Application of orthonormal basis functions for identification of flexible-link manipulators

ARTICLE IN PRESS

Control Engineering Practice 14 (2006) 99–106 www.elsevier.com/locate/conengprac

Application of orthonormal basis functions for identification of flexible-link manipulators Kamyar Ziaei, David W.L. Wang Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 Received 8 November 2001; accepted 30 November 2004 Available online 8 March 2005

Abstract The advantage of generalized orthonormal bases, as compared to other orthonormal basis functions such as the Laguerre and Kautz functions, is the ability to include a priori knowledge of the real and complex pole locations of the model. In this paper, the implementation of generalized orthonormal basis functions (GOBF) for system identification is discussed and applied to identification of the dynamics of a single flexible-link manipulator. The plant is non-minimum phase with real and complex poles. A global optimization strategy for selecting the location of the poles of the basis functions is proposed. This method is evaluated through simulation and experimental studies and shows superior performance as compared to ARMAX and FIR identification. r 2005 Elsevier Ltd. All rights reserved. Keywords: System identification; Optimization; Control oriented models; Applications

1. Introduction The synthesis of many control systems relies on models obtained through system identification. The approximation of stable dynamical systems using orthonormal functions has attracted wide interest over the last decade (Ninness & Gustafsson, 1997; Wahlberg & Makila, 1996). This is because models based on parametric orthonormal bases offer distinct advantages over existing model structures, such as FIR, AR and ARX models. Despite this recent activity, relatively little has appeared involving experimental verification in the system identification context. Specifically, the authors are not aware of any reported implementation and realworld application of system identification based on the GOBF. The objective of this paper is to study and document the implementation of complete orthonormal bases, Corresponding author. Tel.: +1 519 888 4567x3968; fax: +1 519 746 3077. E-mail addresses: [email protected] [email protected] (D.W.L. Wang).

(K.

Ziaei),

0967-0661/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2004.11.020

developed by Ninness and Gustafsson (1997). The latter paper includes only the introduction of GOBF and proof of completeness and accuracy properties. No implementation issues, simulation or experimental results are presented. In the present work, we apply the GOBF base model for the identification of a flexiblelink manipulator which is non-minimum phase and possesses both real and complex poles. These behaviors, together with nonlinear dynamics, create major challenges in system modeling and system identification of flexible-link manipulators (Arciniegas, Eltimsahy, & Cios, 1994). We note that, although we show the identification results for single-link flexible manipulators, the identification method can be easily applied to obtain a linear time invariant (LTI) model for multi-link flexible manipulators at an operating point of interest. Further, the presented identification technique in this paper should be applicable to other industrial sectors. Our simulation and experimental studies show that GOBF is superior to many existing system identification techniques, especially in the presence of nonlinear and large measurement noise. However, if the underlying pole locations of the basis functions are not exact, one

ARTICLE IN PRESS K. Ziaei, D.W.L. Wang / Control Engineering Practice 14 (2006) 99–106

100

has to increase the number of basis functions. Although the time response of the model may improve, we have found that the frequency response plot becomes nonsmooth due to numerical issues and pole-zero cancellations. This is especially important for a resonant system where the pole-zero cancellations occur close to the unit circle. The important observation that we made is that for well-damped systems the correctness of filter pole locations is not as crucial as in the resonant case. Therefore, as an improvement, in order to relax the assumption of the prior knowledge of system dynamics, we propose a global optimization algorithm to determine the fixed pole location of the basis filters. Nevertheless, the a priori approximate knowledge of the system dynamics can be used to define lower and upper bounds in the optimization routine to reduce the computational time. Consider the single-input–single-output LTI system yðkÞ ¼ GðqÞuðkÞ þ vðkÞ;

GðqÞ ¼

1 X

gk qk ,

in a straightforward manner as shown by Ninness and Gustafsson (1997). The objective of this paper is to study the application of system identification based on GOBF for a single flexible-link manipulator where the underlying transfer function model includes both real and complex poles. The advantage of the generalized orthonormal basis function, as compared to other orthonormal systems, is the ability to incorporate knowledge of a variety of poles in the model. The orthonormal basis function proposed by Ninness and Gustafsson (1997) is presented along with a new solution to avoid basis functions with complex-valued impulse responses. A global optimization strategy is implemented to obtain the location of the poles for the basis function. This will result in a lower order and more accurate model. The main motivation for this study is to obtain a reliable frequency response model with uncertainty estimates for robust controller design.

(1)

k¼1

where yðkÞ and uðkÞ are the output and input signals and q denotes the shift operator: q1 uðkÞ9uðk  1Þ: The noise process fvðkÞg is assumed to be a stationary process with zero mean values. The objective of system identification is to find an estimate of GðqÞ from observations of the input and output measurements. One popular method to identify GðqÞ is to approximate it with a finite impulse response model (FIR), which is obtained by truncating the infinite sum in (1): GðqÞ ¼

ng X

gk qk .

(2)

k¼1

However, when the FIR model is used for a system with a very long impulse response, a very large number of parameters are necessary. Thus, the usefulness of this approach is limited by the rate of decay of the impulse response. This difficulty can be overcome by replacing the delay operators qi with more appropriate basis functions Li ðqÞ that include some a priori knowledge of the underlying system. For example, by using a set of basis functions, a model of order n is given by ! n X yðkÞ ¼ yi Li ðqÞ uðkÞ. (3) i¼1

Note that, similar to the FIR model, the model in (3) is linear in parameters yi : In addition, if the set of basis functions, Li ðqÞ; corresponds to orthonormal rational transfer functions, many nice properties of the FIR models including the numerical ones are preserved. Also, the theoretical results of the FIR model, including evaluation of the estimates variances, can be extended to the more general model with orthonormal basis function

2. Orthonormal basis functions The set of functions fLi ðejo Þg; 1pipn; in L2 ð½p; p Þ forms a set of orthonormal functions if the inner product Z 1 p hLm ; Ln i ¼ Lm ðejo ÞLn ðejo Þ do 2p p ( 1 if m ¼ n ¼ ð4Þ 0 otherwise: In general, the poles of the basis transfer functions Li ðqÞ are chosen based on a priori knowledge of the system dynamics. Suppose we chose the poles at fx1 ; :::; xn g: The orthonormal basis functions are then given by (Ninness & Gustafsson, 1997) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! m

1  jxm j2 Y 1  xi q Lm ðqÞ ¼ m ¼ 1; :::; n. q  xi q  xm i¼1 (5) The basis functions (5) are general orthonormal bases since they encompass other orthonormal bases. For example, if we assume all poles are zero (xi ¼ 0), we obtain the FIR model, i.e. Li ðqÞ ¼ qi : The Laguerre model (Wahlberg, 1991) is obtained from (5) by fixing xi ’s to a single real pole (xi ¼ a) describing the xm ¼ a dominant dynamics of the system: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1  a2 Þ 1  aq i1 ; a 2 R; jajo1. Li ðq; aÞ ¼ qa qa (6) If the system under consideration includes resonant behavior, it is reasonable that pairs of complex conjugate poles be incorporated in the basis functions

ARTICLE IN PRESS K. Ziaei, D.W.L. Wang / Control Engineering Practice 14 (2006) 99–106

(Wahlberg, 1994). However, a complex conjugate pole pair, xi and xiþ1 ¼ xi ; results in complex impulse responses for filters Li ðqÞ and Liþ1 ðqÞ which are inappropriate in a system identification setting. A solution, as suggested by Ninness and Gustafsson (1997), is to find two new basis functions, L0i ðqÞ and L0iþ1 ðqÞ; formed by linear combinations of Li ðqÞ and Liþ1 ðqÞ that produce real-valued impulse responses L0i ðqÞ L0iþ1 ðqÞ

! ¼

c1

c2

c3

c4

!

Li ðqÞ Liþ1

! ;

c1 ; . . . ; c4 2 C. (7)

It is apparent that in this definition the orthogonality is preserved, i.e. for all koi; hLi ; Lk i ¼ 0 and hLiþ1 ; Lk i ¼ 0 imply hL0i ; Lk i ¼ 0 and hL0iþ1 ; Lk i ¼ 0: Further, since the new basis functions should be orthonormal to each other, we have the following constraints: jc1 j þ jc2 j ¼ 1,

(8)

jc3 j þ jc4 j ¼ 1,

(9)

c1 c3 þ c2 c4 ¼ 0.

(10)

To find complex-valued parameters c1 to c4 ; we write Li and Liþ1 in a recursive form using (5) and knowing that xiþ1 ¼ xi : sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1  jxi j2 1  xi1 q Li ðqÞ ¼ Li1 ðqÞ, 1  jxi1 j2 q  xi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  jxi j2 Liþ1 ðqÞ ¼ 1  jxi1 j2

(11)



ð1  xi1 qÞð1  xi qÞ Li1 ðqÞ. q2  ðxi þ xi Þq þ jxi j2

101

xT1 Mx2 ¼ 0,

(15)

where x1 9ðm1 ; m2 Þ and x2 9ðm3 ; m4 Þ and ! 1 þ jxi j2 xi þ xi M9 . xi þ xi 1 þ jxi j2 T

T

(16)

Since M is symmetric and positive definite, m1 and m2 lie on an ellipse. In fact, there is an infinite number of solutions for m1 and m2 : Assume that we find a solution x1 satisfying (14), then an x2 that satisfies both (14) and (15) may be found by rotating x1 by 90 in the normalized eigenspace of M as shown in Ninness and Gustafsson (1997). That is,

0 1 x2 ¼ M 1=2 M 1=2 x1 . (17) 1 0 There are infinite solutions for x1 and x2 that satisfy constraints (14) and (15). By selecting m2 ¼ 0; we obtain the following solutions: !T j1  x2i j x1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 0 , (18) 1 þ jxi j2

x2 ¼

xi þ xi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 þ jxi j2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!T 1 þ jxi j2 .

(19)

Similarly, many other solutions can be found. Our numerical studies showed that the choice of these solutions does not have any significant effect on the identification process. It is now straightforward to find the complex parameters, ci ’s, which, substituted in (13), yield the new basis functions. The new basis vectors, L0i and L0iþ1 are then used in (3) instead of the complexvalued impulse response basis vectors Li and Liþ1 :

(12) Applying the above two equations in (7), L0i can be written as 2

m1

m2

3

zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ 6 7 ð1  x qÞ ðc  c x Þ q þ ðc 4 1 2 2  c 1 xi Þ 5 i1 i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  jxi j2 L0i ¼ 1  jxi1 j2 q2  ðxi þ xi Þq þ jxi j2  Li1 ðqÞ.

ð13Þ

Let m1 9c1  c2 xi and m2 9c2  c1 xi : Obviously, m1 and m2 should be real to ensure that L0i has a real-valued impulse response. From a similar equation as (13) for L0iþ1 ; we can define m3 and m4 in terms of c3 and c4 : Now the constraints in (8)–(10) become xTk Mxk ¼ j1  x2i j2 ;

k ¼ 1; 2,

(14)

3. Parameter estimation Having determined the orthonormal basis functions, the next step is to use the input–output data fuðkÞ; yðkÞg to estimate the unknown parameters. Using the standard quadratic criterion J¼

N 1X ^ Þ2 ð yðkÞ  yðkÞ N k¼1

(20)

^ with the estimated plant output, yðkÞ; given in linear regression form ^ yðkÞ ¼ /ðkÞT h, T

(21)

where /ðkÞ ¼ ½L1 ðqÞuðkÞ; . . . ; Ln ðqÞuðkÞ : A state-space realization for the elements of the measurement vector /ðkÞ is presented here, which is useful for numerical implementation. Define the states

ARTICLE IN PRESS K. Ziaei, D.W.L. Wang / Control Engineering Practice 14 (2006) 99–106

102

xi ðkÞ ¼ Li ðqÞuðkÞ; i ¼ 1; . . . ; n: From (11), we obtain the following recursive difference equations: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x1 ðk þ 1Þ ¼ x1 x1 ðkÞ þ 1  jx1 j2 uðtÞ, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  jx2 j2 x1 ðkÞ x2 ðk þ 1Þ ¼ x2 x2 ðk  1Þ þ 1  jx1 j2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  jx2 j2 x1 ðk þ 1Þ,  x1 1  jx1 j2 .. . sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  jxn j2 xn ðk þ 1Þ ¼ xn xn ðkÞ þ xn1 ðkÞ 1  jxn1 j2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  jxn j2 xn1 ðk þ 1Þ. ð22Þ  xn1 1  jxn1 j2

Since E is a lower triangular matrix with detðEÞ ¼ 1; it is 1 1 ¯ ¯ A and B9E B; we always invertible. Defining A9E can rewrite (23) in state-space form:

Defining xðkÞ ¼ ½xð1Þ; . . . ; xðnÞ ; the equation in (22) can be rewritten in matrix form:

where

Exðk þ 1Þ ¼ AxðkÞ þ BuðkÞ,

1 B sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi B 1  jx2 j B x1 B 1  jx1 j2 B B B .. . E¼B B B B 0 B B B @ 0

Assuming that the model includes complex poles, the states xðkÞ in (24) may be complex. For example, suppose the model has n poles fx1 ; . . . ; xn g where the first m poles are real and the rest are complex. Then xi ðkÞ ¼ Li ðqÞuðkÞ; moipn are complex because the impulse response Li ðqÞ are complex. As mentioned earlier, we need to replace complex-valued impulse response basis functions with their real-valued counterparts given in (7). For the state vector, xðkÞ; this replacement is equivalent to the transformation xðkÞ ¼ TxðkÞ which ¯ leads to ¯ 1 xðkÞ ¯ xðk þ T BuðkÞ, ¯ þ 1Þ ¼ T AT ¯

(25)

(23)

0

0

1

0 .

0

..

.



0

.

0

0

0



x2

0

..

.

..

.

..

.

..

.



0

and 1 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p C B B 1  jx1 j2 C C B C B 0 C. B B¼B C .. C B C B . A @ 0

1

C C 0C C C C .. C .. .C . C C C 1 0 C sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 2 C 1  jxn j A 1 xn1 2 1  jxn1 j 

..

..

0



and

0

(24)

(26)

where 0

x1 B sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 1  jx j2 2 B B B 1  jx1 j2 B B .. B . A¼B B B . B .. B B B B @ 0

¯ ¯ xðk þ 1Þ ¼ AxðkÞ þ BuðkÞ.

0



0

..

.. .

.

xn1 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  jxn j2 xn 1  jxn1 j2

1 C C C C C C C C C C C C C C C C A

Here, I mm denotes an m  m identity matrix. Note that the transformation matrix T is block diagonal and it can be verified that all diagonal blocks are non-singular if the basis functions are stable; therefore, T is nonsingular. The vector /ðkÞ in (21) is simply /ðkÞ ¼ xðkÞ: The ¯ optimal least-squares parameter estimates using N input–output measurements is " #1 N N X X T ^hN ¼ /ðkÞ/ðkÞ /ðkÞyðkÞ. (27) k¼1

k¼1

Now the question arises on how to select the fixed pole locations of the orthogonal basis functions in (5). Obviously, a good knowledge of the dominating time constants is required to obtain a feasible low-order model. The filter poles are mostly selected based on some a priori knowledge about the plant to be identified. This knowledge can be attained through mathematical modeling and/or preliminary experimental observations, such as step response test and spectral analysis. The filter coefficients can also be estimated directly, in case no prior knowledge is available or when prior knowledge is not sufficient in obtaining a reasonable low-order model. In this paper, we consider estimating the location of poles of the basis functions by minimizing the cost function (Eq. (20)) with respect to the filter poles. However, the poles appear nonlinearly in the cost function and cannot be found analytically. Thus, we need to resort to numerical search strategies. The approach that we adopted in this paper is described below.

ARTICLE IN PRESS K. Ziaei, D.W.L. Wang / Control Engineering Practice 14 (2006) 99–106

Assume the model has n poles fx1 ; . . . ; xn g where the first m poles are real and the rest are complex and complex conjugate pairs are ordered consecutively. In order to restrict the optimization to real numbers subject to simple bounds, we define the filter parameter vector h iT g ¼ x1 ; . . . ; xm ; b1 ; g1 ; . . . ; bðnmÞ=2 ; gðnmÞ=2 , (28) where bi ¼

2Reðxmþ2i1 Þ 1 þ jxmþ2i1 j2

and

gi ¼ jxmþ2i1 j2 .

(29)

Note that the parameters bi and gi are identical to the ones used in parameterization of the denominator coefficients of Kautz filters (Wahlberg, 1994). Observe also that stability of the orthogonal filters confines all elements of g in the range ð1; 1Þ: Thus, this constraint will be considered in the numerical optimization methods (as simple bounds) so that the estimation of g corresponds to stable filters. From (20) and (21), when replacing /ðkÞ with /ðk; gÞ to indicate the dependence of the processed input data on the filter parameters, the optimization problem becomes g^ N ¼ arg min g

N " #2 1X yðkÞ  /ðk; gÞT h^ N . N k¼1

103

as the filter parameters are varied. Clearly, we can see the occurrence of multiple local minima in the vicinity of the global minimum. Therefore, a global optimization strategy, although computational intensive, is more appropriate as compared to gradient-based methods often used for nonlinear least-squares optimization. In this paper, we use a branch and bound optimization technique proposed by Jones, Perttunen, and Stuckman (1993), the so-called DIRECT algorithm which is an acronym for divide rectangles. This algorithm is for finding the global minimum of a multivariate function subject to simple bounds, i.e., min f ðxÞ subject to x

xL pxpxU ,

(31)

where xL and xU are the lower and upper bounds imposed on the optimization parameter vector. Notice that for our optimization problem (30), we can use any prior knowledge of our system to define suitable lower and upper bounds for the optimization parameters in (28). DIRECT is a modification of the standard Lipschitzian approach that does not need to specify a Lipschitz constant of the objective function. In the multivariate version, the DIRECT algorithm does the following (for details see Jones et al., 1993):

(30)

As mentioned earlier, the minimization of filter parameters must be done numerically. Notice that after each iteration in numerical search strategy, the model parameter h^ N needs to be updated using least-squares technique (27), which gives a global optimal solution for the model parameter with respect to the chosen basis functions. Our numerical studies reveal the potential presence of local minima for the objective function (30). For example, Fig. 1 depicts the error surface for a simple resonant system where the filter parameter vector (28) is given by g ¼ ½b1 ; g1 T : The figure shows the cost function

(a) Normalize the search space into a unit hypercube and evaluate the function f at the center point of the hypercube. Set number of partitions (hyperrectangles) m ¼ 1: (b) Identify the potentially optimal hyperrectangles among m partitions. (c) Sample and divide these rectangles. Update f min and set m ¼ m þ Dm; where Dm is the number of new points sampled. (d) Repeat steps (b) and (c) until the iteration limit has been reached.

4. Identification of a flexible-link manipulator

0.04 0.035 0.03

J

0.025 0.02

0.015 0.01

-0.9988 -0.999 -0.9992 -0.9994 -0.9996 -0.9998

γ

0 0.98

1

0.005

0.985

0.99

β1

0.995

1

-1

Fig. 1. Objective function for a second order resonant system.

Some drawbacks of identification of single flexiblelink (SFL) manipulators using ARMA-type models have been previously reported (Rovner & Cannon, 1987; Tzes & Yurkovich, 1991). Rovner and Cannon (1987) showed that the standard ARMA model in unmodified form cannot satisfactorily describe the resonant modes of the system. To improve the identification results, the authors made several modifications to the model including pre-filtering and fixing some parameters based on spectral analysis. As an alternative to ARMA models, a frequency domain technique has been used to parameterize the transfer function of flexible manipulators by Tzes and Yurkovich (1991). Since only the magnitude response is used, the

ARTICLE IN PRESS K. Ziaei, D.W.L. Wang / Control Engineering Practice 14 (2006) 99–106

yðsÞ bn s2n þ bn1 s2n2 þ bn2 s2n4 þ    þ b0 ¼ 2 2n , uðsÞ s ðs þ an1 s2n2 þ an2 s2n4 þ    þ a0 Þ

(32)

where n is the number of oscillatory modes in the model. For our experimental setup, the first mode dominates the other modes, therefore, in this study, we consider only the first mode of vibration. Hence, our plant model has two poles at origin, a pair of complex poles and two real zeros approximately at 10.8 and 10:5: We note that in our simulation model we also considered structural damping for the beam. 1000 data points were used for the identification. The output is corrupted with white noise (with a variance of 0:0042 ; resulting in a signal to noise ratio (SNR) of  30 dB). For this study, we completely ignored the a priori knowledge about the pole locations and the filter poles were obtained through the optimization algorithm. In Fig. 2 the frequency response of the identified model and the actual simulation model are shown. It is seen that despite the low SNR, the identification model matches very well with the actual model. For comparison, forth-order ARX, OE and ARMAX models are fitted to the data using the MATLABs identification toolbox (Ljung, 1997). The results are shown in Fig. 3. As is seen in the figure, none of the models can identify the plant dynamics satisfactorily. It should be mentioned that, while for a linear time invariant (LTI) system one would expect favorable system identification results using ARX and OE models, the failure of these identification methods in our simulation studies is primarily due to the large noise that was added to the output data. In our studies we found that the identification based on an orthonormal basis function is generally less prone to failure in the presence of relatively high output noise.

Amplitude (dB)

50 0 -50 -100 -150 10 -1

10 0

10 1

10 2

10 3

10 1

10 2

10 3

Phase (deg)

-100 -200 -300 -400 -500 -600 10 -1

10 0

Frequency (rad/sec)

Fig. 2. Bode plot of the mathematical model (solid) compared to identified GOBF model (dashed) from simulation data.

50

Amplitude (dB)

frequency domain identification method given by Tzes and Yurkovich (1991) is merely suitable for identifying minimum-phase transfer functions with slightly damped zeros such as the transfer function from the shaft velocity to tip acceleration. The system identification using orthogonal bases discussed in the previous sections will be applied to data from a simulation example and to experimental data taken from a single flexible-link manipulator. For both simulation and experiment, our objective is to find a transfer function from motor torque to tip deflection. This transfer function is non-minimum phase and includes two real and complex poles. Four poles are selected; two real poles for the rigid body dynamics and two complex poles to model the first mode of vibration. Simulation Example: A pseudo-binary random signal (PRBS) is applied to a simulated model of a SFL, where the input is the hub torque and the output is the manipulator’s tip displacement. Using assumed mode approach and assuming no motor and structural damping, the transfer function of a flexible link is given by (Wang & Vidyasagar, 1991)

0 -50 -100 -150 -1 10

10 0

10 1

10 2

10 3

10 1

10 2

10 3

200

Phase (deg)

104

0 -200 -400 -600 10 -1

10 0

Frequency (rad/sec)

Fig. 3. Bode plot of the mathematical model (solid) compared to the identified models from simulation data: ARX (thin dashed), ARMAX (thick dashed) and OE (dash dotted).

4.1. Experimental results A five-bar manipulator with three degrees of freedom is available for experiments (Trautman, 1995; Trautmann & Wang, 1996). The last link is a flexible link which is a 1 m long aluminum beam with a rectangular cross-section that is 6:35 mm  6:35 mm: The experimental data were collected by applying a PRBS signal to the base motor of the manipulator. The flexible link uses two sets of strain gauges to measure the tip deflection. The input is the motor voltage and the output is the tip position of the manipulator. In this case, one of the real poles is related to the motor dynamics. The data were collected with a sampling frequency of 200 Hz. In Fig. 4 the frequency responses of different identified models of

ARTICLE IN PRESS K. Ziaei, D.W.L. Wang / Control Engineering Practice 14 (2006) 99–106

105

0 0.15

-20 -40

0.1

-60 -80 -100 -1 10

10 0

10 1

10 2

10 3

100

Phase (deg)

0

Tip position (m)

Amplitude (dB)

20

0.05

0

-100 -0.05

-200 -300 -400

-0.1

-500 -600 10 -1

10 0

10 1

10 2

10 3

0

1

2

Frequency (rad/sec)

3

4

5

6

Time (sec)

Fig. 4. Bode plot of the identified models from experimental data: GOBF (solid), ARX (thin dashed), ARMAX (thick dashed) and OE (dash dotted).

Fig. 6. Cross-validation: experimental data (solid) and time response of identified models: ARX (thin dashed), ARMAX (thick dashed) and OE (dash dotted).

0.2

Amplitude (dB)

20

0.15

0.1

0 -20 -40 -60

0.05

10 0

10 1

10 2

10 3

10 2

10 3

Frequency (rad/sec) 0

0

Phase (deg)

Tip position (m)

-80 -100 -1 10

-0.05

-0.1

-100 -200 -300 -400 -500

-0.15

0

1

2

3

4

5

6

Time (sec)

-600 -1 10

10 0

10 1

Frequency (rad/sec)

Fig. 5. Cross-validation: experimental data (solid) and time response of identified GOBF model (dashed).

Fig. 7. Bode plot of identified GOBF model with two standard deviation error bounds.

the same order (4th) are compared. The figure shows that besides the GOBF model, only the OE model finds the resonant mode correctly, although other poles and zeros are not identified properly. In Fig. 5 the time domain response of the identified GOBF model is compared with a set of fresh data, apart from the data used for identification. As can be seen, the GOBF model can describe the dynamics of the system quite well. For comparison a number of ordinary ARX, OE and ARMAX models were fitted to the data, but no model has the ability to capture the dynamics of the system adequately. Cross-validation of these models is shown in Fig. 6. Our identified model is a low-order approximation of the flexible-link manipulator (an infinite dimensional

system in nature) in our frequency range of interest. We can include some quantification of the implied model imperfections. Frequency domain uncertainty estimates have gained considerable interest in the control community (Baillieul, 1992). Unlike the classical identification methods, these new techniques enable the inclusion of both structural model errors (bias) and noise (variance) in the estimated uncertainty bounds. Due to the nature of the GOBF model, one method that is directly applicable for assessing model uncertainty is the stochastic embedding approach. For a detailed presentation, readers are referred to Goodwin, Gevers, and Ninness (1992) and the references therein. The results of uncertainty estimates using the stochastic embedding approach are depicted in Fig. 7. The figure shows the

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bode plot of the identified GOBF model with two standard deviation error bounds. As expected, the model uncertainty is larger at higher frequencies.

5. Conclusions Previously studied orthogonal functions such as Laguerre and Kautz functions are appropriate for modeling well-damped or resonant systems, respectively. In this paper, we outlined a system identification framework using generalized orthogonal basis functions. The basic idea is simply to generalize the FIR model structure by replacing the delay operator by discrete orthonormal basis functions that allow prior knowledge in the form of a variety of poles to be incorporated in the model. For fine-tuning of the pole locations of the basis function, a global optimization strategy is suggested. As an application, the presented identification scheme is applied to a single flexible-link manipulator. Simulation and experimental results show that an identified model based on generalized orthonormal bases can accurately describe the dynamics of a flexible-link manipulator. References Arciniegas, J. I., Eltimsahy, A. H., & Cios, K. J. (1994). Identification of flexible robotic manipulators using neural networks. Integrated Computer-Aided Engineering, 1(3), 195–208.

Baillieul, J. (Ed.), 1992. IEEE Trans. Automatic Control [Special Issue]. System Identification for Robust Control 37(7). Goodwin, G. C., Gevers, M., & Ninness, B. (1992). Quantifying the error in estimated transfer functions with application to model order selection. IEEE Trans. Automatic Control, 37(7), 953–963. Jones, D. R., Perttunen, C. D., & Stuckman, B. E. (1993). Lipschitzian optimization without the lipshitz constant. Journal of Optimization Theory and Application, 79(1), 157–181. Ljung, L. (1997). System Identification Toolbox—User’s Guide. Sherborn, MA: The MathWorks Inc. Ninness, B., & Gustafsson, F. (1997). A unifying construction of orthonormal bases for system identification. IEEE Transaction on Automatic Control, 42(4), 515–521. Rovner, D. M., & Cannon, R. H. (1987). Experiments towards on-line identification and control of a very flexible one-link manipulator. The International Journal of Robotics Research, 6(4), 3–14. Trautman, C. (1995). Modelling and control of flexible link manipulators with nonlinear vibrations and two degrees of vibrational freedom. Master’s thesis, University of Waterloo, Department of Electrical and Computer Engineering. Trautmann, C., Wang, D. (1996). Noncollocated passive control of a flexible link manipulator. In: Proceedings of the IEEE International Conference on Robotics and Automation. (pp. 1107–1114). Tzes, A. P., & Yurkovich, S. (1991). Application and comparison of on-line identification methods for flexible manipulator control. The International Journal of Robotics Research, 10(5), 515–527. Wahlberg, B. (1991). System identification using Laguerre models. IEEE Transaction on Automatic Control, 36(5), 551–562. Wahlberg, B. (1994). System identification using Kautz models. IEEE Transaction on Automatic Control, 39(6), 1276–1282. Wahlberg, B., & Makila, P. M. (1996). On approximation of stable linear dynamical systems using Laguerre and Kautz functions. Automatica, 32(5), 693–708. Wang, D., & Vidyasagar, M. (1991). Transfer functions for single flexible link. International Journal of Robotics Research, 10(5), 540–549.