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Gradient based estimation algorithm for Hammerstein systems with saturation and dead-zone nonlinearities
Applied Mathematical Modelling 36 (2012) 238–243
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Gradient based estimation algorithm for Hammerstein systems with saturation and dead-zone nonlinearities q Jing Chen a,b,⇑, Xiuping Wang a, Ruifeng Ding b,⇑ a b
Wuxi Professional College of Science and Technology, Wuxi 214028, PR China School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, PR China
a r t i c l e
i n f o
Article history: Received 8 March 2011 Received in revised form 20 May 2011 Accepted 22 May 2011 Available online 6 June 2011 Keywords: Iterative method System identification Parameter estimation Hammerstein model
a b s t r a c t This paper focuses on identification problems for Hammerstein systems with saturation and dead-zone nonlinearities. An appropriate switching function is introduced to derive an identification model with fewer parameters and all the unknown parameters can be estimated by using an iterative method. A numerical simulation is carried out to show the effectiveness of the proposed method. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Identification of Hammerstein nonlinear systems (linear time-invariant (LTI) block following a static nonlinear block) has been active in theory and application. In general, existing identification approaches for Hammerstein nonlinear systems can be roughly divided into two categories: the recursive and the iterative algorithms [1–6]. The recursive algorithms are often used in on-line estimation and the iterative algorithms are often used in off-line estimation. The basic idea of the iterative method is to adopt the iterative estimation theory: the parameter estimates relying on the unknown variables are computed by using the estimates of these unknown variables which are obtained from the preceding parameter estimates [7–11]. The iterative algorithms are usually utilized in the identification of Hammerstein systems with hard input nonlinearities [3,4,12–15]. Recently, Ding et al. presented a series of identification methods for Hammerstein nonlinear systems [11,16–18]; Chen et al. proposed an auxiliary model based multi-innovation algorithms for multivariable nonlinear systems [19]; Wang et al. derived an extended stochastic gradient identification algorithm for Hammerstein–Wiener ARMAX systems [20], an auxiliary model-based recursive generalized least squares parameter estimation for Hammerstein OEAR systems [21] and auxiliary model-based RELS and MI-ELS algorithms for Hammerstein OEMA systems [22]. Hard input nonlinearities are common in engineering practice. Identification of Hammerstein systems with hard input nonlinearities is more difficult, because the parameters in the hard input nonlinearities are in cascade with the linear system which makes the output of the hard nonlinear block difficult to be written as an analytical function of the input. Due to this difficulty, there exists only scattered work on identification of Hammerstein systems with hard input nonlinearities [12,15,23–25], e.g., Vörös proposed an appropriate switching function to model and identify a Hammerstein system with backlash [15]. Bai used a deterministic approach and the correlation analysis method to estimate the parameters of systems q
This work was supported by the National Natural Science Foundation of China.
⇑ Corresponding authors. Address: School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, PR China (J. Chen). E-mail addresses: [email protected] (J. Chen), [email protected] (X. Wang), [email protected] (R. Ding). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.05.049
J. Chen et al. / Applied Mathematical Modelling 36 (2012) 238–243
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with hard input nonlinearities [3]. This paper deals with the identification of Hammerstein models with saturation and deadzone nonlinearities by using the iterative method. The basic idea is to transform the Hammerstein model with a hard nonlinearity into an analytic form by using an appropriate switching function and then to estimate all the unknown parameters of the system based on the available measured inputs and outputs by using the iterative method. Briefly, the outline of the paper is as follows. Section 2 derives the identification model of Hammerstein nonlinear systems with the saturation and dead-zone nonlinearities. Section 3 presents an iterative algorithm for the nonlinear Hammerstein systems. Section 4 provides an illustrative example. Finally, concluding remarks are included in Section 5. 2. The system description and the identification model Consider a Hammerstein system with the saturation and dead-zone nonlinearity in Fig. 1, where u(t) and x(t) are the input and output of the nonlinear part, respectively, y(t) is the output of the system, v(t) is a noise with zero mean, A(z) and B(z) are scalar polynomials in the unit backward shift operator z1[z1y(t) = y(t 1)] and
AðzÞ :¼ 1 þ a1 z1 þ a2 z2 þ þ ana zna ; BðzÞ :¼ 1 þ b1 z1 þ b2 z2 þ þ bnb znb : The nonlinear part is shown in Fig. 2, and the x(t) can be expressed as
8 m1 ðr 2 r 1 Þ; > > > > > > < m1 ðuðtÞ r 1 Þ; xðtÞ ¼ 0; > > > m2 ðuðtÞ l1 Þ; > > > : m2 ðl2 l1 Þ;
r2 6 uðtÞ; r1 6 uðtÞ 6 r 2 ; l1 6 uðtÞ 6 r 1 ;
ð1Þ
l2 6 uðtÞ 6 l1 ; uðtÞ 6 l2 ;
where m1 and m2 are the corresponding segment slopes, r1 and l1 are the dead-zone points, and r2 and l2 are the saturation points. The linear part is written as
yðtÞ ¼
BðzÞ 1 xðtÞ þ v ðtÞ or AðzÞyðtÞ ¼ BðzÞxðtÞ þ v ðtÞ: AðzÞ AðzÞ
ð2Þ
Since ri, li and mi are unknown, we must introduce an appropriate switching function [15]
hðtÞ ¼ h½uðtÞ ¼
0; if uðtÞ > 0; 1; if uðtÞ 6 0;
so that the output of the hard nonlinear block can be written as an analytic function of the input. That is, the output x(t) of the nonlinearity can be expressed as
xðtÞ ¼ m1 ðr 2 r 1 Þhðr 2 uðtÞÞ þ m2 ðl2 l1 ÞhðuðtÞ l2 Þ þ m1 hðr 1 uðtÞÞhðuðtÞ r 2 ÞuðtÞ m1 r 1 hðr 1 uðtÞÞhðuðtÞ r2 Þ þ m2 hðl2 uðtÞÞhðuðtÞ l1 ÞuðtÞ m2 l1 hðl2 uðtÞÞhðuðtÞ l1 Þ:
ð3Þ
Eq. (2) can be equivalently written as
yðtÞ ¼ xðtÞ þ ½BðzÞ 1xðtÞ þ ½1 AðzÞyðtÞ þ v ðtÞ:
ð4Þ
Substituting x(t) in (3) into (4) gives an analytic model,
yðtÞ ¼ m1 ðr 2 r 1 Þhðr2 uðtÞÞ þ m2 ðl2 l1 ÞhðuðtÞ l2 Þ þ m1 hðr 1 uðtÞÞhðuðtÞ r2 ÞuðtÞ m1 r 1 hðr 1 uðtÞÞhðuðtÞ r 2 Þ þ m2 hðl2 uðtÞÞhðuðtÞ l1 ÞuðtÞ m2 l1 hðl2 uðtÞÞhðuðtÞ l1 Þ þ ½BðzÞ 1xðtÞ þ ½1 AðzÞyðtÞ þ v ðtÞ: Define the parameter vector h and the information vector u(t) as
Fig. 1. The Hammerstein model.
ð5Þ
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Fig. 2. The saturation and dead-zone characteristic.
h :¼ ½m1 ðr2 r 1 Þ; m2 ðl2 l1 Þ; m1 ; m1 r1 ; m2 ; m2 l1 ; b1 ; b2 ; . . . ; bnb ; a1 ; a2 ; ; ana T 2 Rna þnb þ6 ;
uðtÞ :¼ ½hðr2 uðtÞÞ; hðuðtÞ l2 Þ; hðr1 uðtÞÞhðuðtÞ r2 ÞuðtÞ; hðr1 uðtÞÞhðuðtÞ r2 Þ; hðl2 uðtÞÞhðuðtÞ l1 ÞuðtÞ; hðl2 uðtÞÞhðuðtÞ l1 Þ; xðt 1Þ; xðt 2Þ; . . . ; xðt nb Þ;
ð6Þ
yðt 1Þ; yðt 2Þ; . . . ; yðt na ÞT 2 Rna þnb þ6 : Then the Hammerstein model with saturation and dead-zone nonlinearities can be written as a simple form,
yðtÞ ¼ uT ðtÞh þ v ðtÞ:
ð7Þ
Let k = 1, 2, 3, . . . be an iteration variable and ^ hk be the estimate of h at iteration k. Once h is estimated, i.e. we can obtain the estimates of the parameters m1, m2, m1(r2 r1), m2(l2 l1), m1r1 and m2l1 in h, and further determine/compute the estimates ^ k1 ; m ^ k2 ; ^rk1 ; ^r k2 ; ^lk1 and ^lk2 of the parameters m1, m2, r1, r2, l1 and l2. m 3. The gradient based iterative algorithm Let L represent the data length and define the stacked output vector Y(L) and the stacked information matrix U(L) as
3 yðLÞ 6 yðL 1Þ 7 7 6 7 6 7 6 YðLÞ :¼ 6 yðL 2Þ 7 2 RL ; 7 6 . 7 6 .. 5 4 yð1Þ 2
3 uT ðLÞ 7 6 T 6 u ðL 1Þ 7 7 6 T 7 6 UðLÞ :¼ 6 u ðL 2Þ 7 2 RLðna þnb þ6Þ : 7 6 . 7 6 .. 5 4 2
uT ð1Þ 2
Let lk P 0 be the convergence factor and kXk :¼ tr[XTX]. Using the gradient search and minimizing the quadratic criterion function
JðhÞ :¼ kYðLÞ UðLÞhk2 give the following gradient based iterative algorithm,
^hk ¼ ^hk1 lk grad½Jð^hk1 ¼ ^hk1 þ l UT ðLÞ½YðLÞ UðLÞ^hk1 : k 2
ð8Þ
Because the information matrix in (8) [that is u(t)] contains the unknown inner variables x(t i) and unknown parameters r1, r2, l1 and l2, the above stochastic gradient algorithm in (8) cannot be applied to estimate h in (7). The solution is to use the iterative estimation technique [7] or the auxiliary mode identification idea [26–29]: the unknown variables are replaced with their estimates or with the outputs of the auxiliary model. Define
^ k ðtÞ :¼ ½hð^rk1 u uðtÞÞ; hðuðtÞ ^l2k1 Þ; hð^r 1k1 uðtÞÞhðuðtÞ ^r 2k1 ÞuðtÞ; hð^rk1 uðtÞÞhðuðtÞ ^r k1 2 1 2 Þ; ^k1 uðtÞÞhðuðtÞ ^lk1 Þ; ^x ðt 1Þ; ^x ðt 2Þ; . . . ; ^x ðt n Þ; hð^l2k1 uðtÞÞhðuðtÞ ^lk1 k1 k1 k1 b 1 ÞuðtÞ; hðl2 1 yðt 1Þ; yðt 2Þ; . . . ; yðt na ÞT 2 Rna þnb þ6 ; 3 ^ Tk ðLÞ u 7 6u T 6 ^ k ðL 1Þ 7 7 6 T 7 ^ ^ k ðLÞ :¼ 6 U 6 uk ðL 2Þ 7 2 RLðna þnb þ6Þ ; 7 6 . 7 6 .. 5 4 T ^ k ð1Þ u
ð9Þ
2
ð10Þ
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^ 1k1 ; m ^ 2k1 ; ^r k1 ^k1 ^k1 and ^l2k1 , the estimate Replacing the unknown m1, m2, r1, r2, l1 and l2 in (3) with their estimates m 1 ; r 2 ; l1 ^ xk ðtÞ of x(t) can be computed by
k1 ^lk1 ^lk1 h uðtÞ ^lk1 þ m ^ k1 ^ k1 ^k1 uðtÞÞh uðtÞ ^r k1 ^xk ðtÞ ¼ m ^r 2 ^r 1k1 h ^r 2k1 uðtÞ þ m ^ k1 uðtÞ 1 2 2 1 2 1 hðr 1 2 ^ k1 ^k1 uðtÞ hðuðtÞ ^lk1 ^k1 ^k1 uðtÞ hðuðtÞ ^rk1 ^ k1 m 1 r1 h r1 2 Þ þ m2 h l2 1 ÞuðtÞ k1 k1 k1 k1 ^ 2 ^l1 h ^l2 uðtÞ h uðtÞ ^l1 : m ^ k ðLÞ, we can summarize the gradient based iterative identification algorithm Thus, replacing U(L) in (8) with its estimate U (the GI algorithm for short):
^ ^hk ¼ ^hk1 þ l U ^ ^T k k ðLÞ½YðLÞ Uk ðLÞhk1 ;
ð11Þ
YðLÞ ¼ ½yðLÞ; yðL 1Þ; yðL 2Þ; ; yð1ÞT ; ^ k ðLÞ ¼ ½u ^ k ðLÞ; u ^ k ðL 1Þ; u ^ k ðL 2Þ; ; u ^ k ð1ÞT ; U
ð12Þ
h
ð13Þ
^ k ðtÞ ¼ h ^r 2k1 uðtÞ ; h uðtÞ ^lk1 ; h ^r 1k1 uðtÞ h uðtÞ ^r k1 u uðtÞ h uðtÞ ^r 2k1 ; uðtÞ; h ^rk1 2 2 1
h ^l2k1 uðtÞ h uðtÞ ^lk1 uðtÞ; h ^l2k1 uðtÞ h uðtÞ ^lk1 ; ^xk1 ðt 1Þ; ^xk1 ðt 2Þ; . . . ; ^xk1 ðt nb Þ; 1 1 i ð14Þ yðt 1Þ; yðt 2Þ; . . . ; yðt na Þ ;
k1 k1 ^lk1 ^lk1 h uðtÞ ^lk1 þ m ^ k1 ^ k1 ^ uðtÞ h uðtÞ ^r 2k1 uðtÞ ^r 2 ^r 1k1 h ^r 2k1 uðtÞ þ m ^ k1 ^xk ðtÞ ¼ m 1 2 2 1 2 1 h r1 ^ 2k1 h ^lk1 ^k1 ^k1 uðtÞ h uðtÞ ^r k1 ^ k1 uðtÞ h uðtÞ ^lk1 þm uðtÞ m 1 r1 h r1 2 2 1 ^k1 ^k1 uðtÞ h uðtÞ ^lk1 ; ^ k1 m 2 l1 h l2 1 2 T : ^ k ðLÞ ^ ðLÞU kmax U k
lk 6
ð15Þ ð16Þ
The steps of computing the parameter estimation vector ^ hk by the iterative algorithm in (11)–(16) are listed in the following. 1. 2. 3. 4. 5. 6. 7.
Collect the input and output data {u(t), y(t) : t = 0, 1, 2, . . . , L} and form Y(L) by (12). x0 ðtÞ ¼ 1=p0 with 1 being a column vector whose entries are all unity and p0 = 106. To initialize, let k = 1, ^ h0 ¼ 1=p0 ; ^ ^ ^ Form uk ðtÞ by (14) and Uk ðLÞ by (13). Choose lk according to (16). Update the parameter estimation vector ^ hk by (11). Compute ^ xk ðtÞ by (15). Compare ^ hk and ^ hk1 : if they are sufficiently close, or for some pre-set small e, if k^ hk ^ hk1 k 6 e, then terminate the procedure and obtain the iterative times k and estimate ^ hk ; otherwise, increase k by 1 and go to step 3.
4. Example Consider the following Hammerstein system,
yðtÞ þ 0:93yðt 1Þ ¼ ½1 0:05z1 xðtÞ þ v ðtÞ; the saturation and dead-zone nonlinearities are shown in Fig. 1 with l2 = 0.13, l1 = 0.01, m2 = 1, r2 = 0.44, r1 = 0.24 and m1 = 1.2. For this example, we have
h ¼ ½m1 ðr 2 r 1 Þ; m2 ðl2 l1 Þ; m1 ; m1 r1 ; m2 ; m2 l1 ; a1 ; b1 T ¼ ½0:24; 0:12; 1:2; 0:288; 1; 0; 01; 0:05; 0:93T ;
uðtÞ ¼ ½hðr2 uðtÞÞ; hðuðtÞ l2 Þ; hðr1 uðtÞÞhðuðtÞ r2 ÞuðtÞ; hðr1 uðtÞÞhðuðtÞ r2 Þ; hðl2 uðtÞÞhðuðtÞ l1 ÞuðtÞ; hðl2 uðtÞÞhðuðtÞ l1 Þ; xðt 1Þ; yðt 1ÞT ¼ ½hð0:44 uðtÞÞ; hðuðtÞ þ 0:13Þ; hð0:24 uðtÞÞhðuðtÞ 0:44ÞuðtÞ; hð0:24 uðtÞÞhðuðtÞ 0:44Þ; hð0:13 uðtÞÞhðuðtÞ þ 0:01ÞuðtÞ; hð0:13 uðtÞÞhðuðtÞ þ 0:01Þ; xðt 1Þ; yðt 1ÞT : Here, {u(t)} is taken as a persistent excitation signal sequence with zero mean and unit variance, and {v(t)} as a white noise sequence with zero mean and variance r2 = 0.202. Applying the proposed GI algorithm to estimate the parameters of this
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Table 1 The GI estimates and errors. k
10
30
90
180
270
360
450
True values
a1 b1 m1(r2 r1) m2(l2 l1) m1 m1r1 m2 m2l1 d (%)
0.11977 0.90544 0.24278 0.09836 1.22998 0.33345 0.94764 0.25269 15.26275
0.02138 0.92513 0.24339 0.11332 1.23648 0.30838 0.93385 0.01127 4.64582
0.01557 0.92673 0.24323 0.11851 1.23080 0.30664 0.94367 0.00124 4.09446
0.01562 0.92683 0.24324 0.11899 1.22286 0.30388 0.95918 0.00012 3.28948
0.01566 0.92683 0.24324 0.11900 1.21546 0.30131 0.97446 0.00099 2.60539
0.01569 0.92683 0.24324 0.11900 1.20855 0.29891 0.98951 0.00208 2.12504
0.01572 0.92683 0.24324 0.11900 1.20211 0.29667 1.00433 0.00315 1.96852
0.05000 0.93000 0.24000 0.12000 1.20000 0.28800 1.00000 0.01000
0.6 0.5
δ
0.4 0.3 0.2 0.1 0
0
50
100
150
200
k
250
300
350
400
450
Fig. 3. The parameter estimation errors d versus k (L = 1000).
system, the parameter estimates and their errors are shown in Table 1, the parameter estimation errors d :¼ k^ h hk=khk versus k are shown in Fig. 3. From Table 1 and Fig. 3, we can see that the proposed GI algorithm can work well. With the estimated parameters m1 1, m2 1, we can compute the parameters m1, m2. Then with the computed parameters m1, m2 and the estimated parameters m1r1, m2l1, we can compute the parameters r1, l1. At last, with the computed parameters m1, m2, r1, l1 and the estimated parameters m1(r2 r1), m2(l2 l1), we can compute the parameters r2, l2. 5. Conclusions An approach to identify the Hammerstein systems with saturation and dead-zone nonlinearities is presented in this paper. The nonlinearity is described as one output equation by using an appropriate switching function. Then all the model parameters to be estimated appear explicitly in the identification model. The simulation results verified the effectiveness of the proposed algorithm. The methods in this paper can be extended to other nonlinear systems or missing-data systems outputs [30–35], non-uniformly sampled-data systems [36–38], and multivariable systems [39–41]. It is worth to point out that the hierarchical identification principle [42–45], the multi-innovation identification theory [46–55] and the data filtering techniques [56,57] can be used to study identification problem of this class of nonlinear systems with colored noises. References [1] B. Yu, H. Fang, Y. Lin, Y. Shi, Identification of Hammerstein output-error systems with two-segment nonlinearities: algorithm and applications, J. Control Intell. Syst. 38 (4) (2010) 194–201. [2] E.W. Bai, An optimal two-stage identification algorithm for Hammerstein–Wiener nonlinear systems, Automatica 34 (3) (1998) 333–338. [3] E.W. Bai, Identification of linear systems with hard input nonlinearities of known structure, Automatica 38 (5) (2002) 853–860. [4] V. Cerone, D. Regruto, Parameter bounds for discrete-time Hammerstein models with bounded output errors, IEEE Trans. Autom. Control 48 (10) (2003) 1855–1860. [5] Y. Zhu, Estimation of an N-L-N Hammerstein–Wiener model, Automatica 38 (9) (2002) 1607–1614. [6] M. Ahmadi, H. Mojallali, Identification of multiple-input single-output Hammerstein models using Bezier curves and Bernstein polynomials, Appl. Math. Model. 35 (4) (2011) 1969–1982. [7] F. Ding, P.X. Liu, G. Liu, Gradient based and least-squares based iterative identification methods for OE and OEMA systems, Digital Signal Process. 20 (3) (2010) 664–677. [8] Y.J. Liu, D.Q. Wang, F. Ding, Least-squares based iterative algorithms for identifying Box–Jenkins models with finite measurement data, Digital Signal Process. 20 (5) (2010) 1458–1467.
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Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Gradient based estimation algorithm for Hammerstein systems with saturation and dead-zone nonlinearities q Jing Chen a,b,⇑, Xiuping Wang a, Ruifeng Ding b,⇑ a b
Wuxi Professional College of Science and Technology, Wuxi 214028, PR China School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, PR China
a r t i c l e
i n f o
Article history: Received 8 March 2011 Received in revised form 20 May 2011 Accepted 22 May 2011 Available online 6 June 2011 Keywords: Iterative method System identification Parameter estimation Hammerstein model
a b s t r a c t This paper focuses on identification problems for Hammerstein systems with saturation and dead-zone nonlinearities. An appropriate switching function is introduced to derive an identification model with fewer parameters and all the unknown parameters can be estimated by using an iterative method. A numerical simulation is carried out to show the effectiveness of the proposed method. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Identification of Hammerstein nonlinear systems (linear time-invariant (LTI) block following a static nonlinear block) has been active in theory and application. In general, existing identification approaches for Hammerstein nonlinear systems can be roughly divided into two categories: the recursive and the iterative algorithms [1–6]. The recursive algorithms are often used in on-line estimation and the iterative algorithms are often used in off-line estimation. The basic idea of the iterative method is to adopt the iterative estimation theory: the parameter estimates relying on the unknown variables are computed by using the estimates of these unknown variables which are obtained from the preceding parameter estimates [7–11]. The iterative algorithms are usually utilized in the identification of Hammerstein systems with hard input nonlinearities [3,4,12–15]. Recently, Ding et al. presented a series of identification methods for Hammerstein nonlinear systems [11,16–18]; Chen et al. proposed an auxiliary model based multi-innovation algorithms for multivariable nonlinear systems [19]; Wang et al. derived an extended stochastic gradient identification algorithm for Hammerstein–Wiener ARMAX systems [20], an auxiliary model-based recursive generalized least squares parameter estimation for Hammerstein OEAR systems [21] and auxiliary model-based RELS and MI-ELS algorithms for Hammerstein OEMA systems [22]. Hard input nonlinearities are common in engineering practice. Identification of Hammerstein systems with hard input nonlinearities is more difficult, because the parameters in the hard input nonlinearities are in cascade with the linear system which makes the output of the hard nonlinear block difficult to be written as an analytical function of the input. Due to this difficulty, there exists only scattered work on identification of Hammerstein systems with hard input nonlinearities [12,15,23–25], e.g., Vörös proposed an appropriate switching function to model and identify a Hammerstein system with backlash [15]. Bai used a deterministic approach and the correlation analysis method to estimate the parameters of systems q
This work was supported by the National Natural Science Foundation of China.
⇑ Corresponding authors. Address: School of Internet of Things Engineering, Jiangnan University, Wuxi 214122, PR China (J. Chen). E-mail addresses: [email protected] (J. Chen), [email protected] (X. Wang), [email protected] (R. Ding). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.05.049
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with hard input nonlinearities [3]. This paper deals with the identification of Hammerstein models with saturation and deadzone nonlinearities by using the iterative method. The basic idea is to transform the Hammerstein model with a hard nonlinearity into an analytic form by using an appropriate switching function and then to estimate all the unknown parameters of the system based on the available measured inputs and outputs by using the iterative method. Briefly, the outline of the paper is as follows. Section 2 derives the identification model of Hammerstein nonlinear systems with the saturation and dead-zone nonlinearities. Section 3 presents an iterative algorithm for the nonlinear Hammerstein systems. Section 4 provides an illustrative example. Finally, concluding remarks are included in Section 5. 2. The system description and the identification model Consider a Hammerstein system with the saturation and dead-zone nonlinearity in Fig. 1, where u(t) and x(t) are the input and output of the nonlinear part, respectively, y(t) is the output of the system, v(t) is a noise with zero mean, A(z) and B(z) are scalar polynomials in the unit backward shift operator z1[z1y(t) = y(t 1)] and
AðzÞ :¼ 1 þ a1 z1 þ a2 z2 þ þ ana zna ; BðzÞ :¼ 1 þ b1 z1 þ b2 z2 þ þ bnb znb : The nonlinear part is shown in Fig. 2, and the x(t) can be expressed as
8 m1 ðr 2 r 1 Þ; > > > > > > < m1 ðuðtÞ r 1 Þ; xðtÞ ¼ 0; > > > m2 ðuðtÞ l1 Þ; > > > : m2 ðl2 l1 Þ;
r2 6 uðtÞ; r1 6 uðtÞ 6 r 2 ; l1 6 uðtÞ 6 r 1 ;
ð1Þ
l2 6 uðtÞ 6 l1 ; uðtÞ 6 l2 ;
where m1 and m2 are the corresponding segment slopes, r1 and l1 are the dead-zone points, and r2 and l2 are the saturation points. The linear part is written as
yðtÞ ¼
BðzÞ 1 xðtÞ þ v ðtÞ or AðzÞyðtÞ ¼ BðzÞxðtÞ þ v ðtÞ: AðzÞ AðzÞ
ð2Þ
Since ri, li and mi are unknown, we must introduce an appropriate switching function [15]
hðtÞ ¼ h½uðtÞ ¼
0; if uðtÞ > 0; 1; if uðtÞ 6 0;
so that the output of the hard nonlinear block can be written as an analytic function of the input. That is, the output x(t) of the nonlinearity can be expressed as
xðtÞ ¼ m1 ðr 2 r 1 Þhðr 2 uðtÞÞ þ m2 ðl2 l1 ÞhðuðtÞ l2 Þ þ m1 hðr 1 uðtÞÞhðuðtÞ r 2 ÞuðtÞ m1 r 1 hðr 1 uðtÞÞhðuðtÞ r2 Þ þ m2 hðl2 uðtÞÞhðuðtÞ l1 ÞuðtÞ m2 l1 hðl2 uðtÞÞhðuðtÞ l1 Þ:
ð3Þ
Eq. (2) can be equivalently written as
yðtÞ ¼ xðtÞ þ ½BðzÞ 1xðtÞ þ ½1 AðzÞyðtÞ þ v ðtÞ:
ð4Þ
Substituting x(t) in (3) into (4) gives an analytic model,
yðtÞ ¼ m1 ðr 2 r 1 Þhðr2 uðtÞÞ þ m2 ðl2 l1 ÞhðuðtÞ l2 Þ þ m1 hðr 1 uðtÞÞhðuðtÞ r2 ÞuðtÞ m1 r 1 hðr 1 uðtÞÞhðuðtÞ r 2 Þ þ m2 hðl2 uðtÞÞhðuðtÞ l1 ÞuðtÞ m2 l1 hðl2 uðtÞÞhðuðtÞ l1 Þ þ ½BðzÞ 1xðtÞ þ ½1 AðzÞyðtÞ þ v ðtÞ: Define the parameter vector h and the information vector u(t) as
Fig. 1. The Hammerstein model.
ð5Þ
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Fig. 2. The saturation and dead-zone characteristic.
h :¼ ½m1 ðr2 r 1 Þ; m2 ðl2 l1 Þ; m1 ; m1 r1 ; m2 ; m2 l1 ; b1 ; b2 ; . . . ; bnb ; a1 ; a2 ; ; ana T 2 Rna þnb þ6 ;
uðtÞ :¼ ½hðr2 uðtÞÞ; hðuðtÞ l2 Þ; hðr1 uðtÞÞhðuðtÞ r2 ÞuðtÞ; hðr1 uðtÞÞhðuðtÞ r2 Þ; hðl2 uðtÞÞhðuðtÞ l1 ÞuðtÞ; hðl2 uðtÞÞhðuðtÞ l1 Þ; xðt 1Þ; xðt 2Þ; . . . ; xðt nb Þ;
ð6Þ
yðt 1Þ; yðt 2Þ; . . . ; yðt na ÞT 2 Rna þnb þ6 : Then the Hammerstein model with saturation and dead-zone nonlinearities can be written as a simple form,
yðtÞ ¼ uT ðtÞh þ v ðtÞ:
ð7Þ
Let k = 1, 2, 3, . . . be an iteration variable and ^ hk be the estimate of h at iteration k. Once h is estimated, i.e. we can obtain the estimates of the parameters m1, m2, m1(r2 r1), m2(l2 l1), m1r1 and m2l1 in h, and further determine/compute the estimates ^ k1 ; m ^ k2 ; ^rk1 ; ^r k2 ; ^lk1 and ^lk2 of the parameters m1, m2, r1, r2, l1 and l2. m 3. The gradient based iterative algorithm Let L represent the data length and define the stacked output vector Y(L) and the stacked information matrix U(L) as
3 yðLÞ 6 yðL 1Þ 7 7 6 7 6 7 6 YðLÞ :¼ 6 yðL 2Þ 7 2 RL ; 7 6 . 7 6 .. 5 4 yð1Þ 2
3 uT ðLÞ 7 6 T 6 u ðL 1Þ 7 7 6 T 7 6 UðLÞ :¼ 6 u ðL 2Þ 7 2 RLðna þnb þ6Þ : 7 6 . 7 6 .. 5 4 2
uT ð1Þ 2
Let lk P 0 be the convergence factor and kXk :¼ tr[XTX]. Using the gradient search and minimizing the quadratic criterion function
JðhÞ :¼ kYðLÞ UðLÞhk2 give the following gradient based iterative algorithm,
^hk ¼ ^hk1 lk grad½Jð^hk1 ¼ ^hk1 þ l UT ðLÞ½YðLÞ UðLÞ^hk1 : k 2
ð8Þ
Because the information matrix in (8) [that is u(t)] contains the unknown inner variables x(t i) and unknown parameters r1, r2, l1 and l2, the above stochastic gradient algorithm in (8) cannot be applied to estimate h in (7). The solution is to use the iterative estimation technique [7] or the auxiliary mode identification idea [26–29]: the unknown variables are replaced with their estimates or with the outputs of the auxiliary model. Define
^ k ðtÞ :¼ ½hð^rk1 u uðtÞÞ; hðuðtÞ ^l2k1 Þ; hð^r 1k1 uðtÞÞhðuðtÞ ^r 2k1 ÞuðtÞ; hð^rk1 uðtÞÞhðuðtÞ ^r k1 2 1 2 Þ; ^k1 uðtÞÞhðuðtÞ ^lk1 Þ; ^x ðt 1Þ; ^x ðt 2Þ; . . . ; ^x ðt n Þ; hð^l2k1 uðtÞÞhðuðtÞ ^lk1 k1 k1 k1 b 1 ÞuðtÞ; hðl2 1 yðt 1Þ; yðt 2Þ; . . . ; yðt na ÞT 2 Rna þnb þ6 ; 3 ^ Tk ðLÞ u 7 6u T 6 ^ k ðL 1Þ 7 7 6 T 7 ^ ^ k ðLÞ :¼ 6 U 6 uk ðL 2Þ 7 2 RLðna þnb þ6Þ ; 7 6 . 7 6 .. 5 4 T ^ k ð1Þ u
ð9Þ
2
ð10Þ
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^ 1k1 ; m ^ 2k1 ; ^r k1 ^k1 ^k1 and ^l2k1 , the estimate Replacing the unknown m1, m2, r1, r2, l1 and l2 in (3) with their estimates m 1 ; r 2 ; l1 ^ xk ðtÞ of x(t) can be computed by
k1 ^lk1 ^lk1 h uðtÞ ^lk1 þ m ^ k1 ^ k1 ^k1 uðtÞÞh uðtÞ ^r k1 ^xk ðtÞ ¼ m ^r 2 ^r 1k1 h ^r 2k1 uðtÞ þ m ^ k1 uðtÞ 1 2 2 1 2 1 hðr 1 2 ^ k1 ^k1 uðtÞ hðuðtÞ ^lk1 ^k1 ^k1 uðtÞ hðuðtÞ ^rk1 ^ k1 m 1 r1 h r1 2 Þ þ m2 h l2 1 ÞuðtÞ k1 k1 k1 k1 ^ 2 ^l1 h ^l2 uðtÞ h uðtÞ ^l1 : m ^ k ðLÞ, we can summarize the gradient based iterative identification algorithm Thus, replacing U(L) in (8) with its estimate U (the GI algorithm for short):
^ ^hk ¼ ^hk1 þ l U ^ ^T k k ðLÞ½YðLÞ Uk ðLÞhk1 ;
ð11Þ
YðLÞ ¼ ½yðLÞ; yðL 1Þ; yðL 2Þ; ; yð1ÞT ; ^ k ðLÞ ¼ ½u ^ k ðLÞ; u ^ k ðL 1Þ; u ^ k ðL 2Þ; ; u ^ k ð1ÞT ; U
ð12Þ
h
ð13Þ
^ k ðtÞ ¼ h ^r 2k1 uðtÞ ; h uðtÞ ^lk1 ; h ^r 1k1 uðtÞ h uðtÞ ^r k1 u uðtÞ h uðtÞ ^r 2k1 ; uðtÞ; h ^rk1 2 2 1
h ^l2k1 uðtÞ h uðtÞ ^lk1 uðtÞ; h ^l2k1 uðtÞ h uðtÞ ^lk1 ; ^xk1 ðt 1Þ; ^xk1 ðt 2Þ; . . . ; ^xk1 ðt nb Þ; 1 1 i ð14Þ yðt 1Þ; yðt 2Þ; . . . ; yðt na Þ ;
k1 k1 ^lk1 ^lk1 h uðtÞ ^lk1 þ m ^ k1 ^ k1 ^ uðtÞ h uðtÞ ^r 2k1 uðtÞ ^r 2 ^r 1k1 h ^r 2k1 uðtÞ þ m ^ k1 ^xk ðtÞ ¼ m 1 2 2 1 2 1 h r1 ^ 2k1 h ^lk1 ^k1 ^k1 uðtÞ h uðtÞ ^r k1 ^ k1 uðtÞ h uðtÞ ^lk1 þm uðtÞ m 1 r1 h r1 2 2 1 ^k1 ^k1 uðtÞ h uðtÞ ^lk1 ; ^ k1 m 2 l1 h l2 1 2 T : ^ k ðLÞ ^ ðLÞU kmax U k
lk 6
ð15Þ ð16Þ
The steps of computing the parameter estimation vector ^ hk by the iterative algorithm in (11)–(16) are listed in the following. 1. 2. 3. 4. 5. 6. 7.
Collect the input and output data {u(t), y(t) : t = 0, 1, 2, . . . , L} and form Y(L) by (12). x0 ðtÞ ¼ 1=p0 with 1 being a column vector whose entries are all unity and p0 = 106. To initialize, let k = 1, ^ h0 ¼ 1=p0 ; ^ ^ ^ Form uk ðtÞ by (14) and Uk ðLÞ by (13). Choose lk according to (16). Update the parameter estimation vector ^ hk by (11). Compute ^ xk ðtÞ by (15). Compare ^ hk and ^ hk1 : if they are sufficiently close, or for some pre-set small e, if k^ hk ^ hk1 k 6 e, then terminate the procedure and obtain the iterative times k and estimate ^ hk ; otherwise, increase k by 1 and go to step 3.
4. Example Consider the following Hammerstein system,
yðtÞ þ 0:93yðt 1Þ ¼ ½1 0:05z1 xðtÞ þ v ðtÞ; the saturation and dead-zone nonlinearities are shown in Fig. 1 with l2 = 0.13, l1 = 0.01, m2 = 1, r2 = 0.44, r1 = 0.24 and m1 = 1.2. For this example, we have
h ¼ ½m1 ðr 2 r 1 Þ; m2 ðl2 l1 Þ; m1 ; m1 r1 ; m2 ; m2 l1 ; a1 ; b1 T ¼ ½0:24; 0:12; 1:2; 0:288; 1; 0; 01; 0:05; 0:93T ;
uðtÞ ¼ ½hðr2 uðtÞÞ; hðuðtÞ l2 Þ; hðr1 uðtÞÞhðuðtÞ r2 ÞuðtÞ; hðr1 uðtÞÞhðuðtÞ r2 Þ; hðl2 uðtÞÞhðuðtÞ l1 ÞuðtÞ; hðl2 uðtÞÞhðuðtÞ l1 Þ; xðt 1Þ; yðt 1ÞT ¼ ½hð0:44 uðtÞÞ; hðuðtÞ þ 0:13Þ; hð0:24 uðtÞÞhðuðtÞ 0:44ÞuðtÞ; hð0:24 uðtÞÞhðuðtÞ 0:44Þ; hð0:13 uðtÞÞhðuðtÞ þ 0:01ÞuðtÞ; hð0:13 uðtÞÞhðuðtÞ þ 0:01Þ; xðt 1Þ; yðt 1ÞT : Here, {u(t)} is taken as a persistent excitation signal sequence with zero mean and unit variance, and {v(t)} as a white noise sequence with zero mean and variance r2 = 0.202. Applying the proposed GI algorithm to estimate the parameters of this
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Table 1 The GI estimates and errors. k
10
30
90
180
270
360
450
True values
a1 b1 m1(r2 r1) m2(l2 l1) m1 m1r1 m2 m2l1 d (%)
0.11977 0.90544 0.24278 0.09836 1.22998 0.33345 0.94764 0.25269 15.26275
0.02138 0.92513 0.24339 0.11332 1.23648 0.30838 0.93385 0.01127 4.64582
0.01557 0.92673 0.24323 0.11851 1.23080 0.30664 0.94367 0.00124 4.09446
0.01562 0.92683 0.24324 0.11899 1.22286 0.30388 0.95918 0.00012 3.28948
0.01566 0.92683 0.24324 0.11900 1.21546 0.30131 0.97446 0.00099 2.60539
0.01569 0.92683 0.24324 0.11900 1.20855 0.29891 0.98951 0.00208 2.12504
0.01572 0.92683 0.24324 0.11900 1.20211 0.29667 1.00433 0.00315 1.96852
0.05000 0.93000 0.24000 0.12000 1.20000 0.28800 1.00000 0.01000
0.6 0.5
δ
0.4 0.3 0.2 0.1 0
0
50
100
150
200
k
250
300
350
400
450
Fig. 3. The parameter estimation errors d versus k (L = 1000).
system, the parameter estimates and their errors are shown in Table 1, the parameter estimation errors d :¼ k^ h hk=khk versus k are shown in Fig. 3. From Table 1 and Fig. 3, we can see that the proposed GI algorithm can work well. With the estimated parameters m1 1, m2 1, we can compute the parameters m1, m2. Then with the computed parameters m1, m2 and the estimated parameters m1r1, m2l1, we can compute the parameters r1, l1. At last, with the computed parameters m1, m2, r1, l1 and the estimated parameters m1(r2 r1), m2(l2 l1), we can compute the parameters r2, l2. 5. Conclusions An approach to identify the Hammerstein systems with saturation and dead-zone nonlinearities is presented in this paper. The nonlinearity is described as one output equation by using an appropriate switching function. Then all the model parameters to be estimated appear explicitly in the identification model. The simulation results verified the effectiveness of the proposed algorithm. The methods in this paper can be extended to other nonlinear systems or missing-data systems outputs [30–35], non-uniformly sampled-data systems [36–38], and multivariable systems [39–41]. It is worth to point out that the hierarchical identification principle [42–45], the multi-innovation identification theory [46–55] and the data filtering techniques [56,57] can be used to study identification problem of this class of nonlinear systems with colored noises. References [1] B. Yu, H. Fang, Y. Lin, Y. Shi, Identification of Hammerstein output-error systems with two-segment nonlinearities: algorithm and applications, J. Control Intell. Syst. 38 (4) (2010) 194–201. [2] E.W. Bai, An optimal two-stage identification algorithm for Hammerstein–Wiener nonlinear systems, Automatica 34 (3) (1998) 333–338. [3] E.W. Bai, Identification of linear systems with hard input nonlinearities of known structure, Automatica 38 (5) (2002) 853–860. [4] V. Cerone, D. Regruto, Parameter bounds for discrete-time Hammerstein models with bounded output errors, IEEE Trans. Autom. Control 48 (10) (2003) 1855–1860. [5] Y. Zhu, Estimation of an N-L-N Hammerstein–Wiener model, Automatica 38 (9) (2002) 1607–1614. [6] M. Ahmadi, H. Mojallali, Identification of multiple-input single-output Hammerstein models using Bezier curves and Bernstein polynomials, Appl. Math. Model. 35 (4) (2011) 1969–1982. [7] F. Ding, P.X. Liu, G. Liu, Gradient based and least-squares based iterative identification methods for OE and OEMA systems, Digital Signal Process. 20 (3) (2010) 664–677. [8] Y.J. Liu, D.Q. Wang, F. Ding, Least-squares based iterative algorithms for identifying Box–Jenkins models with finite measurement data, Digital Signal Process. 20 (5) (2010) 1458–1467.
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