Feedback Stabilization of Systems with

Transcription

1 System Tianjin University, China 9th Workshop Distributed Parameter,2015, Beijing

2 System 1 System Examples Classification 2 Question question issue 3 Mathematical Model Existing Results Existing and extended results 4 Question 1 Question 2 Question 3 5

3 delay System Examples Classification Example Using positive feedback stabilizes the Oscillatory system ÿ(t) + k 2 y(t) = γy(t τ) Example The elastic system memory { 2 w(x,t) t 2 = w(x, t) + t 0 g(t s) w(x, s)ds = 0, w(x, t) = 0, x Ω Example The output finite memory { ẋ(t) = Ax(t) y(t) = t t τ Cx(θ)dθ

4 Classification System Examples Classification There are mainly three types delays in systems: interior delays (or called the state delay), input delays (control delays) and output delays (measurement delays). The control system interior delay (or memory) term { dx(t) dt = Ax(t) + Φ(x t ) + Bu(t), t > 0 x(0) = x 0 x(s) = v(s), s [ τ, 0]; Φ(f) = 0 τ dη(s)f(s) where η : [ τ, 0] L(X) bounded variation.

5 Classification System Examples Classification The delay occurs in the signal input phase, it also is called the controller delay. The control system input delay (or memory) term { dx(t) dt = Ax(t) + BΦ(u t ), t > 0 x(0) = x 0 u(s) = ϕ(s), s [ τ, 0]; Φ(f) = 0 τ dη(s)f(s) where η : [ τ, 0] L(U) bounded variation.

6 Classification System Examples Classification The delay occurs in the signal output phase, it also is called the observation delay. Output The control system output delay (or memory) term dx(t) dt = Ax(t), x(0) = x 0 y(t) = Φ(x t ), t > 0 y(s) = ψ(s), s [ τ, 0] Φ(f) = 0 τ dη(s)f(s) where η : [ τ, 0] L(X, Y) bounded variation.

7 System Question question issue Problem and

8 System Question question issue Question1 Let H be a Hilbert space, and A be a unbounded and linear operator in H, A 1 is another linear operator on H. Let A generate a C 0 semigroup on H exponential stability. Let us consider the dynamic system ẋ(t) = Ax(t) + A 1 x(t τ) (1) Under what conditions on A 1 and τ, is the system still stable? Remark Usually, a linear system parabolic-type interior delay has a stability margin. When the delay term is small enough in the sense some norm, the system is still exponentially stable.

9 System Question question issue For the linear conservation system hyperbolic-type, it is sensitive to small delay. proposed the following question: Question Let H be a Hilbert space, and A be a positive definite linear operator in H. Let U be another Hilbert space, and C : U H be a linear operator. Let us consider the dynamic system ẍ(t) + Ax(t) = Bu(t). (2) Suppose that the feedback operator K : H U is bounded, the dynamic system negative feedback control law is uniformly stable. u(t) = Kẋ(t) (3)

10 Question System Question question issue Question Suppose the system is damped and has the following form ẍ(t) + Dẋ(t) + Ax(t) = Bu(t) (4) where D also is a nonnegative operator on H. When u(t) = 0, the system is uniformly stable. If the feedback control law (3) is placed by u(t) = Kẋ(t τ), τ > 0 (5) Whether do it improve the stability characteristics (4)? Reference R., Two Concerning the Boundary Control Certain Elastic, Journal Differential Equations, Vol.92,1991,pp.27-44,

11 Following System Question question issue X. J. Li, K. S. Liu s Result(1993): A is positive definite, B is boundary control ẋ(t) + Ax(t) = BB x(t τ) the system is robustly for small delay. ẍ(t) + Ax(t) = BB x(t τ) the system is sensitively for small delay Reference X. J. Li, K. S. Liu, the effect small time delay in the feedback on boundary stabilization, Science in China (A), Vol.36, No.12, 1993, pp

12 Following System Question question issue Rebarbery and Townley s Result(1998): A is unbounded and B is admissible, K is compact. If BK improves stability e At, then ẋ(t) = Ax(t) BKx(t τ) is robustly for small τ; U is finite dimensional and K is bounded, BK cannot be found to improve exponential stability. Reference R. Rebarbery, S. Townley, Robustness respect to delays for exponential stability distributed parameter systems, SIAM J. Control Optim., Vol. 37, No. 1, 1998, pp

13 Following Question System Question question issue R. X. Wang and Y. T. Wang s Result (2004) w tt (x, t) + w xxxx (x, t) 2αw xxt (x, t) = 0, x (0, 1), t > 0 w(0, t) = w xx (0, t) = 0, w x (1, t) = 0 w xxx (1, t) = w t (1, t ε) w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x), w t (x, s) = ϕ(x, s), s ( ε, 0) (6) If 0 < α < 1 and 0 < ε < ε 0, the system is Robust stable; Reference R. X. Wang, Y. T. Wang, The Well-posedness And Robust Stability Of Flexible Beam With Respect To Small s In The Loop, Master Thesis, Shanxi University, 2004.

14 Following Question System Question question issue Nicaise and Pignotti s Result(2014): A positive definite and unbounded,b j : U j H 1,j = 1, 2, are A-admissible, ẅ(t) = Aw(t) + B 1 B 1ẇ(t) + B 2B2ẇ(t τ), t > 0 w(0) = w 0, ẇ(0) = w 1, (7) B 2ẇ(s) = f(s), s ( τ, 0). Suppose that e (A+B 1B 1 )t is exponentially stable. B 2 satisfies there exists α (0, 1) such that B 2w U2 α B 1w U1, w D(A 1 2 ) then, the system (7) also is the exponentially stable. Reference S. Nicaise, C. Pignotti, second-order evolution equations time delay, Math. Control Signals Syst., Vol.26, 2014, pp

15 Following Question System Question question issue Nicaise and Pignotti s Result (2015) Suppose that e At is exponentially stable; F satisfies Lipschitz condition F x F y L x y ; B is bounded. U t (t) = AU(t) + F (U(t)) + kbu(t τ), t > 0, U(0) = U 0, BU(s) = f(s), s ( τ, 0), where A : D(A) H H generates C 0 semigroup exponential stability; F : H H is a Lipschitz mapping, B is a bounded linear operator on H. They showed that if the C 0 - semigroup describing the linear part is exponentially stable, then the system retains this good property when a suitable smallness condition on the time-delay feedback is satisfied. Reference S. Nicaise, C. Pignotti, Exponential stability abstract evolution equa-

16 Survey System Question question issue Theorem Let A generates exponentially stable and contraction C 0 semigroup, A 1 = BB where B : Z H. and Z be another Hilbert space. Suppose D(A) D(A 1 ). A + A 1 generates a contraction semigroup if and if (A 1 x, x) (A R x, x), x D(A), where A R = A+A 2 is an extension. A + A 1 generates a contraction semigroup exponential stability if and if there exists α (0, 1) such that (A 1 x, x) α(a R x, x), x D(A)

17 Survey System Question question issue Theorem Let A generates exponentially stable and contraction C 0 semigroup, A 1 = BB where B : Z H. and Z be another Hilbert space. Suppose D(A) D(A 1 ). if A + A 1 generates a contraction semigroup exponential stability, then ẋ(t) = Ax(t) ± A 1 x(t τ) also is exponentially stable, and the decay rate λ and τ satisfy relation λ = inf 2 α αeλτ (A Rx, x) = µ 0. x =1

18 Sketch Pro Lyapunov Functional V (x(t), t) = e λt x(t) 2 + t t τ e λ(s+τ) B x(s) 2 ds. System Question question issue dv dt = e λt [λ x(t) 2 + 2(A r x(t), x(t)) + (1 + e λτ )(A 1 x, x)] e λt B x(t) ± B x(t τ) 2 e (A+A 1)t is exponentially stable iff (A 1 x, x) α(a R x, x). dv dt the condition dv dt 0. e λt [λ x(t) 2 + [2 α(1 + e λτ )](A 1 x, x)] λ 2 α αe λτ = inf x =1 (A R x, x) = µ 0 implys

19 Survey System Question question issue Theorem Let A generates exponentially stable and contraction C 0 semigroup, A 1 is linear operator. Suppose D(A) D(A 1). then A + A 1 generates a C 0 semigroup exponential stability, µ 0 = inf x =1 (A Rx, x) ; there exists a constant α (0, 1) such that A 1x 2 α (A Rx, x), x D(A) µ 0 > α. ẋ(t) = Ax(t) ± A 1x(t τ) also is exponentially stable, and the decay rate λ and τ has more complicated relationship. Lyapunov Functional V (x(t), t) = e λt x(t) 2 + t t τ e λ(s+τ) A 1x(s) 2 ds.

20 Solvability Issue System Question question issue Necessariness solvability delayed equations When the differential equation involves time-delay in the e- quation or boundary condition, the study solvability issue is completely necessary. Shang and Xu considered a 1D wave equation the delay depending on position. They showed if the largest time delay equals to 1, the system equation is unsolvable. Reference Y. F. Shang, G. Q. Xu, The stability a wave equation delaydependent position, IMA Journal Mathematical Control and Information Vol. 28, 2011, pp

21 issue System Question question issue The control issue the system state delay Issue The linear system state delay in Hilbert space: ẋ(t) = Ax(t) + A 1 x(t τ) + Bu(t), t > 0 x(0) = x 0, x(s) = f(s), s [ τ, 0) where A = A, and A 1 is a linear operator in H (it might be unbounded), B is A-admissible. (A, B) is exactly controllable in finite time. Find a state feedback control law such that the closed loop system is exponentially stable.

22 issue System Question question issue control law u(t) = B K 1 x(t) B K 2 x(t τ) (A BB K 1 ) generates a C 0 semigroup Exponential stability, and A 1 BB K 2 H,H 1 α is small enough. Theorem The closed loop system in Hilbert space: { ẋ(t) = (A BB K 1 )x(t) + (A 1 BB K 2 )x(t τ), t > 0 x(0) = x 0, x(s) = f(s), s [ τ, 0) is exponentially stable. Reference G. Q. Xu, stabilization the systems delays, in Preparing,

23 System Mathematical Model Existing Results Existing and extended results Problem and Control Design Mathematical Model Existing Results (Collocated ). Control design

24 Mathematical Model Let us consider the following system z(s,t) t = z(s,t) s, s ( τ, 0), t > 0 z(0, t) = u(t), t 0 z(s, 0) = f(s), s ( τ, 0) (8) System Mathematical Model Existing Results Existing and extended results where u(t) is the control input and f(x) is memory function. The output the system is v(t) = 0 τ dν(s)z(s, t) where ν(x) is a bounded variation function and the integral is in the sense the Riemann-Stieltjes integral. Solution to (8) z(s, t) = u(t + s), s ( τ, 0)

25 Mathematical Model System Mathematical Model Existing Results Existing and extended results Hence the output is the following form Therefore, dx(t) dt z(s,t) t v(t) = 0 τ dν(s)u(t + s). = Ax(t) + Bv(t), = z(s,t) s, s ( τ, 0), z(0, t) = u(t), t 0 v(t) = 0 τ dν(s)z(s, t) x(0) = x 0, z(s, 0) = f(s), s ( τ, 0) y(t) = Cx(t) is equivalent to the following dx(t) dt = Ax(t) + B 0 τ dν(s)u(t + s), x(0) = x 0, u(s) = f(s), s ( τ, 0) y(t) = Cx(t)

26 Mathematical Model System Mathematical Model Existing Results Existing and extended results Question for v(t) v(t) = 0 τ dν(s)u(t + s). Usually, u(t) is in L 2 loc. For such a function the integral may not exist. We make requirement: Requirement ν(s) takes the form ν(s) = αh(s) + βh(s + τ) + γ(s)χ [ τ,0] where α, β are constants, H(s) is the Heaviside function, and γ(s) is in Sobolev space H 1 ( τ, 0).

27 Mathematical Model System Mathematical Model Existing Results Existing and extended results Corresponding to the this form, the output is 0 v(t) = αu(t) + βu(t τ) + γ (s)u(t + s)ds τ where γ (s) L 2 [ τ, 0], this means that the output includes the distributed delays. Special cases γ(s) = 0: then v(t) = αu(t)+βu(t τ), that means that the output has partial delay if α 0; γ(s) 0, α = 0, v(t) has the form v(t) = βu(t τ), that means that the output has full delay if β 0. γ(s) 0, β = 0, v(t) has the form v(t) = αu(t) that means that the output has no delay if α 0.

28 Control Issue System Mathematical Model Existing Results Existing and extended results dx(t) dt = Ax(t) + B[αu(t) + βu(t τ) + 0 x(0) = x 0, u(s) = f(s), s ( τ, 0) y(t) = Cx(t) where g(s) = γ (s). τ g(s)u(t + s)ds],

29 Existing Results System Mathematical Model Existing Results Existing and extended results Control Problem 1-D wave equation Boundary control problem 1-d wave equation w tt (x, t) = w xx (x, t), x (0, 1), t > 0 w(0, t) = 0, w x (1, t) = v(t), w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x), x (0, 1). where v(t) is the exterior force. Observation The system observation is (9) y(t) = w t (1, t) (10)

30 Existing results System Mathematical Model Existing Results Existing and extended results Positive result If there is no input delay, i.e., v(t) = u(t), under the collocated feedback control law u(t) = ky(t), k > 0 (11) the closed-loop system decays exponentially. Reference: J.-L. Lions,1982, Also see J.-L. Lions, Exact controllability, stabilization and perturbations for distributed parameter system. SIAM Rev. 30 (1988) 1õ68. Negative results: For any small delay time τ, if the system has the full input delay, i.e., v(t) = u(t τ), then under the collocated feedback control law (11), the closed-loop system is unstable.

31 Existing results System Mathematical Model Existing Results Existing and extended results References: 1. R., J. Lagness and M.P. Poilis, An example on the effect time delays in boundary feedback stabilization wave equations. SIAM J. Control Optim. 24 (1986) 152õ R., Not all feedback stabilized hyperbolic systems are robust respect to small time delay in their feedbacks. SIAM J. Control Optim. 26 (1988) 697õ713. Comparing For any finite-dimensional system, if the system is stable, then for small time delay τ, it also is stable. That means that there is a stable margin. Question: How we design anti-delay control?

32 Existing results System Mathematical Model Existing Results Existing and extended results Case 1: α > 0, β > 0. v(t) = αu(t) + βu(t τ) If there is a input delay, then the system becomes w tt (x, t) = w xx (x, t), x (0, 1), t > 0 w(0, t) = 0, w x (1, t) = αu(t) + βu(t τ) w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x), u(s) = f(s), s ( τ, 0) (12) where α > 0, β > 0. The collocated observation y(t) = w t (1, t). Collocated feedback law take the collocated feedback law u(t) = y(t).

33 Existing results System Mathematical Model Existing Results Existing and extended results Main result: Assume that α > 0, β > 0, under the feedback law (11), the closed-loop system is the following properties 1) when α > β, the closed-loop system is exponentially stable for any τ > 0. 2) when α = β, the closed-loop system at most is asymptotically stable, the stability depends on the τ-value. 3) when α < β, the closed-loop system is unstable. References: G. Q. Xu, S. P. Yung and L. K. Li, wave systems input delay in the boundary control, ESAIM: Control Optim. Calc. Var., Vol.12, (2006), pp

34 1 2-stable rule System Mathematical Model Existing Results Existing and extended results How to understand this result Rewrite v(t) = αu(t) + βu(t τ) into α v(t) = (α + β)[ α + β u(t) + α u(t τ)]. α + β α Then the above result can be read as α+β > 1 2, the closedloop system is stable, and α α+β < 1 2, then the closed-loop system is unstable stable rule In v(t), the weight out delay part control is larger than the delay part, the closed-loop system is exponentially stable, otherwise, the system is unstable.

35 Extended results about 1 2-stable rule System Mathematical Model Existing Results Existing and extended results High-dimensional wave S. Nicaise, C. Pignotti, Stability and instability results the wave e- quation a delay term in the boundary or internal feedbacks, SIAM Journal on Control and Optimization 45 (5) (2006) 1561õ d wave networks S. Nicaise, J. Valein, the wave equation on 1-d networks a delay term in the nodal feedbacks, Networks and Heterogeneous Media 2 (3) (2007) 425õ479. High-dimensional wave distributed delay S. Nicaise and C. Pignotti, the wave equation boundary or internal distributed delay, Differential and Integral Equations, 21 (2008), d Euler-Bernoulli beam J. Y. Park, Y. H. Kang, J. A. Kim, Existence and exponential stability for a Euler-Bernoulli beam equation memory and boundary output feedback control term, Acta Appl. Math., 104 (2008),

36 Extended results about 1 2-stable rule System Mathematical Model Existing Results Existing and extended results Abstract evolution equation E. M. Ait Benhassi, K. Ammari, S. Boulite, L. Maniar, stabilization a class evolution equations delay, Journal Evolution Equations 9 (2009) 103õ121. Abstract evolution equation S. Nicaise and J. Valein, second order evolution equations unbounded feedback delay, ESAIM Control, Optimization and Calculus Variations, 16, 2010, pp Wave equation interior input delay S. Nicaise and C. Pignotti, Interior feedback stabilization wave e- quations time dependent delay, Electron. J. Diff. Equ., Vol.2011, (2011), No.41, pp

37 Extended results about 1 2-stable rule System Mathematical Model Existing Results Existing and extended results 1-d Timoshenko beam Z. J. Han and G. Q. Xu, Exponential stability Timoshenko beam system delay terms in boundary feedbacks, ESAIM: Control, Optimisation and Calculus Variations, 17 (2010), d Timoshenko beam Z. J. Han, G. Q. Xu, Dynamical behavior networks non-uniform Timoshenko beams system boundary time-delay inputs, Networks and Heterogeneous Media, Vol. 6, 2011, pp d Timoshenko beam B. Said-Houari, Y. Laskri, A stability result a Timoshenko system a delay term in the internal feedback, Applied Mathematics and Computation, Vol.217, 2010, pp d Timoshenko beam M. Kirane, B. Said-Houari, M. N. Anwar, Stability result for the Timoshenko system a time-varying delay term in the internal feedbacks, Commun. Pure Appl. Anal., 10(2), 2011, pp

38 Extended results about 1 2-stable rule System Mathematical Model Existing Results Existing and extended results 1-d Timoshenko beam B. Said-Houari and A. Soufyane, Stability result the Timoshenko system delay and boundary feedback, IMA Journal Mathematical Control and Information, 2012, pp d Euler-Bernoulli beam Y. F. Shang, G. Q. Xu and Y. L. Chen, Stability analysis Euler- Bernoulli beam input delay in the boundary control, Asian Journal Control,14 (2012), Notice This research shows that β > 0 is not necessary.

39 Extended results about 1 2-stable rule System Mathematical Model Existing Results Existing and extended results Extending to case 2: v(t) = αu(t) + βu(t τ) + 0 How to understand these extension results v(t) = αu(t) + βu(t τ) + τ 0 τ g(s)u(t + s)ds g(s)u(t + s)ds If α > β + 0 τ g(s) ds, then under the collocated feedback control law, the closed-loop system is stable. This is a sufficient condition, but not necessary. It also is an extension version 1 2-stable rule. 1-d Euler-Bernoulli beam Y. F. Shang, G. Q. Xu and Y. L. Chen, Stability analysis Euler- Bernoulli beam input delay in the boundary control, Asian Journal Control,14 (2012), 1 11.

40 System Mathematical Model Existing Results Existing and extended results All results are under the collocated feedback control law. Therefore we have following questions: Question 1 In the case v(t) = αu(t) + βu(t τ), can one find out a new feedback control law such that, for any α, β R and α + β 0, the closed-loop system is exponentially stable? Question 2 If the system is a coupled system such as Timoshenko beam, the system has many input delays, can one find out a feedback control law that makes the closed loop system stable? Question 3 In the case distributed delay, i.e., v(t) = αu(t) + βu(t τ) + 0 τ α (s)u(t + s)ds, find out a feedback control law and the least conditions that make the closed loop system stable.

41 System Mathematical Model Existing Results Existing and extended results Question 4 If the system has input delay and output delay, can one find out a feedback control law and the least conditions that make the closed loop system stable? Question 5 In the multi-delays case, v(t) = αu(t)+βu(t τ 1 )+γu(t τ 2 ), can one find out a new feedback control law such that, for any α, β, γ R and α + β + γ 0, the closed-loop system is exponentially stable? Question 6 If above problems can be done, can one extend such control design to high-dimensional system? Notice There is a connection between Question 4 and Question 5.

42 Question 1 System Question 1 Question 2 Question 3 Euler-Bernoulli beam Ying Feng Shang, Gen Qi Xu, an EulerõBernoulli beam input delay in the boundary control, System & Control Letter,Vol.61, 2012, pp Euler-Bernoulli beam Z. J. Han, G. Q. Xu, Output-based stabilization Euler- Bernoulli beam time-delay in boundary input, IMA Journal Mathematical Control and Information, (2013), doi: /imamci/dnt030 1-D Wave equation Han Wang, Gen Qi Xu, Exponential stabilization 1-d wave equation input delay, WSEAS Transactions on Mathematics, 12(10), 2013, pp

43 Question 2 System Question 1 Question 2 Question 3 Timoshenko beam, Hongxia Wang, Timoshenko beam system delay in the boundary control, International Journal Control, 2013, INT. J. Control, 86, (2013), p- p Remarks: Note that the order the system is, Euler-Bernoulli beam, 1-d wave equation, Timoskenko beam. What is the difference among them? 1) Euler-Bernoulli beam has larger spectral gap; 2) 1-d Wave equation has a constant spectral gap; 3) Timoshenko beam has coupled spectrum and smaller spectral gap, or has eigenvalue multiplicity 2; The main difficulty appears in the pro stability closed loop system. The detail please see the applicable method below.

44 Question 3 System Question 1 Question 2 Question 3 Euler-Bernoulli beam Ying Feng Shang, Gen Qi Xu, Dynamic feedback control and exponential stabilization a compound system, Journal Mathematical Analysis and Applications, 422, (2015), 858õ879 Timoshenko beam X. F. Liu and G. Q. Xu, Exponential stabilization for Timoshenko beam distributed delay in the boundary control, Abstract and Applied Analysis, (2013), doi: /2013/ Timoshenko beam X. F. Liu and G. Q. Xu, Output-based stabilization Timoshenko beam the boundary control and input distributed delay, Under Review

45 Idea to deal the input delay System Idea Realization Transform realization Generation control signal Comparing Test exponential Step 1: By a transform T, we translate the delay system into a system out delay. Step 2: Control signal generating For the system out delay, we adopt the collocated feedback approach to obtain a control signal. Step 3: stability analysis the closed loop system For the system out delay, under the collocated feedback control law, we prove exponential stability the closed loop system Step 4: Comparing Applying the control signal to original system, we prove this system also is exponentially stable.

46 Transform realization System Idea Realization Transform realization Generation control signal Comparing Test exponential Purpose: Find a transform T that translate the delay system into a system out delay Smith predictor { ẋ(s, t) = Ax(s, t) + Bu(t + s) x(0, t) = x(t) it fits the pure delay problem ẋ(t) = Ax(t) + Bu(t τ), x(0) = x 0. But it does not fit the problem the form ẋ(t) = Ax(t) + B 0 u(t) + B 1 u(t τ)

47 Transform realization System Idea Realization Transform realization Generation control signal Comparing Test exponential Artstein Transform: Artstein Transform z(t) = x(t) + translates the delayed equation into a delay-independent system t t τ e A(t s τ) B 1u(s)ds ẋ(t) = Ax(t) + B 0u(t) + B 1u(t τ) ż(t) = Az(t) + (B 0 + e Aτ B 1)u(t) However, in the infinite-dimensional system, e Aτ might be a unbounded operator. Reference Z. Artstein, Linear ed Controls: A Reduction, IEEE Transaction on Automatic Control, Vol. AC-278, No.4, 1982,pp

48 Transform realization System Idea Realization Transform realization Generation control signal Comparing Test exponential Partial state predictor Purpose: Transform the delay system into a system out delay. Partial state predictor Consider the delayed equation ẋ(t) = Ax(t) + B 0u(t) + B 1u(t τ). We introduce the partial state predictor: { ẋ(s, t) = Ax(s, t) + B1u(t τ + s), s (0, τ) x(0, t) = x(t) Set t p(t) = e Aτ x(t) + e A(t s) B 1u(s)ds. t τ Then we have a delay-independent system ṗ(t) = Ap(t) + (e Aτ B 0 + B 1)u(t)

49 Transform realization System Idea Realization Transform realization Generation control signal Comparing Test exponential Extending to more general case: 0 ẋ(t) = Ax(t) + B 0 u(t) + B 1 u(t τ) + G(s)u(t + s)ds τ where G(s) is operator-valued function defined on [0, τ]. Partial state predictor We introduce the partial state predictor: { [ ẋ(s, t) = Ax(s, t) + B 1u(t + s τ) + ] t G(t + s r)u(r)dr, s (0, t+s τ x(0, t) = x(t) whose solution is x(s, t) = e As x(t) + + s 0 s 0 e A(s ν) B 1u(t + ν τ)dν τ e A(s ν) dν G(r)u(t + ν r)dr ν

50 Transform realization System Idea Realization Transform realization Generation control signal Comparing Test exponential Set Transform t [ τ p(t) = x(τ, t) = e Aτ x(t)+ e A(τ s) B 1 + e A(t s r+τ) G(r) t τ t s Transform realization Under this transform, the equation ẋ(t) = Ax(t) + B 0u(t) + B 1u(t τ) + is translated into a delay-independent system ṗ(t) = Ap(t) + [e Aτ B 0 + B 1 + τ 0 0 τ G(s)u(t + s)ds e A(τ s) G(s)ds]u(t)

51 Transform realization System Idea Realization Transform realization Generation control signal Comparing Test exponential Case study we consider the following compounded dynamical system z t (s, t) = z s (s, t), s ( τ, 0), t > 0 w tt (x, t) + w xxxx (x, t) = 0, x (0, 1), t > 0, w xxx (1, t) = αz(0, t) + βz( τ, t) + 0 τ g(s)z(s, t)ds, z(0, t) = u(t), t 0 w(0, t) = w x (0, t) = w xx (1, t) = 0, t > 0 z(s, 0) = f(s), s ( τ, 0) w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x), x (0, 1) (13) where g L 2 [ τ, 0] g( ) L 2 0, α, β are real constants, and u(t) is the external input.

52 Transform realization System Idea Realization Transform realization Generation control signal Comparing Test exponential Case study By solving the first equation in (13), we know that the system (13) is equivalent to the following: w tt (x, t) + w xxxx (x, t) = 0, x (0, 1), t > 0, w xxx (1, t) = αu(t) + βu(t τ) + 0 τ g(η)u(t + η)dη, w(0, t) = w x (0, t) = w xx (1, t) = 0, t > 0 u(s) = f(s), s ( τ, 0) w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x), x (0, 1). (14) Obviously, this is a system Euler-Bernoulli beam distributed delay in input. Assumption Suppose that the state the system is measurable, that is, (w(x, t), w t (x, t)) can be measured.

53 Transform realization System Partial state predictor design ŵ ss (x, s, t) + ŵ xxxx (x, s, t) = 0, 0 x 1, 0 < s τ, ŵ(0, s, t) = ŵ x (0, s, t) = ŵ xx (1, s, t) = 0, ŵ xxx (1, s, t) = βu(t + s τ) + s τ g(η)u(t + s + η)dη ŵ(x, 0, t) = w(x, t), ŵ s (x, 0, t) = w t (x, t) (15) Idea Realization Transform realization Generation control signal Comparing Test exponential Notice The initial datum are the state original system, the control is only the delay part. Please note that in this auxiliary system we do not use the control information after t.

54 Transform realization System Idea Realization Transform realization Generation control signal Comparing Test exponential Transform Case study T (w) = p(x, t) = w(x, τ, t), T (w t) = q(x, t) = w s(x, τ, t) Using (14), we derived the following system: p t(x, t) = q(x, t) a(x)u(t), 0 < x < 1, t > 0, q t(x, t) + p xxxx(x, t) = b(x)u(t), p(0, t) = p x(0, t) = p xx(1, t) = 0, p xxx(1, t) = βu(t), p(x, 0) = E 0(w 0, w 1)(x) + 0 a0(x, r)f(r)dr, τ q(x, 0) = E 1(w 0, w 1)(x) 0 a1(x, r)f(r)dr. τ (16) where a 0(x, s), a 1(x, s), a(x) and b(x) are measurable functions and E 0 and E 1 are bounded linear operators on H 2 [0, 1] L 2 [0, 1]. Remark: (16) is a system the control acting internal and boundary at same time. But it has no time delay.

55 Generation control signal System Idea Realization Transform realization Generation control signal Comparing Test exponential Generating control signal Purpose: Generating control signal by control design Collocated feedback control design To obtain a control signal, we consider the energy functional system (16): E(t) = 1 1 [ p 2 xx (x, t) + q 2 (x, t) ] dx 2 Direct calculation gives de(t) dt = u(t) 0 [ βq(1, t) q(x, t)b(x)dx + For (16) we adopt the feedback control law u(t) = U(p, q) = βq(1, t) + 1 This is the control signal used later. 0 q(x, t)b(x)dx ] p xx(x, t)a (x)dx p xx(x, t)a (x)dx. (17)

56 Comparing System Idea Realization Transform realization Generation control signal Comparing Test exponential Closed loop system to (16) p t(x, t) = q(x, t) a(x)u(p, q), 0 < x < 1, t > 0, q t(x, t) + p xxxx(x, t) = b(x)u(p, q), p(0, t) = p x(0, t) = p xx(1, t) = 0, p xxx(1, t) = βu(p, q), p(x, 0) = E 0(w 0, w 1)(x) + 0 a0(x, s)f(s)ds, τ q(x, 0) = E 1(w 0, w 1)(x) 0 a1(x, s)f(s)ds. τ Original system (17) w t(x, t) = w t(x, t), 0 < x < 1, t > 0, w tt(x, t) + w xxxx(x, t) = 0, w(0, t) = w x(0, t) = w xx(1, t) = 0, w xxx(1, t) = αu(p, q)(t) + βu(p, q)(t τ) + 0 g(s)u(p, q)(t + s)ds, τ w(x, 0) = w 0(x), w t(x, 0) = w 1(x). (18) (19)

57 Relation between both systems System Idea Realization Transform realization Generation control signal Comparing Test exponential Error e(x, t) = w(x, t + τ) p(x, t), η(x, t) = w t(x, t + τ) q(x, t) Error estimate Stability result p(, t) w(, t + τ) H 2 E (0,1) + q(, t) w t(, t + τ) L 2 (0,1) 4α 2 M 2 1 [E(t) E(t + τ)] 0 +4(M2 2 + M3 2 )τ 2 g(r) 2 dr[e(t τ) E(t + τ)]. τ If the closed loop system (18) is exponentially stable, then the system (19) also is exponentially stable. If the closed loop system (18) is asymptotically stable, then the system (19) also is asymptotically stable.

58 Test exponential stability System Idea Realization Transform realization Generation control signal Comparing Test exponential Difficulty in Stability Test Multiplier method fails to apply to the closed loop system! Difficulty in Stability Test The spectral method fails to apply to the closed loop system! Feasible method The duality method and system theory can be applied to the closed loop system! Basic relation If the system is exactly observable, then the dual system is exactly controllable. Hence the collocated feedback system is exponentially stable.

59 Dual system (16) System The observation system corresponding to (16) is w t (x, t) = v(x, t), 0 < x < 1, t > 0, v t (x, t) = w xxxx (x, t), w(0, t) = w x (0, t) = w xx (1, t) = w xxx (1, t) = 0, w(x, 0) = w 0 (x), v(x, 0) = v 0 (x), y(t) = βv(1, t) w xx(x, t)a (x)dx v(x, t)b(x)dx. Lemma Let the differential operator in L 2 [0, 1] be defined by (20 Idea Realization Transform realization Generation control signal Comparing Test exponential Lz(x) = z (4) (x), D(L) = {z(x) H 4 (0, 1) z(0) = z (0) = 0 z (1) = z (1) = 0 } (21) then eigenvalues L are 0 < µ 1 < µ 2 < < µ n < (22) and corresponding eigenfunctions ϕ n(x) are real functions and form a normalized orthogonal basis for L 2 [0, 1].

60 Test Observability System Idea Realization Transform realization Generation control signal Comparing Test exponential The spectral method for Exact observability Let H be a Hilbert space, and let A 0 be a skew-adjoint operator compact resolvent in H, i.e., A 0 = A 0. Let σ(a 0 ) = {λ k ; k = ±1, ±2, } and {Φ k } k= be the corresponding eigenvectors. Let Y be another Hilbert space, and let C L(D(A 0 ), Y). If A 0 and C satisfy the following conditions: (1) Spectral gap condition: (2) Boundedness condition: inf λ k λ m > 0, k m 0 < m = inf CΦ k Y < sup CΦ k Y = M < k Z k Z then C is an admissible observable operator for A 0, and (A 0, C) is exactly observable in finite time.

61 Main result System Idea Realization Transform realization Generation control signal Comparing Test exponential Theorem Let {µ n, n N} be given as in Lemma 1, and ξ n = τ 0 e i µ nη g(η τ)dη, n N. Then the following assertions hold: 1) If α, β R satisfy the condition that inf n β + αe i µ nτ + ξ n > 0, then the system (18) decays exponentially. 2) If α, β R satisfy condition β + αe i µ nτ + ξ n = 0 for some n N, the system (18) is unstable. 3) If α = β = 0 and ξ n 0 for all n N, the system (18) is asymptotically stable. If for some n, ξ n = 0, then the system is unstable. Remark The condition β + αe i µ nτ + ξ n 0 cannot be improved. This is because if the equality holds, then the control the from e i µ nt is invalid.

62 Thank you! System Idea Realization Transform realization Generation control signal Comparing Test exponential Author: Address: Gen-Qi Xu Department Mathematics Tianjin University Tianjin, , China