1. Introduction and mathematical recalls (lecture: 4 h)
Automatic control and systems. Differential equations. Complex numbers. Eulero's relationships. Vectors. Matrices. Fourier transform.
2. Dynamical systems (lecture: 2 h, ex: 3h)
Input, state and output variables. Input-output (IO) and state variables (SV) models. Causality principle. SV and internal energy. Classification of dynamical systems. Local linearization of non linear systems.
3. Laplace transform (lecture: 4 h)
Laplace transform of: impulse, step, exponential and sinusoidal function. Properties: time an frequency shift; derivation, integration and convolution theorems. Laplace anti-transform.
4. Linear SV models (lecture: 11 h, ex: 8 h)
State transition matrix. Characteristic polynomial, eigenvalues and poles. Lagrange formula: free and forced response. System stability, modes and eigenvalues. Equivalence transformations. Diagonalization. Jordan form. Generalised eigenvectors. Controllability and observability matrices. State-feedback eigenvalues assignment. Luenberger observer. Separation principle.
5. IO models (lecture: 8 h, ex: 4 h)
Transfer matrix. Transfer function. Step response. Harmonic response. Bode diagram. Filtering parameters of the harmonic response. Nyquist diagram.
6. Feedback systems (lecture: 5 h, ex: 3 h)
Control systems structure. Systems with dominant modes. Block algebra. Load effects. Transfer function of connected systems. Stability of feedback systems: Nyquist and Bode criteria. Stability margins. Roots locus: modal analysis of feedback systems.
7. Requirements in control systems (lectures: 7h, exercise: 3h)
Sensitivity function with respect to external disturbances. Tracking error. Steady state tracking error with respect to canonical inputs and disturbances. Transient behaviour and closed loop characteristics. Relationships between step and frequency response. Relationships between open-loop and closed-loop characteristics.
8. Controller design (lectures: 9 h, exercise: 7 h)
Controller design by loop-shaping. Lead, lag, lead-lag compensators. Normalized diagrams for the lead compensator. Control design by means of zero-pole compensation using the roots locus. PID tuning via open-loop and closed-loop Ziegler and Nichols methods. Relay feedback test.
9. Z transform (lectures: 4 h)
Z transform: Kroneker impulse, constant, power, harmonic sequences. Properties:forward and backward shift; initial value, final value and convolution theorems. Z anti-transform.
10. Discrete time systems (lez: 6 ore, es: 2 ore)
SV models. Free evolution, modes and stability. Forced response. Transfer function. Digital control systems. Sampling. ZOH D/A converter. Sampled systems. Aliasing. Filtering characteristics of the ZOH. Correspondence between s-plane and z-plane. Spectrum of sampled signals. Feedback system stability: roots locus analysis. Digital controller design by roots locus.Discretization of continuous-time controllers: zero-pole correspondence, forward and backward difference methods, Tustin method.