Automatic Control with Experiments

1. Introduction

      1.1  An anti-aircraft gun control system 

      1.2  History of automatic control 

      1.3  Didactic prototypes

 

2. Physical system modeling

      2.1  Mechanical systems

             2.1.1  Translational mechanical systems

             2.1.2  Rotative mechanical systems

      2.2  Electrical systems 

      2.3  Transformers

             2.3.1  Electric transformer

             2.3.2  Gear reducer

             2.3.3  Rack and pinion

      2.4  Converters

             2.4.1  Armature of a permanent magnet brushed DC motor

             2.4.2  Electromagnet 

      2.5  A case of study. A DC-to-DC high-frequency series resonant power converter 

      2.6  Exercises

 

3. Ordinary linear differential equations

      3.1  First order differential equation 

             3.1.1  Graphical study of the solution

             3.1.2  Transfer function

      3.2  An integrator

      3.3  Second order differential equation 

             3.3.1  Graphical study of solution

             3.3.2  Transfer function

      3.4  Arbitrary order differential equations 

             3.4.1  Real and different roots 

             3.4.2  Real and repeated roots

             3.4.3  Complex conjugated and not repeated roots 

             3.4.4  Complex conjugated and repeated roots

             3.4.5  Conclusions

      3.5  Poles and zeros in higher-order systems 

             3.5.1  Pole-zero cancellation and reduced order models

             3.5.2  Dominant poles and reduced order models

             3.5.3  Approximating transitory response  of higher-order systems 

      3.6  The case of sinusoidal excitations

      3.7  The superposition principle

      3.8  Controlling first and second order systems

             3.8.1  Proportional control of velocity in a DC motor

             3.8.2  Proportional position control plus velocity feedback for a DC motor

             3.8.3  Proportional-derivative position control of a DC motor

             3.8.4  Proportional-integral velocity control of a DC motor

             3.8.5  Proportional, PI and PID control of first order systems

      3.9  Case of study. A DC-to-DC high-frequency series resonant power electronic converter 

      3.10 Exercises

 

4. Stability criteria and steady state error

      4.1  Block diagrams

      4.2  Rule of signs

             4.2.1  Second degree polynomials 

             4.2.2  First degree polynomials 

             4.2.3  Polynomials with degree greater than or equal to 3

      4.3  Routh's stability criterion

      4.4  Steady state error 

             4.4.1  Step desired output

             4.4.2  Ramp desired output 

             4.4.3  Parabola desired output

      4.5  Exercises

 

5. Time response-based design 

      5.1  Drawing the root locus diagram

             5.1.1  Rules to draw the root locus diagram

      5.2  Root locus-based analysis and design

             5.2.1  Proportional control of position 

             5.2.2  Proportional-derivative (PD) control of position

             5.2.3  Position control using a lead-compensator

             5.2.4  Proportional-integral (PI) control of velocity

             5.2.5  Proportional-integral-derivative (PID) control of position

             5.2.6  Assigning the desired closed-loop poles

             5.2.7  Proportional-integral-derivative (PID) control of an unstable plant 

             5.2.8  Control of a ball and beam system

             5.2.9  Assigning the desired poles for a ball and beam system

      5.3  Case of study. Additional notes on PID control of position for a permanent magnet brushed DC motor

      5.4  Exercises

 

6. Frequency response-based design 

      6.1  Frequency response of some electric circuits

             6.1.1  A series RC circuit: output at capacitance

             6.1.2  A series RC circuit: output at resistance

             6.1.3  A series RLC circuit: output at capacitance

             6.1.4  A series RLC circuit: output at resistance 

      6.2  Relationship between frequency response and time response

             6.2.1  Relationship between time response  and frequency response 

      6.3 Common graphical representations

             6.3.1  Bode diagrams

             6.3.2  Polar plots 

      6.4 Nyquist stability criterion 

             6.4.1 Contours around poles and zeros 

             6.4.2 Nyquist path 

             6.4.3 Poles and zeros 

             6.4.4 Nyquist criterion. A special case

             6.4.5 Nyquist criterion. The general case 

      6.5 Stability margins

      6.6 Relationship between frequency response and time response

             6.6.1  Closed-loop frequency response and closed-loop time response

             6.6.2  Open-loop frequency response and closed-loop time response 

      6.7    Analysis and design examples

            6.7.1 Analysis of a nonminimum phase system 

            6.7.2 A ball and beam system

            6.7.3 PD position control of a DC motor 

            6.7.4 PD position control redesign for a DC motor

            6.7.5 PID position control of a DC motor 

            6.7.6 PI velocity control of a DC motor 

      6.8  Case of study. PID control of an unstable plant 

      6.9  Exercises

 

7. The state variables approach 

      7.1  Definition of state variables

      7.2  Approximate linearization of nonlinear state equations 

             7.2.1  Procedure for first order state equations without input

             7.2.2  General procedure for arbitrary order state equations with arbitrary number of inputs

      7.3  Some results from linear algebra

      7.4  Solution of a linear time invariant dynamical equation

      7.5  Stability of a dynamical equation

      7.6  Controllability and observability

             7.6.1  Controllability

             7.6.2  Observability 

      7.7  Transfer function of a dynamical equation 

      7.8  A realization of a transfer function

      7.9  Equivalent dynamical equations 

      7.10 State feedback control

      7.11 State observers

      7.12 The separation principle

      7.13 Case of study. The inertial wheel pendulum

             7.13.1 Obtaining forms in (7.57)

             7.13.2 State feedback control 

      7.14 Exercises

 

8. Advanced topics in control

      8.1  Structural limitations in classical control

             8.1.1  Open-loop poles at origin

             8.1.2  Open-loop poles and zeros located out of origin 

      8.2  Differential flatness 

      8.3  Describing function analysis 

             8.3.1  The dead zone nonlinearity [3], [4]

             8.3.2  An application example 

             8.3.3  The saturation nonlinearity [3], [4]

             8.3.4  An application example 

      8.4  The  sensitivity  function  and some limitations when controlling unstable plants

 

9. Feedback  electronic circuits

      9.1  Reducing effects of nonlinearities in electronic circuits

             9.1.1  Reducing distortion in amplifiers 

             9.1.2  Dead zone reduction in amplifiers 

      9.2  Analogue controllers with operational amplifiers

      9.3  Design of sinusoidal waveform oscillators

             9.3.1  Design based on an operational amplifier. Wien bridge oscillator

             9.3.2  Design based on an operational amplifier. Phase shift oscillator

             9.3.3  A transistor-based design

      9.4  A regenerative radio-frequency (RF) receiver

 

10. Velocity control of a PM Brushed DC  motor 

      10.1 Mathematical model

      10.2 Power amplifier 

      10.3 Electric current control

      10.4 Identification 

      10.5 Velocity control 

             10.5.1 A modified PI controller

             10.5.2 A two-degrees-of-freedom controller

      10.6 Experimental prototype

             10.6.1 Electric current control

             10.6.2 Power amplifier 

      10.7 Experimental results

      10.8 Microcontrolller PIC16F877A programming

      10.9 Frequency response-based design 

             10.9.1 Model identification 

             10.9.2 Proportional-integral control design

             10.9.3 Prototype construction

 

11. Position control of a PM Brushed DC  motor 

      11.1 Identification

      11.2 Position control when disturbances are not present (Tp = 0)

             11.2.1 Proportional position control with velocity feedback 

             11.2.2 A lead-compensator 

      11.3 Control under effect of external disturbances

             11.3.1 A modified PID controller

             11.3.2 A two-degrees-of-freedom controller

             11.3.3 A classical PID controller 

      11.4 Trajectory tracking

      11.5 Prototype construction

      11.6 Microcontroller PIC16F877A programming

      11.7 Personal computer-based control 

      11.8 Frequency response-based design

             11.8.1 Model identification 

             11.8.2 Proportional-integral-derivative control design 

             11.8.3 Prototype construction

 

12. Control of a servomechanism with flexibility

      12.1 Mathematical model

      12.2 Experimental Identification

      12.3 Controller design

             12.3.1 Multi-loop control

             12.3.2 Direct control of th2

      12.4 Experimental prototype construction

      12.5 Microcontroller PIC16F877A C program

      12.6 Personal computer Builder C++ program

 

13. Control of a magnetic levitation system

      13.1 Complete nonlinear mathematical model 

      13.2 Approximate linear model 

             13.2.1 A state variables representation model 

             13.2.2 Linear approximation

      13.3 Experimental prototype construction

             13.3.1 Ball 

             13.3.2 Electromagnet 

             13.3.3 Position sensor

             13.3.4 Controller

             13.3.5 Electric current loop

             13.3.6 Power amplifier 

      13.4 Experimental identification of model parameters

             13.4.1 Electromagnet internal resistance, R

             13.4.2 Electromagnet inductance, L(y) 

             13.4.3 Position sensor gain, As 

             13.4.4 Ball mass, m 

      13.5 Control system structure 

             13.5.1 Internal current loop

             13.5.2 External position loop

      13.6 Controller design

             13.6.1 PID position controller design using root locus

             13.6.2 Design of the PI electric current controller <

             13.6.3 Some experimental tests

             13.6.4 PWM power amplifier 

             13.6.5 Design of the  PID position controller using the frequency response 

             13.6.6 Some other experimental tests

             13.6.7 An alternative procedure to design the PI electric current controller

 

14. Control of a ball and beam system

      14.1 Mathematical model

             14.1.1 Nonlinear model  <

             14.1.2 Linear approximate model

      14.2 Prototype construction

             14.2.1 Ball position x measurement system 

             14.2.2 Beam angle th measurement system 

      14.3 Parameter identification

             14.3.1 Motor-beam subsystem 

             14.3.2 Ball dynamics

      14.4 Controller design

      14.5 Experimental results

      14.6 Control system electric diagram

      14.7 Builder 6 C++ code used to implement control algorithms

      14.8 PIC C code used to program microcontroller PIC16F877A

      14.9 Control based on a PIC16F877A microcontroller

             14.9.1 Prototype construction

             14.9.2 Controller design

             14.9.3 Experimental results

             14.9.4 PIC16F877A microcontroller programming

 

15. Control of a Furuta pendulum

      15.1 Mathematical model

      15.2 A controller to swing up the pendulum

      15.3 Linear approximate model

      15.4 A differential flatness based model

      15.5 Parameter identification

      15.6 Design of a stabilizing controller

      15.7 Experimental tests

      15.8 Control system construction 

      15.9 Sampling period selection

      15.10 The Builder 6 C++ program

      15.11 The PIC16F877A microcontroller C program

 

16. Control of an inertia wheel pendulum

      16.1 Inertia wheel pendulum description

      16.2 Mathematical model

      16.3 Swing up nonlinear control 

      16.4 Balancing controller 

      16.5 Prototype construction and parameter identification 

      16.6 Controller implementation

      16.7 Experimental results

 

Appendices

 

A   Fourier and Laplace transforms

      A.1  Fourier series 

      A.2  Fourier transform

      A.3  Laplace transform 

 

B    Bode  diagrams 

      B.1  First order terms

             B.1.1  A differentiator 

             B.1.2  An integrator

             B.1.3  A first order pole

             B.1.4  A first order zero

             B.1.5  A second order transfer function 

             B.1.6  A second order zero  

 

C    Decibels, dB

 

D    Magnetically coupled coils

      D.1  Invertance

      D.2  Coil polarity marks

 

E    Euler-Lagrange equations submitted to constraints

 

F    Numerical implementation of controllers

      F.1  Numerical computation of integral

      F.2  Numerical differentiation 

      F.3  Lead compensator

      F.4  Controller in fig. 14.8(a)

             F.4.1  Controllers in (12.37) and (12.40)