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 2017 1 / 35
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 2017 2 / 35
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 2017 3 / 35
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 2017 4 / 35
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 2017 5 / 35
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 2017 6 / 35
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 2017 7 / 35
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 2017 8 / 35
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 2017 9 / 35
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 2017 10 / 35
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 2017 11 / 35
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 2017 12 / 35
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 2017 13 / 35
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 2017 14 / 35
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 2017 15 / 35
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 2017 16 / 35
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 = 1 0 0 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 2017 17 / 35
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 2 1 0 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 2017 18 / 35
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 2 1 0 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 2017 19 / 35 )
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 2017 20 / 35
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 2017 21 / 35
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 2017 22 / 35
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 2017 23 / 35
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 2017 24 / 35
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 2017 25 / 35
Application for Saint Venant model Cascade network of n Saint-Venant hydraulic systems Cascade network ( ) Hi t Q i + 0 1 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 2017 26 / 35
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 2017 27 / 35
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 2017 28 / 35
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 2017 29 / 35
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 = 0.01. Simulations for the output regulation Figure: Output measurements y i (t) Ngoc-Tu TRINH Groupe de travail PIC 24 Mars 2017 30 / 35
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 2017 31 / 35
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 2017 32 / 35
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 2017 33 / 35
Conclusions References G. Bastin and J.-M. Coron, Stability and boundary stabilization of 1-D hyperbolic systems, PNLDE (88), Birkhauser, 2016. 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. 177-187, 2009. 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. 1460-1498, 2008. 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. 1310 1318, 2008. J. de Halleux, C. Prieur, J.M. Coron and G. Bastin, Boundary control in networks of open channels, Automatica 39, pp. 1365-1376, 2003. J.K Hale and S.M Verduyn Lunel. Introduction to functional differential equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. 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. 5654-5659, Las Vegas, USA, 2016. 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), 2017. Ngoc-Tu TRINH Groupe de travail PIC 24 Mars 2017 34 / 35
Conclusions THANK YOU FOR YOUR ATTENTION! Ngoc-Tu TRINH Groupe de travail PIC 24 Mars 2017 35 / 35