 # Linear Optimal Control, 1st edition

• Jeffrey B. Burl

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## Overview

Linear Optimal Control: H2 and Hà Methods is a reader-friendly book that features recent research results on robustness, Hà control, and ...m- synthesis. Linear Optimal Control combines these new results with previous work on optimal control to form a complete picture of control system design and analysis.
A comprehensive book, Linear Optimal Control covers the analysis of control systems, H2 (linear quadratic Gaussian), and Hà to a degree not found in many similar books. Its logical organization and its focus on establishing a solid grounding in the basics be fore tackling mathematical subtleties make Linear Optimal Control an ideal learning tool.

Back Cover

A comprehensive book, Linear Optimal Control covers the analysis of control systems, H2 (linear quadratic Gaussian), and to a degree not found in many texts. Its logical organization and its focus on establishing a solid grounding in the basics before tackling mathematical subtleties make Linear Optimal Control an ideal teaching text. The book's structure also makes it suitable for two-semester, one-semester, and two-quarter courses, as well as for professional use.

Features:

• Provides computer projects in each chapter to challenge readers with "what if" questions concerning the choice of cost functions and specifications.
• Simplifies results and derivations by treating special cases whenever possible without compromising the clarity of the results and methods.
• Familiarizes readers with the use of CAD tools for robust optimal controller design; software use is integrated throughout the book, and is included in almost every example.
• Offers a case study that compares the LQG and the µ-synthesis design methodologies and provides guidelines on when to apply which method.
• Presents LQG and control within a common framework— uses variational methods for both solutions.

List of Symbols.

1. Introduction.

The Fundamental Objectives of Feedback Control. A Brief History of Modern Controller Design. Scope and Objectives. Organization. References

1. ANALYSIS OF CONTROL SYSTEMS.

2. Multivariable Linear Systems.

The Continuous-Time State Model. The Discrete-Time State Model and Simulation. Transfer Functions. Frequency Response.

Frequency Response for SISO Systems. Frequency Response for MIMO Systems. The Singular Value Decomposition. The Principle Gains.

Poles, Zeros, and Modes.

Polesand Zeros for Siso Systems. Poles and Zeros for Mimo Systems. Modes.

Stability.

Internal Stability.

Change of Basis: Similarity Transformations. Controllability and Observability.

Controllability. Observability.

Observer Feedback.

State Feedback. Observers. The Deterministic Separation Principle.

Summary. References. Exercises. Computer Exercises.

3. Vector Random Processes.

The Description of Vector Random Processes.

Second Moment Analysis. Two Random Processes. Wide Sense Stationarity. The Spectral Density. Gaussian Random Processes.

White Noise. Linear Systems with Random Inputs.

The Output Correlation Function. The Output Spectral Density. Approximation of Real Inputs by White Noise. Simulation of Systems with White Noise Inputs.

Colored Noise via Shaping Filters. Summary. References. Exercises. Computer Exercises.

4. Performance.

General Models of Feedback Control Systems. Transient Performance Analysis. Tracking Performance Analysis. Disturbance Rejection Analysis. Cost Functions and Norms.

Norms. Quadratic Cost Functions. Cost Functions for Systems with Random Inputs. The System 2-Norm Cost Function. The System Cost Function. Weighting Functions for System Norms.

Summary. References. Computer Exercises.

5. Robustness.

Internal Stability of Feedback Systems. The SISO Nyquist Stability Criterion. Gain and Phase Margins for SISO Systems. Unstructured Uncertainty.

Unstructured Uncertainty Models. Stability Robustness Analysis.

Structured Uncertainty.

The Structured Uncertainty Model. The Structured Singular Value and Stability Robustness. Bounds on the Structured Singular Value. Additional Properties of the Structured Singular Value.

Performance Robustness Analysis Using The SSV. Summary. References. Exercises. Computer Exercises.

2. H2 CONTROL.

Optimization.

Variations. Lagrange Multipliers.

The Hamiltonian Equations. The Riccati Equation. Computation of the Optimal Cost. Selection of the Weighting Matrices. Perspective.

Computation of the Feedback Gain Matrix. Existence and Uniqueness. Robustness. The Closed Loop Poles.

The Stochastic Regulator.

Cost Computation. H2 Optimal Control.

Summary. References. Exercises. Computer Exercises.

7. The Kalman Filter.

Linear Minimum Mean Square Estimation.

The Orthogonality Principle. The Optimal Estimation Error. Updating an Estimate Given New Data.

The Kalman Filter.

The Kalman Filter Equation. The Kalman Gain Equations. Application Notes.

H2 Optimal Estimation. Duality. Computation of the Kalman Gain. Existence and Uniqueness. Robustness. The Kalman Filter Poles.

Non White Noise Inputs.

Non White Plant Noise. Non White Measurement Noise.

Summary. References. Exercises. Computer Exercises

Combined Estimation and Control: LQG Control.

Performance.

Performance. Robustness.

Loop Transfer Recovery.

Asymptotic Properties. Robustness to Output Multiplicative Perturbations. Frequency Shaped LTR.

Tracking System Design.

Tracking Constant Reference Inputs. Tracking Time-Varying Reference Inputs.

Designing for Disturbance Rejection.

Feedforward Disturbance Cancellation. Integral Control.

Frequency Shaped Control via LQG Methods. Equivalence of LQG and H2 Optimal Control. Summary. References. Exercises. Computer Exercises.

3. CONTROL.

9. Full Information Control and Estimation.

Differential games. Full Information Control.

The Hamiltonian Equations. The Riccati Equation. The Value of the Objective Function. Steady-State Full Information Control. Generalizations.

Estimation.

Summary. References. Exercises. Computer Exercises.

10. Output Feedback.

Controller Structure. Finite-Time Control.

An Alternative Estimator Riccati Equation. Summary.

Performance Limitations. Integral Control. Designing for Robustness.

µ-Synthesis.

D-Scaling and the Structured Singular Value. D-K- Iteration.

Comparison of Design Methodologies. Summary. References. Exercises. Computer Exercises.

11. Controller Order Reduction.

Perturbation Analysis. Frequency Weighting. Removing Poles and Zeros from SISO Controllers. Balanced Truncation.

The Balanced Realization. Balanced Truncation.

Summary. References. Exercises. Computer Exercises.

Appendix: Mathematical Notes.

Calculus of Vectors and Matrices.

Calculus of Vector-Matrix Functions of a Scalar. Derivatives of Vector-Matrix Products.

Useful Relations from Linear Algebra.

Positive Definite and Positive Semidefinite Matrices. Relations Involving the Trace. Determinants of Block Matrices. The Matrix Inversion Lemma. Block Matrix Inversion.

The Singular Value Decomposition. Spectral Theory of Matrices. L2 Stability. Change of Basis (Time-Varying Transformations). Controllability and Observability Grammians. Useful Relations Involving (I+GK) and (I+KG).

Equivalence of the Determinants. The Push Through Theorem. Miscellaneous.

Properties of the System. Bound on the System. The Adjoint System. The Kalman Filter Innovations. The Phase-Gain Relationship. References.