ABSTRACT: This tutorial article discusses some basic issues in the design of

control systems. The concepts of well posedness and total stability are introduced to

deal with noise and disturbance problems. The imple mentable transfer function is developed and is shown to

solve completely pole-and-zero assignment and model matching problems. Two feedback configurations are introduced to realize the implementable transfer func tions, and feedback compensation is ob tained by solving sets of linear algebraic equations.

When the open-loop plant transfer func tion has been specified, there are basically two approaches to carry out design.

In the first approach, we choose a feedback config uration and compensation with undetermined parameters and then adjust the parameters so that the resulting closed-loop system will meet design specifications.

The root-locus and frequency-domain methods are ways to evaluate the adjustments used in this ap proach.

In the second approach, we choose an overall closed-loop system to meet design specifications.

We then choose an appropri ate feedback configuration and compute the required compensation.

The linear quadratic optimal control method and design through pole-zero pattern are examples of the second approach . We call the first approach the outward approach and the sec ond the inward approach.

In the inward approach, the first step is to choose an overall closed-loop transfer func tion to meet a set of specifications. Because of physical constraints, this choice is not entirely arbitrary

In this paper, we introduce four constraints on the choice of the overall transfer function: namely, properness of compensators, well posedness, total stabil ity, and no plant leakage.

An overall transfer function that can be implemented under these four constraints is called an implementable transfer function

The implementable transfer functions shown to solve pole-and-zero assignment and model matching problems.

Once an implementable overall transfer function is chosen, the next step is to choose a control configuration

However, unity feed back can be used to achieve arbitrary pole assignment. We then introduce two more so phisticated configurations, which can be used to

achieve pole assignment and zero assign ment simultaneously: namely, the two-pa rameter and the plant input/output (I/O) feedback configurations.

The compensators are obtained by solving sets of linear alge braic equations.

Example and IssuesFirst, we will use an example to illustrate the issues that may arise in the design

of control systems. Consider the plant with the open-loop transfer function G(s).G(s) = (s - I)/[s(s - 2)] The problem is to design an overall closed loop system such that the plant output

yet will track a reference input r(t). As can be seen, the plant transfer function is unstable and has a non minimum-

phase zero. This is a difficult problem if the root-locus method or frequency-domain method

is used to carry out the design. If the inward approach is used, then the first step is to select an overall closed-

loop transfer function. It is clear that an overall transfer function of unity is the best possible system we

can design. Indeed, if an overall transfer function is unity, then the plant output is identical to

any reference input;

the position and velocity errors are zero; and the rise time, settling time, and overshoot are also all zero.

Therefore, no other transfer function can perform better than a transfer function of unity.

Note that, for a unity transfer function, the power levels at the reference input and plant output are different;

otherwise, the control system would be unnecessary Of course, a transfer function of unity usually cannot be imple mented in practice because we must use pure differentiators as compensator

The actuating control signal may get very large, caus ing the plant to saturate. Therefore, a more realistic overall transfer function must be chosen.

The calculations to obtain a realistic transfer function may be carried out by com puter simulation using existing computer aided design packages.

General Control System

Sensor

Actuator ProcessController ++

Set-point or

Reference input

Actual Outpu

t

ErrorControlled Signal

Disturbance

Manipulated

Variable

Feedback Signal

+

-

++

Control System Design Process

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

Examples

(a) Automobile steering control system.

(b) The driver uses the difference between the actual and the desired direction of travel

to generate a controlled adjustment of the steering wheel.

(c) Typical direction-of-travel response

System – An interconnection of elements and devices for a desired purpose.

Control System – An interconnection of components forming a system configuration that will provide a desired response.

Process – The device, plant, or system under control. The input and output relationship represents the cause-and-effect relationship of the process.

• The interaction is defined in terms of variables.i. System inputii. System outputiii. Environmental disturbances

Control System

• Control is the process of causing a system variable to conform to some desired value.• Manual control Automatic control (involving machines only).• A control system is an interconnection of components forming a system

configuration that will provide a desired system response.

Control System

Output

Signal

Input Signa

l

Energy

Source

Multivariable Control System

Open-Loop Control Systems utilize a controller or control actuator to obtain the desired response.

Closed-Loop Control Systems utilizes feedback to compare the actual output to the desired output response.

Control System Classification

Open-Loop Control System

Missile Launcher System

Control System Classification

Closed-Loop Feedback Control System

Missile Launcher System

Manual Vs Automatic Control• Control is a process of causing a system variable such as

temperature or position to conform to some desired value or trajectory, called reference value or trajectory.• For example, driving a car implies controlling the vehicle to follow

the desired path to arrive safely at a planned destination.i. If you are driving the car yourself, you are performing manual control of

the car.

ii. If you use design a machine, or use a computer to do it, then you have built an automatic control system.

Control System Classification

Desired Output

Response

Measurement

Output Variabl

es

Controller

Process

Multi Input Multi Output (MIMO) System

Purpose of Control Systems

i.Power Amplification (Gain)• Positioning of a large radar antenna by low-power rotation of a knob

ii.Remote Control• Robotic arm used to pick up radioactive materials

iii.Convenience of Input Form• Changing room temperature by thermostat position

iv.Compensation for Disturbances• Controlling antenna position in the presence of large wind disturbance torque

Human System

The Vetruvian Man

Human System

i.Pancreas Regulates blood glucose level

ii.Adrenaline Automatically generated to increase the heart rate and oxygen in

times of flight

iii.Eye Follow moving object

iv.Hand Pick up an object and place it at a predetermined location

v.Temperature Regulated temperature of 36°C to 37°C

Control System Components

i.System, plant or process• To be controlled

ii.Actuators• Converts the control signal to a power signal

iii.Sensors• Provides measurement of the system output

iv.Reference input• Represents the desired output

General Control System

Sensor

Actuator ProcessController ++

Set-point or

Reference input

Actual Outpu

t

ErrorControlled Signal

Disturbance

Manipulated

Variable

Feedback Signal

+

-

++