Symposium on Modelling, Analysis and Simulation

e-
•
CESA'96 IMACS Multiconference.

Computational Engineering in Systems Applications

Lille - France, July 9-12, 1996

Volume 10'2
 CESA'96 IMACS Multiconference

General Chair: P. BORNE (FRANCE)

Secretary: S. EL KHATTABI (FRANCE)

Steering Committee:

J. DESCUSSE (FRANCE) V.M. MATROSOV (RUSSIA)

A. FOSSARD (FRANCE) A. SAGE (USA)
T. FUKUDA GAPAN) M.G. SINGH (ENGLA1'ID)
W. GRUVER (CANADA) A. SYDOW (GERMANY)
M. JAMSHIDI (USA) J. TIEN (u.s.A.)
T. KACZOREK (POLAND) S. TZAFESTAS (GREECE)
V. LAKSHMlKANTHAM (U.S.A.) R. VICHNEVETSKY (USA)
G.Q. LI (CHINA) C. WHITE (USA)

National Organizing Committee:

Chair: P. SERNICLAY (FRANCE)

Members:
A. AZMANI L. DUBOIS W.KORBAA
P.BORNE S. EL KHATT ABI P.KUBIAK
G. CUIENGNET J.GARCIA H.OHL
A.M. DESODT S.HAYAT J.ROZINOER
F.DELMOTTE A.KAMEL A. SCHMITT
J.Y. DIEULOT V.KONCAR X.ZENG
 LINEARIZATION OF INPUT-OUTPUT MAPPING
FOR NONLINEAR DELAY SYSTEMS
VIA STATIC STATE FEEDBACK

A. Germani, C. Manes, P. Pepe

Dipartimento di Ingegneria Elettrica
Universita. degli Studi dell' Aquila
Monteluco di Roio 67040 L'Aquila - ITALY
Fax. ++39 - 862 - 434403
e-mail: manes@dsiaql.ing.univaq.it
ABSTRACT
dimensionality of the state and nonlinear structure of
The problem of controlling systems described by dif- the differential equations [11].
ferential equatwns with delay in the state is investi- In this paper we show that, through the introduc-
gated in this paper. The main difficulty in dealing tion of a suitable mathematical formalism, these diffi-
with such delay systems is that the state should be culties can be overcome for an interesting class of non-
properly considered in an infinite dimensional space. linear delay systems. In a sense this class resembles
While linear delay systems have been extensively stud- the class of those nonlinear systems, without delay,
ied in literature, few attention has been devoted to having relative degree and stable zero dynamics.
nonlinear delay systems, due to the mathematical dif- The paper is organized as follows. In the section
ficulties derived by the infinite dimensionality of the 2 the appropriate notations and definitions are intro-
state and by the nonlinear structure of the differential duced. In section 3 the problem of finding the feed-
equations. In this paper, through the use of a suitable back control law which linearizes the input-output
formalism, we show a solution of the problem of the mapping and "removes the effect of the delay is for-
input-output linearization via static-state feedback for mulated and solved. Finally, in section 4, an example
an interesting class of nonlinear delay systems. Sim- of application is worked out and simulation results
ulation results on an unstable nonlinear delay system are presented.
are reported.
2. PRELIMINARIES
1. INTRODUCTION
The control system under investigation is described
In many applications the process to be controlled is by the following equation
described by differential equations with delay in the
state. This happens, e.g., in many ecological systems, i(t) = f(x(t), x(t - ~»+
industrial processes, in telerobotics, in earth control g(x(t),x(t - ~»u(t), (2.1)
of satellite devices and other significant applications. y(t) = h(x(t», t ~ 0, (2.2)
The main difficulty in dealing with delay systems is
that the state should be properly considered in an where x(i) E IR", u(t) E JR and yet) E JR, the vector
infinite dimensional space. functions f and 9 are Coo with respect to both ar-
The control problem for the case of linear delay guments, and h is a Coo scalar function. The model
systems, both in deterministic and stochastic setting, description is completed by the knowledge of the func-
has been extensively studied and is still under in- tion x(r), r E [-~,O] (initial state).
vestigation. General solutions have been provided In order to deal with compact expressions, a suit-
for some classes of control problems (see e.g. [3-5,7- able notation is needed, and therefore the following
10,13-15]). definitions are required. Let
On the contrary in literature few attention has
been devoted to the problem of nonlinear delay sys-
tems control, and no general solution is so far avail-
x(ix(t)
- .6.) 1 u(tu(t)
-~) 1
XA;(t) = [ : I UA;(t) = [ : '
able. It is evident the enormous mathematical diffi-
culty derived by the simultaneous presence of infinite x(t - k~) u(t - k.o.)
(2.3)
This work is supported by MiniJtero dell'Universita e della be the extended vectors containing delayed state and
Ricerca Scientijioo e Tecnologica. input until time instant t - k.6., with k ~ O. Quite

599
obviously X.i;(t) is defined for t ~ (.I: -1).6. and U.i;(t) It is easy to verify that for a system having delay
for t ~ k.6.. relative degree equal to r the time derivative of the
Moreover, for k ~ 1 let us define output until order r - 1 can be written as

!(x(t),x(t -.6.))
!(x(t - .6.), x(t - 2.6.»)
1 k=0,1, ... ,r-1,
F.i;-l(Xk) =[ . , (2.12)
while the r-th output derivative can be expressed as
!(x(t - (k - 1)~), x(t - k.6.)))
(2.4) y(r){t) = L~lh(Xr(t» + LG._ 1 L~-llh(Xr(t)Ur_l(t),
and the block-diagonal matrix (2.13)

G I: -1 (XI:) = diag [
g(x(t), x(t - .6.»)
:
1 .
or, taking into account the decomposition (2.10),

y(r)(t) = L~]h(X,.(t» + L,L~-l)h(Xr(t»U(t)+

_ g(x(t - (k -1).6.),x(t - k.6.»)
(2.5) + rn,. (X,. (t), u(t - .6.}, ... , u(t - (r - 1}.6.) ).
With these positions one has, for any k ~ I, (2.14)
XI:-l(t) = FI:_l(XI:(t» + G"_1(XI:(t»U"_1(t). Remark 2.2. If the usual definition of Lie deriva-
(2.6) tive is preferred, an integer p should be chosen so
In order to write synthetically the higher order time that Lie derivatives ofthe output function h(x), con-
derivatives of the output yet) it is convenient to define sidered as a function of X p, along Fp can be computed
the following chain of directional derivatives of h as

L}.h(Xp} = (d~p Lt 1h (Xp») Fp(Xp+1)' (2.15)

(2.7)
L~.h(Xp) =h(Xp). k =1,2, ... ,p,
and also
Note that L} h(Xp} is actually a function of X".
The derivativ: of this term along the directions of
the columns of Gp gives

Moreover, introducing the notation

LG,L},h{Xp) = (d~p L}.h(Xp») Gp(X p+1),
k = 1,2, ... ,p-1.
LgL,[1:-1] heX,,) = (d
dX o L,["-1] h(XI:-d ) g(Xd, (2.16)
Note that L~~lh and LGpL~.h depend on X p+1'
(2.9) With these definitions it can be stated that a non-
the following decomposition can be defined linear delay system of the form (2.1) (2.2) has relative
degree r at X,. if in a neighborhood one has
LGk _1 L)"-l]h(X,,(t»U"_l (t) =
=LgL)"-1)h(XI:(t»u(t)+ LG._ 1 L}'_l h(X,.) 0, = (2.17)
+ rnl: (XI: (t), u(t - .6.), ... , u{t - (k - 1}.6.)), =
k 0,1, ... ,r-2
(2.10)
where the function rn" {.} is uniquely determined. and
Now, we are ready to give the following definition.
Definition 2.1. The dynamical nonlinear delay sys-
tem (2.1) (2~ is said to have a delay relative degree
r at a point X,. iffor all X,. in a neighborhood of X,.

LG._IL~-llh(X,,) = 0, Ie= 1,2, ... ,r-l, Remark 2.3. Note that for the nonlinear system

LgL~-llh(X.·_d ::f: O. {2.11}

=
(2.1) (2.2) in the case of.6. 0 the definition of delay
relative degree reduces to the usual concept ofrelative
• degree (see e.g. [2] and references therein). e

600
 3. MAIN RESULTS the set of such eigenvalues and let [0'0 0'1 ... 0',.-1]
be, in increasing order, the coefficients of the monic
Theorem 3.4. If the nonlinear delay system (2.1) polynomial having ..\ as solutions. The control law
(2.2) has delay relative degree r at a point X r, the (3.19) in which
feedback control law
vet) = -O'r_ly,.-I)(t) - ... - O'oy(t), (3.23)
vet) - Lr1h(X,.(t» - m"
u(t) - - - - f - -"'"'":7"---- (3.19)
- LgLr-1]h(X,,(t» assigns the prescribed linear dynamics, without delay,
to the output variable. Note that the derivatives of
the output of order until r - 1 are functions of X,.-I,
where m,. (X,. (t), u(t - Ll), ... , u(t - (r - 1)..1.)) and are given by (2.12).
is such that in a neighborhood of X,. the input-output As in the classical non delayed case, although the
mapping assumes the form input is such to bring the output yet) exponentially
to zero, the same can not be said in general on the
y(I")(t) = vet). (3.20) variable z(t). As a consequence the control technique
here presented is restricted to minimum phase non-
If the system (2.2) (2.2) has delay relative degree r linear delay systems.
at any point X,. E ~.("+1) the control law (3.19) is
global. 4. EXAMPLE OF APPLICATION
Proof. It follows directly from the definition of delay
As an example of application consider the follow-
relative degree and from equation (2.14), in which the
ing system, describing a particular population growth
expression (3.19) is substituted. •
model
Remark 3.5. The control law (3.19) transforms
the input-output mapping given by (2.1) (2.2) in a :rl(t) = aZ2(t) + bz 1(t - ..1.)z2(t - A),
linear one without delay. However, the input u is still :r2(t) = CZ2(t - A) + u(t), (4.24)
a function of the state variables z from instant t to
yet) = ZI(t).
t - rA, and of the input u itself from instant t - A
to t - (r - 1)..1.. Thus, although from an external
The delay relative degree for this system is 2. Note
point of view the system under feedback has finite
that even in this case the minimum phase property of
dimensional dynamics, its internal dynamics is still
the system is not guaranteed, because of the infinite
infinite dimensional. •
dimensional character of the zero dynamics.
The control law (3.19) is such that if at a given The feedback control law assumes the form
time lone has

J: = O,l, ... ,r-l, (3.21)

u(t) = ~( - aOzl(t) - a1 (az 2(t)+

+ bz 1(t - ..1.)z2(t - ~»)-

then
- acz2(t - A) - abz~(t - ..1.)-
vet) = 0, "It ~ l ::} yet) = 0, "It ~ f. (3.22) - b2z 2 (t - ..1.)%l(t - 2~)Z2(t - 2..1.)-
Following a standard notation, the dynamics of the - bCZ1(t - A)z2(t - 2~)-
system when the output is forced to be identically - bz 1(t - A)u(t - ~»).
zero is called zero dynamics.
(4.25)
In figures 1 and 2 simulation results are reported
Definition 3.6. Nonlinear systems of the form (2.1)
in which constants a, band C are set to 1, the delay
(2.2) are said to be minimum phase if they have stable
..1. is set to 0.1, and the feedback gains ao and al are
zero-dynamics. •
such to assign eigenvalues Al = -3 and A2 = -4. The
Of course, when the transformation of the input- initial state is zl(r) = z2(r)= 1, r E [-..1.,0], and
output mapping in the form (3.20) is obtained, the the feedback law is applied starting at the time O.l.
corresponding output control problem becomes a triv- Figure 1 reports the trajectory of the state variable
ial one. For instance, if r is the delay relative degree Zl, that coincides with the output, and the one of the
of the system (2.1) (2.2), then r eigenvalues can be internal state variable Z2' It can be recognized that
imposed, so that yet) can be driven to zero with a in this example the zero dynamics is stable. Figure 2
. prescribed exponential rate. Let A = [>'1 ... Ar] be shows the behavior of the control input.

601
We would like to stress that with the chosen parameters

.. . . . -
1,50
the considered model in open loop is strongly unstable.

t:~~=;~F~=r
1,00
5. CONCLUSIONS
The control problem of nonlinear delay systems is here 0,50
investigated. It is well known that the analysis of such
systems is heavily complicated by the infinite dimension- 0,00
ality of the state and by the nonlinear structure of the .o,so
differential equations. The main contribution of this pa-
per is the definition of a suitable mathematical formal- -1,00
ism which a.llows the analysis of nonlinear delay systems
through classical tools of finite dimensional differential ge- -1,50
ometry. The use of this formalism allows the definition of
an interesting class of nonlinear delay systems, that can -2,00
be said minimum phase. For such class a feedback con-
trollaw is presented that linearizes and mues undelayed -2,50
the input-output mapping. ~10 ~ ~ ~

Figure 1. State variables Xl and x2 (recall that y=xI)'

REFERENCES
[1] A. V. Bala.krishnan, "Applied Functional Analysis," SO,OO
Springer- Verlag, New York, 1981. 40,00
[2] A. !sidori, "Nonlinear Control Systems," Springer- 30,00
Verlag, Berlin, 1989.
20,00
[3] A. Bensoussan, G. Da Prato, Michel C. Delfour and
S. K. Mitter, "Representation and Control of Infinite 10,00
Dimensional Systems," Birkhauser, Boston, 1992. 0,00
[4] H. T. Banu and F. Kappel, "Spline Approximations
·10,00
for Functional Differential Equations," J. Differen-
tial Equations, No. 34, pp. 496-522, 1979. ·20,00
[5] J. S. Gibson, "Linear Quadratic Optimal Control of ·30,00
Hereditary Differential Systems: Infinite-Dimensional
-40,00
Riccati Equations and Numerical Approximations,"
SIAM J. Control Optim., No. 31, pp. 95-139, 1983. .50,00
[6] A. De Santis, A. Germani and L. Jetto, "Approxima- .0,10 0,90 1,90 t 2,90
tion of the Algebraic Riccati Equation in the Hilbert
Figure 2. Control input u(t).
Space of Hilbert-Schmidt Operators," SIAM J. Con-
trol Optim., No.4, pp. 847-874, 1993. [11] I.G. Rosen, "Difference Equation Sta.te Approxima-
[7] A. Germani, L. Jetto, C. Manes and P. Pepe, "The tions for Nonlinear Hereditary Control Problems,"
LQG Control Problem for a Class of Hereditary Sys- SIAM J. Control Optim., No.2, pp. 302-326, March
tems: a Method for Computing its Approximate Solu- 1984.
tion," proc. of 33rd IEEE Conference on Decision [12] H. Mounier, J. Rudolph, M. Petitot, M. Fliess "A
and Control, Vol. 2, pp. 1362-1367, Orlando, Florida, Flexible Rod as a Linear Delay System," proc. of
1994. 3ni European Control Conference, pp. 3676-3681,
[8] A. Germani, C. Manes and P. Pepe, "Implementation Rome, Italy, 1995.
of an LQG Control Scheme for Linear Systems with [13] S.-I. Niculescu, KH-infinity memoryless control with
Delayed Feedback Action," proc. of 3rd European an alpha-stability constraint for time delay systems:
Control Conference, Vol. 4, pp. 2886-2891, Rome, an LMI approach," proc. of 3./th IEEE Conference
Italy, 1995. on Decision and Control, Vol. 2, pp. 1507-1512,
[9] A. Germani, C. Manes and P. Pepe, "Numerical Solu- New Orleans, Louisiana, 1995.
tion for Optimal Regulation of Stochastic Hereditary [14] P. Picard, J. F. Luay, V. Kucera, "Realiza.tion of pre-
Systems with Multiple Discrete Delays," proc. of 3./th compensators for linear systems with delays," proc. of
IEEE Conference on Decision and Control, Vol. 2, 34th IEEE Conference on Decision and Control,
pp. 1497-1502, New Orleans, Louisiana, 1995. Vol. 2, pp. 2035-2040, New Orleans, Louisiana, 1995.
[10) B. Lehman, J. Bentsman, S. V. Lunel and E. I. Ver- [15] M. Dambrine, A. Goubet, J. P. Richard, "New results
riest, "Vibrational Control of Nonlinear Time Lag on constrained stabilizing control of time-dela.y sys-
Systems with Bounded Delay: Averaging The- terns," proc. of 34th IEEE Conference on Decision
ory. Stabilizability, and Transient Behavior," IEEE and Control, Vol. 2, pp. 2052-2057, New Orleans,
Trans. Automai. Contr., No.5, pp. 898-912, May Louisiana., 1995.
1994.

602