Delay Compensation using PID Controller and GA
Delay Compensation using PID Controller and GA Andri Mirzal, PhD Faculty of Computing Universiti Teknologi Malaysia
Introduction • Time delay in a control system can be defined as time interval between an event started in one point and its output in another point within the system • Delays always reduce stability of minimum phase systems • Delays can be caused by – – – –
transportation and communication lag, sensor response delay, time to generate control signals, and system parameters approximation using First Order Lag plus Time Delay (FOLPD).
Introduction f (t T )
f (t ) Delay (T)
f(t)
f(t-T) t T
Fig 1. Delay component in a system
The relationship between f(t) and f(t-T) can be written as:
[ f (t T )u (t T )] f (t T )u (t T )e st dt 0
Introduction The relationship between f(t) and f(t-T) can be written as:
[ f (t T )u (t T )] f (t T )u (t T )e st dt 0
with u(t) denotes unit step (the testing signal), e-st denotes the delay component, [ f (t )] denotes the Laplace transform of f(t), and s denotes the complex plane In complex frequency domain, this relationship can be described with X in (s )
e Ts
X out ( s ) e Ts X in ( s )
Fig 2. Delay in frequency domain.
Introduction • The conventional way to determine the optimal parameters for PID controller is to use tuning methods like the Iterative Method and Ziegler-Nichols rule • But there are some cases where we can’t use these two tuning methods, e.g., the dynamic plants which its parameters are constantly changing • In this kind of systems, we have to do retuning in real time, which can’t be accomplished by the tuning methods because we have to take the systems offline first in order to set the parameters • In this work, we show the using of genetic algorithm (GA) to determine the optimal parameters for PID controller for compensating the influence of delay to system stability
Delay configuration • In a control system, delay components can be found in controlled plant, sensor that measures the output, and/or other parts of the system • In this work, we assume that the system can be modelled using the following structure R(s)
+
G(s)
e
Ts
-
H(s)
Fig 3. Delay configuration of interest
C(s)
Delay configuration • There are other possible configurations, the most common ones are delay at the feedback, the input, and the output • The following figures depict these configurations. R(s)
+
e Ts
G(s)
C(s)
R(s) +
C(s) G(s)
-
-
H(s)
(a)
R(s) +
(b)
e Ts
G(s) -
e Ts
H(s)
C(s)
R(s)
e Ts
C(s)
+ G(s) -
H(s)
H(s)
(c)
(d)
Fig 4. Delay configurations in control systems
Delay configuration • For configuration (c) and (d), the delay components are not in the closed loop, so they do not affect system stability (they only shift the output/input without changing the control signal or the system response) • For configuration (b), since we can transform it into the following equivalent configuration: +
R(s)
1 / H (s)e Ts
G( s ) H ( s )
e
Ts
C (s)
-
• then, the configuration inside the loop which can influence the stability is similar to Fig 4(a), thus for stability analysis it suffices to consider only system in fig. 3.
PID Controller • PID controllers are the main controllers in industries • It has high level of robustness, and is easy to operate and understand because of the structural simplicity • A PID controller can be used to improve stability of time-delay systems because it can increase stability margin and reduce %overshoot and settling time • Here, we will use a PID controller to improve stability of a system with time delay
PID Controller • The following figure shows the schematic of the PID controller and plant with Gp(s) = 1/(1+sTp) 1 Ti s R(s) +
X
+
E(s) Kc
-
+ U(s)
X +
Td s
Fig 6. PID controller and plant
G p ( s )e
s p
Y(s)
GA • Here we use five performance indices as the objective functions of the GA: – – – – –
Mean of the Squared Error (MSE) Integral of Time multiplied by Absolute Error (ITAE) Integral of Absolute Magnitude of the Error (IAE) Integral of the Squared Error (ISE) Integral of Time multiplied by the Squared Error (ITSE)
GA + PID Controller • The PID controller is used to minimize the values of performance indices mentioned above • And because these values are inversely proportional to the fitness of the corresponding chromosomes, we define the fitness of the chromosomes as: fitness value
1 performance index
Standard performance measures • Percent overshoot is defined as the point where the system response reaches the peak • There are several criteria for settling time, for example 1% criterion, 2% criterion, and 5% criterion. Here we use 5% criterion for settling time • Rise time is defined as the time needed by the system to reach the final value. For measurement simplicity, we use 95% of the final value • Peak time is the point where the maximum value is reached (the overshoot value) • And error signal is the difference between the input signal magnitude and system response final magnitude.
Experimental results Percent overshoot 60 Ziegler Nichols Iterative Method MSE IAE ISE ITAE ITSE
50
Percent Overshoot
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delay in second
GA can improve percent overshoots
Experimental results Settling time 5
4.5
Settling time 5% criterion
4
3.5
3
2.5
2
1.5
Ziegler Nichols Iterative Method MSE IAE ISE ITAE ITSE
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delay in second
GA has comparable performance with ZN and IM
Experimental results Rise time 1
Rise time in second (logarithmic scale)
10
Ziegler Nichols Iterative Method MSE IAE ISE ITAE ITSE
0
10
-1
10
-2
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delay in second
GA has comparable performance with ZN and IM
Experimental results Peak time 2
Peak time in second (logarithmic scale)
10
Ziegler Nichols Iterative Method MSE IAE ISE ITAE ITSE
1
10
0
10
-1
10
-2
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delay in second
GA has comparable performance with ZN and IM
Experimental results Stability margin 3
Stability margin (logarithmic scale)
10
Ziegler Nichols Iterative Method MSE IAE ISE ITAE ITSE
2
10
1
10
0
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delay in second
GA has comparable performance with ZN and IM
Experimental results TABLE I. AVERAGE OF STANDARD PERFORMANCE MEASURES
GA
Parameter
ZN
IM MSE
IAE
ISE
ITAE
ITSE
%OV
38%
15%
10%
6%
11%
10%
8%
ST
1.53
1.54
1.47
1.01
1.45
0.745
1.37
RT
0.444
0.912
0.453
0.495
0.455
0.588
0.474
PT
0.613
3.43
0.576
0.622
0.576
0.836
0.597
SM
36.25
37.86
33.68
33.8
24.4
36.8
32
Summary • Delay components are always present in a control system • The main concern in this paper is the influence of the delays to system stability • Tuning the PID parameters using the Iterative Method can improve the stability of the system with delay significantly. • Compared to the uncompensated system, in average the Iterative Method improves stability margin more than threefold, %overshoot more than twofold, settling time more than 30%, and eliminates error signals completely • However, the Iterative Method cannot be used in online fashion
Summary • Thus, other methods that can be employed in online fashion need to be studied • GA is an example of such methods • Experimental results have shown that the performances of GA are comparable or better than the tuning methods (Ziegler Nichols method and the Iterative Method) in the analyzed cases
Introduction • Time delay in a control system can be defined as time interval between an event started in one point and its output in another point within the system • Delays always reduce stability of minimum phase systems • Delays can be caused by – – – –
transportation and communication lag, sensor response delay, time to generate control signals, and system parameters approximation using First Order Lag plus Time Delay (FOLPD).
Introduction f (t T )
f (t ) Delay (T)
f(t)
f(t-T) t T
Fig 1. Delay component in a system
The relationship between f(t) and f(t-T) can be written as:
[ f (t T )u (t T )] f (t T )u (t T )e st dt 0
Introduction The relationship between f(t) and f(t-T) can be written as:
[ f (t T )u (t T )] f (t T )u (t T )e st dt 0
with u(t) denotes unit step (the testing signal), e-st denotes the delay component, [ f (t )] denotes the Laplace transform of f(t), and s denotes the complex plane In complex frequency domain, this relationship can be described with X in (s )
e Ts
X out ( s ) e Ts X in ( s )
Fig 2. Delay in frequency domain.
Introduction • The conventional way to determine the optimal parameters for PID controller is to use tuning methods like the Iterative Method and Ziegler-Nichols rule • But there are some cases where we can’t use these two tuning methods, e.g., the dynamic plants which its parameters are constantly changing • In this kind of systems, we have to do retuning in real time, which can’t be accomplished by the tuning methods because we have to take the systems offline first in order to set the parameters • In this work, we show the using of genetic algorithm (GA) to determine the optimal parameters for PID controller for compensating the influence of delay to system stability
Delay configuration • In a control system, delay components can be found in controlled plant, sensor that measures the output, and/or other parts of the system • In this work, we assume that the system can be modelled using the following structure R(s)
+
G(s)
e
Ts
-
H(s)
Fig 3. Delay configuration of interest
C(s)
Delay configuration • There are other possible configurations, the most common ones are delay at the feedback, the input, and the output • The following figures depict these configurations. R(s)
+
e Ts
G(s)
C(s)
R(s) +
C(s) G(s)
-
-
H(s)
(a)
R(s) +
(b)
e Ts
G(s) -
e Ts
H(s)
C(s)
R(s)
e Ts
C(s)
+ G(s) -
H(s)
H(s)
(c)
(d)
Fig 4. Delay configurations in control systems
Delay configuration • For configuration (c) and (d), the delay components are not in the closed loop, so they do not affect system stability (they only shift the output/input without changing the control signal or the system response) • For configuration (b), since we can transform it into the following equivalent configuration: +
R(s)
1 / H (s)e Ts
G( s ) H ( s )
e
Ts
C (s)
-
• then, the configuration inside the loop which can influence the stability is similar to Fig 4(a), thus for stability analysis it suffices to consider only system in fig. 3.
PID Controller • PID controllers are the main controllers in industries • It has high level of robustness, and is easy to operate and understand because of the structural simplicity • A PID controller can be used to improve stability of time-delay systems because it can increase stability margin and reduce %overshoot and settling time • Here, we will use a PID controller to improve stability of a system with time delay
PID Controller • The following figure shows the schematic of the PID controller and plant with Gp(s) = 1/(1+sTp) 1 Ti s R(s) +
X
+
E(s) Kc
-
+ U(s)
X +
Td s
Fig 6. PID controller and plant
G p ( s )e
s p
Y(s)
GA • Here we use five performance indices as the objective functions of the GA: – – – – –
Mean of the Squared Error (MSE) Integral of Time multiplied by Absolute Error (ITAE) Integral of Absolute Magnitude of the Error (IAE) Integral of the Squared Error (ISE) Integral of Time multiplied by the Squared Error (ITSE)
GA + PID Controller • The PID controller is used to minimize the values of performance indices mentioned above • And because these values are inversely proportional to the fitness of the corresponding chromosomes, we define the fitness of the chromosomes as: fitness value
1 performance index
Standard performance measures • Percent overshoot is defined as the point where the system response reaches the peak • There are several criteria for settling time, for example 1% criterion, 2% criterion, and 5% criterion. Here we use 5% criterion for settling time • Rise time is defined as the time needed by the system to reach the final value. For measurement simplicity, we use 95% of the final value • Peak time is the point where the maximum value is reached (the overshoot value) • And error signal is the difference between the input signal magnitude and system response final magnitude.
Experimental results Percent overshoot 60 Ziegler Nichols Iterative Method MSE IAE ISE ITAE ITSE
50
Percent Overshoot
40
30
20
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delay in second
GA can improve percent overshoots
Experimental results Settling time 5
4.5
Settling time 5% criterion
4
3.5
3
2.5
2
1.5
Ziegler Nichols Iterative Method MSE IAE ISE ITAE ITSE
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delay in second
GA has comparable performance with ZN and IM
Experimental results Rise time 1
Rise time in second (logarithmic scale)
10
Ziegler Nichols Iterative Method MSE IAE ISE ITAE ITSE
0
10
-1
10
-2
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delay in second
GA has comparable performance with ZN and IM
Experimental results Peak time 2
Peak time in second (logarithmic scale)
10
Ziegler Nichols Iterative Method MSE IAE ISE ITAE ITSE
1
10
0
10
-1
10
-2
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delay in second
GA has comparable performance with ZN and IM
Experimental results Stability margin 3
Stability margin (logarithmic scale)
10
Ziegler Nichols Iterative Method MSE IAE ISE ITAE ITSE
2
10
1
10
0
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delay in second
GA has comparable performance with ZN and IM
Experimental results TABLE I. AVERAGE OF STANDARD PERFORMANCE MEASURES
GA
Parameter
ZN
IM MSE
IAE
ISE
ITAE
ITSE
%OV
38%
15%
10%
6%
11%
10%
8%
ST
1.53
1.54
1.47
1.01
1.45
0.745
1.37
RT
0.444
0.912
0.453
0.495
0.455
0.588
0.474
PT
0.613
3.43
0.576
0.622
0.576
0.836
0.597
SM
36.25
37.86
33.68
33.8
24.4
36.8
32
Summary • Delay components are always present in a control system • The main concern in this paper is the influence of the delays to system stability • Tuning the PID parameters using the Iterative Method can improve the stability of the system with delay significantly. • Compared to the uncompensated system, in average the Iterative Method improves stability margin more than threefold, %overshoot more than twofold, settling time more than 30%, and eliminates error signals completely • However, the Iterative Method cannot be used in online fashion
Summary • Thus, other methods that can be employed in online fashion need to be studied • GA is an example of such methods • Experimental results have shown that the performances of GA are comparable or better than the tuning methods (Ziegler Nichols method and the Iterative Method) in the analyzed cases
