EE456: DigitalControlSystems · PDF fileLecture Objectives In these introductory lectures we will study the following: X Control system analysis and design objectives X Signals and

EE 456: Digital Control Systems

Prof. Khoder Melhem

Qassim University

Academic year 2014-2015

College of Engineering

Department of Electrical Engineering

Lectures 1-3

Introduction to Digital

Control Systems

Lecture Objectives

In these introductory lectures we will study the following:

X Control system analysis and design objectives

X Signals and systems dealt with in this course

X Description with examples to digital control systems

X Course description

Digital Control Systems Academic year 2014-2015 4Prof. K. Melhem (Qassim University)

What is control? A real life example . . .

• Water inflow u(t) must be controlled to reach and maintain the desired

temperature. r(t)

• Sensors on skin measure water temperature y(t).

• Water inflow u(t) manipulated so that y(t)≈ r(t) . . .

• . . . in spite of flow and temperature fluctuations d(t).


What is automatic control?

• Operator has been replaced by an electronic circuit.

• Control is now automatic; it is accomplished without human intervention.


What is a control system?

Objective: To make the system OUTPUT and the desired REFERENCE as close as

possible, i.e., to make the ERROR as small as possible.

Key Issues: 1) How to describe the system to be controlled? (Modeling)

2) How to design the controller? (Control)


Response characteristics of a control systemElevator example

When the fourth-floor button of an elevator is pushed on the ground floor, the elevator rises to the

fourth floor with a speed and floor-leveling accuracy designed for passenger comfort.

• Physical entities cannot change their states instantaneously; the elevator undergoes a gradual

change as it rises from the first floor to the fourth floor. This is called the transient response.• After the transient response, a physical system approaches its steady-state responsewhich is an

approximation to the desired response; the elevator reaches the fourth floor.

• The accuracy of the steady-state response could also make the output different from the input.

We call this difference as steady-state error; elevator may not be level with the floor.

Steady-state errors must designed to be zero for some applications.


Control system analysis and design objectivesElevator example

A control system is dynamic: It responds to an input by undergoing a transient response before

reaching a steady-state response that generally resembles to the input.

• Transient responseis important. a slow response makes elevator passengers impatient, whereas

an excessively rapid response makes them uncomfortable. If the elevator oscillates about the

arrival floor for more than a second, a disconcerting feeling can result. Transient response is

also important for structural reason: Too fast a transient response could cause a permanent

physical damage. Thus, we analyze the elevator for its transient response and (if needed) we

adjust parameters or design components to yield desired transient response.

• Steady-state responseof the elevator is its location reached near the fourth floor. An elevator

must be level enough with the floor for the passengers to exit. Thus, the elevator’s steady-state

error should be analyzed and (if needed) design corrective action to reduce the steady-state

error should be taken.

• Discussion of transient response and steady-state error is moot if the system does not have

stability . Actually, the total response of a system is the sum of the natural response and the

forced response. For a control system to be stable, the natural response must eventually

approach zero, thus leaving only the forced response. If the natural response grows without

bound the system is no longer controlled or unstable. Instability could lead to self-destruction

of the physical device if limit stops are not part of the design. In our example, the elevator

would crash through the floor or exit through the ceiling. Thus, a control system must be

analyzed and designed to be stable.


What is important in a control system?

• Stability

• (Transient) response speed

• Accuracy

⊲ dynamic overshooting and oscillation duration

⊲ steady-state error

• Robustness

⊲ errors in models (uncertainties and nonlinearities)

⊲ effects of disturbances

⊲ effects of noises


What is important in a control system?

Solutions for Robustness:

• Plant nonlinearities – nonlinear plants – nonlinear control

• Sensor nonlinearities – nonlinear control

• Control input nonlinearities (control saturations) – nonlinear control

• Plant parameter uncertainties – robust control or adaptive control

• Noise and disturbancerejection – robust control or optimal control

These issues are too complicated to be considered in this undergraduate course . . .

Come back for postgraduate studies if you are interested in these topics.


Other considerations in control system analysis and design

• Factors affecting hardware selection

⊲ motor sizing to fulfill the power requirements

⊲ choice of sensors for accuracy

• Design economic impact

⊲ budget allocation

⊲ competitive pricing


Modeling of dynamic systems

Model: A representation of a system.

Types of Models:

• Physical models (prototypes)

• Mathematical models (e.g., input-output relationships)

⋄ Analytical models (using physical laws)

⋄ Computer (numerical) models

⋄ Experimental models (using input/output experimental data)

Models for physical dynamic systems:

• Lumped-parameter models

• Continuous-parameter models. Example: Spring element (flexibility, inertia,

damping)


How to design a (modern) control system?

• Understand the automation problem:

⊲ what are the specifications to be achieved?

⊲ which variables can be manipulated by actuators?

⊲ what are the output variables of interest?

⊲ what should we measure?

⊲ which are the disturbances?

• Choose sensorsto measure the required feedback signals.

• Choose actuatorsto drive the plant.

• Get a simplified mathematical modelor a reliable simulation model of the plant, sensors, and

actuators.

• Synthesize the control algorithm based on the developed models and the control criteria.

• Test the design analytically by simulation.

• Validate on real process (implementation).

• Iterate this procedure until a satisfactory physical system response results.


Control systems engineer’s skills and knowledge are many

Control systems engineering is an exciting field in which to apply your engineering talents, because

it cuts across numerous disciplines and numerous functions within those disciplines. Many engineers

are engaged in only one area, such as circuit design and software development. However, as a control

systems engineer, you may find yourself working in a broad arena and interacting with people from

numerous branches of engineering and the sciences. A control systems engineer must be good in:

• Mathematics to get a good mathematical model for the process and design a controller that

responds to the desired requirements.

• Physicsto understand the physical phenomenon of the process to be controlled so as a good

mathematical model can be provided.

• Simulation to analyze the control system as well as simulate its performance.

• Being aware of the current available technologyto choose the best hardware and software for

implementation of the proposed controller.

A lot of skills and knowledge to have got from a control systems engineer, which makes the control

engineering one of a kind!


Signal categories for identifying control system types

Continuous-time signal & quantized signal

• Continuous-time signal is defined continuously in the time domain. Figure on the

left shows a continuous-time signal, represented by x(t).

• Quantized signal is a signal whose amplitudes are discrete and limited. Figure on

the right shows a quantized signal.

• Analog signal or continuous signal is continuous in time and in amplitude. The

real word consists of analog signals.



Discrete-time signal & sampled-data signal

• Discrete-time signal is defined only at certain time instants. For a discrete-time signal,

the amplitude between two consecutive time instants is just not defined. Figure on the

left shows a discrete-time signal, represented by y(kh), or simply y(k), where k is an

integer and h is the time interval. Example of a discrete-time signal is blood pressure

readings of a patient every one hour.

• Sampled-data signal is a discrete-time signal resulting by sampling a continuous-time

signal. Figure on the right shows a sampled-data signal deriving from the

continuous-time signal, shown in the figure at the center, by a sampling process. It is

represented by x∗(t).

• Discrete-time signal may be quantized resulting in a digital signal.



Digital signal or binary coded data signal

• Digital signal is a sequence of binary numbers. In or out from a microprocessor, a

semiconductor memory, or a shift register. Figure at the top shows a digital signal.

• In practice, a digital signal, as shown in the figures at the bottom, is derived by two

processes: sampling and then quantizing.


Control system types

• Analog control systems (or Continuous-time systems) in which the associated signals are

continuous-time signals while discrete-time systems in which the associated signals are

discrete-time signals. Sampled-data control systems include both sampled-data and

continuous-time signals, while finally digital control systems contain digital, sampled-data, and

continuous-time signals.

• Examples of analog systems include RLC circuits, mass-spring-damper mechanical systems, DC

motors. While, examples of digital systems include microprocessors, semiconductor memories,

and shift registers.


Systems dealt with in the course

• A static systemis a memoryless system with an output signal at an specific time

depends only on the input signal at this time. A simple continuous-time static system

is a resistor R where i(t) = (1/R)v(t). A dynamic systemis a system with memory with

an output signal at any specified time depends on the value of the input signal at both

the specified time and at other times. A dynamic system contains one or more

energy-storage elements and described by differential or difference equations. An

example of continuous-time dynamic system is a capacitor C whose input-output

relationship is expressed as v(t) = 1C

∫ t−∞ i(τ)dτ.

• A linear system is when the superposition principle applies. A linear system is described

by linear input-output relationship (linear algebraic or differential or difference

equations). An example of a linear continuous-time dynamic system is an RC electric

network whose input-output relationship is described by

RCdvo

dt+ vo = vi

where output vo is the voltage across the capacitor and input vi is the applied source

voltage to the network.


Systems dealt with in the course

• A time-invariant system is a system in which a time shift in the input signal results in

corresponding time shift in the output signal. A time invariant system has parameters

which are independent of time. Otherwise, the system is time-variant. An example of

time-variant system is S[u(t)] = tu(t), where u(t) is a pulse. Figure above shows the

system plot.


Test waveforms used in control systems


Unit impulse or Dirac delta signal

Definition

δ(t)=

∞, for t = 0

0, elsewhere

Properties

• Area or weight:∫ +∞−∞ δ(t)dt = 1

• Laplace transform: L [δ(t)] = 1

• Impulse response: A system G(s) = L [g(t)] with unit impulse δ(t) as input hasimpulse response output y(t) as y(t) = g(t).

• Sifting theorem:∫ +∞−∞ f (t)δ(t −T )dt = f (T )

• Dirac delta function δ(t) is used to sample a continuous-time signal


Analog control system

Note: Analog process G(s) with a PID controller.


Digital control system

A digital computer performs two main functions:

• Control : performs calculation that emulate the replaced physical compensator.

• Supervisory: scheduling tasks, monitoring parameters and variables for

out-of-range values, initiating safety shutdown, or effective user interfaces for

analysis and command.


Digital control system - - different components - -


Why digital control?

Using computers to implement controllers has substantial advantages. Many of the difficulties with

analog implementation can be avoided.

• Flexibility

⊲ Easy to implement complex/nonlinear control algorithms in digital computers than in

analog computer/analog OPAs.

⊲ Easy to change controllers by simple program recoding without rewiring and hardware

modifications.

⊲ Many loops can be controlled by the same computer through time sharing.

⊲ Possible to have effective user interfaces for monitoring, analysis, and command.

• Robustness

⊲ Digital circuits are less sensitive to noise. Indeed, digital signals are represented in terms of

0s and 1s with typically 12 bits or more to represent a single number. This involves a very

small error as compared to analog signals where noise and power supply drift are always

present.

⊲ Digital filter coefficients are more accurate than R, L, C in OPA circuits.

⊲ Digital control permits the use of sensitive control elements with relatively low-energy

signals. The advantages of using a digital transducer is the relative immunity of its digital

signals to distortion by noise and nonlinearities and its high accuracy and resolution as

compared to analog transducers.


Why digital control?

• Speed

⊲ The speed of computer hardware has increased exponentially since the 1980s. This increase

in processing speed has made it possible to sample and process control signals at very high

speeds. Because the interval between samples, the sampling period, can be made very

small, digital controllers achieve performance that is essentially the same as that based on

continuous monitoring of the controlled variable.

• Cost

⊲ Although the prices of most goods and services have steadily increased, the cost of digital

circuitry continues to decrease. Advances in very large scale integration (VLSI) technology

have made it possible to manufacture better, faster, and more reliable integrated circuits

and to offer them to the consumer at a lower price. This has made the use of digital

controllers more economical even for small, low-cost applications.


Disadvantages of digital control

• The mathematical analysis and design of a discrete-time control system is more

complex and tedious as compared to continuous-time control system

development. This is because of the additional analysis and design parameter;

the sampling period.

• Because the A/D converter, D/A converter, and the digital computer in reality

delay the control signal input (sampling period is not zero), the performance

objectives can be more difficult to achieve since the theoretical design approaches

usually do not model this small delay.

• Since the digital computer is working with a quantized amplitude representation

of the analog signal formed from values of the analog signal at discrete intervals

of time, quantization error is produced, which affects accuracy in digital control.


Time delay in control systems

Time delays arise in control systems due to three sources:

1. Delays in the process itself. For example, chemical plants often have processes with time delay

representing the time material takes to flow through pipes.

2. Delays in the processing of the sensed signals. For example, in measuring the attitude of a

spacecraft en route to Mars, there is a significant time delay for the sensed quantity to arrive

back on Earth due to the speed of light.

3. Small delays in any digital control systems due to the cycle time of the computer (sampling

period is not zero) and the fact that data is processed at discrete intervals.

Statement: Time delay always reduces the relative stability of a feedback control system.


Types of sampling operations

There are several types of sampling operations of practical importance:

• Periodic sampling: The sampling instants are equally spaced.

• Random sampling: The sampling instant are random.

• Multiple-order sampling: The difference between two consecutive sampling

instants is repeated periodically.

• Multiple-rate sampling: Different sampling periods are present in different

feedback paths; convenient for control systems with multiple loops having

different time constants.

In this course, we are concerned only with the simple case of periodic sampling.


Controller design in digital control systems

• Implementation of the controller by a digital computer rather than analog

computer or components.

• How to design this digital controller?

• 2 methods of controller design: s-plane design and z-plane design methods.


Controller design in digital control systems - - Digitization(DIG) or discrete control design - -

• Follow whatever we have learnt in EE 351 to design a continuous-time controller

and then discretize it (by using Tustin’s transformation) to obtain an equivalent

digital controller.

• Simulation tests should be performed to check the performance required.

• The above design works very well if sampling period T is sufficiently small.


Controller design in digital control systems - - Direct (DIR)control design - -

• Alternatively, one could discretize the plant first to obtain a discrete-time system

and then apply discrete-time control system design techniques to design a digital

controller.

• Subsequently, it should be shown that G(z) = (1− z−1)Z{

G(s)s

}

.


Mathematical comparison between analog and digitalcontrol systems


A typical digital control system


Most used sensors and actuators in control systems


Sensors and actuators in control systems


Hardware-software for implementing a digital controller

From top to bottom and left to right: digital signal processor (DSP), microcontroller,

LabVIEW software, and programmable logic controller (PLC).


Examples of digital control systemsDrug delivery system


Examples of digital control systemsAircraft turbojet engine


Examples of digital control systemsRobot manipulator


What you will learn in this course

Having successfully completed this course, you will be able to demonstrate

knowledge and understanding of:

• z transform analysis of discrete-time control systems

• Stability analysis in the s- and z- plane

• Classical techniques of root locus and frequency response for digital controller

analysis and design



• Expressing real engineering problems as an exercise in linear digital controller

design

• Choice of appropriate design methodology

• Choice of performance analysis tools


What you will learn in this course (Cont’d)



• Ability to develop algorithms for digital control implementation

• Ability to use Matlab and its Simulink package in the modeling, analysis, and

design of discrete-time control systems

• Formulate a digital control problem, design a solution, and test the result by

simulating it via Matlab


Topics covered in this course

© Introduction to digital control systems (3 lectures)

© Review of continuous-time systems (3 lectures)

© The sampling and holding processes (3 lectures)

© The z-transform and its inverse (3 lectures)

© Modeling, stability, transient response characteristics, and steady-state error of

discrete-time systems (6 lectures)

© Analysis and design of digital control systems (18 lectures)

© Digital controller structures (3 lectures)

© Microcomputer implementation of digital controllers (3 lectures)

© Analysis, design, and simulation of digital control systems by Matlab/Simulink

(3 lectures)


Learning resources

K. Melhem, Lecture notes, (what you’re looking at right now . . . But, it is not

enough!). Soon, available online at college webpage.

C.L. Philips and H.T. Nagle, Digital Control Systems: Analysis and

Design, Third Edition, Prentice Hall, 1995.

(Good classical textbook on digital control, your textbook!)

C.H. Houpis and G.B. Lamont, Digital control systems: theory, hard-

ware, and software, McGraw-Hill, 1991.

(Very good classical textbook on digital control)

K. Ogata, Discrete-time control systems, Prentice Hall, 1995.

(Good classical textbook on digital control)


Logistics

Lecture and tutorial schedules

Sundays and Thursdays: From 14 O’clock to 16 O’clock

Grading

Homework⋆, Quizzes◦, and Attendance 20 %

First Mid-Term Exam 15 %

Second Mid-Term Exam 15 %

Final Exam 50 %

⋆ Deals with analysis, design, and simulation of digital control systems using Matlab/Simulink.

◦ Designed in multiple-choice form to assess students’ understanding of the course.

Office hours for student help: Working days (see Instructor’s schedules for more

details). Also, you may ask your questions after class.


Lectures & Tutorials

Attendance is essential

Ask your tutor any question related to the course at any time during thelecture and tutorial


How to succeed in this course

• Before class:

© Review material (textbook and notes if available) from previous class

© Preview material (textbook and notes if available) to be covered

© Arrive on time

• During class:

© Attend all classes and tutorials - Not everything can be understood by

self-study within an acceptable short time

© Pay attention, take notes, and ask questions if needed

• After class:

© Review material (textbook and notes if available)

© Identify and understand key points

© Do all the problem sets assigned in time

Documents

EE456: DigitalControlSystems · PDF fileLecture Objectives In these introductory lectures we will study the following: X Control system analysis and design objectives X Signals and