Numerically Robust Transfer Function Modeling ...

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[7] F. L. Lewis, “A survey of linear singular systems,” in Circuits Syst. Signal Proc., vol. 5, 1986, pp. 3–36. [8] L. Dai, Lecture Notes In Control and Information Sciences. Berlin, Germany: Springer-Verlag, 1989, vol. 118, ch. Singular Control Systems. [9] A. Kumar and P. Daoutidis, “State–space realizations of linear differential-algebraic-equation systems with control-dependent state space,” IEEE Trans. Autom. Control, vol. 41, no. 2, pp. 269–274, Feb. 1996. [10] D. Wang and C. B. Soh, “On regularizing singular systems by decentralized output feedback,” IEEE Trans. Autom. Control, vol. 44, no. 1, pp. 148–152, Jan. 1999. [11] A. Rhem and F. Allgower, “General quadratic performance analysis and synthesis of differential algebraic equation (dae) systems,” J. Process Control, vol. 12, pp. 467–474, 2002. [12] N. H. McClamroch, “Feedback stabilization of control systems described by a class of nonlinear differential-algebraic equations,” Syst. Control Lett., vol. 15, pp. 53–60, 1990. [13] H. Krishnan and N. McClamroch, “Tracking in nonlinear differential-algebraic control systems with applications to constrained robot systems,” Automatica, vol. 30, pp. 1885–1897, 1994. [14] A. Kumar and P. Daoutidis, “Feedback control of nonlinear differentialalgebraic equations,” AIChE J., vol. 41, pp. 619–636, 1995. [15] X. Liu and S. Celikovsky, “Feedback contol of affine nonlinear singular control systems,” Int. J. Control, vol. 68, pp. 753–774, 1997. [16] Z. Chen and J. Huang, “Solution of output regulation of singular nonlinear systems by normal output feedback,” IEEE Trans. Autom. Control, vol. 47, no. 5, pp. 808–813, May 2002. [17] A. Dervisoglu and C. A. Desoer, “Degenerate networks and minimal differential equations,” IEEE Trans. Circuits Syst., vol. CAS-22, no. 10, pp. 769–775, Oct. 1975. [18] W. Yim and S. N. Singh, “Feedback linearization of differential agebraic systems and force and postion control of manipulators,” in Proc. Amer. Control Conf., San Francisco, CA, 1993, pp. 2279–2283. [19] M.-N. Contou-Carrere and P. Daoutidis, “Output feedback regularization of nonlinear dae systems,” in Proc. 15th Triennial World Congr. International Federation of Automatic Control, Barcelona, Catalonia, Spain, 2002. [20] H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems. New York: Springer-Verlag, 1990.

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Numerically Robust Transfer Function Modeling From Noisy Frequency Domain Data A. Bultheel, M. Van Barel, Y. Rolain, and R. Pintelon Abstract—Using vector orthogonal polynomials as basis functions for the representation of the rational form of a linear time invariant system, in frequency domain identification problems, it is shown that the notorious numerical ill conditioning of these maximum likelihood problems can be overcome completely. For the identification of high-order (100 100) systems operating over a wide frequency band, or even in the situation of overor undermodeling, condition numbers less than ten are reported for real measurements. Index Terms—Discrete rational approximation, frequency domain identification, maximum likelihood, vector orthogonal polynomials.

I. INTRODUCTION Frequency domain identification of linear time-invariant systems has regained some interest during the last years [1], [2]. Periodic excitation signals can give cheap estimations of the frequency response function and a reliable measure of the uncertainty on these spectra. The knowledge of the variance of the measured spectral lines bounds the stochastic frequency response function variation and thus helps to design a fully automatic model order selection procedure [3], [4]. On the dark side, frequency domain methods are known to suffer from a poor numerical conditioning when the frequency span and/or the model order become large (more than two decades and an order more than 20). This numerical problem ruins both the modeling performance and the model order selection capability. Several attempts have been made in the past to circumvent numerical degeneracy. For example, frequency scaling [5] or the use of polynomials orthogonal with respect to the inner product defined by the normal equations of the estimator [2], [6]. Even if these approaches give some improvement, they do not remove numerical problems totally, neither do they extrapolate gracefully to multiple-input–multiple-output (MIMO) systems. The approach proposed here solves the numerical conditioning issue perfectly for all the frequency domain methods whose cost function can be reduced to a linear weighted least squares problem or a weighted (generalized) total least squares problem. The main difference of the approach of [7] with the presented method is that the inner product used to construct the orthogonal basis in [7] is data independent. Consequently the orthogonal basis in [7] does not diagonalize the normal equations resulting in suboptimal condition numbers. It gives reasonable condition numbers (asymptotically 1) under the assumption that: i) the input is white noise, and ii) the data covers the whole unit circle. If either of these two assumptions is violated, then Manuscript received August 23, 2004; revised March 29, 2005. Recommended by Associate Editor E. Bai. The work of A. Bultheel and M. Van Barel was supported in part by the Project OT-00-16 SLAP: Structured Linear Algebra Package of the K.U. Leuven, and the Project G.0078.01 SMA: Structured Matrices and their Applications of the National Science Foundation Flanders. The work of all the authors was supported by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian Federal Science Policy Office. The scientific responsibility rests with the authors. A. Bultheel and M. Van Barel are with the Department of Computer Science, K.U. Leuven, B-3001 Leuven, Belgium (e-mail: adhemar.bultheel@ cs.kuleuven.be; [email protected]). Y. Rolain and R. Pintelon are with the Vrije Universiteit Brussel, Department ELEC, B-1050 Brussels, Belgium (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2005.858651

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the condition number of the normal equations can become arbitrarily large. This is not true for the method presented in this note: The input spectrum may have an arbitrary coloring, and a small part of the unit circle (imaginary axis) may be modeled.

Equation (1) implicitly assumes that the input/output result from steady state experiments with periodic excitation signals. However, all results remain valid for nonperiodic excitation signals if we take in (1) R = AY 0 BU 0 I with I ( ; ) 2 n 21 ; P = [A B I ] 2 n 2(n +n +1) , and S = [Y 3 0 U 3 In ]3 .

II. MODELLING PROBLEM We consider a system whose exact input and output spectra are U0 ( ) 2 n 21 and Y0 ( ) 2 n 21 , where stands for the time lag z 01 (discrete time) or the Laplace variable s (continuous time). We collect these in a vector S0 = [Y03 0 U03 ]3 2 n 21 ; nt = ny + nu , (the star stands for complex conjugate transpose). The system model is [8] G( ; ) = A01 ( ; )B ( ; ), where P = [A B ] 2 n 2(n +n ) is a polynomial matrix1 containing the polynomial matrices A and B . Note that we use the left matrix fraction description which is suited for input–output measurements as well as for measurements of the transfer function. For a right matrix fraction description, the analysis is completely analogous. Our model is parameterized by the vector  which is to be determined. The spectra are however measured in discrete points so that the data available are not S0 ( ), but S ( k ); k = 1; . . . ; F where S is corrupted by additive noise: S ( ) = S0 ( ) + NS ( ). The covariance matrix of the noise 6( ) = cov(NS ( )) 2 n 2n is supposed to be known. Note that the linearized residual is given by R = P S = AY 0 BU 2 n 21 . Our estimator will be obtained as the solution of a least squares problem which minimizes a cost function of the form V

=

F

Rk3 Wk Rk

k=1

= R3 WR

(1)

with Rk = R( k ; ) = P ( k ; )S ( k ); P 6= 0; Wk 2 n 2n positive definite weights, while = [R13 1 1 1 RF3 ]3 2 n F 21 and = diag(W1 ; . . . ; WF ). In the classical least squares problem, the weights Wk are all In , but this results in a very noise sensitive solution. In the (total) least squares cost function, they can be much more general. In this general setting, we choose some n 2n wk 2 (possibly depending on ~) and define M () = F 2 3 2 wk CR wk : with CR = cov(Rk ), thus2 CR2 = Ak CY2 A3k + k=1 2 3 2 Bk CU Bk 0 2Herm(Ak CY U Bk3 ) = P ( k ; )CS2 P 3 ( k ; ), where Ak = A( k ; ); Bk = B ( k ; ); Sk = S ( k ), and 2 = E(NX NZ3 ) is the covariance of the noise on X and Z and CXZ 2 2 CX = CX X . The weights for this generalized least squares problem are then WkWGLS = wk3 M 01 (~)wk , which may depend on ~. In the case of weighted generalized total least squares, the weights are WkWGTLS = wk3 wk =trace(M ()), which may also depend on ~; see [9]. In general, Wk can explicitly depend on the solution  . For example in the maximum likelihood (ML) estimator for the nonlinear problem, 2 ]01 . In this the weight Wk is given by WkML = W ( k ; ) = [CR case, it is a nonlinear problem that can be solved iteratively. However, the cost function is not convex and may have many local minima. Therefore, very good starting values for  are essential. The following procedure is proposed. We start by a simple least squares problem (using weights Wk = In ) or a generalized total least squares problem, using some estimate ~ for the unknown parameters. This gives us a first estimate for  . This estimate is used to compute a better weight and with this weight, a new estimate is obtained, etc., following a Sanathanan–Koerner iteration [10]. (See also [11].) This iterative process will then eventually converge to a close approximation of the ML estimates [1]. Finally, a true nonlinear iteration can be done to minimize the ML cost function.

R

W

1

is the set of scalar polynomials. ) = ( + ) 2 is the Hermitian part of

2Herm(

III. PARAMETERIZATION OF THE PROBLEM Until now, we have not been precise about what the parameters  were. They should somehow be used in the representation n 2n . To this end, we will of the polynomial matrix P 2 stack all the ny nt scalar polynomials in a long vector using the vec-operator. So, defining vec(M ) as the vector which stacks the columns of the matrix M on top of each other. We then have R = vec(R) = vec(P S ) = vec(In P S ) = (S T In )vec(P ). Here, we use the Kronecker product . Given the matrices F and G, then F G is the block matrix whose (i; j )th entry is Fij G. Setting T and Pk = vec(P ( k )), we can write the cost k = Sk In function (1) as V () = k k3 ()Mk k () with Mk = 3k Wk k positive semidefinite. 2 n n 21 . From now on, we have to parameterize the vector Suppose that the maximal degree that appears in , thus the maximal degree that appears in the entries of A and B is n and suppose n n 21 : d = 0; . . . ; n; j = 1; . . . ; ny nt g that fd;j 2 forms a basis for the space of all possible where the first index refers to the degree and the second index to the ny nt independent polynomials for each degree. Thus, we can write () = nd=0 nj=1n d;j d;j . Let us group this in n + 1 blocks of size ny nt as follows d = [d;1 ; . . . ; d;n n ] 2 n n 2n n and correspondingly d = [d;1 ; . . . ; d;n n ]T 2 n n 21 . So we = nd=0 d d = ' , with  = [03 ; . . . ; n3 ]3 and can rewrite ' = [0 ; . . . ; n ] 2 n n 2(n+1)n n . To illustrate this idea, suppose we consider the SISO case, where nu = ny = 1 and thus nt = 2. Let na = @ (A); nb = @ (B ) hence n = max(na ; nb ). The transfer function can then be written as

S

P

P

S

S

P

P

P

P

P

G( ; ) =

Note that there are na

P=

A B

B ( ; ) A( ; )

=

bd d : a d d=0 d n

d=0 n

+ n + 1 free parameters. Now, setting b

= 0 0 + 1 1 1 + 

n

n

n

=

2

d;j d;j : d=0 j =1

Denoting the first element of d;j as Ad;j and the second element of d;j as Bd;j , then we have G( ; ) =

B ( ; ) A( ; )

=

n d=0 n d=0

2 j =1 2 j =1

Bd;j ( )d;j Ad;j ( )d;j

which seems to have 2n +2 parameters. However, suppose for example that G is strictly proper, so that nb < na . The 0 ; . . . n are all in 221 and form 2nb +2 parameters. Since the degree of B is restricted to nb , the second element in d d for d = nb + 1; . . . ; na has to be zero, which gives na 0 nb linear conditions on the remaining parameters. Together with a normalization condition, this leaves us with na 0 nb 0 1 additional free parameters, giving a total of 2nb +2+(na 0 nb 0 1) = na + nb + 1 degrees of freedom, just as in the previous representation. Thus V () = 3 [ k '3 ( k )Mk '( k )] = 3 83 8 with = diag(M1 ; . . . ; MF ) and 8 = ['( 1 )3 ; . . . ; '( F )3 ]3 . This has to be minimized under some degree constraint for P , i.e., certain d;j should be nonzero (e.g., chosen as 1). Without this constraint, the problem would have the trivial solution  = 0. Since the matrix 83 8 is positive semidefinite, we can write it as 3 so that we have to find the least squares solution of  = 0 under the given

M

M

M

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H

HH

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 11, NOVEMBER 2005

H

constraints on  . These columns of that correspond to d;j that are chosen to be 1, can be brought to the right-hand side, giving an r , with a inhomogeneous least squares problem of the form  nontrivial solution. Until now, we have assumed that the parameters  are complex. However, in most practical situations,  is real. In that case the 3 T  and cost function (1) can be replaced by V  and re  rre the corresponding linear systems are re  where for any array M , the 1 re operator is defined as Mre 3 T M T T . Note that M T re . From now re on, we will discuss the case of real  . For a complex vector  , just leave out the 1 re and the 1 from the notation.

J =

( ) = Re(8 M8) H =0 J = () = Re(H H) = H H

[Re( ) Im( ) ] ()

Re( )

IV. VECTOR ORTHOGONAL POLYNOMIALS: CONTROLLING CONDITION NUMBER

THE

J

( ) =

= (min ( ) + max ( )) 2

J

H

Re(J J)

Re(J J) Re(8 M8) = F

Re[3 ( )M  ( )] =  p

k=1

k

k

q

k

p;q

In

n

:

In the complex and real setting, the discrete least squares problem has been discussed in all its generality in [14]–[17]. The result is an algorithm that computes the optimal  , using these vector orthogonal polyare the L; ; @P where nomials. We will denote it as  3 3 with M L31 ; ; LF Lk3 Lk repregiven frequency points, L k sent the measurements and the chosen weights (which may depend on a choice of  ), and @P represents a degree structure of numerator and denominator that one wants to have. The details can be found in the papers cited above. A minimal summary is as follows. Up to a column permutation, which is dictated by the degree structure @P , the algorithm computes an orthogonal similarity transformation of the matrix RjZ where Z In

; F . This tranformation anihilates 1; the lower triangular elements. In fact the orthogonal matrix that does 3 L1 ; ; LF and 3 . the job is the matrix  3  , finding the optimal  constrained by the deSince then V  gree conditions is trivial.

= DLS(

= [ ... ]

=

( )=

diag( . . . ) Q = diag( . . .

)

=



[

)8

We recall that this is a fast algorithm that is numerically stable. It works for continuous systems (where is a purely imaginary number) as well as for discrete systems (where is a number on the unit circle). Moreover, in the latter case, we can process two complex conjugate data simultaneously so that all the computations will be real. The overall algorithm goes as follows.





= 1 ... = 0).

1. Given 2. Choose an initial (e.g., 3. Compute untill convergence

= W (); and L  = DLS (L; ):

Wk

k

k

such thatMk

= S3 W S = L3 L k

k

k

k

k

4. Compute the nonlinear ML solution by Gauss–Newton iteration.

It is a well-known fact that the Jacobian re of the latter least squares problem can be extremely ill conditioned. For example, taking the stand In n , then the dard basis for the orthogonal polynomials: d columns of the re matrix can be badly scaled and very skewed, which results in a matrix that is numerically singular. One of the earlier techniques to overcome this condition problem is = . This to scale the frequency by sc k k k k helps, and we shall implement this throughout our computations, but it is not sufficient. Another, more efficient way of improving the computational aspects of the problem is by making the columns of re (hence, the columns of re ) as orthogonal as possible. In a first attempt, a separable basis of Forsythe polynomials was selected [12], [13]. These are polynomials orthogonal with respect to a discrete inner product. One such basis was associated with the numerator B and one basis was used to represent the denominator A. Despite the optimality of this basis for the representation of numerator and denominator separately, the method failed in bringing the condition number of the overall problem down to 1. Indeed, the matrix 3 consists of four blocks. The two diagonal blocks are identity matrices because of the orthogonality of numerator and denominator bases, but the two off-diagonal blocks mixing the two bases will be dense matrices. The best conditioned matrix for the overall problem will be obtained 3 is the identity matrix, hence if we can choose a basis ' such if 3 I(n+1)n n , i.e., the following discrete block-orthat thogonality holds:

J

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]

Q Q Q = 8 M8

Note that with each call of DLS, the system that is solved has an optimal condition number so that there is almost no degradation because of rounding errors. The difference between the iterations in step 3 and step 4 is that in step 4, the orthogonal basis is fixed as it was at the end of step 3. Since the weights depend on  , which changes through the iteration, orthogonality will be lost. The correction  is obtained by solving a linear system whose condition number will not be 1 anymore. However practical experience has shown that when the initial guess is good enough, the condition number will not grow above ten because the result of step 3 will be a close approximation of the ML estimate. We summarize the advantages of the method developed in this note. 1) It is recursive and fast and numerically stable. 2) In the SISO case, it gives not only a description of all the best least squares approximants of degree n but also all the solutions of lower degree. 3) One can impose an arbitrary degree structure for MIMO systems. 4) The algorithm can be efficiently implemented on a parallel computer.

1

V. SOME FURTHER COMPUTATIONAL ASPECTS

=1

Obviously, the problem simplifies considerably if ny , i.e., in the MISO (hence, a fortiori in the SISO) case because the Kronecker product is then avoided. There are several possibilities to choose the initial weights, different from the one we proposed. See, e.g., [13], and the examples that follow. The determination of the degree of the approximant is part of the general problem. For the moment, assume that it is given. Since we propose a recursive procedure, we will compute not only the best approximant of degree n but also all the approximants of lower degree, so that a right decision on the degree of the approximant can be made. Also updating and downdating (adding or removing a data point) is possible for this procedure. See [18] and [19]. This can be used to remove “outliers” or simulate a sliding window. Because of space limitations we can not go into the details. VI. EXPERIMENTAL RESULTS To show that the proposed method for practical examples which are selected in such a way that most common types of practical nonideal behavior are present. As a first example we mention briefly a very high order mechanical = was obtained vibrating SISO system. A model of order without a problem where in step 4 of the algorithm, the condition number of the Jacobian matrix degrades from 1 to 1.4 during the

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120 120

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Fig. 1. Block diagram of the measurement setup: are the excitation the forces applied signals stored in the arbitrary function generators, by the shakers to the Al-plate, and and the observed accelerations of the Al-plate. ( ) and ( ) are the 2 by 2 transfer function matrices of the Al-plate and shakers, respectively.

ML estimation. This clearly shows the effectiveness of the method proposed. Our second example is a MIMO system. The measurement setup is as follows (see Fig. 1). An aluminum plate (185 mm 2 63 mm 2 1.5 mm) hung by three nylon threads is excited by two mini-shakers via plexi-glass stinger rods. The forces u1 ; u2 applied by the shakers and accelerations y1 ; y2 at the excitations points are measured. The periodic signals r1 ; r2 exciting the shakers are generated by two arbitrary = 12  function generators at a sampling frequency fs : . The output of the arbitrary function generators is lowpass filtered (seventh-order inverse Chebyshev filter with a cut off frequency of 1 kHz) before being applied to the mini-shaker. To reduce the effect of the inductive impedance of the shaker on the generator unit, an = W resistance is put in series with its input. The measurement results: For each MIMO measurement two random phase multisines (= sum of harmonically related sinewaves with deterministic amplitude spectrum and random phase spectrum) with flat amplitude spectrum, interleaved frequency grids, points per period are used as excitation signals. and N Expressed in harmonic numbers w.r.t. the frequency resolution fs =N  : Hz the interleaved frequency grids are for grid f0 1:119:2:617 and for grid 2:120:2:688, which corresponds to F in a frequency band of about [71 Hz, 368 Hz]. To cope with the nonlinear distortions, 25 MIMO experiments with different realizations of the random phase multisines are performed MIMO mea(see [2]). Each MIMO experiment consists of nu surements. Hence, four different random phase multisines r11 (frequency grid 1), r21 (frequency grid 2), r12 (frequency grid 2), and r22 (frequency grid 1) are calculated for each MIMO experiment. In the j th measurement j ; of each experiment signals r1j and r2j are applied to shakers 1 and 2, respectively. Referring the observed input/output DFT spectra to the exactly known reference signals r1 ; r2

= 10 MHz 2

2 44 kHz

18 5

= 4096 = 06

= 500

=2

( = 1 2)

01

(k) = U^ n (k) R n (k) 0 n YR (k) = Y^ n (k) R n (k) ^ n (k); Y^ n (k) and R n (k) are the 2 by 2 input, output en where U (n)

( )

( )

( )

( )

( )

UR

( )

( )

1

( )

reference DFT spectra of the nth MIMO experiment, it is possible to calculate the sample means and sample covariance matrices of the measurements over the 25 MIMO experiments (see [2] and [20])

^ ( ) = X^^ (k) X^^ (k) X (k ) X (k ) N ^ij (k) = 1 X^ijn (k) X N X k

11

12

21

22

^ ( )= 1 N 01

N

N Cxz k

1x

n=1

1x n 1z n ( )

( )

3

= ( ) 0 ^( ) = vec( ) = ^ = ^ = 25 ^ () ^ () ^ () ^() ^() (n)

(n)

^( ) = ^ ( ) ^ ( )

01 k together with its standard measured FRM G j!k YR k U R deviation calculated via (see the Appendix)

^

CvecG ^

= U^R0T In C Y U^R0 In + U^R0T G^ C U U^R0 G^ 3 0 2Herm U^R0T In C Y U U^R0 G^ 3 vec

vec

1

1

vec

vec

1

:

(2) The identification results are as follows. A common denominator nb explains the data very well: It can model of order na be seen from Fig. 2 that the difference between the identified model G j!k ;  and the measured frequency response matrix (FRM) G j!k lies almost everywhere within the 95% uncertainty bound of the FRM measurement. This is confirmed by the actual value of the maximum likelihood (ML) cost function (3867) which is close to the expected value in the absence of model errors ( ny F 0 n = , the number of outputs, F the number of frewith ny 2 0 the number of free model quencies, and n parameters). The vector polynomial basis ( 2 vector polynomials of size 5 2 1) of the bootstrapped total least squares algorithm (see [2]) is used for the ML estimates. This results in a condition for the Jacobian matrix of the ML estimates. number 

( ^)) ( )

=2

=

= 60

848 = = 500 = 61 5 1 = 304 61 5 = 305

2

= 3867

VII. CONCLUDING REMARKS

( )

n=1

^) Fig. 2. Validation of the identified transfer function model ( with the measured frequency response matrix (FRM) ^ ( ). Black line: Measured FRM. Gray line: 95% uncertainty bound FRM measurement. Dashed ^). line: Difference ^ ( ) (

x k x k ;x Z with X; Z YR and/or (n) UR ; Cxx Cx ; N , and where, for example YR22 k is the DFT spectrum of the second output in the second measurement of the nth MIMO experiment. Contrary to U (n) k and Y (n) k , the columns of UR k and YR k are not independently distributed. Fig. 2. shows the

We proposed an algorithm for system identification (MIMO and SISO) that uses a representation with respect to an appropriate orthogonal basis. Consequently the systems to solve in the Sanathanan– Koerner iteration have condition number 1, so that there is no loss of accuracy by rounding errors. The eventual result is then used as an initial guess for a true ML iteration. In these steps, the condition number grows only moderately. APPENDIX For ease of notation, the frequency arguments are dropped. The noisy input–output DFT spectra U; Y resulting form nu independent

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MIMO experiments are related to the true U0 ; Y0 by Y = Y0 + NY , and U = U0 + NU . Using the first order Taylor series expansion 01  U 01 0 U 01 NU U 01 we get G = Y U 01 = (U 0 + N U ) 0 01 = 0G0 + N0Y U 01 0 G0 NU U 01 . Using (Y0 + NY )(U0 + NU ) 0 0 01 01 01 vec(G0 NU U0 ) = (U0 G0 )vec(NU ) and vec(NY U0 ) = 0 1 0 1 vec(In NY U0 ) = (U0 In )vec(NY ) (see [21]), it can easily 01 be verified that GvecG = (U00T In )VvecY (U 0 In ) + (U00T

01 01 U (U 0 G30 ) 0 2Herm((Uo0T In )VvecY vecU (U 0

3 G0 )). Replacing the true values U0 and G0 by the measured values, G0 )Cvec

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[19] G. Ammar, W. Gragg, and L. Reichel, “Downdating of Szegõ polynomials and data-fitting applications,” Linear Alg. Appl., vol. 172, pp. 315–336, 1992. [20] J. Schoukens, G. Vandersteen, R. Pintelon, and P. Guillaume, “Frequency domain system identification using nonparametric noise models estimated from a small number of data sets,” Automatica, vol. 33, no. 6, pp. 1073–1086, 1997. [21] J. Brewer, “Kronecker products and matrix calculus in systems theory,” IEEE Trans. Circuits Syst., vol. CAS-25, no. 9, pp. 772–781, Sep. 1978.

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Control by Interconnection of Mixed Port Hamiltonian Systems Alessandro Macchelli and Claudio Melchiorri Abstract—In this note, the regulation problem for mixed finite and infinite dimensional port Hamiltonian systems (m-pH systems) is discussed. A m-pH system results from the power conserving interconnection of finite and infinite dimensional systems in port Hamiltonian form. In particular, the system given by the interconnection of two finite dimensional systems, one of which is the controller, by means of an infinite dimensional connection is studied. The proposed control methodology is a generalization to the infinite dimensional case of a well-established passivity-based control technique for finite-dimensional port Hamiltonian systems, the control by interconnection and energy shaping, according to which the open-loop energy function is shaped so that a minimum in the desired configuration is introduced. This procedure is possible once the state variable of the controller is related to the state variable of the plant by constraining the state of the closed-loop system on a structural invariant (defined by a set of Casimir functions). In this way, the energy function of the controller, which is freely assignable, becomes a function of the configuration of the plant and, then, it can be easily shaped in order to solve the regulation problem. Index Terms—Casimir function, control by interconnection and energy shaping, mixed port Hamiltonian systems.

I. INTRODUCTION Port Hamiltonian systems [7], [15] are a powerful framework for modeling and controlling nonlinear dynamical systems, [9], [11]. A system described in port Hamiltonian form explicitly expresses its passivity properties under the condition that the Hamiltonian function is smooth and bounded from below. This feature is of great interest both for system analysis and for the development of advanced control strategies. As a matter of fact, global stability can be easily achieved by means of passivity-based control techniques, such as damping injection [13], [15] or energy shaping, [1], [9], [10]. The latter technique aims to develop a passive controller that shapes the total energy function of the plant in order to obtain a closed-loop energy with a minimum in the desired equilibrium configuration and stability can be proved by means of energetic considerations. Furthermore, since controller, plant and, consequently, closed-loop system are passive, stability can be assured even in presence of model uncertainties.

Manuscript received August 1, 2003; revised November 1, 2004. Recommended by Associate Editor P. D. Christofides. This work was supported by the EU project GeoPlex IST-2001-34166. Further information can be found at http://www.geoplex.cc. The authors are with the University of Bologna, CASY – DEIS, 40136 Bologna, Italy (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2005.858656

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