Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

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

Hidden attractors in dynamical systems

Cooperative control of a nonuniform gantry crane with constrained tension

In this paper, a flexible cable with a payload attached at the bottom is considered to be the model of a crane system used for positioning the payload. The dynamics of the flexible cable coupled with the tip payload contribute to a hybrid system represented by partial-ordinary differential equations. An integral-barrier Lyapunov function (IBLF)-based control is proposed to suppress the undesirable vibrations of the flexible crane system with the boundary output constraint. Adaption laws are developed for handling parametric uncertainties. A novel IBLF is adopted to guarantee the uniform stability of the closed-loop systems without the violation of the boundary constraint. All closed-loop signals are ensured to be bounded. Extensive simulations are demonstrated to illustrate the performance of the control system.

Adaptive Control of a Flexible Crane System With the Boundary Output Constraint

This work deals with some sliding mode (SM) and adaptive sliding-mode control (SMC) strategies for a class of nonlinear biotechnological processes. First, a dynamical SM-based feedback strategy is designed in order to ensure the asymptotic output stabilization of nonlinear bioprocesses. The control law design is done by means of a combination between the exact linearization approach and the SMC. Second, an adaptive SMC strategy is derived for this class of bioprocesses. In order to deal with the parametric uncertainties of the bioprocesses, the adaptive form of the SMC law is designed by means of direct, overparameterized adaptive control techniques available for linearizable systems. The paper also presents the implementation of the proposed control strategies for two typical bioprocesses belonging to the studied nonlinear class. The first prototype process takes place into a Continuous Stirred Tank Bioreactor, and the second is a lipase production process that takes place inside a Fed-Batch Bioreactor. Copyright © 2007 John Wiley & Sons, Ltd.

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Sliding mode and adaptive sliding‐mode control of a class of nonlinear bioprocesses

http://www.nt.ntnu.no/users/skoge/prost/proceedings/ifac11-proceedings/data/html/papers/0084.pdf

Stick-slip Oscillations in Oillwell Drilstrings: Distributed Parameter and Neutral Type Retarded Model Approaches

This paper focuses on forced oscillations (periodic and almost periodic) in some dynamical models described by coupled differential and difference equations. The existence and stability result is based on a theorem on invariant manifolds on Banach spaces due to Kurzweil and Halanay. The state-space (based on Liapunov-Krasovskii functionals combined with the S-procedure) approach is considered. An illustrative example is presented and some conclusions are drawn concerning sharpness and ease in applications of the method.

Vladimir Rasvan

Papers

Sliding mode and adaptive sliding‐mode control of a class of nonlinear bioprocesses

Stick-slip Oscillations in Oillwell Drilstrings: Distributed Parameter and Neutral Type Retarded Model Approaches

Oscillations in lossless propagation models: a Liapunov–Krasovskii approach

Stability and Stable Oscillations in Discrete Time Systems

Functional differential equations of lossless propagation and almost linear behavior