Linear Control System Example
Vinnie Frevert
Linear control are control systems and control theory based on negative feedback for producing a control signal to maintain the controlled process variable (PV) at the desired setpoint (SP). There are several types of linear control systems with different capabilities.
Proportional control is a type of linear feedback control system in which a correction is applied to the controlled variable which is proportional to the difference between the desired value (SP) and the measured value (PV). Two classic mechanical examples are the toilet bowl float proportioning valve and the fly-ball governor.
In some systems, there are practical limits to the range of the MV. For example, a heater has a limit to how much heat it can produce and a valve can open only so far. Adjustments to the gain simultaneously alter the range of error values over which the MV is between these limits. The width of this range, in units of the error variable and therefore of the PV, is called the proportional band (PB).
When controlling the temperature of an industrial furnace, it is usually better to control the opening of the fuel valve in proportion to the current needs of the furnace. This helps avoid thermal shocks and applies heat more effectively.
At low gains, only a small corrective action is applied when errors are detected. The system may be safe and stable but may be sluggish in response to changing conditions. Errors will remain uncorrected for relatively long periods of time and the system is overdamped. If the proportional gain is increased, such systems become more responsive and errors are dealt with more quickly. There is an optimal value for the gain setting when the overall system is said to be critically damped. Increases in loop gain beyond this point lead to oscillations in the PV and such a system is underdamped. Adjusting gain to achieve critically damped behavior is known as tuning the control system.
In the underdamped case, the furnace heats quickly. Once the setpoint is reached, stored heat within the heater sub-system and in the walls of the furnace will keep the measured temperature rising beyond what is required. After rising above the setpoint, the temperature falls back and eventually heat is applied again. Any delay in reheating the heater sub-system allows the furnace temperature to fall further below the setpoint and the cycle repeats. The temperature oscillations that an underdamped furnace control system produces are undesirable.
In a critically damped system, as the temperature approaches the setpoint, the heat input begins to be reduced, the rate of heating of the furnace has time to slow and the system avoids overshoot. Overshoot is also avoided in an overdamped system but an overdamped system is unnecessarily slow to initially reach a setpoint response to external changes to the system, e.g. opening the furnace door.
Pure proportional controllers must operate with residual error in the system. Though PI controllers eliminate this error they can still be sluggish or produce oscillations. The PID controller addresses these final shortcomings by introducing a derivative (D) action to retain stability while responsiveness is improved.
On control systems involving motion control of a heavy item like a gun or camera on a moving vehicle, the derivative action of a well-tuned PID controller can allow it to reach and maintain a setpoint better than most skilled human operators. If a derivative action is over-applied, it can, however, lead to oscillations.
The integral term magnifies the effect of long-term steady-state errors, applying an ever-increasing effort until the error is removed. In the example of the furnace above working at various temperatures, if the heat being applied does not bring the furnace up to setpoint, for whatever reason, integral action increasingly moves the proportional band relative to the setpoint until the PV error is reduced to zero and the setpoint is achieved.
Some controllers include the option to limit the "ramp up % per minute". This option can be very helpful in stabilizing small boilers (3 MBTUH), especially during the summer, during light loads. A utility boiler "unit may be required to change load at a rate of as much as 5% per minute (IEA Coal Online - 2, 2007)".[1][failed verification]
It is possible to filter the PV or error signal. Doing so can help reduce instability or oscillations by reducing the response of the system to undesirable frequencies. Many systems have a resonant frequency. By filtering out that frequency, stronger overall feedback can be applied before oscillation occurs, making the system more responsive without shaking itself apart.
Feedback systems can be combined. In cascade control, one control loop applies control algorithms to a measured variable against a setpoint but then provides a varying setpoint to another control loop rather than affecting process variables directly. If a system has several different measured variables to be controlled, separate control systems will be present for each of them.
Control engineering in many applications produces control systems that are more complex than PID control. Examples of such field applications include fly-by-wire aircraft control systems, chemical plants, and oil refineries. Model predictive control systems are designed using specialized computer-aided-design software and empirical mathematical models of the system to be controlled.
A control system is a system that is used to control the behavior of a device or process. It is made up of three main components: a sensor, a controller, and an actuator. The sensor detects a physical quantity such as temperature, pressure, or position and converts it into an electrical signal. The controller processes this signal and generates an output signal that is used to control the actuator. The actuator is a device that translates the output signal from the controller into a physical action, such as opening or closing a valve, turning a motor on or off, or adjusting the speed of a motor.
Control systems are used in a wide range of applications, including manufacturing, transportation, and energy production. They are an essential part of many modern devices and systems and are used to maintain stable and predictable behavior.
An embedded control system is a control system that is integrated into a larger product or system. Embedded control systems are used to automate and control the operation of the product or system in which they are embedded.
Embedded control systems are typically designed to be compact, efficient, and reliable, as they are integrated into products and systems that are expected to operate for extended periods of time without requiring maintenance or repair.
Feedback control can be used to improve the performance of a control system by comparing the desired output of the system to the actual output, and adjusting the input to the system based on the difference between these two signals (called the error). This can help to reduce errors, improve stability, and achieve other desired performance characteristics.
The stability of a control system can be analyzed using techniques such as root-locus analysis or frequency response analysis. These methods allow the designer to understand how the system will respond to different inputs and disturbances, and to identify any potential instability or performance issues. Stability can also be guaranteed by designing the control system to meet certain stability criteria (such as the Routh-Hurwitz criterion) or by using robust control techniques.
Different control design methods can have different trade-offs in terms of performance, complexity, and implementation. For example, PID control is a simple and widely-used method that can achieve good performance in many cases, but it may not be optimal in all situations. On the other hand, more advanced methods such as linear quadratic regulator (LQR) control can provide better performance but may be more complex to implement and require more detailed system knowledge.
Robust control techniques can be used to design control systems that are resistant to uncertainties or variations in the system parameters. This can be achieved by designing the control system to be stable for a range of possible parameter values, or by using control algorithms that are designed to be robust to certain types of uncertainties.
Nonlinear control techniques can be used to design control systems that can handle nonlinearities or other complex behaviors. These techniques may involve using specialized control algorithms, linearizing the system around a particular operating point, or using feedback to cancel out the effects of nonlinearities.
Control systems can be implemented and tested using a variety of tools and methods, including simulation tools, hardware-in-the-loop testing, and prototyping platforms. Testing is an important step in the control design process, as it allows the designer to verify that the control system is behaving as expected and to identify and fix any issues.
Control systems can be optimized for a particular performance criterion (such as minimizing error or maximizing efficiency) by using optimization techniques such as gradient descent or evolutionary algorithms. These methods can help to find the control inputs that result in the best performance for a given system.
Control systems can be integrated with other systems (such as communication networks or software systems) by using interfaces and protocols that allow the systems to exchange data and control signals. This can allow the control system to access information from other systems, or to influence the behavior of other systems.
Control systems can be used to achieve a particular goal by designing the control algorithm and system architecture to produce the desired output or behavior. This may involve defining a performance criterion or a set of constraints and then designing the control system to meet these requirements.
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