Feedback linearization: Difference between revisions

Revision as of 06:27, 7 August 2022

Revision as of 08:20, 8 August 2022

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'''Feedback linearization''' is a common strategy employed in [[nonlinear control]] to control [[nonlinear systems]]. Feedback linearization techniques may be applied to nonlinear control systems of the form

::{{NumBlk|:|<math>\dot{x}(t) = f(x(t)) + \sum_{i=1}^{m}\,g_i(x(t))\,u_i(t)</math><ref>{{cite book |last1=Isidori |first1=Alberto |title=Nonlinear Control Systems |date=1995 |publisher=Springer-Verlag London |isbn=978-1-4471-3909-6 |page=5 |edition=Third}}</ref>|{{EquationRef|1}}}}

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,

:{{NumBlk|:|<math>\dot{z}(t) = A\,z(t) + \sum_{i=1}^{m} b_i\,v(t).</math><ref>H. Nijmeijer and A. van der Shaft, Nonlinear Dynamical Control Systems, Springer-Verlag, p. 163, 2016.</ref>|{{EquationRef|2}}}}

An outer-loop control strategy for the resulting linear control system can then be applied to achieve the control objective.

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Here, consider the case of feedback linearization of a single-input single-output (SISO) system. Similar results can be extended to multiple-input multiple-output (MIMO) systems. In this case, <math>u \in \mathbb{R}</math> and <math>y \in \mathbb{R}</math>. The objective is to find a coordinate transformation <math>z = T(x)</math> that transforms the system (1) into the so-called [[Normal form (abstract rewriting)|normal form]] which will reveal a feedback law of the form

:{{NumBlk|:|<math>u = a(x) + b(x)v\,</math><ref>{{cite book |last1=Isidori |first1=Alberto |title=Nonlinear Control Systems |date=1995 |publisher=Springer-Verlag London |isbn=978-1-4471-3909-6 |page=147 |edition=Third}}</ref>|{{EquationRef|3}}}}

that will render a linear input&ndash;output map from the new input <math>v \in \mathbb{R}</math> to the output <math>y</math>. To ensure that the transformed system is an equivalent representation of the original system, the transformation must be a [[diffeomorphism]]. That is, the transformation must not only be invertible (i.e., bijective), but both the transformation and its inverse must be [[smooth function|smooth]] so that differentiability in the original coordinate system is preserved in the new coordinate system. In practice, the transformation can be only locally diffeomorphic and the linearization results only hold in this smaller region.