Elmer G. Gilbert

Linear systems with state and control constraints: the theory and application of maximal output admissible sets

The initial state of an unforced linear system is output admissible with respect to a constraint set Y if the resulting output function satisfies the pointwise-in-time condition y(t) in Y, t>or=0. The set of all possible such initial conditions is the maximal output admissible set O/sub infinity /. The properties of O/sub infinity / and its characterization are investigated. In the discrete-time case, it is generally possible to represent O/sub infinity / or a close approximation of it, by a finite number of functional inequalities. Practical algorithms for generating the functions are described. In the continuous-time case simple representations of the maximal output admissible set are not available, however, it is shown that the discrete-time results may be used to obtain approximate representations. >

A fast procedure for computing the distance between complex objects in three-dimensional space

An algorithm for computing the Euclidean distance between a pair of convex sets in R/sup m/ is described. Extensive numerical experience with a broad family of polytopes in R/sup 3/ shows that the computational cost is approximately linear in the total number of vertices specifying the two polytopes. The algorithm has special features which makes its application in a variety of robotics problems attractive. These features are discussed and an example of collision detection is given. >

Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: stability and moving-horizon approximations

Stability results are given for a class of feedback systems arising from the regulation of time-varying discrete-time systems using optimal infinite-horizon and moving-horizon feedback laws. The class is characterized by joint constraints on the state and the control, a general nonlinear cost function and nonlinear equations of motion possessing two special properties. It is shown that weak conditions on the cost function and the constraints are sufficient to guarantee uniform asymptotic stability of both the optimal infinite-horizon and moving-horizon feedback systems. The infinite-horizon cost associated with the moving-horizon feedback law approaches the optimal infinite-horizon cost as the moving horizon is extended.

Theory and computation of disturbance invariant sets for discrete-time linear systems

This paper considers the characterization and computation of invariant sets for discrete-time, time-invariant, linear systems with disturbance inputs whose values are confined to a specified compact set but are otherwise unknown. The emphasis is on determining maximal disturbance-invariant sets X that belong to a specified subset Γ of the state space. Such d-invariant sets have important applications in control problems where there are pointwise-in-time state constraints of the form χ(t)∈Γ . One purpose of the paper is to unite and extend in a rigorous way disparate results from the prior literature. In addition there are entirely new results. Specific contributions include: exploitation of the Pontryagin set difference to clarify conceptual matters and simplify mathematical developments, special properties of maximal invariant sets and conditions for their finite determination, algorithms for generating concrete representations of maximal invariant sets, practical computational questions, extension of the main results to general Lyapunov stable systems, applications of the computational techniques to the bounding of state and output response. Results on Lyapunov stable systems are applied to the implementation of a logic-based, nonlinear multimode regulator. For plants with disturbance inputs and state-control constraints it enlarges the constraint-admissible domain of attraction. Numerical examples illustrate the various theoretical and computational results.

Distance functions and their application to robot path planning in the presence of obstacles

An approach to robotic path planning, which allows optimization of useful performance indices in the presence of obstacles, is given. The main idea is to express obstacle avoidance in terms of the distances between potentially colliding parts. Mathematical properties of the distance functions are studied and it is seen that various types of derivatives of the distance functions are easily characterized. The results lead to the formulation of path planning problems as problems in optimal control and suggest numerical procedures for their solution. A simple numerical example involving a three-degree-of-freedom Cartesian manipulator is described.

THE DECOUPLING OF MULTIVARIABLE SYSTEMS BY STATE FEEDBACK

1. Introduction. The objective of this paper is to develop a comprehensive theory for the decoupling of multivariable systems by state feedback. We begin by giving a preliminary formulation of the decoupling problem, discussing certain aspects of its solution, reviewing previous research, and indicating the contributions of this paper.

Nonlinear tracking control in the presence of state and control constraints: a generalized reference governor

This paper proposes a new approach to reference governor design. As in prior literature, the governor accepts input commands and modifies their evolution so that specified pointwise-in-time constraints on state and control variables are satisfied. The new approach applies to general discrete-time and continuous-time nonlinear systems with uncertainties. It relies on safety properties provided by sublevel sets of equilibria-parameterized functions. These functions need not be Lyapunov functions, and the corresponding sublevel sets need not be positively invariant. Technical conditions that capture the bare essentials of what is needed are identified and the usual desirable properties of reference governors are established. The new approach significantly broadens the class of methods available for constructing the nonlinear function that is required in the implementation of the reference governors. This advantage is illustrated in a nonlinear control problem where off-line, computer-based simulation is the basis for constructing the nonlinear function.

Computation of minimum-time feedback control laws for discrete-time systems with state-control constraints

The problem of finding a feedback law that drives the state of a linear discrete-time system to the origin in minimum-time subject to state-control constraints is considered. Algorithms are given to obtain facial descriptions of the M -step admissible sets. These descriptions are then used to characterize the complete class of minimum-time feedback laws. Moreover, the characterization leads to a conceptually simple on-line implementation. The main ideas are illustrated with two simple examples.

Convergence of a Generalized SMO Algorithm for SVM Classifier Design

Convergence of a generalized version of the modified SMO algorithms given by Keerthi et al. for SVM classifier design is proved. The convergence results are also extended to modified SMO algorithms for solving ν-SVM classifier problems.

Computing the distance between general convex objects in three-dimensional space

A methodology for computing the distance between objects in three-dimensional space is presented. The convex polytope is replaced by a general convex set, avoiding the errors caused by the usual polytope approximations and actually reducing the overall computational time. The basic algorithm is a simple extension of the polytope distance algorithm described by E.G. Gilbert et al. (1988). It utilizes the support mappings of the sets representing the objects. A calculus for evaluating these mappings that allows the extended algorithm to be applied to a rich family of nonpolytopal objects is presented. While the convergence of the algorithm is not finite, it is fast and an effective stopping condition that guarantees the accuracy of the numerical solution is available. Extensive numerical experiments support the claimed efficiency. >