general disclaimer one or more of the following statements ...substituting eqs. lb and le into j...
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Computation of Output Feedback Gains for wLinear Stochastic Systems Using the Zangwill-Powell Method
by
Howard KaufmanAssociate Irofessor n,
Electrical & Systems FgAineering Department%°Rensselaer Polytechnic Institute
Troy, New York 12181 ! ^`Ct\^F^,`l
iti yP ^^ ^1lP:^J^•^
Abstract
Because conventional optimal linear regulator theory results in a con-
troller which requires the capability of measuring and/or estimating the entire
state vector, it is of interest to consider procedures for computing controls
which are restricted to be linear feedback functions of a lower dimensional
output vector and which take into account the ;resence of measurement noise s'and process uncertainty. To this effec •: a stochastic linear model has been
developed that accounts for process parameter and initial uncertainty, measure-
ment noise, and a restricted number of measurable outputs. Optimization with
respect to the corresponding output feedback gains was then performed for both
finite and infinite time performance indices without gradient computation by
using Zangwill's modificatic: of a procedure originally proposed by Powell.
Results using a seventh order process show the proposed procedures to be very
effective.
(NASA -CR-146308) ; OMPUTATION OF OUTPUT N76-18861FEEDBACK GAIN .t LINEAR STOCHASTIC SYSTEMSUSING THE ZANG. -LL-FOWELL METHOD (RensselaerPolytechnic Inst.) 15 p HC $3.50 CSCL 12A Unclas
G3/65 18521
This work was supne'rted by NASA Grant IIo. NSG 1188 administered ?v` Langley Research Center, Hampton, Virginia.
i
I ,
1. introduction
F K _ t am ^., ^xz^ k ..
Because conventional optimal linear regulator theory results in a con-
troller which requires the capability of measuring and feeding back the entire
state vector, it is of interest to consider procedures for computing controls
which are restricted to be linear feedback functiors of a lower dimensional
output vector. Such a procedure, however, has its limitations in that the
feedback gains will be functions of the initial state vector. In addition, the
presence of measurement noise and process uncertainty can lead to additional
problems relating to both modelling and computation.
To this effect a stochastic linear model has been developed that accounts
for process parameter and initial uncertainty, measurement noise, and a
restricted number of measurable outputs. Both finite and infinite time per-
formance indices were considered. Optimization with respect to the output
feedback gain was perforn.ed without gradient computation by using Zangwill's1
modification of a procedure originally proposed by Powc'-2 . This procedure is
such that if the cost index were indeed quadratic in the gains, then the search
would be along a set of conjugate directions.
The Zangwill-Powell method, is especially useful for infini+ , time perform-a
ante indices since many of the procedures proposed to date for finding output
feedback gains for such indices cannot be guaranteed to converge to a solution. 3,4
rAdditional problems also arise if an intermediate gain perturbation result in
an unstable system. This can be immediately corrected using the Zangwill- ^i
Powell procedure by setting the index itself to a very large number.3
The effectiveness of the Zangwill-Powell algorithm was evaluated using sixth
order linearized longitudinal equations of motion for an aircraft. Results showed
sa
the algorithms to be capable of converging to a set of gains useful for gust
I
alleviation.
i
F:
F TH'. _ vY ... ..y .n ;..^rvrvr .. e1. yv fy .M mre;, .-..P^ •—nRx-
2. Problem Statement
The system being optimized is of the form:
Process: x = Ax + Bu + AAx + ABu + v(t) (la)
Measurement: y = Cx + n(t) (lb)
Co:htrol: u = Ky (lc)
Initial StateCovariance:
[^(x0G x0 ) - PO
where x = (n x 1) state vector
U = (t x 1) control vector
Y = (m x 1) output vector
K = gain matrix containing both fixed elements
(most likely zero) and variable elements to be
determined
v,n = white noise vectors with respective covariance
matrices V, N.
and AA,AB = uncertainty in A and B respectively.
Two procedures were considered in order to take into account the total
process uncertainty AAx + LBa; namely:
1) Defining
w(t) = AAx(t) + ABu(t) (2)
as an additional white noise vector with zero mean and assigned covariance
matrix W suitably chosen to reflect the uncertainty. The procedures S
developed by J'osh1 5 are then applicable to solution.
2) (3a)Letting AAx = F xi As i
and 13 = L ui Abii
RITRODUCIBILry ()p T%ORIGTNAL PANJ1 (§ $^fl}}
(3b)
where Aai , Abi , the ith columns of AA andAB respectively, are in turn
set equal, to
Aai = Fi w
Abi = Gi w
where w is a white noise vector with covariance matrix W, and F i , Gi
are constant matrices. The optimization procedure cited oy McLane 6 are then3
applicable to the solution.
vGiven the above two formulations, the performance index to be minimized
x
is:
rJ T JT (yT Q y + uT ru) dt (4)
0
where E denotes the statistical expectation operator.
Substituting eqs. lb and le into J gives
J =
P
( ,J T [xTC rQCx + 2xT CT Qn0
+ n Qn + x CT KT R K C x + 2 x C T KT R n + n KT R K n] dt (5)
This formulation of the index can be simplified by noting that
(XT CT Qn) = E(xT CT KT R n) = 0
Furthermore (nT Q n) is a constant term independent of the control and
therefore has no effect on the index. Then minimization of (5) is equivalent
to the minimization of
(T rJ = £ T J x ( CT QC + CT KT R K t) x dt + L.(nT KT R K n) (6)
0
R
+5j
3
C.
UP RODUC11 iA l'Y OF THE
ORIGINAJ, PAGC IS POORr:
v
J{
y
. :^aa
Elimination of the expectation operator is iow possible by recognizing
that J can be rewritten as:
J '= Trace (CT Q C + CT KT R K C) S + Trace (KT R K N) (7)
(T ('•where S = T
1 L (x xT ) dt (8)0
Thus upon computation of the integral of the state covariance matrix Fix xT),
the value of the index can be found from cq. 7.
In particular if T is finite, the covariance
P = E(x J) (9)
can be readily propogated, given a value for K, as follows:
Formulation defined by eq. 2 5.
P= (A+BKC) P + P (A + B K C) T + B K 11 KT BT + (V+W) (10a)
P(0) = S (x(0) x r (0)) (10b)
• Formulation defined by eq. 3 6.
P = (A + B Y. C) P + P(A + B K C) T + M(P, K) + N(K)
+BKNKT BT + y. (lla)
P(0) _ [r (x(0) xT (0)) (llb)
where M(P,K) _ Pij Fi W F (llc)i,j
N(K) _ I Gi(Ki N K i T ) W G,T(lld)i,j
f
^
-I-
Fi = F + OR (Y. C) Ri(lle)R
Kiith row of K
and Pi, = i - ith component of P
For the case in which T = -, the integral S of eq. 8 wi!1 not in
general converge if there is measurement noise (n) and/or process noise (v).
Thus as in refs. 3, 4, 5, in the limit S will be replaced by
(TS = Lim
T1 P(t) dt = Pas (12)0
where Pss is the steady state solution to either eq. 10 or eq. 11.
3. Computational Procedures
Since the performance index (ca. 7) is easily evaluated given a value for
the gain matrix K, the Zangwill-Powell1,2 method which does not require gradient
computation is very attractive for optimization. In particular, the IMSL sub-
routine NPOWL was used for implementati.on. 7 Starting with initial values for
elements of the gain mat. ,ix K, successive perturbations are made in each of
the variable elements and the corresponding value of J computed. Using the
computed indices, perturbation directions are chosen such that convergence t.
the minimum of a quadratic function requires a finite number of iterations.
One particular attractive feature of the algorithms is the ability to correct,
when T = O0 , for a set of unstable gains which do not permit the determination
of a steady state covariance matrix, P. This was done by computing the
eigenvalues of (A + g K C) for each perturbed value of K and setting J
equal to a very large number (i.e., 10 50 ) whenever instability is noted.
4.' Experimental Results
4.1 System definition
A modified 6-dimensional version of the TIFS B aircraft with a
gust input (o a 15 fps) was used for evaluation in the presence of a zero
reference command. The corresponding variable definitions were as follows:
Plant state: q pitch rate
Ae pitch angle
AV velocity
= Aa angle of attackx =
Se elevator deflection
6z direct lift flap deflection
ag gust induced attack angle
Plant control: 6 elevator commandu e
6zC lift flap command
Observations: q pitch rate
Ae pitch angle
a angle of attack
Point 1 vertical accelerationnzl
nz2 Point 2 vertical acceleration
y flight path angular rate
The structural matrices corresponding to climb condition, i.e., h = 1524 m,
V = 106 m /s, which were used for evaluation purposes are (see eq. 1):
-.1686 .000035 .000231 -.486 -4.3778 -.19948 -.486
1. 0. 0. 0. 0. 0. 0.
0. -32.17 -.0143 18.027 0. -3.0933 .0518
A = 1. 0.000013 -.000531 -1.223 -.1273 -.2667 -1.223
0. 0. 0. 0. -20. 0. 0.
0. 0. 0. 0. 0. -4o. 0.
0. 0. 0. 0. 0. 0. -.2784
0. 0.
0. 0.
0. 0.
0.B=(0. 0.
0. 0.
0. 4o.
0.
1. 0. 0. 0. 0. 0. 0.
0. 1. 0. 0. 0. 0. 0.
0. 0. 0. 1. 0. 0. 0.
C =64.63 .00318 .176 444.2 212.1 100.4 h44.2
-61.82 .00580 .193 407.8 -116.2 85.5 407.8
0. .000013 .000531 1.223 .1273 .2667 1.223
'S
Atl
r^ • i
F1., ^tI-; s *^. -I 1i,1'I.Y OF TI3$19 POOR
4
x
^a1
^"ry
Corresponding sensor noise deviations were:
aq = .5 deg/sec
Cr 0.2 deg
on = 0.05 gz
(f Since the only plant disturbance was the excitation for the gust
E(Vi 2) = 0 i = 1-6; to account ,^r a 15 fps standard deviation
E(V72 ) was set equal to .0003.
Values for qii and rii in eq. 4 were chosen to be representative
of the inverse maximum squared value of the weighted variables. In particulara
the following values were used:
q112500.
j q22 = 50.} I} q = 50.
' 33
q )+)4 = 4.
q 4.55 =
q66 = .2500.
ril = 6.
^- r22 3.
Also considered was the situation in which no penalty was placed on nzl' nz2'
and y (i.e., q44 = q55 = q66 = 0)'For the stochastic problem defined by eq. 2, the covariance W
of the plant disturbance was chosen to be
I
Diag (.2, 0., .0007, .0005, 0., 0.)
These elements were chosen by computing for each component of the state
equation
4W(i,i) (As2 ) z (MAX (x 2 )
ij
k Z r ^
where Aaij was approximated to reflect the data in reference 8.
For the problem defined by eq. 3, the Gi 's were set equal to
zero, and the Fi 'a were selected such that the standard deviations in the
corresponding components of A were (in matrix form):
4 0 .002 .2 2.2 .11 0
0 0 0 0 0 0 00 0 .001 .17 0 .025 0
0 0 0 .24 .024 .16 o
0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0
4.2 Results
Using the preceding data, the six gains relating q,AO,Aa to
6e and 6z were determined.c
Of interest were the following observations:
• Convergence time for the infinite time problem was an
order of magnitude less than the time required for optimiza-
tion over a two second interval. This resulted from having
to c.,mpute only the solution to a set of algebraic equations
rather than a set of differential equations.
Assuming that every 6 gain perturbations called for by the
Zangwill-Powell method corresponds to a gradient evaluation,
it was noted that the numhar of gradient evaluations performed
by a steepest descent algorl;;hm was comparable to the number
performed by the Zangwill-Powell method.
• For a finite time index it is indeed possible to obtain gains
that yield an unstable set of eigenvalues. This resulted when
the finite time version of the formulation defined by eq. 2
was used with no weigh'.ing on nzl' nz2' Y-
it
`7 , k ^. wp 4 , ;wF ^ci'^vliWi
^ .:v.,^.y..wvR".4'C^...
In order to test performance with respect to the behavior of the
states and the control signals, the gains were used to regulate the longitudinal
motion subject to the initial condition:
p = .02r/s,AO = .15r, Aa = .15r
Figure 1 depicts responses for q, a, and nZl
which resulted
from applying those gains which resulted from the finite time version of the
formulation defined by eq. 2 when W = 0 and with no weighting applied to
nZl , nz2 , y.
5. Conclusions
Optimal output feedback control gains were determined using the Zangwill-
Powell procedure which does not require gradie;rt computation. Two stochastic
formulations of the process equation were considered in order to take into
account process uncertainty, process disturbances and sensor noise. Results
using a seventh order system show that the Zangwill-Powell method is very
effective for control gain computation.
Continuing efforts are considering such topics as:
• Constraints on the gain magnitudes.
• Effects of reference commands.
• Evaluation of the two procedures for modelling process
uncertainties.
r
He.^
ek 4^
References:
1. Zangwill, W., "Minimizing a Function without Calculating Derivatives",Computer Journal, Vol. 10, 1967, L,p. 293-296.
2. Powell, M.J.P., "An Efficient Method for Finding the Minimum of a Functionof Several Variables without Calculating Derivatives", Computer Journal,Vol. 7, 1964, pp. 155-162.
3. Mendel, J. M., and Feather, J., "On the Design of Optimal Time-InvariantCompensators for Linear Stochastic Time-Invariant Systems", IEEE Trans.Auto. Control, Oct. 1975, PP. 653-657.
4. Basuthakur, S. and Knapp, C. 1I., "Optimal Constant Controllers forStochastic Linear Systems:, IEEE Trans. Auto. Control, Oct. 1,75,pp. 664-666.
5. Joshi, S. W., "Design of O ptimal. Partial State Feedback Controllers forLinear Systems in Stochastic Environments"., 1975 IEEE Southeastcon,Charlotte, PIC, 1975.
6. McLane, P. J., "Linear Optimal Stochastic Control Using InstantaneousOutput Feedback", Int. J. Control, 1971, Vol. 13, No. 2, pp. 383-396.
7. IMSL Library 1, Edition 4, 1975, IMSL, Inc., Houston, Texas.
8. Chen, R. T. et. al., "A Study for Active Control Research and Validationusing the TIFS aircraft", NASA CR-13261h, April 1975.
s
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