# Introduction to Feedback and Control Systems

The purpose of this course is to provide participants with a thorough and rigorous background in the analysis and synthesis of linear, single-loop, feedback control systems. The fundamentals of linear systems form the foundation for the analysis of feedback systems so the first half of the course begins with a review of nth order and state-space descriptions of differential equations. Linearizations about equilibria and trajectories are defined, and solutions to initial value problems are characterized. Connections with input-output descriptions, including the notions of causality, time invariance, and linearity and convolution, also are covered. From the state-space perspective, the properties of controllability and observability, and stability characterization with Lyapunov functions, also are introduced. Frequency domain descriptions of signals and systems are discussed in the context of the Fourier and Laplace transforms.

The second half of the course focuses on applying the preceding material to the analysis and synthesis of feedback control systems. The motivation for using feedback, namely to maintain performance in the presence of uncertainty in the operating environment and uncertainty in the dynamics of the system to be controlled, is provided via numerous examples. A frequency domain loop shaping procedure based on Bode and Nyquist plots provides the basis for synthesizing controllers for a variety of performance objectives, including disturbance rejection and reference command tracking.

##### Coordinator and Lecturer

**Robert M’Closkey, PhD**, Professor and Vice Chair for Undergraduate Studies, Department of Mechanical and Aerospace Engineering, Henry Samueli School of Engineering and Applied Science, UCLA. For the last 10 years, Professor M’Closkey has been collaborating with researchers from Boeing, HRL Laboratories, and the Jet Propulsion Laboratory on the development of inertial sensors for aerospace applications. His other academic interests include the analytical and experimental aspects of controlling jets injected into cross-flows. His research has been supported by NSF, NASA, ARO, and ONR as well has numerous industrial sponsors. He received the National Science Foundation CAREER award in 2000, and has several teaching awards within the School of Engineering. He received his PhD from Caltech in 1995 in the area of non-holonomic control systems.

##### Course Program

**Models of Linear Systems**

- nth order ODEs

—Motivation, examples

—Time invariance, causality

—Solutions to initial value problems

—Inputs of exponential form

—Transfer functions, poles, zeros

—Deriving impulse response, convolution

—Frequency response

—Asymptotic stability

- State-space models

—Motivation, examples

—Time invariance and causality

—Representing nth order ODEs as state-space model

—Canonical forms

—Matrix exponential: e-vals, e-vecs, special cases, relation to nth order ODEs; properties

—Solution to initial value problems

—Deriving impulse response, convolution, for state-space models

—Matrix theory review, change of coordinates

—Transfer functions

—Frequency response

—Asymptotic stability

—Time-varying and time-periodic state-space models

—Numerical solution of IVPs

- Input-output models for LTI systems

—Convolution properties, superposition

—IO models from nth order and state-space models

—Realization theory: nth order and state-space models from IO models

- Nonlinear Differential Equations

—Motivation, examples: second order, more general state-space; some differences with linear; sub- and super-harmonic responses

—Existence, uniqueness to IVPs

—Equilibria and linearizations; examples; stability of equilibria

—Phase planes

—Stability theorems and conditions for boundedness of solutions

**Interconnections of Systems**

- Motivation, examples
- Basic interconnections: series/cascade, parallel, feedback; block diagrams and assumptions
- Stability analysis of interconnections
- General interconnections of linear systems and block diagram algebra

**Frequency Domain Analysis I**

- Periodic signals: Fourier series
- Initial value problems with periodic inputs
- Motivation for non-periodic signals with finite “energy”: Fourier transforms
- Auto- and cross-correlations for “energy signals”
- Initial value problems with “energy signals”
- Energy density spectrum
- Non-periodic signals with infinite energy: revisit auto- and cross-correlations; power spectral density; time-domain interpretation of PSD

**Frequency Domain Analysis II**

- Laplace transform
- Use in solving initial value problems for causal, LTI systems
- Revisiting transfer functions

**SISO Feedback Systems**

- Block diagrams of open-loop and closed-loop systems; signals present in both cases; additional signals in feedback; requirements: asymptotic stability and some measure of performance
- Feedback copes with uncertainty

—Uncertainty in the system being controlled: plant uncertainty

—Uncertainty in the operating environment: modeling input and output disturbances

—Performance metrics for feedback systems

—Time-domain specifications; minimizing RMS outputs to disturbances

- Frequency domain specifications: disturbance rejection; bandwidth and cross-over; sensitivity and complementary sensitivity functions; graphical methods

- Stability in SISO Feedback Systems

—Frequency domain characterization: Nyquist criterion; graphical methods

—Limitations of performance: waterbed effect

- Loop shaping design

—Performance and robustness specifications reflected to loop gain, L

—Controller design via Bode plots

- Root locus design

**Introduction to MIMO Feedback Systems**

- Multivariable Nyquist; differences with SISO case
- System gains
- Nominal performance; robust stability; robust performance
- Analysis examples

**For more information contact the Short Course Program Office: **

shortcourses@uclaextension.edu | (310) 825-3344 | fax (310) 206-2815