Chapter 16 Linear Time-Variant Systems The complexity of the physical phenomena studied cannot be reduced to only one modeling by linear dynamic systems with constant coefficients. These models are sometimes poorly adapted because, for example, they can only deal with magnitudes having an exponentially decreasing correlation. However, in fields full of variety such as hydrology [HUR 65], electronics [VAN 88], traffic [RIE 97, WIL 95], electrical engineering [CHA 81] and mechanics [CLE 98, ZHU 96], there are many situations that generate behaviors that do not obey these quite simple models. Therefore, in the last 30 years, new analysis models and tools have appeared. In this perspective, one of the goals of research is to extend the class of linear dynamic systems by including those for which the coefficients vary in time. These variations can be divided into two classes. The first class concerns the sudden non- stationarities or “failures” characterized by time intervals where the coefficients are constant. The non-stationarity is due only to the presence of instantaneous shifts in their values. This modeling is found, for example, in the field of monitoring and diagnostic [BAS 93]. As such, the problem is to essentially detect the instants of change as well as the amplitude of the parametric shifts. The second class pertains to the systems where the coefficients are functions of time. When these dynamics are “slow” with respect to those of the system, they can be dealt with through adaptive techniques. However, there are also many cases in which the evolution of parameters is “fast” (T-periodic systems [RAB 92], auto-similar systems [GUG 01], etc.). This last category makes the development and the implementation of specific methods in Chapter written by Michel GUGLIELMI. 522 Analysis and Control of linear Systems both the control and identification fields indispensable. Firstly, it is important to have the basic mathematical tools indispensable to their analysis. This chapter is dedicated to the analysis of the dynamic systems required by the linear differential equations with time-variant coefficients. The approach presented consists of an approach parallel to that adopted for the constant coefficient systems. The Laplace transform, even if it can be always applied to the input/output magnitudes of the system, can no longer be used in order to define the transfer function of these systems. However, this transfer concept can, despite everything, be extended to the non- stationary linear differential systems provided they operate on the non-commutative body of rational fractions. This body is isomorphic to the group generated by the non-stationary linear dynamic systems. Hence, we can elaborate the composition rules of these systems with the help of the algebraic rules applied to the transfer functions. The results obtained can be used in order to solve control or/and identification problems. This chapter deals, in the first place, with the construction of the non- commutative polynomial ring and of the body of related rational fractions. As in the traditional case, the relation between the basic properties of dynamic systems (stability, etc.) and the characteristics of the elements of the body of fractions (poles, etc.) can be established. The second part pertains to the construction of the systems: serialization or/and parallelization. It is possible, for each association diagram, to write the transfer function of the system composed with the help of simple algebraic rules. Finally, based on these results, two applications illustrate the use of the results obtained. The first one concerns the modeling of multi-component polynomial phase signals and the second is dedicated to the design of a pole placement control law. 16.1. Ring of non-commutative polynomials Let )(λΠ be the set of polynomials of degree n: { [] } niKtatatataP i n n n n ,1)()()()()()( 0 1 1 ∈∀∈++λ+λ=λ=λΠ − − … Let K be a differential body, i.e. a body on which is defined a derivation operator ∀∈()da dt a t K (noted from now by a  ) which will satisfy the traditional derivation properties: +=+ = +   () andab ab ab abab Linear Time-Variant Systems 523 when coefficient )(ta n is equal to 1, polynomial )(λP is standardized. The set )(λΠ including addition and multiplication which satisfies: )()()(,)( . tatataKta +λ=λ∈∀ has a non-commutative ring structure [ORE 33]. 16.1.1. Division and the right highest divisor (RHD) ∀ )( 1 λP , )()( 2 λΠ∈λP ⊗ )(λΠ  insofar as 21 PP nn ≥ (where X n represents the degree of λ ()X ) there is a unique pair of polynomials λ λ [(),()]QR so that: )()()()( 21 λ+λλ=λ RPQP with R n < 2 P n We can infer Euclid’s division algorithm: )()()()( 3211 λ+λλ=λ PPQP …. … … )()()()( 211 λ+λλ=λ −−− nnnn PPQP The RHD of )( 1 λP , )( 2 λP is then defined as the standardized polynomial resulted from )(λ n P . 16.1.2. Right least common multiple (RLCM) It is then possible to define the RLCM of )( 1 λP , )( 2 λP as the lowest degree standardized polynomial divisible on the right by both )( 1 λP and )( 2 λP . )()()()()( 2211 λλ=λλ=λ PQPQM 524 Analysis and Control of linear Systems Generally, the existence of the Euclidian division implies the existence of the RHD [ORE 33]. 16.1.3. Explicit formulation of RLCM Based on all the factors of the Euclidian division, it is possible to express the RLCM: [ λ αλλ λ λ λλ λ −−− −−− = 111 121231 ( ) [ ( )] [ ( )] [ ( )] [ ( )] [ ( )] ( )] [ ( )] nnn n MPPPP PPP The constant )(tα (which does not depend on λ) is such that polynomial )(λM is normalized. NOTE.– the product of two polynomials 12 () ()PP λ λ cannot generally be divided on the right by )( 1 λP which makes the use of all the terms of the Euclidian division compulsory in the expression above. )(λM is a polynomial and writing it in the inverse form is basically a useful notation. By applying the same approach, we can define the left highest divisor and the left least common multiple. 16.1.4. Factoring, roots, relations with the coefficients Any polynomial )(λP can be factorized in the general form: ))(())(()) (())(()()( 121 tptptptptP nn −λ−λ−λ−λα=λ − The roots of )(λP are provided by the solutions of equation 0)( =λP . The relation between the roots and the coefficients of )(λP is a non-linear differential equation [KAM 88]: 0)()()()()()( 0 1 1 1 1 2 1 =+++ −− − − ∑ tatpStatpStatpS n n n i n in n Linear Time-Variant Systems 525 where S is the operator defined by: 2)()( )()()( )( 1 . 2 1 ≥∀+= += − − iStptpS tptptSp dt pSd i nn i nnn i EXAMPLE 16.1.– let us consider the second degree polynomial: ))())((()()()( 2101 2 tptptataP −λ−λ=+λ+λ=λ From the following relations: )()()( 121 tatptp −=+ and )()()()( 0 2 . 21 tatptptp =− we infer that the roots are solutions of: 0)()()()()( 021 2 . 2 2 =+++ tatptatptp Particular case: 0)(0)()( 2 01 =λ=λ⇒== Ptata 21 22 21 22 11 () () 0 and () , ()pt pt pt pt k tk tk ⇒ == =− =∀∈ℜ ++ Hence, the factoring of )(λP : ℜ∈∀ + −λ + +λ=λ=λ k ktkt P ) 1 )( 1 ()( 2  16.2. Body of rational fractions In general it is not possible to define a body of rational functions from polynomials for which the unknown factor and the parameters cannot be switched 526 Analysis and Control of linear Systems (skew). However, if the polynomials verify the two following conditions called ORE [AMI 54]: 12 12 21 12 (), () () () ˆˆ ˆ ˆ (), () so that () () () () PP PP PP PP λ λ Πλ Πλ λ λλλλλ ∀∈⊗ ∃= (condition on the left) and: λλ λ λ λ λλλλλ ∀∈Π⊗Π ∃=    12 12 11 22 (), () () () (), ()so that () () () () PP PP PP PP (condition on the right) it is possible to consider the set: { } { } 1* 1* () ()() , () (), () () ( ) ( ), ( ) ( ) , ( ) ( ) FPQ P Q QP P Q λλλ λΠλ λΠλ λλ λΠλ λΠλ − − =∀∈∀∈= ∀∈ ∀∈ where λλ Π=Π− * () () {0} which has a body structure [ZHU 89]. NOTE.– the inverse of polynomial )(λP is unique and verifies: λλ λλ −− == 11 () () () () 1PP P P 16.3. Transfer function In the case of linear systems with constant coefficients, the transfer function is defined as the ratio between the Laplace transforms of the output and those of the input. When the coefficients are time functions, it is possible to extend this concept of transfer function which preserves certain properties obtained in the traditional case even if it does not represent any longer the ratio between the Laplace transforms of the pair input/output. Linear Time-Variant Systems 527 Let ∑ )( dt d be the set of n degree single-variable systems described by the linear differential equation with variable coefficients belonging to a derivable body K: −− +++=+++ () ( 1) (0) () ( 1) (0) 101 () () () () () () () () () () nn nn nn ytaty t atytbutbtu t btut with == () () () () , () () iiiiii yt dytdtut dutdt It is easy to show that this set, consisting of two internal operations (serialization and parallelization) is a body. Hence, the application of ∑ )( dt d 1 on )(λF is an isomorphism. Therefore, system Σ can be formally described by its transfer function: )()()( 1 λλ=λ − PQH where: 11 101 ( ) () (), ( ) () () nn nn nn Qat atPbbt bt λλ λ λ λ λ −− =+ ++ = + ++ 16.3.1. Properties of transfer functions THEOREM 16.1.– two transfer functions )( 1 λH and )( 2 λH are equivalent if and only if there is a polynomial )(λD such that [KAM 88]: λ λλ λ λλ == 1212 () () ()and () () ()PDP QDQ The transfer function )( 1 λH is said to be minimal if the numerator and denominator are first on the right. 1 Henceforth, )( dt d Σ will be simply noted by Σ . 528 Analysis and Control of linear Systems 16.3.2. Normal modes If we make the homogenous equation: () ( 1) (0) 1 () () () () () 0 nn n ytaty t atyt − +++= correspond to the polynomial equation 0)( =λQ , we show that if )(tq n is a root of )(λQ , then ∫ ττ t n dq e 0 )( is a normal mode of the system. EXAMPLE 16.2.– for 0)( )2( =ty which corresponds to: λλ +−∀∈ℜ= ++ 11 ()() 0 k tk tk . A root kt + 1 provides the normal mode: kte t dk += ∫ τ+τ 0 )(1 (well known). 16.3.3. Stability The solutions of the homogenous equation form a vector space of size smaller or equal to its n degree [AMI 54, ZHU 89]. The general solution is written: ∑ ∫ = ττ i dq i t i ecty 0 )( )( We infer that the system is stable if: itq t i ∀<ℜ ∞→ 0))((lim where ℜ means real part. Linear Time-Variant Systems 529 16.4. Algebra of non-stationary linear systems A major interest in the transfer function is due to the possibility of easily calculating the transfer function of associated systems, either serially or in parallel. 16.4.1. Serial systems Let 1 ∑ and 2 ∑ be two systems of transfer functions )()()( 1 1 11 λλ=λ − PQH and )()()( 2 1 22 λλ=λ − PQH respectively; the transfer function of the system ∑ obtained by serializing 1 ∑ and 2 ∑ can be calculated as follows: Let )(λM be the RLCM of )( 2 λP and of )( 1 λQ : )(Q)(Q ˆ )()( ˆ )( 1122 λλ=λλ=λ PPM then we have: )()()()( ˆ )( ˆ )( )()()()( ˆ )( )()()()( ˆ )( ˆ )( )()()()()()()( 1 1 111 1 2 1 2 1 1 1 1 2 1 2 1 1 122 1 2 1 2 1 1 12 1 212 λλλλλλ= λλλλλ= λλλλλλ= λλλλ=λλ=λ −−− −−− −−− −− PQQQPQ PQMPQ PQPPPQ PQPQHHH and finally: λ λλ λλ − = 1 22 11 ˆ () [ () ()] () ()HPQ QP EXAMPLE 16.3. 11 1 2 2 2 1 1111 111 2 22 Si () 1 () () and () 1 () () () () () ( 1) () () () ( 1) yt tyt ut y t ty t yt H QP t H QP t λλλλλλλλλ −−−− ∑ += ∑ −= ==+ ==−  we obtain: tPPM 1)(Q)(Q ˆ )()( ˆ )( 1122 +λ=λλ=λλ=λ and 1 )( − λ=λH so )()( tuty =∑  If 530 Analysis and Control of linear Systems 16.4.2. Parallel systems Let 1 ∑ and 2 ∑ be two systems of transfer functions )()()( 1 1 11 λλ=λ − PQH and )()()( 2 1 22 λλ=λ − PQH respectively; the transfer function of the system ∑ obtained by putting 1 ∑ and 2 ∑ in parallel is provided by: Let )(λM be the RLCM of )( 1 λQ and )( 2 λQ : )()( ˆ )(Q)(Q ˆ )( 2211 λλ=λλ=λ QQM  then we simply have: ))()( ˆ )()( ˆ )(()( )()()()()()()( 2 1 21 1 1 1 2 1 21 1 121 λλ+λλλ=λ λλ+λλ=λ+λ=λ −−− −− PQPQMH PQPQHHH EXAMPLE 16.4. )()(1)( 111 tutytty =+∑  and )()(1)( 222 tutytty =− ∑  1 2 1 22 1 1 1 11 )1()()()()1()()()( −−−− −λ=λλ=λ+λ=λλ=λ tPQHtPQH gives: tttttM 21)1)(2()1()( 2 +λ+λ=−λ+λ=+λλ=λ and )1(2)21()( 12 tttH +λ+λ+λ=λ − or: ))( 1 )((2)( 2 )( 1 )( tu t tuty t ty t ty +=++∑  [...].. .Linear Time-Variant Systems 531 16. 5 Applications In this section, two types of usage of this algebra are presented in the field of modeling and control 16. 5.1 Modeling One of the methods of signal and control processing consists of designing models capable of representing the magnitudes in question Hence, models MA, AR, ARMA with... Service Automatique Gif-sur-Yvette, France Patrick BOUCHER Supélec Gif-sur-Yvette, France Philippe CHEVREL IRCCyN CNRS – Ecole des Mines de Nantes France Martial DEMERLE Service Automatique, Supélec Gif-sur-Yvette, France Gilles DUC École Supérieure d’Électricité (Supélec) Service Automatique Gif-sur-Yvette, France 538 Analysis and Control of Linear Systems Didier DUMUR Supélec Gif-sur-Yvette, France Sylvianne... 542 Analysis and Control of Linear Systems error dynamic 271, 277, 285, 287 static 27 1-2 74, 284, 289, 295, 301, 304, 308, 311, 314, 317, 320, 376, 482 Euler’s method 239, 245, 246 F, I filter Kalman 412, 415 non-observable 488 Fourier transform 14, 18, 33, 34, 41, 14 2-1 45, 147, 14 9-1 51 continuous-time 88, 14 2-1 44 discrete-time 88, 96, 14 2-1 44 internal stability 132, 133, 135, 328, 332, 335, 409, 416, ... Discrete Linear Control: the Polynomial Approach, John Wiley and sons, 1979 [ORE 33] ORE O., “Theory of non-commutative polynomials”, Annals of mathematics, vol 34, p 48 0-5 08, 1933 [VAN 88] VAN DER ZIEL, “Unified presentation of 1/f noise in electronic devices: fundamental 1/f noise sources”, Proceedings of IEEE, vol 76, no 3, p 23 3-2 58, 1988 [WIL 95] WILINGER W., TAQQU M.S., LELAND W.E., WILSON V., “Self-similarity... W.E., WILSON V., “Self-similarity in high-speed packed traffic: analysis and modelling of Ethernet traffic measurements”, Statistical Science, vol 10, p 67 6-6 85, 1995 [ZHU 89] ] ZHU J., JOHNSON C.D., “New results in the reduction of linear time-varying dynamical systems , SIAM J Control & Optimization, p 47 6-4 94, 1989 This page intentionally left blank List of Authors Alain BARRAUD Laboratoire d’Automatique... possible to raise and solve traditional control or /and identification problems Obviously, the complexity of calculations is increased with respect to traditional systems (i.e with constant coefficients) and, in practice, it is Linear Time-Variant Systems 535 necessary to use formal calculation tools Finally, the approach presented here was to voluntarily consider the continuous-time systems but an analogous... filtering 168 loop closed 25 3-2 58, 260, 26 4-2 67, 269, 280, 281, 28 3-2 86, 309, 324, 332, 382, 390, 408, 409, 449, 45 8-4 60, 463, 465, 472, 473, 475, 485, 499, 503, 533 open 253, 254, 256, 25 8-2 65, 272, 273, 276, 278, 28 3-2 87, 297, 308, 315, 316, 363, 380, 451, 46 2-4 65, 470, 486 M margin delay 268, 269, 360, 381 gain 267, 270, 284, 287, 380 phase 188, 26 7-2 70, 284, 287, 297, 298, 301, 302, 304, 308, 311, 31 7-3 20,... France Jean-François MAGNI ONERA-CERT, Département de commande des systèmes Toulouse, France Michel MALABRE IRCCYN Ecole centrale de Nantes France List of Authors Houria SIGUERDIDJANE Service Automatique, Supélec Gif-sur-Yvette, France Yves TANGUY Supélec – Service Automatique Gif-sur-Yvette, France Gérard THOMAS Dept EEA Ecole Centrale de Lyon Ecully, France Patrick TURELLE Supélec Gif-sur-Yvette, France... t 2 − 3t + 3 16. 6 Conclusion Due to the use of the algebra defined on the non-commutative body of rational fractions, it was shown that, not only could the concept of transfer function of a linear dynamic system with time variable coefficients be extended, but also the traditional operations on these systems had simple solutions, based on simple algebraic operations defined on the body of the related... 1)3 534 Analysis and Control of linear Systems leads to: t 1 t 2 − 3t + 3 ~ ~ λ+ Q(λ) = λ + and P(λ) = (1 − t ) t −1 (1 − t ) 2 And the condition: ~ ~ Q(λ)P(λ) = P(λ)Q(λ) finally gives: Q(λ) = λ − t 4 − 4t 3 + 6t 2 − 2t + 5 (1 − t )(t 3 − 3t 2 + 3t + 1) t 2 − 3t + 3 ⎛ 2 t 3 − 6t 2 + 15t + 14 ⎞ ⎜ (t − 3t + 3)λ + ⎟ P(λ) = − 3 ⎜ ⎟ t − 3t 2 + 3t + 1 ⎝ t 3 − 3t 2 + 3t + 1 ⎠ which gives the following controller: . )()( ˆ )( 1122 +λ=λλ=λλ=λ and 1 )( − λ=λH so )()( tuty =∑  If 530 Analysis and Control of linear Systems 16. 4.2. Parallel systems Let 1 ∑ and 2 ∑ be two systems of transfer functions )()()( 1 1 11 λλ=λ − PQH. +=++∑  Linear Time-Variant Systems 531 16. 5. Applications In this section, two types of usage of this algebra are presented in the field of modeling and control. 16. 5.1. Modeling One of the. Automatique, Supélec Gif-sur-Yvette, France Gilles DUC École Supérieure d’Électricité (Supélec) Service Automatique Gif-sur-Yvette, France 538 Analysis and Control of Linear Systems Didier