D x )
] (
h x , x ) K( x x
1
4
4
3
1
1
2
1
3 )
(
D x 1) {
x 2
dt
x
1
(22)
h
1
1
[ (
D x
1 ) ( (
h x 1, x 2) K( x 1 x 3))] K( x 2 x 4)} a 4( x 1, x 2 , x 3) (
D x 1) Kx 4
x
2
where for simplicity we define the function a 4 to be that in the definition of y 4 except the
last term, which is
1
D Kx 4 . Note that x 4 appears only in this last term so that a 4 depends
only on x 1, x 2 , x 3 .
As in the single-link case, the above mapping is a global diffeomorphism. Its inverse can be
found by
x
1
y 1
(23)
x
2
y 2
(24)
1
x
3
y 1 K ( (
D y 1) y 3
(
h y 1, y 2))
(25)
1
x
4
K
(
D y 1)( y 4 a 4( y 1, y 2 , y 3))
(26)
The linearizing control law can now be found from the condition
y
4
v
(27)
where v is a new control input. Computing y4 from (22) and suppressing function
arguments for brevity yields
134
Applications of Nonlinear Control
a
4
a 4
1
a 4
d
1
1
1
1
v
x
D ( h K( x x ))
x
[ D ] Kx D K( J K( x x ) J u)
2
1
3
4
4
1
3
x
1
x 2
x 3
dt
(28)
(
a x) (
b x) u
where
a
4
a 4
1
a 4
d
1
1
1
(
a x) :
x
(29)
2
D ( h K( x 1 x 3))
x 4
[ D ] Kx 4 D KJ K( x 1 x 3)
x
1
x 2
x 3
dt
1
1
(
b x) D ( x) KJ
u
(30)
Solving the above expression for u yields
1
u
(
b x)
( v (
a x))
(31)
:
( x) ( x) v
(32)
where
1
( x) JK
(
D x) and
1
( x)
(
b x)
(
a x)
With the nonlinear change of coordinates (19)-(22) and nonlinear feedback (32) the
transformed system has the linear block form
0 I 0 0
0
0 0 I 0 0
y
y
v
(33)
0 0 0 I
0
0 0 0 0
I
:
Ay bv
where I n n identity matrix, 0 n n zero matrix, T
T
T
T
T
4
(
n
y
y
1 , y 2 , y 3 , y 4 )
, and
n
v . The system (33) represents a set of n decoupled quadruple integrators.
4. Outer loop design based on predictive function control
4.1 why use predictive function control
The technique of feedback linearization is important due to it leads to a control design
methodology for nonlinear systems. In the context of control theory, however, one should
be highly suspicious of techniques that rely on exact mathematical cancellation of terms,
linear or nonlinear, from the equations defining the system.
In this section, we investigate the effect of parameter uncertainty, computational error,
model simplification, and etc. We show that the most important property of feedback
linearizable systems is not necessarily that the nonlinearities can be exactly cancelled by
nonlinear feedback, but rather that, once an appropriate coordinate system is found in
which the system can be linearized, the nonlinearities are in the range space of the input.
This property is highly significant and is exploited by the predictive function control
techniques to guarantee performance in the realistic case that the nonlinearities in the
system are not known exactly.
Predictive Function Control of the Single-Link Manipulator with Flexible Joint
135
Consider a single-input feedback linearizable system. After the appropriate coordinate
transformation, the system can be written in the ideal case as
y
1
y 2
(34)
1
y v ( x)[ u ( x)]
n
provided that u is given by (16) in order to cancel the nonlinear terms ( x) and ( x) .
In practice such exact cancellation is not achievable and it is more realistic to suppose that
the control law u in (16) is of the form
u ˆ
ˆ
( x) ( x) v
(35)
where ˆ
( x) , ˆ( x) represent the computed versions of ( x) , ( x) , respectively. These
functions may differ from the true ( x) , ( x) for several reasons. Because the inner loop
control u is implemented digitally, there will be an error due to computational round-off
and delay. Also, since the terms ( x) , ( x) are functions of the system parameters such as
masses, and moments of inertia, any uncertainty in knowledge of these parameters will be
in reflected in ˆ
( x) , ˆ( x) . In addition, one may choose intentionally to simplify the control
u by dropping various terms in the equations in order to facilitate on-line computation. If
we now substitute the control law (35) into (34) we obtain
y
1
y 2
(36)
1
y v ( x)[ ˆ
ˆ
( x) ( x) v ( x)]
n
v ( y 1,, y , v)
n
where the uncertainty is given as
( y ,, y , v)
n
1 ˆ 1
1
v
ˆ
1
| 1
(37)
y T ( X)
The system (36) can be written in matrix form as
y Ay b{ v ( y, v)}
(38)
where A and b are given by (18). For multi-input case, similar to (33), and if
m
v , and
: n
m
m
. Note that the system (38) is still nonlinear whenever 0 . The practical
implication of this is solved by the outer loop predictive function control (PFC).
The system (38) can be represented by the block diagram of Figure 2. The application of the
nonlinear inner loop control law results in a system which is “approximately linear”. A
common approach is to decompose the control input v in (38) into two parts, the first to
136
Applications of Nonlinear Control
stabilize the ‘nominal linear system’ represented by (38) with 0 . In this case v can be
taken as a linear state feedback control law designed to stabilize the nominal system and/or
for tracking a desired trajectory. A second stage control v
is then designed for robustness,
that is, to guarantee the performance of the nominal design in the case that 0 . Thus the
form of the control law is
u ˆ
ˆ
( x) ( x) v
(39)
v Ky v
(40)
v
PFC( y )
(41)
r
where Ky is a linear feedback designed to place the eigenvalues of A in a desired location,
v
represents an additional feedback loop to maintain the nominal performance despite the
presence of the nonlinear term . yr is a reference input, which can be chosen as a signal for
tracking a desired trajectory.
base
function
nonlinear
u
system
x
y
y
v ôf joint
T(x)
( x)
PFC
flexibility
ˆ
( x)
-K
ˆ y
linear model
e
Fig. 2. block diagram for PFC outer loop design
4.2 Predictive function control
All MPC strategies use the same basic approach i.e., prediction of the future plant outputs,
and calculation of the manipulated variable for an optimal control. Most MPC strategies are
based on the following principles:
Use of an internal model
Its formulation is not restricted to a particular form, and the internal model can be linear,
nonlinear, state space form, transfer function form, first principles, black-box etc. In PFC,
Predictive Function Control of the Single-Link Manipulator with Flexible Joint
137
only independent models where the model output is computed only with the present and
past inputs of the process models are used.
Specification of a reference trajectory
Usually an exponential.
Determination of the control law
The control law is derived from the minimization of the error between the predicted output
and the reference with the projection of the Manipulated Variable (MV) on a basis of functions.
Although based on these principles the PFC algorithm may be of several levels of
complexity depending on the order and form of the internal model, the order of the basis
function used to decompose the MV and the reference trajectory used.
4.3 First order PFC
Although it is unrealistic to represent industrial systems by a first order system, as most of
them are in a higher order, some well behaved ones may be estimated by a first order. The
estimation will not be perfect at each sample time, however, the robustness of the PFC will
help to maintain a decent control.
If the system can be modelled by a first order plus pure time delay system, then the
following steps in the development of the control law are taken.
Model formulation
In order to implement a basic first order PFC, a typical first order transfer function equation
(42) is used.
K
y ( s)
M
(
u s)
M
(42)
T S 1
M
Note that the time delay is not considered in the internal model formulation and in this case
KM is equal to one. The discrete time formulation of the model zero-order hold equivalent
is then obtained in (43).
y ( k) y ( k 1) K (1 ) (
u k 1)
M
M
M
(43)
T
where exp(
s
) . If the manipulated variable is structured as a step basis function:
M
T
y ( k H)
H
y ( k)
L
M
(44)
y ( k H) K (1
H
) (
u k)
F
M
(45)
Where, yL and yF are respectively, the free (autoregressive) and the forced response of yM .
Reference trajectory formulation
If yR is the expression of the reference trajectory, then at the coincidence point H:
138
Applications of Nonlinear Control
C( k H) y ( k H)
H
( C( k) y ( k))
R
P
(46)
thus:
y ( k H) C( k)
H
( C( k) y ( k))
R
P
(47)
Predicted process output
The predicted process output is given by the model response, plus a term given the error
between the same model output and the process output:
ˆ y ( k H) y ( k H) ( y ( k) y ( k))
P
M
P
M
(48)
where (
y
k H) y ( k H) y ( k H)
H
y ( k) K (1
H
) (
u k)
M
L
F
M
M
.
Computation of the control law
At the coincidence point H:
y ( k H) ˆ y ( k H)
R
P
(49)
Combining (44), (45), (47) and (48) yields
C( k)
H
( C( k) y ( k)) y ( k) y ( k H) y ( k) P
P
M
M
(50)
Replacing y ( k H)
M
by its equivalent in equations (44) and (45) we obtain:
(
C k)(1
H
) y ( k)(1
H
) y ( k)(1
H
)