A general nonlinear programming algorithm (NICO) is discussed. An academic optimization example is given. The NICO multi-input, multi-output control system design is discussed. NICO applications relative to launch vehicle autopilot design, space shuttle static balance, transient response criteria in the design of a reentry vehicle control system, and waterjet propulsion and lift system components sized to a large surface effect ship are noted

Hauser, F.

English

NASA Technical Reports Server

N87-11735
A NONLINEAR PROGRAMMINGMETHOD FOR SYSTEM DESIGN
WITH RESULTS THAT HAVE BEEN IMPLEMENTED
Frank Hauser
Boeing Aerospace Co.
Seattle, Washington
IPli_CE_NG PAGE BLANK HOT FILMED
267
https://ntrs.nasa.gov/search.jsp?R=19870002302 2020-03-23T14:51:27+00:00Z
OPTIMIZATION THEORY
The field of optimization is as broad and interrelated as the problems it
attempts to solve. However, there appear to be two general categories:
(1) The indirect or classical methods, based on the calculus of variations,
which are often said to involve"parachuting" to the optimum
(2) The direct or programming methods, which involve searching or climbing
to the optimum
To list individual methods in the categories is difficult because many
that involve indirect methods actually "search for places to parachute". For
example, two Lagrangian methods that involve searching are: (i) the Sequential
Gradient Restoration Algorithm (ref. I); and (2) the Projected Lagrangian. The
NICO program (Nonlinear Inequality Constrained Optimization) is definitely a
Gradient Search Method.
Indirect or classical, based on calculus of variations (parachuting)
• Lagrangian methods
• Optimal control e.g, LQR
• Etc.
Direct or programming (searchlng or climbing)
• Simplex algorithm
• Dynamic programming
• Integer programming
• Etc,
• Gradient methods
• Projected gradient algorithm
• Method of constrained derivatives
• NICO
• Etc.
268
NICO - GENERAL NONLINEAR PROGRAMMINGALGORITHM
Nonlinear Programming is perhaps the most powerful category in the field of
optimization. With it, engineering problems can be solved directly; i.e., the
problem does not have to be fit into a canonical form, wherein it can lose some
of its features. The general nonlinear programming problem is to determine values
for n variables, which satisfy m nonlinear constraints:
Gi(X I, ..., Xn)[<, =,>]B i i = I,..., m
and, in addition, maximize (or minimize) a nonlinear objective function
Z = f (X I, ..., Xn)
All engineering problems fit this formulation, though many times the G's and f
are only in the engineer's mind as he solves the problem by trial and error. The
field of nonlinear programming contains systematic ways to solve the problem if
it can be quantified. In most cases, it can be.
NICO can be further classified as a Parameter Optimization method; i.e., it
finds the best values for variables whose "first guess" values are input by the
user.
Determine xn values that satisfy nonlinear constraints
and maximize (minimize) nonlinear objective function
z : ,×.)
Note: n not related to m
269
NICO ITERATIVELY FORMS AND SOLVES
THE FOLLOWING PROBLEM
The fundamental principle in NICO is to iterate to the final solution by
solving a succession of linear programming problems.
NICO has two distinctive features: (1) the "first guess" does not have
to satisfy any of the constraints (in most real engineering problems, the
"first guess" is seldom feasible); and (2) it can converge to local interior as
well as exterior optima.
In NICO, the objective is not as important as the constraints. But in
most, if not all, engineering problems, the solution is dominated by the con-
straints. In fact, the optimum is constrained; i.e., constraints literally
define the best values of the objective.
Matrix of constraint equations
Bo
L
Where z_)<. = X.- X. and k_--X_-.
J J ,Jo
is computed via finite difference
Objective function to be maximized
Yl
j:l o
Variable constraints
270
ACADEMIC OPTIMIZATION EXAMPLE
This figure shows the application of NICO to an academic problem. It shows
how NICO iterated to the solution from three different nonfeasible "first guesseS'.
It also shows another distinctive feature of NICO; i.e., it concentrates on the
constraints. Run 2 shows how the objective function decreases during the first
few iterations in order to reach a feasible solution. NICO moves in a "deflected
gradient" direction; i.e., the objective function gradient is deflected by the
constraints. Often this is exactly what the engineer does as he optimizes, since
most real engineering problems are dominated by the constraints, with the objec-
tive function providing only a general direction.
RUN #3
FIRST GUESSlXlO - 1.6
X20 - 3.0 /
1.0'
.8
X2
.6
0
RUN #l
FIRST GUESS
CONSTRAINT
MI
RUN #2
FIRST GUESS
I'
.5 .0
XI
Academic Optimization Example
i
1.5
Z- x,
@$./,,/ec ÷
271
ACADEMIC OPTIMIZATION EXAMPLE (continued)
The academic example of the previous figure has an unusual situation at
the point Xl = X2 = O. As shown on this figure, the two "feasible regions" of
this example are connected by this single point. NICO had no difficulty passing
through this point, because it uses a "hunting" technique and operates much like
an engineer as he iterates via a succession of trail points.
4
X2 -2
-4
-6
-8
Run #4 First Guess
Nll-
A
, ;. . : • • "
-10 -8 -6 -4 -2 0
X].
Academic Optimization Example
272
NICO APPLICATIONS
(1) NICO was first applied to launch vehicle autopilot design and produced
gains and filters for several (at least two) vehicles that flew. It
has also been applied to a so-called Variable Payload Vehicle where it
also produced results that were installed in software that actually
flew. The latest successful application is on a Variable Trajectory
Vehicle, where its design of the roll-yaw autopilot falls in the
category of multivariable design.
(2) NICO was also applied to the control effector "trim" or static moment
balance problem on an early space shuttle proposal.
(3) Another application included handling quality transient response
criteria in the design of a reentry vehicle control system.
(4) Waterjet propulsion and lift system components were "sized" on a
large Surface Effect Ship.
(5) NICO was used in an iceberg transportation study to select candidate
icebergs, propulsion system size, and the best route.
Launch vehicle autopllot design
• Launch vehicle autopllot commend •
mixer deslgn
• Reentry vehlcle autopllot deslgn •
• Surface Effect Ship Sizing
• Iceberg transportation .
Frequency response
Transient response
Static moment balance
Handllng qualities
Propulsion and lift system for
payload/range
Candidate Icebergs, propulsion
system, route
273
NICO LINKED TO LINEAR ANALYSIS PROGRAM
NICO is currently linked as a subroutine to a program that computes frequency
responses. Previously, the engineer ran the linear analysis program iteratively,
by trial and error selection of control system gains and filters, until the
desired stability margins were achieved. NICO replaces the engineer, who now only
inputs a "first guess". The engineer can now concentrate on the most important
job; i.e., math modelling. NICO does the technically unchallenging job of
iterating.
The interface program searches the system response and passes to NICO the
array of current values of each of the constraints. It also passes the value of
the objective function. NICO passes back new values for the variables that are
to be used in computing the next frequency response.
Engineer ]Problem D finition
1 Math MOdel i Objectlve I" First guess varlables
+ Data Function (z) J (Xno)
_. Constraint requirements
Simulation Program
(e.g. MDELTA
Linear Ana]Ysls
Program)
Variables
Response (e.g,
Frequency
Response)
Interface
Varlables
Constraints (G_
Objective (z)
NICO
"The Gopher"
274
NICO RESULTS THAT HAVE BEEN IMPLEMENTED
NICO was used to design an autopilot for a "so-called" variable payload
vehicle. The gains and filters that were computed by NICO were programmed in
the software of this vehicle on a mission that actually flew. The top figure
shows that the initial user defined (first guess) values for the gains and
filters resulted in an unstable bending mode. After six iterations, NICO
produced values that stabilized this mode and exceeded all required stability
margins. The final iteration resulted in a better system than had been previously
designed by manual techniques, and it is felt that NICO required less engineering
and computer time.
O
UNSTABLEBENDING
-360 -180
PHASE(DEG)
• FIRST GUESS RESULTED IN
UNSTABLE BENDING MODE
/-- BENDINGMODE
/GAIN & PHASE
/ STABILIZED
_ 0
PHASE MAIIt61N
(EXCEEDS
SPEC)
-360 -180
PHASE(DEG)
• 6 ITERATIONS LATER, STABILIZED
WITH ALL MARGINS MET
275
NICO MULTI-INPUT MULTI-OUTPUT CONTROL SYSTEM DESIGN
Therehavealways been discussions about the shortcomings of classical fre-
quency response design methods when dealing with multi-input multi-output (MIMO)
control system design. The difficulties with MIMO systems arise from interactive
effects; i.e., the action of one feedback loop affects the actions of the others.
The standard practice used successfully by the "classical" engineer is simply to
design all loops simultaneously. This iterative design approach includes an
extensive tolerance analysis and yields optimum systems. NICO automates this
classical approach.
This figure is a remarkable example of NICO applied to MIMO.
design of a roll-yaw autopilot on a variable trajectory vehicle.
NICO simultaneously considered:
This was the
On every step,
(1) Stability margins of the roll loop with the yaw loop closed
(2) Stability margins of the yaw loop with the roll loop closed
(3) Shape of the frequency response of roll due to command with all loops
closed
(4) Shape of the frequency response of sideslip due to command with all
loops closed
Disturbance rejection could also have been simultaneously considered by
reducing the peak value of the frequency response of roll, sideslip, and all fin
angles due to a disturbance like the wind.
I /r_ ROLL DUE TO COMMAND_TIME
FINAL
SIDESLIP DUE TO COMMAND_TIME
RESPONSE, ALL LOOPS CLOSED
/
PHASE MARGIN (DEGREES)
INITIAL FINAL
ROLL OPEN, YAW CLOSED 22. 60.
FREQUENCY RESPONSE, PEAK RESONANCE (dB),
ALL LOOPS CLOSED
INITIAL FINAL
ROLL DUE TO COMMAND 9. O.
SIDESLIP DUE TO COMMAND 5. O.
276
SURFACE EFFECT SHIP SIZING
At one point in its development, NICO was linked to a program that computed
the range/speed/payload performance of a Surface Effect Ship. NICO was used to
establish: (1) the pump diameter, inlet area, and nozzle area of the waterjet
propulsion system; (2) the lift system air flow rate; and (3) the vehicle speed
that would maximize the payload for a given range. The constraints were: (I)
maximum pump diameter, inlet area, nozzle area, horsepower, and air flow rate;
(2) minimum speed; and (3) maximum speed possible while avoiding pump cavitation.
Poyload (short tons)
Cruise Speed (knots)
Travel Time (hours)
RESULTS
Engineer NICO Chon_e
3760 4000 +6.4%
43 46 +6.5%
(cavItotlng) (No cov|t.)
60 56.7 +5.4%
• Foster Response
ADVANTAGES
Englneer NICO
Over two weeks of 1 day
engineeringmanhours
• Lower Cost Over 40 runs varying 1 run K $20)
major parometers (,$100)
• Improved Results Feaslble ship found, True Optimum
but not true optimum
REFERENCE
lo Miele, A.; Calastro, A.; Rossi, F.; and Wu, A. K.: A Modification of the Sequen-
tial Gradient Restoration Algorithm for Mathematical Programming Problems with
Inequality Constraints, Aero-Astronautics Report No. 124, Rice Univ., 1975.
277


A nonlinear programming method for system design with results that have been implemented

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