Performance Criteria Research on PSO-PID Control Systems

Yongwei Zhang, Fei Qiao, Jianfeng Lu, Lei Wang, Qidi Wu College of Electronics and Information Engineering

Tongji University Shanghai, P.R.China

zyw19830531@gmail.com

Abstract—Many results have emerged in design of controller based on numerical optimization techniques, along with various system performance criteria. This paper enumerates common cost function, within which four typical criteria are used as cost function to conduct simulation based on particle swarm optimization. The effects of different cost function cast on system performance are analyzed based on simulation results. The conclusion can be extended to other numeral optimization techniques, and possess high practical value.

Keywords-PSO; PID; cost function; numerical Optimization

I. INTRODUCTION PID control technique is simple in structure and easy in

implement, has been widely used in various fields. Recently, some literature [1]-[3] report the synthesis methods for PID controller design based on rigorous theoretical justification. The development of numerical optimization techniques, such as genetic algorithm [4], simulated annealing algorithm [5], particle swarm optimization [6], give birth to the reports of numerical PID controller design [7]-[9].

One common characteristic of numerical optimization is that problems need to be described as a cost function or fitness function. For control systems, general approach is selecting a integral error criterion as cost function [10], and search controller parameters by minimize the cost function. Besides, there is literature [11] that use the sum of weighted performance criteria as cost function. Its principle is to select as many criteria of system performance as possible, and weight criteria differently based on the different focus of design goal. These literatures use various criteria as cost function and obtain satisfied control performances, but did not specify the basis of selecting criteria, which lack of support from theory or experiment. Therefore, this paper uses modified particle swarm optimization [12] (MPSO) as a platform, to study how typical criteria affects system response feature in numerical optimization. The conclusion can be applied to other numerical optimization techniques, has high reference value for numerical design of control systems.

The structure of remain parts is as follow. Section 2 introduces the design method of PSO-PID controller. Where 2.1 provides an overview of PSO, 2.2 is problem description, 2.3 provides algorithm flow. Section 3 discusses the selection of cost function. An illustrative example is studied in section 4 to analyze the effects of different cost functions. Section 5 summarizes the characteristics of typical cost functions.

II. DESIGN OF PSO-PID CONTROLLER

A. Particle Swarm Optimization Kennedy and Eberhart [6] proposed a swarm-

intelligence-based parallel optimization algorithm, Particle Swarm Optimization(PSO) in 1995. PSO shows well performance on pattern classification, optimization and controller parameters design. Based on PSO, Shi[12] et.al proposed Modified Particle Swarm Optimization(MPSO), which introduces the concept of inertia weight and greatly improves the performance of PSO. The description of MPSO is as (1)-(4):

11 1 2 2( ) ( )

id id

g gid id gd idv wv c r p x c r p x+ = + − + − (1)

max max

max max

id id

id id

v v if v vv v if v v

= >⎧⎨ = − < −⎩

(2)

1 1g g gid id idx x v+ += + (3)

max minmax

w ww w g

G−

= − ⋅ (4)

Where (1) updates particle velocity, gidv and g

idx represent the velocity and position of i -th ( 1 ~ )i N= particle of g -th ( 1 ~ )g G= generation in d -th ( 1 ~ )d D= dimension space. w is inertia weight decided by (4); 1c and 2c are accelerate constants, which convergence speed of every particle and often set to 2.0 according to past experiences [13]; 1r and 2r are random numbers uniformly distributed within [0,1]; idp records the best position of i -th particle and

gdp records the best position of whole population. (2) defines the minimum and maximum velocity of each particle;

minw and maxw in (4) is the minimum and maximum value of inertia weight; g is the current generation number, G is maximum iteration number.

2010 International Conference on Intelligent Computing and Cognitive Informatics

978-0-7695-4014-6/10 $26.00 © 2010 IEEE

DOI 10.1109/ICICCI.2010.51

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B. Problem Description The typical feedback PID system is as showed in Figure

1. , where ( )r t is reference input, ( )u t is control signal, ( )y t is output, ( )d t is external disturbance, ( )e t is the error

of output and reference input, and also is the input signal of controller. ( )C s is PID controller as follow form:

( ) ip d

KC s K K s

s= + + (5)

Where pK , iK and dK is proportion constant, integral constant and derivative constant respectively. The domain of these parameters is as follow:

3{( , , ) R : 0, 0, 0}f p i d p i dK K K K K KΔ = ∈ > > > (6)

where R donates real number domain. ( ) ( ) / ( )G s N s D s= is linear time-invariant system, ( )N s and ( )D s are polynomial coprime, and as follow form:

1

1 1 01

1 1 0

( )

( )

n nn nm m

m

N s a s a s a s a

D s s b s b s b

−−

−−

= + + + +

= + + + + (7)

where n m< . The close loop transfer function is:

( ) ( )( )1 ( ) ( )

C s G sT sC s G s

=+

(8)

Figure 1. Typical Control System Diagram

Thus the problem of designing a controller can be described as: searching for parameter combination{ , , }p i dK K K in feasible bounds fΔ , to make the close loop system in equation (8) asymptotic stable.

C. MPSO-PID Controller Design The problem of designing PID controller is to find

parameter combination K in feasible bounds defined in equation(6) that minimize cost function. Set the space coordinate of particles as 1i px K= , 2i ix K= , 3i dx K= , the steps of finding solutions is as follow:

Step1: Randomly initialize the position and velocity of every individual particles within the range max[0, ]x and

max[0, ]v . Randomly initialize the individual best position

idp and global best position gdp , where i is particle index, d is parameter index. Iteration number 1g = .

Step2: For every individual , 1,2,ix i N= in the population, calculate the cost function.

Step3: Compare the cost function value of every individual in the population, if the value is smaller than the evaluation of idp , then update idp as current particle position. Compare current population best evaluation with the evaluation of gp , if smaller, updates gp .

Step4: Updates the velocity of each particle according to (1) and (2), the setting of inertia weight see (4).

Step5: Updates the position of each particle according to (3), iteration number plus 1.

Step6: If the iteration index reaches max iteration number, go to Step7, else return to Step2.

Step7: gdp is the controller parameter vector.

III. CHOSEN OF COST FUNCTION For SISO system, from the perspective of design

purpose, controller design is a multi-object optimization problem, which involves the dynamic and static, time and frequency characteristics. But due to the un-decoupling feature between performance criteria, the affect on one performance criterion will unavoidably affects the others, thus the controller design problem is down to a single object problem that optimize a function who reflects comprehensive system performance. For MIMO system, there is possibility to decouple inputs and outputs, then the controller design problem is equal to a multi-object problem that optimize the comprehensive performance criteria of multiple decoupled subsystems. This paper only discusses SISO system.

Chose of cost function has many options. In numerical optimization field, integral of error is frequently used, which includes Integral of Squared-Error(ISE), Integral of Absolute Error(IAE), Integral of Time-Weighted Squared-Error(ITSE) and Integral of Time-Weighted Absolute Error(ITAE). The definition of these four criteria is as follow:

2 2 21 10

{ ( ) [( ) ( ) ]}y un ny u uj yj j uj j jj j

ISE e e u dtω ω ω∞ Δ

= == + + Δ∑ ∑∫ (9)

1 10

{ ]}y un ny u uj yj j uj j jj j

IAE e e u dtω ω ω∞ Δ

= == + + Δ∑ ∑∫ (10)

2 2 21 10

{ ( ) [( ) ( ) ]}y un ny u uj yj j uj j jj j

ITSE

t e e u dtω ω ω∞ Δ

= =

=

+ + Δ∑ ∑∫ (11)

1 10

{ ]}y un ny u uj yj j uj j jj j

ITAE

t e e u dtω ω ω∞ Δ

= =

=

+ + Δ∑ ∑∫ (12)

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Where yje is the tracking error between j -th output and set point or reference variable, uje is the error of j -th manipulating variable (input variable) and its expectation (if any), juΔ is changing rate of j -th manipulating variable;

yn and un is the number of outputs and inputs; yjω , u

jω and ujωΔ are weights of three components

respectively. For common control systems, there is no expectation for manipulating variables, (9)-(12) can work well with only tracking error component. Under certain circumstances (e.g. slow action of implementation devices or unsuitable for high changing rate conditions), manipulating variable is expected to be slow in changing. Then, besides tracking error, changing rate is also needed to balance system performance.

In addition to above criteria, recently a new cost function is reported [11]. It use rising time rt , settling time st and overshoot %σ represent dynamic performance, use static state error sse represent static state performance, use gain margin mG and phase margin mP represent frequency domain performance. Finally a sum of weighted criteria is formed as cost function:

( )J K W S= ⋅ (13)

where { , , }p i dK K K K= represents vector composed by

controller parameters, 1 2{ , , }nW w w w= is weight vector, S is column vector composed by performance criteria.

To study the effect of different criteria, we selective four representative situation within enumerated criteria, which covers IAE , ITAE , time domain and frequency domain. Time domain cost function is the sum of weighted rising time rt , settling time st , overshoot %σ and static state error sse . Frequency domain cost function is the sum of weighted reciprocal of gain margin mG and phase margin mP .

When conducting experiments, we discovered that searching process will fall into space origin point if only

mG and mP is used in frequency domain cost function, namely the controller parameters are all zero. The cause for this is, when the controller parameters are all zero,

mG and mP will be infinite, and corresponding cost function obtain global minima 0. At this moment, step response of given system is constant 0, and obviously cannot meet expectation. The improvement strategy is adding ITAE criteria into cost function.

In order to satisfy required performance index, weight vector W must be properly selected, this paper follows below principles:

• Normally the magnitude of rt , st and σ is 101, therefore corresponding weight [0,1]iw ∈ .

• The desired magnitude of sse under unit step response condition is 10 n− , where n is a positive integer. To balance the importance of sse in evaluation function, we suggest 10n

iw = 。 • According to past experiences, the magnitude

of ( )mG dB and mP of a stable system is 101 and 102 respectively, thus weight for mG is [0,1]iw ∈ , and weight for mP is 210iw = .

• According to past experiences, the magnitude of ITSE of a well responded system is 101, thus

[0,1]iw ∈ .

IV. SIMULATION EXPERIMENT The response of many systems can be approximated as 1-

order inertia plus dead time in a certain range, the transfer function is as follow:

( )1

sG

G

K eG s

s

τ

τ

−

=+

(14)

Where GK is gain, τ is dead time, Gτ is time constant. Without loss of generality, this example sets 1GK = , 1τ = and 10Gτ = . For SISO system, suppose there is no requirement for manipulating variable, then IAE and ITAE criteria can take following simplified form:

0

( )IAE e t dt∞

= ∫ (15)

0

( )ITAE t e t dt∞

= ∫ (16)

Experiment parameters are as follow: • Sampling time (simulation step) 0.01s • Time horizon 200s • Population size 30 • Iteration times 500 • Inertia weight is calculated according to (4), where

max 0.9w = , min 0.4w = • Feasible bounds [0,500]。 • Maximum velocity max 100v = ， max maxidv v v− < < • Accelerate constant 1 2c = ， 2 2c = Conducts 10 trials, and select best results, the obtained

controller parameters is as showed at TABLE I. . Unit step response is as showed at Figure 2. , corresponding manipulating variables are shown in Figure 3. Performance criteria are given in TABLE II. , where dt is delay time, pt is peak time.

Controller structures are uniformly PI; this result shows the inner unity of four different cost functions, the cause of this phenomenon needs further study.

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From system step response and performance criteria data, we can see the characteristics of four types of cost function are as follow.

TABLE I. CONTROLLER PARAMETERS OF FOUR COST FUNCTIONS

Cost Function Type pK iK dK IAE 4.5508 0.4276 0 ITAE 6.1171 0.6410 0

Time Domain 2.9396 0.3047 0 Frequency Domain 5.7405 0.5053 0

Figure 2. Unit Step Response of Four Cost Functions

Figure 3. Manipulate Variables of Four Cost Functions

1. Integral of Absolute Error (IAE) deals with error in a same manner in the whole time horizon, thus the output curve is smooth. Because of the good tracking ability for step signal, the settling time is the smallest in all cost function, and overshoot is relatively small. It sets smaller restraint to tracking error in the late stage, leads to bigger static state error. The frequent domain criteria reflect that its stability is

relatively high. When there is low requirement on static state error and high requirement on dynamic response, IAE is suitable.

2. Integral of Time-Weighted Absolute Error (ITAE) clamps increasing weights on error in the time horizon, makes system tradeoff early stage dynamic performance for late stage precise tracking. Output curve change rate and overshoot are the biggest, but static state error is the smallest. ITAE has shortest rise time, and settling time is small. The frequency domain criteria are the lowest, which indicates that system has poor robustness for open loop characteristic change. When the requirement of control accuracy is high and dynamic process is low, ITAE is more suitable.

3. Time domain cost function integrates four main time domain criteria, includes settling time, rising time, overshoot and static state error. Its dynamic response process is the poorest but the frequency domain criteria are the best. The advantage of time domain cost function is that we can freely manipulate certain criterion by adjusting its weight. For example, we clamps bigger weight on overshoot in this case, and then obtained the smallest overshoot among all cost functions. Similarly, when there is special requirement for other criteria, it can be satisfied by simply adjust weights. Time domain cost function is easy for using, its physical meaning is clear. When there are special requirements for system response, its more suitable than others.

4. The response of frequency domain cost function is quite satisfactory, partly because ITAE is added. As mG and mP cannot use alone in practical problem solving, they can only be used as supplementation when other cost function cannot satisfy design requirements.

V. CONCLUSION Based on MPSO-PID tuning technique, this paper

analyzed the cost function of MPSO. Without loss of generality, we chose Integral of Absolute Error (IAE), Integral of Time-Weighted Absolute Error (ITAE), time domain and frequency domain cost function to conduct simulation experiment. Results show that integral of error criteria has satisfactory effect for common control systems, and when time weight is added, the response accuracy in late stage is improved, but the dynamic process is inferior to former, there is limitation for implement. Time domain cost function is flexible, but weight chosen is tricky, theoretically applies for all control system. Frequency domain cost function can only be used as supplementation for other cost functions, it is helpful to improve the system overall performance.

TABLE II. PERFORMANCE CRITERIA OF FOUR COST FUNCTIONS

Cost Function Type dt rt pt st %σ sse IAE ITAE ( )mG dB ( )mP ° IAE 2.1410 2.3026 5.5847 4.3628 0.5518 -1.7036×10-6 2.3558 4.6979 7.8244 180 ITAE 1.8481 1.4139 3.9184 5.8271 13.3811 2.5653×10-7 2.1540 3.7062 1.5626 58.3405

Time Domain 2.8277 4.7970 15.9261 8.9780 0.4513 7.2923×10-7 3.3898 8.9555 12.7349 Inf Frequency Domain 1.9087 1.5855 4.0901 8.0489 7.6380 -4.5735×10-6 2.2347 5.6969 4.8688 87.1281

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ACKNOWLEDGMENT This work is supported by the Key Project in the National

Science & Technology Pillar Program of China (No.2008BAH29B06), National Natural Science Foundation major project (No.70531020) and Major research project of Ministry of Education (No.306023).

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