Robust, Fragile, Or Optimal? - Automatic Control, IEEE ... - IEEE Xplore

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 42, NO. 8, AUGUST 1997

Robust, Fragile, or Optimal? L. H. Keel, Member, IEEE, and S. P. Bhattacharyya, Fellow, IEEE

Abstract— In this paper, we show by examples that optimum , l1 , and and robust controllers, designed by using the 2 , formulations, can produce extremely fragile controllers, in the sense that vanishingly small perturbations of the coefficients of the designed controller destabilize the closed-loop control system. The examples show that this fragility usually manifests itself as extremely poor gain and phase margins of the closed-loop system. The calculations given here should raise a cautionary note and draw attention to the larger issue of controller sensitivity which may be important in other nonoptimal design techniques as well.

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Index Terms—Fragility, optimality, robustness.

I. INTRODUCTION

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VER THE last 15 years, control theory has developed several techniques for designing linear time-invariant control systems that are optimum and robust. These rely on the YJBK parameterization [1] of all stabilizing controllers for a fixed linear time-invariant plant, which provides a free parameter over which an appropriate function of a closed-loop transfer function may be minimized. Elegant formulations for minimizing the [2], [1], [3], [4], and [6] norms of various closed-loop transfer functions as well as efficient numerical approaches to minimizing , a structured uncertainty measure [5], have been developed using this technique, the state-space DGKF approach [7], and the MUSYN ToolBox [8]. These formulations reflect robust stability and/or robust performance. An implicit assumption that is inherent to this type of methodology is that the controller that is designed will be implemented exactly. Relatively speaking, this assumption is valid, in the sense that plant uncertainty is clearly the most significant type of uncertainty in a control system, while controllers are generally implemented with high-precision hardware. On the other hand, it is necessary that any controller that is part of a closed-loop system be able to tolerate some uncertainty in its coefficients. There are at least two reasons for this. First, controller implementation is subject to the imprecision inherent in analog-digital and digital-analog conversion, finite word length, and finite resolution measuring instruments and roundoff errors in numerical computations. Thus, it is required that there exists a nonzero (although possibly small) Manuscript received May 3, 1996. Recommended by Associate Editor, L. Qiu. This work was supported in part by NASA under Grant NCCW-0085 and the NSF under Grant ECS-9417004. L. H. Keel is with the Center of Excellence in Information Systems, Tennessee State University, Nashville, TN 37203 USA. S. P. Bhattacharyya is with the Department of Electrical Engineering, Texas A&M University, College Station, TX 77843 USA (e-mail: [email protected]). Publisher Item Identifier S 0018-9286(97)05964-3.

margin of tolerance around the controller designed. Second, every paper design requires readjustment because no scalar index can capture all the performance requirements of a control system. This means that any useful design procedure should generate a controller which also has sufficient room for readjustment of its coefficients. This translates to the requirement that an adequate stability and performance margin be available around the transfer function coefficients, or other characterizing implementation parameters, of the designed nominal controller. With the above background as motivation, we study in this paper the parametric stability margin of several controller designs from the published literature, obtained by using the , , , and approaches. The parametric stability margin calculation is a standard computation in the robust parametric control literature [10]–[13]. For the calculations given in this paper, we used the software package Robust Parametric Control Toolbox developed in conjunction with the text [10]. In each of the examples treated, we obtain the somewhat surprising conclusion that the parametric stability margin of the controller designed is vanishingly small. This means that extremely small perturbations of the coefficients of the controller designed will succeed in destabilizing the loop; in other words, the controller itself is fragile, and so is the control system. We also compute the gain and phase margins of the systems designed and observe that these too are very poor, except in one case. In the last section of the paper, we briefly discuss some of the issues raised by the calculations presented in the paper. It is obvious that it would be unwise to place a controller that is fragile with respect to perturbations of its coefficients in an actual control system without further precautions and analysis. A quick “fix” to the design procedure is to include the parametric stability margin with respect to the controller designed as a side constraint in the optimization algorithm. However, our hope is that this paper will open up research into some of the fundamental issues related to controller sensitivity in optimal and robust designs as currently advocated in the control literature. II. EXAMPLES We analyze four examples of optimal designs taken from the control literature. Each example is a single-input/singleoutput system, and the controller is designed to optimize some closed-loop performance function. In Examples 1, 3, and 6, the procedure consists of fixing the nominal plant model, parameterizing all proper feedback controllers that stabilize the nominal model through the YJBK parameter , which is only required to be stable and proper, and optimally selecting

0018–9286/97$10.00  1997 IEEE

KEEL AND BHATTACHARYYA: ROBUST, FRAGILE, OR OPTIMAL?

Fig. 1. Nyquist plot of

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P (s)C (s).

. In cases where the optimal turns out to be improper, it is divided by a factor to make it proper, and is chosen to be suitably small and positive. For our analysis, we compute in each case the parametric stability margin around the coefficients of the designed optimal controller as well as the gain and phase margins of the closedloop system. Example 1 ( -Based Optimum Gain Margin Controller): This example is found in [9, p. 200]. It uses the YJBK parameterization and the machinery of the -Model Matching Problem to optimize the upper gain margin. The plant to be controlled is

and the controller, designed to give an upper gain margin of 3.5 (the closed loop is stable for the gain interval [1, 3.5]), is obtained by optimizing the norm of a complementary sensitivity function. The controller found is

where

and the poles of the closed-loop system are

and this verifies that the controller is indeed stabilizing. The Nyquist plot of is shown in Fig. 1 and verifies that the desired upper gain margin is achieved. On the other hand, we see from Fig. 1 that the lower gain margin and phase margin are Gain Margin Phase Margin

degree.

This means, roughly, that a reduction in gain of one part in one thousand will destabilize the closed-loop system! Likewise, a vanishingly small phase perturbation is destabilizing. To continue with our analysis, let us consider the transfer function coefficients of the controller to be a parameter vector with its nominal value being

and let be the vector representing perturbations in . We compute the parametric stability margin around the nominal point. This becomes

The normalized ratio of change in controller coefficients required to destabilize the closed loop is

The poles of this nominal controller are

This shows that a change in the controller coefficients of less than one part in a million destabilizes the closed loop. This controller is anything but robust; in fact, we are certainly justified in labeling it as a fragile controller.

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In order to verify this rather surprising result, we construct the destabilizing controller whose parameters are obtained by setting and are

The closed-loop poles of the system with this controller are

which shows that the roots crossover to the right half-plane at , and the perturbed controller is indeed destabilizing. Example 2 (An Arbitrary Controller): For the sake of comparison, we continue with the previous example and try a first-order controller with the same plant

We design a pole placement controller placing closed poles on a circle of radius spaced equidistantly in the left half-plane. The transfer function of this controller is

where

Introduce the parameter vector corresponding to the controller coefficients and with nominal value

We compute the troller to be

P (s)C (s).

The system can tolerate gain reduction of about 21%. This is an improvement over the previous controller by a factor of about 20 000! The phase margin is improved by a factor of about 60. We have already shown the drastic improvement in the parametric stability margin. Therefore, this nonoptimal controller is far less fragile than the optimal controller, on all counts. Example 3 ( Robust Controller): The following example is also taken from [9, p. 192]. This example designs an optimal -robust controller that minimizes , where is the complementary sensitivity function and the weight is chosen as the high-pass function

The plant transfer function is

parametric stability margin for this conand the optimally robust controller found is

The normalized ratio of change in controller coefficients required to destabilize the loop is

which is to be compared to the previous 10 value. This controller can tolerate a change in coefficient values of 7.2% compared to the value of 10 for the optimum controller. The Nyquist plot of the system with this controller is shown in Fig. 2 and gives the lower gain and phase margins Gain Margin Phase Margin


degree.

The poles of the closed-loop system are

and, therefore, the controller does stabilize the nominal plant. For the purposes of our analysis we first took the controller coefficients as a parameter vector and found the parametric stability margin ( norm of the smallest destabilizing perturbation ) to be



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P (s)C (s).

The normalized ratio of change in controller coefficients required for destabilization is

where

which shows that the controller, which by design is maximally robust with respect to perturbations, is quite fragile with respect to controller coefficient perturbations. To continue our analysis the Nyquist plot of the system with this controller is drawn in Fig. 3. From this, we obtain the lower gain and phase margins Gain Margin Phase Margin

degrees

which are quite poor and would probably be unacceptable in a real system. Example 4 ( -Based Design): This example was published in [14, pp. 530–536], wherein a robust controller for an electromagnetic suspension system is designed by using the -synthesis technique. The plant transfer function is

and the controller designed to tolerate prescribed structured plant perturbations is given as


which verifies that the closed-loop system is nominally stable. For our analysis we first took the controller coefficient vector as a parameter vector and determined the parametric stability margin around the nominal controller. This parametric

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The optimal controller designed to minimize the disturbance transfer function is

norm of a

where



P (s)C (s).

stability margin is found to be


which indicates that closed-loop stability is very fragile with respect to controller parameter perturbations. To continue, we draw the Nyquist plot of the plant with this controller . This is shown in Fig. 4, which shows that

and, therefore, the controller does stabilize the nominal closedloop system. For our analysis, we took the controller transfer function coefficients as a parameter vector and computed the parametric stability margin around the nominal controller. Here, we found that


Gain Margin Phase Margin

degree

Comparing this with the previous fragility with respect to coefficient perturbations reminds us that good gain and phase margins are not necessarily reliable indicators of robustness. However, poor gain and/or phase margins are accurate indicators of fragileness! Example 5 ( -Optimal Control): This example is taken from [6, pp. 51 and 342]. The given plant which is a discrete time model of an X-29 aircraft has transfer function

This shows how fragile the system is with respect to controller parameter perturbations: perturbations of less than one part in 10 000 will destabilize the closed loop. Next, the Nyquist plot of is drawn in Fig. 5. This gives the closed-loop gain and phase margins Gain Margin Phase Margin

with

degree

Example 6 ( -Optimal Design): This example is taken from [9, p. 186]. The plant transfer function is

and the optimal controller is determined by minimizing a weighted norm of the disturbance transfer function , where is the sensitivity function. In this example, the optimal YJBK parameter is improper, and a suboptimal controller is picked after dividing by



the factor designed is

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P (z)C (z ). with

and

. The controller

where

To continue, we draw the Nyquist plot of the system with this controller in Fig. 6. It shows that the gain and phase margins are Gain Margin Phase Margin


degrees

which is again rather poor. In this particular example, we found that the poles of the closed-loop system appear to be extremely sensitive with respect to controller coefficient changes. Consider the following : perturbation vector

which verifies nominal closed-loop stability. To proceed with our analysis, we took the controller coefficients as a parameter vector and computed the parametric stability around the nominal. Here we found that


which again shows that the controller is extremely fragile.

whose norm is , which equals the parametric stability margin for this problem. If we add this perturbation to the nominal controller coefficients, we have the following closed-loop poles with the

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P (s)C (s).

perturbed controller:

This shows that one pole is far in the left half-plane. Now we select another small perturbation

whose norm is , which is slightly larger than the previous one. If we add this perturbation to the nominal controller coefficients, we have the following new set of closed-loop poles:

which includes a right half-plane pole at about 100 10 ! This example shows that a slight perturbation in controller coefficients of the optimal controller will result in very large perturbations of the closed-loop poles. This is due to the degree dropping of the characteristic polynomial at the perturbation in the highest coefficient.

III. CONCLUDING REMARKS The calculations presented in this paper show that , , , and designs can lead to fragile controllers. This means that very small perturbations of the controller coefficients can result in instability. This fragility also shows up usually as extremely small gain and/or phase margins of the closedloop system. Moreover, these margins were calculated at the nominal plant; the worst case margins over the set of uncertain plants would certainly be even poorer! The numbers obtained for the parametric margin around the controller coefficients mean that there is practically no freedom left to readjust or tune the controller. Thus, at least in the examples presented here, the control engineer who opts for such an optimal design is forced to either accept a fragile design or reject it altogether. These calculations, therefore, raise a cautionary note regarding the role of optimal and robust designs as developed over the last few decades, in practical applications. It is important to understand the fundamental reasons for controller fragility if we are to successfully overcome it. We make some speculative comments on this issue next. Modern control theory results in higher-order controllers. As we know, the stability regions in the parameter space of higherorder systems have “instability holes” and the optimization algorithm can stuff the controller parameter into tight spots close to these holes, since no margin is asked for with respect to the controller in the optimization procedure. Thus, in a


sense, the design transfers the sensitivity from the plant to the controller. Of course the fragility shown in this paper may not be limited to the techniques discussed here nor to high-order controllers. For example, [10, Example 4.8. p. 194] also shows that one may obtain extremely sensitive low-order controllers. This example shows that the controller that maximizes the plant parameter perturbation tolerance actually puts the closedloop poles on the boundary of the specified D-stability region. This means that while the closed-loop poles may only move inward to the D-stability region with respect to plant parameter perturbations, even a small perturbation in the controller parameters sends them out of the D-stability region. Our experience has shown that the state space-based design methods can result in the same sort of acutely sensitive controllers as shown in this paper. It is also worthwhile to point out that one view of optimal control is that the fixed plant is a constraint on the optimization process. There are “good” and “bad” plants, and some nonminimum phase plants are so bad that no compensator can produce a robust system. Some of the examples given here are indeed bad plants, and this leads to poorly posed optimization problems. The above discussion suggests that there are a number of subtle and sophisticated issues in controller design that need to be looked into more deeply. ACKNOWLEDGMENT The authors gratefully acknowledge discussions with Prof. J. B. Pearson, who shared his insights and his suggestions for improving the paper, as well as Prof. Ya. Z. Tsypkin, who assured the authors that the subject of controller sensitivity had not been addressed in the control literature and was important. REFERENCES [1] M. Vidyasagar, Control System Synthesis. Cambridge, MA: MIT Press, 1985. [2] B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods. Englewood Cliffs, NJ: Prentice Hall, 1989. Control Problem. Englewood Cliffs, NJ: [3] A. A. Stoorvogel, The Prentice-Hall, 1992. [4] H. Kimura, Y. Lu, and R. Kawatani, “On the structure of control systems and related questions,” IEEE Trans. Automat. Contr., vol. 36, pp. 653–667, June 1991. [5] K. Zhou, J. C. Doyle, and K. Glover, Robust Optimal Control. Englewood Cliffs, NJ: Prentice Hall, 1995.

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[6] M. A. Dahleh and I. J. Diaz-Bobillo, Control of Uncertain Systems: A Linear Programming Approach. Englewood Cliffs, NJ: Prentice Hall, 1995. [7] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State control problems,” IEEE Trans. space solution to standard 2 and Automat. Contr., vol. 34, pp. 831–847, Aug. 1989. [8] G. J. Balas, J. C. Doyle, K. Glover, A. Packard, and R. Smith, —Analysis and Synthesis ToolBox. Natick, MA: Mathworks, 1993. [9] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback Control Theory. New York: Macmillan, 1992. [10] S. P. Bhattacharyya, H. Chapellat, and L. H. Keel, Robust Control: The Parametric Approach. Upper Saddle River, NJ: Prentice Hall, 1995. [11] D. Hinrichsen and B. Martensson, Eds., Control of Uncertain Systems. Berlin: Bitkhaäuser, 1990. [12] M. Mansour, S. Balemi, and W. Truöl, Eds., Robustness of Dynamic Systems with Parameter Uncertainties. Berlin: Birkhäuser, 1992. [13] N. K. Bose, “A system-theoretic approach to stability of sets of polynomials,” Contemporary Math., vol. 47, pp. 25–34, 1985. [14] M. Fujita, T. Namerikawa, F. Matsumura, and K. Uchida, “-synthesis of an electromagnetic suspension system,” IEEE Trans. Automat. Contr., vol. 40, pp. 530–536, Mar. 1995.

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L. H. Keel (S’82–M’86) received the B.S. degree in electronic engineering from Korea University, Seoul, Korea, in 1978, and the M.S. and Ph.D. degrees in electrical engineering from Texas A&M University, College Station, TX, in 1983 and 1986, respectively. Since then, he has been with the Center of Excellence in Information Systems at Tennessee State University, where he is now a Professor. His research interests include robust control, system identification, structure and control, and computer-aided design. He has authored and coauthored numerous technical papers in the field of control systems and two books, including Robust Control: The Parametric Approach (Englewood Cliffs, NJ: Prentice-Hall, 1995).

S. P. Bhattacharyya (S’67–M’72–SM’86–F’89) was born in Rangoon, Burma, on June 23, 1946, and educated at the Indian Institute of Technology, Bombay, India, 1962–1967, and Rice University, Houston, TX, 1967–1971. He established the graduate program in Automatic Control at the Federal University, Rio de Janeiro, Brazil, 1971–1980, where he was Chairman of the Department of Electrical Engineering, 1978–1980. From 1974 to 1975, he worked as a National Academy of Sciences Research Fellow at NASA’s Marshall Space Flight Center. In 1980, he joined Texas A&M University, College Station, TX, where he is presently Professor of Electrical Engineering, and served as Director of the Systems and Control Institute, 1990–1992. He has authored three books and more than 150 papers in the field of control systems.