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 Introduction 1 Statement of the problem and main result 2 Lyapunov techniques and the proof of the main result 3 Application for Saint Venant model 4 Conclusions 5 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 ˙ z = 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 ) , ˙ z = 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 Introduction 1 Statement of the problem and main result 2 Lyapunov techniques and the proof of the main result 3 Application for Saint Venant model 4 Conclusions 5 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 ∂ t φ i 1 ( x , t ) + λ i 1 ∂ x φ i 1 ( x , t ) = 0 , x ∈ [0 , L ] , t ∈ [0 , ∞ ) , i = 1 , n ∂ t φ i 2 ( x , t ) − λ i 2 ∂ x φ i 2 ( x , t ) = 0 where two states φ i 1 , φ i 2 : [0 , L ] × [0 , ∞ ) → R and λ i 1 > 0, λ i 2 > 0. Boundary conditions defined at junctions � φ i 2 ( L , t ) = R i 2 φ i 1 ( L , t ) + u i ( t ) , i = 1 , n φ i 1 (0 , t ) = R i 1 φ i 2 (0 , t ) + α i φ ( i − 1)1 ( L , t ) + δ i φ ( i − 1)2 ( L , t ) , where φ 01 = φ 02 = 0. n measured outputs y i ( t ) = a i φ i 1 ( L , t ) + b i φ i 2 ( 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 u i ( t ) = K iP ( y i ( t ) − y ir ) + K iI z i ( t ) , ˙ z i = y i ( t ) − y ir K iP ∈ R and K iI ∈ R to be designed. Consider the state space of closed-loop network : ( L 2 (0 , L )) 2 × R � n � E = with the norm associated n � � || Y || 2 � || φ i 1 ( ., t ) || 2 L 2 (0 , L ) + || φ i 2 ( ., t ) || 2 L 2 (0 , L ) + z 2 E = i ( t ) i =1 where Y = ( φ 11 , φ 12 , z 1 , · · · , φ n 1 , φ n 2 , 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 i 2 � = 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 iI = − µ ( b i + a i R i 1 e µ L )( a i + b i R i 2 ) K iP = − R i 2 , ∀ 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 . ( H 1 (0 , L )) 2 × R � n , Output regulation, i.e � With initial conditions in t →∞ | y i ( t ) − y ir | = 0 , ∀ i = 1 , n . lim 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 i 1 ( L , t ) + b i φ i 2 ( L , t ) + y ir , K iI = − µ ( b i + a i R i 1 e µ L )( a i + b i R i 2 ) K iP = − R i 2 , ∀ i = 1 , n a i a i 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 i 2 � = 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 Introduction 1 Statement of the problem and main result 2 Lyapunov techniques and the proof of the main result 3 Application for Saint Venant model 4 Conclusions 5 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. n � V ( φ 11 , φ 12 , z 1 , · · · , φ n 1 , φ n 2 , z n ) = p i V i i =1 where T φ i 1 e − µ x φ i 1 e − µ x � L 2 2 µ x µ x V i ( φ i 1 , φ i 2 , z i ) = P i dx φ i 2 e φ i 2 e 2 2 0 z i z i with 1 0 q i 3 P i = 0 q i 1 q i 4 q i 3 q i 4 q i 2 Here p i > 0 and q i 1 , q i 2 , q i 3 , q i 4 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 T φ i 1 e − µ x φ i 1 e − µ x � L 1 0 q i 3 n 2 2 � µ x µ x V = p i 0 q i 1 q i 4 dx φ i 2 e φ i 2 e 2 2 0 q i 3 q i 4 q i 2 i =1 z i z i If q i 2 = q i 3 = q i 4 = 0, this is the Lyapunov functionnal of Bastin, Coron and Andr´ ea Novel 2009 for a cascaded network. If n = 1 and q i 3 = q i 4 = 0, this is the Lyapunov functionnal of Bastin and Coron 2016 for a single system. By adding the new terms ( q i 3 , q i 4 � = 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
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