## Robust control

**Instructor:** Alberto Isidori, Andrea Cristofaro

**Course web page:** www.diag.uniroma1.it/~isidori/teaching.htm

**Credits:** 6

**Infostud code:** 1041453

### Objectives

The course is addressed to students willing to expand their knowledge on the design of control systems in presence of model uncertainties. The course covers, in a systematic manner, various fundamental methods of analysis based on the use of linear matrix inequalities and various design methods, to be used in the case of parameter uncertainties (structured uncertainties) as well as in the case of modeling uncertainties (unstructured uncertainties). The course addresses the design of control (possibly multi-input and multi-output) systems, in order to meet two basic design requirements: stability and asymptotic performance in the presence of exogenous inputs. Various analysis and design techniques are presented, to verify and guarantee that the required design performances continue to hold in the presence of parameter variations as well as in the presence of un-modeled parasitic dynamics. Most of the techniques in question repose on a systematic use of linear matrix inequalities.

### Program

Summary of some basic Systems and Control concepts. Stabilizability, detectability, separation principle.
The stability criterion of Lyapunov for linear systems.

The concept of robust stabilization: parametric perturbations and un-structured perturbations.
Normal forms of a linear system. Relative degree, high-frequency gain, transmission zeroes.
Robust stabilization of systems having all zeros with negative real part: the case of relative degree 1 and the general case.

The "gain" of a linear system: possible interpretations in terms of gain in the response to sinusoidal inputs and in terms of gain in the response to finite energy inputs.

The characterization of the gain in terms of dissipation inequalities. The fundamental Lemma for the characterization of the gain. The role of the linear matrix inequalities.
Importance of the Hamiltonian matrices and of the algebraic Riccati equations.

The small gain Theorem for the characterization of the robust stability in the presence of un-structured perturbations.
Use of the small gain Theorem and of the linear matrix inequalities for the design of controllers guaranteeing robust stability.
Analogies and differences with the classical problem of stabilization via output feedback.

The problem of asymptotic regulation: Stability and steady-state performance for classes of exogenous inputs (disturbances and commands).
The geometric approach to the problem of regulation: Design of the regulator in the case of full information and in the case of error feedback.

The problem of robust regulation in the presence of structured perturbations. Synthesis of the internal model and design of the stabilizer.
The problem of robust regulation in the case of uncertainties on the exogenous input. Principles of adaptive control.
Regulation in the presence of sinusoidal disturbance of unknown frequency.

Application of the techniques for robust regulation to the problem of active suppression of harmonic disturbances (such as suppression of vibrations).

**Type of exam:** Written test, Oral test

### Reference texts

- Notes prepared expressly for this course by the instructor. In addition, recommended readings include selected chapters of the textbook: A. Isidori, "Sistemi di Controllo, Volume II", Siderea, 1991

### Further readings

- P. Gahinet, P. Apkarian, "A Linear Matrix Inequality Approach to H-infinity Control," International Journal of Robust and Nonlinear Control, Vol. 4, pp. 421-448, 1994
- S. Boyd, L. Vandenberghe, "Convex Optimization," Cambridge University Press, 2004