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It just completely eludes me as a I try understand the Wikipedia links to these concepts. I'm not sure if it's been written for the general audience as the descriptions are filled with a lot of complex terminology and it's been a very slow and fruitless read so far. So would anyone mind clearing up the difference between linear and nonlinear control for a newbie? :D |
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A "linear" system has the property that the sum of two outputs is the same as the output of their sum, ie, f(x+y) = f(x) + f(y). In two dimensions, this relationship describes a line. In three dimensions, it's a plane and in higher dimensions, a hyperplane, but we still call it a linear relationship. A "nonlinear" system is one that doesn't have this property. More colloquially, linear systems have a direct, proportional input/output relationship: if you add 10% more input, you'll get 10% more output. A common example is your shower: you turn the faucet a lot to make a big change in temperature, but when it's close to what you want, you make only a small change and you get a small change in temperature. On the other hand, a common nonlinear system is an inverted pendulum: balancing a broom vertcally on your hand. Here it's still the case that large movements will make large changes in the broom's angle, but the relationship is no longer linear. Once it leans too far, very large changes are needed to have the same effect. Near the balance point, a 10% change in hand position may cause a 10% change in angle; farther from vertical, a 20% change in hand position may be needed to make a 10% change in the angle. We can distinguish further between the linearity of the system and the linearity of the controller. For example, we may use a nonlinear controller such as a neural net to control a linear system. Or we may use a linear controller to control a nonlinear system by constraining it to work in a narrow region of the nonlinear system. In the case of the inverted pendulum, we can approximate the area near vertical as a linear system, then use a linear controller. This approach works because the system is smooth and relatively flat in the region of interest (ie, near the point of linearization centered on vertical). As long as the controller is able to keep the system in the region where the linear approximation is valid, it works well. Thank you so much!! :)
(Dec 28 '11 at 20:08)
Kaitlyn McMordie
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I'm no expert on this so this might not be a very accurate description but some of my colleagues work on this and your question is quite general. Informally, the goal of control is to direct the behaviour of a dynamical system. A simple example is to keep the temperature in a heating tank constant by adjusting the amount of hot water that is entering it. Linear control assumes that the underlying dynamical system can be described with a state space model for which most properties of interest can be computed analytically. For non-linear systems, one might be forced to use more complex techniques that are not tractable any more or rely on approximations. Thank you!! :)
(Dec 28 '11 at 20:08)
Kaitlyn McMordie
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