Chapter 10 Synthesis of Closed Loop Control Systems 10.1. Role of correctors: precision-stability dilemma The correction methods covered in this chapter refer to considerations of scalar behavior. It is fundamental to understand that the specifications stipulating the closed loop performances will be translated by the constraints on the frequency response of the open loop corrected system. The search for a compromise between stability and rapidity generally leads to imposing, on the closed loop, a behavior similar to that of a second order system having conjugated complex number poles. The choice of the damping value ξ is imposed by the required degree of stability. That is why it is indispensable to have a good knowledge of the relations between the parameters and the behavior of the second order systems in order to be able to use specifications defining the performances required from the closed loop final system. The general principle of a specification list is based on two points: – interpretation of specifications in order to obtain the characteristics of a second order model of the closed loop needed; – search for the constraints on the open loop introducing the behavior sought in closed loop. For reasons of clarity, let us recall the main results of the previous chapter concerning the analysis of the behavior of systems. Chapter written by Houria SIGUERDIDJANE and Martial DEMERLÉ. 284 Analysis and Control of Linear Systems 10.1.1. Analysis of systems’ behavior 10.1.1.1. Static errors For a zero static error: – with respect to a set point step function, there has to be at least one integration in the open loop; – with respect to a set point ramp, there have to be at least two integrations in the open loop; – with respect to an interference step function, there has to be one integration upstream from the input point of the interference in the open loop. 10.1.1.2. Stability The analysis of stability of the looped system, in the current case of a stable system in open loop, is based on the simplified Nyquist theorem: the image of the contour by µ β must not surround the point –1. The distance with respect to point –1, which is expressed in terms of phase margin and gain margin provides a “measurement” of stability. We can associate a “visual” criterion of the stability measurement with the help of the closed loop response overflow with a set point step function. For a system whose closed loop transfer function is of second order, we know how to connect the concept of unit-step response overflow to the damping coefficient ξ of the poles and consequently to the phase margin of the open loop (OL). We note that for a second order system, a sufficient phase margin implies a good gain margin and the only concept of phase margin is thus sufficient. Figure 10.1. Overflow curve based on damping Synthesis of Closed Loop Control Systems 285 Another way to characterize a stability measurement consists of analyzing the frequency response of the closed loop. The instability is characterized by the presence of a resonance peak. For a system whose closed loop transfer function is of second order, we connect this concept of frequency response resonance of the closed loop to the damping coefficient ξ of the poles and hence to the phase margin of the open loop. ξ ∆Φ in degrees Resonance in dB 0.1 12 12 0.2 22 8 0.4 43 2.7 0.6 58 0.35 Table 10.1. Damping, phase and resonance margin There is no more resonance from .7.0= ξ 10.1.1.3. Rapidity The character of rapidity can be perceived in two ways, either directly by observing a time or frequency response, or indirectly through the concept of dynamic error (a rapid system enables the pursuit of an input that rapidly varies and hence a low dynamic error; see the method of the equivalent sine curve described in the previous chapter). Closed loop frequency behavior, bandwidth To evaluate the rapidity of the looped system, a sinusoidal input is used whose frequency can be chosen. More often than not, the transfer function of feedback β is a constant and we can then consider two cases for the behavior in closed loop µβ+ µ 1 . If 1 µβ >> , i.e. if we are in the bandwidth in open loop, we can consider the closed loop behavior equivalent to β /1 , presenting a constant gain irrespective of 286 Analysis and Control of Linear Systems the frequency. For any frequency sinusoidal input in the bandwidth, we can then consider that the output is a faithful image of the input. If 1 µβ << , i.e. we are outside the bandwidth in open loop, we can consider the behavior in closed loop equivalent to µ . In this case, the amplitude of the output is very low compared to the amplitude of the input. A simple evaluation of rapidity consists of considering the bandwidth c ω in closed loop. With a constant feedback β , we can then consider the closed loop gain as constant if 1>> µβ . Hence, we can associate the bandwidth in closed loop with the bandwidth in open loop. The bandwidth in open loop is a qualifier of the rapidity in closed loop. Time behavior in unit-step response We can also qualify the rapidity of a closed loop system by observing its response to a set point step function. For a system without overflow, we can measure the establishing time, but the dilemma precision-stability often leads to a compromise that entails a response with overflow. For a system having such a response, we use as a rapidity criterion the time necessary to reach the first maximum noted by m t . For a second order system we can connect the time notion of first maximum to the concept of bandwidth in open loop 2 1/ ξωπ −= cm t . We can use in general the relation approached: 3≈ mc t ω which is valid for any correct damping () .16.0 << ξ Due to this relation, we can translate a first maximum time constraint of a unit- step response in closed loop into a bandwidth constraint in open loop. Dynamic precision In order to calculate an upper bound of the instantaneous error following the variations of any set point, we use the method of the equivalent sine curve. The result of this analysis leads to a specification of minimum gain in open loop for a certain bandwidth 0 to 0 ω . Therefore, this constraint leads in general to imposing a certain bandwidth to the open loop, even if the constraint in the gain refers only to the bandwidth 0 to 0 ω . Synthesis of Closed Loop Control Systems 287 In short, we will use the following rules: – in order to obtain a good precision, we need to: - in static state, have one or more integrations in direct chain in order to cancel the permanent state errors, - in dynamic state, have a high gain for a frequency band until 0 ω . This makes it possible to limit the dynamic errors; – for a good degree of stability, it is necessary that the phase margins and (or) the gain margins defined near c ω are satisfactory; – for good rapidity, it is necessary that the bandwidth in open loop is large and therefore generally having a high gain. If we use the Bode diagram to represent the scalar characteristics of the open loop to be corrected, we could translate the above constraints depending on three frequency areas as indicated in Figure 10.2. The previous chart underlines the contradictions between the specifications concerning precision and stability. An increase in the gain favors the precision at the expense of stability. Figure 10.2. Bode diagram of an open loop to be corrected The correctors or regulators have the goal to provide a control signal u to the process in order to maintain the requirements of precision and stability. They are inserted into a looped system as represented in Figure 10.3. 288 Analysis and Control of Linear Systems Figure 10.3. Corrector in a looped system The control magnitude can be a function of bx,, ε or e . Based on the signals considered and the type of the function created, there are several types of correctors. Firstly, we will review the topologies of the most widely used correctors. 10.1.2. Serial correction Figure 10.4. Serial corrector This type of corrector is inserted in the direct chain in a serial connection with the process and provides a control signal: ))(()( tftu ε = The control takes into account only the error signal. Among the usable functions f, we can find: – the proportional action noted by P: )()( tktu ε = or )()( pkpU ε = Synthesis of Closed Loop Control Systems 289 – the integral action noted by I: εθ = ∫ 0 1 () ( ) t i ut dt T or pT p pU i )( )( ε = – the derived action noted by D: dt td Ttu d )( )( ε = or )()( ppTpU d ε = – the phase lead action: )( 1 1 )( p aTp Tp KpU ε + + = with 1<a – the phase delay action: )( 1 1 )( p aTp Tp KpU ε + + = with 1>a A serial corrector creates the combinations of these actions more or less perfectly. It should be noted that this corrector acts on the static precision, dynamic precision and stability. For a first approximation: – action I cancels the static error; – action P increases the dynamic precision; – action D or the phase lead tend to stabilize the system. 10.1.3. Parallel correction This type of corrector is grafted in parallel on an element of the direct chain as shown in Figure 10.5. 290 Analysis and Control of Linear Systems Figure 10.5. Parallel corrector 1 µ can possibly be a serial corrector. x is an intermediary magnitude between the control and the output. If v contains a term of the form dt tds )( , we say that we realized a tachymetric correction. It should be noted that this corrector acts essentially on the dynamic stability and precision and not on the static error (this corrector cannot introduce integration). 10.1.4. Correction by anticipation These correction techniques are used only as a complement of looped correction techniques. They rely on the injection of signals in open loop in order to minimize the transitions felt by the main correction loop, following the input variation or an external interference. 10.1.4.1. Compensation of interferences (zero input) Figure 10.6. Corrector by anticipation Synthesis of Closed Loop Control Systems 291 Let us suppose that the interference is measurable. It is then possible – at least theoretically – to eliminate the influence of the interference b(t) with the help of a corrector by anticipation C(p) described by the diagram in Figure 10.6. In fact, it is enough to choose: 2 1 () () Cp p µ =− to cancel the effect of )(tb on the output. We note in this case that the stability is not affected, nor is the precision with respect to the input. Most often, )( 1 2 p µ is not physically feasible, which leads to adopting approximate forms of )( 1 2 p µ . There is no perfect compensation of the transient state of interferences. EXAMPLE 10.1.– if p K p =)( 2 µ , we could take () p p K pC τ + −= 1 1 )( with τ less than the main time constants of 3 µ . 10.1.4.2. Compensation of the input (zero interference) Figure 10.7. Input compensation This diagram can be considered, with respect to the error, as the overlapping of the two diagrams of Figure 10.8. 292 Analysis and Control of Linear Systems Figure 10.8. Diagram decomposition of the input compensation We have: () βµµ βµ βµµ εεε 21 2 21 21 1 )()( 1 )( + − + + =+= pCpE pE In order to cancel the error with respect to the input, it is enough to choose βµ 2 1 )( = pC . It should be noted that in this case the closed loop control system perfectly follows the input law, without introducing the integration in the direct chain. The stability of the system is not modified, nor is the influence of interferences with respect to the error. Most often, βµ 2 1 is not physically feasible. Thus, the compensation will not be perfect in transient states. 10.1.5. Conclusions The feedforward correctors, when they are feasible, do not modify the stability of the loop and compensate either the error due to the input, or the effect of interference. In a complex case (several interferences, some of which are non- [...]... Figure 10. 16 Figure 10. 16 Phase delay network With T = R 2 C and a = R1 + R 2 R2 The phase delay corrector has the Bode diagram represented in Figure 10. 17 300 Analysis and Control of Linear Systems Figure 10. 17 Bode diagram of a phase delay network 1/a 1/4 1/6 1/8 1 /10 1/12 φm –37° –45° –51° –55° –58° Table 10. 3 Maximal phase delay 10. 2.2.2 Action mechanism of these correctors The effect of these... represented in Figure 10. 24 306 Analysis and Control of Linear Systems Figure 10. 24 Bode graph in OL of the corrected system ' We can estimate the phase margin of the corrected system, if ω c is close to the environment of slope segment (–1), by referring to Table 10. 2 of the phase lead ' circuit This would be even more justified in the present case since ω c > 10 0 and ' ω c < 1 / 10T4 10. 4 Proportional... diagram of such a corrector is represented in Figure 10. 21 304 Analysis and Control of Linear Systems Figure 10. 21 Bode diagram of a lead-delay network 10. 3.1.1 Action mechanism of these correctors These correctors combine the actions previously studied They are used when the simple action correctors do not lead to the performances desired EXAMPLE 10. 4 Figure 10. 22 Combined action corrector In Figure 10. 22,... Figure 10. 12 that the phase margin becomes insufficient A correction is thus necessary, for example, a phase lead correction 296 Analysis and Control of Linear Systems Figure 10. 12 Bode graph with insufficient phase margin Figure 10. 13 Bode graph in OL of the corrected system Synthesis of Closed Loop Control Systems 297 Since this corrector is in series with µβ , there is an addition of gain and phase... system saturates, the linearly defined performances are not reached (increased response time and limited precision) On the other hand, irrespective of the linearity of the system, it is not always interesting to have a too extended bandwidth, due to the amplification of noises Synthesis of Closed Loop Control Systems 315 10. 7 Parallel correction 10. 7.1 General principle This type of correction is represented... used 10. 2 Serial correction 10. 2.1 Correction by phase lead 10. 2.1.1 Transfer function Generally speaking, we call a phase lead corrector a corrector whose transfer function has the form: C ( p) = K 1 + Tp with a < 1 1 + aTp This corrector can be physically created by the circuit of Figure 10. 9 Figure 10. 9 Phase lead network with T = R1C and a = R2 R1 + R 2 294 Analysis and Control of Linear Systems. .. modify the phase margin ∆φ The corrected curve in open loop is represented in Figure 10. 28 Figure 10. 28 Bode diagram of the corrected system in OL This corrector provides the desired performances to the closed loop  310 Analysis and Control of Linear Systems 10. 6 Proportional integral proportional (PID) correction 10. 6.1 Transfer function An ideal PID corrector has as transfer function: ⎛ 1 C ( p )... variations of these two transfer functions in the Bode plane The curves are cut in points A and B as indicated in Figure 10. 35 (one of these points can be infinitely rejected) Figure 10. 35 Variation of the two transfer functions In these points we have: µ1 µ 2 µ 3 β = µ1 µ 3 β C i.e.: µ2 = 1 C or µ 2 C = 1 Points A and B are thus the separations of cases 1 and 2 Synthesis of Closed Loop Control Systems. .. a PID in order to control a system whose transfer function is unknown Figure 10. 33 System with unknown transfer function 314 Analysis and Control of Linear Systems In certain cases, the transfer function of a system is unknown; hence the adjustment of a PID cannot use the methods previously presented Thus, there are experimental adjustment methods We shall keep in mind the Ziegler-Nichols method (sustained... presence of an amplifier of gain a The Bode graph is represented in Figure 10. 10 Figure 10. 10 Bode graph of a phase lead network The tabulated values of the maximum phase input φ m (obtained for ω = 1 / T a ) are given for your information (Table 10. 2), in practice we can use the 1− a formula sin φ m = in order to find the value of a corresponding to a desired φ m 1+ a value a 1/4 1/6 1/8 1 /10 1/12 . written by Houria SIGUERDIDJANE and Martial DEMERLÉ. 284 Analysis and Control of Linear Systems 10. 1.1. Analysis of systems behavior 10. 1.1.1. Static errors For a zero static error: – with. 10. 1.3. Parallel correction This type of corrector is grafted in parallel on an element of the direct chain as shown in Figure 10. 5. 290 Analysis and Control of Linear Systems Figure 10. 5 precision and stability. They are inserted into a looped system as represented in Figure 10. 3. 288 Analysis and Control of Linear Systems Figure 10. 3. Corrector in a looped system The control