This paper deals with the class of nonlinear systems described by the equation M(q(t))q(t) = f(t) - N(q(t),qÂ¿(t)) with f(t) a control input. We employ a simple method of control design which has two stages. First, a global linearization is performed to yield a decoupled controllable linear system. Then a controller is designed for this linear system. We provide a rigorous analysis of the effects of uncertain dynamics, which we study using robustness results in the time domain based on a Lyapunov equation and the total stability theorem. Using this approach we are able to give meaningful robustness bounds which justify assumptions that are currently made in the literature in an ad hoc fashion

Abdallah, Chaouki T.

Lewis, F. L.

English

University of New Mexico

University of New Mexico
UNM Digital Repository
Electrical & Computer Engineering Faculty
Publications Engineering Publications
6-15-1988
On Robustness Analysis in the Control of
Nonlinear Systems
Chaouki T. Abdallah
F. L. Lewis
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Recommended Citation
Abdallah, Chaouki T. and F. L. Lewis. "On Robustness Analysis in the Control of Nonlinear Systems." American Control Conference
(1988): 2196-2198. https://digitalrepository.unm.edu/ece_fsp/104
FP4 4:45 ON ROBUSTNESS ANALYSIS IN THE CONTROL OF NONLINEAR SYSTZH8
C. Abdallah and F. L. Lewis
School of Electrical Engineering
Georgia Institute of Technology
Atlanta, GA 30332
404-89 4-299 4
Research Supported by a Grant From The Westinghouse Foundation.
H(q() is invetible for all q.
This paper deals with the class of nonlinear system
described by the equation K(q(t))q(t)= f(t) - I(q(t),4(t)) with
flt) a control input. We eploy a siple mthod of control
design which has two stages. First, a global liearization is
performed to yield a decoupled controllable linear system. Then
a controller is designed for this linar systm.
Ve provide a rigorous analysis of the effects of ucertain
dynamics, which we study using robastns results in the time
domain based on a Lyapunov equation and the total stability
theorem. Using this approach we are able to give eaningful
robustness bounds which justify assuptions that are currently
made in the literature in an ad hoc fashion.
Define a state by x': (q' 471) and the desired trajectory
x. by x.-' 1g' 4']'t Defining the error as e= (x-x), the
error dyaics become
A: {: ;btbe+ a - Aetk , (2.2)
where
ut rxi - I-f +t . (2.3)
lote that we have simply described an iplicit global
lierization of the kind ued in [51 and that the resulting
linear system is decoupled. Since the inpt transformation (2.3)
is one-to-one, f(t) can be recovered from u(t) by using
I. 1CI(
A class of nonlinear system that has been extensively
studied is described by the equation
P1(q)q= f - N(q,4) (1.1)
where f is a control vector, and M(q) and N(q,f4) are a mtrix and
a vector containing system paramters. The argat t has been
dropped for convenience. This cla of system includes robotic
systems in the Lagrange-Euler formlation (71. The contributions
of this paper are a simple control design scheme and a rigorous
analysis of the effects of uncertainties present in the entries
of lI(q) and (q,4).
The proposed control design techniqoe has two steps and is
similar to otber approaches in the literature 18[. A global
linearization, is first perfored to yield a controllable and
decoupled linear system. We then present two controllers, the
first being a pole-placemtt design and the second a linear-
quadratic tracking design. The results are easily interpreted
since the states of the linearized system and those of (1.1) are
the sam.
A rigorous time-domin robust anlysis based on a
Lyapunov equation [111 and the total stability theorem 121 is
then carried out to find bounds on the uncertainties in K1(q) and
N(g,q) for closed-loop stability. Unlike the analysis in 13,91,
our results alloy the inclusion of stagturnd unertainties. We
are also able to vary two design parameter a' and b' which
affect a trade-off between the required accuracy of M1(g) and that
of I(,4). Our result gives bouds for stability on the
uncertainties in the indvid-ual entries of H(q) and (q,4),
yielding results which are easily interpreted from a practical
standpoint. This short paper is a preliminay exposition of
results to appear in wre detail in 1121, to which we defer for
all proofs.
fz X - u) + N. (2.4)
which does not require the inversion of M.
The problem of regulating tbe original nolinear system has
therefore been transformed into the siqpler problem of
stabilizing the (controllable) decoupled linear system (2.2).
The control (2.41 has been called partitioned control, or
co6uted torque control [71. In the following, we give two
methods of finding the control u(t) for (2.2).
First, let the control objective be the pole placement of
the linear system in (2.2). Consider the closed-loop system
IAke
where
(2.5)
u= -Ke (2.6)
so that
A,= k- BK = 0
Let the 2n desired eigenvalues of A be : lt, J½,.
define D and I by
D= diag(pi), 1= diag(pl+d; i=l,..,n
(2.7)
., lt, anl
(2.8)
ksu 21l
The gain 1 (1 Kzu required to place the poles of (2.2)
has K1, K2 diagonal and given by:
K2.= -D
l - (D t E)
IL Ml nut
Let a nonlinear system be described by the-differential
equation
M(q)j. f - I(q,4) (2.1)
with q(t)E RR and the control f(t)i RP. P and N contain the
-vstem Darameters, some of which my be unknown. We assin that
(2.9)
(2.10)
U
Next, consider the control objective to be the minimization of
the linear-qudratic (IO) perforne index
a
1 1!
- (eOetaRu)dt
214
0
(2.11)
2196
or
Q= diag(01), i 1,...,n
e4 Ae . Bu)
= (A - BK + BELK)e t B(LqXi t 8) (3.6)
4: (L t BZLK)e + B(LA4 t 68)
with Q0k, R>01 and Ox, Oz, and R ru diagonal (to obtain simple
explicit fortlas in the sequel). Note that the state of the
nonlinear systes is not changed by the linearization, so that
performance index (2.11) is meaningful in term of real
petformce objectives for (2.1).
It is well known [1,61 that the solution to the 1. problem
described by (2.2) and (2.11) is given by the state feedback
(2.6) where
= R-tBTS
..= A'S t SA - sua-'rS.
(2.12)
The simple structure of our linear system leads to an explicit
expression for 1, as we now show.
The feedbaclk gain K: [(K 121 minimizing (2.11) is given by
Kl- CQ&R-ItP", 2= (2f1 t024-X32 (2.13)
Moreover, the closed-loop poles are described by (2.81) with
Q1R-I= DI2', Q2R-I= D' + 1' (2.14)
e e t RV (3.7)
The objective is to find bouds on a and 6 to keep the
above system stable given that (2.5) is stable. Consider the
system (3.7). It is clear that its stability is dependent on A
and on v. The effect of j is the stability robustness probles
stdiled in [4,10,11). Ve can therefore use the results from
ill) which allow for structured uncertainties, taking advantage
of the special form of the matrices A and B, to define
[0
B5J =
Pi Pa
(3.8)
where
Fs= LE, ; i= 1, 2.
Let
F S jFiSj..-m cis and Eamia Lis.
where Fit denotes the ijflth tera of matrix F.
(3.9)
(3.10)
m
It is therefore possible in this special case to relate the
weighting matrices 0Q, 02 and k to maningful physical parasters
suh as the damping ratios and natural freqencies of the closed-
loop poles. Using the expresion for K given in equations
(2.9),(2.10) in evaluating f as in (2.4) one gets:
f= -N[D'!2 -(D2 + B21le + +1I (2.15)
This nonlinear feedback law will make the nonlinear system (2.1)
follow the desired trajectory ua.
IIIn. In uinss sa nIS
In practice, the system described by
uncertainty in the entries of N(q) and I(q,4).
the calculated control law f. to be different
when N and I are completely known (3,91.
(2.1) suffers from
This will cause
from the one foud
One of our contributions is the capability to deal vith
stiKitzd mucertainties in N(q) and I(q,4) and so obtain tighter
bounds than those in (9). Let the calculated control for the
nonlinear system be given by
Suppose that K has been selected so that A, has a stability
margin of -a, ere the constant wa' is a design parauter which
may be selected to trade off the stability of A, (guranteed by
the feedback K) against the required accuracy in N. and I,C as
will becom clear in our developent.
hsmid3a1
Let (L -al) be asymptotically stable (AS). Then the solution
P of the Lyapunov equation
(L-al)'P + P(L-aI) t 2I= 0
is given by P=
(3.11)
[3P1 z
P. a
where Ps, iz 1,2,3 are diagonal and
Pa= (143I 4a'K, 4 a(122 t 3KL) + K1K12-[K[ t a(2I-KO)
P= (K2 + 2aIP-t(I + P2), Pi= (Ka + 2a1)P2 + PA, (3.12)
f-= N(qd - u) + N. (3.1)
where M. and J. are the calculated values of N and I. N. and N.
differ ftom N and I due to simplifying assumptions and/or
uncertain values of sot parasters. Let u be given by (2.6) so
that
f.= N..Cqd + Xe) + Ic (3.2)
One therefore obtains a calculated version u, of the input u to
the linear system (2.2). Let u. be given by
4 t Nt - N f.e
==aL j + C + (L - I)Ke
a= (I - N-NJ, 8 = N-'( - B.).
In the next theorem, conditions on Li are given for the
eigenvalues of (A.+F) to be less than -b where b is a positive
design parameter (b<a) selected to trade off requireunts on the
accuracy of N. against requiremnts on the accuracy of N., as
shown next. Let us define
eL' maxjF1t + (a-b)l.. (3.13)
hum id3.2
Let A. have eigenvalues witb real parts less than -a. Then
the eigenvalues of CL + P) have real parts less than -b if(3.3)
(3.4)
where:
One is now concerned with the stability of the system
El < (s..P.MU).V 1/S (3.14)
s.(.)= maxim. singular value of (.)
2197
where
or
where
( 3 .5
Pail IP'sI
[0I if Fi 0.
j), othervise.
(PJ.)s - Syitric part of PSU.
Now that the global asymptotic stability of & has been
guranteed with a desired margin of stability -b, w shift our
atteutios to the effect of the disturba wit) on the closed-
loop beavior. Te following lem is neded in the sequel.
Let i = ere = A. t F(t) , with jand satisfying
the conditions of Theorem 3.2. The state transition matrix
I(t,O) is then bounded as follows,
j{I(t,0)jl s ce'tl az -(Znsctptja-bJ)>0 and nl (3.151
where jfI(t,0 )j (2. x1 )2)2,
p- max real part of eigeralues of A, JL-a, (3.16)
c' and s defined in (3.13) and (3.14).
In the next theorem the error is bonded, hen v(t) is not
zero.
Let i(t) jeCt) + Bv(t) ere jis asymptotically
stable with a margin -b and vII(e(t))II s Lr for sm constant L
when lle(ttI r. If jle(OI < Oran/d (La/u)(l there exists
a unie so1utioneit) of (3.71 and
nle(tlit s e-tIIje(C)II + (Lr/)r(1 - et) s r
for all t i0.
(3.17)
If in Theorem 3.4 one has IiIv(e(t))II I Lliefl thn under
the conditions of that Theores
s
1t II)le(O31! where a'- 2nLtr t p. (3.18)
Moreover, eC(t is asymptotically stable if e'>0.
Note that s>-(J+b)/(2nE'tla-bI) is always guaranteed because
of Theorem 3.2. Therefore, (Lu/s)< 1 is satisfied if
L < -e(2nfttl a-bh /(Ita). This gives an per bound on the
mgnitude of v It or stability. In order to get a bomd on v(t)
using a norm of the type defined in (3.10) let us note that
iBvil srnV tere V1r jflja. and V Max V. Iext, we will
assum that nV S Lr and find stability robustness bounds on the
differenes betwen the indvidMal true entries of N and I and
their calculated ones. These bouts are useful in practical
applications.
?e 3.6
Let the computed entries of K and I in (3.1) be3(Lis and{L,j andte tmeones be (K,)anl fit). Ifa: b, then the
error in [2.2) is bouded as in (3.18) if
I n
imus - Ls1 <~7 £ 1"
sk Sal
k= max k±,Ii 11..,2n
kt ith diagonal elemant of 1 if i s a
(i - nlth diagonal elemant of 12 if n0l s i s 2n,
andaS~~~~
IiN - S1 *'E 1Sss1
where dC' maxlG6s`.
IT. SW.UIU
(3.20)
a
Using a global linearization, a nonliar system as
transformed into a decoupled linear system. A controller as
designed for the latter by two utbods, pole-piaceant a! BO0
theory, ad from this as derived the control for the nonlina
systes.
Controllers in robotics, for example, are often desigwd by
ad hoc mans which amount to oitting certain nonlinear tern and
assming a decoupled dynamical description of the system, but
little work has been done on rigorously Justifying the
simplification, notable exceptions being fond in 13,91. In this
paper, the robustes of the clsed-loop system was studied using
a tim-domain Lyapnov approach, and stability robustns bounds
were found in term of mingful physical paramters. Our
approach exloits the strctue of the disturbaces, and presents
bounds on the actual mgnitude of the disturbances rather than on
their L. norm.
[11 A.B. Bryso, Y.-C. Bo, boied t&imal Control. emisphere,
pp. 148-71, evIeid printing, 1975.
121 C. Corduneanu, Prigniples of Differential and Inteal
g tuIm,o 2nd Id,lew York: Chelsea, pp. 90-95.(31 E.G. Gilbert and I.J.Ha, 'Robust tracking in nonlina
system,' I Tra. tot tQol Vol. rC-32, No. 9,
pp. 763-771, 1987.
(41 5. Gutan, 'Uncertain Dynmical-System-A Lyapunov Kin-liar
approach,'" TU Trans. Autgot. Control Vol. AC-24, Io. 3,
pp. 437-443, 1979.
[51 L.R. Hunt and R. SU, and G. Peyer, 'Global transformations
of nolinear system,' IC Irans AtOWt. Control, Vol.
AC-28, to. 1, wp 24-31, 1983(61 P.L. Lewis, QptiL. Control, Dew York: Viley, pp.212-213,
1986.
(71 R.P. Paul, Robot Mni2ulators: MatheNtic:s. PRo inc and
Qnot Mke York: MIT Press, 1911.(8) C. Va and R.P. Paul, 'Rsolved mtion force control of robot
manipulators,' I Trans. s., an, & Cyb.. Vol. S)C-12,
No.3, pp. 266-275, 1982.
[91 N.Y. Spong and N. Vidyasagar, 'Robust linear compensator
design for nonlinear robotic control,' IW J. Robotics and
Auts.. Vol. RA-3, No. 4, pp. 345-351, 1987.
[101 I. Torch, 'Improved Bouds for the tigenvalues of Solutions
of Lyapunow Equations,' 1 Trans. Autmt. Control. Vol.
AC-32, No. 8, pp. 744-747, 1987.
(11] R.K. Yedavali, 'Perturbation bounds for robust stability in
linear state space models," It. 3. Control. Vol.42, No. 6,
pp.1507-1517, 1985.
1121 C. AMallah and F.L. Levis, 'Robustness Analysis for A Class
of lonlinear Systems,' submitted, 1988.
(3.1-5)
where
2198


On Robustness Analysis in the Control of Nonlinear Systems

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On Robustness Analysis in the Control of Nonlinear Systems

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