Operator-valued Kernels and Control of Infinite dimensional Dynamic Systems

This article presents two different approximations to linear infinite-horizon optimal control problems arising in model predictive control. The dynamics are approximated using a Galerkin approach with parametrized state and input trajectories. It is shown that the first approximation represents an upper bound on the optimal cost of the underlying infinite dimensional optimal control problem, whereas the second approximation results in a lower bound. We analyze the convergence of the costs and the corresponding optimizers as the number of basis functions tends to infinity. The results can be used to quantify the approximation quality with respect to the underlying infinite dimensional optimal control problem.

By exploiting min-plus linearity, semiconcavity, and semigroup properties of dynamic programming, a fundamental solution semigroup for a class of approximate finite horizon linear infinite dimensional optimal control problems is constructed. Elements of this fundamental solution semigroup are parameterized by the time horizon, and can be used to approximate the solution of the corresponding finite horizon optimal control problem for any terminal cost. They can also be composed to compute approximations on longer horizons. The value function approximation provided takes the form of a min-plus convolution of a kernel with the terminal cost. A general construction for this kernel is provided, along with a spectral representation for a restricted class of sub-problems.

We examine the variational stability of infinite dimensional optimal control problems governed by non-linear evolution equations. Our tools are the Kuratowski-Mosco convergence of sets and the corresponding τ-convergence of functions. We prove the τ-convergence of cost functionals, the convergence of the values of the problems and we examine the variational stability of the solution and reachable sets. These results are then applied to a sequence of non-linear parabolic distributed parameter optimal control problems.

The study of infinite dimensional optimal control has been a rapidly growing area of research. In this chapter we present preliminaries which we need in order to study turnpike properties of infinite dimensional optimal control problems. We discuss Banach space valued functions, unbounded operators, C 0 semigroups, evolution equations, and admissible control operators.

This paper treats linear-quadratic optimal control of a thermoelastic rod. The main objective of the paper is to approximate the solution to the infinite dimensional optimal control problem by solving a sequence of finite dimensional optimal control problems for a sequence of approximations to the coupled partial differential equations that represent the axial vibrations and temperature distribution in the rod. Numerical results for two approximation schemes are presented.

This paper deals with the reduced order controller design for infinite dimensional port Hamiltonian systems (IDPHS). Firstly, a structure preserving and passive LQG control design equivalent to Control by Interconnection is proposed. Based on this LQG controller, a structure preserving reduction method is used to approximate both the closed loop IDPHS and the LQG controller. This closed loop reduction guarantees that the reduced order controller will ensure acceptable closed loop performances on the infinite dimensional system. The proposed methods is applied to the control of a vibro-acoustic system.

A common approach to solving infinite dimensional optimal control problems, such as in launch vehicle trajectory optimization, is to approximate the optimal control function as a set of parameters. This process, known as the direct method, transcribes the optimal control problem into a finite dimensional parameter optimization problem. Traditionally, subject matter experts and trajectory analysts work manually to find usable trajectories by changing the initial guess for the control parameters. This paper presents a method to capture the feasible design space of a launch vehicle trajectory problem. The result is a region of feasible values for the launch vehicle control parameters. In this context, feasibility is defined by the trajectory reaching the termination criteria, thereby returning a function evaluation. The method is applied to a Delta IV Medium launch vehicle. This vehicle is used because it represents the challenges of trajectory optimization for ETO launch vehicles while not being overly complex. Two ways of sampling the control parameter design space to determine feasibility are compared. Results are represented using design variable limits and the global feasible space is shown directly using scatter-plot matrices.

The main objective of this article is to review the major progress that has been made on controllability of dynamical systems over the past number of years. Controllability is one of the fundamental concepts in the mathematical control theory. This is a qualitative property of dynamical control systems and is of particular importance in control theory. A systematic study of controllability was started at the beginning of sixties in the last century, when the theory of controllability based on the description in the form of state space for both time-invariant and time-varying linear control systems was worked out. Roughly speaking, controllability generally means, that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls. It should be mentioned, that in the literature there are many different definitions of controllability, which strongly depend on a class of dynamical control systems and on the other hand on the form of admissible controls. Controllability problems for different types of dynamical systems require the application of numerous mathematical concepts and methods taken directly from differential geometry, functional analysis, topology, matrix analysis and theory of ordinary and partial differential equations and theory of difference equations. In the paper we use mainly state-space models of dynamical systems, which provide a robust and universal method for studying controllability of various classes of systems. Controllability plays an essential role in the development of modern mathematical control theory. There are various important relationships between controllability, stability and stabilizability of linear both finite-dimensional and infinite-dimensional control systems. Controllability is also strongly related to the theory of realization and so called minimal realization and canonical forms for linear time-invariant control systems such as the Kalmam canonical form, the Jordan canonical form or the Luenberger canonical form. It should be mentioned, that for many dynamical systems there exists a formal duality between the concepts of controllability and observability. Moreover, controllability is strongly connected with the minimum energy control problem for many classes of linear finite dimensional, infinite dimensional dynamical systems, and delayed systems both deterministic and stochastic. Finally, it is well known, that controllability concept has many important applications not only in control theory and systems theory, but also in such areas as industrial and chemical process control, reactor control, control of electric bulk power systems, aerospce engineering and recently in quantum systems theory.

Controllability of dynamical systems with constraints

Rotational Motion Control by Feedback with Minimum L1-Norm

A general perturbation theory of the Kolmogorov-Arnold-Moser type is described concerning the existence of infinite dimensional invariant tori in nearly integrable hamiltonian systems. The key idea is to consider hamiltonians with aspatial structure and to express all quantitative aspects of the theory in terms of rather general weight functions on such structures. This approach combines great flexibility with an effective control of the vrious interactions in infinite dimensional systems.

Optimal Control of Stochastic Systems

Optimal control of stochastic functional neutral differential equations with time lag in control

On Lipschitz conditions of infinite dimensional systems

Rotary inverted pendulum (RIP) system is a nonlinear, non-minimum phase, unstable and underactuated system. Controlling such system can be a challenge and is considered a benchmark in control theory problem. Prior to designing a controller, equations that represent the behaviour of the RIP system must be developed as accurately as possible without compromising the complexity of the equations. Through Takagi-Sugeno (T-S) fuzzy modeling technique, the nonlinear system model is then transformed into several local linear time-invariant models which are then blended together to reproduce, or approximate, the nonlinear system model within local region. A parallel distributed compensation (PDC) based fuzzy controller using linear quadratic regulator (LQR) technique is designed to control the RIP system. The results show that the designed controller able to balance the RIP system.