Journal of Nature and
Science, Vol.1, No.2, e41, 2015
Key Laboratory of Marine Simulation
and Control,
Robust |
closed-loop gain shaping | non-square systems | model perturbation | Padé
approximation Non-square system is a common
industrial process in fields of the practical engineering. The number of the
input variables does not equal to that of the outputs, e.g. the steam generator
level control system, the fuel cell voltage control system [1, 2], etc. The
control method of a non-square system is to transform it into a square system
by adding or deleting the input and output variables. But this conventional
method can not only increase the control cost but also reduce the control
performance, so it is difficult to obtain a satisfactory result. The method of decoupling internal model control for multi-variable
non-square system with time delays was proposed in [3], the controller had good
performance of tracking ability and strong robustness, but as the traditional
internal model control if the model was mismatched, the control effects would
be variation and the simulation result had overshoot also. A modified internal model
control method for non-square systems was proposed in [4], the control effect of the method was satisfactory. A controller was designed by steady gain matrix of
non-square systems in [5]. However the method belonged
to the static decoupling method which needed to regulate two parameters and it was time-consuming. A PI control method for non-square systems was proposed in [6], the control effect of the method was dissatisfactory. In [3~6], the designed
controllers had all pure time delays and were not minimum phase systems. Some
of them had zeros in the right-half plane. The author
presented a kind of closed-loop
gain shaping algorithm (CGSA) to non-square systems
with multiple time delays in 2009. The robust control design algorithm for non-square
systems was given through solving the pseudo-inverse of the nominal plant
without multiple time delays [7, 8]. It
has a simple solving process. The controller designed by this algorithm has
lower order and good robustness. CGSA
has been applied in many different fields [9-12]. Contrary to omitting the time
delays, this paper will study the system further, using Padé stable
approximation and equivalenting
the time delays as the first order component. Then it will obtain the robust
controller with closed-loop gain shaping algorithm. The new robust controller
can keep the set inputs without overshoot and steady state error, and it has
good robustness. The control effects are a little better than those in [7],
which are further verified
the feasibility and validity. Closed-Loop
Gain Shaping Algorithm (CGSA) Inspired
by H CGSA uses the
result of the mixed sensitivity algorithm of H For signal
tracking of MIMO systems, the closed-loop transfer function matrix
of system from input to output is actually the complementary
sensitive matrix
From
the formula above,
we obtain
For MIMO systems
where The definition of generalized inverse is
For the closed-loop system with two inputs and two outputs,
let the non-diagonal elements of
i.e.
the two inputs and the two outputs are completely decoupled. Let Then we can obtain the equation (2).
(2) Therefore, the controller (3) is as
follows.
(3)
Taking
a 2×3 non-square matrix in [4] as an example,
the transfer function matrix
of the system model is
(4) The controller was given by
using the modified internal model control (IMC) method in [4]
The controller has pure time
delays and zeros in the right-half plane, and it is not a minimum phase system.
This paper uses Padé
approximation to linearize the time delay system more precisely, if the time
delay link
(5) where the second
approximation Equation (5) shows that numerator and denominator
have the same order in the approximate formula, but the numerator polynomial
has negative
coefficient, which can cause system unstable when the
closed-loop system is constructed. In order to solve this problem we can derive
more proper Padé approximate formula. The Taylor series of time
delay function
using the advanced mathematics’ formula
then
if these parameter calculation is
used to design the controller directly, the final controller’s form is complex,
which has many parameters and large amount of
calculation. In order to simplify the problem and
reduce the calculation, making the time delay in every row of the controlled
object in (4) equal to its maximum value, then
Let Then it has
that the new matrix inversion equals originalmatrix inversionmultipliy by the changed
inverse matrix, so the solving of the new robust controller will change into a
simple process, that is where
The
final designed controller is a sixth order controller. The third order controller
can be acquired through a new order-reduction
algorithm based on the stability of consideration,
which took advantage of the approximate method
of the Routh [13]. The reduced third-order controller is given in equation (7).
(7) （a） （c） Fig.1.Set-point responses （a） （c） Fig.2.Model
perturbation responses (1) （a） （c） Fig.3.Model
perturbation responses (2) （a） （c） Fig.4.Disturbance
responses Simulation Analysis The simulation results of
set-point response are given in Fig.1, where The simulation
results of model perturbation response (1) are given in Fig.2, where the gains,
the time delays and the time constants of every element in the transfer function matrix
are increased by 10%. The solid line is the unchanged transfer function, and the
dotted line is the changed transfer function. From the analysis of Fig.2, it follows
that the system control effect
is satisfactory without overshoot
and static error, and the new
controller has good robustness. The simulation
results of model perturbation response (2) are given in Fig.3, where the gains,
the time delays and the time constants of every element in the transfer function matrix
are increased by 10%. The solid line is the new controller, and the dotted line
is the previous controller in [7]. From the analysis of Fig.3, it follows that
the previous controller has overshoot and the new controller has a better
robustness. The simulation
results of disturbance responses are given in Fig.4, where we set step
disturbance for the first channel with the magnitude of -0.1 and for the second
channel with 0.1 at 2000s respectively. From the analysis of Fig.4, it follows
that the new controller can get satisfactory results even there is a step
disturbance in the system, and the new controller has good robustness.
CGSA for multiple and large time
delays non-square systems is proposed in this paper using Padé
stable approximation of time delays, its control performance is satisfactory.
The control effects are a little better than those in [7]. Compared with [3-6],
the designing procedure of controller is relatively simple. The designed controller
in this paper has not pure time delays and zeros in the right-half plane, and
is a stable minimum phase system. The simulation results show that this method
has better control performance and good robustness than other control methods
of non-square systems.
This work is supported in part by National Natural Science Foundation of
China (Grant No. 51109020), a grant from the Major State Basic Research
Development Program of China (973 Program) (Grant No. 2009CB320805) The authors
would like to thank Jia Xinle for his insightful
remarks on this note, and anonymous reviewers for their valuable comments to
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of interest: No conflicts declared.
©
2015 by the Journal of Nature and Science (JNSCI).
Xianku Zhang was born in 1968. He received his Ph.D. in 1998 from Dalian
Maritime University (DMU), Hongshuai Pang was born in 1982. He is a lecturer at |