Parametric optimization of a neutral system with two delays and PD-controller

10.2478/acsc-2013-0008 Archives of Control Sciences Volume 23LIX, 2013 No. 2, pages 131 143 Parametric optimization of a neutral system with two elays an PD-controller JÓZEF DUDA In this paper a parametric optimization prolem for a linear neutral system with two elays with an integral quaratic performance inex is formulate an solve. The metho of computing of a performance inex value ases on etermining of a Lyapunov functional efine on a state space such that its value for an initial state is equal to a performance inex value. In the paper a form of a Lyapunov functional is assume an a metho of computing its coefficients is given. An example illustrating the application of iscusse theory is presente. It concerns the system with a PD-controller esigne to control a plant with two elays oth retare an neutral type. For such system a value of consiere performance inex is etermine. Key wors: parametric optimization, Lyapunov functional, time elay system, neutral system 1. Introuction The Lyapunov quaratic functionals are use to: test the staility of the systems, in computation of the critical elay values for time elay systems, in computation of the exponential estimates for the solutions of the time elay systems, in calculation of the roustness ouns for the uncertain time elay systems, to calculate of a quaratic performance inex of quality for a process of parametric optimization for the time elay systems. One constructs the Lyapunov functionals for a system with a time elay with a given time erivative. For the first time such Lyapunov functional was introuce y Repin [14] for a case of the retare time elay linear systems with one elay. Repin [14] elivere also a proceure for etermination of the functional coefficients. Dua [1] use a Lyapunov functional, which was propose y Repin, for the calculation of a quaratic performance inex value in the process of parametric optimization for the systems with a time elay of retare type an extene the results to a case of a neutral type time elay system in [2]. Dua [3] presente a metho of etermining of a The Author is with AGH University of Science an Technology, Faculty of Electrical Engineering, Automatics, Computer Science an Electronics, Department of Automatics. al. Mickiewicza 30, 30-159 Krakow, Polan. E-mail: jua@agh.eu.pl I thank the associate eitor an the reviewers for their suggestions, which have improve the quality of the paper. Receive 14.08.2012. Unauthenticate Downloa Date 1/30/18 3:53 PM

132 J. DUDA Lyapunov functional for a linear ynamic system with two lumpe retare type time elays in a general case with no-commensurate elays an presente a special case with commensurate elays in which a Lyapunov functional coul e etermine y solving a set of the orinary ifferential equations. Dua [4] introuce a metho of etermining of a Lyapunov functional for a linear ynamic system with two elays oth retare an neutral type time elay an in the paper [5] presente a metho of etermining of a Lyapunov quaratic functional for a linear time-invariant system with k-non- commensurate neutral type time elays. Dua [6] consiere the parametric optimization prolem for a neutral system with two elays with a P-controller. Infante an Castelan s [9] construction of a Lyapunov functional is ase on a solution of a matrix ifferential-ifference equation on a finite time interval. This solution satisfies the symmetry an ounary conitions. Kharitonov an Zhako [13] extene the Infante an Castelan s results an propose a proceure of construction of the quaratic functionals for the linear retare type time elay systems which coul e use for the roust staility analysis of the time elay systems. This functional was expresse y means of a Lyapunov matrix which epene on a funamental matrix of a time elay system. Kharitonov [10] extene some asic results otaine for a case of the retare type time elay systems to a case of the neutral type time elay systems, an in [11] to the neutral type time elay systems with a iscrete an istriute elay. Kharitonov an Plischke [12] formulate the necessary an sufficient conitions for the existence an uniqueness of a Lyapunov matrix for a case of a retare system with one elay. The paper eals with a parametric optimization prolem for a system with two elays oth retare an neutral type time elay with a PD-controller. To the est of author s knowlege, such a parametric optimization prolem has not een reporte in the literature. An example illustrating this metho is also presente. 2. Formulation of a parametric optimization prolem Let us consier a linear system with two elays oth retare an neutral type whose ynamics is escrie y equations xt t xt τ D = Axt + Bxt τ +Cut r t xt 0 + = Φ xt ut = Kxt T t where: t t 0, [,0], r τ > 0 an are rationally inepenent; A, B, D R n n, D is nonsingular, xt R n, Φ C 1 [,0],R n, C R n m, K, T R m n, ut R m, C 1 [,0],R n is a space of continuously ifferentiale functions efine on an interval [,0] with values in R n. 1 Unauthenticate Downloa Date 1/30/18 3:53 PM

PARAMETRIC OPTIMIZATION OF A NEUTRAL SYSTEM WITH TWO DELAYS AND PD-CONTROLLER 133 We can reshape an equation 1 to a form xt t xt 0 + = Φ xt τ xt r D +CT = Axt + Bxt τ CKxt r t t 2 where: t t 0, [,0], r τ > 0 an are rationally inepenent, A, B, D R n n, D is nonsingular, C R n m, K, T R m n, Φ C 1 [,0],R n. A solution of the equation 2 is a continuously ifferentiale function efine for t t 0 r except at suitale multiples of the time elays. If the conition Φ0 = D Φ τ Φ CT + AΦ0 + BΦ τ CKΦ 3 hols then the solution of equation 2 has a continuous erivative for all t t 0 r. A ifference equation associate with 2 is given y the following term xt = Dxt τ CT xt r, t t 0. 4 The eigenvalues of a ifference equation 4 play a funamental role in the asymptotic ehavior of the solutions of a neutral equation 2. Definition 1 The spectrum σg is a set of complex numers λ for which a matrix λi G is not invertile. The spectral raius of a matrix G is given y a form γg = sup{ λ : λ σg}. 5 We assume that matrices D an CT satisfy the hypotheses given in Theorem 9.6.1 of Hale an Veruyn Lunel [7] { r τ > 0 an are rationally inepenent sup { γ e i 1 D e i 2 CT : 1, 2 [0,2π] } 6 < 1 an in this case a ifference equation 4 is stale. We introuce a new function y, efine as follows yt = xt Dxt τ +CT xt r f or t t 0. 7 Thus an equation 2 takes the form yt = Ayt + AD + Bxt τ ACT +CKxt r t yt = xt Dxt τ +CT xt r yt 0 = Φ0 DΦ τ +CT Φ xt 0 + = Φ 8 Unauthenticate Downloa Date 1/30/18 3:53 PM

134 J. DUDA The conition 6 guarantees asymptotic staility of the close-loop system 2. The asymptotic staility of the close-loop system 2 implies the asymptotic staility of the close-loop system 8. State of the system 8 forms the folowing vector [ ] yt St = for t t 0 9 where x t C 1 [,0],R n, x t = xt + for [,0]. The state space is efine y a formula x t X = R n C 1 [,0],R n 10 We search for the matrices K an T which minimize the integral quaratic performance inex J = an matrix T fulfills the conition 6. t 0 y T tytt 11 3. A metho of etermining of a performance inex value On the state space X one efines a Lyapunov functional, positively efine, ifferentiale, with erivative compute on a trajectory of the system 3 eing negatively efine. V St = y T tαyt + + y T tβxt + + x T t + δ,σxt + σσ 12 for t t 0 where α = α T R n n, β C 1 [,0],R n n, δ C 1 Ω,R n n, Ω = {,ς : [,0], ς [,0]}. C 1 is the space of continuous functions with continuous erivative. We ientify the coefficients of the Lyapunov functional 12, assuming that its erivative compute on the system 3 trajectory satisfies the following relationship V St t = y T tyt for t t 0. 13 Unauthenticate Downloa Date 1/30/18 3:53 PM

PARAMETRIC OPTIMIZATION OF A NEUTRAL SYSTEM WITH TWO DELAYS AND PD-CONTROLLER 135 If the relationship 13 hols an the close-loop system 8 is asymptotically stale, one can easily etermine the value of square inicator of the quality for the parametric optimization prolem ecause J = t 0 y T tytt = V St 0. 14 4. Determination of the Lyapunov functional coefficients Derivative of the functional 12 is compute on trajectory of the system 3 accoring to the formula V St 0 t = lim sup V yt 0 + h, x t0 +h V y 0, Φ. 15 h 0 h The time erivative of the functional 12 calculate on the asis of 15 is given y the following [ V St = y T t A T α + αa + β0 + ] βt 0 yt+ t 2 +y T t[2αb + AD + β0d]xt τ+ +y T t[ 2αACT +CK β0ct β]xt r+ + + [ y T t A T β β ] + δt,0 xt + + 16 x T t τ[b + AD T β + D T δ T,0]xt + + [ ] x T t r ACT +CK T β + T T CT δ T,0 + δ, xt + + [ δ,σ x T t + + δ,σ ] xt + σσ. σ Unauthenticate Downloa Date 1/30/18 3:53 PM

136 J. DUDA From equations 4 an 13 one otains the following set of equations A T α + αa + β0 + βt 0 2 = I 17 2αB + AD + β0d = 0 18 2αACT +CK + β0ct + β = 0 19 A T β β + δt,0 = 0 20 B + AD T β + D T δ T,0 = 0 21 ACT +CK T β + T T CT δ T,0 + δ, = 0 22 δ, σ for [,0], σ [,0]. From an equation 18 it follows + δ,σ σ = 0 23 β0 = 2α A + BD 1. 24 We now sustitute 24 into 17. After some calculations one otains the result αp + P T α = I 25 where P = BD 1. 26 From an equation 25 one otains the matrix α. Taking into account the relations 19 an 24 we get the formula From equation 21 one otains β = 2αPCT CK. 27 δ T,0 = A + P T β. 28 We now put 28 into 20 receiving after some calculations for [,0]. β = PT β 29 Unauthenticate Downloa Date 1/30/18 3:53 PM

PARAMETRIC OPTIMIZATION OF A NEUTRAL SYSTEM WITH TWO DELAYS AND PD-CONTROLLER 137 Solution of ifferential equation 29 is as follows for [,0]. After putting 27 into 30 one otains Solution of equation 23 is as elow β = exp P T + r β 30 β = 2exp P T + r αpct CK. 31 where φ C 1 [,r],r n n. From equations 32 an 22 one otains δ,σ = φ σ 32 δ, = φ = ACT +CK T β T T CT δ T,0. 33 After putting 28 into 33 one otains the following for [,0]. Hence Taking into account 32 an 35 we otain φ = PCT CK T β 34 φξ = PCT CK T β ξ r. 35 δ,σ = PCT CK T βσ r. 36 Finally taking the relation 31 into account one gets a formula δ,σ = 2PCT CK T exp P T σ αpct CK. 37 In this way we otaine all parameters of a Lyapunov functional. 5. Determination of a performance inex value Accoring to formula 14 the performance inex value is given y a term J = V yt 0,Φ = y T t 0 αyt 0 + + y T t 0 βφ + Φ T δ,σφσσ. 38 After putting the relations 31 an 37 into 38 one gets Unauthenticate Downloa Date 1/30/18 3:53 PM

138 J. DUDA J = y T t 0 αyt 0 + 2 + 2 y T t 0 exp P T + r αpct CKΦ + Φ T PCT CK T exp P T σ αpct CKΦσσ. To otain the optimal values of the parameters K an T we compute the performance inex erivatives with respect to K i j an T ls an then we equal them to zero JK i j,t ls = 0 K i j JK i j,t ls T ls = 0 39 for i m, j n, l m, s n. 40 In this case one otains a set of algeraic equations with unknown variales K opt an T opt. 6. An example Let us consier a system escrie y the equation xt t xt 0 + = Φ xt τ xt r + ct = axt + xt τ ckxt r t t 41 where: t t 0, xt R, [,0], a,, c,, k R, 0, r τ > 0. The elements an ct shoul satisfy the conition 6, which takes the following form + ct < 1. 42 One can rewrite equation 41 to the form yt = ayt + + axt τ act + ckxt r t yt = xt xt τ + ct xt r 43 yt 0 = Φ0 Φ τ + ct Φ xt 0 + = Φ Unauthenticate Downloa Date 1/30/18 3:53 PM

PARAMETRIC OPTIMIZATION OF A NEUTRAL SYSTEM WITH TWO DELAYS AND PD-CONTROLLER 139 where: t t 0, xt R, [,0], a,, c,, k, T R, 0, r τ > 0. We search for the parameters k an T which minimize an integral quaratic performance inex J = A Lyapunov functional V is efine y the formula V St = αy 2 t + t 0 y T tytt = V yt 0,Φ. 44 ytβxt + + where accoring to 25 α is given y Accoring to 31 the coefficient β is given y the formula β = 2αpcT ckexp = xt + δ, σxt + σσ 45 α = 1 2p = 2. 46 + r ct ck exp + r. 47 It follows from 37 that the element δ can e expresse as δ,σ = 2αpcT ck 2 exp = ct 2 ck exp σ σ =. 48 Value of the performance inex is given y J = 2 Φ0 Φ τ + cφt 2 + 49 + + ct ck ct 2 ck Φ0 Φ τ + cφt ΦΦσ exp σ Φ exp σ. + r + A performance inex is a quaratic function with respect to the variales k an T. To attain the optimal values of the parameters k an T we compute the performance inex erivatives with respect to k an T Unauthenticate Downloa Date 1/30/18 3:53 PM

140 J. DUDA Jk,T k = c Φ0 Φ τ + cφt 2c2 0 T k ΦΦσ exp Φ exp σ σ + r + 50 Jk,T = cφ Φ0 Φ τ + cφt + T + cφ0 Φ τ + cφt + c 2 Φ T k 0 Φ exp + 22 c 2 2 T k 0 Φ exp ΦΦσ exp + r + + r + 51 σ σ. Now we equal the erivatives to zero an we otain a set of linear equations with unknown variales k opt an T opt. [ ][ ] [ ] q 11 q 12 k opt h 1 = 52 q 21 q 22 T opt. h 2 where q 11 = 2c ΦΦσ exp σ σ 53 q 12 = cφ + 22 c 2 + r Φ exp + ΦΦσ exp σ σ 54 Unauthenticate Downloa Date 1/30/18 3:53 PM

PARAMETRIC OPTIMIZATION OF A NEUTRAL SYSTEM WITH TWO DELAYS AND PD-CONTROLLER 141 q 21 = cφ 22 c 2 + r Φ exp + ΦΦσ exp σ σ 55 q 22 = cφ2 + 23 c 3 + 2cΦ ΦΦσ exp Φ exp σ σ + r + 56 h 1 = 0 Φ0 Φ τ h 2 = Φ0 Φ τ Φ Presente elow is the numerical solutions. 1. Accoring to following set of parameters Φ exp Φ exp + r 57 + r Φ = constans, = 0.516, c = 1, = 0.4, r = 1 one otains the optimal values for which the conition 42 is satisfie k opt = 1,5451, T opt = 0,4253 + ct = 0.8253 < 1. 2. Accoring to following set of parameters Φ = constans, = 0.516, c = 1, = 0.45, r = 1. 58 Unauthenticate Downloa Date 1/30/18 3:53 PM

142 J. DUDA one otains the optimal values k opt = 2,0391, T opt = 1,2100 for which the conition 42 is not satisfie ecause + ct = 1,66 > 1 59 From 59 it follows that the optimization proceure may lea to instaility of the close-loop system an therefore the constraints for elements an T given y a formula 42 are necessary. 7. Conclusions The paper presents a parametric optimization prolem for a system with two elays oth retare an neutral type time elay with a PD-controller. The metho for computing of the performance inex value ases on etermining a Lyapunov functional efine on the state space which value for initial state is equal to the performance inex. In the paper form of the Lyapunov functional is assume an the metho for computing its coefficients is presente. An example shows that the optimization proceure may lea to instaility of the close-loop system an therefore the matrices D an T nee to e constraine y the formula 6. References [1] J. DUDA: Parametric optimization prolem for systems with time elay. PhD thesis AGH University of Science an Technology, Polan, 1986. [2] J. DUDA: Parametric optimization of neutral linear system with respect to the general quaratic performance inex. Archiwum Automatyki i Telemechaniki, 33 1988, 448-456. [3] J. DUDA: Lyapunov functional for a linear system with two elays. Control an Cyernetics, 39 2010, 797-809. [4] J. DUDA: Lyapunov functional for a linear system with two elays oth retare an neutral type. Archives of Control Sciences, 20 2010, 89-98. [5] J. DUDA: Lyapunov functional for a system with k-non-commensurate neutral time elays. Control an Cyernetics, 39 2010, 1173-1184. [6] J. DUDA: Parametric optimization of neutral linear system with two elays with P-controller.Archives of Control Sciences, 21 2011, 363-372. Unauthenticate Downloa Date 1/30/18 3:53 PM

PARAMETRIC OPTIMIZATION OF A NEUTRAL SYSTEM WITH TWO DELAYS AND PD-CONTROLLER 143 [7] H. GÓRECKI, S. FUKSA, P. GRABOWSKI an A. KORYTOWSKI: Analysis an Synthesis of Time Delay Systems, John Wiley & Sons, Chichester, New York, Brisane, Toronto, Singapore, 1989. [8] J. HALE an S. VERDUYN LUNEL: Introuction to Functional Differential Equations, New York, Springer, 1993. [9] E.F. INFANTE an W.B. CASTELAN: A Liapunov Functional For a Matrix Difference-Differential Equation. J. Differential Equations, 29 1978, 439-451. [10] V.L. KHARITONOV: Lyapunov functionals an Lyapunov matrices for neutral type time elay systems: a single elay case. Int. J. of Control, 78 2005, 783-800. [11] V. L. KHARITONOV: Lyapunov matrices for a class of neutral type time elay systems. Int. J. of Control, 81 2008, 883-893. [12] V.L. KHARITONOV an E. PLISCHKE: Lyapunov matrices for time-elay systems. Systems & Control Letters, 55 2006, 697-706. [13] V.L. KHARITONOV an A.P. ZHABKO: Lyapunov-Krasovskii approach to the roust staility analysis of time-elay systems. Automatica, 39 2003, 15-20. [14] J. KLAMKA: Controllaility of Dynamical Systems. Kluwer Acaemic Pulishers Dorrecht, 1991. [15] YU. M. REPIN: Quaratic Lyapunov functionals for systems with elay. Prikl. Mat. Mekh., 29 1965, 564-566. Unauthenticate Downloa Date 1/30/18 3:53 PM