Mathematical and Computer Modelling 52 (2010) 309–317
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Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Auxiliary model based recursive generalized least squares parameter estimation for Hammerstein OEAR systemsI Dongqing Wang a,∗ , Yanyun Chu a , Guowei Yang a , Feng Ding b a
College of Automation Engineering, Qingdao University, Qingdao 266071, PR China
b
School of Communication and Control Engineering, Jiangnan University, Wuxi 214122, PR China
article
info
Article history: Received 16 August 2009 Received in revised form 27 February 2010 Accepted 1 March 2010 Keywords: Recursive identification Parameter estimation Hammerstein models Key-term separation principle Nonlinear systems Auxiliary model identification
abstract This paper deals with the parameter identification problem of Hammerstein output error auto-regressive (OEAR) systems with different nonlinearities by combining the key-term separation principle and the auxiliary model identification idea. The basic idea is, by using the key-term separation principle, to present auxiliary model based recursive generalized least squares algorithms in terms of the auxiliary model idea. The proposed algorithm can obtain the system model parameter estimates and the noise model parameter estimates, and can be extended to other nonlinear systems. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction The Hammerstein system with the block structure oriented nonlinearity consists of a static nonlinear block followed by a linear dynamic block [1–6]. The parameter estimation problems of such nonlinear systems have been widely studied in system modelling, system identification, signal processing and filtering [3,4,7,8]. Vörös presented the key-term separation principle based estimation algorithm for Hammerstein models with discontinuous and dead-zone nonlinearities [9,10]. In order to state the key-term separation principle in [9,10], we take the following compound functions as an example: y(t ) = g [a1 , a2 , . . . , an , x(t ), z ], x(t ) = f [c1 , c1 , . . . , cm , u(t )], where y(t ) is the system output, u(t ) is the system input, x(t ) is the internal variable, g (∗) is a linear dynamical system with (a1 , . . . , an ) as its parameters, f (∗) is a static nonlinear function of u(t ) with parameters (c1 , . . . , cm ) as its coefficients, z is a unit forward shift operator: zx(t ) = x(t + 1) and z −1 x(t ) = x(t − 1). For some special function g (∗), which can be written as g [a1 , . . . , an , x(t ), z ] = x(t ) + g 0 [a1 , . . . , an , x(t ), z ], we have y(t ) = x(t ) + g 0 [a1 , . . . , an , x(t ), z ], where x(t ) in the above equation is called the key-term. Substituting x(t ) = f [c1 , . . . , cm , u(t )] into the separated key-term x(t ) (the first term on the right-hand side) and keeping the non-separated key-term g 0 [a1 , . . . , an , x(t ), z ] unchanged give y(t ) = f [c1 , . . . , cm , u(t )] + g 0 [a1 , . . . , an , x(t ), z ]. I This work was supported by the Shandong Province Colleges and Universities Outstanding Young Teachers in Domestic Visiting Scholars Project at the Jiangnan University and by the National Natural Science Foundation of China (60973048). ∗ Corresponding address: College of Automation Engineering, Qingdao University (Jiangnan University), Qingdao 266071, PR China. E-mail addresses:
[email protected] (D. Wang),
[email protected] (Y. Chu),
[email protected] (G. Yang),
[email protected] (F. Ding).
0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.03.002
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Fig. 1. The Hammerstein OEAR system.
Assume that y(t ) can be expressed as the linear combination of u(t − i) and x(t − i) with the system parameters (c1 , . . . , cm , a1 , . . . , an ) as their coefficients. If replace the unknown x(t ) in the above equation with its estimate xˆ (t ), then any linear parameter identification algorithm can be used to generate the estimates (ˆc1 (t ), . . . , cˆm (t ), aˆ 1 (t ), . . . , aˆ n (t )) of (c1 , . . . , cm , a1 , . . . , an ) and thus the estimate xˆ (t ) of x(t ) can be computed by xˆ (t ) = f [ˆc1 (t ), . . . , cˆm (t ), u(t )]. The auxiliary model identification idea is a very useful approach for estimating the parameters of output error type systems [4,11]. For Hammerstein output error auto-regressive (OEAR) systems with nonlinearities, this paper presents an auxiliary model based recursive generalized least squares algorithm by using the key-term separation principle and the auxiliary model identification idea. The proposed algorithm can obtain the system model parameter estimates, the noise model parameter estimates and the internal variable estimates. The paper is organized as follows. Section 2 describes the system formulation related to the Hammerstein OEAR models with a simple two-segment piecewise nonlinearity. Section 3 derives an auxiliary model based recursive generalized least squares algorithm by using the key-term separation principle. Section 4 extends the proposed approach for Hammerstein OEAR systems to some complex nonlinearities such as the discontinuous asymmetric piecewise-linear nonlinearity, the preloads and dead-zone nonlinearity, the saturation nonlinearity, the hysteresis and hysteresis-relay nonlinearities. Section 5 provides an illustrative example. Finally, concluding remarks are given in Section 6. 2. The problem formulation Consider the Hammerstein output error auto-regressive (OEAR) system in Fig. 1, where u(t ) and y(t ) are the system input and output, respectively, v(t ) is a stochastic white noise with zero mean and variance σ 2 ; the linear part is an OEAR model: y(t ) =
B(z ) A(z )
x(t ) +
1 C (z )
v(t ),
(1)
A(z ), B(z ) and C (z ) are polynomials in the unit backward shift operator z −1 : A(z ) = 1 + a1 z −1 + a2 z −2 + · · · + ana z −na , B(z ) = b0 + b1 z −1 + b2 z −2 + · · · + bnb z −nb , C (z ) = 1 + c1 z −1 + c2 z −2 + · · · + cnc z −nc . The output x(t ) of the nonlinear block (i.e., the input of the linear block) with a two-segment piecewise nonlinearity can be expressed by x(t ) =
k1 u(t ), k2 u(t ),
u(t ) > 0, u(t ) < 0.
Assume that the orders na , nb and nc are known and y(t ) = 0, u(t ) = 0, x(t ) = 0, r (t ) = 0 and v(t ) = 0 for t 6 0. Set b0 = 1 for the uniqueness of the system parameters. Introduce a switched function: h(t ) = h[u(t )] = 0.5{1 + sgn[u(t )]},
(2)
where sgn(u) =
1, −1,
u > 0, u < 0.
Thus, x(t ) can be expressed as x(t ) = k2 u(t ) + (k1 − k2 )u(t )h(t ).
(3)
This paper studies identification problems for Hammerstein OEAR systems using the auxiliary model identification idea [4,11] and the key-term separation principle [9,10], and also evaluates the accuracy of the parameter estimates by simulations on computers.
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3. The auxiliary model based recursive generalized least squares algorithm From Fig. 1, we have r (t ) =
B(z ) A(z ) 1
w(t ) =
x(t ),
(4)
v(t ).
(5)
C (z ) From (4), we can get
r (t ) = x(t ) + b1 x(t − 1) + b2 x(t − 2) + · · · + bnb x(t − nb ) − a1 r (t − 1) − a2 r (t − 2) − · · · − ana r (t − na ).
(6)
The first term x(t ) (its coefficient is 1) on the right-hand side is taken as a separated key-term, Referring to [9,10], substituting (3) into the separated key-term x(t ) in (6) gives r (t ) = k2 u(t ) + (k1 − k2 )u(t )h(t ) + b1 x(t − 1) + · · · + bnb x(t − nb ) − a1 r (t − 1) − · · · − ana r (t − na ).
(7)
Define
ϕs (t ) θ ∈ Rna +nb +nc +2 , θ := s ∈ Rna +nb +nc +2 , θn ϕn (t ) k u(t ) 2 k1 − k2 u(t )h(t ) a1 −r (t − 1) .. .. . . na +nb +2 n a +n b +2 , , θ s := ϕs (t ) := ana ∈ R −r (t − na ) ∈ R b1 x(t − 1) .. .. . . bn b x ( t − nb ) −w(t − 1) c1 c2 −w(t − 2) nc ∈ Rnc , ϕn (t ) := θ n := .. .. ∈ R , . . −w(t − nc ) cnc ϕ(t ) :=
where subscripts s and n are the first letters of the words ‘‘system’’ and ‘‘noise’’ models, respectively. Then (7), (5) and (1) can be written as r (t ) = ϕTs (t )θ s ,
(8)
w(t ) = −c1 w(t − 1) − c2 w(t − 2) − · · · − cnc w(t − nc ) + v(t ) = ϕTn (t )θ n + v(t ),
(9)
y(t ) = r (t ) + w(t )
(10)
= k2 u(t ) + (k1 − k2 )u(t )h(t ) − a1 r (t − 1) − · · · − ana r (t − na ) + b1 x(t − 1) + · · · + bnb x(t − nb ) − c1 w(t − 1) − c2 w(t − 2) − · · · − cnc w(t − nc ) + v(t ) = ϕTs (t )θ s + ϕTn (t )θ n + v(t ) = ϕT (t )θ + v(t ).
(11)
The difficulty of identification arises here is that the information vector ϕ(t ) on the right-hand side in (11) contains the unknown inner variables r (t − i) and x(t − i), and unmeasurable terms w(t − i). The solution here is based on the auxiliary model identification idea [4,11] to construct an auxiliary model, shown in Fig. 2: these unknown r (t − i) and x(t − i) in ϕs (t ) of ϕ(t ) are replaced with the outputs ra (t − i) of the auxiliary model and the estimates xˆ (t − i), respectively, and the unmeasurable terms w(t − i) in ϕn (t ) of ϕ(t ) are replaced with their estimates w( ˆ t − i). Let
u(t ) u ( t ) h ( t ) −ra (t − 1)
ˆ t ) := ϕ(
ϕˆ s (t ) , ϕˆ n (t )
.. . , ϕˆ s (t ) := − ra (t − na ) xˆ (t − 1) .. . xˆ (t − nb )
−w( ˆ t − 1) ˆ t − 2) −w( , ϕˆ n (t ) := .. .
−w( ˆ t − nc )
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Fig. 2. The Hammerstein OEAR system with an auxiliary model.
θ (t )
hˆ
θ
i
h i
ˆ t) = s and θ( be the estimate of θ = θ s at time t, and n θˆ n (t ) kˆ 2 (t ) ˆ ˆ k1 (t ) − k2 (t ) aˆ 1 (t ) .. . θˆ s (t ) = ˆ , ana (t ) bˆ 1 (t ) .. . bˆ nb (t )
cˆ1 (t ) cˆ2 (t )
θˆ n (t ) = .. . . cˆnc (t )
ˆ s (t ) to be the information vector ϕa (t ) of the auxiliary model, and θˆ s (t ) to be the parameter vector θ a (t ) of Here, we take ϕ the auxiliary model, thus we have ˆ s (t )θˆ s (t ). ra ( t ) = ϕ T
(12)
Replacing ki in (3) with its estimate kˆ i (t ), we can get the estimate xˆ (t ) of x(t ) as follows: xˆ (t ) = kˆ 2 (t )u(t ) + [kˆ 1 (t ) − kˆ 2 (t )]u(t )h(t ). From (10), we have
w(t ) = y(t ) − r (t ). ˆ s (t )θˆ s (t ), the noise estimate w( Replacing r (t ) with ra (t ) = ϕ ˆ t ) can be computed by T
w( ˆ t ) = y(t ) − ra (t ) = y(t ) − ϕˆ Ts (t )θˆ s (t ). Forming and minimizing the quadratic cost function: J (θ) =
t X [y(i) − ϕˆ T (i)θ]2 , i=1
we can obtain the following auxiliary model based recursive generalized least squares algorithm by using the key-term separation principle of estimating θ for Hammerstein OEAR systems (the H-AM-RGLS algorithm for short) [3]:
ˆ t ) = θ( ˆ t − 1) + L (t )[y(t ) − ϕˆ T (t )θ( ˆ t − 1)], θ(
(13)
ˆ t )[1 + ϕˆ (t )P (t − 1)ϕ( ˆ t )]−1 , L (t ) = P (t − 1)ϕ(
(14)
ˆ (t )]P (t − 1), P (t ) = [I − L (t )ϕ
(15)
T
P (0) = p0 I ,
T
ˆ t) = ϕ(
ϕˆ s (t ) , ϕˆ n (t )
ˆ ˆ t ) = θ s (t ) , θ( θˆ n (t )
u( t ) u ( t ) h ( t ) −r a ( t − 1 )
(16)
.. . , ϕˆ s (t ) = − ra (t − na ) xˆ (t − 1) .. . xˆ (t − nb )
−w( ˆ t − 1) ˆ t − 2) −w( , ϕˆ n (t ) = .. .
−w( ˆ t − nc )
(17)
D. Wang et al. / Mathematical and Computer Modelling 52 (2010) 309–317
(a) Preloads and dead-zones.
(b) Saturation.
(c) Hysteresis-relay.
(d) Hysteresis.
313
Fig. 3. Some typical nonlinearities.
ˆ s (t )θˆ s (t ), ra (t ) = ϕ
(18)
xˆ (t ) = kˆ 2 (t )u(t ) + [kˆ 1 (t ) − kˆ 2 (t )]u(t )h(t ),
(19)
w( ˆ t ) = y(t ) − ϕˆ Ts (t )θˆ s (t ).
(20)
T
ˆ 0) = 1/p0 , p0 = 106 , 1 representing a column vector of appropriate size whose The initial values are generally taken to be θ( elements are all 1. 4. The algorithm expansion There exist many other nonlinearities, e.g., preloads and dead-zones, saturation, hysteresis-relay and hysteresis, as shown in Fig. 3. The above H-AM-RGLS algorithm can be extended to deal with the Hammerstein systems with such nonlinearities. Here, we take the preloads and dead-zones nonlinearity in Fig. 3(a) as an example and derive the corresponding H-AM-RGLS algorithm. 4.1. The preloads and dead-zones nonlinearity The output x(t ) of the preloads and dead-zones nonlinearity in Fig. 3(a) can be written as x(t ) = 0.5{1 + sgn[|u(t )| − τ ]}¯x(t ),
(21)
x¯ (t ) = k(t ){u(t ) − τ sgn[u(t )]} + csgn[u(t )],
(22)
k(t ) = k2 + (k1 − k2 )h(t ),
(23)
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where k1 and k2 are the corresponding segment slopes, τ > 0 is the dead-zone and c is the preload constant, h(t ) is a switched function. Define middle variable
ξ (t ) = k(t )u(t ) = k2 u(t ) + (k1 − k2 )h(t )u(t ).
(24)
Separating the key-terms x¯ (t ) and ξ (t ), (21) and (22) can be rewritten as x(t ) = x¯ (t ) − 0.5{1 − sgn[|u(t )| − τ ]}¯x(t ),
(25)
x¯ (t ) = k(t )u(t ) − τ sgn[u(t )]k(t ) + csgn[u(t )] = ξ (t ) − τ sgn[u(t )]
ξ (t ) + csgn[u(t )]. u(t )
(26)
The key-terms in (25) and (26) are x¯ (t ) and ξ (t ), the first terms on the right-hand sides of (25) and (26) are the separated key-terms x¯ (t ) and ξ (t ), respectively. Substituting (24) into the separated key-term ξ (t ) in (26) gives x¯ (t ) = k2 u(t ) + (k1 − k2 )h(t )u(t ) − τ sgn[u(t )]
ξ (t ) + csgn[u(t )]. u(t )
(27)
Substituting (27) into the separated key-terms x¯ (t ) in (25) yields x(t ) = k2 u(t ) + (k1 − k2 )h(t )u(t ) − τ sgn[u(t )]
ξ (t ) + csgn[u(t )] − 0.5{1 − sgn[|u(t )| − τ ]}¯x(t ). u(t )
(28)
4.2. The recursive algorithm For the Hammerstein OEAR system in Fig. 1 with the nonlinearity in (28), assume that the linear block is still an OEAR model, substitute (28) into the separated key-term x(t ) in (6) gives
ξ (t ) + csgn[u(t )] − 0.5{1 − sgn[|u(t )| − τ ]}¯x(t ) u( t ) + b1 x(t − 1) + · · · + bnb x(t − nb ) − a1 r (t − 1) − · · · − ana r (t − na ).
r (t ) = k2 u(t ) + (k1 − k2 )h(t )u(t ) − τ sgn[u(t )]
(29)
Define
ϕs1 (t ) ϕ1 (t ) := ∈ Rna +nb +nc +4 , ϕn (t ) u(t )
θ θ 1 := s1 ∈ Rna +nb +nc +4 , θn
k 2 u(t )h(t ) k − k2 1 −sgn[u(t )] ξ (t ) τ u(t ) c sgn[u(t )] a1 −r (t − 1) na +nb +4 . ∈ Rna +nb +4 , ϕs1 (t ) := , θ s1 := .. .. ∈ R . an a −r ( t − n a ) b1 x(t − 1) . . .. . . bn b x(t − nb ) −w(t − 1) c1 c2 −w(t − 2) nc ∈ R nc , ϕn (t ) := θ n := .. .. ∈ R . . . −w(t − nc )
cnc
Then, we have r (t ) = ϕTs1 (t )θ s1 − 0.5{1 − sgn[|u(t )| − τ ]}¯x(t ),
(30)
w(t ) = ϕ (t )θ n + v(t ).
(31)
T n
Substituting (30) and (31) into (10) gives y(t ) = ϕTs1 (t )θ s1 + ϕn (t )T θ n + v(t ) − 0.5{1 − sgn[|u(t )| − τ ]}¯x(t )
= ϕT1 (t )θ 1 + v(t ) − 0.5{1 − sgn[|u(t )| − τ ]}¯x(t ).
(32)
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Replacing the unmeasured true outputs r (t − i), unknown inner variables ξ (t ), x¯ (t ) and x(t − i) and unknown noise terms w(t − i), with the outputs of an auxiliary model ra (t − i), the estimates ξˆ (t ), xˆ¯ (t ), xˆ (t − i) and w( ˆ t − i), a similar derivation to that of the H-AM-RGLS algorithm for Hammerstein models with the two-segment nonlinear block can result in the following H-AM-RGLS algorithm for estimating θ 1 for Hammerstein models with the preload and dead-zone nonlinearity:
θˆ 1 (t ) = θˆ 1 (t − 1) + L 1 (t )[y(t ) + 0.5{1 − sgn[|u(t )| − τˆ (t )]}xˆ¯ (t ) − ϕˆ T1 (t )θˆ 1 (t − 1)],
(33)
ˆ 1 (t )[1 + ϕˆ (t )P 1 (t − 1)ϕˆ 1 (t )] L 1 (t ) = P 1 (t − 1)ϕ
(34)
T 1
−1
,
ˆ (t )]P 1 (t − 1), P 1 (t ) = [I − L 1 (t )ϕ
(35)
ϕˆ (t ) ϕˆ 1 (t ) = s1 , ϕˆ n (t )
(36)
T 1
θˆ (t ) θˆ 1 (t ) = ˆs1 , θ n (t ) u( t ) kˆ 2 (t ) u(t )h(t ) kˆ 1 (t ) − kˆ 2 (t ) −sgn[u(t )] ξˆ (t ) τˆ (t ) u(t ) ˆ c ( t ) sgn[u(t )] ˆ a ( t ) 1 −r a ( t − 1 ) .. , ˆ s1 (t ) = θ ϕˆ s1 (t ) = , .. . . aˆ (t ) na −r a ( t − n a ) bˆ 1 (t ) xˆ (t − 1) .. . . .. bˆ nb (t ) xˆ (t − nb ) cˆ1 (t ) −w( ˆ t − 1) ˆ t − 2) −w( cˆ2 (t ) , ϕˆ n (t ) = θˆ n (t ) = .. .. , . . −w( ˆ t − nc ) cˆnc (t )
ξˆ (t ) = kˆ 2 (t )u(t ) + [kˆ 1 (t ) − kˆ 2 (t )]h(t )u(t ),
(37)
(38)
(39)
xˆ¯ (t ) = kˆ 2 (t )u(t ) + [kˆ 1 (t ) − kˆ 2 (t )]h(t )u(t ) − τˆ (t )sgn[u(t )]
ξˆ (t ) + cˆ (t )sgn[u(t )], u(t )
xˆ (t ) = kˆ 2 (t )u(t ) + [kˆ 1 (t ) − kˆ 2 (t )]h(t )u(t ) − τˆ (t )sgn[u(t )]
+ cˆ (t )sgn[u(t )] − 0.5{1 − sgn[|u(t )| − τˆ (t )]}xˆ¯ (t ),
(40)
ξˆ (t ) u( t ) (41)
ˆ s1 (t )θˆ s1 (t ) − 0.5{1 − sgn[|u(t )| − τˆ (t )]}xˆ¯ (t ), ra (t ) = ϕ
(42)
w( ˆ t ) = y(t ) + 0.5{1 − sgn[|u(t )| − τˆ (t )]}xˆ¯ (t ) − ϕˆ (t )θˆ s1 (t ).
(43)
T
T s1
5. Example Consider the following Hammerstein OEAR system with the linear block: B(z ) 1 y(t ) = x( t ) + v(t ), A(z ) C (z ) A(z ) = 1 + a1 z −1 + a2 z −2 = 1 − 1.60z −1 + 0.80z −2 , B(z ) = b0 + b1 z −1 + b2 z −2 = 1 + 0.20z −1 − 0.25z −2 , C (z ) = 1 + c1 z −1 = 1 − 0.85z −1 . The nonlinear block takes the following two cases, respectively.
• Case I. A two-segment piecewise nonlinearity: k1 u(t ), u(t ) > 0, x( t ) = k2 u(t ), u(t ) < 0. k1 = 1.50,
k2 = 3.00, θ = [k1 , k2 , a1 , a2 , b1 , b2 , c1 ]T = [1.50, 3.00, −1.60, 0.80, 0.20, −0.25, −0.85]T .
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Table 1 The parameter estimates and their errors of Case I.
σ2
t
k1
k2
a1
a2
b1
b2
c1
δ (%)
0.102
100 200 300 500 1000 1500 2000
1.45170 1.52228 1.53080 1.50655 1.50188 1.50123 1.50250
3.21115 3.11844 3.06550 3.04787 3.01963 3.01902 3.01584
−1.55526 −1.57035 −1.56885 −1.57702 −1.58710 −1.59067 −1.59197
0.78098 0.78500 0.77791 0.78657 0.79274 0.79520 0.79600
0.14211 0.17259 0.18995 0.18816 0.19101 0.19077 0.19343
−0.04448 −0.10188 −0.12652 −0.15821 −0.18961 −0.20258 −0.20880
−0.84574 −0.84707 −0.84833 −0.84870 −0.85008 −0.85087 −0.85087
10.23059 5.55161 3.82118 2.94794 1.74430 1.42985 1.21391
1.52073 1.57697 1.58351 1.53142 1.50746 1.51181 1.51052
3.33099 3.22659 3.10697 3.08489 3.01674 3.02797 3.02159
−1.55321 −1.55662 −1.54912 −1.55403 −1.57408 −1.58051 −1.58290
0.78847 0.77860 0.76352 0.77236 0.78493 0.78952 0.79125
0.06649 0.12523 0.16346 0.16947 0.18275 0.17991 0.18571
0.05916
0.502
100 200 300 500 1000 1500 2000
−0.00710 −0.04099 −0.08028 −0.12798 −0.15114 −0.16131
−0.83116 −0.83518 −0.83583 −0.83511 −0.84113 −0.84345 −0.84146
14.51659 9.60364 6.32617 5.29962 3.28961 2.77385 2.44079
1.50000
3.00000
−1.60000
0.80000
0.20000
−0.25000
−0.85000
True values
Table 2 The parameter estimates and their errors of Case II.
σ2
t
k1
k2
c
a1
a2
b1
b2
c1
0.102
100 200 300 500 1000 1500 2000
1.48965 1.50872 1.51805 1.49712 1.48430 1.49041 1.49111
3.00127 3.05324 3.03641 3.02904 3.00627 3.01217 3.00991
0.42512 0.40087 0.39320 0.39712 0.40613 0.40103 0.40145
−1.56500 −1.57931 −1.58194 −1.58760 −1.59478 −1.59688 −1.59747
0.78183 0.78924 0.78636 0.79179 0.79654 0.79805 0.79854
0.22953 0.22032 0.21932 0.20850 0.20303 0.20001 0.20011
−0.13478 −0.18475 −0.20231 −0.21966 −0.23351 −0.23834 −0.23978
−0.89487 −0.90905 −0.90931 −0.91425 −0.91236 −0.90971 −0.90563
3.46059 2.95934 2.33089 2.16170 1.75761 1.67635 1.54019
1.44308 1.57213 1.63009 1.53301 1.46791 1.49244 1.49072
3.38343 3.27995 3.18278 3.13025 3.01294 3.04137 3.03028
0.26321 0.32164 0.30388 0.35129 0.41207 0.39302 0.39771
−1.53338 −1.54970 −1.54924 −1.55433 −1.57718 −1.58466 −1.58703
0.78217 0.77582 0.76185 0.77016 0.78540 0.79076 0.79269
0.13889 0.19325 0.21992 0.21015 0.20634 0.19695 0.19862
0.02533
0.502
100 200 300 500 1000 1500 2000
−0.04708 −0.08278 −0.12438 −0.16816 −0.18777 −0.19448
−0.95209 −0.92595 −0.90454 −0.91044 −0.90095 −0.89411 −0.88429
17.15690 10.73282 7.23475 5.76535 2.83535 2.58371 2.12490
1.50000
3.00000
0.40000
−1.60000
0.80000
0.20000
−0.25000
−0.85000
True values
δ (%)
• Case II. A two-segment piecewise nonlinearity with preloads c: k1 u(t ) + c , u(t ) > 0, x(t ) = k2 u(t ) − c , u(t ) < 0. k1 = 1.50,
k2 = 3.00,
c = 0.40,
θ = [k1 , k2 , c , a1 , a2 , b1 , b2 , c1 ] = [1.50, 3.00, 0.40, −1.60, 0.80, 0.20, −0.25, −0.85]T . T
The input {u(t )} is taken as an uncorrelated persistent excitation signal sequence with zero mean and unit variance, and {v(t )} as a white noise sequence with zero mean and variances σ 2 = 0.102 and σ 2 = 0.502 , respectively. Applying the H-AM-RGLS algorithm to estimate the parameters of this system, the parameter estimates and their errors are shown in ˆ t ) − θk/kθk versus t are shown in Fig. 4 with different noise Tables 1 and 2, respectively, and the estimation errors δ := kθ( variances, where kX k2 := tr[X X T ] represents the norm of the squares matrix X . From Tables 1 and 2 and Fig. 4, we can draw the following conclusions:
• The parameter estimates given by the H-AM-RGLS algorithm converge to their true values as the noise variance becomes small.
• The parameter estimation errors become (generally) smaller and smaller with the data length t increasing. This shows that the proposed algorithm is effective. 6. Conclusions This paper presents an H-AM-RGLS algorithm for Hammerstein systems with two-segment piecewise nonlinearity by using the key-term separation principle. The proposed algorithms can obtain the system model parameter estimates, the noise model parameter estimates and the internal variable estimates, and can be extended to deal with the complex nonlinear blocks, e.g., preloads and dead-zones, saturation, hysteresis-relay and hysteresis. The proposed method can
D. Wang et al. / Mathematical and Computer Modelling 52 (2010) 309–317
317
Fig. 4. The parameter estimation errors δ versus t of Case I.
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