Controller Design and the Gauss–Lucas Theorem - IEEE Xplore

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Controller Design and the Gauss–Lucas Theorem Michael Jason Knap, L. H. Keel, and Shankar P. Bhattacharyya Abstract—In this technical note, we demonstrate how the classical Gauss–Lucas theorem can aid in the determination of stabilizing controllers for feedback systems. Moreover, we show how to apply the Gauss–Lucas theorem in the construction of new necessary conditions for Schur and Hurwitz stability. The results are used to derive new bounds on the stability margins and performance of control systems. Index Terms—Gauss–Lucas theorem, Hurwitz stability, Schur stability.

I. INTRODUCTION In linear systems theory, the controller synthesis problem is often reduced to finding a set of controller parameters that ensure the stability of a polynomial. For this reason, polynomials play important roles in control system design and analysis. Indeed, the subject is one of the most well-studied in mathematics and also in the area of control systems. The well-known necessary and sufficient conditions for stability of a polynomial are the Routh-Hurwitz criterion [1], [2] for continuoustime systems (Hurwitz stability) and Jury’s criterion [3] for discretetime systems (Schur stability). However, use of these necessary and sufficient conditions is severely limited for control design problems. The main difficulty lies in the fact that the stability conditions for the polynomial are typically highly nonlinear functions of the coefficients of the polynomial. On the other hand, there exist necessary conditions for both Hurwitz and Schur stability that are simple and linear. Unfortunately, these conditions are often too weak to be useful. Recent literature addresses the aforementioned difficulty in a number of ways. Malik et al. [4] constructs sequences of both inner and outer approximations of the stability set for a fixed-structure controller; moreover they show that the union of the inner feasible sets approaches the complete set of stabilizing controllers as the number of feasible sets approach infinity. Henrion et al. [5] develop a method to obtain inner approximations from recent results on positive polynomials and propose an application to controller synthesis using LMI optimization. The more recent work by Cerone et al. [6], though nestled in the framework of system identification, also develops parameter bounds which ensures stability using explicit a priori information, the theory of moments, and LMI relaxation techniques. In this technical note, we present a simple application of a classical result in complex analysis which defines the relationship between the roots of a polynomial and the roots of the derivative of the polynomial. This is used to derive bounds on the stability and performance of control systems. We compose this result with Schur necessary conditions Manuscript received April 19, 2012; revised August 29, 2012; accepted April 08, 2013. Date of publication April 18, 2013; date of current version October 21, 2013. This work was supported in part by the Department of Defense (DOD) Grant W911NF-08-0514 and National Science Foundation (NSF) Grants CMMI-0927664 and CMMI-0927652. Recommended by Associate Editor F. Dabbene. M. J. Knap is with the Department of Mathematics, Tennessee State University, Nashville, TN 37209 USA (e-mail: [email protected]). L. H. Keel is with the Department of Electrical and Computer Engineering, Tennessee State University, Nashville, TN 37209 USA) (e-mail: keel@gauss. tsuniv.edu). S. P. Bhattacharyya is with the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX 77843-3128 USA (e-mail: [email protected]). Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2013.2258795

Fig. 1. Closed-loop system

.

so that we can generate many more sets of linear conditions to be added to the existing set of necessary conditions; however, this is only one way in which this result may be used. In fact it is suggested here that current other approximations and methods such as those in [4]–[6] may also benefit from the Gauss-Lucas theorem. We will illustrate through examples that the proposed necessary conditions for Schur stability are tighter than previous known conditions and can be a good tool for control system design. II. THE GAUSS-LUCAS THEOREM The following classical result which is known as the Gauss–Lucas Theorem was implied in a note of Carl Friedrich Gauss dated 1836, and it was stated explicitly and proved by Félix Lucas in 1874 [7]. Since then, some extensions and remarks have been given in [8]–[11] and references therein. be Theorem 1 (Gauss-Lucas Theorem [7], [12], [13]): Let a polynomial with real or complex coefficients. All the zeros of the lie in the convex hull of the set of zeros of . If derivative are not collinear, no zeros of the derivative lie the zeros of . on the boundary of unless it is a multiple zero of III. APPLICATION TO CONTROL SYSTEMS In this section, we apply the Gauss-Lucas theorem to derive results on the stability and performance of closed-loop control systems. It is worthwhile to mention that the Gauss-Lucas theorem has been used to solve control problems [14]–[16] that are fundamentally different from the application we present here. Consider the feedback system in Fig. 1. Let (1) where and are polynomials of degree and , respectively with real coefficients, and is a real parameter vector. The is closed-loop characteristic polynomial of the system (2) and is said to be stable if all roots of (2) lie in an open convex set , of the complex plane, denoted the stability region1. , , , and (see Fig. 2) be defined Let as follows: for for for for

Let

denote the stabilizing set of parameters for .

for

1Note that special cases of are the open left-half plane and the open unit-disc corresponding to Hurwitz and Schur stability, respectively.

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Fig. 2. Closed-loop system

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.

Theorem 2: is stable only if (a) (b) The stabilizing sets ,

,

are stable. satisfy (3)

is upper bounded by the gain margins (c) The gain margin of , . of is upper bounded by the phase mar(d) The phase margin of , . gins of Proof: The proof of (a) and (b) follow immediately from the Gauss-Lucas theorem and the convexity of . The proof of (c) and (d) follow from applying the Gauss-Lucas theorem to (4)

and

Fig. 3. Hurwitz stability sets of Example 1.

(5) respectively. In the rest of this technical note, we provide some applications of this result to control systems. IV. EXAMPLES Example 1: An example is provided to illustrate Theorem 2. Let the plant be

and a controller be

Fig. 4. Feedback system.

We want to know whether the given plant can be stabilized by a PID controller of the form (6) To answer this question, we apply the Gauss-Lucas theorem to the characteristic polynomial of the closed-loop system

Then and its successive derivatives Its first and second derivative polynomials are

It is easy to see that is free of the design parameters and it is not Hurwitz. Thus, we conclude that the given system is not feedback stabilizable by any controller of the form (6). V. A NEW NECESSARY CONDITION FOR SCHUR STABILITY

The set of stabilizing parameters is found and plotted within parameter space (see Fig. 3). Additionally, the stabilizing parameters are calculated for two derivatives of the given characteristic polynomial, and show outer approximations of the original stability set. An immediate application of the Gauss-Lucas theorem is illustrated in the following example. Example 2: Consider the feedback system in Fig. 4. Let the plant be

In general, the necessary and sufficient conditions obtained from the Jury’s table or the Jury-Raible’s table consist of expressions that are highly nonlinear functions of the coefficients of the polynomial in test. As a result, the conditions are difficult to use for control design problems. On the other hand, there exists a simple necessary condition for Schur stability. Consider a polynomial with real coefficients (7) The following is a well known necessary condition for Schur stability of (7).

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and from again from Lemma 1

(14)

Fig. 5. Feedback system.

Lemma 1: If is Schur, it satisfies the following three conditions: , (1) , (2) . (3) The advantage of such necessary conditions are their simplicity and linearity with respect to coefficients of the polynomial. Nevertheless, the condition is not tight enough to be useful. In this technical note, our aim is to improve Lemma 1 by introducing additional inequalities that are linear with respect to the coefficients of lie a polynomial. From the Gauss-Lucas theorem, the zeros of . For Schur stainside of the convex hull of the set of zeros of must be located inbility, the convex hull of the set of zeros of side the unit circle. Therefore, it is necessary that the derivative polyis also Schur. Therefore, if is Schur stable, so are nomial . By applying the conditions in Lemma 1 to each of these derivative polynomials, we have the following. With a slight of abuse of notation, we have the following necessary of degree and conditions for Schur stability of a polynomial derivative with respect to where . Let its (8)

Essentially, apply the conditions from Lemma 1 to each polynomial as . Each one of these sets are the derivative is taken from 0 to outer approximations of the stabilizing sets. That is (9) is an outer approximation of the stabilizing set. In addition to this set-theoretic notation, we describe a “closed-form” notation as well. Rewriting the conditions in Lemma 1, we have

Note that the second conditions of the above derivations may be simplified with the relationship

By repeating the process of derivation and simplifying, we have the following. in (7) is Schur, then its coefficients Lemma 2: If a polynomial satisfy the following conditions:

(15)

for

and

As seen, the necessary condition given in (15) is a set of linear inequalities in terms of the coefficients of the polynomial (7) that can be evaluated efficiently. In feedback connection with predetermined controller order, the coefficients of a characteristic polynomial become affine linear functions of controller transfer function coefficients. Thus, the necessary condition given here can serve as a good starting point to find the coefficients of a stabilizing controller transfer function. Example 3: In this example, we show that the application of the Gauss-Lucas theorem to Schur stability analysis also presents another advantage. One may see easily whether or not a proposed system is stabilizable by controllers of a given structure through simple observation of the new conditions. Consider the plant

(10)

and controller

(11)

With the unity feedback configuration shown in Fig. 5, the characteristic polynomial becomes

Taking the derivative of (7)

and applying Lemma 1, we have

(12)

The necessary conditions for Schur stability due to Lemma 1 are

Now we take the derivative of (11)

(13)

It is easy to see that there is a solution to these conditions. For example, , , will simultaneously satisfy these linear inequalities. However, applying Lemma 2 results in additional linear

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Fig. 7. Necessary conditions in (16) and (17).

Fig. 6. Necessary conditions in (16).

inequalities to be satisfied. The full list of additional conditions from are the sequence of derivatives of

Fig. 8. Necessary conditions in (16) – (18).

It is easy to see that the set of inequalities contains conditions that are not consistent. Therefore, the given system is not stabilizable by the proposed structure of controllers. Example 4: To illustrate another application of the Gauss-Lucas theorem, consider the discrete-time plant

The second set of conditions from

are

(17) and controller of the form

A third set of conditions from

are

The characteristic polynomial of the closed-loop system becomes (18) Finally, the fourth set from

are

In this example, we illustrate how the necessary conditions improve as more conditions from the derivatives of the polynomial are added. are The first set of linear conditions from (19) (16)

Fig. 6 plots the parameter space which satisfies the necessary conditions in (16). These conditions are unbounded in the and dimensions.

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Fig. 9. Necessary conditions in (16) – (19).

Fig. 7 plots the parameter space which satisfies the conditions in (16) and (17). Fig. 8 considers the conditions in (16), (17), and (18). This region is now bounded in all dimensions, providing a finite volume containing the stability region. Finally, Fig. 9 considers all of the conditions in (16)–(19). This example clearly illustrates that the necessary conditions are improving as we applying Gauss–Lucas Theorem.

[6] V. Cerone, D. Piga, and D. Regruto, “Enforcing stability constraints in set-membership identification of linear dynamic systems,” Autom., vol. 47, no. 11, pp. 2488–2494, Nov. 2011. [7] F. Lucas, “Propriétés géométriques des fractions rationnelles,” C. R. Acad. Sci. Paris, vol. 77, pp. 431–433, 1874. [8] D. K. Dimitrov, “A refinement of the Gauss-Lucas Theorem,” Proc. Amer. Math. Soc., vol. 126, no. 7, pp. 2065–2070, Jul. 1998. [9] A. W. Goodman, “Remarks on the Gauss-Lucas Theorem in higher dimensional space,” Proc. Amer. Math. Soc., vol. 55, no. 1, pp. 97–102, Feb. 1976. [10] T. Craven and G. Csordas, “The Gauss-Lucas Theorem and Jensen polynomial,” Trans. Amer. Math. Soc., vol. 278, no. 1, pp. 415–429, Jul. 1983. [11] J. L. Diaz-Barrero and J. J. Egozcue, “A generalization of the GaussLucas Theorem,” Czechoslovak Math. J., vol. 58, no. 2, pp. 481–486, 2008. [12] F. Lucas, “Statique des polynômes,” Bulletin de la Société Mathématique de France, vol. 17, pp. 2–69. [13] M. Marden, Geometry of Polynomials. Providence, RI: American Mathematical Society, 1966, vol. 3, Mathematical Surveys. [14] J.C. Hamann and B.R. Barmish, “Convexity of frequency response arcs associated with a stable polynomial,” IEEE Trans. Autom. Control, vol. 38, no. 6, pp. 904–915, 1993. [15] N. Cohen and J. Kogan, “Convexity of a frequency response arc associated with a stable entire function,” IEEE Trans. Autom. Control, vol. 41, no. 2, pp. 295–299, 1996. [16] J.V. Burke, A.S. Lewis, and M.L. Overton, “Variational analysis of the abscissa mapping for polynomials via the Gauss-Lucas theorem,” J. Global Optim., vol. 28, no. 3/4, pp. 259–268, 2004. [17] F. Dabbene, B.T. Polyak, and R. Tempo, “On the complete instability of interval polynomials,” Syst. Control Lett., vol. 56, no. 6, pp. 431–438, 2007.

VI. CONCLUSION We introduce some useful results on the stability and performance of feedback systems derived from the classical Gauss–Lucas Theorem. New necessary conditions for Schur stability of a polynomial are also introduced in this technical note through simple, iterative application of the Gauss–Lucas theorem. We have shown that these conditions can be a significant improvement over the existing necessary conditions. The Gauss–Lucas theorem may also yield other useful design information. For instance, bounds on damping margin and ratio, frequency of oscilor and then lation, etc. can be obtained by replacing by applying the Gauss–Lucas theorem to the resulting polynomial in . In addition, reviewers pointed out that several other problems may also benefit from the Gauss–Lucas theorem. In [17], the so-called one-ina-box problem is studied. When an interval polynomial family is not completely unstable, a stable polynomial may be found by using a randomized algorithm based on necessary conditions for Hurwitz stability. Any potentially stable candidate polynomial produced by an algorithm such as one-in-a-box problem [17] can be eliminated if it does not satisfy the conditions of the Gauss–Lucas theorem, namely, stability of all lower order polynomials. Since the Gauss–Lucas theorem provides new necessary conditions, it clearly may improve all polynomial based results. These are fruitful areas of future research.

REFERENCES [1] A. Hurwitz, “On the conditions under which an equation has only roots with negative real parts,” in 65 Selected Papers on Mathematical Trends in Control Theory, R. Bellman and R. Kalaba, Eds. New York: Dover, 1964, pp. 70–82A. Hurwitz, “,” Mathematische Annelen, vol. 46, pp. 273–284, 1895. [2] E. J. Routh, A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion. New York: Macmillan, 1877. [3] E. I. Jury, Inners and Stability of Dynamic Systems. New York: Wiley, 1974. [4] W. A. Malik, S. Darbha, and S. P. Bhattacharyya, “A linear programming approach to the synthesis of fixed-structure controllers,” IEEE Trans. Autom. Control, vol. 53, no. 6, pp. 1341–1352, Jul. 2008. [5] D. Henrion, M. Sebek, and V. Kucera, “Positive polynomials and robust stabilization with fixed-order controllers,” IEEE Trans. Autom. Control, vol. 48, no. 7, pp. 1178–1186, Jul. 2003.

State Sensitivity Evaluation Within UD Based Array Covariance Filters J. V. Tsyganova and M. V. Kulikova Abstract—This technical note addresses the UD factorization based Kalman filtering (KF) algorithms. Using this important class of numerically stable KF schemes, we extend its functionality and develop an elegant and simple method for computation of sensitivities of the system state to unknown parameters required in a variety of applications. For instance, it can be used for efficient calculations in sensitivity analysis and in gradient-search optimization algorithms for the maximum likelihood estimation. The new theory presented in this technical note is a solution to the problem formulated by Bierman et al. in [1], which has been open since 1990s. As in the cited paper, our method avoids the standard approach based on the conventional KF (and its derivatives with respect to unknown system parameters) with its inherent numerical instabilities and, hence, improves the robustness of computations against roundoff errors. Index Terms—Array algorithms, filter sensitivity equations, Kalman filter, UD factorization.

I. INTRODUCTION Linear discrete-time stochastic state-space models, with associated Kalman filter (KF), have been extensively used in practice. Application Manuscript received April 11, 2012; revised November 06, 2012; accepted April 17, 2013. Date of publication April 18, 2013; date of current version October 21, 2013. This work was supported in part by the Ministry of Education and Science of the Russian Federation under 1.919.2011 and by the Portuguese National Fund ( Fundação para a Ciência e a Tecnologia ) within the scope of projects PEst-OE/MAT/UI0822/2011 and SFRH/BPD/64397/2009. Recommended by Associate Editor A. Chiuso. J. V. Tsyganova is with Ulyanovsk State University, Ulyanovsk 432017, Russia (e-mail: [email protected]). M. V. Kulikova is with the Technical University of Lisbon, Instituto Superior Técnico, CEMAT (Centro de Matemática e Aplicações), Lisboa 1049-001, Portugal (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2013.2259093

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