# Linear Systems Prelim Syllabus

##### Updated October 2008*

The core of the linear systems prelim is the graduate course EECS 221A. The following is the syllabus:

1. Linear time varying systems. Properties of the state transition matrix. The adjoint equation and the variational equation.

2. Linear time invariant systems. Eigenvalues and left and right eigenvectors. Dyadic expansions.

3. Conditioning of matrices. Hermitian matrices. The Lyapunov equation X -> AX + XA*=-Q. The singular value decomposition. Direct sum decomposition into invariant subspaces, minimal polynomials, the Jordan form theorem. Functions of a matrix. Spectral mapping theorem.

4. Input-Output and state representation, equivalent states, and representation.

5. Controllability, observability, effects of state feedback and output injection. Characterization of controllability and observability for linear systems. Stabilizability and detectability. Duality, the Kalman decomposition; internal stability and I/O stability.

6. Full order observers, state estimation, and eigenvalue assignment.

7. Stability of linear time varying systems. I/O and internal stability of feedback systems and MIMO Nyquist criterion.

8. Qualitative behavior and phase portraits of linear vector fields in |Rⁿ , n ≤ 3.

9. Dynamics of piecewise-linear systems.

Students Will Be Expected To Apply This Material To Simple Applications Involving:

1. Simple differential equations (linear and piecewise-linear)
2. Systems involving state feedback, output feedback, and output injection
3. Electric circuits.

#### Primary References:

• C.A. Desoer and F.M. Callior, Linear System Theory, Springer Verlag, 2nd Printing, 1994.

#### Supplementary References:

• Kailath, T., Linear Systems, Prentice-Hall, 1980.
• Chen, C. T., Linear System Theory and Design, Holt, Rinehart and Winston, 1984.

NOTE: The following courses may not be used to fulfill the prelim breadth requirement if you take the Linear Systems prelim: EECS 128, 221A, 222, 223, 290N and 290O.

*S. Sanders 1/25/99, R. Fearing 10/9/08

ruthg@eecs.berkeley.edu
02/03/98