Defects of pure differential control
U(s)/E(s)=TdS
After U(s)=TdS*E(s) Laplace transform on both sides, it can be obtained by discretization
When e(t) is a step input
Note : e(t) must change during the sampling process, it cannot be a step input, otherwise the control system will have no meaning to exist. But in practice, not every sampled value is used to find e(t). The system has a control cycle, and the sampling values in the control cycle may be different, but only the first value at the beginning of the control cycle takes effect. If each sampling is executed, it is like the teacher asking you to go to the library, and you will stop before you arrive. Let you go to the teaching building, and let you go to the playground before the teaching building arrives. The system needs a time to execute the command, and this time is the control period, in which e(t) is the step input.
in conclusion:
① The pure differential system only has an output from the controller during the first sampling period
② When T is small, the effect of Td/T is large, which is easy to cause the actuator to enter the saturation region or cut-off region when the control signal is too large, resulting in nonlinear characteristics of the actuator, which in turn leads to excessive overshoot and continuous oscillation of the system. Poor dynamic quality
③ The differential action is more sensitive to noise (high frequency disturbance), which easily leads to oscillation of the control process and reduces the adjustment quality
Improve:
A pure differential link is connected in series with an inertia link, and the action time of the differential is enhanced. Then the structural block diagram of the PID algorithm becomes
The transfer function at this time is
The improved PID algorithm is written in the same way as the standard PID algorithm, and the only thing to pay attention to is how the improved differential term is discretized
The final simplification is Ud(k). It should be noted that Ud(k-1) and U(k-1) cannot be combined when u(k)=Up+Ui+Ud is finally solved.
The solved u(k) is the positional formula, and the incremental formula is also △u(k)=u(k)-u(k-1), which will not be deduced here.
Finally, let's take a look at the difference between complete differentiation and incomplete differentiation in the last picture.
