Auxiliary model based recursive generalized least squares parameter estimation for Hammerstein OEAR systems

Auxiliary model based recursive generalized least squares parameter estimation for Hammerstein OEAR systems

Mathematical and Computer Modelling 52 (2010) 309–317 Contents lists available at ScienceDirect Mathematical and Computer Modelling journal homepage...

379KB Sizes 2 Downloads 61 Views

Mathematical and Computer Modelling 52 (2010) 309–317

Contents lists available at ScienceDirect

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

310

D. Wang et al. / Mathematical and Computer Modelling 52 (2010) 309–317

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.

D. Wang et al. / Mathematical and Computer Modelling 52 (2010) 309–317

311

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 )

312

D. Wang et al. / Mathematical and Computer Modelling 52 (2010) 309–317

Fig. 2. The Hammerstein OEAR system with an auxiliary model.

θ (t )



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)

314

D. Wang et al. / Mathematical and Computer Modelling 52 (2010) 309–317

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)

D. Wang et al. / Mathematical and Computer Modelling 52 (2010) 309–317

315

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 .

316

D. Wang et al. / Mathematical and Computer Modelling 52 (2010) 309–317

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.

be extended to multivariable systems [12–15], dual-rate nonlinear systems [16–19], non-uniformly sampled multirate nonlinear systems [20,21]. References [1] J.L. Figueroa, S.I. Biagiola, O.E. Agamennoni, An approach for identification of uncertain Wiener systems, Mathematical and Computer Modelling 48 (1–2) (2008) 305–315. [2] M. Xu, G.R. Chen, Y.T. Tian, Identifying chaotic systems using Wiener and Hammerstein cascade models, Mathematical and Computer Modelling 33 (4–5) (2001) 483–493. [3] F. Ding, T. Chen, Identification of Hammerstein nonlinear ARMAX systems, Automatica 41 (9) (2005) 1479–1489. [4] F. Ding, Y. Shi, T. Chen, Auxiliary model based least-squares identification methods for Hammerstein output-error systems, Systems & Control Letters 56 (5) (2007) 373–380. [5] F. Ding, Y. Shi, T. Chen, Gradient-based identification methods for Hammerstein nonlinear ARMAX models, Nonlinear Dynamics 45 (1–2) (2006) 31–43. [6] D.Q. Wang, F. Ding, Extended stochastic gradient identification algorithms for Hammerstein–Wiener ARMAX systems, Computers & Mathematics with Applications 56 (12) (2008) 3157–3164. [7] E.W. Bai, M. Fu, A blind approach to Hammerstein model identification, IEEE Transactions on Signal Processing 50 (7) (2002) 1610–1619. [8] F.Z. Chaoui, F. Giri, Y. Rochdi, M. Haloua, A. Naitali, System identification based on Hammerstein model, International Journal of Control 78 (6) (2005) 430–442. [9] J. Vörös, Parameter identification of discontinuous Hammerstein systems, Automatica 33 (6) (1997) 1141–1146. [10] J. Vörös, Recursive identification of Hammerstein systems with discontinuous nonlinearities containing dead-zones, IEEE Transactions on Automatic Control 48 (12) (2003) 2203–2206. [11] L.L. Han, J. Sheng, F. Ding, Y. Shi, Auxiliary model identification method for multirate multi-input systems based on least squares, Mathematical and Computer Modelling 50 (7–8) (2009) 1100–1106. [12] F. Ding, T. Chen, Hierarchical gradient-based identification of multivariable discrete-time systems, Automatica 41 (2) (2005) 315–325. [13] F. Ding, T. Chen, Hierarchical least squares identification methods for multivariable systems, IEEE Transactions on Automatic Control 50 (3) (2005) 397–402. [14] F. Ding, T. Chen, Hierarchical identification of lifted state-space models for general dual-rate systems, IEEE Transactions on Circuits and Systems–I: Regular Papers 52 (6) (2005) 1179–1187. [15] Y.J. Liu, Y.S. Xiao, X.L. Zhao, Multi-innovation stochastic gradient algorithm for multiple-input single-output systems using the auxiliary model, Applied Mathematics and Computation 215 (4) (2009) 1477–1483. [16] F. Ding, T. Chen, Z. Iwai, Adaptive digital control of Hammerstein nonlinear systems with limited output sampling, SIAM Journal on Control and Optimization 45 (6) (2006) 2257–2276. [17] F. Ding, T. Chen, Combined parameter and output estimation of dual-rate systems using an auxiliary model, Automatica 40 (10) (2004) 1739–1748. [18] F. Ding, T. Chen, Parameter estimation of dual-rate stochastic systems by using an output error method, IEEE Transactions on Automatic Control 50 (9) (2005) 1436–1441. [19] F. Ding, P.X. Liu, H.Z. Yang, Parameter identification and intersample output estimation for dual-rate systems, IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans 38 (4) (2008) 966–975. [20] F. Ding, L. Qiu, T. Chen, Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems, Automatica 45 (2) (2009) 324–332. [21] Y.J. Liu, L. Xie, F. Ding, An auxiliary model based recursive least squares parameter estimation algorithm for non-uniformly sampled multirate systems, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 223 (4) (2009) 445–454.