Responses Manual VS PID ~ Vistarobotik

Sabtu, 26 September 2015

Responses Manual VS PID

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Ideal versus standard PID form

The form of the PID controller most often encountered in industry, and the one most relevant to tuning algorithms is the standard form. In this form the

K_p

gain is applied to the

I_{\mathrm{out}}

, and

D_{\mathrm{out}}

terms, yielding:

\mathrm{MV(t)}=K_p\left(\,{e(t)} + \frac{1}{T_i}\int_{0}^{t}{e(\tau)}\,{d\tau} + T_d\frac{d}{dt}e(t)\right)

where

T_i

is the integral time

T_d

is the derivative time

In this standard form, the parameters have a clear physical meaning. In particular, the inner summation produces a new single error value which is compensated for future and past errors. The addition of the proportional and derivative components effectively predicts the error value at

T_d

seconds (or samples) in the future, assuming that the loop control remains unchanged. The integral component adjusts the error value to compensate for the sum of all past errors, with the intention of completely eliminating them in

T_i

seconds (or samples). The resulting compensated single error value is scaled by the single gain

K_p

.

In the ideal parallel form, shown in the controller theory section

\mathrm{MV(t)}=K_p{e(t)} + K_i\int_{0}^{t}{e(\tau)}\,{d\tau} + K_d\frac{d}{dt}e(t)

the gain parameters are related to the parameters of the standard form through

K_i = \frac{K_p}{T_i}

and

K_d = K_p T_d \,

. This parallel form, where the parameters are treated as simple gains, is the most general and flexible form. However, it is also the form where the parameters have the least physical interpretation and is generally reserved for theoretical treatment of the PID controller. The standard form, despite being slightly more complex mathematically, is more common in industry.

Reciprocal gain

In many cases, the manipulated variable output by the PID controller is a dimensionless fraction between 0 and 100% of some maximum possible value, and the translation into real units (such as pumping rate or watts of heater power) is outside the PID controller. The process variable, however, is in dimensioned units such as temperature. It is common in this case to express the gain

K_p

not as "output per degree", but rather in the form of a temperature

1/K_p

which is "degrees per full output". This is the range over which the output changes from 0 to 1 (0% to 100%).

Basing derivative action on PV

In most commercial control systems, derivative action is based on PV rather than error. This is because the digitized version of the algorithm produces a large unwanted spike when the SP is changed. If the SP is constant then changes in PV will be the same as changes in error. Therefore this modification makes no difference to the way the controller responds to process disturbances.

\mathrm{MV(t)}=K_p\left(\,{e(t)} + \frac{1}{T_i}\int_{0}^{t}{e(\tau)}\,{d\tau} - T_d\frac{d}{dt}PV(t)\right)

Basing proportional action on PV

Most commercial control systems offer the option of also basing the proportional action on PV. This means that only the integral action responds to changes in SP. The modification to the algorithm does not affect the way the controller responds to process disturbances. The change to proportional action on PV eliminates the instant and possibly very large change in output on a fast change in SP. Depending on the process and tuning this may be beneficial to the response to a SP step.

\mathrm{MV(t)}=K_p\left(\,{-PV(t)} + \frac{1}{T_i}\int_{0}^{t}{e(\tau)}\,{d\tau} - T_d\frac{d}{dt}PV(t)\right)

Laplace form of the PID controller

Sometimes it is useful to write the PID regulator in Laplace transform form:

G(s)=K_p + \frac{K_i}{s} + K_d{s}=\frac{K_d{s^2} + K_p{s} + K_i}{s}

Having the PID controller written in Laplace form and having the transfer function of the controlled system makes it easy to determine the closed-loop transfer function of the system.

PID pole zero cancellation

The PID equation can be written in this form:

G(s)=K_d \frac{s^2 + \frac{K_p}{K_d}s + \frac{K_i}{K_d}}{s}

When this form is used it is easy to determine the closed loop transfer function.

H(s)=\frac{1}{s^2 + 2\zeta \omega_0 s + \omega_0^2}

If

\frac{K_i}{K_d}=\omega_0^2

\frac{K_p}{K_d}=2\zeta \omega_0

Then

G(s) H(s)=\frac{K_d}{s}

While this appears to be very useful to remove unstable poles, it is in reality not the case. The closed loop transfer function from disturbance to output still contains the unstable poles.

Series/interacting form

Another representation of the PID controller is the series, or interacting form

G(s) = K_c \frac{(\tau_i{s}+1)}{\tau_i{s}} (\tau_d{s}+1)

where the parameters are related to the parameters of the standard form through

K_p = K_c \cdot \alpha

,

T_i = \tau_i \cdot \alpha

, and

T_d = \frac{\tau_d}{\alpha}

with

\alpha = 1 + \frac{\tau_d}{\tau_i}

.

This form essentially consists of a PD and PI controller in series, and it made early (analog) controllers easier to build. When the controllers later became digital, many kept using the interacting form.

Discrete implementation

The analysis for designing a digital implementation of a PID controller in a microcontroller (MCU) or FPGA device requires the standard form of the PID controller to be discretized.^[25] Approximations for first-order derivatives are made by backward finite differences. The integral term is discretised, with a sampling time