A Nonlinear Fuel Optimal Reaction Jet Control Law Page: 4 of 10
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UNCLASSIFIED
ecmd
Fcmd I leaS Tcmd TAcs ton F3 Fjet: r/yy 1
Controller S S
Ocmd Controller FL T S
S------ --------------PWM
Figure-2: Simplified single-axis controllerNonlinear Controller Derivation
A continuous single-axis system may be
described by,Fet r
Iy,(1)
P e
I~Ted
00 w
o t t T
0 ti t2 Tncswhere r is the jet control moment arm, I,, is the
inertia term, and Fet is the fixed-amplitude jet force.
A discrete representation is given by,
eT =0o +wo -TACS --xpz +ac-p-(TACS -ti)
2WT = wo +a - p,
(2)
where TACS is the control cycle period, t1 marks the
beginning of the jet pulse, and p is the pulse width, p
= t2 - t1, where, t2 marks the end of the pulse width.
Figure-3 shows the states, B and co as a function of
time for a jet pulse occurring over a particular time
period, t1 to t2. Note that the beginning of the pulse
is not shown to coincide with the beginning of the
control cycle, and it is this feature of both pulse width
and pulse time that the nonlinear controller relies on
to satisfy the two error criteria of AO -+0, aO ->0,
where AO = 9d - 9 o o =cd - oT and, Bd and
od are the desired end states.
The cost function is defined as,J(ti,p)=(d -T) +k-(-od-WT)2
(3)
which, as seen from Eq.-2, allows for two
degrees of freedom, t1 and p, in order to satisfy the
two criteria of AB -+ 0, Aw -+0 . The parameter k
defines the relative weighting between the position
and rate error. Note that because of the PWPT
solution must lie within the next ACS control
interval, a constrained optimization of Eq. (3) is
needed.Figure-3: Discrete state transition over one ACS
control cycle as a function of jet pulse width and
jet pulse temporal position within the control
cycle.
Referring to Figure-3, there are a total of five cases to
be evaluated:
Case-1: t 0 and t2 < Tacs, where, p = t2 - t
Case-2: t1 < 0 and t2 TACS - p = t2
Case-3: t1 0 and t2 > TACS - p =TAcs - t1
Case-4: -- pmnum pulse - Aco =0, t(- 0
a
Case-5: tt < 0 and t2 > TACS, large angle saturation.
The cost function of Case-1(depicted in Figure-3)
is an unconstraint case and can be evaluated against
t1 and p in order to derive expressions for their
optimal values as follows,
-J = 2- 8 - a -p = 0 => B = for p 0,
t11
so that,
A8=---a-p2+a-p-(TACS-ti)
2t* = TACS--(2+a P)
(4)
We next seek the optimal value of p by taking the
derivative of J w.r.t. p.2
11th AIAA/MDA Technology Conference and Exhibit, Monterey, California, 29 July-2 August 2002
UNCLASSIFIED
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Breitfeller, E. & Ng, L.C. A Nonlinear Fuel Optimal Reaction Jet Control Law, article, June 30, 2002; California. (https://digital.library.unt.edu/ark:/67531/metadc742240/m1/4/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.