Abstract We present a control design method for nonlinear partial differential equations (PDEs) based on a combination of gain scheduling and backstepping theory for linear PDEs. A benchmark first-order hyperbolic âŠ
Abstract We present a control design method for nonlinear partial differential equations (PDEs) based on a combination of gain scheduling and backstepping theory for linear PDEs. A benchmark first-order hyperbolic system with an in-domain nonlinearity is considered first. For this system a nonlinear feedback law, based on gain scheduling, is derived explicitly, and a proof of local exponential stability, with an estimate of the region of attraction, is presented for the closed-loop system. Control designs (without proofs) are then presented for a string PDE and a shear beam PDE, both with KelvinâVoigt (KV) damping and free-end nonlinearities of a potentially destabilizing kind. String and beam simulation results illustrate the merits of the gain scheduling approach over the linearization based design.
In this note we show that a symmetric shock profile of the linearized viscous Burgers equation under high-gain âradiationâ boundary feedback is exponentially stable, though the previously reported numerical eigenvalue âŠ
In this note we show that a symmetric shock profile of the linearized viscous Burgers equation under high-gain âradiationâ boundary feedback is exponentially stable, though the previously reported numerical eigenvalue calculations have reported instability. We also show limitations of the radiation feedback by deriving an analytical bound on the closed-loop decay rate for a given shock profile. We prove that the decay rate goes to zero exponentially as the shock becomes sharper. This limitation in the decay rate achievable by radiation feedback highlights the importance of backstepping designs for the Burgers equation, which achieve arbitrarily fast local convergence to arbitrarily sharp shock profiles.
We consider the problem of boundary stabilization of a 1-D (one-dimensional) wave equation with an internal spatially varying antidamping term. This term puts all the eigenvalues of the open-loop system âŠ
We consider the problem of boundary stabilization of a 1-D (one-dimensional) wave equation with an internal spatially varying antidamping term. This term puts all the eigenvalues of the open-loop system in the right half of the complex plane. We design a feedback law based on the backstepping method and prove exponential stability of the closed-loop system with a desired decay rate. For plants with constant parameters the control gains are found in closed form. Our design also produces a new Lyapunov function for the classical wave equation with passive boundary damping.
We consider the problem of boundary stabilization of a one-dimensional wave equation with an internal spatially varying anti-damping term. This term puts all the eigenvalues of the open-loop system in âŠ
We consider the problem of boundary stabilization of a one-dimensional wave equation with an internal spatially varying anti-damping term. This term puts all the eigenvalues of the open-loop system in the right half of the complex plane. We design a feedback law based on the backstepping method and prove exponential stability of the closed-loop system with a desired decay rate. For plants with constant parameters the control gains are found in closed form. Our design also produces a new Lyapunov function for the classical wave equation with passive boundary damping.
Numerous physical processes are modeled by partial differential equations and are often instrumented with boundary actuators and sensors. A major new effort has been underway in recent years to develop âŠ
Numerous physical processes are modeled by partial differential equations and are often instrumented with boundary actuators and sensors. A major new effort has been underway in recent years to develop constructive designs of boundary control laws for unstable PDE systems. This development draws upon the ideas of "backstepping" synthesis for nonlinear ODEs from the 1990s and is an infinite-dimensional, continuum extension of backstepping. Initial efforts on infinite-dimensional backstepping focused on linear PDEs and produced a set of methodologies, for all of the major classes of PDEs, that results in elegant formulae for the gain functions of the feedback laws, which do not require the solution of operator Riccati equations. The most recent efforts pushed even further, into developing an adaptive control approach for PDEs, with unknown functional parameters and output feedback, and into developing feedback linearizing control designs for nonlinear PDEs. The most significant application-driven results in this expanding area of research have so far been for turbulent and magnetohydrodynamic flows, such as those that arise in aerodynamics applications, and in plasma and liquid metal flow problems in tokamak fusion reactors. This talk will present highlights of various methods and applications of infinite-dimensional backstepping.
We study a problem of output feedback stabilization of complex-valued reaction-advection-diffusion systems with parametric uncertainties (these systems can also be viewed as coupled parabolic PDEs). Both sensing and actuation are âŠ
We study a problem of output feedback stabilization of complex-valued reaction-advection-diffusion systems with parametric uncertainties (these systems can also be viewed as coupled parabolic PDEs). Both sensing and actuation are performed at the boundary of the PDE domain and the unknown parameters are allowed to be spatially varying. First, we transform the original system into the form where unknown functional parameters multiply the output, which can be viewed as a PDE analog of observer canonical form. Input and output filters are then introduced to convert a dynamic parametrization of the problem into a static parametrization where a gradient estimation algorithm is used. The control gain is obtained by solving a simple complex-valued integral equation online. The solution of the closed-loop system is shown to be bounded and asymptotically stable around the zero equilibrium. The results are illustrated by simulations.
In this chapter we extend the designs developed in Chapter 4 to the case of "parabolic-like" plants with a complex-valued state. Such plants can also be viewed as two coupled âŠ
In this chapter we extend the designs developed in Chapter 4 to the case of "parabolic-like" plants with a complex-valued state. Such plants can also be viewed as two coupled PDEs. We consider two classes of such plants: the GinzburgâLandau system (Section 6.2) and its special case, the Schrödinger equation (Section 6.1). For breadth of illustration, the Schrödinger equation is treated as a single complex-valued equation, where as the GinzburgâLandau equation is treated as two coupled PDEs.
Two equations that are popular in the research communities studying chaos, strange attractors in PDEs, and soliton waves are the KuramotoâSivashinsky and Kortewegâde Vries equations. The former is frequently used âŠ
Two equations that are popular in the research communities studying chaos, strange attractors in PDEs, and soliton waves are the KuramotoâSivashinsky and Kortewegâde Vries equations. The former is frequently used as a model of flamefront instabilities and thin film growth instabilities, whereas the latter is used as a model of shallow water waves and ion-acoustic waves in plasmas.
In the previous two chapters we considered hyperbolic PDEs of second order that usually describe oscillatory systems such as strings and beams. We now turn our attention to the first-order âŠ
In the previous two chapters we considered hyperbolic PDEs of second order that usually describe oscillatory systems such as strings and beams. We now turn our attention to the first-order hyperbolic equations. They describe quite a different set of physical problems, such as chemical reactors, heat exchangers, and traffic flows. They can also serve as models for delays.
While the wave equation is the most appropriate "point of entry" into the realm of hyperbolic PDEs, beam equations are considered a physically relevant benchmark for control of hyperbolic PDEs âŠ
While the wave equation is the most appropriate "point of entry" into the realm of hyperbolic PDEs, beam equations are considered a physically relevant benchmark for control of hyperbolic PDEs and structural systems in general.
We consider the problem of stabilization of a one- dimensional wave equation that contains instability at its free end and control on the opposite end. In contrast to classical collocated âŠ
We consider the problem of stabilization of a one- dimensional wave equation that contains instability at its free end and control on the opposite end. In contrast to classical collocated "boundary damper" feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition, the controllers and the associated observers designed in the paper are more complex due to the open- loop instability of the plant. The controller and observer gains are designed using the method of "backstepping," which results in explicit formulae for the gain functions. We prove exponential stability and the existence and uniqueness of classical solutions for the closed-loop system. We illustrate the applicability of our designs using simulation results.
After briefly overviewing the application problems that call for boundary control method and the existing techniques for control of PDEs, the paper introduces the key concepts for spatially continuous backstepping âŠ
After briefly overviewing the application problems that call for boundary control method and the existing techniques for control of PDEs, the paper introduces the key concepts for spatially continuous backstepping control design for PDE systems. After that, a general procedure for parabolic PDEs of a "spatially causal" class is presented, followed by a discussion of the main design equations for gain computations. For PDE systems with boundary sensors a backstepping observer design is introduced. The paper concludes with the application of the backstepping method to the Schrodinger equation and first-order hyperbolic PDEs (the transport equation and its derivatives).
This tutorial paper presents comprehensive new techniques for adaptive control of PDE systems. It overviews three design methods - the Lyapunov design, the passive design, and the swapping design and âŠ
This tutorial paper presents comprehensive new techniques for adaptive control of PDE systems. It overviews three design methods - the Lyapunov design, the passive design, and the swapping design and discusses tradeoffs between them. The basic ideas for each design are introduced through benchmark plants with constant unknown coefficients. This is then expanded to PDEs with spatially varying coefficients and with boundary sensing only, employing adaptive PDE observers. These adaptive controllers are applicable even to PDE systems with an infinite relative degree, infinitely many unknown parameters, and open loop unstable.
In this paper, we continue the development of state feedback boundary control laws based on the backstepping methodology, for the stabilization of unstable, parabolic partial differential equations. We consider the âŠ
In this paper, we continue the development of state feedback boundary control laws based on the backstepping methodology, for the stabilization of unstable, parabolic partial differential equations. We consider the linearized Ginzburg--Landau equation, which models, for instance, vortex shedding in bluff body flows. Asymptotic stabilization is achieved by means of boundary control via state feedback in the form of an integral operator. The kernel of the operator is shown to be twice continuously differentiable, and a series approximation for its solution is given. Under certain conditions on the parameters of the Ginzburg--Landau equation, compatible with vortex shedding modelling on a semi-infinite domain, the kernel is shown to have compact support, resulting in partial state feedback. Simulations are provided in order to demonstrate the performance of the controller. In summary, the paper extends previous work in two ways: (1) it deals with two coupled partial differential equations, and (2) under certain circumstances handles equations defined on a semi-infinite domain.
In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial âŠ
In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equation's well posedness and the kernel's smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.
In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial âŠ
In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equation's well posedness and the kernel's smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.
In this paper we study the problem of boundary feedback stabilization for the unstable heat equation ut(x,t) = uxx(x,t)+a(x) u(x,t). This equation can be viewed as a model of a âŠ
In this paper we study the problem of boundary feedback stabilization for the unstable heat equation ut(x,t) = uxx(x,t)+a(x) u(x,t). This equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused (mathematically due to the diffusive term uxx) but also the destabilizing heat is generating (mathematically due to the term a u with a >0). We show that for any given continuously differentiable function a and any given positive constant $\l$ we can explicitly construct a boundary feedback control law such that the solution of the equation with the control law converges to zero exponentially at the rate of $\l$. This is a continuation of the recent work of Boskovic, Krstic, and Liu [IEEE Trans. Automat. Control, 46 (2001), pp. 2022--2028] and Balogh and Krstic [European J. Control, 8 (2002), pp. 165--176].
For the observation or control of solutions of second-order hyperbolic equation in $\mathbb{R}_t \times \Omega $, Ralstonâs construction of localized states [Comm. Pure Appl. Math., 22 (1969), pp. 807â823] showed âŠ
For the observation or control of solutions of second-order hyperbolic equation in $\mathbb{R}_t \times \Omega $, Ralstonâs construction of localized states [Comm. Pure Appl. Math., 22 (1969), pp. 807â823] showed that it is necessary that the region of control meet every ray of geometric optics that has, at worst, transverse reflection at the boundary. For problems in one space dimension, the method of characteristics shows that this condition is essentially sufficient. For problems on manifolds without boundary, the sufficiency was proved in [J. Rauch and M. Taylor, Indiana Univ. Math. J., 24 (1974)]. The theorems regarding propagation of singularities [M. Taylor, Comm. Pure Appl. Math., 28 (1975), pp. 457â478], [R. Melrose, Acta Math., 147 (1981), pp. 149â236], [J. Sjostrand, Communications in Partial Differential Equations, 1980, pp. 41â94] allows the extension of the latter argument to the problem of interior control [C. Bardos, G. Lebeau, and J. Rauch, Rendiconti del Seminario Mathematico, Universita e Politecnico di Torino, 1988, pp. 11â32]. In this paper, the sufficiency is proved for problems of control and observation from the boundary. For multidimensional problems, the region of control must meet each ray in a nondiffractive point, and a new microlocal lower bound on the trade of solutions at the boundary at gliding points is required. This paper treats linear problems with variable coefficients and solutions of all Sobolev regularities. The regularity of the controls is precisely linked to the regularity of the solutions.
We consider a quasilinear partial integro-differential equation modelling the vibrations of a nonlinear string.In the presence of an appropriate boundary damping, we prove a global existence and uniqueness result, for âŠ
We consider a quasilinear partial integro-differential equation modelling the vibrations of a nonlinear string.In the presence of an appropriate boundary damping, we prove a global existence and uniqueness result, for "small" initial data.By the use of suitably chosen Lyapunov functionals, we also show the exponential decay of the energy for any strong solution and we give precise estimates of the rate of decay.
This paper is an assessment of the current state of controllability and observability theories for linear partial differential equations, summarizing existing results and indicating open problems in the area. The âŠ
This paper is an assessment of the current state of controllability and observability theories for linear partial differential equations, summarizing existing results and indicating open problems in the area. The emphasis is placed on hyperbolic and parabolic systems. Related subjects such as spectral determination, control of nonlinear equations, linear quadratic cost criteria and time optimal control are also discussed.
The energy in a string subject to positive viscous damping is known to decay exponentially in time. Under the assumption that the damping is of bounded variation, we identify the âŠ
The energy in a string subject to positive viscous damping is known to decay exponentially in time. Under the assumption that the damping is of bounded variation, we identify the best rate of decay with the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup. We analyze the spectrum of this nonselfadjoint operator in some detail. Our bounds on its real eigenvalues and asymptotic form of its large eigenvalues translate into criteria for over/underdamping and a proof that the decay rate achieves its (negative) minimum over those dampings whose total variation does not exceed a prescribed value.
We consider a one-dimensional wave equation, which governs the vibrations of a damped string with spatially nonhomogeneous density and damping coefficients. We introduce a family of boundary conditions depending on âŠ
We consider a one-dimensional wave equation, which governs the vibrations of a damped string with spatially nonhomogeneous density and damping coefficients. We introduce a family of boundary conditions depending on a complex parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Corresponding to different values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the problem describes either vibrations of a finite string or propagation of elastic waves on an infinite string. Our main object of interest is the family of non-selfadjoint operators <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript h"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">A_h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the energy space of two-component initial data. These operators are the generators of the dynamical semigroups corresponding to the above boundary-value problems. We show that the operators <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript h"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">A_h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are dissipative, simple, maximal operators, which differ from each other by rank-one perturbations. We also prove that the operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A 1 left-parenthesis h equals 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>h</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A_1 (h=1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coincides with the generator of the Lax-Phillips semigroup, which plays an important role in the aforementioned scattering problem. The results of this work are applied in our two forthcoming papers both to the proof of the Riesz basis property of the eigenvectors and associated vectors of the operators <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript h"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>h</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">A_h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and to establishing the exact and approximate controllability of the system governed by the damped wave equation.
We consider adaptive distributed linear quadratic (LQ) control of a parabolic distributed system with spatially varying parameters. The control law is convolutional rather than multiplicative, and we give a new âŠ
We consider adaptive distributed linear quadratic (LQ) control of a parabolic distributed system with spatially varying parameters. The control law is convolutional rather than multiplicative, and we give a new adaptive controller and discuss its stability using averaging analysis.
We prove that under rather general assumptions an exactly controllable problem is uniformly stabilizable with arbitrarily prescribed decay rates. Our approach is directand constructive and avoids many of the technical âŠ
We prove that under rather general assumptions an exactly controllable problem is uniformly stabilizable with arbitrarily prescribed decay rates. Our approach is directand constructive and avoids many of the technical difficulties associated with the usual methods based on Riccati equations. We give several applications for the wave equation and for Petrovsky systems.
<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This note addresses the stabilization of a one-dimensional wave equation with control at one end and noncollocated observation at another end. A simple exponentially convergent observer is âŠ
<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This note addresses the stabilization of a one-dimensional wave equation with control at one end and noncollocated observation at another end. A simple exponentially convergent observer is constructed. The dynamical stabilizing boundary output feedback is designed via the observed state. While the closed-loop system is nondissipative, we show its exponential stability using Riesz basis approach. </para>
This book uses techniques of Fourier and functional analysis to deal with certain problems in differential equations. The Fourier and functional analysis are merely tools; the authors' real interest lies âŠ
This book uses techniques of Fourier and functional analysis to deal with certain problems in differential equations. The Fourier and functional analysis are merely tools; the authors' real interest lies in the differential equations that they study. It has been known since 1967 that a wide variety of sets {ewikt} of complex exponential functions play an important role in the control theory of systems governed by partial differential equations. However, this book is the first serious attempt to gather all of the available theory of these nonharmonic Fourier series in one place, combining published results with new results by the authors, to create a unique source of such material for practicing applied mathematicians, engineers and other scientific professionals.
This paper studies the quadratic optimal control problem for second order (linear) hyperbolic partial differential equations defined on a bounded domain $\Omega \subset R^n $ with boundary $\Gamma $. Both âŠ
This paper studies the quadratic optimal control problem for second order (linear) hyperbolic partial differential equations defined on a bounded domain $\Omega \subset R^n $ with boundary $\Gamma $. Both the finite interval case $[0,T]$, $T < \infty $, and the infinite interval case $T = \infty $ (regulator problem) are considered. The distinguishing feature of the paper, which differentiates it from previous (scarce!) literature on the subject, is that the controls are only $L_2 (0,T;L_2 (\Gamma ))$-functions which act in the Dirichlet B.C. and that the corresponding solutions are penalized in the $L_2 (0,T;L_2 (\Omega ))$-norm (smoother controls, particularly in space, were taken in the few previous works on this subject). The well-posedness of this formulation stems from recent results by the authors about regularity of second order hyperbolic mixed problems [L - T.1], [L - T.3]. Under minimal assumptions, the optimal control is synthesized, in a pointwise feedback form, through an operator which is shown to satisfy in a suitable sense a Riccati differential equation for $T < \infty $ and a Riccati algebraic equation for $T = \infty $. Unlike most, if not all, of the literature on quadratic control problems (for different dynamics!), the algebraic Riccati equation is not derived as a limit process as $T \uparrow \infty $ of the Riccati differential (or integral) equation on $[0,T]$. This has a special advantage in the case of hyperbolic dynamics. Rather, the approach followed for the control problems is direct, in the sense that first an operator is defined in terms of the hyperbolic dynamics and only subsequently shown to satisfy a Riccati equation (differential for $T < \infty $, algebraic for $T = \infty $). Regularity results of the optimal pair are also included. A functional analytic model, based on cosine operator theory and introduced by the authors in [L - T.1], [L - T.3], is used throughout to describe the hyperbolic dynamics.
We study the wellposedness and the main features of a class of feedback control systems. The involved control system is composed of the generator of a strongly continuous group for âŠ
We study the wellposedness and the main features of a class of feedback control systems. The involved control system is composed of the generator of a strongly continuous group for the free part and of an unbounded control operator, so that the results can be applied to boundary or point control problems for partial differential equations of hyperbolic or Petrowski type. The feedback operator is explicit and one can achieve an arbitrary large decay rate for the closed-loop system. These results are proved under a controllability assumption and the proofs rely on general results about the algebraic Riccati equation associated with the linear quadratic regulator problem.
In this paper, we continue the development of state feedback boundary control laws based on the backstepping methodology, for the stabilization of unstable, parabolic partial differential equations. We consider the âŠ
In this paper, we continue the development of state feedback boundary control laws based on the backstepping methodology, for the stabilization of unstable, parabolic partial differential equations. We consider the linearized Ginzburg--Landau equation, which models, for instance, vortex shedding in bluff body flows. Asymptotic stabilization is achieved by means of boundary control via state feedback in the form of an integral operator. The kernel of the operator is shown to be twice continuously differentiable, and a series approximation for its solution is given. Under certain conditions on the parameters of the Ginzburg--Landau equation, compatible with vortex shedding modelling on a semi-infinite domain, the kernel is shown to have compact support, resulting in partial state feedback. Simulations are provided in order to demonstrate the performance of the controller. In summary, the paper extends previous work in two ways: (1) it deals with two coupled partial differential equations, and (2) under certain circumstances handles equations defined on a semi-infinite domain.
The problem of robustly stabilizing a linear system subject to Hâ-bounded perturbations in the numerator and the denominator of its normalized left coprime factorization is considered for a class of âŠ
The problem of robustly stabilizing a linear system subject to Hâ-bounded perturbations in the numerator and the denominator of its normalized left coprime factorization is considered for a class of infinite-dimensional systems. This class has possible unbounded, finite-rank input and output operators, which include many delay and distributed systems. The optimal stability margin is expressed in terms of the solutions of the control and filter algebraic Riccati equations.
In this note, a feedback boundary controller for an unstable heat equation is designed. The equation can be viewed as a model of a thin rod with not only the âŠ
In this note, a feedback boundary controller for an unstable heat equation is designed. The equation can be viewed as a model of a thin rod with not only the heat loss to a surrounding medium (stabilizing) but also the heat generation inside the rod (destabilizing). The heat generation adds a destabilizing linear term on the right-hand side of the equation. The boundary control law designed is in the form of an integral operator with a known, continuous kernel function but can be interpreted as a backstepping control law. This interpretation provides a Lyapunov function for proving stability of the system. The control is applied by insulating one end of the rod and applying either Dirichlet or Neumann boundary actuation on the other.
1R9. Mathematical Control Theory of Coupled PDEs. - I Lasiecka (Univ of Virginia, Charlottesville VA). SIAM, Philadelphia. 2002. 242 pp. Softcover. ISBN 0-89871-486-9. $60.00.Reviewed by GC Gaunaurd (Code AMSRL-SE-RU, Army âŠ
1R9. Mathematical Control Theory of Coupled PDEs. - I Lasiecka (Univ of Virginia, Charlottesville VA). SIAM, Philadelphia. 2002. 242 pp. Softcover. ISBN 0-89871-486-9. $60.00.Reviewed by GC Gaunaurd (Code AMSRL-SE-RU, Army Res Lab, 2800 Powder Mill Rd, Adelphi MD 20783-1197).The mathematics of control theory for a single Partial Differential Equation has been studied for some time. However, for more complex systems governed by systems of coupled PDEs, the available works are fewer. The tools developed for single PDE systems are usually inadequate for the analysis of coupled systems. New questions have been formulated, including how can one take advantage of the coupling in the model to improve the system performance? The propagation of some components of a system into some other, originate new phenomena via the coupling. The present book describes classes of coupled PDE models displaying the above-mentioned coupled properties and presents tools to analyze the resulting control problems. The ultimate goal of the book is said to be to provide a mathematical theory to guide the solution of three main problems: a) well-posedness and regularity, b) stabilization and stability, and c) optimal control and existence and uniqueness of some associated Riccati equations. The structural acoustics model is used toward the end of the book as the choice âexampleâ to illustrate various coupling phenomena that appear in interconnected systems. There are wave equations in an acoustic medium that appear coupled to the plate or shell equations from which they are separated by an interface that is part of the boundary for the acoustic medium. The book has six chapters. It is only possible here to give their titles. The analysis starts with the well-posedness of 2nd-order nonlinear equations with boundary damping. It continues with a study of the stabilizability of nonlinear waves and plates. There is then a chapter on the uniform stability of structural acoustic models and another on Semi Group and PDE models for structural acoustic control problems. The final two chapters deal with feedback noise control for finite and infinite time-horizon problems. This results in detailed studies of certain pertinent Riccati equations. Mathematical Control Theory of Coupled PDEs has 242 pages, with no figures and no computed results. This is a pure mathematical treatment full of theorems, lemmas, assumptions, propositions, and countless corollaries. It is the authorâs stated hope that it will be of use to applied acousticians and theoretical engineers, but this is an unrealistic expectation. The connection to structural acoustics is buried in a sea of theorems with little applicability. There are over 200 references, but about half are by the author herself and one of her associates. The couple of works mentioned which are authored by well-known acousticians, such as C Fuller and the textbook by P Morse and U Ingard, are the only ones that come from the regular acoustic literature. Therefore, this mathematical document will be mostly of interest to other mathematicians carrying on research on these obscure topics. This is certainly not a textbook, but perhaps could be a reference mathematics book for some institutional libraries. With the current emphasis on relevance required by the Army Research Office (ARO), now a part of the Army Research Lab, this reviewer was surprised to learn that they sponsored this work.
Feedback controls based on the receding horizon method have proven to be a useful and easy tool in stabilizing linear ordinary differential systems. In this paper the receding horizon method âŠ
Feedback controls based on the receding horizon method have proven to be a useful and easy tool in stabilizing linear ordinary differential systems. In this paper the receding horizon method is applied to linear systems with delay in the control. An open-loop optimal control which minimizes control energy subject to certain side constraints is first derived and then transformed to a closed-loop control via the receding horizon concept. The resulting feedback system is shown to be asymptotically stable under a complete controllability condition. It is also shown how the receding horizon control suggests a more general class of stabilizing feedback control laws. The control laws of this paper are perhaps the easiest way to stabilize a linear system with delay in the control.
The instability mechanisms, related to the implementation of distributed delay controllers in the context of finite spectrum assignment, were studied in detail in the past few years. In this note âŠ
The instability mechanisms, related to the implementation of distributed delay controllers in the context of finite spectrum assignment, were studied in detail in the past few years. In this note we introduce a distributed delay control law that assigns a finite closed-loop spectrum and whose implementation with a sum of point-wise delays is safe. This property is obtained by implicitly including a low-pass filter in the control loop. This leads to a closed-loop characteristic quasipolynomial of retarded type, and not one of neutral type, which was shown to be a cause of instability in previous schemes.
We consider the Korteweg-de Vries-Burgers equation on the interval [0,1]. Motivated by simulations resulting in modest decay rates with recently proposed control laws by Liu and Krstic, which keeps some âŠ
We consider the Korteweg-de Vries-Burgers equation on the interval [0,1]. Motivated by simulations resulting in modest decay rates with recently proposed control laws by Liu and Krstic, which keeps some of the boundary conditions as homogeneous, we propose a strengthened set of feedback boundary conditions. We establish stability properties of the closed-loop system, prove well-posedness and illustrate the performance improvement by a simulation example.
We consider distributed parameter systems where the underlying dynamics are spatially invariant, and where the controls and measurements are spatially distributed. These systems arise in many applications such as the âŠ
We consider distributed parameter systems where the underlying dynamics are spatially invariant, and where the controls and measurements are spatially distributed. These systems arise in many applications such as the control of vehicular platoons, flow control, microelectromechanical systems (MEMS), smart structures, and systems described by partial differential equations with constant coefficients and distributed controls and measurements. For fully actuated distributed control problems involving quadratic criteria such as linear quadratic regulator (LQR), H/sub 2/ and H/sub /spl infin//, optimal controllers can be obtained by solving a parameterized family of standard finite-dimensional problems. We show that optimal controllers have an inherent degree of decentralization, and this provides a practical distributed controller architecture. We also prove a general result that applies to partially distributed control and a variety of performance criteria, stating that optimal controllers inherit the spatial invariance structure of the plant. Connections of this work to that on systems over rings, and systems with dynamical symmetries are discussed.
This tutorial paper presents comprehensive new techniques for adaptive control of PDE systems. It overviews three design methods - the Lyapunov design, the passive design, and the swapping design and âŠ
This tutorial paper presents comprehensive new techniques for adaptive control of PDE systems. It overviews three design methods - the Lyapunov design, the passive design, and the swapping design and discusses tradeoffs between them. The basic ideas for each design are introduced through benchmark plants with constant unknown coefficients. This is then expanded to PDEs with spatially varying coefficients and with boundary sensing only, employing adaptive PDE observers. These adaptive controllers are applicable even to PDE systems with an infinite relative degree, infinitely many unknown parameters, and open loop unstable.
Feedback stabilization of unstable parabolic equations is of great interest. The fact that it is not necessarily possible to stabilize the equations by means of static feedback schemes when both âŠ
Feedback stabilization of unstable parabolic equations is of great interest. The fact that it is not necessarily possible to stabilize the equations by means of static feedback schemes when both observation and control can be realized only through the boundary is illustratively shown by a simple example. In view of this, a functional observer of Luenberger type is derived and then utilized in order to stabilize unstable parabolic equations for which observation of the state and control can be carried out only through the boundary.
Presents the synthesis of adaptive identifiers for distributed parameter systems (DPS) with spatially varying parameters described by partial differential equations (PDEs) of parabolic, elliptic, and hyperbolic type. The features of âŠ
Presents the synthesis of adaptive identifiers for distributed parameter systems (DPS) with spatially varying parameters described by partial differential equations (PDEs) of parabolic, elliptic, and hyperbolic type. The features of the PDE setting are utilized to obtain the not directly intuitive parameter estimation algorithms that use spatial derivatives of the output data with the order reduced from that of the highest spatial plant derivative. The tunable identifier parameters are passed through the integrator block, which forms their orthogonal expansions. The latter are shown to be pointwise plant parameter estimates. In this regard, the approach of the paper is in the spirit of finite-dimensional observer realization in integrating rather than differentiating the output data, only applied to the spatial rather than temporal domain. The constructively enforceable identifiability conditions, formulated in terms of the sufficiently rich input signals referred to as generators of persistent excitation, are shown to guarantee the existence of a unique zero steady state for the parameter errors. Under such inputs, the tunable parameters in the adaptive identifiers proposed are shown to converge to plant parameters in L/sub 2/ and the orthogonal expansions of these tunable parameters-pointwise.
Motion planning and design of feedforward and feedback tracking control are studied for a tubular reactor modeled by nonlinear parabolic diffusionâconvectionâreaction equations. The approach is based on formal power series âŠ
Motion planning and design of feedforward and feedback tracking control are studied for a tubular reactor modeled by nonlinear parabolic diffusionâconvectionâreaction equations. The approach is based on formal power series parametrizations of the system states and inputs, whereby the domain of convergence of the formal series solutions can be greatly extended using appropriate summation methods. As a result, feedforward controls can be determined for a wide range of trajectories and system parameters. Furthermore, on the basis of a reinterpretation of the formal power series parametrization, state feedback with asymptotic tracking control and an observer can be designed for the tubular reactor. On the other hand, following the 2-degrees-of-freedom approach, the feedforward control can be supplemented by a standard output feedback to obtain robust tracking control for tubular reactors with parameter sets reflecting the unpacked tubular reactor up to the real fixed-bed reactor.
The interest in control of nonlinear partial differential equation (PDE) sys tems has been triggered by the need to achieve tight distributed control of transport-reaction processes that exhibit highl
The interest in control of nonlinear partial differential equation (PDE) sys tems has been triggered by the need to achieve tight distributed control of transport-reaction processes that exhibit highl
A new edition in a single volume Over the past decade, more and more sophisticaced mathematical tools and approaches have been incorporated in the ?eld of Control of in?nite dim- âŠ
A new edition in a single volume Over the past decade, more and more sophisticaced mathematical tools and approaches have been incorporated in the ?eld of Control of in?nite dim- sional systems. This
This paper is concerned with the boundary controllability of entropy weak solutions to hyperbolic systems of conservation laws. We prove a general result on the asymptotic stabilization of a system âŠ
This paper is concerned with the boundary controllability of entropy weak solutions to hyperbolic systems of conservation laws. We prove a general result on the asymptotic stabilization of a system near a constant state. On the other hand, we give an example showing that exact controllability in finite time cannot be achieved, in general.