Normal Form for Linear Infinite-Dimensional Systems in Hilbert Space and Its Role in Direct Adaptive Control of Distributed Parameter Systems

2017 ◽  
Author(s):  
Mark J. Balas ◽  
Susan A. Frost
Keyword(s):  
Hilbert Space ◽  
Adaptive Control ◽  
Normal Form ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Direct Adaptive Control

Adaptive Tracking Control for Linear Infinite Dimensional Systems

2016 ◽  
Author(s):  
Mark J. Balas ◽  
Susan A. Frost
Keyword(s):  
Hilbert Space ◽  
Adaptive Control ◽  
Finite Basis ◽  
Bounded Operators ◽  
Linear Diffusion ◽  
Adaptive Tracking Control ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Direct Adaptive Control ◽  
Diffusion Systems

Tracking an ensemble of basic signals is often required of control systems in general. Here we are given a linear continuous-time infinite-dimensional plant on a Hilbert space and a space of tracking signals generated by a finite basis, and we show that there exists a stabilizing direct adaptive control law that will stabilize the plant and cause it to asymptotically track any member of this collection of signals. The plant is described by a closed, densely defined linear operator that generates a continuous semigroup of bounded operators on the Hilbert space of states. There is no state or parameter estimation used in this adaptive approach. Our results are illustrated by adaptive control of general linear diffusion systems.

Augmentation of Fixed Gain Controlled Infinite Dimensional Systems With Direct Adaptive Control

2020 ◽  
Author(s):  
Mark J. Balas
Keyword(s):  
Hilbert Space ◽  
Adaptive Control ◽  
Adaptive Optics ◽  
Open Loop ◽  
Loop System ◽  
Linear Diffusion ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Transmission Zeros ◽  
Direct Adaptive Control

Abstract Linear infinite dimensional systems are described by a closed, densely defined linear operator that generates a continuous semigroup of bounded operators on a general Hilbert space of states and are controlled via a finite number of actuators and sensors. Many distributed applications are included in this formulation, such as large flexible aerospace structures, adaptive optics, diffusion reactions, smart electric power grids, and quantum information systems. In this paper, we focus on infinite dimensional linear systems for which a fixed gain linear infinite or finite dimensional controller is already in place. We augment this controller with a direct adaptive controller that will maintain stability of the full closed loop system even when the fixed gain controller fails to do so. We prove that the transmission zeros of the combined system are the original open loop transmission zeros, and the point spectrum of the controller alone. Therefore, the combined plant plus controller is Almost Strictly Dissipative (ASD) if and only if the original open loop system is minimum phase, and the fixed gain controller alone is exponentially stable. This result is true whether the fixed gain controller is finite or infinite dimensional. In particular this guarantees that a controller for an infinite dimensional plant based on a reduced -order approximation can be stabilized by augmentation with direct adaptive control to mitigate risks. These results are illustrated by application to direct adaptive control of general linear diffusion systems on a Hilbert space that are described by self-adjoint operators with compact resolvent.

scholarly journals Regional Boundary Gradient Detectability in Distributed Parameter Systems

2018 ◽  
Vol 14 (2) ◽  
pp. 7818-7833 ◽  
Author(s):  
Raheam Al Saphory ◽  
Mrooj Al Bayati
Keyword(s):  
Hilbert Space ◽  
Domain Boundary ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
System State ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Unknown System ◽  
Regional Boundary ◽  
Boundary Gradient

The aim of this paper is study and explore the notion of  the regional boundary gradient detectability in connection with the choice of strategic gradient sensors on sub-region of the considered system domain boundary. More precisely, the principal reason behind introducing this notion is that the possibility to design a dynamic system (may be called regional boundary gradient observer) which enable to estimate the unknown system state gradient. Then for linear infinite dimensional systems in a Hilbert space,  we give various new results related with different measurements. In addition, we provided a description of the regional boundary exponential gradient strategic sensors for completion the regional boundary exponential gradient observability and regional boundary exponential gradient detectability. Finally, we present and illustrate the some applications of sensors structures which relate by regional boundary exponential gradient detectability in diffusion distributed parameter systems.

scholarly journals REGIONAL EXPONENTIAL REDUCED OBSERVABILITY IN DISTRIBUTED PARAMETER SYSTEMS

2015 ◽  
Vol 11 (4) ◽  
pp. 5058-5074 ◽  
Author(s):  
Shahad AL-MULLAH ◽  
Raheam Al-Saphory
Keyword(s):  
Sufficient Conditions ◽  
Distributed Parameter Systems ◽  
Distributed Parameter ◽  
Usual Sense ◽  
Two Phase ◽  
Linear Dynamical Systems ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
New Concepts ◽  
Necessary And Sufficient

The regional exponential reduced observability concept in the presence for linear dynamical systems is addressed for a class of distributed parameter systems governed by strongly continuous semi group in Hilbert space. Thus, the existence of necessary and sufficient conditions is established for regional exponential reduced estimator in parabolic infinite dimensional systems. More precisely, the introduced approach is developed by using the decomposed system and reduced system in connection with various new concepts of (stability, detectability, estimator, observability and strategic sensors). Finally, we also show that there exists a dynamical system for two-phase exchange system described by the coupled parabolic equations is not exponentially reduced observable in usual sense, but it may be regionally exponentially reduced observable.

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