Proceedings of the 7th ICEENG Conference, 25-27 May, 2010
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7th International Conference
on Electrical Engineering
ICEENG 2010
Military Technical College
Kobry El-Kobbah,
Cairo, Egypt
Non-Linear Local Optimal Control for Multivariable Systems
By
Mahmoud Ashry*
Gamal Elnashar*
Tim Breikin**
Abstract:
To control non-linear systems along the whole range of operations, Non-linear control
techniques are used. Neural network and generalized minimum variance controllers are
two of the most common non-linear control techniques.
In this paper, Local optimal controller is generalized to deal with non-linear systems.
This new non-linear controller is used to deal with multivariable systems as well as
single-input single-output systems.
The effectiveness of the non-linear local optimal controller when dealing with
multivariable systems is represented in this paper based on the results obtained from the
high purity distillation column model.
Keywords:
Non-linear control, local optimal control, multivariable systems.
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* Egyptian Armed Forces
** Control systems centre, University of Manchester, Manchester, UK
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1. Introduction:
Non-linear control is the branch of control engineering that deals with non-linear
systems. As non-linear systems are difficult to be analyzed, linear approximations are
used to analyze those systems. However, such approximations are only valid for certain
ranges of operation. This approximation process is known as linearization. Along the
whole range of operation of the non-linear system, certain operating points are defined.
Based on the linearization technique, linear model is representing the non-linear system
around these operating points. As such, the linear control techniques can be used to
control the non-linear system for each region of operation [1, 2].
However, the controlled system performance degrades as the system subjected to rapid
and large changes around its operating point. As a result, non-linear controllers are
proposed to deal with the non-linear models [3]. Neural network and generalized
minimum variance controllers are two of the most common non-linear control
techniques [4–8].
Local Optimal Controller (LOC) is one of the advanced control techniques first
introduced by Lyantsev et al. in 2004 [9]. However, this LOC approach is incapable of
dealing with non-minimum phase systems. A modified LOC approach for multivariable
systems is proposed By Ashry, et al. in 2008 [10]. The proposed method provides closed
loop stability when dealing with non-minimum phase plant, which is a considerable
advantage over the original LOC. As discussed in [9, 10], LOC is a model-based
controller for multivariable systems as well as Single-Input Single-Output (SISO)
systems.
The modified LOC is generalized to deal with non-linear SISO systems [11]. In this
paper, LOC is used to control nonlinear multivariable systems as well as SISO systems.
High purity distillation column model [12, 13] is used as a highly non-linear
multivariable system to show the effectiveness of the proposed non-linear LOC.
An introduction to linear LOC is introduced in section 2. The proposed generalization of
the linear LOC to become a non-linear controller is deduced in section 3. The non-linear
LOC is designed in section 4 for a high purity distillation column as a highly non-linear
multivariable system. The simulated results are introduced. Finally, the conclusion
remarks are represented in section 5.
2. Linear local optimal control:
As mentioned, the local optimal control approach for linear model is described in [9, 10,
14]. This can be summarized as follows for a system model represented in discrete-time
form. Consider the system model as in (1).
1
1
(1)
A( z ) y (t ) B ( z )u (t 1)
-1
where z is the back shift operator, and
Proceedings of the 7th ICEENG Conference, 25-27 May, 2010
A( z
B( z
1
1
) 1 a z
1
1
)b b z
o
1
a z
2
1
2
b z
2
L L a
2
na
z
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na
L L b z
nb
nb
(2)
(3)
where na and nb represent the order of the system model polynomials with na > nb. A
model of the system in (1) can be presented in compact form as:
T
(4)
y (t ) (t 1)
where y(t) is the reference trajectory at sample t, is a vector of the identified system
parameters:
T
(5)
a1 ,L L , ana ,b0 ,b1 ,L L ,bnb
and φ is a data vector includes the past measured values of inputs and outputs
T
(t 1) y (t 1),L L , y (t na ),u (t 1),L L ,u (t nb 1)
(6)
A modified form for the output deviation is derived using (4) as:
T
y (t 1) y (t ) (t )
(7)
where Δ φ(t) is the measured input and output data deviations between successive
samples at sample t and can be defined as:
T
(8)
(t ) y (t ),L L , y (t na 1), u (t ),L L , u (t nb )
where
(9)
y (t ) y (t ) y (t 1) ,
and
(10)
u (t ) u (t ) u (t 1)
The model given in (4) presents an accurate description of the system. However, in this
expression the vector of the system parameters is approximately identified. This leads
to a residual error between the targeted trajectory, which can be defined as y*(t+1) and
the actual related measured output y(t). A weighting coefficient h is used to derive the
reference trajectory deviation from (5), (6) and (7) as:
na
nb
*
y (t 1) y (t 1) y (t ) h ai y (t i 1) b j u (t j )
i 1
j 0
(11)
The controller output rate of change Δ u(t) can be derived from (11) as:
u (t )
nb
1 y* (t 1) na
ai y (t i 1) b j u (t j )
b0
h
i 1
j 1
(12)
It’s clear from (12) that controller output deviation is function of system parameters,
past measured input and output data, reference trajectory, and the weighting coefficient
h (the only tunable parameter). For digital implementation, the controller output u(t) is
derived from (12) using backward Euler discrete integration [10, 14].
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3. Non-linear local optimal control:
The non-linear LOC design procedure is deduced based on the general NARMAX
model structure represented in (13) [12, 15].
ny
nu
ny
ny
y (t ) [ 0 y (t i ) u (t i ) i3, j y (t i ) y (t j )
1
i
i 1
nu
2
i
i d
i 1 j 1
ny
nu
nu
i4, j u (t i )u (t j ) i5, j y (t i )u (t j )
i d j d
i 1 j d
ny
ne
higher order termsupto deg ree l ] [i6, j y (t i )e(t j )
i 1 j 1
nu
ny
ne
nu
ne
u (t i )e(t j ) i8, j ,k y (t i )u (t j )e(t k )
7
i, j
i d j 1
(13)
i 1 j d k 1
all possiblecombinations of y (t ), u (t ), e(t ) upto deg ree l ]
ne
ne
ne
[ e(t i ) i10,j e(t i )e(t j )
9
i
i 1
i 1 j 1
higher order termsupto deg ree l ]
where:
ny is the output dynamic order,
nu is the input dynamic order,
ne is the error dynamic order,
l
is the degree of non-linearity,
d
is the input time delay.
This model structure can be presented in a compact form as in the following equation.
y (t ) 0 T (t 1) ,
(14)
where θ is the vector of the identified model parameters and ϕ is the data vector includes
the regressors. These two vectors are given in (15).
Proceedings of the 7th ICEENG Conference, 25-27 May, 2010
M
1
i
M
2
i
M
3
i , j
M
4
i , j
M
5
i , j
M ,
i6, j
M
i7, j
M
8
i , j ,k
M
9
i
M
10
i, j
M
M
y (t i )
M
u (t i )
M
y (t i ) y (t j )
M
u (t i )u (t j )
M
(
)
(
)
y
t
i
u
t
j
(t 1)
M
y (t i )e(t j )
M
u (t i )e(t j )
M
y (t i )u (t j )e(t k )
M
e(t i )
M
e(t i )e(t j )
M
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(15)
A modified form for the output deviation is derived using (14) as:
y (t 1) y (t ) T (t ) ,
(16)
where Δ ϕ(t) is the measured input and output data deviations between successive
samples at sample t and can be defined as:
Proceedings of the 7th ICEENG Conference, 25-27 May, 2010
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M
y (t i 1)
M
u (t i 1)
M
[ y (t i 1) y (t j 1)]
M
[u (t i 1)u (t j 1)]
M
y
t
i
u
t
j
[
(
1)
(
1)]
(t )
M
[ y (t i 1)e(t j 1)]
M
[u (t i 1)e(t j 1)]
M
[ y (t i 1)u (t j 1)e(t k 1)]
M
(17)
e(t i 1)
M
[e(t i 1)e(t j 1)]
M
and:
y (t i 1) y (t i 1) y (t i ),
u (t i 1) u (t i 1) u (t i ),
[ y (t i 1) y (t j 1)] y (t i 1) y (t j 1) y (t i ) y (t j ),
[u (t i 1)u (t j 1)] u (t i 1)u (t j 1) u (t i )u (t j ),
[ y (t i 1)u (t j 1)] y (t i 1)u (t j 1) y (t i )u (t j ),
(18)
[ y (t i 1)e(t j 1)] y (t i 1)e(t j 1) y (t i )e(t j ),
[u (t i 1)e(t j 1)] u (t i 1)e(t j 1) u (t i )e(t j ),
[ y (t i 1)u (t j 1)e(t k 1)] y (t i 1)u (t j 1)e(t k 1) y (t i )u (t j )e(t k ),
e(t i 1) e(t i 1) e(t i ),
[e(t i 1)e(t j 1)] e(t i 1)e(t j 1) e(t i )e(t j ).
Proceedings of the 7th ICEENG Conference, 25-27 May, 2010
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The model given in (14) presents an accurate description of the system. However, in this
expression the vector of the system parameters is identified (estimated). This leads to a
residual error between the targeted trajectory, which can be defined as y*(t + 1) and the
actual related measured output y(t). A weighting coefficient h is used to derive the
reference trajectory deviation from (15) and (16) as:
y* (t 1) y* (t 1) y (t ) h( T (t ) )
(19)
The controller output u(t) that minimizes Δ y*(t + 1) in (19) can be obtained as a
function of the system parameters, the past measured input and output data, the
reference trajectory, and the weighting coefficient h, which is the only tuneable
parameter.
4. Non-linear LOC for multivariable systems:
Identification and control of highly non-linear processes pose a challenging problem to
the process industry. In the absence of a reasonably accurate model, these processes are
fairly difficult to be controlled. High-purity distillation column is considered as an
example of non-linear multivariable system in this section [12, 13]. The NARMAX is
the model structure chosen for this case.
The non-linear multivariable LOC is designed in similar method to that for the linear
multivariable LOC but based on the relations deduced in Section 3 for NARMAX
models.
4.1 Mathematical model:
The process that is considered in this section is a methanol-ethanol distillation column
simulation model with 27 trays. The temperature on tray 21 (near the top) and tray 7
(near the bottom) are to be controlled by manipulating the reflux flow rate and the
reboiler vapor rate. The process variables are given in the following dimensionless
forms [12, 13].
T21 T21ss
,
30.0
L Lss
u1
,
6.0
y1
T7 T7 ss
,
30.0
V Vss
u2
,
6.0
y2
(20)
Proceedings of the 7th ICEENG Conference, 25-27 May, 2010
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where:
T21, T7 are temperature on tray 21 and tray 7 (oR), respectively;
T21ss is the steady state temperature on tray 21 (614.10oR);
T7ss is the steady state temperature on tray 7 (636.77oR);
L is the reflux flow rate (mol/min);
Lss is the steady state reflux flow rate (3.384 mol/min);
V is the reboiler flow rate (mol/min);
Vss is the steady state reboiler flow rate (3.856 mol/min).
The inputs used for model identification in this case are random variables taken from a
uniform distribution with appropriate range [16]. This range, which cover the operating
interval of interest for the given column has higher and lower, limits of 0.05 and -0.05
respectively [12,13]. Each input is allowed to change at each sampling time (every
minute in this case) with a predetermined probability (switching probability). I/O data
are generated and the model has been identified using Multi-Input Single-Output
(MISO) algorithm in which, each output is modeled separately. The identified model
given in [12, 13] is represented in (21).
y1 (k 1) 0.0012 0.98 y1 ( k ) 0.18u1 ( k )
1.1 y1 (k 2)u2 (k ) 1.8 y2 (k )u1 (k )
y2 (k 1) 0.0018 0.92 y2 (k ) 0.22u1 ( k )
(21)
30.4 y22 (k )u2 (k ) 1.7u22 (k )
4.2 Non-linear controller design:
The non-linear LOC is designed for the high purity distillation column using the nonlinear model given in (21). Similar designing technique to that of the linear LOC is used
but based on the non-linear model obtained. The non-linear LOC outputs (u1(k), u2(k))
are deduced in (22) and (23).
u1 (k )
1 1
[ (r1 (k ) y1 (k )) 0.98 y1 (k ) 0.98 y1 (k 1)
0.18 h1
0.18u1 (k 1) 1.1 y1 (k 2)u2 (k ) 1.1 y1 (k 3)u2 (k 1)
1.8 y2 (k )u1 (k ) 1.8 y2 (k 1)u1 (k 1)]
(22)
Proceedings of the 7th ICEENG Conference, 25-27 May, 2010
u2 (k ) (30.4 y22 (k ) 1.7u2 (k )) 1[
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1
(r2 (k ) y2 (k )) 0.92 y2 (k )
h2
0.92 y2 (k 1) 0.22u1 (k ) 0.22u1 (k 1)
(23)
30.4 y22 (k 1)u2 (k 1) 1.7u22 (k 1)]
The system controlled by non-linear LOC is simulated using Simulink [17]. Two
reference inputs are applied to the controlled system, one at 16.67 min and the other at
50 min. The reference inputs and the simulated output responses of the high purity
distillation column model controlled using non-linear LOC are shown in Figure (1).
Also, the non-linear LOC outputs (plant input) are shown in Figure (2).
From these figures, the reference input trajectories are achieved. It should be noted that
the two outputs are completely decoupled.
Comparing these results with the results presented in [13] using non-linear model
predictive controller described in [18], faster output responses and better decoupling
between the outputs are achieved using the non-linear LOC.
Figure (1): The reference inputs and the simulated output responses of the controlled
system.
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Figure (2): The simulated controller outputs (input to the plant).
5. Conclusions:
In this paper, a generalized version of the LOC is introduced as a non-linear controller
that deals with non-linear models presented in NARMAX model structure.
The non-linear LOC is designed to control non-linear multivariable systems. The results
obtained based on the high purity distillation column model show the effectiveness of
the non-linear LOC when dealing with multivariable systems. Faster response and high
degree of decoupling are the advantages of the new non-linear LOC.
As a model-based controller, the non-linear LOC can be used in automatic adaptation
mode based on the on-line system identification.
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Gamal Elnashar