where r-, = I; and h, solves the optimization problem of maxi- ... quickly by a search procedurc. ..... 11um gasoline engines with contiouously varir\bte tiilnsimissioiil.. in Prrri., ... suing his doctoral degree, he has been a consultant to the.
nlinear stems: A Case Study
By Robert H. Miller, llya Kolmanovsky, Elmer G. Gilbert, and Peter D. Washabaugh HIS ARTICLE PRESENTS A Lyapunov function approach to the control of nonlinear systems that are subject to pointwise-intime constraints on state and control. This approach is applied to an electromechanical system that serves a s a prototype for tile first mode of an electrostaticaiiy shaped membrane. Electrostatically shaped mrmhranes have been proposed as mirrors and antennas since theearly 1960s [1]-[4] becausetheycanbeused as lightweight reflectors for radar. radio, and optics applications. Lightweight reflectors are in demand, for example, in spacecraft applications whcre launch weight is a significant constraint, A thin, electrically conducting membrane is formed into a desired shape by
23
electrostatic forces that are controlled by varying the electrical potential between the membrane and an electrode mounted below it. Becausc the membrane is under lateral in-plane tensionand auniformnornialstressdueto theelectrostatic potential, it assumes a paraboloidal shape for optics applications. Since the focal length can be varied by changing the gap distance, electrostatically controlled membranes are particularly suitable for adaptive optics applications 151. Small focal lengths needed for many applications can be achieved if the gap distance between the membrane and the fixed plate is made sufficiently small.
Concurrent with the membrane work, experiments on a simpler physical system that illustrates similar control difficulties were deemed important. This led to the system considered in this article: an electroinagnetically actuated mass-sprlng damper (EAMSD) that exhibits the same general behavior as the first mode of an electrostatically controlled membrane [S]-[S], It has been implemented in the control laboratory of D.S. Beriistein (see Fig. 1). Results presented here includc the description of an effective nonlinear control scheme antl experiments involving its use with the laboratory hardware. Naturally, the objective of this article falls short of developing . .a successful control scheme forelectrostatically controlled membranes; however, light is shed on the basic control scheme and Its effectiveness in dealing with the key problems of electrostatic membrane control. The highly nonlinear behavior of t h e EAMSD s u g g e s t s t h e u s e of Lyapunov function methods. Although these methods are popular for designing lionlinear controllers, their application to systems with general reference conimands and demanding state antl control constraints iS problematir. Theapproach described here takes its inspitation from prior work on reference governors [9]-[ 151. Reference governors are auxiiary nonlinear systems that filter reference commands to closed-loop systems in such a way that constraints on their internal variables are satisfied. Unlike input preplanning schemes, such as those used in robotics applications to generate constraint-admissible motions, reference governors operate online, responding immediately to reference coinmands as they occur. Maximal constraint-admissit,le positively invariant sets play a dominant role in the prior literature. Our approach exploits a family of Lyapunov functions that are parametrized by set-l,oint-tletermiiied equilibria of the closed-loop system. The resulting reference governor generates a piecewise constant output that, subject to future constraint satisfaction, tracks as closely as possible the reference command. The design process has considerable flexibility and is conceptually simple. It should be emphasized that the exploitation of parametrized equilibria is distinctly differentthan in gain-scheduling of linear designs. In the reference governor approach, it is assumed a priori that a control scheme (perhaps a gain-scheduled one) has been devised that provides good performance in a neighborhood of each set-point determined equilibrium. The central purpose of the reference governor Is to guarantee constraint satisfaction in the presence of a very general class of input commands. This article is organized as follows. Thc next section describes the nonlinear control algorithm in general terms. Then an einpirically based model for the EAMSD is cieveloped that includes inequality constraints on magnet cur-
se
Also, the voltage required to maintain the equilibria with a small gap distance decreases as the gap distance decreascs 141. Consequently, steady-state operation at eqiiiiibria with sinall gap distance values is highly desirable. However, these equilibria are open-loop unstable [4], and active control is needed to achieve their stabilization for a suitable range of gap set points. Unfortunately, this technology was essentially abandoned due to difficultiesin stabilizing the membrane at desired open-loop unstable equilibria. Thus, it is a n application where control may be a critical enabling teclinology. P.D. Washabaugh has developed an cxperimental testbed for studyingthe application of advanced membrane control algorithms (see photograph on previous page). While the deflection of the membrane is described by a partial differentialequation, the control voltage is scalar. The laboratorymembrane exhibits the open-loop unstable equilibria and is subject to severe state antl control constraints. Theconstraintsareduetotheliinits on themaximalvoltage the amplifier can deliver. membrane collisions with the fixed electrode, arid electric field breakdown.
24
IEEE Control Systems Magazine
February 2000
rent and mass displacement. Equilibria for the modcl are then characterized, after which equations that implement the control algorithm for the EAMSD are considered If the current constraint is sufficiently binding, the set of achirvable equilibria is disconnected. Then, as described in tlie next section, the overall control algorithm must be augmented to include a temuorarv. unstable ohase for imidementing transfer between equilibria that belong to separate components of the set of achievable equilibria. Finally, some experiments with tlie actual hardware are described The conclusion summarizes the principal featurcs of the reierence governor scheme.
.
Basic Control Algorithms
where x is an n-vector and U Is an mvector. Our reference governor approach is based on parametrized equilibria of this system that aregenerated by means of astabiliziiigand performance-enhancingcontroller U = U ( x , r). The resulting closed-loop system is =
f(x,r)
=
F(x,U(x,r)),
is bounded; ([I) for all x c n(r)
Then, under an additional LaSallc invariance assumption [17], it follows tliatn(r)is apositivelyinvariant set that is a
-
Closed.Loop System
Figure 2. Refr,ren O and x(t)+x,,(r) as t + - ] . Suitable Lyapuiiov functions are often associatecl with controller synthesis methods, or they can 11e generated by recursive proccdures [ 161. Another approach is to feedback linearize (1) by U(x, r). V(x, r) can tlicn be generated by solving a Lyapunov equation associated with the resulting linear system (2). See [ 181 and [ 191 for examples of this approach in tlie context of automotive control problems. In most applications, such as the EAMSD, state aiid control constraints are Imposed by physical limitations of the hardware or safety considerations. Usually these constraints can be represented by a system of nonlinear inequalities:
(2)
where r is an mvector reference command which, when held constant, becomes a set point. Set points are restricted to a s e t s which determines the desired equilibria (i.e., r t S impilesf(x,(r), r) = O).Notctiiatiorsomer t S,theremay be several equilibria; the functional notation x,(r) means that one of them has been chosen. Avariety of effectivedesign tools exist for determining the controller, l i ( ~r),, at each equilibrium point; see [16] for a general discussion of Control Lyapunov Function (CLF)-based design procedures. Often sucli controllers can be shown to be optimal with respect to certain types of Integral performance criteria and to have an associated Lyapunov function. Dynamic controllers can also be iritroduccd. In this case, the state i n (2) is augmented to include controller states. The key to our approach is an r-parametrized Lyapunov function for (Z), V(x, r). Specifically, we s u p ~ m s ethat V(x, r) is smooth and for each r E S satlslies the following assumptions: (a) V(x,r) tO; (ti) V(x,r) = 0 only for x = x@(r);(c) there is a9(r) > Osuch that the sublevel set
rciIrnaryZOIIII
Reference Governor
I .
Suppose the plant motlei is of the form
X
's
H(X,ll) are determined is coinputationaiiy straightforward. We want r(t) t o move toward q,(t) = r, from its current value. This requirement is implemented by setting, f o r k = O , l , .., ,
r,, = rI3-)+ Mr,,(W -rh
(8)
where r-, = I; and h, solves the optimization problem of maxiniizinghsubjecttoo < h < l s i d x ( k T ) t n ( r , - , + h(r, -r,+,)), Equivalently, h,#solves the problem: niaximiic h subjcct tu
0 s h < l , F ( x ( k T ) , r ; , ~ , + h ( r , ( k T ) ~ r ~ ~ , ) ) ($1) s0, where
F ( x , r ) = V ( x , r) - 9 ( r ) .
(10)
Since h is a scalar, the optimization problem can be solved quickly by a search procedurc. A theoretical justification of the control scheme is givcn in [ZO]. In addition t o assumptions (a)-(e), three more are required. While technical in character, they are intuitively reasonable. Roughly speaking, they have the foliowhig objectives: (f) res t r i c t s r s o t h a t P(xfk7". . . ,. r,+, + h(r,,(kT)-r,*~,))isdefinedforall h E [0,l]; (9) establishes in a precise way the required smoothness of the there functions f . V , andn: . (ii) . , implies . is a decrease in F for each reference governor time step. Specifically, the assumptions are: (f) S is compact and convex; (g) f ( x ,r ) and V ( x , r ) are continuously differentiable on R"+"', and q(r)is continuous o n 3 (h) for every fixed r E S , V(x, r)satisfies a LaSaile-like condition i n [20] on Il(r), T h e verification of conditions (f)-(h) for the EAMSD is discussed later. There. as in most practical situations, the conditions occur naturally as a cooseqiience of constructing a V ( x , r ) that satisfies assumptions (a)-(e), Under assumptions (a)-(h), the reference governor is guaranteed [20]to behave nicely. It is not necessary t o assumethatx(O)=x,(r,)orthatr,,(t) =r,forallf 2O.Suppose that r,,(t)
E
S f o r t 2 0 and that
Choose an r_, so that x(0) E Il(r-,).Then, from the above argumentsand(I)), itfollowsthatther,, belongtoSanclarcwell defined. Moreover, x(t) ~ n ( r , )for t t [ k T , ( / z + l ) k~ 20. , Thus, the constraints h(x(t), r(t)) < 0 , t 2 0 , are satisfied. Suppose further that there is a f t 0 such that r,,(t) = r, E S for ail I 2 f. Then [20]shows that tiie r, have a finite settling t 0 such that r(t) = r, for all time (there is an integer I th~)andx(t)~x,(r,)ast~-. Note that the condition (11) does not depend on r, and that theset Xmaybequiteiarge.Tiius,bychoosingr_,sothat x(0) E n(r.J, it is possible to extend significantiythe rangeof
a
Figure 3. Mhtalionfiir the EAMSB.
26
IEEE Control Systems Magazine
rebruary 2uuu
> I
initial conditions that may be handled. There are several 0'45 practical ways for linding r It is only necessary that rl, t S 0.4 and F(x(O), r-,) < 0. One api. proach is to choosc a finite grid ol points E S, j t J , such that X = u,Gln(?,)c approximates X . Then, for x(n) E 2,r I can b e found by cvaluating F(x(O),7,) f o r j t J until F(x(O), 7,) < 0. Another allUnstable Equilibria proach is to apply an iterative procedure for the minimization of F(x(O), r) for r t S. in general, it is not necessary to obtain the minimum; the iterative proccss is terminated wlicn F(x(O),r ) 5 0. T h e general idea of t h e control algorithm 'is not new 0 0 0.001 0.008 0.00: 3.004 0.005 0.006 0.007 0.008 0.009 0.01 and, in fact, underlies niucli of tile research on reference d' Position (m) dmax < governors (see, e.g., [9]-[15]) gure 4. bquiiihrin positions. and, morc generally, t h e use of positively invariant sets for constraint satisfaction Reformulating the equations of motion into the form (1) (see, e.g., [Zl] for an extensive literature review). Sec also gives [22], which exploits nested families of I.yapunov functions to develop control laws for robots moving in an ohstacle-restricted space. What is new is the emphasis on implementation of a reference governor in the context of general Lyapunov-based designs for continuous-time nonlinear systems. where d = x,.d = x2,and the control input is U = I "
,,
5.
x
The control Is constrained to be nonnegative,
Model for the EAMSD Fig. 3 illustrates the basic features of the EAMSD. Idealized equations of motion are
where m is tlie mass, k is the spring constant, cis the damping constant, i ( t j is the current in the electromagnet coil, d ( l j is the position of the mass, z,is the distance between tlie electromagnet and the inass when i = 0, and c1 is a current-to-force constant. Experimental measurements lead to a somewhat different form for the last term. Specifically, in MKS and ampere units,
d.'= - - kc / - - d c+ - - a i(t)" in in m (d,, - x)' ' ~
(131
U(t) t o .
(15)
This is a pointwise-in-time control constraint that renders t h e system underactuated in a physical sense. Specifically, t h e control can pull tlie mass closer to t h e electromagnet coil, but it cannot push t h e mass away from t h e ciectromagnet coil. The maximal current is also limited, thereby leading to another pointwise-in-time control constraint,
n(t)
llllx
= (i,,,,,,)",
(161
Collisions of the mass with the fixed electromagnet must b e avoided. Thus, d ( t ) < 2,. I n fact, it may be wise to provide a margin of safety or limit peak swings of tran< zo sient responses. For this reason, we Introduce dnlay and require
wiierc a = 4 . S x I O s , [3=1.92, y=1.99, c=0.0659, k=38.94. z o = 0.0086, d,, = 0.0102, and m = 1.54.
FCIXLW~
zonn
IEEE Control Systems Magazine
d ( t ) < d,..>..
(1 7) 27
Open-Loop Equilibria For constant U, the equilibria are obtained by equating the right-hand side of (14) t o zero. For x , =de,,, x i =0, and ~ ( t =) U , this gives
The resulting relationship between dFqand i is shown in Fig. 4. The maximum current allowed by the relationship is i' = 0.3631. Denote the corresponding values of U and d by U' = (i.)" =0.1430 and d =0.0034. For 0 5 U < t i . , there are two equilibria for d: 0 9 $(U) < d and dd:,(U) < do;for U = u ' , there is one equililxium value: = d . Thus, for U,,,,2 u~~the set of constraint-admissible steady state displacements, dFq,is the interval 10, d,,,,,]. Let u,,,,~ be determined byd:o(u ,,,,,)= d,,,,,. Then foru ,,,,, < U n,i, < U ' , the set of steady-state constraint-admissible displacements is the union of two disjoint intervals, [O,d,!,,(u,,,,,)lU [d:o(u,nax),d,,,,]; f o r 0 < U,,,,, < 11 ,,,,1 , it is the interval [O, d
d
Implementation of the Reference Governor We now apply the ideas discussed under "Basic Control Algorithms" t o the EAMSD. The main steps are the choice of U ( x , r) and V ( x , r), application of the constraint conditions t o the determinatioii of 9 ( r ) , verification of assumptions (a)-(h), anti definition of the cnntroi equations. We choose U ( x , r ) so that it stabilizes the equilibria and linearizes the resulting closed-loop system:
U ( x , r ) = d ( d 0-x,)'(kr-c,,x,).
(2 1)
Then (2) becomes
a mass-spring d a m p e r w h o s e equilibrium s t a t e is
x,(r) = ( r , 0). While the controller aiiuws an increase in systemdamping,itis"iowgain"inthesensethatthenaturalfrequency of the open-loop system is not increased by feedback. Forsimpiicity, we consider first the case where c,, = Oand i,,,,, is significantly larger than i". This eliminates constraint (16). Constraint (15) together with (21) requires r > 0, acondition that is satisfied by choosing S so that it contains no negative elements. The remaining constraint, (17), gives h ( x , r ) : h , ( x , r ) = x , -d,,,,. From h ( x e ( r ) , r ) 9 0, it follows t h a t constraint-admissible set points must satisfy r < d,,,,. Thus, S c [0,dm2J A natural choice for V ( x , r ) is the "energy" of tile system (22) relative t o
x,,(r): m V ( x ,r) = r x i 2
+ -k( x ,
- 1)'.
2
(23)
Since c > 0, we understand that the necessary and suffiSince = - ( c + c,()x:, it follows that assumptions (a)-(d) cient condition for an equilibrium point t o be locally asymptotically stable is that the stiffness coefficient, K , is are satisfied positive: To find tile most effectiveq(r) that satisfies (e), we maximizc 9 ( r ) subject t o condition (7). This optimization problem has a simple geometric interpretation: n ( r ) touches the set { x : / i , ( x , r ) t o } , but there is no x E n ( r ) such that h , ( x , r ) > 0. Thus, It is easy to confirm that constraint-admissible equilibria 9 ( r ) = m i n V(x,r)s~ibjectio/i,(~,r)=O. are stable f o r 0 < dc,,< d' and unstable f o r d < u,~,,,, the set of constraint-admissible open-loop equilibria splits It follows that into two intervals; a stable interval and an unstable interval. If the control current constaint is tight (U,,,,, < U ' ) , these intervals are disconnected.
IEEE Control Syslems Magazine
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February 2000
Clearly, (e) is satisfied if r is restricted so thatq(r) > 0. This is where assumption (0 comes in; S must be a closed interval, chosen so that it excludes r = d,, . The obvious choice for S is S = [0, d,,,,, -E] where E > 0. Since (f) and (g) are now satisfied, it remains to verify (h). This condition [20] requires that for all constant r E S,the solutions of (Z),starting in the set [ x : 0 < V ( x , r ) < q(r), p ( x , r ) = 01 immediately leave the set. The property is easily confirmed. l o complete the statement of the control law, it Is necessary to solve the h-optimization problem (9). Normally, the solution must be obtained by an iterative procedure. Hnwever,inthiscase,F(x, r)islinearinrandthereis aformulaforh:ifF(x(kT),r,,(kT)) S1,thenh =1; Y othcrwise,
*
where r, t S,.To achieve this objective, it is only necessary, for somep, E S,,to invent a constraint-admissible bridging controller, U,,(x), that causes the state of the resulting closed-loop system to move from initial states in a neighborhood, N , of x,(p,) to states in X,. The overall control strategy consists of the following steps: (1) apply the reference governor on S,with r,,(t) = p, and continue its operation until there is an integer k , such that x ( k , T ) t N ; (2) starting at I = k,T, apply the bridging controller and increase f until thereis ak, such thatx(kJ') E X,; antl (.?)startingat t = kJ', apply the reference governor on S, with r_, determined by 1 0 ~ ~
.. .. .. .. , .. .
. .. . . .
,
. .. .. .. . . .
. .. .. . . . . .
6-
where
Position
4-
When i,,,,, c i' and c,, # 0, the derivation of the control equations becomes more complex. Then, all three of the constraints (15)-(17) become active and h ( x , r) has three components. Each of them is handled separately in the manner described in the preceding paragraphs. Let the resultingoccurrencesofq andh bedenotedbyq'(r) -6 anti h'. i =1,2,3. Then, q(r)=min[q'(r)&r), q3(r)) 0 5 10 and h=min(h',h2,h3}.Note that whiieq(r)isnolonTime (s) ger differentiable, it satisfies assumption (g) because it is continuous. Similarly, the requirement 'igure 5. Rridging controller with refewice gov'vno,: on S becomes more complex. For example, if c(, = 0 it follows that the set of constraint-admissiin-3 ble set points is E , U E 2 where E , = [O, d:o(un,ax)] and E, = [ ~ / ~ ~ ( u , , , , ) , d ,Since , , , , ] . the two intervals are disjointed and.^ c E , UE, isconvex,Sis given by b y e i t h e r S,=[0,dLg(u,,,.,)-~] o r S, = [d:,,(u,,,.,) + E, d,... - E ] . Thus, the reference governor cannot achieve transfers between all equilibria determined by r E S,US,.
I 15
A Control Scheme for Disconnected Equilibria We now discuss, in general terms, "bridging strategies" that allow automatic set-point transfers between disconnected sets such ass, antl S , in the preceding paragraph. Suppose reference governors have been designed for each of the two intervals and r., is determined byx(0) E Il(r-,), as described in "Basic 0 1 2 3 4 5 Control Algorithms." Let X , and X , denote the Time (s) corresponding sets defined by (1 1). Supposc we want to move from a state x(0) E X , to Xc(r,) Figure 6. / h i t i o n r q x m s r without r q % r m c ~governor. ,
znnn
F C ~ W W ~
ILEE Control Systems Magazine
6
7
28
Here,C/(x,G)isgivcn hy(Zl)wIthc,$ =-2C < - c . The parameter 6 is a sinall positive number that guarantees that thc motions generated by U,, move away from x(0) E N. Note that U,(x) t O whenx, > 0. Thus, (28) automatically imposes the c o n s t r a i n t s (15) and (16). T h e c o n s t r a i n t x,(t) < d,,,,),is confirmed dircctiy by simulations. Furx(0) E N, it was also confirmed that x(0)eventually enters n(O.0075) c X,. Figure 5 Illustrates the result. At t = 7 5 , the reference governor for S, is engaged with r,!(t) = p l =0.0075, and it brings x(t) to the corresponding equilibrium (p2, 0). Notethat positlonconstraintx,(t) 2 0.008issatislied lor all t.
0.7 0.6 -
-< + C
a,
5
0.5 -
0.4 0.3 -
0.2
0.1
I
- 0 0
1
2
3
4
5
6
7
Experimental Results
Ancillary equipment associatcd with the EAMSD Time (s) consisted of an Intel-based coniputer with a DSPACE controller board, a differential-transformer position sensor, and a current amplifier. Position data were sampled at a I-kHz rate. The x 104 mass velocity was estimated using a finite-differI ence filter based on the bilinear transform and approximating the transler function C(s) = S(TS + I)-]. A break frequency of20 Mz provided accurate velocity data with a dynamic lag negligible compared t o tlie system response times. Colitroller cock was generated from a Siinuiink block diagram and cmex file using the Real-Time Workshop In MATLAB. The inherent damping in the mass-spring damper is very small, so extra damping was added by setting c,( = 3. Since the maximum amplilicr current exceeds i' by a wide margin, constraint (16) is inactive, even though cC,# 0. Thus, h is given by (26) and 'i, by (8). Each full computation of r;, takes approxi100 p.It is implicit in the dcscription of mately I the reference governorthat ri, is computed instan0 1 2 3 4 5 6 7 Time ( s ) tancously from x(kT). Practically, this assumption is met if T >> Il)-'. In fact, it was possible t o operate the system successfully with T = 1W3,the basic sampling period of the interface. Figs. 6-9 illustrate some of the experimental results. In both cases considered, the reference x(k,73 E Il(r-,) and r,,(t) z r,. Clearly, this strategy leads t o command of position inuves lrom r,, = 0 (an open-loop stable tlie set point r, anti then, as t + -, x(t) --f x,,(r,). equilibrium) to r,, = 0.0051 (an open-loop unstable equilibFig. 5 illustrates the simulated response of a bridging 6 and 7 show thc position and ciirreiit responses rium). Figs. cootroller lor the EAMSD. It destabilizes motions for initial for the controller (21) with no reference governor. ?'he sysconditions in a neighborhood N ofx,,(O) = 0. leads them into X,,anti satisfies tile current constraints. The controller is tem is lightly damped, resulting in a Iargc overshoot in both position and current. In fact, the peak value in the position defined by d ( t ) is almost 0,0077,which is very close t o the mass-magnet ci,(x) = min{C/(x,8),un,nx} whenx, 2 0 collision limit z , = 0.0086. Figs. 8 and 9 show the responses =
30
o
wllcnx, < n.
(28)
with the reference govcrnor and the position constraint
IEEE Control Systems Magazine
I:A~~ zoo0 w~
d,,,,,=0.0056, which is 10% larger than the set point. The peak position almost reaches but does not exceed tlie constraint.The output ofthereference governor has a sizable jump at the time of change in the set-point command, hut then increases smoothly for about 1.0 s until it reaches the set-point command. This delay is small compared with the overall settling time; howevcr, the minimum distance between the mass and the magnet is increascd from0.0009 to0.0031. It is interesting to note that in Fig. 9. tlie peak current is less than0.4G A, Thus, the position constraint has also produced an appreciable reduction of tlie peak current.
0.5 0.45 0.4 0.35 0.3 0.25-
e
2
0.20.015 0.1 0.05
Conclusion This article has described a coiiceptually sim"0 ple and practical approach to a difficult class of problems: set-point control of nonlinear systems that are subjcct to hard constraints on state and control variables. It exploits a family of Lyapnoov functions parametrized by t h e s e t points and constructed to meet an intuitively sensible requirement: for each set-point-determined eqnllibrinm state, t h c Lyapunov function defines a constraint-admissihle domaiii of attraction. This property is t h e basis for a finite settling-time reference governor that filters tlie s e t p o i n t commands and generates constraint-admissihle motions. Online implen~entationof t h e governor is straightforward, requiring at most the solution uf several nonlinear root-finding problems in asinglevariable. The approaches taken in tlie prior literaturc have several advantages over our 1.yapunov function approach. Since they are based on maximal constraint-admissible invariant sets, the resulting reference governors respond more rapidly and function over the larger set of initial conditions. Further, for discrete-time linear systems, it is often possible to obtain explicit representations for the invariant sets, thus avoiding the problem of finding an appropriate family of Lyapunov functions. However, they also have disadvantages: the theory is more complex. and it does not apply to continuous-time norilinear systems. The purpose of the article has bcen to describe the details of the approach and illustrate their application usins an interesting example, tiic EAMSD. I.aboratory experiments demonstrate the effectiveness of the resulting reference governor. Prom their success with this and another example application, the authors believe that the described methodology has much to offer. The EAMSD is a prototype prohlem for a morc complex system, an electrostatically actuated membrane, The necessity of dealing with inultlple modes of the meinbranc is essential to successful design of an experimental control scheme for this more coinpiex application; this design will he pursued in our future work.
moo
F ~ ~ ~ L I Z W ~
1
2
4
3
5
6
7
Time (s)
References [ 11 C.S.Clailiu imd N. Rercket. "Coniigoringnoelectrostatic membrane mirmr
iby least-squares fitting with analytically dcrivcd influencc fimctions,".l. Opti~ cnl.Sociely ofArrierica. "01. :I. 110. 11, pp. 1833-18JH. 19R6. [2l R.P. Grosso ant1 M. Yellin, "Thr mcmbranc niiiriir as an adaptive optical element," J. Ol~limlSocielyofAnwico, vol. 67. 110. 3, pp. 399406, 1977. [3l J.H. I.nng ant1 D.H. Stsrlio. "F.lcr:trostati~allyfigured reflccting imembrane BlltrnrlaS for SatelliteS;' I Trms. Aahmal. Corm, vol. AC-27, nu. 3. 1))). 666-6711. 1982. [4] D.J. Mihoret and P.J. Redmond, "Elretrostatiri~llyformed ttiiteiiiiils." .I. .spncacrarr. 19x0. [SI R.K. ' ~ y s ~ , , , , P ~ iofAdooptior , ~ ~ , ~ l ~Opricr. ~ s NcwYark Acadcmic Pres.;. 1991. [GI .I.Hong. I.A. Cummings, P.D. Wiirhabaugh. ancl D.S. Bernstcin. "Stiibilization Of an Plrctroinngiictically controlled oScilliitoi." iii Proc. Americtrn C m mic ~ ~ rilii&ipiLia, i , PA. 1998. p p 2775-2778. (71 I. Kolmaoovsky, R.H. Millcr, P.U. Washahaupti. ant1 F.O. Gilbert. "Nonlinear w r i t i d o i ~ l e ~ t i ~ s t i l t i ~slrapcd a l l y mrmixancs with state iintl control cumtraints.'' 10 Pro?, Americon Cnnlrlll Corif. Phiisdeiphia, PA, 199R. p p 2997-3002. [RI 1K11. Miller. "ClectmStatic membrane control project," Technical Report, IJrpartmcnt Of Aerospace Engincerisg, Univcisity of Michigan 1998. [91 A. Bemporttd. "Control 01 constrhiried nonlinear s y s t e m via refwrme managemeiit,"in Pm'rAstericne Conlroi C w i , Albuqocrqur. NM. 1997. ipi). 3343-3347. [ I O ] A. Bempor;id. "Refcrcnce goucmor for constrained iionli~~rili systems." 1EFI;"~ron.sAiilom
