Control of 2 2 Linear Hyperbolic Systems: Backstepping-Based Trajectory Generation and PI-Based Tracking

Transcription

1 Control of Linear Hyperbolic Systems: Backstepping-Based Trajectory Generation and PI-Based Tracking Pierre-Olivier Lamare, Nikolaos Bekiaris-Liberis, and Aleandre M. Bayen Abstract We consider the problems of trajectory generation and tracking for general systems of first-order linear hyperbolic PDEs anti-collocated boundary input and output. We solve the trajectory generation problem via backstepping. The reference input, which generates the desired output, incorporates integral operators acting on advanced and delayed versions of the reference output kernels which were derived by Vazquez, Krstic, and Coron for the backstepping stabilization of linear hyperbolic systems. For tracking the desired trajectory we employ a PI control law on the tracking error of the output. We prove eponential stability of the closed-loop system, under the proposed PI control law, when the parameters of the plant and the controller satisfy certain conditions, by constructing a novel non-diagonal Lyapunov functional. I. INTRODUCTION Control of systems of first-order hyperbolic PDEs is an active area of research since numerous processes can be modeled this class of PDE systems. Among various applications, systems model the dynamics of traffic [15], [17], hydraulic [], [8], [1], [13], as well as gas pipeline networks [18], and the dynamics of transmission lines [7]. Several articles are dedicated to the control and analysis of linear [], [9], [1], [] [9], [3], [31] and nonlinear [4], [5], [6], [19], [5], [6] systems. Results for the control of n n systems also eist [1], [11], [1]. Algorithms for disturbance rejection in systems are recently developed [1], [8]. The motion planning problem is solved in [14], [3], for a class of systems and in [16], [4] for a class of wave PDEs. Perhaps the most relevant results to the present article are the results in [1], dealing the Lyapunov-based output-feedback control of linear systems, the results in [3], dealing the backstepping stabilization of linear systems, and the results in [3], dealing the motion planning for a class of systems. In this paper, we are concerned the trajectory generation and tracking problems for general systems of first-order linear hyperbolic PDEs anti-collocated boundary input and output. We solve the motion planning problem for this class of systems employing backstepping Section II. Specifically, we start from a simple transformed system, namely, a cascade of two first-order hyperbolic P.-O. Lamare is Laboratoire Jean Kuntzmann, Université de Grenoble, BP 53, 3841 Grenoble, France, pierre-olivier.lamare@imag.fr. N. Bekiaris-Liberis, and A. M. Bayen are the Departments of Electrical Engineering & Computer Sciences and Civil & Environmental Engineering, University of California Berkeley, Berkeley, CA 947, USA, bekiaris-liberis@berkeley.edu, bayen@berkeley.edu. This research was partially supported by the LabE PERSYVAL-Lab ANR-11-LABX-5. PDEs coupled only at the boundary, for which the motion planning problem can be trivially solved. We then apply an inverse backstepping transformation to derive the reference trajectory and reference input for the original system. The reference output is assumed to be continuously differentiable and uniformly bounded. Our approach is different than the one in [3], in that we use backstepping for trajectory generation rather than stabilization, and the one in [3], in that we employ a different conceptual idea to a different class of systems. Although the idea of the backstepping-based trajectory generation for PDEs was conceived in [], and applied to a beam PDE [7] and the Navier-Stokes equations [3], this approach has neither been systematized nor been applied to the class of systems considered in the present article. We then employ a PI control law for the stabilization of the error system, namely, the system whose state is defined as the difference between the state of the plant and the reference trajectory. We prove eponential stability in the L norm of the closed-loop system by constructing a Lyapunov functional which incorporates cross-terms between the PDE states of the system and the ODE state of the controller, when the parameters of the system and the controller satisfy certain conditions Section III. Our result differs than the result in [1] in that we employ PI control on an output of the system in the Riemann coordinates and we construct a non-diagonal Lyapunov functional for proving closed-loop stability. II. TRAJECTORY GENERATION FOR LINEAR HYPERBOLIC SYSTEMS USING BACKSTEPPING We consider the following system z 1 t ε 1 z 1 = c 1 z 1 c z 1 z t ε z = c 3 z 1 c 4 z, under the boundary conditions z 1, t = qz, t 3 z 1, t = St 4 z, t = yt, 5 t [, is the time variable, [, 1] is the spatial variable, y is the output of the system, q is a constant parameter, and V is the control input. The functions ε 1, ε belong to C [, 1] and satisfy ε 1, ε >, for all [, 1], and the functions c i, i = 1,, 3, 4 belong to C 1 [, 1]. Defining the change of variables see, for

2 eample, [] χ 1 = ep χ = ep and the new coordinates c 1 s ε 1 s ds c 4 s ε s ds 6 7 χ = χ 1 χ, 8 u = χ 1 z 1 9 v = χ z, 1 system 1 5 is transformed into the following system u t ε 1 u = γ 1 v 11 v t ε v = γ u, 1 The boundary conditions become γ 1 = χc 13 γ = χ 1 c u, t = qv, t 15 v1, t = Ut 16 v, t = yt, 17 the original control variable satisfies U = χ 1V. 18 Theorem 1: Let y r C 1 R be uniformly bounded. The functions u r, t = qy r t Φ 1 q v r, t = y r t Φ q U r t = y r t Φ 1 q Φ 1 = Φ = L αα, y r t Φ 1 d L αβ, y r t Φ d 19 L βα, y r t Φ 1 d L ββ, y r t Φ d L βα 1, y r t Φ 1 d L ββ 1, y r t Φ d, 1 1 d ε 1 1 d, ε 3 and L αα, L αβ, L βα, L ββ are the solutions of the following equations ε L βα ε L ββ ε 1 L αα ε 1 L αβ ε 1 L βα = ε 1L βα γ L αα 4 ε L ββ = ε L ββ γ L αβ 5 ε 1 L αα = ε 1L αα γ 1 L βα 6 ε L αβ = ε L αβ γ 1 L ββ, 7 the boundary conditions L βα γ, = ε 1 ε 8 L βα, = ε qε 1 Lββ, 9 L αβ, = qε 1 ε Lαα, 3 L αβ, = γ 1 ε 1 ε, 31 are uniformly bounded and solve the boundary value problem 11, 1, 15, 16. In particular, v r, t = y r t. Proof: First note that since ε 1, ε C [, 1] ε 1, ε >, for all [, 1] and γ 1, γ C 1 [, 1], system 4 31 has a unique solution L αα, L αβ, L βα, L ββ C 1 T T = {, : 1} [6]. Hence, from 19 1 and the uniform boundness of y r it follows that u r, v r, and U r are bounded for all t and [, 1]. Taking the time and space derivatives of u r we get u r t ε 1 u r = q ε 1 qε 1 L αα, y r t Φ 1 d L αβ, y r t Φ d L αβ, y r t Φ d L αα, y r t Φ 1 d ε 1 L αβ, y r t Φ qε 1 L αα, y r t Φ 1. 3 Integrating by parts the first two integrals we get u r t ε 1 u r = q ε1 L αα, ε 1 L αα, ε 1L αα, y r t Φ 1 d ε 1 L αβ, ε L αβ, ε L αβ, y r t Φ d qε 1 L αα, ε L αβ, y r t ε 1 L αβ, ε L αβ, y r t Φ. 33 Due to the fact that L αβ and L αα are the solutions of 6 and 7 the boundary conditions 3 and 31 one gets,

3 by using, that u r satisfies 11. The proof that v r satisfies 1 follows analogously. Setting = in 19, and using, 3, we get that u r and v r satisfy 15. Setting = 1 in it follows that 1 satisfies 16. Setting in = and using 3 we get v r, t = y r t. Remark 1: The approach for the trajectory generation introduced here is inspired from backstepping. Consider the following system boundary conditions It is shown that the functions α t ε 1 α = 34 β t ε β =, 35 α, t = qβ, t 36 β1, t = y r t Φ α, t = qy r t Φ 1 38 β, t = y r t Φ, 39 Φ 1 and Φ are defined in and 3 respectively, satisfy and, in particular, β, t = y r t. Using the inverse backstepping transformation introduced in [3] u r, t = αt, v r, t = βt, L αα, α, td L αβ, β, td 4 L βα, α, td L ββ, β, td, 41 relations 38, 39 and the fact that the functions L αα, L αβ, L βα, and L ββ satisfy 4 31, one can conclude that the functions u r, v r, and U r = v r 1 solve the trajectory generation problem for system 11, 1, Eample 1: We consider the following system boundary conditions z 1 t ε 1 z 1 = 1 τ z1 4 z t ε z = 1 τ z1, 43 z 1, t = qz, t 44 z 1, t = V t, 45 τ is a positive parameter. Among various systems that can be modeled by 4 45 for instance, the Saint- Venant equations, see [1], [8], system 4 45 can be viewed as a linearized version of the Aw-Rascle-Zhang ARZ macroscopic model of traffic flow in the Riemann coordinates z 1 = w V s s 46 z = w, 47 w and s correspond to the velocity and density of the vehicles at time t and location, respectively. The variable V s is the nominal velocity of the cars and s is the nominal density. The opposite transport velocities in 4, 43 correspond to traffic flow in a congested mode. The parameter 1 τ is an indicator of the convergence rate of the velocity w of the cars to the nominal velocity V s. For more details the reader is referred to [15]. The boundary condition 44 in the original variables is written as w = V s s 1 q. 48 Hence, the boundary condition 44 dictates that there is a static relation, at the entrance of the road, between the density and the velocity similarly to the static relation between the nominal velocity V s and the density of the cars in the road. The change of variables 9, 1, 13, and 14 transform system 4 45 to u t ε 1 u = 49 v t ε v = 1 τ ep 1 u 5 τε 1 u, t = qv, t 51 v1, t = Ut, 5 Ut is given by 18. Observing that γ 1 =, relations 4 31 can be solved eplicitly as L αα, = 53 L αβ, = 54 ep 1 ε1ε L βα τε 1 ε 1ε, = 55 τ ε 1 ε qε 1 ep 1 ε1 ε 1 L ββ τε 1 ε 1ε, =. 56 τε ε 1 ε Therefore, for system 4 45, the reference input which generates the desired output y r t is V r t = y r t 1 q ε τ ε 1 ε ep 1 τε 1 qε 1 τε ε 1 ε ε1 ε ε 1 ε 1 τε 1 y r t ε1 d ε1 ε 1 ep ε 1 ε y r t d. 57 ε III. TRAJECTORY TRACKING USING PI CONTROL For stabilizing the system around the desired trajectory for any initial condition u,, v,, rather than only for u,, v, = u r,, v r,, we employ a PI-feedback control law. We first write the dynamics of the tracking errors ũ, t = u, t u r, t and ṽ, t = v, t v r, t as

4 M 1 = M = [ q β kp eµ 1 κγ βk P k I e µ γ eν k P 1 ρ βk P k I e µ γ eν k P 1 ρ βki eµ γe ν k I γ [ µ θ ε 1 γ 1 ε 1 e µ β γ e µ γ γ θρ γ ε ε eµ γ θρ γ γ ε eν ν θ ε e ν ] γ 1 ε 1 e µ β γ ε eµ γ γ ν θ e ν θρ γ ε ε β µ θ e µ γ e ν e ε κ ε γ ν θ ν θρ γ ε ] ũ t ε 1 ũ = γ 1 ṽ 6 ṽ t ε ṽ = γ ũ 61 ũ, t = qṽ, t 6 ṽ1, t = Ũt, 63 Ũ = U U r and U r is the reference input generating the desired reference trajectory. We employ the controller Ũt = k P ṽ, t k I ηt, 64 ηt = ṽ, t. 65 Theorem : Consider system 6 63 together the control law 64, 65. Let the positive constants µ, β, ρ, γ, ν, κ, and θ be such that the matrices 58, 59, shown at the top of the net page, are positive semi-definite for all [, 1], and the inequalities βρ > γ e ν µ, [, 1] 66 ε γ > θρ, 67 hold. Then, there eist positive constants λ and κ such that the following holds for all t Ωt = Ωt κe λt Ω, 68 ũ, t ṽ, t d η t. 69 Proof: In order to analyze the stability of system 6 65 we propose the following Lyapunov functional t t V t = ũ, ṽ, t P ũ, ṽ, t d ηt ηt = R 1 t R t R 3 t R 4 t, 7 and e µ ε 1 P = β eµ ε R 1 t = R t = β γe ν ε γe ν ε ρ, 71 ũ, t e µ d 7 ε 1 R 3 t = γ ηt ṽ, t eµ d 73 ε ṽ, t eν d 74 ε R 4 t = ρ η t. 75 Let us introduce the constants λ = min [,1] λ minp 76 λ = ma [,1] λ map. 77 Inequality 66 ensures that P is positive definite and symmetric for all [, 1], and hence, using the fact that ε 1, ε C [, 1] ε 1, ε >, for all [, 1], one can conclude that, λ, λ >. Therefore, λ ũ, t ṽ, t d η t V t, 78 and V t λ ũ, t ṽ, t d η t. 79 Using 7 75 we get along the solutions of system 6 65 that Ṙ 1 t = ũ, tũ, te µ d ũ, tṽ, t γ 1 ε 1 e µ d = q ṽ, t e µ ũ 1, t µ Ṙ t = β β ũ, te µ d ũ, tṽ, t γ 1 ε 1 e µ d 8 ṽ, tṽ, te µ d ũ, tṽ, t γ ε eµ d = β k P e µ ṽ, t k P k I e µ ṽ, t ηt k I e µ η t ṽ, t µβ β Ṙ 3 t = γ ηt γṽ, t γ ηt ṽ, te µ d ũ, tṽ, t γ ε eµ d 81 ṽ, te ν d ṽ, t eν ε d ũ, t γ ε eν d

5 γ ηt e ν k P ṽ, t k I ηt ṽ, t νγ ηt γ κ γ ηt ṽ, te ν d κγ ṽ, t ṽ, t eν ε d ũ, t γ ε eν d 8 Ṙ 4 t = ρṽ, t ηt, 83 we used integration by parts in the first terms of 8 8 and Young s inequality in the second term of 8. Using 7, 8 83 we get [ṽ, ] [ṽ, ] t t V t M ηt 1 ηt [ũ, ] ũ, t t ṽ, t ηt M ṽ, t d ηt e µ ũ 1, t θv t, 84 M 1 is given by 58 and [ ] A B M =, 85 B C [ ] A1 A A =, 86 A 3 A 4 A 1 = µ θ e µ ε 1 87 A = γ 1 ε 1 e µ β γ ε eµ 88 A 3 = γ 1 ε 1 e µ β γ ε eµ 89 A 4 = β µ θ e µ γ e ν ε κ ε 9 B = [ γ C = γ θρ γ ε eν γ ν θ ε e ν ] Using the Schur complement of C in M and 67, 9 one has that M for all [, 1], if and only if M = A B C 1 B. 93 Thus, if M 1 and M, for all [, 1], one has V t e µ ũ 1, t θv t, 94 and hence, V t e θt V, for all t. Combining this relation 78, 79 the proof is complete. Remark : A control law an integral action is designed in [1] for hyperbolic systems. Stability of the closed-loop system is proved using a diagonal Lyapunov functional. Here the non-diagonal term in the Lyapunov functional is needed for proving stability using a quadratic Lyapunov functional. Indeed, let us assume that the Lyapunov functional is diagonal. We can write it as V t = q1 ũ, t q ṽ, t d ρ η t, 95 the functions q 1 and q belong to C 1 [, 1] q 1, q >, for all [, 1]. The time derivative of V along the solutions of system 6, 61 boundary conditions 6 65 is given by [ṽ, ] [ ] [ṽ, ] t D1 D V t = t ηt [ũ, t ṽ, t D 3 D 4 ηt ] [ũ, ] t E d ṽ, t q 1 1ε 1 1ũ 1, t, 96 D 1 = q 1 ε 1 q q ε q 1ε 1k P 97 D = 1 q 1ε 1k P k I ρ 98 D 3 = 1 q 1ε 1k P k I ρ 99 D 4 = q 1ε 1kI [ 1 E =. 11 ] q 1ε 1 q 1γ 1q γ q 1γ 1q γ q ε Using 96 and 1 one can conclude that when k I the inequality V can not be satisfied for any [ ũ ṽ η ]. As eplained in Remark the non-diagonal term in the Lyapunov functional is crucial for proving stability using a quadratic Lyapunov functional. However, this term adds considerable compleity in verifying analytically that the matrices 58, 59 are positive definite and that 66 holds. Net, we numerically verify the conditions of Theorem for the system from Eample 1. Eample Eample 1 Continued: We set in ε 1, ε, τ, q = 3, 6, τ, 5,., 1 and choose U in 63 according to 64 k P =.1 13 k I = 1.583, 14 in order to stabilize the zero equilibrium of We verify numerically that the conditions of Theorem are satisfied β, κ, µ, ν, θ, ρ, γ =.7,.,.5,.,.7,,. 15 From 58 we get that [ ].4485 M 1 = > The verification of the positive definiteness of matri 59 is more delicate due to its dependence on. Fig. 1 shows the evolution of the eigenvalues of M and the determinant of matri 71, which remain positive for all [, 1].

6 Fig. 1. Evolution of the eigenvalues of 59 as a function of square and cross markers, and of the determinant of P in 71 star marker for Eample. IV. CONCLUSIONS We presented solutions to the trajectory generation and tracking problems for general systems of first-order linear hyperbolic PDEs. We solved the motion planning problem backstepping and the trajectory tracking problem PI control. We proved eponential stability of the closed-loop system by constructing a Lyapunov functional. ACKNOWLEDGMENTS The authors are deeply grateful to Antoine Girard and Christophe Prieur for many constructive suggestions. REFERENCES [1] O. M. Aamo, Disturbance rejection in linear hyperbolic systems, IEEE Trans. Autom. Contr., vol. 58, pp , 13. [] G. Bastin and J.-M. Coron, On boundary feedback stabilization of non-uniform linear hyperbolic systems over a bounded interval, Systems & Control Letters, vol. 6, pp. 9 96, 11. [3] J. Cochran and M. Krstic, Motion planning and trajectory tracking for the 3-D Poiseuille flow, Journal of Fluid Mechanics, vol. 66, pp , 9. [4] J.-M. Coron, B. d Andréa-Novel, and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Autom. Contr., vol. 5, pp. 11, 7. [5] J.-M. Coron, G. Bastin, and B. d Andréa-Novel, Dissipative boundary conditions for one dimensional nonlinear hyperbolic systems, SIAM Journal on Control and Optimization, vol. 47, pp , 8. [6] J.-M. Coron, R. Vazquez, M. Krstic, and G. Bastin, Local eponential H stabilization of a quasilinear hyperbolic system using backstepping, SIAM Journal on Control and Optimization, vol. 51, pp. 5 35, 13. [7] C. Curro, D. Fusco, and N. Manganaro, A reduction procedure for generalized Riemann problems application to nonlinear transmission lines, Journal of Physics A: Mathematical and Theoretical, vol. 44, paper 3355, 11. [8] J. de Halleu, C. Prieur, J.-M. Coron, B. d Andréa-Novel, and G. Bastin, Boundary feedback control in networks of open channels, Automatica, vol. 39, pp , 3. [9] A. Diagne, G. Bastin, and J.-M. Coron, Lyapunov eponential stability of linear 1-D hyperbolic systems of balance laws, Automatica, vol. 48, pp , 1. [1] F. Di Meglio, M. Krstic, R. Vazquez, and N. Petit, Backstepping stabilization of an underactuated 3 3 linear hyperbolic system of fluid flow transport equations, American Control Conference, 1. [11] F. Di Meglio, R. Vazquez, and M. Krstic, Stabilization of a system of n 1 coupled first-order hyperbolic linear PDEs a single boundary input, IEEE Transactions on Automatic Control, vol. 58, pp , 13. [1] V. Dos Santos, G. Bastin, J.-M. Coron, and B. d Andréa-Novel, Boundary control integral action for hyperbolic systems of conservation laws: Stability and eperiments, Automatica, vol. 44, pp , 8. [13] V. Dos Santos and C. Prieur, Boundary control of open channels numerical and eperimental validations, IEEE Transactions on Control Systems Technology, vol. 16, pp , 8. [14] F. Dubois, N. Petit, and P. Rouchon, Motion planning and nonlinear simulations for a tank containing a fluid, European Control Conference, Karlsruhe, Germany, [15] S. Fan, M. Herty, and B. Seibold, Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Networks and Heterogeneous Media, vol. 9, pp , 14. [16] M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph, A distributed parameter approach to the control of a tubular reactor: A multi-variable case, IEEE Conference on Decision and Control, Tampa, FL, [17] P. Goatin, The Aw-Rascle vehicular traffic flow model phase transitions, Math. Comp. Modeling, vol. 44, pp , 6. [18] M. Gugat, M. Dick, and G. Leugering, Gas flow in fan-shaped networks: Classical solutions and feedback stabilization, SIAM Journal on Control and Optimization, vol. 49, pp , 11. [19] M. Gugat and M. Herty, Eistence of classical solutions and feedback stabilisation for the flow in gas networks, ESAIM: Control, Optimisation and Calculus of Variations, vol. 17, pp. 8 51, 11. [] M. Krstic and A. Smyshlyaev, Boundary Control of PDEs. Adv. Des. Control 16, SIAM, Philadelphia, 8. [1] T.-T. Li, Classical solutions for quasilinear hyperbolic systems, John Wiley & Sons. Chichester, [] X. Litrico and V. Fromion, Boundary control of hyperbolic conservation laws using a frequency domain approach, Automatica, vol. 45, pp , 9. [3] N. Petit and P. Rouchon, Dynamics and solutions to some control problems for water-tank systems, IEEE Transactions on Automatic Control, vol. 47, pp ,. [4] N. Petit and P. Rouchon, Flatness of heavy chain systems, SIAM Journal on Control and Optimization, vol. 4, pp , 1. [5] C. Prieur, Control of systems of conservation laws boundary errors, Networks & Heterogeneous Media, vol. 4, pp , 9. [6] C. Prieur, J. Winkin, and G. Bastin, Robust boundary control of systems of conservation laws, Mathematics of Control, Signals, and Systems, vol., pp , 8. [7] A. Siranosian, M. Krstic, A. Smyshlyaev, and M. Bement, Motion planning and tracking for tip displacement and deflection angle for fleible beams, Journal of Dynamic Systems, Measurement, and Control, vol. 131, paper 319, 9. [8] B. S. Tang and M. Krstic, Sliding mode control to stabilization of linear hyperbolic systems boundary input disturbance, American Control Conference, 14. [9] R. Vazquez and M. Krstic, Marcum Q-functions and eplicit kernels for stabilization of linear hyperbolic systems constant coefficients, Systems & Control Letters, vol. 68, pp. 33 4, 14. [3] R. Vazquez, M. Krstic, and J.-M. Coron, Backstepping boundary stabilization and state estimation of a linear hyperbolic system, IEEE Conference on Decision and Control, Orlando, FL, 11. [31] R. Vazquez, M. Krstic, J.-M. Coron, and G. Bastin, Collocated output-feedback stabilization of a quasilinear hyperbolic system using backstepping, American Control Conference, 1.