8.6 Linearization of Nonlinear Systems In this section we show how to perform linearization of systems described by nonlinear differential equations.
The procedure introduced is based on the Taylor series expansion and on knowledge of nominal system trajectories and nominal system inputs. We will start with a simple scalar rst-order nonlinear dynamic system
Assume that under usual working circumstances this system operates along the
trajectory
while it is driven by the system input
. We call
and
, respectively, the nominal system trajectory and the nominal system input.
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On the nominal trajectory the following differential equation is satised
Assume that the motion of the nonlinear system is in the neighborhood of the nominal system trajectory, that is
where
represents a small quantity. It is natural to assume that the system
motion in close proximity to the nominal trajectory will be sustained by a system input which is obtained by adding a small quantity to the nominal system input
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For the system motion in close proximity to the nominal trajectory, we have
Since
and
are small quantities, the right-hand side can be expanded
into a Taylor series about the nominal system trajectory and input, which produces
Canceling
higher-order
terms
(which
contain
very
small
quantities
), the linear differential equation is obtained
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The partial derivatives in the linearization procedure are evaluated at the nominal points. Introducing the notation
the linearized system can be represented as
In general, the obtained linear system is time varying. Since in this course we study only time invariant systems, we will consider only those examples for which the linearization procedure produces time invariant systems. It remains to nd the initial condition for the linearized system, which can be obtained from
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Similarly, we can linearize the second-order nonlinear dynamic system
by assuming that
and expanding
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into a Taylor series about nominal points
, which leads to
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where the corresponding coefcients are evaluated at the nominal points as
The initial conditions for the second-order linearized system are obtained from
Example 8.15: The mathematical model of a stick-balancing problem is
where
is the horizontal force of a nger and
represents the sticks angular
displacement from the vertical.
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This second-order dynamic system is linearized at the nominal points , producing
The linearized equation is given by
important to point out that the same linearized model could have been obtained by setting , which is valid for small values of .
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Note that
since
0 1!!%" ( & $
0 1)'%# ( & $ " 2 0 14!3# ( & $ " !
. It is
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We can extend the presented linearization procedure to an
-order nonlinear
dynamic system with one input and one output in a straightforward way. However, for multi-input multi-output systems this procedure becomes cumbersome. Using the state space model, the linearization procedure for the multi-input multi-output case is simplied. Consider now the general nonlinear dynamic control system in matrix form
5
where vector, the
, and
are, respectively, the
-dimensional system state space -dimensional vector function. is known and that
6
-dimensional input vector, and the
Assume that the nominal (operating) system trajectory
the nominal system input that keeps the system on the nominal trajectory is given
6
by
.
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Using the same logic as for the scalar case, we can assume that the actual system dynamics in the immediate proximity of the system nominal trajectories can be approximated by the rst terms of the Taylor series. That is, starting with
7 7 7 E D B @ 9 !%IH'8 E D !%B @ F 7 7 7 E D B @ 9 !%CA48 E D !GB @ F 7
and
we expand the right-hand side into the Taylor series as follows
7 7 7
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Higher-order terms contain at least quadratic quantities of and
and
. Since
are small their squares are even smaller, and hence the high-order terms
can be neglected. Neglecting higher-order terms, an approximation is obtained
The partial derivatives represent the Jacobian matrices given by
g srd e i pd g hfd e q fpd i g hfd e g i pd
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V U !%T
V U T !%CR
Q 'P
i pd
i pd
e rd
e fd
V U !%T R W V U T R Q !YXS'P t urd e v wpd i R q fpd i e rd q R g g
V U !%T R W V U T R Q '%IS4P i pd i pd e rd ` cb` a V U !YT R W V U T R Q !3#S4P e fd
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Note that the Jacobian matrices have to be evaluated at the nominal points, that
is, at
and
. With this notation, the linearized system has the form
g g g
The output of a nonlinear system satises a nonlinear algebraic equation, that is
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!3
!3
y x
sr r r ur e fd
sr r
p r r cb !%# y '%#S'x
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This equation can also be linearized by expanding its right-hand side into a
h n m !3l k h
Taylor series about nominal points
h h h
and
. This leads to
n m l k j !YCS4i n m !3l k o
linearized part of the output equation is given by
where the Jacobian matrices
and
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Note that
cancels term
By neglecting higher-order terms, the
satisfy
n m l k j '3#S4i o
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z y !%x z y !%x w w { v 'u | } p| d| } p| ~ | } 1| z y !%x w { v 4u w rp q ~ | p1| } ~ d| ~ ~ } p| | } 1| d| | } 1| z y !%x 1p| } d| ~ } p| z y !Yx w { v u w w z y !Yx 1| p| } X1| ~ pp| } #1| z y x !%#w { z y x w v 'YXA4u s trp q ~ fp| ~ d1| } ~ 1| } p| 1| ~ p| } } X1| Xp| w w fp| } 1| } 1| 1| ~ d1| }
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Example 8.16: Let a nonlinear system be represented by
Sfu f f
Assume that the values for the system nominal trajectories and input are known and
f
given by
and
system is obtained as
f
Having obtained the solution of this linearized system under the given system input , the corresponding approximation of the nonlinear system trajectories is
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. The linearized state space equation of this nonlinear
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Example 8.17: Consider the mathematical model of a single-link robotic manipulator with a exible joint given by
respectively, link mass and length, and the change of variables as
the manipulators state space nonlinear model is given by
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where
are angular positions,
are moments of inertia,
is the link spring constant. Introducing
and are,
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are
pCCX
Assuming that the output variable is equal to the links angular position, that is , the matrices
and
are given by
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Take the nominal points as
, then the matrices
and
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