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 (sociology).'' 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 (control theory) tries to manipulate the inputs of the system to realize this behaviour at the output of the system. == 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.
''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: :

Y(s) = P(s) U(s)\,\!

U(s) = C(s) E(s)\,\!

E(s) = R(s) - Y(s)\,\!

Control theory#Appendix A in terms of ''R''(''s''), we obtain: :

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

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 Pole (complex analysis)s of its transfer function must * lie in the left half of the complex plane if the Laplace transform is used OR * lie inside the unit circle if the Z-transform is used 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 :

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

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

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

which has a pole at

0.5 + j 0

(zero imaginary number). This system is BIBO stable since the pole is ''inside'' the unit circle. However, if the impulse response was changed to :

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

then the Z-transform is :

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

which has a pole at

1.5 + j 0

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 marginal stability, and have non-repeated poles along the vertical axis (i.e. the real component is zero). == State space representation == ''See State space (controls).'' == Controllability and observability == ''See controllability and observability.'' ==See also== * Adaptive control * Control engineering * H infinity * Intelligent control * Non-linear control * Optimal control * Process control * PID controller * Robotic unicycle * Servomechanism * State space (controls) * Fractional order control * Stable polynomial == Appendix A == Derivation of transfer function:

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

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Control theory

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Control theory

==Cruise control== I fail to see the correctness of :"(Does not apply to manual transmission vehicles.)" What specifically does not apply? Holding the throttle fixed works the same for a automatic and manual transmission. Removing sentence and call for further clarification. User:Cburnett 00:07, 6 Jan 2005 (UTC) Locking the throttle on an automatic transmission does not lock speed, due to losses in the torque convertor. --User:Sponge 04:10, 12 Jan 2005 (UTC) Locking the throttle on an engine does not lock in speed; it doesn't matter what the transmission is. Change in angle changes the torque on the engine, which changes the speed. So I still don't see how the quote is relevant... User:Cburnett 05:15, 12 Jan 2005 (UTC) It is good to put an edit summary when you write something. When I saw that this page was modified, and no summary was put, I thought it was vandalism (it happens all too often unfortunately). Besides, putting an edit summary helps people who have this article on their watchlist understand what you are up to. This is just a thought. User:Oleg Alexandrov 05:25, 12 Jan 2005 (UTC) By the way, do you know anything about optimal control? That page needs some work. See that page and its history. User:Oleg Alexandrov 05:25, 12 Jan 2005 (UTC) :Ugh, of all the control classes I've taken....optimal control was my least favorite. I'll add it to my list. User:Cburnett 05:42, 12 Jan 2005 (UTC) == Clarification == The stability section needs to be explained better. 128.112.86.171 ---- (sig added by Cburnett; please use ~~~~ to sign your posts) :What exactly do you find can be improved. For someone, like me, that knows this stuff...it's a bit harder to know exactly what you think needs to be better explained. I'll try though. User:Cburnett 03:46, Jun 9, 2005 (UTC)

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