# Module Description

## Module: Optimal and Robust Control

### Courses:

Title | Type | Hrs/Week | Period |
---|---|---|---|

Optimal and Robust Control | Lecture | 2 | Summer Semester |

Optimal and Robust Control | Recitation Section (small) | 2 | Summer Semester |

### Module Responsibility:

Prof. Herbert Werner

### Admission Requirements:

None

### Recommended Previous Knowledge:

- Classical control (frequency response, root locus)
- State space methods
- Linear algebra, singular value decomposition

### Educational Objectives:

#### Professional Competence

##### Theoretical Knowledge

- Students can explain the significance of the matrix Riccati equation for the solution of LQ problems.
- They can explain the duality between optimal state feedback and optimal state estimation.
- They can explain how the H2 and H-infinity norms are used to represent stability and performance constraints.
- They can explain how an LQG design problem can be formulated as special case of an H2 design problem.
- They can explain how model uncertainty can be represented in a way that lends itself to robust controller design
- They can explain how - based on the small gain theorem - a robust controller can guarantee stability and performance for an uncertain plant.
- They understand how analysis and synthesis conditions on feedback loops can be represented as linear matrix inequalities.

##### Capabilities

- Students are capable of designing and tuning LQG controllers for multivariable plant models.
- They are capable of representing a H2 or H-infinity design problem in the form of a generalized plant, and of using standard software tools for solving it.
- They are capable of translating time and frequency domain specifications for control loops into constraints on closed-loop sensitivity functions, and of carrying out a mixed-sensitivity design.
- They are capable of constructing an LFT uncertainty model for an uncertain system, and of designing a mixed-objective robust controller.
- They are capable of formulating analysis and synthesis conditions as linear matrix inequalities (LMI), and of using standard LMI-solvers for solving them.
- They can carry out all of the above using standard software tools (Matlab robust control toolbox).

#### Personal Competence

##### Social Competence

Students can work in small groups on specific problems to arrive at joint solutions.

##### Autonomy

Students are able to find required information in sources provided (lecture notes, literature, software documentation) and use it to solve given problems.

### ECTS-Credit Points Module:

6 ECTS

### Examination:

Oral exam

### Workload in Hours:

Independent Study Time: 124, Study Time in Lecture: 56

## Course: Optimal and Robust Control

### Lecturer:

Herbert Werner

### Language:

English

### Period:

Summer Semester

### Content:

- Optimal regulator problem with finite time horizon, Riccati differential equation
- Time-varying and steady state solutions, algebraic Riccati equation, Hamiltonian system
- Kalman’s identity, phase margin of LQR controllers, spectral factorization
- Optimal state estimation, Kalman filter, LQG control
- Generalized plant, review of LQG control
- Signal and system norms, computing H2 and H∞ norms
- Singular value plots, input and output directions
- Mixed sensitivity design, H∞ loop shaping, choice of weighting filters

- Case study: design example flight control
- Linear matrix inequalities, design specifications as LMI constraints (H2, H∞ and pole region)
- Controller synthesis by solving LMI problems, multi-objective design
- Robust control of uncertain systems, small gain theorem, representation of parameter uncertainty

### Literature:

- Werner, H., Lecture Notes: "Optimale und Robuste Regelung"
- Boyd, S., L. El Ghaoui, E. Feron and V. Balakrishnan "Linear Matrix Inequalities in Systems and Control", SIAM, Philadelphia, PA, 1994
- Skogestad, S. and I. Postlewhaite "Multivariable Feedback Control", John Wiley, Chichester, England, 1996
- Strang, G. "Linear Algebra and its Applications", Harcourt Brace Jovanovic, Orlando, FA, 1988
- Zhou, K. and J. Doyle "Essentials of Robust Control", Prentice Hall International, Upper Saddle River, NJ, 1998