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**Discrete Event Control**

11/04/2017 Discrete Event Control CONTENTS 1. Introduction 2. State Diagram 3. Boolean Logical Equation Lecture note 7 discrete event control

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**Discrete Event Control**

11/04/2017 Discrete Event Control Concept Representation DEC controller design DEC controller implementation Lecture note 7 discrete event control

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**Discrete Event Control: introduction**

11/04/2017 Discrete Event Control: introduction DEC: All control variables are discrete variables, and their change is 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. Discrete Logic Control system Y1 I2 Ip I1 Y2 Ym Discrete Event Control system Figure 1 Lecture note 7 discrete event control

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**Lecture note 7 discrete event control**

11/04/2017 Introduction Figure 2 Level Limit Switch LLS Controller Input valve Level of Water Tank Figure 3 Lecture note 7 discrete event control

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**Lecture note 7 discrete event control**

11/04/2017 Introduction We need a method to represent system dynamics, i.e. control system (including both plant and controller). However, in a discrete event driven system, plant (transient) dynamics is ignored. This means that when the valve is open, the water level seems to rise to a level instantly. Therefore, the control system for a discrete event driven system reduces to the controller only. goal controller plant plant output State and state diagram is the method to represent system dynamics. Lecture note 7 discrete event control

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**Lecture note 7 discrete event control**

11/04/2017 State Diagram States: indicators that system changes State Variables: assign a name to each independent class of states. EX 1: Switch. The switch is a state variable. It has two states (1, 0), where 1=on and 0=off. State change has a cause. State diagram (Fig. 4, Fig. 5) represents the state change with cause;I n 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 Lecture note 7 discrete event control

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**Lecture note 7 discrete event control**

State diagram 11/04/2017 Figure 4 Example system Figure 5 Lecture note 7 discrete event control

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**Lecture note 7 discrete event control**

11/04/2017 State Diagram System has input, output, and itself. Fluid is a part of the system or total system. System itself includes components, e.g., valve, pump. System itself is represented by a set of state variables. The total system has fluid and device, and the device manipulates the fluid. Level of the fluid in the tank is the output of the plant (e.g., tank) or the plant control system (including both the plant and controller) and the input to the controller. In the time continuous system, the goal or reference variable such as L (in the tank example) is an input to the control system, while the level of the fluid is the output of the control system. Lecture note 7 discrete event control

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**State Diagram: how to identify the state variable**

11/04/2017 State Diagram: how to identify the state variable Remain to see what is X and what is the output (to controller). Level of fluid: LLS Control system (X: state variable) Lecture note 7 discrete event control

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**Lecture note 7 discrete event control**

11/04/2017 State Diagram Level of fluid: LLS Control system (X: state variable) X: state variable: valve. Output (for controller): X as well. So we have: output = state variable of the system Lecture note 7 discrete event control

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**Lecture note 7 discrete event control**

11/04/2017 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: Pump Lecture note 7 discrete event control

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**Lecture note 7 discrete event control**

11/04/2017 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

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**State Diagram (for controller)**

11/04/2017 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. Turn the pump off if it is on and LLS=1, or keep the pump off if LLS=1. Lecture note 7 discrete event control

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**Lecture note 7 discrete event control**

11/04/2017 State Diagram 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

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**Lecture note 7 discrete event control**

11/04/2017 State Diagram Figure 6 Put all state variables of the system in one circle Lecture note 7 discrete event control

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**Lecture note 7 discrete event control**

11/04/2017 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 or state. A state variable can be the cause of changes for other state variables. Lecture note 7 discrete event control

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**Lecture note 7 discrete event control**

11/04/2017 State Diagram Fig. 7 Remark: The meaning that the pump can never be on if the valve is closed is not represented by the state diagram. This shows a limitation of the state diagram Lecture note 7 discrete event control

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**Lecture note 7 discrete event control**

11/04/2017 State Diagram Level of fluid: LLS Control system (X: state variable) X1: state variable: pump X2: state variable: valve Output (for controller): X1, X2 So we have: output = state variable Lecture note 7 discrete event control

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**State Diagram: Summary**

11/04/2017 State Diagram: Summary I O Control system (X: state variable) I: a vector of inputs (for controller) O: a vector of outputs (for controller) 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

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**State Diagram: Summary**

11/04/2017 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 of another way to represent the controller or control system. Lecture note 7 discrete event control

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**Boolean Logic Equations**

11/04/2017 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). Lecture note 7 discrete event control

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**Boolean Logic Equation – operational property**

11/04/2017 Boolean Logic Equation – operational property A+B means that either A or B is true. Examples: A+B=0 when A=0 and B=0. A+B=1 otherwise. AB means that both A and B are true. Examples: AB=1 when A=1 and B=1. AB=0 otherwise. Not operation, by when A=0 when A=1 Lecture note 7 discrete event control

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