The course aims to present the fundamentals of systems theory and control and the main techniques of analysis and design of control systems. To this end, after having abundantly exemplified some of the main problems that this discipline proposes to solve, the basic methodological tools necessary are presented, accompanied by examples both numerical and inspired by concrete cases. Analytical tools will be provided to determine the main structural properties of invariant linear dynamical systems, with particular emphasis on the stability of equilibrium solutions. Dynamic systems will be described both in terms of state variables (internal representation) and input-output variables (external representation). In the latter approach, in the so-called frequency domain, the problem of stability, analysis and synthesis of simple automatic control systems will be addressed. The final objective is to enable students to analyze simple dynamic systems and to set and solve simple problems of designing automatic control systems.

The student will gain knowledge on the fundamentals of signal theory and transforms and will be able to apply that knowledge to:

• Identify the constituent elements of a control problem.
• Formalize a dynamic system both in continuous and discrete time.
• Determine valid linearized models in the neighborhood of equilibrium points.
• Analyze the stability of continuous and discrete time linear stationary systems.
• Address the design of a regulator in the frequency domain, with particular emphasis on servomechanisms.
• Implement digital control systems from the discretization of analog controllers.

1. Introduction.

1.1. The control problem.
1.2. Open and closed loop control.
1.3. Instrumentation (notes).
2. Dynamic systems
2.1. Definition of dynamic system.
2.2. Examples of dynamic systems. Classification of dynamical systems. Distributed parameter systems (elements).
2.3. Motion. Free and forced motion. Principle of superposition. Motion calculations for continuous time linear systems. Motion calculations for discret time systems.
2.4. Linearization.
3. Signals and transforms.
3.1. Recalls of signal theory.
3.2. Laplace transform.
3.3. Heaviside method.
3.4. Z transform
4. Transfer function.
4.1. Transfer function for continuous-time systems. Equilibrium and gain.
4.2. Transfer function for discrete-time systems. Equilibrium and gain.
5. Stability.
5.1. Stability of equilibrium.
5.2. Linearization and stability.
6. Block diagrams.
7. Response to canonical signals.
7.1. Step response of first and second order systems.
8. Frequency response.
8.1. Bode plots. Asymptotic diagrams of magnitude and phase.
8.2. Dynamical systems as filters.
8.3. Polar plots.
9. Formulation of the control problem.
9.1. Nyquist criterion.
9.2. Gain and phase margin.
9.3. Bode criterion.
9.4. Dynamic analysis.
9.5. Static analysis.
9.6. Specifications assignment.
10. Sythesis of the regulator
10.1. Sythesis of the regulator.
10.2. PID controllers
11. Digital control
11.1. Sampling
11.2. Digital implementation of analog controllers.
12 Control of servomechanisms 
12.1 Current control.
12.2 P/PI control.
12.3 Model of compliant joints.
12.4 Position control of compliant joints.

Elementary knowledge of mathematical analysis and geometry, with particular reference to operations on complex numbers, differential and integral calculus, linear differential equations and matrix algebra. All these topics are included in the programs of Mathematical Analysis I and II and Linear Algebra and Geometry.

The exam will consist of a written test, in which the student must demonstrate:

• To know how to formalize a dynamic system in continuous and/or discrete time.
• To know how to calculate simple movements of linear and stationary dynamic systems.
• To know how to calculate linearized systems around equilibrium points.
• To know how to calculate the transfer functions and analyze their properties.
• To evaluate the stability properties of linear systems in both continuous and discrete time.
• To know how to process simple block diagrams.
• To know how to analyze control systems (linear and continuous time) in closed loop in the frequency domain, estimating their static characteristics (transient error exhausted, noise attenuation ...) and dynamics (response and settling time, over-elongation ...)
• To know how to face the synthesis of a control system so that it satisfies precise static and dynamic specifications, proposing a suitable regulator.
• To know how to implement digital control algorithms by discretizing the corresponding designed analogic regulators.