# Basic control theory

## Basic concepts

Systems and modelsTypes 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 equationsModelling 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 analysisTransient response and steady state

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

Effects of adding a zero

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 problemBenefits 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

Steady state error

Feedback loop types

Error coefficients

## Formulation of specifications

Relation between time and frequencyTracking 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 systemsProportional, 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

The lead controller

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 schemesInversion-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