# Basic control theory

## Basic concepts

Systems and models
Types of models
Signals, states, inputs and outputs
Blocks and descriptions
Feedback and feedforward
Open loop and closed loop
Positive and negative feedback
Disturbances and noise
Classical examples of control systems

## External description of linear systems

Overview of differential equations
Modelling of mechanical systems
Modelling of electrical systems
Modelling of fluid systems
Modelling of thermal systems
Linearization of nonlinear systems
Review of complex variable
Laplace transform
Laplace transform of basic functions (I)
Laplace transform of basic functions (II)
Properties of the Laplace transform (I)
Properties of the Laplace transform (II)
Inverse Laplace transform
Partial fraction decomposition (I)
Partial fraction decomposition (II)
Solving differential equations with the Laplace transform
Transfer function
Impulse response
Interpretation of the convolution
Alternative expressions of the transfer function
Blocks in the s-domain
Block algebra
Signal flow graphs
Mason formula

## System analysis

Standard functions for system analysis
Equilibrium
Stability of an LTI system
Operating points
Stability of simple systems
Stability of arbitrary order systems
Routh stability criterion
Routh stability criterion: special cases
Step response parameters
Step response of first order systems
Step response of second order systems I
Step response of second order systems II
Higher order systems and order reduction
Dominance and cancellation
Frequency response
Graphical representations of the frequency response
Bode diagram construction I
Bode diagram construction II
Crossover frequency and bandwidth
Polar or Nyquist diagram
Magnitude vs phase or Nichols diagram
Unstable and non minimum phase systems
Transport delay

## Feedback systems

The control problem
Benefits of feedback
Basic control schemes
Relation between open loop and closed loop
Root locus
Magnitude and angle conditions
Rules for root locus construction I
Rules for root locus construction II
Nyquist stabilty criterion
Nyquist stabilty criterion: special cases
Relative stability
Gain margin and phase margin
Stability in the Nichols plot
Conditionally stable systems
Feedback loop types
Error coefficients

## Formulation of specifications

Relation between time and frequency
Tracking on the root locus
Disturbance rejection on the root locus
Tracking on the frequency domain
Disturbance rejection on the frequency domain
Noise rejection specifications
Independent specifications
Control degrees of freedom

## Design of one degree of freedom control systems

Limitations of the one degree of freedom control systems
Proportional, integral and derivative actions
Basic controller structures
The PD controller
Design of PD controllers on the root locus
Design of PD controllers on the bode diagram
The PI controller
Design of PI controllers on the root locus
Design of PI controllers on the bode diagram
Design of lead controllers on the root locus
Design of lead controllers on the bode diagram (I)
Design of lead controllers on the bode diagram (II)
The lag controller
Design of lag controllers on the root locus
Design of lag controllers on the bode diagram (I)
Design of lag controllers on the bode diagram (II)
Analytical compensation: the Guillemin-Truxal procedure
Experimental compensation: Ziegler-Nichols (I)
Experimental compensation: Ziegler-Nichols (II)
Guidelines for solving control problems

## Design of two degree of freedom control systems

Equivalence among two degree of freedom schemes
Inversion-based tracking
Inversion-based disturbance rejection
Limitations of PID control: derivative kick-off
Limitations of PID control: integral wind-up
PI-D and I-PD controllers
Regions on the Bode diagram
Dealing with uncertainty
General design: loop-shaping