Subspace Identification for Nonlinear Systems That are Linear in Unmeasured States 1

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1 Subspace Identification for Nonlinear Systems hat are Linear in nmeasured States 1 Seth L Lacy and Dennis S Bernstein 2 Abstract In this paper we apply subspace methods to the identification of a class of multi-input multi-output discrete-time nonlinear time-varying systems Specifically, we study the identification of systems that are nonlinear in measured data and linear in unmeasured states We present numerical simulations to demonstrate the efficacy of the method 1 Introduction System identification is the process of estimating system dynamics using measured data hese identified models can then be used for control and estimator design, model validation, system analysis, and output prediction Linear system identification has been well studied [1, 3, 8 11, 13 15, and nonlinear system identification has received increasing attention [5, 7, 16 18, 2 Subspace methods have been applied to linear systems with great success [1, 1, 15, 19 hree of the most developed and widely used algorithms, CVA, N4SID, and MOESP have been implemented in Matlab packages [2, 6, 12 hese methods are computationally simple and naturally applied to MIMO systems Recently, consistency of one subspace approach under modest conditions has been shown [4 Nonlinear and time-varying subspace identification methods have been devised in [17, 18, 2 In this paper we study systems that are nonlinear in measured data and linear in unmeasured states, see Figure 1 Our u z N y L y Figure 1: Block diagram of a system linear in unmeasured states: N is a nonlinear function of current and past data, L is a linear dynamic system method consists of three steps First, we approximate the nonlinearities in the system as finite sums of known basis functions with unknown coefficients hen we estimate the state sequence Finally we use the state sequence to estimate the unknown system coefficients Our method allows for the incorporation of prior knowledge through the selection of the basis functions his paper is organized as follows In Section 2 we define the problem and the notation to be used throughout the paper In Section 3 we derive the main results of the paper in the zero noise case In Section 4 we estimate the system order and the state sequence in the 1 Supported in part by NASA under GSRP NG and AFOSR under grant F {sethlacy,dsbaero}@umichedu, Aerospace Engineering Department, niversity of Michigan, Ann Arbor MI hp6-7 presence of noise In Section 5 we use the state estimate to calculate the system coefficients In Section 6 we summarize the identification procedure Finally, in Section 7 we demonstrate the method with numerical examples 2 Nonlinear Subspace Identification Here we study systems of the form xk +1=Axk+wk +F k, uk,,uk b,yk,,yk b, 21 yk =Cxk+Ewk+vk +Gk, uk,,uk b,yk 1,,yk b 22 his structure models multi-input, multi-output systems that depend linearly on the unmeasured states, but nonlinearly on the measured quantities k Z +, u, and y For convenience we rewrite 21, 22 as xk +1=Axk+wk + F k, uk b : k,yk b : k, 23 yk =Cxk+Ewk+vk + Gk, uk b : k,yk b : k 1, 24 where we use the convention ai : j = [ ai aj, i j, 25 and b is the number of delays to consider in the model, x R n, A R n n, F : R R m b+1 R p b+1 R n, uk R m, yk R p, C R p n, G : R R m b+1 R p b R p In addition wk R n and vk R p represent state and output noise, respectively he presence of E in 24 is used to model correlated state and measurement noise he model 23, 24 includes classical model structures as special cases For example, to capture a Hammerstein system, in which the nonlinearities are functions of the input, we write 23 and 24 as xk +1=Axk+F uk + wk 26 yk =Cxk+Guk + Ewk+vk 27 For a nonlinear-feedback system, in which the input is a nonlinear function of the output, we write 23 and 24 as xk + 1 = Axk + Buk+ F yk + wk 28 yk =Cxk+Duk+Ewk+vk 29 We assume F and G can be exactly represented as linear combinations of a finite set of known basis functions f i : R R m b+1 R p b+1 R, g i : R R m b+1 R p b R, and h i : R R m b+1 R p b R with unknown matrix coefficients B 1 R n r, D 1 R p s, B 2 R n t, and D 2 R p t sing this notation we can write F k, uk b : k,yk b : k = B 1 fk+b 2 hk, 21 Gk, uk b : k,yk b : k 1 = D 1 gk+d 2 hk, 211 where fk R r, gk R s, and hk R t are given by

2 fk = gk = hk = f 1 k, uk b : k,yk b : k f r k, uk b : k,yk b : k g 1 k, uk b : k,yk b : k 1 g s k, uk b : k,yk b : k 1 h 1 k, uk b : k,yk b : k 1 h t k, uk b : k,yk b : k 1, 212, Note that g and h are functions of delayed outputs only, while f is a function of both delayed and current outputs If the functions F and G cannot be exactly represented as a finite sum of basis functions, the expansions 21 and 211 can be regarded as approximations he basis functions in f, g, and h are sorted according to whether they appear in the expansion of F, G, or both Specifically, fk is a list of the basis functions that appear only in the expansion of F ; gk is a list of the basis functions that appear only in the expansion of G; and hk is a list of the basis functions that appear in the expansions of both F and G Without this convention, Z q k defined below would not have full row rank With the notation 21, 211 we can rewrite 23 and 24 as xk +1=Axk+B 1 fk+b 2 hk+wk, 215 yk =Cxk+D 1 gk+d 2 hk+ewk+vk 216 sing 215 and 216, we construct the blockmatrix equation Y q k =Γ q xk : k+l 2q+Λ q Z q k+υ q N q k, 217 where q is a user-defined window length denoting the number of block rows in 217, and l is a second userdefined window length which denotes the number of columns 2q + 1 in 217 l is often used to denote the number of measurements of u and y available For notational convenience, we define σ = r + s + t Next, we define the block-hankel matrix Y q k R pq l 2q+1 as Y q k = yk : k + l 2q yk + q 1:k + l q 1, the extended observability matrix Γ R pq n as C CA Γ q =, CA q 1 the block-oeplitz matrix Λ q R pq qσ as D 1 D 2 CB 1 CB 2 Λ q = CAB 1 CAB 2 CA q 2 B 1 CA q 2 B 2 D 1 D 2 CB 1 CB 2, CA q 3 B 1 CA q 3 B 2 D 2 the block-hankel matrix Z q k R qσ l 2q+1 as zk : k + l 2q Z q k =, zk + q 1:k + l q 1 the column vector zk R σ as fk zk = gk, hk the block-oeplitz matrix Υ q R pq qn+p as E I C E I Υ q = CA C E I, CA q 2 CA q 3 CA q 4 I and the block-hankel matrix N q k R qn+p l 2q+1 as wk : k + l 2q vk : k + l 2q N q k = wk + q 1:k + l q 1 vk + q 1:k + l q 1 Finally, we define the data matrix q k R qp+σ l 2q+1 as q k = Yq k 224 Z q k 3 State Reconstruction In this section, we give conditions under which the state sequence can be exactly reconstructed from noise free data For V R n m let RV denote the range column space of V hen RV is the row space of V Let V L = V V 1 V and V R = V VV 1 denote left and right inverses of V, respectively We also define the projection Π V = V R V = V VV 1 V and Π V = I Π V Note that V Π V = V and V Π V = heorem 1 Assume the following conditions are satisfied: i rank Γ q = n ii wk and [ vk are zero for all k Z + Zq k iii rank =2qσ for all k Z Z q k + q + iv rankxk : k +l 2qΠ Z qk = rank xk : k +l 2q for all k Z

3 v rank xk : k + l 2q =n for all k Z + hen q n/p, 31 l 2qσ +1+n 1, 32 n = rank 2qσ for all k Z +, 33 Rxk +q : k +l q =R q k R q k +q for all k Z + 34 Proof 31 follows directly from i o prove 34 we first show that 34 holds with = replaced with hen we show that the left and right hand sides of 34 have the same dimension Let k Z + First note that ii implies that 217 has the form Y q k =Γ q xk : k + l 2q+Λ q Z q k 35 and Y q k + q =Γ q xk + q : k + l q+λ q Z q k + q 36 By i we can solve 35 and 36 for the state matrices xk : k + l 2q = Γ L q Γ L q Λ q, 37 xk + q : k + l q = Γ L q Γ L q Λ q q k + q 38 sing 215 and 37 the state matrix xk+q : k+l q can also be written as xk + q:k + l q= A q xk : k + l 2q+Ψ q Z q k = A q Γ L q Γ L q Λ q +Ψ q Z q k = A q Γ L q Ψ q A q Γ L q Λ q, where Ψ q R n qσ is given by 39 [ Ψ q = A q 1 B 1 B 2 A q 2 B 1 B 2 B1 B and 39 imply Rxk + q : k + l q R q k R 311 Next we show that the left and right hand sides of 34 have the same dimension By iii, Π Z exists We multiply 35 on the right by Π Z qk qk and use Z q kπ Z qk = to obtain Y q kπ Z =Γ qk qxk : k + l 2qΠ Z 312 qk sing i, iv, and v yields ranky q kπ Z = rankγ qk qxk : k + l 2qΠ Z qk = rankxk : k + l 2qΠ Z qk = rankxk : k + l 2q = n 313 sing iii we can calculate [ Yq k Yq kπ Z rank q k = rank = rank qk Z q k Z q k = rankz q k + ranky q kπ Z qk =qσ + n 314 Since 314 hold at time k + q rank = qσ + n 315 Similarly, rank 2q k = 2qσ + n, 316 Zq k Yq k Writing Z 2q k = and Y Z q k + q 2q k = Y q k + q we see rank = rank 2q k=2qσ + n, 317 proving 33 Since 2q k R 2qp+σ l 2q+1 it follows that 32 We can now calculate the dimension of the right hand side of 34, dimr q k R = rank q k + rank rank = n follows from v Now we calculate the intersection of the row spaces of q k and and thus a representation of the state matrix xk + q : k + l q Proposition 2 Assume i - v of heorem 1 Let M, L 11, L 22, L 11 L 12 L 1 22 L 21, and L 22 L 21 L 1 11 L 12 be nonsingular, where M R l 2q+1 l 2q+1, L 11 R qp+σ qp+σ, L 12 R qp+σ qp+σ, L 21 R qp+σ qp+σ, and L 22 R qp+σ qp+σ Consider the singular value decomposition 11 L M = 12 S11 V, 319 [ L11 L where L = 12 R L 21 L 2qp+σ 2qp+σ, 22 S 11 R 2qσ+n 2qσ+n, 11 R qp+σ 2qσ+n, 12 R qp+σ 2pq n, 21 R qp+σ 2qσ+n, 22 R qp+σ 2pq n, and V R l 2q+1 l 2q+1 Also let r R 2pq n n, S r R n n, and V r R 2qσ+n n be defined through the singular value decomposition 12 L L 21 L11 L 12 L 1 22 L S L = [ S r r V r s 11 L 12 Vs = r S r V r 32 hen there exists nonsingular R n n such that xk + q : k + l q =r 12 L L 21 = [ S r Vr V M 1 = r 12 L L 22 q k + q 321 Proof Rewriting the singular value decomposition 319 as L11 L 12 S M = V, L 21 L 22 [ L 11 q k+l L 11 q k+l L 21 q k+l = L 21 q k+l 22 S11 V M 1, 323 we see that 12 L L 21 = 12L L 22 q k + q has 2pq n rows, where 2pq n n by heorem 1 In heorem 1 we proved that only n of these rows are linearly independent We now select n linear combinations of these rows to find a minimal set of vectors to span the intersection of the row spaces of q k and q k+q and thus find a representation of the state matrix xk + q : k + l q Rewriting 319 we have 352

4 = L S11 V M 1 [ q k + q L11 L = 12 L 1 22 L L 21 L11 L 12 L 1 22 L L 11 L 1 [11 12 S L22 L 21 L 1 11 L 11 1 V M S 11 L11 L 12 L 1 22 L S 11 = 11 L 11 L S L 21 L11 L 12 L 1 22 L S 11 + L 22 L 21 L 1 11 L S 11 hen V M L L 21 = 12L L [ 21 L11 L 12 L 1 22 L S L 11 L S 11 V M 1 12 L L 21 L11 L 12 L 1 = 22 L L 11 L S 11 V M 1 = [ r S r V r V M 1 = r [ Sr V r V M hus r 12 L L 21 = [ S r Vr V M 1 = r 12 L L 22 q k + q 328 is a basis for the row space of 12L L 21 = 12L L 22 q k + q, and a realization of the state matrix xk + q : k + l q 4 Noise Effects Since wk and vk are present in real systems, our goal is to apply the above results without ii Referring to 33 and 321, we define the following approximations ˆn = rank L M 2qσ, 41 ˆxk + q : k + l q = r 12 L L 21 = r 12 L L 22 q k + q, 42 for n and xk + q : k + l q, where we have arbitrarily set = I in 321 to obtain 42 can be used to select the basis for the realization of the state sequence he problem of rank determination is central to estimating the order of the unknown system 41 he presence [ of wk and vk will generally add rank to L M For computational purposes, we use [ the following technique to estimate the rank of L M Define the singular value decomposition L M = SV 43 Let sk be defined as sk = Sk,k 2qσ k<min2qp + σ,l 2q +1 Sk+1,k+1, and Sk +1,k+1 else hen numrank L M = k, 45 where k is given by sk = max sk 46 In 41 we know that L M has rank 2qσ, so we define sk only for k 2qσ 5 Coefficient Estimation Now that we have an estimate ˆxk + q : k + l q R n l 2q+1 of the state sequence xk + q : k + l q we proceed to estimate A, B 1, B 2, C, D 1, and D 2 as well as the covariance matrices of the noise sequences Q = E[ww and R = E[vv, and the correlation matrix E Consider the cost function JÂ, ˆB 1, ˆB 2, Ĉ, ˆD 1, ˆD 2 = [ˆxk + q +1:k + l q yk + q : k + l q 1 [ˆxk + q : k + l q zk + q : k + l q 1 [Â ˆB1 ˆB2 Ĉ ˆD1 ˆD2 F = J 1 Â, ˆB 1, ˆB 2 +J 2 Ĉ, ˆD 1, ˆD 2 where J 1 Â, ˆB 1, ˆB 2 = ˆxk + q +1:k + l q [ Â ˆB 1 ˆB2 R1 k + q : k + l q 1 F, J 2 Ĉ, ˆD 1, ˆD 2 = yk + q : k + l q 1 [ Ĉ ˆD 1 ˆD2 R2 k + q : k + l q 1 F, and F is the Frobenius matrix normand ˆxk ˆxk R 1 k = fk, R 2 k = gk hk hk Proposition 3 he matrices [Â ˆB1 ˆB2 =ˆxk + q +1:k + l q [Ĉ ˆD1 R 1 k + q : k + l q 1 R, 52 = ˆD2 yk + q : k + l q 1 R 2 k + q : k + l q 1 R, 53 minimize the cost functions J 1 and J 2 Proposition 4 E and the noise covariance matrices can be estimated as ˆQ = Σ 11, 54 where Ê = Σ 21 Σ 1 11, 55 ˆR = Σ 22 Σ 21 Σ 1 11 Σ 21, 56 Σ 11 =ˆxk + q +1:k + l q Π 1k+q:k+l q 1ˆxk R + q +1:k + l q, 57 Σ 21 = yk + q : k + l q 1 I Π R1k+q:k+l q 1 Π R2k+q:k+l q 1 +Π R1k+q:k+l q 1Π R2k+q:k+l q 1 ˆxk + q +1:k + l q, 58 Σ 22 = yk + q : k + l q 1 Π R yk + q : k + l q 2k+q:k+l q he Algorithm Here we list the steps in the nonlinear subspace identification algorithm 1 Collect input-output data Choose the window lengths q and l l must be less than or equal to the number of data pairs available 3521

5 1 Van der Pol Oscillator Validation Error = 2 4 Phase Portrait of the Van der Pol Oscillator Validation Data uk x1k x2k x2k Figure 2: ime races of the Van der Pol Oscillator with 4 basis functions, vertical line indicates end of ID data set and beginning of validation data set, solid line indicates signal from true continuous time system, dashed line indicates signal from estimated discrete time system 2 Select the time frame b and basis functions f i, g i, and h i to model the system 3 Construct q k and as in Select weighting matrices L and M 5 Calculate[ the primary singular value decomposition of L M in 319 and obtain ˆn in 41 as well as 11, 12, 21, 22, and S 11 6 Calculate the second singular value decomposition in 32 and obtain r 7 Estimate the state matrix in 42 8 Estimate the system matrices Â, ˆB1, ˆB2, Ĉ, ˆD1, and ˆD 2 in 52 and 53 Estimate Ê and the noise covariance matrices ˆQ and ˆR in 54, 55, and 56 7 Examples Here we apply the nonlinear subspace identification algorithm to several nonlinear systems We define the validation error as e = y1 : l ŷ1 : l F 71 y1 : l F 71 Van der Pol Oscillator Here we consider the system ẋ 1 = x 2 72 ẋ 2 = ω 2 x 1 + ɛω 1 µ 2 x 2 1 x2 + u 73 with ω = ɛ = µ = 1 We excite this continuous time system with a zero-order-held sequence of l = 1 inputs with time interval τ =5s, and measure l outputs x 1 and x 2 with sampling rate 1/τ = 2Hz We choose all polynomials up to third order in x 1, x 2, and u as our basis functions We then estimate a discrete time system 215, 216 o validate, we continue the input sequence and measure outputs of both the true continuous time system and the estimated discrete time system and compare the results See Figure 2 and Figure 3 k x 1 k Figure 3: x 1 vs x 2 Validation Signals for the Van der Pol Oscillator with 2 basis functions, solid line indicates signal from true continuous time system, dashed line indicates signal from estimated discrete time system 72 Planar Articulated Spacecraft We model the planar motion of two flexible bodies linked by a hinge For simplicity, we model only the first flexure mode of each link he equations of motion are sin θ θ2 1 cos θ λ 2 θ2 2 θ 1 = λ 1 λ 2 + cos 2 + 2δ 1 ζ1 +, 74 θ sin θ θ2 2 cos θ λ 1 θ2 1 θ 2 = λ 1 λ 2 + cos 2 + 2δ 2 ζ2, 75 θ = ζ 1 +2c 1 ω 1 ζ1 + ω1ζ δ 1 θ1, 76 = ζ 2 +2c 2 ω 2 ζ2 + ω2ζ δ 2 θ2, 77 where λ 1 = J 1 /η, λ 2 = J 2 /η, η = d 1 d 2 m, J 1 = I 1 +md 2 1, J 2 = I 2 + md 2 2, m = m1m2 m 1+m 2 is the reduced mass, d 1 and d 2 are the distances from the hinge point to the center of mass of each body, m 1 and m 2 are the masses of the two bodies, θ = θ 1 θ 2 is the angle between the two bodies, θ 1 and θ 2 are the angular positions of the two bodies with respect to an inertial frame, ζ 1 and ζ 2 are the fundamental flexible modes of each body, ω 1 and ω 2 are the modal frequencies, δ 1 and δ 2 are the coupling coefficients, and is the control torque applied between the two bodies Since λ 1 λ 2 > 1 these equations do not have a singularity We measure l = 2 data points with sampling rate 1/τ = 2Hz We take measurements of θ 1, θ 2, θ 1, and θ 2, and set uk 1 θ 1k 1 2 sinθ 1 k 1 θ 2 k 1 zk=hk= θ 2k 1 2 sinθ 1 k 1 θ 2 k 1 θ 1k 1 2 sin 2θ 1 k 1 θ 2 k 1, 78 θ 2k 1 2 sin 2θ 1 k 1 θ 2 k 1 a function of delayed data, with the nonlinearities periodic in θ We obtain a fourth order discrete time model and plot the identification and validation data in Figure 4

6 θ1k θ1k θ2k θ2k uk Planar Articulated Spacecraft Error = 25 k Figure 4: ime race of Measured Variables and Estimates, vertical line indicates end of ID data set and beginning of validation data set, solid line indicates signal from true continuous time system, dashed line indicates signal from estimated discrete time system 8 Conclusion We presented a subspace-based identification method for identifying nonlinear time-varying systems that are nonlinear in measured data and linear in unmeasured states We applied the algorithm to two numerical examples Future work will focus on experimental applications, extending the class of identifiable systems, and methods for basis function selection References [1 M Deistler, K Peternell, and W Scherrer Consistency and relative efficiency of subspace methods Automatica, 3112: , 1995 [2 H Haverkamp and M Verhaegen Subspace Model Identification oolbox Systems and Control engineering Group, Delft niversity of echnology [3 J-N Juang Applied System Identification Prentice Hall, 1993 [4 Knudsen Consistency analysis of subspace identification methods based on a linear regression approach Automatica, 37:81 89, 21 [5 S L Lacy, R S Erwin, and D S Bernstein Identification of wiener systems with known noninvertible nonlinearities In 21 American Control Conference, pages , Arlington VA, June 21 [6 W E Larimore ADAP X sers Manual Adaptics, Inc, McLean VA [7 W E Larimore Generalized canonical variate analysis of nonlinear systems In Proceedings of the 27th Conference on Decision and Control, pages , Austin, X, December 1988 IEEE [8 L Ljung System Identification: heory for the ser Prentice Hall Information and System Sciences Series Prentice Hall, 2nd edition, January 1999 [9 L Ljung and Mckelvey Subspace identification from closed loop data Signal Processing, 52:29 215, 1996 [1 M Moonen, B De Moor, L Vandenberghe, and J Vandewalle On- and off-line identification of linear state-space models International Journal of Control, 491: , 1989 [11 Soderstrom and P Stoica System Identification Prentice Hall, 1989 [12 Matlab System Identification oolbox Mathworks, Natick, MA [13 P Van Overschee and B De Moor N4SID: Subspace algorithms for the identification of combined deterministic-stochastic systems Automatica, 31:75 93, January 1994 [14 P Van Overschee and B De Moor A unifying theorem for three subspace system identification algorithms Automatica, 3112: , 1995 [15 P Van Overschee and B De Moor Subspace Identification for Linear Systems: heory, Implementation, Applications Kluwer, 1996 [16 Van Pelt and D S Bernstein Nonlinear system identification using hammerstein and nonlinear feedback models with piecewise linear static maps - part I: heory - part II: Numerical examples In Proceedings of the American Control Conference, pages , , Chicago, IL, June 2 [17 M Verhaegen and D Westwick Identifying MIMO Hammerstein systems in the context of subspace model identification methods Int J Control, 632: , 1996 [18 M Verhaegen and X Yu A class of subspace model identification algorithms to identify periodically and arbitrarily time-varying systems Automatica, 312:21 216, 1995 [19 M Viberg Subspace-based methods for the identification of linear time-invariant systems Automatica, 3112: , 1995 [2 D Westwick and M Verhaegen Identifying MIMO Wiener systems using subspace model identification methods Signal Processing, 52: ,