Performance Robustness Criterion of PID Controllers

PID is one of the earliest and most popular controllers. The improved PID and classical PID have been applied in various kinds of industry control fields, as its tuning methods are developing. After the PID controller was first proposed by Norm Minorsky in 1922, the various PID tuning methods were developing and the advanced and intelligent controls were proposed. In the past few decades, Z-N method which is for first-order-plus-timedelay model was proposed by Ziegler and Nichols (Ziegler & Nivhols, 1943), CHR method about generalized passive systems was proposed by Chien, Hrones and Reswick (Chien et al., 1952), and so many tuning methods were developed such as pole assignment and zeropole elimination method by Wittenmark and Astrom, internal model control (IMC) by Chien (Chien & Fruehauf, 1990). The gain and phase margin (GPM) method was proposed by Åström and Hägglund (Åström & Hägglund, 1984), the tuning formulae were simplified by W K Ho (Ho et al., 1995).


Introduction
PID is one of the earliest and most popular controllers.The improved PID and classical PID have been applied in various kinds of industry control fields, as its tuning methods are developing.After the PID controller was first proposed by Norm Minorsky in 1922, the various PID tuning methods were developing and the advanced and intelligent controls were proposed.In the past few decades, Z-N method which is for first-order-plus-timedelay model was proposed by Ziegler and Nichols (Ziegler & Nivhols, 1943), CHR method about generalized passive systems was proposed by Chien, Hrones and Reswick (Chien et al., 1952), and so many tuning methods were developed such as pole assignment and zeropole elimination method by Wittenmark and Astrom, internal model control (IMC) by Chien (Chien & Fruehauf, 1990).The gain and phase margin (GPM) method was proposed by Åström and Hägglund (Åström & Hägglund, 1984), the tuning formulae were simplified by W K Ho (Ho et al., 1995).
In classical feedback control system design, the PID controller was designed according to precise model.But the actual industrial models has some features as follows: 1.The system is time variant and uncertain because of the complex dynamic of industrial equipment.2. The process is inevitably affected by environment and the uncertainty is introduced.3. The dynamic will drift during operation.4. The error exists with the dynamic parameter measurement and identification.
So there are two inevitable problems in control system designing.One is how to design robust PID controller to make the closed-loop system stable when the parameters are uncertain in a certain range.The other is the performance robustness which must be considered seriously when designing PID controllers.The performance robustness is that In power system, Monte-Carlo method was applied in reliability assessment of generation and transmission system, the software was design and the application was successful (Ding & Zhang, 2000).

Performance robustness criterion based on Monte-Carlo method
Consider the SISO system as follows: (1) In this system, N(s) and D(s) are coprime polynomials, and D(s)'s order is greater than or equal N(s)'s order, L is rational number greater than or equal to zero.The controlled model is some uncertain, and the parameters of N(s) and D(s) are variable in bounded region.So, the model is a group of transfer function denoted by {G(s)}.The control system is shown in figure 1.

PID G
ry u e The controller is PID controller: 1 () ( 1 )() The parameters K p , K i , K d are positive number, and all of the PID controllers compose a controller group denoted by {PID}.
The PID tuning methods are used on the nominal controlled models, and the closed-loop systems are obtained.The overshoot %  and adjustment time T s are considered as dynamic performance index.Because the controlled models are a group of transfer function, the dynamic performance index is a collection, denoted by: Obviously, it is a collection of two-dimension vector an area in plane plot.The distance between this area and origin reflects the quality of control system, and the size of this area shows the dispersion of performance index, that is the performance robustness of control system.
The comparison study on PID tuning methods should follow the steps below: 1. Confirm the controlled model transfer function and parameter variety interval, and the transfer function group is obtained.2. Confirm the compared PID tuning methods, and choose the appropriate experiment times N to ensure the dispersion of performance index invariable when the N is larger.3. Tuning PID controller for the nominal model.4. In every experiment, a specific model is selected from the transfer function group by a rule (random in this paper).With the PID controller obtained in step three, the step response of closed-loop PID control system is tested, and the overshoot and adjustment time could be measured.5. Repeat the step 4 N times, and plot the performance index on coordinate diagram.So, the N points compose an area on the coordinate diagram.6. Repeat the step 3-5 by different tuning methods.7. Compare the performance index of different tuning methods.
In next section, performance robustness is applied on PID control system comparison.

Performance robustness comparison of typical PID control systems
In this section, we consider four typical models as follows: 1. First-order-plus-time-delay model (FOPTD) If the tuning object is zero overshoot, the selection of IMC method free parameter T f will only correlate to delay-time L. We fit the approximate relation between L and T f .
The different transfer function models can be simplified and transferred to FOPTD model (Xue, 2000).
Calculate the first and second derivative and then we obtain We can get L and T from equation above, and the system gain can be obtained directly by k=G(0).
So, in actual application, if we have the transfer functions, the more accurate FOPTD equivalent models will be get.
For example, the transfer function is The approximate FOPTD model is The step response is shown in figure 2.
For FOPTD model (4), the L/T is very important.So, there are three cases to be discussed L<T, L≈T and L>T.The parameters and simulation results are shown in table 2, 3, figure 3, 4 and 5.  .The nominal parameters are T 1 =T 2 =20, L=90.The simulation results are shown in table 4 and figure 6 For High-order model ( 6), we choose [16, 24] T  and [0.8,1.2]k  . The nominal parameters are T=20, k=1 and n=3.The simulation results are shown in table 5 and figure 7   From the simulation results above, it is clear that the GPM method and IMC method are superior to other compared tuning methods.

Performance robustness comparison of DDE and IMC
The desired dynamic equation method (DDE) is proposed for unknown models.This twodegree-of-freedom (2-DOF) controller designing can meet desired setting time, and has physical meaning parameters (Wang et al., 2008).
In this section, we consider 15 transfer function models as follows.In order to compare the two methods easily, we divide them into four types shown in table 8.

No.
Type Model with integral G 6 、G 8 、G 14 Table 8.Four types of models The Normal model is simple and easy to control.The simulation results are shown in table 9 and 10.

Model Controller
Step  Most of High-order model is series connection of inertial element in industry field (Quevedo, 2000).But, the simple PID is hard to control them because of the delay cascaded by inertial elements.The simulation results are shown in table 11 and 12.

Model Controller
Step    For Non-minimum model, the two method has similar step response, but the undershoot is smaller with DDE method.DDE method also has good performance robustness.
Integral is the typical element in control system.If a system contains an integral, it will not be a self-balancing system.It is open-loop unstable and easy to oscillate in close-loop.So it is hard to obtain a good control effect.The simulation results are shown in table 15 and 16.
The simulation results of Model with integral shows that the overshoot of IMC method is much larger than DDE method, and DDE method is much quicker than IMC method.The performance robustness of DDE method is better than IMC method.
The comprehensive comparison is shown in table 17.

Model Controller
Step response Performance robustness DDE IMC   Simulation results show that DDE method has better performance robustness than GPM method generally.Apparently, the points on overshoot ~ adjustment time plane of DDE method concentrate more together near the bottom left corner than GPM method.Except the G P3 result, the points on gain margin ~ phase margin plane of DDE method are more concentrated than GPM method.

Types of models DDE method
GPM method Settings PID parameters Settings PID parameters

Fig. 6 .
Fig. 6.Simulation results of SOPTD model (the abscissa represents overshoot and the ordinate represents adjustment time)

Fig. 7 .
Fig. 7. Simulation results of High-order model (the abscissa represents overshoot and the ordinate represents adjustment time) CHR IMC Pole assignment GPM IST 2 E Cohen Z-N

Table 2 .
Parameters of FOPTD model

Table 4 .
. Performance index of SOPTD model www.intechopen.comPID Controller Design Approaches -Theory, Tuning and Application to Frontier Areas

Table 6 .
Performance index of Non-minimum model

Table 7 .
Controller parametersThe DDE and IMC method are used on them to compare the performance robustness.The controller parameters are shown in table 7. 10%  parameter perturbation is taken for performance robustness experiment with 300 times. www.intechopen.com

Table 9 .
Simulation results of Normal model

Table 11 .
Simulation results of High-order model

Table 12 .
Performance index of High-order modelIt is clear that DDE method is as fast as IMC method on High-order model, but the overshoot is almost zero.DDE method also has good performance robustness especially on G 3 and G 5 .The Non-minimum model has the zeros and poles on right half complex plane or time delay.The simulation results are shown in table13 and 14.

Table 13 .
Simulation results of Non-minimum model www.intechopen.comPID Controller Design Approaches -Theory, Tuning and Application to Frontier Areas

Table 14 .
Performance index of Non-minimum model

Table 15 .
Simulation results of Model with integral

Table 17 .
Comparison of DDE method and IMC method

Performance robustness comparison of DDE and GPM In
this section, we also consider the four typical models shown in table 18.

Table 18 .
Four types of typical model According to desired adjustment time and prospective gain margin ~ phase margin to design controller in each DDE and GPM methods.Within nominal parameter, design PI controller for FOPTD model, design PID controller for SOPTD model, high-order model and non-minimum model.Proceed performance robustness experiment within 10%  parameter perturbation.In order to keep the comparison impartial, select adjustment time of GPM method as the desired adjustment time.Controller parameters are shown in table 19, results of Monte-Carlo simulation are shown in table 20, comparison of performance indices is shown in table 21.

Table 21 .
Comparison of performance index www.intechopen.com