Separation theorem for linearly constrained LQG optimal control

We solve the linearly constrained linear-quadratic (LQ) and linear-quadratic-Gaussian (LQG) optimal control problems. Closed-form expressions of the optimal controls are derived, and the Separation Theorem is generalized.     

Numerical solutions for constrained time-delayed optimal control problems

In this paper, time-delayed optimal control problems governed by delayed differential equation are solved. Two different techniques based on integration and differentiation matrices are considered. The time-delayed term of the problem has been approximated by Chebyshev interpolating polynomials. On this basis, the optimal control problem can be solved as a mathematical programming problem. The example illustrates the robustness, accuracy and efficiency of the proposed numerical techniques.     

A canonical structure for constrained optimal control problems

We consider a general class of optimal control problems with regional pole and controller structure constraints. Our goal is to show that for a fairly general class of regional pole and controller structure constraints, such constrained optimal control problems can be transformed to a new one with a canonical structure. A three-step transformation procedure is used to achieve our goal, which essentially amounts to repeated augmentations of plant dynamics and repeated reductions of the controller. The transformed problem is one of the standard optimal static output feedback with a decentralized and repeated structure.     

An interior penalty method for inequality constrained optimal control problems

This paper presents a penalty function approach to the solution of inequality constrained optimal control problems. The method begins with a point interior to the constraint set and approaches the optimum from within, by solving a sequence of problems with only terminal conditions as constraints. Thus, all intermediate solutions satisfy the inequality constraints. Conditions are given which guarantee that the un "constrained" problems have solutions interior to the constraint set and that in the limit these solutions converge to the constrained optimum. For linear systems with convex objective and concave inequalities, the unconstrained problems have the property that any local minimum is global. Further, under these conditions, upper and lower bounds in the optimum are easily available. Three test problems are solved and the results presented.     

A constrained optimal control problem

In this note the problem of optimizing a standard quadratic cost functional subject to a hard constraint is solved (i.e., the poles of the resulting system must lie in a specified region of the left-halfs-plane). This can he viewed as a generalization of the problem of finding the optimal controller that yields a system with specified relative stability. The problem is phrased in terms of a design procedure using coprime fractional representations.     

A nonsmooth Newton’s method for control-state constrained optimal control problems

We investigate optimal control problems subject to mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer–Burmeister function the minimum principle is transformed into an equivalent nonlinear and nonsmooth equation in appropriate Banach spaces. This nonlinear and nonsmooth equation is solved by a nonsmooth Newton’s method. We will show the local quadratic convergence under certain regularity conditions and suggest a globalization strategy based on the minimization of the squared residual norm. A numerical example for the Rayleigh problem concludes the article.     

Solution for state constrained optimal control problems applied to power split control for hybrid vehicles

This paper presents a numerical solution for scalar state constrained optimal control problems. The algorithm rewrites the constrained optimal control problem as a sequence of unconstrained optimal control problems which can be solved recursively as a two point boundary value problem. The solution is obtained without quantization of the state and control space. The approach is applied to the power split control for hybrid vehicles for a predefined power and velocity trajectory and is compared with a Dynamic Programming solution. The computational time is at least one order of magnitude less than that for the Dynamic Programming algorithm for a superior accuracy.     

On the application of sequential quadratic programming to state constrained optimal control problems

In this paper a numerical method for the solution of state constrained optimal control problems is presented. The method is derived from an infinite dimensional analogue of sequential quadratic programming. The main purpose of the paper is to present some theoretical aspects of the method. An experimental numerical implementation of the method is discussed.     

On the optimality of the boundary control for the non-linear control system with controls appearing linearly†

In this paper, the optimization of a non-linear control system is studied, where both the system and the performance index are linear in the control variables. This study is concerned mainly with the optimality of the totally singular control of the maximum principle.

The problem is reduced to the study of & non-regular case of the classical calculus of variations, for which anew criterion is found concerning the property of its extremals

By means of this new criterion, it is proved that the totally singular control cannot be optimal for the higher dimensional case. In other words, the optimal control should be the boundary control     

An unconstrained optimization technique for large-scale linearly constrained convex minimization problems

Consider the problem of minimizing a smooth convex functionf subject to the constraintsAx=b andx≥0, whereA∈? ^p×n . This constrained optimization problem is shown to be equivalent to a differentiable unconstrained optimization problem with 2n+p variables. This formulation of the convex constrained optimization problem can be of great advantage ifn andp are large. Some preliminary numerical results are reported.     

State-space support for path-based reward variables

Many sophisticated formalisms exist for specifying complex system behaviors, but methods for specifying performance and dependability variables have remained quite primitive. To cope with this problem, modelers often must augment system models with extra state information and event types to support particular variables. This often leads to models that are non-intuitive, and must be changed to support different variables. To address this problem, we extend the array of performance measures that may be derived from a given system model, by developing new performance measure specification and model construction techniques. Specifically, we introduce a class of path-based reward variables, and show how various performance measures may be specified using these variables. Path-based reward variables extend the previous work with reward structures to allow rewards to be accumulated based on sequences of states and transitions. To maintain the relevant history, we introduce the concept of a path automaton, whose state transitions are based on the system model state and transitions. Furthermore, we present a new procedure for constructing state spaces and the associated transition rate matrices that support path-based reward variables. Our new procedure takes advantage of the path automaton to allow a single system model to be used as the basis of multiple performance measures that would otherwise require separate models or a single more complicated model.     

A segmented algorithm for solving a class of state constrained discrete optimal control problems

A segmented algorithm for solving state constrained optimal control problems is presented. Its main feature is that it may be implemented on an array of digital computers. Such an implementation exhibits two attractive properties. The first one is that the computational time is reduced due to the parallelism of computations achieved. The second property is that a graceful degradation of performance is realized, i.e., the array produces approximate solutions in the presence of hardware faults.     

On the orthogonal collocation and mathematical programming approach for state constrained optimal control problems

The orthogonal collocation approach is now well known to solve, effectively, the state constrained optimal control problems. Mathematical programming technique was also used as an effective tool to construct the optimal trajectories. In this paper, a study is done on the efficiency and accuracy requirements of the combined orthogonal collocation and mathematical programming approach, as regarding the employed optimization algorithm, and the number of orthogonal collocation points. It is shown, by experimentation with numerical examples that Fletcher-Powell optimization algorithm is much more faster to produce convergence than Fletcher-Reeves algorithm. The efficiency can be a ratio of six-to-one. The results are compared with an alternative approach to solve the same problem. It is shown that the present algorithm is less costly than the alternative approach, although requiring more computation time. The choice is then a compromise one. As the number of orthogonal points increases, the resulting solutions are more accurate, but the convergence speed decreases. Experimentation with N, shows a save of five-to-one in computing time can be achieved with almost the same cost function. Finally, it is shown, by a numerical example, that uniformly distributed collocation points result in non-optimal solutions, which also violate the problem constraints. It is a numerical proof of the superiority of the orthogonal collocation approach.     

Application of multiple shooting to the numerical solution of optimal control problems with bounded state variables

Algorithms for the numerical solution of optimal control problems with bounded state variables are developed. Two main cases are considered: either the control variable appears nonlinearly or the control variable appears linearly. In the first case, an extremal are touching the boundary or containing a boundary arc, is shown to satisfy a suitable two-point boundary value problem. In the second case, a numerical idea for solving the problem in statespace is presented which dispenses with the Lagrange-multipliers. Three numerical examples are discussed illustrating the efficiency of the different algorithms. The encountered two-point boundary value problems are solved with the method of multiple shooting.     

On constrained infinite-time linear quadratic optimal control

This work presents a solution to the infinite-time linear quadratic optimal control (ITLQOC) problem with state and control constraints. It is shown that a single, finite dimensional, convex program of known size can yield this solution. Properties of the resulting value function, with respect to initial conditions, are also established and are shoen to be useful in determining the aforementioned problem size. An example illustrating the method is finally presented.     

Decomposition of problems of optimal control and estimation for discrete systems with fast and slow variables

Consideration was given to the linear-quadratic problem of optimal control for the discrete linear system with fast and slow variables under incomplete information about system state. Decomposition of the discrete matrix Riccati equations was carried out. The proposed decomposition algorithm relies on a geometrical approach using the properties of the invariant manifolds of slow and fast motions of the nonlinear multirate discrete systems as basis. The splitting transformation was constructed in the form of asymptotic decomposition in the degrees of a small parameter.     

A constrained optimal control program in APL

This note formulates a dynamic control model which optimizes a quadratic objective functional subject to linear constraints and solves in APL. Besides giving the optimal objective function value, the APL program computes values of the gain matrices, the Riccati equations, the state variables and the control variables. The power, compactness and flexibility of the APL program are undoubted when compared with other software. The most useful result of the exercise is that the APL program can be easily modified to the exact requirements of the individual researcher, faculty member, consultant and student.     

A constrained optimal control of nonlinear systems

A new method for optimal control of nonlinear systems with input constraint is discussed in this paper. The system is optimized by minimizing a quadratic performance index. Considering this problem as a nonlinear programming problem, the necessary and sufficient conditions for optimal control are derived, which are later simplified to an integral equation. This integral equation becomes a necessary condition. The existence of a solution of this integral equation is studied, and a method of solving it is discussed. A numerical example is worked out at the end.