A PID system refers to a control system with a proportional, differentiator, integrator gain respectively. The PID measures the process variables and compares them with the set point. The controller then attempts to reduce the error accordingly, based on the predesigned algorithm. Proportional, integrator, and differentiator are independent parameters. However, they are designed to work separately or together to improve the stability of the system. These parameters reduce improve the stability by reducing the overshoot and the error within the system. This project intends to explore major applications of a PID controller and the approach that is used to improve the stability of the system.
STEP 1. Design using an open loop system
Figure 1: Open loop system.
Figure 2: Armature circuit and motor dynamics
Step 2: Using a closed loop transfer function
STEP 3
>> Sys=linmod('order')
> In dlinmod at 172
In linmod at 60
Sys =
a: [4x4 double]
b: [4x1 double]
c: [0.5000 -100 0 0]
d: 0
StateName: {4x1 cell}
OutputName: {'order/Out1'}
InputName: {'order/In1'}
OperPoint: [1x1 struct]
Ts: 0
>> Tss=ss(SYS.a,SYS.b,SYS.c,SYS.d)
??? Undefined function or variable 'SYS'.
>> Tss=ss(Sys.a,Sys.b,Sys.c,Sys.d)
a =
x1 x2 x3 x4
x1 0 0 0 4.37
x2 0.5 -100 0 0
x3 0.5 -100 -243.1 -3.74
x4 0 0 1 0
b =
u1
x1 0
x2 0
x3 12
x4 0
c =
x1 x2 x3 x4
y1 0.5 -100 0 0
d =
u1
y1 0
Continuous-time model.
>> [n,d]=tfdata(Tss,'V');
>> T=tf(n,d)
Transfer function:
26.22 s
s^4 + 343.1 s^3 + 2.431e004 s^2 + 371.8 s - 6.526e-013
Step 4: PD controller Kp/Kd=4
Figure 3: the design and corresponding curve of the PD circuit when the ratio of Kp:Kd is 4.
Designing the PD controller
The PD Controller
Figure 4: The PD design when the overshoot is 0.2%
The value of Kd is 50, which is the optimum value that is used to achieve an overshoot of 0.2%
Graph 1: When Kd is 50 and the overshoot is 0.2%
Step 5
The system was successful. I was able to achieve an overshoot of 0.2% when Kd of 50 through a series of adjustments. Increasing Kd beyond had a slight impact on the rise time, a decrease in overshoot, and a decrease in the settling time. However, adjustments on Kd did not have any effect on the steady state error. This is a closed loop system or an open loop system, and the derivative gain corrects the overshoot and settling time.
Step 6
The PD control gain reduces rise time, overshoot, and settling time. However, the error does not eliminate the steady state error. From the design, the error and the overshoot could only be reduced to 0.2%, but could not be eliminated.
Step 7: PID design
PID
Kd=50
Kp=1
Ki=0
The design
Figure 5: After modifying the design
The output
Graph 2: the system after introducing a PID controller system
The above graph in step 7 shows a system, which has no overshoot, reduced rise time, and no SSE error. From the graph, it is evident that the introduction of an integrator eliminates the steady state error.
Discussion
The system was upgraded gradually by adjusting the value of K d with the aim of achieving an optimum overshoot for the system. Adding an integrator into the system reduced the SSE (steady state error) to zero. The implementation of the integrator into the system was easy to implement. Under normal operational circumstances of any system (mainly analogue systems), the integral control system integrates the error to produce a control signal. Practically, a proportional controller is used as an analogue integrator. Digital integrators used a preinstalled standard algorithm to correct the error signal within the system. However, introduction of an integrator into the system reduced the response time of the system.
