Reduction of Dimensionality, Dynamic Programming, and Control Processes

1961 ◽  
Vol 83 (1) ◽  
pp. 82-84 ◽  
Author(s):  
Richard Bellman ◽  
Robert Kalaba
Keyword(s):  
Dynamic Programming ◽  
Time Lag ◽  
Successive Approximations ◽  
Time Lags ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Control Processes ◽  
Finite Dimensional ◽  
Finite Set ◽  
And Control

A major difficulty in the way of a successful systematic approach to the study of control processes by way of the theory of dynamic programming is the occurrence of processes having state vectors of high dimension. However difficult the problem is for systems ruled by a finite set of differential equations, it is several orders of magnitude more complex for systems of infinite dimensionality and for systems with time lags. By combining a technique presented earlier for dealing with finite dimensional systems and various methods of successive approximations and quasi-linearization, certain classes of control processes associated with infinite dimensional systems can be treated. The ideas are illustrated by discussing control of a system involving a time lag and control of a thermal system.

Active Control of Mechanical Vibrations in a Circular Disk

1992 ◽  
Vol 114 (1) ◽  
pp. 104-112 ◽  
Author(s):  
C. Y. Kuo ◽  
C. C. Huang
Keyword(s):  
Mechanical Vibration ◽  
Circular Disk ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Sensors And Actuators ◽  
Mechanical Vibrations ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Finite Dimensional ◽  
And Control

Mechanical vibration is a common phenomenon observed in the operation of many machines and arises from the inertia effect of machine parts in motion. While many control system design methods for distributed parameter systems have already been proposed in the literature, generally they are either based on truncated models and, as a result, suffer from computational and “spillover” difficulties or require distributed parameter actuators which are rarely available in reality. Therefore, there is a definite need for the development of a class of controllers which can be realized by spatially discrete sensors and actuators and whose design specifically includes stabilization and control of all the higher frequency vibration modes. To address this need, we propose the design of linear compensators whose design is based on root locus arguments for infinite dimensional systems. Since the design is not based on finite dimensional models of the plant to be controlled, we expect it to perform well for those distributed parameter systems for which sufficiently accurate data on pole and zero locations can be obtained. In this paper we apply this approach to control mechanical vibrations in those physical systems which can be accurately modeled as a flexible circular disk. Computer simulation results indicate that all the predominant lower frequency vibrations can be efficiently eliminated by just a few pairs of colocated sensor and actuator.

Delay-Adaptive Linear Control

2020 ◽  
Author(s):  
Yang Zhu ◽  
Miroslav Krstic
Keyword(s):  
Nonlinear Differential Equations ◽  
Stability Study ◽  
Delay Systems ◽  
Engineering Practice ◽  
Delay Compensation ◽  
Infinite Dimensional ◽  
Controlled Systems ◽  
Finite Dimensional ◽  
And Control ◽  
Parameter Values

Actuator and sensor delays are among the most common dynamic phenomena in engineering practice, and when disregarded, they render controlled systems unstable. Over the past sixty years, predictor feedback has been a key tool for compensating such delays, but conventional predictor feedback algorithms assume that the delays and other parameters of a given system are known. When incorrect parameter values are used in the predictor, the resulting controller may be as destabilizing as without the delay compensation. This book develops adaptive predictor feedback algorithms equipped with online estimators of unknown delays and other parameters. Such estimators are designed as nonlinear differential equations, which dynamically adjust the parameters of the predictor. The design and analysis of the adaptive predictors involves a Lyapunov stability study of systems whose dimension is infinite, because of the delays, and nonlinear, because of the parameter estimators. This book solves adaptive delay compensation problems for systems with single and multiple inputs/outputs, unknown and distinct delays in different input channels, unknown delay kernels, unknown plant parameters, unmeasurable finite-dimensional plant states, and unmeasurable infinite-dimensional actuator states. Presenting breakthroughs in adaptive control and control of delay systems, the book offers powerful new tools for the control engineer and the mathematician.

Modeling and boundary control of infinite dimensional systems in the Brayton–Moser framework

2017 ◽  
Vol 36 (2) ◽  
pp. 485-513
Author(s):  
Krishna Chaitanya Kosaraju ◽  
Ramkrishna Pasumarthy ◽  
Dimitri Jeltsema
Keyword(s):  
Boundary Control ◽  
Zero Energy ◽  
Line System ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Finite Dimensional ◽  
Boundary Control Problem ◽  
Energy Flows ◽  
Admissible Pairs ◽  
The Stability

Abstract It is well documented that shaping the energy of finite-dimensional port-Hamiltonian systems by interconnection is severely restricted due to the presence of dissipation. This phenomenon is usually referred to as the dissipation obstacle. In this paper, we show the existence of dissipation obstacle in infinite dimensional systems. Motivated by this, we present the Brayton–Moser formulation, together with its equivalent Dirac structure. Analogous to finite dimensional systems, identifying the underlying gradient structure is crucial in presenting the stability analysis. We elucidate this through an example of Maxwell’s equations with zero energy flows through the boundary. In the case of mixed-finite and infinite-dimensional systems, we find admissible pairs for all the subsystems while preserving the overall structure. We illustrate this using a transmission line system interconnected to finite dimensional systems through its boundary. This ultimately leads to a new passive map, using this we solve a boundary control problem, circumventing the dissipation obstacle.

Analysis and Control of Infinite-Dimensional Systems

2009 ◽  
pp. 319-368
Author(s):  
Vincent Duindam ◽  
Alessandro Macchelli ◽  
Stefano Stramigioli ◽  
Herman Bruyninckx
Keyword(s):  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
And Control
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