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In many physical applications, the system's state varies with spatial variables as well as time. The state of such systems is modeled by partial differential equations and evolves on an infinite-dimensional space. Systems modeled by delay-differential equations are also infinite-dimensional systems. The full state of these systems cannot be measured. Observer design is an important tool for estimating the state from available measurements. For linear systems, both finite- and infinite-dimensional, the Kalman filter provides an estimate with minimum-variance on the error, if certain assumptions on the noise are satisfied. The extended Kalman filter (EKF) is one type of extension to nonlinear finite-dimensional systems. In this article we provide an extension of the EKF to semilinear infinite-dimensional systems. Under mild assumptions we prove the well-posedness of equations defining the EKF. Next, local exponential stability of the error dynamics is shown. Only detectability is assumed, not observability, so this result is new even for finite-dimensional systems. The results are illustrated with implementation of finite-dimensional approximations of the infinite-dimensional EKF on an example.

In this work, the multiplier method is extended to obtain a general lower bound on the exponential decay rate for port-Hamiltonian systems in one space dimension with boundary dissipation. The physical parameters of the system may be spatially varying. It is shown that, under assumptions of boundary or internal dissipation, the system is exponentially stable. This is established through a Lyapunov function defined through a general multiplier function. Furthermore, an explicit bound on the decay rate in terms of the physical parameters is obtained. The method is applied to a number of examples.

This paper is concerned with the use of \BBH2-control performance metrics in finding optimal actuator locations for a class of infinite-dimensional systems. Conditions that guarantee convergence of the optimal actuator location for the finite-dimensional representation of the infinite-dimensional system to the optimal actuator location of the original infinite-dimensional system are provided. Minimizing an \BBH2-performance index, parameterized by the possible actuator locations and disturbance functions, leads to optimal locations that in addition to enhanced performance, also provide robustness with respect to the spatial distribution of disturbances. Numerical studies for a cantilever beam and a diffusion problem are used to demonstrate the effects of the spatial distribution of disturbances on the optimal actuator location. The effect of the diffusivity parameter on the best actuator locations is also considered. [ABSTRACT FROM AUTHOR]

In many control applications, there is freedom to place the actuators. The actuator location should be chosen to optimize certain performance objectives. In this paper, \cal H\infty-performance with state-feedback is considered. That is, both the controller and the actuator locations are chosen to minimize the effect of disturbances on the output. A framework for calculating \cal H\infty-optimal actuator locations is developed. Conditions for well-posedness of the \cal H\infty-optimal actuator location problem are presented. Many optimal actuator problems involve systems modelled by partial differential equations and conditions under which approximations yield reliable results are given. A derivative-free optimization algorithm to calculate \cal H\infty-optimal actuator locations is described. The results are illustrated using several examples. [ABSTRACT FROM AUTHOR]

Accelerometers are often used to measure the output of second-order systems, such as structural vibrations. Conditions under which these systems are well-posed are obtained. We also establish conditions under which these systems have minimum-phase transfer functions. [ABSTRACT FROM AUTHOR]