Design of Robust Mixed H2/H∞ PID Controller Using Particle Swarm Optimization · 2017-11-11 · Design of Robust Mixed H2/H∞ PID Controller Using Particle Swarm Optimization Ramzy

International Journal of Advancements in Computing Technology

Volume 2, Number 5, December 2010

Design of Robust Mixed H2/H∞ PID Controller Using Particle Swarm

Optimization

Ramzy S. Ali Al-Waily

Electrical Engineering Department, College of Engineering, University of Basrah,

Basrah, Iraq.

Email: [email protected] doi:10.4156/ijact.vol2. issue5.5

Abstract An approach is proposed to tune the parameters of a mixed H2/H∞ PID controller via particle swarm

optimization. The main target is to find a suitable controller that minimizes the performance index of error

signal under the robust stability and robust performance conditions. Different types of performance indexes

are used to have optimal PID controller such as IAE, ISE, and ITAE. In this paper, all the three indexes

are used to estimate the performance of the mixed H2/H∞ PID controller using PSO under the uncertainty

and disturbance. The results show that the ISE was the best to use with disturbance condition to get

robustness.By testing two different control systems with the typical characteristics such as time delays and

system with RHS pole and zero, the proposed algorithm has been demonstrated to have good results with

parameters uncertainty and disturbances.

Keywords: Mixed H2/H∞ PID Controller, Particle Swarm Optimization, Robust Stability

1. Introduction

Proportional-integral-derivative (PID) controller has been widely used in the most industrial processes

despite continual advances in control theory. The PID controllers have found extensive industrial

applications for several decades because of its simple structure. This is not only due to the simple structure

which is theoretically easy to understand but also to the fact that the tuning techniques provide adequate

performance in the wide majority of applications. Most of the PID tuning rules developed in the past years

use the conventional methods such as Ziegler and Nichols which is often hard to determine optimal PID

parameters. Recently, a lot of efforts have been made to develop systematic methodologies for tuning PID

controller parameters, resulting in numerous strategies [1-4]. Particle Swarm Optimization (PSO) has

attracted a lot of attention in recent years because of the following reasons[5-8]:

It requires only a few lines of computer code to realize the basic PSO algorithm, which leads to an easy

implementation.

Its search technique using not the gradient information but the values of the objective function makes it

an easy to use algorithm.

It is computationally inexpensive, since its memory and CPU speed requirements are very low.

It is a stochastic approach, and thus does not require a considerably strong assumption made in

conventional deterministic methods such as linearity, differentiability, convexity, separability or

nonexistence of constraints in order to solve the problem efficiently.

Their solution doses hardly depend on initial states of particles, which could be a great advantage in

engineering design problems based on optimization approaches.

In this paper, a tuning method based on PSO method is suggested for robust PID controller design. The

suggested method provides the PID parameters that realize the expected step response of the plant. The

numerical results show the effectiveness of the suggested method.

2. Particle Swarm Optimization

Particle Swarm Optimization, first introduced by Kennedy and Eberhart, is one of optimization

algorithms. It was developed through simulation of simplified social system, and has been found to be

robust in solving continuous nonlinear optimization problems [5]. The PSO technique can generate a high

- 53 -

mailto:[email protected]

Design of Robust Mixed H2/H∞ PID Controller Using Particle Swarm Optimization


quality solution within shorter calculation time and stable convergence characteristic than other stochastic

methods.

PSO is a population based search process where individuals, referred to as particles, are grouped into a

swarm. Each particle in swarm represents a candidate solution to the optimization problem. In PSO

technique, each particle is “flown” through the multidimensional search space, adjusting its position in

search space according to its own experience and that of neighboring particles. A particle therefore makes

use of best position encountered by itself and that of its neighbors to position itself toward an optimal

solution. The effect is that particles “fly” toward a minimum, while still searching a wide area around the

best solution. The performance of each particle (i.e., the “closeness” of a particle to a global optimum) is

measured using a predefined fitness function, which encapsulates the characteristics of the optimization

problem.

The procedure of PSO is to iterate the following equation:

(1)

(2)

Where i is a particle number, j is the PID parameter specie number, k is a iteration number, x is the PID

parameter, v is a moving vector, pbest is a personal best of particle i , gbest is a global best of all particles,

w, c1, and c2 are weight parameters, rand() is a uniform random number from 0 to 1.

The description of PSO algorithm is as follows:

Begin

t→0 //iteration number//

Initialize X(t) //X(t): Swarm for iteration t//

Evaluate f(X(t)) //f(.): fitness function//

While (not termination condition) do

Begin

t→t+1

//Process of PSO//

Update velocity v(t) and position of each particle x(t) based on (1) and (2)

if v(t)<vmax

v(t)=vmax

end

if v(t)>-vmax

v(t)=-vmax

end

//end of the process of PSO//

Reproduce a new X(t)

Evaluate f(X(t))

End

End

In the above description , X(t) denotes a swarm at the tth iteration. First, the particles of the swarm are

initialized and then evaluated by a defined fitness function. The objective of the PSO is to iteratively

minimize the fitness values of particles. The swarm evolves from iteration t to t+1 by repeating the above

procedure.

3. Designing Robust PID Controller

Consider a MIMO control system with ni inputs and no outputs as shown in Fig. 1, where P(s) is the

plant perturbation, C(s) is the controller, r(t) is the reference input, u(t) is the control input, e(t) is the

tracking error, d(t) is the external disturbance, and y(t) is the output of the system.

- 54 -



Figure 1. Control system with plant perturbation and external disturbance

If a controller C(s) is designed so that:

The nominal closed loop system (∆P(s)=0 and d(t)=0) is asymptotically stable and

The robust stability performance satisfies the following inequality:

1 (3)

and

The disturbance attenuation performance satisfies the following inequality:

then the closed loop system is also asymptotically stable with ∆P(s) and d(t). Where W2(s) is a stable

weighting function matrix specified by designers. S(s) and T(s)=I-S(s) are the sensitivity and

complementary sensitivity functions of the system, respectively

and the H∞-norm in (3) and (4) is defined as

∞

A balanced performance criterion to minimize both Ja and Jb simultaneously is to minimize J∞ [9]

For advancing the system performance, robust stability and disturbance attenuation are often not enough in

the control system design. The minimization of tracking error J2 (i.e., H2 norm) should be taken into

account

where e(t)=r(t)-y(t) is the error which can be obtained from the inverse Laplace transformation of E(s)

with ∆P(s)=0 and d(t)=0

The objective function of the investigated problem of designing mixed H∞/H2 optimal controllers is as

follows:

The order of the derived optimal controller is very high when using conventional methods, making it hard

to implement. To overcome this difficulty, the mixed H∞/H2 optimal PID controller using PSO is proposed.

C(s) P(s)(I+∆P(s)) r e u

d

+ + +

- - y

- 55 -



In practical industrial applications, most of conventional controllers used in control systems have

fundamental structures such as PID which is given in the following configuration [10]:

A PID controller has 27 design parameters. A PI controller with 18 design parameters is a special case of a

PID controller where B2=0.

3.1. Performance Estimation of PID Controller

In general, there are different types of performance estimation of the PID controllers such as the

integrated absolute error (IAE), or the integral of squared error (ISE), or integral of time absolute error

(ITAE). The above integral performance criteria in the frequency domain have their own advantages and

disadvantages. For example, a disadvantage of the IAE and ISE criteria is that its minimization can results

in response with relatively small overshoot but a long settling time because the ISE performance criterion

weights all errors equally independent of time [11,12]. The IAE, ISE, and ITAE performance criterion

formulas are as follows:

In this paper, all the three indexes are used to estimate the performance of the mixed H2/H∞ PID controller

using PSO under the uncertainty and disturbance.

4. Simulations and Discussions

This section presents numerical examples to demonstrate the effectiveness of the proposed robust PID

controller tuning method based on PSO algorithm.

4.1. System with Delay

The system shown below has a delay effect

It assumed that the gain, the delay and the integral parameter are uncertainty parameters with the

nominal values 2, 1 and 5, respectively, the uncertainty is described for both k, and as (3,1), (0.2,1.8),

respectively, while the uncertainty is described in percentage for as (-40, 60). The PID controller would

be designed to cope only the uncertainty problem ( and the tracking performance .

The PID controller would be designed, by making the performance index , and the tracking

performance .

This limitation assimilates by time delay so the weighting will be

The weightings in the optimal problem in (11) are calculated using PSO algorithm and we take the best

values for , , the problem will take 90% as a portion of the optimal problem

while the tracking problem will take 10% of it.

Based on the above scenario the equation (11) will be rewritten in the following way:

- 56 -



Five runs have been done for each type of integral and step response of the best run for each one of the

integration above are shown in Figure 2. The values of and are as follows:

, and 61. These values give the optimal PID with the following values:

To examine the robustness of the three integrals, each parameter has been perturbed to give the worst

case that will be occur, , , , and . The results are shown in

Figure 3.

Although that the transient response of the system yielded using the controller that is found using ISE

mixed with disturbance rejection performance, has a settling time reach to 5sec.

, but this controller made the system more robust to any parameter perturbation as shown in Figure 3.

This mean that ISE get the complete reasonable reason to use in this research. The resulting sensitivity

and complementary sensitivity are shown in Figure 4.

To test the disturbance attenuation within the range of the frequencies less than the bandwidth of the

system. As it seems in Figure 4, the sensitivity of the resultant feedback system which is represented by the

transfer function between disturbance and the output has an attenuation reached to at frequencies

near the cutoff frequency and it's an acceptable value. To ensure that a disturbance signal would be inserted and the step response in this case is shown in Figure 5. Its explicit that signal

attenuate to approximately it's half value.

In the next step another system with another limitation will be put under investigation using the same

way.

Figure 2. Step response for the three types of the integration of error mixed with disturbance rejection

performance

Figure 3. Step response of the worse- case for each type of the integration of error mixed with

disturbance rejection performance

0 1 2 3 4 5 6 7 8 9 10 -6

-4

-2

0

2

4

6

Time (sec)

Amplitude

ISE ITAE IAE

0 1 2 3 4 5 6 7 8 9 10 -0.5

0

0.5

1

1.5

Amplitude

Time (sec)

ISE ITAE IAE

- 57 -



Figure 4. Singular value plot of the resulting system using ISE

Figure 5. Step response of the resulting system after inserting a disturbance

4.2. System with RHS zero and RHS pole

Consider the system shown below:

Where and are the uncertainty parameters with the nominal values 1, respectively both of them

are bounded by the ranges , respectively.

To decide what will the value of , the singular value of was drawn and upon its shape the decision

will make . The value of and has been chosen equal to [13], but since the work is based on

, can be rewritten as follows[13]

The value of , and had the same value in the previous section. After many time of runs, the results

was , and , with the following PID controller:

Figure 6 shows singular value plot of , and for the resultant feedback system. They have a perfect

reasonable shape, if the higher ultimate (which is come from the existence of the RHS pole) is ignored.

Figure 7, illustrates the step response of the feedback system while Figure 8 shows step response when

the RHS zero and pole has been perturbed to their maximum values. As it clear there is no impalpable

change occur when the parameters are perturbed.

Figure 9 shows the step response when a signal d(t)=0.2sin(0.1t) has been inserted in the feedback

system.

0 5 10 15 20 25 30 -0.5

0

0.5

1

1.5

Time (sec)

1

0 -

2 1

0 -

1 1

0 0 1

0 1 1

0 2 -

35

-

30

-

25

-

20

-

15

-

10

-

5

0

5

T S

Singular

Values

(dB)

- 58 -



5. Conclusion

In this paper, we proposed a mixed H2/H∞ PID controller based PSO. PSO can offer an effective and

simple method to tune the proposed controller which is difficult treated by conventional techniques.

Simulation results of two different control systems with parameters uncertainty and disturbances give good

performance and stability.

By comparing different types of integration errors, it can be seen that the ISE is the best to use with the

disturbance condition to get robustness, , the robust stability had been subscribed by disturbance

rejection condition mixed with the condition of minimizing of the square of error.

Figure 6. Singular value plot of , and for the resultant system in (20)

Figure 7. Step response for the resultant feedback system in (20)

Figure 8. Step response of the resultant feedback system in (20) when the uncertain parameters perturbed

to their maximum value

0 5 10 15 20 25 30 -0.5 -0.25

0 0.25

0.5 0.75

1 1.25

1.5 1.75

2 2.25

2.5

Time (sec)

Amplitude

0 5 10 15 20 25 30 -0.5 -0.25

0 0.25 0.5 0.75

1 1.25 1.5

1.75 2

2.25 2.5

Time (sec)

10 -2 10 -1 10 0 10 1 10 2 -30

-25

-20

-15

-10

-5

0

5

10

Frequency (rad/sec)

T S

- 59 -



Figure 9. The step response when a signal has been inserted in the feedback system

optimization problem in (20)

6. References [1] P. J. Van Rensburg, I. S. Shaw, and J. D. Van Wyk, "Adaptive PID-Control Using a Genetic

Algorithm," 2nd

International Conference on Knowledge –Based Intelligent Electronics Systems,

Adelaide, Australia, 21-23, pp. 133-138, April 1998.

[2] Y. G. Wang, H. H. Shao, and J. Wang, "PI Tuning for Processes with Large Dead Time," Proceedings

of the American Control Conference, Chicago, Illinois, pp. 4274-4278, June 2000.

[3] R. A. Krohling and J. P. Rey, "Design of Optimal Disturbance Rejection PID Controllers Using

Genetic Algorithms," IEEE Trans. Evol. Comput., vol. 5, no. 1, pp. 78-82, Feb. 2001.

[4] W. M. Qi, W. Y. Cai, Q. L. Ji, and Y. C. Cheng, " A Design of Nonlinear Adaptive PID Controller

Based on Genetic Algorithm," Proceedings of the 25th Chinese Control Conference, Harbin,

Heilongjiang, 7-11 August, pp. 175-178, 2006 .

[5] J. Kennedy and R. Eberhart, "Particle Swarm Optimization," in Proceedings IEEE Int. Conf. Neural

Networks, vol. IV, Perth, Australia, pp.1942-1948, 1995.

[6] Z. L. Gaing, "A Particle Swarm Optimization Approach for Optimum Design of PID Controller in

AVR System," IEEE Trans. Energy Conversion, vol. 19, no. 2, pp. 384-391, June 2004.

[7] T.H. Kim, I. Maruta and T. Sugie, "Particle Swarm Optimization based Robust PID Controller Tuning

Scheme, "Proceedings of the 4th IEEE Conf. On Decision and Control, New Orleans, LA, USA, Dec.

12-14, pp. 200-205, 2007.

[8] Oi, et al., "PID Optimal Tuning Method by Particle Swarm Optimization, "SICE Annual Conference,

Japan, August 20-22, pp. 3470-3473, 2008.

[9] S. J. Ho, S. Y. Ho and L. S. Shu, "OSA: Orthogonal Simulated Annealing Algorithm and Its

Application to Designing Mixed H2/H∞ Optimal Controllers," IEEE Trans. Systems, Man, And

Cybernetics- Part A: Systems and Humans, vol. 34, no. 5, pp. 588-600, Spe. 2004.

[10] M-H. Hung, et. al., "A Novel Intelligent Multiobjective Simulated Annealing Algorithm for Designing

Robust PID Controllers," IEEE Trans. System, Man, And Cybernetics- Part A: Systems and Humans,

vol. 38, no. 2, pp. 319-330, March 2008.

[11] Z-L. Gaing," A Particle Swarm Optimization Approach for Optimum Design of PID Controller in

AVR System," IEEE Trans. On Energy Conversion, vol.19, no. 2, pp. 384-391, June 2004.

[12] Q. Zeng and G. Tan," Optimal Design of PID Controller Using Modified Ant Colony System

Algorithm", 3rd

International Conference on Natural Computation ICNC 2007.

[13] S. Scogestad and I. Postlethwaite , Multivariable Feedback Control. NewYork, Wiley, 1996.

0 10 20 30 40 50 60 -0.5 -0.25

0 0.25

0.5 0.75

1 1.25

1.5 1.75

2 2.25

2.5

Amplitude

- 60 -

Documents

Design of Robust Mixed H2/H∞ PID Controller Using Particle Swarm Optimization · 2017-11-11 · Design of Robust Mixed H2/H∞ PID Controller Using Particle Swarm Optimization Ramzy