Way Of Finding The Optimal PID Controller Engineering Essay

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Use of proportional integral derivative (PID) controller for commercial purpose is increasing as it can provide various benefits. At the same time tuning of PID is a challenging issue for the researcher as well as the plant operators. Various methods have been proposed to meet this essential. In this paper optimization method based on Chaos algorithm will be used for tuning the PID controller gains for an Automatic voltage regulator (AVR) system. Comparing with the other methodologies used for optimization purpose chaotic algorithm provides quick system response as it can directly search the global optimum. This criteria are used for designing an AVR system with optimized PID controller. Simulation studies on the AVR system using Matlab are demonstrated and hence found that chaotic algorithm provides better PID parameter to get good system performance.

Index Terms- Chaotic optimization, PID controller, AVR


Stability of power system is one of the most important areas in electric system operation. Power system may often undergo faults, load variation or many types of disturbances which in turn reduce the stability of the whole system. In order to have a sharp response over these disturbances an automatic voltage regulator (AVR) is used. To have a control over the AVR the use of (PID) controller is gaining its interest by providing lots of benefits in commercial uses. PID controller provides a simple structure which is easy to be understood and by tuning the gain parameters desired system response can be obtained.

Peter Van Overschee, Christiaan Moons, and Wim Van Brempt have presented a report that [1] almost 90% of controller in commercial use are PID type and among them 80% are poorly or not optimally tuned.

Many techniques that have been already existed to tune the PID controller are Ziegler/Nichols, gain-phase margin method, minimum variance method, predictive control [2-5]. These techniques have some drawbacks like complex procedure to set the gain, transient closed loop response and difficult mathematical model [6]. So to determine the PID controller parameter suitably researcher has come up with the new techniques such as evolutionary algorithm [7], artificial neural network [8], particle swarm optimization [9], and fuzzy system [10]. The use of chaotic algorithm for optimizing the PID controller is another holistic approach, which can provide better results with less computation time.

Chaos is aperiodic long term behavior in a deterministic system that exhibits sensitive dependence on initial condition [11]. Using chaotic sequence to tune the PID controller provides the benefit that it can avoid results to local extreme [12].

The aim of this work is to optimize the parameter of proportional, integral and derivative gain of a PID controller connected with an AVR system using chaotic algorithm.


A synchronous generator connected with an infinite bus system may show various kinds of unwanted behaviors like oscillation of output voltage caused by load variation, line faults or unexpected disturbances, hampering the stability of the whole system. So a control methodology is required which can provide dynamic response to these kinds of sudden disturbances. Automatic Voltage Regulator (AVR) is widely used to control the output voltage by comparing the output voltage with a reference value and provide desired response. A simple AVR consists of amplifier, exciter, generator and sensor. The generalized model of each part is discussed below [12] [16].

Amplifier model: An amplifier used in AVR can be presented with the gain Ka and time constant Ta. The open loop transfer function of amplifier circuit is then

Typical values of constant Ka vary from 10 to 400 and Ta from 0.02 to 0.1s.

Field Exciter model: The generator field exciter is needed the gain parameters Ke and time constant Te to present open loop transfer function

Where the value of Ke is kept in between 10 to 400 and Te is 0.5 to 1.0 s.

Generator model: The equivalent model of a generator has a proportional gain Kg and time constant Tg. If the input and output of the generator are field voltage from the exciter Vf and terminal voltage Vg then transfer function stands

The constants of a generator model depend on different loads, Kg can vary from 0.7 to 1 and Tg from 1 to 2s.

Sensor model: Sensor takes the terminal voltage from the generator and gives the response to the input terminal. Transfer function of the sensor model is

Where time constant Ts vary from 0.001 to 0.06s [12] [17].

AVR system with a PID controller can be represented by the following block diagram, Figure 1 [3].

Figure 1.Block Diagram of an AVR with PID controller

The closed loop transfer function of the AVR without controller is,

Transfer function with PID controller is,


PID controller has three gain components, proportional gain Kp, integral gain Ki and derivative gain Kd. Kp reduces the error, Ki eliminates the steady state error and Kd decrease the overshoot of the step responses. Block diagram of PID controller is shown below in Figure 2 [12].


Control Signal


Figure 2.Block Diagram of PID Controller

Output from a system is sensed with a sensor and it is fed as the input of the PID controller.PID controller can compare the output with the reference value set and provide new controller gains for the system to maintain the desired output. Mathematical expression for the PID controller is as bellow

Where Ti and Td are integral time constant and derivative time constant respectively. After Laplace transformation the system has the transfer function like this:

PID controller values are chosen based on the system behavior which provides the optimal solution for the system response.


Chaos can be described as a common nonlinear phenomenon whose behavior is similar to randomness. The characteristic of chaos is that a small change in initial value can yield widely diverging outcomes on future behavior of the system like stable fixed points, periodic oscillations or ergodicity. Details of the behavior of chaos can be found in [11].

Chaos can move on every states of a certain range without repetition. This shows the property of disordered randomness, according to laws of iteration. Hence characteristics of chaos is faster than other algorithm like stochastic which is related to probabilities. So this property can be applied in optimization calculation [13].

A system with multiple variables sensitive to each other on initial condition is dependent on multiple elements with nonlinear interactions [14].

Many types of equations have been introduced in literature for application in optimization method. Namely Tent map, Gauss map, Lozi map, sinusoidal iterator, Mackey-Glass system, Lorenzo system, Ikeda map [12]. In this paper the chaotic phenomena will be proposed based on Lozi map [15].

Lozi function is the simplified representation of Henon [20] and it admits strange attractors. This chaotic function also involves non-differentiable equations which are difficult to model in time domain. The function for Lozi map is given by

Where k is the number of iteration. Here values for y are normalized between 0 to 1. The normalized function can be represented as

Where y vary from -0.6418 to 0.6716 and constant c and d are specified as -0.6418 and 0.6716 respectively in equation (11). The parameters used in equation (9) and (10) are a=1.7 and b=0.5 , these values are suggested by [19]. By using these equations a lozi map is shown in Figure 3.

Main objective of this work is to optimize the value of PID controller gains kp, kd and ki. Hence the minimize function can be written as:

f(X), X = [x1, x2, x3] = [ kp kd ki]

holding the constraint xi Ñ” [Li, Ui]; i = 1, 2, 3

Where X is the decision variable vector bounded by lower (Li=0) and upper (Ui=1.5) limits.

lozi map_200 samples

Figure 3: Lozi map with 200 samples

Algorithm for the chaotic optimization search:


k=Iteration no

λ=Step size (Convergence behavior of optimization)


X* = Feasible solution for current Search

f* = Best objective function

Step1: Initializing the variables: Set iteration=1. Set the initial values for y1, y, a, and b for generating Lozi Map. Set f*

Step2: Algorithm for Chaotic Search


While iteration ≤ k do

For i=1 to n(here n=3)

If r < 0.5 then (r is a uniformly distributed random variable with range [0, 1])

Xi(k)=xi*+ λ.zi(k).|Ui-Xi*|

Else if

Xi(k)=xi*-λ.zi(k).| Xi*- Li|

End if

End for

If f (X(k)<f*) then

X*=X (k)

f*= f(X(k))

End If

iteration= iteration +1;

End While


Here λ is the convergence factor. For this work its value has been taken as 0.01, 0.05 and 0.1.


This optimization algorithm was simulated Matlab version R2010a (Math works).Three case studies were done. In each case study 30 individual samples were taken and for each value of λ eigen value analysis was done by plotting the root locus of the transfer function. From there the best set of values for kp, kd and ki were selected with a criteria based on minimum overshoot. Table 1-3 summarizes the optimization value for the different cases and also the best set among the cases. For Case-1 and Case-2 best solution was found out for λ=0.01and Case-3 for λ=0.05. Figure 4. displays the open loop step response of the AVR system. Figure 5, 6, 7 displays the step response of the AVR system with the optimized value of PID parameter found in each cases. Root locus plot for first cases is shown in Figure 8, 9. From the Figure 5, 6, 7 it is seen that optimized PID controller provides better response in terms of overshoot, settling time and steady state error.

Table 1.Case -1: Optimized PID gains of AVR System with Kg=0.7 and tg=1.0

Convergence Factor

Numerical Analysis of Overshoot

Optimized PID Gains

Lamda (λ)

Minimum Overshoot (best)

Maximum Overshoot (worst)


















Table 2.Case -2: Optimized PID gains of AVR System with Kg=0.7 and tg=1.5

Convergence Factor

Numerical Analysis of Overshoot

Optimized PID Gains

Lamda (λ)

Minimum Overshoot (best)

Maximum Overshoot (worst)


















Table 3.Case -3: Optimized PID gains of AVR System with Kg=1 and tg=2

Convergence Factor

Numerical Analysis of Overshoot

Optimized PID Gains

Lamda (λ)

Minimum Overshoot (best)

Maximum Overshoot (worst)


















Openloop response

Figure 4.Open loop response with Kg=0.7,tg=1


Figure 5. Output response of an AVR system

(Kg=0.7,tg=1) using PID with λ=0.01

T_3_step Figure 6. Output response of an AVR system (Kg=0.7,tg=1.5)using PID with λ=0.01

T_3_step Figure 7. Output response of an AVR system

(Kg= 1, tg=2) using PID with λ=0.05

T_1_root locus_all Figure 8. Root Locus plot for the AVR system (Kg=0.7,tg=1) using PID with λ=0.05

C:\Documents and Settings\foisal\Desktop\project\Project_lisa lana\Graphs\T_1_lamda001_al.jpgFigure 9. Root Locus plot (dominant pole location) for the AVR system (Kg=0.7, tg=1) using PID with λ=0.01


In this paper PID controller parameter tuning based on chaos optimization theory was used and applied to Automatic Voltage Regulation system. By using the optimized PID controller the system shows a very small steady state error and also provides a good dynamic response with a less overshoot for terminal voltage.