Boundary Control of an Anti-Stable Wave Equation With Anti-Damping on the Uncontrolled Boundary

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1 9 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 1-1, 9 WeC5. Boundary Control of an Anti-Stable Wave Equation With Anti-Damping on the Uncontrolled Boundary Andrey Smyshlyaev Miroslav Krstic Abstract Much of the boundary control of wave equations in 1D is based on a single principle passivity under the assumption that control is applied through Neumann actuation on one boundary the other boundary satisfies a homogeneous Dirichlet boundary condition. We have recently eped the scope of tractable problems by allowing destabilizing antistiffness (a Robin type condition) on the uncontrolled boundary, where the uncontrolled system has a finite number of positive real eigenvalues. In this paper we go much further develop a methodology for the case where the uncontrolled boundary condition has anti-damping, which makes the real parts of all the eigenvalues of the uncontrolled system positive arbitrarily high, i.e., the plant is anti-stable (eponentially stable in negative time). Using a conceptually novel integral transformation, we obtain etremely simple, eplicit formulae for the gain functions. For the case with only boundary sensing available (at the same end with actuation), we design backstepping observers which are dual to the backstepping controllers have eplicit output injection gains. We then combine the control observer designs into an outputfeedback compensator prove eponential stability of the closed-loop system. I. INTRODUCTION We consider the problem of stabilization of a onedimensional wave equation which is controlled from one end contains instability at the other (free) end (see [1], [], [4], [7] for classical passivity-based designs for wave equations). The nature of instability (negative damping) is such that all of the open-loop eigenvalues are located on the right h side of the comple plane, thus the plant is not just unstable, it is anti-stable. Our control design is based on the method of backstepping [6], [5], [3], which results in eplicit formulae for the gain functions. In a recent paper [3], backstepping method was used to design controllers observers for an unstable wave equation with destabilizing boundary condition at the free end. However, in that paper the destabilizing term was proportional to displacement, while in this paper it is proportional to velocity, in the form of anti-damper, which results in all eigenvalues being unstable instead of just a few. The concept of a boundary anti-damper is not of huge physical relevance, however, the design that we develop for this anti-stable system is a methodological breakthrough in boundary control of wave equations. For the case when only boundary sensing is available (at the same end with actuation), we design backstepping observers which are dual to the backstepping controllers Andrey Smyshlyaev Miroslav Krstic are with the Department of Mechanical Aerospace Engineering, University of California at San Diego, La Jolla, CA 993, USA krstic@ucsd.edu asmyshly@ucsd.edu have eplicit output injection gains. Both setups are considered: Neumann actuation/dirichlet sensing Dirichlet actuation/neumann sensing. We then combine the control observer designs into dynamic compensators prove eponential stability of the closed-loop system. We consider the plant II. PROBLEM FORMULATION u tt (, t) = u (, t) (1) u (, t) = qu t (, t) () u (1, t) = U(t), (3) where U(t) is the control input q is a constant parameter. For q =, equations (1) (3) model a string which is free at the end = is actuated on the opposite end. For q < the system (1) (3) is stabilized trivially with the feedback law U = au(1, t), a >. In this paper we assume q >, so that the free end of the string is negatively damped, with all eigenvalues located on the right h side of the comple plane (hence the open-loop plant is anti-stable ). We also assume q 1, since for q = 1 the plant is uncontrollable. The objective is to eponentially stabilize the system (1) (3) around the zero equilibrium. The case of Dirichlet actuation is considered in Section VI. III. CONTROL DESIGN Consider the transformation w(, t) = u(, t) k(, y)u(y, t)dy s(, y)u t (y, t)dy m(, y)u (y, t)dy, (4) where the gains k(, y), s(, y), m(, y) are to be determined. We want to map the plant (1) (3) into the following target system w tt (, t) = w (, t) (5) w (, t) = cw t (, t) (6) w (1, t) = c w(1, t), (7) which is eponentially stable for c > c > (see, e.g., [3]). We also assume c 1. As will be shown later, the transformation (4) is invertible in a certain norm, so that stability of the target system ensures stability of the closedloop system /9/$5. 9 AACC 1511

2 Compared with the backstepping transformations for parabolic PDEs, there are two additional terms in (4) the second the third integrals in (4). The term with u t is natural because hyperbolic systems are second order in time therefore the state variable is (u, u t ) instead of just u. The need for the term with u is less obvious it is in fact the main conceptual novelty of the paper. Substituting (4) into (5) (7) we obtain w tt (, t) = w (, t) u(, t) d k(, ) d u t (, t) d d s(, ) u (, t) d m(, ) d (k (, y) k yy (, y))u(y, t)dy (s (, y) s yy (, y))u t (y, t)dy (m (, y) m yy (, y))u (y, t)dy k y (, )u(, t) [qm y (, ) s y (, ) qk(, )]u t (, t) [m(, ) qs(, )]u (, t). (8) Matching all the terms we get the following PDE for k(, y): k (, y) = k yy (, y), (9) k y (, ) =, (1) d k(, ) =, d (11) two coupled PDEs for s(, y) m(, y): s (, y) = s yy (, y), (1) s y (, ) = qm y (, ) qk(, ), (13) d s(, ) =, d (14) m (, y) = m yy (, y), (15) m(, ) = qs(, ), (16) d m(, ) =. d (17) Substituting (4) into the boundary condition (6) we get = w (, t) cw t (, t) = [qm(, ) s(, ) q c]u t (, t) k(, )u(, t), (18) which gives two more conditions: k(, ) =, (19) qm(, ) = s(, ) q c. () The solution to (9) (11), (19) is simply k(, y). This is a consequence of not changing stiffness of the system (i.e., if we were to add the term proportional to w(, t) to the boundary condition at =, the gain k(, y) would not be zero). To solve the PDEs for s m, we note that a general solution to (1) (14) is s(, y) = φ( y) similarly for (15), (17) we have m(, y) = ψ( y) for arbitrary functions φ ψ. Using (13) (16) we obtain φ () = qψ (), (1) ψ() = qφ(). () Integrating (1) from to using (), we obtain (q )(φ() φ()) =. (3) Since q 1, we get that both φ() ψ() are constant functions of. Finally, using the relationship () together with (), we get s(, y) q c q, (4) q(q c) m(, y) q. (5) The transformation (4) can therefore be written in one of the two forms, either as or as q(q c) w(, t) = u(, t) q u (y, t)dy q c q u t (y, t)dy, (6) w(, t) = qc q(q c) q u(, t) q u(, t) q c q u t (y, t)dy. (7) Differentiating (7) with respect to, setting = 1, using the boundary condition (7), we obtain the following controller U(t) = c q(q c) u(, t) c u(1, t) 1 qc q c 1 qc u t(1, t) c (q c) 1 qc u t (y, t)dy. (8) Our main result on stabilization is given by the following theorem. Theorem 1. For any initial data (u(, ), u t (, )) H = H (, 1) H 1 (, 1), compatible with the boundary conditions, closed-loop system (1) (3), (8) has a unique classical solution (u, u t ) C 1 ([, ), H), which is eponentially stable in the sense of the norm ( ) 1/ u (, t) d u t (, t) d u(1, t). (9) Proof. First, let us establish stability of the target system. With the Lyapunov function V 1 (t) = 1 δ w (, t) d 1 w t (, t) d c w(1, t) ( )w (, t)w t (, t)d, (3) 151

3 where δ is sufficiently small (so that V 1 is positive definite), we obtain V 1 = δ (w (, t) w t (, t) )d [ c δ(1 c ) ] w t (, t) δ [ wt (1, t) c w(1, t) ] ωv 1, (31) where ω >. Since there eist a 1 >, a > such that where V (t) = we obtain a 1 V V 1 a V, (3) w (, t) d w t (, t) d w(1, t), (33) V (t) a a 1 e ωt V (). (34) Let us differentiate the transformation (7) with respect to t. We get w t (, t) = qc q u t(, t) q c q u (, t), (35) w (, t) = qc q u (, t) q c q u t(, t). (36) From (35) (36) it easily follows that (w (, t) w t (, t) )d (c 1) (q ) (u (, t) u t (, t) )d. (37) Using the transformation (6) with = 1, we get w(1, t) u(1, t) 4(q c) (q ) From (37) (38) we obtain where V 3 (t) = (u (, t) u t (, t) )d. (38) V 4 (q c) (c 1) (q ) V 3, (39) u (, t) d u t (, t) d u(1, t). (4) One can easily show that the inverse transformation to (35) (36) is u t (, t) = qc c w t(, t) q c c w (, t), (41) u (, t) = qc c w (, t) q c c w t(, t). (4) Setting = 1 in (6) using (41), (4) to epress u(1, t) in terms of w(1, t), w t, w, we obtain V 3 4 (q c) (q 1) (c ) V. (43) From (34), (39), (43) one gets V 3 (t) 16a (q c) (c 1) a 1 (c ) (q ) (q c) (q 1) (c ) (q ) e ωt V 3 (). (44) The eistence uniqueness of the solution follow by stard arguments as in [3]. First, the abstract operator describing the system is introduced, it is dissipative due to estimates above; then it is shown that it has a bounded inverse; the result follows from Lumer-Phillips theorem. IV. OBSERVER DESIGN In this section, we design an observer for the plant (1) (3) when only boundary measurements are available. We assume that displacement velocity at the end = 1 are measured (i.e., u(1, t) u t (1, t)). Since we epect this observer to be dual to the controller designed in the previous section, it is natural to assume that the observer gains are also constant. We propose the following observer û tt (, t) = û (, t) p 1 [u(1, t) û(1, t)] p [u t (1, t) û t (1, t)] (45) û (, t) = qû t (, t) p 3 [u(1, t) û(1, t)] p 4 [u t (1, t) û t (1, t)] (46) û (1, t) = U(t) p 5 [u(1, t) û(1, t)] p 6 [u t (1, t) û t (1, t)]. (47) The observer error ũ = u û satisfies ũ tt (, t) = ũ (, t) p 1 ũ(1, t) p ũ t (1, t) (48) ũ (, t) = qũ t (, t) p 3 ũ(1, t) p 4 ũ t (1, t) (49) ũ (1, t) = p 5 ũ(1, t) p 6 ũ t (1, t). (5) Consider the transformation ũ(, t) = w(, t) α w(y, t)dy β w t (y, t)dy γ w (y, t)dy. (51) Note that unlike in the control transformation (6), here the integrals run from to 1. This is because the input the output are collocated. We map the observer error system into the system w tt (, t) = w (, t) (5) w (, t) = c w t (, t) (53) w (1, t) = c w(1, t), (54) which is eponentially stable for c >, c 1 c >. 1513

4 First, we differentiate the transformation (51) twice w.r.t. time (note that ũ(1, t) = w(1, t)), ũ tt (, t) = ũ (, t) α w (1, t) β w t (1, t) γ w tt (1, t) = ũ (, t) αc ũ(1, t) βc ũ t (1, t) γũ tt (1, t). (55) Comparing the above with (48) we get γ =, p 1 = αc, p = βc. From (5) we get ũ (1, t) = w (1, t) α w(1, t) β w t (1, t) = (c α)ũ(1, t) βũ t (1, t), (56) which gives p 5 = c α, p 6 = β. Then, from (49) ũ (, t) qũ(, t) = αw(, t) qα w t (y, t)dy w t (, t)[ c β q qβ c] qβc ũ(1, t) (57) therefore p 3 = qβc, p 4 =, α =, c q = β(1 q c). Finally, we get the following observer û tt (, t) = û (, t) c (q c) 1 q c [u t(1, t) û t (1, t)] (58) û (, t) = qû t (, t) c q(q c) [u(1, t) û(1, t)] (59) 1 q c û (1, t) = U(t) c [u(1, t) û(1, t)] q c 1 q c [u t(1, t) û t (1, t)]. (6) the transformation is ũ(, t) = w(, t) q c 1 q c w t (y, t)dy. (61) Note the duality of 4 observer gains in (58) (6) to 4 control gains in (8) (for c = c), even though the control observer transformations are different. Theorem. For any initial data (ũ(, ), ũ t (, )) H = H (, 1) H 1 (, 1) compatible with the boundary conditions, the observer error system (48) (5) has a unique classical solution (ũ, ũ t ) C 1 ([, ), H), which is eponentially stable in the sense of the norm ( )1/ ũ (, t) d ũ t (, t) d ũ(1, t). (6) Proof. With the Lyapunov function V 4 = 1 w (, t) d 1 w t (, t) d c w(1, t) δ ( ) w (, t) w t (, t)d, (63) where δ is sufficiently small, eactly the same calculation as in (31) shows that where ω >. V 4 ωv 4, (64) Differentiating the transformation (61) we get ũ (, t) = w (, t) q c 1 q c w t(, t), (65) ũ t (, t) = w t (, t) q c 1 q c w (, t) c (q c) w(1, t). 1 q c (66) Note also that ũ(1, t) = w(1, t). Therefore, ũ ũ t ũ(1) M 6 ( w w t w(1) ), (67) where M 6 = 3 3 ma{c, 1}(q c) /(1 q c). The inverse to (65), (66) is w (, t) = (1 q c) ũ (, t) (q c)(1 q c)ũ t (, t) (q )(c ) c (q c) (q )(c t) (68) )ũ(1, w t (, t) = (1 q c) ũ t (, t) (q c)(1 q c)ũ (, t) (q )(c ) c (q c)(1 q c) (q )(c ũ(1, t). (69) ) Therefore, w w t w(1) ) M 7 ( ũ ũ t ũ(1) ), (7) where M 7 = 3 ma{c, 1}((q c) (1 q c) ) (q 1) ( c ). From (64), (67), (7) we get that the norm (6) decays eponentially. The eistence uniqueness of the solution of the observer error system are obtained as in the proof of Theorem 1. V. OUTPUT FEEDBACK In this section we combine the controller the observer designed in previous two sections to solve the outputfeedback problem. Theorem 3. Consider the plant (1) (3) with the observer (58) (6) the controller U(t) = c q(q c) û(, t) c u(1, t) 1 qc q c 1 qc u t(1, t) c (q c) 1 qc û t (y, t)dy. (71) For any initial data (u(, ), u t (, ), û(, ), û t (, )) H = H (, 1) H 1 (, 1) H (, 1) H 1 (, 1) compatible with the boundary conditions, the closed-loop system has a unique classical solution (u, u t, û, û t ) C 1 ([, ); H), which is eponentially stable in the sense of the norm ( u (t) u t (t) u(1, t) û (t) û t (t) û(1, t) ) 1/. (7) Proof. Consider two transformations: (61) ŵ(, t) = qc q(q c) q t) û(, q û(, t) q c q û t (y, t)dy. (73) 1514

5 It is straightforward to show that these transformations along with the control law (71) map the system (û, ũ) into (ŵ, w)- system, where w part is given by (5) (54) ŵ satisfies the following PDE: ŵ tt (, t) = ŵ (, t) A(q w t (1, t) w tt (1, t)) (74) q(1 qc) ŵ (, t) = cŵ t (, t) A w(1, t) (75) q c ŵ (1, t) = c ŵ(1, t) c c 1 q c w t(1, t), (76) where A = c (q c)(q c) (q )(1 q c). (77) Note that the PDE (74) (76) contains the terms proportional to w t (1, t) w tt (1, t), which are in H H 3 respectively, while our Lyapunov functions used in control observer designs are H 1 norms. To overcome this difficulty, we introduce a new variable w(, t) = ŵ(, t) A w(1, t), which eliminates the term w tt (1, t). The remaining w t (1, t)- terms are hled using the term w t (1, t) in V 4, which was simply discarded in the estimate (64) (see (31)). The variable w(, t) satisfies the following PDE: w tt (, t) = w (, t) Aq w t (1, t) (78) w (, t) = c w t (, t) c c(q c) w(1, t) (1 q c) (79) w (1, t) = c w(1, t) c c 1 q c w t(1, t) Using the Lyapunov function V 5 = 1 (c 1)A w(1, t). (8) w (, t) d 1 w t (, t) d c w(1, t) δ 1 ( ) w (, t) w t (, t)d KV 4 (81) one can show that for large enough K small enough δ 1 δ V5 ω 1 V 5 α w t (1, t), ω 1 >, α >. (8) Going back to the old variable ŵ, using (8) we obtain eponential stability in ( w, ŵ) variables. From the transformations (61) (73) their inverses we obtain ep. stability in (ũ, û) variables, therefore in (u, û) variables. The eistence uniqueness of the solutions is proved as in Theorem 1. VI. DIRICHLET ACTUATION AND NEUMANN SENSING In this section we use the control observer transformations derived in Sections III IV to design the controller the observer for the case of Dirichlet actuation Neumann sensing. Consider the plant u tt (, t) = u (, t) (83) u (, t) = qu t (, t) (84) u(1, t) = U(t). (85) Using the transformation (7), we map this plant into the target system w tt (, t) = w (, t) (86) w (, t) = cw t (, t) (87) w(1, t) =, (88) which is eponentially stable for c >, c 1. The controller is obtained by setting = 1 in (7): U(t) = q(q c) q c u(, t) 1 qc 1 qc u t (y, t)dy. (89) Theorem 4. For any initial data (u(, ), u t (, )) H = H (, 1) H 1 (, 1) compatible with the boundary conditions, the closed-loop system (83) (85), (89) has a unique classical solution (u, u t ) C 1 ([, ), H), which is eponentially stable in the sense of the norm ( u (t) u t (t) ) 1/. (9) Proof. Starting with the Lyapunov function V 6 (t) = 1 δ w (, t) d 1 where δ is sufficiently small, we obtain V 6 = δ w t (, t) d ( )w (, t)w t (, t)d, (91) (w (, t) w t (, t) )d [ c δ(1 c ) ] w t (, t) δ w (1, t) ωv 6, ω > (9) The rest of the proof is very similar to the proof of Theorem 1. When only measurements of u (1, t) u t (1, t) are available, we design the observer û tt (, t) = û (, t) p 1 [u (1, t) û (1, t)] p [u t (1, t) û t (1, t)] (93) û (, t) = qû t (, t) p 3 [u (1, t) û (1, t)] p 4 [u t (1, t) û t (1, t)] (94) û(1, t) = U(t), (95) where p 1, p, p 3, p 4 are constants to be chosen. The observer error system is ũ tt (, t) = ũ (, t) p 1 ũ (1, t) p ũ t (1, t) (96) ũ (, t) = qũ t (, t) p 3 ũ (1, t) p 4 ũ t (1, t) (97) ũ(1, t) =. (98) We use the observer transformation (61), derived for the case of the Dirichlet sensing to map the observer error system into the following system: w tt (, t) = w (, t) (99) w (, t) = c w t (, t) (1) w(1, t) =. (11) 1515

6 Substituting (61) into (96) (98), we get the following conditions: p 1 = p =, p 3 (1 q c) = q(q c), p 4 (1 q c) = q c. Therefore, the observer is û tt (, t) = û (, t) q c 1 q c [u t(1, t) û t (1, t)] (1) û (, t) = qû t (, t) q(q c) 1 q c [u (1, t) û (1, t)] (13) û(1, t) = U(t). (14) Note the duality of two observer gains in (1), (13) to the two control gains in (89) (for c = c). Theorem 5. Consider the plant (83) (85) with the observer (1) (14) the controller U(t) = q(q c) q c û(, t) 1 qc 1 qc û t (y, t)dy. (15) For any initial data (u(, ), u t (, ), û(, ), û t (, )) H = H (, 1) H 1 (, 1) H (, 1) H 1 (, 1) compatible with the boundary conditions, the closed-loop system has a unique classical solution (u, u t, û, û t ) C 1 ([, ); H), which is eponentially stable in the sense of the norm ( u (t) u t (t) û (t) û t (t) u (t) û (t) u t (t) û t (t) ) 1/. (16) Proof. The transformations (61) ŵ(, t) = qc q(q c) q t) û(, q û(, t) q c q û t (y, t)dy (17) map (96) (98), (1) (14) into (99) (11) the following system ŵ tt (, t) = ŵ (, t) q c 1 q c w t(1, t) (q c)(q c) (q )(1 q c) w tt(1, t) (18) q(1 qc)(q c) ŵ (, t) = cŵ t (, t) (q )(1 q c) w (1, t) (19) ŵ(1, t) =. (11) First, we establish eponential stability of the system (99) (11) with the Lyapunov function V 7 (t) = 1 1 w (, t) d 1 w (, t) d 1 w t (, t) d δ 1 ( ) w (, t) w t (, t)d w t (, t) d δ ( ) w (, t) w t (, t)d. (111) It is straightforward to show that V 7 ωv 7 α w t (1, t), ω >, α >. (11) Note that, unlike in the case of Dirichlet sensing, here the Lyapunov function has to contain H norms for us to be able to show eponential stability of the observer error system. This is due to the H nature of the terms ũ (1, t) appearing in (96) (98). To eliminate the term proportional to w tt (1, t) in (18) we introduce a new variable (q c)(q c) w(, t) = ŵ(, t) (q )(1 q c) w (1, t). (113) With the Lyapunov function V 8 (t) = 1 we obtain δ w (, t) d 1 w t (, t) d ( ) w (, t) w t (, t)d KV 7 (t), V 8 ω V 8, ω > (114) for sufficiently large K sufficiently small δ, δ 1, δ. The rest of the proof is similar to the proof of Theorem 3. VII. CONCLUSIONS In this paper we introduced a new integral transformation for wave equations used it to obtain eplicit controllers observers for a wave equation with negative damping at the boundary. The application of the presented approach to other hyperbolic systems is very promising will be the subject of future work. REFERENCES [1] G. Chen, Energy decay estimates eact boundary value controllability for the wave equation in a bounded domain, J. Math. Pure Appl., 58, pp. 4973, [] V. Komornik E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pure Appl, 69, 33-54, 199. [3] M. Krstic, B.-J. Guo, A. Balogh, A. Smyshlyaev, Outputfeedback stabilization of an unstable wave equation, Automatica, vol. 44, pp , 8. [4] I. Lasiecka R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optim, vol. 5, pp , 199. [5] A. Smyshlyaev M. Krstic, Backstepping observers for a class of parabolic PDEs, Systems Control Letters, vol. 54, pp , 5. [6] A. Smyshlyaev M. Krstic, Closed form boundary state feedbacks for a class of 1-D partial integro-differential equations, IEEE Trans. on Automatic Control, 49(1)(4), [7] C. Bardos, L. Halpern, G. Lebeau, J. Rauch, E. Zuazua, Stabilization of the wave equation by Dirichlet type boundary feedback, Asymptotic Analysis, vol.4, no.4, pp ,