A formulation of dissipativity on pencil model

14th World Congress oflFAC

A FORMULATION OF DTSSIPATIVITY ON PENCIL MODEL.. .

D-2b-lO-2

Copyright © 1999 IFAC 14th Triennial World Congress, Beijing, P .R. China

A FORMULATION OF DISSIPATIVITY ON PENCIL MODEL Koki Shibasato· Tetsuo Shiotsuki·* Shigeyasu Kawaji ***

* Graduate School of Science and Technology, Kumamoto University, 2-39-1 Kurokami, Kumamoto-city, Kumamoto, 860-8555, Japan, [email protected] ** Department of Computer Science, Kumamoto University, 2-39-1 Kurokami, Kumamoto-city, Kumamoto, 860-8555, Japan, [email protected] *** Department of Computer Science, Kumamoto University, 2-39-1 Kurokami, Kumamoto-city, Kumamoto, 860-8555, Japan, [email protected]

Abstract: The concept of dissipativity and passivity is of interest from a theoretical as well as a practical point of view. In this paper, a formulation of dissipativity for linear time-invariant system on pencil model is derived. According to the structural properties of canonical form for pencils, it can be shown that a storage function is written as a quadratic form of the solution of the LMI which consists of the original system parameters. Copyright © 1999 1FAC Keywords: Implicit systems, Behaviour, Linear systems, Time-invariant systems, Passive elements, Modes

1. INTRODUCTION

decide what are causes and what are effects. Thus it has an advantage for describing non-proper systems and/or the systems whose inputs and outputs are not defined.

In control engineering a dynamical system is often considered as a set of differential/algebraic equations, which is used for analyzing the system behaviour that progress with time. In most cases, however, the reduced model which only consists of dynamical relations by canceling static relations out is used. Throughout this paper the pencil model is used in order to describe the system that is one of the system representation in the context of the behavioral approach proposed by J.C. Willems (Willems, 1986; Willems and Polderman, 1998; Kuijper, 1994). It is very natural form in the sense of no need of the reduction. And it is often useful to analyze the dynamical systems, when it is not clear which of the variables should be regarded as inputs and which as outputs. For example, in economical systems it is difficult to

Clearly the algebraic Riccati equation is an important concept in control theory. It is well known that the Riccati equation is derived from the dissipation inequality which expresses the fact that the system is dissipative; the energy stored inside the system does not exceed the amount of supply which flows into the system (Trentelman and Willems, 1991). In this paper the dissipativity condition and the passivity condition that relate the Riccati equation are derived in terms of the original system parameters, i.e. these conditions are given as a LMI with the coefficients of a pencil model.

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A FORMULA nON OF DTSSIP ATTVTTY ON PENCIL MODEL...

14th World Congress ofTFAC

The outline of the paper is as follows. First, the notation for the pencil model is introduced. Next, some kind of pencils are defined in order to discuss the canonical form for the pencil (Shiotsuki et al., 1997). It is well known that the matrix pencil can always be put into the canonical quasidiagonal form (Gantmacher, 1959; Dooren, 1979). The bidiagonal sub-pencil with row full rank is called under-determined mode. In the pencil model, the most essential properties of the re-

A criterion for strict equivalence of two pencils is established, next a strictly equivalent canonical form for each pencil is determined (Gantmacher, 1959).

Definition 2. Any pencil M - >'N E Rqxn[..\.] is transformed into a canonical form, after multiplication on the left by an appropriate regular matrix p E ~qxq and on the right by Q E IRnxn :

lated dynamical systems relevant to the behavior of the under-determined mode, while the square pencil which expresses the static relation does not play an important role in the system behaviour. Next, main result of this paper is given. For linear time-invariant finite-dimensional systems with quadratic supply rates, it is shown that the dissipativity of the system leads to a solution of a LMI. Finally, as numerical example, the algebraic Riccati equation for an electrical circuit is illustrated.

(4)

where

Pu(>') Ps(..\.) Py(>.) Po(>')

P.(>')

=

Mu - >"Nn , Ms - >"N", Mc - >"Ny. Mo - >..No , Mi - >.NI · .

2. PRELIMINARIES These pencils (Pu (>"), P z (>'), Py(>..) , Po (>..) , PiC>"» are called sub-pencils. The sizes of these matrices are shown in Table l.

2.1 Pencil model in beha'llioral approach

A need for a framework at a more abstract level gave rise to the behavioral approach. In this approach, dynamical systems are defined in terms of trajectories of the external and the internal variables. The present section reviews the definition of pencil model from the behavioral point of view (Kuijper, 1994). Furthermore, definitions of the sub-pencil and that of canonical form are introduced.

Table 1. Sizes of sub-pencils mode under-determined zero finite over-determined infinite

Note that the size of Mu and Nu (Mo and No) is admissible to be 0 x nu(qo x 0). In addition,

The pencil representation has first order in the internal variables and zeroth order in the external variables, such that

[M - VN] z=O,

(1)

w=Sz.

Definition 1. Two pencil M - >"N and

(2)

[M -

>.N] Q =

M - >,N.

Mu and Nu are full row rank,

(H)

Mo and No are full column rank, M." and Ni are nilpotent matrices, Nz, Mr, Nr and Mi are regular matrices.

The regularity of sub-pencils is summarized as follows. Table 2. Properties of pencils mode

M

of the same dimension (JRqxn[>..]) are said to be strictly equivalent, when there exist constant regular matrices P and Q such that

P

(i) (Hi) (iv)

Here M, N and S are constant real valued matrices whose sizes are JRqxn, Rqxn and IRrxn , respectively. The internal variables z E Rn and the external variables w E IR r are time continuous function. The letter V denotes a differential operator in the sense of distribution.

>"N

size qu x nu q,. x n z qf x nf qo x no qi X n;.

sub-pencil Mu->"Nu M"->"N.,, Mr->..Nr Mo->..No M; - "\'Ni

(3)

where P E jRqxq and Q E lRnxn.

regularity full row rank CO' >. E C U { 00 } ) regular (of>.. E C/O), the rank defects at >. = 0 regular at >.. = 0 and 00, the rank defects for some>.. (>.. E C/O) full column rank (If>.. E C U {oo}) regular ('If>. E q, the rank defects at >.. = 00 regular (It>.. E C u {co})

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ISBN: 008 0432484

A FORMULA nON OF DTSSIP ATTVTTY ON PENCIL MODEL...

14th World Congress ofTFAC

In detail, the infinite mode Pi (A) is composed of impulse Illode Pol (A) and purely static Illode P;2{A).

environment - - - - .

JOT &(w(t))dt supply rote

Theorem 3. The purely static mode can be described as

Because the mode consists of only the algebraic relation, Mi2 is regular matrix and (6) It is clear that

Zi2

=

dynamical system Fig. 1. Dissipative system matrices in the pencil model. Let the storage function be given by a quadratic function of wet) such as

o.

3. DEFINITION OF DISSIPATIVITY

s(w(t)) = wT(t)Rw(t).

In this section the definition of dissipativity of dynamical system is given. In dissipative systeIll, the energy income and expenditure are denoted by a supply rate and a storage function, respectively. These functions are connected by a dissipation inequality. The inequality says that the energy stored inside the system does not exceed the amount of supply which flows into the system. The dissipativity is defined as follows (Trentelman and Willems, 1991; Trentelman and Willems, 1997; Van der Geest and Trentelman, 1997).

Define

V(z(O» - V(z(T))

+

I

~ 0,

(7)

V(O) = O.

(8)

s(w{t))dt

=

r(X)

_MTXN-NTXM+STRS. (10)

Then the following strong dissipativity theorem holds (Shibasato et al.• 1997).

Theorem 5. The pencil system is dissipative with respect to the supply rate s(w(t}}, if there exists X which satisfies r(x) 2:

Definition 4. The time-invariant dynamical system is called dissipative with respect to supply rate s, if there exists a function V which satisfies T

(9)

o.

(11)

Then, zT(t)N T XNz(t) is a storage function of the system.

PROOF. When the system is given as the pencil model (1) and (2), (7) and (8) in definition 4 are restated as

o

I {-! T

Then V(z(t)) is called a storage function.

V{z(t»

+ S(w(t»} dt 2: 0,

(12)

V(O) = O.

(13)

o

The function s is the rate at which energy flows into the system, then it is defined as a function with external variables wet). Thus s(w{t»dt is equal to the total energy that flows into the system. The storage function V means the amount of supply that is stored inside the system. Then V is described as a function with internal variables z(t).

t:

Let

Substitute (1). (9) and (14) in the left side of (12), then

4. MAIN RESULTS _NT X M

4.1 Dissipativity condition based on pencil model

+ ST RS) z(t)dt.

(15)

Equation (15) is positive semi-definite for v z(t), if r(X) 2: O. Now there exists a storage function V(z(t») with respect to the supply rate s(w(t», the system is dissipat ive.

The definition of dissipativity in section 3 has been given in terms of the external variable and the internal variable. In the present section the dissipativity is discussed in terms of coefficient

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ISBN: 008 0432484

A FORMULA nON OF DTSSIP ATTVTTY ON PENCIL MODEL...

14th World Congress ofTFAC

Even if the mathematical models describe the same system, the evaluated value of r(X) are different. Example 6. Consider the electrical circuit of Fig. 2 consisting of a resistor. The variable vet) is the voltage across and the the variable i(t) is the current through the element. The variables v(t) and i (t) are related as

(16)

vet) = Ri(t). model 1 (z(t) "" [vet) i(t)JT)

At first, define the internal variable as follows:

z(t)

=

[ v(t)] i(t) .

(17)

dynamical element regarding the internal variable z(t). Evaluate (10) for the same supply rate (20), then

Since (11).

r

is positive definite, this model satisfies

Remark 7. Example 6 explains the fact that the LMI (11) is only sufficient condition for the dissipativity. Therefore, when the model includes the purely static mode, (10) is very tight condition. Model 1 in example 6 is composed of 0 x 1 underdetermined mode and 1 x 1 infinite mode. r{X) is not often positive definite, nevertheless the system is dissipative.

Then the circuit is described as

[1 -RJ z(t) =0,

(18)

wet) = z(t).

As mentioned above, any matrix pencil has, under strict equivalence, a diagonal form:

(19)

p

[M -

AN] Q

Let a storage function for the system be an electric power:

1 [0 1]

s(w(t») = 2"wT(t)

1 0 wet).

= [Ne - AN] =

SQ

(20)

[Afl ~ AN1 ~J ,(25)

= S = [81 82 ] ,

(26)

where P and Q are Then reX) becomes constant matrix r:

r =

ST RS =

p=

~ [~ ~] .

(21)

r

(27)

E Rqx q ,

Q2]

Q = [Q~

It is easily verified that eigenvalue of is equal to ± This is the case not satisfied (11 ) .

t.

[;~]

(28)

E lRnxn.

P and Q transform the model into the purely static mode and the other modes. Define it. and i as follows,

model 2 (z(t) = i(t»

x= [~11T ~12] = p-TXp-l,

Next, define the other internal variable as z(t)

= i(t).

X 12

(22)

i(t) =

Then the circuit in Fig. 2 is modeled as follows

wet)

= [ ~] z(t).

X

[~~m]

(29)

22

= Q-I z(t).

(30)

Then (15) is rewritten as

(23)

J T

Note that the size of coefficient matrices M and N is 0 x 1 in model 2, since there exists no

zT(t)QT (-MTXN

o

-NTXM + STRS) Qi(t)dt.

i(t)

(31)

Substitute (29) into (31), then T

vet)

R

f

ZT(t) (_QTMTPTXPNQ

o _QT N T pT X P MQ Fig. 2. Electrical circuit with resistor

+ QT ST RSQ) z(t)dt.(32)

By using the notations rewritten as

M, N

and

S,

(32) is

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Copyright 1999 IF AC

ISBN: 008 0432484

A FORMULA nON OF DTSSIP ATTVTTY ON PENCIL MODEL..

I

14th World Congress ofTFAC

0

4.2 Passivity on the pencil model

T

zT(t) (_UT XlV The dissipativity is a generalized concept of passivity. A concerned system is passive, if there exists a non-negative storage function. Thus the following lemma is derived from theOl.'em 8.

o

Assume that

Lemma 9. The given system is passive, if there exists a solution X of LMI:

(34)

Define

r(X) for the model [M - XNJi = 0: -

- T

-

-

- T

-

-

-T-

QI T r(X)Q1 2!. 0,

(44)

~O,

(45)

r(X)=-M XN-N XM+S R5

=

[ru (-!u) r

T

12 {X12 )

r12 (-!12)]

F 2 2{X2:<1)

.

X l1 (35)

with

The internal variable i2 expresses the purely static mode. fl-om theorem 3,

(46)

(36)

The size of X 11 is as same as the row size of Plo

Example 10. Consider the electrical circuit shown in Fig. 3. The variable V2 is the voltage across the resistor. From Kirchhoff's lows (47) and (48) are obtained.

Then iT r(X)i

= [i1 TO] r(.X) [ z{] rll(XU)Zl,

=.%1 T

(37)

VI (t)

where

= V2(t)

+

L:

(47)

i(t),

V2(t) = Ri(t). -

-

T

-

-

-T-

-

rll(X ll ) = -M1 X ll N l

-NI XllMI

Define the internal variable z and the external variable w as follows:

-T-

+ 51

(48)

RS I . (38)

Therefore, it holds that

i(t) ] wet) = [ Vl{t) . V2(t)

(39)

for the model: Then the circuit is modeled as (49) and (50).

[M - AN] Z = w

0,

(40)

=5£.

(41)

[[ 001~1] -1 R

The following theorem is the result on the weak dissipativity.

wet)

Theorem 8. The model is dissipative with respect to a supply rate s (w (t», if there exists a solution

=

A [LOO]] 0 0 0 z(t) = 0,

(49)

[100]

(50)

0 1 0 001

z(t),

for positive Rand L. Let R = 1 and L = 1, then

X ofLMI: i{t)

(42)

PROOF. Any matrix pencil is transformed into a diagonal form like as (40) and (41). Then there exists X which satisfies

R

T

/ zT(t)r(X)z(t)dt o

~ 0,

(43) L

if QITr(X)Ql :2: O. Then, zT(t)NTXNz(t) is a storage function of the system.

Fig. 3, Electrical circuit

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ISBN: 008 0432484

A FORMULA nON OF DTSSIP ATTVTTY ON PENCIL MODEL...

14th World Congress ofTFAC

5. SUMMARY

M=[~~=n,

N=[~~~]' S=Is · Here, assume the supply rate sew) is equal to wTSTRSw:

In this paper a formulation of the dissipativity based on the pencil model is derived. The main result (Th. 8) implies that the dissipativity of the systern.s is characterized by a simple LMI, and that the purely static mode has no effect to the dissipativity and the LMI should be reduced to the dynamical part of the system.

1 00] R= [ 020 . 003

REFERENCES

Then P and Q which transform the pencil into a canonical form ace calculated as follows: P =

Dooren, P. Van (1979). The computation of Kw.. necker's canonical form of a singular pencil. Linear algebra and its applications 27, 103140. Gantmacher, F.R. (1959). THE THEORY OF MATRICES. Vol. 1.. 2. CHELSEA PUBLISH .. ING COMPANY. New York. Kuijper, M. (1994). First-order Representations of Li.near Systems. Birkhiiuser. Boston. Shibasato, K., T. Shiotsuki and S. Kawaji (1997). Dissipativity, passivity and Riccati equation in pencil modeL Proceedings of The IEEE Singapore International Symposium on Con .. f:rol Theory and Applications 1, 409-413. Shiotsuki, T., K. Shibasato and T. Sato (1997). A behavioral approach by using pencil based system models. Proceedings of the 11th IFAC Symposium on System Identification 3, 12311236. Trenteiman, H.L. and J.C. Willems (1991). The dissipation inequality and the algebraic Ric.. cati equation. In: The Riccati Equation (S. Bittanti, A.J. Laub and J.C. Willems, Eds.). pp. 197-242. Springer.. Verlag. Berlin. Trentelman, H.L. and J.C. Willems (1997). Every storage function is a state function. Systems and Control Letters 32, 249-259. Van der Geest, R. and H. Trentelman (1997). The Kalman-Yakubovich-Popov lemma in a behavioural framework. Systems and Control Letters 32, 283-290. Willems, J.C. (1986). From time series to lineae system part 1. Finite dimensional linear time invariant systems. Automatica 22(5), 561580. Willems, J.C. and J.W. Polderman (1998). Introduction to Mathematical Systems Theory. Springer. New York.

[·~:·~~~~··~~·:·~~ii·] ,

Q= [

-0.1803 -0.5774 0.7071] 0.3607 -0.5774 0.0000 . -0.1803 -0.5774 -0.7071

It is clear that the model is composed of the underdetermined IIlode and the infinite mode whose sizes are 1 x 2 and 1 x 1, respectively. Solve the next LMI: minimize X 11, subject to Q1 T r(X)Q1 2:

o.

There exists a solution X l l = 1.4641. Since X l1 is positive definite, the system is passive from theorem 8.

Remark 11. After eliminating V2 by substituting (48) in (47) in example 10, the differential equation relating the port voltage Vl(t) to the port CUl"rent i(t}

is obtained. This is a minimal model of the circuit, however, the implicitly related model (49) is used. This example shows that the LMI (42) can evaluate the dissipativity by using the model that includes the purely static mode. This means that theorem 8 does not need the reduced model.

Remark 12. The symmetric solution Y Riccati equation (52) is equal to X l1 .

of the

ATY+YA+DTQD -(Y B

+ CT)(DT RD)-l(BTy + C) =

0,(52)

where A, B, C, D are system matrices of Fig. 3:

with the current i(t) as the state and the voltage Vl(t) as the output.

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ISBN: 008 0432484