design of p and pi controllers for quasi linear systems
TRANSCRIPT
-
7/25/2019 design of p and pi controllers for quasi linear systems
1/12
Com.~~uters
t hem. Engn g ,
ol. 14, No. 4/S, pp. 415426, 1990
Printed in Great Britain. All rights reserved
0098-I 354/90 $3.00 + 0.00
Copyright (0 1990 Pergamon Press plc
DESIGN OF P AND PI STABILIZING CONTROLLERS
FOR QUASI LINEAR SYSTEMS
J.-P.
CALVET
and Y.
ARKUN~
School of Chemical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, U.S.A.
(Received 23 Oc tobe r 1989; rece ived
or
pubk a r ion 29 Novembe r 1989 )
Abstract-The systems studied in this paper are nonlinear systems perturbed by disturbances which are
feedback transformable to quasi-linear systems [i.e. i = AZ + Bv + [ ( r ) d with A, B) controllable]. We
consider the problem of designing stabilizing controllers for perturbed nonlinear systems through their
equivalent quasi-linear systems. With the addition of integral action, we can guarantee not only ultimate
d-stabilization but also zero steady state offset for both the output of the quasi-linear system u = Cz)
and for the equivalent output [y = h x)] of the nonlinear system. Moreover, when the so-called
disturbance matching condition is satisfied, it is shown that all the states of the quasi-linear system and
the nonlinear system) will exhibit zero steady state offset. All the results presented here are for single
control input systems.
1. INTRODUCTION
In recent years, the use of differential geometry to
transform nonlinear systems into linear systems has
received much attention
in the control literature.
Several methods compete and differ in their defini-
tions and the issues they address. One approach is to
transform a nonlinear system in an input/output
sense. This was first investigated by Gilbert and Ha
(1984) and Ha and Gilbert (1987). Another approach
is to transform the nonlinear state equations into an
equivalent linear system without specifying any out-
put. This technique is known as the feedback linear-
ization and has been investigated in particular by Su
(1982) and Hunt er al. (1983). The transformations
required are state and input transformations with a
nonlinear state feedback. Finally, feedback lineariz-
ation with decoupling of outputs was addressed by
Isidori et al. (1981). Various control applications
have appeared in aerospace engineering (Meyer et al.,
1984), robotics (Tarn et al., 1984), power systems
(Marina, 1984) and chemical engineering (Hoo and
Kantor, 1987; Kravaris and Chung, 1987; Calvet and
Arkun, 1988a).
It should be noted that there is very limited knowl-
edge about the robustness characteristics of these
techniques with respect to modeling errors and dis-
turbances. For example, the pioneering work of
Kravaris and Palanki (1988a, b) shows that under
matching conditions it is possible t o design robust
controllers for a transformed (in the input/output
sense) nonlinear system with modeling errors. How-
ever, the design of controllers guaranteeing stability
of the transformed nonlinear system influenced by
modeled disturbances has not yet been investigated.
Our recent work (Calvet and Arkun, 1988a, b) has
___
~__
tTo whom all correspondence should be addressed.
shown that under feedback linearization a perturbed
nonlinear system is in general no longer transformed
to a linear system but to a so-called quasi-linear
system, QLS. The QLS is then affected by nonlinear-
ities only due to a state-dependent perturbation
coupled with the modeled disturbance. In this paper
we address the following problem:
Given a bound on the set of disturbances affect-
ing the nonlinear system, we want to design a
proportional (P) controller for the QLS which will
ultimately stabilize the original nonlinear system.
Furthermore, if an output of the non-linear system
is specified, we want to design a stabilizing propor-
tional integral (PI) controller for the QLS which
will guarantee zero steady state offset for the
output.
The paper is organized as follows. The origin of QLSs
is given in Section 2. In Section 3, a theorem gives the
sufficient condition for the design of (P) stabilizing
gains. The extension to PI stabilization is addressed
in Section 4. Because of the inherent conservation due
to the sufficiency of the theorem, a design procedure
to compute the least conservative stabilizing gains
possible is given in Section 5. In Section 6, a steady
state analysis shows that zero steady state offset can
be guaranteed for
a l l
the states of the original non-
linear system under some disturbance matching con-
ditions. Finally, in Section 7 we apply the design
procedure to control an unstable reactor influenced
by various process disturbances.
2. QUASI-LINEAR SYSTEM
Definition I-A quasi-linear system (QLS) is a
system which is linear or affine with respect to
its control input and the modeled disturbances.
415
-
7/25/2019 design of p and pi controllers for quasi linear systems
2/12
416
J.-P.
CALVE?
and Y.
ARKUN
It is described by:
i =Az+Bv+
z d,
1)
where z E
R
are the states, v E
R
is the control input,
d E RP
is a set of modeled disturbances and c(z) is an
n x p matrix with sm ooth scalar entries. The pair
(A , B)
is a controllable matrix pair. It is usually given
in the Brunovsky canonical form (BCF):
R
It is important to realize that a QLS has the nice
property to be truly linear w hen free of disturbances
(d = 0) .
Also, note that the perturbation term
c(z )d
is
s ta te
dependen t and not pa rame te r dependen t [i.e.
I I < D V t > t o } ,
where /I. 11 h
S t e usual Euclidean norm and
D
is the
know n bound. Note that the set of disturbances could
be time varying but should be bounded at all time by
D .
We now giv e the definitions of &-stabilization
(Schmitendorf, 1988) and ultimate bound edness
(Corless and Leitman n, 1981) as adapted for QLS.
efinition
2-A solution z(-):[t,,
co]+R, z(t,) = z,,
(initial condition) of a QLS is said to be ultimately
bound ed with respect to a closed set
B(S) =
{z E R; ) l z [ I
-
7/25/2019 design of p and pi controllers for quasi linear systems
3/12
Design of P and PI
This assumption is the most crucial one but can
always be satisfied in closed balls B(F) for a finite F.
However, if P tends to infinity, sometimes p and/or cc
may not be finite implying that Assumption 2 will not
be satisfied. In other words, this means that Assump-
tion 2 can always be satisfied locally [in B(6)] but not
globaly (when i-00). This choice of F is thus very
important and involves searching for the pair (p, p)
not in the whole space R but in an adequate ball B(F)
which must be specified judiciously. Also, this as-
sumption introduces some inherent conservatism in
the design of 6 -stabilizing controllers. This point will
be discussed and illustrated later.
We now give an important theorem where the
following notation is used: an eigenvalue of a square
matrix W is denoted by 1( W). If
W
is strictly positive
definite, then 1 *(W) and 1 *(W) are its maximum
and minimum
eigenvalues,
respectively. Also
yw = E, (
W)/ , l * ( W
is its condition number.
Theorem-Given 6 such that J 2 6 B 0; if there
exists a P > 0 (strictly positive definite) solving the
algebraic Riccati equation (ARE):
PA + Ar P - PB(2R- - I )B TP + H /E =O,
for some 6 > 0,
N=HT )O
and
R=Rr>O
satisfying
with i , (H ) - I_c~D%~ > 0, then the control law:
v(l)= -R- B=Pz( t ) ,
(7 )
is a a-stabilizing control for the QLS, for all initial
conditions in
B ( r / & )
with r -c F.
Proof-See Calvet and Arkun (1989). The proof of
the theorem is based on the inspection of the sign of
the derivative of a candidate Lyapunov function
V(z) = zTPz; where P is the solution of an algebraic
Riccati equation whose form is different depending
on whether or not the disturbance matching condi-
tion is satisfied.
Note that when the QLS is a single control input
system, the parameter R is a scalar. Assumption 1 is
a disturbance matching condition similar to the one
given in Corless and Leitmann (1981). However, this
assumption can be relaxed. Indeed, if Assumption 1
is not satisfied, one must verify the following assump-
tion instead of Assumption 2:
Assump t i o n ?-The perturbation term itself must
be bounded in a ball B(F) :
3 (p , p ) E R2 (both finite) s.t.
lli(z)lj2
T* .
(8 )
4 .1 . Selec t ion o f the ou tpu t
It is known that in general an a p r i o r i specified
output of the original nonlinear system (2):
Y(l) = h]x(r)l,
(9)
will not depend linearly on the transformed states of
the QLS, z,, , z,.
This has been recognized as a
disadvantage of the feedback linearization transfor-
mation (Kravaris and Chung, 1987). In the proposed
method, however, the output of the QLS must be a
linear combination of the states as:
y(t)=Cz(t)
CER~.
(10)
We also consider that the output C[z(t)] will render
the unforced QLS observable. Hence without loss of
generality, we can set the output of the QLS to be
Z, ,
i.e. C = [I, 0, . . , 01. Indeed, the unforced QLS can
be easily transformed to such a form through linear
transformation and feedback (Kailath, 1980).
The critical requirements for the output of the QLS
to satisfy simultaneously (9) and (10) can be tackled
in different ways:
Given a desired output (9) for the original
nonlinear system, choose a manipulated vari-
able u which will admit feedback transformation
(4, 5) that gives a QLS with a linear output (10).
This is called an a p r i o r i output selection.
Apply the feedback linearization (4, 5) and then
select a linear output of the QLS (IO) which has
a physical meaning in the original nonlinear
system. This is called an a pos te r io r i output
selection.
If these two approaches fail, one should resort to
partial linearization (Krener et a l . , 1984) or input/
output linearization (Kravaris and Chung, 1987)
which will not be addressed here. The option of
selecting an output a p r i o r i (before applying the
feedback transformation) or Q po s t e r i o r i (after) has
been investigated in Kantor (1986) and applied by the
authors in Calvet and Arkun (1988a). In the reactor
application here, the selection of the output will be
done according to the procedure given above.
-
7/25/2019 design of p and pi controllers for quasi linear systems
4/12
418
J. P. CALVET and Y. ARKIJN
4 .2 . Th e fo rm u l a t i on o f the con t r o l l er design p rob lem
The design problem we investigate is then the
following:
Given the bound D of the set of disturbances, and
specifying 8, construct a control law which will:
1. a-stabilize the QLS (i.e. the norm of the states
will be ultimately bounded by 6).
2. Guarantee zero steady state offset of the output
of the QLS [under ultimately time invariant
perturbations (8)].
It is also important to realize that a-stability for a
QLS will automatically guarantee stability for the
original nonlinear system in a closed contour called
a 6-contour. This contour is obtained through the
inverse transformation [X = T-(z)] of the ball B(6).
Therefore, if stability of the original nonlinear system
is addressed through its equivalent QLS, the choice of
6 should be done judiciously so that it corresponds to
a desired region of stability, the 8-contour for the
original states x [see an application in Calvet and
Arkun (1989)].
The approach we suggest is to augment the dimen-
sion of the QLS and apply the P stabilization results
given in the previous section. First, we introduce an
additional state as:
z,+,(t) =
5
z,( t)dz.
(11)
10
Next, we obtain an augmented QLS described by:
t = ;ir + Bv + c(z )d ,
(12)
with i = [zl..
. . .z,, z,,,]~.
and
i
0 1
. . . 0 0
:
: . :
_ij= (j (j
.I. ; 0
ERf+I,Xc+I>
0 0
. . . 0 0
I 0
. . 0 0
I
and
Finally, the new control law (PI controller) can be
constructed:
v(t) = -&Z(t) = -[K,, . . . , K,, K,+, ]g ( t )
,=n
c
= - c Kjz i ( t ) -K ,+ ,
z , ( t )d T,
(13)
r-1
.I@
where (K, , . . . , K,) and K,+,
are, respectively, the
proportional and integral gains.
The theorem given for P stabilization applies to PI
stabilization as well after replacing A,
B, c(z )
by
1, B, c(z). For example, if the disturbance matching
condition is satisfied by the augmented system (12);
then, one will have to solve the following algebraic
Riccati equation ARE:
P; i + ; i=P - PB (2R - - Z)z P + H /e2 = 0.
Otherwise the following Riccati equation ARE must
be solved:
Pa + ;i=P - P(2BR - BT - Z)P + H Jc2 = 0 .
5. DESIGN PROCEDURE
In this section, we give guidelines which will ease
the search of the b-stabilizing gains. This procedure
applies for both P and PI stabilization of a QLS. It
is a graphical approach and is not analytical mainly
because a closed expression for the condition number
yp as a function of R, H and L is not available when
one solves the above algebraic Riccati equations.
5.1.
Sol v i n g ARE and ARE
One can see that the gains of the s-stabilizing
control K = R - BTP do not depend explicitly on the
constants p and ZJ ntroduced in Assumption 2 (As-
sumptions 2 or 2) but rather on the parameters t,
H
and R of the algebraic Riccati equation. Assumption
1 will determine which Riccati equation needs to be
solved. Then all the possible gains can be computed
as a function of E (parametrized by R and H ) and
independently of p and p. In other words, Assump-
tion 2 need not to be checked before computing the
gains.
If the disturbance matching condition (Assumption
1) is satisfied, then, there always exists a
un ique P > 0
solving the ARE for all H = H > 0 , R = RT with
2R -- - Z > 0 and t > 0. Indeed, one can recognize
that in this case the Riccati equation is similar to the
one encountered in the LQR optimal control prob-
lem (Kwakemaak and Sivan, 1972). As mentioned
earlier, the matrix pair (A,
B )
is usually in the BCF.
As an example, for a 2-D BCF and for R = 1 and
H = Z , we can plot the gains R - BT P as a function
of L as shown in Fig. 1. Also, in Fig. 2, we give the
gains obtained when the QLS is augmented with the
integral action. Note that we can obtain the gains for
all values of es, and they tend to infinity as L tends
to zero.
However, if the disturbance matching condition is
not satisfied, we may not be able to solve the ARE
for all the values of the parameters t, H and R .
Indeed, we can show that for a given H and R there
exists a value of e, say clim elow which ARE does not
have a positive solution P . The value of +,,, depends
on the parameters H and R . In Fig. 3, we give the
-
7/25/2019 design of p and pi controllers for quasi linear systems
5/12
Design of P and PI stabilizingcontrollers
419
Fig. 1. Stabilizing gains from the ARE (for Q S with
disturbancematchingcondition).
gains we calculated for a 2-D BCF for n = I and
R = 0.1.
Also, in Fig. 4 are the gains for the aug-
mented QLS. The parameter R = 0.1 was chosen
because it gives the lowest value of c for which ARE
has a positive solution P with N fixed to the identity
matrix (the value of R was obtained by trial and
error). In comparison with the gains obtained from
the ARE, these gains tend to infinity as t tends to cltm.
Below Ellrn
o gains can be computed.
Clearly, it is more attractive to solve the classical
algebraic Riccati equation (ARE) than the ARE.
Therefore the matching condition (Assumption 1) is
a desired property. In the case that Assumption 1 is
not satisfied one may wonder if either a similarity
transformation (change of state coordinate a = Qz)
or another suitable difleomorphism [z = T(x)]
would render Assumption 1 satisfied. However, this
is not true as one can easily show that the matching
condition Assumption 1 is independent of the
200,
Fig. 2. Stabilizing ains from the ARE (for augmented QLS
Fig. 4. Stabilizing gains from the ARE (for augmented
with disturbance matching condition).
QLS without disturbance matching condition).
0;
A 1
1
0 urn 2 4
s 8
EPS
Fig. 3. Stabilizing gains from the ARE (for QLS without
disturbance matching condition).
diffeomorphism and the choice of coordinate system
in the z-domain.
5.2. Ver i f v i n g Assump t i o n 2
Assumption 2 is the most crucial assumption. In
fact, this assumption is the only one dealing with the
nonlinearities of the QLS. It quantifies in a way the
magnitude of the nonlinear state-dependent per-
turbation through the values of the two constants p
and p. Therefore, the challenge in our design method
is not to solve the algebraic Riccati equation (ARE
or ARE) which is usually an easy task; but rather to
find the constants p and p which are obviously not
unique and are problem dependent. In the applica-
tion we will refer to an algorithm which computes
these constants.
5.3. Conse rva t i sm
In general, assuming that the bound of the distur-
bances D is available and the desired region of
of
I
2
4
EL
2
-
7/25/2019 design of p and pi controllers for quasi linear systems
6/12
420
J. P. CALVET and Y. ARKUN
8 values
E
Fig. 5. Graphical procedure to select the least conservativegains(with disturbancematchingcondition).
stabilization (i.e. 6) is specified, the proposed method
will give higher stabilizing gains than it is necessary.
Hence, given two pairs of constants @,, p,) and
&, p2) satisfying Assumption 2, we will say that one
pair gives less conservative results, if for a given set
of parameters t, n and
R ,
it gives smaller S-stabiliz-
ation gains than the other pair. We then have the
following result that will become clearer later on. If:
a,:.,,(~, H, R) < ap:,,,z(~ H, R),
where 6* is given by (6); then, the pair @, , p,) will
give less conservative (smaller) gains than b2, pr). In
other words (for H and R fixed) if the curve SF,,,, vs
.E s below the curve 6,:. p2; then, the pair @, , p, ) will
be preferred. The curve 6 * vs will have the following
shape:
.
.
5.4.
If the disturbance matching condition is satis-
fied, the 6* curve is strictly increasing with 6.
Also as L tends to zero, S* tends to zero as well
(see the schematic plots in Fig. 5).
If the disturbance matching condition is not
satisfied; then, the S* curve has a minimum
(3E s.t. as*/&),_,=
0) and the 6 curve goes to
infinity as c tends to tlim see schematic plots in
Fig. 6).
Procedu re
In light of the above results, we give two different
procedures depending on whether or not the
disturbance matching condition is satisfied. These
procedures help the designer to get the smallest
stabilizing gains once 6 and D are specified. The gains
obtained will guarantee that the QLS will be ulti-
mately stable in B(8) , and with no steady sate offset
for the output if the QLS is augmented.
With matching condition:
This procedure is schematically illustrated in Fig. 5
where each step number is circled.
1.
2.
3.
4.
5.
Plot the gain curve (K vs C) with usually Ei = I
and
R = 1.
Find @&, , past) so that a&(e) is the lowest
possible (this will avoid conservatism).
Locate S and pick 6 * so that S r 6 * (as required
by the theorem).
Get E (from 6* vs e curve).
Get the stabilizing gains K (from K vs L curves).
One can see in Fig. 5 that for 6 given, we get smaller
gains as 6* tends to 6. Also it is now clear that if
another pair @, JJ) gives a 6 * curve above the one
depicted in Fig. 5 it will give higher gains. Hence Step
2 in the procedure is the most important step which
avoids conservatism. It is important to notice that as
6 tends to zero, then 5* must tend to zero and
henceforth L. The stabilizing gains will then all tend
to infinity. As a result, asymptotic stability of all the
states of the QLS can be guaranteed as K tends to
infinity. This important result is not true when the
disturbance matching condition is not satisfied.
-
7/25/2019 design of p and pi controllers for quasi linear systems
7/12
Design of P and PI stabilizing controllers
421
gains
Fig. 6. Graphical procedure to select the least conservative gains (without disturbance matching
condition).
Also for P controllers only, as c increases, K i
decreases and the amplitude of the overshoot (if it
exists) and the steady state offset will increase. This
can be seen from Fig. 1. Since the offset is related to
S, the steady state offset will decrease as e decreases
(the K i s increase). Also, the addition of integral
action in the control law will increase the values of the
gains. A comparison of Figs 1 and 2 shows that for
a given c, higher gains are obtained with the use of
integral action.
Without matching condition:
This procedure is schematically illustrated in Fig. 6.
1.
2.
3.
4.
5.
6.
Plot the gain curves K vs 6 ) with usually N = I
and
R
is tuned so that slim is the smallest
possible.
Find (Pbeat,~,_,) so that 6 ,(s) is the lowest
possible (thus will avoid conservatism).
Obtain 6 zin
as*
[from 3C s.t. X ~_~
= 0 then S&, = S*(C)].
Select 6* > ~5% and 6 > 6* from the theorem).
Get L (from 6* vs c curve).
Get the stabilizing gains K (from K vs 6 curves).
Hence, for all S > 6 * z=- tin such gains are 6 -stabiliz-
able. Once again, when 6* tends to S, the gains will
be smaller. Also the existence of ~5% shows that 6
cannot be arbitrarily chosen as small as we wish. AS
a result asymptotic stability of all the states of the
QLS cannot be guaranteed contrary to results with
matching conditions.
Also, if the disturbance matching condition is
satisfied but is not detected by the designer, stabihz-
ing controller gains can still be obtained through the
procedure without the matching condition, but in
general, the results will be more conservative.
6. STEADY STATE ANALYSIS OF PI STABILIZATION
The reason we introduce integral action in the
control law is to achieve zero steady state offset of the
controlled output under ultimately time invariant
disturbances. In this section, we show that when the
disturbance matching condition is satisfied, the con-
trol law will lead to zero steady state offset, not only
for the output but also for the other states of the
QLS. Equivalently, by virtue of the state transform-
ation [the diffeomorphism (4)], this means that UN the
states of the original nonlinear system will exhibit
zero steady state offset. We illustrate this by perform-
ing a steady state analysis. With the control law:
,=1
s
aa = - c Kiz,(t) - Km+,
21 (rW7,
r-l
0
the augmented closed loop QLS is given by
t=af+r:
-
7/25/2019 design of p and pi controllers for quasi linear systems
8/12
422
J . P.
CALVET and Y. ARKUN
with
7. APPLICATION
As an illustration, we first show that the standard
CSTR model perturbed by fluctuations in the feed
temperature and the feed concentration falls into the
category of a nonlinear system described by (2, 3) and
can be transformed into a QLS by feedback lineariza-
tion. We will then stabilize the CSTR under the
influence of these disturbances using PI control. We
will make use of the plots of the stabilizing gains
obtained in Section 5.1 and then follow the proce-
dures given in Section 5.4. The only knowledge
required about the set of (ultimately time invariant)
disturbance(s) will be the bounds.
7.1.
T r a n s f or m a f i o n o f a
CSTR model i n t o a QLS
The dimensionless model of a first-order exo-
thermic irreversible reaction taking place in a CSTR
is given by Uppal ef a l . (1976) and Ray (1981):
zp+ 2 &_,&-)d,=O k -2,. . ..n.
i=l
i=n
i=p
K.zPq -
K
1 1
n+ ,zz+ +
C
LiWM = 0,
i=,
zp=o.
(14)
Since the disturbances are all ultimately time invari-
ant, all the terms df , i = 1 , . . . ,p are considered
constant in the steady state analysis. Note that the
last equation is a natural consequence of the fact that
the control was designed to guarantee zero steady
state offset of the output
z, ,
The above equations do
not simplify further. In particular z: for i = 2, . . . , n
cannot be obtained explicitly in terms of the
disturbances d is . However, if we assume that c(z )d
satisfies the disturbance matching condition with
respect to B as:
3~ (z) E R xp s.t. g ). (17)
where (
_ .)
denotes the inner product. Note also
that this state space coordinate transformation maps
xP to the origin in the z-state space [i.e. T(xP) = 01.
-
7/25/2019 design of p and pi controllers for quasi linear systems
9/12
Design of P and PI stabilizingcontrollers
423
Then under these transformations, the CSTR model
is transformed to the QLS:
and the algebraic Riccati equation to solve is the
ARE where we picked n = I and R = I ( see gain
plots in Fig. 2).
In order to verify Assumption 2, we inspect the
inequality in the ball B ( f = 0.2). Among all the
with
where the inverse transformation x = T- (z ) is ob-
tained through the one-to-one mapping of the state
transformation:
x, = Z, + x;
[
2, +
z, + x;)p
x2 = v In
Da(1 -z, - ~7)
II
[
n
z2 ,xy
Da(1 -z, -x9)
I>
20)
Remark-We will consider X, (i.e. the dimension-
less composition) to be the output of the nonlinear
system. According to the state transformation (16)
such output corresponds to z,. Hence, the required
conditions (9) and (10) from Section 4.1 are satisfied
simultaneously and PI stabilization can he applied
with z, as output. However, if we would have chosen
x2 as output for the CSTR, then the residence time
should have been the new manipulated variable r en -
dering the nonlinear system transformable to a QLS
with an output z, corresponding to the dimensionless
temperature. Here, the flexibility to choose a new
manipulated variable as mentioned in Section 4.1
indeed exists.
7.2. P I s t a b i l i z a t i o n unde r f eed t em pe r a t u r e p e r t u r b a -
t i o n s d , = 0 )
Consider that the CSTR is subject to feed temper-
ature perturbations only. With a PI stabilizing con-
troller we then have the augmented QLS described
by:
-1
[
II
2
1 -Daexp ~
1 +x2/v
X T- 1(Z)
(19)
possible pairs @, p) satisfying Assumption 2, the pair
giving the lowest 6* curve is:
(PM, C(M) = (0.45, 1.51).
An algorithm to compute such a pair was developed
by the authors and is available in Calvet and Arkun
(1989). a:__,,.,,
curves as a function of L and
parametrized by various bounds D of the disturbance
d , are displayed in Fig. 7. According to the procedure
(with matching condition) we can now compute the
stabilizing gains of the PI controller. Let d = 0.2 and
6* = 0.199; also we consider ultimately time invariant
disturbance bounded by
D =
0.3. The a-stabilizing
gains are then:
[K , , KZ, K,] = [4.93,3.93,2.37].
A simulation in Fig. 8 with these gains in the control
law show that as expected b o t h states of the CSTR
exhibit zero steady offset under a step disturbance of
d, t ) = 0.3. As a performance criterion we can also
obtain the integral below the curve z, (t) = x,(t) - xyp
vs time. Indeed according to the construction of the
augmented QLS (1 I, 12) and the steady state equa-
tions (15) we have:
z? = lim
C
z, IT) d r = x od , = 0.048
I-CC
Jo ' JG
Remark Note that under the condition of distur-
bance matching condition, it is not necessary to
and the control law is U(Z) = -X:1: K i z i t ) where
(K,,
K2 )
and
K 3
are, respectively, the proportional
specify an a p r i o r i output of the nonlinear system (i.e.
and integral gains to be determined. In this case, the
here dimensionless composition or temperature).
state-dependent perturbation term satisfies the distur-
Indeed, integral action on z, {no matter what its
bance matching condition with:
relationship with the original nonlinear system may
l
x) Da exp[*]]_ r_,(Z1
mean) will guarantee zero steady state offset of all the
X(z)= (1 +&Iv>2
states of the QLS and henceforth of the states of the
original nonlinear system (i.e. here the CSTR).
-
7/25/2019 design of p and pi controllers for quasi linear systems
10/12
424
J.-P. CAL~ET and Y.
ARKUN
0.24
-
0.20
-
0.16
-
*
00 0.12
-
0.06
-
EPS
Fig. 7. 6 * curves vs L parametrized by D (with disturbance
matching condition).
7.3. PI st a b i l i z a t i o n unde r f eed composi t i o n per t u r b a -
t i ons (d , = 0)
If we consider feed composition perturbation only,
then, with a PI stabilizing controller, the augmented
QLS is described by:
0.520
0.515
OH0
z
z
0.505
k
i
3.06.10.06.16.14.12 -
c
x
a495 I 1
I 1
0 2 4
6
3D2
a
Time
Fig. 8. PI stabilization (with disturbance matching condi-
tion).
admissible value of 0.006. Then the stabilizing gains
are:
[K,, K,, &] =
[2.44,
1.86, 1.21.
and the control law is again u(t)= -%I: K,z,(t) .
One can easily see that the disturbance matching
condition is not satisfied. Then the algebraic Riccati
equation to solve is ARE where we picked n = I and
R = 0.1 (see gain plots in Fig. 4). With the algorithm
given in Calvet and Arkun (1989) we get the pair
(p. p) satisfying Assumption 2 in B (P = 0.2) that
gives the lowest 6* curve (for n = I and R = 0.1) vs
L. The pair is:
(P
bcs,,p~brrt) (2.4, 13.22).
dFbt._ curves as a function of L and parametrized by
various bounds
D
of the disturbance
d 2
are displayed
in Fig. 9. One can see that, as a result of the absence
of disturbance matching condition, all the 6* curves
parametrized by
D
have a minimum
S& (D ) .
There-
fore, 6 cannot be as small as we may wish. For
example, a-stabilization with 6 = 0.2 cannot be guar-
anteed for disturbance having a bound larger than
0.0065. This can be seen in Fig. 10 where we plotted
S ,& (D ) as a function of D . Indeed, such curve
gives regions where a-stabilization can or cannot be
implemented.
According to the procedure (without matching
condition), we can now compute the stabilizing gains
of the PI controller. Let d = 0.2 and 6 = 0.199. As
a bound D for the disturbance dZ we picked an
A simulation in Fig. 11 with these gains imple-
mented in the control law shows that, as expected,
on ly
the output of the QLS i.e. z, will exhibit zero
steady state offset. However, the other state z2 will
exhibit a steady state offset.
By virtue of the
diffeomorphism T , this corresponds to zero steady
a30
*
00
t
0.25
0.20
0.15
I
0 10
I
I I
I
I I
1 2 3
Ek
5 6 7
Fig. 9. 6 curves vs c parametrized by D (without distur-
bance matching condition).
-
7/25/2019 design of p and pi controllers for quasi linear systems
11/12
Design of P and PI stabilizing controllers
425
0 .30 -
0 .25 -
8
-stabilization
posaibls
0.20 -
0.15 -
8 stabilization
0.10 not possible
0.00 1
I I
I I IILLI
0.001
0 01
D
Fig. 10. Region of d-stability (without matching condition).
state offset for the dimensionless composition x,
and a steady state offset for x2 the dimensionless
temperature.
8. CONCLUSION
The theory and application of the P and PI stabiliz-
ation of quasi-linear systems (QLS) is presented. The
origin and practical importance of QLSs is intro-
duced, and the concept of ultimate boundedness and
&-stabilization is adapted for such systems. A
theorem and a procedure are given to compute the
stabilizing gains in the least conservative sense for the
proposed methodology. The results permit to stabil-
ize the class of nonlinear systems with bounded
disturbances which are feedback transformable to
QLSs. If the so-called disturbance matching con-
dition is satisfied, we show that PI stabilization will
guarantee zero steady state offset, not only for the
output but also for all the other states of the QLS
(and henceforth for the original nonlinear system).
Simulation results on an open-loop unstable (and
perturbed) CSTR mode1 illustrate and agree with the
theory.
NOMENCLATURE
y = h(x) = (Single) output of a system
R = Set of real numbers
d E RP = Disturbance vector
R = Set of real n vectors
I / x I =
1. = Absolute value of elements in R
24= (Single) control input of a system
Euclidean
Xl_ ,
xf
(usually nonlinear)
norm for x E R , I I x I I =
R
m =
All n x m real matrices
I E R x = Identity matrix
x E
R =
States of a system (usually nonlinear)
z E
R =
States of a system (usually linear or
quasi-linear)
zi= (Single) control input of a system
(usually linear or quasi-linear)
K E R = Stabilizing gains
Fig. Il. PI stabilization (without disturbance matching
condition).
(A, B) = A controllable matrix pair (usually the
BCF)
f(x), g(x) = Smooth vector fields in R (infinitely
differentiable i.e. C)
Y(x) E WXp = Perturbation matrix associated with the
disturbances
z = T(x) = A
nonlinear one-to-one mapping
(diffeomorphism)
dT / a x =
Jacobian matrix of T
u = S(.X, U) = A (single) input nonlinear transform-
ation with state feedback
B(6) = Ball of radius 6
r, P = Real positive constants denoting the
radii of balls E(r), B(T)
6,6* = Real positive constants denoting the
radii of balls B(d), B d * )
p . p = Real positive constants satisfying As-
sumption 2
D =
Real positive number, bound of the
disturbance(s)
R, c
RP =
Set of disturbances
c(z) = aT/a x Y x) = Perturbation of the QLS
X(Z)ER
rp = Row vector satisfying Assumption I
L = Real positive number (parameter of the
ARE)
R, W = Matrices of the ARE (parameters)
,I(~) = Eigenvalue of a square matrix
,I*(.), A* (~) = Maximum and minimum eigenvalues
of a positive definite matrix
v(.) = 1 *(-)/A * (.) = Condition number
P z 0 =
Solution of an algebraic Riccati equa-
tion (ARE or ARE)
Abb re v i a t i o n s
ARE = Algebraic Riccati equation
BCF = Brunovski canonical form
QLS = Quasi-linear system
P = Proportional
PI = Proportional integral
REFERENCES
Calvet J.-P. and Y. Arkun, Feedforward and feedback
linearization of nonlinear systems and its implementation
Calvet J.-P. and Y. Arkun, Feedforward and feedback
using IMC. Znd . Engng Chem. Rex . 27, 1822-1831
linearization of nonlinear systems with disturbances. Zn t .
(1988a).
J . Con t r o l 4 8 , 1551~1559 (1988b).
-
7/25/2019 design of p and pi controllers for quasi linear systems
12/12