Induced ℓ∞- Optimal Gain-Scheduled

INDUCED ℓ∞ OPTIMAL GAIN-SCHEDULED FILTERING OF T−AKAGI-SUGENO FUZZY SYSTEMS

Isaac Yaesh Control Department, IMI Advanced Systems Div., P.O.B. 1044/77, Ramat–Hasharon, 47100, Israel

Uri Shaked School of Electrical Engineering, Tel Aviv University, Tel Aviv, 69978, Israel

Keywords: Takagi-Sugeno fuzzy systems, Polytopic uncertainties, H∞-optimization, ℓ∞-optimization, S -procedure, Lin- ear matrix inequalities.

Abstract: The problem of designing gain-scheduled filters with guaranteed induced ℓ∞ norm for the estimation of the state-vector of finite dimensional discrete-time parameter-dependent Takagi-Sugeno Fuzzy Systems systems is considered. The design process applies a lemma which was recently derived by the authors of this paper, characterizing the induced ℓ∞ norm by Linear Matrix Inequalities. The suggested filter has been successfully applied to a guidance motivated estimation problem, where it has been compared to an Extended Kalman Filter.

1 INTRODUCTION response and an upper-bound on the ℓ1-norm of its transfer function (see (Dahleh and Pearson, 1987)).

The theory of optimal design of estimators for linear In the present paper, the problem of discrete- discrete-time systems in a state-space formulation has time optimal state-estimation in the minimum induced ℓ∞ norm sense is considered for a class of been first established in (Kalman, 1960). The original − problem formulation assumed Gaussian white noise Takagi-Sugeno fuzzy systems. The plant model for models for both the measurement noise and the ex- the systems considered, is described by a collection ogenous driving process. For this case, the results of ’sample’ finite-dimensional linear-time-invariant of (Kalman, 1960) provided the Minimum-Mean- plants which possess the same structure but differ Square Estimator (MMSE). The Kalman filter has in their parameters. All possible plant models are found since then many applications (see e.g. (Soren- then assumed to be convex combinations of these spe- son, 1985) and the references therein). Following the cific plant models (namely a polytopic system where the ’sample’ plant models are denoted as its vertices introduction of H∞ control theory in (Zames, 1981), (Boyd et al., 1994)). The solution of the estima- a method for designing discrete-time H∞ optimal estimators within a deterministic framework has been tion problem is characterized by LMIs (Linear Ma- developed in (Yaesh and Shaked, 1991), where the ex- trix Inequalities) based on the quadratic stability as- ogenous signals are of finite energy. The case where sumption. We note that more recent developments of the driving signal is of finite energy (e.g. piecewise (Geromel et al., 2000) include gain scheduled filter constant for a finite time) ,whereas the measurement synthesis for the cases of linear (Tuan et al., 2001) noise is white has been recently considered in (Yaesh and nonlinear (Hoang et al., 2003) dependence on the and Shaked, 2006). However, in some cases the min- parameters with parameter dependent Lyapunov functions. We also note that in (Salcedo and Martinez, imization of the maximum absolute value of the es- 2008) related results appear where the continuous- timation error (namely the ℓ∞ norm) rather than the error energy is required where− the exogenous signals time fuzzy output feedback and filtering were considered in parallel to the discrete-time results of the are also of finite ℓ∞ norm. In such cases, an induced − present paper. ℓ∞ norm is obtained which is often referred to as an − ℓ1 problem due to the fact that the induced-ℓ∞ norm The paper is organized as follows. In Section 2, − for a linear system is just the ℓ1-norm of its impulse the problem is formulated and a key lemma character-

67 Yaesh I. and Shaked U. INDUCED âDˇ ¸Sâ´Ld¯ â´LŠ OPTIMAL GAIN-SCHEDULED FILTERING OF TAKAGI-SUGENO FUZZY SYSTEMS. DOI: 10.5220/0002169500670073 In Proceedings of the 6th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2009), page ISBN: 978-989-674-001-6 Copyright c 2009 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

izing the induced ℓ∞ norm in terms of LMIs is pre- external disturbances and/or time (Tanaka and Wang, sented. In Section 3− the filter design inequalities are 2001) and rewrite (1) as : obtained. Section 4 considers a numerical example IF s1 is Mi1 and s2 is Mi2 and ... sp is Mip THEN dealing with a robust gain scheduled tracking prob- x k 1 A k x k B w k x 0 x lem. Finally, Section 5 brings some concluding re- ( + ) = i( ) ( ) + i ( ), ( ) = 0 y(k) = C (k)x(k) + D w(k) marks. i i (6) z k L k x k Notation: Throughout the note the superscript ( ) = i( ) ( ) i 1 2 N ‘T ’ stands for matrix transposition, R n denotes the = , ,..., n m n dimensional Euclidean space, R × is the set of where Mi j is the fuzzy set and N is the all n m real matrices, and the notation P > 0, for number of model rules. Defining s(t) = ×n n P R × means that P is symmetric and positive def- col s1(t),s2(t),...,sp(t) , ∈ { } inite. The space of square summable functions over ω Πp i(s(t)) = j 1Mi j(s j(t)) [0 ∞] is denoted by l2[0 ∞], and . 2 stands for the = Σ∞ T || ||1/2 standard l2-norm, u 2 = ( k=0uk uk) . We also use and || || 2 T ωi(t) . ∞ for the l∞-norm namely, u ∞ = supk(u uk). f (s(t)) = || || || || k i ΣN ω The convex hull of a and b is denoted by C o a, b , i=1 i(s(t)) { } In is the unit matrix of order n, and 0n,m is the n m we readily get the representation of (1). We, there- × zero matrix and Im,n is a version of In with last n m fore, assume indeed that the p premise scalar vari- − rows omitted. ables si(t),i = 1,2,..., p, and, consequently fi are exactly known and consider the following filter: xˆ(k + 1) = Axˆ(k) + K(k)(y Cxˆ), zˆ(k) = Lxˆ(k) 2 PROBLEM FORMULATION − (7) AND PRELIMINARIES where the filter gain is given by the following: N We consider the following linear system: K = ∑ Ki fi (8) i=1 x(k + 1) = A(k)x(k) + Bw(k), x(0) = x0 y(k) = C(k)x(k) + Dw(k) (1) N N and where A = ∑ Ai fi and L = ∑ Li fi. We will dif- z(k) = L(k)x(k) i=1 i=1 ferently treat, in the sequel, the case where C is con- n r where x R is the system states, y R is the mea- N surement,∈ w R q includes the driving∈ process and stant and the case where C = ∑ Ci fi. the measurement∈ noise signals and it is assumed to i=1 Our aim is to find the filter parameters Ki so that ∞ R m have bounded ℓ norm. The sequence z is the the following induced ℓ∞ norm condition is satisfied. state combination− to be estimated and A, B∈, C, D and − γ L are matrices of the appropriate dimensions. supw ℓ∞ z zˆ ∞/ w ∞ < (9) ∈ || − || || || We assume that the system parameters lie within To solve this problem we will first define another the following polytope polytopic system : Ω := ABCDL (2) Ω¯ := A¯ B¯ C¯ D¯ (10) which is described by its vertices. That is, for which is described by the vertices: Ω i := Ai Bi Ci Di Li (3) Ω¯ i := A¯i B¯i C¯i D¯ i , i = 1,...,N (11) we have The system of (10)-(11) will represent, in the sequel, Ω = C o Ω1,Ω2,...,ΩN (4) the dynamics of the estimation error for the system { } where N is the number of vertices. In other words: (1). The following technical lemma will be needed N N in order to provide convex characterization of the induced ∞ norm of the estimation error system: Ω = ∑ Ωi fi , ∑ fi = 1 , fi 0. (5) ℓ ≥ − i=1 i=1 Lemma 1. The system Assuming that f are exactly known, the above sys- i x¯(k + 1) = A¯(k)x¯(k) + Bw¯ (k), x(0) = x tem is just a Tagaki-Sugeno fuzzy system. To see this, 0 (12) z(k) = C¯(k)x¯(k) + D¯ w¯(k) one may introduce new parameters si(t),i = 1,2,..., p (so called premise variables, see (Tanaka and Wang, satisfies γ 2001)) possibly depending on the state-vector x(t), supw ℓ∞ z ∞/ w ∞ < (13) ∈ || || || ||

68 INDUCED - OPTIMAL GAIN-SCHEDULED FILTERING OF TAKAGI-SUGENO FUZZY SYSTEMS

if the following matrix inequalities are satisfied for and i = 1,2,...,N: λ T P 0 Li ¯T ¯ λ ¯T ¯ 0 (γ µ)I 0 > 0, λ < 1 (22) Ai PAi + P P Ai PBi  −  T − T < 0 (14) Li 0 γI B¯ PA¯i µI + B¯ PB¯i i − i We, therefore, obtain the following result: and λ ¯T P 0 Ci Theorem 1. Consider the estimator of (12) for the 0 (γ µ)I D¯ T > 0 (15) system of (1) with C = C,D = D,i = 1,2,...,N. The  − i  i i C¯i D¯ i γI estimation error satisfies (9) if (21) and (22) are satis- λ so that P > 0, µ > 0 and λ < 1.  fied for i = 1,2,...,N so that P > 0, µ > 0 and < 1. The proof of this lemma is given in (Shaked and We next address the problem where C and D are Yaesh, 2007) and is also provided, for the sake of vertex dependent. To this end we consider a version Ω completeness, in Appendix A. of Lemma 1 which can be written in terms of rather than Ωi, namely we replace (16) and (14) by: Remark. Note that (14) can be written, using λ ¯T Schur complements ((Boyd et al., 1994)), as follows: P P 0 A P −0 µI B¯T P > 0 (23) λ ¯T  PAP¯ BP¯  P P 0 Ai P − ¯T 0 µI Bi P > 0 (16) and     T PA¯i PB¯i P λP 0 C¯ T or equivalently as  0 (γ µ)I D¯ > 0 (24)  C¯ −D¯ γI  P λP 0 A¯T i and substitute A¯ = ∑N (A  K C ) f f ,B¯ = −0 µI B¯T > 0 (17) i, j=1 i i j i j i i N N−  ¯ ¯ 1  ∑ (Bi KiD j) fi f j and C¯ = ∑ Li fi. We obtain Ai Bi P− i, j=1 − i=1 defining Yi = PKi :   ¯ ¯ The fact that the inequality (16) is affine in Ai and Bi N will be utilized in the sequel to obtain convex charac- ∑ Gi j fi f j > 0 (25) terization (i.e. in LMI form) of the filter parameters i, j=1 Ki. where λ T T T P P 0 Ai P Cj Yi − T − T T Gi j := 0 µI B P D Y 3 GAIN SCHEDULED FILTERING  i − j i  PAi YiCj PBi YiD j P − −  (26) Defining the state estimation error to be: and λ T e(k) = x(k) xˆ(k) (18) N P 0 Li − ∑ 0 (γ µ)I 0 fi > 0 (27)  −  we readily have for the case where fi are available for i=1 Li 0 γI the estimation process, that Since, however (see (Tanaka and Wang, 2001)) e(k + 1) = (A K(k)C)e(k) + (B K(k)D)w(k) equation (25) can be also written as − − N N N (19) Gi j + G ji ∑ G f f = ∑ G f 2 + 2 ∑ ∑ f f and i j i j ii i 2 i j z(k) zˆ(k) = Le(k) (20) i, j=1 i=1 i=1i< j − (28) ¯ ¯ We substitute Ai = Ai KiC, Bi = Bi KiD and Defining a simple transformation of the convex co- C¯ L in (14) and (15) where− we restrict our− attention 2 = i ordinates fk so that for k = 1,2,...,N we set hk = f to the case where C and D are not vertex dependent k where as the remaining hk for k = N +1,N+2,...,N + (i.e. Ci = C,Di = D,i = 1,2,...,N). In this case, we N(N 1) 2− are defined by hk = 2 fi f j, j = 1,2...,N,i < j. define Yi = PKi and readily obtain from (16) and (15) N(N 1) N+ 2− that Since obviously ∑ hk = 1 where hk 0 they k=1 ≥ λ T T T can serve as convexcoordinates. We, therefore, define P P 0 Ai P C Yi −0 µI BT P −DTY T > 0 the following LMIs inspired by (Tanaka and Wang,  i − i  2001), PAi YiC PBi YiDP − −   (21) Gii > 0,i = 1,2,...N and Gi j + G ji > 0,i < j (29)

69 ICINCO 2009 - 6th International Conference on Informatics in Control, Automation and Robotics

and obtain the following result: x˙˜ = νcos(ψ˜ ) + w1 Theorem 2. Consider the estimator (12) for the y˙˜ = νsin(ψ˜ ) + w 2 (31) system (1). The estimation error satisfies (9) if (29) ψ˜˙ = φ˜ and (22) for i = 1,2,...,N are satisfied so that P > 0, φ˙ = φ/τ + u/τ µ > 0 and λ < 1. − The solution offered above, for the case where C wherex ˜ andy ˜ are the first two coordinates of a flight and D are uncertain and are known to reside in a given vehicle cruising in a constant altitude, in a local level ψ polytope, seeks a single matrix P that solves the LMIs north-east-down system, ˜ is the vehicle body angle with respect to the north (i.e. azimuth angle) for N(N+1) vertices, instead of the N vertices that were 2 and φ is the vehicle’s roll angle assumed to be gov- solved for in the case of known C and D. A solu- erned by a first-order low-pass filter dynamics hav- tion for such large number of vertices by a single P ing a time-constant of τ seconds, driven by the roll- entails a significant overdesign. Even the relaxation angle command u. The wind velocities at the north offered by e.g. (Shaked, 2003) to reduce the overde- and east directions respectively are denoted by w1 and sign by allowing different P i 1 2 N(N+1) for i, = , ,..., 2 w2 whereas ν is the true-air-speed. Our aim is to filter N(N+1) φ the 2 vertices still suffers from a considerable the noisy measurements ofx ˜,y ˜ and and to estimate conservatism. Moreover, the computational complex- ψ˜ . Defining, ity of the solution also rapidly increases as a function x = col x˜,y˜,vsin(ψ˜ ),vcos(ψ˜ ),φ of the number of vertices. { } In many cases, C resides in some uncertainty poly- the measurements vector which consists of noisy tope, while D is fixed and known. In such a case, an measurements of the position componentsx ˜ andy ˜ and alternative way to deal with the problem is to define the roll angle φ is given by ξ(k) = col x(k), y(k) andw ˜(k) = col w(k), w(k + 1/2 1) so that the{ augmented} system becomes:{ y = Cx + R v } ξ(k + 1) = A˜(k)x(k) + B˜w˜(k) where v is the measurement noise which is taken in ˜ ξ ˜ the simulations in the sequel as a 3-vector of zero- y(k) = C(k) (k) + Dw˜(k) mean unity variance white noise sequences but for all z(k) = L˜(k)ξ(k) practical purposes is assumed to be v ℓ∞. The noise where level is set by ∈ A 0 B 0 A˜ = B˜ = , C˜ = 0 I , CA 0 CBD r R = diag 25,25,0.1 { } L˜ = L 0 , and D˜ = D 0 and the measurement matrix is (30) 1 0 0 0 0 In (30) the uncertainties appear in A¯ and B¯ only and, C = 0 1 0 0 0 therefore, Theorem 1 above may be invoked. We,  0 0 0 0 1  therefore, obtain the following result which offers   reduced conservatism with respect the correspond- Note that ing continuous-time results of (Salcedo and Martinez, x˙1 = x4 2008): x˙2 = x3 x˙3 = x4x5 (32) Theorem 3. Consider the estimator of (12) for the x˙4 = x3x5 − system of (1) for Di = D,i = 1,2,...,N. The estima- x˙5 = x5/τ + u/τ γ √ γ − tion error satisfies (9) with replaced by 2 if (21) namely we have a bilinear system rather than a lin- and (22) are satisfied for i = 1,2,...,N so that P > 0, ear one. Following (Tanaka and Wang, 2001) with λ ˜ ˜ ˜ ˜ µ > 0 and < 1 with A,B,C,L replaced by A,B,C,L a series of simple manipulations, this system can be of (30). represented as a Takagi-Sugeno fuzzy system, namely as a convex combination of linear systems where the convex coordinates are online measured. To achieve 4 EXAMPLE such a representation we recall that x5 = φ is measured on line, and define s1 = x5 while neglecting We consider the dynamic model of guidance in a the small enough noise in measuring φ. The valid- plane: ity of the latter assumption will be verified in the sequel by the estimation quality we will obtain. Assum- φ φ s1 s1,min ing x5 [ max, max] we define f1 = s − s = ∈ − 1,max− 1,min

70 INDUCED - OPTIMAL GAIN-SCHEDULED FILTERING OF TAKAGI-SUGENO FUZZY SYSTEMS

x5+φmax φ , f2 = 1 f1, α1 = φmax and α2 = φmax. We γ = γ0 were obtained: 2 max − − readily see that s1 = φmax f1 φmax f2 := α1 f1 + α2 f2. − 0.7574 0.0007 0.0000 We note then that the system is then governed by 0.0018− 0.7593 0.0000 ˙ ξ = Ac(ξ)ξ+Bcw where ξ = col x1,x2,x3,x4,x5 and  −  { } K1 = 0.0000 0.0000 0.0000 0 0 0 1 0  0.0000 0.0000 0.0000   0 0000 −0 0000 0 0003  0 0 1 0 0  . . .   − −  Ac =  0 0 0 s1 0  0.7558 0.0024 0.0000 −  0 0 s1 0 0  0.0021 0.7613 0.0000  0 0− 0 0 1/τ  − −   K2 =  0.0000 0.0000 0.0000   −  − − Therefore, A = A α + A α where A is ob-  0.0000 0.0000 0.0000  c c,1 1 c,2 2 c,1  0 0000 0 0000 0 0003  tained from A by replacing s by α and A is sim-  . . .  c 1 1 c,2  −  ilarly obtained from Ac by replacing s1 by α2. We This ℓ∞ filter will be compared to an Extended also define w = col w1,w2,ν1,ν2,ν3 and complete Kalman Filter (EKF, see (Jazwinsky, 1970)) based on the remaining matrices{ needed for the} representation the nonlinear model of (31). Note that higher com- of our problem (1)-(3) by applying a zero-order-hold plexity filters such as the particle filter (e.g. (Osh- discrete-time equivalent of our continuous-time sys- man and Carmi, 2006) are out the scope of the tem, where we have chosen a sampling time of h = present paper. For the simulations we take ν = 0.02. Due to the small enough h we have chosen, we 100m/s and try to control the vehicle to follow a have eAh = I + Ah + O(h2) and we, therefore, readily constant command at y = 5m, in spite of a wind obtain that the system is governed by (1) and (3)-(5) step at w2 of 10m/s. The estimation results are 2 where A = A1α1 + A2α2 + O(h ) where used to control the vehicle, using the simple law T u = 0.0200 4.0000 yˆ 5 ψˆ where all − − 1.0000 0.0002 0.0200 0 components of the initial state-vector are taken as − 0 1.0000 0.0200 0.0002 0 zero, besides y0 = 20m. The EKF and the ℓ∞ esti- A1 = 0 0 0.9998 0.0209 0  matedx ˜ y˜ trajectory results are compared in Fig. 1 0 0 0 0209 0 9998 0 −  . .  to the true trajectory. One can notice the bias in the  0 0− 0 0 0.9231    EKF estimate. In Fig. 2, the true ψ˜ and the estimated   1.0000 0 0.0002 0.0200 0 values for ψ˜ that are obtained by using the EKF and 0 1.0000 0.0200 0.0002 0 the ℓ∞ filter are depicted. We clearly see in this figure − A2 = 0 0 0.9998 0.0209 0  that the ℓ∞ filter outperforms the EKF which assumes 0 0 0.0209− 0.9998 0   a white noise w2 but leads to a bias when w2 has a  0 0 0 0 0.9231    bias. In contrast, the ψ˜ estimate of the ℓ∞ filter is  0.2000 0 0 0 0  barely separable from the true values. Moreover, the 0 0.2000 0 0 0 ℓ∞ filter does not require the on-line numerical solu-   B1 = B2 = B = 0 0 0 0 0 tion of a Riccati equation of order 4 and the gains are 0 0 0 0 0    0 0 0 0 0  obtained there by a simple convex interpolation on K1     and K2. The fact that K1 and K2 are close to each and other is somewhat surprising. Our experience shows 0 0 2.2361 0 0 that for a larger γ (i.e. suboptimal values), a larger D1 = D2 = D = 0 0 0 2.2361 0 K1 K2 is obtained.  0 0 0 0 0.1000  || − ||   We note at his point that, in order to mini- mize the design conservatism stemming from the 5 CONCLUSIONS quadratic stability assumption, we applied a parameter dependent Lyapunov function (Boyd et al., The problem of designing robust gain-scheduled fil- T T 1994), max(x P1x,x P2x). Minimization of γ sub- ters with guaranteed induced ℓ∞ norm has been con- ject the the LMis that are obtained with this func- sidered. The solution has been− derived using a re- tion to replace (21) and (22) (see Appendix B), us- cently developed bounded-real-lemma like condition ing fminsearch.m from the optimization toolbox of for bounding the induced ℓ∞ norm of a system. This MAT LABTM and (Lagarias et al., 1998) to search result has been applied to derive the robust induced λi, i = 1,2,3,4, ρ1, ρ22, θ1 and θ2, has resulted in ℓ∞ filter (or equivalently robust ℓ1 filter) in terms of 7 − γ = γ0 = 10.2732 and λ = 2.41 10 . The follow- LMIs. These LMIs have been applied to a guidance × − ing gain matrices K1 and K2 have been obtained for motivated estimation example. In this example, the

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72 INDUCED - OPTIMAL GAIN-SCHEDULED FILTERING OF TAKAGI-SUGENO FUZZY SYSTEMS

T T Collecting terms we have and if xk P2x > xk P1xk then T T T T ¯ ξk = x (A¯ PA¯ + λP P)xk + x (A¯ PB¯)wk ξ 2 ,A T ¯T ¯ λ T ¯T ¯ k − k k{ } := xk (A P2A + P2 P2)xk + xk (A P2B)wk +wT (B¯T PA¯)x + wT ( µI + B¯T PB¯)w . T T T − T k k k k +w (B¯ P2A¯)xk + w ( µI + B¯ P2B¯)wk < 0. − k k − ξ Therefore, (14) guarantees k < 0 for all wk and xk. Since these conditions are required to be satisfied ζ T Defining k = xk Pxk and assuming x0 = 0 and throughout the polytope, we readily obtain, using T ζ ζ λζ T wk wk < 1 we have that k+1 k + k µwk wk < 0. again the S-procedure, that in addition to the con- ¯ ¯ − −¯ T λ Namely, ζk < ζk where ζk 1 = (1 λ)ζk + µw wk. stant > 0, the existence of six additional constants + − k However, using ρ := 1 λ we have λi > 0, i = 1,2,3,4, θ1 > 0, θ2 > 0 establishes a suf- − k 1 k 1 ficient condition for the above inequalities to hold, if ¯ − k j 1 T k 1 − 1 j T ζk = ∑ ρ − − µw w j = ρ − ∑(ρ− ) µw w j j j 1 ,A¯i j=0 j=0 ξ{ } λ1(P1 P2) > 0,i = 1,2 − k − − ρ 1 k ρk and k 1 (1 − ) 1 ¯ <ρ µ=µ . A.1 1 ,Ai − − ρ 1 − ρ ξ{ } λ2(P2 P1) > 0,i = 1,2. 1 − 1 − k − − − − Following the lines of proof of Theorem 1 above, we readily obtain the following LMIs for i = 1,2 to re- From (15) we have using Schur complements that place (21) and (22): T T T λP 0 1 C¯ x λ λ T T T [x w ]( γ− C¯ D¯ ) > 0 P1 P i(P1 P2) 0 Ai P1 C Yi 0 (γ µ)I D¯ T w − − − T − T T − 0 µI B P1 D Y >0, −  i − i  P1Ai YiCP1Bi YiDP1 Namely, − −   T T T T ¯ λP1 θ1(P1 P2) 0 L z zk < γ[(γ µ)w wk +λx Pxk] < γ[(γ µ)+λζk] A.2 i k − k k − − 0 − (γ µ)I 0 > 0, λ < 1  − γ  Substituting A.1 we readily see that Li 0 I and  1 ρk zT z < γ[(γ µ)+(1 ρ)µ − ] = γ[γ µ+µ µρk]. λ λ T T T k k ρ P2 P i+2(P2 P1) 0 Ai P2 C Yi − − 1 − − − − − T − T T − 0 µI B P2 D Y >0,  i − i  ρ P2Ai YiCP2Bi YiDP2 Since 0 < < 1 we obtain that − −   T γ γ γ2 λ θ T zk zk < [ µ + µ] = . P2 2(P2 P1) 0 Li − − 0 − (γ µ)I 0 > 0, λ < 1.  −  Li 0 γI   APPENDIX B - PARAMETER We note that we have also replaced (A.2) with: T γ γ T λ T T DEPENDENT RESULTS zk zk < [( µ)wk wk+ max(xk P1xk,xk P2xk)]. − × T T Namely, if x P1x > x P2x we require In order to reduce conservatism, we replace in the T γ γ T λ T proof of Lemma 1 in Appendix A, the parameter- zk zk < [( µ)wk wk + xk P1x], T − independent Lyapunov function V(x,P) = xk Pxk by T T the parameter-dependent Lyapunov function ((Boyd whereas if x P2x > x P1x we require et al., 1994)) V(x,P ,P ) = max(xT P x ,xT P x ). To T γ γ T λ T 1 2 k 1 k k 2 k zk zk < [( µ)wk wk + xk P2x]. T − ensure V(xk,P1,P2) > 0 we have to satisfy xk P1xk > 0 T T T Using again the S -Procedure with additional tuning whenever x P1x > x P2x and x P2x > 0 whenever k k k k k k constants θ > 0 and θ > 0, which add up to the pre- xT P x < xT P x . Applying the S-procedure (Boyd 1 2 k 1 k k 2 k viously introduced 7 tuning constants ρ > 0, ρ > 0, et al., 1994) we readily obtain that a sufficient condi- 1 2 λ > 0 and λ > 0,i = 1,2,3,4 the above results are tion or these requirements to hold, is the existence of i obtained. Note that a similar approach can be applied constants ρ > 0 and ρ > 0 so that 1 2 also on the continuous-time results of (Salcedo and P1 ρ1(P1 P2) > 0 and P2 ρ2(P2 P1) > 0. Martinez, 2008) to reduce conservatism. − − − − T T We also require that if xk P1x > xk P2xk then ξ 1 ,A¯ T ¯T ¯ λ T ¯T ¯ {k } := xk (A P1A + P1 P1)xk + xk (A P1B)wk T T T − T +w (B¯ P1A¯)xk + w ( µI + B¯ P1B¯)wk < 0, k k −