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Numerical solution of optimal control problems for parabolicsystems
 in Parallel Algorithms and Cluster Computing. Implementations,Algorithms, and Applications
, 2006
"... Summary. We discuss the numerical solution of optimal control problems for instationary convectiondiffusion and diffusionreaction equations. Instead of viewing this problem as a largescale unconstrained optimization problem after complete discretization of the corresponding optimality system, we ..."
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Summary. We discuss the numerical solution of optimal control problems for instationary convectiondiffusion and diffusionreaction equations. Instead of viewing this problem as a largescale unconstrained optimization problem after complete discretization of the corresponding optimality system, we formulate the problem as abstract linearquadratic regulator (LQR) problem. Using recently developed efficient solvers for largescale algebraic Riccati equations, we show how to numerically solve the optimal control problem at a cost proportional to solving the corresponding forward problem. We discuss two different optimization goals: one can be seen as stabilization of the plant model, the second one is of tracking type, i.e., a given (optimal) solution trajectory is to be attained. The efficiency of our approach is demonstrated for a model problem related to an optimal cooling process. Moreover, we discuss how the LQR approach can be applied to nonlinear problems. 1
A ModelBased Fault Detection and Diagnosis Scheme for Distributed Parameter Systems: A Learning Systems Approach,” ESAIM−Control Optimisation and
 Calculus of Variations
, 2002
"... Abstract. In this note, fault detection techniques based on nite dimensional results are extended and applied to a class of innite dimensional dynamical systems. This special class of systems assumes linear plant dynamics having an abrupt additive perturbation as the fault. This fault is assumed to ..."
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Abstract. In this note, fault detection techniques based on nite dimensional results are extended and applied to a class of innite dimensional dynamical systems. This special class of systems assumes linear plant dynamics having an abrupt additive perturbation as the fault. This fault is assumed to be linear in the (unknown) constant (and possibly functional) parameters. An observerbased model estimate is proposed which serves to monitor the system’s dynamics for unanticipated failures, and its well posedness is summarized. Using a Lyapunov synthesis approach extended and applied to innite dimensional systems, a stable adaptive fault diagnosis (fault parameter learning) scheme is developed. The resulting parameter adaptation rule is able to \sense " the instance of the fault occurrence. In addition, it identies the fault parameters using the additional assumption of persistence of excitation. Extension of the adaptive monitoring scheme to incipient faults (time varying faults) is summarized. Simulations studies are used to illustrate the applicability of the theoretical results.
ROBUST MRAC OF A LINEAR TIMEVARYING PARABOLIC SYSTEM WITH BOUNDED DISTURBANCE
"... Abstract: In this paper, a robust MRAC (model reference adaptive control) for a linear parabolic partial differential equation with unknown timevarying parameters and bounded disturbance is investigated. In the adaptive control of timevarying plants, the derivative of a Lyapunov function candidate ..."
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Abstract: In this paper, a robust MRAC (model reference adaptive control) for a linear parabolic partial differential equation with unknown timevarying parameters and bounded disturbance is investigated. In the adaptive control of timevarying plants, the derivative of a Lyapunov function candidate, which allows the derivation of adaptation laws, is not negative semidefinite in general. Under the assumption that the disturbance is uniformly bounded, the proposed robust MRAC scheme guarantees the boundedness of all signals in the closed loop system and the convergence of the state error near to zero. With an additional persistence of excitation condition, the parameter estimation errors are shown to converge near to zero as well. Simulation results are provided. Copyright © 2001 IFAC.
Model Reference Adaptive Control of a Flexible Structure
, 2001
"... In this paper, the model reference adaptive control for a flexible structure is investigated. Any mechanically flexible structures are inherently distributed parameter systems whose dynamics are described by partial, rather than ordinary, differential equations. Adaptation laws are derived by the Ly ..."
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In this paper, the model reference adaptive control for a flexible structure is investigated. Any mechanically flexible structures are inherently distributed parameter systems whose dynamics are described by partial, rather than ordinary, differential equations. Adaptation laws are derived by the Lyapunov redesign method on an infinite dimensional Hilbert space. Combined state and parameter estimator equations are constructed as an initial value problem of an infinite dimensional evolution equation in weakform. The wellposedness of the nonlinear closed loop system is shown. It is then shown through the Lyapunov redesign approach that the state error actually converges asymptotically to zero. With the additional persistence of excitation assumption, the parameter errors are shown to converge to zero as well.
Delay Identification in Linear Differential Difference Systems
"... Abstract: Two algorithms for the identification of multiple time delays in linear differential difference systems are proposed. The identification problem is posed as a minimization problem in Hilbert spaces and necessitates the computation of system sensitivity to delay perturbations. Delay identif ..."
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Abstract: Two algorithms for the identification of multiple time delays in linear differential difference systems are proposed. The identification problem is posed as a minimization problem in Hilbert spaces and necessitates the computation of system sensitivity to delay perturbations. Delay identifiability conditions are derived in terms of the properties of the associated strong derivative of the adopted cost function with respect to perturbations in the delays. Delay identifiability is of a local type and is shown to be related to system controllability. The steepest descent and generalized Newton type algorithms are then developed. Convergence of the identifier algorithms is rigorously assessed. Key–Words: delay, differential difference systems, parameter identification, sensitivity to delay parameters. 1
SUMMARY
"... While adaptive control of finite dimensional systems is an advanced field that has produced adaptive control methods for a very general class of LTI systems, adaptive control techniques have been developed for only a few of the classes of PDEs for which nonadaptive controllers exist. We present a c ..."
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While adaptive control of finite dimensional systems is an advanced field that has produced adaptive control methods for a very general class of LTI systems, adaptive control techniques have been developed for only a few of the classes of PDEs for which nonadaptive controllers exist. We present a catalog of approaches for the design of adaptive controllers for PDEs controlled from a boundary and containing unknown destabilizing parameters affecting the interior of the domain. We differentiate between two major classes of schemes: Lyapunov schemes and certainty equivalence schemes. Within the certainty equivalence class, two types of identifier designs are pursued: passivitybased and swapping designs. Each of those designs is applicable to two types of parametrizations: the plant model in its original form (which we refer to as the ‘umodel’) and a transformed model to which a backstepping transformation has been applied (which we refer to as the ‘wmodel’). Hence, a large number of control algorithms result from combining different design tools}Lyapunov schemes, wpassive schemes, uswapping schemes, etc. Our method builds upon the explicitly parametrized control formulae that we introduced in our earlier work on nonadaptive backstepping control for PDEs. These formulae allow us to develop tunable controllers that avoid solving Riccati or Bezout equations at each time step. This paper is primarily a tutorial. Its purpose is to provide structure that helps the future reader of five