The Space COntrol Laboratory Experiment (SCOLE) is a challenge for control engineering applications. This is a result of the system dynamics, the available measurement information, the actuator capabilities and finally the specified performance requirements set. Results on the use of Model Reference Adaptive Control were reported. In view of the necessity for rapid response, this work deals with an optimal control formulation, with a minimum time requirement and constrained input. A mathematical statement of the problem is presented. The time optimal control formulation is presented and the reasons that make such an approach not promising are discussed. As a result, a pseudo time-optimal control algorithm is discussed. The proposed approach is tested to see if it satisfies the design specifications, and finally a discussion and suggestions for further research are provided

Kaufman, H.

Kyriakopoulos, K. J.

English

NASA Technical Reports Server

,I
Appendix A
A PRACTICAL APPROACH FOR MINIMUM TIME CONTROL
OF THE SPACECRAFT CONTROL LABORATORY EXPERIMENT
(SCOLE)
H. Kaufman and K.J. Kyriakopoulos
Electrical, Computer and Systems Engineering Dept.
Rensselaer Polytechnic Institute
Troy, NY 12180-3590, USA
(NASA-CK-Id600_) A PRACIIC_tL AoP_u_CH FOR
MINIMUM TIME CGNTR_JL r!F THL S_'ACECRAFT
CONT_L LA_]L]KaTni<Y FXPC_IMth_I (_COLF),
APPFN,P[X A (Rcnsse|,l_r Polyt,'chtljc Inst.)
26 L> CSCI 22P, c,_/i_
NgO-134qb
https://ntrs.nasa.gov/search.jsp?R=19900004179 2020-03-19T23:51:22+00:00Z
A PRACTICAL APPROACH FOR MINIMUM TIME CONTROL
OF THE SPACECRAFT CONTROL LABARATORY EXPERIMENT
(SCOLE)
H.Kaufman and K.J.Kyriakopoulos
Electrical,Computer and Systems Engineering Dept.
Rensselaer Polytechnic Institute
Troy, NY 12180-3590, USA
1. INTRODUCTION
The Space COntrol Labaratory Experiment (SCOLE) is a challenge for control
engineering applications. This is a result of the system dynamics, the available mea-
surement information, the actuator capabilities and finally the specified performance
requirements set.
Results on the use of Model Reference Adaptive Control have already been re-
ported [1]. In view of the necessity for rapid response, this work deals with an optimal
control formulation, with a minimum time requirement and constrained input.
In Section 2 a mathematical statement of the problem is presented. The time
optimal control formulation is presented in Section 3, and the reasons that make such
an approach not promising are discussed. As a result, a pseudo time-optimal control
algorithm is presented in Section 4. In Section 5, the proposed approach is tested
to see if it satisfies the design specifications, and finally in Section 6, discussion and
suggestions for luther research are provided.
2. BRIEF STATEMENT OF THE PROBLEM
The antenna-beam system to be controlled is modelled as:
= A. x + B • u A E _12x12,B E _12x3
Only the 12th order approximation of the distributed parameter system is considered
here. The states are organized in the form of modes, that is x21_l:position, x2i:velocity,
i = 1,2, ..., 6. The modes are dynamically decoupled, as can be seen from the structure
of matrix A:
A ._.
•/1 0 0 ...
o .;2 o ...
: : "'. 0
0 0 0 J6
J_
However, the coupling is introduced by matrix B where all three controls affect
all the modes:
B= Bi =
' B1 "
I°°°}
Z Bil Bi2 B,3
,B6
The output to be controlled is
y = = Cp ° x Cp E _:_2×12
Y
where yl, Y2 are the X and Y coordinates of the tip of the antenna, and
?Cpl 1 0 Cpl 3 0 ... Cp16 O)G={G21_ o cp23 o ... c_26 o
The actual measured output is
ym = Cm • x C,, E R3x12.
where
Cm _-
! Cmn 0 Cm,3 0 ... Cm:6
Cm21 0 Cm23 0 ... Cm26
Cm31 0 C_33 0 ... Cm36
Obviously, the output matrix Cp incorporates at the desired output only the position
component states, while Cm maps at the output only the velocity components of the
modes.
Given an initial output, such that the displacement Jly(O)[I = do = 1 ft, it is
desired to transfer the state vector to x(tf) = 0 in minimum time t], subject to the
constraints
luil < U i=1,2,3
where U = 6.25. The numerical values of matrices A, B, Cp, Cm can be found in [1].
Starting with an initial state at time t = 0:
x_(0)=[-3.3013 0 -0.71870 ... 01T
corresponding to the initial output
y(0) = x (0) = ]ly(0)ll= do = 1 It
11.3842
the output of the free response of the system is shown in fig. 1. It is obvious that the
settling time is t I > 1000 sec.
In order to improve the response of the system, the following design specifications
were considered:
• The decision should be a feedback controller.
• If possible, explicit use of observers should be avoided.
• The decision scheme should be implementable in an ON-LINE fashion.
3
• The designshouldprovide a controller robust with respectto initial state and
to parameteruncertainties.
In the sequel,a descriptionof the waysthat the problemwasattackedis presented,
and simulation results are given and analyzedfor performanceevaluation. Finally
suggestionsfor improvementsaregiven.
3. THE MINIMUM TIME PROBLEM
The first approachtowardstrying to solvethe aboveproblemwas to formulate it
asthe simpleminimum time control problem:
rain t l
u
s.t
2,=A.x+B.u
• (o)= x(tD =o
[uil<U i = 1,2,3
Numerical implementation of this optimal control problem requires its transfor-
mation to a Two Point Boundary Value Problem (TPBVP). Such a transformation
is obtained by introducing the scalar variable
t
tf
thus transforming the problem to the equivalent
n_n z • dr
s.t
x' = (A. x + B. u). z,
zl=O
(.)'_ a(.)
dr
x(0)=xo x(1)=0
[uil<U i=1,2,3
Solution of the above problem using multiple shooting methods, was not possible due
to numerical difficulties. The major problem is the almost marginal stablitiy of the
system that makes necessary the use of very small sampling intervals, and thus a very
large number of intervals and parameters.
Another approach to solving the above problem was that of [2], which in fact
transforms the infinite dimensional problem to a finite dimensional one. A series of
Linear Programs have to be solved. Similarly, the large number of variables along with
the resulting constraints that are extremely restricting, made the solution impossible.
4. THE PSEUDO-MIMIMUM TIME PROBLEM
All the above problems that led to numerical instabilities convinced us that find-
ing a minimum time control with the optimal control formulation presented above, is
a very difficult task. Its application would be even harder due to the fact that it is
open-loop control and therefore non-robust. Additionally, minimum time control tra-
jectories obtained for reduced order subsystems showed that the controller saturation
levels were changing signs every 3ms making the implementation impossible. There-
fore another approach was adopted; namely, the LQ Regulator with fixed function of
final state [3]. Such an approach is defined as follows:
SYSTEM
xk+l = A- xk + B. uk
PERFORMANCE INDEX
i T IN-I
J = _x N. S. XN + _ _(x T" Q" xk + u T" R. uk)
k=0
SN>_O, Q>_o, R>0
5
FINAL STATE CONSTRAINT
OPTIMAL CONTROL LAW
Cp'xN=O
Kk = (B T" Sk+l " B + R) -1 • B T. S/c+t • A SN : given
Sk = A T. Sk+l " (A - B. K/C) + Q
v/c = (A- B . g/c) T. V/c+l VN = C T
PI, = Pie+, - vkT+, " B. (B T . Sk+, " B + R)-' .
If_' = (B T. S/C+1 • B + R) -1
ak = -(K/c- V/c+,.pfl
B T. Vk+ 1
•B T
.v[)
PN=0
u/c = Gk • xk
Obviously this is a state feedback controller. The measurements matrix Cm con-
tains linear combinations of the velocities of the modes. Therefore state x can be
derived only by using an observer. An effort to avoid observer utilization is reported
in the next section.
The approach, described above, allows much flexibility in the selection of the
parameters of the matrices S, Q, R. Since the output constraint considers only the
linear combination of the position components of the states, the state can be reduced
by penalizing the final state, i.e by having a large S matrix. In the present set of
experiments, S was 50000 * I. The selection of the R matrix directly affects the
maximum value of the control trajectories. However since the bounds in all controls
are the same, R was chosen diagonal with the same elements for all controls. In the
present set of simulations R was I. Finally, the selection of Q matrix, provides some
valid flexibility since it is obvious from the control structure that the control values
6
are directly proportional to the states. An iterative adjustment was made using
maxte[o.t/](x_)
q" -- EiL1 maxte[o,tj](x 2)
Finally, the number of steps N that define the final time if, have to be found exper-
imentally based on the selected strategy.
As an example, consider again the initial state
x_---- [-3.3013 0 - 0.71870 ... O]T
corresponding to Ily(0)ll = x ft. Using the proposed controller, the system's re-
sponce can be shown by the trajectories of the two components of the output yk =
Cp.xk that are shown in fig.2. The corresponding optimal control trajectories
ui_ i = 1,2 are shown in fig. 3 (because u3 is of the order of 10 -1, it was not
shown). Inspection shows that the output is lead to zero while the input satisfies
the constraints. The settling final time was found to be t I = 130 sec which is
considerably less than the settling time of t s > 1000 sec of the free system of fig.1.
5. DESIGN SPECIFICATIONS TESTING
The proposed approach might be considered an ON-LINE procedure, because the
set of feedback gains Gk can be computed fast. This is so, because the computation of
Gk's is not iterative, but it is straightforward. It is also a feedback policy because the
control input is given by uk = Gk. xk. However this is state feedback, and not output
feedback as required. Additionally, a problem with this approach is that although tile
output is going to be lead to zero for every initial state, the input uk is not guaranteed
to be bounded by U. If the only considered measurements are ymk = Cm.xk, (i.e linear
combinations of velocities), then either the explicit or the implicit use of observers is
unavoidable.
Assuming that the desired ouput yk = Cp.xk is available, an attempt to obtain an
approximate estimate of the state, was the following: Matrix Cp is not square, and
7
thereforean invertible submatrix C'p C N2×2 of Cp was sought, so that
' = (c'p)-' ykX k
and
_k=[x_(1) 0 x_(2) 0 ... 0] T
could be used as an approximation to xk. By considering the first two modes of the
system as dominant, matrix C_ was constructed as
cL = [ca,, ca,,].
Still, this approach did not seem to work as can be seen by the following example:
Assume that at time t = 0 the system is at state
x0_=[0 0 0 0 0.3018 0 56.0972 0 ... 01r.
corresponding to an ouput, y0 = Cp-xg = [0.7589 2.2768] T with Ily011= 2.4 in.
Obviously, the magnitude of this initial deviation is smaller than the 12 in require-
ment, but this was done because of memory constraints of the computing system
used. According to the above scheme, the estimated initial state corresponding to
Y0 was
_1 -.. (CL)-l.y0 ._ [-0.6603 0 - 0.14370 ... 0] T.
This was the state that the controller actually "observed" at t = 0. Using _01 as an
initial estimate of the state, a set of gain matrices Gk k = 1,2,...N was obtained by
the algorithm of section 4. N was chosen in a such a way that the input
Uk = Gk " xk
of the fictitious system:
Xk+l = A.xk +B. uk Xo = _
are boundedby U. Since boundedness depends on the initial state, we denote the set
of gains obtained for this fictitious system by Gk(ko x) k = 1,2, ...N, in order to show
that these gains were selected based on k01.
The response of the output of the fictitious system is shown in fig. 4, and the
inputs ui i = 1,2, 3 of the fictitious system which are always less than U = 6.25 are
shown in figures 5,6 and 7 respectively. The final time for the fictitious system was
found to betl= 13 sec.
However, the initial state was not k_ but actually x_ which was essentially un-
known to the controller. The control gains Gk(3c_) were calculated based on the esti-
mate _0_ of the initial state, and unfortunately, the control inputs uk = Gk-[(C'p) -1 "Yk]
that were actually fed to the system, exceeded by far the bound U = 6.25, while the
actual output yk was not lead to zero. Even an increase of the final time ty from
13 sec to tl = 100 .sec at the fictitious system (which meant the calculation of a
whole new set of gains Gk(k0 x) k = 1,2, .... N') did not give better results, as can be
seen in figures 8 and 9 for y(1),y(2),u(1) and u(2) respectively.
Finally, the robustness properties to parameters variation were tested by increas-
ing the resonant frequencies of the system by 10%. Using as initial state the previously
used x02, the set of gains obtained for the nominal system for tl = 13 sec and finally
assuming state feedback, the "perturbed" system's output was lead to zero as shown
in fig. 10, however the control (fig. 11) exceeded U. The responces for this perturbed
system do not seem to be much different than these of the nominal (fig. 4,5,6). An
increase of the final time to tl = 15 sec seems to resolve the problem since the
output is lead to zero (fig. 12) and the control is now bounded (fig. 13). These can
be compared to the corresponding reponses of the nominal system for tl = 15 sec
cited at (fig.14 and 15).
6. DISCUSSION
It is obvious, that the proposed scheme requires knowledge of the full system state
for efficient implementation. In this case the system performs acceptably since the
output is lead to zero, while the state, because of the penalty imposed by matrix S
is lead close enough to zero. Additionally, the properties of the control scheme with
respect to system parameters variation are satisfactory if a larger final time is chosen
for the nominal system. The on-line calculation of G(t) will not be a big issue if
current parallel computation schemes are utilized [4].
In the case that the only available measurement is y,,, = Cm • x, which is a
linear combination of the velocities of the system, then no obvious way to guarantee
performance exist other than use of an observer.
REFERENCES
[1] A. Ansari, "Model Reference Control for the SCOLE", M.S. Thesis, ECSE,
RPI, December 1989.
[2] A.F. Fath, "Approximation to the time optimal control of linear state con-
strained systems", 1968 JACC, pp.962-969, Ann Arbor, Mich.
[3] F.L. Lewis, "Optimal Control", John Wiley, 1986.
[4] R.Travassos, H.Kaufman, "Parallel Algorithms For Solving Nonlinear Two
Point Boundary Value Problems Which Arise in Optimal Control", Journ. of Opt.
Theory and Appl., Jan. 1980, p.p 53-71.
10
yl
10_
8
6
4
2
0
-2
-4
-6
-8
-10
0 1_ 2_ 3_ 400 5_ 6_ 7_ 8_ 9_ 1000
y2
1°l
8
6
4
2
0
-2
-4
-6
-8
-10
0
i
100 200 300
Fig. ]
I I ! I I I
400 500 600 700
Output yl and y2 of the FREE
response of the system
8OO 9OO IOOO
! |
yl
2
0
-2
20
I I
40 60 80 100 120 140
15
y2
10
0
I
20
Fig. 2.
I I I I I
40 60 80 100 120
Output yl and y2 of the system under feedback
control
140
9
LA,
ul 0
I
2O
I
4O
I
6O
I
8O
I
100
I
120 140
4
3
2
I
u2 0
Fig. 3.
I I I I I
20 40 60 80 100
Feedback inputs ul and u2 of the system
I
120 140
k..
yl
0.8
o.6!I
J _ i ! !
-0.8 ' , , , , ,
0 2 4 6 8 10 12 14
y2
2.5
2
1.5
I
0.5
0
-0.5
-I
-1.5
-2
-2.5
0
I ! j ! i I
I I I I I I
2 4 6 8 10 12 14
Fig. 4. Ouput yl and y2 of EXPECTED response
7
42
ul 0
_8 I I I I I I
0 2 4 6 8 10 12
Fig. 5. EXPECTED feedback input ul
14
i
A
/ 4
ul
1..5
1
0..5
0
-0.5
-1
-1.5
-2
-2,5
0 2 4 6 8 10 12 14
Fig. 6. EXPECTED feedback input u2
0.025 .....
u3
0.02
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
-0.02
-0.025
0
I I I I I
2 4 6 8 10
Fig. 7. EXPECTED feedback input u3
12 14
,yl,
yl
2
-I
-2
-3
-4
-5
0
i i i i ! ,
i i i I i I I i I
10 20 30 40 50 60 70 80 90 100
y2
I ' ' ' ' ' ' ' I I
0
-2
-4
-6
-8
-10
-12 ' ' ' ' ' ' ' ' '
0 I0 20 30 40 50 60 70 80 90 100
Fig. 8. ACTUAL response of output yl and y2
ul
0
-2000
-6000
-8000
-12000 ' ' ' ' ' ' ' ' '
0 10 20 30 40 50 60 70 80 90 100
4500
U2
4000
3500
3000
2500
2000
1500
1000
500
1 i
0 10 20
Fig. 9.
I I I I I
30 40 50 60 70
ACTUAL feedback inputs ul and u2
I
80
I
90 I00
,8 I I ! i i i i
L,
yl
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
V
A
J a
V
-0.8 ' '
0 2 4
I I I I
6 8 10 12 14
2.5
y2
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
0
A
I
2
Fig. 10.
I I I I I
4 6 8 10 12
Output response yl and y2 of system with tf:13 sec
14
ul
4
2
0
i ! i i i
[ A nA f , ;; ,",, A
v V V
I I ! I i
2 4 6 8 10 12
i
14
2.5 i J i i !
A A A .A
u2
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
0
Fig. II.
iI
v v/i
2 4 6 8 10
Inputs ul and u2 of system with tf : 13 sec
A
12 14
.8 i ! ! ! ! T |
J
yl
0.6
0.4
0.2
0
-0.2
-0.4
-0.6 V
A
-08. ' ' ' ' ' ' '
0 2 4 6 8 10 12 14 16
2.5
y2
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
0
/1
I
2
Fig. 12.
A
J J
I I I i I I
4 6 8 10 12 14 16
Output response yl and y2 of system with tf = 15 sec
8 i i ! i i i |
ul
4
-2
.4
-6
-8
0
I
4
A A A
V
I
8
I
I0
V
i
12
i
14 16
2.5
LJ
u2
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5 _
0
A
I
2
Fig. 13.
A A
f
v V
f
I
A
I I I I I
4 6 8 10 12
Inputs ul and u2 of system with tf = 15 sec
A
I
14 16
.8 i I ! i I i I
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
0
I
2
A
I I I i I I
4 6 8 10 12 14 16
2.5
¢.-q
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
0
A
n I
I I I I I 1 I
2 4 6 8 10 12 14
Fig.14 Output response yl and y2 of nominal system with tf=15 sec
16
0i I i i i
A A A A
I
i
2 4 6 8 10 12 14 16
2
C'q 0
A _ A
I
I I
4 6 8 10
A
I
12
Fig. 15 Inputs ul and u2 of nominal system with tf=15 sec
!
14 16



A practical approach for minimum time control of the Spacecraft Control Laboratory Experiment (SCOLE), appendix A

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server