1. Serial Pi network:
- Connection method: The Pi network connects the three components (one resistor and two capacitors) in series.
- Filtering Characteristics: Series Pi networks are primarily used for high-frequency filtering, that is, transmitting high-frequency signals by filtering out low-frequency signals. In the low-frequency range, the capacitor acts as an open circuit, blocking low-frequency signals. However, in the high-frequency range, the capacitor's impedance is relatively low, and the circuit behaves almost like an ideal resistor.
- Transfer function: The transfer function of a series Pi network depends on the values of the resistors and capacitors, and can be selected and adjusted according to requirements.
2. Parallel Pi network:
- Connection method: The parallel Pi network connects three components (one resistor and two capacitors) in parallel.
- Filtering Characteristics: Parallel Pi networks are mainly used for low-pass filtering, that is, transmitting low-frequency signals by filtering out high-frequency signals. Capacitors have higher impedance in the low-frequency range, blocking high-frequency signals. Conversely, in the low-frequency range, the impedance of the capacitor is lower, and the circuit behaves almost like an ideal capacitor.
- Transfer function: The transfer function of a parallel Pi network also depends on the values of the resistors and capacitors, which can be selected and adjusted according to requirements.
In general, series Pi networks and parallel Pi networks have somewhat opposite filtering characteristics. Series Pi networks are mainly used for high-pass filtering, transmitting high-frequency signals while blocking low-frequency signals; while parallel Pi networks are mainly used for low-pass filtering, transmitting low-frequency signals while blocking high-frequency signals. The specific network to choose depends on the required filtering characteristics and application requirements.
Series PI controller parameter design
The general steps for designing parameters for a series PI controller are as follows:
1. Determine the system type: First, determine the type of system to be controlled, including first-order systems, second-order systems, etc. The system type will determine the methods and strategies for parameter design.
2. Determine control objectives: Determine the objectives of the controller, such as fast response, stability, and anti-interference capability, and select parameters based on the trade-offs of these objectives.
3. Design the proportional gain (Kp): Select an appropriate proportional gain based on the system's response characteristics and steady-state error requirements. Increasing the proportional gain can improve the system's response speed, but may lead to overshoot and oscillation; decreasing the proportional gain can reduce overshoot and oscillation, but may slow down the system response.
4. Designing the Integral Time Constant (Ti): The integral time constant is a crucial parameter determining the response of the integral element, affecting the system's ability to correct steady-state errors. By adjusting the integral time constant, the steady-state error of the system can be controlled. Generally, a suitable integral time constant is selected through experiments or by adjusting the system based on its characteristics.
5. Design the differential time constant (Td) (optional): If the system has significant inertia, lag, or jitter, a differential element can be introduced for compensation. The differential time constant affects the system's response speed and stability; an appropriate differential time constant should be selected through experimentation or based on experience.
6. Parameter Adjustment and Optimization: Based on the actual system performance and requirements, adjust and optimize the parameters. Parameters can be optimized through experimentation or by using adaptive control to achieve better control results.
It is important to note that designing the parameters of a series PI controller is an iterative process that requires debugging and optimization through actual testing and feedback. The characteristics and requirements of the system will also influence the selection and adjustment of the parameters. Therefore, when designing controller parameters, it is necessary to comprehensively consider and debug the specific application scenario and system characteristics.
In a PI controller, P and I represent proportional control and integral control, respectively, and they each play different roles in the control system.
P (Proportional Control)
The function of proportional control (P) is to adjust the control quantity proportionally to the system's deviation signal. When a deviation occurs in the system, the proportional controller immediately generates an adjustment action to reduce the deviation, ensuring a fast system response. The output of the proportional controller is proportional to the deviation signal. Generally, the higher the proportional gain, the faster the system response speed; however, excessively high proportional gain may lead to system instability.<sup>12</sup>
I (Integral Control)
Integral control (I) adjusts the control quantity by integrating the deviation, primarily used to eliminate steady-state error in the system. The integral controller integrates the deviation signal, generating a control signal proportional to the accumulated deviation, thereby eliminating steady-state error. Integral control has a relatively slow response time, but its function is to eliminate persistent errors in the system and improve the system's accuracy.<sup>12</sup>
Combined effects of PI controllers
PI controllers combine the advantages of proportional and integral control, enabling better system regulation and performance. Proportional control handles rapid response to system changes, while integral control eliminates steady-state errors. By adjusting the proportional gain and integral time constant, the system's response characteristics can be optimized, such as reducing overshoot and shortening settling time.<sup>13</sup> Proportional Action (P)
A proportional controller is essentially an amplifier with an adjustable gain, i.e., ΔP = Kp × e, where Kp is the proportional gain, which can be greater than or less than 1; e is the input of the controller, which is the difference between the measured value and the setpoint, also known as the deviation.
It should be noted that most analog controllers do not use the proportional gain Kp as the scale, but rather the proportional gain, i.e., δ=1/Kc×100%. In other words, the proportional gain is proportional to the reciprocal of the controller's amplification factor; the smaller the proportional gain of the controller, the larger its amplification factor and the greater its ability to amplify deviations, and vice versa.
Understanding the above relationships reveals that: the larger the proportional gain, the smaller the amplification factor of the controller, and the smoother the curve of the controlled parameter; conversely, the smaller the proportional gain, the larger the amplification factor of the controller, and the more volatile the curve of the controlled parameter.
Proportional control has a drawback: it generates residual error. To overcome residual error, integral action must be introduced.
Integral action (I)
The integral function of a controller is designed to eliminate the steady-state error of an automatic control system. Integral means accumulation over time; that is, when a deviation input e exists, the integral controller continuously accumulates the deviation over time. In other words, the rate of integral accumulation is directly proportional to the magnitude of the deviation e and the integral speed. As long as a deviation e exists, the output of the integral controller must change; that is, the integral is always active, and it only stops when the deviation disappears.
For a constant deviation, adjusting the integral action essentially changes the rate of change of the controller output. This rate is measured by the time required for the integral action output to equal the proportional action output. A shorter integral time indicates a faster integral speed and a stronger integral action; conversely, a longer integral time indicates a weaker integral action. If the integral time is infinite, it indicates no integral action, and the controller becomes a pure proportional controller.
In practice, integral action is rarely used alone. It is usually used in conjunction with proportional action, so that it has both the proportional action that amplifies (or reduces) the deviation and the integral action that accumulates the deviation over time, and their directions of action are consistent. In this case, the controller output is: ΔP = Ke + ΔPi, where ΔP is the change in the controller output value; Ke is the output caused by the proportional action; and ΔPi is the output caused by the integral action.
Differential action (D)
Differential action is mainly used to overcome the hysteresis of the controlled object and is commonly used in temperature control systems. Besides employing differential action, when using a control system, attention must be paid to the hysteresis of measurement transmission, such as the selection and installation location of the temperature measuring element.
In conventional PID controllers, the output change of the derivative action is proportional to the derivative time and the rate of change of the deviation, regardless of the magnitude of the deviation. The greater the rate of change of the deviation and the longer the derivative time, the greater the output change of the derivative action. However, if the derivative action is too strong, it may cause oscillations due to its rapid change, resulting in obvious "spiking" or "jumping" in the controller output. To avoid this disturbance, the derivative-first PID algorithm can be used in PID controllers and DCS, that is, only the measured value PV is differentiated. When the setpoint SP of the controller is manually changed, it will not cause a sudden change in the controller output, avoiding the disturbance to the control system caused by the instantaneous change of SP. For example, the TDC-3000 adds a soft switch to the conventional PID algorithm, allowing the user to choose whether the controller differentiates the deviation or the measured value during configuration.
When a step signal is input, the ratio of the maximum change in the output of the differentiator at the beginning to the change in the output after the differential action disappears is the differential amplification factor Kd, i.e., the differential gain. The unit of differential gain is time. Setting the differential time (or differential gain) to zero will cancel the differential function.
To help you remember the functions of proportion, integral, and differential, I've copied down three mnemonic rhymes for your reference.
Rhyme about proportionality
The scale regulator is like an amplifier;
An error occurs, is amplified, and then transmitted.
To determine the magnification, carefully examine the knob.
Larger scale factor results in lower magnification.
A rhyme about the function of integrals
Reset the regulator, accumulate skills;
As long as the deviation exists, the accumulation will not stop;
Accumulate speed, pay close attention to the knob;
The points accumulation time is long and the accumulation speed is slow.
A mnemonic for the action of differentials
Differentiators are not mysterious at all;
A step input results in a jump output.
To adjust the descent speed, carefully observe the knob.
The longer the differentiation time, the slower the descent.
Explanation of reset controllers: Resetting means resetting, because the integral action in the controller is responsible for this resetting process. Previously, proportional-integral controllers were called reset controllers.
