** Stability Region Analysis of PID and Series **

**Leading Correction PID Controllers for the **

**Time Delay Systems **

D. RAMA REDDY*

Department of ECE, Jawaharlal Nehru Technological University Kakinada (JNTUK), Kakinada, E.G.Dt, Andhra Pradesh, India, Pin:-533003

dwrama50@gmail.com

Dr. M.SAILAJA

Associative professor,

Department of ECE, Jawaharlal Nehru Technological University Kakinada (JNTUK), Kakinada, E.G.Dt, Andhra Pradesh, India, Pin:-533003

* sailajahece@gmail.com *

**Abstract: **

This paper describes the stability regions of PID (Proportional +Integral+ Derivative) and a new PID with series leading correction (SLC) for Networked control system with time delay. The new PID controller has a tuning parameter ‘β’. The relation between β, KP, KI and KD is derived. The effect of plant parameters on stability region of PID controllers and SLC-PID controllers in first-order and second-order systems with time delay are also studied. Finally, an open-loop zero was inserted into the plant-unstable second order system with time delay so that the stability regions of PID and SLC-PID controllers get effectively enlarged. The total system is implemented using MATLAB/Simulink.

**Keywords: Stability region, PID controller, Series Leading Correction (SLC), Time delay systems. **

**1.** **INTRODUCTION **

In Networked Control System (NCS), the communication network is used to exchange the information between sensors, actuators and controllers. The NCS increases the system agility and ease to maintenance. The Fig1 derives the structure of NCS. The sensor generates the control signal to the controller. The controller generates the error signal to actuator. The NCS is effected by delay called the Network induced delay ( ) which is given by equation.1

(1)

Figure.1 Networked Control System

Generally PID controller is the most common controller used in many control applications. To design the effective controller, the stability region of PID controller must be studied. Equation.2 shows the mathematical model of PID controller.

C(s) = K (2)

The method introduced in [1] to stabilize the PI controller requires the polynomials of transfer function and (KP, KI) plot. This method is extended to PID controller where it requires (KP, KI), (KI, KD) and (KP, KD) plots. An another method is developed in [2] to stabilize the PID controller with LTI system, which does not require the knowledge of parameters of plant transfer function, but only its frequency response. Another method where KI is proportional to KD is proposed in [3] to stabilize the PID controller with time delay. The impact of time delay on system performance and the stability region of PID controller for first and second order systems with time delay are analysed in [4]. The recent designing approaches are proposed in [6, 7]. To improve the system performance, a new PID controller with series leading correction is described in [5]. Many new PID control strategies are discussed in [8-14] based on some intelligent algorithms. But these algorithms require lot of prior knowledge about control objects and difficult to use in actual applications. The SLC-PID controller model has only one adjustable parameter. The mathematical model of SLC-PID controller is given by equation.3

= K (3)

Here β is the tuning parameter. The series leading correction can reduce the overshoot. The impact of β on system performance is discussed in [5]. In this project, stability region of PID controller and SLC-PID controller for time delay systems is analysed and compared. Finally, the superiority of the SLC-PID controller is shown.

**2.** **SYSTEM PERFORMANCE **

Consider the second order single input and single output (SISO) system of Fig.1 with transfer function given by equation.4

_{ } (4)

Consider the transfer function of PID controller shown in equation.5

40 90 (5)

And consider β=0.6 for series leading correction. The β must be in the range of 0.45<*β<1 [5]. When *β=1
then the SLC-PID controller is equivalent to PID controller. Fig.2 and 3 shows the closed loop control system
diagrams of NCS with PID and SLC-PID controllers.

Figure.2 Time Delay System with PID Controller

Figure.3 Time delay system with SLC-PID Controller

TABLE.I PERFORMANCE OF TIME DELAY SYSTEM WITH PID CONTROLLER

S.No

### τ

Settlingtime

Rise

time Overshoot

1 0 0.74 0.35 2.19

2 0.1 1.47 0.24 24.29

3 0.25 19.55 0.22 87.40

TABLE.II PERFORMANCE OF TIME DELAY SYSTEM WITH SLC-PID CONTROLLER

S.No

### τ

Settlingtime

Rise

time Overshoot

1 0 2.96 0.41 0

2 0.1 2.75 0.26 14.63

3 0.25 10.99 0.23 68.74

When time delay ( ) increases then the Root locus becomes closer and closer to imaginary axis [4]. Fig.7-9
gives the root locus curves of above time delay system with PID controller and SLC PID controllers at **=**0, 0.05,
0.2. From Fig.7-9, the root locus of the system is improved with series leading correction at any value of . So the
stability region of given system must be improved with series leading correction.

**3.** **PARAMETER COMPUTATION **

The Fig.2 shows the general unity feedback time-delay system. Consider Gp (s) is the plant with time delay which is given by equation.6 and C(s) is the PID controller shown in equation.7. Here τ is the time-delay. The parameters of the stability PID controller are computed [15]. Also, the parameters of SLC-PID controller will be described. The transfer function of system is given in equation.8

(6)

C(s) = K (7)

(8)

Then the closed loop transfer function is as shown in eqution.9

⁄ 1 (9)

Substitute * in equation.8 and decomposing the numerator and denominator into real and imaginary *
parts as equation.10.

⁄ (10)

From equation.9 the closed loop characteristic polynomial can be derived as shown in equation.11.

(11)

From Euler’s formulae it is known that cos

Then equation.11 can be modified as given in 12

(12)

Where

cos sin (13)

cos sin (14)

Consider 0, 0 then

Here

sin cos cos sin

cos sin

cos sin cos sin sin cos

(16)

In classical tuning we could assume

(17)

Where n is the proportional factor consider. Substituting equation.17 into equation.15 then the coefficients KP, KI and KD are derived as shown in equation.18.

(18)

By using equation.15 the parameters of PID controller and SLC-PID controllers with first and second order time delay systems are computed.

**I.****First order time delay system **

The general first order system with time delay is

(19)

Here k is the open loop gain and T is time constant and assume n=4. From eqution.17 and 18, the equations.20 and 21 are derived.

*i.* *With PID controller *

The space coordinates (KP, KI, KD) PID controller with first order time delay system is derived by equation.20.

cos ⁄

4 sin ⁄ 4

⁄4 (20)

*ii.With SLC-PID controller *

Here the relation between β and parameters of PID controller with first order time delay system is derived from eqution.17 and eqution.18 and that is shown eqution.21.

1 1

1 1 . _{4}

**II.** **Second order time delay system **

The general second order system with time delay is shown in eqution.22.

(22)

Here k is open loop gain, T1, T2 are parameters of system and is the time –delay. Consider n=4, then from eqution.17 and equation.18, the eqution.23 and eqution.24 are derived.

i. *With PID controller *

The space coordinates (KP, KI, KD) PID controller with second order time delay system is derived by equation.23.

cos ⁄

4 sin ⁄ 4

_{4} (23)

*ii.* *With SLC-PID controller *

The relation between β, KP, KI and KD for second order time delay system is given in equation 24. 1

1 . ⁄4

* * _{4} (24)

In equation.21 and equation.24 sin cos

cos sin

**4.** **STABILITY REGION ANALASIS OF PID AND SLC- PID CONTROLLERS FOR TIME-DELAY **

**SYSTEMS **

If the stability frequency ( w*sf *) is known,then it is easy to derive the PID control parameters and (KP, KI)
stability region. Here w*sf=min {wg, wo} from [15] where wg* is the phase cross over frequency. It is the minimum

positive frequency among the frequencies at which F(w) plot and tan(w ) plot are intersecting as shown in
equation.25 and w*o* is the stability critical frequency. The minimum positive value of w which satisfies

eqution.26 is w*o*.

tan (25) 0 (26)

Computation of w*sf* is obtained from [15]. Substitute w from zero (o) to w*sf* in equation.18, a spatial curve

is plotted in space coordinates (KP, KI, KD), the spatial dimension is divided into stable and unstable regions. Stability region is the area between KI—KP boundary and the line KI=0.

**I.** **First order time delay systems **

From equation.19, the first order time delay system can be effected by open loop gain (k), time-delay (τ) and time constant (T). By using equation.20 and eqution.21, the plots for first order time delay systems with PID controller and SLC-PID are derived.

*i.* *Effect of Open loop gain (k) *

*ii.* *Effect of T *

From equation.16, the plant is unstable when T >0 and the plant is stable when T<0 and the stability region
is increased when T increase. Consider k=1, *τ=0.5, β=0.6; Figure.12 shows the effect of T on stability region *
*(KP, KI) of PID controller and SLC-PID controllers. Fig.12 gives the stability regions of first order system with *

PID and SLC-PID controllers at T=1 and T=3.The stability region for SLC-PID controller is more at any value of T as compare to PID controller. The stability region increases with increasing |T| [4].

*iii.* *Effect of time delay (τ)*

When τ increases then the (KP, KI) stability region is decreased and the control capacity PID of controller is diminishes [4].

At any of T, the stability region of SLC-PID controller is higher than stability region of PID controller. We can observe this analysis from fig.13. Here k=1, T=1 and β=0.6;

So from Figure.11-13, the stability region of first order time delay system can be improved by using series leading correction value to PID controller.

**II.****Second order time delay system **

From eqution.19 the (KP, KI) stability region of second order time delay system can be effected by open loop gain (k), system parameters T1,T2 and time delay(T).

By using equation.20, equation.21 we can derive the stability region for second order time delay systems. The effect of k and are already derived in First order so here, the effects of T1, T2 parameters on time delay system which is given by equation.22 with PID controller and SLC-PID controller are discussed.

i. *Effect of T1*

Here k=1,τ=0.5,T2=2,β=0.6 and T1=1,3. Fig.14 shows that the increase in T1 leads to increase the (KP, KI) stability regions of second order time delay systems with PID controller and SLC-PID controllers.

ii. *Effect of T2*

Here k=1,τ=0.5,T1=1,β=0.6 and T2=1,3.Fig.15 shows that the increase in T2 leads to increase the (KP, KI) stability region of second order time delay system with PID controller and SLC-PID controllers. In Fig.14 and Fig.15 the Series leading correction leads to improve the stability region.

**III.** **Effect of Open loop Zero **

The insertion of an open-loop zero into the forward path is leads to improve system qualities. Consider an unstable second order system with time delay as

_{.} . _{ (27) }

Here w*sf* is calculated from [15]. The (K*P, KI) curve of equation.27 can be obtained, shown in figure.20. It is *

clear that KP ranges in interval (1, 3.2) and KI ranges in interval (0, 0.4). Insert an open-loop zero in system (27) then

_{.} . _{ (28) }
Here KP ranges from (1, 3.8) and KI ranges from (0, 1.9)

Consider Series Leading Correction for (25) then KI ranges from (0, 2.2) and KP ranges from (1.5, 4.2). The total area under this curve is large as compare to others.

**5.** **SIMULINK **

By using MATLAB Program, the stability region is analysed for first order and second order time delay systems with PID and SLC-PID controllers at different conditions. The MATLAB Program is written by using Equations.20, 21, 23 and 24.

The 2X1 multiplexer is used to take inputs from the outputs of closed loop system with PID controller and closed loop system with SLC-PID controller, and gives one output to scope. The scope displays both curves.

Figure.4 Step response comparison of PID, SLC-PID controllers

**6.** **SIMULATION RESULTS **

Here the tuning parameter of series leading correction is considered as *β=0.6. . The step response and the *
root loci are derived for the system given by equation.4.Figure. 5-7 represents and compares the step responses
of time delay system with PID controller and SLC-PID controller at different time delays τ=0,0.1 and 0.25. Here
X-axis represents time and Y-axis represents step output. In Figure.5, the overshoot of the time delay system
with PID controller is reduced to Zero (0) by using SLC-PID.

Figure.5 Comparison of Step response at τ=0

Figure.6 Comparison of Step response at τ=0.1

Figure.7 Comparison of Step response at τ=0.25

Figure.7-9 represents the root loci of time delay system with PID controller and SLC-PID controller at different time delays. Here X-axis represents Real axis and Y-axis represents Imaginary axis.

Figure.9Comparison of Root locus at τ=0.05

Figure.10 Comparison of Root locus at τ=0.2

In figure 11-13, X-axis represents KP and Y-axis represents KI. Here the stability region (KP, KI) is analysed for first order time delay system which is given by equation.19 with PID controller and SLC-PID controller are compared in figure.11-13.

Figure.12 Comparison of stability region (KP, KI) at T=1,3

Figure.13 Comparison of Stability region at τ=0.35,0.5

The stability region is analysed for second order time delay system which is shown in eqution.22 with PID controller and SLC-PID controller and compared at different conditions in figure.14-15. Here X-axis represents KP and Y-axis represents KI.

Figure.15 Comparison of Stability region at T2=3, 1

By inserting open loop zero, the (KP, KI) stability regions of an unstable system which is given by equation.27 with PID controller and with SLC-PID controller are enlarged in figure.16.

Figure.16 Effect of open-loop Zero on stability region for an unstable system

**CONCLUSION **

In this paper the dynamic performance of the time delay system compared in both cases by simulation. The
SLC-PID controller has another adjustable parameter compared with traditional PID controller. Obviously,
parameter adjustment improves the system dynamic performance. The relation between *β, KP, KI and KD* is

derived. The stability region also analysed and compared for both PID controller and SLC-PID controller with time delay systems.

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**AUTHORS **

D.Rama Reddy* received the B.Tech degree in Electronics and Communication Engineering in 2009 from the JNT University, Kakinada and pursuing M.Tech degree Instrumentation and Control Systems from JNT University Kakinada. His areas of interest include Instrumentation, Control Systems, and MATLAB and communication systems.