Estimations are given for solutions of hybrid systems with delays, by using Lyapunov-Krasovsky functionals

Kuzmych, O.

Langerak, R.

English

Langerak, Romanus

University of Twente Research Information

ESTIMATIONS OF SOLUTIONS CONVERGENCE OFHYBRID SYSTEMS WITH DELAYLangerak R.,Department of Computer Science, University of Twente, P.O.Box 217, 7500AE Enschede, The Netherlands. E-mail: langerak@cs.utwente.nl.Kuzmych O.,Department of Complex System Modeling, Faculty of Cybernetics, TarasShevchenko National University of Kiev,Address: Vladimirskaya Str., 64, 01033, Kiev, Ukraine.Tel.: +380 44 259 02 37 (office), E-mail: lenamaks79@mail.ruIntroductionLet consider the hybrid dynamic system with switches consist of a set ofsubsystems which are linear differential-difference equations•x(t) = Aix(t) + Bix(t − τ), i = 1, n, x(t) ∈ Rn, t > 0. (1)Each of subsystems describes dynamic on a fixed finite time interval ti−1 6t < ti, i = 1, N, t0 = 0. Subsystems can be stable or unstable. It isrequired to estimate the size of deviation of solutions x(t) of logic-dynamicalsystem (1) from origin at the final moment t = tN if it is known the initialindignation.Here and further the following vector and matrix norms are used|A| = {λmax(AT A)}12 (2)|x(t)| ={n∑i=1x2i (t)}1/2‖x(t)‖τ = max−τ6s60 {|x(s + t)|}‖x(t)‖τ,β =0∫−τeβs |x(t + s)|2ds1/2,λmax (·) , λmin (·) −the greatest and least eigenvalues of the correspondingsymmetric, positively certain matrixes.In this paper we use Lyapunov-Krasovsky functional in square form. Themultiplier of exponential view at the integral allows to get estimation forderivative with an element containing an integral.1Preliminary ResultIn this section we obtain the estimation of convergence of solution x(t) of alinear stationary subsystem with delay•x(t) = Ax(t) + Bx(t − τ), (3)determined on an interval t0 6 t 6 t1. For obtaining of estimation of solutionswe use a functional in such viewV [x(t), t] = eγtxT (t)Hx(t) +0∫−τeβsxT (t + s)Gx(t + s)ds . (4)Let us denoteϕ11 (H) =λmax (H)λmin(H),ϕ12(G, H) =λmax(G)λmin(H),ϕ21(G, H) =λmax(H)λmin(G),ϕ22(G) =λmax(G)λmin(G).,S [G, H] =[−AT H − HA − G −HB−BT H G]. (5)We obtain the following result.Theorem 1. Let there exist positively certain matrixes G and H at whichthe matrix S [G, H] also is positively determined. Then the system (3)isasymptotic stable and for its solutions x(t) are fair following top exponentialestimations of convergence|x(t)| 6[√ϕ11(H) |x(0)| +√ϕ12(G, H) ‖x(0)‖τ,β]exp{−12ςt}, (6)t > 0,and‖x(t)‖τ,β 6[√ϕ21 (G, H) |x(0)| +√ϕ22(G) ‖x(0)‖τ,β]exp{−12ςt}, (7)t > 02ς (β, γ) = min{λmin (S [G, H])λmax(H), βλmin(G)λmax(G)+ γ[1 − λmin(G)λmax(G)]}, (8)The value β > 0 can be arbitrary forλmin (S [G, H]) > λmax(G).Andβ 61τln{λmax(G)λmax(G) − λmin (S [G, H])},ifλmin (S [G, H]) < λmax (H) .The value γ satisfies to a condition γ 6 β.Proof. For the proof we’ll use Lyapunov-Krasovsky functional of kind(4)with positively certain matrixes G and H. It satisfies to the followingbilateral estimationseγt{λmin(H) |x(t)|2 + λmin(G) ‖x(t)‖2τ,β}6 V [x(t), t] 66 eγt{λmax(H) |x(t)|2 + λmax(G) ‖x(t)‖2τ,β}(9)We’ll find an estimation for its derivative in force of system (3). We’ll makereplacement t + s = ξ. Then functional will transform toV [x(t), t] = eγtxT (t)Hx(t) +t∫t−τe−β(t−ξ)xT (ξ)Gx(ξ)dξ . (10)We’ll calculate a full derivative of transformed functional (10) along solutionsx(t) of system (3).We obtainddtV [x(t), t] = −eγt(β − γ)t∫t−τe−β(t−ξ)xT (ξ)Gx(ξ)dξ−−eγt(xT (t), xT (t − τ)) [ −AT H − HA − G −HB−BT H G](x(t)x(t − τ))++ γeγtxT (t)Hx(t) + eγt(1 − e−βτ)xT (t − τ)Gx(t − τ). (11)We suppose, as follows from conditions of the theorem 1, there are positivelydetermined matrixes G and H at which the matrix S [G, H] also is positivelydetermined and β > γ > 0. Then we obtainddtV [x(t), t] 6 −eγt {λmin (S [G, H]) − γλmax(H)} |x(t)|2 −3−eγt{λmin (S [G, H]) −(1 − e−βτ)λmax(G)}|x(t − τ)|2 −− eγt (β − γ) λmin(G) ‖x(t)‖2τ,β (12)If the parameters of system and functional areλmin (S [G, H]) > λmax(G)than from an inequality (12) follows, thatddtV [x(t), t] 6 −eγt {λmin (S [G, H]) − γλmax(H)} |x(t)|2 −− eγt (β − γ) λmin(G) ‖x(t)‖2τ,β (13)for any β > 0.Ifλmin (S [G, H]) < λmax(G),that inequality (13)will be carried out for0 6 β <1τln[λmax(G)λmax(G) − λmin (S [G, H])].We’ll transform right part of the inequality of square-law forms (9) to− eγtλmax(H) |x(t)|2 − eγtλmax(G) ‖x(t)‖2τ,β 6 −V [x(t), t] . (14)Let’s consider two cases.Let’s transform the inequality (14) as−eγt |x(t)|2 6 − 1λmax(H)V [x(t), t] + eγtλmax(G)λmax(H)‖x(t)‖2τ,βand we’ll substitute it in the first composed of inequalities (13). We getddtV [x(t), t] 6 −λmin (S [G, H]) − γλmax(H)λmax(H)V [x(t, t)]−−eγt{(β − γ) λmin(G) − [λmin (S [G, H]) − γλmax(H)]λmax(G)λmax(H)}‖x(t)‖2τ,β .4If the parameters are(β − γ)λmin(G) > [λmin (S [G, H]) − γλmax(H)]λmax(G)λmax(H), (15)thanddtV [x(t), t] 6 −λmin (S [G, H]) − γλmax(H)λmax(H)V [x(t), t] .Solving the received differential inequality, we obtainV [x(t), t] 6 V [x(0), 0] e−αt,α =λmin (S [G, H]) − γλmax(H)λmax(H), t > 0. (16)From hereζ = α + γ =λmin (S [G, H])λmax(H)We’ll transform an inequality (14) to the following view−eγt ‖x(t)‖2τ,β 6 −1λmax(G)V [x(t), t] + eγtλmax(H)λmax(G)|x(t)|2and again we’ll substitute it in the second composed of inequalities (13). WereceiveddtV [x(t), t] 6 −(β − γ) λmin(G)λmax(G)V [x(t), t]−−eγt{λmin (S [G, H]) − γλmax(H) − (β − γ) λmin(G)λmax(H)λmax(G)}|x(t)|2And, if parameters areλmin (S [G, H]) − γλmax(H) − (β − γ) λmin(G)λmax(H)λmax(G)> 0, (17)thanddtV [x(t)] 6 −(β − γ) λmin(G)λmax(G)V [x(t)] .Having integrated the received expression, we’ll obtain5V [x(t), t] 6 V [x(0), 0] e−αt,α = (β − γ) λmin(G)λmax(G), t > 0 . (18)From here we haveζ = α + γ = βλmin(G)λmax(G)+ γ[1 − λmin(G)λmax(G)]For reception of required result we’ll return to bilateral estimations ofLyapunov - Krasovsky functional (9). Using expressions (16), (18), we shallwrite downeγt{λmin(H) |x(t)|2 + λmin(G) ‖x(t)‖2τ,β}66 V [x(t), t] 6 V [x(0), 0] e−αt 66 e−αt{λmax(H) |x(0)|2 + λmax(G) ‖x(0)‖2τ,β}From here it is possible to obtain two estimations. First, we get|x(t)|2 6[λmax(H)λmin(H)|x(0)|2 + λmax(G)λmin(H)‖x(0)‖2τ,β]e−(α+γ)t.And, using designations ϕ11(H), ϕ12(G, H), we receive|x(t)| 6[√ϕ11(H) |x(0)| +√ϕ12(G, H) ‖x(0)‖τ,β]exp{−12(α + γ)t},t > 0,Further it is possible to write down‖x(t)‖2τ,β 6[λmax(H)λmin(G)|x(0)|2 + λmax(G)λmin(G)‖x(0)‖2τ,β]e−(α+γ)t.And, using designations ϕ21(G, H), ϕ22(G), we receive an inequality‖x(t)‖τ,β 6[√ϕ21 (G, H) |x(0)| +√ϕ22(G) ‖x(0)‖τ,β]exp{−12ςt},t > 0.6As follows from consideration of both cases we haveς =λmin (S [G, H])λmax(H)(19)for β λmin(G)λmax(G)+ γ[1 − λmin(G)λmax(G)]> λmin(S[G,H])λmax(H)ς =βλmin(G)λmax(G)+ γ[1 − λmin(G)λmax(G)](20)for β λmin(G)λmax(G)+ γ[1 − λmin(G)λmax(G)]< λmin(S[G,H])λmaxH)Uniting these expressions, we obtain the statement of the theorem 1.We shall consider a case when it is not possible to find a matrix G andHat which the matrix S [G, H] is positively determined.Let’s denoteS [G, H, γ] =[−AT H − HA − γH − G −HB−BT H G]. (21)Obviously, due to a choice of a scalar value γ < 0 the matrix S [G, H, γ]can be made positively certain.Lemma. Let the matrixes G, H are positively determined and suchinequality also is carried outγ <λmin[−AT H − HA − G − HBG−1BT H]λmax(H)(22)Then the matrix S [G, H, γ] also is positively determined.Using the lemma, we obtain the following result.Theorem 2. Let a positively certain matrixes G, H at which the matrixS [G, H] also is positively determined are not exist. If the value γ is chosenaccording to an inequality (22) and β > γ then for the solutions x(t) ofsystem (3)are fair top exponential estimations of convergence (6), (7).|x(t)| 6[√ϕ11(H) |x(0)| +√ϕ12(G, H) ‖x(0)‖τ,β]exp{−12ςt},t > 0,7‖x(t)‖τ,β 6[√ϕ21 (G, H) |x(0)| +√ϕ22(G) ‖x(0)‖τ,β]exp{−12ςt},t > 0ς (β, γ) = min{λmin (S [G, H])λmax(H)+ γ, βλmin(G)λmax(G)+ γ[1 − λmin(G)λmax(G)]}, (23)The value β can be arbitrary ifλmin (S [G, H, γ]) > λmax(G)Andβ 61τln{λmax(G)λmax(G) − λmin (S [G, H, γ])},ifλmin (S [G, H, γ]) < λmax (H) .The proof to be conducted analogically to theorem 1.Remark 1. As for value ‖x(t)‖2τ,β the top estimations are fair‖x(t)‖2τ,β =0∫−τeβs |x(t + s)| ds 6 max−τ6s60{|x (t + s)|2} 0∫−τeβsds 661β(1 − e−βτ)‖x(t)‖2τ 6 τ ‖x(t)‖τwhere‖x(t)‖τ = max−τ6s60 {|x(t + s)|} ,than it is possible the inequality (6) transforms to following|x(t)| 6[√ϕ11(H) |x(0)| +√ϕ12(G, H) ‖x(0)‖τ]e−12ςt, t > 0,or, even,8|x(t)| 6[√ϕ11(H) +√ϕ12(G, H)]‖x(0)‖τ e− 12ςt, t > 0, (24)Remark 2. As estimations of majorant type than they contain two freeparameters β and γ, and in the second theorem γ can be negative. If to puta task of a finding of “an optimum estimation” in given class of functionals,the parameters β and γ it is possible to calculate precisely.Main ResultIn the previous sections majorant estimations of solutions of stable andunstable subsystems are separately received. now we shall consider wholehybrid system (1).Let on each of intervals ti−1 6 t < ti, i = 1, N for research Lyapunov -Krasovsky functional of a kind (4) with positively certain matrixes Hi, Gi,i = 1, N is chosen. If there are positively certain matrices Hi, Gi i ∈ I,such that matricesSi [Gi, Hi] =[−ATi Hi − HiAi − Gi −HiBi−BTi Hi Gi], i ∈ Iare positively certain, than we designateNi =[√ϕ11 (Hi) +√ϕ12 (Gi, Hi)]exp {ςi (βi, γi) τ} ,where value βi > 0 can be arbitrary atλmin (S [Gi, Hi]) > λmax(Gi)andβi 61τln{λmax(Gi)λmax(Gi) − λmin (S [Gi, Hi])},ifλmin (S [Gi, Hi]) < λmax (Hi) .The value γ satisfied to a condition γ 6 β.If such matrices Hi, Gi , j ∈ J are not exist than we assumeγj <λmin[−ATj Hj − HjAj − Gj − HjBjG−1j BTj Hj]λmax(Hj),and we designate9S [Gj, Hj, γj] =[−ATj Hj − HjAj − γjHj − Gj −HjBj−BTj Hj Gj],Nj =[√ϕ11 (Hj) +√ϕ12 (Gj, Hj)]exp {ςj (βj, γj)} ,whereςj (βj, γj) = min{λmin (S [Gj, Hj, γj])λmax(Hj)+ γj, βjλmin(Gj)λmax(Gj)+ γj[1 − λmin(Gj)λmax(Gj)]}The value βj can be arbitrary atλmin (S [Gj, Hj, γj]) > λmax(Gj)andβj 61τln{λmax(Gj)λmax(Gj) − λmin (S [Gj, Hj, γj])},ifλmin (S [Gj, Hj, γj]) < λmax (Hj) .Theorem 3. Let the initial state of the logic-dynamical Hybrid system(1) satisfied to the condition ‖x(0)‖τ < δ. Then at t = tN inequality will beexecuted‖x(tN)‖ 6N∏i=1Ni exp{−12N∑i=1ςi (ti − ti−1)}.Proof.Let consider the first time interval t0 6 t 6 t1, t0 = 0. If thereare positively certain matrices G1, H1, at which matrix S [G1, H1] also posi-tively certain, than as follows from expression (24) of remark 1, inequality isexecuted‖x(t1)‖ 6[√ϕ1(H1) + ϕ (G1, H1)]‖x(t0)‖τ e− 12ς1(t1−τ).If there are not such matrices, for arbitrary positively certain matricesG1, H1, there exists a constant γ1, at which a matrix S [G1, H1, γ1] also willbe positively certain. Again using epression (24) of remark 1, we get‖x(t1)‖ 6[√ϕ1(H1) + ϕ (G1, H1)]‖x(t0)‖τ e− 12ς1(t1−t0).10And for the time moment t = t1 takes place‖x(t1)‖τ 6 N1 ‖x(t0)‖τ e− 12ς1(t1−t0).Let we consider a next interval t1 6 t 6 t2. As for the second interval asimilar estimation takes place‖x(t2)‖τ 6 N2 ‖x(t1)‖τ e− 12ς2(t2−t1),we obtain‖x(t2)‖τ 6 N1N2 ‖x(t0)‖τ exp{−12[ς1 (t1 − t0) + ς2 (t2 − t1)]}.Continuing a process farther, for the time moment t = tN we get‖x(tN)‖ 6N∏i=1Ni exp{−12N∑i=1ςi (ti − ti−1)},that it was required to prove.Literature[2] Branicky M. S. Stability of Switched and Hybrid Systems. In: Proc.3-rd IEEE CDC, 1994, pp.3498-3503, [Ang].[3] Langerak R., Polderman J.W., Krilavicius T. Stability Analysis forHybrid Automata using Conservative Gains. Twente, 2004, [Ang].[4] Kheil D. Theory functional differential equations. Moskow. Mir, 1984,421pp., [Rus].[7] Krasovsky N.N. Some tasks of theory of motion stability. Moskow,Fizmatgiz, 1959, 212pp., [Rus].[8] Korenevskiy D.G. Stability of the dynamical systems at casual in-dignations of parameters. Algebraic criteria. Kiev, Naukova Dumka, 1989,208pp., [Rus].[9] Slinko V.I. About the conditions of motion stabilityof the linear impul-sive systems with delay. Applied mechanics, Vol.41, (no 6). 2005, 130-138pp.,[Rus].[10] Khusainov D.Y., Kuzmych O.I. The Estimates of stability of Logic-Dynamical Systems with time switching. Bulletin of the Kiev University.Series: Physical-Mathematical sciences, (no 1), 2005. 230-237pp., [Ukr].[11] Kuzmych O.I. The Estimates of stability of Hybrid Systems withfinite number time switchings. Bulletin of the Kiev University. Series: Cy-bernetics, (no 6), 2005. 230-237pp., [Ukr].11

Estimations of solutions convergence of hybrid systems with delay

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Estimations of solutions convergence of hybrid systems with delay

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