# Fondamenti di Automatica I (6 CFU) (Feedback control systems - Modules 1)

Ingegneria Elettronica A.A. 2008/2009 and A.A. 2009/2010

### Course Outline

First Part

• Basic concepts: an introduction to feedback control. Definitions of system, signals, inputs and outputs. Control system goals. Inputs and disturbances. The reference signal. The error signal. Control and regulation. Introduction to basic control techniques: open-loop control, closed-loop control. Examples. The fundamental components of a feedback control system: input filter, controller, actuator, sensor and plant.
• Modelling of systems: Mathematical models taxonomy. Static systems, dynamic systems. Causality. Linear and non-linear systems. Time-variant and time-invariant systems. Mathematical models of basic systems: mechanic systems (mass-spring damper model, quarter car), electrical systems, water tank systems, thermal systems. Linearization of a non-linear system around the equilibrium (water tank system example). Linear time invariant systems (LTI): linear differential equations with constant coefficients. Solution of linear diff. equations with constant coefficients: free evolution and forced response. The initial conditions problem.
• Laplace Transform: Test signals: step, ramp, parabolic ramp, Dirac delta (impulse), finite duration impulse, sinusoid. Laplace transform and properties. Convolution integral. Anti-transformation techniques. L-transform to solve differential equations.
• Basic block diagrams reduction techniques: Basic interconnections: series, parallel, feedback. Moving summing junctions. Moving branch points. Examples.
• Examples of LTI systems modelling. Electrical networks: integrator, derivative, lead, lag, lead-lag. Armature control of DC electric motor.
• Analysis in the complex variable domain. Free evolution and the initial value problem. The forced response and the transfer function. The pole-zero map and the impulse response time evolution. First-order and second-order systems: forced response to impulse, step and ramp. Influence of a zero on the step response of a first order and second order system. Transient response specifications: delay time, rise time, settling-time, peak time, maximum overshoot.

Second Parte

• Stability analysis. Definitions and fundamental theorems of stability of LTI systems. BIBO stability. Routh Lemma (necessary condition) and Routh-Hurwitz criterion. Special cases in the construction of the Routh table. Examples.
• Properties of closed-loop systems. Steady state error in feedback control systems. Error coefficients (position, velocity, acceleration). Disturbance rejection in open-loop and closed-loop systems. Feed-forward compensation. Sensitivity to parametric variations in the transfer function of the forward branch and feedback branch. The sensitivity function.
• Root locus. Root locus and complementary root locus. General rules to construct the root loci. Root locus for control system design (basic concepts). Examples.

# Fondamenti di Automatica I (6 CFU) (Feedback control systems - Modules 1)

Ingegneria Elettronica A.A. 2008/2009 and A.A. 2009/2010

### Course Outline

First Part

• Basic concepts: an introduction to feedback control. Definitions of system, signals, inputs and outputs. Control system goals. Inputs and disturbances. The reference signal. The error signal. Control and regulation. Introduction to basic control techniques: open-loop control, closed-loop control. Examples. The fundamental components of a feedback control system: input filter, controller, actuator, sensor and plant.
• Modelling of systems: Mathematical models taxonomy. Static systems, dynamic systems. Causality. Linear and non-linear systems. Time-variant and time-invariant systems. Mathematical models of basic systems: mechanic systems (mass-spring damper model, quarter car), electrical systems, water tank systems, thermal systems. Linearization of a non-linear system around the equilibrium (water tank system example). Linear time invariant systems (LTI): linear differential equations with constant coefficients. Solution of linear diff. equations with constant coefficients: free evolution and forced response. The initial conditions problem.
• Laplace Transform: Test signals: step, ramp, parabolic ramp, Dirac delta (impulse), finite duration impulse, sinusoid. Laplace transform and properties. Convolution integral. Anti-transformation techniques. L-transform to solve differential equations.
• Basic block diagrams reduction techniques: Basic interconnections: series, parallel, feedback. Moving summing junctions. Moving branch points. Examples.
• Examples of LTI systems modelling. Electrical networks: integrator, derivative, lead, lag, lead-lag. Armature control of DC electric motor.
• Analysis in the complex variable domain. Free evolution and the initial value problem. The forced response and the transfer function. The pole-zero map and the impulse response time evolution. First-order and second-order systems: forced response to impulse, step and ramp. Influence of a zero on the step response of a first order and second order system. Transient response specifications: delay time, rise time, settling-time, peak time, maximum overshoot.

Second Parte

• Stability analysis. Definitions and fundamental theorems of stability of LTI systems. BIBO stability. Routh Lemma (necessary condition) and Routh-Hurwitz criterion. Special cases in the construction of the Routh table. Examples.
• Properties of closed-loop systems. Steady state error in feedback control systems. Error coefficients (position, velocity, acceleration). Disturbance rejection in open-loop and closed-loop systems. Feed-forward compensation. Sensitivity to parametric variations in the transfer function of the forward branch and feedback branch. The sensitivity function.
• Root locus. Root locus and complementary root locus. General rules to construct the root loci. Root locus for control system design (basic concepts). Examples.