Dead-Time Compensation for Wave/String PDEs
Smith predictorlike designs for compensation of arbitrarily long input delays are com- Miroslav Krstic monly available only for finite-dimensional systems. Only very few examples exist, where Professor such compensation has been achieved for partial differential equation (PDE) systems, Department of Mechanical and Aerospace including our recent result for a parabolic (reaction-diffusion) PDE. In this paper, we Engineering, address a more challenging wave PDE problem, where the difficulty is amplified by University of California, allowing all of this PDE’s eigenvalues to be a distance to the right of the imaginary axis. San Diego La Jolla, CA 92093-0411 Antidamping (positive feedback) on the uncontrolled boundary induces this dramatic e-mail: [email protected] form of instability. We develop a design that compensates an arbitrarily long delay at the input of the boundary control system and achieves exponential stability in closed-loop. We derive explicit formulae for our controller’s gain kernel functions. They are related to the open-loop solutions of the antistable wave equation system over the time period of input delay (this simple relationship is the result of the design approach). ͓DOI: 10.1115/1.4003638͔
1 Introduction not as challenging a problem as wave/hyperbolic equations since this class of PDEs does not exhibit a Datko-type loss of delay 1.1 Background on Delay Compensation and Predictor robustness margin. Feedback. Despite decades of development, the area of delay systems remains a seemingly endless source of research chal- 1.3 Contribution of the Present Paper. Our focus in the lenges and continues to be an active area of research, especially in present paper is on wave equations with input delay, as displayed mechanical engineering ͓1–14͔. in Fig. 1. We pursue a problem similar to Ref. ͓45͔ but with two One problem in control of delay systems that has received con- differences, one methodological and one in the system being con- tinuous attention over the last 5 decades is the problem of systems sidered. The methodological difference is that we approach the with input or output delay, or, simply, systems with dead-time. problem with a backstepping-based technique for compensation of Starting with the Smith predictor ͓15͔, ODE systems with dead- the input delay and arrive at a compensator, whose gain functions time have been studied successfully from numerous angles, lead- we derive explicitly, which achieves exponential stabilization in ing even to adaptive designs and designs for nonlinear ODEs the presence of arbitrarily long actuator delay. The other differ- ͓16–42͔. ence, in the class of systems, is that we consider a wave equation of an unconventional type. The standard wave equation has all of 1.2 Delay Compensation for PDEs. Control of PDEs with its infinitely many eigenvalues on the imaginary axis. In this pa- input delays, or, to be more precise, predictor-based compensation per, we consider a wave equation that contains an antidamping of actuator or sensor delays in control of PDEs, is a new area that effect on the boundary opposite to the controlled boundary. This is just opening up for research. The motivation partly comes from antidamping effect results in all of the infinitely many eigenvalues the classical examples by Datko et al. ͓43͔ and Datko ͓44͔ who of the uncontrolled system being in the right-half plane, and pos- identified some classes of hyperbolic PDE systems, which, al- sibly being arbitrarily far to the right of the imaginary axis. though exponentially stabilizable, have zero robustness to insert- One of the key tools employed in the design in this paper is a ing a delay in the feedback loop, i.e., arbitrarily small dead-time new type of backstepping transformation recently introduced by leads to instability. Smyshlyaev and Krstic ͓58͔ for wave equations with boundary In recent work, Guo and Xu ͓45͔ presented an observer-based antidamping. The rest of the tools are a collection of techniques compensator of sensor delay for a wave equation, which Guo and we recently introduced in Refs. ͓24,55,59,60͔. Chang ͓46͔, then, followed by an extension to an Euler–Bernoulli The difference between the present paper and Ref. ͓58͔ is that beam problem. The wave/string PDE is the subject of continuing the presence of the delay requires the construction of an additional advances in boundary control ͓47–50͔. It serves as a benchmark backstepping transformation for the delay state, which has to also and a stepping stone for the more complex beam equation. While incorporate the displacement and velocity states of the wave equa- few experiments have been conducted with stringlike systems, tion. As a result, the construction of the transformation kernels in several experimental applications of boundary control of beams this paper is much more complex than for the wave equation alone have been reported ͓51–54͔. in Ref. ͓58͔. The results of the present paper are of interest even in In a recent manuscript ͓55͔, we initiated an effort for develop- the absence of antidamping in the wave equation ͓58͔; in that ing predictor feedbacks for PDEs with input delay, via the method case, one obtains a novel control law for the conventional un- of backstepping ͑see Refs. ͓56,57,68,69͔ for other uses of back- damped wave/string system with input delay. The difference be- stepping in boundary control of PDEs͒. The PDE considered in tween the present paper and Ref. ͓55͔ is that the second-order Ref. ͓55͔ is a parabolic ͑reaction-diffusion͒ system, which, al- character of the PDE plant makes not only the backstepping trans- though open-loop unstable and with a long delay at the input, is formation of the delay state more complex but it also makes the stability analysis much more involved, requiring, among other things, that the input delay state be quantified not in the L2 norm Contributed by the Dynamic Systems Division of ASME for publication in the ͑ ͒ ͑ as appropriate when the plant is an ODE in the H1 norm as JOURNAL OF DYNAMIC SYSTEMS,MEASUREMENT, AND CONTROL. Manuscript received ͒ December 6, 2008; final manuscript received April 5, 2010; published online March appropriate when the plant is a parabolic PDE but in the H2 23, 2011. Assoc. Editor: Michael A. Demetriou. norm.
Journal of Dynamic Systems, Measurement, and Control MAY 2011, Vol. 133 / 031004-1 Copyright © 2011 by ASME
Downloaded 23 Mar 2011 to 137.110.36.98. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm anti-stable anti-damping U(t) delay force wave PDE
control Fig. 1 Antistable wave PDE system with input delay
1.4 Physical Motivation for the Wave Equation With Fig. 3 A diagram of a string with control applied at boundary Antidamping. The 1D wave equation is a good model for acous- x=1 and an antidamping force acting at the boundary x=0 tic dynamics in ducts and vibration of strings. The phenomenon of antidamping that we include in our study of wave dynamics rep- resents injection of energy in proportion to the velocity field, akin to a damper with a negative damping coefficient. Such a process u͑1,t͒ = U͑t − D͒͑3͒ arises in combustion dynamics, where the pressure field is dis- where for qϾ0 and U͑t͒ϵ0, one obtains an “antistable” wave turbed in proportion to varying heat release rate, which, in turn, is equation, with all of its infinitely many eigenvalues in the right- proportional to the rate of change of pressure. Figure 2 shows the half plane and given by Ref. ͓56͔,p.82 classical Rijke tube control experiment ͓61͔. For the sake of our study, we assume that the system is controlled using a loud- 1 1 1+q n + , 0 Յ q Ͻ 1 speaker, with a pressure sensor ͑microphone͒ near the speaker. ͯ ͯ 2 ͑ ͒ n = ln + j Ά · 4 Both the actuator and the sensor are placed as far as possible from 2 1−q n, q Ͼ 1 the flame front, for their thermal protection. The process of anti- damping is located mostly at the flame front. In a Rijke tube, the The structure of the system is such that the destabilizing force acts flame front may not be at the end of the tube, however, the leaner on the opposite boundary from the input, as represented by the the fuel/air mixture, the longer it takes for the mixture to ignite, string example in Fig. 3. and the further the flame is from the injection point. In the limit, By denoting v͑x,t͒U͑t+x−1−D͒, the delay-wave systems for the leanest mixture for which combustion is sustained, the ͑1͒–͑3͒ are alternatively written as transport-wave PDE cascades flame is near the exit of the tube, which corresponds to a situation ͑ ͒ ͑ ͒ ͑ ͒͑͒ with boundary antidamping. utt x,t = uxx x,t , x 0,1 5 An input delay can arise from a computational delay, as de- u ͑0,t͒ =−qu ͑0,t͒͑6͒ picted in Fig. 2. Alternatively, if the system is controlled not using x t loudspeaker actuation but using modulation of the fuel injection u͑1,t͒ = v͑1,t͒͑7͒ rate ͓62–64͔, a long delay may result both from the delay associ- ated with the servovalve and feed line operation and from the v ͑x,t͒ = v ͑x,t͒, x ͓1,1 + D͒͑8͒ transport process within the combustor from the injection point to t x the ignition location. The leaner the mixture, the longer the input v͑1+D,t͒ = U͑t͒͑9͒ delay in this configuration. ͑ ͒ ͑ ͒ Another physical example involving antidamping in string dy- where U t is the overall system input and u,ut ,v is the state. namics is discussed in Sec. 6. Hence, we formulate the control problem as boundary control of a cascade of a transport PDE with a wave equation. For an example 1.5 Organization of This Paper. This paper is organized as of a control problem involving a nonscalar transport PDE with follows. In Sec. 2, we present a control design and state the sta- reaction effects and nonlinearities, see Ref. ͓65͔. bility result. In Sec. 3, we derive the gain functions of this control The spatial variable x, the time variable t, and the delay D are law explicitly. In Sec. 4, we prove exponential stability of the non dimensional. This is achieved by a standard nondimensional- target system resulting from the backstepping construction. In ization procedure. Consider a more general wave equation Sec. 5, we establish our main result, through a stability analysis 2 uˇˇtˇt͑xˇ ,ˇt͒= uˇ xˇxˇ͑xˇ ,t͒ on the domain xˇ ͑0,L͒, where L is the length for the system in its original variables. In Sec. 6, we offer some closing comments. of the domain in meters, is the wave propagation speed in meters per second, and ˇt is time in seconds. We introduce a new 2 Control Design for Antistable Wave PDE With Input spatial variable x=xˇ /L and a new time variable t=ˇt/L, which are ͑ ͒ Delay both nondimensional. Then, the new state variable u x,t =uˇ͑Lx,Lt/͒ satisfies the wave equation u ͑x,t͒=u ͑x,t͒ for x We consider the delay-wave cascade system tt xx ͑0,1͒. Similarly, if the boundary input of the wave equation is ͑ ͒ ͑ ͒ ͑ ͒͑͒ utt x,t = uxx x,t , x 0,1 1 uˇ͑L,ˇt͒=Uˇ ͑ˇt−Dˇ ͒, the new control is defined as U͑t͒=Uˇ ͑Lt/͒ and ͑ ͒ ͑ ͒͑͒ the actuator state is defined as v͑x,t͒=Uˇ ͑L͑t+x−1−D͒/͒, where ux 0,t =−qut 0,t 2 D=Dˇ /L. The quantity 1+D in Eqs. ͑8͒ and ͑9͒ is nondimen- x sional. 1 0 We consider the backstepping transformation x q͑q + c͒ q + c loudspeaker ACOUSTICS ͑ ͒ ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ w x,t = u x,t − u 0,t + ut y,t dy 10 1+qc 1+qc 0 pressure flame front sensor x injection of z͑x,t͒ = v͑x,t͒ − ͵ p͑x − y͒v͑y,t͒dy − ͑x͒u͑0,t͒ fuel/air mixture 1 CONTROLLER (computational delay) 1 1 ͵ ␥͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ ͑ ͒ − x,y u y,t dy − x,y ut y,t dy 11 0 0 Fig. 2 Control of a thermoacoustic instability in a Rijke tube ͑ ͒ †61‡„a duct-type combustion chamber… where Eq. 10 was introduced by Smyshlyaev and Krstic in Ref.
031004-2 / Vol. 133, MAY 2011 Transactions of the ASME
Downloaded 23 Mar 2011 to 137.110.36.98. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm ͓ ͔ ␥ 58 , the kernels p, , , need to be chosen to transform the ϖ −(−−59→) ϑ −(−−45→) ς −(−−43→) ρ cascade PDE system into the target system ͑ ͒ ͑ ͒ ͑ ͒͑͒ wtt x,t = wxx x,t , x 0,1 12 Fig. 4 The process of finding the solution to the PDEs „18…–„22… for the gain function „x,y…. First, the undamped wave ͑ ͒ ͑ ͒͑͒ equation with homogeneous boundary conditions, Eqs. wx 0,t = cwt 0,t 13 „60…–„62… or Eqs. „63…–„65…, is solved for using Lemmas 4 w͑1,t͒ = z͑1,t͒͑14͒ and 5. Second, Lemma 3 yields the solution to the wave equation with domainwide antidamping , Eqs. „47…–„49… or ͑ ͒ ͑ ͒ ͓ ͒͑͒ Eqs. „50…–„52…. Third, Lemma 2 yields the solution ς to the wave zt x,t = zx x,t , x 1,1 + D 15 equation with boundary antidamping q, Eqs. „38…–„42…. Finally, using Eq. 43 , the gain functions , , , which are needed in z͑1+D,t͒ =0 ͑16͒ „ … x y the controller Eq. „37…, are found in Proposition 6. and c is a positive gain, used in the gain kernels of the control law 1+D ͑ ͒ ͵ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ U t = p 1+D − y v y,t dy + 1+D u 0,t =wxx x,t ,wx 0,t =cwt 0,t ,w 1,t =0, whose eigenvalue Eq. 1 ͑26͒ was determined in Ref. ͓56͔,p.82. 1 1 The inverse backstepping transformation, which is needed in ͵ ␥͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ our analysis, is given by + 1+D,y u y,t dy + 1+D,y ut y,t dy 0 0 x c͑q + c͒ q + c ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ 17 u x,t = w x,t − w 0,t − wt y,t dy 27 1+qc 1+qc 0 The target system ͑w,z͒, which is a transport-wave cascade inter- connected through a boundary, is exponentially stable in an ap- x propriate norm, as we shall establish in Sec. 4. v͑x,t͒ = z͑x,t͒ − ͵ ͑x − y͒z͑y,t͒dy − ͑x͒w͑0,t͒ By differentiating Eq. ͑11͒ once with respect to t and once with 1 respect to x, equating the two expressions, substituting the PDEs 1 1 for u and v, and integrating by parts with respect to y, we obtain − ͵ ␦͑x,y͒w͑y,t͒dy − ͵ ͑x,y͒w ͑y,t͒dy ͑28͒ conditions that the kernels p,,␥, need to satisfy in order for the t 0 0 ͑u,v͒-system and the ͑w,z͒-system to be equivalent. The kernel ␦ is governed by the PDE where the kernels , , , are defined next. The kernel is governed by the PDE ͑x,y͒ = ͑x,y͒͑18͒ xx yy ͑ ͒ ͑ ͒͑͒ xx x,y = yy x,y 29 ͑x,0͒ =−q ͑x,0͒͑19͒ y x ͑ ͒ ͑ ͒͑͒ y x,0 = c x x,0 30 ͑x,1͒ =0 ͑20͒ ͑x,1͒ =0 ͑31͒ where x͓1,1+D͔ and y͑0,1͒. Note that this system has a ͓ ͔ ͑ ͒ structure identical to the uncontrolled wave equation plant. Since where x 1,1+D and y 0,1 . Again, note that the structure Eqs. ͑19͒ and ͑20͒ play the role of boundary conditions, the vari- of this system is identical to the target system w. The initial con- able x can be viewed as a timelike variable, even though it plays dition of this PDE is the role of a spatial variable in the kernel of the transformation q + c Eq. ͑11͒. The initial condition of Eq. ͑18͒ is ͑1,y͒ = ͑32͒ 1+qc q + c ͑1,y͒ =− ͑21͒ ͑ ͒ ͑ ͒ 1+qc x 1,y =0 33 After solving for ͑x,y͒, the kernels ,,␦ are obtained as ͑ ͒ ͑ ͒ x 1,y =0 22 ͑s͒ =− ͑1+s,1͒, s ͓0,D͔͑34͒ After solving for ͑x,y͒, the kernels p,,␥ are obtained as y ͑ ͒ ͑ ͒ ͓ ͔͑͒ ͑x͒ = c͑x,0͒͑35͒ p s =− y 1+s,1 , s 0,D 23 ␦͑ ͒ ͑ ͒͑͒ ͑x͒ =−q͑x,0͒͑24͒ x,y = x x,y 36 ␥͑ ͒ ͑ ͒͑͒ x,y = x x,y 25 We present a detailed Lyapunov stability analysis in Secs. 4 and 3 Explicit Gain Functions 5, however, we state first a result on closed-loop eigenvalues ͑for In the standard predictor feedback form, the controller is writ- the target system ͑w,z͒͒. ten as PROPOSITION 1. The finite part of the spectrum of systems 1 (12)–(16) is given by ͑ ͒ ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ U t =−q 1+D,0 u 0,t + x 1+D,y u y,t dy 1 0 1 1+c n + , 0 Յ c Ͻ 1 ͯ ͯ 2 ͑ ͒ 1 t n =− ln + j Ά · 26 2 1−c ͑ ͒ ͑ ͒ ͑ ͒ ͑͒ n, c Ͼ 1 + ͵ 1+D,y ut y,t dy − ͵ y t − ,1 U d 0 t−D where n Z. ͑ ͒ Proof. Systems ͑12͒–͑16͒ are a cascade connection of the trans- 37 ͑ ͒ port PDE zt͑x,t͒=zx͑x,t͒,z͑1+D,t͒=0, whose eigenvalues have and so, our task is to find the solution x,y and its first deriva- real parts equal to negative infinity ͑see, for example, Ref. ͓66͔, tives with respect to both x and y. The procedure of solving for ͑ ͒ Appendix C͒, and of the boundary-damped wave PDE wtt͑x,t͒ x,y is outlined in Fig. 4.
Journal of Dynamic Systems, Measurement, and Control MAY 2011, Vol. 133 / 031004-3
Downloaded 23 Mar 2011 to 137.110.36.98. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm 0 ͑0,t͒ =0 ͑51͒ −0.5 ͑1,t͒ + ͑1,t͒ =0 ͑52͒ −1 t Proof. By deriving and using the fact that the transformation λ w yieldsۋu −1.5 −2 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒͑͒ x x,t = cosh x x x,t + sinh x t x,t 53 −2.5 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒͑͒ t x,t + x,t = sinh x x x,t + cosh x t x,t 54 −3 0 0.5 1 1.5 2 2.5 3 u yieldsۋq and that the transformation w
͑x,t͒ = cosh͑x͒ ͑x,t͒ − sinh͑x͒͑ ͑x,t͒ + ͑x,t͒͒ ͑55͒ Fig. 5 The graph of the function „q… x x t ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒͑ ͑ ͒ ͑ ͒͒ t x,t = − sinh x x x,t + cosh x t x,t + x,t ͑56͒ We seek the solution of Eqs. ͑18͒–͑22͒ by seeking the solution of the ͑space-reversed͒ PDE system The reader should note that the relationship between q and a, ͑ ͒ ͑ ͒ ͑ ͒͑͒ which is given by Eq. 46 , yields tt x,t = xx x,t 38 1 1−q x/2 1+q x/2 ͑0,t͒ =0 ͑39͒ sinh͑x͒ = ͩͯ ͯ − ͯ ͯ ͪ ͑57͒ 2 1+q 1−q ͑ ͒ ͑ ͒͑͒ x 1,t = q t 1,t 40 1 1−q x/2 1+q x/2 with initial conditions cosh͑x͒ = ͩͯ ͯ + ͯ ͯ ͪ ͑58͒ 2 1+q 1−q q + c ͑x,0͒ =− ͑41͒ 1+qc To make the -system easily solvable, we introduce another transformation, given in the next lemma. This lemma completely ͑x,0͒ =0 ͑42͒ eliminates the in-domain antidamping but using time-dependent t scaling of the state variable. from which we shall obtain LEMMA 3. Let ͑x,t͒=e t͑x,t͒, i.e., ͑x,y͒ = ͑1−y,x −1͒, y ͓0,1͔,x Ն 1 ͑43͒ 1+q t/2 We present the construction of ͑x,t͒ through a series of lem- ͑x,t͒ = ͯ ͯ ͑x,t͒͑59͒ 1−q mas. Most lemmas that we state are reasonably straightforward to prove, or the proofs can be obtained by direct ͑albeit possibly For q͓0,1͒, the function ͑x,t͒ satisfies Eqs. (47)–(49) if and lengthy͒ verification. Hence, we omit most proofs. only if the function ͑x,t͒ satisfies The first lemma introduces a backstepping-style transformation, ͑ ͒ ͑ ͒͑͒ which moves the antidamping effect from the boundary condition tt x,t = xx x,t 60 x͑1,t͒=qt͑1,t͒ into the domain, where it can be handled ͑for the ͒ purpose of solving the PDE more easily. ͑0,t͒ =0 ͑61͒ LEMMA 2. Consider systems (38)–(40) and the transformations x ͑ ͒ ͑ ͒ x 1,t =0 62 ͑x,t͒ = cosh͑x͒͑x,t͒ + ͵ sinh͑y͒͑ ͑y,t͒ − ͑y,t͒͒dy t Ͼ ͑ ͒ 0 For q 1, the function x,t satisfies Eqs. (50)–(52) if and only if the function ͑x,t͒ satisfies ͑44͒ ͑ ͒ ͑ ͒͑͒ x tt x,t = xx x,t 63 ͑ ͒ ͑ ͒͑ ͒ ͵ ͑ ͒͑ ͑ ͒ ͑ ͒͒ x,t = cosh x x,t − sinh y t y,t +2 y,t dy 0 ͑0,t͒ =0 ͑64͒ ͑45͒ ͑1,t͒ =0 ͑65͒ where denotes the function t Furthermore, if the initial conditions of systems (38)–(40) are − tanh−1͑q͒, q ͓0,1͒ ͑q͒ = ͭ ͮ ͑46͒ given by Eqs. (41) and (42), then the initial conditions −1 ͑ ͒͑ ͒ ͑ ͒ ͑ ͒ − coth ͑q͒, q Ͼ 1 0 x x,0 and 1 x t x,0 for the -system are given which is shown in Fig. 5. For q͓0,1͒, the function ͑x,t͒ satis- by fies Eqs. (38)–(40) if and only if the function ͑x t͒ satisfies , q + c ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 2͑ ͒ ͑ ͒͑͒ 0 x =− 66 tt x,t +2 t x,t + x,t = xx x,t 47 1+qc ͑ ͒ ͑ ͒ 0,t =0 48 ͑ ͒ ͑ ͒ 1 x =0 67 ͑ ͒ ͑ ͒ x 1,t =0 49 The -systems are now readily solvable in explicit form. Their For qϾ1, the function ͑x,t͒ satisfies Eqs. (38)–(40) if and only if solutions, for arbitrary initial conditions, are stated in the next two lemmas. the function ͑x,t͒ satisfies LEMMA 4. For q͓0,1͒, the solution of systems (60)–(62) with ͑ ͒ ͑ ͒ 2͑ ͒ ͑ ͒͑͒ tt x,t +2 t x,t + x,t = xx x,t 50 arbitrary initial conditions is
031004-4 / Vol. 133, MAY 2011 Transactions of the ASME
Downloaded 23 Mar 2011 to 137.110.36.98. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm ϱ 1 ͑ ͒ ͩ͑ ͒ ͪ ͵ ͩ͑ ͑x,y͒ =−͑x,y͒ +e−͑x−1͒ͫcosh͑͑1−y͒͒ ͑1−y,x −1͒ x,t =2͚ sin 2n +1 x ͫ sin 2n x t n=0 2 0 1−y +1͒ yͪ ͑y͒dy cosͩ͑2n +1͒ tͪ ͵ ͑ ͒͑ ͑ ͒ ͑ ͒͒ 0 − sinh s s,x −1 + a s,x −1 dsͬ 2 2 tt t 0 1 2 ͑73͒ + ͵ sinͩ͑2n +1͒ yͪ ͑y͒dy sinͩ͑2n ͑2n +1͒ 2 1 0 ͑ ͒ −͑x−1͓͒ ͑͑ ͒͒͑ ͒ ͑͑ y x,y =e − sinh 1−y 1−y,x −1 − cosh 1 − y͒͒ ͑1−y,x −1͒ − sinh͑͑1−y͒͒͑ ͑1−y,x −1͒ +1͒ tͪͬ ͑68͒ x t 2 + a͑1−y,x −1͔͒͒ ͑74͒ Ͼ For q 1, the solution of systems (63)–(65) with arbitrary initial where, for q͓0,1͒ , the functions x͑x,t͒,t͑x,t͒,tt͑x,t͒ are conditions is given by ϱ ϱ q + c 1 ͑x,t͒ =− 2͚ cosͩ͑2n +1͒ xͪcosͩ͑2n +1͒ tͪ ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ ͑ ͒ x 1+qc 2 2 x,t =2͚ sin n x ͫ sin n y y dy cos n t n=0 0 n=1 0 ͑75͒ 1 1 + ͵ sin͑ny͒ ͑y͒dy sin͑nt͒ͬ ͑69͒ ϱ 1 q + c n ͑ ͒ ͩ͑ ͒ ͪ ͩ͑ ͒ ͪ ͑ ͒ 0 x,t = 2͚ sin 2n +1 x sin 2n +1 t 76 t 1+qc 2 2 For the initial condition Eqs. ͑66͒ and ͑67͒, we obtain the ex- n=0 plicit solutions ͑x,t͒ as follows. ϱ q + c LEMMA 5. For q͓0,1͒, the solution of systems (60)–(62) with ͑ ͒ ͑ ͒ ͩ͑ ͒ ͪ ͩ͑ ͒ ͪ tt x,t = 2͚ 2n +1 sin 2n +1 x cos 2n +1 t initial conditions given by Eqs. (66) and (67) is 1+qc n=0 2 2 2 ͑77͒ Ͼ ͑ ͒ ͑ ͒ ͑ ͒ ϱ sinͩ͑2n +1͒ xͪcosͩ͑2n +1͒ tͪ and for q 1, the functions x x,t , t x,t , tt x,t are given by q + c 2 2 ͑x,t͒ =− 2 = ϱ ͚ q + c 1+qc n=0 ͑ ͒ ͑ ͒ ͑ ͒͑͒ ͑2n +1͒ x x,t =− 2͚ cos 2m x cos 2m t 78 2 1+qc m=1
ϱ ϱ sinͩ͑2n +1͒ ͑x + t͒ͪ − sinͩ͑2n +1͒ ͑t − x͒ͪ q + c q + c 2 2 ͑x,t͒ = 2͚ sin͑2mx͒sin͑2mt͒͑79͒ − ͚ t 1+qc 1+qc m=1 n=0 ͑2n +1͒ 2 ϱ q + c ͑ ͒ ͑ ͒ ͑ ͒͑͒ ͑70͒ tt x,t = 2͚ m sin 2m x cos 2m t 80 1+qc m=1 For qϾ1, the solution of systems (63)–(65) with initial conditions given by Eqs. (66) and (67) is 4 Stability of the Target System „w,z… ϱ Having completed the derivation of explicit expressions for the q + c sin͑2mx͒cos͑2mt͒ ͑x,t͒ =− 2 control gains in Sec. 3, we now turn our attention to the exponen- ͚ 1+qc m=1 m tial stability analysis of the target system ϱ w ͑x,t͒ = w ͑x,t͒, x ͑0,1͒͑81͒ q + c sin͑2m͑x + t͒͒ − sin͑2m͑t − x͒͒ tt xx =− ͚ ͑ ͒ ͑ ͒͑͒ 1+qc m=1 m wx 0,t = cwt 0,t 82 ͑ ͒ 71 w͑1,t͒ = z͑1,t͒͑83͒ Now, we return to the gain formula ͑43͒ and, using Lemmas ͑ ͒ ͑ ͒ ͓ ͒͑͒ 2–5, obtain the following explicit expression for the gain kernel zt x,t = zx x,t , x 1,1 + D 84 ͑x,y͒, as well as for the functions ͑1+D,y͒ and ͑t−,1͒, x y ͑ ͒ ͑ ͒ which are used in the control law Eq. ͑37͒. z 1+D,t =0 85 PROPOSITION 6. The solution of systems (18)–(22) is given by The analysis employs two new transformations, whose role is de- picted and explained in Fig. 6. 1−y We first denote ͑ ͒ ͑x,y͒ =e− x−1 ͫcosh͑͑1−y͒͒͑1−y,x −1͒ − ͵ sinh͑s͒ −1 tanh ͑c͒, c ͓0,1͒ ͑ ͒ ͭ ͮ ͑ ͒ 0 a c = 86 coth−1͑c͒, c Ͼ 1 ϫ͑ ͑s,x −1͒ + a͑s,x −1͒͒dsͬ ͑72͒ and then introduce the transformations t ͑ ͒ ͑ ͑ ͒͒ ͑ ͒ ͑ ͑ ͒͒ ͑ ͒͑ ͒ x x,t = cosh a 1−x wx x,t − sinh a 1−x wt x,t 87 where for q͓0,1͒, the function ͑·,·͒ is given by Eq. (70) and ͑ ͒ ͑ ͒ ͑ ͑ ͒͒ ͑ ͒ ͑ ͑ ͒͒ ͑ ͒ for qϾ1, the function ͑·,·͒ is given by Eq. (71). Furthermore, t x,t + a x,t = − sinh a 1−x wx x,t + cosh a 1−x wt x,t the partial derivatives of Eq. (72) are ͑88͒
Journal of Dynamic Systems, Measurement, and Control MAY 2011, Vol. 133 / 031004-5
Downloaded 23 Mar 2011 to 137.110.36.98. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm unstable U(t) boundary (v,u) −(−−−−−−10), (11→) (z,w) −(−−93→) (z,ϕ) −(−−−101→) (z,ψ) v interconnection u Fig. 7 The transformations among the four system represen- ¾ tations in Fig. 6. Only the transformation „v,u… „z,w… is of backstepping type, whereas the other two have the role of mov- boundary ing the damping and moving the transport-wave interconnec- interconnection tion from a boundary into the domain, which is a form that z w facilitates the analysis.
damping at x = 0 ͑1,t͒ = z͑1,t͒͑97͒ boundary For cϾ1, the transformation converts the w-system into interconnection ͑ ͒ ͑ ͒ 2͑ ͒ ͑ ͒͑͒ z φ tt x,t +2a t x,t + a x,t = xx x,t 98 domain-wide ͑ ͒ ͑ ͒ ͑ ͒ t 0,t + a 0,t =0 99 damping ͑1,t͒ = z͑1,t͒͑100͒ in-domain Even though the z-system is the exponentially stable transport interconnection ͑ ͒ ͑ ͒ ͑ ͒ z ψ equation zt x,t =zx x,t , z 1,t =0, the stability analysis for the -system cannot proceed in this form because of z͑1,t͒ entering domain-wide the -system through a boundary condition, which makes the re- damping sulting input operator unbounded, and the resulting gain from z͑1,t͒ to ͑x,t͒͑in any suitable norm͒ unbounded. Fig. 6 A sequence of system transformations employed in the We first perform a transformation that shifts z͑1,t͒ into the analysis. The original plant is „v,u…, whereas the backstepping interior of the domain ͑0,1͒. The next lemma introduces this trans- transformation converts the closed-loop system into the au- formation and presents a Lyapunov function for the resulting sys- tonomous, exponentially stable system z,w . Due to the „ … tem ͑see Fig. 7 for a summary of transformations used in the boundary interconnection and the boundary damping at x=0, paper͒. the stability of the system „z,w… is hard to analyze. The damp- ing is moved from the boundary to the domain using the trans- LEMMA 7. Consider the change of variable ¾ formation „z,w… „z, …. Finally, the boundary interconnection, ͑x,t͒ = ͑x,t͒ − x2z͑1,t͒͑101͒ in which an unbounded operator arises, is converted into an easier-to-analyze in-domain interconnection using the transfor- and the resulting system, which, for c͓0,1͒,is mation z, ¾ z, . The transformations between the four „ … „ … ͑ ͒ ͑ ͒ 2͑ ͒ ͑ ͒ ͑ ͒͑͒ representations are indicated in Fig. 7. The stability analysis of tt x,t +2a t x,t + a x,t = xx x,t + g x,t 102 the „z, …-system is outlined in Fig. 8. Stability of the ͑0,t͒ =0 ͑103͒ „z, …-system is studied in Lemmas 7–10, using the Lyapunov x function Ω„t…. Then, stability of the „z, …-system is shown in Lemma 11 and Proposition 12. ͑1,t͒ =0 ͑104͒ and for cϾ1 is ͑ ͒ ͑ ͒ 2͑ ͒ ͑ ͒ ͑ ͒͑͒ tt x,t +2a t x,t + a x,t = xx x,t + g x,t 105 ͑1,t͒ = z͑1,t͒͑89͒ ͑0,t͒ + a͑0,t͒ =0 ͑106͒ and t ͑ ͒ ͑ ͑ ͒͒ ͑ ͒ ͑ ͑ ͒͒͑ ͑ ͒ ͑ ͒͒ ͑1,t͒ =0 ͑107͒ wx x,t = cosh a 1−x x x,t + sinh a 1−x t x,t + a x,t ͑90͒ and where ͑ ͒ ͑ ͒ 2͑ ͑ ͒ ͑ ͒ 2 ͑ ͒͒ ͑ ͒ ͑ ͒ ͑ ͑ ͒͒ ͑ ͒ ͑ ͑ ͒͒͑ ͑ ͒ ͑ ͒͒ g x,t =2z 1,t − x zxx 1,t +2azx 1,t + a z 1,t 108 wt x,t = sinh a 1−x x x,t + cosh a 1−x t x,t + a x,t Then, for the Lyapunov functional ͑91͒ 1 1 ͑ ͒ ͑ ͒͑͒ ͑ ͒ ͵ ͑͑ ͑ ͒ ͑ ͒͒2 2͑ ͒͒ ͑ ͒ w 1,t = z 1,t 92 V t = t x,t + a x,t + x x,t dx 109 2 0 By integrating in x, these transformations are also written as the following holds: 1 ͑x,t͒ = cosh͑a͑1−x͒͒w͑x,t͒ + ͵ sinh͑a͑1−y͒͒͑w ͑y,t͒ 1 t ˙ ͑ ͒ ͑ ͒ ͵ ͑ ͑ ͒ ͑ ͒͒ ͑ ͒ ͑ ͒ x V t =−2aV t + t x,t + a x,t g x,t dx 110 0 − aw͑y,t͒͒dy ͑93͒ for all tՆ0. 1 Proof. Most of this lemma is obtained by direct verification. ͑ ͒ ͑ ͑ ͒͒͑ ͒ ͵ ͑ ͑ ͒͒͑ ͑ ͒ The expression w x,t = cosh a 1−x x,t − sinh a 1−y t y,t x ͑ ͒ ͑ ͒ ͑ ͒ 2͑ ͒ ͑ ͒ ͑ ͒ g x,t = tt x,t +2a t x,t + a x,t − xx x,t = tt x,t ͑ ͒͒ ͑ ͒ +2a y,t dy 94 ͑ ͒ 2͑ ͒ ͑ ͒ ͑ ͒ 2͑ ͑ ͒ +2a t x,t + a x,t − xx x,t +2z 1,t − x zxx 1,t ͓ ͒ For c 0,1 , the transformation converts the w-system into ͑ ͒ 2 ͑ ͒͒ ͑ ͒ 2͑ ͑ ͒ ͑ ͒ +2azx 1,t + a z 1,t =2z 1,t − x zxx 1,t +2azx 1,t ͑x,t͒ +2a ͑x,t͒ + a2͑x,t͒ = ͑x,t͒͑95͒ tt t xx + a2z͑1,t͒͒ ͑111͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ x 0,t =0 96 involves zx 1,t and zxx 1,t . These quantities are obtained as fol-
031004-6 / Vol. 133, MAY 2011 Transactions of the ASME
Downloaded 23 Mar 2011 to 137.110.36.98. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm lows. By setting x=1 in z =z , one obtains z ͑1,t͒=z ͑1,t͒.By ψ t x t x z-system: −−−−−−−→Lemma 8 -system: differentiating zt =zx with respect to t, one obtains ztt=zxt=zxx. Lemma 9 Lemma 7 Setting x=1 yields ztt͑1,t͒=zxx͑1,t͒. Substituting zt͑1,t͒=zx͑1,t͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ and ztt 1,t =zxx 1,t into Eq. 111 , one obtains Eq. 108 . The Lemma 10 derivative of the Lyapunov function is obtained as 1 Fig. 8 The outline of the proof of exponential stability of the ˙ ͑ ͒ ͵ ͑͑ ͑ ͒ ͑ ͒͒͑ ͑ ͒ ͑ ͒͒ cascade system „z, …. The Lyapunov functional for the autono- V t = t x,t + a x,t tt x,t + a t x,t mous z-system is constructed in Lemma 9. The Lyapunov func- 0 tional for the -system is constructed in Lemma 7. The input- 1 to-state stability „ISS… of the -system with respect to the + ͑x,t͒ ͑x,t͒͒dx = ͵ ͑͑ ͑x,t͒ + a͑x,t͒͒͑− a ͑x,t͒ z-system is shown in Lemma 8. The Lyapunov stability of the x xt t t overall z, -system is shown in Lemma 10. 0 „ … 2͑ ͒ ͑ ͒ ͑ ͒͒ ͑ ͒ ͑ ͒͒ − a x,t + xx x,t + g x,t + x x,t xt x,t dx = 1 1 LEMMA 10. For systems (84), (85), and (102)–(108), the follow- ͵ ͑ ͑ ͒ ͑ ͒͒2 ͵ ͑͑ ͑ ͒ ͑ ͒͒ ing holds: − a t x,t + a x,t dx + t x,t + a x,t 0 0 ⍀͑ ͒ Յ⍀ −min͕a,b͖t ∀ Ն ͑ ͒ t 0e , t 0 118 1 Ͼ ϫ͑ ͑ ͒ ͑ ͒͒ ͑ ͒ ͑ ͒͒ ͵ ͑ ͑ ͒ for all b 0, where xx x,t + g x,t + x x,t xt x,t dx =−a t x,t 1+D 0 3͑5+a2͒ ⍀͑t͒ = V͑t͒ + ͵ eb͑x−1͒͑z2͑x,t͒ + z2͑x,t͒ + z2 ͑x,t͒͒dx 1 4a x xx ͑ ͒͒2 ͵ ͑ ͑ ͑ ͒ ͑ ͒͒ ͑ ͒ 1 + a x,t dx + − xt x,t + a x x,t x x,t 0 ͑119͒ ͑ ͒ ͑ ͒͒ ͑ ͑ ͒ ͑ ͒͒ ͑ ͉͒1 Proof. With Lemmas 8 and 9, we obtain + x x,t xt x,t dx + t x,t + a x,t x x,t 0 1 3 1 ⍀˙ ͑ ͒ Յ ͑ ͒ ͩ͑ 2͒ 2͑ ͒ 2 2͑ ͒ 2 ͑ ͒ͪ ͵ ͑ ͑ ͒ ͑ ͒͒ ͑ ͒ ͑ ͒ ͑ ͑ ͒ t − aV t + 5+a z 1,t + a zx 1,t + zxx 1,t + t x,t + a x,t g x,t dx =−2aV t + t x,t 4a 5 0 3͑5+a2͒ 1 ͑ 2͑ ͒ 2͑ ͒ 2 ͑ ͒͒ − z 1,t + zx 1,t + zxx 1,t ͑ ͒͒ ͑ ͉͒1 ͵ ͑ ͑ ͒ ͑ ͒͒ ͑ ͒ 4a + a x,t x x,t 0 + t x,t + a x,t g x,t dx 1+D 0 3͑5+a2͒ ͵ b͑x−1͒͑ 2͑ ͒ 2͑ ͒ 2 ͑ ͒͒ Յ ͑112͒ − b e z x,t + zx x,t + zxx x,t dx 4a 1 where integration by parts is used in the second to last steps. 1+D Equation ͑110͒ is obtained by substituting either of the boundary 3͑5+a2͒ ͑ ͒ ͵ b͑x−1͒͑ 2͑ ͒ 2͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ − aV t − b e z x,t + zx x,t conditions Eq. 103 or Eq. 106 into Eq. 112 . 4a The following lemma is readily verifiable with several applica- 1 tions of Young’s inequality. 2 ͑ ͒͒ ͑ ͒ + zxx x,t dx 120 LEMMA 8. If functions V͑t͒ and g͑x,t͒ satisfy Eqs. (108)–(110), which yields then they satisfy the following bounds: ⍀˙ ͑ ͒ Յ ͕ ͖⍀͑ ͒͑͒ 1 1 t −min a,b t 121 V˙ ͑t͒ Յ − aV͑t͒ + + ʈg͑t͒ʈ2 ͑113͒ 2a 4a and leads to the result of the lemma. Even though we have obtained exponential stability in the ͑,z͒ 1 variables, we have yet to establish exponential stability in the ʈg͑t͒ʈ2 Յ 3ͩ͑5+a2͒z2͑1,t͒ + a2z2͑1,t͒ + z2 ͑1,t͒ͪ ͑114͒ ͑ ͒ x 5 xx w,z variables. We have to consider the chain of transformations ͒͑͒ ͑ ۋ ͒ ͑ ۋ ͒ ͑ Next, we turn our attention to the z-systems ͑84͒ and ͑85͒ and w,z ,z ,z 122 state the following directly verifiable result. as well as their inverses, to establish stability of the ͑w,z͒-system. LEMMA 9. For systems (84) and (85), the following are true: The following lemma, which establishes the equivalence be- 1+D 1+D tween the Lyapunov function ⍀͑t͒ and the appropriate norm of the d ͵ eb͑x−1͒z2͑x,t͒dx =−z͑1,t͒2 − b͵ eb͑x−1͒z2͑x,t͒dx ͑w,z͒-system, is the key to establishing exponential stability in the dt 1 1 ͑w,z͒ variables. ͑115͒ LEMMA 11. Consider the transformation Eqs. (93), (94), and (101), along with the energy function ⍀͑t͒ defined in Eq. (119). 1+D 1+D The following holds: d ͵ b͑x−1͒ 2͑ ͒ ͑ ͒2 ͵ b͑x−1͒ 2͑ ͒ e zx x,t dx =−zx 1,t − b e zx x,t dx ␣ ⌶͑ ͒ Յ⍀͑ ͒ Յ ␣ ⌶͑ ͒͑͒ 1 t t 2 t 123 dt 1 1 where ͑116͒ 1 1+D 1+D 1+D ⌶͑ ͒ ͵ ͑ 2͑ ͒ 2͑ ͒͒ ͵ ͑ 2͑ ͒ 2͑ ͒ t = wx x,t + wt x,t dx + z x,t + zx x,t d b͑x−1͒ 2 2 b͑x−1͒ 2 ͵ e z ͑x,t͒dx =−z ͑1,t͒ − b͵ e z ͑x,t͒dx 0 1 dt xx xx xx 1 1 2 ͑ ͒͒ ͑ ͒ + zxx x,t dx 124 ͑117͒ and for any bϾ0. 3͑5+a2͒ 1 With Lemmas 8 and 9, we obtain the following result, as dis- ␣ =minͭ , ͮ ͑125͒ played in Fig. 8. 1 4a 8 cosh͑2a͒max͕1,2a͖
Journal of Dynamic Systems, Measurement, and Control MAY 2011, Vol. 133 / 031004-7
Downloaded 23 Mar 2011 to 137.110.36.98. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm 8 3͑5+a2͒ 8 ␣ =maxͭ2 cosh͑2a͒, ͑1+a2͒ + ebDͮ ͑126͒ V͑t͒ Յ ʈ ͑t͒ + a͑t͒ʈ2 + ʈ ͑t͒ʈ2 + ͑1+a2͒͑ʈz͑t͒ʈ2 + ʈz ͑t͒ʈ2 2 5 4a t x 5 x ʈ ʈ ʈ ͑ ͒ʈ2͒͑͒ Proof. To save on notation, in this proof, we use the symbol · to + zxx t 139 mean both ʈ·ʈ ͓ ͔ and ʈ·ʈ ͓ ͔. We first consider the transfor- L2 0,1 L2 1,1+D Now, we first focus on Eq. ͑138͒ w. Squaring up Eq. ͑87͒,wegetۋ and ۋmations w 4 w2͑x,t͒ Յ 2 cosh2͑a͑1−x͒͒2͑x,t͒ + 2 sinh2͑a͑1−x͒͒͑ ͑x,t͒ ʈ ͑t͒ + a͑t͒ʈ2 + ʈ ͑t͒ʈ2 Յ 4ͩV͑t͒ + ͑1+a2͒͑ʈz͑t͒ʈ2 + ʈz ͑t͒ʈ2 x x t t x 5 x + a͑x,t͒͒2 ͑127͒ 4 Doing the same with Eqs. ͑88͒, ͑90͒, and ͑91͒, integrating from 0 + ʈz ͑t͒ʈ2͒ͪ Յ 4ͩV͑t͒ + ͑1 xx 5 to 1, and majorizing cosh2͑a͑1−x͒͒ and sinh2͑a͑1−x͒͒ over ͑0,1͒ under the integrals, we get 1+D 2͒͵ b͑x−1͒͑ 2͑ ͒ 2͑ ͒ ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 Յ ͑ 2͑ ͒ 2͑ ͒͒͑ʈ ͑ ͒ʈ2 ʈ ͑ ͒ + a e z x,t + zx x,t wx t + wt t 2 cosh a + sinh a x t + t t 1 + a͑t͒ʈ2͒͑128͒ + z2 ͑x,t͒͒dxͪ ͑140͒ and xx ʈ ͑t͒ʈ2 + ʈ ͑t͒ + a͑t͒ʈ2 Յ 2͑cosh2͑a͒ + sinh2͑a͒͒͑ʈw ͑t͒ʈ2 x t x Invoking Eq. ͑119͒,weget ʈ ͑ ͒ʈ2͒͑͒ + wt t 129 4 From Eq. ͑101͒,weget ͑1+a2͒ 5 ʈ ͑ ͒ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 Յ ⍀͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 2͑ ͑ ͒ ͑ ͒͒ t t + a t + x t 4 max 1, t t x,t + a x,t = t x,t + a x,t + x zx 1,t + az 1,t Ά 3͑5+a2͒ · ͑130͒ 4a ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒͑͒ 141 x x,t = x x,t +2xz 1,t 131 ͑ ͒ ͑ ͒ With a few steps of majorization, it is easy to see that where we have used the fact that zt 1,t =zx 1,t . Taking the L2 norm of both sides of both equations, we obtain 4 ͑1+a2͒ 2 5 16a ʈ ͑ ͒ ͑ ͒ʈ2 Յ ʈ ͑ ͒ ͑ ͒ʈ2 ͑ ͑ ͒ ͑ ͒͒2 Յ Ͻ 2a ͑142͒ t t + a t 2 t t + a t + zx 1,t + az 1,t ͑ 2͒ 5 3 5+a 15 ͑132͒ 4a Hence 2 2 2 2 ʈ ͑ ͒ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 Յ ͕ ͖⍀͑ ͒͑͒ ʈ ͑t͒ʈ Յ 2ʈ ͑t͒ʈ + z ͑1,t͒͑133͒ t t + a t + x t 4max1,2a t 143 x x 3 Recalling Eq. ͑128͒,weget as well as ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 Յ ͑ 2͑ ͒ 2͑ ͒͒ ͕ ͖⍀͑ ͒ wx t + wt t 8 cosh a + sinh a max 1,2a t 2 ʈ ͑t͒ + a͑t͒ʈ2 Յ 2ʈ ͑t͒ + a͑t͒ʈ2 + ͑z ͑1,t͒ + az͑1,t͒͒2 = 8 cosh͑2a͒max͕1,2a͖⍀͑t͒͑144͒ t t 5 x where we have used the fact that cosh2͑a͒+sinh2͑a͒=cosh͑2a͒. ͑134͒ Furthermore, from Eq. ͑119͒,weget 2 a ʈ ͑ ͒ʈ2 Յ ʈ ͑ ͒ʈ2 2͑ ͒͑͒ 2 2 2 4 x t 2 x t + z 1,t 135 ʈz͑t͒ʈ + ʈz ͑t͒ʈ + ʈz ͑t͒ʈ Յ ⍀͑t͒͑145͒ 3 x xx 3͑5+a2͒
Using the fact that z͑1,t͒ϵ0 and zx͑1,t͒ϵ0, where the latter With Eqs. ͑144͒ and ͑145͒, we obtain the left side of inequality ͑ ͒ϵ ͑ ͒ ␣ ͑ ͒ follows from the fact that zt 1,t 0, with Agmon’s inequality, we 123 with 1 given by Eq. 125 . Now, we turn our attention to get that Eq. ͑139͒ and to proving the right-hand-side of inequality ͑123͒. From Eqs. ͑129͒ and ͑139͒,weget ʈ ͑ ͒ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 Յ ͑ʈ ͑ ͒ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2͒ t t + a t + x t 2 t t + a t + x t 8 16 4 2 V͑t͒ Յ 2͑cosh2͑a͒ + sinh2͑a͒͒͑ʈw ͑t͒ʈ2 + ʈw ͑t͒ʈ2͒ + ͑1+a2͒ + ʈz ͑t͒ʈ2 +4ͩ a2 + ͪʈz ͑t͒ʈ2 x t 5 5 xx 5 3 x ϫ͑ʈz͑t͒ʈ2 + ʈz ͑t͒ʈ2 + ʈz ͑t͒ʈ2͒ = 2 cosh͑2a͒͑ʈw ͑t͒ʈ2 ͑136͒ x xx x 8 ʈ ͑ ͒ʈ2͒ ͑ 2͒͑ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2͒ ʈ ͑ ͒ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 Յ ͑ʈ ͑ ͒ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2͒ + wt t + 1+a z t + zx t + zxx t t t + a t + x t 2 t t + a t + x t 5 16 4 2 ͑ ͒ ʈ ͑ ͒ʈ2 ͩ 2 ͪʈ ͑ ͒ʈ2 146 + zxx t +4 a + zx t 5 5 3 Then, with Eqs. ͑119͒ and ͑146͒, we obtain ͑137͒ 8 ⍀͑t͒ Յ 2 cosh͑2a͒͑ʈw ͑t͒ʈ2 + ʈw ͑t͒ʈ2͒ + ͑1+a2͒͑ʈz͑t͒ʈ2 With further majorizations, we achieve simplifications of expres- x t 5 sions 1+D 3͑5+a2͒ 16 + ʈz ͑t͒ʈ2 + ʈz ͑t͒ʈ2͒ + ͵ eb͑x−1͒͑z2͑x,t͒ ʈ ͑t͒ + a͑t͒ʈ2 + ʈ ͑t͒ʈ2 Յ 4V͑t͒ + ͑1+a2͒͑ʈz͑t͒ʈ2 + ʈz ͑t͒ʈ2 x xx 4a t x 5 x 1 ʈ ͑ ͒ʈ2͒͑͒ + z2͑x,t͒ + z2 ͑x,t͒͒dx Յ 2 cosh͑2a͒͑ʈw ͑t͒ʈ2 + ʈw ͑t͒ʈ2͒ + zxx t 138 x xx x t
031004-8 / Vol. 133, MAY 2011 Transactions of the ASME
Downloaded 23 Mar 2011 to 137.110.36.98. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm 8 3͑5+a2͒ x ͩ ͑ 2͒ bDͪ͑ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2͒ + 1+a + e z t + zx t + zxx t z͑x,t͒ = v͑x,t͒ + ͵ ͑1+x − y,1͒v͑y,t͒dy + q͑x,0͒u͑0,t͒ 5 4a y 1 ͑ ͒ 147 1 1 ͑ ͒ ͵ ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ ͑ ͒ This completes the proof of the right side of inequality 123 with − x x,y u y,t dy − x,y ut y,t dy 157 ␣ ͑ ͒ 2 given by Eq. 126 . 0 0 With Lemmas 10 and 11, we prove the following result on exponential stability of the target system ͑w,z͒. x ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ PROPOSITION 12. For systems (81)–(85), the following inequality zx x,t = vx x,t + y 1,1 v x,t + xy 1+x − y,1 v y,t dy holds for the norm Eq. (124) for all bϾ0: 1 1 ␣ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ⌶͑t͒ Յ 2 ⌶ e−min͕a,b͖t, ∀ t Ն 0 ͑148͒ + q x x,0 u 0,t − ͵ xx x,y u y,t dy ␣ 0 1 0 1 ͵ ͑ ͒ ͑ ͒ ͑ ͒ − x x,y ut y,t dy 158 0 5 Stability in the Original Plant Variables „u,v… x We now return to the backstepping transformations ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͵ ͑ u,v͒ in Sec. 2. After the substitution zxx x,t = vxx x,t + y 1,1 vx x,t + xy 1,1 v x,t + xxy 1+x͑ۋw,z͒ and ͑w,z͒͑ۋu,v͒͑ of the gain kernels expressed in terms of ͑x,y͒ and ͑x,y͒, the 1 backstepping transformation is written as 1 ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ − y,1 v y,t dy + q xx x,0 u 0,t − xxx x,y u y,t dy x q͑q + c͒ q + c 0 w͑x,t͒ = u͑x,t͒ − u͑0,t͒ + ͵ u ͑y,t͒dy t 1 1+qc 1+qc 0 − ͵ ͑x,y͒u ͑y,t͒dy ͑159͒ ͑ ͒ xx t 149 0
x The normed quantities appearing in ⌼͑t͒ are given by ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ z x,t = v x,t + y 1+x − y,1 v y,t dy + q x,0 u 0,t q + c 1 u ͑x,t͒ = w ͑x,t͒ − w ͑x,t͒͑160͒ x x 1+qc t 1 1 ͵ ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ ͑ ͒ − x x,y u y,t dy − x,y ut y,t dy 150 q + c 0 0 u ͑x,t͒ =− w ͑x,t͒ + w ͑x,t͒͑161͒ t 1+qc x t and the inverse backstepping transformation is written as x x c͑q + c͒ q + c v͑x,t͒ = z͑x,t͒ + ͵ ͑1+x − y,1͒z͑y,t͒dy − c͑x,0͒w͑0,t͒ ͑ ͒ ͑ ͒ ͑ ͒ ͵ ͑ ͒ y u x,t = w x,t − w 0,t − wt y,t dy 1 1+qc 1+qc 0 1 1 ͑151͒ ͵ ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ ͑ ͒ − x x,y w y,t dy − x,y wt y,t dy 162 0 0 x ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ x v x,t = z x,t + y 1+x − y,1 z y,t dy − c x,0 w 0,t 1 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ vx x,t = zx x,t + y 1,1 z x,t + xy 1+x − y,1 z y,t dy 1 1 1 ͵ ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ ͑ ͒ 1 − x x,y w y,t dy − x,y wt y,t dy 152 0 0 ͑ ͒ ͑ ͒ ͵ ͑ ͒ ͑ ͒ − c x x,0 w 0,t − xx x,y w y,t dy Since the stability result in Proposition 12 is given in terms of 0 1 ⌶͑ ͒ ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 t = wx t + wt t + z t + zx t + zxx t ͵ ͑ ͒ ͑ ͒ ͑ ͒ − x x,y wt y,t dy 163 ͑153͒ 0
and we want to establish stability in terms of x v ͑x,t͒ = z ͑x,t͒ + ͑1,1͒z ͑x,t͒ + ͑1,1͒z͑x,t͒ + ͵ ͑1 ⌼͑t͒ = ʈu ͑t͒ʈ2 + ʈu ͑t͒ʈ2 + ʈv͑t͒ʈ2 + ʈv ͑t͒ʈ2 + ʈv ͑t͒ʈ2 xx xx y x xy xxy x t x xx 1 ͑ ͒ 154 ͒ ͑ ͒ ͑ ͒ ͑ ͒ + x − y,1 z y,t dy − c xx x,0 w 0,t we need to derive the expressions for all of the five normed quan- 1 1 ⌶͑ ͒ tities appearing in t and all of the five normed quantities ap- − ͵ ͑x,y͒w͑y,t͒dy − ͵ ͑x,y͒w ͑y,t͒dy ͑164͒ ⌼͑ ͒ xxx xx t pearing in t . 0 0 The normed quantities appearing in ⌶͑t͒ are given by In establishing a relation between ⌼͑t͒ and ⌶͑t͒, first we estab- ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 q + c lish a relation between ux t + ut t and wx t + wt t . ͑ ͒ ͑ ͒ ͑ ͒͑͒ wx x,t = ux x,t + ut x,t 155 LEMMA 13. Consider the transformation Eqs. (149) and (151), 1+qc along with the norm Eqs. (153) and (154). The following are true: ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 Յ ͑ 2͒͑ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2͒ Յ ͑ 2͒⌶͑ ͒ q + c ux t + ut t 2 1+ wx t + wt t 2 1+ t w ͑x,t͒ = zu ͑x,t͒ + u ͑x,t͒͑156͒ t 1+qc x t ͑165͒
Journal of Dynamic Systems, Measurement, and Control MAY 2011, Vol. 133 / 031004-9
Downloaded 23 Mar 2011 to 137.110.36.98. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 Յ ͑ 2͒͑ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2͒ Յ ͑ 2͒⌼͑ ͒ 1+D 1+D wx t + wt t 2 1+ ux t + ut t 2 1+ t ␦ =6maxͭ1,2͑1,1͒ + D͵ 2 ͑x,1͒dx,4ͩq͵ 2͑x,0͒dx ͑166͒ 1 y xy x 1 1 where 1+D 1 1+D 1 + ͵ ͵ 2 ͑x,y͒dydxͪ,͵ ͵ 2͑x,y͒dydxͮ ͑178͒ q + c xx x = ͑167͒ 1 0 1 0 1+qc ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 ʈ ͑ ͒ʈ2 ⌶͑ ͒ Next, we focus on relating v t + vx t + vxx t to t ␦ =7maxͭ1,2͑1,1͒,2 ͑1,1͒ 2 2 2 2 y xy and ʈz͑t͒ʈ +ʈzx͑t͒ʈ +ʈvxx͑t͒ʈ to ⌼͑t͒. LEMMA 14. For the transformation Eq. (152) along with the 1+D 1+D norm Eq. (153), the following are true: + D͵ 2 ͑x,1͒dx,4ͩq͵ 2 ͑x,0͒dx xxy xx ʈ ͑ ͒ʈ2 Յ ␥ ⌶͑ ͒͑͒ 1 1 v t 0 t 168 1+D 1 1+D 1 + ͵ ͵ 2 ͑x,y͒dydxͪ,͵ ͵ 2 ͑x,y͒dydxͮ ʈ ͑ ͒ʈ2 Յ ␥ ⌶͑ ͒͑͒ xxx xx vx t 1 t 169 1 0 1 0 ͑179͒ ʈv ͑t͒ʈ2 Յ ␥ ⌶͑t͒͑170͒ xx 2 Proof. We only prove inequality ͑170͒. All of the other inequali- where ties are easier to prove. Starting from Eq. ͑164͒,weget
1+D 1+D 2 2 2 2 2 2 2 2 ʈv ͑t͒ʈ Յ 7ͩʈz ͑t͒ʈ + ͑1,1͒ʈz ͑t͒ʈ + ͑1,1͒ʈz͑t͒ʈ ␥ =5maxͭ1+D͵ ͑x,1͒dx,4ͩc͵ ͑x,0͒dx xx xx y x xy 0 y 1 1 1+D 1+D 1 1+D 1 ͵ 2 ͑ ͒ ʈ ͑ ͒ʈ2 2 D x dx z t + ͵ ͵ ͑x,y͒dydxͪ,͵ ͵ 2͑x,y͒dydxͮ + xxy ,1 x 1 1 0 1 0 ͑ ͒ 1+D 171 ͵ 2 ͑ ͒ 2͑ ͒ + c xx x,0 dxw 0,t 1 1+D 1+D 1+D 1 ␥ =6max 1,2͑1,1͒ + D͵ 2 ͑x,1͒dx,4ͩc͵ 2͑x,0͒dx ͭ 2 1 y xy x + ͵ ͵ ͑x,y͒dydxʈw͑t͒ʈ2 1 1 xxx 1 0 1+D 1 1+D 1 1+D 1 + ͵ ͵ 2 ͑x,y͒dydxͪ,͵ ͵ 2͑x,y͒dydx ͑172͒ ͮ 2 xx x + ͵ ͵ ͑x,y͒dydxʈw ͑t͒ʈ2ͪ ͑180͒ 1 0 1 0 xx t 1 0 where we have used the fact that ␥ =7maxͭ1,2͑1,1͒,2 ͑1,1͒ 1+D x−1 1+D 2 y xy ͵ ͵ 2 ͑ ͒ ͵ ͑ ͒2 ͑ ͒ xxy 1+s,1 dsdx = 1+D − x xxy x,1 dx 1+D 1+D 1 0 1 + D͵ 2 ͑x,1͒dx,4ͩc͵ 2 ͑x,0͒dx 1+D xxy xx 2 1 1 Յ ͵ ͑ ͒ ͑ ͒ D xxy x,1 dx 181 1+D 1 1+D 1 1 + ͵ ͵ 2 ͑x,y͒dydxͪ,͵ ͵ 2 ͑x,y͒dydxͮ Employing Agmon’s inequality, we get xxx xx 1 0 1 0 ͑173͒ ʈv ͑t͒ʈ2 Յ 7ͫʈz ͑t͒ʈ2 + 2͑1,1͒ʈz ͑t͒ʈ2 + ͩ2 ͑1,1͒ xx xx y x xy For the transformation Eq. (150) along with the norm Eq. (154), the following are true: 1+D 1+D + D͵ 2 ͑x,1͒dxͪʈz͑t͒ʈ2 +4ͩc͵ 2 ͑x,0͒dx xxy xx ʈ ͑ ͒ʈ2 Յ ␦ ⌼͑ ͒͑͒ z t 0 t 174 1 1 1+D 1 ʈ ͑ ͒ʈ2 Յ ␦ ⌼͑ ͒͑͒ + ͵ ͵ 2 ͑x,y͒dydxͪʈw͑t͒ʈ2 zx t 1 t 175 xxx 1 0 1+D 1 ʈz ͑t͒ʈ2 Յ ␦ ⌼͑t͒͑176͒ xx 2 + ͵ ͵ 2 ͑x,y͒dydxʈw ͑t͒ʈ2ͬ ͑182͒ xx t where 1 0
1+D 1+D from which Eq. ͑170͒ follows with Eq. ͑173͒. ␦ =5maxͭ1+D͵ 2͑x,1͒dx,4ͩq͵ 2͑x,0͒dx With Lemmas 13 and 14, we get the following relation between 0 y ⌼͑ ͒ ⌶͑ ͒ 1 1 t and t . LEMMA 15. The following relation holds between the norm Eqs. 1+D 1 1+D 1 (153) and (154): + ͵ ͵ 2͑x,y͒dydxͪ,͵ ͵ 2͑x,y͒dydxͮ x ␣ ⌼͑ ͒ Յ⌶͑ ͒ Յ ␣ ⌼͑ ͒͑͒ 1 0 1 0 3 t t 4 t 183 ͑177͒ where
031004-10 / Vol. 133, MAY 2011 Transactions of the ASME
Downloaded 23 Mar 2011 to 137.110.36.98. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm 5 Lemma 11 Lemmas 13–1 ϒ(v,u) ⇐⇒ Ξ(z,w) ⇐⇒ Ω(z,ψ)
Fig. 9 The norms used in the stability analysis of the closed- loop system. Their equivalence is established in the lemmas indicated in the figure.
Fig. 10 A pickup on an electric guitar. Based on Faraday’s law of induction, it converts the string velocity into voltage. Con- ␣ /͓ ͑ 2͒ ␥ ␥ ␥ ͔͑͒ 3 =1 2 1+ + 0 + 1 + 3 184 nected into a high-gain amplifier, this system results in „do- mainwide… antidamping, which manifests itself as a swell in vol- ␣ ͑ 2͒ ␦ ␦ ␦ ͑ ͒ 4 =2 1+ + 0 + 1 + 3 185 ume, up to a saturation of the amplifier, which guitarists refer to, simply, as feedback. Finally, we obtain our main result on exponential stability of the ͑u,v͒-system with the help of the lemmas, as indicated in Fig. 9. THEOREM 16. Consider the closed-loop system consisting of the v͑2,t͒ = U͑t͒͑192͒ plant Eqs. (5)–(9) and the control law Eq. (37). The following where D and q are unknown. An adaptive design for this system holds for all bϾ0: would indirectly address the Datko ͓44͔ question but in a more ␣ ␣ challenging setting, where D is not small but it is large and has a ⌼͑t͒ Յ 2 4 ⌼ e−min͕a,b͖t, ∀ t Ն 0 ͑186͒ ␣ ␣ 0 large uncertainty and where the wave equation plant is not neu- 1 3 trally stable but antistable. The q-adaptive problem for the anti- ⌼͑ ͒ where the system norm t is defined in Eq. (154) and a is stable wave equation with D=0 was solved in Ref. ͓60͔. The defined in Eq. (86). delay-adaptive control problem for ODE plants was recently Since the coefficient b is arbitrary, we can choose it as b=a. solved in Ref. ͓67͔. Then, we obtain the following corollary. It is worth mentioning that, even though we have considered COROLLARY 17. Consider the closed-loop system consisting of only a state feedback case in this paper, there is no problem to the plant Eqs. (5)–(9) and the control law Eq. (37). The following extend the results to output feedback. For example, the observers holds: ͑Ref. ͓58͔, Eqs. ͑110͒–͑112͒͒ ␣ ␣ t/2 2 4 1−c q + ˜c ⌼͑t͒ Յ ⌼ ͯ ͯ , ∀ t Ն 0 ͑187͒ ͑ ͒ ͑ ͒ ͓ ͑ ͒ ͑ ͔͒ ͑ ͒ ␣ ␣ 0 uˆ tt x,t = uˆ xx x,t − uxt 1,t − uˆ xt 1,t 193 1 3 1+c 1+qc˜ ␣ ␣ ␣ ␣ It should be noted that all of the coefficients 1 , 2 , 3 , 4 de- ͑ ͒ ␣ ␣ q͑q + ˜c͒ pend on a c . In addition, 3 and 4 depend on q. Finally, it is ˆ ͑ ͒ ˆ ͑ ͒ ͓ ͑ ͒ ˆ ͑ ͔͒ ͑ ͒ ␣ ␣ ␣ ux 0,t =−qut 0,t − ux 1,t − ux 1,t 194 important to observe that 2, as well as 3 and 4, are nondecreas- 1+qc˜ ing functions of D. uˆ͑1,t͒ = U͑t − D͒͑195͒ where ˜cϾ0 is an observer gain, can be combined with the predic- 6 Conclusions tor feedback law The result of this paper offers an alternative approach to ad- 1 1 dressing the problem of input delays in control of wave PDEs, ͑ ͒ ͑ ͒ˆ͑ ͒ ͵ ͑ ͒ˆ͑ ͒ ͵ ͑ U t =−q 1+D,0 u 0,t + x 1+D,y u y,t dy + 1 brought up by the Datko challenge. Stabilization in the presence 0 0 of arbitrarily long delay is achieved for a wave equation plant with antidamping, which has all of its infinitely many open-loop t ͒ˆ ͑ ͒ ͵ ͑ ͒ ͑͒ ͑ ͒ eigenvalues arbitrarily far to the right of the imaginary axis. + D,y ut y,t dy − y t − ,1 U d 196 Some interesting open problems arise from the considerations t−D in this paper. First, one would want to consider the problem of to achieve output feedback stabilization using the measurement robustness to small errors in D that is employed in the predictor ux͑1,t͒, which is collocated with the control u͑1,t͒=U͑t−D͒,as feedback. While we were successful in establishing robustness to in Fig. 2. The proof of this fact is more complicated than in the delay error for an ODE with predictor feedback ͓24͔, we are not present paper due to the need to incorporate the observer error very hopeful that such a robustness result would hold for a wave system in the analysis. Furthermore, a predictor feedback can be PDE plant. designed for the architecture, where the control and measurement Second, the foremost problem would be to consider adaptive are switched, namely, where ux͑1,t͒=U͑t−D͒ is a Neumann input stabilization of the antistable wave equation with unknown delay, and u͑1,t͒ is a Dirichlet output, as in Ref. ͓58͔, Eqs. ͑63͒–͑65͒ and namely, of the systems ͑76͒. ͑ ͒ ͑ ͒ ͑ ͒͑͒ utt x,t = uxx x,t , x 0,1 188 Another example of a wave system with antidamping is that of electrically amplified stringed instruments ͑for example, electric ͑ ͒ ͑ ͒͑͒ ͒ ux 0,t =−qut 0,t 189 guitar . Such instruments employ an electromagnetic pickup, shown in Fig. 10, where the voltage on the terminals of the pickup u͑1,t͒ = v͑1,t͒͑190͒ is proportional, according to Faraday’s law of induction, to the velocity of the string above it. The pickup’s voltage is then am- ͑ ͒ ͑ ͒ ͓ ͒͑͒ Dvt x,t = vx x,t , x 1,2 191 plified using an electric amplifier. The loudspeaker of a high-gain
Journal of Dynamic Systems, Measurement, and Control MAY 2011, Vol. 133 / 031004-11
Downloaded 23 Mar 2011 to 137.110.36.98. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm amplifier, when played at high volume, is capable of producing an 2930–2935. ͓ ͔ acoustic excitation of such intensity that its force acts to mechani- 25 Krstic, M., 2008, “On Compensating Long Actuator Delays in Nonlinear Con- trol,” IEEE Trans. Autom. Control, 53, pp. 1684–1688. cally excite the string. This is a positive feedback loop, where the ͓26͔ Krstic, M., and Smyshlyaev, A., 2008, “Backstepping Boundary Control for string velocity is converted to voltage, multiplied by high-gain, First Order Hyperbolic PDEs and Application to Systems With Actuator and and then applied back as a force on the string. The phenomenon is Sensor Delays,” Syst. Control Lett., 57, pp. 750–758. not the same as our antidamping at the boundary but it appears in ͓27͔ Kwon, W. H., and Pearson, A. E., 1980, “Feedback Stabilization of Linear ͑ ͒ ͑ ͒ ͑ ͒ Systems With Delayed Control,” IEEE Trans. Autom. Control, 25, pp. 266– the form utt x,t =uxx x,t +gut p,t , where g is the gain and p 269. ͑0,1͒ is the location of the pickup along the length of the string. ͓28͔ Manitius, A., and Olbrot, A., 1979, “Finite Spectrum Assignment Problem for The instability described here manifests itself as a loud, sustain- Systems With Delays,” IEEE Trans. Autom. Control, 24, pp. 541–552. ͓29͔ Michiels, W., and Niculescu, S.-I., 2003, “On the Delay Sensitivity of Smith ing, tone, even when the string is not being “plucked,” although, Predictors,” Int. J. Syst. Sci., 34, pp. 543–551. when used in a control manner, it can be employed musically ͓30͔ Michiels, W., and Niculescu, S.-I., 2007, Stability and Stabilization of Time- ͑leading to “swells” of sound that the musician can induce on Delay Systems: An Eigenvalue-Based Approach, SIAM, Philadelphia. chosen notes͒. Among electric guitarists, this phenomenon is re- ͓31͔ Mirkin, L., 2004, “On the Approximation of Distributed-Delay Control Laws,” ͓ ͔ Syst. Control Lett., 51, pp. 331–342. ferred to, simply, as feedback 68,69 . ͓32͔ Mondie, S., and Michiels, W., 2003, “Finite Spectrum Assignment of Unstable Time-Delay Systems With a Safe Implementation,” IEEE Trans. Autom. Con- trol, 48, pp. 2207–2212. References ͓33͔ Niculescu, S.-I., and Annaswamy, A. M., 2003, “An Adaptive Smith- ء ͓1͔ Asl, F. M., and Ulsoy, A. G., 2003, “Analysis of a System of Linear Delay Controller for Time-Delay Systems With Relative Degree n Ն2,” Syst. Con- Differential Equations,” ASME J. Dyn. Syst., Meas., Control, 125͑2͒, pp. trol Lett., 49, pp. 347–358. 215–223. ͓34͔ Olbrot, A. W., 1978, “Stabilizability, Detectability, and Spectrum Assignment ͓2͔ Fazelinia, H., Sipahi, R., and Olgac, N., 2007, “Stability Robustness Analysis for Linear Autonomous Systems With General Time Delays,” IEEE Trans. of Multiple Time-Delayed Systems Using “Building Block” Concept,” IEEE Autom. Control, 23, pp. 887–890. Trans. Autom. Control, 52, pp. 799–810. ͓35͔ Richard, J.-P., 2003, “Time-Delay Systems: An Overview of Some Recent ͓3͔ Olgac, N., Ergenc, A. F., and Sipahi, R., 2005, “Delay Scheduling: A New Advances and Open Problems,” Automatica, 39, pp. 1667–1694. Concept for Stabilization in Multiple Delay Systems,” J. Vib. Control, 11, pp. ͓36͔ Tadmor, G., 2000, “The Standard Hϱ Problem in Systems With a Single Input 1159–1172. Delay,” IEEE Trans. Autom. Control, 45, pp. 382–397. ͓4͔ Olgac, N., and Sipahi, R., 2002, “An Exact Method for the Stability Analysis ͓37͔ Watanabe, K., 1986, “Finite Spectrum Assignment and Observer for Multivari- of Time Delayed LTI Systems,” IEEE Trans. Autom. Control, 47, pp. 793– able Systems With Commensurate Delays,” IEEE Trans. Autom. Control, 31, 797. pp. 543–550. ͓5͔ Olgac, N., and Sipahi, R., 2004, “A Practical Method for Analyzing the Sta- ͓38͔ Watanabe, K., and Ito, M., 1981, “An Observer for Linear Feedback Control bility of Neutral Type LTI-Time Delayed Systems,” Automatica, 40, pp. 847– Laws of Multivariable Systems With Multiple Delays in Controls and Out- 853. puts,” Syst. Control Lett., 1, pp. 54–59. ͓6͔ Olgac, N., and Sipahi, R., 2005, “The Cluster Treatment of Characteristic ͓39͔ Zaccarian, L., and Nesic, D., “A Cascade Interpretation of the Smith Predictor Roots and the Neutral Type Time-Delayed Systems,” ASME J. Dyn. Syst., and the Arising Enhanced Scheme,” American Control Conference, 2006. Meas., Control, 127, pp. 88–97. ͓40͔ Zhong, Q.-C., 2004, “On Distributed Delay in Linear Control Laws—Part I: ͓7͔ Olgac, N., Sipahi, R., and Ergenc, A. F., 2007, “Delay Scheduling, an Uncon- Discrete-Delay Implementation,” IEEE Trans. Autom. Control, 49, pp. 2074– ventional Use of Time Delay for Trajectory Tracking,” Mechatronics, 17, pp. 2080. 199–206. ͓41͔ Zhong, Q.-C., 2006, Robust Control of Time-Delay Systems, Springer, New ͓8͔ Sipahi, R., and Olgac, N., 2006, “A Unique Methodology for the Stability York. Robustness of Multiple Time Delay Systems,” Syst. Control Lett., 55, pp. ͓42͔ Zhong, Q.-C., and Mirkin, L., 2002, “Control of Integral Processes With Dead 819–825. Time—Part 2: Quantitative Analysis,” IEE Proc.: Control Theory Appl., 149, ͓9͔ Sipahi, R., and Olgac, N., 2006, “Stability Robustness of Retarded LTI Sys- pp. 291–296. tems With Single Delay and Exhaustive Determination of Their Imaginary ͓43͔ Datko, R., Lagnese, J., and Polis, M. P., 1986, “An Example on the Effect of Spectra,” SIAM J. Control Optim., 45, pp. 1680–1696. Time Delays in Boundary Feedback Stabilization of Wave Equations,” SIAM ͓10͔ Sipahi, R., and Olgac, N., 2006, “Complete Stability Analysis of Neutral-Type J. Control Optim., 24, pp. 152–156. First Order Two-Time-Delay Systems With Cross-Talking Delays,” SIAM J. ͓44͔ Datko, R., 1988, “Not All Feedback Stabilized Hyperbolic Systems Are Robust Control Optim., 45, pp. 957–971. With Respect to Small Time Delays in Their Feedbacks,” SIAM J. Control ͓11͔ Yi, S., Nelson, P. W., and Ulsoy, A. G., 2007, “Survey on Analysis of Time Optim., 26, pp. 697–713. Delayed Systems via the Lambert W Function,” Dyn. Contin. Discrete Impul- ͓45͔ Guo, B. Z., and Xu, C. Z., 2007, Boundary Stabilization of a One-Dimensional sive Syst.: Ser. A - Math. Anal., 14͑S2͒, pp. 296–301. Wave Equation With Time Delay, Academy of Mathematics, Academia Sinica, ͓12͔ Yi, S., Nelson, P. W., and Ulsoy, A. G., 2008, “Controllability and Observabil- preprint. ity of Systems of Linear Delay Differential Equations via the Matrix Lambert ͓46͔ Guo, B.-Z., and Yang, K.-Y., 2009, “Dynamic Stabilization of an Euler– W Function,” IEEE Trans. Autom. Control, 53, pp. 854–860. Bernoulli Beam Equation With Boundary Observation Time Delay,” Auto- ͓13͔ Yi, S., Nelson, P. W., and Ulsoy, A. G., 2010, “Eigenvalue Assignment via the matica, 45, pp. 1468–1475. Lambert W Function for Control of Time Delayed Systems,” J. Vib. Control, in ͓47͔ Chen, G., 1979, “Energy Decay Estimates and Exact Boundary Value Control- press. lability for the Wave Equation in a Bounded Domain,” J. Math. Pures Appl., ͓14͔ Yi, S., Nelson, P. W., and Ulsoy, A. G., “Robust Control and Time-Domain 58, pp. 249–273. Specifications for Systems for Delay Differential Equations via Eigenvalue ͓48͔ Guo, B. Z., and Xu, C. Z., 2007, “The Stabilization of a One-Dimensional Assignment,” American Control Conference, 2008. Wave Equation by Boundary Feedback With Non-Collocated Observation,” ͓15͔ Smith, O. J. M., 1959, “A Controller to Overcome Dead Time,” ISA, 6, pp. IEEE Trans. Autom. Control, 52, pp. 371–377. 28–33. ͓49͔ Krstic, M., Guo, B.-J., Balogh, A., and Smyshlyaev, A., 2008, “Output- ͓16͔ Artstein, Z., 1982, “Linear Systems With Delayed Controls: A Reduction,” Feedback Stabilization of an Unstable Wave Equation,” Automatica, 44, pp. IEEE Trans. Autom. Control, 27, pp. 869–879. 63–74. ͓17͔ Evesque, S., Annaswamy, A. M., Niculescu, S., and Dowling, A. P., 2003, ͓50͔ Luo, Z.-H., Guo, B.-Z., and Morgul, O., 1998, Stability and Stabilization of “Adaptive Control of a Class of Time-Delay Systems,” ASME J. Dyn. Syst., Infinite Dimensional Systems With Applications, Springer, New York. Meas., Control, 125, pp. 186–193. ͓51͔ Dadfarnia, M., Jalili, N., Xian, B., and Dawson, D. M., 2004, “Lyapunov- ͓18͔ Fiagbedzi, Y. A., and Pearson, A. E., 1986, “Feedback Stabilization of Linear Based Vibration Control of Translational Euler–Bernoulli Beams Using the Autonomous Time Lag Systems,” IEEE Trans. Autom. Control, 31, pp. 847– Stabilizing Effect of Beam Damping Mechanisms,” J. Vib. Control, 10, pp. 855. 933–961. ͓19͔ Gu, K., and Niculescu, S.-I., 2003, “Survey on Recent Results in the Stability ͓52͔ de Queiroz, M. S., Dawson, D. M., Agarwal, M., and Zhang, F., 1999, “Adap- and Control of Time-Delay Systems,” Transactions of ASME, 125, pp. 158– tive Nonlinear Boundary Control of a Flexible Link Robot Arm,” IEEE Trans. 165. Rob. Autom., 15͑4͒, pp. 779–787. ͓20͔ Jankovic, M., 2006, “Forwarding, Backstepping, and Finite Spectrum Assign- ͓53͔ de Queiroz, M. S., and Rahn, C. D., 2002, “Boundary Control of Vibration and ment for Time Delay Systems,” American Control Conference. Noise in Distributed Parameter Systems: An Overview,” Mech. Syst. Signal ͓21͔ Jankovic, M., 2007, “Control of Cascade Systems With Time Delay—The Process., 16, pp. 19–38. Integral Cross-Term Approach,” IEEE Conference on Decision and Control. ͓54͔ Zhang, F., Dawson, D. M., de Queiroz, M. S., and Vedagarbha, P., “Boundary ͓22͔ Jankovic, M., 2008, “Recursive Predictor Design for Linear Systems With Control of the Timoshenko Beam With Free-End Mass/Inertia,” IEEE Confer- Time Delay,” American Control Conference. ence on Decision and Control, 1997. ͓23͔ Klamka, J., 1982, “Observer for Linear Feedback Control of Systems With ͓55͔ Krstic, M., 2009, “Control of an Unstable Reaction-Diffusion PDE With Long Distributed Delays in Controls and Outputs,” Syst. Control Lett., 1, pp. 326– Input Delay,” Syst. Control Lett., 58, pp. 773–782. 331. ͓56͔ Krstic, M., and Smyshlyaev, A., 2008, Boundary Control of PDEs: A Course ͓24͔ Krstic, M., 2008, “Lyapunov Tools for Predictor Feedbacks for Delay Systems: on Backstepping Designs, SIAM, Philadelphia. Inverse Optimality and Robustness to Delay Mismatch,” Automatica, 44, pp. ͓57͔ Meurer, T., and Kugi, A., 2009, “Tracking Control for Boundary Controlled
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Downloaded 23 Mar 2011 to 137.110.36.98. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Parabolic PDEs With Varying Parameters: Combining Backstepping and Dif- ͓64͔ Krstic, M., Krupadanam, A. S., and Jacobson, C. A., 1999, “Self-Tuning Con- ferential Flatness,” Automatica, 45, pp. 1182–1194. trol of a Nonlinear Model of Combustion Instabilities,” IEEE Trans. Control ͓58͔ Smyshlyaev, A., and Krstic, M., 2009, “Boundary Control of an Unstable Syst. Technol., 7, pp. 424–436. Wave Equation With Anti-Damping on the Uncontrolled Boundary,” Syst. ͓65͔ Aksikas, I., Fuxman, A., Forbes, J. F., and Winkin, J. J., 2009, “LQ Control Control Lett., 58, pp. 617–623. Design of a Class of Hyperbolic PDE Systems: Application to Fixed-Bed ͓59͔ Krstic, M., 2009, “Compensating a String PDE in the Actuation or Sensing Reactor,” Automatica, 45, pp. 1542–1548. Path of an Unstable ODE,” IEEE Trans. Autom. Control, 54, pp. 1362–1368. ͓66͔ Sano, H., 2003, “Exponential Stability of a Mono-Tubular Heat Exchanger ͓60͔ Krstic, M., 2009, “Adaptive Control of an Anti-Stable Wave PDE,” 2009 Equation With Output Feedback,” Syst. Control Lett., 50, pp. 363–369. American Control Conference. ͓67͔ Bresch-Pietri, D., and Krstic, M., 2009, “Adaptive Trajectory Tracking Despite ͓61͔ Annaswamy, A., and Ghoniem, A., 1995, “Active Control in Combustion Sys- Unknown Input Delay and Plant Parameters,” Automatica, 45, pp. 2074– tems,” IEEE Control Syst. Mag., 15, pp. 49–63. 2081. ͓62͔ Banaszuk, A., Ariyur, K. B., Krstic, M., and Jacobson, C. A., 2004, “An ͓68͔ Smyshlyaev, A., and Krstic, M., 2004, “Closed Form Boundary State Feed- Adaptive Algorithm for Control of Combustion Instability,” Automatica, 14, backs for a Class of 1D Partial Integro-Differential Equations,” IEEE Trans. pp. 1965–1972. Autom. Control, 49͑12͒, pp. 2185–2202. ͓63͔ Krstic, M., and Banaszuk, A., 2006, “Multivariable Adaptive Control of Insta- ͓69͔ Smyshlyaev, A., and Krstic, M., 2005, “Backstepping Observers for a Class of bilities Arising in Jet Engines,” Control Eng. Pract., 14, pp. 833–842. Parabolic PDEs,” Syst. Control Lett., 54, pp. 613–625.
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