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Responses Manual VS PID

06.43

Unknown

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

\Delta t

,as follows,

\int_{0}^{t_k}{e(\tau)}\,{d\tau} = \sum_{i=1}^k e(t_i)\Delta t

The derivative term is approximated as,

\dfrac{de(t_k)}{dt}=\dfrac{e(t_k)-e(t_{k-1})}{\Delta t}

Thus, a velocity algorithm for implementation of the discretized PID controller in a MCU is obtained by differentiating

u(t)

, using the numerical definitions of the first and second derivative and solving for

u(t_k)

and finally obtaining:

u(t_k)=u(t_{k-1})+K_p\left[\left(1+\dfrac{\Delta t}{T_i}+\dfrac{T_d}{\Delta t}\right)e(t_k)+\left(-1-\dfrac{2T_d}{\Delta t}\right)e(t_{k-1})+\dfrac{T_d}{\Delta t}e(t_{k-2})\right]

T_i = K_p/K_i, T_d = K_d/K_p

Derivatife term

06.35

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The derivative of the process error is calculated by determining the slope of the error over time and multiplying this rate of change by the derivative gain K_d. The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain, K_d.

The derivative term is given by:

D_{\mathrm{out}}=K_d\frac{d}{dt}e(t)

Derivative action predicts system behavior and thus improves settling time and stability of the system.^[11]^[12] An ideal derivative is not causal, so that implementations of PID controllers include an additional low pass filtering for the derivative term, to limit the high frequency gain and noise.^[13] Derivative action is seldom used in practice though - by one estimate in only 25% of deployed controllers^[13] - because of its variable impact on system stability in real-world applications.

Integral Term

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The contribution from the integral term is proportional to both the magnitude of the error and the duration of the error. The integral in a PID controller is the sum of the instantaneous error over time and gives the accumulated offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain (

K_i

) and added to the controller output.

The integral term is given by:

I_{\mathrm{out}}=K_{i}\int_{0}^{t}{e(\tau)}\,{d\tau}

The integral term accelerates the movement of the process towards setpoint and eliminates the residual steady-state error that occurs with a pure proportional controller. However, since the integral term responds to accumulated errors from the past, it can cause the present value to overshoot the setpoint value (see the section on loop tuning).

Proportional term

06.32

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The proportional term produces an output value that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant K_p, called the proportional gain constant.

The proportional term is given by:

P_{\mathrm{out}}=K_p\,{e(t)}

A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable (see the section on loop tuning). In contrast, a small gain results in a small output response to a large input error, and a less responsive or less sensitive controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances. Tuning theory and industrial practice indicate that the proportional term should contribute the bulk of the output change.

PID controller theory

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The PID control scheme is named after its three correcting terms, whose sum constitutes the manipulated variable (MV). The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. Defining