An interactive software package which provides design solutions for both standard linear quadratic regulator (LQR) and suboptimal linear regulator problems is described. Intended for time-invariant continuous systems, the package is easily modified to include sampled-data systems. LQR designs are obtained by established techniques while the large class of suboptimal problems containing controller and/or performance index options is solved using a robust gradient minimization technique. Numerical examples demonstrate features of the package and recent developments are described

Fleming, P. J.

English

NASA Technical Reports Server

NASA ContractorReport 172536
NAsA-cR-_72536
19850011392ICASEREPORT NO. 85-6
ICASE
SUBOPT - A CAD PROGRAM FOR SUBOPTIMAL LINEAR REGULATORS
Peter J. Fleming
Contract No. NASI-17070
February 1985
INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGINEERING
NASA Langley Research Center, Hampton, Virginia 23665
Operated by the Universities Space Research Association
;5:;,. j ".i :r,.5"o
National Aeronautics and - "
Space Administration LA,N.t3LEYRES£ARCHCEI_,'TEIF_
Langley Research Center LI_'RARY.NASA
Hampton,Virginia23665 I-L'-,:,.;PTOt4,VIRGII','!A
https://ntrs.nasa.gov/search.jsp?R=19850011392 2020-03-20T20:15:11+00:00Z

SOBOPT - A CAD PROGRAMFOR SUBOPTIHALLINEAR REGULATORS
Peter J. Fleming
University College of North Wales
Abstract
This interactive software package provides design solutions for both
standard linear quadratic regulator (LQR) and suboptimal linear regulator
problems. Intended for time-invariant continuous systems the package is
easily modified to include sampled-data systems. LQR designs are obtained by
established techniques while the large class of suboptimal problems containing
controller and/or performance index options is solved using a robust gradient
minimization technique. Numerical examples demonstrate features of the
package and recent developments are described.
Appearedin "Proc.of the Instituteof Measurementand ControlWorkshopon Computer_AidedControl
SystemDesign,"Brighton,England,September1984.
Research was supported in part by the National Aeronautics and Space
Administration under NASA Contract No. NASI-17070 while the author was in
residence at the Institute for Computer Applications in Science and
Engineering, NASA Langley Research Center, Hampton, VA 23665.

Introduction
In addition to providing CAD solutions to the linear quadratic regulator
(LQR) problem for time-invariant continuous systems this package offers the
designer a range of attractive "suboptimal" alternatives. These alternatives
were first suggested in the early seventies to offset certain disadvantages of
the LQR approach which include its reliance on full state feedback or state
reconstruction techniques and its sensitivity to plant specifications. The
so-called "suboptimal" designs optimize problems which are related to the LQR
formulation but which are not bound by its restrictive framework. In addition
to handling the basic LQR design SUBOPT accommodates options which permit both
alternative controller configurations and performance index (PI) descriptions.
The design options made available are:
Controller Options
*Full state feedback
Output feedback
*Dynamic compensation
PI Options
*Sensitivity reduction
*Virtual model-followlng
*Implicit model following
One controller option is selected by the designer and may be used simply with
the conventional PI containing quadratic terms in state and control or an
extended PI which includes sensitivity reduction and/or model-followlng
\
-2-
terms. With this more flexible design approach it is easy to compare
different design philosophies within the package. A further feature, the
"gain-fixing," permits the designer to fix one or more controller parameters
to have zero or non-zero values. This has proved useful, for example, (i) in
eliminating redundant gain elements, (ii) for investigating certain failure
modes and (iii) for realizing minimal parameter compensators.
Methodology
Plant parameters are input according to the system description
x = A x + B u , v = C x
--p p--p p--p =-p p--p
and the desired controller configuration is selected:
(i) u = K x - full state feedback
s --p
(ii) --PU= K0 _p - output feedback
= A + B x - dynamic compensation(iii) Up C _ C--C
where --cx= Cc _ + Dc Xc, _c(0) = 0,
x is the plant state vector,
--p
_p is the output state vector,
u is the control vector and
--p
x is the compensator state vector.
--c
-3-
Elements of controller gain matrices Ks, K0 or Ac, Bc, Cc and Dc are the
minimizing parameters of the PI.
The conventional quadratic infinite-time PI
oo
is augmented with a quadratic compensator measure if controller configuration
(lii) is selected.
It may be augmented further if any of the PI options are selected. If
sensitivity reduction is an objective a quadratic measure of sensitivity is
included, based on the sensitivity vector, x = _x /_a, where c is a varying
--s --p
system parameter. If model-following is to be achieved a quadratic measure of
the difference between plant and model states for virtual model-following
(VMF) or derivatives for implicit model-following (IMF) replaces the usual
state measure in the PI, where the model is defined thus:
x =A = C
--m m -_m' Y--m m_m'
x is the model state vector and
--tll
Y--m is the model output vector.
The general suboptimal linear regulator problem (and LQR problem) is
formulated as follows:
Given the system
x = A x, x(0) = _0
-4-
find the feedback controller, u , of specified configuration which minimizes
--p
the PI
J=f x Q xdt.u --
0
Vector x may simply contain x or depending on the option(s) selected
-- --p
may be automatically augmented to include _c' _s' and/or x . Accordingly
the program builds the appropriate A and Q.
Option choices which lead to the basic LQR design are identified by the
program and the resulting matrix Riccati equation is solved using Kleinman's
iterative techinque [I]. Otherwise, the optimization problem is solved using
a quasi-Newton gradient minimization algorithm to minimize J. Both
approaches require an initial controller estimate which results in a stable
system. Such an estimate is usually available; if not, one can be obtained
within the program package employing the method suggested by Koenigsberg and
Frederick [2].
The program realization of the problem formulation permits the
specification of fixed elements within the controller structure so that not
all of the controller gain elements need be minimizing parameters: this
introduces additional flexibility to controller configurations.
The minimization algorithm fails only if the user-selected convergence
termination criterion is not satisfied after a prespecified number of PI
evaluations. Such a failure will occur
(i) if an insufficient number of PI evaluations have been made in pursuit
of the solution, or
(ii) if the termination requirement demands a numerical precision for which
the particular combination of program package and host machine is
unsuited.
-5-
A "soft" failure mode has been adopted thus permitting
(a) immediate re-entry to the routine in the event of (i), using the most
recent controller estimate, or
(b) specification of a less stringent termination criterion in the event of
(ii).
High precision in the solution is rarely needed when the design goal is viewed
within the control engineering context.
An attractive feature of the LQR design approach is that its solution is
independent of plant initial conditions. If these are unknown the initial
condition dependency of the suboptimal solution is adequately accommodated by
permitting the designer to
(a) minimize J for an average set of initial conditions,
(b) minimize J for a worst-case set of initial conditions, or
(c) minimize J, given some statistical knowledge of the initial
conditions.
Software Details
The program package contains robust, efficient numerical algorithms which
have been thoroughly tested in industrial and academic environments. It is a
conversational-mode package, having versions in FORTRAN-10 and FORTRAN 77 with
the graphics component based on GHOST. Source code is fully commented and the
software documented [3], [4]. Array sizes in code are parameterized in order
that the user may tailor the program to fit design and host machine
requirements. Typically, an overall system of order 30 will require 190K
bytes of executable code (excluding graphics). A typical run-time on a DEC-10
-6-
machine for a tenth-order suboptimal linear regulator problem having 4
variable gain elements is 17 seconds.
Design evaluation tools include closed-loop eigenvalue determination, a
COST routine to permit investigation of individual components of the PI, e.g.,
/ xT Qs x dt or f u_ dr,--S --s0 0
and a simulation facility for graphical display of system responses. A series
of design solutions may be obtained through modification of weighting
matrices, controller/Pl options, etc. The evaluation tools, therefore,
provide a ready comparison between designs, e.g., contrasting an LQR solution
with an output feedback design.
Design Examples
Helicopter Regulation
Helicopter longitudinal dynamics are described in [5] for a plant which
[ _]T, _P [ ]r Anhas four states, --PXT = Bx' _z' 0, and two controls, T = Ul' u2 .
LQR design, incorporating full state feedback, obtained for
Qp = diag(0.1,O.l,0,O) and Rp = diag(0.Ol,O.Ol), yields a satisfactory model
response:
K* =
L = (Ap + Bp s)_m Am x_m,
where u = K x . However, the goal is to satisfactorily regulate _x and
--p s --p
_z (forward and vertical velocities) employing feedback from only two states,
0 and 0 (pitch angle and pitch rate), since the use of airspeed sensors for
Bx and _z is undesirable.
-7-
Such a design may be effected by minimization of the PI:
T uTJ = (_ - Xm ) Qp(Xp - _m ) + R u dt--p p--p
which includes a virtual model-following term and by specifying controller
feedback from 8 and 8 alone. The VMF term forces this new reduced
feedback design to closely match the response of the original LQR design.
Setting Qp and Rp as given above yields controller A:
[ I°010.146 0.074U = -0.279 0.134
and from the resulting responses (see Figure 1) it is clear that reasonable
regulation can be achieved with comparable control effort to the LQR design
without resorting to use of velocity sensors.
Assume now a possible pitch rate sensor failure, in which ease we wish to
maintain reasonable regulation with pitch angle feedback alone. Setting the
gain elements corresponding to pitch rate feedback to zero and minimizing J
with respect to the two remaining control gains reveals that although there is
some degradation in system performance using the resulting controller B:
u= 0
_- .301J
the quality of regulation is still acceptable. Now if we accept this as a
base or "worst-case" controller an improved controller can be obtained by
fixing the pitch angle gain elements at these values and minimizing J with
respect to the pitch rate gain elements. The resulting "robust" controller,
p
-8-
u1:1570.05411:]-- 301 .114J
since it will tolerate a pitch rate sensor failure, yields responses which are
very similar to the responses for controller A.
Desensitizing a Gas Turbine Engine Controller
Consider now the regulation of the two spool speeds (x3 and x4) of a
twin-spool gas turbine engine. An output feedback control design which
conforms to rigid engineering constraints, has already been effected for this
2-control 5-state problem via a nonlinear optimization technique (for details
and model data, see [6]). Particular attention has been paid to control
magnitudes and the step responses of x3 and x4, which must have little or
no overshoot since overspeeding of the engine rotors will lead to blade
failure. In this design, however, no account could be taken of plant
parameter variations.
The engine is particularly susceptible to variations in the low pressure
spool speed time constant, TLp (nominal value rLP0) , and its effect on x3
is shown in Figure 2(a); variations have little effect on x4. _nen
TLp = 0.5 TLP0, the low pressure spool speed, x3, has an unacceptable
overshoot, and for TLp = 1.5 rLp0 the response is rather sluggish. The
object, then, is to design a controller which is less sensitive to variations
in rLP and which will produce a similar nominal response. This nominal
response is characterized by a model matrix, Am, derived from the original
design.
One approach which suggests itself is to include a sensitivity term in
the PI; however, experience has indicated that simple inclusion of this term
-9-
leads to design difficulties since it can prove difficult to choose suitable
weighting matrices to simultaneously yield sensitivity reduction and a
desirable system response. It has been found easier to replace the state term
by a model-following term thus:
J = f0 {( - x-m) - ) + --sxT Qs x--s+ --pUTRp__pU}dr.
IMF has been employed here because, although it is less accurate than VMF, it
leads to a lower-order problem and has the added advantage that Rp may be
set to zero and PI minimization will still lead to a finite controller,
thereby releasing the designer from the chore of balancing Qp, Qs' and Rp
to achieve the desired result.
Setting Qp = i000 15 and Qs = diag(0,0,105'0'0) leads to the
controller:
u[00 [xj-13.70 61.44J x4
and the desensitized response for x3 is shown in Figure 2(b). State
dispersion of x3 is considerably reduced, and the overshoot when
TLp = 0.5 TLp 0 has been eliminated using a controller of comparable size to
that used in the original design.
The performance has been further improved by employing a dynamic
compensator (see Figure 2(c)):
[ [] 14.50j--2.28 8.61 x3 -0.37U = + X--P -8.49 37.28 x4 -13.60 7.54 --c
where
-i0-
111++l[x31jx4774 2°Jx = +--c -5.76" _c' _c (0) = 0
which also results in a controller of comparable size. IMF is unsuitable in
connection with dynamic compensation, so VMF has been implemented in this case
where the PI is
co
J = / {(_ - y_m)T Qp( X_c}0 YP - y-m) + XT uT--s Qs x + R u + x R dt.--s --p p--p --c c
The controller was found by setting Qp = diag(3000,3000), Qs =
diag(0,0,100,0,0), Rp = I00 12 and Rc = 12. (It is necessary for Rc to be
positive definite although its contribution here to the final design is
negligible.) It will be noted from the controller description above that a
minimal parameter canonical form has been adopted: using the gain fixing
facility, elements (i,I) and (2,1) of Cc are set to unity and elements
(1,2) and (2,1) of Dc are set to zero prior to minimization.
Recent Developments
Both the LQR and suboptimal regulator approaches rely on a trial-and-
error approach to design in which weighting matrices are adjusted iteratively
in a design loop until the final goal is realized. In a new approach [7], the
linear quadratic constrained regulator (LQCR), the design is recast as a
constrained optimization problem in which both design objective and
constraints are quadratic cost functions. This formulation permits the
separate consideration of, for example, model-following errors, sensitivity
-11-
measures and control energy as objectives to be minimized or limits to be
observed. The result is a one-pass design procedure whose solution, which
includes identification of active constraints at the minimum, reveals useful
insights into the design problem (see [8]).
The LQCR technique is available as research code, the software having
been constructed from the SUBOPT package. Present work is engaged in
investigating multi-objective optimization strategies in which multiple goals
are specified and design solutions sought through nonlinear programming
techniques.
Acknowledgement
The author is indebted to Mr. A. G. Shutler (Rolls Royce) for valuable
discussions in connection with gas turbine engine design.
-12-
References
I. Kleinman, D. L., "On an iterative technique for Riccati equation
computations," IEEE Trans. Automat. Control, Vol. AC-13, 1968, pp. 114-
115.
2. Koenigsberg, W. D. and D. K. Frederick, "Output feedback control with
application to unstable linear systems," Proc. JACC, 1970, pp. 674-682.
3. Fleming, P. J., The Suboptimal Linear Regulator CAD Program User Manual,
University College of North Wales, 1980.
4. Fleming, P. J., The Suboptimal Linear Regulator CAD Program Implementation
Document, University College of North Wales, 1981.
5. Michael, F. J. and F. A. Farrar, "The use of model-following methods to
generate suboptimal feedback control," Internat. J. Control, 17, 1973,
pp. 1325-1333.
6. Jones, C. W., Controller Design for a Gas Turbine Engine, Ph.D. Thesis,
University College of North Wales, 1981.
7. Fleming, P. J., "CAD of control systems: application of nonlinear
programming to a linear-quadratic formulation," Proc. IFAC Workshop on
Application of Nonlinear Programming to Optimization and Control, San
Francisco, June 1983.
-13-
8. Fleming, P. J., "An application of nonlinear programming to the design of
regulators using a linear-quadratic formulation," NASA CR-172253, ICASE
Report No. 83-60, October 1983.
-14-
1 I
0 " "--,_._-"-_" -="°-".......... 0
• ....
-0"25 -0-25
0 10 20 0 _0 20
t (s=c) t. (s¢c_
2 2
Ul .
r,.."_:...
r, _ /o %,
0 _ _ -:-:'_"_........ 0 ; :""*'".=.- :-_' - - - -
,q
It,//
o 0 20 o lO 20
t (sec) t (_ec)
Figure 1
-15-
://
, X 3 it I
x3 , ,/
o o
0 t(sec) 5 0 t (sq:c) 5
(a) (b)
F 'TLp=0.5"rLp°
X 3 ti q'LP= ,TLPO
.? "_L P =1 '5 'TL Po
.:P
_11
!-
o
0 t (_¢c1 5
(o)
Figure 2


I nepo_tt_o. NASA CR-!72536 2. Gove, nment Acces4,on No. 3. RKmpqtni'! CJUlog No.
ICASE Report No. 85-6
4 Title and Subtitle _. Repocl Datl
SUBOPT - A CAD PROGRAM FOR SUBOPTIMAL LINEAR REGULATORS February 1985
6. Parfo,ming OrganizationCoc_
7. Author(s) 8. PerformingOrganizJtionReportNO.
Peter J. Fleming 85-6
10. WorkUnll No.
9. PerformingO,ganizationNameandAdd,eu
Institute for Computer Applications in Science
and Engineering 11.Con.JetorGrantNo
Mail Stop 132C, NASA Langley Research Center NASI-17070
'i
Hampton, VA 23665 13. Typeof Reporland PeriodCove,_
12. SponsoringAgencyNameandAdd,e_$ Contractor Report
National Aeronautics and Space Administration 14. SponsoringAgencyCoc_
Washington, D.C. 20546 505-31-83-01
15. ,C_JpplementaryNotes
Langley Technical Monitor: J. C. South, Jr.
Final Report
16. Abstract
This interactive software package provides design solutions for both standard
linear quadratic regulator (LQR) and suboptimal linear regulator problems. Intended
for time-invariant continuous systems the package is easily modified to include
sampled-data systems. LQR designs are obtained by established techniques while the
large class of suboptimal problems containing controller and/or performance index
options is solved using a robust gradient minimization technique. Numerical
examples demonstrate features of the package and recent developments are described.
17. Key Words (Sugg_ted by Author(s)) 18. Distribution Statement
linear quadratic regulator problem 61 - Computer Programming & Software
suboptimal linear regulator problem 64 - Numerical Analysis
optimization
Unclassified - Unlimited
19. S_ity _a_if_(p_thi_re_rt) 20. S_urit¥Cla_Jpf _is pe_) 21. N_._fPages 22. Diceunciassz_zea unclasszzzea A02
For sale by Ihe Nalional Technical Inlo_malionService, Sp.np.lield, V.I_inia 2216!
NASA-Langley,1985
i
1
LANGLEY CENTER
3 1176 00520 8468


SUBOPT: A CAD program for suboptimal linear regulators

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