On Gramians Properties of Continuous Control Systems

4th IFAC Symposium on System, Structure and Control Università Politecnica delle Marche Ancona, Italy, Sept 15-17, 2010

On Gramians Properties of Continuous Control Systems Igor B. Yadykin V.A. Trapeznikov Institute of Control Sciences, 65, Profsoyuznaya, Moscow, 117997 GSP, Russia (Tel.: +(7) (095) 3349031. e-mail: [email protected]) Abstract. A new approach to the controllability and observability properties, based on controllability and observability matrices links, is considered. The approach is based on matrices A, Am integrals expansion Lemma and properties of the matrices function convolution in complex plane. Coefficients of characteristic equations and Faddeev’s matrices in the resolvent expansion of the system dynamic matrix play a dominant role in matrices integrals expansion forming. The gramians expansion on dynamic system matrix spectrum provide a new notion for the solutions of the Lyapunov and Sylvester equations. The results are illustrated by the example for two-zones furnace control. Kwakernaak and Sivan (1991) have used the resolvent of the dynamics control system matrix with multiple-input multipleoutput (MIMO) plants for many problems of analysis and synthesis of such systems.

1. INTRODUCTION The structural properties of control systems (Kalman and Bucy, 1960, Kalman, 1963 and Luenberger, 1964) have invariably attracted the attention of researchers in theory of dynamic systems, including control systems. The study of structural properties is of key importance for a deeper insight into fundamental properties of the systems and may lead to the new principles in design of control systems and new algorithms of state estimation. Handbooks on control theory (Levin, 1996, Krasovsky, 1987) seem to offer a most exhaustive overview of references in this field. Historically, the first papers in this field analyzed structural properties of controllability and observability of linear control systems featuring constant and variable parameters (Andreyev, 1976, Kailath, 1980, Kwakernaak and Sivan, 1991, Hanzon and Peeters, 1996, Polyak and Shcherbakov, 2002, Poznyak, 1991, 2008, Balandin and Kogan, 2007).

During last decade, methods of constructive algebra were successfully applied to the polynomial equations decision and algebraic optimization problems, simulation of complex dynamic systems, and problems of differential algebra and differential geometry, used in a systems theory (Constructive Algebra and System Theory, 2006). Peeters and Rapisardra (2006) have developed a polynomial approach to solving a new class of Lyapunov and Sylvester polynomial equations. Balandin and Kogan (2007) have suggested to use controllability and observability gramians for the decision of problems of control systems optimal synthesis using linear matrix inequalities.

For linear dynamic systems, the criteria of controllability and observability have been received by Kalman in the form of controllability and observability rank criteria of matrices, represents the block matrices, composed of products of input and output matrices and degrees of a system dynamics matrix (Kalman, 1963).

The paper will outline a new approach to studying these properties in linear continuous time invariant systems. This approach is based on expansion of the dynamic system operator’s resolvent on the system operator spectrum. The new notion for resolvent expansion, made up by means of Faddeev’s series is used in the paper. Gramians with infinite time being the decisions of Lyapunov and Sylvester algebraic equations have received much consideration in technical and science literature. Gramians with finite time also being the decisions of Lyapunov and Sylvester differential equations have not received nearly so much consideration.

Ranks of controllability and observability matrices are the system invariants. Algebraic methods, in particular the methods of linear algebra, are well positioned in the mathematical systems theory (Gantmacher, 1990, Fuhrmann, 1996, Datta, 2004, Poznyak, 2008). In the paper resolvent decomposition of system dynamics matrices plays the central role in research of gramians and controllability and observability matrices properties. Faddeev has developed an efficient algebraic method for the decomposition of the matrix resolvent of the system dynamics matrix as well as effective computational algorithm for calculating the matrix polynomial numerator and the characteristic polynomial of the denominator of this expansion (Faddeev and Faddeyeva, 1963).

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The current techniques for the decision of the differential equations obtaining are based on the differential equation theory. On the other hand, the decisions of Lyapunov and Sylvester algebraic equations are based on linear algebra techniques. Below we suppose the alternative approach to the problem, based on a spectral expansion of gramians in frequency and time domain.

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2. LEMMA ON EXPANSION OF MATRIX INTEGRALS T

∫e

Aτ

T

R e Amτ ,

0

T

∫e

AmTτ

where L denotes the computation of the Laplace transform of an expression [ f1(τ ) f 2 (τ )] . Let us apply to the relation (4) the theorem of the product of two functions in the real domain.

S e Aτ , ON SPECTRA

0

OF SQUARE MATRICES A[n×n ], Am[r×r ]

With the above assumptions the function F (s ) is a proper fractional rational matrix function and so

Lemma. Let A[n×n ], Am[r×r ], R[n×r ], S[r×n ] be real matrices

L[ f1 (τ ) f 2 (τ )] =

and eigenvalues of A[n×n ], Am[r×r ] are denoted as sk , smλ , be simple, with

sk + smλ ≠ 0 . Then at all times ∀T ∈ [ 0, ∞ )

∫e

Aτ

n n−1 nm −1

R eAmτ dτ = ∑∑∑ T

0

nm n−1 nm −1

nm n −1 nm −1

Aτ Aτ ∫ e m S e dτ = ∑∑∑ T

λ =1 j =0 η =0

0

sηmλ (−smλ ) j AmTη SAj + Nm' (sλ ) N (−sλ )

T

j =0

i =0

i = j +1

nm

n

nm −1

j

η=0

skj sηmλ 1 ⋅ Aj RAmTη , ' k=1 λ=1 j=0 η=0 (smλ + sk )N '(sk )Nm(smλ + sk ) s − smλ − sk n nm n−1 nm −1

+∑∑∑∑

∀T :0 ≤ T <∞ whence follows the identity (1). By having f1 (τ ) and f 2 (τ ) change places we prove in a similar way the identity (2). If the A and Am matrices are Hurwitz ones, in the identities (1) and (2) we can go to a limit

n

∑ ai Ai − j +1 ,

(Is − Am ) = ( ∑ s Amη )(∑ami si )−1, Amη = −1

i=0

∑

i =η + 1

a mi Ami −η +1 ,

T

lim ∫ e Aτ R e Am dτ =

T →∞

(3)

Am . We have used upper designation (.)

Proof. Let f1 (τ ) = e Aτ R ,

0

= ∑∑∑

Τ

k =1 j =0 η =0

for transposed matrix (.).

T

f 2 (τ ) = e

AmΤτ

skj (−sk )η Aj RAmTη = Ω1 , N '(sk ) Nm (−sk )

(6)

lim ∫ e Amτ Se Aτ dτ =

T →∞

.

T

0

n n−1 nm −1

Then the Laplace transformation of the function f1 and f 2 have the form

= ∑∑∑ λ =1 j =0 η =0

L[ f1(τ )] = ( Is − A) −1 R, L [ f 2 (τ ) ] = (Is − AmT ) −1 ,

sηmλ (−sηmλ )η AmTη SAj = Ω2 . Nm '(smλ ) N (−smλ )

(7)

Remark. In deriving the identities (1) and (2) the matrices A and Am were not assumed be Hurwitz.

By virtue of the properties of Laplace transformations using we have

⎡ ⎤ T L ⎢ ∫ e Aτ R e Am dτ ⎥ = s −1 L [ f1 (τ ) f 2 (τ ) ] , ⎣0 ⎦

T

n n−1 nm −1

where ai , ami − coefficients of the characteristic equations of matrices A and

Τ

0

d where N ' (s) = N (s), A j and Amη are Faddeev’s matrices, ds generated by expansion of resolvents of the matrices A and Am (Faddeev, Faddeyeva, 1963, and Hanzon, Peeters, 1996). n

skj (−sk )η Aj RAmTη + sN '( s ) N ( − s ) k=1 j=0 η=0 k m k n n−1 nm −1

L[∫ eAτ S eAmτ dτ ] = ∑∑∑

(2)

n −1

⎞ ⎟, ⎠

the simple poles, si ≠ 0 , we obtain

skj sηmλ +∑∑∑∑ AmTη SAj e( sk +smλ )T ' ' k =1 λ =1 j =0 η =0 ( smλ + sk ) Nm ( sλ ) N ( sk + smλ )

( Is − A) −1 = ( ∑ s j A j )( ∑ ai si ) −1 , A j =

(5)

where f ( s ) / F ( s ) is a fractional rational function, that has

nm n −1 nm −1

n

∑

q ⎛ f (s ) f (s) f (0) 1 i = + ∑ ⎜ sF ( s ) sF (0) i =1⎝ si F ' ( si ) s − si

(1) T

N ' ( s k ) η =0 N m ( s − s k )

∑∑

Transforming the second multiplier in the right-hand part of the equality and using the identity,

skj sηmλ +∑∑∑∑ Aj RAmTηe(sk +smλ )T , ' k =1 λ =1 j =0 η=0 (sk + smλ )N '(sk )Nm (sk + smλ ) n

k =1 j =0

⎡ T Aτ ⎤ n n −1 skj A j R nm −1 ( s − sk )η AmTη AmT . L ⎢ ∫ e R e dτ ⎥ = ∑ ∑ ∑ ⎣0 ⎦ k =1 j =0 N ' ( sk ) η =0 sN m ( s − sk )

η

s (−sk ) Aj RAmTη + N '( s ) N ( − s ) k =1 j =0 η=0 k m k j k

T skj A j R nm −1 ( s − sk )η Am η

Substituting (5) into (4) we have

the following identities are true: T

n n −1

Matrices Ω1 , Ω 2 represent the weighted sum of products of the direct and transposed Faddeev’s matrices, generated by matrices A, Am resolvent decomposition. Weighted factors of the specified sums depend on weight matrices R, S, from

T

(4)

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factors and roots of the characteristic equations of matrices A, Am . Let us introduce the notation for the second group of summands in the identities (1)-(2),

where x ∈ R n , u ∈ R r , y ∈ R r . Assume also that the system is completely controllable and observable. A matrix transfer function is known to be represented in the form

Ξ1 (T ) =

W (s) = C( Is − A)−1 B =

n nm n−1 nm −1

j η k λ

ss ⋅Aj RAmTη e(smλ +sk )T , ' mλ + sk )N '(sk )Nm (smλ + sk )

∑∑∑∑ λ η (s k=1 =1 j=0 =0

∀T :0 ≤ T <∞

skj sηmλ ⋅AmTη SAj e(smλ +sk )T . ∑∑∑∑ ' ' s s N s N s s ( + ) ( ) ( + ) k =1 λ=1 j=0 η=0 mλ k m k mλ k n nm n−1 nm −1

Matrices Ξ1 (T ), Ξ 2 (T ) also represent the weighted sum of products of the direct and transposed Faddeev’s matrices, generated by decomposition of resolvents of matrices A, Am .

Corollary

λ =1

s = smλ

⋅S ⋅ ( Is − A)−1

an I n

0n

...

...

a2 I1

...

1

eigenvalues

0n ⎤ 0n ⎥⎥ , ... ⎥ ⎥ an I n ⎦

of

the

Lemma.

Let

T

of

the

matrix

λi , λ j

the

condition

Re(λi + λ j ) ≠ 0 be true. Let us apply the identities (1) and (2) to the controllability gramian computing (Hanzon and Peeters, 1996, Polyak and Shcherbakov, 2003, Poznyak, 1991, Balandin and Kogan, 2007).

The another identities are true

nm

0n

Am = A, R = BB , S = C C , nm = n , the pair (A, B) be controllable, the pair (A, C) be observable, eigenvalues of A[n×n] matrix, denoted as sk , smλ , be simple and for all

T →∞

Ω 2 = ∑ Res ( Is − AmΤ ) −1

0n

T

lim Ξ1 (T ) = 0, lim Ξ 2 (T ) = 0.

k =1

i =1

(10)

It is obvious, that for a Hurwitz matrices A, Am

× R × ( Is − AmΤ ) −1

i = j +1

Aj = Qn − jV0 , Qn − j = row ⎡⎣ an − j −1 I n an − j I n ...an I n 0...0 ⎤⎦ .

Each coefficient in the above sum contains the multiplier, which is a complex-values exponential function of the current time T, which index is the sum of the values of the roots of the matrices A, Am characteristic equations, being multiplied by the current time.

s = sk

j =0

(9)

U 0 = V0T = row ⎡⎣ I n AA2 ... An −1 ⎤⎦ , U = U 0 B, V = CV0 ,

Weighted factors of the specified sums depend on weight matrices R, S from factors and roots of the characteristic equations of matrices A, Am , and, moreover, depend on the current time.

n

n

⎡ an I n ⎢a I Q = ⎢ n −1 n ⎢ ... ⎢ ⎣ a1 I n

∀T :0 ≤ T <∞

Ω1 = ∑ Res (Is − A) −1

n

where the matrix factors of the polynomial in the numerator Aj are members of the Faddeev’s sequence and satisfy the relations (Faddeev and Faddeyeva, 1963, Hanzon and Peeters, 1996):

Ξ 2 (T ) =

T →∞

n−1

= ∑C ∑ ai Ai − j −1Bs j (∑ ai si )−1.

s =− sk

,

∞

n n−1 nm −1

skj (−sk )n Aj BBT AηT = N s N s − '( ) ( ) k =1 j =0 η =0 k k

Pc = ∫ eAτ BBTeA τ dτ = ∑∑∑ s =− smλ

Τ

0

.

n

= ∑Res (Is − A)−1

Above a sign Res denotes the matrix function residue.

k =1

s=sk

×BBT × (Is − AΤ )−1

s=−sk

. (11)

The proof of the above identities is given in APPENDIX. 3. EXPANSION OF THE RESOLVENT OF THE DYNAMIC SYSTEM’S OPERATOR AND STRUCTURAL PROPERTIES OF ITS CONTROLLABILITY AND OBSERVABILITY GRAMIANS

Similar relationships hold for the observability gramian

sηλ (−sλ ) j P = ∫ e C Ce dτ = ∑∑∑ AηΤСTСAj = λ=1 j =0 η=0 N '(sλ )N(−sλ ) 0 ∞

0

Let a linear stationary stable dynamic system be described by an equation

x& = Ax + Bu, y = Cx,

x(0) = 0,

ATτ

T

n n−1 nm −1

Aτ

n

= ∑Res (Is − AΤ )−1

(8)

k =1

s=sλ

×CTC × (Is − A )−1

s=−sλ

. (12)

The gramians (4) and (5) are known to be solutions to Lyapunov equations (Faddeev and Faddeyeva, 1963, 180

SSSC 2010 Ancona, Italy, Sept 15-17, 2010

Andreyev, 1976, Kwakernaak and Sivan, 1991, Hanzon and Peeters, 1996, Polyak and Shcherbakov, 2003, Poznyak, 1991, Balandin and Kogan, 2007)

n

k =1 j = 0 η = 0

AP + PAT + BBT = 0,

(13)

AT P + PA + C T C = 0.

(14)

n

n −1

j =0

j =0

4. AN ILLUSTRATIVE EXAMPLE. Let us take up the case of controlling a two-zones furnace (Andreyev, 1976). The control plant is described by a model

M c ( s ) = ∑ s j A j B, M o ( s ) = ∑ s j CA j .

−0.5 x& = Ax + Bu , A = ⎡⎢ ⎣ 0

Then the formulae (11) and (12) may be rearranged as

∑ [N ' (sk ) N (−sk )]−1 M c (sk )M cT (−sk ),

0.5⎤ ⎡1 0⎤ C=⎢ ⎥, ⎥ 2⎦ ⎣0 1 ⎦

y = Cx.

(15)

(19)

As the matrices B and C are nonsingular square ones, we have

n

∑ [N ' (sk ) N (−sk )]−1 M 0T (sk )M o (−sk ).

⎡1 0 ⎤ , B=⎢ ⎥ −1.0 ⎦ ⎣0.5

x(0) = 0,

n

k =1

P0 =

n −1 n −1

k =1 j = 0 η = 0

Introduce the notation. n −1

(18)

= ∑∑∑ ϕ jη ( sk )(CQn −η V0 )T (CQn − jV0 ).

Formulae (11) and (12) provide solutions to Lyapunov equations (13), (14) as an expansion of controllability and observability gramians on the spectrum of the matrix A.

Pc =

n −1 n −1

P o = ∑∑∑ ϕ jη ( sk ) AηT C T CAj =

(16)

rank ⎣⎡C Τ

k =1

AΤ C Τ ⎦⎤ = 2, rank [ B

AB ] = 2.

Let us assume further, that all roots of the characteristic equation of the system are simple (not multiple).

So, the system is completely controllable and completely observable.

From equalities (15)-(18) the theorem follows.

For this case we have

Theorem. For linear time invariant continuous system of a general form (8) completely controllable and observable, eigenvalues of A[ n×n] , denoted as sk , sλ , be simple, for all

det( Is − A) = N 0 ( s ) = a2 s 2 + a1 s + a0 =

eigen

values

of

the

matrix si , s j

the

= ( s − s1 )( s − s2 ), s1 = −0.5, s2 = −1, a2 = 1, a1 = 1.5, a0 = 0.5, N 0 ( s ) = 2a2 s + a1 ,

condition

−1 (Is − A)−1 = (As 1 + A0 )N0 (s),

Re( si + s j ) ≠ 0 is being fulfilled, controllability gramian is determined by the expressions n

−1

⎡s + 0.5 0 ⎤ ⎡s +1 0 ⎤ 2 = ×(s +1.5s + 0.5)−1 , ⎢ 0 s +1⎥⎦ ⎢⎣ 0 s + 0.5⎥⎦ ⎣

n −1 n −1

P c = ∑∑∑ ϕ jη ( sk ) Aj BB T AηT = k =1 j = 0 η = 0

n

⎡1 0⎤ ⎡1 0 ⎤ ⎡1.25 1.5 ⎤ T A1 = ⎢ ⎥ , A0 = ⎢0 0.5⎥ , BB = ⎢ 1.5 4.25⎥ , 0 1 ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(17)

n −1 n −1

N 0 ( s1 ) N '0 (− s1 ) = 0.75,

= ∑∑∑ ϕ jη ( sk )(Qn − jV0 B)(Qn −ηV0 B) , T

value

of

is

fractional-rational

η

s (− s) ϕ jη ( s ) = ' N (s) N (− s) j

ϕ jη ( sk )

in roots

(21)

N 0 ( s2 ) N '0 (− s2 ) = −1.5,

k =1 j = 0 η = 0

where the complex-values factor

(20)

'

For this example the expression of the controllability gramian takes the form

represents the function

2

Pc = ∑

sk of characteristic

k =1

polynomial expression, V0 is «a structural block kernel» of the observability matrices (3), Qn − j – block row of matrix Q .

1

∑ j =0

(sk ) j (−sk )η ~ Τ ~ A j Aη , A j = Aj BBΤ , ∑ ' η =0 N (sk ) N (−sk ) 1

(22) where in right hand part of the equality one can see bilinear ~

Similarly observability gramian is defined by the formula

Τ

forms, formed by the weighted Faddeev’s matrices A j , Aη . Let us compute the values of the matrices in the numerator of the expression (22)

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~ ⎡0.6125 1.5 ⎤ ~ Τ ⎡1.25 A0 A0Τ = ⎢ ⎥ , A1 A0 = ⎢ 1.5 ⎣ 0.375 2.125⎦ ⎣ ~ ~ ⎡1.25 1.5 ⎤ ⎡1.25 A0 A1Τ = ⎢ , A1 A1Τ = ⎢ ⎥ ⎣0.75 2.125⎦ ⎣ 1.5

0.75 ⎤ , 2.125⎥⎦ 1.5 ⎤ , 4.25⎥⎦

Lyapunov equations (13) and (14), the gramian expansions on the spectrum of the dynamic system’s matrix provide a new representation for solutions of Lyapunov equations. Lemma 1 may be shown to lead to similar new representations for solutions of Sylvester equations (Sorensen and Antoulas, 2002).The coefficients of characteristic equations and Faddeev’s matrices in the resolvent expansion have a central role to play in obtaining gramians. The Lemma on expansion of matrix integrals may also be used for estimating of the ellipsoids boundaries in obtaining attraction regions of the dynamic system (Polyak and Shcherbakov, 2002). The identities (21), (22) give us the new canonical forms for controllability and observability gramians notion.

(23)

Substitute the expressions (21) and (23) into the formula (22), we obtain ~

~

~

~

( A − 0.5 A1 )(0.5 A1Τ + A0Τ ) ( A0 − A1 )( A1Τ + A0Τ ) P = 0 − = 0.75 1.5 (24) 1 ⎤ ⎡1.25 =⎢ . (24) 2.125⎥⎦ ⎣ 1 c

The structures of both types of gramians are similar. They are weighted sums of products of direct and transposed matrices of controllability and observability, respectively.

Let us check up, whether the expression (24) be a solution to a Lyapunov equation of the form (13).

For a discrete case the criteria of identifiability based on the analysis of identifiability matrices rank properties has been received by Peeters and Hanzon (2005). In the work it is shown, that a necessary and sufficient conditions of local and global identifiability is a nonsingularity of Fisher’s information matrix which can be defined via decisions of two Lyapunov matrix equations and two Sylvester equations. A comparison between the results and above theorem points out the deep link, which exists between structural properties of controllability, observability and identifiability. The paper’s results may also be applied to the problems of H2-reduction of LTI dynamic systems models (Sorensen and Antoulas, 2002), and to the design of adaptive algorithms for controllers with fixed structure and researches the adaptability structural properties (Yadykin, 2009).

1 ⎤ ⎡ −0.5 0 ⎤ ⎡1.25 + AP c + P c AT = ⎢ 2.125⎥⎦ ⎢⎣ 0 −1⎥⎦ ⎣ 1 1 ⎤ ⎡ −0.625 −0.5 ⎤ ⎡ −0.5 0 ⎤ ⎡1.25 +⎢ = + ⎥ ⎢ −1⎦ ⎣ 1 2.125⎥⎦ ⎢⎣ −1 −2.125⎥⎦ ⎣ 0 ⎡ −0.625 −0.5 ⎤ ⎡ −1.25 −1.5 ⎤ Τ +⎢ ⎥ = ⎢ −1.5 −4.25⎥ = − BB . − 1 − 2.125 ⎣ ⎦ ⎣ ⎦

The matrix P c is symmetrical and positive-definite Δ1 = 1.25 > 0, Δ 2 = 1.65625 > 0.

So, gramian spectrum expansion with infinite time is being the sum

6. ACKNOWLEDGEMENT

4 2 2 1 Рс = A0 BBΤ A0Τ + A0 BBΤ A1Τ − A1 BBΤ A0Τ − A1 BBΤ A1Τ − 3 3 3 3 2 2 2 2 - A0 BBΤ A0Τ − A0 BBΤ A1Τ + A1 BBΤ A0Τ + A1 BBΤ A1Τ . 3 3 3 3

This work was supported in part by the Federal Agency on Science and Innovations (Russia) and European Commission in the framework of collaborative project ICOEUR of the Programme FP7 – ENERGY-2008-RUSSIA The discussion with colleagues (N.I. Voropai, O.A. Soukhanov, E. Bompard), taken place in Irkutsk at workshop “Liberalization and Modernization of Power Systems: Coordinated Monitoring and Control towards Smart Grids (LMPS ‘09)”, are gratefully acknowledged.

The sum becomes a bilinear form of the matrices Aj B, B Τ AηΤ , j = 0,1;η = 0,1. The first 4 summands of the sum are corresponding eigenvalue s1 = −0.5, the second 4 summands of the sum are corresponding eigenvalue

s2 = −1.0.

REFERENCES 5. CONCLUSIONS

Andreyev Yu.N. (1976). Control of Finite Dimension Linear Plants. Moscow, Russia: Nauka Publishers. (in Russian) Balandin D.V. and Kogan M.M. (2007). LMI-based Control System Design. Moscow, Russia: Nauka Publishers. (in Russian) Datta B.N. (2004). Numerical Methods for Linear Control Design (Design ans Analysis). USA, San Diego, California: Elsevier. Faddeev D.K and Faddeyeva V.N. (1963). Computing Methods of Linear Algebra. Moscow, Russia: Physical and Mathematical Literature Publishers. (in Russian) Fuhrmann P.A. (1996). Polinomial Approach to Linear Algebra. Berlin, Germany: Springer-Verlag. Gantmacher F. (1959). The theory of matrices. 4th Edition. N.Y.: Chelsea Publishing Company.

Controllability and observability of linear dynamic systems are among their fundamental properties. The paper suggests the alternative approach to studying these properties by analyzing an interdependence of matrices and their controllability and observability gramians by using a Lemma on expansion of matrix integrals through using expansions of square matrix A, Am resolvents and the theorem on convolution of matrix functions in a complex field. The identities (1) and (2) lead to a way to compute matrix integrals on matrix spectra in the shape of expansion on the matrices spectra of the system’s dynamics matrices. As the controllability and observability gramians are explicitly expressed in terms of matrix integrals that are solutions to the 182

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Hanzon B., Peeters R.L.M. (1996). A Faddeeev Sequence Method of Solving Lyapunov and Sylvester Equations. Linear Algebra Appl, vol. 241-243, pp. 401-430. Hanzon B. and Hazewinkel M. (Ed.). Constructive Algebra and System Theory (2006). Amsterdam: Royal Netherlands Academy of Arts and Sciences. Kalman R.E., Bucy R.S. (1960). New Results in Linear Filtering and Prediction Theory. J. Basic Engineering Trans ASME, Series D, no. 83, pp. 95-108. Kailath T. (1980). Linear Systems. New York, USA: Prentice Hall, Englewood Cliffs. Kalman R.E. (1963). Mathematical Description of Linear Systems. SIAM Journal of Control, no. 1, pp. 152-192. Krasovsky A.A. Ed. (1987). Control Theory Handbook. Moscow, Russia: Nauka Publishers, Chief Editorial Board of Physical and Mathematical Literature. (in Russian) Kwakernaak H., Sivan P. (1991). Modern Signals and Systems. New York, USA: Prentice Hall, Englewood Cliffs. Levin W.S., Ed. (1996). The Control Handbook. CRC Press & IEEE Press, vol. I, pp. 461-469. ISBN: 9780849385704. Luenberger D.G. (1964). Observing the State if Linear Systems. IEEE Transactions on Military Electronics, no. 8, pp. 74-80. Polyak B.T. and Shcherbakov P.S. (2002). Robust Stability and Control. Moscow, Russia: Nauka Publishers. (in Russian) Peeters R.L.M and Hanzon B. (2005). Identifiability of homogeneous systems using state isomorphism approach. Automatica, vol. 41, pp. 513-529. Peeters R. and Rapisardra P. (2006). Solution of Polinomial Lyapunov and Sylvester equations. In: Constructive Algebra and System Theory. B. Hanzon and M. Hazewinkel (Ed.). Royal Netherlands Academy of Arts and Sciences, pp.151-166. Petrov B.N., Terajev E.D., and Shamrikov B.M. (1977). Conditions of parametric identifiability in open and closed control systems. Izv. of Technical Cyberneutics, 1977, no. 2, pp. 160-175. (in Russian) Poznyak A.S. (1991). Fundamentals of Robust Control ( H ∞ Theory). Moscow, Russia: Moscow Institute of Physics and Technology Press. (in Russian) Poznyak A.S. (2008) Advanced Mathematical Tools for Automatic Control Engineers. Volume 1: Deterministic Techniques. Elsevier. 774 p. Sorensen D. and Antoulas A. (2002). The Sylvester equation and approximate balanced reduction. Linear Algebra Appl. vol. 351/352, pp. 671-700. Yadykin I.B. (2009). On Adaptability Grammians. Proceedings of the International Conference on Mathematical Theory of Systems, Moscow, Russia: Published by ISA of the Russian Academy of Sciences, pp. 127-141. (in Russian)

APPENDIX Taking into account, that roots of denominator are simple, we have for matrices A, Am the resolvent expansion n −1

( Is − A )

−1

∑s

j k Aj Zk j =0 =∑ , Zk = ' , N ( sk ) k =1 s − sk n

N ' ( sk ) =

(A.1)

dN ( s ) , ds s = sk nm −1

( Is − A )

Τ −1 m

j smk AmΤη ∑ Z mk =∑ , Z mk = η =0 ' , N m ( smk ) k =1 s − smk nm

N m' ( sk ) =

(A.2)

dN m ( s ) . ds s = smk

On the other hand, matrices Z k , Z mk are residues of the Τ

matrices A, Am

resolvents in the poles of polynomials

N −1 ( s ), N −1m ( s ) Zk = Res ( Is - A)

−1 s = sk

, Zmk = Res ( Is - AmΤ )

−1 s = smk

. (A.3)

Τ

For matrices A, Am resolvents we have

( Is - A )

Τ −1 m

s =− sk

=

nm −1

A η sη ⋅ N ∑ η =0

Τ m

−1 m

( s)

,

(A.4)

.

(A.5)

s =− sk

In similar manner we obtain:

( Is - AΤ )

n −1

−1 s =− smk

= ∑ AΤj s j ⋅ N −1 ( s ) j =0

s =− smk

Substituting the expressions(A.1)-(A.5) in the identities (6) and (7), we finally obtain n

Ω1 = ∑ Res (Is − A) −1 k =1

s = sk

⋅R ⋅ ( Is − AmΤ ) −1

nm

Ω 2 = ∑ Res ( Is − AmΤ ) −1 λ =1

183

s = smλ

s =− sk

⋅S ⋅ ( Is − A)−1

s =− smλ

.