State Inequality Constraints Research Articles

Sequential conjugate gradient‐restoration algorithm for optimal control problems with non‐differential constraints and general boundary conditions, part I

AbstractIn this paper, a new member of the family of sequential gradient‐restoration algorithms for the solution of optimal control problems is presented. This is an algorithm of the conjugate gradient type, which is designed to solve two classes of optimal control problems, called Problem P1 and Problem P2 for easy indentification.Problem P1 involves minimizing a functional I subject to differential constraints and general boundary conditions. It consists of finding the state x(t), the control u(t), and the parameter pi so that the functional I is minimized, while the constraints and the boundary conditions are satisfied to a predetermined accuracy. Problem P2 extends Problem P1 to include non‐differential constraints to be satisfied everywhere along the interval of integration.The approach taken is a sequence of two‐phase cycles, composed of a conjugate gradient phase and a restoration phase. The conjugate gradient phase involves one iteration and is designed to decrease the value of the functional, while the constraints are satisfied to first order. The restoration phase involves one or more iterations; each restorative iteration is designed to force constraint satisfaction to first order, while the norm squared of the variations of the control, the parameter, and the missing components of the initial state is minimized. The resulting algorithm has several properties: (i) it produces a sequence of feasible solutions; (ii) each feasible solution is characterized by a value of the functional I which is smaller than that associated with any previous feasible solution; and (iii) for the special case of a quadratic functional subject to linear constraints, the variations of the state, control, and parameter produced by the sequence of conjugate gradient phases satisfy various orthogonality and conjugacy conditions.The algorithm presented here differs from those of References 1‐4, in that it is not required that the state vector be given at the initial point. Instead, the initial conditions can be absolutely general. In analogy with References 1‐4, the present algorithm is capable of handling general final conditions; therefore, it is suitable for the solution of optimal control problems with general boundary conditions.The importance of the present algorithm lies in that many optimal control problems either arise naturally in the present format or can be brought to such a format by means of suitable transformations.5 Therefore, a great variety of optimal control problems can be handled. This includes: (i) problems with control equality constraints, (ii) problems with state equality constraints, (iii) problems with state‐derivative equality constraints, (iv) problems with‐control inequality constraints, (v) problems with state inequality constraints, (vi) problems with state‐derivative inequality constraints, and (vii) Chebyshev minimax problems.Several numerical examples are presented in Part 2 (Reference 6) in order to illustrate the performance of the algorithm associated with Problem P1 and Problem P2. The numerical results show the feasibility as well as the convergence characteristics of the present algorithm.

A transformation technique for optimal control problems with partially linear state inequality constraints

This paper considers optimal control problems involving the minimization of a functional subject to differential constraints, terminal constraints, and a state inequality constraint. The state inequality constraint is of a special type, namely, it is linear in some or all of the components of the state vector. A transformation technique is introduced, by means of which the inequality-constrained problem is converted into an equality-constrained problem involving differential constraints, terminal constraints, and a control equality constraint. The transformation technique takes advantage of the partial linearity of the state inequality constraint so as to yield a transformed problem characterized by a new state vector of minimal size. This concept is important computationally, in that the computer time per iteration increases with the square of the dimension of the state vector. In order to illustrate the advantages of the new transformation technique, several numerical examples are solved by means of the sequential gradient-restoration algorithm for optimal control problems involving nondifferential constraints. The examples show the substantial savings in computer time for convergence, which are associated with the new transformation technique.

Necessary conditions for problems with higher derivative bounded state variables

This paper combines the separate works of two authors. Tan proves a set of necessary conditions for a control problem with second-order state inequality constraints (see Ref. 1). Russak proves necessary conditions for an extended version of that problem. Specifically, the extended version augments the original problem by including state equality constraints, differential and isopermetric equality and inequality constraints, and endpoint constraints. In addition, Russak (i) relaxes the solvability assumption on the state constraints, (ii) extends the maximum principle to a larger set, (iii) obtains modified forms of the relationH =Ht and of the transversality relation usually obtained in problems of this type, and (iv) proves a condition concerning μα(t1), the derivative of the multiplier functions at the final time.

A Transformation Method With Two Parameters for Constrained Optimal Control Problems

A transformation method with two parameters is described which can be applied to nonconvex optimal control problems with constraints, does not present numerical difficulties, and has a rapid convergence rate. The basic algorithm of the method is summarized for optimal control problems with equality and inequality constraints, the function of the parameters in the method is considered, one modified algorithm is proposed, and its theoretical results are proved. Finally, the method is applied successfully to both convex and nonconvex optimal control problems with a state inequality constraint.

Transformation Methods of an Exponential Function Type for the Optimal Control Problems with Constraints : 1st Report, A Method with One Correcting Parameter

The purpose of this paper is to present a numerical method for the optimal control problems with inequality constraints. In this method such problems are solved by converting them to a sequence of problems without inequalities and by searching for the saddle point of the unconstrained function. The features in this method are as follows. 1. The optimal value of parameter is closely related to the optimal Lagrange multiplier. 2. The optimal solution is obtained as a finite value of the parameter. Thus, the fact does not yield an ill-conditioning for the penalty function in this method, as opposed to the penalty methods. 3. On account of the existence of the saddle point this method is superior to the ordinary Lagrange method. 4. This method can approach the optimal solution from the interior or exterior of constraints by choosing a parameter corresponding to the Lagrange multiplier. This method has been successfully applied to the problem with state inequality constraint.

Lagrange Duality Theory for Convex Control Problems

The Lagrange dual of control problems with linear dynamics, convex cost and convex inequality state and control constraints is analyzed. If an interior point assumption is satisfied, then the existence of a solution to the dual problem is proved; if there exists a solution to the primal problem, then a complementary slackness condition is satisfied. A necessary and sufficient condition for feasible solutions in the primal and dual problems to be optimal is also given. The dual variables p and v corresponding to the system dynamics and state constraints are proved to be of bounded variation while the multiplier corresponding to the control constraints is proved to lie in $\mathcal{L}^1 $. Finally, a control and state minimum principle is proved. If the cost function is differentiable and the state constraints have two derivatives, then the state minimum principle implies that a linear combination of p and v satisfy the conventional adjoint condition for state constrained control problems.

Optimal controls with equality state constraints

In this paper, we derive necessary conditions for a measurable controlu with values in a compact convex subsetU ofRP to minimize the integral $$\int_a^b {b^0 (s,x(s),u(s))} ds$$ subject to equality and inequality state constraints. The general conditions take the form of a maximum principle which involves an integration with respect to a finitely additive set function, but conditions are given under which this reduces to the classical maximum principle. These results are similar to results derived in Ref. 1 for relaxed controls.

Minimum weight design of an orthogonally stiffened waffle cylindrical shell with buckling constraint

The minimum weight design of an orthogonally stiffened waffle cylindrical shell with simple supported edges is studied. The minimum eigenvalues and the corresponding mode numbers for buckling are searched by the sectioning method or the gradient method. The formulations for the buckling analysis include two loading cases: uniform axial compressive stress; and bending, as well as uniform, axial stresses. The optimization method used is the steepest descent method with state inequality constraint. Six design variables are considered: thickness of the shell; depth and width of the longitudinal stiffeners; depth and width of the circumferential stiffeners; and spacing between the stiffeners. Three design examples are performed: two with uniform compression and one with bending load. When compared with three previously published designs, the reductions in weight are found to be 32.7, 60.2, and 54.7%, respectively. Such results suggest that tremendous weight reduction can be achieved by appropriate wall stiffening and that this method is highly effective for minimum weight design.

Recent advances in gradient algorithms for optimal control problems

This paper summarizes recent advances in the area of gradient algorithms for optimal control problems, with particular emphasis on the work performed by the staff of the Aero-Astronautics Group of Rice University. The following basic problem is considered: minimize a functionalI which depends on the statex(t), the controlu(t), and the parameter π. Here,I is a scalar,x ann-vector,u anm-vector, and π ap-vector. At the initial point, the state is prescribed. At the final point, the statex and the parameter π are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations. First, the sequential gradient-restoration algorithm and the combined gradient-restoration algorithm are presented. The descent properties of these algorithms are studied, and schemes to determine the optimum stepsize are discussed. Both of the above algorithms require the solution of a linear, two-point boundary-value problem at each iteration. Hence, a discussion of integration techniques is given. Next, a family of gradient-restoration algorithms is introduced. Not only does this family include the previous two algorithms as particular cases, but it allows one to generate several additional algorithms, namely, those with alternate restoration and optional restoration. Then, two modifications of the sequential gradient-restoration algorithm are presented in an effort to accelerate terminal convergence. In the first modification, the quadratic constraint imposed on the variations of the control is modified by the inclusion of a positive-definite weighting matrix (the matrix of the second derivatives of the Hamiltonian with respect to the control). The second modification is a conjugate-gradient extension of the sequential gradient-restoration algorithm. Next, the addition of a nondifferential constraint, to be satisfied everywhere along the interval of integration, is considered. In theory, this seems to be only a minor modification of the basic problem. In practice, the change is considerable in that it enlarges dramatically the number and variety of problems of optimal control which can be treated by gradient-restoration algorithms. Indeed, by suitable transformations, almost every known problem of optimal control theory can be brought into this scheme. This statement applies, for instance, to the following situations: (i) problems with control equality constraints, (ii) problems with state equality constraints, (iii) problems with equality constraints on the time rate of change of the state, (iv) problems with control inequality constraints, (v) problems with state inequality constraints, and (vi) problems with inequality constraints on the time rate of change of the state. Finally, the simultaneous presence of nondifferential constraints and multiple subarcs is considered. The possibility that the analytical form of the functions under consideration might change from one subarc to another is taken into account. The resulting formulation is particularly relevant to those problems of optimal control involving bounds on the control or the state or the time derivative of the state. For these problems, one might be unwilling to accept the simplistic view of a continuous extremal arc. Indeed, one might want to take the more realistic view of an extremal arc composed of several subarcs, some internal to the boundary being considered and some lying on the boundary. The paper ends with a section dealing with transformation techniques. This section illustrates several analytical devices by means of which a great number of problems of optimal control can be reduced to one of the formulations presented here. In particular, the following topics are treated: (i) time normalization, (ii) free initial state, (iii) bounded control, and (iv) bounded state.

The Ritz–Trefftz Method for State and Control Constrained Optimal Control Problems

The Ritz–Trefftz method is analyzed for control problems with quadratic cost, linear dynamics and linear inequality state and control constraints. Although the dual function is only semi-definite, a solution to the Ritz–Trefftz problem is proved to exist. First order convergence of the Ritz–Trefftz method in piecewise polynomial subspaces is easily proved; however, higher order convergence requires not only that higher order spaces be used, but also that the grid points in a neighborhood of changes in binding constraints be left as free parameters in the maximization procedure. The convergence proofs are based on a theorem on the regularity of the solution to the control problem and results on the approximation of either nonnegative or monotone functions by nonnegative or monotone polynomials. The paper concludes with a bound on the error in the free boundary (or time when a constraint becomes binding) for the Ritz–Trefftz problem.

Second-order necessary conditions for general problems with state inequality constraints

This paper is a sequel to a previous article by the author, concerned with a certain canonical problem in optimal control involving constraints of the typeψα(t, x)⩽0, α=1,...,m. In that article, a set of second-order conditions necessary for a solution arc was obtained. In this paper, those results are extended to a general control problem involving the above type of constraints.

A hybrid approach to optimal control problems with bounded state

This paper considers the numerical solution of optimal control problems involving a functional I subject to differential constraints, a state inequality constraint, and terminal constraints. The problem is to find the state x(t) , the control u(t) , and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy. The approach taken is a sequence of two-phase processes or cycles, each composed of a gradient phase and a restoration phase. The gradient phase involves a single iteration and is designed to decrease the functional, while the constraints are satisfied to first order. The restoration phase involves one or several iterations and is designed to restore the constraints to a predetermined accuracy, while the norm of the variations of the control and the parameter is minimized. The principal property of the algorithm is that it produces a sequence of feasible suboptimal solutions: the functions x(t), u(t) , π obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the functionals of any two elements of the sequence are comparable. The technique employed is of the hybrid type, in an attempt to combine some of the best features of both the indirect and direct approaches. While a predetermined number and sequence of subarcs are assumed (a feature of direct methods), enforcement of the state inequality constraint is obtained through a Valentine-type representation (a feature of indirect methods). By properly choosing the analytical form of certain non-differential constraints to be satisfied by the augmented control along each subarc composing the extremal arc (these nondifferential constraints are arrived at through the Valentine-type transformation), one ensures satisfaction of the state inequality constraint everywhere. Specifically, strict inequality is enforced for the subarcs internal to the state boundary and strict equality is enforced for the subarcs lying on the state boundary. To facilitate the numerical solution on digital computers, the actual time θ is replaced by the normalized time t , defined in such a way that each subarc composing the extremal arc has a normalized time length Δ t = 1. In this way, variable-time corner conditions and variable-time terminal conditions are transformed into fixed-time corner conditions and fixed-time terminal conditions. The actual times θ 1 , θ 2 , τ at which (i) the state boundary is entered, (ii) the state boundary is exited, and (iii) the terminal manifold is reached are regarded to be components of the parameter π being optimized. The numerical examples illustrating the theory demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.

Second Order Necessary Conditions for Problems with State Inequality Constraints

A set of second order necessary conditions is obtained for solution to a basic problem in optimal control involving the state constraints \[\psi ^\alpha (t,x) \leqq 0,\quad alpha = 1, \cdots ,m.\] For the case of a normal solution arc, the methods of Hestenes are used to show that the second variation of a certain functional is nonnegative on a class of variations of the solution arc. The results for this problem are basic in the sense that they can be extended to include analogous results for much more complicated problems.

状態に制限のある最短時間制御系の操作量決定法

状態に制限のある最短時間制御系の操作量決定法

The linear quadratic cost problem with linear inequality state constraints

The linear quadratic cost problem with linear inequality state constraints

On difference-differential systems with a state inequality constraint

This correspondence proposes a simple degradation method for the solution of optimal control problems associated with difference-differential systems having a state inequality constraint.

Modified quasilinearization algorithm for optimal control problems with bounded state

This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, a state inequality constraint, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy.

Sequential gradient-restoration algorithm for optimal control problems with bounded state

This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, a state inequality constraint, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy.

An optimal feedback control law for regulator problems with linear state inequality constraints

An optimal feedback control law for regulator problems with linear state inequality constraints

Studies of human locomotion via optimal programming

Research to date toward an understanding of human biped locomotion has been primarily experimental in nature, largely due to the complexity of the process. In view of the new, exciting possibilities of programmed electrostimulation of paralyzed extremities to restore locomotion, a critical study at the theoretical level is greatly warranted. Optimal programming and modern control theory offer a new approach to the study. First, it is proposed that normal walking obeys a certain “principle of optimality.” Next, at the dynamic level, modern control theory is used to derive the optimal moment profiles that actuate the locomotor elements to synthesize the observed patterns of the normal gait. Development of the problem structure relies closely on the functional characteristic of the biped gait, particularly the ideas of distinct phasic activities and the associated temporal patterns of a walking cycle. The result is a multiarc programming problem with three stages. Each stage involves dynamic constraints that reflect the particular nature of the phasic activity. Activity in the stance phase is described by equality constraints on the “states,” whereas the swing phase is characterized by inequality state constraints. A novelty of the approach is that the theory could be used to study walking behavior under different environmental conditions, such as walking upstairs or over a hole. Joining of the arcs is arranged in such a way as to maintain continuity of certain trajectories as well as repeatability of motion. A distinct feature of the present approach that differs from other studies is the presence of a minimizing performance criterion. Based on external characteristics of muscles and certain assumptions regarding normal locomotion, a simple quadratic type of performance index is proposed. This performance criterion is meaningful in that it is shown to be proportional to the mechanical work done during normal walking. Invoking the necessary conditions of optimal control theory, a multipoint boundary value problem is obtained. A penalty function technique is then employed for iterative numerical solution on a digital computer. Using the parameters available in the literature, simulation results are obtained that agree well with the experimental studies performed by Eberhart and his associates. Furthermore, certain refined features are obtained that are not available in previous studies. Success in applying optimal programming techniques to human locomotion could yield better design procedures for prostheses and could allow eventual realization of the dream of programmed stimulation of many paralyzed persons.