George N. Saridis

Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation

In this paper we study the convergence of the Galerkin approximation method applied to the generalized Hamilton-Jacobi-Bellman (GHJB) equation over a compact set containing the origin. The GHJB equation gives the cost of an arbitrary control law and can be used to improve the performance of this control. The GHJB equation can also be used to successively approximate the Hamilton-Jacobi-Bellman equation. We state sufficient conditions that guarantee that the Galerkin approximation converges to the solution of the GHJB equation and that the resulting approximate control is stabilizing on the same region as the initial control. The method is demonstrated on a simple nonlinear system and is compared to a result obtained by using exact feedback linearization in conjunction with the LQR design method.

An Approximation Theory of Optimal Control for Trainable Manipulators

A theoretical procedure is developed for comparing the performance of arbitrarily selected admissible controls among themselves and with the optimal solution of a nonlinear optimal control problem. A recursive algorithm is proposed for sequential improvement of the control law which converges to the optimal. It is based on the monotonicity between the changes of the Hamiltonian and the value functions proposed by Rekasius, and may provide a procedure for selecting effective controls for nonlinear systems. The approach has been applied to the approximately optimal control of a trainable manipulator with seven degrees of freedom, where the controller is used for motion coordination and optimal execution of object-handling tasks.

EMG Pattern Analysis and Classification for a Prosthetic Arm

This paper deals with the statistical analysis and pattern classification of electromyographic signals from the biceps and triceps of a below-the-humerus amputated or paralyzed person. Such signals collected from a simulated amputee are synergistically generated to produce discrete lower arm movements. The purpose of this study is to utilize these signals to control an electrically driven prosthetic or orthotic arm with minimum extra mental effort on the part of the subject. The results show very good separability of classes of movements when a learning pattern classification scheme is used, and a superposition principle seems to hold which may provide a means of decomposition of any composite motion to the six basic primitive motions, e.g., humeral rotation in and out, elbow flexion and extension, and wrist pronation and supination. Since no synergy was detected for the hand movements, different inputs have to be provided for a grip. The method described is not limited by the location of the electrodes. For amputees with shorter stumps, synergistic signals could be obtained from the shoulder muscles. However, the presentation in this paper is limited to bicep-tricep signal classification only.

Analytic formulation of the principle of increasing precision with decreasing iNtelligence for intelligent machines

Abstract Intelligent machines, like intelligent robots are capable of performing autonomously in uncertain environments, and have imposed new design requirements for modern engineers. New concepts, drawn from areas like artificial intelligence, operations research and control theory, are required in order to implement anthropomorphic tasks with minimum intervention of an operator. This work deals with the analytic formulation of the principle of increasing precision with decreasing intelligence; the fundamental principle of hierarchically intelligent control. A three-level structure representing the organization, coordination and execution has been developed as a probabilistic model of such a system and the approaches necessary to implement each one of them on an intelligent machine are discussed. The principle is derived also from a probabilistic model and can be expressed in terms of entropies. It is compatible with the current formulation of the hierarchically intelligent control problem, the mathematical programming solution of which minimizes the total entropy. The derivation and design of parallel architectures for artificial intelligence, like the Boltzmann machine is obtained from such formulation.

Analytical design of intelligent machines

Abstract The problem of designing “intelligent machines” operating in uncertain environments with minimum supervision or interaction with a human operator is examined. The structure of an “intelligent machine” is defined to be the structure of a Hierarchically Intelligent Control System, composed of three levels hierarchically ordered according to the principle of “increasing precision with decreasing intelligence”, namely: the organizational level, performing general information processing tasks in association with a long-term memory, the coordination level, dealing with specific information processing tasks with a short-term memory, and the control level, which performs the execution of various tasks through hardware using feedback control methods. The behavior of such a machine may be managed by controls with special considerations and its “intelligence” is directly related to the derivation of a compatible measure that associates the intelligence of the higher levels with the concept of entropy, which is a sufficient analytic measure that unifies the treatment of all the levels of an “intelligent machine” as the mathematical problem of finding the right sequence of internal decisions and controls for a system structured in the order of intelligence and inverse order of precision such that it minimizes its total entropy. A case study on the automatic maintenance of a nuclear plant illustrates the proposed approach.

Toward the realization of intelligent controls

This is a paper of expository nature reflecting the authors past experiences, his current research efforts, and his aspirations about the future of automatic-control systems. It is not intended to give a quantitative analysis of modern control methodologies, which may be found in the bibliography at the end of the text, but rather emphasize the importance of a growing area in control engineering. Reviewing the classical, optimal, and stochastic control systems, the reader is led into the uncertainties and controversies of adaptive and learning controls. While self-organizing control was proposed for a systematic unification of these most advanced control methodologies, intelligent control--a discipline capable of high-level decision making and task execution--is predicted as the next level of sophistication in the hierarchy of control systems. A case study on a hierarchically intelligent controlled prosthesis, summarized herein, establishes the feasibility of the suggested methodologies. Future applications to other larse scale systems of general or specific scientific interest may prove the importance of such a discipline.

Minimum jerk path generation

A simple method of trajectory generation of robot manipulators is presented. It is based on an optimal control problem formulation. The jerk, the third derivative of position, of the desired trajectory, adversely affect the efficiency of the control algorithms and therefore should be minimized. assuming joint position, velocity and acceleration to be constrained, a cost criterion containing jerk is considered. Initially, the simple environment without obstacles and constrained by the physical limitations of the joint angles only is examined. For practical reasons, the free execution time has been used to handle the velocity and acceleration constraints instead of the complete bounded state variable formulation. The problem of minimizing the jerk along an arbitrary Cartesian trajectory is formulated and given analytical solution, making this method useful for real-world environments containing obstacles.<<ETX>>

Approximate solutions to the time-invariant Hamilton-Jacobi-Bellman equation

In this paper, we develop a new method to approximate the solution to the Hamilton–Jacobi–Bellman (HJB) equation which arises in optimal control when the plant is modeled by nonlinear dynamics. The approximation is comprised of two steps. First, successive approximation is used to reduce the HJB equation to a sequence of linear partial differential equations. These equations are then approximated via the Galerkin spectral method. The resulting algorithm has several important advantages over previously reported methods. Namely, the resulting control is in feedback form and its associated region of attraction is well defined. In addition, all computations are performed off-line and the control can be made arbitrarily close to optimal. Accordingly, this paper presents a new tool for designing nonlinear control systems that adhere to a prescribed integral performance criterion.

Entropy formulation of optimal and adaptive control

The use of entropy as the common measure to evaluate the different levels of intelligent machines is reported. At the execution level, the design of the desirable control can be expressed by the uncertainty of selecting the optimal control that minimizes a given performance index. By choosing a density function over the set of admissible controls to minimize the differential control entropy, it can be shown that the optimal control problem is equivalent to the problem of minimization of the assigned entropy function with respect to the association control. The adaptive control problem can be analyzed by considering the same entropy over extended space that includes the uncertain parameters. It is shown that the optimal entropy is decomposed into three terms: the optimal control term with given parameters, the parameter identification term, and the equivocation term which accounts for the active transition of dual control. The equivocation when calculated can serve as a measure of optimality of the adaptive control algorithms that involve only distinct identification and optimal control algorithms. An upper bound can be used instead, when the equivocation is hard to calculate. An example illustrates the method. >

The control of a prosthetic arm by EMG pattern recognition

An electromyographic (EMG) signal pattern recognition system is constructed for real-time control of a prosthetic arm through precise identification of motion and speed command. A probabilistic model of the EMG patterns is first formulated in the feature space of integral absolute value (IAV) to describe the relation between a command, represented by motion and speed variables, and location and shape of the corresponding pattern. The model provides the sample probability density function of pattern classes in the decision space of variance and zero crossings based on the relations between IAV, variance, and zero crossings established in this paper. Pattern classification is carried out through a multiclass sequential decision procedure designed with an emphasis on computational simplicity. The upper bound of probability of error and the average number of sample observations are investigated. Speed and motion predictions are used in conjunction with the decision procedure to enhance decision speed and reliability. A decomposition rule is formulated for the direct assignment of speed to each primitive motion involved in a combined motion. A learning procedure is also designed for the decision processor to adapt long-term pattern variation. Experimental results are discussed in the Appendix.