ME 460 Notes

General Introduction to Control for ME 460 - Chris Zhang

1. By control it is meant automated control that is carried out by devices called controllers, as opposed to manual control that requires human intervention. The devices could be hardware, software, or mixture of the two.

2. The controller comprises of both hardware and software. The implementation of a controller can be made possible using

- Microprocessors, digital signal processing (DSP) chips, programmable logic controllers (PLCs), personal commuter (PCs), or workstations.

- A control system contains: plant or process, sensor, actuator, and controller.- The input to a control system is a reference signal or goal signal (goal: a plant needs to

achieve), and the output of a control system is the plant output.

3. A key component of control technology is the design of the controller. Common types of controllers include proportional-plus-integral (PI), lead-lag, model predictive controllers (MPCs), adaptive controllers, neural net controllers, and fuzzy-logical controllers. However, controller design is actually a relatively small part of the overall control problem. The following is a list of control problems:

- signal conditioning and signal processing such as filtering and sampling,- Use of real-time and historical databases to store information,- Interfaces between controllers and sensors, devices, and equipment,- Operator interfaces including graphics user interfaces (GUIs),- Communication networks such as field-buses and local-area networks (LANs),- Computer operating system such as Windows NT,- Object-oriented (OO) control software such as Java-based applications.

4. When applied to manufacturing, the word of control is used on each of several different levels in the hierarchy shown in Figure 1.

Enterprise Level

Factory Floor Level

Cell/Line Level

Machine/Process Level

Figure 1 Levels of Different Sub-systems in an Enterprise Systems

General Concept of Control 1

ME 460 Notes

In this class, we will focus on the lowest level, i.e., machine or process level but we will briefly discuss the control at the higher levels.

5. General control configuration

Figure 2 shows a closed-loop configuration.

The following concepts are used throughout this class:

(1) c1, c2, …, cm are called process or plant inputs(2) y1, y2, …, ym are called process or plant outputs(3) The process is said to be of the closed-loop type since the process outputs yi are fed back

through the controller to generate the process inputs ci to the process.(4) The process inputs ci and the process outputs yi may be continuous variables or discrete

variables. Here “continuous” means that the variables may take on values over some range of real numbers. Examples of continuous a variable are voltage, position coordinates, pressures, flow rates, etc. Continuous variables are also said to be analog variables. “Discrete” means that the variable may have only a finite number of different values. An example of a discrete valuable is the variable M, which represents the on/off status of a motor, defined by such that M=0 when the motor is off, and M=1 when the motor is on.

(5) Ci and yi may be related to time t; i.e., they are continuous functions of time t. Such a system may be called continuous time system.

(6) Ci and yi may change values due to the occurrence of events, in which case the system is said to be event triggered. For example, suppose that the system is a motor that is controlled by a switch. The status of the motor is denoted by M, where M=1 when the motor is on and M=0 when the motor is off, and the status of the switch is denoted by S, where S=0 when the switch is off and S=1 when the switch is on. If we take the control input to be S and the control output to be M, the control system (the motor and switch) is defined by the equation M=S, so that the motor is on if and only if the switch is on. This is an event-driven system since the input S depends on the event of turning the switch on or off. There are only a finite number of values of the input and output; such a system is referred to as discrete event system.

(7) A control system includes a plant, a controller, sensor, and actuator. It is very important to pay attention to the prefix before word “input” and word “output”. Ci are a process’s or plant’s inputs and are controller’s outputs as well, and yi are process’s or plant’s outputs and are controller’s inputs as well. To a control system, in particular feedback control system, its input is the reference signal, and its output is the process’s or plant’s output.

General Concept of Control 2

ME 460 Notes

Figure 2 General Configuration Diagram for a Control System

6. Examples

Look at Figure 3 which describes a tank system. Figure 4 shows the plant system and controller together. It also shows the control input or controller’s input is the same as plant output or plant’s output.

Figure 3. Tank plant or process

Figure 4. General control configuration for the tank process or plant

General Concept of Control 3

y2

ym

… …

c2

c1 y1

Reference SignalsControl Recipe

…

Plant orProcess

Controller

cm

…

ME 460 Notes

The following variables are defined to represent the dynamics of the tank system:

(1) Status of two pumps: P1, P2:

Domain of (P1,P2): (0,1)

Off On

(2) Positions or status of valve: iv(t), ov(t):

They are continuous variables but they can be re-defined as discrete variables; see below

Domain (IV, OV) = (0,1)

Closed Open

It is noted that when a continuous variable is re-defined into a discrete variable, the original meaning of this variable is partially lost.

(3) Flow (rate and temperature): fin(t), fout(t)

Domain (f) is a continuous variable.

(4) Constraint:

Semantics: when there is no fluid, pump must be off.Expression: IV<ε, P1=0

OV<ε, P2=0

where ε is a very small number; so “IV<ε” and OV<ε mean the no fluid situation.

Pumps are “on” only when there is fluid flowing; the pumps will not burn out in this case.

(5) Control problem 1:

Fill tank 2 to L and keep at L.

(6) Control problem 2:

- Fill Tank to L1

- Heat Liquid to T1

- Keep (L1, T1) for time t1

General Concept of Control 4

ME 460 Notes

After that (t1)

- Fill Tank 2 from L1 to L2

- …- …

The above like a “recipe”. So it is also called “recipe” control or “sequence control”. Such a control problem is also called “trajectory tracking control” problem. The control problem 1 is also called “set point control”.

The previous discussion also showed that we can “convert” a continuous problem into a discrete problem with loss of some meaning of the original problem or some accuracy of a description of the problem.

For example (level of fluid):

Similarly,

IV=0 Value closedIV=1 Value open

Now, the issue of designing a controller (discrete) is to find a “function”, which can tell what should be the value of IV, 0 or 1, given the values of Y, Y=0 or 1.

In the above, Y: process’s or plant’s output; IV: process’s input or plant’s input.

The simplest form of the controller for the tank control problem 1 (Figure 4) can be represented by

General Concept of Control 5

y(t) < Ly(t) L

y=0y=1

Continuous

Controller

Y IV

y(t) < Ly(t) L

y=0y=1

Discrete

ME 460 Notes

Remark 1: In the above example, if pump is on, the value of IV must be open; otherwise, the pump will burn out. These results in the modification of the controller as follows:

If Y=1, then P1=0 ∧ IV=0

In the above, ‘∧’: ‘and’.

Now, we ask if we can treat the tank problem as a continuous problem. The answer is definitely yes. In the following, we present the system of Figure 4 as a continuous control problem.

7. Continuous-Variable Control for the Tank Problem

First, we need to know how the plant or process or plant works, in particular what principle is behind the plant or process. For the process of Figure 4, the principle behind this process is as follows:

Principle: “opening more the valve leads to more liquid in the tank.”

The mathematical representation of such principle knowledge is called “model”; in this case called “plant model” or “process model”. Sometimes, we also use the term “dynamic model of the plant or process” to refer to “plant model” or “process model”.

To the tank system, the model should describe the relationship among y(t) and iv(t), ov(t)

where y: plant’s or process’s output, iv and ov: plant’s or process’s inputs.

Now, we focus on tank 2 (Figure 3), and we can obtain the following equations:

(1)

(2)

(3)

(4)

We further assume:

General Concept of Control 6

If Y=0, then IV=1If Y=1, then IV=0

ME 460 Notes

where, and are just constant, depending on particular types of values; td: time delay of value. Therefore, we have:

(7)

Now, we try to develop a controller which is expected to solve the control problem (let us consider the set point control problem only; see discussion before).

In this case, the controller is represented below.

Control input with respect to the plant output

(8)

where, r(t)=L, reference signal, or desired performance of the plant system.

In the above, we also assume the output value, ov(t) is not interesting. It is noted that there could be different control strategies or methods which are as a result of different expressions as Eq. (8).

The above equation, Eq. (8), is also called “control law”.

The effect of a control law needs to be evaluated. This can be done by combining Eq (7) and Eq. (8) (notice is removed from Eq. (7)). The result is:

(9)

General Concept of Control 7

Controller

L

ov(t)iv(t)

y(t)

ME 460 Notes

One should notice that the item of (y-r) (where r is L for the tank problem) is also called “error”. We take Laplace transform for Eq. (9), we can get:

(10)

(11)

From Eq. (11), we can examine the (y → L) with respect to the time, t. This is really what we want to see about the effects of the control law, namely Eq. (8). Figure 5 shows the curve of y ~ t.

Figure 5. Time Response of the Control System

From Figure 5, we can see how the level of liquid will approach L with respect to time.

In particular, it shows that when the time goes sufficient longer, the y(t) can reach L. This means that the control law will make the whole control system stable.

Several terminologies are introduced further to explore the nature of feedback control. We will call Eq. (7) “plant or process system model”, Eq. (8) “control law”. Integration of “plant system model” and “control law” leads to the controlled plant system, i.e., Eq. (9). In the control text, the controlled plant system is also called “control system”.

From Eq. (9), which describes the behavior of the controlled plant system or control system, we can see that this equation contains elements from both plant or process (i.e., ) and controller (i.e., Kp). It is also seen that a different control law will lead to a different control system. For example, if we assume

General Concept of Control 8

y(t) (meter)

10L

0 120 t (sec)

ME 460 Notes

(12)

Substitute Eq. (12) into Eq. (7) (remove ov(t)) leads to

(13)

(14)

We can get another y(t), and a curve can be drawn for y ~ t. The (y-t) in this case differs from the curve as shown in Figure 5. This implies that different control laws will lead to different control results.

From the above discussion, we can see both plant or process parameters ( ) and controller parameters (Kp) affect the behavior of the controller plant or process system (or control system for short).

Now, the issue is how to determine and to meet the required performance of a control system (notice: its input is a reference signal and its output is a plant’s or process’s output). Determining and is also called “design of a controller”.

As we can see here, the design is an iterative process. It is difficult to get a mapping f such that

f: Requirement → design parameters ( and )

Now if we go back to the curve y-t (Figure 5), we can define a quantity which measures the performance of a control system in some way, that is, (time constant),

and is particularly defined by the equation below

From this equation, we can obtain

(15)

So, Kp ↑, ↓: means the time to reach the described L is shorter if we increase Kp. Ideally, we hope that a shorter τ is achieved. This means that we hope to have a large Kp. However, there is a restraint to the increase of Kp; see the analysis below.

General Concept of Control 9

ME 460 Notes

When Kp ↑ → iv(t) ↑, but iv(t) has a limit; see below,

where, e(t): error. The maximum error is as follows:

So,

(16)

(17)

The above example has demonstrated how to design a controller – P controller in this case according to . It is further demonstrated that a desired has lower bound or limit, i.e.,

. That is to see that one cannot expect τ as small as one desires.

It is very interesting to see that the design of a control system is usually about determining controller’s parameters such as Kp in the case of P controller. However, the above example with its analysis has shown that plant or process parameter, e.g., is also related to the performance of a control system. This has given the evidence for the need of an integrated approach to design a control system, i.e., design of the controller and design of the plant should be done together. The seminal paper on the integrated design and control can be found from the following:

Zhang W.J., Li Q., and S.L. Guo, 1999. Integrated Design of Mechanical Structure and Control Algorithm for a Programmable Four-Bar Linkage, IEEE/ASME Trans. on Mechatronics, 4(4), 354-362.

General Concept of Control 10