Unified Frequency Domain Inequalities With Design Applications

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 1, JANUARY 2005

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Generalized KYP Lemma: Unified Frequency Domain Inequalities With Design Applications Tetsuya Iwasaki, Senior Member, IEEE, and Shinji Hara, Senior Member, IEEE

Abstract—The celebrated Kalman–Yakuboviˇc–Popov (KYP) lemma establishes the equivalence between a frequency domain inequality (FDI) and a linear matrix inequality, and has played one of the most fundamental roles in systems and control theory. This paper first develops a necessary and sufficient condition for an -procedure to be lossless, and uses the result to generalize the KYP lemma in two aspects—the frequency range and the class of systems—and to unify various existing versions by a single theorem. In particular, our result covers FDIs in finite frequency intervals for both continuous/discrete-time settings as opposed to the standard infinite frequency range. The class of systems for which FDIs are considered is no longer constrained to be proper, and nonproper transfer functions including polynomials can also be treated. We study implications of this generalization, and develop a proper interface between the basic result and various engineering applications. Specifically, it is shown that our result allows us to solve a certain class of system design problems with multiple specifications on the gain/phase properties in several frequency ranges. The method is illustrated by numerical design examples of digital filters and proportional-integral-derivative controllers. Index Terms—Control design, digital filter, frequency domain inequality, Kalman–Yakuboviˇc–Popov (KYP) lemma, linear matrix inequality (LMI).

I. INTRODUCTION

O

NE OF THE most fundamental results in the field of dynamical systems analysis, feedback control, and signal processing, is the Kalman–Yakuboviˇc–Popov (KYP) lemma [1]–[3]. Various properties of dynamical systems can be characterized by a set of inequality constraints in the frequency domain. The KYP lemma establishes equivalence between such frequency domain inequality (FDI) for a transfer function and a linear matrix inequality (LMI) for its state space realization. The basic roles of the KYP lemma are two fold: it provides 1) insights into analytical approaches to systems theory, and 2) a framework for numerical approaches to systems analysis and synthesis.

Manuscript received September 4, 2003; revised May 25, 2004 and September 11, 2004. Recommended by Associate Editor Y. Ohta. This work was supported in part by the National Science Foundation under Grant 0237708, by The Ministry of Education, Science, Sport, and Culture, Japan, under Grant 14550439, by CREST of the Japan Science and Technology Agency (JST), and by the 21st Century COE Program on Information Science and Technology Strategic Core. T. Iwasaki is with the Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904-4746 USA (e-mail: [email protected]). S. Hara is with the Department of Information Physics and Computing, Graduate School of Information Science and Engineering, The University of Tokyo, Tokyo 113-8656, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2004.840475

The KYP lemma [3] states that, given matrices Hermitian matrix , the FDI

,

, and a

(1) holds for all

if and only if the LMI (2)

admits a Hermitian solution . Thus, the infinitely many inequalities (1) parametrized by can be checked by solving the finite-dimensional convex feasibility problem (2). Appropriate choices of in (1) allows us to represent various system properties including positive-realness and bounded-realness. While the KYP lemma has been a major machinery for developing systems theory, it is not completely compatible with practical requirements. In particular, each design specification is often given not for the entire frequency range but rather for a certain frequency range of relevance. For instance, a closed-loop shaping control design typically requires small sensitivity in a low frequency range and small complementary sensitivity in a high frequency range. Thus a set of specifications would generally consists of different requirements in various frequency ranges. On the other hand, the standard KYP lemma treats FDIs for the entire frequency range only. The current state of the art for fixing the incompatibility is to introduce the so-called weighting functions. A low/band/highpass filter would be added to the system in series as a weight that emphasizes a particular frequency range and then the design parameters are chosen such that the weighted system norm is small. The weighting method has proven useful in practice, but there remains some room for improvement. First, the additional weights tend to increase the system complexity (e.g., controller order), and the amount of increased complexity is positively correlated with the complexity of the weights. Second, the process of selecting appropriate weights can be time-consuming, especially when the designer has to shoot for a good tradeoff between the complexity of the weights and the accuracy in capturing desired specifications. We remark that some of these deficiencies may be addressed by recent developments for new characterizations of disturbance signals [4], [5]. An alternative approach is to grid the frequency axis. In this case, infinitely many FDIs are approximated by a finite number of FDIs at selected frequency points. This approach has a practical significance especially when the system is well damped and the frequency response (after the design) is expected to be “smooth” (i.e., no sharp peaks). The resulting computational

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burden often seems moderate. However, it is difficult to determine a priori how fine the grid should be to achieve a certain performance. The method also suffers from the lack of a rigorous performance guarantee in the design process as the violation of the specifications may occur between grid points. Another approach that avoids both weighting functions and frequency gridding is to generalize the fundamental machinery, the KYP lemma, in such a way that FDIs in finite frequency ranges can be treated directly. Although the time-consuming design iterations may not be completely eliminated with this approach, it is expected that the ability to directly deal with finite frequency ranges without weighting functions would provide a more user-friendly platform for systems design. There are several results in the literature along this line. The finite frequency , KYP lemma obtained in [6] states that, given a scalar matrices , , and a Hermitian matrix , the FDI (1) holds in the low frequency range if and only if the LMI

(3) admits Hermitian solutions and . The nonstrict inequality version of this result has been obtained in [7] in the context of integrated design of structure/control systems. Other generalizations of the KYP lemma include those in [8]–[10] where FDIs for polynomials are considered in the discrete-time setting. Specifically, algebraic conditions are given to characterize polynomials that are positive on an arc of the unit circle in the complex plane. Some of these results are developed for the purpose of digital filter design. The main objective of this paper is to further generalize the KYP lemma and to unify all the previous extensions previously mentioned by a single theorem. There are two basic aspects in our generalization—the frequency range and the system class. In particular, we consider the frequency intervals characterized by two quadratic forms of the frequency variable . This characterization captures curve(s) on the complex plane and encompasses low/middle/high frequency conditions for both continuous-time and discrete-time systems. Thus, the result includes the previous results on the low/middle frequency conditions for continuous-time systems [7] as special cases, and adds new contributions to the cases involving high frequency conditions and discrete-time systems. The system class is generalized by, roughly speaking, replacing the term in (1) with a new term of the form . It turns out that this allows us to capture FDIs for descriptor systems as well as polynomials. The generalized KYP lemma will be obtained within the framework of the -procedure [11], [12] that has been extensively used in the systems and control literature (e.g. [13]–[16]). It converts an inequality condition with multiple constraints to an unconstrained inequality condition with multipliers. The -procedure is conservative in general, i.e., the latter condition implies the former, but the converse is not always true. Fradkov showed in 1973 that the -procedure with scalar multipliers is lossless (i.e., nonconservative) if and only if a certain rank-one property holds for an associated separating hyperplane [17].

This type of rank-one property has been used as a sufficient condition for the -procedure to be exact in a more general setting of matrix-valued multipliers [3], [6], [18]–[21]. In this paper, we will prove necessity of the rank-one property, providing a necessary and sufficient condition for the -procedure to be lossless in the general setting. Our result on the -procedure will be used to generalize the KYP lemma, but it will also be of independent interest as a basis for systems analysis in general. Another contribution of the paper is to develop a useful interface between the generalized KYP lemma and various engineering applications. We will show that the design specifications encompassed by our result can be summarized, in the case of single-input–single-output (SISO) transfer function designs, as follows. The Nyquist plot of the transfer function within a prescribed frequency interval must lie in the intersection of prescribed conic sections on the complex plane. It will also be shown that several classes of important engineering problems can be formulated in terms of such specifications, and can be solved exactly through LMI optimizations. Finally, design examples of digital filters and proportional-integral-derivative (PID) controllers will illustrate the procedures and advantages of our approach in comparison with existing ones. This paper is organized as follows. We will first discuss in Section II system design problems that are naturally described by FDI specifications in various frequency ranges. These problems motivate our generalization of the KYP lemma to treat finite frequency ranges. Section III develops an extension of the -procedure and obtains a necessary and sufficient condition for losslessness. Section IV gives a characterization of general frequency ranges (curves on the complex plane) in terms of quadratic forms of the frequency variable. Section V presents our main results that generalize the standard KYP lemma. We will then show in Section VI how various FDI specifications for system design can be described within our framework. Finally, Section VII shows what classes of design problems can be solved via convex optimizations, where analytical discussions will be followed by design examples. We use the following notation. For a matrix , its transpose, complex conjugate transpose, and the Moore–Penrose inverse , , and , respectively. The real and are denoted by are denoted by and . The imaginary parts of stands for the set of Hermitian matrices. For symbol , inequalities and a matrix denote positive (semi)definiteness and negative (semi)definiteis deness, respectively. The set of matrices means the Kronoted by . For matrices and , and , a function necker product. For is defined by

(4) The convex hull and the interior of a set are denoted by and , respectively. Given a positive integer , let be the . set of nonnegative integers up to , i.e., Finally, denotes the set of positive integers.

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II. MOTIVATION: SYSTEM DESIGN We will first motivate our research through several examples of design problems for which our generalized KYP lemma is naturally suited and can be useful. Later in the paper, some of these problems will be discussed in more detail with solution procedures and numerical design examples. Digital filter design: The problem is to find a stable SISO transfer function satisfying a set of frequency domain specifications. Typical design requirements for band-pass filters are given by the following Chebyshev approximation setting (see, e.g., [22]–[25]): : : : : all

(5)

where and is a given function that . An has desired gain/phase properties in the pass-band would have the unity magnitude and the ideal response linear phase in the pass-band, i.e., . The first three conditions specify the stop-bands and the pass-band, while the last condition is added to suppress overshoot in the transition and . bands Sensitivity-shaping: This is a typical control design with specifications on the closed-loop transfer functions [26], [27]. and a controller , the sensitivity and the For a plant and are defined complementary sensitivity functions by

Fig. 1.

Open-loop shaping specifications.

the low frequency range for the sensitivity reduction. The servo bandwidth requirement in the middle frequency range might be represented by an elliptic constraint as shown in the figure, where we may maximize the bandwidth. The two straight lines provide gain/phase lying to the right of the point stability margins, and the small circle centered at the origin relates to the roll-off requirement in the high frequency range for robust stability. Structure/control design integration: Control performance of mechanical structures can be significantly enhanced if the designs of the structure and the controller are integrated [7], [28]–[30]. It has been shown that the finite frequency positivereal (FFPR) property in the low frequency range (6)

The sensitivity-shaping problem typically consists of the following type of requirements: : : : all : all where . The first constraint may represent good tracking of a reference signal with spectral contents in the low frequency range, while the second may be interpreted as a robustness requirement against unmodeled high frequency dynamics of the plant. The last two constraints tend to avoid oscillatory time responses. Open-loop shaping: This is a classical control design problem of determining the controller parameters to meet specifications on the open-loop transfer function. Given a , a set of specifications on marginally stable SISO plant is given in terms of the Nyquist plot of the controller the open-loop transfer function . Fig. 1 shows typical design requirements, where the shaded regions indicate where the Nyquist plot should lie in various frequency at the lower right ranges. The half plane constraint on part of the figure corresponds to the high-gain requirement in

is an important requirement for mechanical structure design to guarantee the existence of controllers that achieve high servo-bandwidth [7], [31], where is the transfer function of the mechanical system to be designed. These references demonstrated that structures of practical significance can be designed to achieve the FFPR property by solving LMI feasibility problem with the aid of an existing version of the finite frequency KYP lemma [6], [7]. In particular, the method has been applied to a shape design of a swing arm for hard disk drives [7] and a smart arm design using piezo-electric film [31]. III.

-PROCEDURE

We will first develop a generalized version of the -procedure that converts a constrained inequality to an unconstrained inequality with multiplier(s) [12], [17], [32], [33]. The result is instrumental to the proof of the generalized KYP lemma to be stated later, and will be of independent interest as a tool for developing systems theory. A. Classical Form and Its Generalization The classic version of the -procedure [12], [33] is the following. Given , , we have the equivalence such that such that

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where the regularity, version

, is assumed. The strict inequality such that

such that also holds, and for this equivalence the regularity assumption is is negative semidefinite in particular, the not required. When statement reduces to the Finsler’s theorem [34]. The -procedure is to replace the condition on the left of “ ” by the one on the right. It may be difficult to check the former condition directly because the inequalities impose nonconvex constraints on in general. On the other hand, the latter condition is easier to verify as it is a search for the scalar multiplier satisfying convex constraints. To generalize the classical -procedures, let us rewrite them with different notation. First note that the set of matrix-valued multipliers associated with the classical -procedures can be defined as

a) admissible if it is a nonempty closed convex cone2 and ; ; b) regular if c) rank-one separable if . It can readily be verified by a separation theorem (Lemma is 11 in Appendix II) that a nonempty closed convex cone admissible if and only if the set is nonempty. Note that is the set of the hyperplanes that separate from , motivating the term “rank-one separable” in c). The definition of regularity is consistent with the standard notion. For example, the set in . It will turn out that regularity (7) is regular if and only if corresponds to controllability when we consider a special class that is useful for systems analysis. of Some examples of admissible, regular, rank-one separable in (7) with and sets include

(11)

(7) Also note that a general Hermitian form can be expressed as where satisfies and . Then alternative expressions for the aforementioned -procedures are given by1 (8) (9) where

is a set specified by

as follows:

rank

(10)

on in For brevity, we will omit the dependence of and the development below. Clearly, the -procedures are completely specified by the set through (8)–(10). We consider a generalization of the -procedure to the case where is an arbitrary subset of Hermitian matrices, rather than the specific set in (7). In this case, the equivalence in (8) or (9) no longer holds in general. In fact, it is easy to verify that the implication “ ” always holds, but the converse “ ” may or may not, depending upon and . The -procedure is said to be lossless if not only “ ” but also “ ” hold. B. Characterization of Lossless -Procedure The objective of this section is to show under what condithe associated -procedure is lossless regardless of tion on . The previous result in [6] provided a sufficient condition for the strict version of the -procedure (9) to be lossless. Our result below shows that the previous condition turns out to be also necessary, and is valid for the nonstrict case (8) as well under a regularity assumption. To this end, let us introduce the following. is said to be Definition 1: A set

tr(2S )  0

tr(2 )  0

1Notation means that S holds for all S type of notation will be used throughout this section.

2S

It is straightforward to verify that these sets are admissible and regular. The rank-one separability has been proven for (7) in in [3], [35], and for in [6]. The set re[12], for lates to the entire frequency range in the continuous-time setassociated with via (10) is ting. Specifically, the set with , given by matrices of the form for some whenever [3]. On the other hand, the set relates to a finite frequency range by and [6], [19]. The multiplier associated with the ( , ) scaling that gives a real upper bound [36] has been shown to be rank-one . Other results in separable [19], and is closely related to the literature (e.g. [18] and [37]) may also be interpreted within the framework of rank-one separability. The following result provides a necessary and sufficient condition for the S-procedure to be lossless. Theorem 1 (S-procedure): Let an admissible set be given and define by (10). Then, the strict S-procedure is , if and only if lossless, i.e., (9) holds, for an arbitrary is rank-one separable. Moreover, assuming that is regular, the nonstrict -procedure is lossless, i.e., (8) holds, for any , if and only if is rank-one separable. Proof: We shall prove the result for the nonstrict inequality case. The strict inequality case can be proven similarly. is rank-one separable. Let us first prove sufficiency. Suppose Note that the implication “ ” in (8) is obvious. To show the . Then, from Lemma converse, suppose that guarantees that there exists 10 in Appendix II, regularity of such that has no solution . From an a separating hyperplane argument (Lemma 11 in Appendix II), such that there exists a nonzero

Since

. This 2A

set

is a convex cone containing the origin, we have

M is a cone if M 2 M implies  M 2 M for all   0.

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Thus, trices

. Since is rank-one separable, there exist masuch that . Note that implies that must be true for some index . This means that the condition on the left in (8) does not hold, and we conclude that “ ” holds in (8). We now prove necessity. The main idea follows a result by Fradkov [17, Th. 19] that proved necessity for a different case is of the type given by (7). Suppose is not rank-one where such that and separable. Then there exists because . Note that the latter condition is disjoint from . Since means that the point is a closed convex cone and , the infimum of the disand is strictly positive. Consequently, tance between a separation theorem [38, Th. 11.4] infers that the two sets are strongly separable by a hyperplane, i.e., there exists a Hermitian matrix such that and

Clearly, the second condition implies . On the other hand, the first condition implies that because otherwise we have the following contradiction: For such that

where we noted that , and due to and . Thus, we have shown that if is not rank-one separable, then there exists for which the left-hand side (LHS) of (8) holds but the right-hand side (RHS) does not. Hence, for such , (8) does not hold, proving necessity. Theorem 1 shows that the rank-one separability is necessary and sufficient for the -procedure to be lossless regardless of the choice of . It should be noted that the rank-one separability may not be necessary for a particular choice of . Indeed, a recent result by Ebihara et al. [39] provides a sufficient condition for losslessness when is negative semidefinite, and their condition is weaker than the notion of rank-one separability. A class of rank-one separable sets can be generated by a transformation of a particular rank-one separable set as follows. The result will be used to derive a generalized KYP lemma in a later section. be a rank-one separable set. Then Lemma 1: Let the set is rank-one separable for any matrix and subset of positive–semidefinite matrices containing the origin. be a nonzero positive–semidefinite Proof: Let matrix such that . Since , we have . Since is rank-one separable, there exist such that an integer and vectors

and . By [34, Th. 2.3.1], the former implies existence of such that

Then, noting that

hold for all

, it can be verified that

and

IV. FREQUENCY RANGE CHARACTERIZATION One of the key developments in this paper is a unified characterization of finite frequency ranges for both continuous-time and discrete-time systems. In general, a frequency range is visualized as a curve (or curves) on the complex plane. For instance, a low frequency range in the continuous-time setting is a line segment on the imaginary axis containing the origin. A frequency range in the discrete-time setting would be an arc of the unit circle. The objective of this section is to propose a general framework for representing various curves (frequency ranges), which will later be used for characterizing FDIs. Let us first clarify what we mean by curves on the complex plane. Definition 2: A curve on the complex plane is a collection of continuously parametrized by infinitely many points for where , and .A is said to represent a curve (or set of complex numbers curves) if it is a union of a finite number of curve(s). We consider the following set of complex numbers that represents a certain class of curves: (12) are given matrices and the function is where , defined in (4). The dependence of on and may be made when no confusion implicit and is used in place of should arise. We will explain what type of curves can be repthrough what choices of and . The resented by has been extensively studied in the literaspecial case ture (e.g. [40]) with the notion of “ -stability”, and the previous work provides a basis for the analysis in the next paragraph. is the intersection of and Note that the set . It can readily be verified that the set represents . If , then it is a curve if and only if an empty set, the entire complex plane, or a set with a single , then it is either a circle or a straight line. element. If Conversely, every circle or a straight line can be represented by with such that . In particular

defines the circle of radius Without loss of generality, we may assume that to be zero. Define lowing for some be a full rank factorization. Then . By [34, Th. 2.3.8], there exists such that

. This completes the proof.

with center at

, and

by aland defines a straight line with the normal vector . The set is a region on the complex plane with the boundary

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whenever ; it is the inside or outside of a , then it is the entire complex circle, or a half plane. If and , it is plane. The remaining case is trivial; if either an empty set or a set with a single element. represents curve(s) only if We now see that the set and either or , in which case, it can be partial or whole segment(s) of a circle or a straight line. When the necessary condition is satisfied, it is possible that the and is empty or intersection of the two sets a single point, rendering the condition not sufficient. To state to represent a necessary and sufficient condition for curve(s), let us introduce a simultaneous matrix factorization result. A proof is given in Appendix I. Lemma 2: Let , be given. Suppose . Then there exists a common congruence transformation such that (13) (14) where , , and . In particular, and can . be ordered to satisfy With the factorizations in (13), the essential part of the set may be captured by the canonical set . Specifically, the former can be generated by the bilinear transformation of the latter. The following lemma formally states and proves this fact in the general setting where , are arbitrary. and nonsingular be Lemma 3: Let , by given. Define scalars , , , and and function (15) Then the following holds true:

Conversely, let that

be an element of the set on the LHS. Note

Hence, is well defined and it can be verified that . Since then follows from the previous identity, we see that belongs to the set on the RHS of (16). with and deLet us now examine the set fined in (14). Note that holds if and only if for some . Moreover, for such , we have . Hence, the set is the entire imaginary if , and is a partial segment of if axis ; otherwise, it is either empty or a single point. Conis conceived as the image of (a segsequently, the set ment of) the imaginary axis mapped through the bilinear transis an invertible, continuous mapping alformation. Since most everywhere on , represents curve(s) if and only does so. if The following result summarizes the aforementioned development. Its formal proof is omitted for brevity. be given and define the set Proposition 1: Let , by (12). Then the set represents curve(s) on the complex plane if and only if the following two conditions hold: ; • or ; • either where and are defined by the factorization in (13). In this case, there are two possibilities. I) is a circle or a line specified by since . is a partial segment (or II) segments) of a circle or a line specified by . By an appropriate choice of and in (12), the set can be specialized to define a certain range of the frequency variable . For the continuous-time setting, we have

(16) Proof: Let be an element of the set on the RHS of (16). such that and Then there exists . Note that the following identity holds for any and if :

From this identity and and have

, it readily follows that holds and, hence, we . Moreover

due to nonsingularity of . Hence, we see that seton the LHS of (16).

belongs to the

where is a subset of real numbers specified by an additional choice of , for instance, as follows:

where , and LF, HF, and MF stand for low, high, and middle frequency ranges, respectively. Similarly, for the discrete-time setting, we have

IWASAKI AND HARA: GENERALIZED KYP LEMMA

where

47

is a subset of real numbers specified by

where

, , and . Detailed derivations of these tables are omitted here due to space limitation, but can be found in a conference version of this paper [41]. The set can also capture a portion of the real axis as

This variation is not fully explored in this paper but will be useful for robustness analysis with respect to real parametric uncertainties (e.g., [42]). It is worth noting that an interval on the real axis can also be treated by the unit circle [43] through the Bliman’s transformation . V. GENERALIZED KYP LEMMA A. Main Theorem This section derives a generalized KYP lemma using the -procedure in (9) or (8); an FDI will be specified by the LHS of each statement, and the RHS then provides an LMI captures condition for satisfaction of the FDI. In particular, the input–output graph of a system in a certain frequency range. To elaborate on this point, let us first discuss the standard KYP would be given by lemma. For the FDI in (1), the set

and to show that the as in (10) by an appropriate choice of possesses the properties in Definition 1 so that the chosen set -procedure is lossless. We will take these steps in the “canonical coordinates” via the bilinear transformation. In particular, can be reduced we first show that the case with general through the simultaneous factorization in to that with (13), and then take the aforementioned two steps for the latter case. and a nonsingular matrix Lemma 4: Let , be given and define , by (13). Consider in in (12), and in (19). Suppose (18), represents curve(s). The following conditions on a given vector are equivalent. holds for some . i) holds for some . ii) Proof: When statement i) holds, there are three possible , b) , , or c) , , cases: a) where the entries of matrix are defined by (15). Hence, in view is of Lemma 3, we see that i) holds if and only if: a) , b) , , unbounded and and , or c) for some such . Through some algebraic manipulations and using that Lemma 13 in Appendix II, it can be verified that the condition is , is unbounded, and equivalent to: a) ; or , , and , b) , is unbounded, and , or for some such that . c) We now see that the condition is equivalent to for some . The previous lemma allows us to convert the -procedure to a standard form. In particular for some is equivalent to

This set can also be characterized (in the spirit of [44]) as for some (17) where

and (18)

We now generalize this characterization of the FDI with respect to the frequency range. Recall that the set in (12) captures finite frequency ranges of our interest. Hence, we consider the general frequency range in (12) for the definition of in (17) where is defined as (if is bounded) (otherwise).

(19)

Finally, the system description is also generalized by considering a completely arbitrary matrix rather than the one with the structure as in (18). It can now be seen that the main technical steps to arrive at in (17) the generalized KYP lemma are to express the set

where , and similarly for the nonstrict inequality case. Note that, if the factorization in (13) is used, the is (a segment of) the imaginary axis, in which set case some prior results already exist [3], [6], [7], [45]. Here, we provide the aforementioned two steps by two lemmas. and , be given Lemma 5: Let by (19) such that in (12) represents curves. Define and and (18), respectively. Then, the set defined in (17) can be characterized by (10) with

(20) Proof: We will first prove that the following two conditions are equivalent: for some ; i) for all , , ; ii) and are defined in (14). First note that i) holds where holds for some such if and only if either a) that , or b) is unbounded (i.e. )

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and Since

, where , are defined by . represents curve(s), there are two possibilities: and . In the former case, [3, Cor. 4] proves . It is then straightforward that i) is equivalent to to show that this condition is further equivalent to ii). In the latter case, [6, Lemma 2] can be used to show the equivalence between i) and ii). be defined by (17), and be defined to be in Now, let (10) with (20). Then, for a nonzero vector for some for some for all

where the first and fourth equivalences easily follow from the definitions, and the second and third hold due to Lemma 4 and ii), respectively. the previous equivalence i) is a closed convex cone. When , the set Clearly, is nonempty and hence is admissible. From Lemma 2, we have

is regular and the matrix is a minimal realb) The set in the sense that ization of the set

(21) if and only if (22) We are now ready to state and prove a general theorem. , Theorem 2 (Generalized KYP): Let matrices , and , be given and define and by (12) and (19), respectively. Suppose represents curves on the null space of where the complex plane. Denote by is defined in (18). The following statements are equivalent. . i) ii) There exist , such that and (23) Moreover, if the rank condition in (22) is satisfied, then the following statements are equivalent. . i) such that and ii) There exist , (24)

where case, defining

or

can be assumed. In the former

the set can be characterized as with defined in (11). Since is rank-one separable, it then follows from Lemma 1 that is rank-one separable. Similarly, , the set can be characterized as in the latter case for some matrix and for a subset of positive semidefinite matrices. Again, Lemma 1 infers that is rank-one separable. These arguments are summarized in statement a) of the following lemma. Statement b) gives a condition for regularity, and is proven in Appendix III. Lemma 6: Let matrices , and ( , ) be given such that in (12) represents curves. Define by (20) and the matrix-valued mapping by (18). the set The following statements hold true. a) The set is admissible and rank-one separable.

Proof: We will prove the strict inequality case. The nonstrict inequality case can be shown similarly. Note that (i) holds if and only if holds where is defined in (17). can be characterized by (10) with From Lemma 5, the set in (20). By Lemma 6, the set is admissible and rank-one separable. Hence, from the generalized -procedure (Theorem 1), satisfying condition is equivalent to the existence of , or equivalently, the existence of , such and (23) hold. Since the inequality in (23) is strict, that we can strengthen the positivity of as without loss of generality. In view of the developments in Section IV, the term in statement ii) of Theorem 2 can be specialized to the low/middle/high frequency ranges in the continuous/discretetime settings as shown in the equation at the bottom of the page. When , , , and are all real matrices, and in statement ii) of Theorem 2 can be restricted to be real without loss of generality. This follows basically from the fact that the real part of a complex Hermitian positive–definite matrix is positive definite. Note that, if the frequency region is not symmetric about the real axis, then one needs to search for complex and

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even when and are real. In such case, the LMI in complex variables can be converted to an LMI of larger dimension in real variables through the following equivalence:

is controllable. Let be the set of eigenvalues b) the pair of in . Then, the following statements are equivalent. , we have i) For each

ii) There exist where

and

,

such that

and

are real square matrices. (27)

B. Specific Extensions of the KYP Lemma The general result in Theorem 2 can be specialized to several versions of the KYP lemma. There are two aspects of specialization: the frequency range and the underlying system. The former has been briefly discussed at the end of the previous section. We will devote this section to the latter. In particular, we consider FDIs for descriptor and state space systems as well as for polynomial functions. Theorem 3 (Descriptor, Strict Inequality Version): Let ma, , , , and , trices , be given and define and by (12) and (19), respectively. Suppose that: a) represents curves on the complex plane, b) for all , and c) either is nonsingular or is bounded. Then, the following statements are equivalent. i) For , we have for all . such that and ii) There exist , (25) Proof: The result follows from Theorem 2 by choosing (26) If is bounded, then the result is immediate by noting that the is given by for all null space of under the invertibility assumption on . If is unbounded, coincides with the null we need to make sure that the above at as well. This is indeed the case if is space of nonsingular. In Theorem 3, statement ii) implies that for , provided that the upper left block of , denoted all , is positive semidefinite. To see this, let us assume ii) and by for some . Let be a vector in the null space of , i.e., . Denote by the upper block of the matrix on the LHS of (25). Then left

which cannot be true as every terms are nonnegative. By contrafor any diction, we conclude that ii) implies . Based on this observation, Theorem 3 can be modified as follows: Replace supposition b) with , and add the for all to statement i). condition Theorem 4: (State–space, nonstrict inequality version) Let , , , and , matrices be given and define and by (12) and (19), respectively. Suppose that a) represents curves on the complex plane, and

Proof: The result basically follows from the nonstrict inequality case of Theorem 2 by choosing as in (26) with and . Note that the rank condition in (22) translates to and nonsingularity of when . controllability of The only crucial step in this proof is to show that statement i) is equivalent to (28) where is the null space of Theorem 3, can be given by Hence, i) is equivalent to

. As discussed in the proof of if .

By Lemma 12 in Appendix II, this condition is equivalent to (28) under the controllability assumption. Theorems 3 and 4 capture certain properties of dynamical systems expressed in terms of state space, possibly descriptor, realizations. The results include, as special cases, the standard version of the KYP lemma [3], its finite frequency extensions [6], [7], and a unified (continuous/discrete-time) FDI charac) [46]. A terization for the entire frequency range (i.e., and may also be special case of Theorem 3 with obtained through an application of a recent, independent result by Scherer [21]. The following result characterizes certain properties of polynomials, and is particularly useful for the design of digital filters as discussed later. Theorem 5 (Polynomial version): Let matrices , , and , be given and define by (12). Suppose that represents curves on the complex plane. Define (29) Then, the following statements are equivalent. i) holds for all . such that and ii) There exist ,

(30) where .

,

,

, and

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The equivalence also holds when the three inequalities are replaced by strict inequalities with the modification that condiis added to statement i) if is unbounded, tion where is the upper left block of . , define for Proof: For a vector . Then and hold where and . Noting that , we see that, for a fixed , condition holds if and only if for all

such that

in different frequency ranges this type, with different sets where . We will address such design problems in the next section. The objective of this section is to show what classes of can be treated within the framework of our generalized KYP lemma. Recall that the FDIs arising in various versions of the KYP lemma (theorems in Section V) are given in terms of quadratic forms. Thus, a set can be completely captured within our framework if it can be described by a quadratic form. The idea is and , a transfer function to construct, from given and a Hermitian matrix such that (31)

If is bounded, the result then directly follows from Theorem 2. When is unbounded, Theorem 2 infers equivalence of i) and ii) if condition for all

such that

for each fixed frequency. If is a polynomial, then the FDI in a given frequency range can be characterized by an LMI using Theorem 5. If it is a transfer function , then the FDI can be expressed as

is added to statement i). This additional condition is equivalent . We claim that this is already implied by the to original statement i) when is unbounded. To see this, note that

This implies that has a positive eigenvalue for with a sufficiently large magnitude, whenever . This completes the proof for the nonstrict inequality case. Finally, the strict inequality case can be proved in the same is not immanner. Note, however, that condition for all in general and, hence, plied by has to be added to statement i). Versions of positive-real lemma for polynomials have been obtained in the standard (unrestricted) frequency setting [47]–[49]. However, results in the restricted frequency setting are limited. References [8]–[10] gave characterizations of scalar-valued quasipolynomials that are nonnegative on the unit circle. Our results extend and unify these prior results to the case of matrix-valued positive definite polynomials in both continuous-time and discrete-time settings with both strict and nonstrict inequality cases. In particular, a convex parametrization of positive quasipolynomials as in [9], [10] can be obtained from Theorem 5; see [50] for details.

VI. FREQUENCY RESPONSE SPECIFICATIONS Consider a transfer function that depends on the deinclude the opensign parameter vector . Examples of , the closed-loop sensitivity funcloop transfer function tions and , and the digital filter as discussed in Section II. The design problem is to find such that

where is a prescribed subset of the complex matrices. More generally, may be required to satisfy multiple constraints of

(32) allows us to treat the FDI Thus, the appropriate choice of within the framework of Theorem 3 (or Theorem 4 when ). Hence, the question is how to construct and so that (31) holds. We will answer this for different classes of in the following sections. A. Half Plane and Circular Regions The simplest choice of is characterized by the FDI in (31) is

. In this case, the set

(33) This set captures some fundamental properties. For instance, the small gain condition is given by

and the positive-real condition is given by

To discuss a more general specification on the Nyquist represents a plot, let us consider the case where single-input–single-output (SISO) system. In this case, for all means that the segment of the lies in the Nyquist plot specified by the frequency range region on the complex plane. As discussed in Section IV, the , region defined by (33) is nontrivial if and only if in which case, it is given by a half plane or a region inside or outside of a circle. Clearly, the small gain (disk) and the positive-real (right-half plane) conditions are characterized by special cases of such regions.

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B. General Conic Sections

TABLE I 10 CASES OF VARIOUS REGIONS

is a scalar (SISO) Next we consider the case where transfer function and is a general conic section on the complex plane.3 Recall that a conic section can be characterized by (34) . The following lemma shows that for some real matrix this characterization can be converted to another which is compatible with our description of the frequency domain inequality. be given and consider Lemma 7: Let a real matrix the set of complex numbers defined in (34). This set can also be characterized by

where

is defined by

Proof: Note that

Then, it is straightforward to see that

from which the result follows. Lemma 7 allows us to establish the equivalence in (31) when is a conic section by choosing appropriate and . In parbelongs to the conic section ticular, the transfer function defined in (34) if and only if is chosen as shown in Lemma 7 and is specified as

where

is defined by condition for all for the continuous-time case for the discrete-time case. Thus, for and , the conic section condition can each frequency with the above augmentation, and be converted to the latter can further be converted to an LMI via theorems in Section V. The conic sections captured by (34) are summarized later. The proof is omitted for brevity, and can be found in [50]. be given and conProposition 2: Let a real matrix sider the set in (34). Partition as , e.g.,

Fig. 2.

Essential regions for 4 .

which is parametrized by a vector

satisfying

In particular, it is obtained from one of the ten regions in the -plane (see Table I and Fig. 2) through a rotation by and a . shift by VII. APPLICATIONS TO SYSTEM DESIGN

and denote the spectral decomposition of by where and . Then, the set is nontrivial and has an interior if and only if is indefinite. In this case, is given by the set of such that

This section is devoted to synthesis problems in engineering applications that can be solved using the LMI characterization of the FDI in the generalized KYP lemma. A. Basic Idea for Synthesis

3The developments below can be extended for multiple-input–multiple-output (MIMO) square plants, but the physical meaning is not completely clear.

From the discussion in the previous section, a general synthesis problem can be formulated as follows. For each , let transfer function , Hermitian matrix , and frequency range be given where depends on the

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parameter vector for some ,

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to be designed, and is specified by (12) . The problem is to find such that (35)

for all . We show how the problem can be solved exactly under certain assumptions. For brevity of exposition, let , i.e., we have just one specification. us consider the case The case with multiple specifications can be handled similarly. In the sequel, we will drop the subscript “ ” in (35). A standing assumption is that the design parameter vector appears affinely in . If is a polynomial, then its is a proper rational coefficients are affine functions of . If transfer function, then it has a state–space realization where and are affine functions of while and are constant matrices. Another assumption is such that is that the weighting matrix (36) where and are the numbers of inputs and outputs of . This condition makes the set of satisfying convex. Note in particular that, when , the design specification (35) defines a convex region in the complex is desired to lie. This framework capplane in which tures such standard specifications as the bounded-real and positive-real properties. Under these assumptions, the condition in (35) is clearly convex in terms of the design parameter . Furthermore, theorems in Section V can be used to reduce the problem to an LMI. We will explicitly show how this can be done using Theorem is a proper transfer function. Other 4 for the case where is a polynomial or a nonproper (descriptor) cases where transfer function can be treated similarly using Theorem 5 or 3, except that the inequality in (35) has to be made strict in the latter case. Lemma 8: For a transfer function , the condition , holds if and only if such that there exist Hermitian matrices and (37) holds, where

provided (36) holds. Proof: This result follows directly from Theorem 4 with in (32) and the Schur complement. When and are affine functions of the design parameter vector , (37) defines an LMI in terms of the variables , , and . Therefore, can be computed via convex programming.

The reduction of the synthesis problem (35) to LMIs is rather straightforward once the proper assumptions are imposed. Attention must be paid, however, to when the assumptions are satisfied, and how they can be enforced through reparametrization of the design parameters. We will discuss these points for several engineering problems in the sequel. Structure/control design integration: The specification is given by the FFPR condition in (6), which can be converted to an LMI via Theorem 4. The design parameters would apfor certain problem settings, pear only in the numerator of including actuator or sensor placement. The design space may is affine in the new design then be reparametrized such that parameter vector . In this case, the design process is reduced to an LMI problem, which can be solved via semidefinite programming. This situation occurs in some practical design problems [7], [31]. Open-loop shaping: The control design via open-loop shaping discussed in Section II falls into the aforementioned framework if the poles of the controller are fixed a priori. An important practical problem that naturally fits in here is the PID control design. The controller transfer function is described by

(38) is a small parameter introduced to approximate where the differentiator by a proper transfer function. If is fixed, all the design parameters ( , , and ) appear linearly in the open-loop transfer function . Various specifications on the Nyquist plot of may be expressed as confor all where straints of the form is a frequency interval and is the corresponding convex region on the complex plane such as a disk, half plane, and a conic section. The idea described in Section VI can be used to in terms of quadratic forms of some (possibly represent as in (35). The design paramaugmented) transfer function eters will enter linearly and, hence, the PID controller can be designed by solving an LMI as in Lemma 8. Closed-loop control design: This is a fundamental problem of designing a controller to satisfy various constraints on the , including the sensiclosed-loop transfer functions tivity-shaping problem discussed in Section II as a special case. With the Youla parametrization of all stabilizing controllers [51], each closed-loop transfer function depends on the free in an affine manner. Thus the search for is parameter a convex problem, provided the constraints are convex in terms of the closed-loop transfer functions [52]. Since the infinite dimensionality of the parameter space is problematic, one often approximates the space by a finite dimensional space spanned by a selected set of basis functions. The problem then becomes the search for the coefficients of the basis functions, which can be converted to an LMI problem within the above framework satisfying if the specifications are given by (35) with each (36). Digital filter design: This is another problem discussed in Section II. Let us first consider the finite impulse response (FIR) filter synthesis problem. The denominator of the FIR filter

IWASAKI AND HARA: GENERALIZED KYP LEMMA

is fixed to

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, and a state–space

realization of (39) may be given by the pair form and

in the controllable canonical

where are the coefficients of . Note that and are fixed matrices and the design parameters appear linearly in and . Moreover, it is not difficult to see that . Therefore, the constraints in (5) are convex in terms of Lemma 8 can be directly applied. Next, we consider the infinite impulse response (IIR) filter and are design variables. The design where both problem is fundamentally more difficult than the FIR case because the above specifications are not convex in terms of , and the simple-minded approach described previously does not yield LMI problems due to the dependence of on , the coefficients of . However, using the ideas of [9], [10], [53], and [54], the problem can be made tractable when the specifications are given by inequality constraints only. For instance, one can handle the on the gain specifications (5) if the third constraint is replaced by

This type of gain specifications can be converted to FDIs in terms of polynomials based on the Fejer–Riesz result [54, Th. 8.4.5]. Hence, the generalized KYP lemma for polynomials (Theorem 5) will be useful here. Finally, in certain applications, design objectives may be reasonably well described by gain constraints only, but the IIR filter design with both gain and phase constraints still remains as an important practical problem that needs to be addressed. B. Design examples 1) PID Controller: We consider the design of PID controllers (38) for a mechanical plant with a lightly damped flexible mode

The design objective is to have a good tracking performance with a reasonable stability margin and robustness against unmodeled dynamics. Our approach is to shape the Nyquist plot . In parof the open-loop transfer function ticular, we consider the following specifications: a) b) c) Specification a) with small ensures robustness against unmodeled dynamics which typically exists in the high frequency

range. Specification b) is meant to guarantee a certain stability margin. It may be more natural to require the Nyquist plot to be . Howoutside of a circle with its center at the point ever, such requirement leads to a nonconvex region on the complex plane and our design method cannot be applied as condition (36) is not satisfied. On the other hand, the above Spec ification b) is a half-plane requirement which can be handled within our framework. The frequency range corresponding to Specification b) would be the entire imaginary axis, but a small lower bound has been introduced to exclude from the range and to avoid a possible numerical difficulty due to the controller ensures pole at the origin. Specification c) with a large sensitivity reduction in the low frequency range by making high gain. Again, would be more direct for the purpose but does not define a convex constraint on . Specification c) is a reasonable alternative, given that the phase angle of apas approaches 0 from above. proaches We have designed a PID controller by minimizing subject to Specifications a)–c) where the other parameters are fixed as

The resulting optimal PID controller is found to be

with the optimal value . The Nyquist plot, sensitivity functions, and the step response are depicted in Figs. 3–5, respectively. Also plotted for comparison are the result of a popular heuristic PID tuning rule (Ziegler–Nichols ultimate sensitivity method) [55] where the PID parameters are given by

The stars and the (small) circles in Fig. 3 indicate the points on the Nyquist plots at frequencies and , respectively. We see from Fig. 3 that the constraints in Specifications b) and c) are active for the optimal PID control, and the optimization clearly has improved the stability margin and the sensitivity reduction in the low frequency range. The latter effect is transparent in Fig. 4, which shows that the optimization added more damping to the closed-loop poles than the Ziegler–Nichols design. This can also be seen directly from the oscillatory nature of the step responses plotted in Fig. 5. Overall, this example illustrates that our method allows for the tuning of the PID parameters to achieve desired frequency responses in prescribed frequency ranges. 2) FIR Filter: We consider the design of low-pass FIR filters to meet the following specifications: a) b) where is the filter transfer function to be designed. Spec, that the ification a) requires, within the pass-band has frequency characteristics similar to the desired filter function having unity gain and the linear phase with group

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Fig. 3. Nyquist plots. (Left) Optimized. (Right) Ziegler-Nichols.

Fig. 4.

Sensitivity functions. (Left) Optimized. (Right) Ziegler-Nichols.

Fig. 5. Step responses. (Solid) Optimized. (Dashed) Ziegler-Nichols.

delay , where is a positive integer.4 Specification b) ensures has a low gain in the stop-band . Let be that the order of the FIR filter to be designed. For fixed , , , , and , we design an FIR filter by minimizing subject to the above constraints. This is a Chebyshev approximation problem where the peak value of the frequency response is used as the design criteria. Most of the currently available methods for solving the Chebyshev approximation problem are based on the frequency 4A transfer function H (z ) is said to have a linear phase if it can be represented as H (z ) = jH (z )jz on the unit circle for some real positive scalar  . The parameter  is called the group delay.

gridding [22], [23], [25]. Some recent methods avoid frequency gridding using KYP lemmas [9], [10], [53], but these methods can deal only with a subclass of the problem where all the FDIs . No methods have been are given in terms of the gain of known, to our knowledge, to solve the FIR Chebyshev approximation problem with phase constraints as formulated above, without gridding the frequency axis. Our method naturally handles such problem. A standard approach in the literature to achieve the linear phase for FIR filters is to make the FIR coefficients symis even, it can be metric [22], [56]. For instance, when shown that the constraint on the FIR coefficients in (39) enforces the linear phase with group delay . With this symmetry constraint, Specification a) can be equivalently written as a condition on the gain only a) where we used the fact that holds on the unit with . Thus, the circle for a symmetric FIR filter design problem is exactly solvable by existing methods [9], [10] under the symmetry constraint. A drawback of this approach is that some design freedom is wasted to enforce the unnecessary constraint of the linear phase property in the stop-band. By removing the unnecessary constraint, the design freedom may be used to improve the overall

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Fig. 6.

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FIR responses. (a) n = 30, d = 15. (b) n = 30, d = 10. (c) n = 20, d = 10. (b’) n = 30, d = 10.

performance. We will illustrate this point by numerical examples in the sequel. We fix the frequency ranges and the gain bound in the stop-band as follows:

advantage of our method that allows us to treat the linear phase constraint in a finite frequency range, as opposed to the existing methods that enforce the constraint for all frequencies by symmetry of the FIR coefficients. VIII. CONCLUSION

and design several FIR filters for different choices of and . Fig. 6(a) shows the gain response of a typical symmetric FIR and , where the frequency is filter with normalized by . This filter is obtained as the optimal solution to the above Chebyshev approximation problem. The optimal . Now, consider performance value is found to be a situation where the group delay of time steps is too large and is not acceptable. Such small delay requirement would be important for certain real-time applications [25], [57]. Supis the desired group delay. Then, the standard pose that framework of symmetric FIR filters requires that the filter order . The gain response of the optimal FIR filter for be this case is shown in Fig. 6(c). The optimal performance value which is significantly larger than the previous is design, as seen by the nonflatness of the gain response in the pass-band. On the other hand, our design method is free from and, for instance, the group delay of the constraint may be (approximately) achieved with a filter of order equal to the original design, i.e., , as shown in Fig. 6(b) and (b’). We see that, within the pass-band, the linear phase requirement is practically met and the gain response is almost as good as the original . This example shows the

We have generalized the KYP lemma to allow for more flexibility in system classes and various frequency ranges. In particular, the system class is described by the null space of a certain frequency-dependent matrix, capturing descriptor systems and polynomials in addition to the standard state space systems. The frequency range is specified by two quadratic forms that define segment(s) of a circle or a line on the complex plane. In this way, an FDI condition can be considered in the low/middle/high frequency ranges for both continuous-time and discrete-time settings in a unified manner. The generalized KYP lemma converts a certain FDI in a finite frequency range to a numerically tractable LMI condition. When system design specifications are expressed in terms of such FDIs, the generalized KYP lemma may be used to reduce the problem to an LMI optimization. We have given some technical conditions under which this reduction is possible, and discussed various engineering applications that fit into the framework, including the design of digital filters, feedback controllers with fixed poles, and mechanical structures. Further developments have been reported at conferences, addressing such issues as the gain feedback design [58], controller order reduction, a robust generalized KYP lemma [59], and a GKYP design

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toolbox with MATLAB [60]. General control problems with dynamic output feedback, however, still remain open for further research. APPENDIX I SIMULTANEOUS MATRIX FACTORIZATION In this section, we prove Lemma 2. The following result is useful for this purpose. Lemma 9: Let be given. Then, admits the following factorization: (40) where , ,

and

with

In particular, and are the eigenvalues of Proof: Let the entries of be defined by

, then Lemma 9 guarantees it can holds. Let . Since and are the be decomposed as in (40) with , they can be ordered so that . eigenvalues of Then, the factorizations of and in (13) are obtained from (42) and (40) by noting the relation (41) and defining . APPENDIX II TECHNICAL LEMMAS and be given. Suppose Lemma 10: Let is an admissible, regular set. Then the following statethat ments are equivalent. such that . i) There exists , there exists such that ii) For each . ii) is obvious. To show the Proof: The implication i) converse, consider the following. such that , where iii) There exists

. We will show ii)

iii) i). Suppose iii) does not hold. Then, for all . Since is closed, is compact and, hence, there exists such that Then Let The spectral factorization of

be arbitrary and define

gives

Then, we have be verified that where the columns of are eigenvectors and and are eigenvalues. Note that , , and are all real. Moreover, since is real symmetric, can be chosen to satisfy . Thus, belongs to . Now, equating the two ex, we have pressions for

Finally, it can readily be verified that (41) holds for any . Substituting this expression into the second term, we obtain the result. Proof of Lemma 2: Since , there exists a nonsuch that singular matrix (42)

since is a cone, and it can readily . Therefore

Since is arbitrary, we see that ii) does not hold. Thus, we have shown ii) iii). Now, suppose iii) holds and let be such that . If , then and regularity implies , contradicting . So , in which case, is an element of and , proving i). satisfies Lemma 11: [19], [61] Let be a convex subset of , and be an affine function. The following statements are equivalent. i) The set is empty. ii) There exists nonzero such that is nonnegative for all . and , Lemma 12: Let matrices and sets of complex numbers and be given such that and the closure of coincides with . Denote by the null space of where is defined in (18). Suppose holds for all . Then, holds rank for all if and only if it holds for .

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Proof: The necessity is obvious. To prove the sufficiency, such that does suppose that there exists such that not hold. Then, there exists a vector and . Let . Then, and holds for all . we have is continuous on due to the rank assumption, Note that holds if is sufficiently close to and hence . Such can be chosen from because is . Thus, we have shown that if an element of the closure of holds for all , then it must also hold for . all in (12). Suppose it repLemma 13: Consider the set is resents curves on the complex plane. Then the set and . unbounded if and only if and . Then, is Proof: Suppose a straight line, while is a half plane or the outside of a , is clearly unbounded. circle. Hence, their intersection, This proves the sufficiency. The necessity can be proved by con, then is a circle and its subset tradiction. If must be bounded. If , then for with sufficiently large magnitude and hence is bounded. must be bounded. Therefore, its subset

APPENDIX III PROOF OF LEMMA 6 Statement a) has been proven in the paragraph above the lemma. We will prove statement b). It is tedious but straightforward to verify that (22) holds if and only if

holds where is an arbitrarily fixed nonsingular matrix. Then, b) is equivalent to the statement obtained by replacing , , , and in b). Hence, without and by , loss of generality, we prove b) for the case where and where and are given by (14). Note that is regular and (21) holds if and only if

(43) We first show the “only if” part of the proof. Suppose there exists such that rank . Then there exists . Let if a nonzero vector such that or and if . It can be verified that the is a counter example to (43). This choice proves necessity. To prove the converse, suppose that the rank condition (22) is has full row rank and hence there exists a satisfied. Then nonsingular matrix that transforms into the following form:

Note that (

,

) is controllable because

Let , be such that and . We will show that ( , ) is controllable, there exists has no eigenvalues on . Then

. Since such that

If or , then it follows from [3], [45] that implies under controllability of . implies , eliminating So we will show that with the remaining possibility. For , we have

Since represents curve(s), implies . , given by In this case, the relative interior of , is nonempty. We then have for all . Since , this implies has infinitely many roots in , and hence that must vanish identically. It then follows from Theorem 1 due to controllability (statement 12) in [62, Sec. 34] that of . ACKNOWLEDGMENT The would like to thank I. Yamada and H. Hasegawa at the Tokyo Institute of Technology (TIT), Japan, for their helpful comments on filtering design, and Y. Iwatani at TIT for his help with the PID design example. They would also like to thank C. Scherer at Delft University of Technology, The Netherlands, for providing them with a preprint of [21] accompanied by his insightful comments, to K. Murota at the University of Tokyo for helpful discussions on the -procedure, and to A. L. Fradkov at the Institute for Problems of Mechanical Engineering for directing their attention to the connection between [6] and [17], which has lead to the necessity proof of Theorem 1. REFERENCES [1] B. D. O. Anderson, “A system theory criterion for positive real matrices,” SIAM J. Control, vol. 5, pp. 171–182, 1967. [2] J. C. Willems, “Least squares stationary optimal control and the algebraic Riccati equation,” IEEE Trans. Autom. Control, vol. AC-16, no. 6, pp. 621–634, Dec. 1971. [3] A. Rantzer, “On the Kalman-Yakubovich-Popov lemma,” Syst. Control Lett., vol. 28, no. 1, pp. 7–10, 1996. [4] S. Boyarski and U. Shaked, Discrete-Time and Control of systems with structured disturbances under probability requirements, 2003.

H

H

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IWASAKI AND HARA: GENERALIZED KYP LEMMA

Tetsuya Iwasaki (S’89–M’94–SM’01) received the B.E. and M.E. degrees in electrical and electronic engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 1987 and 1990, respectively, and the Ph.D. degree from the School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN, in 1993. He held a postdoctoral position at Purdue University. After holding a faculty position for five years at the Tokyo Institute of Technology, he moved to the University of Virginia, Charlottesville, in May 2000, where he is currently a Professor. His research interests include robust and optimal control, and modeling and control of biological sensing, locomotion, and oscillation. Dr. Iwasaki received the 2002 Pioneer Prize from the Society of Instrument and Control Engineers, and the 2003 National Science Foundation CAREER Award. He is a past Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, and currently is on the Editorial Boards of Automatica and Systems and Control Letters.

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Shinji Hara (M’87–SM’04) was born in Izumo, Japan, in 1952. He received the B.S., M.S., and Ph.D. degrees in engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 1974, 1976, and 1981, respectively. From 1976 to 1980, he was a Research Member of Nippon Telegraph and Telephone Public Corporation, Japan. He served as a Research Associate at the Technological University of Nagaoka, Japan, from 1980 to 1984. In 1984, he joined the Tokyo Institute of Technology as an Associate Professor and has served as a Full Professor for ten years. Since 2001, he has been a Full Professor in the Department of Information Physics and Computing, The University of Tokyo. His current research interests are in robust control, sampled-data control, hybrid control, learning control, quantum control, and computational aspects of control system design. He is a Member of SICE and ISCIE. Dr. Hara was a BoG member of the IEEE Control Systems Society, the General Chair of the CCA04, and an Associate Editor of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and Automatica. He received Best Paper Awards from SICE (the Society of Instrument and Control Engineers, Japan) in 1987, 1991, 1992, 1997, and 1998, from the Japan Society for Simulation Technology in 2001, and from ISCIE in 2002.