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Feedback linearization should not be confused with Input-Output Feedback Linearization. Feedback linearization techniques involve finding feedback transformations that exactly linearize some portion, or all of, the dynamics. It may or may not involve a system with a predefined output. Instead, let us consider the more general problem in the description, and then discuss the special cases in the body. |
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where <math>x(t) \in \mathbb{R}^n</math> is the state, <math>u_1(t), \ldots, u_m(t) \in \mathbb{R}</math> are the inputs. The approach involves transforming a nonlinear control system into an equivalent linear control system through a change of variables and a suitable control input. In particular, one seeks a change of coordinates <math>z = \Phi(x)</math> and control input <math>u = a(x) + b(x)\,v,</math> so that the dynamics of <math>x(t)</math> in the coordinates <math>z(t)</math> take the form of a linear, controllable control system,
:<math>\dot{z}(t) = A\,z(t) + \sum_{i=1}^{m} b_i\,v(t).</math><ref>H. Nijmeijer and A. van der Shaft,
An outer-loop control strategy for the resulting linear control system can then be applied to achieve the control objective.
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