Analysis and Control of Linear Systems - Chapter 16 ppt

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.. In this perspec

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

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 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 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

non-signals and the second is dedicated to the design of a pole placement control law

16.1 Ring of non-commutative polynomials

Let Π(λ) be the set of polynomials of degree n:

when coefficient a n (t) is equal to 1, polynomial P(λ) is standardized

The set Π(λ) including addition and multiplication which satisfies:

)()()(,) K a t a t a. t

t

a ∈ λ = λ+

∀

has a non-commutative ring structure [ORE 33].

16.1.1 Division and the right highest divisor (RHD)

∀ P1(λ),P2(λ)∈Π(λ) ⊗ Π(λ)

insofar as n P1 ≥n P2 (where n X represents the degree of X( )λ ) there is a unique pair of polynomials [ ( ), ( )]Qλ Rλ so that:

)()()()

16.1.2 Right least common multiple (RLCM)

It is then possible to define the RLCM of P1(λ),P2(λ) as the lowest degree standardized polynomial divisible on the right by both P1(λ) and P2(λ)

)()()()()

(λ =Q1 λ P1λ =Q2 λ P2 λ

M

Generally, the existence of the Euclidian division implies the existence of the RHD [ORE 33]

16.1.4 Factoring, roots, relations with the coefficients

Any polynomial P(λ) can be factorized in the general form:

))(())((

))

(())((()

The roots of P(λ) are provided by the solutions of equation P(λ)=0

The relation between the roots and the coefficients of P(λ) is a non-linear differential equation [KAM 88]:

0)()()()())

1 2

− p t ∑a t S p t a t S p t a t

n i n

n

where S is the operator defined by:

2)

()

(

))()

(

) ( 1

2

1

≥

∀+

p

S

t p t p t

Sp

dt p S d i n n

i

n n n

i

EXAMPLE 16.1.– let us consider the second degree polynomial:

))())(

(())

−λ++λ

=λ

=

k t k t

P( ) 2 ( 1 )( 1 ) 

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

(skew) However, if the polynomials verify the two following conditions called ORE

it is possible to consider the set:

where Π*( )λ = Π( ) {0}λ − which has a body structure [ZHU 89]

NOTE.– the inverse of polynomial P(λ) is unique and verifies:

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

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

Therefore, system Σ can be formally described by its transfer function:

)()()

16.3.1 Properties of transfer functions

THEOREM 16.1.– two transfer functions H1(λ) and H2(λ) are equivalent if and only if there is a polynomial D(λ) such that [KAM 88]:

is a normal mode of the system

EXAMPLE 16.2.– for y(2)(t)=0 which corresponds to:

λ+1 + λ−1 + ∀ ∈ℜ =

A root 1t+k provides the normal mode: e t k

t d k

(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:

i

d q i

t i

e c t

) (

)

(

We infer that the system is stable if:

i t

(

lim

where ℜ means real part

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 H1(λ)=Q1(λ)−1P1(λ) and

)()

(

)

2 λ =Q λ − P λ

H respectively; the transfer function of the system ∑ obtained

by serializing ∑1 and ∑2 can be calculated as follows:

LetM(λ) be the RLCM of P2(λ) and of Q1(λ):

)(Q)(Qˆ)()(ˆ

)()()()(ˆ(

)()()()(ˆ(ˆ(

)()()()()()()

(

1 1 1 1 1 1 2 1

2

1

1 1

1 2

1

2

1

1 1 2 2

1 2

1

2

1 1 1 2 1 2 1

2

λλλλλλ

=

λλλλλ

=

λλλλλλ

=

λλλλ

=λλ

P Q M P Q

P Q P P P Q

P Q P Q H

16.4.2 Parallel systems

Let ∑1 and ∑2 be two systems of transfer functions H1(λ)=Q1(λ)−1P1(λ) and

)()

(

)

2 λ =Q λ − P λ

H respectively; the transfer function of the system ∑ obtained

by putting ∑1 and ∑2 in parallel is provided by:

Let M(λ) be the RLCM of Q1(λ) and Q2(λ):

)()(ˆ)(Q)(Qˆ

()

(

)()()()()()()

(

2 1 2 1 1 1 1

2 1 2 1 1 1 2

1

λλ+λλλ

=

λ

λλ+λλ

=λ+λ

H

P Q P Q H

H

H

EXAMPLE 16.4

))(1)

1 2 2

1 1

1 1

t

t

y + + = +

∑

to characterize a sufficiently large variety of dynamic behaviors Their properties were the object of numerous studies and their applications are extremely diversified However, these models are sometimes insufficient A current approach is, similarly

to the one that facilitated the traditional models MA, AR and ARMA, the research for new models taking into account the highly non-stationary character

We can illustrate this requirement for the modeling of polynomial phase signals, which we frequently encounter in physics, especially for processing signals coming from radar, sonar, etc These signals are non-stationary with frequency characteristics that continuously evolve in time with variation speeds that may be significant The dynamic model of the single-component signal is very easy to establish, whereas that of the multi-component signal is very complex However, this is indispensable when we deal, for example, with the problem of multiple trajectories due to reflections The algebra developed here makes it possible to create the complete dynamic model

Let us consider the following multi-component signal:

) (

1 1

2 2)

()

i i n

i

y t

y t t y t

α

−

However, obtaining the one that governs the sum y (t) requires the use of the results presented in the previous section

(

)

(λ =A−1 λ Bλ

H , is there a looping defined by Q−1(λ)P(λ) so that the closed

loop transfer has a dynamics set a priori by a polynomial C(λ)?

From A(λ)y=B(λ)u, Q( )λ r P= ( )λ y and u = − r + yc we obtain:

c

y Q y P u

Q

u B

y

A

)()()

(

)()

(

λ+λ

which is connected to the initial factorization by:

)(

~)()

A( )~( ) ( )~( )) ( )~( )

In order for the closed loop poles to be given by C(λ), it is necessary that:

)(

~)()(

~)(

)

(λ =AλQλ +Bλ Pλ

C

with: P(λ)Q~(λ)=Q(λ)P~(λ)

Which leads to the following algorithm:

– solution of the Diophantine equation;

leads to:

1)

(

~

−+λ

1)

1(

33)

(

~

t t

t t P

−+λ

−

+

−

=λAnd the condition:

)(

~)()

1(

5264)

++

−

−

+

−+

−

−λ

=

λ

t t t t

t t t t Q

−

++

−+λ+

−++

14156)

33(133

33)

t t t

t t t t

t t t t

t t P

which gives the following controller:

necessary to use formal calculation tools Finally, the approach presented here was

to voluntarily consider the continuous-time systems but an analogous approach can

be followed for the discrete-time systems (see, for example, Kamen’s works)

Cliffs, New Jersey, 1993

[CLE 98] CLEMENT A., “An ordinary differential equation for the Green function of

time-domain free-surface hydrodynamics”, Journal of Engineering Mathematics, vol 33, 1998

[GUG 01] GUGLIELMI M., NORET E., “Une classe des systèmes auto-similaires et à

mémoire longue”, Techniques et sciences informatiques, vol 20, no 9, 2001

[HUR 65] HURST H.E., BLACK R.P., SINAIKA Y.M., “Long term storage in reservoirs An

experimental study”, Constable, London, p 1153-1173, 1965

[KUC 79] KUCERA V., 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 480-508, 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 233-258, 1988

[WIL 95] WILINGER W., TAQQU M.S., LELAND W.E., WILSON V., “Self-similarity in

high-speed packed traffic: analysis and modelling of Ethernet traffic measurements”,

Statistical Science, vol 10, p 676-685, 1995

[ZHU 89] ] ZHU J., JOHNSON C.D., “New results in the reduction of linear time-varying

dynamical systems”, SIAM J Control & Optimization, p 476-494, 1989

Alain BARRAUD

Laboratoire d’Automatique de Grenoble

Ecole Nationale Supérieure d’Ingénieurs Electriciens de Grenoble ENSIEG Institut National Polytechnique de Grenoble

Didier DUMUR

Supélec

Gif-sur-Yvette, France

Sylvianne GENTIL

Laboratoire d’Automatique de Grenoble

Ecole Nationale Supérieure d’Ingénieurs Electriciens de Grenoble ENSIEG Institut National Polytechnique de Grenoble

Laboratoire d’Automatique de Grenoble

Ecole Nationale Supérieure d’Ingénieurs Electriciens de Grenoble ENSIEG Institut National Polytechnique de Grenoble

parallel 293

PD 306

PI 307-309, 312 PID310, 312, 313 serial 289, 290, 293

D

decoupling 133-136, 445, 448, 451,

452, 460, 462, 467, 470, 475 differential equation linear 197, 228, 231, 232, 235, 527 non-linear 197, 199, 524

Dirac impulse 4-6, 9, 34, 37, 143,

144, 151

E

equation Bezout 329, 341, 362 differential 15, 16, 46, 103, 199,

231, 234, 236, 238, 496, 531, 533 Diophantus 335, 336, 377 Lyapunov 227, 231, 233, 406,

407, 410, 421, 426, 438 Riccati 171, 172, 174, 175, 178-

180, 189, 191, 410, 420, 421, 426,

488, 489

AR 152 ARMA 150 ARMAX 153-156 ARX 153, 154 behavioral 196, 198, 214 deterministic

functional 195 main 21 signal 4 statistical 141 structural 195

N, O, P

Nyquist criterion 258-260, 263, 265,

486 optimization non-linear 214, 217, 221 quadratic 171, 173, 177, 179, 216 Padé approximant 230, 231, 234, 247 polynomial

invariant 114,115 minimal 114 Prony’s method 152

R

representation external 57, 88 internal 43, 44, 57, 60, 89 response

fixed 110, 135, 136 forced 90, 376, 392 free 90, 387, 407 frequency 14, 15, 17, 18, 21, 24,

30, 31, 96, 98, 103, 187, 197, 256-258, 261, 262, 285

harmonic 204, 267 closed loop 255

393, 394 transfer function 12-21, 23, 27, 28,

44, 57, 58, 60, 61, 96, 97, 101, 102,

105, 130, 150, 151, 153-155, 186, 187,

197, 202-209, 211, 212, 214, 244, 246-248, 254, 255, 257-261, 263, 269,