Feedback linearization
Encyclopedia
Feedback linearization is a common approach used in controlling nonlinear systems. The approach involves coming up with a transformation of the nonlinear system into an equivalent linear system through a change of variables and a suitable control input. Feedback linearization may be applied to nonlinear systems of the form
where
is the state vector, 
is the vector of inputs, and 
is the vector of outputs. The goal is to develop a control input
that renders a linear input–output map between the new input
and the output. An outer-loop control strategy for the resulting linear control system can then be applied.
and 
. We wish to find a coordinate transformation 
that transforms our 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
to the output 
. 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
so that differentiability in the original coordinate system is preserved in the new coordinate system. In practice, the transformation can be only locally diffeomorphic, but the linearization results only hold in this smaller region.
We require several tools before we can solve this problem.
and its first 
derivatives. To understand the structure of this target system, we use the Lie derivative
. Consider the time derivative of (2), which we can compute using the chain rule
,
Now we can define the Lie derivative of
along 
as,
and similarly, the Lie derivative of
along 
as,
With this new notation, we may express
as,
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,
and
and its first 
derivatives, we must understand how the input 
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 
at a point 
if,
in a neighbourhood
of
and all 
Considering this definition of relative degree in light of the expression of the time derivative of the output
, we can consider the relative degree of our system (1) and (2) to be the number of times we have to differentiate the output 
before the input 
appears explicitly. In an LTI system, the relative degree is the difference between the degree of the transfer function's denominator polynomial (i.e., number of poles) and the degree of its numerator polynomial (i.e., number of zero
s).
. In this case, after differentiating the output 
times we have,
where the notation
indicates the 
th derivative of 
. Because we assumed the relative degree of the system is 
, the Lie derivatives of the form 
for 
are all zero. That is, the input 
has no direct contribution to any of the first 
th derivatives.
The coordinate transformation
that puts the system into normal form comes from the first 
derivatives. In particular,
transforms trajectories from the original
coordinate system into the new 
coordinate system. So long as this transformation is a diffeomorphism
, smooth trajectories in the original coordinate system will have unique counterparts in the
coordinate system that are also smooth. Those 
trajectories will be described by the new system,
Hence, the feedback control law
renders a linear input–output map from
to 
. The resulting linearized system
is a cascade of
integrators, and an outer-loop control 
may be chosen using standard linear system methodology. In particular, a state-feedback control law of
where the state vector
is the output 
and its first 
derivatives, results in the LTI system
with,
So, with the appropriate choice of
, we can arbitrarily place the closed-loop poles of the linearized system.
. However, the normal form of the system will include zero dynamics (i.e., states that are not observable
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 stable or at least controllable, and so measures can be taken to ensure these states do not cause problems in practice.
where
is the state vector, is the vector of inputs, and is the vector of outputs. The goal is to develop a control inputthat renders a linear input–output map between the new input
and the output. An outer-loop control strategy for the resulting linear control system can then be applied.Feedback Linearization of SISO Systems
Here, we 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, and . We wish to find a coordinate transformation that transforms our system (1) into the so-called normal formNormal form
Normal form may refer to:* Normal form * Normal form * Normal form * Normal form In formal language theory:* Beta normal form* Chomsky normal form* Greibach normal form* Kuroda normal form...
which will reveal a feedback law of the form
that will render a linear input–output map from the new input
to the output . To ensure that the transformed system is an equivalent representation of the original system, the transformation must be a diffeomorphismDiffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
. That is, the transformation must not only be invertible (i.e., bijective), but both the transformation and its inverse must be smooth
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
so that differentiability in the original coordinate system is preserved in the new coordinate system. In practice, the transformation can be only locally diffeomorphic, but the linearization results only hold in this smaller region.
We require several tools before we can solve this problem.
Lie derivative
The goal of feedback linearization is to produce a transformed system whose states are the output and its first derivatives. To understand the structure of this target system, we use the Lie derivativeLie derivative
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
. Consider the time derivative of (2), which we can compute using the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
,
Now we can define the Lie derivative of
along as,and similarly, the Lie derivative of
along as,With this new notation, we may express
as,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,
and
Relative degree
In our feedback linearized system made up of a state vector of the output and its first derivatives, we must understand how the input 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 at a point if, in a neighbourhoodNeighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
of
and all Considering this definition of relative degree in light of the expression of the time derivative of the output
, we can consider the relative degree of our system (1) and (2) to be the number of times we have to differentiate the output before the input appears explicitly. In an LTI system, the relative degree is the difference between the degree of the transfer function's denominator polynomial (i.e., number of poles) and the degree of its numerator polynomial (i.e., number of zeroZero (complex analysis)
In complex analysis, a zero of a holomorphic function f is a complex number a such that f = 0.-Multiplicity of a zero:A complex number a is a simple zero of f, or a zero of multiplicity 1 of f, if f can be written asf=g\,where g is a holomorphic function g such that g is not zero.Generally, the...
s).
Linearization by feedback
For the discussion that follows, we will assume that the relative degree of the system is. In this case, after differentiating the output times we have,where the notation
indicates the th derivative of . Because we assumed the relative degree of the system is , the Lie derivatives of the form for are all zero. That is, the input has no direct contribution to any of the first th derivatives.The coordinate transformation
that puts the system into normal form comes from the first derivatives. In particular,transforms trajectories from the original
coordinate system into the new coordinate system. So long as this transformation is a diffeomorphismDiffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
, smooth trajectories in the original coordinate system will have unique counterparts in the
coordinate system that are also smooth. Those trajectories will be described by the new system,Hence, the feedback control law
renders a linear input–output map from
to . The resulting linearized systemis a cascade of
integrators, and an outer-loop control may be chosen using standard linear system methodology. In particular, a state-feedback control law ofwhere the state vector
is the output and its first derivatives, results in the LTI systemwith,
So, with the appropriate choice of
, 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. However, the normal form of the system will include zero dynamics (i.e., states that are not observableObservable
In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...
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 stable or at least controllable, and so measures can be taken to ensure these states do not cause problems in practice.
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
- ECE 758: Modeling and Nonlinear Control of a Single-link Flexible Joint Manipulator – Gives explanation and an application of feedback linearization.
- ECE 758: Ball-in-Tube Linearization Example – Trivial application of linearization for a system already in normal form (i.e., no coordinate transformation necessary).