Modern Control Engineering OGATA 5th Ed

(Parte 4 de 6)

Outline of the Chapter.Section 2–1 has presented an introduction to the mathematical modeling of dynamic systems.Section 2–2 presents the transfer function and impulse-response function.Section 2–3 introduces automatic control systems and Section 2–4 discusses concepts of modeling in state space.Section 2–5 presents state-space representation of dynamic systems.Section 2–6 discusses transformation of mathematical models with MATLAB.Finally,Section 2–7 discusses linearization of nonlinear mathematical models.

2–2TRANSFER FUNCTION AND IMPULSERESPONSE FUNCTION

In control theory,functions called transfer functions are commonly used to characterize the input-output relationships of components or systems that can be described by linear,time-invariant,differential equations.We begin by defining the transfer function and follow with a derivation of the transfer function of a differential equation system. Then we discuss the impulse-response function.

Transfer Function.The transfer functionof a linear,time-invariant,differential equation system is defined as the ratio of the Laplace transform of the output (response function) to the Laplace transform of the input (driving function) under the assumption that all initial conditions are zero. Consider the linear time-invariant system defined by the following differential equation:

whereyis the output of the system and xis the input.The transfer function of this system is the ratio of the Laplace transformed output to the Laplace transformed input when all initial conditions are zero,or

By using the concept of transfer function,it is possible to represent system dynamics by algebraic equations in s.If the highest power of sin the denominator of the transfer function is equal to n,the system is called an nth-order system.

Comments on Transfer Function.The applicability of the concept of the transfer function is limited to linear,time-invariant,differential equation systems.The transfer function approach,however,is extensively used in the analysis and design of such systems.In what follows,we shall list important comments concerning the transfer function.(Note that a system referred to in the list is one described by a linear,time-invariant, differential equation.)

= Y(s)

Transfer function = G(s) = l[output] l[input] 2 zero initial conditions

16Chapter 2/Mathematical Modeling of Control Systems

1.The transfer function of a system is a mathematical model in that it is an operational method of expressing the differential equation that relates the output variable to the input variable.

2.The transfer function is a property of a system itself,independent of the magnitude and nature of the input or driving function.

3.The transfer function includes the units necessary to relate the input to the output; however,it does not provide any information concerning the physical structure of the system.(The transfer functions of many physically different systems can be identical.)

4.If the transfer function of a system is known,the output or response can be studied for various forms of inputs with a view toward understanding the nature of the system.

5.If the transfer function of a system is unknown,it may be established experimentally by introducing known inputs and studying the output of the system.Once established,a transfer function gives a full description of the dynamic characteristics of the system,as distinct from its physical description.

Convolution Integral.For a linear,time-invariant system the transfer function G(s) is whereX(s)is the Laplace transform of the input to the system and Y(s)is the Laplace transform of the output of the system,where we assume that all initial conditions involved are zero.It follows that the output Y(s)can be written as the product of G(s)and X(s),or

Note that multiplication in the complex domain is equivalent to convolution in the time domain (see Appendix A),so the inverse Laplace transform of Equation (2–1) is given by the following convolution integral:

where both g(t)andx(t)are 0 for t<0.

Impulse-Response Function.Consider the output (response) of a linear timeinvariant system to a unit-impulse input when the initial conditions are zero.Since the Laplace transform of the unit-impulse function is unity,the Laplace transform of the output of the system is

(2–2)Y(s) = G(s) g(t)x(t - t)dt x(t)g(t - t)dt

Y(s) = G(s)X(s)

G(s) = Y(s)

X(s)

Section 2–3/Automatic Control Systems17

The inverse Laplace transform of the output given by Equation (2–2) gives the impulse response of the system.The inverse Laplace transform of G(s),or is called the impulse-response function.This function g(t)is also called the weighting function of the system.

The impulse-response function g(t)is thus the response of a linear time-invariant system to a unit-impulse input when the initial conditions are zero.The Laplace transform of this function gives the transfer function.Therefore,the transfer function and impulse-response function of a linear,time-invariant system contain the same information about the system dynamics.It is hence possible to obtain complete information about the dynamic characteristics of the system by exciting it with an impulse input and measuring the response.(In practice,a pulse input with a very short duration compared with the significant time constants of the system can be considered an impulse.)

2–3AUTOMATIC CONTROL SYSTEMS

A control system may consist of a number of components.To show the functions performed by each component,in control engineering,we commonly use a diagram called the block diagram.This section first explains what a block diagram is.Next,it discusses introductory aspects of automatic control systems,including various control actions.Then,it presents a method for obtaining block diagrams for physical systems,and, finally,discusses techniques to simplify such diagrams.

Block Diagrams.Ablock diagramof a system is a pictorial representation of the functions performed by each component and of the flow of signals.Such a diagram depicts the interrelationships that exist among the various components.Differing from a purely abstract mathematical representation,a block diagram has the advantage of indicating more realistically the signal flows of the actual system.

In a block diagram all system variables are linked to each other through functional blocks.The functionalblock or simply blockis a symbol for the mathematical operation on the input signal to the block that produces the output.The transfer functions of the components are usually entered in the corresponding blocks,which are connected by arrows to indicate the direction of the flow of signals.Note that the signal can pass only in the direction of the arrows.Thus a block diagram of a control system explicitly shows a unilateral property.

Figure2–1 shows an element of the block diagram.The arrowhead pointing toward the block indicates the input,and the arrowhead leading away from the block represents the output.Such arrows are referred to as signals.

Figure 2–1

Element of a block diagram.

18Chapter 2/Mathematical Modeling of Control Systems

Summingpoint Branch point

Figure 2–3 Block diagram of a closed-loop system.

Note that the dimension of the output signal from the block is the dimension of the input signal multiplied by the dimension of the transfer function in the block.

The advantages of the block diagram representation of a system are thatit is easy to form the overall block diagram for the entire system by merely connecting the blocks of the components according to the signal flow and that it is possible to evaluate the contribution of each component to the overall performance of the system.

In general,the functional operation of the system can be visualized more readily by examining the block diagram than by examining the physical system itself.A block diagram contains information concerning dynamic behavior,but it does not include any information on the physical construction of the system.Consequently,many dissimilar and unrelated systems can be represented by the same block diagram.

It should be noted that in a block diagram the main source of energy is not explicitly shown and that the block diagram of a given system is not unique.A number of different block diagrams can be drawn for a system,depending on the point of view of the analysis.

Summing Point.Referring to Figure2–2,a circle with a cross is the symbol that indicates a summing operation.The plus or minus sign at each arrowhead indicates whether that signal is to be added or subtracted.It is important that the quantities being added or subtracted have the same dimensions and the same units.

Branch Point.Abranch pointis a point from which the signal from a block goes concurrently to other blocks or summing points.

Block Diagram of a Closed-Loop System.Figure2–3 shows an example of a block diagram of a closed-loop system.The output C(s)is fed back to the summing point,where it is compared with the reference input R(s).The closed-loop nature of the system is clearly indicated by the figure.The output of the block,C(s)in this case, is obtained by multiplying the transfer function G(s)by the input to the block,E(s).Any linear control system may be represented by a block diagram consisting of blocks,summing points,and branch points.

When the output is fed back to the summing point for comparison with the input,it is necessary to convert the form of the output signal to that of the input signal.For example,in a temperature control system,the output signal is usually the controlled temperature.The output signal,which has the dimension of temperature,must be converted to a force or position or voltage before it can be compared with the input signal. This conversion is accomplished by the feedback element whose transfer function is H(s), as shown in Figure2–4.The role of the feedback element is to modify the output before it is compared with the input.(In most cases the feedback element is a sensor that measures a – b

Figure 2–2 Summing point.

the output of the plant.The output of the sensor is compared with the system input,and the actuating error signal is generated.) In the present example,the feedback signal that is fed back to the summing point for comparison with the input is B(s) =H(s)C(s).

Open-Loop Transfer Function and Feedforward Transfer Function.Referring to Figure 2–4,the ratio of the feedback signal B(s)to the actuating error signal E(s)is called the open-loop transfer function.That is,

The ratio of the output C(s)to the actuating error signal E(s)is called the feedforward transfer function,so that

If the feedback transfer function H(s)is unity,then the open-loop transfer function and the feedforward transfer function are the same.

Closed-Loop Transfer Function.For the system shown in Figure 2–4,the output C(s)and input R(s)are related as follows:since

eliminatingE(s)from these equations gives or (2–3)

The transfer function relating C(s)toR(s)is called the closed-loop transfer function.It relates the closed-loop system dynamics to the dynamics of the feedforward elements and feedback elements. From Equation (2–3),C(s)is given by

C(s) = G(s)

1 + G(s)H(s) R(s)

C(s)

R(s) =

G(s)

C(s) = G(s)CR(s) - H(s)C(s)D

= R(s) - H(s)C(s)

Feedforward transfer function= C(s)

E(s) = G(s)

Open-loop transfer function= B(s)

E(s) = G(s)H(s)

Section 2–3/Automatic Control Systems19

Figure 2–4 Closed-loop system.

20Chapter 2/Mathematical Modeling of Control Systems

C(s) C(s)

(a) (b)

(c)

Figure 2–5 (a) Cascaded system; (b) parallel system; (c) feedback (closedloop) system.

Thus the output of the closed-loop system clearly depends on both the closed-loop transfer function and the nature of the input.

Obtaining Cascaded, Parallel, and Feedback (Closed-Loop) Transfer Functions with MATLAB.In control-systems analysis,we frequently need to calculate the cascaded transfer functions,parallel-connected transfer functions,and feedback-connected (closed-loop) transfer functions.MATLAB has convenient commands to obtain the cascaded,parallel,and feedback (closed-loop) transfer functions.

Suppose that there are two components G1(s)andG2(s)connected differently as shown in Figure2–5 (a),(b),and (c),where

To obtain the transfer functions of the cascaded system,parallel system,or feedback (closed-loop) system,the following commands may be used:

[num, den] = series(num1,den1,num2,den2) [num, den] = parallel(num1,den1,num2,den2)

[num, den] = feedback(num1,den1,num2,den2)

As an example,consider the case where

MATLAB Program 2–1 gives C(s)/R(s)=num!den for each arrangement of G1(s) andG2(s).Note that the command printsys(num,den) displays the num!denCthat is,the transfer function C(s)/R(s)Dof the system considered.

den1

Section 2–3/Automatic Control Systems21

Automatic Controllers.An automatic controller compares the actual value of the plant output with the reference input (desired value),determines the deviation,and produces a control signal that will reduce the deviation to zero or to a small value. The manner in which the automatic controller produces the control signal is called thecontrol action.Figure2–6 is a block diagram of an industrial control system,which

MATLABProgram 2–1 num1 = [10]; den1 = [1 2 10]; num2 = [5]; den2 = [1 5]; [num, den] = series(num1,den1,num2,den2); printsys(num,den) num/den =

[num, den] = parallel(num1,den1,num2,den2); printsys(num,den) num/den =

[num, den] = feedback(num1,den1,num2,den2); printsys(num,den) num/den =

Automatic controller Error detector

Amplifier Actuator Plant Output

Sensor

Reference input

Actuating error signal

Setpoint" #

Figure 2–6 Block diagram of an industrial control system, which consists of an automatic controller, an actuator,a plant, and a sensor (measuring element).

22Chapter 2/Mathematical Modeling of Control Systems consists of an automatic controller,an actuator,a plant,and a sensor (measuring element).The controller detects the actuating error signal,which is usually at a very low power level,and amplifies it to a sufficiently high level.The output of an automatic controller is fed to an actuator,such as an electric motor,a hydraulic motor,or a pneumatic motor or valve.(The actuator is a power device that produces the input to the plant according to the control signal so that the output signal will approach the reference input signal.)

The sensor or measuring element is a device that converts the output variable into another suitable variable,such as a displacement,pressure,voltage,etc.,that can be used to compare the output to the reference input signal.This element is in the feedback path of the closed-loop system.The set point of the controller must be converted to a reference input with the same units as the feedback signal from the sensor or measuring element.

Classifications of Industrial Controllers.Most industrial controllers may be classified according to their control actions as:

1.Two-position or on–off controllers 2. Proportional controllers 3. Integral controllers 4. Proportional-plus-integral controllers 5. Proportional-plus-derivative controllers 6. Proportional-plus-integral-plus-derivative controllers

(Parte 4 de 6)