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 non-linear distributed parameter systems by a finite-element collocation technique

A finite-element collocation technique is proposed for solving non-linear distributed parameter system (DPS) optimal control problems. On each element, two Gaussian collocation points and Hermite approximation functions are used in the finite-element collocation technique. The numerical experience for two non-linear DPS, one involving distributed control and the other involving spatially-independent control, is reported. Optimal control of DPS can be computed with relatively low-order models when this finite-element collocation technique is used.     

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.     

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.     

Distributed controller design and performance optimization for discrete‐time linear systems

This article addresses the problem of distributed controller design for linear discrete‐time systems. The problem is posed using the classical framework of state feedback gain optimization over an infinite‐horizon quadratic cost, with an additional sparsity constraint on the gain matrix to model the distributed nature of the controller. An equivalent formulation is derived that consists in the optimization of the steady‐state solution of a matrix difference equation, and two algorithms for distributed gain computation are proposed based on it. The first method consists in a step‐by‐step optimization of said difference matrix equation, and allows for fast computation of stabilizing state feedback gains. The second algorithm optimizes the same matrix equation over a finite time window to approximate asymptotic behavior and thus minimize the infinite‐horizon quadratic cost. To assess the performance of the proposed solutions, simulation results are presented for the problem of distributed control of a quadruple‐tank process, as well as a version of that problem scaled up to 40 interconnected tanks.     

Distributed optimal consensus control for nonlinear multiagent systems based on DISOPE algorithm

In this note, a distributed dynamic integrated system optimization and parameter estimation (DISOPE) algorithm for nonlinear multiagent system with general local finite time horizon performance index is proposed, which integrates the model optimization and parameter estimation techniques. Two difficult issues for optimal consensus problem, nonlinear multiagent system and nonlinear quadratic performance index, are considered. With the help of DISOPE algorithm, a distributed optimal control policy is designed to minimize the local performance index in finite time horizon. Furthermore, the convergence of distributed DISOPE algorithm is given, such that the convergence solution satisfies the optimality conditions. Finally, a simulation is employed to show the effectiveness of the distributed DISOPE algorithm.     

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 control of HVAC systems using DDP and NLP techniques

The objective of this study is to apply the differential dynamic programming (DDP) technique of optimal control to heating, ventilating and air-conditioning (HVAC) systems and to compare its performance with a non-linear programming (NLP) technique using the sequential quadratic programming method. The DDP technique is briefly described and studied. Limitations of the technique are noted. Three cases of a system that has been treated previously in the literature are optimized by the two techniques and the computational times compared. The study shows DDP to be efficient compared with NLP for the example problems. NLP is, however, more robust and general and can treat constraints on the state variables directly. Further investigation is needed for larger-scale problems to fully explore the features of the two methods.     

Distributed discrete‐time optimization algorithms with applications to resource allocation in epidemics control

This work presents and analyzes 2 novel distributed discrete‐time nonlinear algorithms to solve a class of decentralized resource allocation problems. The algorithms allow an interconnected group of agents to collectively minimize a global cost function under inequality and equality constraints. Under some technical conditions, it is shown that the first proposed algorithm converges asymptotically to the desired equilibrium, while the second one converges to the solution in a practical way as long as the stepsize chosen is sufficiently small. Of particular interest is that the algorithms are designed to be robust to temporary errors in communication or computation. In addition, agents do not require global knowledge of total resources in the network or any specific procedure for initialization and the cost function is not required to be separable. The convergence of the algorithms is established via nonsmooth Lyapunov analysis. Finally, we illustrate the applicability of our strategies on a virus mitigation problem over computer and human networks.     

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 production and marketing systems with distributed time lags

This paper deals with the problem of production and advertising policy for a decaying item whose gradual loss of potential or utility is associated with the passage of time, such as grain, photographic film or electronic components. The delay effects of the advertisement on the sales rate are also considered in the problem. This leads to a control problem where the state equations are represented by integro-differential equations. The objective is to use optimal control theory to determine the production and advertising rates of the decay item which maximize the current value of net profit stream. For the model developed we present a computational procedure of finding the optimal policy and then carry out sensitivity analysis through an example problem. The properties of the optimal solution are discussed for the case of constant rate of delay effect.     

Randomized optimal consensus of multiagent systems based on a novel intermittent projected subgradient algorithm

In this article, a novel intermittent projected subgradient algorithm is presented to solve the randomized optimal consensus problem for heterogeneous multiagent systems with time-varying communication topologies. The multiagent systems achieve the consensus meanwhile minimizing the global objective function via the proposed algorithm, where f_i(x) is the convex objective function of agent i itself. Due to the common Bernoulli distribution adopted in the existing random optimization algorithm without considering the different computing capability of each agent. An individual projection probability is assigned for each agent based on computing capabilities so that either making projection or taking average is chosen according to the above probability which can effectively avoid overload for some agents with lower computing capabilities and improve the reliability of the overall systems. A new sufficient step-size condition is given to ensure all agents converge to the optimal solution with probability one. Finally, a numerical example is also given to validate 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.     

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.     

Distributed adaptive consensus tracking for uncertain high-order nonlinear multiagent systems with event-triggered communication

This article investigates the distributed adaptive consensus tracking control problem for a class of high-order nonlinear multiagent systems with unmodeled system dynamics, uncertain external disturbances, and event-triggered communication. Under an undirected graph condition, a robust distributed adaptive consensus control scheme is proposed. To deal with the lumped system uncertainties involved in each agent, adaptive compensation techniques are adopted. Moreover, by designing a triggering condition for each agent, which only relies on individual states' changing rates, continuous monitoring of neighboring states can be avoided. It is shown that with the proposed control scheme, all the closed-loop signals are globally uniformly bounded and all the agents' outputs can track the desired trajectory with adjustable tracking errors. Meanwhile, Zeno behavior is excluded in each agent.     

Optimal fixed‐levels control for nonlinear systems with quadratic cost‐functionals

This paper is devoted to general optimal control problems (OCPs) associated with a family of nonlinear continuous‐time switched systems in the presence of some specific control constraints. The stepwise (fixed‐level type) control restrictions we consider constitute a common class of admissible controls in many real‐world engineering systems. Moreover, these control restrictions can also be interpreted as a result of a quantization procedure appglied to the inputs of a conventional dynamic system. We study control systems with a priori given time‐driven switching mechanism in the presence of a quadratic cost functional. Our aim is to develop a practically implementable control algorithm that makes it possible to calculate approximating solutions for the class of OCPs under consideration. The paper presents a newly elaborated linear quadratic‐type optimal control scheme and also contains illustrative numerical examples. Copyright © 2015 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.     

Optimal design of simply supported columns subject to distributed axial load and stress constraint

The optimum designs are given for columns, which are simply supported and under distributed and concentrated axial loads. The objective is to maximize the buckling load subject to volume and maximum stress constraints. The area of the minimum cross‐section under a stress constraint is not known a priori as it depends on the maximum buckling load which in turn depends on the optimum shape. This minimum cross‐sectional area is computed as part of an iterative procedure. An iterative solution method is developed based on finite elements and the results are obtained for n=1, 2, 3, defined as I =α_nAⁿ with I being the moment of inertia, and A the cross‐sectional area. Numerical results show that the optimal areas become larger in the direction of the distributed load. Results are given for uniformly and triangular distributed loads, which are shown to have distinct effects on the optimal column shape. Copyright © 2009 John Wiley & Sons, Ltd.