This is a beginning-graduate course in linear, time-invariant, single-input, single-output (SISO) systems.
- Continuous-time state
equations (examples, state variable diagrams,
variable changes, diagonal form).
- Linearization about
constant operating points.
- Zero-input solution.
- Properties of matrix
exponentials.
- Zero-state and
complete solutions.
- LTI properties of
solutions
- Discrete-time linear
state equations and solution. (nth-order
difference equation and sampled-data examples).
- Laplace transform
representation, solution formula, and further
properties of matrix exponentials.
- Transfer functions,
SISO response properties including steady-state
frequency response and response characterizations
of poles and zeros.
- Internal stability
(asymptotic stability of zero-input response). (Eigenvalue
and linear Lyapunov equation characterizations,
but not Lyapunov stability.)
- External stability (BIBO
stability of zero-state response).
- Discrete-time
stability.
- Reachability and
observability (including rank condition, PBH
tests, and SISO canonical forms).
- Discrete-time case.
- Realization theory and
minimal realization construction.
- Equivalence of
asymptotic and BIBO stability.
- General properties of
linear state/output feedback.
- Transfer function
analysis of feedback: well posedness and internal
stability.
- Stabilization and
eigenvalue placement by state feedback
(single-input case).
- Decoupling (noninteraction)
by state feedback.
- State observers, both
full and reduced dimension.
- Eigenvalue placement
by dynamic output (observed state) feedback.
- Output regulation
problem: asymptotic tracking of constant inputs
with rejection of constant disturbances.
Prerequisites
Undergraduate courses in control systems and in linear algebra.
Text
None. The following books have been put on
reserve at MSEL:
- J.S. Bay,
Fundamentals of Linear State Space Systems,
McGraw Hill, 1999.
- J.D. Aplevich,
Essentials of Linear State Space Systems,
Wiley, 1999.
- C.T. Chen, Linear
System Theory and Design, 3rd edition,
Oxford, 1999.
Homeworks
In a graduate course such as this, homework should be an individual effort. On the other hand, students should be encouraged to discuss the course material and help each other with obscurities and difficulties. The following policy is an attempt to fairly delineate the boundaries of homework collaboration. Discussion of particular aspects of the homework assignment is permitted for clarification of the problems, but no notes should be carried away from the discussion. The written work you hand in should be your own work.
Be extremely neat, precise, and concise. It is important that you learn what to include and what to omit from your solutions. Staple your homework in the upper left corner, and begin each problem, in correct order, at the top of a new page (or side). (Sorry about the trees.)
All of the problems that will be assigned can be solved using material that we have discussed in class. Do not solve a problem by quoting a theorem in some reference, or by stating that the solution is an easy consequence of Theorem 5.5 in a book you found. All problems can and should be solved using the approaches and tools we have discussed in class.
Grading Policy
There will be roughly one homework assignment every two weeks. Homework assignments will count towards 30% of the final grade. There will be two in class midterms (March 5 and April 25), and one final exam (Thursday May 10 2:00-5:00 PM). Each midterm will count towards 20% of the final grade, and the final will count 30%.