Classical Feedback Control with MATLAB
Boris J. Lurie and Paul J. Enright, Marcel Dekker, NY, 2000
Reprinted from Classical Feedback Control pp 387397 by courtesy of Marcel Dekker, Inc. Copyright © 2000 Marcel Dekker, Inc. These materials may be copied for nonprofit use, so long as they are unaltered and accompanied by this header. If the materials are used for classroom instruction, mention is to be made of the source.
The function of the "Discussions" sections is to anticipate questions and objections, and also to address certain persistent misconceptions. In most cases, these are condensed transcripts of conversations which took place with the authors' colleagues and students, to whom we extend our sincere gratitude.
Q: 
In the Preface, it is stated that modern technology somehow relates to highorder compensators. What does the word "modern" have to do with the compensator's order? Isn't it used here just as an advertisement gimmick? 
A: 
It is more than the gimmick. The old way of compensator implementation used the same technology as that for the actuation. There exist funny mechanical, hydraulic, and pneumatic gadgets whose outputs are proportional to the integral or to the derivative of the inputs. These devices were combined to make the desired transfer function of the compensator. Since the devices were relatively expensive and rather difficult to tune, the compensators remained loworder. The most complicated among them were the PID controllers using a proportional device, an integrator device, and a differentiator device, all connected in parallel. 
Q: 
The specifications on the closedloop performance are often formulated in the time domain. Is it worthwhile to convert them into frequencydomain specifications and then design the compensator with frequencydomain methods? 
A: 
Definitely. The benefits of frequencydomain design far outweigh the trouble of converting the timedomain specifications. The conversion of most practical specifications is simple and transparent. 
Q: 
For initial design, is it advisable to optimize the performance while ignoring some details like noise, nonlinearities, and plant parameter variations? 
A: 
That sort of approach typically produces disastrous results in engineering practice. Even in the initial design, the system should be addressed in its entirety, or else the time and resources allocated for design will be wasted on dead ends. 
Q: 
Does the minus sign at the summer mean that the feedback is negative? 
A: 
No. At any specified frequency, the sign of the feedback also depends on the modulus and phase of the elements in the loop. 
Q: 
From the transfer functions CAP and B, the closedloop transfer function CAP/(1 + CAPB) can be calculated, and viceversa. Is that all there is to it? 
A: 
It's not quite this simple. When the feedback is large, only the first calculation will be accurate. Even rather large errors in the knowledge of C, A, and P do not cause much uncertainty in the closedloop transfer function. The result is that the inverse problem is illconditioned, with small errors in the observed or calculated closedloop response possibly mapping into large errors in the calculation of the loop transmission. Therefore, the calculations or measurements of the closedloop response are commonly inadequate to determine the loop transfer function. 
Q: 
But isn't a common definition of a tracking system that the feedback path transmission is 1? 
A: 
B = 1 is required for frequencies much lower than the crossover, and B can be identically 1 in a system with a prefilter. However, with B = 1 and without a prefilter, the disturbance rejection and the compensator design need to be compromised to provide acceptable transient responses for the nominal plant. (This may be unavoidable for homing type systems.) 
Q: 
Should the feedback path and the compensator be analog or digital? 
A: 
There are many factors to consider here. A digital compensator has the advantage that it can be reprogrammed in response to known changes in the plant; however, it should be pointed out that changing software is often neither trivial nor cheap. A primary disadvantage of digital compensators is that they reduce the available feedback when the computational delay in the loop is significant, as it often is. For this reason, analog compensators are the first choice for fast feedback loops. If necessary, the parameters of an analog compensator can be made "programmable" by employing multiplying D/A converters. The summing device and the feedback path can be analog as long as the required relative accuracy is not better than about 0.0001. To achieve much higher accuracy, they should probably be digital. Many other factors will affect this decision, including power requirements, cost, production quantity, etc. 
Q: 
The compensator is optimized for disturbance rejection, and the feedback link (and/or prefilter) is designed to provide the desired closedloop response with the nominal plant. But what happens to the closedloop response when the plant deviates from the nominal? 
A: 
A methodology has been developed, called quantitative feedback theory (QFT), that designs the system so as to satisfy the closedloop performance specifications in the presence of the worstcase parameter variations. This procedure is discussed in Chapter 8. Although the methods involve a significant amount of calculation, they are useful when parameter variations are very large, i.e. more than 10 dB. For most applications, it is sufficient to first design the closedloop response with the nominal plant, and then determine the changes caused by the plant variations, and, if necessary, modify the prefilter. 
Q: 
What is wrong with design methods that use the closedloop response as the objective for the compensator design, like the pole placement method? 
A: 
With such methods, the appropriate value of feedback and the stability margins are not observed and preserved. This can cause many problems: the system stability may be only conditional, or windup might occur, the disturbance rejection may not be optimized, and the plant parameter variations may cause larger closedloop response variations than what is achievable with feedback maximization methods based on the openloop frequency response. 
Q: 
How accurate must the gain response approximation be for the accuracy of the related phase response to be 5°? 
A: 
About 1/2 dB. 
Q: 
The compensator transfer function must be high order to approximate well the required transcendental response. But, the poles and zeros of a highorder transfer function may be very sensitive to the polynomial coefficients. Then, rounding errors will make the system not robust. Is this correct? 
A: 
Not exactly. Variations in the values of poles and zeros is not important as long as the frequency response doesn't change much. The sensitivity of the frequency response to polynomial coefficients in typical cases is not high, and the compensator order is in this aspect irrelevant. For example, in analog telecommunication systems, equalizers have been routinely employed with 20th order transfer functions, with the accuracy of implementation of the desired response of ± 0.001 dB. The requirements for the control system compensators are by a long way easier. 
Q: 
What if the plant is nonminimumphase? 
A: 
The nonminimumphase component of the loop phase lag must be compensated for by reducing the minimumphase component of the phase lag. This can be done by reducing the frequency f_{b} to increase the length of the Bode step. Simple formulas for these calculations are given in Chapter 4. 
Q: 
The command, the prefilter, and the feedback path are known, and the sensor is measuring the plant's output. This data suffices to calculate CAP. Further, if C is known, AP can be determined exactly, and this information can be used to modify C such that the loop transmission function is as desired. Right? 
A: 
Before deciding to do this, let's consider the errors in the calculation. The sensor is not ideal (noisy), the command is not wellsuited to the goal of characterizing the plant, and the problem is illconditioned for all frequency components where the feedback is large. This approach might be used for the frequency band where the feedback is positive, i.e. in the neighborhood of the crossover frequency f_{b}. However, for this application, the high frequency components of the command need to be sufficiently large, larger than the noise, which is not usually the case. 
Q: 
Can the effects of the plant resonances on the potentially available feedback be calculated in advance? 
A: 
Sure. See Chapter 4, Problem 4 and the answer to the problem. 
Q: 
Is the distance of the poles of the closedloop transfer function from the jaxis a better robustness measure then the stability margins on the Nyquist diagram? 
A: 
No. A practical counterexample is an active RC notch filter. Its closedloop poles are very close to the jaxis, but the stability margins in this feedback system are sufficient, the system is globally stable, reliable and widely used. Conversely, placing the poles of a closedloop system far away from the jaxis does not guarantee global stability, process stability, or robustness. 
Q: 
Is it convenient to define the boundaries on the Nyquist diagram to reflect necessary and sufficient stability requirements? 
A: 
Yes. 
Q: 
How are the values for the stability margins determined? 
A: 
Global and process stability must be assured in the presence of the plant and actuator parameter variations. (Often, process stability is not strictly achieved, but the satisfaction of a certain norm on the errors in the nonlinear state of operation is acceptable.) Also, the sensitivity of the closed loop response to plant parameter variations must be limited. 
Q: 
Why should the area surrounding the critical point be defined by gain and phasemargins, and not by mathematically simpler circular boundary? 
A: 
The shape of the margin boundary is defined by global and process stability requirements and by the plant parameter variations. In the majority of plants, variations of the plant gain and plant phase in the neighborhood of the crossover are not well correlated. This necessitates defining gain and phasemargins independently. 
Q: 
Is much performance lost if we simplify the design methods and use circular stability margin boundary? 
A: 
Changing the stability margins from those appropriate to circular would substantially reduce the integral of positive feedback in this frequency range and, therefore, the integral of negative feedback in the functional frequency band. 
Q: 
Is the definition of Nyquiststable the same as that of conditionally stable? 
A: 
The Lyapunov definition for conditional stability is that the nonlinear system stability depends on the initial conditions. A Nyquiststable system with the usual actuator saturation nonlinearity may be conditionally stable but can be rendered globally stable by the addition of nonlinear dynamic compensation. 
Q: 
Whatever the internal (output) impedance of the actuator, we can always apply to its input such a signal that the system output will be as required. Therefore, it doesn't matter what is the actuator output impedance. Right? 
A: 
Wrong. For the nominal plant, the statement is right, but the effects of the plant parameter deviations from the nominal are to a large extent dependent on the actuator output impedance. 
Q: 
Compound feedback produces some finite output impedance for the driver. But, this can be achieved more simply, by making voltage feedback and placing a resistor in series. 
A: 
The result will be the same as long as the system remains linear. This method can be used in the laboratory for testing various control schemes. However, there will be power losses on this series resistor, and as a result, the actuator needs to be more powerful, with a larger saturation level. 
Q: 
But, isn't the power dissipated anyway on the internal impedance of the actuator when the impedance is made finite by compound feedback? 
A: 
No. Compound feedback causes no power losses and doesn't reduce the actuator efficiency. 
Q: 
At what frequencies is the feedback positive? 
A: 
The feedback is positive within the circle of unit radius centered at (–1,0) on the Tplane. When the loop gain decreases monotonically with frequency, the major part of the area of positive feedback falls in the band 0.6f_{b} to 4f_{b}. The integral of feedback at much higher frequencies is negligibly small (see bibl. to [9]). 
Q: 
Well, the feedback makes the system better at some frequencies, but at others even worse than it was before. So, what is gained from the application of feedback? 
A: 
Negative feedback is used to cover the frequency range where the plant parameter variations and the effects of disturbances are critical, and positive feedback is confined to the frequency range where the noise and the disturbances are so small that even after being increased by the positive feedback, they still will be acceptable. 
Q: 
How many Bode relations are, total? 
A: 
There are about 30 different integral relations in Bode's book. 
Q: 
Have new relations or expansions of these relations been made since then? 
A: 
Yes. The most important are the R. Fano expansions of the Bode integral of the real part, which use not one but several terms in the Laurent expression. They are usable for RF and microwave circuits design. Also, Bode relations were expanded on unstable systems, discrete systems, multivariable systems, parallel paths of the signal propagation, and the transfer function for disturbance isolation. An example of an expansion on certain nonlinear systems is given in Appendix 10. However, these expansions are less important for practice, compared with the four fundamental Bode relations described in this book. 
Q: 
Why is the slope of the Bode diagram expressed here in dB/oct, not in dB/dec? 
A: 
The same reason that supermarkets sell milk by gallons, not barrels. Gain samples one octave apart typically suffice to calculate the phase response with less than 5° error as needed for sound feedback system design. And it is certainly convenient that the slope of a welldesigned Bode diagram is nearly 10 dB/oct. 
Q: 
Does it make sense to consider a loop gain response having a constant slope of 10 dB/oct? How can such a response be implemented in a physical system? 
A: 
It can be closely approximated by a rational function of s. 
Q: 
Are the Bode integrals applicable to transcendental plant transfer functions? 
A: 
Yes. 
Q: 
At a university where I have taught, the teaching relies heavily on the root locus method. Afterward I started to work in the aircraft industry and was surprised that here the engineers prefer Bode diagrams. Why is it so? 
A: 
Bode noticed that for an engineer striving to make the best of the design, it is difficult to cope with three scalar variables: gain, phase, and frequency, and he reduced the number of the variables to only two scalars: gain and frequency. This approach allows handling highorder systems. In contrast, applying the root locus method to an nth order system increases the number of the variables to n complex variables (roots) crawling all over the splane in a strange, threatening, and unmanageable manner. 
Q: 
What are the physical factors which limit the feedback? 
A: 
The major factors are: the uncertainties in the plant transfer function, the clipping of higher frequency components of the sensor noise in the actuator, and the level of noise in the system's output. 
Q: 
Does it pay to exploit as much as possible the available knowledge about the plant when designing the control law? 
A: 
Certainly. This increases the available feedback. The available feedback is infinite for hypothetical fullstate feedback, and no feedback is available for a completely unpredictable plant. 
Q: 
A typical concern of the project manager: the system design tradeoffs require knowledge of the available performance of its subsystems, and these subsystems often employ feedback. How can the system tradeoffs be made without designing the feedback subsystems? 
A: 
The manager should request data about available subsystem performance, i.e. the performance that can be achieved with the optimal compensator. Using the Bode approach this can be calculated without actually designing the compensator. 
Q: 
What kind of data do we need for this estimation? What kind of mathematics is involved? 
A: 
The following data is generally required: the sensor noise spectral density over three octaves in the vicinity of the planned feedback bandwidth, the available (nondistorted) output signal amplitude from the actuator, ranges of frequencies where plant structural modes can fall, up to four octaves over the estimated feedback bandwidth, and sometimes more. This sort of data is typically available, although it may not always be very precise. The available disturbance rejection can then be estimated using the Bode integrals. 
Q: 
I want to develop and market a product. How can I be sure that my competitor will not enter the market, soon after me, with a superior product that uses the same actuator, plant, and sensor, but bigger and faster feedback? 
A: 
The Bode integral approach allows the determination of the best theoretically available system performance. 
Q: 
Is it necessary to approximate a physical plant transfer function (which is sometimes transcendental) by a rational function and/or to perform plant "model reduction" when designing a feedback system with the Bode method? 
A: 
No. The calculated or measured plant and actuator transfer functions are subtracted from the desired loop response to obtain the desired compensator response, and the rational function approximation is introduced only in the final stage of the compensator design. 
Q: 
What is the required accuracy of the approximation of the desired loop gain response? 
A: 
The accuracy must be such that the related phase response will approximate the desired phase response with an accuracy of about 5°. 
Q: 
Why 5° ? 
A: 
In this case, the average loop phase lag must stay 5° away from the stability margin boundary. Then, the average slope of the Bode diagram will be less than the maximum acceptable by 1/3 dB/oct (from the proportion: 90° for 6 dB/oct). Therefore, over each of the 3 octaves of the cutoff, 1 dB of feedback is lost. Typically, these losses are marginally acceptable. The small remaining possible feedback increase may not justify a further increase in the complexity of the compensator. 
Q: 
How accurate must the gain response approximation be for the accuracy of the related phase response to be 5° ? 
A: 
About 1/2 dB. 
Q: 
The compensator transfer function must be high order for accurate approximation of the required transcendental response. But, poles and zeros of a highorder transfer function may be very sensitive to the polynomial coefficients. Then, the rounding errors will make the system not robust. Is this correct? 
A: 
Not exactly. Variations in the values of poles and zeros are not important as long as the frequency response doesn't change much. The typical compensator response has no sharp peaks and notches; the sensitivity of the such function modulus to the polynomial coefficients is typically less than 1, and the compensator order is in this aspect irrelevant. Therefore, in order for the compensator to be accurate within 0.5 dB, i.e. 6% in the magnitude of the transfer function, the polynomial coefficient accuracy need not be better than 1 to 5%, and the effect of rounding is insignificant. 
Q: 
What is the best available accuracy of analog compensators? 
A: 
Equalizers for analog telecommunication systems have been routinely designed with up to 20th order transfer function, with accuracy of implementation of the desired response of ± 0.001 dB. 
Q: 
How were these equalizers designed? 
A: 
By interpolation, by cutandtry procedures, by using asymptotic Bode diagrams, by adjustments in the element domain, by using Chebyshev polynomial series, by using the Second Remez Algorithm, by using the Simplex Method. 
Q: 
What if the plant is nonminimumphase? 
A: 
The nonminimumphase component of the loop phase lag must be compensated for by reducing the minimumphase component of the phase lag. This can be done by reducing the frequency f_{b} to increase the length of the Bode step. Simple formulas for these calculations are given in Chapter 4. 
Q: 
Only analog systems have been discussed. What is the difference with regard to digital controllers? 
A: 
Sampled data systems, which include an analogtodigital converter, a digital filter, and a digitaltoanalog converter are timevariable and have an inherent delay that reduces the available feedback. 
Q: 
What about a proportionalintegralderivative (PID) compensator having the transfer function C(s) = k_{p} + k_{i }/s + k_{d}s ? 
A: 
With appropriate gains, the PID compensator roughly approximates the response of the optimal controller; however, the approximation error can be quite large. PID control falls short of the performance of a system with an appropriate highorder compensator. 
Q: 
But how much better off than a PID's is the optimal controller? 
A: 
It depends on the frequency response of the plant transfer function and the disturbance spectrum density. For smooth plant responses, the expected improvement in disturbance rejection is 5 to 10 dB when a PID controller is replaced by a higherorder controller. For a plant with structural resonances, especially when it is used in conjunction with a nonlinear dynamic compensator, a 5 to 30 dB improvement can be expected from the highorder controller. 
Q: 
Why the ZieglerNichols conditions for tuning PID controllers are not presented in this book? 
A: 
This method does not use a prefilter (or a command feedforward), and is suitable only for specific plants. More general and better results can be achieved with the frequency domain approach. 
Q: 
What about loop response shaping for multiloop systems? 
A: 
Basically, the technique is the same. Multiloop systems can be designed one loop at a time, as long as loop coupling is taken care of by adjusting the responses and providing extra stability margins within certain frequency bands. 
Q: 
Does this technique work well even when the number of the loops is large? 
A: 
Yes, typically. 
Q: 
Is there a case where it is advantageous to apply compensators with nonminimumphase transfer functions? 
A: 
Probably not. Sometimes, when the plant has highfrequency structural modes, a compensator must introduce a phase delay in the loop to make the system stable. However, this phase delay can always be achieved with a minimumphase lowpass filtertype function  with the extra advantage of increasing the amplitude stability margin and reducing the noise effect at the actuator's input. 
Q: 
Isn't it awkward to go back and forth from the Nyquist diagram on the Lplane to the Bode diagram? Isn't one type of diagram enough? 
A: 
It's a case of the right tool for the right job: global and process stability characterization, stability margin definitions, and nonlinear dynamic compensation design all require the use of the Lplane, but the tradeoff resolution and the compensator design are simplified by using the Bode diagram due to a reduced number of variables. 
Q: 
Do the design phases discussed in the following Appendix 12 apply for the design of singleloop or multiloop systems? 
A: 
Singleloop. But multiloop system design is a natural extension of this procedure. 
Q: 
Still, if you have to design, say, a threeinput, threeoutput system, will it take 3, 9, or 27 times longer than a singleloop system of comparable complexity for each channel? 
A: 
3 times longer, typically. 
Q: 
Is it convenient to use Lyapunov functions in the design of high performance control systems? 
A: 
Devising appropriate Lyapunov functions is yet unmanageable for systems with a highorder linear part and several nonlinear elements. 
Q: 
At the 1st Congress of the International Federation of Automatic Control (IFAC) in 1960 in Moscow, researchers from Russian academia presented important theoretical results using timedomain and state variables. This research impressed American professors attending the conference and shifted to a large extent the direction of research in American academia. Were these methods the methods Russian engineers employed to design control laws for their rockets and satellites? 
A: 
No. Russian rockets' control systems have been designed with frequency domain methods in very much the same way as American rockets (see A13.9). 
Q: 
DF analysis fails to predict instability in certain systems. Would it be right therefore to discard the DF approach altogether? 
A: 
No. It would be foolish to discard a Phillips screwdriver just because it cannot drive all screws. Similarly, although DF analysis is not a universal foolproof tool, it is fairly accurate when employed for analysis and synthesis of well designed control loops with steep monotonic lowpass responses. 
Q: 
What is the purpose of using DF now, when nonlinear systems can be simulated well with computers? 
A: 
The DF advantage is seen when it is used not for the purpose of analysis only, but for the purpose of design, and especially, conceptual design of control systems with several nonlinear links. In many cases, the accuracy of the analysis need not be high. The phase of the DF has an error of up to 20° compared with the phase calculated with exact analysis  but, this phase is not of critical importance since NDCs can easily provide the required phase advance. 
Q: 
DF analysis does not account for additional phase shifts created by higher harmonics interacting in the nonlinear link. So does the DF design guarantee robustness? 
A: 
Interaction of highorder harmonics can typically yield up to 15–20° of extra phase lag for the fundamental. The NDC DF phase advance must be increased by this amount, which is rather easy to do since the typical NDC phase advance exceeds 100 200°. 
Q: 
How much improvement in disturbance rejection can be expected from using an NDC? 
A: 
It depends on the desired frequency shaping of the disturbance rejection  in some cases, up to 30 dB. 
Q: 
Is it advantageous to use NDCs in MIMO systems? 
A: 
Yes. In addition to providing the same advantages as for SISO systems, NDCs reduce the effects of nonlinear coupling between the loops on the system stability. 
Q: 
Why is it the number of saturation elements that matters when defining what is the multiloop system? 
A: 
Because the DF of a saturation link changes from 1 down to the inverse of the loop gain coefficient, i.e. 100 or 1000 times. This effect needs to be accounted for during the stability analysis and the loop transfer functions synthesis. Compared with this effect, even the effects of the plant parameter variations are small. 
Q: 
Where is the better place to implement the decoupling matrix  in the forward path or in the feedback path? 
A: 
In the forward path, since in this case the decoupling matrix doesn't need to be precise. In this case, commands are formulated in the sensors' readings. When commands should be formulated in actuators' actions (and sensors are not aligned with the actuators), the decoupling matrix should be placed in the feedback path. 
Q: 
Is the design of MIMO and multiloop systems difficult? 
A: 
It is not simple, but several oftignored techniques greatly simplify the design: 


Q: 
How does coupling between the loops affect the system stability and robustness? 
A: 
The effects of loop coupling can be analyzed by considering the change in one loop transfer function caused by the plant and the actuator transfer function variations in another loop. The robustness can be provided for by correspondingly increasing the stability margins. 
Q: 
At what frequencies is the coupling most dangerous? 
A: 
In the nonlinear mode, at lower frequencies, where the loop gain is large and changes in it caused by saturation could affect other loops. However, this can be taken care of by using an NDC. In the linear mode, the coupling is more dangerous near the crossover frequencies where the feedback in the loops is positive. 
Q: 
What about terminology in Bode's book? 
A: 
Bode developed several powerful approaches valid for a wide range of applications, and he optimized the terminology to serve this wide range of applications. 