5, we obtainafter simplification K Controller Design with Regional Pole Constraints: Hyperbolic and Horizontal Strip Regions 23 where E=Ek is the complete elliptic integral of the second
Trang 1Making use of Eqs 310.02 and 310.04 of Ref 5, we obtain
after simplification
K
Controller Design with Regional Pole Constraints: Hyperbolic and Horizontal Strip Regions
(23)
where E=E(k) is the complete elliptic integral of the second
kind Note that £2 = 0 implies a = b=Q and K = E, from
which x 2 = 0, as expected, since this corresponds to the center
at (*,*) = (0,0)
Thus, we have reduced the study of Eq (16) with p = e to the
study of the averaged slow flow described by
where x 2 is given at Eq (23) and ( )' = d( )/d/x This equation
is valid to 0(e) as long as the flow stays in region 2a (Fig 1),
and it can be shown that if jii(0)>0 and e>0, then trajectories
originating in region 2a remain in that region.9 Numerical
comparisons show that solutions to Eq (24) agree quite well
with the "exact" solution to Eqs (2) and (3)
In this example there is only one region of phase space
where the unperturbed solution is periodic In case there is
more than one such region [e.g., when V(x; ju) is quartic], then
the form of the e = 0 solution and, hence, the form of the
right-hand side of Eq (14) are different in different regions
When the slow flow passes from one region to another, the
averaged equation may lose validity This is because the
transi-tion may involve crossing an instantaneous separatrix of the
unperturbed system At a separatrix, the period of the e = 0
solution becomes infinite, so that the average computed in Eq
(13) is over an infinite time interval, violating the conditions of
the averaging theorem (see Ref 9 for further discussion of
separatrix crossing)
Conclusions
We have presented a general formulation for application of
the method of averaging to a specific class of nonlinear
equa-tions The method exploits the existence of an energy integral
(the Hamiltonian) for the unperturbed system and leads to a
single first-order equation for the slow evolution of the
Hamil-tonian By using the canonical coordinate x as the fast
vari-able, the need to identify the rapidly varying phase angle (as in
Kruskal's method) is eliminated As shown in the example,
application is relatively straightforward when the form of
the potential leads to an explicit solution to the unperturbed
problem
References
*Rand, R H., and Armbruster, D., Perturbation Methods,
Bifurca-tion Theory and Computer Algebra, Springer-Verlag, New York,
1987.
2Meirovitch, L., Methods of Analytical Dynamics, McGraw-Hill,
New York, 1970, pp 302-309
3Rand, R H., "Using Computer Algebra to Handle Elliptic
Func-tions in the Method of Averaging," Proceedings of Symposium on
Symbolic Computations and Their Impact on Mechanics, American
Society of Mechanical Engineers, New York, 1990
4Lichtenberg, A J., and Lieberman, M A., Regular and Stochastic
Motion, Springer-Verlag, New York, 1983, Chap 2.
5Byrd, P F., and Friedman, M D., Handbook of Elliptic Integrals
for Engineers and Scientists, 2nd ed., Springer-Verlag, Berlin, 1971.
6Verhulst, F., Nonlinear Differential Equations and Dynamical
Sys-tems, Springer-Verlag, New York, 1985, Chap 11.
7Hall, C D., and Rand, R H., "Spinup Dynamics of Axial
Dual-Spin Spacecraft," Astrodynamics 1991, Vol 76, Advances in the
As-tronautical Sciences, Univelt, Inc., San Diego, CA, 1992, pp 641-660
8Rand, R H., Kinsey, R J., and Mingori, D L., "Dynamics of
Spinup through Resonance," International Journal of Nonlinear
Me-chanics, Vol 27, No 3, 1992, pp 489-502.
9Hall, C D., "An Investigation of Spinup Dynamics of Axial
Gy-rostats Using Elliptic Integrals and the Method of Averaging," Ph.D
Thesis, Dept of Theoretical and Applied Mechanics, Cornell Univ.,
Ithaca, NY, 1992
Y William Wang* and Dennis S Bernstein!
University of Michigan, Ann Arbor, Michigan 48109
Introduction
IN Ref 1 , fixed-structure synthesis techniques were used to design feedback controllers that place the closed-loop poles within specified regions in the open left half -plane Specifi-cally, circular, elliptic, parabolic, vertical strip and sector re-gions were considered with both static and dynamic output feedback controllers The purpose of the present Note is to extend the results of Ref 1 by considering two regions that were not considered in Ref 1, namely, hyperbolic and hori-zontal strip regions In practice, the hyperbolic region, which was considered in Refs 2-9, imposes a lower bound on the damping ratio of the closed-loop poles, whereas the horizontal strip region, briefly discussed in Ref 10, imposes an upper bound on the damped natural frequencies of the closed-loop poles The complicating aspect of both of these regions is that each region is reflected into the right half-plane Hence, it is necessary to exclude from consideration the right-half portion
of the constraint region The proofs of the following theorems are lengthy and hence are omitted in this paper Details are given in Ref 11
Characterization of the Hyperbolic Constraint Region
To begin, consider the two-sided hyperbolic region 3C(#,Z?) defined by
X < E (ReX)2 (ImX)2
b 2 >1
where a and b are positive real numbers To specify the
left-half region that is of interest for stability, we focus on the
subset 3£ L (a,b) = [X 6 3C(a,&): ReX<0], which corresponds
to the left branch of the hyperbola It is often convenient to write X= - fcow +700^, where 0<f < 1 and u d = o^Vl - f2 It is also known that the settling time is related to ReX In practice, design criteria may involve the damping ratio f and the
recip-rocal of the settling time rj = fa n The constraint f > fmin and
T? ^ fJmin can be enforced by the hyperbola parameters a and b
by choosing a =r/min and 6 =(r?min/fmin)Vl-f£in Next it can
be shown that the region JC(a,Z?) can be equivalently charac-terized by
^ [ \ € C : 1 +2«5(ReX2) + 7|X| 2<0]
where
4a 2 b 2 ' 7 =
a 2 -b 2
This leads to the following result Let "spec" denote spec-trum
Proposition 1: Let ,4 € (R/7X",let V h € (R"xn be positive
def-inite, and let d and 7 be real numbers such that <5<0 and 26<7< -26 Then, if there exists an n xn positive definite matrix Q h satisfying
0 = Qh Q h A 2T ) + yAQ H A T + V h (2)
Received Jan 25, 1992; revision received June 15, 1992; accepted for publication July 1, 1992 Copyright © 1992 by the American Institute of Aeronautics and Astronautics, Inc All rights reserved
*Graduate Student, Department of Aerospace Engineering tAssociate Professor, Department of Aerospace Engineering
Trang 2then spec 04 )C 3C(#,£), where
(3)
Note that 3C(a,Z>) includes regions lying in the open left
half-plane C~ and in the open right half-plane C +
Prop-osition 1 applies to all of 3C(#,&), not just 3CL (#,&)
Con-sidering stability, we now combine the standard Lyapunov
equation with Eq (2) Thus, the characteristic roots will be
constrained to lie inside the left hyperbolic constraint region
Theorem 2: Let A, F, V h , Q, and Q h € (R"xn and Fand V h
be positive definite matrices Then, if there exist
positive-defi-nite matrices Q and Q h and real numbers d and 7 such that d < 0
and 26<7< -26 satisfying
0 = Q h + d(A 2 Q h + Q H A 2T ) + yAQ H A T + V h (4)
then spec(y4)C JCL (#,&), where a and 6 are given by Eq (3).
Let ft and d € (R2"x2* be defined by
4 7 + 5(A2 ®A 2 ) + 7.4
where (x) and © denote Kronecker product and sum
Proposition 3: Let d and 7 be real numbers such that 6 < 0,
and 26 < 7 < - 26, and let a and b be given by Eq (3) Then the
following statements hold
1) Suppose a >b Then Q and A are asymptotically stable
if and only if spec(,4)C 3£ L (a,b).
2) Suppose a < b Then <f and A are asymptotically stable
if and only if spec(,4)C W L (a,b).
Lemma 4: Let spec(^4 ) C JCL (a, b), where a and b are given
by Eq (3), and let Fand V h € (R nxn be positive-definite
ma-trices Let 6 and 7 be given by Eq (1) Then there exist unique
n x n positive-definite matrices Q and Q h satisfying
(7)
Controller Synthesis
Based on Eqs (6) and (7), we can now perform controller
synthesis Here we consider the linear time-invariant system
(9)
where x(t), u(t), w(/), andy(t) are «-, m- 9 d-, and
/-dimen-sional vectors, and A, B, C, and D l are corresponding
con-stant matrices With static output feedback of the form
it is our goal to select K such that the closed-loop system has
the following properties:
1) The closed-loop poles are constrained to lie in the
hyper-bolic constraint region 3CL(#,Z?)
2) The performance index
1
/ 4 l i m S
-is minimized
The closed-loop system (8-10) is given by
(11)
(12)
where A s =A +BKC To determine a feedback gain K
satis-fying properties 1 and 2, we begin by defining an open set of feedback gains JC5 4 [K : spec(A s )c3C L (a,b)] 9 which place
the closed-loop poles in 3£ L (a,b) We assume that JC5 is not empty Equation (11) can be written as
Furthermore, by defining the nonnegative-definite state co-variance
1 f '
the system (8-11) combined with criterion 2 will be as follows:
Minimize J(K) = trQR S9 where R 5 ^R 1 +R 12
+ (KC) T R 2 KC subject to
where V s =DiD? However, to impose criterion 2, we may
overbound the desired performance index as shown in Lemma
5 so that a minimization procedure can be carried out later
Lemma 5: Let K € JC5 and let V s and V h € (R"x" be
positive-definite matrices Then there exist nxn positive-positive-definite ma-trices Q and Q h satisfying
0 = Q H + 4- Q^f ) + yA s Q H A T8 + (15)
(16)
Furthermore, y(Ar)<^(A'), where S(K) tr(Q/2s
We can now formulate the auxiliary minimization problem:
determine K € 3C5 that minimizes $(#) where the
positive-def-inite matrices Q h and Q satisfy Eqs (15) and (16)
Theorem 6: Let K € 3C5 minimize $(/0 Then there exist
positive-definite matrices Q h , Q, PA, andP5 € (R"xn satisfying
0 = Q h A 2T )
d(A 2T P h + P^2) + + P h
(17)
(18)
(19)
(20)
where, under the assumption that II defined next is nonsingu-lar,
Q)C
+ (CB®B T P h Q h C T )U mxl ] +y(CQ h C T ®B T P H B)
such that the feedback gain K is given by
Trang 3Let us now design a full-order dynamic compensator
satisfy-ing pole constraints with regulator/estimator separation
Con-sider the linear time-invariant system
weighted estimator cost is given by
(23)
where x(t), u(t), w(0, and>>(/) are «-, m-, d- 9 and
/-dimen-sional vectors, and A 9 B, C, D\ 9 and D 2 are corresponding
constant matrices Now the goal is to choose A C9 B c , C c such
that the dynamic compensator
(25) satisfies properties 1 and 2
The closed-loop system and performance criterion of Eq
(11) can be restated as follows:
Minimize
(26) subject to
where
r A BCCI r *, *12cc i
» r* /4 r r T T* T r T n r
|_ZJ C C ^c J L ^ c ^ l 2 C c ^2Wj
\ V, VnB
LB C VT 2 B C V 2 B
The set of dynamic compensators that places the closed-loop
poles in JCL(a,Z?) is defined by
K d 4 [& C9 B C9 C c ) : spec(A*)C3CL (*,&)]
The following result is analogous to Lemma 5
Lemma 7: Let the triple (A C9 B C , C c ) € 3C</, and let V d and
V h € (Rw x n be positive-definite matrices Then there exist
posi-tive-definite matrices Q and Q h € (R nxn satisfying
0 = Q h + b(AlQ h + Q h A 2dT ) + jA d Q H A Td + V h (27)
Furthermore, J(A c ,B c ,C c )<d(A C9 B c ,C c ), where 3(A C ,B C9 C C )
Here we enforce regulator/estimator separation for
deter-mining (A C ,B C ,C C ) Thus, the dynamic compensator is
as-sumed to be of the form
x c = Ax c + Bu + Bc (y - Cx c )
u = C c x c
(29) (30)
such that A C ^A +BC C -B C C To exploit this, it is useful to
design the estimator by defining the tracking error e 4 x -x c
such that
A+BC C -BC c
0 A^
Then the goal is to separately place the eigenvalues of the error
dynamics and regulator in the hyperbolic constraint region
3CL(a,&) From Eq (31), it is noticed that there are in fact
two separate problems for determining B c and C c The
sub-problem for the estimator can be formulated such that the
where W is a given n x n positive-definite matrix However,
Eq (32) can be rewritten as
Note that Q e satisfies the Lyapunov equation
(33)
(34)
where A e =A+B c C and V e = V^-B.V^- V n B c +B c V 2 B*.
For the regulator, we consider
(35) (36)
(37)
=trQ r R r
(38)
x = Ax +Bu
u = C c x
which implies that
x = A r x+D l w
The corresponding cost is
where Q r satisfies
where A r =A+BC c andR r =R l +R n C c
Now let 3C5be defined as 3C5 4 [K: spec(A +B c C)CW L (a,b)]
to characterize a dual set of gains for the closed-loop pole assignment To place the eigenvalues of error dynamics and regulator in 3CL (#,&), it is also required that A e and A r be stable as guaranteed by Proposition 1
Lemma 8: Let B c € 325 and C c € 3C5, and let V e , V he , V l9
and V hr € (R nxn be positive-definite matrices Then there exist
positive-definite matrices Q he and Q e € (R"x/2 satisfying
0 = Q he
(40)
such that J e (B c ) < $ e (B c ) = tr (Q e W + QAe) Furthermore, there exist positive-definite matrices <2/jr and Q r 6 (Rwxn satis-fying
0 = A r Q r + Qrv4^ + Ki (42)
and such that /r(Q)<&(Q) = tr(QrJRr + Q^r)
Theorem 9: Let £c 6 3?5 and Cc € 3C5 where ^Je(Bc) and
3 r (C c ) are minimized, and let V e9 V he , V l9 and F^ € (RnXAZ be positive-definite matrices Then there exist positive-definite
matrices P he , P e , Q e , and Q/je € (Rnx" satisfying
4- K (43)
Trang 40 = Q he + d(A 2e Q he + Q he A 2eT ) + yA e Q he A Te + V he (44)
0 = A Te P e + P e A e + W (45)
0 = / + P he + d(A 2eT P he + P he A 2e ) + 7^/V4 e (46)
and positive-definite matrices P hr , P r , Q r , and Q hr 6 (RWX/I
satisfying
(47)
(48)
0 =
rv4?) + yA Tr P hr A r (50)
where, under the assumption that U e and nr defined next are
nonsingular matrices,
0 = A^Pr
0 = 7 + Phr +
1 vec Or
-4 6 [[ / n x / + [(CQhePhe) <8> C7 ]
= RnQr + d(B T A T P hr Q hr + B T P hr Q hr A T )
nr 4 Q r ®R 2 + d[(B ®B T P hr Q hr )U mxn
+ (QhrPhrB ®B T )U mxn ]+ jQhr ®B T P hr B
such that the compensator is given by
A c = A - B C C + BC C
B c = -vec"1!!"1 vec tt e
C c = -vec^n^vecG,.
(51)
(52) (53) Finally, we briefly discuss regional pole placement within
the horizontal strip region To guarantee stability, we are only
interested in the region that is in the open left half-plane The
left portion of the horizontal strip region can be characterized
as
3C5(w) 4 [X € C : ReX<0, (ImX)2<oj2]
where oj is the upper bound on the damped natural frequency
Lemma 10: The set JC5(co) is equivalent to
where
1
2co2
By comparing 3C(#,&) with 3C5(o>), we immediately notice that the constraint inequalities are similar The differences only arise at the coefficients of the inequalities Thus, the major results derived so far for the hyperbolic constraint re-gion can be carried over to be the results for the horizontal strip region with only slight modifications of the coefficients
Conclusion
In this Note we established an upper bound for the cost that can be minimized subject to a pair of matrix root-clustering equations These equations were used to constrain the poles of the closed-loop system to lie in a hyperbolic or horizontal strip region contained in the left half-plane The left hyperbolic region was chosen because of its ability to set desired bounds
on the damping ratio and settling time Because of the similar-ity between root-clustering equations of hyperbolic and hori-zontal strip regions, the results obtained for the left hyperbolic region can be applied to the left horizontal strip region with minor coefficient changes Future research will focus on nu-merical techniques for solving the matrix algebraic equations
Acknowledgments
This research was supported in part by the Air Force Office
of Scientific Research under Grant F49620-92-J-0127 The au-thors wish to thank Shaul Gutman and Wassim Haddad for helpful discussions and suggestions
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