Nonlinear Stabilization of Shock-Like Unstable Equilibria in the Viscous Burgers PDE REFERENCES

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1 678 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 7, AUGUST 8 assessment. The main result lies in the development of a sufficient condition from which conve semidefinite programming can be used by bounding the terms transducing the noncoincidence between the measurement and the reality. A numerical eperiment supports the efficiency of the results. Further developments could be made for various types of parameter dependence, such as, for instance, the linear fractional dependence. REFERENCES [] P. Apkarian and P. Gahinet, A conve characterization of gain-scheduled H controllers, IEEE Trans. Autom. Control, vol. 4, no. 5, pp , May 995. [] G. J. Balas, Linear parameter varying control and its application to a turbofan engine, Int. J. Non-linear and Robust Control, Special issue on Gain Scheduled Control, vol., no. 9, pp ,. [3] J. M. Biannic and P. Apkarian, Missile autopilot design via a modified LPV synthesis technique, Aerosp. Sci. Technol., pp. 53 6, 999. [4] C. Courties, Sur la commande robuste et LPV de systèmesà paramètres lentement variables, Ph.D. dissertation, l Institut National des Sciences Appliquées, Toulouse, France, 999, Rep. LAAS-CNRS. [5] M. Dettori and C. W. Scherer, LPV design for a CD player: An eperimental evaluation of performance, in Proc. IEEE CDC,, pp [6] P. Gaspa, I. Szaszi, and J. Bokor, Active suspension design using linear parameter varying control, Int. J. Vehicle Auton. Syst., vol., no., pp. 6, 3. [7] L. Giarré, D. Bauso, P. Falugi, and B. Bamieh, LPV model identification for gain scheduling control: An application to rotating stall and surge control problem, Control Eng. Practice, vol. 4, no. 4, pp , 6. [8] L. H. Keel and S. P. Bhattacharyya, Robust, fragile or optimal?, IEEE Trans. Autom. Control, vol. 4, no. 8, pp. 98 5, Aug [9] I. E. Kose and F. Jabbari, Control of LPV systems with partly measured parameters, IEEE Trans. Autom. Control, vol. 44, no. 3, pp , Mar [] G. Milleriou, L. Rosier, G. Bloch, and J. Daafouz, Bounded state reconstruction error for LPV systems with estimated parameters, IEEE Trans. Autom. Control, vol. 49, no. 8, pp , Aug. 4. [] R. A. Nichols, R. T. Reichert, and W. J. Rugh, Gain scheduling for H controllers: A flight control eample, IEEE Trans. Control Syst. Technol., vol., no., pp , Jun [] R. T. Reichert, Robust autopilot design using -synthesis, in Proc. American Control Conf., 99, pp [3] M. Sato, Robust H problem for LPV systems and its applications to model-following controller design for aircrafts motions, in Proc. IEEE Control Applications, 4, pp [4] H. D. Tuan and P. Apkarian, Monotonic relaations for robust control: New characterizations, IEEE Trans. Autom. Control, vol. 47, no., pp , Feb.. [5] V. Verdult, M. Lovera, and M. Verhaegen, Identification of linear parameter varying state space models with application to helicopter rotor dynamics, Int. J. Control, vol. 77, no. 3, pp , 4. [6] W. G. Wassink, M. V. d. Wal, C. Scherer, and O. Bosgra, LPV control for a wafer stage: Beyond the theoretical solution, Control Eng. Practice, pp. 3 45, 5. Nonlinear Stabilization of Shock-Like Unstable Equilibria in the Viscous Burgers PDE Miroslav Krstic, Lionel Magnis, and Rafael Vazquez Abstract We stabilize the unstable shock-like equilibrium profiles of the viscous Burgers equation using control at the boundaries. These equilibria are not stabilizable (even locally) using the standard radiation feedback boundary conditions. Using a nonlinear spatially-scaled transformation (that employs three ingredients, of which one is the Hopf Cole nonlinear integral transformation) and linear backstepping, we design an eplicit nonlinear full-state control law that achieves eponential stability, with a region of attraction for which we give an estimate. The region of attraction is not the entire state space since the Burgers PDE is known not to be globally controllable. Inde Terms Burgers PDE. I. INTRODUCTION We study nonlinear stabilization for the viscous Burgers equation. We consider a family of shock-like stationary profiles [9] which are unstable and not stabilizable (even locally) by simple radiation boundary conditions. We achieve eponential stabilization (in spatial L norm) of the shock profiles using two control inputs (one at each boundary) by full-state feedback. Early effort on linear static collocated output feedback (a.k.a. radiation boundary conditions) proved local L eponential stability in [6], with etension to L in []. Semi-global version of [6] and regulation to small boundary set points was proved in [7]. Global stabilization is achieved in [3], using nonlinear boundary conditions, and etended to KdV, Kuramoto Sivashinsky, adaptive control, and other problems in [], [], and [5] [8]. Reference [] derives bounds on minimal time for null controllability with one and two inputs. In [3] stabilization is approached using nonlinear model reduction, with in-domain actuation. Control of the inviscid Burgers equation, a challenging problem and different than the one studied here, was studied in [4]. Our design for nonlinear (non-convective) parabolic PDEs [3], [4] is based on a feedback linearizing nonlinear Volterra series transformation. We follow a similar idea here and construct a nonlinear transformation (based on three ingredients, one of which is the Hopf Cole [8], []), that transforms the system (with the help by one of the two boundary controls) into a linear reaction-diffusion PDE, which is then stabilized using linear backstepping [9], yielding a control that is nonlinear in the original state. We provide an estimate of the region of attraction, which is finite because the Burgers system unfortunately is not globally controllable []. II. BURGERS EQUATION AND ITS SHOCK PROFILES Consider the viscous Burgers equation ut = u uu; [; ] () 8-986/$5. 8 IEEE Manuscript received September 8, 7; revised May 5, 8. First published August, 8; current version published September, 8. Recommended by Associate Editor Denis Dochain. M. Krstic is with the Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA USA ( krstic@ucsd.edu). L. Magnis is with the Ecole des Mines de Paris, 756 Paris, France ( lionel.magnis@ensmp.fr). R. Vazquez is with the Departmento de Ingeniería Aeroespacial, Universidad de Sevilla, 49 Sevilla, Spain ( rvazquez@ucsd.edu). Digital Object Identifier.9/TAC.8.98

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 7, AUGUST Fig.. Value of the reaction coefficient in (8). The coefficient is positive and large in the region around ==, potentially destabilizing the system. Fig.. Three eample shock profiles for different values of. with u(; t) as the state, with boundary conditions u (;t)=! (t); u (;t)=! (t); () obtaininig t = + cosh ( ( =)) (8) () = tanh (=) () (9) () = tanh (=) (): () and with! (t) and! (t) as the control inputs. To save space, we will drop the arguments (; t) whenever the contet allows. We are interested in the family of symmetric shock-like stationary solutions [5] parameterized by U () = tanh ( ( =)) (3) (see Fig. ). The result in this paper can be derived for other equilibria, but (3) is of interest as the most unstable case. While the Burgers PDE is often studied as u t = "u u u, to reduce notation we focus on " =, as U () = " tanh ( ( =)) scales with ", so eigenvalues scale with " (and so does the estimate of the region of attraction). Continuing with (3), we first observe that U () = tanh ( ( =)), hence U () = U () = tanh (=), so from () we obtain that! =! = tanh (=) (4) produce the equilibrium (3). Let us denote the fluctuation variable around the shock profile as ~u(; t) = u(; t) U (), along with ~! (t) =! (t) U (); ~! (t) =! (t) U (), which yields ~u t =~u U ()~u U ()~u ~u ~u (5) ~u (;t)=~! (t); ~u (;t)=~! (t): (6) III. INSTABILITY OF SHOCK PROFILES To study the stability of the origin of the open-loop system (5), we linearize it, obtaining t = + (tanh ( ( =)) ) ; () = () =, and then eliminate the advection term in this PDE using the invertible transformation (; t) =G()(; t), where For = the system is neutrally stable. For >, the boundary conditions are destabilizing and so is the reaction term in (8), in the vicinity of ==, see Fig.. The larger, the more positive the first eigenvalue of (8) (). For eample, for =5 the first eigenvalue is +:6. IV. INSTABILITY UNDER RADIATION FEEDBACK With radiation boundary feedback [6], [7] ~! (t) =k~u(;t); ~! (t) =k~u(;t); k > () system (8) () changes only in boundary conditions () = (k tanh (=)) (); () () = (k tanh (=)) (): (3) Stability of the system (8), (), and (3) improves as k! +.However, eigenvalue computation shows that for > 3, where 3, the system always has eactly one unstable eigenvalue, so no value of k eists that stabilizes the system, see Fig. 3. Thus, semiglobal stability in [7] holds for sufficiently small set points but not for large set points. So, a more sophisticated feedback (full-state or dynamic) is needed to stabilize shock-like equilibria (even locally). V. FULL STATE FEEDBACK We use a state transformation and feedback for ~! to linearize the transformed PDE and its boundary condition at =; then design a feedback for ~! to stabilize the resulting linear system using backstepping. A. Linearizing Transformation and the Design of ~! Define v(; t) =~u(; t)e variable, written using (7) as [~u(y;t)+u(y)]dy, which is a new state cosh ( ( =)) G() = cosh (=) (7) v(; t) =G()~u(; t)e ~u(y;t)dy : (4)

3 68 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 7, AUGUST 8 Fig. 4. Target system reaction coefficient in (4), compared with the coefficient of the v plant in (), for =5. Notice that the coefficient only has been changed close to ==, where it was destabilizing. The new coefficient is negative everywhere. we obtain the following linear reaction-diffusion system akin to (8): v t = v + cosh ( ( =)) v () v () = tanh (=) v() () v () = tanh (=) v() + ~! ~u() v(y) dy : () G(y) B. Design of ~! Using the Backstepping Method Fig. 3. Top: Seven rightmost eigenvalues of (8), (), and (3) for growing k and fied =5. All are real and move leftward as k increases. Bottom: Detail for first eigenvalue, positive for all k. Transformation (4) is a composition of the Hopf Cole [8], [] transformation on ~u(; t) and the scaled by the gauge transformation G(). The transformation (4) from ~u to v is invertible: To find the feedback ~! to stabilize () (), we use backstepping parabolic PDEs [9]. We define a new state w(; t) =v(; t) which we require to satisfy the target PDE k(; y)v(y;t)dy (3) ~u(; t) = v(; t)=g() : (5) v(y;t) G(y) dy Substituting (4) into (5) (6), we obtain w t = w + cosh ( ( =)) cw (4) w () = tanh (=) w() (5) w () = tanh (=) w() (6) w v t = v U () + v ~! U ()~u() ~u () + v (6) v () = ~! (~u() + U ()) ~u() (7) v () = ~! (~u() + U ()) ~u() e [~u(y;t)+u(y)]dy : (8) Setting the feedback law where c is a control parameter. In (4) we do not eliminate the reaction term from the original system () but only lower it to eliminate its positive part, without wasting control to change its negative part, see Fig. 4. The coefficient c can be set to zero whenever >. When =, we need c > because the system w t = w, w (;t) = w (;t)=is only neutrally stable. Following [9], we find that the kernel k has to verify k = k yy + tanh ( (y =)) + tanh ( ( =)) k + ck (7) ~! = U ()~u() + ~u () =tanh (=) ~u() + ~u () (9) k(; ) = [tanh ( ( =)) + tanh (=)] c (8) k y (; ) = tanh (=) k(; ) (9)

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 7, AUGUST 8 68 which is a linear hyperbolic PDE in the domain T = f(; y) : y g. In [9] it is shown that (7) (9) is well-posed and that k C (T ). The kernel k can be computed from (7) (9) numerically or symbolically using procedures outlined in [9]. From (6) and (3) we obtain the condition v () = (k (;y)+tanh (=) k(;y)) v(y)dy + (k(; ) tanh (=)) v(). Substituting () we find the control law VII. PROOF OF THE MAIN RESULT Lemma : Given ~u L and v defined in terms of ~u as given in (4), then v L and kvk L g (k~uk L ), where g(r) =r=e r=. Moreover, if ~u H (resp. H ) then v H (resp. H ). Proof: From the transformation (4) we obtain kvk L ma jg()jk~uk L e j~u(y;t)jdy k~uk L e k~uk (34) [;] ~! (t) = ~u(;t) + G(y)e +(k(; ) tanh (=)) ~u(;t) (k (;y)+ tanh (=) k(;y)) ~u(;t)d ~u(y; t)dy (3) where we have used (5) to epress v in terms of ~u. As in [9], the transformation (3) can be inverted to obtain an epression for v in terms of w as follows: v(; t) =w(; t) + l(; y)w(y; t)dy: (3) since ma [;] jg()j =. The last part of the lemma is proved analogously, by taking derivatives of (4). Lemma : Given v L such that kvk L < = and ~u defined in terms of v as given in (5), then ~u L and k~uk L cosh (=) kvk L kvk L : (35) If v H (resp. H ) then ~u H (resp. H ). Proof: We first show that, from the assumption kvk L =, it follows that v(y) dy < ; which implies that G(y) the denominator of (5) is nonzero, so ~u is well-defined. Using Cauchy Schwartz The inverse kernel l(; y) satisfies a well-posed hyperbolic linear partial differential equation like (7) (9). v(y) G(y) dy kvk L G(y) dy VI. STABILITY UNDER FULL-STATE FEEDBACK Given a function f (), we define the norm kfk L (;) = f d = and the space L () (; ) = ff : kfk L (;) < g. Similarly we define the norms kfk H = kf k L + kfk L, kfk H = kf k L + kf k L + kfk L, and the function spaces H = ff : kfk H < g and H = ff : kfk H < g, where H H L. For a function f (; t), its time-varying spatial norms kf(t)k are defined as above. Norms in time and space are given by kfk H = T kf(t)k H dt =, kfk H = kf tk H + kfk H, and we denote H ; = H ; and H ; = H ; We denote the initial condition as ~u () = ~u(; ), and define a class K [] function g(r) = r e. Theorem : Assume that ~u H is such that cosh (=) = kvk L cosh ( (y =)) dy cosh (=) = kvk L = Hence ~u is well defined in (5) and tanh (=)) kvk L < : (36) k~u()k L = ma [;] kvk L v()=g() G () kvk L v(y) dy G(y) dy G(y) d k~u k L <g m (3) = cosh (=) kvk L kvk L (37) and that it verifies compatibility with feedback boundary conditions (6), (9), (3). Then the equilibrium ~u of system (5) (6) with feedback laws (9) and (3) is eponentially stable in the L norm, i.e., for all t> k~u(t)k coth (=) m g (k~uk L ) g (k~uk L ) et (33) where m = + ma (;y)t jk(; y)j + ma (;y)t jl(; y)j, and = c + minf=4;tanh(=)g. Moreover, the solution ~u belongs to H ;. since the maimum of =G() is cosh =. The lemma s last part is proved analogously, by taking derivatives of (5), and noting that the denominator is always the same, hence the condition kvk L < = is enough to guarantee that the derivatives of ~u are welldefined. We now establish Theorem. Consider the system (4) (6) with initial condition w () =G()~u ()e ~u (y)dy k(; y) G(y)~u (y)e ~u ()d dy: (38)

5 68 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 7, AUGUST 8 Since ~u H and k C (T ), by taking twice the derivative with respect to in (38), ~w H. Since w verifies the boundary conditions (5) (6), from [4], we obtain that there is a unique solution w H ; T [i.e., the solution is defined at least for some interval t (;T)]. Consider now the following Lyapunov functional L(t) = w(; d t) =. We have that kw(t)k L _L = w tanh + hw; w ickwk d q kw k + 4 (w() + w() ) ckwk (39) where q = minf; 4 tanh(=)g. Since w(t) H for some T, we can use Poincare s inequality [3], obtaining d L(t) dt c + q d 4 w = c + q L. Hence, for = c + q=4 = 4 c + minf=4;tanh(=)g, we obtain that Fig. 5. Finite time blow-up of open-loop system (with constant inputs (4)) for =3and u () =U () + + (U () U () 4). kw(t)k L kw k L e t : (4) Similar estimates can be derived in the H and H norms (see [] for eamples on how to obtain those estimates), which can be used to deduce that w H ;, i.e., that the solution is defined for all t (; ). From the w estimate of (4) we can derive estimates for v as follows. The transformation (3) and its inverse (3) determine that kw(t)k L + ma jk(; y)j kv(t)k L (4) (;y)t kv(t)k L + ma jl(; y)j kw(t)k L (4) (;y)t hence kv(t)k L + ma (;y)t jl(; y)j kwk L e t mkvk L e t. Note that since k; l C (T ), estimates in the H and H norms are also derived [], and it follows that v H ;. From Lemma VII, we get that kv(t)k L mg (k~uk L ) e t. Using assumption (3) in the Theorem, we obtain that kv(t)k L < =, thus the conditions of Lemma VII are verified and ~u belongs to L (and to H and H, hence ~u H ; ), and satisfies the estimate k~u(t)k L cosh (=) kv(t)k L and from (43) the estimate (33) follows. kv(t)k L cosh (=) mg (k~u k L ) e t (43) mg (k~u k L ) e t VIII. SIMULATIONS The shock-like equilibrium of the open-loop system () (), (4) is unstable for any >. In Fig. 5 one can see it undergoing a finite time blow-up for =3. Before showing the results with our nonlinear design, we show the results with a linearized full-state backstepping controller ~! (t) = tanh (=) ~u(;t) (44) We use Theorem 9. on page 343, which is proved for Dirichlet boundary conditions, but also stated to be valid under Robin boundary conditions (which we have here) in [4, pp. 34 and 35]. Fig. 6. The backstepping kernel () (left) and the closed-loop solution under the linear backstepping feedback (right) for =5. ~! (t) = 3 tanh (=) + c ~u(;t) (y)g(y)~u(y; t)dy: (45) The kernel () and a closed-loop solution with the controller (44) and (45) are shown in Fig. 6. In Fig. 7 we show solutions from various initial conditions under nonlinear controller (9), (3).

6 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 53, NO. 7, AUGUST Fig. 7. Convergence of closed-loop system under the nonlinear full-state feedback for various initial conditions and =5. [5] J. A. Burns, A. Balogh, D. S. Gilliam, and V. I. Shubov, Numerical stationary solutions for a viscous Burgers equation, J. Math. Syst., Estim., Control, vol. 8, no., 998. [6] C. I. Byrnes, D. S. Gilliam, and V. I. Shubov, On the global dynamics of a controlled viscous Burgers equation, J. Dynam. Control Syst., vol. 4, no. 4, pp , 998. [7] C. I. Byrnes, D. S. Gilliam, and V. I. Shubov, Boundary control, stabilization and zero-pole dynamics for a non-linear distributed parameter system, Int. J. Robust Nonlin. Control, vol. 9, pp , 999. [8] J. D. Cole, On a quasilinear parabolic equation occurring in aerodynamics, Q. Appl. Math., vol. 9, pp. 5 36, 95. [9] S. Engelberg, The stability of the shock profiles of the Burgers equation, Appl. Math. Lett., vol., no. 5, pp. 97, 998. [] E. Fernandez-Cara and S. Guerrero, Null controllability of the Burgers equation with distributed controls, Syst. Control Lett., vol. 56, pp , 7. [] E. Hopf, The partial differential equation u +uu = u, Comm. Pure Appl. Math., vol. 3, pp. 3, 95. [] H. K. Khalil, Nonlinear Systems, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall,. [3] M. Krstic, On global stabilization of Burgers equation by boundary control, Syst. Control Lett., vol. 37, pp. 3 4, 999. [4] O. Ladyzhenskaya, V. Solonnikov, and N. UralÕtseva, Linear and Quasilinear Equations of Parabolic Type. Providence, RI: AMS Translations, 968, vol. 3. [5] W.-J. Liu and M. Krstic, Backstepping boundary control of Burgers equation with actuator dynamics, Syst. Control Lett., vol. 4, pp. 9 33,. [6] W.-J. Liu and M. Krstic, Stability enhancement by boundary control in the Kuramoto-Sivashinsky equation, Nonlin. Anal., vol. 43, pp ,. [7] W.-J. Liu and M. Krstic, Adaptive control of Burgers equation with unknown viscosity, Int. J. Adapt. Control Signal Process., vol. 5, pp ,. [8] W.-J. Liu and M. Krstic, Global boundary stabilization of the Korteweg-De Vries-Burgers equation, Comput. Appl. Math., vol., pp ,. [9] A. Smyshlyaev and M. Krstic, Closed form boundary state feedbacks for a class of partial integro-differential equations, IEEE Trans. Autom. Control, vol. 49, pp. 85, 4. [] H. V. Ly, K. D. Mease, and E. S. Titi, Distributed and boundary control of the viscous Burgers equation, Numer. Funct. Anal. Optim., vol. 8, pp , 997. [] R. Vazquez and M. Krstic, A closed-form feedback controller for stabilization of the linearized -D Navier-Stokes Poisseuille system, IEEE Trans. Autom. Control, vol. 5, pp. 98 3, 7. [] R. Vazquez and M. Krstic, Eplicit output feedback stabilization of a thermal convection loop by continuous backstepping and singular perturbations, in Proc. Amer. Control Conf., 7. [3] R. Vazquez and M. Krstic, Control of -D parabolic PDEs with Volterra nonlinearities Part I: Design, Automatica, to be published. [4] R. Vazquez and M. Krstic, Control of -D parabolic PDEs with Volterra nonlinearities Part II: Analysis, Automatica, to be published. REFERENCES [] A. Balogh and M. Krstic, Burgers equation with nonlinear boundary feedback H stability, well posedness, and simulation, Matt. Prob. Eng., vol. 6, pp. 89,. [] A. Balogh and M. Krstic, Boundary control of the Korteweg De vries Burgers equation: Further results on stabilization and numerical demonstration, IEEE Trans. Autom. Control, vol. 45, pp ,. [3] J. Baker, A. Armaou, and P. D. Christofides, Nonlinear control of incompressible fluid flow: Application to burgers equation and D channel flow, J. Math. Anal. Appl., vol. 5, no., pp. 3 55,. [4] J.-P. Aubin, A. M. Bayen, and P. Saint-Pierre, Computation and control of solutions to the burgers equation using viability theory, in Proc. 5 Amer. Control Conf., 5, pp