Modern Control Engineering OGATA 5th Ed

Modern Control Engineering OGATA 5th Ed

(Parte 5 de 6)

Most industrial controllers use electricity or pressurized fluid such as oil or air as power sources.Consequently,controllers may also be classified according to the kind of power employed in the operation,such as pneumatic controllers,hydraulic controllers, or electronic controllers.What kind of controller to use must be decided based on the nature of the plant and the operating conditions,including such considerations as safety, cost, availability, reliability, accuracy, weight, and size.

Two-Position or On–Off Control Action.In a two-position control system,the actuating element has only two fixed positions,which are,in many cases,simply on and off.Two-position or on–off control is relatively simple and inexpensive and,for this reason,is very widely used in both industrial and domestic control systems.

Let the output signal from the controller be u(t)and the actuating error signal be e(t).

In two-position control,the signal u(t)remains at either a maximum or minimum value, depending on whether the actuating error signal is positive or negative,so that whereU1andU2are constants.The minimum value U2is usually either zero or –U1. Two-position controllers are generally electrical devices,and an electric solenoid-oper- ated valve is widely used in such controllers.Pneumatic proportional controllers with very high gains act as two-position controllers and are sometimes called pneumatic twoposition controllers.

Figures2–7(a) and (b) show the block diagrams for two-position or on–off controllers. The range through which the actuating error signal must move before the switching occurs

Section 2–3/Automatic Control Systems23 is called the differential gap.A differential gap is indicated in Figure 2–7(b).Such a differential gap causes the controller output u(t)to maintain its present value until the actuating error signal has moved slightly beyond the zero value.In some cases,the differential gap is a result of unintentional friction and lost motion;however,quite often it is intentionally provided in order to prevent too-frequent operation of the on–off mechanism.

Consider the liquid-level control system shown in Figure2–8(a),where the electromagnetic valve shown in Figure 2–8(b) is used for controlling the inflow rate.This valve is either open or closed.With this two-position control,the water inflow rate is either a positive constant or zero.As shown in Figure2–9,the output signal continuously moves between the two limits required to cause the actuating element to move from one fixed position to the other.Notice that the output curve follows one of two exponential curves,one corresponding to the filling curve and the other to the emptying curve.Such output oscillation between two limits is a typical response characteristic of a system under two-position control.

(a) (b)

U2 ue

Differential gap

Figure 2–7

(a) Block diagram of an on–off controller; (b) block diagram of an on–off controller with differential gap.

115 V Float

R C h

(a) (b) qiMovable iron core Magnetic coil

Figure 2–8 (a) Liquid-level control system; (b) electromagnetic valve.

Differential gap

Figure 2–9 Level h(t)-versus-t curve for the system shown in Figure 2–8(a).

24Chapter 2/Mathematical Modeling of Control Systems

From Figure 2–9,we notice that the amplitude of the output oscillation can be reduced by decreasing the differential gap.The decrease in the differential gap,however,increases the number of on–off switchings per minute and reduces the useful life of the component.The magnitude of the differential gap must be determined from such considerations as the accuracy required and the life of the component.

Proportional Control Action.For a controller with proportional control action, the relationship between the output of the controller u(t)and the actuating error signal e(t) is or,in Laplace-transformed quantities, whereKpis termed the proportional gain. Whatever the actual mechanism may be and whatever the form of the operating power,the proportional controller is essentially an amplifier with an adjustable gain.

Integral Control Action.In a controller with integral control action,the value of the controller output u(t)is changed at a rate proportional to the actuating error signal e(t).That is, whereKiis an adjustable constant.The transfer function of the integral controller is

Proportional-Plus-Integral Control Action.The control action of a proportionalplus-integral controller is defined by u(t) = Kpe(t) + Kp e(t)dt

U(s)

E(s)

= Ki s e(t)dt du(t) dt = Kie(t)

U(s)

E(s) = Kp u(t) = Kpe(t)

Section 2–3/Automatic Control Systems25 or the transfer function of the controller is where is called the integral time.

Proportional-Plus-Derivative Control Action.The control action of a proportionalplus-derivative controller is defined by and the transfer function is where is called the derivative time.

Proportional-Plus-Integral-Plus-Derivative Control Action.The combination of proportional control action,integral control action,and derivative control action is termed proportional-plus-integral-plus-derivative control action.It has the advantages of each of the three individual control actions.The equation of a controller with this combined action is given by or the transfer function is whereKpis the proportional gain,is the integral time,and is the derivative time. The block diagram of a proportional-plus-integral-plus-derivative controller is shown in

TdTi

U(s)

Tis + Tdsb u(t) = Kpe(t) + Kp e(t)dt + KpTd de(t)

U(s)

E(s) = KpA1 + TdsB u(t) = Kpe(t) + KpTd de(t)

U(s)

Tis

Tis Figure 2–10

Block diagram of a proportional-plusintegral-plusderivative controller.

26Chapter 2/Mathematical Modeling of Control Systems

Figure 2–1 Closed-loop system subjected to a disturbance.

Closed-Loop System Subjected to a Disturbance.Figure2–1 shows a closedloop system subjected to a disturbance.When two inputs (the reference input and disturbance) are present in a linear time-invariant system,each input can be treated independently of the other;and the outputs corresponding to each input alone can be added to give the complete output.The way each input is introduced into the system is shown at the summing point by either a plus or minus sign.

Consider the system shown in Figure 2–1.In examining the effect of the disturbanceD(s),we may assume that the reference input is zero;we may then calculate the responseCD(s)to the disturbance only.This response can be found from

On the other hand,in considering the response to the reference input R(s),we may assume that the disturbance is zero.Then the response CR(s)to the reference input R(s) can be obtained from

The response to the simultaneous application of the reference input and disturbance can be obtained by adding the two individual responses.In other words,the response C(s)due to the simultaneous application of the reference input R(s)and disturbance D(s)is given by

Consider now the case where |G1(s)H(s)|"1 and |G1(s)G2(s)H(s)|"1.In this case,the closed-loop transfer function CD(s)/D(s)becomes almost zero,and the effect of the disturbance is suppressed.This is an advantage of the closed-loop system.

On the other hand,the closed-loop transfer function CR(s)/R(s)approaches1/H(s) as the gain of G1(s)G2(s)H(s)increases.This means that if |G1(s)G2(s)H(s)|"1,then the closed-loop transfer function CR(s)/R(s)becomes independent of G1(s)andG2(s) and inversely proportional to H(s),so that the variations of G1(s)andG2(s)do not affect the closed-loop transfer function CR(s)/R(s).This is another advantage of the closed-loop system.It can easily be seen that any closed-loop system with unity feedback,

H(s)=1,tends to equalize the input and output.

CR(s)

CD(s)

Section 2–3/Automatic Control Systems27

Procedures for Drawing a Block Diagram.To draw a block diagram for a system,first write the equations that describe the dynamic behavior of each component. Then take the Laplace transforms of these equations,assuming zero initial conditions, and represent each Laplace-transformed equation individually in block form.Finally,assemble the elements into a complete block diagram.

As an example,consider the RCcircuit shown in Figure2–12(a).The equations for this circuit are

(2–5) The Laplace transforms of Equations (2–4) and (2–5),with zero initial condition,become

Equation (2–6) represents a summing operation,and the corresponding diagram is shown in Figure 2–12(b).Equation (2–7) represents the block as shown in Figure 2–12(c). Assembling these two elements,we obtain the overall block diagram for the system as shown in Figure 2–12(d).

Block Diagram Reduction.It is important to note that blocks can be connected in series only if the output of one block is not affected by the next following block.If there are any loading effects between the components,it is necessary to combine these components into a single block.

Any number of cascaded blocks representing nonloading components can be replaced by a single block,the transfer function of which is simply the product of the individual transfer functions.

Eo(s)= I(s) eo = 1idt C i = ei - eo R

(d)

(c)

Ce oei i

Figure 2–12 (a) RC circuit; (b) block diagram representing Equation (2–6); (c) block diagram representing Equation (2–7); (d) block diagram of the RC circuit.

28Chapter 2/Mathematical Modeling of Control Systems

R G3 C

(b)

(c) (d)

(e)

+ –Figure 2–13

(a) Multiple-loop system; (b)–(e) successive reductions of the block diagram shown in (a).

A complicated block diagram involving many feedback loops can be simplified by a step-by-step rearrangement.Simplification of the block diagram by rearrangements considerably reduces the labor needed for subsequent mathematical analysis.It should be noted,however,that as the block diagram is simplified,the transfer functions in new blocks become more complex because new poles and new zeros are generated.

EXAMPLE 2–1Consider the system shown in Figure2–13(a).Simplify this diagram.

By moving the summing point of the negative feedback loop containing H2outside the positive feedback loop containing H1,we obtain Figure 2–13(b).Eliminating the positive feedback loop, we have Figure 2–13(c).The elimination of the loop containing H2/G1gives Figure 2–13(d).Finally, eliminating the feedback loop results in Figure 2–13(e).

Section 2–4/Modeling in State Space29

Notice that the numerator of the closed-loop transfer function C(s)/R(s)is the product of the transfer functions of the feedforward path.The denominator of C(s)/R(s)is equal to

(The positive feedback loop yields a negative term in the denominator.)

2–4MODELING IN STATE SPACE

In this section we shall present introductory material on state-space analysis of control systems.

Modern Control Theory.The modern trend in engineering systems is toward greater complexity,due mainly to the requirements of complex tasks and good accuracy.Complex systems may have multiple inputs and multiple outputs and may be time varying.Because of the necessity of meeting increasingly stringent requirements on the performance of control systems,the increase in system complexity,and easy access to large scale computers,modern control theory,which is a new approach to the analysis and design of complex control systems,has been developed since around 1960.This new approach is based on the concept of state.The concept of state by itself is not new,since it has been in existence for a long time in the field of classical dynamics and other fields.

Modern Control Theory Versus Conventional Control Theory.Modern control theory is contrasted with conventional control theory in that the former is applicable to multiple-input,multiple-output systems,which may be linear or nonlinear, time invariant or time varying,while the latter is applicable only to linear timeinvariant single-input,single-output systems.Also,modern control theory is essentiallytime-domain approach and frequency domain approach (in certain cases such as H-infinity control),while conventional control theory is a complex frequency-domain approach.Before we proceed further,we must define state,state variables,state vector, and state space.

State.The state of a dynamic system is the smallest set of variables (called state variables) such that knowledge of these variables at t=t0,together with knowledge of the input for t!t0,completely determines the behavior of the system for any time t ! t0. Note that the concept of state is by no means limited to physical systems.It is appli- cable to biological systems,economic systems,social systems,and others.

State Variables.The state variables of a dynamic system are the variables making up the smallest set of variables that determine the state of the dynamic system.If at

30Chapter 2/Mathematical Modeling of Control Systems leastnvariablesx1,x2,p,xnare needed to completely describe the behavior of a dy- namic system (so that once the input is given for t!t0and the initial state at t=t0is specified,the future state of the system is completely determined),then such n variables are a set of state variables.

Note that state variables need not be physically measurable or observable quantities.

Variables that do not represent physical quantities and those that are neither measurable nor observable can be chosen as state variables.Such freedom in choosing state variables is an advantage of the state-space methods.Practically,however,it is convenient to choose easily measurable quantities for the state variables,if this is possible at all,because optimal control laws will require the feedback of all state variables with suitable weighting.

State Vector.Ifnstate variables are needed to completely describe the behavior of a given system,then these nstate variables can be considered the ncomponents of a vectorx.Such a vector is called a state vector.A state vector is thus a vector that deter- mines uniquely the system state x(t)for any time t!t0,once the state at t=t0is given and the input u(t)fort!t0is specified.

State Space.The n-dimensional space whose coordinate axes consist of the x1

State-Space Equations.In state-space analysis we are concerned with three types of variables that are involved in the modeling of dynamic systems:input variables,output variables,and state variables.As we shall see in Section 2–5,the state-space representation for a given system is not unique,except that the number of state variables is the same for any of the different state-space representations of the same system. The dynamic system must involve elements that memorize the values of the input for t!t1.Since integrators in a continuous-time control system serve as memory devices, the outputs of such integrators can be considered as the variables that define the inter- nal state of the dynamic system.Thus the outputs of integrators serve as state variables. The number of state variables to completely define the dynamics of the system is equal to the number of integrators involved in the system. Assume that a multiple-input,multiple-output system involves nintegrators.Assume also that there are rinputsu1(t),u2(t),p,ur(t)andmoutputsy1(t),y2(t),p,ym(t).

Definenoutputs of the integrators as state variables:x1(t),x2(t),p,xn(t)Then the system may be described by

(Parte 5 de 6)

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