1

Design of Control SystemsCascade Root Locus Design

This is the first lecture devoted to the control system design. In the previous lectures we laid the groundwork for design techniques

based on root locus analysis. At the end, root locus design will not prove to be the best technique. That honor is reserved for the Bode

design method.

HoweverRoot locus design is very intuitive. The root locus

provides a portrait for 0 K for all possible closed-loop pole locations.

Proportional plus integral plus derivative (PID) control is widely used in industrial applications. PID control is

best understood from the root locus perspective

2

Compensation

• The design of a control system is concerned with the arrangement of the system structure and the selection of a suitable components and parameters.

• A compensator is an additional component or circuit that is inserted into a control system to compensate for a deficient performance.

• Types of Compensation– Cascade compensation

– Feedback compensation

– Output compensation

– Input compensation

3

PID Controllers

• PID control consists of a proportional plus derivative (PD) compensator cascaded with a proportional plus integral (PI)

compensator.

• The purpose of the PD compensator is to improve the transient response while maintaining the stability.

• The purpose of the PI compensator is to improve the steady state accuracy of the system without degrading the stability.

• Since speed of response, accuracy, and stability are what is needed for satisfactory response, cascading PD and PI will suffice.

4

The Characteristics of P, I, and D ControllersNote that these correlations may not be exactly accurate, because Kp, Ki, and Kd are

dependent of each other. In fact, changing one of these variables can change the effect of the other two. For this reason, the table should only be used as a reference when you are

determining the values for Ki, Kp and Kd.

Response Rise Time Overshoot Settling Time

SS Error

KP Decrease Increase

Small Change Decrease

KI Decrease Increase Increase Eliminate

KD

Small Change Decrease Decrease

Small Change

5

The Simplest form of compensation is gain compensation

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6

Root Locus for Simple Gain Compensator

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7

Lead/Lag Compensation

• Lead/Lag compensation is very similar to PD/PI, or PID control.

• The lead compensator plays the same role as the PD controller, reshaping the root locus to improve the transient response.

• Lag and PI compensation are similar and have the same response: to improve the steady state accuracy of the closed-loop system.

• Both PID and lead/lag compensation can be used successfully, and can be combined.

8

Lead Compensation Techniques Based on the Root-Locus Approach

• From the performance specifications, determine the desired location for the dominant closed-loop poles.

• By drawing the root-locus plot of the uncompensated system ascertain whether or not the gain adjustment alone can yield the desired closed-loop poles. If not calculate the angle deficiency. This angle must be contributed

by the lead compensator.

• If the compensator is required, place the zero of the phase lead network directly below the desired root location.

• Determine the pole location so that the total angle at the desired root location is 180o and therefore is in the compensated root locus.

• Assume the transfer function of the lead compensator.

• Determine the open-loop gain of the compensated system from the magnitude conditions.

9

Lead Compensator using the Root Locus

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10

Adding Lead CompensationThe lead compensator has the same purpose as the PD compensator:

to improve the transient response of the closed-loop system by reshaping the root locus. The lead compensator consists of a zero and a pole with the zero closer to the origin of the s plane than the pole. The

zero reshapes a portion of the root locus to achieve the desired transient response. The pole is placed far enough to the left that it does

not have much influence of the portion influenced by the zero.

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11

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13

Adding a Lag Controller

• A first-order lag compensator can be designed using the root locus. A lag compensator in root locus form is given by

• where the magnitude of zo is greater than the magnitude of po. A phase-lag compensator tends to shift the root locus to the right, which is undesirable. For this reason, the pole and zero of a lag compensator must be placed close together (usually near the origin) so they do not appreciably change the transient response or stability characteristics of the system.

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14

How does the Lag Controller Shift the Root Locus to the

Right? • Recall finding the asymptotes of the root locus that lead to the zeros at

infinity, the equation to determine the intersection of the asymptotes along the real axis is:

• When a lag compensator is added to a system, the value of this intersection will be a smaller negative number than it was before. The net number of zeros and poles will be the same (one zero and one pole are added), but the added pole is a smaller negative number than the added zero. Thus, the result of a lag compensator is that the asymptotes' intersection is moved closer to the right half plane, and the entire root locus will be shifted to the right.

zerospoles

zerospoles

15

Control ModesThere are many ways by which a control unit can react to an error

and supply an output for correcting elements.

• The two-step mode: The controller is just a switch which is activated by the error signal and supplies just an on-off correcting signal. Example of such mode is the bimetallic thermostat.

• The proportional mode (P): This produces a control action that is proportional to the error. The correcting signal thus becomes bigger the bigger the error. Therefore, the error is reduced the amount of correction is reduced and the correcting process slows down. A summing operational amplifier with an inverter can be used as a proportional controller.

• The derivative mode: This produces a control action that is proportional to the rate at which the error is changing. When there is a sudden change in the error signal the controller gives a large correcting signal. When there is a gradual change only a small correcting signal is produced. An operational amplifier connected as a differentiator circuit followed by another operational amplifier connected as an inverter make an electronic derivative controller circuit.

16

• The integral mode (I): This produces a control action that is proportional to the integral of the error with time. Therefore, a constant error signal will produce an increasing correcting signal. The correction continues to increase as long as the error persists.

• Combination of modes: Proportional plus derivative modes (PD), proportional plus integral modes (PI), proportional plus integral plus derivative modes (PID). The term three-term controller is used for PID control.

• The controller may achieve these modes by means of pneumatic circuits, analog electronics involving operational amplifiers or by the programming of a microprocessor or computer.

17

DC Motor Speed ModelingThe DC motor has been the workhorse in industry for many reasons including good torque speed characteristics. It is a common actuator in control systems.

It directly provides rotary motion and, coupled with wheels or drums and cables, can provide transitional motion. The electric circuit of the armature and

the free body diagram of the rotor are shown in the following Figure.We develop here the transfer function of a separately excited armature

controlled DC motor.

Motor

Generator

Mechanical energy(T, )

VA

RA IA

VF

RF

LF

IF L

k

J motor & load

T

Field circuit Armature circuit

18

Physical Parameters

• Electrical Resistance R= 1 • Electrical Inductance L = 0.5 H

• Input Voltage V

• Electromotive Force Constant K = 0.01 nm/A

• Moment of Inertia of the Rotor J = 0.01 kg.m2/s2

• Damping Ratio of the Mechanical System b = 0.1 Nms

• Position of the Shaft • The rotor and shaft are assumed to be rigid

• The motor torque T is related to the armature current by a constant Kt

• The back emf, e, is related to the rotational velocity

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19

Speed Control• Speed Control by Varying Circuit Resistance: The operating speed can only

be adjusted downwards by varying the external resistance, Rext

• Speed Control by Varying Excitation Flux:

• Speed Control by Varying Applied Voltage: Wide range of control 25:1; fast acceleration of high inertia loads.

• Electronic Control.

rad/s

22

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20

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21

Data Measurement

• Once we have identified the transfer function of the system we may proceed to the final two phases of the design cycle, the design of a suitable controller and the implementation of the controller on the actual system. In the case of speed control of the DC motor, the control will prove to be quite easy.

• An important point to be highlighted here is that if we have a good model of the plant to be controlled, and we already have identified the parameters of the model, then the design of the controller is easy.

Computer D/A Converter Power Amplifier DC Motor

OscilloscopeArmature voltage

Tachometer

22

Design Needs• The uncompensated motor may only rotate at 0.1 rad/sec with an input

voltage of 1 V. Since the most basic requirement of a motor is that it should rotate at the desired speed, the steady-state error of the motor speed should be less than 1%.

• The other performance requirement is that the motor must accelerate to its steady-state speed as soon as it turns on. In this case, we want it to have a settling time of 2 seconds for example. Since a speed faster than the reference may damage the equipment, we want to have an overshoot of less than 5%. If we simulate the reference input (r) by a unit step input, then the motor speed output should have:

• Settling time less than 2 seconds• Overshoot less than 5%• Steady-state error less than 1%

Use the MATLAB to represent the open loop response

23

PID Design technique for DC Motor Speed Control

• Design a PID controller and add it into the system.• Recall that the transfer function for a PID controller is:

• See how the PID controller works in a closed-loop system using the

previous Figure. The variable (e) represents the tracking error, the difference between the desired input value (R) and the actual output (y). This error signal (e) will be sent to the PID controller, and the controller

computes both the derivative and the integral of this error signal. • The signal (u) just past the controller is equal to the proportional gain (Kp)

times the magnitude of the error plus the integral gain (Ki) times the integral of the error plus the derivative gain (Kd) times the derivative of the

error.

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24

The PID Adjustment Steps

• Use a proportional controller with a certain gain. A code should be added to the end of m-file.

• Determine the closed-loop transfer function.• See how the step response looks like.

• You should get certain plot.• From the plot you may see that both the steady-state error and the

overshoot are too large. • Recall from the PID characteristics that adding an integral term will

eliminate the steady-state error and a derivative term will reduce the overshoot. Let us try a PID controller with small Ki and Kd.

• The settling time is too long. Let us increase Ki to reduce the settling time.

• Large Ki will worsen the transient response (big overshoot). Let us increase Kd to reduce the overshoot.

• See the plot now and see if design requirements will be satisfied.

25

Root Locus Design Method for DC Motor Speed ControlDrawing the Open-Loop Root Locus

• The main idea of root locus design is to find the closed-loop response from the open-loop root locus plot. Then by adding zeros and/or

poles to the original plant, the closed-loop response can be modified.• We need the settling time and the overshoot to be as small as

possible. Large damping corresponds to points on the root locus near the real axis. A fast response corresponds to points on the root locus

far to the left of the imaginary axis. • The system may be overdamped and the settling time will be about

one second, so the overshoot and settling time requirements could be satisfied.

• The only problem we may see from the generated plot is the steady state error. If we increase the gain to reduce the steady-state error,

the overshoot becomes too large. We need to add a lag controller to reduce the steady-state error.