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Unformatted text preview: IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 34, NO. 3. MARCH 1989 293 LQG Control with an H0° Performance Bound:
A Riccati Equation Approach DENNIS S. BERNSTEIN, MEMBER, IEEE, AND WASSIM M. HADDAD, MEMBER, IEEE Abstract—An LQG controldesign problem involving a constraint on
H. disturbance attenuation is considered. The H, performance con
straint is embedded within the optimization process by replacing the
covariance Lyapunov equation by a Riccati equation whose solution leads
to an upper bound on L1 performance. In contrast to the pair of separated
Riccati equations of standard LQG theory, the lI.constrained gains are
given by a coupled system of three modified Riccati equations. The
coupling illustrates the breakdown of the separation principle for the H..
constrained problem. Both full and reducedorder design problems are
considered with an H... attenuation constraint involving both state and
control variables. An algorithm is developed for the fullorder design
problem and illustrative numerical results are given. I. INTRODUCTION HE fundamental differences between Wiener—Hopf—Kalman (W HK) control design (for example, LQG theory [1]) and Ho.
control theory [2]—[4] can be traced to the modeling and treatment
of uncertain exogenous disturbances. As explained by Zames in
the classic paper [2], LQG design is based upon a stochastic noise
disturbance model possessing a ﬁxed covariance (power spectral
density), while Hm theory is predicated on a deterministic
disturbance model consisting of bounded power (square—integra—
ble) signals. Since LQG design utilizes a quadratic cost criterion,
it follows from Plancherel’s theorem that WHK theory strives to
minimize the L2 norm of the closed—loop frequency response,
while Hon theory seeks to minimize the worstcase attenuation.
For systems with poorly modeled disturbances which may possess
signiﬁcant power within arbitrarily small bandwidths, H0, is
clearly appropriate, while for systems with wellknown distur
bance power spectral densities, WHK design may be less
conservative. In addition to the fact that Hon design embodies many classical
design objectives [5], it also presents a natural tool for modeling
plant uncertainty in terms of normed Ho, plant neighborhoods. In
contrast, the H2 topology has been shown in [6] to be too weak for
a practical robustness theory, while the He, norm is not only
suitable for robust stabilization but is also conveniently submulti
plicative. Within the WHK statespace theory, however, the
appropriate robustness model appears not to be a nonparametric
normed plant neighborhood as in Ha. theory, but rather a
parametric uncertainty model. The principal technique for bound
ing the effects of real parameters within statespace models is
Lyapunov function theory (see, e.g., [7]—[16] and the references
therein). Such structured uncertainties are difﬁcult to capture
noneonservatively within Hm theory except with specialized
refinements [17]. ' Manuscript received November 24, 1987; revised August 1, 1988 and April
20, 1988. Paper recommended by Associate Editor, A. C. Antoulas. This
work was supported in part by the Air Force Ofﬁce of Scientiﬁc Research
under Contracts F4962086—C0002 and F4962087C0108. D. S. Bernstein is with the Government Aerospace Systems Division,
Harris Corporation, Melbourne, FL 32902. W. M. Haddad is with the Department of Mechanical and Aerospace
Engineering, Florida Institute of Technology, Melbourne. FL 32901. IEEE Log Number 8824525. In spite of the fundamental differences between WHK design
and H,n theory, a signiﬁcant connection was discovered in [18].
Speciﬁcally, Petersen observed that a modiﬁed algebraic Riccati
equation developed for parameterrobust fullstate—feedback con
trol can be reinterpreted to yield controllers satisfying Hm
disturbance attenuation bounds. This relationship was further
explored in [19] where it was shown that the Hanoptimal static
fullstatefeedback controller is also optimal over the class of
dynamic full—statefeedback controllers. The results of [18]—[20]
thus solve the standard problem considered in [3] and [4] for the
fullstate—feedback case. The extension of these results to the standard problem for
dynamic outputfeedback compensation, however, was not given
in [18]—[20]. Within the realm of quadratic robust stabilization,
the dynamic outputfeedback problem was addressed in [7]. The
results of [7] involve a pair of decoupled modiﬁed Riccati
equations along with an auxiliary inequality. Using different
techniques, a more complete solution was obtained in [13] and
[14] involving a coupled system of three modiﬁed Riccati
equations for fullorder dynamic compensation and a coupled
system of four modiﬁed Riccati and Lyapunov equations in the
ﬁxed—order (i.e., reducedorder) case as in [21]. The results of
[13] and [14] thus raise the following question: What is the
relevance of this system of coupled design equations to the
problem of H... disturbance attenuation via ﬁxedorder compensa
tion? To begin to address this question, the goal of the present paper
is to develop a design methodology for ﬁxedorder, i.e. , full— and
reducedorder, L2 optimal control which includes as a design
constraint a bound on Ho, disturbance attenuation. There are three
principal motivations for developing such a theory. First, the
results of [18]—[20] present full—statefeedback controllers whose
form is directly analogous to the standard LQR solution.
However, no L2 interpretation was provided in [181—[20] to
explain this similarity. The present paper thus provides an L2
interpretation within the context of an H... design constraint, A
novel feature of this mathematical formulation is the dual
interpretation of the disturbances. That is, within the context of L2
optimality the disturbances are interpreted as white noise signals
while, simultaneously, for the purpose of Ho. attenuation the very
same disturbance signals have the alternative interpretation of
deterministic L2 functions. This dual interpretation is unique to
the present paper since stochastic modeling plays no role in [18]—
[20]. We also note recent results obtained in [22] which essentially
show that the H2 plant topology can be induced by imposing L2
and Lo. topologies on the disturbance and output spaces, respec
tively. For further investigation into the relationships between L;
and H... control, see [22a]. The second motivation for our approach is the simultaneous
treatment of both L2 and Hon performance criteria which
quantitatively demonstrates design tradeoffs. Speciﬁcally, in
order to enforce the Hm constraint we derive an upper bound for
the L2 criterion. Minimization of this upper bound shows that the
enforcement of an He. disturbance attenuation constraint leads
directly to an increase in the L2 performance criterion. The third motivation for our approach is to provide a
characterization of ﬁxedorder dynamic outputfeedback control— 00189286/89/03000293$01.00 © 1989 IEEE 294 lers yielding speciﬁed disturbance attenuation. Existing optimal
Ho. design methods tend to yield highorder controllers. Intui
tively, solving the ﬁxed—order design equations for progressively
smaller Ha. disturbance attenuation constraints should, in the
limit, yield an Hwoptimal controller over the class of ﬁxedorder
stabilizing controllers. Although our main result gives sufﬁcient
conditions, we also state hypotheses under which these conditions
are also necessary (Proposition 4.1). It should also be noted that
the inherent coupling among the modiﬁed Riccati equations shows
that the classical separation principle of LQG theory is not valid
for the Hmconstrained full— and reduced—order design problems. In the fullorder case involving equalized L2 and Ha. perform
ance weights, we also show that the Honconstrained gains are
given by two rather than three equations (Section V). These two
equations are precisely those given in [26] for the pure H,
problem without an L2 interpretation. Since the results of [26] are
necessary as well as sufﬁcient, these connections show that our
sufﬁcient conditions (at least in this special case) are also
necessary. The authors are indebted to Prof. J. C. Doyle for
pointing out these relationships and to D. Mustafa for providing a
preprint of [45] which further clariﬁes these connections. Besides establishing connections with robust stabilizability in
state—space systems, an immediate beneﬁt of the modiﬁed Riccati
equation characterization of Hm—constrained controllers is the
opportunity to develop novel computational algorithms for con
troller synthesis. To this end a continuation algorithm has been
developed for solving the coupled system of three modiﬁed
Riccati equations. In a numerical study (see Section VHI) we have
demonstrated convergence of the algorithm and reasonable
computational efﬁciency. Homotopy methods were suggested for
the coupled Riccati equations because of their demonstrated
effectiveness in related problems which also involve coupled
modiﬁed Riccati equations [23]—[25]. Since H, control problems
are solvable by established numerical methods [4], it should be
stressed that the objective of these numerical studies is net to make
direct comparisons with existing H, synthesis algorithms, but
rather to demonstrate solvability of the coupled modiﬁed Riccati
equations. The contents of the paper are as follows. After presenting
notation at the end of this section, the statement of the Ho,—
constrained LQG control problem is given in Section H. The
principal result of Section H (Lemma 2.1) shows that if the
algebraic Lyapunov equation for the closedloop covariance is
replaced by a modiﬁed Riccati equation possessing a nonnegative—
deﬁnite solution, then the closedloop system is asymptotically
stable, the Hon disturbance attentuation constraint is satisﬁed, and
the L2 performance is bounded above by an auxiliary cost
function. The problem of determining compensator gains which
minimize this upper bound subject to the Riccati equation
constraint is considered in Section III as the auxiliary minimiza
tion problem. Necessary conditions for the auxiliary minimization
problem (Theorem 3.1) are given in the form of a coupled system
of three modiﬁed Riccati equations. In Section IV the necessary
conditions of Theorem 3.1 are combined with Lemma 2.1 to yield
sufﬁcient conditions for closedloop stability, H0° disturbance
attenuation, and bounded L2 performance. In Section V we derive
alternative forms of the design equations and specialize the results
to the simpler case in which the LQG weights are equal to the H...
weights. To achieve further design ﬂexibility, the reduced—order
controldesign problem is considered in Section VI. A simpliﬁed
qualitative analysis of the fullorder design equations is given in
Section VII to highlight important features with regard to
existence and multiplicity of solutions. Finally, a numerical
algorithm is presented in Section VIII along with illustrative
numerical results. A series of designs is obtained to illustrate
tradeoffs between the L2 and H... aspects and the conservatism of
the L2 performance bound. Although in the present paper the
numerical results are limited to the case of full—order dynamic compensation, reducedorder designs have been obtained in [27]
using Theorem 6.1. IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 34, NO. 3. MARCH 1989 Notation Note: All matrices have real entries. IR], 12’“, lR’, E Real numbers, r X 5 real matri
ces, RV“, expected value r X r identity matrix, transpose,
r X s zero matrix, 0,,” In ( )T! OIXS) Or tr, p( ) Trace, spectral radius 8’, INF, P’ r X r symmetric, nonnegative
deﬁnite, positive—deﬁnite matrices Z,SZ;,Z,<Z; 22—21 E N’, Zz ‘ Z] G Pr, Zr, 22 e 3’
Positive integers; n + n,(nc s n)
n, m, 1, nc, ﬁdimensional vectors [2:] n. "1.1, nap, «1.11m; '7
x! u, y! x0, x )2 A,B,C an,nxm,Ixnmatrices Ac. Be. Cc n, x nc, n, x I, m X It, matrices  A BC, A [3,0 A, ] w() p—dimensional standard white
noise D1,D2 nxp,lxpmatrices;D,DzT=0 V1, V2 DlDlT, V2 6 PI  D
D [Ba] 7 V1 onxnc
Oncxn Ba EiiEz qxn,q><mmatrices;Esz = 0 E [5. £20.] R1, R2 ErEleZTEziRz E 9'"  R. ’ 0...,   R =ETE
[oncxn Elana E2... q... X n, q... x m matrices;
Eer2w=0 Ea [En E...C.1 Rim: RZm ETmElw, EIQEZQ, ~ Rlao onxn   R, E :57 a,
[orian CZRZonCc] on 2,2 BRz'lBT, CTVZ“C B, 'Y Nonnegative constant, positive constant II. STATEMENT OF THE PROBLEM In this section we introduce the LQG dynamic outputfeedback
control problem with constrained Ha. disturbance attenuation
between the plant and sensor disturbances and the state and
control variables. Without the L1 performance criterion, the
problem considered here essentially corresponds to the standard
problem of [3] and [4]. For simplicity we restrict our attention to
controllers of order II, = It only, i.e., controllers whose order is
equal to the dimension of the plant. This constraint is removed in
Section VI where controllers of reduced order are considered.
Hence, throughout Sections II—V the controller dimension HE and
closed—loop plant dimension ﬁ A n + nC should be interpreted as
n and Zn, respectively. Controllers of order greater than n are
generally of less interest in practice and thus are not considered in
this paper. BERNSTEIN AND HADDAD: LQG CONTROL [faConstrained LQG Control Problem: Given the nthorder
stabilizable and detectable plant X(t)=Ax(t)+Bu(t)+D,w(t), (2.1)
y(t)=Cx(t)+D2w(t) (2.2) determine an nthorder dynamic compensator
X'c(t)=AcXc(t)+ch(t), (23)
um = chc(t) (2.4) which satisﬁes the following design criteria: i) the closed—loop system (2.1)—(2.4) is asymptotically stable,
i.e., A is asymptotically stable; ii) the (1,, X p closedloop transfer function H(s) a 54515—2045 (2.5)
from w(t) to E,,,x(t) + EZqu) satisﬁes the constraint
lH(s)msv (2.6) where 'y > 0 is a given constant; and
iii) the performance functional J(Ac, BC, CC) 2 lim E[XT(I)R,X(1)+uT(t)R2u(t)](2.7) 1H” is minimized.
Note that the closedloop system (2.1)—(2.4) can be written as £(t)=xi;2(t)+15w(t) (2.8)
and that (2.7) becomes
J (A.. B“ C.)=}gm E[(E'i(t))T(Ef(t))]
=lim E[)ET(I)R)?(I)]. (2.9) t—‘o: Remark 2.1: Since (A, B, C) is assumed to be stabilizable and
detectable the set of nthorder stabilizing compensators is non
empty. Remark 2.2: It is easy to show that the performance functional
(2.7) is equivalent to the more familiar expression involving an
averaged integral, i.e., l
J(Ac, BC, Cc)= lim — E Imo t [ [x’(s)R1x(S) +uT(s)Rzu(s)] d5} . Remark 2.3: For convenience we assume D,DZT = 0, which
effectively implies that the plant disturbance and sensor noise are
uncorrelated. Remark 2. 4.‘ One may also consider a general L2 optimization
problem of the form min [l T — UQV 2, where Q is a
parameterization of stabilizing controllers. In this case, without a
constraint on the MacMillan degree of Q, it may be possible to
satisfy (2.6) with smaller values of 7. Note that the problem statement involves both L2 and Han
performance weights. In particular, the matrices R, and R2 are the
L2 weights for the state and control variables. By introducing L2
weighted variables Z(l)=Elx(t). v(t)=Ezu(f) the cost (2.7) can be written as J(An Bea Cc): lim Elzr(t)z(t)+ vT(t)v.(t)l. I—NN 295 For convenience we thus deﬁne R, 9. E (E, and R2 2 E 27 E;
which appear in subsequent expressions. Although an L2 cross
weighting term of the form 2x7(t)R,2u(t) can also be included,
we shall not do so here to facilitate the presentation. For the H,o performance constraint, the transfer function (2.5)
involves weighting matrices E,“ and Ego, for the state and control
variables. The matrices R,cm 3 E mew and R2,, 9. EZTmEzm are
thus the H, counterparts of the L2 weights R, and R2. Although
we do not require that R,“ and R2,. be equal to R, and R7, we
shall require in the next section that R2,, = BZRZ, where the
nonnegative scalar ﬂ is a design variable. Finally, the condition
E LE2" = 0 precludes an Ha, crossweighting term which again
facilitates the presentation. _ Before continuing, it is useful to note that if A is asymptotically
stable for a given compensator (Ac , BC , Cc), then the performance
(2.7) is given by J(A,, BC, c,)=¢r Q15 (2.10) where the steadystate closedloop state covariance deﬁned by Q’ Q 11m E[£(t)ir(t)] (2.11)
satisﬁes the if x I? algebraic Lyapunov equation
0=A"Q+Q"/iT+ 17. (2.12) Remark 2.5: Using (2.10) and (2.12) it can be shown that the
L2 cost criterion (2.7) can be written in terms of the L2 norm of
the impulse response of the closedloop system. Speciﬁcally, by
writing Q satisfying (2.12) as Q= S0 e’l’Ve’iT’ d! (2.10) becomes
J(A.. 8.. C.)= IIE'eWIIidr where I]  N F denotes the Frobenius matrix norm The key step in enforcing the disturbance attenuation constraint
(2.6) is to replace the algebraic Lyapunov equation (2.12) by an
algebraic Riccati equation which overbounds the closed—loop
steadystate covariance. Justiﬁcation for this technique is pro
vided by the following result. Lemma 2.1: Let (Ac, BC, Cc) be given and assume there exists
Q E RIM" satisfying Q E W (2.13)
and
0=AQ+QAT+y2QR,.Q+ V. (2.14)
Then
(1, 15) is stabilizable (2.15)
if and only if
11 is asymptotically stable. (2.16)
In this case,
lH(s)le.S~1 (2.17)
and
Q5 6t. (2.18) 296 Consequently, J(Ac, Be. CJSSMu BC. Cos Q.) (219) where 3(Acr BC! Cc, é tr Proof: It follows from_[28, Theorem 3.6] that (2. 15) implies
that (A, [y‘zinQ + V] 1a) is also stabilizable. Using the
assumed existence of a nonnegativedefinite solution to (2.14) and
[28, Lemma 12.2], it now follows that A is asymptotically stable.
The converse is immediate. The proof of (2.17) follows from a
standard manipulation of (2.14); for details see [29, Lemma 1].
To prove (2.18), subtract (2.12) from (2.14) to obtain 0=A'(QQ~)+(Q—Q~)A'T+T_ZQILQ (2.21)
which, since A is asymptotically stable,'is equivalent to Q—Q= e/I’h‘ZQR'kaAT’ dtzo. (2.22)
Finally, (2.19) follows immediately from (2.18). D Remark 2.6: Note that (2.15) is actually a closed—loop
disturbability condition which is not concerned with control as
such. This condition guarantees that the system does not possess
undisturbed unstable modes. Of course, if Vis positive deﬁnite or
(A, D) is controllable, then (2.15) is satisﬁed. Lemma 2.1 shows that the Ho, disturbance attenuation con—
straint is automatically enforced when a nonnegative—deﬁnite
solution to (2.14) is known to exist and A is asymptotically stable.
Furthermore, all such solutions pr_ovide upper bounds for the
actual closed—loop state covariance Q along with a bound on the L2
performance criterion. Next, we present a partial converse of
Lemma 2.1 which guarantees the existence of a unique minimal
nonnegativedeﬁnite solution to (2.14) when (2.17) is satisﬁed.
The minimal solution is desirable since it yields the least
performance bound in (2.19). This was ﬁrst pointed out in [45]. Lemma 2.2: Let (AC, Br, CC) be given, suppose A is
asymptotically stable, and assume the disturbance attenuation
constraint (2.17) is satisﬁed. Then there exists a unique nonnega
ﬁvedefinite solution Q satisfying (2.14) and such that A +
'y’zQﬁo. is asymptotically stable. Furthermore, this solution is
also minimal. Proof: The result is an immediate consequence of [30, pp.
150 and 167], using Theorems 3 and 2, along with the dual
version of [28, Lemma 12.2]. The proof of minimality is given in
[29]. El Remark 2. 7: To further clarify the relationships between the L2
and H,, aspects of the problem, we note that the closedloop
system can be represented by two possibly different transfer
functions. Speciﬁcally, with respect to the L2 cost criterion, the
closedloop transfer function between disturbances and controlled
variables is given by the triple (A, D, E) while for the H...
constraint the closed—loop transfer function (2.5) corresponds to
the triple (A, D, Em). Finally, it can be shown that the closed—loop Riccati equation
(2.14) also enforces a constraint on the norm of the §an~kel
operator corresponding to the closedloop system (A, D, E...)
when Q is positive deﬁnite. Thus, let 15 E IN!" denote the solution
to 0=A7ﬁ+ﬁA+1€w (2.23)
and note that I5 and Q [satisfying (2.12)] are the observabil_ity_and
controllability Gramians, respectively, of the system (A, D, Ea, ).
As showy in [31], the norm of the~I!ankel operator corresponding
to (A, D, E...) is given by A¥§X(PQ). _
Proposition 2.1: Suppose there exists Q E P" satisfying IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 34, N0. 3, MARCH 1989 (2.14) and such that (2.15) or, equivalently, (2. 16) holds. Then KﬁUSQKv Proof: Since Q is assumed to be invertible, (2.14) is
equivalent to (2 .24) 0=7214TQ“+72Q"A+72Q"7Q“ +15... (2.25) Subtracting (2.23) from (2.25) shows that 7262," — 15 2 0, or,
equivalently, 721,7 2 Ql’zPQl/z. Thus, 722 )‘max(Q‘1/2p'Ql/2) = )‘max(P"l/2Qp'l/2) Z )‘mx(P"l/2 Q1310) which yields (2.24). HI. THE AUXILIARY MINIMIZATION PROBLEM AND NECESSARY
CONDITIONS FOR OPI‘IMALITY As discussed in the previous section, the replacement of (2.12)
by (2.14) enforces the Hm disturbance attenuation constraint and
yields an upper bound for the L2 performance criterion. That is,
given a compensator (Ac, BE, Cc) for which there exists a
nonnegativedeﬁnite solution to (2.14), the actual L2 performance
J (Ac, BC, 0,) of the compensator is guaranteed to be no worse
than the bound given by 3(Ac, BC, Cc, Q). Hence, 5(Ac, BC, Cc,
Q) can be interpreted as an auxiliary cost which leads to the
following mathematical programming problem. Auxiliary Minimization Problem: Determine (Ac, 3,, Cc, Q)
which minimizes 3(Ac, BC, Cc, Q) subject to (2.13) and (2.14). It follows from Lemma 2.1 that the satisfaction of (2.13) and
(2.14) along with the generic condition (2.15) leads to: 1) closed—
loop stability; 2) prespeciﬁed H... performance attenuation; and 3)
an upper bound for the L2 performance criterion. Hence, it
remains to determine (Ac, BC, CE) which minimizes J(Ac, 13,,
CC, Q), and thus provides an optimized bound for the actual L2
performance J (Ac, BC, Cc). Rigorous derivation of the necessary
conditions for the auxiliary minimization problem requires addi—
tional technical assumptions. Speciﬁcally, we restrict (Ac, BC,
Cc, Q) to the open set Sr 3 {(At) Br) Cc: 3 Q. E Pﬂsﬁ+7—2QRou is asymptotically stable, and (Ac, BC, Cc) is controllable and observable}. (3.1) Remark 3.1: The set ﬁx constitutes sufﬁcient conditions under
which the Lagrange multiplier technique is applicable to the
auxiliary minimization problem. Speciﬁcally, the requirement
that Q be positive definite replaces __(2.13) by an open set
constraint, the stability of A + 'y‘ZQR... serves as a normality
condition, and (Ac, BC, Cc) minimal is a nondegeneracy condi
tion. Note that the stabilizability condition (2.15) and stability
condition (2.16) play no role in determining solutions of the
auxiliary minimization problem. The following result presents the necessary conditions for
optimality in the auxiliary minimization problem. The proof of
this result is given in the Appendix as a special case of the
corresponding result for reducedorder dynamic compensation
considered in Section VI. As mentioned previously, we assume
that R2,, = BZRZ, where B 2 0. Furthermore, for arbitrary Q, P
E W" deﬁne S a (In+[327’2QP)". (3.2)
Since the eigenvalues of QP coincide with the eigenvalues of the
nonnegativedeﬁnite matrix Pl’ZQPl/Z, it follows that QP has
nonnegative eigenvalues. Thus, the eigenvalues of 1,, + ﬁ’y‘ZQP
are all greater than one so that the above inverse exists. BERNSTEIN AND HADDAD: LQG CONTROL Theorem 3.1: If (Ac, BC, CC, (2,) E E! solves the auxiliary minimization problem then there exist Q, P, Q 6 [1‘0" such that
(3.3)
(3.4) A,=A—Q>‘:—2Ps+~r2QR,,,,
BC=QCTV2“,
0,: VRg‘BTPS,
= QtQ Q
Q i Q Q]
and such that Q, P, Q satisfy 0=AQ+ QAT+ V1 +7‘2QRlaQ—QiQ. 0=<A+rth+Q1RmTP+P<A+721Q+Q1R1..)
+R1— STPEPS, (3.5) (3.6) (3.7) (3.8)
0=(A —2PS+7‘2QR1,,)Q+Q(A —>:PS+y2QR,,,)T + v ‘2Q(R1.. + 525 TPEPS)Q+ QSQ. (3.9)
Furthermore, the auxiliary cost is given by £10403“ Cc: Q)=tr [(Q+Q)R1+QSTPEPS]. (310) Conversely, if there exist Q, P, Q G N" satisfying (3.7)—(3.9),
then (Ac, Be, Cc, Q) given by (3.3)—(3.6) satisﬁes (2.13) and
(2.14) with auxiliary cost (2.20) given by (3.10). Remark 3.2: If Q and Q are nonnegative deﬁnite, then the fact
that the deﬁniteness condition (2.13) is satisﬁed can easily be seen
by writing Q as "1/2 All2 T were]  As mentioned in Section II, it is desirable to determine solutions Q
and Q which yield the minimal solution to (2. 14). Remark 3.3: Setting {3 = 0, or equivalently, E20, = 0,
specializes Theorem 3.1 to me cheap Ho, control case in which
H, attenuation between disturbances and controls is not con
strained. In this case S = I,,, Q, is given by (3.6), and (3.3)—(3.5)
become A,=A—QS—>:P+yZQR,,,, (3.11)
BC=QCTV2’1, (3.12)
0,: —R,‘BTP (3.13)
where Q satisﬁes (3.7), and (3.8) and (3.9) become
0=(A+7_2[Q+Q]le)TP+P(A +7’2[Q+Q]R1m)
+R1—P2P, (3.14) 0=(A —EP+7‘ZQR10.)Q+Q(A —EP+7'2QR,,,)T
+7’2QR1..Q+Q:Q. (3.15) Finally, the auxiliary cost reduces to sumac, Cc, Q)=tr[(Q+Q)R1+QPEP1. (3.16)
Numerical solution of (3.7), (3.14), and (3.15) is discussed in
Section VIII. Remark 3.4: Note that if both B = 0 and R1,, = 0, then
Theorem 3.1 specializes to the standard LQG result. Theorem 3.1 presents necessary conditions for the auxiliary
minimization problem which explicitly synthesize extremal con
trollers (Ac, 3,, C,). These necessary conditions comprise a 297 system of three modiﬁed algebraic Riccati equations in variables
Q, P, and Q. The Q and P equations are similar to the estimator
and regulator Riccati equations of LQG theory, while the Q
equation has no counterpart in the standard theory. Note that the
Q equation is decoupled from the P and Q equations and thus can
be solved independently. The P equation, however, depends on
Q. Thus, regulator/estimator separation holds in only one
direction which clearly shows that the certainty equivalence
principle is no longer valid for the Lz/H, design problem.
Furthermore, since the P and Q equations are coupled, they must
be solved simultaneously. Finally, note that if the Ho, disturbance
attenuation constraint is sufﬁciently relaxed, i.e., 'y > 0°, then the
If equation becomes decoupled from the Q equation and thus the
Q equation becomes superﬂuous. Furthermore, the remaining Q
and P equations separate and coincide with the standard LQG
result. IV. SUFFICIENT CONDITIONS FOR Ho, DISTURBANCE ATTENUATION In this section we combine Lemma 2.1 with the converse of
Theorem 3.1 to obtain our main result guaranteeing closedloop
stability, Hm disturbance attenuation, and an optimized L2 per
formance bound. A Theorem 4.1: Suppose there exist Q, P, Q 6 [1‘8" satisfying
(3_,7):(3.9) and let (A,, 3,, CC, (5),) be given by (13)—(3.6). Then
(A, D) is stabilizable if and only if A is asymptotically stable. In
this case, the closedloop transfer function H (5) satisﬁes the Ha,
attenuation constraint IIH(5)lwS'Y (41)
and the L2 performance criterion (2.7) satisﬁes the bound
J(A,, 3,, Cc).<_tr [(Q+ Q)R1+ QS TPEPS]. (4.2) Proof: The converse portion of Theorem 3.1 implies that Q,
given by (3.6) satisﬁes (2.13) and (2.14) with auxiliary cost given
by (3.10). It now follows from Lemma 2.1 that the stabilizability
condition (2. 15) is equivalent to the asymptotic stability of A, the
Hm disturbance attenuation constraint (2.17) holds, and the
performance bound (2.19), which is equivalent to (4.2), holds. _ [1 Remark 4.1: In applying Theorem 4.1 it is not actually
necessary to check (2.15) which holds generically. Rather, it
sufﬁces to check the stability of A directly which is guaranteed to
be equivalent to (2.15). In applying Theorem 4.1 the principal issue concerns condi—
tions on the problem data under which the coupled Riccati
equations (3.7)—(3.9) possess nonnegativedeﬁnite solutions.
Clearly, for 7 sufﬁciently large, (3.7)—(3.9) approximate the
standard LQG result so that existence is assured. The important
case of interest, however, involves small 7 so that signiﬁcant H...
disturbance attenuation is enforced. Thus, if (4.1) can be satisﬁed
for a given 7 > 0, it is of interest to know whether one such
controller can be obtained by solving (3.7)—(3.9). Lemma 2.2
guarantees that (2.14) possesses a solution for any controller
satisfying (2.17). Thus, our sufﬁcient condition will also be
necessary as long as the auxiliary minimization problem possesses
at least one extremal over 511. When this is the case we have the
following immediate result. Proposition 4.1: Let 7* denote the infimum of H(s)H 0, over
all stabilizing nthorder dynamic compensators and suppose that
the auxiliary minimization problem has a solution for all 'y > 'y *.
Then for all 7 > 7* there exist Q, P, Q E N" satisfying (3.7)
(3.9). Unlike the standard LQG result involving a pair of separated
Riccati equations, the new result enforcing Ha, disturbance
attenuation involves a nonstandard coupled system of three
modiﬁed Riccati equations. The asymmetry of these equations
suggests the possibility of a dual result in which the modiﬁcations 298 to the standard P and Q Riccati equations are reversed. Such a
dual result will generally be different from Theorem 4.1 and thus
will yield an improved bound for particular problems. This point
was demonstrated in [16] for the problem of robust performance
analysis. Due to space limitations, however, we give only a brief
outline of the dual H... results. Note that J (Ac, 3,, Ce) given by
(2.10) is also given by J(Aca Be. CC)=tr P7 (4.3) where P E N" is the unique solution to (2.23) with 15,, replaced
by 1?. Next, utilizing (4.3) in place of (2.10), the Han disturbance
attenuation constraint (2.6) can now be enforced by replacing
(2.23) by the Riccati equation 0=AT0+0A+7—2017m0+§ (4.4) where 17,. has the same form as I7 but may involve weights V1,,
and V2,... Note that (4.4) is merely the dual of (2.14). We also
require the condition dual to (2.15) given by (E, A) is detectable (4.5) and that xi + 7—217“? be asymptotically stable. Once again, the
sufﬁcient conditions for H” disturbance attenuation involve a
coupled system of three modiﬁed Riccati equations dual to (3.7)
(3.9). Similar remarks apply to ~the reducedorder case given by
Theorem 6.1 below; Finally, if R0, = R and Von = V, then it can
be shown that tr QR = tr (P Vand thus the solutions to the primal
and dual problems coincide. V. ALTERNATIVE FORMS OF THE RICCATI EQUATIONS In this section we develop alternative forms of the Riccati
equations (3.7)—(3.9). These alternative forms provide further
insight into the structure of (3.7)—(3.9) and, in certain cases, are
simpler and thus are easier to solve computationally. This section
also provides connections between our approach and [26]. First we note that the gains (3.3), (3.5), and (3.10) do not
depend upon P and Q individually, but rather only upon the term
Z é PS. Thus, it is of interest to know whether (3.8) and (3.9)
can be transformed to yield an equation which characterizes Z
directly. The following result summarizes useful properties of Z. Lemma 5.1: Let P, Q E N" and deﬁne Z 2 PS. Then Z =
ZT = STP, where ST = (1,, + BH‘WQ)", and Z is
nonnegative deﬁnite. If, in addition, P is positive deﬁnite, then Z
is positive deﬁnite and Z=(P’1+627‘2Q)‘1 (51) Proof: The result (5.1) is immediate. The remaining results can be obtained by replacing P by P + 61", where e > 0, and
taking the limit as e —* 0. El Proposition 5.1: Let Q E N” and suppose there exist P E E9”
and Q E N" satisfying (3.8) and (3.9). Then Z 2 PS satisﬁes 0=(A+7‘ZQR1..+7“ZQIRluBZR1DTZ
+Z(A+vZQR1..+w2Q[R1..—62Rli)
+R.—Z(2+s274Q[R...—82R11Q)z +627‘ZZQ2QZ (5.2)
and (3.9) is equivalent to
0=(A—EZ+7‘2QR,,.)Q+ Q(A—zz+72QR1..)T
+7‘2Q(R1,.+BZZEZ)Q+QEQ. (5.3)
Furthermore, (3.3), (3.5), and (3.10) become
AC=A7Qi—EZ+7’ZQR1,,, (5.4) IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 34, NO. 3, MARCH I989 Cc: —R2“BTZ,
3(Ac, Bu CC! Q)=tr[(Q+ Proof: Using the identities
P:(1n—327‘ZZQT1Z=Z(In*52‘r_2QZ)'l
which follow from (5.1), equation (5.2) can be obtained by
forming the new equation (In_327_ZZQ)(38)(In_3272QZ)+ 627—22(39)Z Finally, (5.3)—(5.5) are restatements of (3.9), (3.3), and (3.5)
with Z = PS. I:
Having obtained a single equation (5.2) for Z = PS by
combining (3.8) and (3.9) for P and Q, it is of interest to know
whether (3.8) for P can be recovered from (5.2) and (5.3).
Proposition 5.2: Let Q E ", [3 > 0, suppose there exist Z 6
I?” and Q E IN" satisfying (5.2) and (5.3), and assume that p(ZQ)<B'272 (58) Then P é (Z“ ’ 627‘2Q)‘1 is positive deﬁnite and satisﬁes
(3.8). Furthermore, P satisﬁes Z = PS. Proof: If (5.8) holds, then it can be shown that P as deﬁned
above is positive deﬁnite. Reversing the proof of Proposition 5.1,
(3.8) can be recovered by forming (InBlv‘le)“[(52)ﬂzv‘ZZ(53)Zl(1nB2'v'ZQZ)‘1 C Although Proposition 5.2 allows us to reconstruct (3.8) for P, it
can only be utilized when (5.8) holds. This fact raises a question
as to the sufﬁciency of (3.7), (5.2), and (5.3) in the absence of
(3.8). It turns out that the matrices P and Z need not actually
satisfy (3.8) and (5.2) to enforce the H... performance constraint
(2.17) since only the Q and Q equations are required. Rather, P
can be viewed as a pararneterization of Z which, in turn, is a
parameterization of the gains A, and Cc given by (5.4) and (5.5)
which yield a controller satisfying the desired Ha, performance.
These observations are summarized by the following result which
does not require that Z be obtained by solving (5.2). A Proposition 5.3: Let Z G EM" and suppose there exist Q, Q E
W" satisfying (3.7) and (5.3). Then (AC, 13,, CC, 6!) given by
(5.4), (3.4), (5.5), and (3.6) satisfy (2.13) and (2.14). Thus,
(2.15) and (2.16) are equivalent, and, in this case, (2.17) and
(2.19) hold. Proof: The result follows by direct veriﬁcation of (2.14).: Proposition 5.3 shows that the H... constraint (2.17) is enforced
for arbitrary Z 6 [W as long as (3.7) and (5.3) can be solved for
Q and The price we pay for using arbitrary Z is that we no
longer are assured that Z is obtained from (5.2) or from Z = PS
where P satisfies (3.8). Since P arises from the Lagrange
multiplier for the constraint (2.14) [see (A.3)], it follows that an
arbitrary choice of P (or Z) may fail to minimize the L2 auxiliary
cost (2.20). Thus, regarding P and Z as free parameters
effectively ignores the L2 aspect of Theorem 4.1. It is also of interest to introduce yet another transformation of
(3.7)—(3.9) by deﬁning Y e (z‘+6271Q)'=(P1+327—2[Q+Q1)1 (5.9) when P is positive deﬁnite. As in Lemma 5.1, Y is also positive
deﬁnite. Proposition 5.4: Let Q E N" and suppose there exist P E P" and Q 6 0‘0” satisfying (3.8) and (3.9). Then Ydeﬁned by (5.9)
satisﬁes 0=(A+v2[Q+QllRla—aleVY
+Y(A+r2[Q+Q][R...—BZR11)
+R1+621F1YVl Y— YEY ~627 ‘4 Y(Q+ Q)(Rlau ~BZR1)(Q+ Q)Y (5.5)
(5.6) (5.7) (5.10) BERNSTEIN AND HADDAD: LQG CONTROL
and (3.9) is equivalent to
0=(A—E[Y“—Bzv‘2Q]“+7‘2QRim)Q +Q(A—2[Y"‘—627’2Ql"+7’2QRim)T
+7‘2Q(Ri~+l32[Y“327‘2Q1"
‘E[Y"627’2Q1“)QA+Q5EQ‘ Furthermore, (3.3), (3.5), and (3.10) become ‘ Ac=A—QiE(Y"—627’ZQ)"+7‘ZQRim (5.12) (5.11) Ct: R{lBT(Y"327’ZQ)"i
304:. Be: Cc, Q.)=tr [(Q+Q)Ri
+Q(Y‘1—327'2Q)‘12(Y"—62‘Y’2Q)‘1] (5.14)
Proof: To obtain (5.10), form (5.13) Y[Z"(5.2)Z"+[32'y‘2(3.7)] Y. D The following result allows us to recover (3.8) for P from
(5.10) and (5.11). Proposition 5.5: Let Q E N”, B > 0, suppose there exist Y E
E9” and Q E N" satisfying (5.10), (5.11), and assume that p(Y[Q+Ql)<I3'272 Then P é (Y’l — Bz'y‘z[Q + Q])“ is positive deﬁnite and
satisﬁes (3.8).
Proof: The result follows by reversing the proof of
Proposition 5.4. I]
By specializing further, it is possible to achieve even greater
simpliﬁcation. Speciﬁcally, we consider the case in which the L2
and H... weights are equalized, i.e., (5.15) le=Rlii3=L (5.16) In this case it is always possible to eliminate (5.3) and (5.11) by
noting that they are satisﬁed by Q = 7121 and Q = 'yzY“ 
Q, respectively. However, although this solution enforces the H...
constraint, it can be seen from the resulting form of 3 that this
solution does not correspond to the minimal solution Q of (2.14).
Hence, we impose additional assumptions which allow us to
directly characterize the solution which yields the minimal
performance bound. We are indebted to D. Mustafa for clarifying
this point in [45] where it is also shown that the auxiliary cost
(2.20) is equivalent to an entropy integral. Proposition 5.6: Assume (5.16) is satisﬁed, suppose there
exist Q E 1N1” and Z... 6 E9" satisfying 0=AQ+ QAT+ Vi +7—2QleQ_Q2Q2
O=(A +74%...)sz +Z.,,(A +7’2QR10.)
+R,.,—z,.sz..+yZz,.Qin.. (5.18) (5.17) and such that
A + 'y ‘ZQRI... + ('y ‘2Q2Q— E)Z.., is asymptotically stable (5 . l9)
and (A +7‘2QRim+Z;‘Rim 7"[Riu+Z»EZ~1"2) is observable
(5.20)
Furthermore, let (Au 8,, Cc) be given by Ac=A—QE*ZZ...+7‘ZQR1,., (5.21) 299 BC: QCTVZ‘l , (5.22) Cc: —R;;BTZ,.. (5.23)
Then (A, D.) is stabilizable if and only if if is asymptotically
stable. In this case, the closedloop transfer function H(s)
satisﬁes the Ho, disturbance attenuation constraint H(s)..5y (5.24)
and the L2 performance criterion (2.7) satisﬁes the bound
J(AL‘) BC! CE)S"[QR1m+Q2QZm] Proof? First note that it follows from (5.18) that —(A +7—2QR1m +Z;1le)=zm[A +7—ZQR1m
+(7‘2QEQ— E)Z...]Z;,l (5.26) and thus (5.19) implies that —(A + 'y'ZQRm, + Z;'R1,,) is
asymptotically stable. It now follows from (5.20) that there exists
N El?" satisfying 0= —(A+1‘2QR1w+Z;1R1.,)TN—N(A +7 ’ZQRIO.
+Z;1Rm)+'y'2(R1m+Z¢,2ZW). (5.27) It can now be shown that Q = 'yZZ: — N " satisﬁes (5.3) with
B = l and Z = Z... Furthermore, (5.8) is satisﬁed so that the
hypotheses of Theorem 4.1 are veriﬁed. The expression (5.25)
now follows by direct substitution. I]
Finally, we consider a simpliﬁed version of Proposition 5.4.
Proposition 5.7: Assume (5.16) is satisﬁed and suppose there
exist Q E IN" and Y... E I?" satisfying I (5.28) 0=AQ+QAT+ Vl+‘Y_2QR1mQ_Q:Q)
0=ATYW+Y°°A+R1°°+T_2YD¢ VIYon_Ym2Ym, 0(QY....)<1(2 (530)
and such that
A +(7 ’2 V1 —2) Y... is asymptotically stable (5.31)
and
(A+ Y;‘R..,.. y‘lRim+(Y;'—7‘2Q)“
 E(Y;‘—7'2Q)]”2) is observable. (5.32)
Furthermore, let (Ac, BC, C.) be given by
AC=A—Q2—E(Ygl—y'ZQ)“+'y'2QR1m (5.33)
B,=QCTV21, (5.34)
ct: _R;;BT(Y;I—y2Q)1. (5.35) Then (xi, 5) is stabilizable if and only if A is asymptotically
stable. In this case, the closedloop transfer function H (3) satisﬁes
the H“ disturbance attenuation constraint H(s)llmsv (5.36) and the L2 performance criterion (2.7) satisﬁes the bound
J(AC!BC!Cc)St1[QR1D+QEQ(Y;l_7—2Q)—]]' (537) Proof: The proof is similar to the proof of Proposition 5.6
with Q = 72h, — Q — N“, where Nsatisﬁes o= —(A+ Y;1R...)TN—N(A + Y;1R...)
+7—2lRi..+(Y;‘—7’2Q)“E(Y;'v‘2Q)“1. D 300 Remark 5.1: The solutions Q and Y... of (5.28) and (5.29) are
analogous to the matrices Y... and X... Of [26], while (5.30)
corresponds to condition 5.2(iii) of [26]. Note that by letting 7 —v
00, (5.25) and (5.37) coincide with 577a of [1] and the LQG
result is recovered. Remark 5.2: It is interesting to note that (5.17) and (5.18) with
controller gains (5.21)—(5.23) are already known since they are
identical to the optimality conditions for the linearexponential—of—
quadratic—Gaussian problem treated in [33] (see also [34] and
[35]). Speciﬁcally, see (3.1) and (4.1) on pp. 603 and 609,
respectively. As shown in [33], minimizing the criterion J: lim Eueu/2(XTRx+u7R2u) 1"!!! leads to the pair of modiﬁed Riccati equations (5.17) and (5.18)
with 7‘2 replaced by it. This implies that the exponential—of
quadratic design problem effectively enforces a bound of u’ 1’ 2 on
the H“, norm of the closedloop transfer function. There also exist fundamental connections with the problem of entropy maximiza
tion [43]—[45]. VI. EXTENSIONS TO REDUCEDORDER DYNAMIC COMPENSATION In this section we extend Theorem 4.1 by expanding the
formulation of Section III to allow the compensator to be of ﬁxed
dimension n, which may be less than the plant order n. Hence, in
this section deﬁne r7 = n + nc, where n, s n. As in [21] this
constraint leads to an oblique projection which introduces
additional coupling in the design equations along with an
additional equation. The following lemma is required. Lemma 6.1: Let Q, P E W" and suppose rank QP = nc. Then
there exist n, X n G, I‘, and n: X nc invertible M, unique except
for a change of basis in @"c, such that QP=GTMT, (6.1)
rcr=1.,. (6.2)
Furthermore, the n X n matrices
1 é GTI‘, (6.3)
T] e I,,—1 (6.4) are idempotent and have rank n, and n —— n3, respectively.
Proof: Conditions (6.1)—(6.4) are a direct consequence of
[36, Theorem 6.2.5]. A A D
Theorem 6.1: Let MC 5 n, suppose there exist Q, P, Q, P E
W" satisfying 0=AQ+QAT+ V1+y'2QR1mQ—Q$Q+n QEQTL (6.5)
0=(A+7_2[Q+Q]Rl°°)TP+P(A+7—2[Q+Q]le)+Rl
— s TPZPS+ 1: S TPEPSTi , (6.6)
0=(A—2PS+7’2QR1¢,)Q+Q(A—2PS+y  2QR...)T
+72Q(R1,.+BZSTP2PS)Q+QiQ—ngigr:, (6.7)
0=(A — Q: +yZQR,,,)T13+ 13(A  Q2 + 'y‘ZQRm)
+STP2PS—rf STPEPSTU (6.8)
rank Q=rank 15=rank Q15=nc
and let (Ac, BC, CC, (52,) be given by (6.9) A,=I‘(A —QE—2PS+7‘2QR1,,)GT, (6.10)
BC=FQCTV2", (6.11)
Cc: —R2—‘BTPSGT, (6.12) IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 34, NO. 3. MARCH 1989 ' A T
Q: [QIfQQ l“ch‘FJ . (6.13) Then, (A, D) is stabilizable if and only if [I is asymptotically
stable. In this case, the closed—loop transfer function H (5) satisﬁes
the Ho. disturbance attenuation constraint H(s)l~57 and the L2 performance criterion (2.7) satisﬁes the bound (6.14) J(Ac, BC, CC)SH[(Q+Q)R1+QSTP2PS]. (6.15) Remark 6.1: It is easy to see that Theorem 6.1 is a direct
generalization of Theorem 4.1. To recover Theorem 4.1, set n, =
n sothatr = G = I‘ = 1,, and r, = 0. Inthiscasethelastterm
in each of (6.5)—(6.8) can be deleted and (6.8) becomes
superﬂuous. Furthermore, (6.5)—(6.7) now reduce to (3.7)—(3.9),
as expected. If, furthermore, 6 = 0 then S = 1,. so that (6.5)—
(6.7) now reduce tO the cheap Hm control case given by (3.7),
(3.14), and (3.15). Alternatively, setting 7 = on and retaining the
reducedorder constraint n, < n yields the result of [21]. Remark: 6.2: By introducing a new variable Z = PS = (P—1
+ Bzy‘zQ)“‘ as in Section V, (6.6) becomes 0=(A+7'ZQR1m+Y‘leRiw(VRIDTZ
+Z(A+1’2QR1m+7’2Q[R1mBZR1D
+R1Z(2+327‘4Q[R1mBZRIIQ)Z
+1ﬂZEZr,+ﬂzy'2Z(QEQ—1]Q2QTZ)Z (6.16) which specializes to (5.2) when no =
(5.16) holds, (6.16) becomes n,i.e.,1l = 0. When 0=(A +y’2QR1m)TZm+Z,,(A +7 ZQR,,.)
+R1,—Z,,.EZ,,+1':Z.,2Z,,T]
+7‘ZZW(QiQ—71 QEQT: )Zoo Analogous equations for Y deﬁned by (5.9) can also be
developed. (6.17) VII. ANALYSIS OF THE DESIGN EQUATIONS Before developing numerical algorithms, it is instructive to
analyze the design equations to determine existence and multiplic—
ity of nonnegative—deﬁnite solutions. Although a detailed theoreti
cal analysis remains an area for future research, in this section we
present a simpliﬁed treatment which highlights important asymp
totic properties of the equations. it turns out that several key
properties are discernible by considering the scalar case n 2 1.
Although many of these properties can be developed for general
n, the simpliﬁed scalar case sufﬁces for obtaining a useful
qualitative analysis. Here we consider only (3.7), (3.14), and
(3.15). Since the Q equation (3.7) is decoupled from (3.14) and (3.15),
it can be analyzed separately. It is easy to_ see that there exists a
unique nonnegative solution for 'y > (RI/E) W as in the case of a
Standard Riccati equation with Stabilizability and detectability
hypotheses. Furthermore, it can be seen that for (RI/[51+ (AZ/V1)])l/2<7<(R1/:)V2 there exist two nonnegative solutions when A is stable and zero
nonnegative solutions whenA is unstable. Below this lower bound
for 7, nonnegative solutions Q do not exist. This result thus
indicates (as in LQG theory [42]) a lower bound to the achievable
Hg. disturbance attenuation as determined by the sensor noise
intensity V; appearing in 2. Since the P and Q equations (3.14) and (3.15) are coupled, they BERNSTEIN AND HADDAD: LQG CONTROL must be analyzed jointly. Since (3.15) is a standard Riccati
equation it follows under generic hypotheses that it possesses
exactly one nonnegative:deﬁnite solution for all values of Q and
Q. The analysis of the Q equation is, however, more involved. It
can be shown that the existence of real solutions is a complicated
function of 7, Q, and P. When real solutions do exist, it follows
that there exist either zero or two nonnegative—deﬁnite solutions.
To obtain further qualitative insight into the solutions P and Q, we
ﬁx 7 and allow R2 —+ 0, that is, the cheap L2 control case. It thus follows that P ~ (R12)“2 and that either Q ~ MHZ/RI)“ or
Q ~ 1/22Q2(2R,)‘“2, which correspond to the previously
mentioned pair of solutions satisfying (3.15). This result thus
indicates that an arbitrarily small Hm disturbance attenuation
constraint 7 can be achieved [subject to the solvability of (3.7)] by
sufﬁciently increasing the L2 controller authority. That is, since
solutions exist in the cheap L2 control case, the Ho, disturbance
attenuation constraint is achievable. The ability to achieve small y
is also attributable to the fact that since 6 = 0, Hon disturbance
attenuation to the control variables is not limited in (3.7), (3.14),
and (3.15) as in Theorems 3.1 and 6.1. Of course, as is wel_1
known, it is not possible to make 7 > 0 by letting E * on and E
> 00 when the system possesses nonminimum phase zeros. Also,
note that both of the asymptotic solutions to (3.15) are guaranteed
to yield the bound (4.1). The solution of interest, however, is Q
= 0(2'112) since it clearly yields a lower value of 5(Ac, BC, Cr,
Q) than Q = 0(21/2). VIII. NUMERICAL ALGORITHM AND ILLUSTRATIVE RESULTS In this section we describe a numerical algorithm which has
been developed and implemented for solving the coupled Riccati
equations (3.7), (3.14), and (3.15). We also present numerical
results for an illustrative example. Coupled modiﬁed Riccati equations arise in a variety of
problems and homotopic continuation methods have been shown
to be particularly successful [23]—[25]. To solve (3.7), (3.14), and
(3.15) we have implemented a simpliﬁed continuation method
involving the constraint constant 7. The idea is to exploit the fact
that for large 7 the problem is approximated by LQG which
provides a reliable starting solution. The continuation parameter 7
is then successively decreased until either a desired value of 'y is
achieved or no further decrease is possible. This algorithm is now
summarized. Let e > 0 denote a convergence criterion. Algorithm 8.1: To solve (3.7), (3.14), and (3.15), perform the
following steps: Step I .' Initialize 'y > 0. Step 2: Solve (3.7) fqr Q. Step 3: Let k = 0, Q0 = 0. A A Step 4: Solve (3.14) for 13k“ = Iiwith Q = Qt. Step 5: Solve (3.15) for Qk+l = Q with P = Pk“. A
“Step 6: Ifk 2 lcheck for “Pk” — Pk” < E and "th —
Qk ll < 6 Step 7: If convergence is not achieved in Step 6 (or k = 0) let k
(— k + 1 and go to Step 4; otherwise decrease 'y and go to Step 2. Steps 2, 4, and 5 were carried out using a standard Riccati
solver [37] which proved to be reliable even when the quadratic
term was indeﬁnite or nonnegative deﬁnite. For instance, for the
example considered below, the term 7 ‘ 2R1 — E was indeﬁnite for
all ﬁnite 'y. The crucial step in the algorithm is the decreasing of 'y
in Step 7. Signiﬁcant effort was devoted to providing a smooth
transition to smaller values of 7 without sacriﬁcing computational
efﬁciency. The development of more sophisticated continuation
algorithms remains an area for future research. The example considered was formulated in [38] and was
considered extensively in [24], [25], and [39] to compare reduced
order design methods. The example is interesting since it
possesses a complex pair of nonminimum phase zeros due to the
fact that the physical system (coupled rotating disks) has noncol
ocated sensors and actuators. The plant is of eighth order and has 301 two neutrally stable poles. The problem data are as follows: n:nc:8, m:l=1, q=p:2, —0.1611000000
—6.004 0 1 0 0 0 0 0
~0.58220010000
—9.98350001000
A: —o.40730000100
—3.9320000010
0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0
0
0
0.0064
0.00235
B: 0.0713 0‘“ 01”]
1.0002
0.1045
0.9955 0.55 11 1.32 18
0 0 0 O ’ 52: . E206: . 3:0, D1=lB 08x1], 172:“) 1] With the problem data as given, the LQG controller was found
to yield a closedloop Hon performance of 1.39 (i.e., 2.87 dB
above unity gain). Using Algorithm 8.1 we obtained a solution for
7 = 0.52 for a net Hon performance improvement of 8.7 dB (see
Fig. 1). Note that this result is consistent with [3, Proposition 8.1]
which implies that the maximum ratio of the H5, performance of
the optimal L2 controller to the Ho, performance of the optimal
Ho, controller can be no more than twice the number of right—half—
plane zeros, which for the present problem with two nonminimum
phase zeros corresponds to a factor of 4 (i.e., 12 dB). Our numerical experience revealed two interesting features.
First, the loop between Steps 4 and 6 converged reliably.
However, a critical value 7",,“ of 7 was invariably found below
which solutions could not be computed. This value 7min appears to
represent the best achievable Ho. performance for the given L2
weights. Second, for each value of 'y 2 7",,“ for which a solution
was computed, the actual Hg, performance was close to this value
revealing that the H, bound is tight. For example, the actual
worstcase attenuation of the 'y = 0.52 design shown in Fig. 1 is
0.511. Controller characteristics are given in Table I and are
plotted in Fig. 2 for several values of 7. Note that in each case the
L2 performance bound is within 30 percent of the actual L2
performance. IX. FURTHER EXTENSIONS The results obtained herein can readily be extended in several
directions. These include the treatment of parameter uncertainties
[13]—[15], [46], extensions to controllers with static feedthrough
[32], and the inclusion of cross—weighting terms (xT(t)R12u(t))
and noise correlation (DIDZT at 0). Finally, as mentioned in
Remark 5.2, connections with the exponentialofquadratic cost
criterion [33]—[35] and entropy maximization [43]—[45] are of
interest. 302 FRED IHZl
lD‘3 1, G lD'2 lU'l 1 10
g, 0.0
CI
\— “a
10 .0‘
ul
3
.I
a: 200
>
K
(I 3IJ 0
.1
3
O
z 4o.u
(0
5lJ0
50ﬂ
Fig. 1.
TABLE I
H6° Attenuation Actual H“, L; Performance Actual L;
Constraint '1 Attenuation Bound Performmce
HHMIIw J(AchnC¢)Q) J(A.,B.,C.)
co (DQG) 1.39 — .143
2 1.18 .159 .146
1.5 1.06 .171 .151
1.0 .855 .204 .168
.9 .797 .217 .176
.8 .732 236 .187
.7 .661 .262 .203
.52 .511 .299 .262
APPENDIX PROOF OF THEOREM 6.1 To optimize (2.20) over the open set S! subject to the constraint
(2.14), form the Lagrangian £(Ac, BC. CC, (2,, (P, k) a tr{>\qﬁ+[ﬁq+qfiT
+7’2QREQ+ 17m} (A.1) where the Lagrange multipliers k 2 O and (P 6 Eli” are not
both zero. We thus obtain %=(/i+y2Q1€m)TG>+<P(A+y2QI€,.)+>\R. (A.2)
Setting BJZ/BQ = 0 yields
0=(A+7’2Q15m)T(P+0(1+7‘2QRm)+k§. (A.3) Since/i + y’ZQR... is assumed to be stable, >\ = 0 implies (P =
0. Hence, it can be assumed without loss of generality that )\ = 1.
Furthermore, (P is nonnegative deﬁnite. Now partition I? x if QG’ into n x n, n x nc, and nc >< nc
subblocks as _ Q1 Q12 _ P1 P12
Q‘i in’y'ipa Pzi' Thus, with )x' = 1 the stationarity conditions are given by 63=M+r2q§wﬂo+@(£+72Qﬁm)+1€=0, 3Q (A.4) IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 34, NO. 3, MARCH 1989 653 aAC=PITZQ12+P2Q2=0, (A.5)
as T T CT—O A 6
ch:PZBcV2+(PnQ1+P2Q12)  t ( ~)
8°C
6C =R2CcQ2+l327‘2R2Cc(P1Q12+P12Q2)TQ12
C
+BT(P1Q12+P12Q2)=0o (A7) Expanding (2.14) and (A.4) yields
0=AQ1+QIAT+BCcQS+Q120337+7'2Q1R1mQ1
+BZ’Y_2Q12CZR2CCQ1T2+Vh (As)
0=AQ12+QizAZ+BCcQ2+QICTBZ+7_2Q1R1uQI2
+327—2Q12C3R2CcQ2, (A9)
0=AcQz+ Q2A6T+BcCQ12+ QITZCTBCT‘F‘Y _2Q{2RlooQ12
+ﬁ2‘y'2Q2CcTRzCLQ2+BCVZBCT, (A.10)
0=ATP. +P1A+CTBCTPTZ+PIZBCC
+7’2R1w(P1Q1+P12Q.T2)T +7_2(P1Q1+P12Q1T2)R1oo+R1. (All)
0=A TPl2+P12Ac+ CTBCTPZ+PIBCC +7’2R1m(P112Q1+P2Q{2)T +327_2(P1Q12+P12Q2)C3R2Cu (A12) O=ACTP2+P2AC+PITZBCC+CIBTP12+CZR2CC. (A.13) Lemma AJ: Q2 and P2 are positive deﬁnite.
Proof: By a minor extension of results from [40], (A. 10) can
be rewritten as 0=(Ac+BcCQuQ2*)Q2+Q2(Ac+BcCQ12Q;)T+‘1’ where ‘1’ g 7—2Q1T2R1wQ12‘l'ﬁzT'zchszccQzﬂBcVsz and Q; is the Moore—Penrose or Drazin generalized inverse of
Q2. Next note that since (AC, Be) is controllable it follows from
[28, Lemma 2.1 and Theorem 3.6] that (Ac + BECQIZQZ“ , ‘1’“)
is also controllable. Now, since Q2 and \I' are nonnegative
deﬁnite, [28, Lemma 12.2] implies that Q; is positive deﬁnite.
Using (A.13), similar arguments show that P; is positive
deﬁnite. El Since R2 , V2, Q2, P2 are invertible, (A.5)—(A.7) can be written
as —P2—1,P1T2Q12Q2_1=Incy (A.14) Br: ‘PJ'U’E 1+P2Q1T2)CTV;‘, (A.15)
Ccllnc+527‘2(erzP1+Q2P§)Q,2Q{] = _R;IBT(P1QIZ+P12Q2)Q2_l (A.l6) Now deﬁne the n X n matrices
Q g QI_Q12Q2.l {2, P éPl—PuPzIPITZ,
Q 3 leQz'l 1T2. Is é PlnglPiTz, T 2 Q12Q2‘1P{'Plrz BERNSTEIN AND HADDAD: LQG CONTROL [L‘D 1.30 i .20 21.", m l  l0 1 .500
I E
z t .01:
E
v
r; u .m

: I goo
'7
x n.au a
070
0.00 [SD 00 303 IC'UIL L2 6031' (I 10'! 1 L2 VEIFBIII‘OCI V. MINFIII'V ﬁTYE'ﬂJﬂ‘HON and the 72, x n, n, X nc, and n, x n matrices
G 2 Q;1 M 2 Qsz, 1‘ 2 —P2'1P172.
Note that 1 = GTI‘.‘ Q 1" onfxn A Clearly, Q, P, Q, and 15 are symmetric and Q and 15 are
deﬁnite, note that Q is the ~upper lefthand block of the
— QizQz— ‘
1,, c '
Similarly, P is nonnegative deﬁnite.
(6.2) and that (6.1) holds. Hence, 1 = 0711 is idempotent, i.e., 12
= r nonnegative deﬁnite. To show that Q and P are also nonnegative
nonnegative deﬁnite matrix QQQT, where
Next note that with the above deﬁnitions (A.14) is equivalent to
It is helpful to note the identities Q=QnG=GTQr2=GTQza P= —P12P= —rTP.T,=PTP2r. (A.17)
TGT=GT, I‘7=I‘, (A.18)
Q=1Q",P=131, (A.19)
QP=  anPl’z (A.20) Using (A.14) and Sylvester’s inequality, it follows that
rank G = rank I‘ = rank Q12: rank Pa = nc.
Now using (A.17) and Sylvester’s inequality yields
nc=rank Qu+rank G—ncsrank Q5 rank le=nc
which implies that rank Q = nc. Similarly, rank P = 11,, and rank Q13 = no follows from (A.20).
The components of Q, and (9 can be written in terms of Q, P, Fig. 2. Q, 15, G, and I‘ as Q1=Q+Q, P1=P+15. (A.21)
Q12= QT T, P12 = — PG T, (A.22)
Q2=I‘QI‘ 7. P2= 6150’. (A23) Next note that by using (A.21)—(A23) it can be shown that the
righthand coefﬁcient of C, in (A.16) is given by s a 1,c+3272eror.
To prove that S is invertible use (A.19) and (6.3) and note that
Inc+ﬁ27 ‘ZI‘QPGT=I,,C+327‘ZI‘QTTPGT
=Inc+827’2(I‘QI‘ T)(GPGT). Since FQPT and GPGT are nonnegative deﬁnite, their product
has nonnegative eigenvalues “(see Lemma 5.1). Thus, each
eigenvalue of I c + {*Izy’ZI‘QPGT is real and is greater than
unity. Hence, is invertible. Now note that by using (6.2) and
(6.3) it can be shown that GT§“=SGT. The expressions (6.11), (6.12), and (6.13) follow from (A.15),
(A. 16), and the deﬁnition of Q. Next, computing either I‘(A.9)
(A.10) or G(A.12) + (A.13) yields (6.10). Substituting (A.21)—
(A.23) into (A.8)—(A.13) and the expression for A, into (A.9),
(A.10), (A.12), and (A.13) it follows that (A.10) = I‘(A.9) and
(A. 13) = G(A.12). Thus, (A. 10) and (A. 13) are superﬂuous and
can be omitted. Thus, (A.8)(A.13) reduce to 0=AQ+QA 7+ V. +72(Q+Q‘)R1..<Q+Q)
+627‘2QSTP2PSQ +(A—2PS)Q+Q(A —2PS)T, (A.24) 304 0: [(A —EPS)Q+Q(A —2PS)7+ QSQ +7—2(Q+Q)Rleo(Q+Q)_7—2QR1¢Q + 327 2QS TPEPSQ‘ ]r T, (A25) 0=(A+7’2[Q+Q]R1m)TP+P(A+y’2[Q+Q]R1m)+Ri
+(AQ§+7‘2QR1w)TP+P(AQ§+7'ZQR1.»), (A26)
0=[(A_Q2+7_ZQle)Tp+p(A_Q:+72QR100)
+STPEPS]GT. (A.27) Next, using (A.24) + GTI‘(A.25)G — (A.25)G — [(A.25)G]T
and GTI‘(A.25)G — (A.25)G — [(A.25)G]T yields (6.5) and
(6.7). Similarly, using (A26) + I‘TG(A.27)I‘ — (A.27)I‘ —
[(A.27)I‘]T and I‘TG(A.27)I‘ — (A.27)I‘ — [(A.27)I‘]T yields
(6.6) and (6.8). Finally, to prove the converse we use (6.5)—(6.13) to” obtain
(2.14) and (A.4)—(A.7). LetAc, 3,, CC, C, I‘, 7', Q, P, Q, P, Q,
be as in the statement of Theorem 6.1 and deﬁne Q, , Q12, Q2, P1,
P12, P2 by (A.21)—(A.23). Using (6.2), (6.11), and (6.12) it is
easy to verify (A6) and (A.7). Finally, substitute the deﬁnitions
of Q, P, Q, P, G, I‘, and 1 into (6.5)—(6.8) using (6.2), (6.3), and
(A.19) to obtain (2.14) and (A4). Finally, note that _ Q Onxnc In T
Q— 0% ]+ [F] QII. P l
which shows that Q, 2 0. :l ACKNOWLEDGMENT The authors wish to thank Prof. P. P. Khargonekar for several
helpful discussions, J. Straehla for transforming the original
manuscript into TEX, A. Daubendiek, S. Greeley, S. Richter, and
A. Tellez for developing the numerical algorithm and performing
the calculations of Section VIH, D. Hyland, E. Collins, and L.
Davis for helpful discussions and suggestions, Dr. A. N.
Madiwale for providing simpliﬁcations of (6.6)—(6.8), the review—
ers for several helpful comments, Prof. J. C. Doyle for helpful
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[17]
[13] [19] [201 [21] [22]
[22a]
[23] [24] [25] [26] [27]
[23]
[29] [30]
[31] New York: [32]
[33]
[34]
[35]
[36] [37]
[38] [39] [40] [41] BERNSTEIN AND HADDAD: LQG CONTROL [421 [43] [44] [45]
[46] performance bound: A Riccati equation approach," in Proc. Amer.
Conlr. Conf. Atlanta, GA, June I988, pp. 7967802. H. Kwakernaak and R. Sivan, “The maximally achievable accuracy of
linear optimal regulators and linear optimal filters,“ IEEE Trans.
Auromar. Comm, vol. ACl7, pp. 7986, 1972. K. Glover and .l. C. Doyle, “State—space formulae for all stabilizing
controllers that satisfy an Hognorm bound and relations to risk
sensitivity," Syst. Comr. Lett.. vol. 11, pp. 167—172, 1988. D. Mustafa and K. Glover, “Controllers which satisfy a closedloop
H... norm bound and maximize an entropy integral,” in Proc. [EEE
Conf. Decision Contr., Austin, TX, Dec. 1988. D. Mustafa, “Relations between maximum entropy/Hm control and
combined Hm/LQG control." preprint. A, N. Madiwale, W. M. Haddad, and D. S. Bernstein, “Robust Hm
control design for systems with structured parameter uncertainty,” in
Proc. IEEE Conf. Decision Comr., Austin. TX, Dec. 1988, pp.
965—972. Dennis S. Bernstein (M'82) received the Sc.B.
degree in applied mathematics from Brown
University, Providence, RI, and the M.S.E. and
Ph.D. degrees from the Computer, Information and
Control Engineering Program at the University of
Michigan, Ann Arbor. After spending two years at Lincoln Laboratory,
Massachusetts Institute of Technology. Lexington.
he joined the Controls Analysis and Synthesis
Group of the Government Aerospace Systems
Division, Harris Corporation, Melbourne, FL. At Harris Corporation his research interests are directed primarily toward the 305 control of spacecraft with ﬂexible appendages with particular emphasis on
robust and nonlinear control techniques.
Dr. Bernstein is a member of SIAM. Wassim M. Haddad (S’S7M’87) was born in
Athens, Greece, on July 14, 1961. He received the
8.5., MS, and PhD, degrees in mechanical
engineering in 1983, 1984. and 1987, respectively,
from the Florida Institute of Technology,
Melbourne. Since 1987 he has been a consultant for the
Controls Analysis and Synthesis Group of the
Government Aerospace Systems Division, Harris
Corporation, Melbourne. FL, and he is currently a
faculty member in the Department of Mechanical and Aerospace Engineering, Florida Institute of Technology. His research
interests are in the area of robust estimation and control for aerospace systems. Dr. Haddad is a member of Tau Beta l’i. ...
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