Classical control theory
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Classical control theory is a branch of control theory that deals with the behavior of dynamical systems with inputs, and how their behavior is modified by feedback, using the Laplace transform as a basic tool to model such systems.
The usual objective of control theory is to control a system, often called the
Classical control theory deals with
Feedback
To overcome the limitations of the
Closed-loop controllers have the following advantages over open-loop controllers:
- disturbance rejection (such as hills in a cruise control)
- guaranteed performance even with model uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact
- unstable processes can be stabilized
- reduced sensitivity to parameter variations
- improved reference tracking performance
In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control is termed feedforward and serves to further improve reference tracking performance.
A common closed-loop controller architecture is the
Classical vs modern
A Physical system can be modeled in the "time domain", where the response of a given system is a function of the various inputs, the previous system values, and time. As time progresses, the state of the system and its response change. However, time-domain models for systems are frequently modeled using high-order differential equations which can become impossibly difficult for humans to solve and some of which can even become impossible for modern computer systems to solve efficiently.
To counteract this problem, classical control theory uses the Laplace transform to change an Ordinary Differential Equation (ODE) in the time domain into a regular algebraic polynomial in the frequency domain. Once a given system has been converted into the frequency domain it can be manipulated with greater ease.
Laplace transform
Classical control theory uses the Laplace transform to model the systems and signals. The Laplace transform is a frequency-domain approach for continuous time signals irrespective of whether the system is stable or unstable. The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), which is a unilateral transform defined by
where s is a complex number frequency parameter
- , with real numbers σ and ω.
Closed-loop transfer function
A common feedback control architecture is the servo loop, in which the output of the system y(t) is measured using a sensor F and subtracted from the reference value r(t) to form the servo error e. The controller C then uses the servo error e to adjust the input u to the plant (system being controlled) P in order to drive the output of the plant toward the reference. This is shown in the block diagram below. 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
If we assume the controller C, the plant P, and the sensor F are
Solving for Y(s) in terms of R(s) gives
The expression is referred to as the closed-loop transfer function of the system. The numerator is the forward (open-loop) gain from to , and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If , i.e., it has a large norm with each value of s, and if , then is approximately equal to and the output closely tracks the reference input.
PID controller
The
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
Applying Laplace transformation results in the transformed PID controller equation
with the PID controller transfer function
There exists a nice example of the closed-loop system discussed above. If we take
PID controller transfer function in series form
1st order filter in feedback loop
linear actuator with filtered input
- , = const
and insert all this into expression for closed-loop transfer function , then tuning is very easy: simply put
and get identically.
For practical PID controllers, a pure differentiator is neither physically realisable nor desirable[3] due to amplification of noise and resonant modes in the system. Therefore, a phase-lead compensator type approach is used instead, or a differentiator with low-pass roll-off.
Tools
Classical control theory uses an array of tools to analyze systems and design controllers for such systems. Tools include the
See also
- Minor loop feedback a classical method for designing feedback control systems.
- State space (control)
References
- ISBN 978-1-4020-7880-4.
The classical controller design methodology is iterative, and is effective for single-input, single-output linear time-invariant system analysis and design.
- ISBN 978-0-13-615673-4.
modern control theory, based on time-domain analysis and synthesis using state variables
- ^ Ang, K.H., Chong, G.C.Y., and Li, Y. (2005). PID control system analysis, design, and technology, IEEE Trans Control Systems Tech, 13(4), pp.559-576.
- ISBN 978-1-1385-4114-6.