BLOCK-ORIENTED CONTINUOUS-TIME MODELING
FOR NONLINEAR SYSTEMS UNDER SINUSOIDAL INPUTS
D. Zhai1, D. K. Rollins, Sr. 2, and N. Bhandari3
Department of Chemical Engineering1, 2, 3 and Department of Statistics1, 2
Iowa State University, Ames, IA 50011, USA
Emails: dzhai@iastate.edu1, drollins@iastate.edu2
2
Corresponding author
Abstract
Discrete-time modeling (DTM) dominates the systems engineering literature in the applications
of block-oriented modeling. The discrete environment of computer-based process control
systems and discrete sampling are the two major reasons [1]. Also, a DTM is easier to obtain
because all input changes are approximated by piecewise step input sequences. Nonetheless,
DTM has (potentially) two critical drawbacks relative to continuous-time modeling (CTM).
DTM requires constant and frequent sampling and can only predict at those points. DTM is not
potentially as accurate as CTM since, at best, it can only approximate continuous-time processes.
In addition, if the error term for a model is stochastically continuous (as in Brownian motion)
and must be treated as such, then DTM is not useful. For Hammerstein and Wiener CTM, our
research group [2, 3] introduced a non-compact CTM approach without restriction as well as a
compact approach with restriction to piece-wise step inputs. This article extends their work and
proposes compact CTM algorithms under sinusoidal input sequences for Hammerstein and
Wiener modeling. The proposed algorithm depends only on the most previous input change and
2
provides an exact solution to true Hammerstein and Wiener systems. The proposed method is
applicable to multiple-input, multiple-output systems as demonstrated.
Key Words
Nonlinear systems, continuous-time modeling, sinusoidal input sequence, Hammerstein and
Wiener models
1. Introduction
Many physical processes are nonlinear and dynamic in nature. Few of those nonlinear dynamic
problems can be solved analytically because usually there is no closed-form algorithm for the
nonlinear descriptive equations and the rigorous analytical techniques to analyze nonlinear
processes are usually thought to have limited practical applications. Instead, numerical analysis
and discrete modeling are widely used because data are sampled at discrete times and stored in
computer databases. The variables are assumed to remain fixed at their sampled values between
one sample instant and the next, though variables in most physical processes are continuously
changing. Thus, information between the sampling time points is missing in discrete sampling.
Furthermore, in DTM, the sampling conditions, such as sampling time and sampling frequency,
play an important role in prediction. In systems engineering, CTM has seen limited applications
even though it has the advantage of prediction at any time, and not just at discrete times. Other
advantages of CTM over DTM include fewer model coefficients and parameters with physical
meaning. With DTM, the continuous process input has to be approximated as piece-wise step
changes. For a continuous model, it is possible to have a compact closed-form algorithm [3],
which does not require iterative calculation. Another advantage of CTM is the ability to apply
3
analytical treatment when require such as in D-optimal experimental design [4]. Also, CTM
identification can be easier than DTM identification due to fewer parameters and a clearer
analytical algorithm form. In addition, when the error term of a model is considered to be
stochastically continuous and must be treated as such, DTM is not appropriate to use and CTM
has to be employed.
Hammerstein and Wiener models are two simple and popular block-oriented model structures
with relatively few dynamic model parameters. Due to their simple structures and efficient
parameterization, the Hammerstein and Wiener models have many applications in practice and
are becoming more popular. For example, the Hammerstein and Wiener models have been
shown to represent many nonlinear chemical processes very well, such as pH neutralizations,
distillation columns, and continuous-stirred tank reactors (CSTR). Much work has been done to
investigate the extent of the applications of this model. Almost all of the work involves use of
discrete-time models [5-12]. Noted exceptions include Greblicki [13-15], Rollins et al. [3], and
Bhandari and Rollins [2, 16]. Greblicki [13-15] introduced a nonparametric continuous-time
approach with the dynamic block identified by impulse response methods, and our research
group has proposed parametric continuous-time Hammerstein and Wiener modeling methods.
The closed-form algorithms for step input changes for Hammerstein and Wiener systems with
various dynamics have been determined by Rollins et al. [3] and Bhandari and Rollins [2] in a
compact form, referred to as H-BEST and W-BEST, respectively, and demonstrate exact
agreement with true Hammerstein or Wiener systems. H-BEST was applied to a household
dryer [3] and a distillation column [16]. Application of W-BEST includes a CSTR [2].
4
In all the studies involving H-BEST and W-BEST techniques, only step input changes were
considered. Under real conditions, input changes are often gradual or periodical [17], which can
be described as a ramp or a sinusoidal function, respectively, in some cases. For example, any
device operating by AC current can potentially induce periodic variability into the process. Also,
cooling water temperatures can fluctuate with ambient conditions and exhibit day-to-night-to-day
fluctuations. These cyclic process changes can often be approximated as sinusoidal functions.
For these kinds of periodical changes, it is important to have a high sampling frequency to obtain
adequate information for the system and to avoid aliasing. However, sufficiently frequent
sampling is not always possible or not always available, especially for some variables such as
concentration measurements of distillation columns. Of course, the periodical input changes can
be approximated as piece-wise step changes. Either DTM or the current H-BEST methodology
could then be employed. However, these approaches may perform unacceptably with inadequate
sampling. Note that noisy measurements can often be described as summations of sinusoidal
waves with an additive noise term [18]. Once the spectral decomposition is done and the
amplitudes and frequencies of the sinusoidal waves are identified, the proposed algorithm can be
employed to obtain the process outputs efficiently for even noisy input signals.
However, to our knowledge, no closed-form algorithm has been presented for Hammerstein
and Wiener systems under the sinusoidal input sequences. Kalafatis et al. [11] made use of
sinusoidal inputs to excite a pH process, which was modeled as a Wiener system; however, they
used a frequency sampling filter (FSF) method for the linear dynamics of a Wiener model with
periodical excitation of the system, which is quite different from the approach proposed here.
5
Rollins et al. [3] presented a non-compact algorithm without restriction and a compact
algorithm for piece-wise step input sequences. This paper extends their work to a compact CTM
algorithm for sinusoidal input changes in Hammerstein and Wiener modeling. The compact,
closed-form algorithms given by process analysis, along with some simulation results are
presented for Hammerstein and Wiener systems with first- and second-order process dynamics
when they are excited by different sinusoidal input changes. Our proposed algorithms can be
exploited for creating a methodology for block-oriented predictive modeling.
After briefly describing the Hammerstein and Wiener systems in Section 2, Section 3 presents
general algorithms to the Hammerstein and Wiener systems when the inputs follow sinusoidal
functions. The algorithms are presented for single-input, single-output (SISO) systems, which
hold analogously for multiple-input, multiple-output (MIMO) systems. This section also includes
an extension to systems with time delay. Two sinusoidal input sequence cases are considered for
each model. Applications to theoretical Hammerstein or Wiener systems are also presented to
verify the closed-form compact algorithms. In Section 4, the algorithms are applied to a MIMO
system. Section 5 gives the concluding remarks of the proposed method.
2. The Hammerstein and Wiener systems
a)
u(t )
f (u(t ))
Static
Nonlinear
Mapping
v (t )
b)
g(t )
Linear
Dynamics
y(t )
u(t )
g(t )
v (t )
Linear
Dynamics
f (v (t ))
Static
Nonlinear
Mapping
Figure 1. a) General Hammerstein model structure and b) General Wiener model structure
y(t )
6
A Hammerstein system [19] consists of a static nonlinear mapping or gain followed by a linear
dynamic block, as shown in Fig. 1a, where u(t) is the input vector, v(t) is the intermediate vector,
which is not measurable, and y(t) is the output vector; f(u(t)) represents the nonlinear static gain
functions, and g(t) describes the linear dynamic block. Note that v(t) = f(u(t)) and each element
of y(t) can be obtained by convolution of v(t) and g(t). For simplicity, it is assumed that all these
variables are deviations variables. u(t), v(t), and y(t) are all vectors, and f(u(t)) and g(t) could be
several different nonlinear mappings and linear dynamic relations, respectively.
A Wiener system consists of the same two blocks but in a reverse order, which is a dynamic
block followed by a static nonlinear mapping or gain, as shown in Fig. 1b. Each element of v(t)
can be obtained by convolution of u(t) and g(t) and y(t) = f(v(t)).
3. Hammerstein and Wiener algorithms
In this section, we present closed-form, compact algorithms for the Hammerstein and Wiener
systems with first- and second-order dynamics for specific forms, without loss of generality, of
sinusoidal input sequences. The algorithms are in closed-form and only depend on the most
recent input changes (i.e. are compact).
Once the changing point is identified, and the
information on amplitude, frequency, phase angle, and the step change is obtained, the outputs
can be predicted by employing the results in this section. In this section, all processes are
initially at steady state and only deviation variables from this steady state are used.
3.1 The Hammerstein system with first-order dynamics
The following algorithms are based on a SISO Hammerstein system with first-order dynamics, as
7
described by (1) and (2) below:
τ
dy (t )
+ y (t ) = v (t ) = f (u (t ))
dt
(1)
giving the transfer function in the Laplace domain as:
G (s ) =
Y (s )
1
=
V (s ) τ s + 1
(2)
where g(t) is its corresponding function in the time domain with a unit step forcing function.
Here, τ is the time constant.
Case I. The sinusoidal input change introduced to the Hammerstein system is imposed on the
step input changes and can be described mathematically as:
u (t ) = bn + An sin(ω n (t − t n −1 ))
for t n −1 < t ≤ t n
(3)
For the nonlinear polynomial static mapping relationship shown in (4), the algorithm (i.e., series
of equations) for the Hammerstein system, for the interval t n −1 < t < t n , is given by (5) to (8).
f (u (t )) = a1u (t ) + a 2 u 2 (t ) ,
(4)
1
⎛
2
2⎞
y (t ) = ⎜ a1bn + a 2 bn + a 2 An ⎟ ⋅ g 0 (t − t n −1 ;τ ) + (a1 An + 2a 2 bn An ) ⋅ g s (t − t n −1 ; ωn ,τ )
2
⎝
⎠
1
2
− a 2 An ⋅ g c (t − t n −1 ;2ωn , τ ) + y (t n −1 ) ⋅ e −(t −tn −1 ) /τ
2
(5)
where g 0 (t; τ ) , g s (t; ω,τ ) , and g c (t; ω,τ ) are defined as:
g 0 (t;τ ) = 1 − e − t / τ
g s (t; ω,τ ) =
g c (t; ω,τ ) =
(6)
1
1 + (ω t )
2
1
1 + (ω t )
2
[ωτ ⋅ e
[− e
−t / τ
−t / τ
]
− ωτ ⋅ cos(ω t ) + sin(ω t )
]
+ ωτ ⋅ sin(ω t ) + cos(ω t )
(7)
(8)
8
The simulation is done for quadratic nonlinear static mapping on a theoretical Hammerstein
process. The dataset of y-true is simulated (i.e., artificially generated) from a Hammerstein
process that can be described as follows:
v (t ) = 1.0 ⋅ u (t ) + 2.0 ⋅ u 2 (t ) and 5.0
dy (t )
+ y (t ) = v (t ) with y (0) = 0
dt
As shown in Fig. 2, the predicted outputs based on the provided formula and the true process
values show perfect agreement for the arbitrary input sequence given in Fig. 2b.
a) 30
b) 5
y-predicted
y-true
25
1
15
u
y
20
3
-1
10
-3
5
0
-5
0
100
time
200
300
0
100
time
200
300
Figure 2. a) Simulated outputs (y-true) and predicted outputs (y-predicted) on a theoretical
Hammerstein process for Case I when forced by b) sinusoidal input change sequence (u) with
different ωi, bi, and Ai values.
Case II. The sinusoidal input change with changing phase described by (9) is introduced to the
Hammerstein system:
u (t ) = sin(ω n (t − t n −1 ) + φn )
for t n −1 < t ≤ t n
For different polynomial static mapping relationships, the algorithms in the interval
(9)
t n−1 < t ≤ t n
for
the Hammerstein system are given below. In (A), the summation is up to an even order, while it
is up to an odd order in (B). The same Hammerstein process is employed here but with a
different input sequence. The simulation results are shown in Fig. 3 for quadratic nonlinear
9
static mapping on a true Hammerstein model. As shown, the predicted outputs and the true
process values overlap exactly.
2m
(A) f (u (t )) = ∑ a i u i (t ) ,
(10)
i =1
m
⎛2 j⎞ 1
y (t ) = ∑ a 2 j ⎜⎜ ⎟⎟ 2 j g 0 (t − t n −1 ;τ )
j =1
⎝ j ⎠2
m a
m
k −1 ⎛ 2 j − 1⎞
2 j −1 ⎡
+ ∑ 2 j −2 ⎢∑ (− 1) ⋅⎜⎜
⎟⎟ ⋅ (cos((2k − 1) φn ) ⋅ g s (t − t n −1 ; (2k − 1) ωn , τ )
−
j
k
j =1 2
k
=
1
⎝
⎠
⎣
+ sin ((2k − 1) φn ) ⋅ g c (t − tn −1; (2k − 1) ωn ,τ ))]
a2 j ⎡ m
2j ⎞
(− 1)k ⎛⎜⎜
⎟⎟(cos(2kφn ) ⋅ g c (t − t n −1 ;2kωn , τ ) − sin(2kφn ) ⋅ g s (t − t n −1 ;2kωn , τ ))]
2 j −1 ⎢ ∑
−
j
k
j =1 2
k
=
1
⎝
⎠
⎣
− (t −tn −1 ) / τ
+ y (t n −1 ) ⋅ e
(11)
m
+∑
2 m −1
f (u (t )) = ∑ a i ⋅ u i (t ) ,
(B)
(12)
i =1
m
⎛2 j⎞ 1
y (t ) = ∑ a 2 j ⎜⎜ ⎟⎟ 2 j g 0 (t − t n −1 ;τ )
j =1
⎝ j ⎠2
a 2 j −1 ⎡ m
2 j − 1⎞
(− 1)k −1 ⋅⎛⎜⎜
⎟⎟ ⋅ (cos((2k − 1) φn ) ⋅ g s (t − t n −1 ; (2k − 1) ωn ,τ )+ sin((2k − 1) φn ) ⋅ g c (t − t n −1 ; (2k − 1) ωn ,τ ))]
2 j −2 ⎢∑
j =1 2
⎝ j−k ⎠
⎣ k =1
m
+∑
a 2 j ⎡ m −1
2j ⎞
(− 1)k ⎛⎜⎜
⎟⎟(cos(2kφn ) ⋅ g c (t − t n −1 ;2kωn , τ ) − sin(2kφn ) ⋅ g s (t − t n −1 ;2kωn , τ ))]
2 j −1 ⎢ ∑
j =1 2
⎝ j − k⎠
⎣ k =1
+ y (tn −1 ) ⋅ e − (t −tn −1 ) /τ
(13)
m −1
+∑
b) 1.5
a) 1.8
1
0.5
1
u
y
1.4
0
-0.5
0.6
y-predicted
y-true
-1
-1.5
0.2
0
100 time
200
300
0
100
time
200
300
Figure 3. a) Simulated outputs (y-true) and predicted outputs (y-predicted) by (11) on a
theoretical Hammerstein model described above for Case II (A) with b) the input sequence (u);
the ωi varies from 0.4 to 1.5, and φi varies arbitrarily.
10
3.2 The Hammerstein system with 2nd-order dynamics
The following results are for a SISO Hammerstein system with second-order dynamics as
described in (14) and (15):
τ 1τ 2
d 2 y (t )
dy (t )
+ (τ 1 + τ 2 ) ⋅
+ y (t ) = v (t ) = f (u (t ))
2
dt
dt
(14)
with the transfer function in the Laplace domain as:
G (s ) =
Y (s )
1
=
V (s ) (τ 1 s + 1)(τ 2 s + 1)
Case I.
(15)
The sequence with sinusoidal input changes imposed on the step input changes
described in (3) is introduced to the above Hammerstein system. For the quadratic nonlinear
static mapping relationship given in (4), the algorithm for the Hammerstein system are written in
(16) to (21) for the time interval t n −1 < t ≤ t n :
1
⎛
2
2⎞
y (t ) = ⎜ a1bn + a 2 bn + a 2 An ⎟ ⋅ g 20 (t − t n −1 ;τ 1 , τ 2 ) + (a1 An + 2a 2 bn An ) ⋅ g 2 s (t − tn −1 ; ωn , τ 1 , τ 2 )
2
⎝
⎠
(16)
1
2
− a 2 An ⋅ g 2 c (t − t n −1 ;2ωn ,τ 1 , τ 2 ) + y (t n −1 ) ⋅ g 02 (t − t n −1 ;τ 1 , τ 2 ) + y ' (t n −1 ) ⋅ g12 (t − t n −1 ;τ 1 , τ 2 )
2
where:
g 2 s (t; ω ,τ 1 , τ 2 ) =
(1 − ω 2τ 1τ 2 )sin(ω t ) − ω (τ 1 + τ 2 ) ⋅ cos(ω t )
ωτ 12 ⋅ e − t /τ
ωτ 22 ⋅ e − t /τ
+
+
(1 + ω 2τ 12 )(1 + ω 2τ 22 )
(τ 1 − τ 2 )(1 + ω 2τ 12 ) (τ 2 − τ 1 )(1 + ω 2τ 22 )
1
2
(17)
g 2 c (t; ω ,τ 1 , τ 2 ) =
(1 − ω 2τ 1τ 2 )cos(ω t ) − ω (τ 1 + τ 2 ) ⋅ sin(ω t )
− τ 1 ⋅ e − t /τ1
− τ 2 ⋅ e − t /τ 2
+
+
(1 + ω 2τ 12 )(1 + ω 2τ 22 )
(τ 1 − τ 2 )(1 + ω 2τ 12 ) (τ 2 − τ 1 )(1 + ω 2τ 22 )
(18)
g 20 (t;τ 1 ,τ 2 ) = 1 +
τ1
τ 2 −τ1
e −t / τ 1 −
τ2
τ 2 −τ1
e −t / τ 2
(19)
11
τ 1e − t / τ − τ 2 e − t / τ
g 02 (t;τ 1 ,τ 2 ) =
τ1 −τ 2
1
g 12 (t;τ 1 ,τ 2 ) =
2
(20)
τ 1τ 2 ⋅ e − t / τ − τ 1τ 2 e − t / τ
τ1 −τ 2
1
2
(21)
The simulation results are shown in Fig. 4 for quadratic static mappings on a theoretical
Hammerstein model. The dataset of y-true is simulated from a nonlinear process that can be
described as follows:
v (t ) = 1.0 ⋅ u (t ) + 2.0 ⋅ u 2 (t ) and 15.0
d 2 y (t )
dy (t )
+ 8.0
+ y (t ) = v (t ) with y (0 ) = 0 , y ' (0 ) = 0
2
dt
dt
As shown by Fig. 4, the predicted process outputs and the simulated process values have
exact agreement for nonlinear static mappings. In this example, the nonlinear Hammerstein
system enlarges the oscillation considerably and shows larger deviations from the steady state
than the input variable. Even though the behavior of the nonlinear system is highly complex,
exact prediction is obtained from the proposed algorithm.
a) 25
b) 5
y-predicted
y-true
20
3
15
u
y
1
10
-1
5
-3
0
-5
0
100
time
200
300
0
100
time
200
300
Figure 4. a) Simulated outputs (y-true) and predicted outputs (y-predicted) by (16) on a
theoretical Hammerstein process described above with τ1 = 5.0, τ2 = 3.0, a1 = 1.0, a2 = 2.0 for b) a
sinusoidal input sequence with ωi varying from 0.4 to 3.0
12
Case II. The sinusoidal input change with changing phase as described by (9) is introduced to
the Hammerstein system. The algorithm for the Hammerstein system can be written as (22) for
the static mapping given in (4) in the time interval t n −1 < t ≤ t n .
2
a A
1
2
y (t ) = a 2 An ⋅ g 20 (t − t n −1 ;τ 1 , τ 2 ) + a1 An cos φn ⋅ g 2 s (t − t n −1 ; ωn , τ 1 , τ 2 ) + 2 n sin 2φn ⋅ g 2 s (t − t n −1 ;2ωn , τ 1 , τ 2 )
2
2
1
2
+ a1 An sin φn ⋅ g 2 c (t − t n −1 ; ωn , τ 1 , τ 2 ) − a 2 An cos 2φn ⋅ g 2 c (t − t n −1 ;2ωn , τ 1 , τ 2 )
2
'
+ y (t n −1 ) ⋅ g 02 (t − t n −1 ;τ 1 , τ 2 ) + y (t n −1 ) ⋅ g12 (t − t n −1 ;τ 1 , τ 2 )
(22)
The same Hammerstein process described for Case I is used in this case but with a sinusoidal
input sequence with phase changes. The simulation results are shown in Fig. 5 for nonlinear
(quadratic) static mappings on a true Hammerstein model. As before, the predicted response and
the true response agree exactly.
a) 4.4
2.5
y-predicted
u
y-true
1.5
3.3
0.5
2.2
-0.5
4
1.5
y
u
u
y
3
y-predicted
y-true
0.5
2
-0.5
1.1
-1.5
-2.5
0
b) 2.5
1
-1.5
0
0
100
100
200
time
time 200
300
300
0-2.5
0 0
100100
200
time 200
time
300
300
Figure 5. a) Simulated outputs (y-true) and predicted outputs (y-predicted) by (22) on a
theoretical Hammerstein process described above for Case II with τ1=5.0, τ2=3.0, a1=1.0, a2=2.0
and b) input sequence u with ωi, Ai, and φi varying arbitrarily.
3.3 The Wiener system with first-order dynamics
The following algorithms are for a SISO Wiener system with first-order dynamics, as described
by (23) and (24) below:
13
τ
dv (t )
+ v (t ) = u (t )
dt
(23)
which gives the transfer function in the Laplace domain as:
G (s ) =
V (s )
1
=
U (s ) τ s + 1
(24)
g(t) is its corresponding function in the time domain. And y (t ) = f (v (t )) gives the nonlinear
static mapping, which can be any nonlinear relation.
Case I. When the sinusoidal input change described by (3) is introduced into the Wiener system,
the algorithm for the interval t n −1 < t < t n is given by (25):
v (t ) = bn ⋅ g 0 (t − tn −1 ;τ ) + An ⋅ g s (t − tn −1 ; ωn ,τ ) + v (tn −1 ) ⋅ e − (t −tn −1 ) /τ
(25)
where g 0 (t; τ ) , and g s (t; ω,τ ) are defined as before.
a) 26
y-predicted
y-true
22
18
14
10
6
2
-2
b) 5
3
u
y
1
-1
-3
-5
0
100
time
200
300
0
100
time
200
300
Figure 6. a) Simulated outputs (y-true) and predicted outputs (y-predicted) on a theoretical
Wiener process for Case I when forced by b) a sinusoidal input change sequence (u) with
different ωi, bi, and Ai values.
The simulation is done for quadratic nonlinear static mapping on a theoretical Wiener process
as shown in Fig. 6. The dataset of y-true is simulated from a Wiener process that can be
described as follows:
14
5.0
dv (t )
+ v (t ) = u (t ) with v (0) = 0 and y (t ) = 1.0 ⋅ v (t ) + 2.0 ⋅ v 2 (t )
dt
As shown by Fig. 6a, the predicted outputs and the true process show perfect agreement for the
arbitrary input sequence given in Fig. 6b.
Case II. The same Wiener process as given in the previous subsection is employed here but with
a different input sequence. More specifically, the sinusoidal input change with changing phase
described by (7) is considered here. The algorithms in the interval t n −1 < t ≤ t n for the Wiener
system are given below:
v (t ) = An sin(φn ) ⋅ g c (t − t n −1 ; ωn ,τ ) + An cos(φn ) ⋅ g s (t − t n −1 ; ωn , τ ) + v (t n −1 ) ⋅ e − (t −tn −1 ) /τ
(26)
The simulation results are shown in Fig. 7 for quadratic static mapping on a true Wiener
system. The predicted outputs and the process values overlap exactly.
a) 1.2
b) 1.5
y-predicted
y-true
1
0.8
1
0.5
u
y
0.6
0.4
0
-0.5
0.2
0
-1
-0.2
-1.5
0
100
time
200
300
0
100
time
200
300
Figure 7. a) Simulated outputs (y-true) and predicted outputs (y-predicted) by (26) on a true
Wiener model described above for Case II; b) the input sequence (u) has ωi varying from 0.4 to
1.5, and φi varying arbitrarily.
15
3.4 The Wiener system with 2nd-order dynamics
The results in this section are based on a SISO Wiener system with second-order dynamics as
described by (27) and (28):
τ 1τ 2
d 2 v (t )
dv (t )
+ (τ 1 + τ 2 ) ⋅
+ v (t ) = u (t )
2
dt
dt
(27)
with the transfer function in the Laplace domain as:
G (s ) =
V (s )
1
=
U (s ) (τ 1 s + 1)(τ 2 s + 1)
(28)
Case I. The sinusoidal input change sequences described in (3) is introduced to the Wiener
system with second-order dynamics. The algorithm for the Wiener system is written in (29)
below for the time interval t n −1 < t ≤ t n :
v(t ) = bn ⋅ g 20 (t − t n −1 ;τ 1 ,τ 2 ) + An ⋅ g 2 s (t − t n−1 ; ω n ,τ 1 ,τ 2 )
+ v(t n−1 ) ⋅ g 02 (t − t n−1 ;τ 1 ,τ 2 ) + v' (t n −1 ) ⋅ g12 (t − t n −1 ;τ 1 ,τ 2 )
(29)
The simulation results are shown in Fig. 8 for quadratic static mappings on a true Wiener
model. The dataset of y-true is simulated from a nonlinear process that can be described as
follows:
d 2 v (t )
dv (t )
15.0
+ 8.0
+ v (t ) = u (t ) with v (0) = 0 , v ' (0) = 0 and y (t ) = 1.0 ⋅ v (t ) + 2.0 ⋅ v 2 (t )
2
dt
dt
As seen from this figure, the predicted process outputs and the simulated process values have
exact agreement for the nonlinear static mappings. In this example, the output sometimes shows
16
much larger deviations from the initial steady state than the input.
a) 23
b) 5
y-predicted
y-true
18
1
y
u
13
3
8
-1
3
-3
-2
-5
0
100
time
200
0
300
100
time
200
300
Figure 8. a) Simulated outputs (y-true) and predicted outputs (y-predicted) by (29) on a true
Wiener process described above for Case I with τ1 = 5.0, τ2 = 3.0, a1 = 1.0, a2 = 2.0, and b) the
input sequence (u) has ωi varying from 0.4 to 3.0, Ai, and bi varying arbitrarily.
Case II. The sinusoidal input change with changing phase as described by (7) is introduced to
the Wiener system. The algorithm for the Wiener system can be written as (17) in the time
interval t n −1 < t ≤ t n .
v(t ) = An cos φ n ⋅ g 2 s (t − t n −1 ; ω n ,τ 1 ,τ 2 ) + An sin φ n ⋅ g 2c (t − t n −1 ; ω n ,τ 1 ,τ 2 )
(30)
+ v(t n −1 ) ⋅ g 02 (t − t n −1 ;τ 1 ,τ 2 ) + v' (t n −1 ) ⋅ g12 (t − t n −1 ;τ 1 ,τ 2 )
a)
1
b) 2.5
y-predicted
y-true
0.8
1.5
0.6
y
0.5
u
0.4
-0.5
0.2
-1.5
0
-2.5
-0.2
0
100 time
200
300
0
100 time
200
300
Figure 9. a) Simulated outputs (y-true) and predicted outputs (y-predicted) by (30) on a
theoretical Wiener process described above for Case II with τ1=5.0, τ2=3.0, a1=1.0, a2=2.0 for b)
a sinusoidal input sequence with ωi, Ai, and φi varying arbitrarily,.
17
The same Wiener process described for Case I is used in this case but with a sinusoidal input
sequence with phase changes.
The simulation results are shown in Fig. 9 for nonlinear
(quadratic) static mappings on a true Wiener model. As previously, the predicted response and
the true response agree exactly.
3.5 Systems with time delay
Often a high order system can be approximated by lower order dynamics (either first-order or
second-order) with dead time [17]. Thus, algorithms for a system with time delay can be useful
and are therefore, needed in practice.
Once the dead time θ for a process is identified, it can be used with our proposed algorithms
with the following modification: for each time interval, replacing t in the formulas for the system
without time delay (as given in the previous sections) with (t-θ) gives the formulas for the
system with time delay.
4. Applications
Though the algorithms provided in the previous section are all for SISO Hammerstein and
Wiener systems, it is not difficult to apply them to the MIMO systems as shown below.
4.1 Application to MIMO Hammerstein system
To illustrate this application, suppose that a process is modeled by a two-input, two-output
(TITO) Hammerstein system (see Fig. 10) and the simulation results and predicted outputs are
18
presented in this section.
The nonlinear static mapping function with the interaction term can be written as:
v (t ) = f (u1 (t ), u 2 (t )) = a1 ⋅ u1 (t ) + a 2 ⋅ u1 (t ) + a 3 ⋅ u 2 (t ) + a 4 ⋅ u 2 (t ) + a 5 ⋅ u1 (t ) ⋅ u 2 (t )
2
2
(31)
where v (t ) goes through first-order dynamics to give y1 (t ) and through second-order dynamics
to give y 2 (t ) . The coefficient matrix is arbitrarily chosen as [1 1 − 1 − 1 1].
u1 (t )
f (u(t ))
u2 (t )
v(t )
Static
Nonlinear
Map
g1 (t )
y1 (t )
First Order
Dynamics
g2 (t )
y2 (t )
Second Order
Dynamics
Figure 10. A TITO Hammerstein system
a)
10
b) 10
y1-predicted
y1-true
7
y2-predicted
7
y2
y1
4
1
y2-true
4
1
-2
-2
-5
-5
0
50
100
150
time
200
250
300
0
50
100
150
time
200
250
300
Figure 11. The predicted TITO Hammerstein process outputs (y1-predicted and y2-predicted)
and the corresponding theoretical outputs (y1-true and y2-true) for first- and second-order
dynamics, respectively, agree exactly.
Arbitrary sinusoidal sequences are introduced into this TITO Hammerstein system. The
corresponding outputs for first- and second-order dynamics are plotted in Fig. 11. The predicted
responses and the true process responses overlap exactly, as shown. The system with second-
19
order dynamics has a less oscillatory response than that with first-order dynamics, which
confirms the frequency analysis results.
4.2 Application to MIMO Wiener system
Assume that a process can be modeled by a TITO Wiener system (see Fig. 12) and sinusoidal
input changes are introduced into it. Furthermore, assume that this process can be represented by
the following TITO Wiener system theoretically as shown in Fig. 12. One input u1 (t ) follows
the first order dynamics to give v1 (t ) and the other input u 2 (t ) follows the second order
dynamics to give v 2 (t ) . Thus, the final responses are obtained by the following nonlinear static
function (32) and (33) with the interaction term.
y1 (t ) = f 1 (v1 (t ), v 2 (t )) = a11 ⋅ v1 (t ) + a12 ⋅ v1 (t ) + a13 ⋅ v 2 (t ) + a14 ⋅ v 2 (t ) + a15 ⋅ v1 (t ) ⋅ v 2 (t )
(32)
y 2 (t ) = f 2 (v1 (t ), v 2 (t )) = a 21 ⋅ v1 (t ) + a 22 ⋅ v1 (t ) + a 23 ⋅ v 2 (t ) + a 24 ⋅ v 2 (t ) + a 25 ⋅ v1 (t ) ⋅ v 2 (t )
(33)
2
2
2
2
where a11 = 1, a12 = 1, a13 = −1, a14 = −1, and a15 = 1; a21 = −1, a22 = 1, a23 = 1, a24 = 0, and a25
= −1 are chosen arbitrarily.
u1 (t )
g1 (t )
v1 (t )
f 1 (v (t ))
y1 (t )
First Order
Dynamics
u2 (t )
g2 (t )
v 2 (t )
Second Order
Dynamics
f 2 (v (t ))
y2 (t )
Different Static
Nonlinear Mapping
Figure 12. A TITO Wiener system
The process outputs with different static mappings after the sinusoidal input change
sequences with arbitrary amplitudes and frequencies are introduced into the TITO Wiener system
20
are shown in Fig. 13. As shown, the predicted outputs by the closed-form compact algorithm
agree with the true Wiener system outputs exactly. Since each input has its own dynamic block,
to include more inputs is straightforward once the dynamic relations, static nonlinear mapping,
and the corresponding parameters are identified.
a)
b)
7
8
y1-true
y1-predicted
5
3
y2-true
y2-predicted
6
y2
y1
1
-1
4
2
-3
0
-5
-7
-2
0
100
time
200
300
0
100
time
200
300
Figure 13. The predicted TITO Wiener process outputs (y1-predicted and y2-predicted) and the
corresponding theoretical outputs (y1-true and y2-true) for different static mappings agree
exactly.
5. Concluding remarks
The process dynamics analysis of the nonlinear systems under sinusoidal input changes is
performed for first-order and second-order overdamped dynamics of block-oriented
Hammerstein and Wiener systems. The closed-form compact algorithms for sinusoidal input
changes considering the amplitude, frequency and phase changes are provided. By simulation on
theoretical Hammerstein and Wiener systems, the predicted responses by these algorithms
demonstrate exact agreement with the true process responses.
Only the previous input
information, the output response, and its derivative (for the second-order dynamics) at the time
of change are needed for the algorithm. The single-input, single-output (SISO) algorithms can
be applied to multiple -input, multiple-output (MIMO) systems as demonstrated in the two-input,
21
two-output (TITO) example. The predictions and the simulated theoretical system responses
agree exactly in all cases.
The proposed first-order and second-order overdamped algorithms, and their extensions to the
dynamics with dead time, can cover a wide range of dynamic processes that can be modeled as
Hammerstein and Wiener systems.
As pointed out by Hajjari and Eloutassi [18], noisy
measurements can be described as summations of sinusoidal waves with an additive noise term.
Once the sine wave parameters for the input sequence and the model parameters are estimated,
the closed-form compact algorithms provided in this work can be applied to obtain the system
outputs. See Hajjari and Eloutassi [18] for a method to obtain periodic functions from noisy
signal in practice. In Zhai et al. [20], we extend this methodology to second-order-plus-lead
processes and apply it to a continuous-stirred tank reactor (CSTR) that was determined to
approximate a Wiener system by Bhandari and Rollins [2]. This process has seven (7) inputs and
five (5) outputs and in [20] we present the results of one of the outputs in detail to sinusoidal
input changes when these changes are signal variations and when they are noise variations. This
work demonstrates the advantage this method has when input sampling is too slow to capture
periodic signal behavior effectively and its ability to effectively capture periodic noise in the
outputs when it is due to process periodic noise in the inputs. Currently this article is available at
the website: http://www.public.iastate.edu/%7edrollins/.
6. Acknowledgements
The authors would like to thank Professor Huaiqing Wu of Department of Statistics at Iowa State
University for his kind help.
22
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24
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Biographies
Dongmei Zhai is a graduate student in the Department of Chemical Engineering and Department
of Statistics at Iowa State University. Zhai worked at the Lanzhou Institute of Chemical Physics,
the Chinese Academy of Sciences for more than one year. Zhai holds a BS and ME in Chemical
Engineering from Zhejiang University, China. She has an MS in Chemical Engineering and an
MS in Statistics from Iowa State University also. She is now working on her PhD co-major
degree in Statistics and Chemical Engineering at Iowa State University.
Derrick K. Rollins, Sr. is an Associate Professor with a joint appointment in the Chemical
Engineering and Statistics Departments at the Iowa State University, Ames, IA. He has been in
his current position since 1990. Prior to graduate school, Rollins worked for seven years for Du
Pont in process engineering at three different sites. He has worked as a consultant for several
industrial companies, including Dow, 3M, and Shell, and has taught shorts courses in statistics
for Amoco and IMC Agrico. Rollins holds a BS in chemical engineering and an MS and PhD in
chemical engineering and an MS in statistics from the Ohio State University.
Nidhi Bhandari was a post-doc research associate in the Department of Chemical Engineering at
Iowa State University when this research was completed. She is now an Assistant Professor in
the Department of Chemical Engineering at Indian Institute of Technology, Roorkee, India. Her
research is in the area of predictive modeling and model predictive control. She has a bachelor
degree in Chemical Engineering from University of Roorkee, Roorkee, India. Bhandari worked
25
briefly at Reliance Industries Limited, India before joining graduate school at Iowa State
University. She holds a PhD in Chemical Engineering from Iowa State University.