Identification of Quadratic in-the-state System Using Nonlinear Programming

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1 Identification of Quadratic in-the-state System Using Nonlinear Programming Wael SULEIMAN and André MONIN LAAS - CNRS, 7 avenue du Colonel Roche, 3177 oulouse Cedex 4 France Abstract he identification of the Quadratic in-the-state Systems QSS is the objective of this paper. his class, on its own, enjoys a useful model for many nonlinear dynamic systems. Moreover, it represents a bilinear system in a feedback loop. In order to identify QSS, we propose to minimize the output error with respect to the system s parameters using a local gradient search. In order to reduce the amount of gradient search to the minimal value, we propose an efficient implementation of the algorithm by projecting out the directions that do not change the output error function. Furthermore, we extend the proposed method to deal with multiple short experiments. u t Fig H Bilinear system C K x t x t v t + + Quadratic system as a bilinear system in a feedback loop. y t I. INRODUCION During the last years, the attention of researchers has been pointed to the identification of nonlinear systems. Such attention is the natural result of the intrinsic limits of the linear systems in many fields of sciences such as ecology, physiology, biology and socio-economics [1], [2], [3], [4]. In fact, the interest of linear systems is due to their simple analytical properties. However, dealing with complex systems which are multivariable and as well nonlinear has pushed the identification theory and the efforts of researchers are focused to overcome the analytical difficulties of nonlinear systems. Consider the following nonlinear finite dimensional system x t+1 = f x t,u t y t = hx t,u t + v t 1 u t R m is the input signals, y t R p is the output signals and the measurement noise v t is assumed to be a white-noise that is independent of the input signal and with zero mean. In the literature, we find that the bilinear systems is one of the classic models [5], [6], [7] which is proposed to approximate the nonlinear system 1. his class of systems is linear separately with respect to the state x t and the input signal u t, and can be characterized by the following model x t+1 = A b x t + B b u t + N b u t x t y t = C b x t + v t 2 denotes the Kronecker product. It is proven that this class of systems is dense, in L 2 sense, in the class of analytical nonlinear systems. For that, studying the properties and the identification of bilinear systems has been subject of active research during the last decade [8], [9], [1]. In this paper, we will consider the identification of bilinear system in a feedback loop which is presented in Fig. 1. Using the property FG HJ = F HG J, the representation of the equivalent model is given by x t+1 = Ax t + Bu t + N u t x t + Qx t x t y t = Cx t + v t 3 A = A b B b K B = B b H N = N b H I n Q = N b K I n he class of systems given by the model 3 is called Quadratic in-the-state Systems QSS. In fact, his class enjoys a useful model more than a representation of a bilinear system in a feedback loop. For example, in ecology and biology the population dynamics of two interacting species may be described by the praypredator Volterra equation [11], [12], which has the exactly structure of QSS. In the engineering area, there are also several examples of QSS, e.g. the dynamics of a general AC machine [13]. Note that, the realization 3 is not unique. Assume the following transformation of the state z t = x t 5 R n n is a nonsingular matrix. hen, we obtain the following structure 4 z t+1 = Āz t + Bu t + N u t z t + Qz t z t y t = Cz t + v t 6 Ā = A B = B N = N I m Q = Q C = C As a consequence, the relation input/output does not change by transforming the state x t according to 5. 7

2 II. IDENIFICAION PROCEDURE he identification of Structure 3 relies upon the estimation of the matrices A,B,C,N,Q. Without any assumption on the structure of these matrices, we assume that all matrices are fully parameterized. Let us define veca vecb vecn vecq vecc vec. denotes the vectorization operator defined as follows vec : M R m n R m n vecm = vec [ ] [ m 1 m 2 m n = m 1 m ] 2 m n Suppose that we have an estimation ˆθ of θ, and a set of input/output data from the real system {u t,y t : t =,1,,L}. he estimated output ŷ t ˆθ can be given as follows 8 ˆx t+1 = A ˆθ ˆx t + B ˆθu t + N ˆθu t ˆx t + Q ˆθ ˆx t ˆx t ŷ t ˆ C ˆθ ˆx t 9 our goal is achieved if ŷ t ˆθ approximates the output y t accurately enough. his criterion can be transformed into the minimization of the output error with respect to the parameter θ which leads to a output-error identification problem. Considering the following output-error cost function J L 1 L L k=1 y k ŷ k θ 2 2 = 1 L E Lθ E L θ 1 E L [ e1 e2 el ] 11 is the error vector in which ek = y k ŷ k θ. he minimization of 1 is a nonlinear, nonconvex optimization problem. he numerical solution of this problem can be calculated by different algorithms. In the literature of output-error identification methods, we find that the gradient search method is the popular adopted method. his iterative method is based on the updating of the system parameters θ as follows A. Computing the iterative parameter update In order to compute the update rule 12, the following quantities E L θ and ψ L θ must be computed. Computing the vector E L θ can be done by simulating the system 9 that corresponds to θ i. Note that this simulation brings out the state ˆx t and ŷ t. In order to simulate ψ L θ i, we should compute the derivative of ŷ t with respect to θ i. Let us define ζ j t = ˆx t 14 θ j is the [ j th element of the vector θ. he computation of ŷ t ŷt ŷ θ t 1 θ ], l is the number of l parameters in θ, can be made using the following model ζ j t+1 =Aζ t j + A ˆx t + B u t + Nu t ζt j + N u t ˆx t + Q ζt j ˆx t + Q ˆx t ζt j + Q ˆx t ˆx t ŷ t =Cζt j + C ˆx t 15 III. LOCAL PARAMEERIZAION As we mentioned the structure 3 is not unique, as consequence the minimization of J L θ does not have a unique solution. Indeed, the optimal solution can be made unique by choosing a suitable parameterization. Such parameterization for QSS does not exist up to our knowledge. However, a local parameterization is proposed for the bilinear systems [14], the directions that do not change the cost function J L θ are identified and projected out at each iteration, for that only the active parameters are updated. Analogously to previous method, we will develop a local parameterization of QSS. Recall that two realizations of QSS in the state space representation are similar if their coefficient matrices are related by Equation 7, the transformation matrix parameterizes the subset of equivalent models. Note that, this subset defines a manifold. In order to identify the tangent plane of the manifold, we linearize the relation 7 around the identity matrix = I n. Considering a small perturbation = I n +, by using the approximation I n + I n and neglecting all second-order terms, we obtain θ i+1 = θ i ψ L θ i ψ L θ i + λ i+1 I ψ L θ i E L θ i 12 Where λ i+1 is the regularization parameter and ψ L θ E Lθ θ 13 is the Jacobian of the error vector E L θ. Ā = A A + A B = B B N = N N + N I m Q = Q Q + QI n + Q I n C = C +C 16 If we consider the following model s vectors of parameters

3 hen θ + By defining veca vecb vecn vecq vecc and vecā vec B vec N vec Q vec C vec A + A vec B vec N + N I m vec Q + QI n + Q I n vecc Π i,k [ n in I n n K in ] it is possible to write I m as follows I m = m Π i,m Π i,m 2 On the other hand, I n can be written as follows I n = n n j=1 Π i,nπ j,n i, j 21 i, j is the element of the row i and column j of. Using the property vecabc = C A vecb, we obtain n n vecq I n = Π j,n Q vec Π i,n i, j j=1 =Γ Q vec 22 Γ Q = [γ 1,1 γ 2,1 γ n,1 γ 1,2 γ 2,2 γ n,2 γ 1,n γ n,n ] and γ i, j = Π j,n Q vec Π i,n 23 Using analogous logic, the relation between θ and θ becomes M θ + M θ vec 24 A I n + I n A B I n N I n + m Π i,m NΠ i,m Q I n + n Π i,n QΠ i,n + Γ Q I n C 25 Lemma 1: he left null space of M θ 25 contains the directions in which the parameters should be modified to lead a change in the value of cost function J N θ 1. Proof: he equation 24 shows that the tangent space of the manifold of all systems similar to A,B,N,Q,C is equal to the column space of the matrix M θ 25. Since the left null space of the matrix M θ is orthogonal complement to the column space, the directions in which the value of the cost function 1 changes are those related to the left null space of M θ. Note that, the left null space of M θ can be efficiently obtained by a QR decomposition M [ Q 1 Q 2 ] [ R ] 26 hen a basis of the left null space of M θ is Q 2. he new update rule becomes θ i = θ i Q 2 Q 2 ψ L ψ L Q 2 + λ i I Q 2 ψ L E L 27 Q 2 and ψ L depend on θ i. Note that since Q 2 depends on the past parameter θ i the QR 26 must be computed at each iteration. A. Summary of the implementation algorithm We can resume the algorithm of implementation as follows 1 Calculate the state ˆx t and ŷ t by simulating the system 9 with θ i. 2 Compute E L θ using Calculate the matrix M θ using Calculate the QR decomposition of M θ 26, from which we obtain Q 2. 5 Calculate the matrix ψ L using 15, we suppose that {ζ j =, j = 1,2,,l}. 6 Calculate the update rule of the gradient search algorithm using Perform the termination test for minimization. If true, the algorithm stops. Otherwise, return to Step 1. i.e. compute the values J L θ i and J L θ i using 1 and test if J L θ i J L θ i 2 is small enough. IV. DEALING WIH MULIPLE DAA SES In the purpose of identifying practical applications, we apply various sets of input. he objective of each set is exciting one or more modes of the system. In real experiments, sometimes we obtain short sets of input/output data due to a large time sampling period or short tracked phenomena. In these cases, we should exploit all the data sets which we have to obtain an accurate model of the system. Moreover, in the case of short experiments the effect of initial conditions can not be neglected, so we should estimate them. Assume that we have the sets { u i t,y i t : t = 1,2,,L i and i = 1,2,,q }. If we consider the initial conditions, the vector of

4 parameters θ becomes ˆx j t+1 = A ˆθ ˆx j veca vecb vecn vecq vecc x 1 x 2. x q t + B ˆθu t j + N ˆθ + Q ˆθ ˆx t j ˆx t j 28 x j is the initial value of the state corresponded to j th data set. he estimated output ŷt j ˆθ of the data set number j is given by the following model ut j ˆx t j ŷt j ˆ C ˆθ ˆx t j 29 he output error function for all data sets can be done as follows J q 1 q q L 1 j j=1 L j k=1 y j k ŷ j k θ 2 = 1 2 q E qθ E q θ E q [E ] 1 L1 θ E 2 L2 θ E q Lq θ and E i L i 1 Li [ e i 1 e i 2 e i L i ] is the error vector in which e i k = y i k ŷi k θ. he minimization of 3 can be calculated by using the gradient search method as follows θ i+1 = θ i ψ q θ i ψ q θ i + λ i I ψ q θ i E q θ i 33 and ψ q A. Local parameterization ψ 1 L 1 θ ψ 2 L 2 θ. ψ q L q θ 34 ψ i L i θ Ei L i θ θ 35 Similar to the case of a single experiment, we will look for a local parameterization in order to define the direction in which the cost function 3 is invariant. Recall that, two realization of QSS are similar and their vectors of parameters are θ and θ respectively. veca vecā vecb vec B vecn vec N vecq vec Q vecc x 1 and vec C x 2 z 1 36 z 2.. x q z q By considering a small perturbation = I n + and analogously to the case of a single experiment we obtain M θ + M θ vec 37 A I n + I n A B I n N I n + m Π i,m NΠ i,m Q I n + n Π i,n QΠ i,n x 1 In. x q In + Γ Q 38 then, we calculate the QR decomposition of M θ similar to 26, from which we obtain Q 2. he new update rule becomes θ k = θ k Q 2 Q 2 ψq ψ q Q 2 + λ I k Q 2 ψq E q 39 B. Computing the iterative parameter update In order to compute the update rule 27, the following quantities E q θ and ψ q θ must be computed. For that, we simulate the systems 29 that corresponds to θ k. Note that this simulation brings out the state ˆx i t and ŷ i t for i = 1,2,,q. In order to simulate ψ q θ i, we should compute the derivative of { ŷ i t : i = 1,2,,q } with respect to θ k. Let us define ζ i, j t = ˆxi t 4 θ j is the[ j th element of the] vector θ. he computation of ŷi t ŷ i t ŷt θ i can be made using the 1 θ l following model ζ i, j t+1 ŷ i t i, j =Aζt + A + Q ˆx t i + B u t + Nu t ζt i, j ˆx t i + Q ˆx t i ζt i, j ζ i, j t + N u t ˆx i t + Q ˆx i t ˆx i t =Cζt i, j + C ˆx t i 41 Note that, as the initial condition is included into the vector of parameters then ζ i, j = ˆxi.

5 V. COMPUING AN INIIAL ESIMAION he complexity of the iterative gradient search algorithm and the stability of the obtained structure depends on the choice of an initial estimation of the vector of parameters θ. One could first estimate a stable bilinear model, then use the matrices A,B,N,C of the model as initial guesses, and the matrix Q can be initialed as zero matrix. In the case of multiple experiments, we use a single data set to calculate an initial estimation of the vector of parameters θ. hen by using this model we can estimate the initial conditions of each data set. VI. ILLUSRAIVE EXAMPLES In this section, first we identify a QSS using a single experiment. Second we point out that the proposed algorithm to deal with multiple short data sets provides a model more precisely than treating each experiment separately. We define the Model accuracy as the Percent Variance Accounted For %VAF %VAF 1 L t=1 y t ŷ t y t ŷ t t=1 L y t ȳ t 1 y t ȳ t ŷ t denotes the estimated output signal and ȳ t is the mean value of y t. A. Identifying QSS using single experiment In this case, we consider that the length of the input and output signals is L = 12 samples, and we have used id = L 2 samples for the identification purpose, and the rest of them val = L id = L 2 samples for the validation purpose. we consider the following example A =, B =, C = N = Q = he input is two-dimensional uniform white noise notation and the standard deviation of independent noises v t is equal to.1. he initial estimated model is a bilinear ABLE I ACCURACY OF HE INIIAL ESIMAED BILINEAR SYSEM AND OPIMIZED QSS Identification task Accuracy %VAF Bilinear system Optimized QQS First output Second output Validation task First output Second output system which is obtained by using the Matlab M package CUEDSID [15]. We consider the output signal without the measurement noise when we calculate the accuracy of the identified models in validation task. he accuracies reported in able I show that the proposed method has efficiently identified the QSS structure. o deal with a clear graphical representation, we represent the data over a window of 15 samples in Fig. 2. B. Identifying QSS using multiple experiments In this case, we consider that we have executed 1 experiments q = 1 with random initial conditions. { hat means we have obtained the sets u i t,yt,x i i : t = 1,2,,L i and i = 1,2,,q } and we suppose that L 1 = L 2 = L 3 = L 4 = L 5 = 2 and L 6 = L 7 = L 8 = L 9 = L 1 = 1. he input for each experiment is an uniform white noise and the standard deviation of independent noises v t is equal to.1. we consider the same example 42 of the single experiment case. In order to show that the obtained model by considering all data sets is more accurate than that obtained by using a single experiment, we have proceeded as follows. First, we have used a single experiment the first one to identify the model of the system and without considering the estimation of initial conditions. Second, we have used q 1 = 9 data sets to identify the model of system and the initial conditions. ABLE II ACCURACY OF HE IDENIFIED QSS USING A SINGLE DAA SE AND HE IDENIFIED QSS USING MULIPLE DAA SES Identification task Accuracy %VAF Using single using mutiple data set data sets First output Second output Validation task First output Second output he validation of the two models has been done on the rest one of data sets Fig. 3. Note that, we suppose that the initial conditions of the data set used for validation are known. It is clear from able II that in this case of short experiments, using multiple experiments leads to a more accurate model. VII. CONCLUSION In this paper, we have proposed a method to identify the Quadratic in-the-state Systems QSS. he method is based on a local parameterization of the state-space representation of the QSS and subsequent gradient search in the resulting local parameter space. A bilinear model is used to initialize the optimization task, in the reported examples this procedure seems to work properly. he algorithm has successfully applied to identify the QSS in the case of single experiment as well as the case of multiple short experiments. his case is infrequently treated in the literature of identification theory.

6 y error y error Fig. 2. he real outputs without noise solid line with the outputs of the identified QSS dashed line and bilinear system dash-dotted line and their errors are superimposed y error y2.4.2 error Fig. 3. he real outputs without noise solid line with the outputs of the identified QSS using all data sets dashed line and the identified QSS using a single data set dash-dotted line and their errors are superimposed. REFERENCES [1]. Söderström and P. Stoica, System identification. New York: Prentice Hall, [2] S. Sastry, Nonlinear Systems: Analysis, Stability and Control. Springer, [3] M. Korenberg and I. Hunter, he identification of nonlinear biological systems: Wiener kernel approaches. Ann Biomed Eng., vol. 18, no. 6, pp , 199. [4] V. Schuppen and H. Jan, System theory for system identification, J. Econometrics, vol. 118, no. 1-2, pp , 24. [5] W. J. Rugh, Nonlinear System heory: he Volterra/Wiener Approach. he Johns Hopkins, [6] R. Mohler, Nonlinear system, II: Applications to bilinear control. Prentice Hall, [7] A. Isidori, Nonlinear Control Systems. Springer-Verlag, [8] A. Monin and G. Salut, I.I.R. Volterra filtering with application to bilinear systems, IEEE rans. Signal Processing, vol. 44, no. 9, pp , September [9] H. Chen and J. Maciejowski, Subspace identification method for deterministic bilinear systems, IFAC System Identification Conf., June 2. [1], An improved subspace identification method for bilinear systems, in 39th IEEE Conf. Decision Control, Sydney, Australia, December 2, pp [11] V. Volterra, Leçons sur la héorie Mathématique de la lute pour la Vie. Paris: Gauthier-Villars, [12] F. E. hau, On the feedback control of nonlinear population dynamics, IEEE rans. Syst. Man, Cybern. Corresp, vol. SMC- 2, pp , July [13] M. M. A. De Carli and A. Ruberti, Speed control of induction motors by frequency variations, in Proc. 3rd IFAC Congrees, London, [14] N. B. Vincent Verdult and M. Verhaegen, Identification of fully parameterized linear and nonlinear state-space systems by projected gradient serach, in Proceedings of the 13th IFAC Symposium on System Identification, he Netherlands, August 23. [15] H. Chen and J. Maciejowski, CUEDSID oolbox User s Guide, Cambridge University System Identification oolbox, Cambridge University Engineering Dept, May 2.

On Identification of Cascade Systems 1

On Identification of Cascade Systems 1 Bo Wahlberg Håkan Hjalmarsson Jonas Mårtensson Automatic Control and ACCESS, School of Electrical Engineering, KTH, SE-100 44 Stockholm, Sweden. (bo.wahlberg@ee.kth.se