Engineering Professional Education

Lumped System Theory


Credit Hours: 3

Learning Objective:
1. Promote in-depth understanding of basic calculations, applications, & theoretical concepts of state variable approach to system theory from basic vocabulary & examples to general solution of linear & time varying state model, to critical material on controllability, observability, & stability.
2. In order to achieve "1", the student must
(i) understand the basic vocabulary of system theory & state variable model formulation
(ii) apply basic principles & techniques to simple and complex modeling & analysis problems
(iii) theoretically analyze complex state variable problems, re: (a) computing solutions, (b) controllability, (c) observability, and (d) stability
(iv) critically evaluate procedure used to obtain accurate solutions.
3. Foster developmental thinking & pressured creativity in problem solving, i.e. by giving someone a fish, one provides a single meal; by teaching someone to fish, one provides food for a lifetime.

Set forth the basic theory, numerical calculations, and applications of linear time invariant and time-varying state model formulations of continuous and discrete time systems. This includes some transfer function matrix material when defined, along with some nonlinear material, e.g., linearization of a nonlinear state model. Develop analytic solutions of state models including the notion of the state transition matrix and the complete solution to the linear state model. In the time invariant case this includes an extensive eigenvalue-eigenvector discussion of the matrix exponential. Finally the course develops the analytic and numerical content of the critical concepts of controllability, observability, and stability. Students should be able to apply the material to determine the controllability, observability and stability of complex systems in state variable form. Depending on the progress, advanced topics such as state feedback control, output based observer, linear quadratic regulation, and model order reduction will also be covered.

Topics Covered:
1. Systems and state variables
2. State-space models of lumped linear systems
3. Linear algebra review
4. Functions of square matrices
5. Matrix exponential
6. Solution of linear time-invaraint systems
7. Solution of linear time-varying systems
8. Stability
9. Lumped nonlinear systems
10. Quadratic forms and singular value decomposition
11. Controllability
12. Observability
13. Minimality, BIBO stability and canonical forms
14. State feedback control
15. Output feedback observer
16. Linear quadratic regulation
17. Model order reduction

1. Basic knowledge of signals and systems: Laplace transforms, Z-transforms, impulse response, transfer functions, convolution, difference equations, differential equations, etc.
2. A minimal knowledge of basic circuit theory.
3. Firm knowledge of linear algebra:
(i) Matrices and Vectors
(ii) Eigenvalues and Eigenvectors
(iii) Rank, column space, determinants, matrix inversion, etc.
4. Some knowledge of complex variables including partial fraction expansions.
5. A willingness to stretch your current knowledge and thinking patterns.

Applied/Theory: 20/80

Web Address:

Assignments handed out on a biweekly basis, and may involve Matlab programming. You are strongly advised to solve independently as much the homework as you can. This will serve you well come exam time. Late homework will not be accepted.


Two midterms. Makeup exams NOT given and missed exam counts zero.

Official textbook information is now listed in the Schedule of Classes. NOTE: Textbook information is subject to be changed at any time at the discretion of the faculty member. If you have questions or concerns please contact the academic department.
Tentative: Recommended Text: "Linear System Theory and Design" by Chi-Tsong Chen, Oxford University Press, 1999. ISBN: 0195117778.

Computer Requirements:
ProEd minimum computer requirements; student edition of MATLAB or its equivalent (e.g., Mathematica).

ProEd Minimum Requirements: view

Tuition & Fees: view

Other Requirements:


Fall 2018
Fall 2020
Fall 2022


T. B. D.



West Lafayette, IN 47907