Feedback linearisation

FEEDBACK LINEARISATION
Dr.R.Subasri
Professor, Kongu Engineering College,
Perundurai, Erode, Tamilnadu, INDIA
Dr.R.Subasri, KEC, INDIA

Algebraically transform a nonlinear system dynamics into
a (fully or partly) linear one, so that linear control
techniques can be applied.
Achieved by exact state transformations and feedback,
rather than by linear approximations of the dynamics.
Feedback linearization techniques can be viewed as ways
of transforming original system models into equivalent
models of a simpler form.
Feedback linearization has been used successfully to
address some practical control problems such as
helicopters, high performance aircraft, industrial robots,
and biomedical devices.
Feedback Linearization

Feedback Linearization and The Canonical Form
The idea of feedback linearization is of cancelling the
nonlinearities and imposing a desired linear dynamics
Applied to a class of nonlinear systems described by
the so-called companion form, or controllability
canonical form.
A system in companion form is written as
(n)
x f(x) b(x)u= +
where u is the scalar control input, X is the state vector
and f(x) and g(x) are nonlinear functions of the states.

we can cancel the nonlinearities and obtain the simple
input-output relation
For systems which can be expressed in the controllability
canonical form, using the control input (assuming b to be
non-zero)
 
1
u v f
b
 −
(n)
x v=
In state space form it is represented as

Thus, the control law
(n 1)
0 1 n 1v k x k x ..... k x −
−= − − − − −
with the ki chosen so that the polynomial
Pn + kn-1 pn-1+ .... + k0 has all its roots strictly in the left-half
complex plane, leads to the exponentially stable dynamics
which implies that x(t) —> 0. For tasks involving the
tracking of a desired output Xj(t), the control law
(where e(t) = x(t) – xd (t) is the tracking error) leads to
exponentially convergent tracking.
(n) (n 1)
n 1 0x k x ....k x 0−
−+ + =
(n) (n 1)
d 0 1 n 1v x k e k e ....k e −
−= − − −

Consider the following second-order SISO nonlinear system:
where f and g are known nonlinear functions. x f(x,t) g(x,t)u= +
Let xd denote the desired trajectory, and e= xd-x
Based on linearization feedback technique, controller is
designed as
v f (x,t)
u
g(x,t)
−
=
where v is the auxiliary controller
expressed as x v=
v is designed as
d 1 1v x k e k e= + +
where k1 and k2are all positive constants.
From the above two equations, we get 1 1e k e k e 0+ + =
Therefore, 1e 0→ 2e 0→ as t → 
In designing controller f and g must be known.Dr.R.Subasri, KEC, INDIA

Consider the problem of designing the control input u for
a single-input nonlinear system of the form x f(x,u)=
First step is to find a state transformation z = z(x) and an
input transformation u = u(x, v) so that the nonlinear
system dynamics is transformed into an equivalent linear
time-invariant dynamics, in the familiar form
Second step is to use standard linear techniques (such
as pole placement) to design v.
z Az bv= +
Input-State Feedback Linearization

•Linear control design can stabilize the system only in a
small region around the equilibrium point (0, 0),
•A specific difficulty is the nonlinearity in the first equation,
which cannot be directly cancelled by the control input u.
Example : Consider the system
1 1 2 1
2 2 1 1
x 2x ax sin x
x x cos x u cos(2x )
= − + +
= − +
Input-State Feedback Linearization

consider the new set of state variables
1 1
2 2 1
z x
z ax sin x
=
= +
then, the new state equations are
1 1 2
2 1 1 1 1 1
z 2z z
z 2z cosz cosz sinz a u cos(2z )
= − +
= − + +
Note that the new state equations also have an
equilibrium point at (0, 0).
( )1 1 1 1
1
1
u v cosz sin z 2z cosz
a cos(2z )
= − +
The nonlinearities can be cancelled by the control law
of the form

where v is an equivalent input to be designed
(equivalent in the sense that determining v amounts to
determining u, and vice versa), leading to a linear input-
state relation
1 1 2
2
z 2z z
z v
= − +
=
Thus, through the state transformation and input
transformation , the problem of stabilizing the original
nonlinear dynamics using the original control input u has
been transformed into the problem of stabilizing the new
dynamics using the new input v.
Since the new dynamics is linear and controllable, it is
well known that the linear state feedback control law
can place the poles anywhere with
proper choices of feedback gains.
1 1 2 2v k z k z= − −

The closed-loop system under the above control law is
represented in the block diagram.
There are two loops in this control system, with the
inner loop achieving the linearization of the input-state
relation, and the outer loop achieving the stabilization
of the closed-loop dynamics. The control input u is seen
to be composed of a nonlinearity cancellation part and a
linear compensation part.

The input-state linearization is achieved by a combination
of a state transformation and an input transformation,
with state feedback used in both.
Thus, it is a linearization by feedback, or feedback
linearization.
This is fundamentally different from a Jacobian
linearization for small range operation, on which linear
control is based.
In order to implement the control law, the new state
components (z1, z2) must be available. If they are not
physically meaningful or cannot be measured directly, the
original state x must be measured and the new states are
to be computed from x

Thus, in general, the system model must be known both
for the controller design and for the computation of z.
If there is uncertainty in the model, e.g.,uncertainty on
the parameter a, this uncertainty will cause error in the
computation of both the new state z and of the control
input u,
Tracking control can also be considered. However, the
desired motion then needs to be expressed in terms of
the full new state vector.
Complex computations may be needed to translate the
desired motion specification (in terms of physical output
variables) into specifications in terms of the new states.

The dynamic equation of the inverted pendulum is
( ) ( )
1 2
2
1 2 1 1 c 1 c
2 2 2
1 c 1 c
x x
g sinx mlx cos x sin x / (m m) cos x / (m m)
x
l 4 / 3 mcos x / (m m) l 4 / 3 mcos x / (m m)
=
− + +
= +
− + − +
Where x1 and x2 are the oscillation angle and the
oscillation rate respectively. g= 9.8 m/s2 , mc is the
vehicle mass mc = 1 kg, m is the mass of pendulum
bar, m= 0.1 kg, l is one half of pendulum length, l =0.5
m, u is the control input
The desired trajectory is xd (t)= 0.1sin (t). Controller
gains are k1 =k2 =5, The initial state of the inverted
pendulum is [ / 60 0].

Feedback Linearisation
function [sys,x0,str,ts] =
spacemodel(t,x,u,flag)
switch flag,
case 0,
[sys,x0,str,ts]=mdlInitializeSizes;
case 1,
sys=mdlDerivatives(t,x,u);
case 3,
sys=mdlOutputs(t,x,u);
case {1,2,4,9}
sys=[];
otherwise
error(['Unhandled flag = ',
num2str(flag)]);
end
function [sys,x0,str,ts]
=mdlInitializeSizes
sizes = simsizes;
sizes.NumContStates = 0;
sizes.NumDiscStates = 0;
sizes.NumOutputs = 1;
sizes.NumInputs = 5;
sizes.DirFeedthrough = 1;
sizes.NumSampleTimes = 0;
sys = simsizes(sizes);
x0 = [];
str = [];
ts = [];

function sys=mdlOutputs(t,x,u)
r=0.1*sin(pi*t); xd (t)= 0.1sin (t)
dr=0.1*pi*cos(pi*t);
ddr=-0.1*pi*pi*sin(pi*t);
e=u(1);
de=u(2);
fx=u(4);
gx=u(5);
k1=5;k2=5;
v=ddr+k1*e+k2*de;
ut=(v-fx)/(gx+0.002);
sys(1)=ut;
dx
dx
v f (x,t)
u
g(x,t)
−
=d 1 1v x k e k e= + +

Consider the system
Input-output Feedback Linearization
The objective is to make the output y(t) track a desired
trajectory yd(t) while keeping the whole state bounded,
where yd(t) and its time derivatives up to a sufficiently high
order are assumed to be known and bounded.
An apparent difficulty with this model is that the output y
is only indirectly related to the input u, through the ,state
variable x and the nonlinear state equations .
Therefore, it is not easy to design the control u for
tracking the output y The difficulty can be reduced a
direct and simple relation between the system output y
and the control input u is found.
This idea constitutes the basis for the input-output
linearization approach to nonlinear control design.
x f(x,u) and y h(x)= =

Consider the third-order system
( )1 2 2 3
5
2 1 3
2
3 1
1
x sin x x 1 x
x x x
x x u
y x
= + +
= +
= +
=
To generate a direct relationship between the output y
and the input u, let us differentiate the output y
1 2 2 3y x sinx (x 1)x= = + +
( )2 1y x 1 u f (x)= + +
where f1(x) is a function of the state defined by
( )5 2
1 1 3 3 2 2 1f (x) (x x )(x cosx ) x 1 x= + + + +
Since y is still not directly related to the input u, let us
differentiate again to obtain

The above equation represents an explicit relationship between y
and u. If the control input is in the form ( )1
2
1
u v f
x 1
= −
+
where v is a new input to be determined, the nonlinearity is
cancelled, and we obtain a simple linear double-integrator
relationship between the output and the new input v,
The design of a tracking controller for this double-integrator
relation is simple.
For instance, letting e = y(t) - yd(t) be the tracking error, and
choosing the new input v as
with k1 and k2 being positive constants, the tracking error of the
closed loop system is given by
which represents an exponentially stable error dynamics.
d 1 2v y k e k e= − −
1 2e k e k e 0+ + =
y v=

The control law is defined everywhere, except at the
singularity points such that x2 = - 1 .
Full state measurement is necessary in implementing the control
law, because the computations of both the derivative of y and the
input transformation require the value of x.
The above control design strategy of first generating a linear
input-output relation and then formulating a controller based on
linear control is referred to as the input-output linearization
approach.
If we need to differentiate the output of a system r times to
generate an explicit relationship between the output y and input
u, the system is said to have relative degree r. Thus, the system in
the above example has relative degree 2. relative degree in
linear systems refers to excess of poles over zeros.Dr.R.Subasri, KEC, INDIA

For any controllable system of order n, it will take at most n
differentiations of any output for the control input to appear, i.e.,
r<n.
If it took more than n differentiations, the system would be of
order higher than n; if the control input never appeared, the
system would not be controllable.
But only accounts for part of the closed-loop dynamics, because it
has only order 2, while the whole dynamics has order 3
Therefore, a part of the system dynamics (described by
one state component) has been rendered "unobservable"
in the input-output linearization. This part of the
dynamics will be called the internal dynamics, because it
cannot be seen from the external input-output
relationship.

For the above example, the internal state can be chosen to
be x3 (because x3 , y , and constitute a new set of states), and the
internal dynamics is represented by the equation
y
( )2
3 1 d 1 2 1
2
1
x x y (t) k e k e f
x 1
= + − − +
+
If this internal dynamics is stable (by which we actually mean that
the states remain bounded during tracking, i.e., stability in the
BIBO sense), our tracking control design problem has indeed been
solved.
Otherwise, the above tracking controller is practically
meaningless, because the instability of the internal dynamics
would imply undesirable phenomena such as the burning-up of
fuses or the violent vibration of mechanical members.
Therefore, the effectiveness of the above control design depends
upon the stability of the internal dynamics.Dr.R.Subasri, KEC, INDIA

Consider the nonlinear system
3
1 2
2
1
x x u
x u
y x
 + 
=   
   
=
Assume that the control objective is to make y track yd(t).
Differentiation of y simply leads to the first state equation.
V, the auxiliary controller expressed as d 1v y k e= +
Based on linearization feedback technique,
controller is designed as
v f (x,t)
u
g(x,t)
−
=
1x v y= =
f(x) g(x)=1
3
d 1 2u y k e x= + −
e=y-yd
Which yields exponential convergence of e to zero
1e k e 0+ =
The same control input is also applied to the second dynamic
equation, leading to the internal dynamics 3
2 2 d 1x x y k e+ = −

The above equation is characteristically, non-autonomous
and nonlinear.
However, in view of the facts that e is guaranteed to be
bounded and yd is assumed to be bounded, it represents
a satisfactory tracking control law for the system , given
any trajectory yd(t) and its derivative is bounded.
Conversely, it can be easily shown that if the second state
equation in the given system is replaced by , then
the resulting internal dynamics is unstable.
2x u= −

To summarize, control design based on input-output
linearization can be made in three steps:
• differentiate the output y until the input u appears
• choose u to cancel the nonlinearities and guarantee
tracking convergence
• study the stability of the internal dynamics
If the relative degree associated with the input-output
linearization is the same as the order of the system, the
nonlinear system is fully linearized and this procedure
indeed leads to a satisfactory controller (assuming that the
model is accurate).
If the relative degree is smaller than the system order, then the
nonlinear system is only partly linearized, and whether the controller
can indeed be applied depends on the stability of the internal
dynamics. Dr.R.Subasri, KEC, INDIA

Thank you

Feedback linearisation

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