Classical control theory: Difference between revisions

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{{Being merged|spacetype=|Control theory|Talk:Control theory#Proposed merge of Classical control theory into Control theory|section=|inactive=|date=March 2023|nocat=|dir=to}}

'''Classical control theory''' is a branch of [[control theory]] that deals with the behavior of [[dynamical system]]s with inputs, and how their behavior is modified by [[feedback]], using the [[Laplace transform]].

{{Multiple issues|{{more citations needed|date=May 2016}}{{expert needed|1=Engineering|date=May 2016|reason=Need more sources and attention of experts in field for information verification.}}{{one source|date=May 2016}}}}

'''Classical control theory''' is a branch of [[control theory]] that deals with the behavior of [[dynamical system]]s 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 ''[[Plant (control theory)|plant]]'', so its output follows a desired control signal, called the ''[[reference]]'', which may be a fixed or changing value. To do this a ''[[Controller (control theory)|controller]]'' is designed, which monitors the output and compares it with the reference. The difference between actual and desired output, called the ''error'' signal, is applied as [[feedback]] to the input of the system, to bring the actual output closer to the reference. Some topics studied in control theory are [[Stability theory|stability]] (whether the output will converge to the reference value or oscillate about it), [[controllability]] and [[observability]].

The usual objective of control theory is to control a system, often called the ''[[Plant (control theory)|plant]]'', so its output follows a desired control signal, called the ''[[reference]]'', which may be a fixed or changing value. To do this a ''[[Controller (control theory)|controller]]'' is designed, which monitors the output and compares it with the reference. The difference between actual and desired output, called the ''error'' signal, is applied as [[feedback]] to the input of the system, to bring the actual output closer to the reference. Some topics studied in control theory are [[Stability theory|stability]] (whether the output will converge to the reference value or oscillate about it), [[controllability]] and [[observability]].

Extensive use is usually made of a diagrammatic style known as the [[block diagram]]. The [[transfer function]], also known as the system function or network function, is a mathematical representation of the relation between the input and output based on the [[differential equation]]s describing the system.

Classical control theory deals with [[linear time-invariant system|linear time-invariant]] (LTI) [[single-input single-output]] (SISO) systems.<ref>{{cite book|last1=Zhong|first1=Wan-Xie|title=Duality System in Applied Mechanics and Optimal Control|url=https://archive.org/details/dualitysystemapp00zhon_389|url-access=limited|date=2004|publisher=Kluwer|isbn=978-1-4020-7880-4|page=[https://archive.org/details/dualitysystemapp00zhon_389/page/n295 283]|quote=The classical controller design methodology is iterative, and is effective for single-input, single-output linear time-invariant system analysis and design.}}</ref> The Laplace transform of the input and output signal of such systems can be calculated. The [[transfer function]] relates the Laplace transform of the input and the output.

==Feedback==

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Closed-loop controllers have the following advantages over [[open-loop controller]]s:

* disturbance rejection (such as hills in thea [[cruise control example above]])

* guaranteed performance even with [[mathematical model|model]] uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact

* [[instability|unstable]] processes can be stabilized

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==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 transformfrequency ___domain. Once a given system has been converted into the transformfrequency ___domain it can be manipulated with greater ease.

[[Modern control theory]], instead of changing domains to avoid the complexities of time-___domain ODE mathematics, converts the differential equations into a system of lower-order time ___domain equations called [[state space (control)|state equations]], which can then be manipulated using techniques from linear algebra.<ref>{{cite book|last1=Ogata|first1=Katsuhiko|title=Modern Control Systems|date=2010|publisher=Prentice Hall|isbn=978-0-13-615673-4|page=2|edition=Fifth|quote=modern control theory, based on time-___domain analysis and synthesis using state variables}}</ref>

==Laplace transform==

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==Closed-loop transfer function==

{{details|closed-loop transfer function}}

TheA common feedback control architecture is the servo loop, in which the output of the system ''y(t)'' is fedmeasured back throughusing a sensor measurement ''F'' toand subtracted from the reference value ''r(t)'' to form the servo error ''e''. The controller ''C'' then takesuses the servo error ''e'' (difference) between the reference and the output to changeadjust the inputsinput ''u'' to the plant (system underbeing controlcontrolled) ''P'' in order to drive the output of the plant toward the reference. This is shown in the figure[[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 [[coordinate vector|vectors]] instead of simple [[scalar (mathematics)|scalar]] values. For some [[distributed parameter systems]] the vectors may be infinite-[[Dimension (vector space)|dimensional]] (typically functions).

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: <math>Y(s) = \left( \frac{P(s)C(s)}{1 + F(s)P(s)C(s)} \right) R(s) = H(s)R(s).</math>

The expression <math>H(s) = \frac{P(s)C(s)}{1 + F(s)P(s)C(s)}</math> is referred to as the ''closed-loop transfer function'' of the system. The numerator is the forward (open-loop) gain from ''<math>r''</math> to ''<math>y''</math>, and the denominator is one plus the gain in going around the feedback loop, the so-called loop gain. If <math>|P(s)C(s)| \gg 1</math>, i.e., it has a large [[norm (mathematics)|norm]] with each value of ''s'', and if <math>|F(s)| \approx 1</math>, then ''<math>Y(s)''</math> is approximately equal to ''<math>R(s)''</math> and the output closely tracks the reference input.

==<math>u(t)</math>PID controller==

{{details|PID controller}}

The [[PID controller]] is probably the most-used (alongside much cruder [[Bang-bang control]]) feedback control design. ''PID'' is an initialism for ''Proportional-Integral-Derivative'', referring to the three terms operating on the error signal to produce a control signal. If ''u(t)'' is the control signal sent to the system, ''<math>y(t)''</math> is the measured output and ''<math>r(t)''</math> is the desired output, and tracking error <math>e(t)=r(t)- y(t)</math>, a PID controller has the general form

:<math>u(t) = K_P e(t) + K_I \int e(t)\text{d}t + K_D \frac{\text{d}}{\text{d}t}e(t).</math>

The desired closed loop dynamics is obtained by adjusting the three parameters <math> K_P</math>, <math> K_I</math> and <math> K_D</math>, 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 [[multiple-input multiple-output systems (MIMO]]) systems are considered.

Applying Laplace transformation results in the transformed PID controller equation

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linear actuator with filtered input

:<math>P(s) = \frac{A}{1 + sT_p}</math>, <math>A</math> = const

and insert all this into expression for closed-loop transfer function <math>H(s)</math>, then tuning is very easy: simply put

:<math>K = \frac{1}{A}, T_i = T_f, T_d = T_p</math>

and get <math>H(s) = 1</math> identically.

For practical PID controllers, a pure differentiator is neither physically realisable nor desirable<ref>Ang, K.H., Chong, G.C.Y., and Li, Y. (2005). [httphttps://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1453566 PID control system analysis, design, and technology, ''IEEE Trans Control Systems Tech'', 13(4), pp.559-576].</ref> 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 [[root locus]], the [[Nyquist stability criterion]], the [[Bode plot]], the [[gain margin]] and [[phase margin]]. More advanced tools include [[Bode's sensitivity integral|Bode integral]]s to assess performance limitations and trade-offs, and describing functions to analyze nonlinearities in the frequency ___domain.<ref>{{cite book |author1=Boris J. Lurie |author2=Paul J. Enright |title=Classical Feedback Control with Nonlinear Multi-loop Systems |edition=3 |year=2019 |publisher=CRC Press |isbn=978-1-1385-4114-6 }}</ref>

==See also==

* [[Minor loop feedback]] a classical method for designing feedback control systems.

* [[State space (control)]]

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{{reflist}}

[[Category:Classical control theory| ]]

[[Category:Control engineering]]

[[Category:Mathematical modeling]]