thesis methods to the ACTEX flight experiment. We show that the decentralized
static output feedback formulation of fixed-structure controller synthesis can ...
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A99936580 CONTROL
DESIGN FOR THE ACTEX FLIGHT ROBUST, FIXED-STRUCTURE
AMA-99-3952 EXPERIMENT TOOLBOX
Scot L. Osburn*, Joe Corradot, Scott Erwint, Dennis S. Bernstei#,
Abstract
organization.
USING
THE
and Wassim M. Haddadll”
most general form, fixed-structure synthesis has been developed for affinely parameter&xl closed-loop dynamics which corresponds to a decentralized static output formulation [S].
In this paper, we implement fixed-structure controller synthesis methods to the ACTEX flight experiment. We show that the decentralized static output feedback formulation of fixed-structure controller synthesis can directly account for the control-structure constraints of the ACTEX flight experiment. Finally, we show that the ACTEX controller structure can be written as a decentralized static output feedback problem and obtain feedback controllers for suppressing broadband disturbances.
In the present paper we show that the decentralized static output feedback formulation of fixed-structure controller synthesis can directly account for the control-structure constraints of the ACTEX flight experiment. Specifically, we show that the ACTEX controller structure can be written as a decentralized static output feedback problem. Having done this, we then proceed to apply the techniques of [S] to obtain feedback controllers for suppressing broadband disturbances.
Introduction Structural Dynamics Modeling of the ACTEX Flight Experiment
The ACTEX flight experiment provides a unique opportunity for users to implement and test controllers on a space based platform. In this regard, the hardware environment has several features that must be accounted for in specifying control algorithms.
The ACTEX flight experiment consists of a plate connected to a satellite by 3 struts. Each strut is equipped with its own control piezo-actuator as well as a colocated and nearly colocated sensor. A disturbance can be introduced to the experimental package through each of the 3 control actuators, or through a disturbance actuator on the plate. In addition, each of the 3 control actuators has an independent decentralized analog controller.
First, the feedback control algorithms that can be implemented on ACTEX are fixed gain, and thus adaptive controllers cannot be used. Furthermore, these fixed-gain controllers are analog, which avoids sampling effects. Finally, the implementable analog controllers have a prespecified structure in which only filter gains and natural frequencies can be modified. Since this constraint does not permit implementation of dynamic compensators of arbitrary structure, standard LQG and ZH, methods cannot be applied.
The dynamics of the ACTEX experiment can be represented by the continuous-time system
k(t) = AZ(~) + Bu(t) + Dlw(t), (1) In this paper we apply fixed-structure controller synthesis y(t) = Cx(t) + D?.&(t) + L%w(t), methods to the ACTEX flight experiment. Fixed-structure (2) methods have been extensively developed in 3-1s and 7-tHa where x E I?!“, u E 77?‘, y E I$, and w E Rd are the state, /I-& settings in continuous and discrete time [l]-[8]. In its input, measurement, and disturbance, respectively. The disturbance w is a standard zero-mean white noise process. *AerospaceEngineering Department, The University of Michigan, Ann Arbor, MI, U.S.A. The performance variables are given by tSchoo1of AerospaceEngineering, GeorgiaInstitute of Technology, Atlanta, GA, U.S.A. z(t) = &x(t) + E*u(t) + Dow(t). (3) $Air Force ResearchLaboratory, SpaceVehiclesDirectorate, Kitland AFB, NM, U.S.A. Several experiments have been run on the ACTEX package, §Professor,AerospaceEngineering Department, The University of Michigan, Ann Arbor, MI, U.S.A. with telemetry returned for identification purposes. MeaqTbis research WBS supported in part by the Air Force Office of Scientbic Resurements were taken from onboard accelerometers, sensor search under grant F49620-96-I-0037. inputs, and actuator outputs, sampled at 4 kHz. Using IlProfessor,School of AerospaceEngineering, Georgia Institute of this data, identification was performed on the plant from Technology,Atlanta, GA, U.S.A. **This research wassupportedin part by the Air Force 05~x3 of Scientilic Re the strut 1 actuator to the strut 1 nearly:colocated sensor. search under grant F4962B.96I-0125. For frequencies below 200 Hz, these dynamics can be repre‘Copyright @ 1999 The American Institute of Aeronautics and Astronautics Inc. All rights reserved.
18
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sented by (l)-(3) where. -1.56 -77.98
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0 0 -0.25 -24.99
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or published
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I II
-1.10 -0.21 -0.11 0.03
’ B =
hi(t)
=
[ 1 -1.10
D1 =
0.70 1.83 6.88 ] ,
-0.21 0 -0.11 0 0.03 0
’ D2=[0
0.11,
DO=
i = 1: 2: 3, (5)
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=
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(11
and
[ 1 i-‘i
.
&,, : [ 0 In, ] ,
In this paper, we wih consider this SISO problem, with input at the strut 1 sensor, and output at the strut 1 actuator. In the next sectionwe review the decentralized static output feedback framework, and-express the ACTEX model in this representation.
c,, b [ c
0 ] 1
c,,
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Next, defining
In this section we use the fixed-structure control framework given in [S] to transform the &optimal ACTEX control problem to a decentralized static output feedback setting. Consider the Cvector input, 4-vector output decentralized
Bu
Z b
c,
c .I 1 b
2
[ au,
Bu,,
5:.
I &~
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[q!u,
q!q
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] ,v$
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v w3
93
and rewriting the decentralized control signals (7) in the compact form f?(t) = kc(t), (8) where
Ee
[
1
40 0 0 Bc 0 , 0 0 cc
(9)
the system matrices A, fi,, and E in the closed-loop dynamics i+(t) = AZ(t) + hII( z(t) = h(t), can now be written as :
Figure 1: Decentralized Static Output Feedback Framework
izJ = & + BuKL@J/y, E = c.i + l&ukLJ&,
system shown in Figure 1, where 6(s) represents the linear, time-invariant dynamic system where i(t)
=
dqt)
+ 6
i%&(t)
+ LLw(t),
A = A + B,,~LEi-‘C,,
t E [O, m)(4)
i=l
19
t E [O,co)
(10) (11)
,
(cl11999 American Institute of Aeronautics
& Astronautics
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with permission
Fixed-Structure Synthesis for the ACTEX Flight Experiment
b-4
=
s2 + .3wrs + wl”
G3c(S)
=
G4c(S)
=
ksw,2 S2 +
.3W2S
S2 +
.3W2S
S2 +
X
+
Wz”
+
W;
b-4
+
.%2S
+
1
2 Y9
W2
WlS
kw;
4 x
s2 +
.3wlS
+
k& G5dS)
=
s2 +
X
.3W3s
+
52 + 34s + wz”
X
To obtain the partial derivative of the Lagrangian with respect to the free parameters in the controller gains, we iirst specify the controller configuration. As an example, we consider controller configuration 1, (12). The settings for each of the other controller configurations are given in the Appendix.
WlS
s2 + .3wlS + w,2y7
-16 _< k3 5 16,
For controller configuration 1, with (17) the block-diagonal matrix k has the form (9). Note that this controller has four free optimization parameters, namely, wl, ws, k3, and k4. We construct the matrix
-8 5 k4 5 8.
(16) Now, we consider the controller associated with strut 1, and express the controller in the decentralized static output feedback framework. Controller conIiguration 1, given by (12), in dynamic compensator form, can be expressed as 0
4,=
-;;"
[
0
1 -ob3"1
0
0
0
;
;
-wz"
-0.3!d2
Cc = [ k3wf
0 k4w;
(21)
ac as = ATP+FA+R, ac aP = A&+ijAT+V.
and where wr, wz, w3, k3, and k4 are subject to the constraints 0 _< wr, w2, w3 5 1024,
= tr Qii,
where P E lRirxir is a Lagrange multiplier and v = fiDT. The partial derivatives with respect to 0 and p in (22) are given by
(12)
(15)
w$
W2
J(k)
where s _> 0 satisfies, (19). The necessary conditions for optimality can be derived by forming the Lagrangian - -C(P, 0, I?) = tr ofi + tr P(AQ + OAT + v), (22)
(14)
wTY1
organization.
In order t,o design ‘&-optimal controllers for the ACTEX experiment; _ we pose the following optimization problem: Determine K that minimizes
(13)
52 + .3wrs + w12y,
and/or author(s)’ sponsoring
and RI g ETE1, R2 fi EZE,, and Rlz 4 ETE, = 0.
Since continuous-time controllers are implemented on ACTEX, hardware constraints place a limit on the form that the controllers may take. Each of the three struts on ACTEX has a decentralized controller, available in second order, third order, fourth order, and fifth order, in the following configurations: G(s)
of author(s)
K=
W00l0 [ 1 0 0 0
w2 0 0 0
0 k3 0 0 k4
’
0 ] .
Controller configurations 2 through 4, given by (13)-(15), can be expressed similarly, and are given in the Appendix. Having specified the form of the controller, we consider the Ns synthesis problem. The ‘l-L2norm of G,,(s) is given by IIG,,(s) II; = tr OR, where 0 is the unique, fi x ii nonnegative-definite to the algebraic Lyapunov equation o=AQ+QAT+imT,
(18)
0
solution
L1=
(19)
where
20
0
-0.3 0 0 0 0 -0.3 0 0 0 0 0 0 0 0 0 0
00
0 0 00 0 0 00 00 00 00 00
0
, RI=
0
1000 ‘0 0 ‘0100 0 0 0 0 0 0 0 0 0 0
0
0
0 0 0 0 0 0 0
0 0 0 0 0
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‘1
0 000 -10 0 0 0 000 0 -1 O./O 0 000 0 0 00 0 o-o 0 0 000 0 0 00
Lz=
0 0 0 0 L3=
0 0 0 0
0 0 0 0
0 0 0 0
,R2=
,M3=
0
0
0
0
0
0
0
0
0
0
0
0
0 0 0 0100 0 0 0000 0 0 0 0 0 0 0 0
or published
0 0 0 0
with p&mission
and/or author(s)’ sponsoring
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for large values of w,: i = 1,2,3, the cost function depended very weaMy upon these values. Therefore, WJ, i = 1,2: 3: were initially chosen to be 48, 72, and 96, respectively. Furthermore, since the open-loop system was stable, an initial stabilizing design could be obtained by setting k3 = k4 = 0. With these values set: the BFGS quasi-Newton algorithm was applied to find the x2-optimal solution for a given fixedstructure controller configuration.
TV
0 0 , 0 0 0 0
of author(s)
0 0 0 0
I
[ 1
,
0 0 1 0
0000
1100
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0 0 0000
0 1
0
0
0
1.
0
A
1
000000000 R=OOOOOOOOO 3
4
000001000 [ 000000010 Now with r?- in terms of K, we can take the derivative of the Lagrangianwith respect to K. Define the foIIowing notation 7fiI+kLpvvur
6
Figure 2: Open-Loop Impulse Response
a%TBp,sC,TLk-T,
fi E 7T@j3bvT .L _ -T u yw K 1 leTTz?T,E:QC;LW-? The derivative of the Lagrangian with respect to K, for each controller configuration, is given by
Simulation
Results
The ACTEX experiment is a lightly damped flexible structure, as can be seen by the impulse response showrrin Figure 2, where z(O)=[O 0 0 11’.
Quasi-Newton
For each of the four different controller configurations, three different controllers were designed by setting the control weighting matrix R2 to 1, 0.01, or 0.0001. The ‘Hs cost of these controllers can be seen in Table 1. It is seen that the ‘H2 cost is generally best for the hrst controller configuration. In fact, the other controller configurations ran into stability constraints, as explained in Section . Therefore, the optimization routine terminated due to a boundary
Algorithm.
To solve the nonlinear optimization problem posed in Section , a general-purpose BFGS quasi-Newton algorithm [9] is used. The line-search portions of the algorithm were modified to include a constraint-checking subroutine to verify that the search direction vector lies entirely within the set of parameters that yield a stable closed-loop system as well as lying within the allowable limits of the free parameters given by (16). This modification ensures that the cost function J remains defined at every point in the line-search process.
Rz= 1 R2 = 0.01 R2 = 0.0001
Config. 1 0.6457 0.2444 0.1910
Config. 2 0.7076 0.5012 0.5942
Config. 3 0.5833 0.6264 0.6246
Table 1: 7-12 Costs for Various Controller Weightings
One requirement of gradient-based’optimization algorithms is an initial stabilizing design. Initial designs showed that
21
Config. 4 0.7216 0.3769 0.5618
Configurations
and
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Conclusions
h
JR,=1 tI 31.081 t 70.702 1 0.184 I-O.125 t 1 36.443 99.118 0.320 0.402 R; = 0.01 R2 = 0.0001 1 37.366 123.840 0.208 0.631
In this paper we showed that the control algorithm structure implemented on the ACTEX Ilight experiment has the form of a decentralized static output feedback controller. This controller structure is amenable to fixed-structure optimization methods. Using the Robust, Fixed-Structure Toolbox, we synthesized several feedback controllers and we assessed their performance under a variety of conditions.
Table 2: Optimal Controller Parameters for Controller Configuration 1 constraint rather than a small gradient condition. However, the optimization routine worked well for the lirst controller conllguration, and it can be seen from Table 1 that the 3ts cost of the closed-loop system decreases with increasing controller authority.
P b
The optimal controller parameters for this controller configuration are given in Table 2. In fact, with R2 = 1, we can see from the output signals of the impulse response in Figure 3 that this controller does not attenuate the vibrations significantly. However, as the authority is increased, the attenuation becomes greater, as seen in Figure 4 and Figure 5, though this does increase the control effort expended, as shown in Figures 6-8, which could lead to actuator saturation. Finally, Figure 9 shows the Bode plots of the open-loop and closed-loop systems. It is seen that the size of the first peak in the closedloop response is decreased as the controller authority is increased. Another important feature of the closed-loop frequency response is the high-frequency rolloff. Since the ACTEX system model has a relative degree of zero, the system possesses gain at all frequencies, thus unmodeled high-frequency dynamics could destabilize the open-loop system, whereas these dynamics would be attenuated with a strictly proper dynamic controller.
Figure 4: Closed-Loop Impulse Response, R2 = 0.01
References [l] D. C. Hyland and D. S. Bernstein, “The Optimal Projection Equations for Fixed-Order Dynamic Compensation,” IEEE Trans. Autom. Conk., Vol. AG29, pp. 10341037, 1984.
l-
[2] D. S. Bernstein and W. M. Haddad, “LQG Control With an H, Performance Bound: A Riccati Equation Approach,” IEEE nans. Autom. Conk., Vol. 34, pp. 293-305, 1989.
:,i
[3] D. S. Bernstein and W. M. Haddad, ‘Robust Stability and Performance via Fixed-Order Dynamic Compensation with Guaranteed Cost Bounds,” Math. Contr. Sig. Sys., Vol. 3, pp. 139-163, 1990. [4] W. M. Haddad and D. S. Bernstein, “Generalized Riccati Equations for the Full- and Reduced-Order MixedNorm Hz/H, Standard Problem,” Sys. Conk L&t., Vol. 14, pp. 185-197, 1990.
5 Figure 3: Closed-Loop Impulse Response, Rz = 1
[5] Y. Ge, L. T. Watson, E. G. Collins, Jr., and D. S. Bernstein, “Probability-One Homotopy Algorithms for
22
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Figure 5: Closed-Loop Imp&e
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[S] R. S. Erwin, A. G. Sparks, and D. S. Bernstein, “FixedStructure Robust Controller Synthesis via Decentralized Static Output Feedback,” Int. J. Robust Nonlinear Con&., Vol. 8, pp. 499-522, 1998.
10 r010000000l
I -0.3 01 -0.3 0
L1=)
0
0
[
2
1 -ob3”1 Wl
[I
0 ; -w;
0 ; -0.3w2
0
&=
;
,
Cc = [ 0 0 k3w;
1.
0
I
0 0 0 0
the controller given by (13) can be
0 -;;‘:
A,=
000000000 000100000 000000000 ooooooogo, 000010000 000000000 000’000000 -0 0 0 o-o
K(J=
Appendix For this configuration, expressed as
0 ks]
and
[9] J. E. Dennis, Jr. and R. B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, 1983.
Configuration
organization.
Note that there are three free parameters, namely, til, ~2, and 4. Thus
[7] R. S. Erwin and D. S. Bernstein, “Fixed-Structure DiscreteTime Mixed H2/H, Controller Synthesis Using the Delta-Operator,” submitted to Int. J. Co&r. ._ [8] R. S. Erwin, D. S. Bernstein, and D. A. Wilson, “FixedStructure Synthesis of Induced-Norm Controllers,” submitted to Int. J. Con&.
Controller
and/or author(s)’ sponsoring
Figure 6: Closed-Loop Control Effort; Ra = 1
Response, Rz = 0.0001
Full and Reduced Orde; Hz/H” Controller Synthesis,” Optimal Cob. Appl. Meth., Vol. 17, pp. 187-208, 1996.
of author(s)
Lz =
0 ] .
0
23
0
0
-1 0 0 0 0 -1 000 0 0 000 000 0 0
0 0 0 0
0
0 0 0 0 0
0
0 0 0 0 0
:
0 0 0 0, ‘OOO-T 01 01 0
,R1=
000 0 0 0 0 0 0 0 0
, 0 0 0 0
100000000 0 0 1 0 0 0 0 0 0 , 0 0 0 0~0 0 0 0 0 I
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Ls=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0
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,
000 1 1
A&=
0 0 1 0 0 0
000000000 000000000. 000000010 Controller
Configuration
For this configuration, expressed as
or published
,
with permission
0
Lz =
I
1
0
01 y
-;wp; -wz"
0
-&SW2
Cc = [ 0 0 k3w;
,
and/or author(s)’ sponsoring
0 0 0 0
I
100000000 001000000)
,&=
[ 000000000
0 0 0 0
000 0 0 0 0 0 0 0 0 0 0 10
MS=
000000000
000000000. 000000010
I
organization.
01
0 0 0 0 0
000 LB=
the controller given by (14) can be
4
0
-1 0 0 0 1 -1 0 0 0 0
0 0 0 0 0 01 0 0 0 0
3
r0
of author(s)
000 [ I 0001,0 0
I
0 ] .
Wl 00 i 1
Note that there are three free parameters, namely, ~1, ~2, and ks. Thus ,
00
K= and
0 W2
ks 0
010000000 000000000
000100000 000000000 000000000, 000010000 000000000 000000000 000000000
K,,=
0
0
000
-0.3 0 0 0 1 -0.3 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
1 0 0 0 0 1 000 0 0 0 0 0 0 0 0
0
L1=
Figure 7: Close&Loop
, RI=
T 0 0 0 , 0 0 0 0
24
Control Effort, R2 = 0.01
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Controller
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Configuration
For this configuration, expressed as
A,=
0 -wl” 0 0 0 -0
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4
the controller given by (15) &I
1 -0.3w1 0 Wl 0 0
with permission
0 0 0 -w; 0 4
0 0 1
0 0 0 -0.3wz 0 0 0 0 -w;
Cc= [ 0 0 0 0 ksw;
be
0 0 0 0 1 -0.3w3
01. I IO'
500 ' 10'
Figure 9: Bode Plots for Open-Loop and Closed-Loop Sys. tems
Note that there are four free parameters, namely, ~1, ~2, 03, and k3. Thus
rwl 0 0 01
and -0 1 0 0 0 0 0 0 0 0.0 ~00000000.00000
0.0’
oooiooooooooo 0 0 0 0 or0 0 0 0 0 0 0.0
0000010000000 0000000000000 Ko= 0 0 0 0 0 0.0 -0000001000000 0000000000000 oooooooooooo,,o 000000000000’0 0000000000000 0000000000000 r.
L1= Figure 8: Closed-Loop Control Effort, & = 0.0001
25
-0.3 0 1 0 0 0 0 0 0 0 0 0
0 0 0 -0.3 0 0 0 0 0 0 0 0 0
00 0 00 0 00 -0.3 0 00 00 00 00 00 00
0 0 0,O
0 0
,
0 0 0 0
,
RI='
0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0000
0 0 0 0 0 1 0 0
O-T 0 0 0 0 , 0 0 0
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-0 -1 0 0 0 0
0 0 0 -1 0 1
0 0 0 0 0 0 0
0 0 0 0 0 0 0
Lz=
or published
00 0 0 00 0 0 00 -1 0
00, 00 00 00 00 00 00
1
1000000000000
R2= S
L3 =
R3 =
0000100000000’ 0000000000000 0010000000000 0 0 0 0 0 0 0 0 0 0 0000 0 0 0 0 0 0 0 0 0 0 0 0 0 ‘0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0 1
0 0 0 0 0 0 0
0000000000000 0000000000000 0000000000000 0000000000010
with permission
’
M3=
0 0 0 0 0001 [ 0 0
0 0
0 0
0
0
1 *
26
of author(s)
and/or author(s)’ sponsoring
organization.
