Feedback linearization
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Feedback linearization is a common strategy employed in
nonlinear control Nonlinear control theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of dyn ...
to control
nonlinear systems In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
. Feedback linearization techniques may be applied to nonlinear control systems of the form where x(t) \in \mathbb^n is the state, u_1(t), \ldots, u_m(t) \in \mathbb 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 z = \Phi(x) and control input u = a(x) + b(x)\,v, so that the dynamics of x(t) in the coordinates z(t) take the form of a linear, controllable control system, An outer-loop control strategy for the resulting linear control system can then be applied to achieve the control objective.


Feedback Linearization of SISO Systems

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, u \in \mathbb and y \in \mathbb. The objective is to find a coordinate transformation z = T(x) that transforms the system (1) into the so-called normal form which will reveal a feedback law of the form that will render a linear input–output map from the new input v \in \mathbb to the output y. To ensure that the transformed system is an equivalent representation of the original system, the transformation must be a
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
. That is, the transformation must not only be invertible (i.e., bijective), but both the transformation and its inverse must be
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...
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. Several tools are required to solve this problem.


Lie derivative

The goal of feedback linearization is to produce a transformed system whose states are the output y and its first (n-1) derivatives. To understand the structure of this target system, we use the
Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
. Consider the time derivative of (2), which can be computed using the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, :\begin \dot = \frac &=\frac\dot\\ &= \fracf(x) + \fracg(x)u \end Now we can define the Lie derivative of h(x) along f(x) as, :L_h(x) = \fracf(x), and similarly, the Lie derivative of h(x) along g(x) as, :L_h(x) = \fracg(x). With this new notation, we may express \dot as, :\dot = L_h(x) + L_h(x)u Note that the notation of Lie derivatives is convenient when we take multiple derivatives with respect to either the same vector field, or a different one. For example, :L_^h(x) = L_L_h(x) = \fracf(x), and :L_L_h(x) = \fracg(x).


Relative degree

In our feedback linearized system made up of a state vector of the output y and its first (n-1) derivatives, we must understand how the input u enters the system. To do this, we introduce the notion of relative degree. Our system given by (1) and (2) is said to have relative degree r \in \mathbb at a point x_0 if, :L_L_^h(x) = 0 \qquad \forall x in a
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of x_0 and all k \leq r-2 :L_L_^h(x_0) \neq 0 Considering this definition of relative degree in light of the expression of the time derivative of the output y, we can consider the relative degree of our system (1) and (2) to be the number of times we have to differentiate the output y before the input u appears explicitly. In an
LTI system In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined ...
, the relative degree is the difference between the degree of the transfer function's denominator polynomial (i.e., number of
pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...
s) and the degree of its numerator polynomial (i.e., number of
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
s).


Linearization by feedback

For the discussion that follows, we will assume that the relative degree of the system is n. In this case, after differentiating the output n times we have, :\begin y &= h(x)\\ \dot &= L_h(x)\\ \ddot &= L_^h(x)\\ &\vdots\\ y^ &= L_^h(x)\\ y^ &= L_^h(x) + L_L_^h(x)u \end where the notation y^ indicates the nth derivative of y. Because we assumed the relative degree of the system is n, the Lie derivatives of the form L_L_^h(x) for i = 1, \dots, n-2 are all zero. That is, the input u has no direct contribution to any of the first (n-1)th derivatives. The coordinate transformation T(x) that puts the system into normal form comes from the first (n-1) derivatives. In particular, :z = T(x) = \beginz_1(x) \\ z_2(x) \\ \vdots \\ z_n(x) \end = \beginy\\ \dot\\ \vdots\\ y^ \end = \beginh(x) \\ L_h(x) \\ \vdots \\ L_^h(x) \end transforms trajectories from the original x coordinate system into the new z coordinate system. So long as this transformation is a
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
, smooth trajectories in the original coordinate system will have unique counterparts in the z coordinate system that are also smooth. Those z trajectories will be described by the new system, :\begin\dot_1 &= L_h(x) = z_2(x)\\ \dot_2 &= L_^h(x) = z_3(x)\\ &\vdots\\ \dot_n &= L_^h(x) + L_L_^h(x)u\end. Hence, the feedback control law :u = \frac(-L_^h(x) + v) renders a linear input–output map from v to z_1 = y. The resulting linearized system :\begin\dot_1 &= z_2\\ \dot_2 &= z_3\\ &\vdots\\ \dot_n &= v\end is a cascade of n integrators, and an outer-loop control v may be chosen using standard linear system methodology. In particular, a state-feedback control law of :v = -Kz\qquad, where the state vector z is the output y and its first (n-1) derivatives, results in the
LTI system In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined ...
:\dot = Az with, :A = \begin 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & 1 \\ -k_1 & -k_2 & -k_3 & \ldots & -k_n \end. So, with the appropriate choice of k, we can arbitrarily place the closed-loop poles of the linearized system.


Unstable zero dynamics

Feedback linearization can be accomplished with systems that have relative degree less than n. However, the normal form of the system will include
zero dynamics In mathematics, zero dynamics is known as the concept of evaluating the effect of zero on systems. History The idea was introduced thirty years ago as the nonlinear approach to the concept of transmission of zeros. The original purpose of intr ...
(i.e., states that are not
observable In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...
from the output of the system) that may be unstable. In practice, unstable dynamics may have deleterious effects on the system (e.g., it may be dangerous for internal states of the system to grow unbounded). These unobservable states may be controllable or at least stable, and so measures can be taken to ensure these states do not cause problems in practice.
Minimum phase In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable. The most general causal LTI transfer function can be uniquely factored into a series of a ...
systems provide some insight on zero dynamics.


See also

*
Nonlinear control Nonlinear control theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of dyn ...


Further reading

* A. Isidori, ''Nonlinear Control Systems,'' third edition, Springer Verlag, London, 1995. * H. K. Khalil, ''Nonlinear Systems,'' third edition, Prentice Hall, Upper Saddle River, New Jersey, 2002. * M. Vidyasagar, ''Nonlinear Systems Analysis'', second edition, Prentice Hall, Englewood Cliffs, New Jersey, 1993. * B. Friedland, ''Advanced Control System Design'', facsimile edition, Prentice Hall, Upper Saddle river, New Jersey, 1996.


External links

* {{Scholarpedia, title=Feedback linearization, urlname=Feedback_linearization, date=2009, curator=Fabio Celani and Alberto Isidori, accessdate=31 December 2022
ECE 758: Modeling and Nonlinear Control of a Single-link Flexible Joint Manipulator
nbsp;– Gives explanation and an application of feedback linearization.
ECE 758: Ball-in-Tube Linearization Example
nbsp;– Trivial application of linearization for a system already in normal form (i.e., no coordinate transformation necessary). *
Wolfram language The Wolfram Language ( ) is a general multi-paradigm programming language developed by Wolfram Research. It emphasizes symbolic computation, functional programming, and rule-based programming and can employ arbitrary structures and data. It is ...
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feedback linearization
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zero dynamics
Nonlinear control