Basic Properties of Feedback


 Augustus Byrd
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1 4 Basic Properties of Feedback A Perspective on the Properties of Feedback A major goal of control design is to use the tools available to keep the error small for any input and in the face of expected parameter changes. Although in this book we will focus on the selection of the controller transfer function, the control engineer must be aware that changes to the plant may be possible that will greatly help control of the process. It is also the case that the selection and location of a sensor can be very important. These considerations illustrate the fact that control is a collaborative enterprise and control objectives need to be considered at every step of the way from concept to finished product. However, in this book, we consider mainly the case of control of dynamic processes and begin with models that can be approximated as linear, timeinvariant, and described by transfer functions. Discussion of the theoretical justification of this assumption is deferred until Chapter 9, in which the theories of Lyapunov are introduced. Given a model, the next step in the design is formulation of specifications of what it is that the control is required to do. While maintaining the essential property of stability, the control specifications include both static and dynamic requirements such as the following: 66 The permissible steadystate error in the presence of a constant or bias disturbance signal. The permissible steadystate error while tracking a polynomial reference signal such as a step or a ramp. The sensitivity of the system transfer function to changes in model parameters.
2 The permissible transient error in response to a step in either the reference or the disturbance input. Openloop and closedloop control The two fundamental structures for realizing controls are openloop control (Fig. 4.) and closedloop control, also known as feedback control (Fig. 4.3). Openloop control is generally simpler, does not require a sensor to measure the output, and does not, of itself, introduce stability problems. Feedback control is more complex and may cause stability problems but also has the potential to give much better performance than is possible with openloop control. If the process is naturally (openloop) unstable, feedback control is the only possibility to obtain a stable system and meet any performance specifications at all. Before specific design techniques such as the root locus are described, it is useful to develop the equations of systems in general terms and to derive expressions for the several specifications in order to have a language describing the objectives toward which the design is directed. As part of this activity, a comparison of openloop to closedloop control will expose both the advantages and the challenges of feedback control. Chapter Overview This chapter begins with consideration of the basic equations of feedback and the comparison of a feedback structure with openloop control. In Section 4. the equations are presented first in general form and used to discuss the effects of feedback on disturbance rejection, parameter sensitivity, and command tracking. In Section 4.2 the steadystate errors in response to polynomial inputs are analyzed in more detail. As part of the language of steadystate performance, control systems are frequently classified by type according to the maximum degree of the input polynomial for which the steadystate error is a finite constant. In Section 4.3 the issue of dynamic tracking errors is introduced by considering a modification of the closedloop characteristic equation using a classical structure of proportional, integral, and differential control, the PID controller. This study will illustrate the interaction of steadystate with transient performance and will set the tone for the more sophisticated design techniques to be described in later chapters. Finally, in Section 4.4, several extensions of the material of the chapter are presented that are interesting and important, but something of a distraction from the main issues of the chapter. Issues discussed there are digital controllers, tuning PID controllers, Truxel s formula for error constants, and timedomain sensitivity. The most important of these is the implementation of controllers in digital form, introduced in Section If time permits, consideration of this section is highly recommended because almost all modern controllers are realized by digital logic. A more complete discussion of this important issue is given in Chapter 8. 67
3 68 Chapter 4 Basic Properties of Feedback Figure 4. Openloop control system W Input shaping H r Controller D ol U Plant G y 4. The Basic Equations of Control We begin by collecting the basic equations and transfer functions that will be used throughout the rest of the text. For the openloop system of Fig. 4., if we take the disturbance to be at the input of the plant, the output is given by ol = H r D ol G + GW (4.) and the error, the difference between reference input and system output, is given by E ol = ol (4.2) = [H r D ol G + GW] (4.3) = [ H r D ol G] GW (4.4) = [ T ol ] GW. (4.5) The openloop transfer function in this case is H r D ol G, for which we will use the generic notation T ol (s). For feedback control, Figure 4.2 gives the basic structure of interest, but with the disturbance and the sensor noise entering in unspecific ways. We will take these signals to be at the inputs of the process and the sensor, respectively, as shown in Figure 4.3. The sensor transfer function is H y and may show important dynamics. However, the sensor can often be selected to be fast and accurate. If this is the case, its transfer function can be taken to be a constant H y, with units of volts/unitofoutput. The reference input r has the same units Figure 4.2 Feedback control system W Input shaping H r Controller D cl Plant G Sensor H y V
4 Section 4. The Basic Equations of Control 69 Figure 4.3 Basic feedback control block diagram Input shaping H r Controller D cl u W Plant G Sensor H y V as the output, of course, and the input filter s transfer function is H r, also with units of volts/unitofoutput. An equivalent block diagram is drawn in Fig. 4.4, with controller transfer function D(s) = H r D cl and with the feedback transfer function as the ratio H = H y H r. It is standard practice, especially if H y is constant, to select equal scale factors so that H r = H y and the block diagram can be drawn as a unity feedback structure as shown in Figure 4.5. We will develop the equations and transfer functions for this standard structure. When we use these equations later, it will be important to be sure that the preceding assumptions actually apply. If the sensor has dynamics that cannot be ignored, for example, then the equations will need to be modified accordingly. For the feedback block diagram of Figure 4.5, the equations for the output and the control are cl = DG + DG + G + DG W DG V, + DG (4.6) D U = + DG DG + DG W D V. + DG (4.7) Perhaps more important than these is the equation of the error, E cl = cl : [ DG E cl = + DG + G + DG W DG ] + DG V (4.8) = + DG G + DG W + DG V. (4.9) + DG Figure 4.4 Equivalent feedback block diagram with H r included inside the loop H r D cl D W G H y H r H V
5 70 Chapter 4 Basic Properties of Feedback Figure 4.5 Unity feedback system when H r = H y and letting D = H r D cl Controller D u W Plant G V This equation is simplified by the definition of the sensitivity function the complementary sensitivity function T as and and = + DG T = = (4.0) DG + DG. (4.) In terms of these definitions, the equation for the closedloop error is E cl = GW + TV. (4.2) For future reference, it is standard to define the transfer function around a loop as the loop gain, L(s). In the case of Fig. 4.4, we have L = DGH, for example. 4.. Watt s Problem of Disturbance ejection One of the early uses of the steam engine in Britain was in mining, to pump water out of mines and to haul wagons loaded with coal. In carrying out these tasks, the steadystate speed of early engines would change substantially when presented with added torque caused by a new load. To correct the problem, Watt s company introduced the flying ball governor shown in Fig.., whereby the speed of the engine was fed back to the steam chest to change the torque of the engine. We will illustrate the principles of operation of this feedback innovation through study of the simple equations of motion of an engine with speed ω e and external load torque τ l. Equation (4.3) describes the dynamics of an engine with inertia J, viscous friction b, control u, and load torque τ l (t): J ω e + bω e = A u + A 2 τ l. (4.3) The reason for the name, coined by H. W. Bode, will be given shortly.
6 Section 4. The Basic Equations of Control 7 If we take the Laplace transform of Eq. (4.3), let the velocity transform be e (s) and the transform of the load torque be T l (s), we obtain the transformed equations of openloop speed control as sj e (s) + b e (s) = A U(s)+ A 2 T l (s), (4.4) sj e (s) + b e (s) = A [U(s)+ A 2 A T l (s)], (4.5) sτ e (s) + e (s) = A[U + W]. (4.6) In deriving Eq. (4.6), we have defined the parameters τ = J/b,A = A /b, and the disturbance variable to be W = A 2 A T l. In transfer function form the equation is e (s) = A (τs + ) U(s)+ A W(s) (4.7) (τs + ) = G(s)[U(s)+ W(s)] (4.8) = G(s)W(s) if U(s) = 0. (4.9) In the feedback case, with no reference input and with control proportional to error as U = K cl e, the equations of proportional feedback control are sτ e (s) + e (s) = A[ K cl e + W ], (4.20) e (s) = G(s)K cl e (s) + G(s)W, (4.2) [ + GK cl ] e (s) = GW, (4.22) e (s) = G + GK cl W. (4.23) In the openloop case, if the control input is U(s) = 0 and W = w o s, the final value theorem gives 2 ω ss = G(0)w o = Aw o. (4.24) To make the comparison with the closedloop case, suppose that G(0) =, w o =, and just for fun, we take the controller gain to be K cl = 99. The steadystate output in the openloop case is ω ss =, and in the closedloop case it is s 0 ω ss = +GK = cl +99 = 0.0. Thus the feedback system will have an error to disturbance that is 00 times smaller than in the openloop case. No wonder Watt s engine was a success! 2 We assume for the moment that G(0) is finite.
7 72 Chapter 4 Basic Properties of Feedback This result is a particular case of application of the error equations. From Eq. (4.4) the error in the openloop case is E ol = GW, and from Eq. (4.2) in the feedback case the error is E cl = GW = E ol.thus, in every case, the error due to disturbances is smaller by a factor in the closedloop case compared with the openloop case. Major advantage of feedback System errors to constant disturbances can be made smaller with feedback than they are in openloop systems by a factor of =, where DG(0) +DG(0) is the loop gain at s = Black s Problem: Sensitivity of System Gain to Parameter Changes During the 920s, H. S. Black was working at Bell Laboratories to find a design for an electronic amplifier suitable for use as a repeater on the long lines of the telephone company. The basic problem was that electronic components drifted and he needed a design that maintained a gain with great precision in the face of these drifts. His solution was a feedback amplifier. To illustrate the advantages he found, we compare the sensitivity of openloop control with that of closedloop control when a parameter changes. The change might come about because of external effects such as temperature changes, because of aging, or simply from an error in the value used for the parameter from the start. Suppose that the plant gain in operation differs from its original design value of A to be A + δa, which represents a fractional change of δa A. The openloop controller gain is taken to be fixed at D ol (0) = K ol. In the openloop case the nominal overall gain is T ol = K ol A, 3 and the perturbed gain would be T ol + δt ol = K ol (A + δa) = K ol A + K ol δa = T ol + K ol δa. Sensitivity Thus, δt ol = K ol δa. To give a fair comparison, we compute the fractional change in T ol, defined as δt ol /T ol for a given fractional change in A. Substituting the values, we find that δt ol T ol = K olδa K ol A = δa A. (4.25) This means that a 0% error in A would yield a 0% error in T ol. H. W. Bode called the ratio of δt/t to δa/a the sensitivity of the gain with respect to the parameter A. In the openloop case, therefore, =. The same change in A in the feedback case (Eq. (4.23)) yields the new steadystate feedback gain T cl + δt cl = (A + δa)k cl + (A + δa)k cl, 3 We use Tol and T cl for the openloop and closedloop transfer functions, respectively. These are not to be confused with the transform of the disturbance torque T ol used earlier.
8 Section 4. The Basic Equations of Control 73 where T cl is the closedloop gain. We can compute the sensitivity of this closedloop gain directly using differential calculus. The closedloop steadystate gain is T cl = AK cl + AK cl. The firstorder variation is proportional to the derivative and is given by δt cl = dt cl da δa. The general expression for sensitivity of a transfer function T to a parameter A is thus given by ( ) δt cl A dt cl δa = T cl T cl da A = (sensitivity) δa A. From this formula the sensitivity is seen to be so T cl A = sensitivity of T cl with respect to A = A T cl dt cl da, T cl A = A ( + AK cl )K cl K cl (AK cl ) AK cl /( + AK cl ) ( + AK cl ) 2 = + AK cl. (4.26) This result, which explains our use of the name sensitivity earlier, exhibits another major advantage of feedback: Advantage of feedback In feedback control, the error in the overall transfer function gain is less sensitive to variations in the plant gain by a factor of = compared with +DG errors in openloop control. As with the case of disturbance rejection, if the gain is such that + DG = 00, a 0% change in plant gain A will cause only a 0.% change in the steadystate gain. The openloop controller is 00 times more sensitive to gain changes than the closedloop system with loop gain of 00.
9 74 Chapter 4 Basic Properties of Feedback The results in this section so far have been computed for the steadystate error in the presence of constant inputs, either reference or disturbance. Very similar results can be obtained for the steadystate behavior in the presence of sinusoidal reference and disturbance signals. This is important because there are times when such signals naturally occur, as with a disturbance of 60 Hz due to powerline interference in an electronic system, for example. The concept is also important because more complex signals can be described as containing sinusoidal components over a band of frequencies and analyzed using superposition of one frequency at a time. For example, it is well known that human hearing is restricted to signals in the frequency range of about 60 to 5,000 Hz. A feedback amplifier and loudspeaker system designed for highfidelity sound must accurately track any sinusoidal (pure tone) signal in this range. If we take the controller in the feedback system shown in Fig. 4.5 to have the transfer function D(s) and we take the process to have the transfer function G(s), then the steadystate openloop gain at the sinusoidal signal of frequency ω o will be D(jω o )G(jω o ) and the error of the feedback system will be E(jω o ) = (jω o ) + D(jω o )G(jω o ). Thus, to reduce errors to % of the input at the frequency ω o, we must make +DG 00 or D(jω o )G(jω o ) > 00, and a good audio amplifier must have this loop gain over the range 2π60 ω 2π5,000. We will revisit this concept in Chapter 6 as part of the design based on frequency response techniques The Conflict with Sensor Noise Finally, it must be noticed that the feedback system error has a term that is missing from the openloop case. This is due to the sensor, which is not needed in the openloop case. The error due to this term is E cl = TV and will be small if T is small. Unfortunately, keeping both error due to W and error due to V small requires that in the one case be small and in the other case T be small. However, Eq. (4.) shows that this is not possible. The standard solution to this dilemma is frequency separation. The reference and the disturbance energies are typically concentrated in a band of frequencies below some limit let s call it ω c. On the other hand, the sensor can usually be carefully designed so that the sensor noise V is held small in the lowfrequency band below ω c, where the energy in and W are substantial. 4 Thus the design should have small where and W are large and where V is small, and should then make T be small (and necessarily larger) for higher frequencies, where sensor noise is unavoidable. It is compromises such as this that will occupy most of our attention in the design of controllers in later chapters. 4 The moral of this is that money spent on a good sensor is usually money well spent.
10 Section 4. The Basic Equations of Control The ADA Problem: Tracking a Time Varying eference In addition to rejecting disturbances, many systems are required to track a moving reference, for which the generic problem is that of a tracking ADA. In a typical system, electric pulses are sent from a parabolic antenna, the echoes from the target airplane are received, and an error between the axis of the antenna and the vector pointing to the target is computed. The control is required to command the antenna pointing angles in such a way as to keep these vectors aligned. The dynamics of the system are of central importance. A constantgain openloop controller has no effect on the dynamics of the system for either reference or disturbance inputs. Only if an openloop controller includes a dynamic input filter, H r (s), can the dynamic response to the reference signal be changed, but the plant dynamics will still determine the system s response to disturbances. On the other hand, feedback of any kind changes the dynamics of the system for both reference and disturbance inputs. In the case of openloop speed control, Eq. (4.7) shows that the plant dynamics are described by the (openloop) time constant τ. The dynamics with proportional feedback control are described by Eq. (4.23), and the characteristic equation of this system is + GK cl = 0, (4.27) + AK cl = 0, (4.28) τs + τs + + AK cl = 0, (4.29) s = + AK cl. (4.30) τ Therefore, the closedloop time constant, a function of the feedback gain K cl, is given by τ cl = τ, and is decreased as compared with the openloop + AK cl value. It is typically the case that closedloop systems have a faster response as the feedback gain is increased and, if there were no other effects, this is generally desirable. As we will see, however, the responses of higher order systems typically become less well damped and eventually will become unstable as the gain is steadily increased. Thus a definite limit exists on how large we can make the gain in our efforts to reduce the effects of disturbances and the sensitivity to changes in plant parameters. Attempts to resolve the conflict between small steadystate errors and good dynamic response will characterize a large fraction of control design problems. The conclusion is as follows: Property of feedback Feedback changes dynamic response and often makes a system both faster and less stable.
11 76 Chapter 4 Basic Properties of Feedback 4.2 Control of SteadyState Error: System Type In the speedcontrol case study in Section 4. we assumed both reference and disturbances to be constants and also took D(0) and G(0) to be finite constants. In this section we will consider the possibility that either or both of D(s) and G(s) have poles at s = 0. For example, a wellknown structure for the control equation of the form u(t) = k p + k I t e(τ) dτ + k D de(t) dt (4.3) is called proportional plus integral plus derivative (PID) control, and the corresponding transfer function is D(s) = k p + k I s + k Ds. (4.32) Definition of system type Figure 4.6 Signal for satellite tracking In a number of important cases, the reference input will not be constant but can be approximated as a polynomial in time long enough for the system to effectively reach steadystate. For example, when an antenna is tracking the elevation angle to a satellite, the time history as the satellite approaches overhead is an Sshaped curve as sketched in Fig This signal may be approximated by a linear function of time (called a ramp function or velocity input) for a significant time relative to the speed of response of the servomechanism. In the position control of an elevator, a ramp function reference input will direct the elevator to move with constant speed until it comes near the next floor. In rare cases, the input can be approximated over a substantial period as having a constant acceleration. In this section we consider steadystate errors in stable systems with such polynomial inputs. The general method is to represent the input as a polynomial in time and to consider the resulting steadystate tracking errors for polynomials of different degrees. As we will see, the error will be zero for input polynomials below a certain degree, and will be unbounded for inputs of higher degrees. A stable system can be classified as a system type, defined to be the degree of the polynomial for which the steadystate system error is a nonzero finite constant. In the speedcontrol example, proportional control was used and the system had a constant finite error to a step input, which is an input polynomial of zero degree; therefore this system is called a type zero (type 0) system. If the error to a ramp or firstdegree polynomial is a finite nonzero constant, such a system is called type one (type ), and so on. System types can be defined with regard to either reference inputs or disturbance inputs, and in this section we will consider both u s Time (sec)
12 Section 4.2 Control of SteadyState Error: System Type 77 classifications. Determining the system type involves calculating the transform of the system error and then applying the Final Value Theorem. As we will see, a determination of system type is easiest for the case of unity feedback, so we will begin with that case System Type for eference Tracking: The Unity Feedback Case In the unity feedback case drawn in Fig. 4.5, the system error is given by Eq. (4.9). If we consider only the reference input alone and set W = V = 0, then, using the symbol for loop gain, the equation is simply E = =. (4.33) + L To consider polynomial inputs, we let r(t) = t k (t), for which the transform is = sk+. As a generic reference nomenclature, step inputs for which k = 0 are called position inputs, ramp inputs for which k = are called velocity inputs, and if k = 2, the inputs are called acceleration inputs, regardless of the units of the actual signals. Application of the Final Value Theorem to the error gives the formula lim t e(t) = e ss = lim s 0 E(s) (4.34) = lim s 0 s = lim s 0 s (s) (4.35) + L. (4.36) + L k+ s We consider first a system for which L has no pole at the origin and a step input for which (s) = s. In this case, Eq. (4.36) reduces to e ss = lim = s 0 s + L s (4.37) + L(0). (4.38) We define such a system to be type 0 and we define the constant L(0) = K p as the position error constant. If L has one pole at the origin, we could consider both step and ramp inputs, but it is quite straightforward to evaluate Eq. (4.36) in a general setting. For this case, it is useful to be able to describe the behavior of the controller and plant as s approaches 0. For this purpose, we collect all the terms except the pole(s) at the origin into a function L(s), which is thus
13 78 Chapter 4 Basic Properties of Feedback finite at s = 0 so that we can define the constant L o (0) = K n and write the loop transfer function as L(s) = L o(s) s n. (4.39) For example, if L has no integrator, then n = 0. If the system has one integrator, then n =, and so forth. Substituting this expression into Eq. (4.36), we have e ss = lim s 0 s + L o(s) s n s k+ (4.40) s n = lim s 0 s n + K n s k. (4.4) From this equation we can see at once that if n>k, then e = 0, and if n<k, then e.ifn = k = 0, then e ss = +K, and if n = k 0, then e ss = 0 K n.if n = k = 0, the input is a zerodegree polynomial otherwise known as a step or position, the constant K o is called the position constant, written as K p, and the system is classified as type 0, as we saw before. If n = k =, the input is a firstdegree polynomial, otherwise known as a ramp or velocity, the constant K is called the velocity constant, written as K v, and the system is classified type. In a similar way, systems of type 2 and higher types may be defined. The type information can be usefully gathered in a table of errors as follows: TABLE 4. Errors as a Function of System Type Input Type Step (Position) amp (Velocity) Parabola (Acceleration) Type 0 + K p Type 0 K v Type K a Type 2 systems The most common case is that of simple integral control leading to a type system. In this case, the relationship between K v and the steadystate error to a ramp input is shown in Fig Looking back at the expression given for D c G in Eq. (4.39), we can readily see that the several error constants can be calculated by counting the degree n of the poles of L at the origin (the number of integrators in the loop with unity gain feedback) and applying the appropriate one of the following simple formulas K p = lim s 0 L(s), n = 0, (4.42) K v = lim s 0 sl(s), n =, (4.43) K a = lim s 0 s 2 L(s), n = 2. (4.44)
14 Figure 4.7 elationship between ramp response and K v r, y Section 4.2 Control of SteadyState Error: System Type e ss 5 K r y Time (sec) EXAMPLE 4. System Type for Speed Control Determine the system type and the relevant error constant for the speedcontrol example shown in Fig. 4.4, with proportional feedback given by D(s) = k p. The plant transfer function is G = A τs+. Solution. In this case, L = kpa, and applying Eq. (4.42), we see that n = 0, as there is τs+ no pole at s = 0. Thus the system is type 0, and the error constant is a position constant given by K p = k p A. EXAMPLE 4.2 System Type Using Integral Control Determine the system type and the relevant error constant for the speedcontrol example shown in Fig. 4.4, with PI feedback. The plant transfer function is G = A, and in τs+ this case the controller transfer function is D c = k p + k I s. Solution. In this case, the transfer function is L(s) = A(kps+k I ), and as a unity feedback s(τs+) system with a single pole at s = 0, the system is immediately seen as type. The velocity constant is given by Eq. (4.43) to be K v = lim sl(s) = Ak I. s 0 obustness of system type The definition of system type helps us to identify quickly the ability of a system to track polynomials. In the unity feedback structure, if the process parameters change without removing the pole at the origin in a type system, the velocity constant will change, but the system will still have zero steadystate error in response to a constant input and will still be type. Similar statements can be made for systems of type 2 or higher. Thus, we can say that system type is a robust property with respect to parameter changes in the unity feedback structure. obustness is the major reason for preferring unity feedback over other kinds of control structure.
15 80 Chapter 4 Basic Properties of Feedback Figure 4.8 Block diagram reduction to an equivalent unity feedback system E D H U G System Type for eference Tracking: The General Case If the feedback H = H y H r in Fig. 4.4 is different from unity, the formulas given in the unity feedback case do not apply, and a more general approach is needed. There are two immediate possibilities. In the first instance, if one adds and subtracts.0 from H, as shown by block diagram manipulation in Fig. 4.8, the general case is reduced to the unity feedback case and the formulas can be applied to the redefined loop transfer function L =, for which the DG +(H )DG error equation is again E = +L =. Another possibility is to develop formulas directly in terms of the closedloop transfer function, which we call the complementary sensitivity function T(s). From Fig. 4.4, the transfer function is and therefore the error is (s) (s) = T(s) = DG + HDG, (4.45) E(s) = (s) (s) = (s) T(s)(s). The referencetoerror transfer function is thus and the system error transform is E(s) (s) = T(s), E(s) = [ T(s)](s) =. We assume that the conditions of the Final Value Theorem are satisfied, namely that all poles of se(s) are in the left half plane. In that case the steadystate error is given by applying the Final Value Theorem to get e ss = lim t e(t) = lim s 0 se(s) = lim s 0 s[ T(s)](s). (4.46)
16 Section 4.2 Control of SteadyState Error: System Type 8 With a polynomial test input, the error transform becomes E(s) = [ T(s)], sk+ and the steadystate error is given again by the Final Value Theorem: e ss = lim s T(s) T(s) s 0 s k+ = lim s 0 s k. (4.47) The result of evaluating the limit in Eq. (4.47) can be zero, a nonzero constant, or infinite. If the solution to Eq. (4.47) is a nonzero constant, the system is referred to as type k. For example, if k = 0 and the solution to Eq. (4.47) is a nonzero constant equal, by definition, to +K p, then the system is type 0. Similarly, if k = and the solution to Eq. (4.47) is a nonzero constant, then the system is type and has a zero steadystate error to a position input and a constant steadystate error equal, by definition, to/k v to a unit velocity reference input. Type systems are by far the most common in practice. A system of type or higher has a closedloop DC gain of.0, which means that T(0) =. EXAMPLE 4.3 System Type for a Servo with Tachometer Feedback Consider an electric motor positioncontrol problem, including a nonunity feedback system caused by having a tachometer fixed to the motor shaft and its voltage (which is proportional to shaft speed) is fed back as part of the control. The parameters corresponding to Fig. 4.4 are G(s) = s(τs + ), D(s) = k p, H(s) = + k t s. Determine the system type and relevant error constant with respect to reference inputs. Solution. The system error is E(s) = (s) (s) = (s) T(s)(s) DG(s) = (s) + H DG(s) (s) + (H (s) )DG(s) = (s). + H DG(s)
17 82 Chapter 4 Basic Properties of Feedback The steadystate system error from Eq. (4.47) is e ss = lim s 0 s(s)[ T(s)]. For a polynomial reference input, (s) = /s k+, and hence [ T(s)] s(τs + ) + ( + k t s )k p e ss = lim s 0 s k = lim s 0 s k s(τs + ) + ( + k t s)k p = 0, k = 0, = + k tk p k p, k =. Therefore the system is type and the velocity constant is K v =. Notice that + k t k p if k t > 0, this velocity constant is smaller than the unity feedback value of k p.the conclusion is that if tachometer feedback is used to improve dynamic response, the steadystate error is increased. k p System Type with espect to Disturbance Inputs In most control systems, disturbances of one type or another exist. In practice, these disturbances can sometimes be usefully approximated by polynomial time functions such as steps or ramps. This would suggest that systems also be classified with respect to the system s ability to reject disturbance inputs in a way analogous to the classification scheme based on reference inputs. System type with regard to disturbance inputs specifies the degree of the polynomial expressing those input disturbances that the system can reject in the steady state. Knowing the system type, we know the qualitative steadystate response of the system to polynomial disturbance inputs such as step or ramp signals. Because type depends on the transfer function from disturbance to error, the system type depends on exactly where the disturbance enters into the control system. The transfer function from the disturbance input W(s) to the error E(s) is E(s) W(s) = (s) W(s) = T w(s), (4.48) because, if the reference is equal to zero, the output is the error. In a similar way as for reference inputs, the system is type 0 if a step disturbance input results in a nonzero constant steadystate error and is type if a ramp disturbance input results in a steadystate value of the error that is a nonzero constant. In general, following the same approach used in developing Eq. (4.4), we assume that a constant n and a function T o,w (s) can be defined with the properties
18 that T o,w (0) = K n,w written as Section 4.2 Control of SteadyState Error: System Type 83 and that the disturbancetoerror transfer function can be T w (s) = s n T o,w (s). (4.49) Then the steadystate error to a disturbance input that is a polynomial of degree k is [ ] y ss = lim st w (s) s 0 s k+ = lim [T o,w (s) sn s 0 s k ]. (4.50) From Eq. (4.50), if n>k, then the error is zero, and if n<k, the error is unbounded. If n = k, the system is type k and the error is given by K n,w. EXAMPLE 4.4 Satellite Attitude Control Consider the model of a satellite attitude control system shown in Fig. 4.9(a), where J = moment of inertia, W = disturbance torque, H y = sensor gain, and D c (s) = the compensator. With equal input filter and sensor scale factors, the system with PD control can be redrawn with unity feedback as in Fig. 4.9(b) and with PID control drawn as in Fig. 4.9(c). Assume that the control results in a stable system and determine the system types and error responses to disturbances of the control system for (a) System Fig. (4.9)(b) PD control (b) System Fig. (4.9)(c) PID control Solution. (a) We see from inspection of Fig. 4.9(b) that, with two poles at the origin in the plant, the system is type 2 with respect to reference inputs. The transfer function from disturbance to error is T w (s) = Js 2 + k D s + k p (4.5) = T o,w (s), (4.52)
19 84 Chapter 4 Basic Properties of Feedback Figure 4.9 Model of a satellite attitude control: (a) basic system; (b) PD control; (c) PID control K D(s) U W Js u s u K (a) W k p k D s Js 2.0 (b) W k p k I s k D s Js 2.0 (c) for which n = 0 and K o,w = k p. The system is type 0 and the error constant is k p, so the error to a unit disturbance step is k p. (b) With PID control, the forward gain has three poles at the origin, so this system is type 3 for reference inputs, but the disturbance transfer function is T w (s) = T o,w (s) = s Js 3 + k D s 2 + k p s + k I, (4.53) n =, (4.54) Js 3 + k D s 2 + k p s + k I, (4.55) from which it follows that the system is type and the error constant is k I,sothe error to a disturbance ramp of unit slope will be k I.
20 Section 4.2 Control of SteadyState Error: System Type 85 EXAMPLE 4.5 System Type for a DC Motor Position Control Consider the simplified model of a DC motor in unity feedback as shown in Fig. 4.0, where the disturbance torque is labeled W(s). (a) Use the proportional controller D(s) = k p, (4.56) and determine the system type and steadystate error properties with respect to disturbance inputs. (b) Let the control be PI, as given by D(s) = k p + k I s, (4.57) and determine the system type and the steadystate error properties for disturbance inputs. Solution. (a) The closedloop transfer function from W to E (where = 0) is B T w (s) = s(τs + ) + Ak p = s 0 T o,w, n = 0, K o,w = Ak p B. Applying Eq. (4.50), we see that the system is type 0 and the steadystate error to a unit step torque input is e ss = B Ak p. From the earlier section, this system is seen to be type for reference inputs and illustrates that system type can be different for different inputs to the same system. Figure 4.0 DC motor with unity feedback B A W(s) D(s) A s(ts ).0
21 86 Chapter 4 Basic Properties of Feedback (b) If the controller is PI, the disturbance error transfer function is T w (s) = Bs s 2 (τs + ) + (k p s + k I )A, (4.58) n =, (4.59) K n,w = Ak I B, (4.60) and therefore the system is type and the error to a unit ramp disturbance input will be e ss = B Ak I. (4.6) 4.3 Control of Dynamic Error: PID Control The PID (proportionalintegralderivative) controller We have seen in Section 4. basic properties of feedback control, and in Section 4.2 we examined the steady state response of systems to polynomial reference and disturbance input. At the end of Section 4. we observed that proportional control changed the time constant of the simple speedcontrol system. In this section the impact of more sophisticated controls on system characteristic equations is examined in the context of a standard controller structure. The most basic feedback is a constant Proportional to error. As we saw in Section 4.2, addition of a term proportional to the Integral of error has a major influence on the system type and steadystate error to polynomials. The final term in the classical structure term proportional to the Derivative of error. Combined, these three terms form the classical PID controller, which is widely used in the process and robotics industries Proportional Control (P) When the feedback control signal is linearly proportional to the system error, we call the result Proportional feedback. This was the case for the feedback used in the controller of speed in Section 4., for which the controller transfer function is U(s) E(s) = D c(s) = k p. (4.62) As we saw in Section 4..4, the time constant of the feedback system was reduced by a factor + Ak p by proportional control. If the plant is second order, as, for example, is a DC motor with nonnegligible inductance, then the transfer function can be written as G(s) = A s 2 + a s + a 2. (4.63)
22 Section 4.3 Control of Dynamic Error: PID Control 87 In this case, the characteristic equation with proportional control is + k p G(s) = 0, (4.64) s 2 + a s + a 2 + k p = 0. (4.65) The designer can control the constant term and the natural frequency, but not the damping of this equation. If k p is made large to get adequate steadystate error, the damping may be much too low for satisfactory transient response. Proportional plus Integral control Proportional plus Integral Control (PI) Adding an integral term to the controller results in the Proportional plus Integral (PI) control equation u(t) = k p e + k I t t 0 e(τ) dτ, (4.66) for which the D c (s) in Fig. 4.5 becomes U(s) E(s) = D c(s) = k p + k I s. (4.67) This feedback has the primary virtue that, in the steadystate, its control output can be a nonzero constant value even when the error signal at its input is zero. This comes about because the integral term in the control signal is a summation of all past values of e(t). In fact, the integral term will not stop changing until its input is zero, and therefore if the system reaches a stable steady state, the input signal to the integrator will of necessity be zero. This feature means that a constant disturbance w (see Fig. 4.4) can be canceled by the integrator s output even while the system error is zero. If PI control is used in the speed example, the transform equation for the controller is ref m U = k p ( ref m ) + k I, (4.68) s and the system transform equation with this controller is (τs + ) m = A(k p + k I s )( ref m ) + TW. (4.69) If we now multiply by s and collect terms, we obtain (τs 2 + (Ak p + )s + Ak I ) m = A(k p s + k I ) ref + AsW. (4.70) Because the PI controller includes dynamics, use of this controller will change the dynamic response in more complicated ways than the simple speedup we saw with proportional control. We can understand this by considering the
23 88 Chapter 4 Basic Properties of Feedback characteristic equation of the speed control with PI control, as seen in Eq. (4.70). The characteristic equation is τs 2 + (Ak p + )s + Ak I = 0. (4.7) The two roots of this equation may be complex and, if so, the natural frequency Ak is ω n = I τ, and the damping ratio is ζ = Ak p+ 2τω n. These parameters are both determined by the controller gains. If the plant is second order, then the characteristic equation is + k ps + k I s A s 2 + a s + a 2 = 0, (4.72) s 3 + a s 2 + a 2 s + Ak p s + Ak I = 0. (4.73) In this case, the controller parameters can be used to set two of the coefficients, but not the third. For this we need derivative control ProportionalIntegralDerivative Control (PID) The final term in the classical controller is derivative control, D, and the complete threeterm controller is described by the transform equation we will use, namely, D c (s) = U(s) E(s) = k p + k I s + k Ds, (4.74) or, equivalently, by the equation often used in the process industries, or D c (s) = k p [ + T I s + T Ds], (4.75) where the reset rate T I in seconds, and the derivative rate, T D, also in seconds, can be given physical meaning to the operator who must select values for them to tune the controller. For our purposes, Eq. (4.74) is simpler to use. The effect of the derivative control term depends on the rate of change of the error. As a result, a controller with derivative control exhibits an anticipatory response, as illustrated by the fact that the output of a PD controller having a ramp error e(t) = t(t) input would lead the output of a proportional controller having the same input by k D kp = TD seconds, as shown in Fig. 4..
24 Figure 4. Anticipatory nature of derivative control u(t) Section 4.3 Control of Dynamic Error: PID Control 89 PD Proportional T D Time (sec) Because of the sharp effect of derivative control on suddenly changing signals, the D term is sometimes introduced into the feedback path as shown in Fig. 4.2(a), which would describe, for example, a tachometer on the shaft of a motor. The closedloop characteristic equation is the same as if the term were in the forward path, as given by Eq. (4.74) and drawn in Fig. 4.2(b), if the derivative gain is k D = k p k t but the zeros from the reference to the output are different in the two cases. With the derivative in the feedback path, the reference is not differentiated, which may be a desirable result if the reference is subject to sudden changes. With the derivative in the forward path, a step change in the reference input will, in theory, cause an intense initial pulse in the control signal, which may be very undesirable. To illustrate the effect of a derivative term on PID control, consider speed control, but with the secondorder plant. In that case, the characteristic equation is Collecting terms results in s 2 + a s + a 2 + A(k p + k I s + k Ds) = 0, s 3 + a s 2 + a 2 s + A(k p s + k I + k D s 2 ) = 0. (4.76) s 3 + (a + Ak D )s 2 + (a 2 + Ak p )s + Ak I = 0. (4.77) Figure 4.2 Alternative ways of configuring rate feedback G(s) k p k t,d s (a) k p k D s G (b)
25 90 Chapter 4 Basic Properties of Feedback The point here is that this equation, whose three roots determine the nature of the dynamic response of the system, has three free parameters in k P, k I, and k D, and by selection of these parameters, the roots can be uniquely and, in theory, arbitrarily determined. Without the derivative term, there would be only two free parameters, but with three roots, the choice of roots of the characteristic equation would be restricted. To illustrate the effect more concretely, a numerical example is useful. EXAMPLE 4.6 PID Control of Motor Speed Consider the DC motor speed control with parameters 5 J m = Nm sec 2 /rad, b=0.028 Nmsec/rad, L a =0 henry, a =0.45 ohms, K t =0.067 Nm/amp, K e =0.067 Vsec/rad. (4.78) Use the controller parameters k p = 3, k I = 5 sec, k D = 0.3 sec. (4.79) Discuss the effects of P, PI, and PID control on the responses of this system to steps in the disturbance and steps in the reference input. Let the unused controller parameters be zero. Solution. Figure 4.3(a) illustrates the effects of P, PI, and PID feedback on the step disturbance response of the system. Note that adding the integral term increases the oscillatory behavior but eliminates the steadystate error, and that adding the derivative term reduces the oscillation while maintaining zero steadystate error. Figure 4.3(b) illustrates the effects of P, PI, and PID feedback on the step reference response, with similar results. The step responses can be computed by forming the numerator and denominator coefficient vectors (in descending powers of s ) and using the step function in MATLAB. For example, after the values for the parameters are entered, the following commands produce a plot of the response of PID control to a disturbance step: numg = [La a 0]; deng = [Jm*La a*b + Ke*Ke + Ke*kD a*ke*ke + Ke*kp Ke*ki]; sysg = tf(numg,deng); y = step(sysg). 5 These values have been scaled to measure time in milliseconds by multiplying the true La and J m by 000 each.
26 Section 4.4 Extensions to the Basic Feedback Concepts P PID PI Amplitude PID PI Amplitude P Time (msec) (a) Time (msec) (b) Figure 4.3 esponses of P, PI, and PID control to (a) step disturbance input and (b) step reference input 4.4 Extensions to the Basic Feedback Concepts 4.4. Digital Implementation of Controllers As a result of the revolution in the costeffectiveness of digital computers, there has been an increasing use of digital logic in embedded applications, such as controllers in feedback systems. With the formula for calculating the control signal in software rather than hardware, a digital controller gives the designer much more flexibility in making modifications to the control law after the hardware design is fixed. In many instances, this means that the hardware and software designs can proceed almost independently, saving a great deal of time. Also, it is easy to include binary logic and nonlinear operations as part of the function of a digital controller. Special processors designed for realtime signal processing and known as digital signal processors, or DSPs, are particularly well suited for use as realtime controllers. While, in general, the design of systems to use a digital processor requires sophisticated use of new concepts to be introduced in Chapter 8, such as the ztransform, it is quite straightforward to translate a linear continuous analog design into a discrete equivalent. A digital controller differs from an analog controller in that the signals must be sampled and quantized. 6 A signal to be used in digital logic needs to be sampled first, and then the samples need to be converted by an analogtodigital converter, or A/D converter, 7 into a quantized digital number. Once the digital computer has calculated the proper next control signal value, this value needs to be converted back into a voltage and held constant or otherwise extrapolated by a digitalto 6 A controller that operates on signals that are sampled but not quantized is called discrete, while one that operates on signals that are both sampled and quantized is called digital. 7 Pronounced A to D.
27 92 Chapter 4 Basic Properties of Feedback analog converter, or D/A, in order to be applied to the actuator of the process. The control signal is not changed until the next sampling period. As a result of the sampling, there are more strict limits on the speed or bandwidth of a digital controller than on analog devices. Discrete design methods that tend to minimize these limitations are described in Chapter8.Areasonable rule of thumb for selecting the sampling period is that during the rise time of the response to a step, the input to the discrete controller should be sampled approximately six times. By adjusting the controller for the effects of sampling, the sampling can be adjusted to 2 to 3 times per rise time. This corresponds to a sampling frequency that is 0 to 20 times the system s closedloop bandwidth. The quantization of the controller signals introduces an equivalent extra noise into the system, and to keep this interference at an acceptable level, the A/D converter usually has an accuracy of 0 to 2 bits. For a first analysis, the effects of the quantization are usually ignored. A simplified block diagram of a system with a digital controller is shown in Figure 4.4. For this introduction to digital control, we will describe a simplified technique for finding a discrete (sampled, but not quantized) equivalent to a given continuous controller. The method depends on the sampling period T s being short enough that the reconstructed control signal is close to the signal that the original analog controller would have produced. We also assume that the numbers used in the digital logic have enough accurate bits so that the quantization implied in the A/D and D/A processes can be ignored. While there are good analysis tools to determine how well these requirements are met, here we will test our results by simulation, following the well known advice that The proof of the pudding is in the eating. Finding a discrete equivalent to a given analog controller is equivalent to finding a recurrence equation for the samples of the control which will approximate the differential equation of the controller. The assumption is that we have the transfer function of an analog controller and wish to replace it with a discrete controller that will accept samples of the controller input, e(kt s ), from a sampler and, using past values of the control signal, u(kt s ), and present and past samples of the input, e(kt s ), will compute the next control signal to be sent to the actuator. As an example, consider a PID controller with the transfer function U(s) = (k p + k I s + k Ds)E(s), (4.80) Figure 4.4 Block diagram of a digital controller A/D e(kt ) Digital controller u(kt ) D/A D(z) T Clock U Plant G Sensor H
28 Section 4.4 Extensions to the Basic Feedback Concepts 93 which is equivalent to the three terms of the timedomain expression t u(t) = k p e(t) + k I e(τ) dτ + k D ė(t) (4.8) 0 = u P + u I + u D. (4.82) Using the fact that the system is linear, the next control sample can be computed term by term. The proportional term is immediate: u P (kt s + T s ) = k p e(kt s + T s ). (4.83) The integral term can be computed by breaking the integral into two parts and approximating the second part, which is the integral over one sample period, as follows: kts +T s u I (kt s + T s ) = k I e(τ) dτ (4.84) 0 kts kts +T s = k I e(τ) dτ + k I 0 kt s e(τ) dτ (4.85) = u I (kt s ) +{area under e(τ) over one period} (4.86) = u I (kt s ) + k I T s 2 {e(kt s + T s ) + e(kt s )}. (4.87) In Eq. (4.87) the area in question has been approximated by that of the trapezoid formed by the base T s and vertices e(kt s + T s ) and e(kt s ), as shown by the dashed line in Fig The area can also be approximated by the rectangle of amplitude e(kt s ) and width T s, shown by the solid blue in Fig. 4.5, to give u I (kt s + T s ) = u I (kt s ) + k I T s e(kt s ). These and other possibilities are considered in Chapter 8. In the derivative term, the roles of u and e are reversed from integration, and the consistent approximation can be written down at once from Eq. (4.87) and Eq. (4.8) as T s 2 {u D(kT s + T s ) + u D (kt s )}=k D {e(kt s + T s ) e(kt s )}. (4.88) Figure 4.5 Graphical interpretation of numerical integration x x f(x, u) x(t i ) t x dt 0 0 t i t i t
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