Continuous Time Dynamical Systems

Continuous Time Dynamical Systems: State Estimation and Optimal Control with Orthogonal Functions

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  • Summarizes key developments in area of orthogonal functions in optimal control and state estimation
  • Focuses on development of recursive algorithms
  • Discusses optimal control of varieties of systems e.g. linear/non-linear, singular, large scale systems, and more
  • Includes descriptions of both recursive and non-recursive algorithms
  • Describes Block pulse functions and Legendre polynomials


Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. An optimal control is a set of differential equations describing the paths of the control variables that minimize the cost functional.

This book, Continuous Time Dynamical Systems: State Estimation and Optimal Control with Orthogonal Functions, considers different classes of systems with quadratic performance criteria. It then attempts to find the optimal control law for each class of systems using orthogonal functions that can optimize the given performance criteria.

Illustrated throughout with detailed examples, the book covers topics including:

  • Block-pulse functions and shifted Legendre polynomials
  • State estimation of linear time-invariant systems
  • Linear optimal control systems incorporating observers
  • Optimal control of systems described by integro-differential equations
  • Linear-quadratic-Gaussian control
  • Optimal control of singular systems
  • Optimal control of time-delay systems with and without reverse time terms
  • Optimal control of second-order nonlinear systems
  • Hierarchical control of linear time-invariant and time-varying systems

Table of Contents

Optimal Control Problem
Historical Perspective
Organization of the Book

Orthogonal Functions and Their Properties
Block-Pulse Functions (BPFs)
Legendre Polynomials (LPs)
Shifted Legendre Polynomials (SLPs)
Nonlinear Operational Matrix
Rationale for Choosing BPFs and SLPs

State Estimation
Inherent Filtering Property of OFs
State Estimation
Illustrative Examples

Linear Optimal Control Systems Incorporating Observers
Analysis of Linear Optimal Control Systems Incorporating Observers
Illustrative Example

Optimal Control of Systems Described by Integro-Differential Equations
Optimal Control of LTI Systems Described by Integro-Differential Equations
Illustrative Example

Linear-Quadratic-Gaussian Control
LQG Control Problem
Unified Approach
Illustrative Example
Recursive Algorithms

Optimal Control of Singular Systems
Recursive Algorithms
Unified Approach
Illustrative Examples

Optimal Control of Time-Delay Systems
Optimal Control of Multi-Delay Systems
Optimal Control of Delay Systems with Reverse Time Terms

Optimal Control of Nonlinear Systems
Computation of the Optimal Control Law
Illustrative Examples

Hierarchical Control of Linear Systems
Hierarchical Control of LTI Systems with Quadratic Cost Functions
Solution of Hierarchical Control Problem via BPFs
Extension to Linear Time-Varying Systems
Computational Algorithm
Illustrative Examples

Editorial Reviews

"… provides a good introduction of using orthogonal function approaches for state estimation and optimal control problems. … the first book I’ve seem that puts it all together in one text. … The authors provide several detailed examples that clearly explain how the shown theory can be applied. This makes it much easier to understand the basic algorithms."
—John L. Crassidis, University at Buffalo, State University of New York

"The approach and selection of topics are very appropriate, because the book has considered all the important components of optimal control problems using orthogonal functions. … Overall, the book is quite good and comprehensive."
—Anish Deb, University of Caluctta, India

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