International
Carpathian Control
Conference ICCC’ 2002
MALENOVICE,
CZECH REPUBLIC
May 27-30, 2002
CLOSED LOOP CONTROL OF THE SYSTEM WITH THE MODES
OF DIFFERENT DYNAMICS AND DAMPING
Petr NOSKIEVIČ
Department of Control Systems and Instrumentation,
VSB - Technical University of Ostrava,
Ostrava, Czech Republic, petr.noskievic@vsb.cz
Abstract: The paper deals with the design and tuning of the PID controller for the closed
loop control of the system with modes of different dynamics and different damping. The
influence of the chosen controller on the properties of the controlled system is shown using
the critical gain and step responses. The use of the presented rules is demonstrated on the
tuning of the controller for the position control of the electrohydraulic drive.
Key words: Controller tuning, damping, hydraulic drive.
1 System analysis
Closed loop control of the five order system with two second order terms with the
different dynamics and damping Figure 1 is the topic of this paper. This structure of the
controlled system is typical for the actuator systems and position control especially for the
closed loop position control using the hydraulic drive. But the presented results may be
used also by the tuning of the controller of the closed loops of the similar structure and
different kind.
The controlled 5th order system is characterised by the two second order terms and one
integrator. The first second order term is well damped, the relative damping ration is
approximately 0.9, the second term is badly damped with the relative damping ratio from
0.05 to 0.2, the following derivations were made for the value 0.1. The dynamics of the is
determined by two values – the time constant Tsv and the time constant TM, which can be
calculated from the natural frequencies
Tsv =
1
ω sv
=
1
1
1
, TM =
.
=
2πf sv
ω M 2πf M
235
(1)
low
good damping
1
2 2
Tsv s + 2ξ svTsv s + 1
x2
1
TM2 s 2 + 2ξ M TM s + 1
x1
KO
s
x
Figure 1. The analysed and controlled system
The dynamics of the open loop, the choice of the right controller and the step response
of the closed loop controlled systems depend on the ratio of the natural frequencies
κ=
f sv
.
fM
(2)
This ratio is greater if the well damped term is faster than the second system with the
low damping. In the practice the ratio κ could be also lower than 1. The Figure 2 shows the
step responses of the closed loop for three different ratios κ=0.25, κ=0.7, κ=3 for the
increasing gain of the proportional controller. In the case of κ=0.7 it is possible using only
the proportional controller to make a necessary compromise between the swiftness of the
response and the allowable overshot and to achieve a damped step response without the
overshot. In the other cases the system starts to oscillate by the increasing gain or the step
response is too slow [NOSKIEVIČ 1995, DIETER 1998].
2 Controller structure
To achieve a good behaviour of the closed loop system it is necessary to select suitable
controller in dependence on the ratio κ - the proportional controller (P), the proportional
controller with derivative part (PDT1)
G ( s) = K R
236
Td s + 1
,
Ts + 1
(3)
the proportional controller with the time constant (PT1)
G(s) = K R
1
Ts + 1
(4)
or the state controller.
Fig.2 Normalised step responses of the closed loop systems with proportional
controller for κ=0.25, κ=0.7, κ=3 (from left).
Let the critical gain – the marginal stability is the criteria for the choice of the
controller. The suitable controller which enables as fast as possible but well damped
behaviour with high amplification will be chosen for the closed loop system characterised
by different ratios κ. The time constants of the controllers can be dimensioned using the
frequency response and using the numerical calculation of the margin stability.
Proportional controller allows to
set the high gain only if the ratio κ is in
the range from 0.5 to 1, Figure 3.
Otherwise the step response is slow or
the system starts to oscillate. For the
ratio κ lower that 0.5 and greater than 1
it is necessary to find a different
controller.
If we set the time constant Td as a
multiply of the time constant Tsv of the
good damped term and parameter a
Td = a * Tsv ,
Figure 3 Critical gain of the 5th order closed
loop system with the proportional controller
we can set the parameter a in dependence on the critical gain. Figure 4 shows the
calculated normalised critical gain for the PDT1 controller and varying time constant Td
and ratio κ in the range 0÷3. The highest influence of this type of controller on the margin
stability can be observe for the systems characterised by the ratio κ in the range 0 ÷ 0.5 and
for the value of parameter a = 1. In this range of κ and a it is the allowable critical gain
237
higher than the highest gain when we use the proportional controller. The time constant T is
set T=0.1*Td.
Figure 4. Normalised critical gain for the PDT1 controller (left), for the PT1 controller (right)
Figure 5. Maximum critical gain surface for three
controllers and different time constant setting
Figure 4 (right) shows the
normalised critical gain for the
different setting of the time
constant T of the PT1
controller. For the time constant
T=2*TM is the critical gain
nearly constant for κ ≥ 2. For
higher time constant T is the
critical gain higher but the step
response is not so swift because
the controller is working like
the low pass filter with very
low bandwidth.
As a conclusion we can
create the maximum value
surface shown in Figure 5 and
summarize the rules for the
controller choice and tuning in the Table 1 and Figure 6.
Table 1. Transfer functions of the controllers and recommended values of the ration κ.
PDT1 controller
P controller
PT controller
State controller
G( s ) = K R
TD = Tsv =
TD s + 1
,
Ts + 1
G( s ) = K R
1
, T = 0.1TD
2πf sv
κ = 0 ÷ 0 .5
κ = 0 .5 ÷ 1
238
1 ,
Ts + 1
1
T = 2TM =
πf M
G( s ) = K R
κ = 1÷ 3
u = sw − r T x
κ ≥3
Figure 6 shows the curves of the critical gain normalized to the cylinder natural
frequency for the three types of controller and different frequency ratios κ. The maximum
value of the critical gain determines the type of the appropriate controller. After the choice
of the right controller dependence on the ratio κ by the use of the plot in Figure 6, we can
set the time constants according to the Table 1 and for the final tuning of the gain apply the
root locus method supported by MATLAB designer for example.
Figure 6. Normalized critical gain for controller of a different kind
and different ratio κ.
3. Application examples of closed-loop position control
Closed-loop position control is a very frequent industrial application of linear
hydraulic servovalve controlled drives. The dynamic behaviour of the closed loop
controlled linear hydraulic in Figure 7 depends on the ratio κ of the natural frequencies of
the servovalve fsv and cylinder fM .
F
K F (TF s + 1)
cylinder
servovalve
w
u
e
G( s )
K sv K Q
KM
TM2 s 2 + 2ξ M TM s + 1
Tsv2 s 2 + 2ξ sv Tsv s +
v
Figure 7. Block diagram of the closed-loop position control system
239
1
s
x
The result of the controllers design for two valve/cylinder drives characterised by the
parameters Ksn = 66.67 Vm-1, KM= 1.0575e+3 m-2, Ksv = 0.1 V-1, Kq = 5.0e-4 m3s-1 , ξm=0.1,
ξsv =0.9, and the different dynamics given by the time constants Tsv = 0.008 s, Tm = 0.002 s,
κ=0.25 and Tsv = 0.008 s, Tm = 0.02 s, κ = 2.5 is shown in Fig.8. The first system is
characterised by the ratio κ = 0.25 and the appropriate controller is the PDT1- controller
according to Table 1. The fastest step response 1 in Fig.8 without the overshot was obtained
with the controller gain Kr = 13. The P controller with the same gain gives the step response
2 with the overshoot, and the slowest step response 3 without overshot is for the lower gain
of the P-controller. To achieve the best response of the second system characterised by
κ=2.5 it is necessary to use PT1 controller – step response 1 in Figure 8 right, response 2 is
by the use of the P controller with the same gain, response 3 – P-controller with the lower
gain.
Fig.8 Step responses for the two drives and different controller
Conclusions
Controller tuning of the 5th order systems with second order subsystems characterised
by different dynamics and different damping was described in the paper. The results have
general importance and may be applied to the control of fluid power systems, mechatronic
systems and other systems. The damping ratio of the both systems has influence on the
shape of the curve of the critical gain. The change of the lower damping ratio causes the
modification of the curve of the critical gain of the PDT1 controller, the change of the
higher damping ratio has influence on the curve of the critical gain by the use of the PT1
controller. If the actual values of the analysed system are different from the supposed
values, the critical gain curves can be recalculated in the same way.
References
NOSKIEVIČ, P., 1995. Auswahlkriterium der Reglerstruktur eines lagegeregelten elektrohydraulischen Antriebes. Ölhydraulik und Pneumatik, 39 ( 1995 ), č.1, str. 49 –
51.ISSN 0341-2660.
DIETER,M., 1998. Ein Beitrag zur systematischen Reglerauslegung positionsgeregelter
Ventil - Zylinder - Antriebe.1.Internationales Fluidtechnisches Kolloquium. Band1,
S.331-344, 17.-18.3.1998, Aachen, ISBN 3-89653-242-1.
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