is applied to controller synthesis of coupled differential-difference equations. The use of the differential-difference formulation has the advantage that the size of the resulting decision variables is reduced, thereby allowing for control of systems with larger numbers of states. These methods are illustrated by designing a stabilizing controller for a system with 6 states and 2 delay channels.. REFERENCES K. Gu. Discretized LMI set in the stability problem of linear uncertain time-delay systems. Int. J. Control, 68: , K. Gu. A further refinement of discretized Lyapunov functional method for the stability of time-delay systems. Int. J. Control, 74: , 21. M.M. Peet, A. Papachristodoulou, and S. Lall. Positive forms and stability of linear time-delay systems. SIAM J. Control Optim, 47: , 29. K. Gu and Y. Liu. Lyapunov-Krasovskii functional for uniform stability of coupled differential-functional equations. Automatica, 45:798 84, 29. K. Gu. Stability problem of systems with multiple delay channels. Automatica, 46: , 21. Y. Zhang, M.M. Peet, and K. Gu. Reducing the compleity of the sum-of-square test for stability of delay linear systems.ieee Transactions Automatic Control, 56: , 211. M.M. Peet. SOS methods for multi-delay systems: a dual form of Lyapunov-krasovskii functional. IEEE Transactions on Automatic Control, submitted, 216. E. Fridman and U. Shaked. An improved stabilization method for linear time-delay systems. IEEE Transactions on Automatic Control, 47:253 27, 22. M.M. Peet and A. Papachristodoulou. Inverse of positive linear operator and state feedback design for time-delay system. In 8th IFAC Workshop on Time-delay System, pages , 29b. M.M. Peet. Full state Feedback of delayed systems using SOS: a new theory of duality. IFAC Proceedings Volumes, pages 24 29, February 213. M.M. Peet. LMI Parameterization of Lyapunov Functions for Infinite-Dimensional Systems: A Toolbo. I Proceedings of the American Control Conference, June 4-6, 214. H. Li. Discretized LKF method for stability of coupled differential-difference equations with multiple discrete and distributed delays. International Journal of Robust and Nonlinear Control, 22: , 212. G. Miao, M.M. Peet and K. Gu. Inversion of separable kernel operators in coupled differential-functional equations and application to controller synthesis. ariv online access,