Stability and output regulation for a cascaded network of 2 2 hyperbolic systems with PI control

Transcription

1 Stability and output regulation for a cascaded network of 2 2 hyperbolic systems with PI control Ngoc-Tu TRINH, Vincent ANDRIEU and Cheng-Zhong XU Laboratory LAGEP, Batiment CPE, University of Claude Bernard Lyon 1, 43 Boulevard du 11 novembre 1918, F-69622, Villeurbanne Cedex, France 24 Mars 2017 Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

2 Plan 1 Introduction 2 Statement of the problem and main result 3 Lyapunov techniques and the proof of the main result 4 Application for Saint Venant model 5 Conclusions Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

3 Introduction 1 Introduction PDE hyperbolic systems and cascaded networks Boundary control problem Output regulation problem PI control design Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

4 Introduction PDE hyperbolic systems and cascaded networks Engineering applications of PDE hyperbolic systems Hydraulic engineering - Saint Venant models Road traffic - Burgers equation Gas pipeline Heat exchanger process Homogeneous first-order hyperbolic systems Let φ R n, A(φ) R n n, x [0, L],t R +, φ t + A(φ) φ x = 0, φ(0, x) = φ 0 (x) A has n real eigenvalues, i.e λ i R i = 1, 2,..n. If A is independent on φ, system is linear. If not, it is quasi-linear. Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

5 Introduction PDE hyperbolic systems and cascaded networks Cascaded network Popular in practical applications (channels of rivers, gas, ) n PDE hyperbolic sub-systems n + 1 junctions, 2 free junctions and n 1 mixed junctions. Figure: Cascaded network of n systems A cascaded network can be considered a large PDE hyperbolic system with complex boundary conditions! Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

6 Introduction Boundary control problem Boundary conditions ( ) f φ(0, t), φ(l, t), U(t) = 0 U(t) is control action on the boundary. Static control, i.e U(t) = g(φ(0, t), φ(l, t)). Dynamic control, i.e U(t) = g(φ(0, t), φ(l, t)) + other dynamic parts. Boundary control problem Find boundary conditions such that : The PDE hyperbolic system has a unique solution in the corresponding state space. The PDE hyperbolic system is (globally/locally) asymptotically/exponentially stable w.r.t some equilibrium point. Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

7 Introduction Boundary control problem Static control laws Literatures : (Li Tatsien 1994, Coron et al. 2015) A sufficient boundary condition for the zero-point stability of quasi-linear systems in C 1 norm. (Coron et al. 2008) A sufficient boundary condition for the zero-point stability of quasi-linear systems in H 2 norm. (Hale and Verduyn Lune 1993) A necessary and sufficient boundary condition for the zero-point stability of linear systems in L 2 norm. Limits : Not robust with constant perturbations. Dynamic control laws with integral actions Literatures with works of Pohjolainen, Xu, Dos Santos, C. Prieur, D. Georges,... Advantages : Robust to constant perturbations. Limits : Become a coupling systems of PDE and ODE, difficult to prove stability. Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

8 Introduction Output regulation problem Given a system one wants to ensure that outputs y(t) follow references y r despite disturbances, i.e y(t) y r Figure: Example of Disturbances Figure: Static error Disturbances in real model : error of the modelisation, linearisation, sensors, Static error between the measurement output and the set-point. Solution : using the integral action to eliminate the static error. Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

9 Introduction Output regulation problem Example : A very trivial system : φ = u + d y = φ State φ R, control u R, unknown constant disturbance d R, measure y R. Objective : Given a reference y r in R, design u such that y y r. If u = (y y r ) equilibrium is stable but y y r. If u = (y y r ) z, where ż = y y r equilibrium is stable and y y r. Conclusion : The integral term added rejects the constant disturbance. Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

10 Introduction PI control design PI controller is a type of dynamic boundary control law : u(t) = K P (y(t) y r ) + K I z(t), ż = y(t) y r Measured output on the boundary y(t) = g(φ(0, t), φ(l, t)) Input u(t), reference y r Gain parameter matrices K p, K I. Schema of closed-loop system : Objective : Design PI controller (determine K P and K I ) such that : Stability of closed-loop system Output regulation : y(t) y r Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

11 Statement of the problem and main result Plan 1 Introduction 2 Statement of the problem and main result 3 Lyapunov techniques and the proof of the main result 4 Application for Saint Venant model 5 Conclusions Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

12 Statement of the problem and main result Network model n PDE hyperbolic systems tφ i1 (x, t) + λ i1 x φ i1 (x, t) = 0 tφ i2 (x, t) λ i2 x φ i2 (x, t) = 0, x [0, L], t [0, ), i = 1, n where two states φ i1, φ i2 : [0, L] [0, ) R and λ i1 > 0, λ i2 > 0. Boundary conditions defined at junctions { φi2 (L, t) = R i2 φ i1 (L, t) + u i (t) φ i1 (0, t) = R i1 φ i2 (0, t) + α i φ (i 1)1 (L, t) + δ i φ (i 1)2 (L, t),, i = 1, n where φ 01 = φ 02 = 0. n measured outputs y i (t) = a i φ i1 (L, t) + b i φ i2 (L, t) + y ir Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

13 Statement of the problem and main result PI structure and state space Design n PI controllers at each juctions K ip R and K ii R to be designed. u i (t) = K ip (y i (t) y ir ) + K ii z i (t), Consider the state space of closed-loop network : with the norm associated Y 2 E = E = ( (L 2 (0, L)) 2 R ) n z i = y i (t) y ir n ) ( φ i1 (., t) 2 L 2 (0,L) + φ i2(., t) 2 L 2 (0,L) + z2 i (t) i=1 where Y = (φ 11, φ 12, z 1,, φ n1, φ n2, z n) E Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

14 Statement of the problem and main result Main result Two hypothesises H 1 : a i 0 i = 1, n H 2 : a i + b i R i2 0 i = 1, n Theorem (Trinh-Andrieu-Xu 2017) There exists µ > 0 such that, if two hypothesises H 1 and H 2 are satisfied, for each µ (0, µ ) and K ip = R i2, K ii = µ (b i + a i R i1 e µl )(a i + b i R i2 ), i = 1, n a i a i Then, we have : Existence and uniqueness of solutions in E The exponential stability of zero point in E. With initial conditions in ( (H 1 (0, L)) 2 R ) n, Output regulation, i.e lim t y i (t) y ir = 0, i = 1, n. Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

15 Statement of the problem and main result About the theorem y i (t) = a i φ i1 (L, t) + b i φ i2 (L, t) + y ir K ip = R i2 a i, K ii = µ (b i + a i R i1 e µl )(a i + b i R i2 ) a i, i = 1, n Two output conditions (two hypothesises) for our PI control design : H 1 for existence of our PI controller. a i 0 i = 1, n H 2 for having dynamic feedback (by integral action), i.e K ii 0. a i + b i R i2 0 i = 1, n Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

16 Lyapunov techniques and the proof of the main result 1 Introduction 2 Statement of the problem and main result 3 Lyapunov techniques and the proof of the main result 4 Application for Saint Venant model 5 Conclusions Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

17 Lyapunov techniques and the proof of the main result Lyapunov candidate functional Use Lyapunov techniques construct a candidate Lyapunov function. where with V(φ 11, φ 12, z 1,, φ n1, φ n2, z n) = n p i V i i=1 L φ i1 e µx T 2 V i (φ i1, φ i2, z i ) = φ i2 e µx φ i1 e µx 2 2 P i φ i2 e µx 2 dx 0 z i z i P i = q i1 q i3 q i4 q i3 q i4 q i2 Here p i > 0 and q i1, q i2, q i3, q i4 need to be designed. Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

18 Lyapunov techniques and the proof of the main result Lyapunov candidate functional n L φ i1 e µx T 2 V = p i φ i2 e µx q i3 0 q i1 q i4 φ i1 e µx 2 φ i2 e µx 2 dx i=1 0 z i q i3 q i4 q i2 z i If q i2 = q i3 = q i4 = 0, this is the Lyapunov functionnal of Bastin, Coron and Andréa Novel 2009 for a cascaded network. If n = 1 and q i3 = q i4 = 0, this is the Lyapunov functionnal of Bastin and Coron 2016 for a single system. By adding the new terms (q i3, q i4 0) and n positive parameters p i, it allows to deal with dynamic feedback of cascaded network of n systems. Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

19 Lyapunov techniques and the proof of the main result Design of Lyapunov functional L φ i1 e µx T 2 V i (φ i1, φ i2, z i ) = φ i2 e µx q i3 0 q i1 q i4 φ i1 e µx 2 φ i2 e µx 2 dx 0 z i q i3 q i4 q i2 z i Lemma (For sub-functional V i ) Let q i1, q i2, q i3, q i4 be defined as follows : q i1 > 3λ i1ri1 2, q i2 = µe µl λ i2 q i1, q i3 = µe 3µL 2 λ i2 a i λ i2 q i1 λ i1, q i4 = µe 3µL 2 a i R i1 q i1. Then there exists µ > 0, M i > 0 and γ i > 0 such that for all µ (0, µ ) 1 1 V i (φ i1, φ i2, z i ) φ i1 (., t) 2 M L 2 (0,L) + φ i2(., t) 2 L 2 (0,L) + z2 i (t) M i V i (φ i1, φ i2, z i ) i 2 V i (t) γ i V i (t) F i (t) + G i 1, where F i (t) = 1 4 z2 i (t) k2 i λ i2q i1 e µl + φ 2 i1 (L, t) λ i1e µl G i 1 = φ 2 (i 1)1 (L, t)λ i1α 2 i ( 3 + 4λ2 i1 q2 i3 e µl k 2 i λ i2q i1 ) 2, + z 2 i 1 (t)λ i1β 2 i ( 3 + 4λ2 i1 q2 i3 e µl k 2 i λ i2q i1 Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35 )

20 Lyapunov techniques and the proof of the main result Sketch of proof V i is definite positive L φ i1 e µx T 2 V i (φ i1, φ i2, z i ) = φ i2 e µx φ i1 e µx 2 2 P i φ i2 e µx 2 dx 0 z i z i With µ small enough, prove that P i is symmetric positive definite (SDP) Consider V i φ i1 (x, t)e µx T 2 φ i1 (x, t)e µx 2 L V i = φ i2 (x, t)e µx 2 0 z i (t) Q i φ i2 (x, t)e µx 2 z i (t) dx F i (t) + G i 1 φ i1 (L, t) φ i1 (L, t) With µ small enough, prove that Q i R 4 4 is SDP t R +, γ i > 0, V i (t) γ i V i (t) F i (t) + G i 1. Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

21 Lyapunov techniques and the proof of the main result Design of Lyapunov functional Lemma (For global functional V) V(φ 11, φ 12, z 1,, φ n1, φ n2, z n) = n p i V i Let q i1, q i2, q i3, q i4 be defined in Lemma of sub functional V i, and p i be defined as follows p 1 > 0, p i+1 = ɛp i Then there exists ɛ > 0 and µ > 0 such that for every µ (0, µ ), we have : 1 There exists M > 0 such that 1 M V(φ 11, φ 12, z 1,, φ n1, φ n2, z n) (φ 11, φ 12, z 1,, φ n1, φ n2, z n) 2 E i=1 M V(φ 11, φ 12, z 1,, φ n1, φ n2, z n). 2 There exists γ > 0 such that V(t) γv(t). Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

22 Lyapunov techniques and the proof of the main result Sketch of proof V is definite positive n i=1 V(φ 11, φ 12, z 1,, φ n1, φ n2, z n) = n p i V i i=1 p i M i V i (φ 11, φ 12, z 1,, φ n1, φ n2, z n) 2 E n p i M i V i Employing the definite positive property of Lemma for sub functional V i, one finds the proof. i=1 V is definite negative V(t) n p i γ i V i (t) i=1 n zi 2 (t) (p i A i p i+1 B i ) i=1 n φ 2 i1 (L, t) (p i C i p i+1 D i ) i=1 With p i+1 = ɛp i, choosing ɛ enough small, we have t R +, γ > 0, V(t) γv(t). Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

23 Lyapunov techniques and the proof of the main result Proof of Theorem Unique solution and zero stability (Existence and uniqueness of solutions) Choosing initial condition ( φ 0 11 (x), φ 0 12 (x), z0 1,, φ0 n1 (x), φ0 n2 (x), ) z0 n E, x [0, L] Closed-loop system with PI controller has a unique solution in E ( using idea in [Coron and Bastin 2008]). (Exponential stability of zero point in E) Directly deduced from the Lemma for global functional V. Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

24 Lyapunov techniques and the proof of the main result Proof of Theorem Output regulation With initial condition ( φ 0 11 (x), φ 0 12 (x), z0 1,, φ0 n1 (x), φ0 n2 (x), ) ( z0 n (H 1 (0, L)) 2 R ) n, x [0, L] and the exponential stability of zero in E With zero stability in ( (H 1 (0, L)) 2 R ) n by closed graph theorem lim φ i1 t H 1 (0,L) = 0, and Sobolev embedding theorem, we have lim φ i1(x, t) = 0, t lim φ i2 t H 1 (0,L) = 0 lim φ i2(x, t) = 0 x [0, L] t Therefore, lim y i (t) y ir = 0 t Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

25 Application for Saint Venant model Plan 1 Introduction 2 Statement of the problem and main result 3 Lyapunov techniques and the proof of the main result 4 Application for Saint Venant model 5 Conclusions Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

26 Application for Saint Venant model Cascade network of n Saint-Venant hydraulic systems Cascade network ( ) Hi t Q i B i Q 2 i B i H i + gb i 2Q i B i H i x y i (t) = H i (L, t) (output measurement) ( Hi Q i ) = 0, Boundary conditions : Q 2 i (L, t) = α i ( ) H i (L, t) U i (t) i = 1, n and Q 1 (0, t) = Q 0 (constant) Conservation law of discharges : Q j (L, t) = Q j+1 (0, t), j = 1, n 1 Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

27 Application for Saint Venant model Linearized model Linearized network with h i = H i Hi, q i = Q i Q 0 ( ) hi 0 + t q i Q0 2 ) 2 + gb i Hi B i (H i 1 B i 2Q 0 B i H i x ( hi q i ) = 0 y i (t) = h i (L, t) + H i Boundary conditions : ) 2Q 0 q i (L, t) = α i (h i (L, t) u i (t) i = 1, n Conservation law of discharges : q 1 (0, t) = 0 q j (L, t) = q j+1 (0, t), j = 1, n 1 Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

28 Application for Saint Venant model Network in characteristic form Using the change of coordinates h i = φ i1 + φ i2, q i = (B i ghi + Q 0 Hi )φ i1 (B i ghi Q 0 Hi )φ i2 Network in new coordinates tφ i1 (x, t) + λ i1 x φ i1 (x, t) = 0 tφ i2 (x, t) λ i2 x φ i2 (x, t) = 0 y i (t) = φ i1 (L, t) + φ i2 (L, t) + Hi, where λ i1 = ghi + Q 0 B i Hi > 0, λ i2 = ghi Q 0 B i Hi Boundary conditions at junctions > 0. φ i2 (L, t) = R i2 φ i1 (L, t) + u i (t) φ i1 (0, t) = R i1 φ i2 (0, t) + α i φ (i 1)1 (L, t) + δ i φ (i 1)2 (L, t), Here R i1, R i2,α i, δ j are constants Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

29 Application for Saint Venant model Application of PI control design PI controller design where t u i (t) = K ip (y i (t) Hi ) + K ii (y i (s) Hi )ds 0 K ip = 2Q 0 (B i gh i + Q 0 Hi ) + α i 2Q 0 (B i gh i Q 0 Hi ) + α i gh i + Q 0 K ii = µ (1 + e µl B i Hi gh i Q 0 B i Hi ) 4Q 0 (B i gh i 2Q 0 (B i gh i Q 0 Hi ) + α i, i = 1, n µ is tuning parameter chosen small enough. Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

30 Application for Saint Venant model Numerical simulations Numerical application for 3 channels (n=3), Length L = 100 m, base width B = 4 m. Set-points H 1 = 10 m, H 2 = 8 m, H 3 = 6.5 m, constant discharge Q0 = 7 m3 /s. Output disturbances w 1o = 0.1, w 2o = 0.2, w 2o = 0.15 ; and control disturbances w 1c = 0.02, w 2c = 0.03, w 2c = Simulations for the output regulation Figure: Output measurements y i (t) Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

31 Application for Saint Venant model Numerical simulations Simulations for the stability Figure: H 1(x, t) Figure: H 2(x, t) Figure: H 3(x, t) Figure: Q 1(x, t) Figure: Q 2(x, t) Figure: Q 3(x, t) Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

32 Conclusions Plan 1 Introduction 2 Statement of the problem and main result 3 Lyapunov techniques and the proof of the main result 4 Application for Saint Venant model 5 Conclusions Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

33 Conclusions Conclusions Obtained results Study a network class of n linear 2 2 hyperbolic systems. Design n boundary PI controllers at each junction. Prove the stability of the closed-loop system in L 2 norm and output regulation based on Lyapunov direct method. Apply the control design for a practical network of n fluid flow Saint Venant systems. Perspectives Extend the PI control design for networks of 2 2 nonlinear hyperbolic PDE systems. Study the problem of optimal PI controllers (eg. the optimal value of µ). Submitted to Automatica Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

34 Conclusions References G. Bastin and J.-M. Coron, Stability and boundary stabilization of 1-D hyperbolic systems, PNLDE (88), Birkhauser, G. Bastin, J.-M. Coron and B. d Andrea Novel, On Lyapunov stability of linearised Saint-Venant equations for a sloping channel, Networks and heterogeneous media, vol.4, pp , J.-M. Coron, G. Bastin and B. d Andrea Novel, Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems, SIAM Journal on Control and Optimization 47, No.3, pp , V. Dos Santos, G. Bastin, J.-M. Coron and B. d Andréa Novel, Boundary control with integral action for hyperbolic systems of conservation laws : stability and experiments, Automatica, IFAC 44(5), pp , J. de Halleux, C. Prieur, J.M. Coron and G. Bastin, Boundary control in networks of open channels, Automatica 39, pp , J.K Hale and S.M Verduyn Lunel. Introduction to functional differential equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, N-T. Trinh, V. Andrieu and C-Z. Xu. Multivariable PI controller design for 2 2 systems governed by hyperbolic partial differential equations with Lyapunov techniques, Proceeding of 55th IEEE Conference on Decision and Control, pp , Las Vegas, USA, N-T. Trinh, V. Andrieu and C-Z. Xu. Design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations, Appear in IEEE Transaction on Automatic Control (Full paper), Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35

35 Conclusions THANK YOU FOR YOUR ATTENTION! Ngoc-Tu TRINH Groupe de travail PIC 24 Mars / 35