Closed-loop controller

==Closed-loop transfer function==

The output of the system y(t) is fed back through a sensor measurement F to a comparison with the reference value r(t). The controller C then takes the error e (difference) between the reference and the output to change the inputs u to the system under control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller. This is called a single-input-single-output (SISO) control system; MIMO (i.e., Multi-Input-Multi-Output) systems, with more than one input/output, are common. In such cases variables are represented through vectors instead of simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically functions). If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e., elements of their transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace transform on the variables. This gives the following relations: : Y(s) = P(s) U(s) : U(s) = C(s) E(s) : E(s) = R(s) - F(s)Y(s). Solving for Y(s) in terms of R(s) gives : Y(s) = \left( \frac{P(s)C(s)}{1 + P(s)C(s)F(s)} \right) R(s) = H(s)R(s). The expression H(s) = \frac{P(s)C(s)}{1 + F(s)P(s)C(s)} is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If |P(s)C(s)| \gg 1, i.e., it has a large norm with each value of s, and if |F(s)| \approx 1, then Y(s) is approximately equal to R(s) and the output closely tracks the reference input. ==PID feedback control==

of a PID controller in a feedback loop; is the desired process value or "set point", and is the measured process value. A proportional–integral–derivative controller (PID controller) is a control loop feedback mechanism control technique widely used in control systems. A PID controller continuously calculates an error value as the difference between a desired setpoint and a measured process variable and applies a correction based on proportional, integral, and derivative terms. PID is an initialism for Proportional-Integral-Derivative, referring to the three terms operating on the error signal to produce a control signal. The theoretical understanding and application dates from the 1920s, and they are implemented in nearly all analogue control systems; originally in mechanical controllers, and then using discrete electronics and later in industrial process computers. The PID controller is probably the most-used feedback control design. If is the control signal sent to the system, is the measured output and is the desired output, and is the tracking error, a PID controller has the general form :u(t) = K_P e(t) + K_I \int^t e(\tau)\text{d}\tau + K_D \frac{\text{d}e(t)}{\text{d}t}. The desired closed loop dynamics is obtained by adjusting the three parameters , and , often iteratively by "tuning" and without specific knowledge of a plant model. Stability can often be ensured using only the proportional term. The integral term permits the rejection of a step disturbance (often a striking specification in process control). The derivative term is used to provide damping or shaping of the response. PID controllers are the most well-established class of control systems: however, they cannot be used in several more complicated cases, especially if MIMO systems are considered. Applying Laplace transformation results in the transformed PID controller equation :u(s) = K_P \, e(s) + K_I \, \frac{1}{s} \, e(s) + K_D \, s \, e(s) :u(s) = \left(K_P + K_I \, \frac{1}{s} + K_D \, s\right) e(s) with the PID controller transfer function :C(s) = \left(K_P + K_I \, \frac{1}{s} + K_D \, s\right). As an example of tuning a PID controller in the closed-loop system , consider a 1st order plant given by :P(s) = \frac{A}{1 + sT_P} where and are some constants. The plant output is fed back through :F(s) = \frac{1}{1 + sT_F} where is also a constant. Now if we set K_P=K\left(1+\frac{T_D}{T_I}\right), , and K_I=\frac{K}{T_I}, we can express the PID controller transfer function in series form as :C(s) = K \left(1 + \frac{1}{sT_I}\right)(1 + sT_D) SDSD Plugging , , and into the closed-loop transfer function , we find that by setting :K = \frac{1}{A}, T_I = T_F, T_D = T_P . With this tuning in this example, the system output follows the reference input exactly. However, in practice, a pure differentiator is neither physically realizable nor desirable due to amplification of noise and resonant modes in the system. Therefore, a phase-lead compensator type approach or a differentiator with low-pass roll-off are used instead. ==References==