PID Control

Closed-Loop transfer function basic form

In a closed feedback loop, if C(s) is output signal, R(s) is reference input, E(s) is error between reference and output, G(s) is a transfer fuction that has error,E(s), as input and C(s), as output, and H(s) is a transfer function to convert the output C(s) to the same unit as reference R(s),

\(\frac{C(s)}{R(s)} = \frac{G(s)}{1 + G(s)H(s)}\)

C(s) = \(\frac{G(s)}{1 + G(s)H(s)}\) R(s)

Hence, the output of closed loop system depends on the reference input, R(s), and the chracteristics of the trasnfer function G(s) that converts the error to the control output.

Mathematical representation

When u(t) is controller output and e(t) is actuating error signal

Proportional controller

• Time domain

        u(t) = \(K_{p}\) e(t)

• Laplace form transfer function

        \(\frac{U(s)}{E(s)}\) = \(K_{p}\)

Intergral controller

• Time domain

        u(t) = \(K_{i}\) \(\int_0^t e(t)\mathrm{d}t\)

• Laplace form transfer function

        \(\frac{U(s)}{E(s)}\) = \(\frac{K_{i}}{s}\)

Proportional Integral Differential (PID) controller

• Time domain

        u(t) = \(K_{p}\) e(t) + \(K_{i}\) \(\int_0^t e(t)\mathrm{d}t\) + \(K_{d}\) \(\frac{\mathrm{d}e(t)}{\mathrm{d}t}\)
        or
        u(t) = \(K_{p}\) e(t) + \(\frac{K_{p}}{T_{i}}\) \(\int_0^t e(t)\mathrm{d}t\) + \(K_{p}\) \(T_{d}\) \(\frac{\mathrm{d}e(t)}{\mathrm{d}t}\)
        = \(K_{p}\) (e(t) + \(\frac{1}{T_{i}}\) \(\int_0^t e(t)\mathrm{d}t\) + \(T_{d}\) \(\frac{\mathrm{d}e(t)}{\mathrm{d}t}\) )

• Laplace form transfer function

        \(\frac{U(s)}{E(s)}\) = \(K_{p}(1 + \frac{1}{T_{i}s} + T_{d}s)\)

Physical significance of PID parameters

Proportional gain constant (\(K_{p}\))

Proportional gain output is directly in proportion to error size. Typically, it should be the first paramter to adjust ahead of intergal and differntial terms. However, using only gain will lead to permant offset error. This is because gain needs error to be nonzero. This offset should be addressed by applying integral control or by manually applying offset. In HVAC valve or damper control, it is ‘controlled system output / throttling range’. So the greater the gain, the narrower the throttling range.

• Too much proportional gain : oscillate
• Too little proportional gain : too slow response

Integral gain constant (\(K_{i}\))

Intergal control output is in proportion to accumulated error that it can effetively handles the steady state offset error of proportion-only controllers. The unit is reciprocal time [1/time], which is the number of times that the integral term is calculated per unit time. It is also called ‘reset rate’ or ‘reset control’. The value of 1/\(T_{i}\) would be usually 1 - 60 min.

• Too much intergral gain : oscillate
• Too little integral gain : offset is not effectively addressed.

Differential gain constant (\(K_{d}\))

Differential control output is in proportion to the rate of error. This is used to speed up the PI control. It does little to remove the steady state offset error of proportional only controller because the rate of error is zero in that phase, However, it can provide quick correction initially. This is always used in combination with proportional and intergal controller and never used alone. Since it responds to error rate, the system adoping differential control can proactively address error before it becomes to large. In other words, it adds sensitivity to control. Derivative control is less frequently used in HVAC control systems that usuall do not require fast response. The value of \(T_{d}\) is typically 0.2 - 15 min.

References

• T.A. Reddy, et al (2016) Heating and cooling of buildings. Principles and practice of energy efficiency deisgn. 3rd edition
• K. Ogata, (2010) Modern control engineering. 5th edition.

Back