Lec 1 introduction

Control Systems
LECT. 1 INTRODUCTION
BEHZAD FARZANEGAN
3/1/2020 PROVIDED BY: BF(B.FARZANEGAN@AUT.AC.IR) 1

Outline
 Introduction
Stability
o Internal Stability
o Routh-Hurwitz criteria
o Nyquist stability criteria
Design Controller via Root locus
o Lag
o Lead
o Lag-lead
Design Controller via Frequency response
o Lag
o Lead
o Lag-lead
PID
Different control structures

References

Who Can Take This Class?
The course is for graduate or advanced undergraduate
students with working knowledge in
• Differential calculus
• Root loci
• Bode diagram
• Classic linear systems/Control theory

What is a system?
What is a signal?
A signal is a locally integrable function 𝑍: 𝑅+ → 𝑅 𝑘.
• The notion of “local integrability” means simply that the function can
be safely and meaningfully integrated over finite intervals.
Systems are objects producing signals (called output
signals), usually depending on
• other signals (inputs)
• some other parameters (initial conditions).

Example

History of Control
Routh 1884, Hurwitz 1895, algebraic stability criterion
Classical feedback control (1930-1960)
• 1927, Bode and Nichols, frequency response analysis
• 1932, Nyquist, steady state frequency response techniques
• 1948, Evans, root-locus theory, A.M. Lyapunov, stability theory
• 1949, Wiener, optimum design
• 1957, Bellman, Dynamic programming
• 1962, Pontryagin , Maximum principle
Nonlinear control (1950)
• “Feedforward thinking”, mostly mechanical systems
Optimal control (1960-1980) (Kalman)
Adaptive control (1970-1985)(Astrom)

History of Control (cont’d)
Robust control (1980-1995) (Zames, J.C. Doyle)
• Combine classical and optimal control
• Optimal design of controllers with guaranteed robustness
Intelligent control theory
• Artificial Neural Networks
• Fuzzy Control
• Expert System
• GA, GP, Chaos, etc.
Discrete event and hybrid systems (current)
• Automata theory
• Computer science

Overview of Control Theory
CLASSICAL CONTROL THEORY
SISO, linear, time invariant
system
Laplace transform
Time, frequency domain
Input-Output Relation
(Transfer function)
MODERN CONTROL THEORY
MIMO, nonlinear, time-
varying system
matrix, vector, linear
algebra
Time domain
Internal Information
(State)

Open-loop Vs. Closed-loop
 Simple control is often open-loop
◦ user has a goal and selects an input to a system to try to achieve this

Open-loop Vs. Closed-loop
More sophisticated arrangements are closed-loop
◦ user inputs the goal to the system

Automatic Control Systems
Examples of automatic control systems:
◦ temperature control using a room heater

Example 2
◦ Cruise control in a car

Example 3
◦ Position control in a human limb

Example 4
◦ Level control in a dam

Control system?
◦ An interconnection of components forming a system configuration that will
provide a desired system response
Reference input (=desired output)
◦ Excitation applied to a control system using an external source
 Disturbance input
◦ A disturbance input signal to the system that has an unwanted effect on the
system output
Output (controlled variable)
◦ The quantity that must follow the command input without responding to
disturbance inputs

Feedback
◦ The output of a system that is returned to modify the input
Actuating signal (error signal)
◦ The signal that is the difference between the reference input and the
feedback signal (it is the input to the control unit that causes the output to
have the desired value)
Open-loop control system
◦ A system in which the output has no effect on the actuating signal (no
feedback)
Closed-loop control system
◦ A system in which the output has an effect on the actuating signal

In a closed loop system we refer to these as:
Sensor
Controller
e
Actuator
c
Process
u CV
Disturbance
+
-
Setpoint
Command
Input
Controlled
Input
Actuating Signal
Error Signal
Reference
Input
Controlled Variable
Output
FEEDBACK

Linear Time Invariant Systems (LTI)
S
Linearity:
Time invariance:

Control Systems
Natural Man-made
Manual Automatic
Open-loop Closed-loop
Non-linear linear
Time variant Time invariant
Non-linear linear
Time variant Time invariant
Types of Control System

Linear differential Systems
Transfer function (With zero initial conditions. Why?)
Two ways to describe the systems:
Time domain
Frequency domain
Laplace
transform
inverse
transform

Spring Mass Example
Time domain
M
Time Domain!
Frequency Domain!

Water tank example
As an example, consider a very simple industrial process where it is desired to
maintain specific water level and water temperature in a tank
Inputs:
• Hot water
• Cold water
Outputs:
• Water level
• Water Temperature
Water Level: y1
Hot
u1 u2
Water temperature: y2

Controller Design Objective
 The principal objective of a feedback controller is typically either disturbance
rejection or setpoint tracking
 Load disturbance is primary function of the control system in chemical processes
 Setpoint tracking is primary function of the control system in aerospace systems
◦ A missile following a certain trajectory
◦ A plane tracking a certain path
 In the sense described above, the control problem can be classified as
◦ Tracking
◦ Regulation

Regulation
Regulation is a control problem to maintain the output at a constant set
point regardless of the disturbances
 Example of a load disturbance in a chemical process
◦ A process is running smoothly and suddenly the flow rate changes due to
valve stiction
◦ Desired throughput change requires change in heat rate, flow rate, etc.
◦ Feed quality change for any reason
◦ Change in ambient conditions
 Chemical processes are desired to operate around their designed
steady state values

Tracking (Servo)
Tracking: design a feedback controller such that
while ensuring boundedness of all state variables
Regulation: 𝑟 𝑡 = 𝑟 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
 Path following (subset of tracking): Following a predefined path which
does NOT involve time as a constraint
◦ If you are on the path and following it with whatever speed, then you have
reached your goal
 Trajectory tracking involves time as a constraint
◦ You have to be at a certain point at a certain time
    lim 0
t
y t r t

 

Regulation Versus Tracking
Consider the following closed loop system with disturbance 𝛿
The closed loop transfer function is
y
gP
u
+
gc
─
er
+
+
g 𝛿
𝛿
     y s g s s  
                1 p c p cy s g s g s g s s g s g s r s   
Disturbance
        p cg s g s r s y s   pg s u s

Regulation Versus Tracking
We obtained
So
For tracking problems we are interested in the 1st term
For regulator problems we are interested in the 2nd term
                1 p c p cy s g s g s g s s g s g s r s   
 
   
   
 
 
   
 
Regulator TFServo TF
1 1
p c
p c p c
g s g s g s
y s r s s
g s g s g s g s

 
 
6 4 4 7 4 4 8 6 4 4 7 4 4 8

The Control Problem
Solving in control problem generally involves;
• Choosing sensors to measure the plant output
• Choosing actuators to drive the plant
• Developing the plant, actuator, and sensors equations
• Designing the controller
• Evaluating the design analytically by simulation, and finally by testing the
physical system.
• If the physical tests are unsatisfactory, iterating these steps.

1. Establish control goals
2. Identify the variables to control
3. Write the specifications for the variables
4. Establish the system configuration and identify the actuator
5. Obtain a model of the process, the actuator and the sensor
6. Describe a controller and select key parameters to be adjusted
7. Optimize the parameters and analyze the performance
If the performance meet the specifications, then finalize design
If the performance does not
meet specifications, then
iterate the configuration and
actuator
Control System Design Process

Given the control plant, the procedure of controller design to satisfy the
requirement is called system compensation.
What is system compensation?
 The closed-loop system has the function of self-tunning. By selecting a
particular value of the gain K, some single performance requirement
may be met.
Is it possible to meet more than one performance requirement?
 Sometimes, it is not possible.K
e
Process+
-
Setpoint u CV

 Design：Need to design the whole controller to satisfy the system
requirement.
 Compensation：Only need to design part of the controller with
known structure.
 Three elements for compensation
• Original part of the system
• Performance requirement
• Compensation device
Control system design and compensation

 Time domain criteria（step response）
◦ Overshoot
◦ settling time
◦ rising time
◦ steady-state error
 Frequency domain criteria
o Open-loop frequency domain criteria
• crossover frequency
• phase margin
• gain margin
o Closed-loop frequency domain criteria
• maximum value 𝑀𝑟
• resonant frequency
• bandwidth
Performance Requirement

2
1
2 1
rM
 


Resonant frequency 2
21   nr
Bandwidth 1)21(21 222
  nb
Gain crossover frequency
4 2
4 1 2c n     
Phase margin 24
214
2




 arctg
Resonant peak
Percentage overshoot %100%
2
1
 



 e
Settling time
3 4
,S
n n
t
 

Frequency domain and time domain criteria

 According to the way of compensation, the compensator can be classified
into following categories:
 Cascade compensation
 Feedback compensation
 Cascade and feedback compensation
 Feed-forward compensation
 Disturbance compensation
Structure of Compensator
Compensator
e
Process+
-
R(s) u C(s)
C(s)+
-
+
-
Process
Compensator
R(s) C(s)+
-
+
-
Process
Compensator
2
Compensator
1
R(s) C(s)++
-
Process
Compensator
+R(s) C(s)+
+
-
Process
Compensator
+
Controller
+
N(s)
+

 Frequency Response Based Method
Main idea：By inserting the compensator, the Bode diagram of the original
system is altered to achieve performance requirements.
Original open-loop Bode diagram＋Bode diagram of
compensator＋alteration of gain
＝open-loop Bode diagram with compensation
 Root Locus Based Method
Methods for Compensator Design
Main idea: Inserting the compensator introduces new open-loop zeros and
poles to change the closed-loop root locus to satisfy the requirement.

Lec 1 introduction

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