Discrete Event Control

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Discrete Event Control. CONTENTS 1. Introduction 2. State Diagram 3. Boolean Logical Equation. 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

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Discrete Event Control

CONTENTS

1. Introduction

2. State Diagram

3. Boolean Logical Equation

Lecture note 7 discrete event control

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.

Lecture note 7 discrete event control

Introduction

Figure 1

Y1

I1

Discrete

Logic

Controller

I2

Y2

Ip

Ym

Lecture note 7 discrete event control

Introduction

Figure 2

Figure 3

Level Limit LLS V

Switch Controller Valve

Lecture note 7 discrete event control

We need a method to represent system dynamics. Control system: including both plant and controller. However, in discrete event driven system, plant dynamics is ignored. Therefore, the control system for discrete event driven system reduces to the controller only.

• State and state diagram is the method.
• Next, we discuss state and state diagram.

Lecture note 7 discrete event control

Is LLS state variable?

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) represents the 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

NO

Lecture note 7 discrete event control

State Diagram

State -> System state.

System: components: valve, pump.

Fluid: is not a part of the system, though the fluid has its state as well. Level of fluid in the tank can be considered as a kind of state, but not in the sense of system state.

Since we concern system state, the level of fluid is not considered as a state variable. Level of fluid is input variable in this case; see the next slide.

In future, state refers to system state, but we omit word ‘system’ here.

Lecture note 7 discrete event control

State Diagram

Level of fluid: LLS

Control system

(X: state variable)

Remain to see what is X and what is output.

Lecture note 7 discrete event control

State diagram

Figure 4

Figure 5

Lecture note 7 discrete event control

State Diagram

Level of fluid: LLS

Control system

(X: state variable)

X: state variable: valve

Output: X as well

So we have: output = X

Lecture note 7 discrete event control

State Diagram

It is noted that the two circles represent different states of one 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:

Lecture note 7 discrete event control

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

Lecture note 7 discrete event control

State Diagram (for controller)
• 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.
• Close the valve if it is open and the level of liquid in the tank is equal to or greater than the desired level L (LLS=1), or keep the valve closed if LLS=1.
• Turn the pump on if it is off and the valve is open and LLS=0, or keep the pump on if it is already on and the valve is open and LLS=0.

Lecture note 7 discrete event control

State Diagram

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

Lecture note 7 discrete event control

State Diagram

Figure 6

Put all state variables of the system in one circle

Lecture note 7 discrete event control

State Diagram
• Fig. 7 shows another way to represent the state diagram for EX 2. The features of Fig. 7 are:
• Each node represents one state variable with its value.
• A state variable can be the cause of changes for other state variables.

Lecture note 7 discrete event control

State Diagram

Fig. 7

The meaning that the pump can never be on if the valve is closed has not been represented by the state diagram. This shows some limitation of the state diagram

Lecture note 7 discrete event control

State Diagram

Level of fluid: LLS

Control system

(X: state variable)

X1: state variable: pump

X2: state variable: valve

Output: X1, X2

So we have: output = X

Lecture note 7 discrete event control

State Diagram: Summary

I

O

Control system

(X: state variable)

I: a vector of inputs

O: a vector of outputs

X: a vector of state variables

I and O are in general function of X. In a special case, O=X or I=X.

Lecture note 7 discrete event control

State Diagram: Summary
• State diagram involves logical variables that take 0 or 1 as their values. State diagram has nodes and edges.
• Each edge represents one cause or event for the state change in the corresponding nodes. The cause is also a representation of the logical variables. For instance, in Fig. 7, the cause can be written as: X2=1 and LLS =0.
• The state diagram has some limitation to express the meaning of the 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.

Lecture note 7 discrete event control

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

Lecture note 7 discrete event control

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, by

when A=0

when A=1

Lecture note 7 discrete event control