Unlike the many soft computing applications where it suffices to achieve a "good approximation most of the time," a control system must be stable all of the time. As such, if one desires to learn a control law in real-time, a fusion of soft computing techniques to learn the appropriate control law with hard computing techniques to maintain the stability constraint and guarantee convergence is required. The objective of the paper is to describe an adaptive dynamic programming algorithm (ADPA) which fuses soft computing techniques to learn the optimal cost (or return) functional for a stabilizable nonlinear system with unknown dynamics and hard computing techniques to verify the stability and convergence of the algorithm. Specifically, the algorithm is initialized with a (stabilizing) cost functional and the system is run with the corresponding control law (defined by the Hamilton-Jacobi-Bellman equation), with the resultant state trajectories used to update the cost functional in a soft computing mode. Hard computing techniques are then used to show that this process is globally convergent with stepwise stability to the optimal cost functional/control law pair for an (unknown) input affine system with an input quadratic performance measure (modulo the appropriate technical conditions). Three specific implementations of the ADPA are developed for 1) the linear case, 2) for the nonlinear case using a locally quadratic approximation to the cost functional, and 3) the nonlinear case using a radial basis function approximation of the cost functional; illustrated by applications to flight control.

Adaptive dynamic programming

This survey is concerned with the definition and main properties of sets of specific values of complex frequency, called poles and zeros, which may be associated with a matrix-valued function of a complex frequency variable. The main emphasis is on the physical interpretation of invariant zeros in terms of the general zero-output behaviour of a linear dynamical system. A discussion is given of the relationship between this definition of a zero and the various other forms adopted in the current literature. The relationship is considered between poles and zeros defined by algebraic means and the standard complex variable theory of algebraic functions. It is shown that the poles and zeros of a square matrix-valued function of a complex variable G(s) are the same as the poles and zeros of an associated algebraic function g(s).

Poles and zeros of linear multivariable systems : a survey of the algebraic, geometric and complex-variable theory

A comprehensive discussion is given of the background to the generalized Nyquist stability criterion for linear multivariable feedback systems. This leads to a proof based on the use of the Principle of the Argument applied to an algebraic function defined on an appropriate Riemann surface. It is shown how the matrix-valued functions of a complex variable which define the loop transmittance, return-ratio and return-difference matrices of feedback systems analysis may be associated with a set of characteristic algebraic functions, each associated with a Riemann surface. These characteristic functions enable the characteristic loci, which featured in previous heuristic treatments of the generalized Nyquist stability criterion, to be put on a sound basis. The relationship between the algebraic structure of the matrix-valued functions and the appropriate complex-variable theory is carefully discussed. These extensions of the complex-variable concepts underlying the Nyquist criterion are then related to an app...

The generalized Nyquist stability criterion and multivariable root loci

An iterative learning control algorithm for a class of linear periodic systems is proposed in which parameter estimation is performed in the domain of an iterative sequence of operations with the time frozen. A sufficient condition for the convergence of the proposed algorithm is given. It is also shown that the proposed learning controller can be applied to the continuous path control of robot manipulators. It is noted that the proposed algorithm works reasonably well only for the case of small perturbations with respect to a nominal trajectory. >

An iterative learning control method with application to robot manipulators

In this paper we examine the stability of a proportional derivative (PD) controller for the trajectory-following problem of a robot manipulator. We use Lyapunov's second method to derive a uniform boundness result for the PD controller. We show that if the PD controller gains are chosen greater than a specific bound and if the initial tracking error is zero, the velocity and position tracking errors are uniformly bounded. We then develop two additional controllers that use auxiliary control inputs along with the PD controller. Both of these controllers are shown to yield a uniform ultimate boundness property for the tracking error.

Robust control for the tracking of robot motion

https://link.springer.com/content/pdf/10.1057%2Fjors.1972.95.pdf

Matrices in control theory

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Matrices in control theory