Note 7 Control

7/29/2019 Note 7 Control

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Lecture note 7 discrete event control 1

Discrete Event Control

CONTENTS

1. Introduction

2. State Diagram

3. Boolean Logical Equation


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Discrete Event Control

1. Introduction

DEC: both input and output control variable are discrete

variables that change values as a result of the occurrence

of events

Multiple-input/multiple-output (MIMO) discrete logical

controller, see Figure 1, where Ii is discrete value-based

input variable, and Yi discrete value-based output.

Ii and Yi only take value 0 (off) or 1 (on).

Input and output devices are usually located at a distance

from the controller.


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Introduction

Figure 1

Discrete

Logic

Controller

Y1

I2

Ip

I1

Y2

Ym


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Figure 2

Figure 3

Level Limit LLS VSwitch Controller Valve

Introduction


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We need a method to represent system

dynamics. Control system: including bothplant and controller. However, in discrete

event driven system, plant dynamics is

ignored. Therefore, the control system fordiscrete event driven system reduces to

the controller only.

State and state diagram is the method.

Next, we discuss state and state diagram.


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State Diagram

States: indicators that system changes

State Variables: assign a name to each independent class of states

EX 1: Switch: on and off. (1,0).

State change has a cause. State diagram (Fig.4, Fig.5) representsthe cause-state change. In particular, node: state; edge: cause.

In this example, we define:

LLS=0 for the level of liquid is below L

LLS=1 for the level of liquid is above L

Is LLS state variable?

NO


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State Diagram

1. State -> System state.

2. System: components: valve, pump.

3. Fluid: is not a part of the system, though the fluid has

its state as well. Level of fluid in the tank can beconsidered as a kind of state, but not in the sense of

system state.

4. Since we concern system state, the level of fluid is not

considered as a state variable. Level of fluid is inputvariable in this case; see the next slide.

5. In future, state refers to system state, but we omit

word system here.


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State Diagram

Control system

(X: state variable)

Level of fluid: LLS

Remain to see what is X and whatis output.


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State diagram

Figure 4

Figure 5


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State Diagram

Control system

(X: state variable)

Level of fluid: LLS

X: state variable: valve

Output: X as well

So we have: output = X


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State Diagram

It is noted that the two circles represent different states ofone state variable (i.e., valve). The system in EX 1 has

only one state variable.

EX 2: In EX 1, if we introduce also the pump in the

system. In particular, there is a piece of knowledge:

when the valve is closed the pump must be off. We can

sum up the desired control actions as follows:


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State Diagram

State variables: X1: pump; X2: valve.

X1: X1=0: pump off

X1=1: pump on

X2: X2=0: valve is closed

X2=1: valve is open


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State Diagram (for controller)

1. Open the valve if it is closed and the level of liquid in the

tank is less than the desired level L (LLS=0), or keep

the valve open if LLS=0.

2. Close the valve if it is open and the level of liquid in thetank is equal to or greater than the desired level L

(LLS=1), or keep the valve closed if LLS=1.

3. Turn the pump on if it is off and the valve is open andLLS=0, or keep the pump on if it is already on and the

valve is open and LLS=0.


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State Diagram

4. Turn the pump off if it is on and LLS=1, or keep the pumpoff if LLS=1.

The above expressions of control action can be represented

by two state variables, namely X1 (for pump) and X2 (for

valve)

X1=0, X2=0 (pump off, valve closed)

X1=0, X2=1 (pump off, valve open)

X1=1, X2=1 (pump on, valve open)

Fig.6 shows the state diagram for EX 2.


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State Diagram

Figure 6

Put all state variables

of the system in onecircle


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State Diagram

Fig. 7 shows another way to represent the state

diagram for EX 2. The features of Fig. 7 are:

1. Each node represents one state variable with its

value.

2. A state variable can be the cause of changes for

other state variables.


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State Diagram

Fig. 7The meaningthat the pump

can never be on

if the valve is

closed has notbeen

represented by

the state

diagram. This

shows some

limitation of the

state diagram


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State Diagram

Control system

(X: state variable)

Level of fluid: LLS

X1: state variable: pump

X2: state variable: valve

Output: X1, X2

So we have: output = X


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State Diagram: Summary

Control system

(X: state variable)I

I: a vector of inputs

O: a vector of outputs

X: a vector of state variablesI and O are in general function of X. In a special

case, O=X or I=X.

O


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State Diagram: Summary

1. State diagram involves logical variables that take 0 or 1 as theirvalues. State diagram has nodes and edges.

2. Each edge represents one cause or event for the state change inthe corresponding nodes. The cause is also a representation of thelogical variables. For instance, in Fig. 7, the cause can be writtenas: X2=1 and LLS =0.3. The state diagram has some limitation to express the meaning ofthe desired control action.

A formal way or mathematical way to represent the meaning:

If X2=1 AND LLS=0, X1 changes from 0 to 1. This desire

leads us to think of Boolean algebra. The idea is to think

another way to represent the controller or control system.

The next will discuss Boolean algebra.


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Boolean Logic Equations

Boolean Logic Equations

Let A and B be binary variables; that is, A, B=0, or 1.

When A =1 (B=1) means that A is true (resp., B is true).A =0 (B=0) means that A is false (resp., B is false).

(1) A+B means that either A or B is true

A+B=0 when A=0 and B=0

A+B=1 otherwise


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Boolean Logic Equation

(2) AB means that both A and B are true

AB=1 when A=1 and B=1

AB=0 otherwise

(3) Not operation, byA

1A

0A

when A=0

when A=1

Note 7 Control

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