The reduction of system response overshoot through adjustment of PI controller's parameters.

1. INTRODUCTION

Even though there are many new types of controllers, based on theories such as fuzzy logic or neural networks, traditional controllers (PI, PID, etc.) are still widely in use (Astrom & Hagglund, 2006). Because of the variety of the systems controlled by these kinds of controllers, the results achieved through traditional approaches developed for calculating these controller parameters (for instance, technical or symmetrical optimum), may in some cases prove to be less then acceptable.

The traditional approach in generating parameters for the PI controller starts with linear mathematical model of the controlled system calculated around predefined stationary state. While the system is operating around that stationary state, controlling the system with PI controller designed for that specific stationary state usually produces satisfactory results. The problems with system response overshoot usually arise when that system is forced to reach a new stationary state (Haugen, 2004; O'Dwyer, 2006).

To overcome this problem, without replacing the controller that is already integrated inside the controlled system, two sets of parameters for the PI controller should be calculated: first ones around original stationary state, and second ones around new stationary state. By doing this, calculating errors, which can not be avoided while building mathematical model of the system, could be minimized.

2. MATHEMATICAL MODEL

To explore the connection between mathematical model of the system and system response overshoot, a two tank system was used. Building of the system's mathematical model starts with comparation of the input and output flow:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where V is volume, A is cross-section, h is height of fluid, [Q.sub.i] is input and [Q.sub.o] is output flow.

Flow of the fluid inside the system's pipes and through the output valve, represented by parameters [Q.sub.i] and [Q.sub.o], are described by Bernoulli's equation:

[FIGURE 1 OMITTED]

[Q.sub.i] = [K.sub.c] x [square root of [rho] x g x [DELTA]h], [Q.sub.o] = [K.sub.o] x [square root of 2 x g x h] (2)

where g is gravitational constant, [rho] is fluid's density, [K.sub.c] is coefficient of the pipes, Ko is coefficient of the output valve and [DELTA]h = [h.sub.1] - [h.sub.2] is the difference between heights of the fluid inside the system's two tanks.

From equations (1) and (2) coefficients [K.sub.c] and [K.sub.o] could be represented as functions of the initial conditions defined by fluid height h2 inside the tank (Durovic, 1997).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

With equations from (1) to (3) a non-linear mathematical model can be constructed. From that mathematical model, it is possible to derive a linear mathematical model around stationary state defined by initial conditions h10 and h20.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Control signal [Q.sub.p], shown in the Figure 1., is defined by the characteristics of the pump (Durovic, 1997). Based on the presented equations (1) to (4) and block diagram of the linear mathematical model shown in Figure 1., transfer function of the system together with the transfer function of the pump gives:

G(s) = K/(1 + [T.sub.1]s) x (1 + [T.sub.2]s) x (1 + [T.sub.3]s) (5)

where K = [K.sub.P] x [K.sub.3], [K.sub.P] = 1/[K.sub.1], [K.sub.3] = 987, [T.sub.1] [T.sub.2] = [A.sup.2]/[K.sub.1][K.sub.2], [T.sub.1] + [T.sub.2] = A([K.sub.1] + 2[K.sub.2])/[K.sub.1][K.sub.2] and [T.sub.3] = 1.85 [s].

3. PI CONTROLLER

Transfer function of a PI controller is

[G.sub.R] (s) = [K.sub.R] 1 + [T.sub.R]S/[T.sub.R]S. (6)

When there is one dominant time constant TD in system's transfer function, parameters [K.sub.R] and [T.sub.R] of the PI controller can be calculated in the following fashion (Surina, 1991):

[T.sub.R] = [T.sub.D], [K.sub.R] = [T.sub.R]/2K[T.sub.S], (7)

where [T.sub.S] is the sum of all time constants in system's transfer function, except time constant [T.sub.D].

Based on the equations (7), parameters [T.sub.R] and [K.sub.R] of PI controller could be described as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

As it can be seen from the equations (8), parameters [T.sub.R] and [K.sub.R], which define the way PI controller controls the system, are functions of parameters a and b, which are functions of parameters in equations (3) and (4). This correlation between parameters of the PI controller and parameters whose values depend directly on the initial conditions, enables us to calculate and use two different sets of parameters [T.sub.R] and [K.sub.R] around two different stationary states.

If only one set of parameters is to be used, the set defined around initial stationary state, the following parameters are calculated using equations (8):

[T.sub.R] = 185,33[s],[K.sub.R] = 64,86. (9)

With only one set of parameters, PI controller will control the system with them, no matter how big the step change is required from the system.

3.1 Using two sets of parameters

To reduce system response overshoot, the second set of PI controllers parameters around new stationary state should be calculated. These sets of parameters should control the system after the system response to a step change of the control signal reaches one third of the step change value.

In Table 1. are presented different values of parameter [K.sub.R] calculated by using equations (8). It is evident from this table that parameter KR will change with every different step change in a way that with a greater step change the difference between value of parameter KR calculated in (9) and new value of KR would be bigger.

4. COMPUTER SIMULATION

Since the difference between values of parameter KR is bigger when the value of the step change is greater, it is obvious that two sets of parameters should be used when a greater stationary state change is required. Because of this, the computer simulation was conducted for step change from [h.sub.2] = 0,935 [m] to [h.sub.2] = 1,2 [m].

The traditional approach would be to use only one set of parameters for PI controller. These parameters are calculated using initial conditions and then used as the first set of parameters for PI controller. Second set of parameters is calculated around new stationary state, and new value for [K.sub.R] is as shown in Table 1. for step value of 1,2 [m].

Once the system reaches one third of the desired new stationary state, second set of parameters for PI controller is used. These new parameters replace the first set of parameters and they control the system until it reaches its new stationary state.

[FIGURE 2 OMITTED]

In Figure 2. the system responses for both described cases, together with control signals, are shown.

As shown in the Figure 2., when only one set of PI controllers parameters is used, the system has reached its new stationary state with overshoot of 4,11 [%], while when two sets of PI controllers parameters are used, the system response overshoot is 1,35 [%]. Also, less time is needed for the system to reach its new stationary state.

In both situations, the control signals don't unnecessarily force the pump (which is this systems actuator) when the system response reaches its new stationary state.

5. CONCLUSION

By using two different sets of parameters calculated around two different stationary states, it was possible to significantly reduce system response overshoot. Also, with proposed approach the system has reached its new stationary state more quickly compared to the traditional approach with only one set of parameters.

The mathematical model of the system is developed around predefined initial conditions and is a starting point for calculation of PI controllers parameters. When the system is forced to reach a new stationary state that is different then the one defined by initial conditions, the mathematical model incorporates calculation errors which are greater when the system stationary state change is bigger.

If two sets of different parameters for PI controller were calculated around two different stationary states, the one system starts from and the one that system is trying to reach, these calculation errors could be reduced, thus causing the significant reduction of the system response overshoot.

6. REFERENCES

Astrom, J. K., Hagglund, T. (2006). Advanced PID control, ISA, ISBN 978-1556179426, USA

Durovic, G. (1997). State space controller employing fuzzy logic, Bachelor's degree, FER Zagreb, Croatia

Haugen, F. (2004). PID control, Tapir Academic Press, ISBN 978-8251919456, Norway

O'Dwyer, A. (2006). Handbook of PI and PID controller tuning rules, Icp, ISBN 1-860-94342-X, London, UK

Surma, T. (1991). Automatska regulacija, Skolska knjiga d.d. Zagreb, ISBN 86-03-00321-1, Croatia
```Tab. 1. Value of KR for different step changes

Step val. 1,20 1,15 1,10 1,05 1,00 0,95
[K.sub.R] 57,26 58,50 59,80 61,22 62,73 64,36
```