Optimal control of neutral systems†

A method is presented whereby an optimal control may be obtained for a linear time-varying neutral system. The performance measure is quadratic with fixed finite upper limit. A form is derived for the coat functional, and the optimal control is obtained as the solution to a set of differential equations. A numerical technique for the solution of the equations is given and existence and uniqueness are proven.     

Stabilization of linear autonomous systems of differential equations with distributed delay

Consideration is given to the problem of optimal stabilization of differential equation systems with distributed delay. The optimal stabilizing control is formed according to the principle of feedback. The formulation of the problem in the functional space of states is used. It was shown that coefficients of the optimal stabilizing control are defined by algebraic and functional-differential Riccati equations. To find solutions to Riccati equations, the method of successive approximations is used. The problem for this control law and performance criterion is to find coefficients of a differential equation system with distributed delay, for which the chosen control is a control of optimal stabilization. A class of control laws for which the posed problem admits an analytic solution is described.     

On sampled-data optimization in distributed parameter systems

A system of parabolic partial differential equations is transformed into ordinary differential equations in a Hilbert space, where the system operator is the infinitesimal generator of a semigroup of operators. A sampled-data problem is then formulated and converted into an equivalent discrete-time problem. The existence and uniqueness of an optimal sampled-data control is proved. The optimal control is given by a linear sampled-states feedback law where the feedback operator is shown to be the bounded seff-adjoint positive semidefinite solution of a Riccati operator difference equation.     

Computer algebra for the calculus of variations,the maximum principle,and automatic control

This paper describes how a computer-algebra system can solve variational optimization problems analytically. For a calculus-of-variations problem, users provide functional integrands and constraints. A program derives corresponding Euler-Lagrange equations, together perhaps with first integrals. Other programs attempt analytic solution of these equations. For an optimal control problem, users provide analytic expressions for the differential constraints on the state variables. A program determines the corresponding Hamiltonian and differential equations for the auxiliary variables, together with solutions to any trivial auxiliary equations. Other programs attempt analytic solution of the remaining equations while maximizing the Hamiltonian.This material is based upon work supported by the National Science Foundation under Grant No. MCS75-22983 A01.     

Time optimal control of a linear hyperbolic system†

The time optimal control of transmission lines with amplitude constraints on the control is considered as a typical problem involving systems governed by hyperbolic partial differential equations. Using a Laplace transformation formulation to yield a time ‘optimal’ solution, it is shown how this sub-optimal control which is bang-bang develops into an optimal control which is not always at its limiting values—demonstrating the effect which the nature of the differential equation has on the form of the optimal control. A simple physical interpretation of the results is given.     

Acceleration-constrained time-optimal control inndimensions

The minimum-time acceleration of a generalized point mass from an initial position and velocity to the origin inndimensions is solved by transforming the problem to an equivalent problem in two dimensions and analytically integrating the system differential equations. Computation of the optimal control is thereby reduced to the solution of five simultaneous nonlinear equations. A numerical continuation method is presented for solving these equations by starting at the known solution of a related single-dimensional problem and progressing incrementally to the desired solution. The problem and solution method are illustrated by a numerical example.     

Formulation and multigrid solution of Cauchy-Riemann optimal control problems

The formulation of optimal control problems governed by Cauchy-Riemann equations is presented. A distributed control mechanism through divergence and curl sources is considered with the boundary conditions of mixed type. A Lagrange multiplier framework is introduced to characterize the solution to Cauchy-Riemann optimal control problems as the solution of an optimality system of four first-order partial differential equations and two optimality conditions. To solve the optimality system, staggered grids and multigrid methods are investigated. It results that staggered grids provide a natural collocation of the optimization variables and second-order accurate solutions are obtained. The proposed multigrid scheme is based on a coarsening by a factor of three that results in a nested hierarchy of staggered grids. On these grids a distributed-Gauss-Seidel and gradient-based smoothing scheme is employed. Results of numerical experiments validate the proposed optimal control formulation and demonstrate the effectiveness of the staggered-grids multigrid solution procedure.     

Optimal control problems for stochastic delay evolution equations in Banach spaces

We consider the optimal control for a Banach space valued stochastic delay evolution equation. The existence and uniqueness of the mild solution for the associated Hamilton–Jacobi–Bellman equations are obtained by means of backward stochastic differential equations. An application to optimal control of stochastic delay partial differential equations is also given.     

Optimal tracking control for large-scale interconnected systems with time-delays

This paper considers an infinite-horizon optimal tracking control problem for a class of large-scale interconnected systems with state time-delays. By using the successive approximation approach, two iteration sequences of vector differential equations are constructed. Meanwhile the large-scale interconnected system is decomposed into finite decoupled subsystems. The existence and uniqueness of the optimal solution is proved, as well as the convergence of the solution sequence. By finite iterations of the solution sequence, a suboptimal tracking control law is obtained. A reduced-order reference input observer is designed to make the feedforward term of the optimal tracking control law physically realizable. A numerical example shows that the presented algorithm is effective and easy to implement.     

A numerical algorithm for optimal control of a class of hybrid systems: differential transformation based approach

A novel numerical algorithm based on differential transformation is proposed for optimal control of a class of hybrid systems with a predefined mode sequence. From the necessary conditions for optimality of hybrid systems, the hybrid optimal control problem is first converted into a two-point boundary value problem (TPBVP) with additional transverse conditions at the switching times. Then we propose a differential transformation algorithm for solving the TPBVP which may have discontinuities in the state and/or control input at the switching times. Using differential transformation, the hybrid optimal control problem reduces to a problem of solving a system of algebraic equations. The numerical solution is obtained in the form of a truncated Taylor series. By taking advantage of the special properties of the linear subsystems and a quadratic cost functional, the differential transformation algorithm can be further simplified for the switched linear quadratic optimal control problem. We analyse the error of the numerical solution computed by the differential transformation algorithm and some computational aspects are also discussed. The performance of the differential transformation algorithm is demonstrated through illustrative examples. The differential transformation algorithm has been shown to be simple to be implemented and computationally efficient.     

Delay optimal control and viscosity solutions to associated Hamilton–Jacobi–Bellman equations

In this article, optimal control problems of differential equations with delays are investigated for which the associated Hamilton–Jacobi–Bellman (HJB) equations are nonlinear partial differential equations with delays. This type of HJB equation has not been previously studied and is difficult to solve because the state equations do not possess smoothing properties. We introduce a new notion of viscosity solutions and identify the value functional of the optimal control problems as the unique solution to the associated HJB equations. An analytical example is given as application.     

线性时滞系统基于观测器的最优输出跟踪控制器近似设计

研究线性时滞系统的最优输出跟踪控制器近似设计过程.首先根据原系统最优输出跟踪控制问题构造了两个分别具有已知初始条件和终端条件的微分方程迭代序列,并证明它们一致收敛于原问题的最优解.然后通过对该解序列的有限次迭代,得到最优输出跟踪控制问题的一个近似解,进一步给出一个计算近似最优输出跟踪控制律的算法.最后通过构造降维参考输入观测器解决了最优输出跟踪控制器中前馈项的物理可实现问题.仿真结果表明该方法是有效的,且易于实现.     

Multiple shooting algorithms for jump-discontinuous problem inoptimal control and estimation

Multiple shooting algorithms are developed for jump-discontinuous two-point boundary value problems arising in optimal control and optimal estimation. Examples illustrating the origin of such problems are given to motivate the development of the solution algorithms. The algorithms convert the necessary conditions, consisting of differential equations and transversality conditions, into algebraic equations. The solution of the algebraic equations provides exact solutions for linear problems. The existence and uniqueness of the solution are proved     

Optimal bang-bang control of linear stochastic systems with a small noise parameter

This study considers the problem of determining optimal feedback control laws for linear stochastic systems with amplitude-constrained control inputs. Two basic performance indices are considered, average time and average integral quadratic form. The optimization interval is random and defined as the first time a trajectory reaches the terminal regionR. The plant is modeled as a stochastic differential equation with an additive Wiener noise disturbance. The variance parameter of the Wiener noise process is assumed to be suitably small. A singular perturbation technique is presented for the solution of the stochastic optimization equations (second-order partial differential equation). A method for generating switching curves for the resulting optimal bang-bang control system is then developed. The results are applied to various problems associated with a second-order purely inertial system with additive noise at the control input. This problem is typical of satellite attitude control problems.     

Weighted residual methods in optimal control

Weighted residual methods (WRM) afford a viable approach to the numerical solution of differential equations. Application of WRM results in the transformation of differential equations into systems of algebraic equations in the modal coefficients. This suggests that WRM can be used as a tool for reducing optimal control problems to mathematical programming problems. Thereby, the optimal control problem is replaced by the minimization of a cost function of static coefficients subject to algebraic constraints. The motivation for this approach lies in the profusion of sophisticated computational algorithms and digital computer codes for the solution of mathematical programming problems. In this note the solution of optimal control problems as mathematical programming problems via WRM is illustrated. The example presented indicates that reasonable accuracy is obtained for modest computational effort. While the simplest types of modes-polynomials and piecewise constants-are employed in this note, the ideas delineated can be applied in conjunction with cubic splines for the generation of computational algorithms of enhanced efficiency.     

Trigonometric approximation of optimal periodic control problems for distributed-parameter systems

The optimal periodic control problem for a system described by first order partial differential equations is approximated by a sequence of discretized optimization problems. Trigonometric polynomials in two variables are used in the latter problems to approximate the state trajectory, the control and functions appearing in differential equations and in the criterion of the basic problem. The state equations and the instantaneous constraints on the state and the control are taken into account by the mixed exterior-interior penalty function. Sufficient conditions are given for the convergence of solutions of discretized problems to the optimal solution of the basic problem. The possibility of applying the method to a class of optimal periodic control problems in chemical engineering is emphasized.     

Solution of optimal control problem in class of piecewise-constant functions

A numerical method of solution of problems of optimal control of objects that may be described by systems of ordinary differential equations in the class of piecewise-constant controls is proposed in the article. At the same time, both the piecewise-constant values of the controls as well as the constancy intervals of these values are optimized. Analytic formulas of the gradient of the functional with respect to the optimized parameters are obtained. The formulas for the gradient of the functional obtained for an initial continuous optimal control problem and the corresponding discretized optimal control problem are compared and results of numerical experiments are presented.     

Feedback control of linear systems with distributed delay

The problem of optimal control of linear systems containing lumped delay, given by differential-difference equations, has been pursued by several authors. However, transportation-lags are better described by distributed delays giving systems that are described by a set of coupled partial and ordinary differential equations. The lumped part of the system is described by ordinary differential equations and the distributed part of the system is described by partial differential equations. The lumped as well as distributed parts are subject to control. The present paper discusses the control of such systems with quadratic performance measures. Riccati-like equations are derived and a technique for their numerical solution is presented.     

Feedback control on Nash equilibrium for discrete-time stochastic systems with Markovian jumps: Finite-horizon case

In this paper, we consider the feedback control on nonzero-sum linear quadratic (LQ) differential games in finite horizon for discrete-time stochastic systems with Markovian jump parameters and multiplicative noise. Four-coupled generalized difference Riccati equations (GDREs) are obtained, which are essential to find the optimal Nash equilibrium strategies and the optimal cost values of the LQ differential games. Furthermore, an iterative algorithm is given to solve the four-coupled GDREs. Finally, a suboptimal solution of the LQ differential games is proposed based on a convex optimization approach and a simplification of the suboptimal solution is given. Simulation examples are presented to illustrate the effectiveness of the iterative algorithm and the suboptimal solution.     

On a game problem involving systems with time delay

A dynamic programming approach is used to obtain feedback solutions to a game problem involving linear systems having a time delay. As is known, the optimal feedback gains in linear time delay systems with quadratic criterion are determined by partial differential equations [1]. In the particular case studied here, the solution to these equations can be obtained by first separating variables and then solving the resulting simpler equations. The optimal feedback control also has the advantage that it can be realized on-line.