Proceedings of the 3rd International Conference on Systems and Control, Algiers, Algeria, October 29-31, 2013

WeBA.5

Markov parameters in the digital systems approximation problem

Arne! Baha Houda ADAMOU-MITICHE, Lahcene MITICHE, and Sa!iha YOUNSI

Abstract- Three recent methods for model reduction of linear discrete systems are presented. They are based on the impulse response gramian which contains information on the input-output behavior of the system. The corresponding low order approximants retain the first r Markov parameters and the first r X r elements of the impulse response gramian of the original system. To see the efficiency of each algorithm, these methods are implemented and several simulations are done. From an illustrative example, we can better appreciate their performances and the closeness of each reduced order model to original system.

I. INTRODUCTION

Model reduction has attracted considerable attention in the past two decades and many methods have been proposed. In this work, we present three model reduction algorithms, for stable discrete systems, based on the impulse response gramian [ 1, 2, 3].

This gramian is given as a function of the impulse response gramian of the system, it is positive definite and satisfies the Lyapunov equation for the system, in controllability canonical form. It contains information on the stability and input-output behavior of the system.

The paper is organized as follows: in section II, some preliminary results regarding impulse response gram ian are presented. In section III, we present three algorithms for linear discrete systems model reduction. An illustrative example is given in section IV.

II. D EFINITIONS

Consider a stable single input single output (SISO) discrete system described by the following minimal realization [4]

{XCk + 1) = AxCk) + buCk), yCk) = cxCk) + duCk), (I)

with xCk) E ',Rn. The impulse response gramian for such a system is defined as follows.

Definition 1: The impulse response gramian for a stable discrete SISO system is given by [5, 6]

with:

P: the impulse response gramian,

hk : the impulse response of system,

hk = cAk-1b,

hk+\hk+n] , h�+n (2)

(3)

Lemma: If Wa, Wc, and Weo are respectively, the observability, the controllability, and the cross gramians, of the system [6], we have

(4)

P = C[om WoCeom' (5)

P = OWcoCeom' (6)

where Ceom and 0 are the controllability and the observability matrices.

Definition 2: The Markov parameters of the original system are given by Mi, [7, 8, 9],

(7)

and the r x r elements of the impulse response gramian of the original system are given by

(8)

where X denotes the steady state covariance of the original system

(9)

and w is the covariance of the input signal and is a scalar for SISO systems.

A. B. H. ADAMOU-MITICH E, L. MTICH E and S. YOUNSI are with Science and Technology Department, University of DJELF A, ALG ERIA. Emails: (amelmitiche@yahoo.fr.l mitiche@yahoo.fr.andpgss 2011@yahoo.fr).

978-1-4799-0275-0/13/$31.00 ©2013 IEEE

Proceedings of the 3rd International Conference on Systems and Control, Algiers, Algeria, October 29-31, 2013

III. MOD EL R EDUCTION

The reduced-order model is given by the following realization

{Xr (k + 1) = Arxr (k) + bru (k) ,

Yr (k) = crxr (k) + du Ck) , (10)

Definition 3: The Markov parameters of the reduced model are given by

(11)

The r x r elements of the impulse response gramian of the reduced model are given by

(Rr)i = crA�XrcJ,

with

(12)

( 13)

where Xr denotes the steady state covariance of the reduced model system.

In this section, we present three algorithms for obtaining reduced order models.

A. Procedure of type A [1 J Input: Given an original complete order discrete system (A, B, C, O,n).

Step 1: Compute the controllability matrix of the original system Ccom.

Step 2: Compute the new realization (A, '6, c,O,n) using

the following equations:

b = C��b,

c= cCcom.

(14)

(15)

(16)

Step 3: Obtain the impulse response gramian by solving the Lyapunov equation

' 1 ' A PA-P=-Q, (17)

where

(18)

Step 4: Compute the unit upper triangular matrix U which diagonalizes P with UDUT [ 10] decomposition

UTpU = D,

and

(19)

Step 5: Construct the new realization

S = (Ad, bd, Cd, 0, n) as

Ad = u-1Au,

bd = U-1Jj,

Cd = cU.

Step 6: Partition the new

Ad = [All AZ1

A1z] A22 '

bd = [��], Cd = [ c1 cz],

D = [�1 �J.

system

WeBA.5

S

(20)

(21)

(22)

(23)

as

(24)

(25)

(26)

(27)

Remark: The intermediate model (All' b1, Cv 0, r) obtained is stable for a stable original system and retains the first r Markov parameters of the original system. However, it does not match the first r x r elements of the impulse response gramian. In the rest of this section, a method is shown to modify the intermediate model obtained to get the required model which matches the first r Markov parameters and the first r x r elements of the impulse response gramian.

Step 7: Compute

(28)

with

all E :R(r-1)X\ L E :R(r-1)x(r-l), a1n E :R1x(r-l), and ann: is a scalar.

Step 8: Select d1, such that

D1 = diag(dv ... , dr),

Step 9: Select dr+1 such that

Dz = diag(dz, ... , dr),

Step 10: Set Dr-1 as

Dr-1 = diag(dz, ... , dr) ·

(29)

(30)

(31)

978-1-4799-0275-0/13/$31.00 ©2013 IEEE

Proceedings of the 3rd International Conference on Systems and Control, Algiers, Algeria, October 29-31, 2013

Step II: Solve the quadratic equation

[d1 + (ailL-TD;!lL-lall)dil �Xf + [a1nd1 - dlannClalljxl-

�dr+1 = 0 . (32)

Step 12: Compute Xr-1

Step 13: Construct the vector x,

x= [Xl ] Xr-1 .

Step 14: Construct the matrix Ax,

A = [� x : o

Step 15: Compute the matrix An Ar = All + Ax

Step 16: Set br

Output: The approximant model (An br , Cr ).

(End of procedure)

B. Procedure of type B

(33)

(34)

(35)

(36)

Input: Given an original full order discrete system (A, B, C, 0, n) .

Step 1: The same steps 1-6 as in the type A procedure.

Step 2: Using the perturbation technique [2], compute

Ar = All + A12 (I -A22) -1 A211 br = b1 + A12 (I -A22)-lb2, Cr = C1 + C2 (I -A22)-lA21·

(37)

(38)

(39)

Output: The approximant model is given by (An br , Cr ).

(End of procedure)

C. Procedure of type C [3]

WeBA.5

Input: Given an original full order discrete system CA, B, C, 0, n) .

Step 1: Compute the Markov parameters hr and output covariance Rr.

Step 2: Compute the matrix Q using (18) .

Step 3: Construct the matrix Pq Plr ]

Pl.r-l

P12 P�l Step 4: Construct the matrix Mq

0 0 0 0 0 hl 0 0 0 0 h2 hl 0 0 0 M = q h3 h2 hl 0 0

hr-1 hr-2 hr-3 hl 0

(40)

(41)

Step 5: Compute the matrix P (gramian of the impulse response)

Step 6: Partition of the matrix Pas P = [Pll Pi2] P12 P22 ' where

P12 = [P12,P13' ... ... Pvf, [P22 P23 P22 = P�3 P�3

P2r P3r and Pll a scalar.

P2r] P3r

. � . P�r

Step 7: Compute P-Q and detIP-QI.

(42)

(43)

(44)

Step 8: If P-Q is positive detIP-QI=O, there is a unique inverse solution to the Lyapunov equation [3],

a = [a�=J = [pilCP1� - QIJ,

where

P1r = [Pln P2n ... ... Pr-1.r f,

Qlr = [qlnq2r, ... ... qr-vf·

(45)

(46)

(47)

978-1-4799-0275-0/13/$31.00 ©2013 IEEE

Proceedings of the 3rd International Conference on Systems and Control, Algiers, Algeria, October 29-31, 2013

Step 9: If P-Q is positive detlP-QI i- deWI the inverse solution of the Lyapunov equation yields two systems. The unknown elements of the systems matrices are given by

d�('_Q)1 ,

det(P) J (48)

(49)

Step 10: If detlP-QI = detlPI, the inverse solution of the Lyapunov equation yields to two unstable system matrices.

Step 11: Construct the realization of the reduced order model [ 0 1

o 0 Ar = . .

�n a�-i

IV. ILLUSTRATIV E EXAMPL E

Consider the 8th order discrete time system [11], given by the following realization:

A= 0.06307 0.4185 -0.0780 0.0570 -0.1935 -0.0983 0.0165 -0.0022 1.00000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000

1 0.1652 T

0 0.1250 0 -0.0025 0 0.0053

b = 0 , c= -0.0226 0 -0.009 0 0.0030 0 -0.0004

The presented method based on the procedure type A gives two reduced 4th order models; the first is given by [0.9311 -0. 0662

Ai _ 1. 0 0 0 0 0.4768 r - 0. 0 0 0 0 1. 0 0 0 0 0. 0 0 0 0 0. 0 0 0 0

bl � [n -0. 0 063 0.4410

-0.2724 1. 0 0 0 0

-0. 0 020 ] 0. 0226

-0. 0766 ' - 0.9130

WeBA.5

c;: = [ 0.1652 0. 0754 - 0. 0272 0. 0 015],

and the second model is [0.9311 -0. 0662

A2 = 1. 0 0 0 0 0.4768 r 0. 0 0 0 0 1. 0 0 0 0 0. 0 0 0 0 0. 0 0 0 0

-0. 0 063 0.4410

-0.2724 1. 0 0 0 0

0. 0 0 03 ] -0. 0232 0. 0124 0.9739

c; = [ 0.1652 0. 0754 - 0. 0272 0. 0 015].

The presented method based on the type B procedure gives

A = r

[0.9311 -0. 0662 1. 0 0 0 0 0.4768 0. 0 0 0 0 1. 0 0 0 0 0. 0 0 0 0 0. 0 0 0 0

m -0. 0 063 -0. 0 048 ] 0.4410 0. 0 055

-0.2724 -0.1815 ' 1. 0 0 0 0 -0.1361

Cr = [ 0.1652 0. 0754 - 0. 0272 0. 0 085].

The presented method based on the type C procedure gives two reduced order models, the first, given by [J02

1. 0 0 0 0 0

LoL l Ai = 0 1. 0 0 0 0 q 0 0

- 0.8686 -0.8 058 2.1 090 [0 1652 ] bi = 0.2292 ci = [1 o 0 0]. q 0.2112 ' q

0.2215

In another hand, the second model is given by [J602

1. 0 0 0 0 0

Loto l A2 - 0 1. 0 0 0 0 q- 0 0

-0. 0707 1.1 064 0.3178 [0,'652 ] b2 = 0.2292

c� = [1 o 0 0]. q 0.2112 ' 0.2215

The first r (r=4) Markov parameters of the original system are

Mi = [ 0.1652 0.2292 0.2112 0.2215]

while the Markov parameters of the reduced models (from the three algorithms) are equal to

Mr = [ 0.1652 0.2292 0.2112 0.2215].

The impulse response gramian of the original system is given by

978-1-4799-0275-0/13/$31.00 ©2013 IEEE

Proceedings of the 3rd International Conference on Systems and Control, Algiers, Algeria, October 29-31, 2013

0.3132 0.2916 0.2507 0.2023 0.1462 0.0888 0.0331 -0.0145 0.2916 0.2859 0.2537 0.2158 0.1657 0.1136 0.0589 0.0111 0.2507 0.2537 0.2333 0.2053 0.1651 0.1205 0.0722 0.0283

p= 0.2023 0.2158 0.2053 0.1887 0.1585 0.1235 0.0823 0.0440 0.1462 0.1657 0.1651 0.1585 0.1397 0.1149 0.0834 0.0527 0.0888 0.1136 0.1205 0.1235 0.1149 0.1008 0.0793 0.0571 0.0331 0.0589 0.0722 0.0823 0.0834 0.0793 0.0682 0.0551

-0.0145 0.0111 0.0283 0.0440 0.0527 0.0571 0.0551 0.0503

while the impulse response gramians of its approximants

are all equal to [0.3132 0.2916 Po = 0.2916 0.2859

r 0.2507 0.2537 0.2023 0.2158

0.2507 0.2023] 0.2537 0.2158 0.2333 0.2053 ' 0.2053 0.1887

We present the impulse responses and the bode responses of the original system and its low order approximants in Figure I and Figure 2, respectively. We notice the behavior of the second reduced order approximant obtained via the type C Procedure represented by

(A�, b�, c�, 0, 4) is the closest to the complete order

system. It is verified either in time and frequency domains. In Figure 3, the stability plane is plotted. The location of the poles shows that the stability of all the reduced order models is always preserved.

V. CONCLUSION

Three algorithms for model order reduction of asymptotically stable, linear, discrete SISO systems based on the gramian combined to Markov parameters of impulse response are proposed to point out the efficiency and performance of each algorithm. The gramian contains information on stability and the input/output behavior of the system. Basically, the main idea of these approaches is a low approximants construction based on the preservation of the most significant properties. The reduced order models of order r are obtained by retaining the first r Markov parameters and the first r X r elements of the impulse response gramian of the original system. From the different approaches, the best approximant is whose behavior fitted the original complete order system. It is chosen according to the domain of interests (time and frequency domain). The stability is always preserved. This is shown by various simulations.

REF ER ENCES

[1] P. Agathoklis and V. Sreeram, "Discrete system reduction via impulse response gramian and its relation to q-Markov covers," IEEE Trans. Automat. Cont., Vol. 37 No. 5, pp. 653-658, 1992

[2] P. Agathoklis and V. Sreeram, "The discrete time q-Markov cover models with improved low-frequency approximation," EEE Trans.

Automat. Cont., Vol. 39, No. 5, pp. 1102-1105, 1994.

[3] V. Sreeram, P. Agathoklis and M. Mansour, "The generation of discrete time q-Markov covers via inverse solution of the

WeBA.5

Lyapunov equation," IEEE Trans. Automat. Cont, Vol. 39 No. 2, pp. 381-385,1994.

[4] P. Agathoklis and V. Sreeram, "Identification and model reduction from impulse response data," International Journal of

Systems Science 21 (8), 1541-1552, 1990.

[5] V. Sreeram and P. Agathoklis," Linear phase IIR filter design," Signal Processing, IEEE Transactions on 40 (2), 389-394, 1992.

[6] V. Sreeram and P. Agathoklis, "Model reduction of linear discrete systems via weighted impulse response Gramians," International Journal of Control 53 (1), 129-144, 1991.

[7] D. A. Wagie and R. E Skelton, "A projection approach to covariance equivalent realization of discrete systems," IEEE Trans.

Automat. Cont., Vol. AC-31,pp. 1l14-1120, 1986.

[8] R. E. Skelton and E. G. Collins, "Set of q-Markov covariance equivalent models of discrete systems," Int. J. Cont., Vol. 46, pp. 1-12,1987.

[9] B. D. O. Anderson and R. E. Skelton, "The generation of all q­Markov covers," IEEE Trans. Circuit Syst. Vol. 35, pp. 375-384, 1988.

[10] G. H. Golub and C. F. Van Loan, Matrix computations, Baltimore, M. D. John Hopkins Univ. Press, 1983.

[11] S. N. Deepa and G. Sugumaran, "MPSO based Model Order Formulation Scheme for Discrete Time Linear System in State Space Form ", European Journal of SCientific Research ISSN 1450-216X Vol. 58 No.4 (2011), pp. 444-454.

-Awo<iIlOO:B Awo<iIl ... Cl1

-.wo, ...... CI?

Figure I. Impulse responses of the original model of order 8 and its approximants of order 4.

978-1-4799-0275-0/13/$31.00 ©2013 IEEE

Proceedings of the 3rd International Conference on Systems and Control, Algiers, Algeria, October 29-31, 2013

� 8. '0 t � iij

=,�--�----�----�--�----�----�--�

0.8

0.6

Figure 2. Frequency responses of the original model and its approximants.

+

:::;.-, ..

-- Unit circle

� Original model -

+ Approximant Al1

Approximant AJ2

I Approximant B Approximant C/1

. � -0.2 + l'

- -0.4

-0.6

-0.8

_1L-�--�--���--�-=��--�--�� -1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8

Real part of poles

Figure 3. Pole locus of the original model and its approximants.

978-1-4799-0275-0/13/$31.00 ©2013 IEEE

WeBA.5