# Control theory

This article is about an engineering theory called control theory. There is also a sociological theory of deviant behavior that is called control theory.

In engineering and mathematics, control theory deals with the behaviour of dynamical systems over time. The desired output of a system is called the reference variable. When one or more output variables of a system need to show a certain behaviour over time, a controller tries to manipulate the inputs of the system to realize this behaviour at the output of the system.

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## An example

As an example, consider cruise control. In this case, the system is a car. The goal of cruise control is to keep the car at a constant speed. Here, the output variable of the system is the speed of the car. The primary means to control the speed of the car is the amount of gas being fed into the engine.

A simple way to implement cruise control is to lock the position of the throttle the moment the driver engages cruise control. This is fine if the car is driving on perfectly flat terrain. On hilly terrain, the car will slow down when going uphill and accelerate when going downhill; something its driver may find highly undesirable.

This type of controller is called an open-loop controller because there is no direct connection between the output of the system and its input. One of the main disadvantages of this type of controller is the lack of sensitivity to the dynamics of the system under control.

## Classical control theory

To avoid the problems of the open-loop controller, control theory introduces feedback. The output of the system y is fed back to the reference value r. The controller C then takes the difference between the reference and the output, the error e, 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.

Missing image
Simple_feedback_control_loop.png
Image:simple_feedback_control_loop.png

A simple feedback control loop

If we assume the controller C and the plant P are linear, time-invariant and all single input, single output, we can analyze the system above by using the Laplace transform on the variables. This gives us the following relations:

[itex]Y(s) = P(s) U(s)\,\![itex]
[itex]U(s) = C(s) E(s)\,\![itex]
[itex]E(s) = R(s) - Y(s)\,\![itex]

Solving for Y(s) in terms of R(s), we obtain:

[itex]Y(s) = \left( \frac{PC}{1 + PC} \right) R(s)[itex]

The term PC/(1 + PC) is referred to as the transfer function of the system. If we can ensure PC >> 1, then Y(s) is approximately equal to R(s). This means we control the output by simply setting the reference.

## Stability

Stability (in control theory) means that for any bounded input over any amount of time, the output will also be bounded. This is known as BIBO stability (see also Lyapunov stability). If a system is BIBO stable then the output cannot "blow up" if the input remains finite. Mathematically, this means that for a continuous-time system to be stable all of the poles of its transfer function must

OR

This is not a contradiction! The Laplace transform is in Cartesian coordinates and the Z-transform is in circular coordinates and it can be shown that

• the negative-real part in the Laplace domain can map onto the interior of the unit circle
• the positive-real part in the Z domain can map onto the exterior of the unit circle

If the system in question has an impulse response of

[itex]x[n] = 0.5^n u[n][itex]

and taking the Z-transform (see Z-transform#Example 2 (causal ROC)) yields

[itex]X(z) = \frac{1}{1 - 0.5z^{-1}}\ [itex]

which has a pole at [itex]0.5 + j 0[itex] (zero imaginary part). This system is BIBO stable since the pole is inside the unit circle.

However, if the impulse response was changed to

[itex]x[n] = 1.5^n u[n][itex]

then the Z-transform is

[itex]X(z) = \frac{1}{1 - 1.5z^{-1}}\ [itex]

which has a pole at [itex]1.5 + j 0[itex] and is not BIBO stable since the pole is outside the unit circle.

Lastly, if the system response neither decays nor grows over time are referred to as marginally stable, and have non-repeated poles along the vertical axis (i.e. the real component is zero).

## Controllability and observability

See controllability and observability.

## Appendix A

Derivation of transfer function:

 [itex]Y(s) = P(s) U(s)\,\![itex] (1) [itex]U(s) = C(s) E(s)\,\![itex] (2) [itex]E(s) = R(s) - Y(s)\,\![itex] (3) (1) + (2) [itex]Y = P C E\,\![itex] (4) (4) + (3) [itex]Y = P C ( R - Y )\,\! [itex] [itex]Y = P C R - P C Y\,\! [itex] Expanding out ( R − Y ) [itex]Y + P C Y = P C R\,\! [itex] Moving P C Y to the left hand side [itex]Y ( 1 + P C ) = P C R\,\! [itex] Consolidating the common term Y [itex]Y = \frac{P C R}{1 + P C}[itex] Isolating out the term Y [itex]Y = \frac{P C}{1 + P C} R[itex] (5)
de:Kontrolltheorie
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