Predictor Feedback of Uncertain Multi-Input Systems

2020 ◽  
pp. 267-292
Author(s):  
Yang Zhu ◽  
Miroslav Krstic
Keyword(s):  
Linear System ◽  
Output Feedback ◽  
State Feedback ◽  
System Matrix ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Unknown System ◽  
Distributed Actuator

This chapter discusses the predictor feedback for uncertain multi-input systems. This is based on the predictor feedback framework for uncertainty-free multi-input systems in the tenth chapter. The chapter addresses four combinations of the five uncertainties that come from a finite-dimensional multi-input linear system with distributed actuator delays. These uncertainties include the following types: unknown and distinct delays, unknown delay kernels, unknown system matrix, unmeasurable finite-dimensional plant state, and unmeasurable infinite-dimensional actuator state. The chapter then examines the adaptive state feedback under unknown as well as uncertain delays, delay kernels, and parameters. It also explores robust output feedback under unknown delays, delay kernels, and PDE or ODE states.

Predictor Feedback of Uncertain Single-Input Systems

2020 ◽  
pp. 235-266
Author(s):  
Yang Zhu ◽  
Miroslav Krstic
Keyword(s):  
Output Feedback ◽  
State Feedback ◽  
System Matrix ◽  
Unknown Parameters ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Linear Plant ◽  
Single Input ◽  
Distributed Actuator

This chapter presents the predictor feedback for uncertain single-input systems. This is based on the predictor feedback framework for uncertainty-free single-input systems in the previous chapter. The chapter addresses the five combinations of the five uncertainties that come from a single-input linear plant with distributed actuator delay. These uncertainties include the following types: unknown delay, unknown delay kernel, unknown parameters in the system matrix, unmeasurable finite-dimensional plant state, and unmeasurable infinite-dimensional actuator state. The chapter then studies adaptive state feedback under unknown delay, delay kernel, and parameter. It also assesses robust output feedback under unknown delay, delay kernel, and PDE or ODE state.

scholarly journals Adapting the range of validity for the Carleman linearization

2016 ◽  
Vol 14 ◽  
pp. 51-54 ◽  
Author(s):  
Harry Weber ◽  
Wolfgang Mathis
Keyword(s):  
Differential Equation ◽  
Linear System ◽  
Nonlinear Differential Equation ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Liouville Problem ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Linear Differential

Abstract. In this contribution, the limitations of the Carleman linearization approach are presented and discussed. The Carleman linearization transforms an ordinary nonlinear differential equation into an infinite system of linear differential equations. In order to transform the nonlinear differential equation, orthogonal polynomials which represent solutions of a Sturm–Liouville problem are used as basis. The determination of the time derivate of this basis yields an infinite dimensional linear system that depends on the considered nonlinear differential equation. The infinite linear system has the same properties as the nonlinear differential equation such as limit cycles or chaotic behavior. In general, the infinite dimensional linear system cannot be solved. Therefore, the infinite dimensional linear system has to be approximated by a finite dimensional linear system. Due to limitation of dimension the solution of the finite dimensional linear system does not represent the global behavior of the nonlinear differential equation. In fact, the accuracy of the approximation depends on the considered nonlinear system and the initial value. The idea of this contribution is to adapt the range of validity for the Carleman linearization in order to increase the accuracy of the approximation for different ranges of initial values. Instead of truncating the infinite dimensional system after a certain order a Taylor series approach is used to approximate the behavior of the nonlinear differential equation about different equilibrium points. Thus, the adapted finite linear system describes the local behavior of the solution of the nonlinear differential equation.

scholarly journals Global Stabilization of Nonlinear Finite Dimensional System with Dynamic Controller Governed by 1 − d Heat Equation with Neumann Interconnection

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 227
Author(s):  
Mohsen Dlala ◽  
Abdallah Benabdallah
Keyword(s):  
State Feedback ◽  
Linear Part ◽  
Coupled System ◽  
High Gain ◽  
Dimensional System ◽  
Nonlinear Part ◽  
Infinite Dimensional ◽  
Dynamic Controller ◽  
Finite Dimensional ◽  
Linear Growth Condition

This paper deals with the stabilization of a class of uncertain nonlinear ordinary differential equations (ODEs) with a dynamic controller governed by a linear 1−d heat partial differential equation (PDE). The control operates at one boundary of the domain of the heat controller, while at the other end of the boundary, a Neumann term is injected into the ODE plant. We achieve the desired global exponential stabilization goal by using a recent infinite-dimensional backstepping design for coupled PDE-ODE systems combined with a high-gain state feedback and domination approach. The stabilization result of the coupled system is established under two main restrictions: the first restriction concerns the particular classical form of our ODE, which contains, in addition to a controllable linear part, a second uncertain nonlinear part verifying a lower triangular linear growth condition. The second restriction concerns the length of the domain of the PDE which is restricted.

scholarly journals Boundary feedback control of an anti-stable wave equation

2020 ◽  
Vol 37 (4) ◽  
pp. 1367-1399
Author(s):  
Pierre APKARIAN ◽  
Dominikus NOLL
Keyword(s):  
Wave Equation ◽  
Output Feedback ◽  
Closed Loop ◽  
Dimensional System ◽  
Torsional Vibrations ◽  
Feedback Controllers ◽  
Boundary Feedback Control ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Infinite Dimensional System

Abstract We discuss boundary control of a wave equation with a non-linear anti-damping boundary condition. We design structured finite-dimensional $H_{\infty }$-output feedback controllers that stabilize the infinite-dimensional system exponentially in closed loop. The method is applied to control torsional vibrations in drilling systems with the goal to avoid slip-stick.

scholarly journals An FDA-Based Approach for Clustering Elicited Expert Knowledge

Stats ◽  
2021 ◽  
Vol 4 (1) ◽  
pp. 184-204
Author(s):  
Carlos Barrera-Causil ◽  
Juan Carlos Correa ◽  
Andrew Zamecnik ◽  
Francisco Torres-Avilés ◽  
Fernando Marmolejo-Ramos
Keyword(s):  
Functional Data ◽  
Expert Knowledge ◽  
Probability Distributions ◽  
Knowledge Elicitation ◽  
Parallel Analysis ◽  
Data Sets ◽  
Dimensional Problem ◽  
Prior Distributions ◽  
Infinite Dimensional ◽  
Finite Dimensional

Expert knowledge elicitation (EKE) aims at obtaining individual representations of experts’ beliefs and render them in the form of probability distributions or functions. In many cases the elicited distributions differ and the challenge in Bayesian inference is then to find ways to reconcile discrepant elicited prior distributions. This paper proposes the parallel analysis of clusters of prior distributions through a hierarchical method for clustering distributions and that can be readily extended to functional data. The proposed method consists of (i) transforming the infinite-dimensional problem into a finite-dimensional one, (ii) using the Hellinger distance to compute the distances between curves and thus (iii) obtaining a hierarchical clustering structure. In a simulation study the proposed method was compared to k-means and agglomerative nesting algorithms and the results showed that the proposed method outperformed those algorithms. Finally, the proposed method is illustrated through an EKE experiment and other functional data sets.

scholarly journals The Farkas lemma of Shimizu, Aiyoshi and Katayama

1985 ◽  
Vol 31 (3) ◽  
pp. 445-450 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Convex Programming ◽  
Infinite Dimensional ◽  
Farkas Lemma ◽  
Finite Dimensional

Shimizu, Aiyoshi and Katayama have recently given a finite dimensional generalization of the classical Farkas Lemma. In this note we show that a result of Pshenichnyi on convex programming can be used to give a generalization of the result of Shimizu, Aiyoshi and Katayama to infinite dimensional spaces. A generalized Farkas Lemma of Glover is also obtained.

DUAL FORMULATION OF OPTIMAL PROBLEM FOR CONTINUOUS-TIME SYSTEMS

2005 ◽  
Vol 02 (03) ◽  
pp. 251-258
Author(s):  
HANLIN HE ◽  
QIAN WANG ◽  
XIAOXIN LIAO
Keyword(s):  
Objective Function ◽  
Continuous Time ◽  
Dimensional Space ◽  
Finite Dimensional Space ◽  
Error Response ◽  
Infinite Dimensional ◽  
Dual Formulation ◽  
Finite Dimensional ◽  
Continuous Time Systems ◽  
Time Systems

The dual formulation of the maximal-minimal problem for an objective function of the error response to a fixed input in the continuous-time systems is given by a result of Fenchel dual. This formulation probably changes the original problem in the infinite dimensional space into the maximal problem with some restrained conditions in the finite dimensional space, which can be researched by finite dimensional space theory. When the objective function is given by the norm of the error response, the maximum of the error response or minimum of the error response, the dual formulation for the problems of L1-optimal control, the minimum of maximal error response, and the minimal overshoot etc. can be obtained, which gives a method for studying these problems.

scholarly journals Free non-associative principal train algebras

1984 ◽  
Vol 27 (3) ◽  
pp. 313-319 ◽  
Author(s):  
P. Holgate
Keyword(s):  
Finite Number ◽  
Ground Field ◽  
Linear Forms ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The Senses ◽  
Special Train ◽  
Complex Coefficients ◽  
Infinite Linear

The definitions of finite dimensional baric, train, and special train algebras, and of genetic algebras in the senses of Schafer and Gonshor (which coincide when the ground field is algebraically closed, and which I call special triangular) are given in Worz-Busekros's monograph [8]. In [6] I introduced applications requiring infinite dimensional generalisations. The elements of these algebras were infinite linear forms in basis elements a0, a1,… and complex coefficients such that In this paper I consider only algebras whose elements are forms which only a finite number of the xi are non zero.

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