Identification of LPV Models for Spatially Varying Interconnected Systems

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1 21 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July 2, 21 ThB216 Identification of LPV Models for Spatially Varying Interconnected Systems Mukhtar Ali, Saulat S Chughtai and Herbert Werner Abstract This paper presents a method for system identification of two dimensional parameter varying systems The idea of linear parameter varying (LPV) identification for one dimensional causal systems in input-output form is extended to identify two dimensional (2-D) parameter varying systems in input-output form which may be causal, semi-causal or noncausal The identification method is general in the sense that it can be used to identify systems which may be separable or non-separable Furthermore, the algorithm explicitly takes boundary conditions into consideration The effectiveness of the approach is demonstrated by applying it to identify a spatially varying model for a non-uniform beam I INTRODUCTION In the last decade there has been a renewed interest in distributed control of complex engineering systems that are multidimensional and composed of similar subsystems, which interact with their closest neighbors Such systems include unmanned aerial vehicles and satellites flying in formations, vehicle platoons and automated highway systems as well as flexible structures, fluid flow and systems that are also characterized by the same class of partial differential equations The control synthesis of two dimensional or multidimensional systems is a rich field and there are a number of methods proposed to design controllers for such systems Distributed controllers are designed for spatially invariant system in Fourier domain in [1]; a method that can be used to design simple controllers for distributed interconnected systems by using the separability condition between the temporal and spatial dimension of the plant is presented in [2] LMI-based control design methods have been developed for spatially invariant distributed systems using multidimensional optimization, where the implementation of the controller is almost decentralized in nature like in [3], [4], [5], [6], [7] To synthesize optimal and/or robust control schemes for spatially interconnected systems, it is important to have a sufficiently accurate model of the system One possibility to obtain such a model is to construct it based on the governing partial differential equations (PDE), followed by the experimental identification of the physical parameters, see eg[8] However, for many practical applications such PDEs are either not available or too complex, and attempts to linearize these might result in unwanted dynamic behavior M Ali, SS Chughtai and H Werner are with the Institute of Control Systems, Hamburg University of Technology, Eissendorfer Str 4, 2173 Hamburg, Germany, {mukhtar,saulatchughtai,hwerner}@tu-harburgde /1/$26 21 AACC 3889 In system identification of 2-D or multidimensional systems few results are available In [9], identification of a 2-D causal system transfer function identification from input output data is presented The identification of non-causal multidimensional systems is presented in [1], [11] Subspace based methods to identify 2-D state-space models for separable-in-denominator causal filters are discussed in [12] and [13] Methods to identify spatially distributed interconnected systems are proposed in [14] and [15] As practically distributed parameter systems may have spatially and temporally varying parameters, recently a distributed gain scheduling control approach has been proposed for spatially and temporally varying distributed interconnected systems in [16], [17] To the authors best knowledge, so far no method to identify parameter varying-models for spatially interconnected systems have been proposed The main contribution of this paper is to extend the idea presented in [18] to the identification of 2-D spatially linear parameter-varying systems The method is general in the sense that it has no assumption on separability, furthermore it is equally applicable to causal, semi-causal (spatially interconnected) and non-causal two-dimensional systems This method has also the advantage that one can easily include various boundary conditions The paper is organized as follows: in section II the method to identify 2-D systems using a least squares technique and the treatment of boundary conditions are described The extension of this method to identify parameter varying models for 2-D spatially varying systems is given in section III Section IV presents the application of the proposed method to a practical example, and conclusions are drawn in section V II PRELIMINARIES Now let u(n 1,n 2 ) be the two dimensional discrete input signal to a linear invariant 2-D system Then its output y(n 1,n 2 ) can be represented in difference equation form, which is a linear combination of weighted input-output values [19] (i 1,i 2 ) M y,(i 1,i 2 ) (,) y(n 1 i 1,n 2 i 2 ) + (i 1,i 2 ) M u u(n 1 i 1,n 2 i 2 ) (1) where M y and M u denote the support regions (masks) for output and input terms, respectively, and n 1 and n 2 are

2 independent variables (usually time and space) The support region (mask) is defined as a subset of the two-dimensional space in which the indices of the coefficients of input and output terms in the difference equation lie A general support region for 2-D systems lies in 2-D plane For causal systems support region is subset of first quadrant of 2-D plane, for semi-causal the support region is subset of right half plane and for non-causal systems the support region lies in all four quadrant of 2-D plane Let us consider a general support region (mask) for a 2-D system as shown in Fig 1 If M represents this region, it consists of a union of intervals where l 1 M = m(i 1 ) (2) i 1 =k 1 m(i 1 )={(i 1,i 2 ) : k 2 (i 1 ) i 2 l 2 (i 1 )} (3) l 1 = max{i 1 :(i 1,i 2 ) M} k 1 = min{i 1 :(i 1,i 2 ) M} l 2 (i 1 )=max{i 2 :(i 1,i 2 ) m(i 1 )} k 2 (i 1 )=min{i 2 :(i 1,i 2 ) m(i 1 )} After introducing the above notation, and considering M y and M u as support region for output and input respectively the difference equation (1) can be written as l y 1 l y 2 (i 1) i 1 =k y 1 i 2 =k y 2 (i 1) l1 u l2 u(i 1) i 1 =k1 u i 2 =k2 u(i 1) + y(n 1 i 1,n 2 i 2 ) u(n 1 i 1,n 2 i 2 ) where for the first term of (4) on the right hand side we have also the condition that (i 1,i 2 ) (,) Now define column-wise data vectors and the corresponding system parameters that are attached to the support regions (masks) M y and M u The Output data corresponding to M y is i 1 [ k y ] 1,ly 1 (4) u m u (i 1 )(n 1,n 2 )= [ u(n 1 i 1,n 2 i 2 ) ] i 2 =k u 2 (i 1),,l u 2 (i 1) and finally b m u (i 1 ) = [ ]i 2 =k u 2 (i 1),,l u 2 (i 1) U M u(n 1,n 2 )= [ u m u (i 1 )(n 1,n 2 ) ] i 1 =k1 u,,lu 1 B M u = [ ] b m u (i 1 ) i 1 =k1 u,,lu 1 Then (1) can be written as (6) y(n 1,n 2 )=ϕ (n 1,n 2 )Θ (7) The regressors vector ϕ is constructed from output input data as [ ] YM y(n ϕ(n 1,n 2 )= 1,n 2 ) (8) U M u(n 1,n 2 ) The parameter vector Θ is given accordingly as [ ] AM y Θ= B M u Once we have the data vectors constructed, we can obtain a parameter vector by the method discussed in [18], which we briefly summarize here If measured data of size N 1 N 2 is available, we can construct the output vector as Y = [ y(1,1) y(n 1,1) y(1,2) Similarly the regressors are arranged as ϕ (1,1) ϕ (N 1,1) Φ= ϕ (1,2) ϕ (N 1,N 2 ) (9) y(n 1,N 2 ) ] (1) (11) then using least square the parameter vector can be given as Θ=(Φ Φ) 1 Φ Y (12) y m y (i 1 )(n 1,n 2 )= [ y(n 1 i 1,n 2 i 2 ) ] i 2 =k y 2 (i 1),,l y 2 (i 1),(i 1,i 2 ) (,) and finally a m y (i 1 ) = [ ]i 2 =k y 2 (i 1),,l y 2 (i 1),(i 1,i 2 ) (,) Y M y(n 1,n 2 )= [ y m y (i 1 )(n 1,n 2 ) ] i 1 =k y 1,,ly 1 A M y = [ ] a m y (i 1 ) i 1 =k y 1,,ly 1 The input data corresponding to M u is i 1 [ k1 u ],lu 1 (5) A Boundary Conditions Consider a 2-D spatio-temporal spatially interconnected system as shown in Fig 2, where there are interacting signals between subsystems The signals at the boundary can be generated by boundary conditions as follows: In the case of Dirichlet boundary conditions, these signals are taken as zero However, arbitrary boundary conditions can be dealt with by considering fictitious elements and thus extending the system as shown in Fig 2, in which black circles correspond to original subsystems and white circles correspond to fictitious elements generated by boundary 389

3 (i 1,k 2 (i 1 )) i 1 l 1 (i 1,l 2 (i 1 )) temporal or spatial signals, the so called scheduling variable p :Z P The compact set P R n p denotes the admissible region in the scheduling space A two dimensional LPV-ARX model structure for SISO systems can be defined as k 1 i 2 (i 1,i 2 ) M y \(,) (p(n 1,n 2 ))y(n 1 i 1,n 2 i 2 ) + (i 1,i 2 ) M u (p(n 1,n 2 ))u(n 1 i 1,n 2 i 2 ) (16) The coefficient functions, : P R have static dependence on p(n 1,n 2 ) and are parameterized as Fig 1 Boundary 1 Fig 2 Two-Dimensional Lattice, filled circles show the Mask M u 1 y 1 y 2 y3 y 4 u 2 u 3 Boundary 2 Boundary effects in spatially interconnected system conditions Generally the boundary conditions are given by the set Ω={B p y(i 1,i 2 )= i 1 = 1 or N 1, and i 2 = 1 or N 2 }, (13) where B is a linear difference operator having order p, which is always less than the order of the system The interacting signals from fictitious elements are obtained from the boundary conditions For example, if we have B of order one and the boundary conditions only depend on the spatial dimension, we have or y(i 1,i 2 ) y(i 1,i 2 )= (14) y(i 1,i 2 )=y(i 1,i 2 ) (15) From this we get an interacting signal for the subsystem at the boundary, which can be readily included in the above system identification technique Generally, boundary conditions are continuous in space and time We can convert them into discrete conditions in space and in time by using a central difference approximation, for details see [18] III TWO DIMENSIONAL PARAMETER-VARYING IDENTIFICATION In linear parameter varying (LPV) systems the model parameters are assumed to be a function of measurable u 4 (p(n 1,n 2 ))=,+ (p(n 1,n 2 ))=,+ s j=1 s j=1, jψ i1,i 2, j(p(n 1,n 2 )) (17), jψ i1,i 2, j(p(n 1,n 2 )) (18) Here ψ i1,i 2, j(p()) : P R are user-defined functions of the scheduling variables, and s represents the number of functions, which can be different for different coefficients but here we are assuming that all coefficients depend on same number of functions Assuming a general support region as in Fig 1, and applying the procedure discussed above, we can write the difference equation in (16) as + l y 1 l y 2 (i 1) i 1 =k y 1 i 2 =k y 2 (i 1) l1 u l2 u(i 1) i 1 =k1 u i 2 =k2 u(i 1) (p(n 1,n 2 ))y(n 1 i 1,n 2 i 2 ) (p(n 1,n 2 ))u(n 1 i 1,n 2 i 2 ) (19) where for the first term of (19) on the right hand side we have the condition that (i 1,i 2 ) (,) The coefficients, are given in (17) and (18) The data column vectors and the corresponding system parameter vectors which are associated with the support region M y and M u are constructed as follows: Output data corresponding to M y is i 1 [ k y ] 1,ly 1 y m y (i 1 )(n 1,n 2 )= [ [ ] 1 y(n 1 i 1,n 2 i 2 ) ψ i1,i 2, j() ] ] a m y (i 1 ) =[[ ai1,i 2, j j=:s j=1:s ] i 2 =k y 2 (i 1),,l y 2 (i 1) i 2 =k y 2 (i 1),,l y 2 (i 1) (2) in (2) we exclude on the right hand side the terms corressponding to point (,) finally Y p M y(n 1,n 2 )= [ y m y (i 1 )(n 1,n 2 ) ] i 1 =k y 1,,ly 1 (21) 3891

4 A p M y =[ ] a m y (i 1 ) i 1 =k y 1,,ly 1 Input data corresponding to M u is i 1 [ k1 u ],lu 1 u m u (i 1 )(n 1,n 2 )= [ [ ] 1 u(n 1 i 1,n 2 i 2 ) ψ i1,i 2, j() ] ] b m u (i 1 ) =[[ bi1,i 2, j j=:s j=1:s ] i 2 =k u 2 (i 1),,l u 2 (i 1) i 2 =k u 2 (i 1),,l u 2 (i 1) (22) TABLE I SPECIFICATIONS Length, L 58 m Max thickness, t b 8 mm Thickness ratio, c t 5 Width, w b 254 mm Mass density, ρ v 271 kg/m 3 Young s Modulus, E Pa y u(x,t) finally U p M u(n 1,n 2 )= [ u m u (i 1 )(n 1,n 2 ) ] i 1 =k u 1,,lu 1 (23) t b t L B p M u =[ ] b m u (i 1 ) i 1 =k1 u,,lu 1 z L x Then (16) can be written as y(n 1,n 2 )=ϕ p (n 1,n 2 )Θ p (24) Fig 3 Non-uniform beam where the regressor vector ϕ p is given as [ Y p ϕ p (n 1,n 2 )= M y(n ] 1,n 2 ) U p M u(n 1,n 2 ) and the parameter vector Θ p is given accordingly as [ A p ] Θ p = M y B p M u (25) (26) If measured data of size N 1 N 2 is available as input, output and scheduling signal, then output and regressors are constructed as Y p = [ y(1,1) y(n 1,1) y(1,2) ϕp (1,1) ϕ p (N 1,1) Φ p = ϕp (1,2) ϕp (N 1,N 2 ) and the parameter vector is given by y(n 1,N 2 ) ] (27) (28) Θ p =(Φ p Φ p) 1 Φ p Y p (29) IV PRACTICAL EXAMPLE A non-uniform flexible beam, taken from [16], which has the physical properties given in table - I and is shown as in Fig 3, is selected as example for illustration Let y(x,t) represent the transverse deflection (in y- direction) of the beam and u(x,t) be the distributed force on the beam Here x represents the spatial and t represents the temporal dependence of the applied force and corresponding deflection If we are in discrete domain, then transverse deflection of the beam is represented by y(n 1,n 2 ) and distributed force on the beam is represented as u(n 1,n 2 ), where n 1 and n 2 are indices for spatial and temporal dimensions, respectively If we divide this beam into N 1 segments along the spatial dimension, and take N 2 temporal data at each segment, then we have N 1 N 2 data points for output y(n 1,n 2 ), input u(n 1,n 2 ) and the scheduling signal The scheduling parameter is selected to depend only on the spatial dimension (n 1 ) and is independent of time (n 2 ); it is defined as δ(n 1 )= 2n 1 L 1 The user-defined scheduling functions are defined as ψ i1,i 2,1()=δ(n 1 ), ψ i1,i 2,2()=δ 2 (n 1 ) (3) We are assuming similar scheduling functions for all (i 1,i 2 ), thus we can write 3892

5 ψ 1 ()=δ(n 1 ), ψ 2 ()=δ 2 (n 1 ) (31) 25 Fourth Element The input and output masks selected are M u ={(,1)} M y ={( 2,1),( 1,1),(,),(,1),(,2),(1,1),(2,1)} this leads to the following regressor vector, using (2), (21), (22), (23) and (25): y(n 1 + 2,n 2 1) y(n 1 + 1,n 2 1) y(n 1,n 2 1) ϕ p (n 1,n 2 )= y(n 1,n 2 2) 1 ψ 1 () (32) y(n 1 1,n 2 1) ψ 2 () y(n 1 2,n 2 1) u(n 1,n 2 1) The output vector and the regressors matrix are then formed using (27) and (28); the parameters of the model are obtained from (29) The simulated beam model is spatially discretized into nine subsystems The sampling time is taken as T s = seconds For excitation, a two-dimensional zero-mean white noise with normal distribution is used If - for comparison - we ignore the parameter dependence and identify a linear model, we get the validation as shown in Fig (4) At first the dependence of the coefficients of the input-output beam model are taken to depend only on ψ 1, but this leads a model with a relatively poor fit A model providing a better fit is obtained by assuming that the coefficients depend not only on ψ 1 but also on ψ 2 as in (32) The validation results for constant values of the spatial variable are shown in Fig (5) to (7) The identified beam model is represented in the following input-output form Sixth Element Fig 4 (red) identified and (blue) actual response when spatial variation is neglected y(n 1,n 2 )+a 2,1 (p())y(n 1 + 2,n 2 1)+a 1,1 (p()) y(n 1 + 1,n 1 1)+a,1 (p())y(n 1,n 2 1)+a,2 (p()) y(n 1,n 2 2)+a 1,1 (p())y(n 1 1,n 2 1)+a 2,1 (p()) y(n 1 2,n 2 1)= b,1 (p())u(n 1,n 2 1) (33) where (p())=,+,1ψ 1 ()+,2ψ 2 () (i 1,i 2 ) M y,(i 1,i 2 ) (,) x Second Element (p())=,+,1ψ 1 ()+,2ψ 2 () V CONCLUSIONS (i 1,i 2 ) M u A method to identify LPV input-output model of twodimensional spatially varying system is proposed The method can be used for the identification of a general class of two-dimensional LPV systems which may be separable 6 8 Fig 5 Red is the actual response, black is the response of the identified model when only ψ 1 is selected as basis function, and blue is the model response when ψ 1 and ψ 2 are selected as basis functions 3893

6 1 5 5 Fourth Element 1 Fig 6 Red is the actual response, black is the response of the identified model when only ψ 1 is selected as basis function, and blue is the model response when ψ 1 and ψ 2 are selected as basis functions Sixth Element Fig 7 Red is the actual response, black is the response of the identified model when only ψ 1 is selected as basis function, and blue is the model response when ψ 1 and ψ 2 are selected as basis functions [3] R D Andrea, Linear matrix enequalities, multidimensional system optimization, and control of spatially distributed systems an example, in Proceedings of the American Control Conference IEEE, 1999, pp pp [4] R D Andrea and G E Dullerud, Distributed control design for spatially interconnected systems, IEEE Transactions on Automatic Control, vol 48, no 9, pp , 23 [5] S S Chughtai and H Werner, Fixed structure controller design for a class of spatially interconnected systems, in Proceedings IFAC Symposium on Large Scale Systems, 27 [6], Distributed control for a class of spatially interconnected discrete-time systems, in Proc of IFAC World Congress, 28, pp [7], New dilated lmis to synthesize controllers for a class of spatially interconnected systems, in Proc of IFAC World Congress, 28, pp [8] H Banks and K Kunish, Estimation Techniques for Distributed Parameters Systems Boston Birkhäuser, 1989 [9] C Chen and Y Kao, Identification of two-dimensional transfer function from finite input-output data, IEEE Transactions on Automatic Control, vol 24, no 5, pp , 1979 [1] K Arun, J Krogmeier, and L Potter, Identification of 2-d noncausal systems, in Proc of the IEEE Conference on Decision and Control, Los Angeles, CA, 1987 [11] J Krogmeier and K Arun, A comparative study of causal and noncausal models for multidimensional spectrum estimation, in Maple Press, 1989 [12] B Lashgari, L M Silverman, and J Abramatic, Approximation of 2- d separable in denominator filters, IEEE Trans Circuits and Systems, vol 3, no 2, pp , 1983 [13] J Ramos, A subspace algorithm for identifying 2-d separable in denominator filters, IEEE Transaction on Circuits and Systems, vol 41, no 1, pp 63 67, 1994 [14] R Fraanji and M Verhaegen, A spatial canonical approach to multidimensional state-space identification for distributed parameter systems, in Fourth International Workshop on Multidimensional systems NDS 25, Wuppertal, Germany, 25 [15] C Buttura and A Zorrico, Decentralized multivariate identification of interconnected systems by stochastic subspace method, in Proc IEEE Conference on Decision and Control, 22 [16] F Wu and E Yildizoglu, A distributed parameter dependent control design for a flexible beam problem, in Proceedings of the American Control Conference, 22 Anchorage, Alaska: IEEE, 22, pp pp [17] S Chughtai and H Werner, Fixed structure control of interconnected lpv systems, Institute for Control Systems, Arbeitsbereich Regelungstechnik, Tech Rep 7, 27 [18] M Ali, S Chughtai, and H Werner, Identification of spatially interconnected systems, in Proc of CDC 9, 29 [19] G Glentis, C H Slump, and o E Herrmann, Efficient twodimensional arx modeling, in International Conference on Image Processing (ICIP), 1994, or non-separable The method is equally valid for causal, semi-causal and non-causal systems Furthermore, boundary conditions can be easily included in the algorithm The effectiveness of the method is demonstrated by applying it to a distributed parameter varying cantilever beam model In continuation of this work, the modification of the algorithm to identify spatially varying interconnected models with noisy data is currently being investigated REFERENCES [1] B Bamieh, F Paganini, and M Dahleh, Distributed control of spatially invarient systems, IEEE Transaction on Automatic Control, vol 47, no 7, pp , 22 [2] D Gorinevsky, S Boyd, and G Stein, Design of low-bandwidth spatially distributed feedback, IEEE Trans on Automatic Control, vol 53, no 1, pp ,