The objective of this course is to present advanced methods for the analysis of dynamical systems and for the synthesis of multi-input, multi-output controllers. First, the Lyapunov stability theory will be introduced and applied to nonlinear system analysis. Then, the structural properties of linear dynamical systems (poles, invariant zeros, principal gains) will be described together with performance and robustness indices and with the proper selection of the regulator structure guaranteeing static and dynamic performance. Pole placement, Linear Quadratic and predictive control synthesis methods will be introduced and applied to a number of application examples. Further topics will concern state estimators based on pole assignment and Kalman filtering, LQG control, gain scheduling regulators and synthesis techniques relying on the robust control synthesis H2 and Hinf approaches.

Lectures and exercise sessions will allow the students to:

• Perform the stability analysis of dynamical systems and the design of nonlinear controllers with the Lyapunov stability theory
• Analyze multivariable linear systems in terms of gain, singular values, poles, and zeros
• Select the proper regulator structure
• Design multivariable control systems with pole-placement, LQ, LQG , H2, Hinf controllers and dynamic observers
• Formulate Model Predictive Control problems based on different plant descriptions and in presence of constraints on the plant variables

The laboratory training sessions will make use of computer simulation tools and will allow students to learn how to:

• Simulate a dynamic system
• Compute and analyze the equilibria
• Numerically compute the linearized model of the (nonlinear) plant
• Design the controller with the algorithms described in the course
• Test the performance of the controller applied to the plant
• Analyze the sensitivity of the solution with respect to the design choices and tuning parameters

1. Nonlinear systems: equilibria, stability, Lyapunov theorems. Synthesis of the control law with control Lyapunov functions and the backstepping procedure.
2. Norms and gain of dynamical systems. Small gain theorem.
3. Structural properties of linear systems: reachability, observability, poles, and invariant zeros.
4. Nominal and robust stability of multivariable feedback systems. Static and dynamics performance. Schemes with integral action.
5. Pole placement control of single-input, single-output and multivariable systems. State observers and dynamic output feedback. Disturbance estimation with state observers.
6. Optimal control. Linear Quadratic control: definitions, properties, examples of application. Kalman filtering and LQG control. H₂ and H_∞ control (hints), model order reduction.
7. Model predictive control: formulations in the state space and with input-output models. Stabilizing properties of MPC.
8. Control design examples: control of a chemical reactor, a distillation column, an inverted pendulum, an aircraft.

Students are required to know:

• Basics of System Theory: state space and input-output description of dynamic systems, stability, canonical forms
• Main control synthesis methods for single-input, single output systems based on frequency domain analysis and root locus.
• Digital control methods based on discretization and direct synthesis approaches for discrete time systems

The final assessment will be a written exam consisting of numerical exercises and theoretical questions.

During the exam the student must prove to be able to design a controller for multivariable systems, and in particular to:

• perform a preliminary analysis, specifying the control and controlled variables and the adopted control structure
• evaluate the limits to the achiavable performance, define the control specifications
• apply modern control design techniques
• critically evaluate the achieved results and their sensitivity to the adopted design choices
• describe the achieved results in a clear and convincing way