Designing of the PID and FOPID Controllers using Conventional Tuning Techniques Amlan Basu
Sumit Mohanty
Rohit Sharma
Research Scholar (M.Tech) Dept. of Electronics and communication ITM University Gwalior, Madhya Pradesh 474001, India
[email protected]
Assistant Professor Dept. of Electronics and communication ITM University Gwalior, Madhya Pradesh 474001, India
[email protected]
Research Scholar (M.Tech) Dept. of Electronics and communication ITM University Gwalior, Madhya Pradesh 474001, India
[email protected]
Abstract— This paper focuses on the designing of Integer order & fractional order PID controllers. Various tuning techniques are used for the designing of the PID controllers and the Fractional order PID controllers (FOPID). Ziggler Nicholas method, Cohencoon, Astrom-Hagglund (AMIGO), Chien-HronesReswick methods are used to find proportional, derivative & integral parameters of PID & FOPID. Nelder Mead optimization technique is used to determine the specialized fractional order parameters for FOPID. The complete algorithm or the tuning process of each and every method has been explained and discussed. All the designs & their step response is specified & all the characteristics of the systems are discussed. Keywords— Ziegler-Nichols, Cohen Coon, Astrom-Hagglund (AMIGO), Chien-Hrones-Reswick (CHR), PID, FOPID.
I. INTRODUCTION The PID controller is basically a device that is being used in a closed loop along with the plant or the system which is to be made stable or whose parameters are to be improved. The PID controller came into existence in the year 1939 and till today it has remained irreplaceable and it is almost used in the 90% of the industrial sector. The PID controller basically has three tuning parameters which are Kp, Ki and Kd. The work of FOPID (fractional order PID) controller is the same as that of the PID controller however it gives better response than the PID controller as it has five tuning parameters instead of three which are Kp, Ki, Kd, λ and μ. The various tuning parameters mentioned can be found using different tuning techniques which are going to be discussed further in this paper. The different techniques that are going to be discussed are the Ziegler-Nichols method, Cohen Coon method, AstromHagglund method (AMIGO) and the Chris-Hrons-Reswick (CHR) method. II. PID CONTROLLER PID stands for the proportional integral derivative which is mathematically defined as, 𝑡
𝑢 𝑡 = 𝐾𝑝 𝑒 𝑡 + 𝐾𝑖
𝑒 𝜏 𝑑𝜏 + 𝐾𝑑 0
𝑑𝑒 𝑑𝑡
(1)
Where, Kp is the gain of proportionality, Ki is the gain of Integral, Kd is the gain of Derivative, e is the Error (SP-PV), t
is the instantaneous time and τ is the variable of integration that takes on the values from time 0 to the present t. On performing the Laplace transform of the equation (1) which is the PID controller equation is, 𝐾 𝐿 𝑠 = 𝐾𝑝 + 𝑖 + 𝐾𝑑 𝑠 (2) 𝑠 The reasons behind the use of PID controller are that it produces less settling time and overshoot, very reliable and simple. III. FOPID CONTROLLER FOPID stands for the fractional order proportional integral derivative. The equation of the FOPID in Laplace domain is [1], 𝐾 𝐿 𝑠 = 𝐾𝑝 + 𝜆𝑖 + 𝐾𝑑 𝑠 𝜇 (3) 𝑠 Where, Kp is the gain of proportionality, Ki is the gain of Integral, Kd is the gain of Derivative and λ and μ are the differential-integral’s order for FOPID controller [1]. The reasons behind the use of FOPID controller are that it does not have any steady state error, has gain and phase cross over frequency and also the gain and phase margin and gives robustness from the variations in the phase cross over frequency and the high frequency noise [1]. IV. ZIEGLER-NICHOLS METHOD To obtain controller parameters, in the year 1940 Ziegler and Nichols formed two empirical methods which are, the non-first order plus dead time circumstances and indulged intense manual computations [2]. To calculate the tuning parameters we use the following procedure: For feedback loop or closed loop, Integral and derivative action must be removed. Integral time (Ti) must be set to 999 or larger and derivative controller (Td) must be set to 0. By changing the set point creates small disturbance in the loop. Until the oscillations have common amplitude keeps adjusting the proportional by increasing or decreasing the gain. The gain value (Ku) and the period of oscillation (Pu) must be recorded [3].
The necessary settings of the controller must be determined by inserting the appropriate values in the Ziegler-Nichols value [3].
TABLE I. CALCULATION OF KP, KI AND KD IN CLOSED LOOP [3] Kp Ti Td PID Ku/1.7 Pu/1.2 Pu/8 PI Ku/2.2 Pu/2 P Ku/2
Advantages in this tuning process are that we only need to change the P controller which justifies that it is easy to experiment and moreover it provides a much accurate scenario of how the system is working by including the complete dynamics of the system. Whereas the disadvantage related to the same is that the experiments being carried out are very time consuming and the other one is that it can cause the system to become uncontrollable by speculating into the unstable regions while the P controller is being tested. For feed forward loop or open loop, The method is also known as Process Reaction method because it has the ability of testing the open-loop reaction of the process so as to bring about the change in the control variable output [4]. The steps are as follows, (i).Open loop step test must be performed. (ii).By studying the process reaction curve dead time or transportation lag (τdead), time for the response to change or the time constant (τ), and the value at which the system reaches the steady state (M0) for a step change X0. 𝑋 τ 𝐾0 = 0 ∗ (4) 𝑀𝑢
τ 𝑑𝑒𝑎𝑑
To calculate the tuning parameters of the controller insert the values of reaction time and lag rate into the Ziegler-Nichols open loop tuning equation. TABLE II. CALCULATION OF KP, KI AND KD IN OPEN LOOP [3] Kp Ti Td PID 1.2Ko 2τdead 0.5τdead PI 0.9Ko 3.3τdead P K0
The advantages of the above method or steps are that the method is quicker and easier to use than other methods, the method discussed above is robust and popular and the method is least disruptive and easiest to implement. The disadvantages related to the same are the dependency on pure proportional measurements so as to estimate I and D controllers, the approximate values of Kc, Ti and Td for different systems might not be accurate and it is not applicable for I, D and PD controllers. V. COHEN COON METHOD OF TUNING Cohen-Coon tuning method is mainly used to overcome the slow, steady state response which occurs in the ZieglerNichols tuning method [5]. This method is generally used for
the first order systems or models having time delay as the controller does not spontaneously responds to the disturbances. It is an offline method that is when a it is at steady state then a step change can be introduced at the input. Now on the foundation of time constant and the time delay the output can be calculated and the initial control parameters can be found out using the response. To obtain the offset and standard decay ratio of minimum value there are a set of predestined settings for the CohenCoon method, Where, P is the percentage in the input, N is percentage change of output/ τ, L is τdead and R is (τ𝑑𝑒𝑎𝑑 τ). We can use Ko in place of (𝑃 (𝑁𝐿)). The procedure of the method is as follows, Wait for the complete process to reach the steady state. Step change is to be introduced at the input. Approximate first order constant with time constant τ which is delayed by τdead units which is based on the output, from the time the step input was introduced. By recording the following time instances the value of τ and τdead can be found, t0=input step start up point, t2=half point time and t3= time at 63.2%. Calculate the process parameters τ, τdead and Ko by using the measurements done at t0, t2, t3, A and B. On the basis of τ, τdead and K0 the parameters of controller can be found [5]. The advantages of the Cohen-Coon method are that the response time of the closed loop is quick or fast and this method can be used in the systems with time delay. Whereas the disadvantages of this method are that it can only be used for the first order systems which include large process delay, it is an offline method, closed loop systems are unstable and the approximated value of τ, τdead and K0 might not be compulsorily accurate for different systems. Kp
Ti
Td
PID
(P/NL)*(1.33+ (R/4))
L*(30+3R)/(9+20R)
0.5τdead
PI
(P/NL)*(0.9+(R/12))
L*(30+3R)/(9+20R)
4L/(11+2R)
P
(P/NL)*(1+(R/3))
TABLE III. CALCULATION OF KP, KI AND KD [5]
VI. CHIEN-HRONE-RESWICK METHOD OF TUNING The modified method of the Ziegler-Nichols method is the Chien-Hrone-Reswick method. There are basically two forms of CHR which are Chien-Hrone-Reswick (set point regulation) also known as CHR-1 and the Chien-Hrone-Reswick (disturbance rejection). The development of this tuning was done in the year 1952 by Chien-Hrone-Reswick. For process control application this method provides a better way of
selecting the compensator [6]. On the basis of this method the controller parameters are often tuned in the industrial processes. The parameters of the controller for the method for 0% and 20% overshoot is summarized in table 4 and table 5
TABLE IV. CHR 1 METHOD OF CALCULATING KP, KI AND KD [6] Overshoot 0% 20% Controller
Kp
Ki
Kd
Kp
Ki
Kd
PID
0.6/a
T
0.5L
0.95/a
1.4T
0.47L
PI
0.35/a
1.2T
-
0.6/a
T
-
P
0.3/a
-
-
0.7/a
-
-
TABLE V. CHR 2 METHOD OF CALCULATING KP, KI AND KD [6] Overshoot 0% 20% Controller
Kp
Ki
Kd
Kp
Ki
Kd
PID
0.95/a
2.4L
0.42L
1.2/a
2L
0.42L
PI
0.6/a
4L
-
0.7/a
2.3L
-
P
0.3/a
-
-
0.7/a
-
-
VII. ASTROM-HAGGLUND OR AMIGO METHOD OF TUNING The Astrom-Hagglund method is the approximate that completes the processing a very simple way. The other name for this tuning method is AMIGO which stands for approximate M-constrained integral gain optimization method for tuning [7]. The procedure of the tuning method is almost similar to the Ziegler-Nichols method of tuning. The tuning procedure of the AMIGO is as follows [7], 1 𝑇 a. 𝐾𝑝 = (0.2 + 0.45 ) (5) 𝐾
b.
𝐾𝑖 = (
c.
𝐾𝑑 =
Order : On the basis of values at the vertices, f(x1) ≤ f(x2) ≤ …………. ≤ f(xn+1) Calculate the centroid of all points (x0) except xn+1. Reflection: Calculate xr= x0+ α (x0 – xn+1). If the reflected point is not better than the best and is better than the second worst, that is, f(x1) ≤ f(xr) < f(xn). After this by replacing the worst point x n+1 with reflected point xr to get a new simplex and go to the first step. Expansion: If we have the best reflected part then f(xr) < f(x1), then solve the expanded point xe=x0+γ(x0-xn+1). If the reflected point is not better than expanded point, that is, [f(xe)
