Applied Optimal Control

Optimization, Estimation and Control

A. E. Bryson

• Publisher: Routledge
• Year: 2018
• Print ISBN: 9780815351306
• eBook ISBN: 9781351465915
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Electrical Engineering & Telecommunications
• Index: Technology & Engineering

This best-selling text focuses on the analysis and design of complicated dynamics systems. CHOICE called it ""a high-level, concise book that could well be used as a reference by engineers, applied mathematicians, and undergraduates. The format is good, the presentation clear, the diagrams instructive, the examples and problems helpful...References and a multiple-choice examination are included.

Parameter optimization problems

1

1.1 Problems without constraints

The simplest class of parameter optimization problems involves finding the values of m parameters u₁, …, u_m that minimize a performance index which is a function of these parameters,

$L(u_1,\dots ,u_m).$

For convenience, we shall use a more compact nomenclature; let

$u=\left[\begin{array}{c}{u}_1\\ .\\ .\\ .\\ u_m\end{array}\right]=\text{decision}\text{vector}$

(1.1.1)

and write the performance index then as

$L(u).$

(1.1.2)

If there are no constraints on possible values of u and if the function L(u) has first and second partial derivatives everywhere, necessary conditions for a minimum are

$\frac{\partial L}{\partial u}=\mathrm{0,}$

(1.1.3)

by which we mean that $\partial L/\partial u_i=\mathrm{0,}$, i = 1, …, m and

$\frac{{\partial }^{2}L}{\partial u^2}\ge \mathrm{0,}$

(1.1.4)

by which we mean that the (m × m)-matrix whose components are ${\partial }^{2}L/\partial u_i\partial u_j$ must be positive semidefinite, i.e., have eigenvalues that are zero or positive.

All points that satisfy (1.1.3) are called stationary points.

Sufficient conditions for a local minimum are (1.1.3) and

$\frac{{\partial }^{2}L}{\partial u^2}>\mathrm{0;}$

(1.1.5)

that is, all the eigenvalues must be positive.

If (1.1.3) is satisfied but ${\partial }^{2}L/\partial u^2=0$, that is, the determinant of the matrix is zero (meaning that one or more of its eigenvalues is zero), additional information is needed to establish whether or not the point is a minimum. Such a point is called a singular point. Note that, if L is a linear function of u, then ${\partial }^{2}L/\partial u^2=0$ everywhere, and, in general, a minimum does not exist.

Examples for $L=L\text{(}{u}_1,u_2\text{)}$.

(a) minimum: both eigenvalues of ${\partial }^{2}L/\partial u_i\partial u_j>0$

$L=\left[u_1{u}_2\right]\left[\begin{array}{cc}1,& -1\\ -1,& 4\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]$

Figure 1.1.1. A minimum point.

(b) saddlepoint: one po...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• 1. Parameter optimization problems
• 2. Optimization problems for dynamic systems
• 3. Optimization problems for dynamic systems with path constraints
• 4. Optimal feedback control
• 5. Linear systems with quadratic criteria: linear feedback
• 6. Neighboring extremals and the second variation
• 7. Numerical solution of optimal programming and control problems
• 8. Singular solutions of optimization and control problems
• 9. Differential games
• 10. Some concepts of probability
• 11. Introduction to random processes
• 12. Optimal filtering and prediction
• 13. Optimal smoothing and interpolation
• 14. Optimal feedback control in the presence of uncertainty
• Appendix A — Some basic mathematical facts
• Appendix B — Properties of linear systems
• References
• Multiple-choice examination
• Index