SA307, Science Building I, NCTU

(交通大學科學一館 307室)

Modern Control Theory: Algebraic Riccati Equations

Eric King-wah Chu (Monash University)

Course Design

‧Try to quickly deliver a basic feel of optimal control theory to students. We shall consider the algebraic and dierential Riccati equations (ARE, DRE) for the solution of the linear quadratic Gaussian (LQG) optimal control problem. Numerical algorithms for the AREs, linking numerical analysis and control system design, will be presented.

‧Background required: Basic linear algebra and multivariable calculus

‧Language: Chinese, supplemented by English?

‧Useful Books: (I refer to the rst and have electronic copies for all three)

1. William L. Brogan, Modern Control Theory, 3rd Edition, Prentice-Hall, Englewood Clis, New Jersey, 1974. (Old classical book)

2. Biswa N. Datta, Numerical Methods for Linear Control Systems | Design and Analysis, Elsevier, New York, 2004. (Modern and easy to read)

3. Shu-Fang Xu, Matrix Computations in Control Theory, Higher Education Press, Beijing, 2011. (Well written, In Chinese)

Aim

To deliver modern, interesting but unfamiliar materials on the numerics of control system design informally, and to arouse interests in associated research areas. The lectures cover a wide area intersecting many elds, such as applied and computational mathematics as well as applications in engineering.

Outline

Roughly following the chapters in Brogan: (with * against chapters ignored in lectures)

(1)* Background and Review (system and control theory, modelling, classication, representation)

(2)* Highlight of Classical Control Theory (Fourier transform and frequency domain, feedback)

(3) State Space (state equations)

(4)* Matrix Algebra (operations, algebra, inverse, partitions, matrix calculus)

(5)* Vectors and Linear Vector Spaces (linear dependence, span, dimension, orthogonality, subspaces, projections, transformations)

(6)* Linear Equations (Gaussian elimination, homogeneous equations, under- and over-determined cases, Lyapunov equations)

(7)* Eigenvalues and Eigenvectors (denitions, basic properties, numerical estimation, spectral decomposition, invariant subspaces)

(8) Functions of Matrices and Cayley-Hamilton Theorem (powers, polynomials, series, characteristic polynomial)

(9) Continuous- and Discrete-time Linear State Equations (constant coecient and time-variant cases, transition matrix)

(10) Stability (linear system, Lyapunov equation)

(11) Controllability and Observability (Kalman canonical form, stabilizability and detectability)

(12)* Relationship between State Variable and Transfer Function Descriptions of Systems

(13)* Design of Linear Feedback Control Systems

(14) LQG Optimal Control Theory (Riccati equations)

(15) Solution of Algebraic Riccati Equations (Newton's method, Schur method, structure-preserving doubling algorithm)

(16)* Stochastic and Nonlinear Control Systems