Optimal control of linear distributed parameter systems by finite-element Galerkin's technique

The paper considers a finite-element discretization for solving two distributed parameter system optimal control problems. Linear and quadratic finite-element approximations are used in conjunction with Galerkin's method. The first problem involves heating a massive body in a furnace and takes into account furnace dyanmics. The input (fuel flow rate) is magnitude constrained, and the performance index is the terminal cost. The second problem pertains to a plug-flow heat exchanger and is solved for end-point control and control along the entire heat exchanger. Uniform wall flux and wall temperature serve as the control variables. The results are compared with earlier results, and advantages of the present approach are discussed.     

Optimal control of a parabolic distributed parameter system via orthogonal polynomials

A class of optimal control of systems with distributed parameters is considered. The process of the systems under consideration is governed by a linear parabolic partial differential equation. By use of the modal space technique, the optimal control of a distributed parameter system is simplified into the optimal control of a linear time-invariant lumped-parameter system. Next, a direct computational method for evaluating the modal optimal control and trajectory of the linear time-invariant lumped-parameter is suggested. The method is based on using finite interpolating orthogonal polynomials to approximate modal state variables. The formulation is straightforward and convenient for digital computation. An illustrative example is given to demonstrate the advantage of this method. © 1998 John Wiley & Sons, Ltd.     

Optimal control of temperature and fuel consumption for soaking pits in a steel plant—a distributed parameter approach

An optimal control theory for a class of non-linear distributed parameter control systems in the general setting involving more than one spatial co-ordinate is developed for use on control of a typical soaking pit in a steel industry. The system is described by a set of non-linear partial differential equations in multidimensional spatial co-ordinates and non-linear boundary conditions with time-dependent boundary controls as well as a spatially independent parameter vector which is governed by its own set of dynamic equations. The set of necessary conditions for optimality obtained from the theoretical development is directly applied to the optimal heating control of the soaking pit with rectangular ingots. The numerical solution by iterations demonstrates the success of the technique and the algorithm leading to the optimal policy for heating control of a typical soaking pit with minimum fuel consumption.     

Optimal weighting design for distributed parameter systems estimation

This paper presents a method which aims at improving parameter estimation in dynamical systems. The general principle of the method is based on a modification of the least‐squares objective function by means of a weighting operator, in view to improve the conditioning of the identification problem. First we recall a previous work using variational calculus in order to obtain the weighting operators through a linear equation. Then we propose a new approach which consists of determining the weights by formulating an optimization problem including positive semidefinite constraints (linear matrix inequalities, LMI). Copyright © 2001 John Wiley & Sons, Ltd.     

Dynamic disturbance minimization control for discrete non-linear stochastic systems

A dynamic feedback control scheme is proposed to reduce the effect of process and measurement disturbances of known wave-form for a class of discrete non-linear stochastic systems. A Lyapunovbased non-linear filter developed previously by the authors is utilized for the estimation of the disturbance state and combined with a least-squares-type control action in a certainty equivalence manner for disturbance reduction. In order to illustrate the performance of the proposed dynamic disturbance rejection technique, three examples which address some important cases are provided.     

Optimal topology design and distributed formation control of multiagent systems with complex-weights directed networks

In this article, we consider the optimal topology design and distributed formation control problem of multiagent systems (MAS) with complex-weighted directed topology. First, a framework is proposed to associate optimal topology of MAS to a constrained optimization problem with a complex Laplacian matrix, which is independent of the agent dynamics. The main contribution of the proposed approach compared with existing results is that the proposed approach does not require the calculation of the stabilizing matrix such that the closed-loop system is asymptotically stable, and a unique set of complex weights that satisfy associated cooperative conditions can be chosen. Then, a distributed formation control protocol is proposed to enable all agents to achieve the control goal. Finally, some numerical example results are provided to demonstrate the effectiveness of the proposed scheme.     

Finite-dimensional approximation for optimal fixed-order compensation of distributed parameter systems

In controlling distributed parameter systems it is often desirable to obtain low-order, finite-dimensional controllers in order to minimize real-time computational requirements. Standard approaches to this problem employ model/controller reduction techniques in conjunction with LQG theory. In this paper we consider the finite-dimensional approximation of the infinite-dimensional Bernstein/Hyland optimal projection theory. Our approach yields fixed-finite-order controllers which are optimal with respect to high-order, approximating, finite-dimensional plant models. We illustrate the technique by computing a sequence of first-order controllers for one-dimensional, single-input/single-output parabolic (heat/diffusion) and hereditary systems using a spline-based, Ritz-Galerkin, finite element approximation. Our numerical studies indicate convergence of the feedback gains with less than 2% performance degradation over full-order LQG controllers for the parabolic system and 10% degradation for the hereditary system.     

Robust approximate non-interacting control design for a class of non-linear stochastic systems

A technique for the approximate decoupling of a class of non-linear stochastic systems which has parameter variation in the plant is presented. First, nominal decoupling is obtained. Then a sensitivity model representing the plant variation is formulated and incorporated into a performance index. Minimization of this index with respect to the free parameters in the feedback matrix results in a robust feedback controller which can reduce the effect of noise and perturbations on the output. An example demonstrating the feasibility of the approach for practical problems is presented.     

Optimal feedback control of power systems governed by non-linear dynamics

Linear-quadratic regulator theory is commonly used in feedback optimization problems for power system stabilization. In this paper an extension of Gamkrelidze's maximum principle, which applies directly to non-linear systems, is used to find an optimal feedback control law from a given practically implementable class of parametrized control structures. An algorithm for finding the optimal set of parameters is presented. The results obtained, using the optimal feedback control law determined by the given algorithm, indicate a substantial improvement of the transient performance of the system.     

Optimal control of linear discrete systems via hahn polynomials

This paper presents a method of finite series expansion using Hahn polynomials for the finite time optimal control of linear discrete time systems with a quadratic performance index. The method converts the discrete two-point boundary value Euler—Lagrange difference equations into a set of linear algebraic equations. As a result, the computation is simple and straightforward.     

LMI-based cooperative distributed model predictive control for Lipschitz nonlinear systems

In this paper, a distributed model predictive control is proposed to control Lipschitz nonlinear systems. The cooperative distributed scheme is considered where a global infinite horizon objective function is optimized for each subsystem, exploiting the state and input information of other subsystems. Thus, each control law is obtained separately as a state feedback of all system's states by solving a set of linear matrix inequalities. Due to convexity of the design, convergence properties at each iteration are established. Additionally, the proposed algorithm is modified to optimize only one control input at a time, which leads to a further reduction in the computation load. Finally, two application cases are studied to show the effectiveness of the proposed method.     

Globally optimal control of self-adjoint distributed systems

The partial differential equations of motion for an uncontrolled distributed structure can be transformed into a set of independent modal equations in terms of natural co-ordinates. It is common practice to design control forces that recouple the modal equations so that the natural co-ordinates for the open-loop (uncontrolled) system cease to be natural co-ordinates for the closed-loop (controlled) system. This approach is referred to as coupled control. In contrast, the independent modal-space control method is a natural control method, i.e. natural co-ordinates for the open-loop system remain natural co-ordinates for the closed-loop system. Moreover, natural control provides a unique and globally optimal closed-form solution to the linear optimal control problem for the distributed structure. Indeed, discretization is not necessary. The optimal control forces are ideally distributed. The distributed control can be approximated by finite-dimensional control, a process that does not require truncation of the plant. Two numerical examples are presented.     

Existence of optimal control for non-linear systems with an implicit derivative

The existence of an optimal control for non-linear systems having an implicit derivative with quadratic performance is proved. The results are established through the notions of condensing map and measure of non-compactness of a set and using a fixed point theorem due to Sadovskii.     

Optimal control of time-delay systems by dynamic programming

The use of iterative dynamic programming employing systematic region contraction and accessible grid points is investigated for the optimal control of time-delay systems. At the time of generating the grid points for the state variables, the corresponding delayed variables at each time stage are also generated and stored in memory. Then, when applying dynamic programming, a linear approximation is used to obtain the initial profile for the delayed variables during integration. This procedure was tested with four problems of different complexity. In each case the optimal control policy is easily obtained and the results compare very favourably with those reported in the literature using other computational procedures.     

Reduced‐order suboptimal control design for a class of nonlinear distributed parameter systems using POD and θ–D techniques

A new computational tool is presented in this paper for suboptimal control design of a class of nonlinear distributed parameter systems (DPSs). In this systematic methodology, first proper orthogonal decomposition‐based problem‐oriented basis functions are designed, which are then used in a Galerkin projection to come up with a low‐order lumped parameter approximation. This technique has evolved as a powerful model reduction technique for DPSs. Next, a suboptimal controller is designed using the emerging θ–D technique for lumped parameter systems. This time domain control solution is then mapped back to the distributed domain using the same basis functions, which essentially leads to a closed form solution for the controller in a state‐feedback form. We present this technique for the class of nonlinear DPSs that are affine in control. Numerical results for a benchmark problem as well as for a more challenging representative real‐life nonlinear temperature control problem indicate that the proposed method holds promise as a good optimal control design technique for the class of DPSs under consideration. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimal spatial field control for controlled release

Most distributed parameter control problems involve manipulation within the spatial domain. Such problems arise in a variety of applications including epidemiology, tissue engineering, and cancer treatment. This paper proposes an approach to solve a state‐constrained spatial field control problem that is motivated by a biomedical application. In particular, the considered manipulation over a spatial field is described by partial differential equations (PDEs) with spatial frequency constraints. The proposed optimization algorithm for tracking a reference spatial field combines three‐dimensional Fourier series, which are truncated to satisfy the spatial frequency constraints, with exploitation of structural characteristics of the PDEs. The computational efficiency and performance of the optimization algorithm are demonstrated in a numerical example. In the example, the spatial tracking error is shown to be almost entirely due to the limitation on the spatial frequency of the manipulated field. The numerical results suggest that the proposed optimal control approach has promise for controlling the release of macromolecules in tissue engineering applications.     

A primal‐dual active‐set method for distributed model predictive control

We present a novel distributed primal‐dual active‐set method for model predictive control. The primal‐dual active‐set method is used for solving model predictive control problems for large‐scale systems with quadratic cost, linear dynamics, additive disturbance, and box constraints. The proposed algorithm is compared with dual decomposition and an alternating direction method of multipliers. Theoretical and experimental results show the effectiveness of the proposed approach for large‐scale systems with communication delays. The application to building control systems is thoroughly investigated. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal control of discrete-time nonlinear heterogeneous multi-agent systems via a distributed DISOPE algorithm

A distributed offline DISOPE algorithm for optimal state synchronization of leader-follower systems with nonlinear discrete-time dynamics is considered, which integrates the model optimization idea and parameter estimation technique together. It can be seen that the convergent solutions of modified linear optimal control problems satisfy the optimality conditions of the original nonlinear optimization problem with non-LQ performance indices. The heterogeneous agents can cooperate and exchange information via network communication. Based on DISOPE algorithm, a distributed optimal control policy is obtained to assure state synchronization and minimize performance indices in finite time horizon. Finally, a simulation example is provided to illustrate the effectiveness of the distributed DISOPE algorithm.     

Optimal non-linear feedback regulation of spacecraft angular momentum

Optimal non-linear feedback laws are presented for the regulation of spacecraft angular momentum by using flywheels and by using fixed on-off reaction jets. The control law for the flywheels takes the form of a linear combination of functions of the state, each positively homogeneous of degree 2p-1 (p1/2); whereas the control law for the reaction jets is of bang-bang form. The range (of the initial disturbance) over which the flywheels are capable of regulating the system and the problem of momentum dumping are also discussed with regard to flywheel saturation constraints. Finally, a control scheme combining the merits of reaction jets and flywheels is proposed.     

Optimal control of continuous systems with impulse controls

Impulse control problems, in which a continuously evolving state is modified by discrete control actions, have applications in epidemiology, medicine, and ecology. In this paper, we present a simple method for solving impulse control problems for systems of differential equations. In particular, we show how impulse control problems can be reformulated and solved as discrete optimal control problems. The method is illustrated with two examples. Published 2014. This article has been contributed to by US Government employees and their work is in the public domain in the USA.