Symbol R R + C Table : Notation Meaning Set of all real nmbers Set of positive real nmbers Set of all complex nmbers A(s) T, conjgate transpose A(s) Λ

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1 EESystems Department, University of Sothern California. March 000. Mltiplier IQCs for Uncertain Time-delays Myngsoo Jn and Michael G. Safonov Dept. of Electrical Engineering Systems University of Sothern California Los Angeles, CA , USA Key Words: Time-delay system, stability criteria, mltiplier, robst control, IQC Abstract This paper describes a set of delay-dependent IQC's for time-delay ncertainty. The set is linearly parameterized in terms of the freqency-response of a complex scalar-valed mltiplier. Using LMI optimization techniqes, one may compte optimal mltipliers and thereby obtain less conservative IQC stability robstness bonds for systems with ncertain time-delays. Introdction Stability criteria for time-delay systems tend to fall into one of two categories according to their dependence pon delay size: delay-dependent or delay-independent. Delay-independent criteria provide conditions for stability withot regard for the size of the time delays. They tend to be more conservative than delaydependent criteria which may exploit prior knowledge of pper-bonds on the amont of time-delay. The robst stability methodology is sefl in dealing with strctred ncertainties (see [],[4]). Time-delays can be considered as strctred ncertainties and timedelay systems can be analyzed sing these robst control theories[]. Many of methods that have been developed within the area of robst control dring the last decade have been shown to be reformlated to fall within the framework of the integral qadratic constraints (IQC's)[8]. F et al.[5] provided two delaydependent reslts for robst stability sing this IQC approach and the linear matrix ineqalities (LMI's) techniqe, which give an estimate of the maximm time-delay which preserves robst stability. Some recent papers on time-delay systems, like [9], [3] and [7], derive sfficient conditions for stability in the form of LMI sing Lyapnov fnctionals. Others have sed inpt-otpt stability theories sch as the smallgain criterion [] and its generalizations, like the IQC stability analysis method [8]. The inpt-otpt meth- Research sponsored in part by AFOSR Grant F Corresponding athor; msafonov@sc.ed; fax ods seem to offer advantages over Lyapnov methods in facilitating the decomposition of the stability robstness analysis problem for complex systems having several sorts of ncertain sbsystems into sbproblems of finding an IQC for each of the sbsystems e.g., for each ncertain time-delay and each nonlinearity or other ncertainty. Once the sbsystem IQC's are in hand, stability analysis for the composite system is then a relatively straightforward matter of optimizing IQC scalings (and, sometimes, other free parameters like Popov mltipliers) in an effort to identify a single aggregate IQC for the system. This paper considers robst stability analysis of systems with time delay based on an IQC approach. It is organized as follows: Notation and preliminary backgrond are provided in Section. The problem formlation is given in Section 3. The main reslt is derived in Section 4 where it is shown that the class of known delay-dependent IQC's for time delays can be generalized to a larger linearly-parameterized class. Discssion and comparison with other reslts are in Section 6. Finally, reslts are smmarized and conclsions are stated in Section 7. Backgrond and Preliminaries Definition (cf. [8]) Consider the feedback system in Figre where G, are casal operators and G has transfer fnction G(s). We say that the interconnection G and is well-posed if the operator I G I has a casal inverse. The interconnection is stable if, additionally, the inverse is bonded. Theorem (The IQC Theorem) [8, 6] Let G(s) RH l m, and let : Ll e[0; ) 7! L m e[0; ) be a bonded casal operator. Assme that: i) for every ff [0; ], the interconnection of G and ff is well-posed where ff is a parameterization of which satisfies a) ffj ff,

2 Symbol R R + C Table : Notation Meaning Set of all real nmbers Set of positive real nmbers Set of all complex nmbers A(s) T, conjgate transpose A(s) Λ herm(m) (m + mλ ) skew(m) (m mλ ) Re( ) Real part of ( ) Im( ) Imaginary part of ( ) ^x(j!) Forier transform of the signal x(t) <x;y> R y(t)t x(t)dt ß R ^y(j!)λ^x(j!)d! kxk p <x;x> b) ff is bonded and casal for ff [0; ], c) there exists >0sch that k ff (y) ff (y)k jff ff j kyk () for all ff ;ff [0; ], ii) the interconnection of G and ffj ff0 is stable, iii) for every ff [0; ], the IQC defined by Π is satisfied by ff, that is, fi y y Π ; 0; () ff(y) ff(y) iv) there exists f>0sch that Λ G(j!) G(j!) Π(j!) fi; 8! R: (3) I I Then, the feedback interconnection of G and is stable. I 0 0 I The vales and of Π(j!) represent the 0 I I 0 small gain theorem and the positivity theorem. The positivity theorem with mltiplier can be reformlated with IQC defined by Π(j!) 0 M Λ (j!) 3 Problem Formlation : (4) Consider a time-delay ncertainty (s)e fis where fi [0; μfi) and fi is assmed to be constant bt nknown. In this paper, we provide a class of Π's for time delay ncertainty with an appropriate mltiplier M(j!) and transformation S(j!). Problem Given a time-delay ncertainty (s) e fis, find a class of Π which satisfies the IQC (). v - h 6 - G(s) ff y? hff Figre : Basic feedback configration 4 Main Reslts Lemma Sppose Π ii R;i ;and Π C. If det Π < 0 and Π 6 0, then the locs (j!) which satisfy the qadratic eqation Λ Π Π (j!) Π Λ 0 (5) Π (j!) is the circle C(Π(j!)) f jj (j!) cj r g where c ΠΛ Π and r q jπj Π Π Π p det Π jπ j. Λ Π Π 0 (j!) Π Λ Π (j!) Π +Π + ΛΠ Λ + ΛΠ Π Λ +Π ΛΠ Λ +Π Π +Π Λ Π +Π ΠΛ +Π ΠΛ + Π Π Π Λ Π Π j +Π ΠΛ j +Π Π Π Λ Π Π j cj r If Π 0 and Π 6 0, the set of (j!)'s which satisfy Eq. (5) is the line which can be represented by the eqation (Π +Π Λ ) (j!)+ (j!)λ + (Π Π Λ (j!) (j!)λ ) + Π 0; (6) that is, Re(Π ) Re( (j!)) Im(Π ) Im( (j!)) w + Π 0: (7) We define ~y(j!) and ~(j!) to satisfy the relation ~y y S(j!) with appropriate transformation ma- ~ trix S(j!). We also define (j!) 4 (j!)y(j!) and p.

3 ~ (j!) 4 ~(j!)~y(j!). We will se the symbol! Λ to denote! Λ 4 μfi!. The Lemma talks abot a transformation S(j!) which maps an arc to the positive real line. Lemma Let the transformation matrix S(j!) be S(j!) 4 cos!λ + j sin! Λ cos! Λ + j sin! Λ : (8) sin! Λ sin! Λ Then, (j!) f j e j' ; ' [0;! Λ ) g and! Λ [0;ß) if and only if ~ (j!) R + [f0g. ~ (j!) (only if) sin! Λ sin! Λ (j!) (cos! Λ + j sin! Λ ) + (cos! Λ + j sin! Λ ) (j!) cot! Λ + j + j (j!) (j!) sin ' + j( + cos ') cot! Λ + cos ' + j sin ' cot! Λ + sin ' cos ' cot! Λ + tan ' cot! Λ + cot ' : We can see that ~ (j!) R + [f0g since ' <! Λ and cot! Λ is strictly increasing when! Λ [0;ß). (if) This can be easily proved from the fact that S(j!) is invertible for 8! Λ [0;ß) and ~ (j!) has all vales in R + [f0g. Now we state or main theorem which provides a linearly parameterized set of Π's for time delay ncertainty. Theorem (Main Theorem) Let (j!) e j!fi 4 ; fi [0; μfi),! Λ μfi!. Then, (j!) satis- fies the IQC () for Π(j!) 8 ψ " herm M(j!) >< ψ >: herm M(j!) # h i! e j!λ e j!λ ;! Λ [0;ß) " 0 0 with Re(M(j!)) 0. #! ; otherwise When! Λ [0;ß), (j!) is represented as (j!)f j e j' ; ' [0;! Λ ) g: (0) (9) And Π(j!) can be said to be Π(j!)S(j!) Λ 0 M(j!) Λ S(j!) () where S(j!) is the same transformation matrix as in Eq. (8). Then, we can say that Λ 0 M(j!) Λ 0 () ~ (j!) ~ (j!) for! Λ [0;ß) since ~ (j!) R + [f0gbylemma and Re(M(j!)) 0. This is eqivalent tosaying that Λ y(j!) y(j!) Π(j!) 0 (3) (j!) (j!) ~y y for! Λ [0;ß)by the fact S(j!). ~ When! Λ [0;ß), the Π(j!) ρ ff 0 Π(j!) herm M(j!) 0 (4) represents a circle with center at the origin and radis by Lemma. And since (j!) for! Λ [0;ß) is also a circle with center at the origin and radis, it can be said that (j!) for Re(M(j!)) 0. Λ 0 M(j!) (j!) 0 (5) Ths, this proves that the Π(j!) in Eq. (9) satisfies the IQC () for 8! R. 5 Parameterization of ff We can see that the IQC () does not hold for all ff [0; ] if we se a simple linear parameterization of, that is, ff ff. So, we have to find other parameterizations than linear parameterization in order to satisfy the condition a), b) and c) in Theorem. p. 3

4 C(Π) φ + φ + (jω) C(Π) l Figre : The graph of (j!) and C(Π(j!)) when!λ [0;ß). C(Π) is the circle with center on the line `. The phase angle ffi 4 M(j!) determines which circle. Theorem 3 The parameterization of (j!) defined by ff(j!)e j!fiff (6) satisfies the IQC () for all ff [0; ] and conditions a),b) and c) in Theorem. It is easy to check that the conditions a) and b) hold with the parameterization (6). Let s check the condition c). If ff ff, can haveanyvale to satisfy the condition c). Now let s prove the case when ff 6 ff. k ff ff k ke j!fiff e j!fiff k ke j!fi k ke ff e ff k ke ff e ff k e ff e ff (ff ff ) ff ff e ff e ff ff ff kff ff k e jff ff j since 0 < eff e ff ff ff e :78 for ff ;ff [0; ]. Ths, if we set e, we can see that the condition c) is satisfied with this vale of. 6 Discssion It can be shown that the IQC's for time-delays introdced by Megretski et al. [8] and by Scorletti [] are special cases of the more parameterized class of IQC's given by or Theorem. In this section, we show that each of these special cases corresponds to a particlar choice for or mltiplier parameter M(j!). The ineqality sed by Megretski et al. [8] to find an IQC for time delay fi [0; μfi] is ψ (! Λ )(jj! Λ (j!)+y(j!)j ( +! Λ)jy(j!)j ) ψ (! Λ )jy(j!) (j!)j ) (7) where ψ ; are the fnctions defined by ψ (!) ψ (!) ( sin!! ; j!j ß 0; j!j>ß ; ( cos!; j!j ß 0; j!j>ß : By Lemma this IQC corresponds to a circle with its center at m (cos! Λ j sin! Λ ) and radis p m (! Λ sin! Λ + cos! Λ )m where m cos! Λ! Λ sin! Λ. If m>0, Eq. (7) is the inside of the circle and otherwise, it is the otside of the circle. Ths by Lemma, Megretski et al.'s IQC for the time delay is eqivalent to the IQC that arises from cos!λ! Π Λ sin! Λ cos! Λ + j sin! Λ cos! Λ j sin! Λ cos! Λ +! Λ sin! Λ (8) Comparing the Eq. (8) with the Eq. (9), we can see that we have the Eq. (8) when M(j!) sin! Λ + j! Λ sin! Λ. p. 4

5 Now let s consider the IQC in []. The ncertainty for time delay in [] is defined as (j!) e j!fi. Ths, if we reformlate the IQC in [] with time ncertainty (j!)e j!fi, then we have Π cot!λ cot! Λ + j cot! Λ j 0 (9) We can easily see that we can get Eq. (9) from Eq. (9) with M(j!)+jcot! Λ. Ths, or reslt incldes the reslts by [8] and []. Frthermore, the Eq. (9) is linear with respect to M(j!). This ensres that the problem of finding the optimal mltiplier M(j!) which provides least conservative bond of delay margin is, at each freqency!, a linear matrix ineqality (LMI); conseqently, the optimal mltiplier M(j!) may be readily compted see, for example, Boyd et al. [] and Safonov et al. [0]. By Lemma, it is possible to interpret or mltiplierbased IQC's for time-delays in terms of a circle that passes throgh the two end points of the arc (j!) f j e j' ; ' [0;! Λ ) g, and e j!λ,inthe Nyqist plane. The phase angle ffi 4 M(j!) determines which one of the many circles passing throgh these two points. See Figre. Notice that the centers of circles lie on the line ` which passes the origin and slope! Λ regardless of the vale of M(j!). And we can see that the ineqality () implies the inside or otside of a circle according to the sign of Π. When Π < 0 it is the inside of a circle by Lemma and otside when Π > 0. If the center of a circle lies on the dashed ray in Figre, the region which the ineqality () represents is the inside of the circle. On the other hand when the center lies on the dash-dotted ray, it is the otside. Jst one of the many possible circles determined by or mltipliers was considered in [8] and []. A line can be thoght as a circle with infinite radis which occrs when ffi ß! Λ. Or mltiplier IQC's for time-delay se the mltiplier M(j!) to parameterize the entire class of all circles which pass throgh those two points. References [] S. Boyd, L. El Ghaoi, E. Feron, and V. Balakrishnan. Linear Matrix Ineqalities in System and Control Theory. SIAM, Philadelphia, PA, 994. [] J. C. Doyle. Analysis of feedback systems with strctred ncertainties. In IEE Proc., volme 9-D(6), pages 450, November 98. [3] S. H. Esfahani and I. R. Petersen. An LMI approach to the otpt-feedback garanteed cost control for ncertain time-delay systems. In Proc. of IEEE Conf. Decision and Control, pages , Tampa, FL, December 998. [4] M. K. H. Fan, A. L. Tits, and J. C. Doyle. Robstness in the presence of mixed parametric ncertainty and nmodeled dynamics. IEEE Trans. on Ato. Control, AC- 36():538, Janary 99. [5] M. F, H. Li, and S. I. Niclesc. Robst stability and stabilization of time-delay systems via integral qadratic constraint approach, volme 8 of Lectre Notes in Control and Information Science, chapter 4. Springer, 998. [6] U. Jönsson and A. Megretski. The Zames-Falb IQC for critically stable systems. In Proc. of the American Control Conference, pages 36366, Philadelphia, PA, 998. [7] K. H. Lee, Y. S. Moon, and W. H. Kwon. Robst stability analysis of parametric ncertain time-delay systems. In Proc. of IEEE Conf. Decision and Control, pages 34635, Tampa, FL, December 998. [8] A. Megretski and A. Rantzer. System analysis via integral qadratic constraints. IEEE Trans. on Ato. Control, AC-4(6):89830, Jne 997. [9] S. Phoojarenchanachai and K. Frta. Memoryless stabilization of ncertain linear systems inclding timevarying states delays. IEEE Trans. on Ato. Control, AC- 37(7):006, Jly 99. [0] M. G. Safonov and R. Y.Chiang. Real/complex kmsynthesis withot crve fitting, volme 56 of Control and Dynamic Systems, pages Academic Press, New York, NY, 993. [] G. Scorletti. Robstness analysis with time-delay. In Proc. of IEEE Conf. on Decision and Control, pages , San Diego, CA, December 997. [] G. Zames. On the inpt-otpt stability of timevarying nonlinear feedback systemsparts I & II: Conditions derived sing concepts of loop gain, conicity, and positivity. IEEE Trans. on Ato. Control, AC-( & 3):8 38 & , April & Jly Conclsion Working from the IQC perspective, we provide a generalized class of Π(j!)'s for time delays which are linearly parameterized in terms a scalar-valed complex mltiplier M(j!) with Re(M(j!)) > 0. The mltipliers may be readily optimized for each specific application sing LMI techniqes. The reslts are less conservative than if a particlar mltiplier were to be prespecified. In particlar, we have shown that the timedelay IQC's of Megretski et al. [8] and of Scorletti [] correspond to two sch particlar choices for the mltiplier, which means that or mltiplier-parameterized IQC's will generally prodce better reslts. p. 5