## Robust Control, 2018

### Description:

The course Robust Control is an advanced course in 'post-modern' control theory for linear systems. We start the course with a brief review of basic concepts from the theory of finite-dimensional, linear, time-invariant systems like controllability, observability, stabilizability and detectability, and the problem of internal stabilization by measurement feedback. The next subject is the design of feedback controllers that make the influence of the unknown external disturbance inputs of the system on the to be controlled system outputs as small as possible. The influence of the disturbances on the to be controlled outputs can be measured in several ways. One possibility is to take the H-2 norm of the closed loop transfer matrix from the external disturbances to the to be controlled outputs (here, the 'H' refers to the famous English mathematician G.H. Hardy, 1877-1947). This gives rise to the so called H-2 optimal control problem: find a stabilizing dynamic feedback controller that minimizes the H-2 norm of the closed loop transfer matrix. The solution to this problem extensively uses the theory of algebraic Riccati equations that also appeared in the context of linear quadratic optimal control (see the bachelor course Calculus of Variations and Optimal Control). In order to arrive at the solution of the H-2 optimal control problem we will briefly review the most important facts on the algebraic Riccati equation. A second possibility to measure the influence of the disturbances on the to be controlled outputs is by taking the H-infinity norm of the associated closed loop transfer matrix. The H-infinity control problem is then to find a dynamic feedback controller that makes the H-infinity norm of the closed loop transfer matrix as small as possible. The solution to this problem that we will give in this course extensively uses the theory of linear matrix inequalities (LMI's) in combination with the famous bounded real lemma from systems and control theory. Next, the results on the H-infinity control problem will be applied to the problem of optimal robust stabilization. The problem here is to compute feedback controllers that not only stabilize a given nominal system model, but also all system models in a maximal neighbourhood of the nominal system model. A controller is called 'robust' if it also works well for all system models in a neighbourhood of the nominal model. By combining our results on the H-infinity control problem with the small gain theorem, we will solve the problem of optimal robust stabilization for three classes of uncertainty; additive uncertainty, coprime factor uncertainty, and multiplicative uncertainty. Finally we will deal with networked multi-agent systems and study the problem of designing distributed controllers (so called protocols) that achieve state synchronization in the network. It will be shown that this problem is equivalent to a simultaneous stabilization problem, i.e. the problem of stabilizing a single linear system by means of multiple, related, feedback controllers. To conclude the course, we take a look at the case that the agent dynamics is uncertain, and study the design of robustly synchronizing distributed controllers.

### Learning goals:

1. The student is able to reproduce and explain the geometric characterizations of the systems properties of controllability, stabilizability, observability and detectability.
2. The student is able to formulate the H-2 optimal control problem, and is able to explain the relevance of the algebraic Riccati equation in the solution to this problem. He/she is able to formulate the basic facts on the existence of solutions to algebraic Riccati equations. 3. The student is able to reproduce the formulation of the H-infinity suboptimal control problem, both in the time-domain and in transfer function terms.
4. The student is able to reproduce the formulation and the proof of the bounded real lemma.
5. The student is able to reproduce the necessary and sufficient conditions for solvability of the general H-infinity control problem in terms of linear matrix inequalities (LMIs).
6. The student is able to independently derive solutions to particular special cases of the H-infinity control problem using the theory treated in the course. 7. The student can reproduce the formulation of the optimal robust stabilization problem for additive and multiplicative perturbations. He/she is able to reproduce the equivalence of the robust stabilization problem with particular special cases of the H-infinity suboptimal control problem using the small gain theorem
8. The student is able to reproduce the solution of the optimal robust stabilization problem, and is able to interpret the optimal stability radius in terms of solutions of certain ARE' s .
9. The student is able to reproduce the solution of the problem of network synchronization using distributed diffusive dynamic feedback protocols.
10. The student is to apply the material presented in the course to formulate and prove extensions of results from the course, and present these results written in a mathematically sound way.

### Literature:

H.L. Trentelman, A.A. Stoorvogel and M.L.J. Hautus, "Control Theory for Linear Systems", Springer Verlag, 2001. Unfortunately, the first edition is sold out, and a second edition is not available. However, the complete pdf file of the book can be downloaded from the website www.math.rug.nl/~trentelman under 'Publications'. Additional lecture notes will be handed out during the course.

### Organization:

Lectures: Tuesday 13:00 -15:00, EA 5159.0114 (34), Thursday 11:00-13:00, NB 5111.0080 (136). The course will start in the week of Tuesday February 6, 2018.

### Examination

During the the course, there will be two `takehome-exams', each consisting of a number of problems on the theory treated in the course. Solutions to the problems of these take-home exams should be handed in before a given date (no exceptions!). After the course there will be an oral examination. The final grade is determined on the basis of the two take-home exams and the oral examination.