ECSE-501A Linear Systems

Fall 2003

Electrical and Computer Engineering

Instructor: Prof. Benoit Boulet

boulet@cim.mcgill.ca

McConnell Building, room 509

Course Outline

1. Linear Vector Spaces

· Sets, product sets, fields

· Linear vector space over a field, linear subspaces

· Linear independence, basis, span, dimension of a linear space

· Subspaces: sums and complements, quotient space

2. Functions and Linear Transformations

· Definition of a function

· Domain, co-domain, graph of a function

· Composition of functions, properties of functions: one-to-one (injective), onto (surjective), invertible (bijective), inverse function

· Restrictions, extensions of functions, null set, support set

· Linear transformations (or operators)

· Range space and nullspace of a linear operator

· Structure of a linear operator

· Matrix representation, change of basis, rank and nullity of a matrix

· Equivalence transformations, elementary row and column operations

· Triangular and echelon matrices, finding the range of matrix and solving.

3. Normed Linear Spaces

· Inner product, inner-product spaces

· Normed linear spaces

· Metrics, Cauchy sequences and convergence

· Banach spaces

· Schwartz's inequality

· Hilbert spaces, , Parseval's identity, Paley-Wiener Theorem

· The Banach spaces

· Induced norms of linear operators, gain of a system, bounded linear operators

· The spaces and of stable LTI systems, the -norm and -norm of a system

· Orthogonality in Hilbert space, the projection theorem

4. Differential Equations and Dynamical Systems

· Differential and difference equations

· Existence of solutions, uniqueness, Lipschitz condition

· Bellman-Gronwall Lemma

· Existence theorems for differential equation; Cauchy-Peano Theorem

· Linear dynamical systems, fundamental matrix, state transition matrix

· Properties of the matrix exponential

· Definitions of dynamical systems

· Nonhomogeneous differential equations

· Stability

· Linear matrix differential equations

· Input-output representation of systems

· Properties of systems: memorylessness, invertibility, causality, BIBO stability, linearity, time-invariance

· The Dirac delta distribution (the impulse function), impulse responses of LTI and LTV systems

· State description of linear discrete-time systems

5. Reachability and Controllability

· Forced continuous-time and discrete-time linear systems: state reachability, state controllability, complete state controllability, completely controllable system

· LTI discrete-time systems, controllability matrix

· Linear transformations in inner-product spaces, the adjoint operator, structure of the adjoint

· Solving

· Moore-Penrose pseudoinverse, quadratic optimality of the pseudoinverse

· Optimal solution to the discrete-time and continuous-time controllability problems, the controllability Grammian, matched filter example

· Differentially controllable and instantaneously controllable systems

· State feedback, forward controllability operator and backward controllability operator

· Bellman's Principle of Optimality, forward optimality principle, backward optimality principle, state feedback control law

· Feedback via Riccati equations

· Least-squares feedback control

· LTI controllability, Cayley-Hamilton theorem

· Response to singularity functions

6. Decomposition Theory for LTI systems

· Decomposition into controllable and uncontrollable parts, invariant subspaces

· Eigenvalues, eigenvectors of matrices

· Modes of LTI systems

· Projection operators

· Decomposition theorem , application to differential equations

· State-space realizations

7. Observability and State Reconstruction

· Completely observable system

· Observability operator

· Optimal observer problem in

· Causal (forward) observer

· Generalized pseudoinverse

· Duality and observability

· LTI observers by output feedback on the state

Homework Assignments

There will be 5 or 6 hand-in homework assignments that, taken together, will be worth 20% of your final grade. Only a random subset of the problems in each assignment will be marked.

Midterm Test

There will be one midterm test worth 30% of your final grade.

Final Exam

You will be responsible to review all material covered in the course for the final exam. It will be worth 50% of your final grade.

References

The main source consists of the course notes (accessible via webCT) originally developed by Professor George Zames.

Other suggested references include:

· Notes for a Second Course on Linear Systems by C.A. Desoer, Van Nostrand Reinhold, 1970.

(this one is out of print, but photocopies may be available)

· Linear System Theory and Design by C.T. Chen, HRW, 1984, ISBN 0-03-060289-0

(for state-space representation)

Revision 0, Sep. 3, 2003

Benoit Boulet