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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 control-design 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 l-I.-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 reduced-order design problems are considered with an H... attenuation constraint involving both state and control variables. An algorithm is developed for the full-order 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 worst-case attenuation. For systems with poorly modeled disturbances which may possess significant power within arbitrarily small bandwidths, H0, is clearly appropriate, while for systems with well-known 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 state-space 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 state-space 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 F49620-86—C-0002 and F49620-87-C-0108. 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 parameter-robust full-state—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 Han-optimal static full-state-feedback controller is also optimal over the class of dynamic full—state-feedback controllers. The results of [18]—[20] thus solve the standard problem considered in [3] and [4] for the full-state—feedback case. The extension of these results to the standard problem for dynamic output-feedback compensation, however, was not given in [18]—[20]. Within the realm of quadratic robust stabilization, the dynamic output-feedback 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 full-order dynamic compensation and a coupled system of four modified Riccati and Lyapunov equations in the fixed—order (i.e., reduced-order) 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 fixed-order compensa- tion? To begin to address this question, the goal of the present paper is to develop a design methodology for fixed-order, i.e. , full— and reduced-order, 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—state-feedback 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 fixed-order dynamic output-feedback control— 0018-9286/89/0300-0293$01.00 © 1989 IEEE 294 lers yielding specified disturbance attenuation. Existing optimal Ho. design methods tend to yield high-order controllers. Intui- tively, solving the fixed—order design equations for progressively smaller Ha. disturbance attenuation constraints should, in the limit, yield an Hw-optimal controller over the class of fixed-order 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 Hm-constrained full— and reduced—order design problems. In the full-order case involving equalized L2 and Ha. perform- ance weights, we also show that the Hon-constrained 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 closed-loop covariance is replaced by a modified Riccati equation possessing a nonnegative— definite solution, then the closed-loop 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 closed-loop 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 control-design problem is considered in Section VI. A simplified qualitative analysis of the full-order 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, reduced-order 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, fi-dimensional 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 output-feedback 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 [fa-Constrained LQG Control Problem: Given the nth-order 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 nth-order 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 closed-loop transfer function H(s) a 54515—2045 (2.5) from w(t) to E,,,x(t) + EZqu) satisfies the constraint l|H(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) 1-H” is minimized. Note that the closed-loop 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 nth-order 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 I-mo 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, cross-weighting 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 steady-state closed-loop 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 closed-loop 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 steady-state 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+y-2QR,.Q+ V. (2.14) Then (1, 15) is stabilizable (2.15) if and only if 11 is asymptotically stable. (2.16) In this case, l|H(s)l|e.S~1 (2.17) and Q5 6t. (2.18) 296 Consequently, J(Ac, Be. CJSSMu BC. Cos Q.) (2-19) 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 nonnegative-definite 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'(Q-Q~)+(Q—Q~)A'T+T_ZQILQ (2.21) which, since A is asymptotically stable,'is equivalent to Q—Q= e/I’h‘ZQR'kaA-T’ 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 nonnegative-definite 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- five-definite 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 closed-loop system can be represented by two possibly different transfer functions. Specifically, with respect to the L2 cost criterion, the closed-loop 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 closed-loop 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 nonnegative-definite 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—2QR-ou 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 reduced-order 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 nonnegative-definite 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+7-21Q+Q1R1..) +R1— STPEPS, (3.5) (3.6) (3.7) (3.8) 0=(A —2PS+7‘2QR1,,)Q+Q(A —>:PS+y-2QR,,,)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]. (3-10) 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+y-ZQR,,,, (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 closed-loop 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 closed-loop transfer function H (5) satisfies the Ha, attenuation constraint IIH(5)l|wS'Y (4-1) 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 nonnegative-definite 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 nth-order 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 reduced-order 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- (5-1) 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“ZQIRlu-BZR1DTZ +Z(A+v-ZQR1..+w-2Q[R1..—62Rli) +R.—Z(2+s27-4Q[R...—82R11Q)z +627‘ZZQ2QZ (5.2) and (3.9) is equivalent to 0=(A—EZ+7‘2QR,,.)Q+ Q(A—zz+7-2QR1..)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)(3-8)(In_327-2QZ)+ 627—22(3-9)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- (5-8) 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 (In-Blv‘le)“[(5-2)-flzv‘ZZ(5-3)Zl(1n-B2'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-‘+627-1Q)-'=(P-1+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+v-2[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—Qi-E(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 = 712-1 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..+y-Zz,.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 closed-loop 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,=QCTV2-1, (5.34) ct: _R;;BT(Y;I—y-2Q)-1. (5.35) Then (xi, 5) is stabilizable if and only if A is asymptotically stable. In this case, the closed-loop 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 5-77a 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 linear-exponential—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(XTR|x+u7-R2u) 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 closed-loop transfer function. There also exist fundamental connections with the problem of entropy maximiza- tion [43]—[45]. VI. EXTENSIONS TO REDUCED-ORDER 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 +7-2Q(R1,.+BZSTP2PS)Q+QiQ—ngigr:, (6.7) 0=(A — Q: +y-ZQR,,,)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 reduced-order 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[R1m-BZR1D +R1-Z(2+327‘4Q[R1m-BZRIIQ)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.,1-l = 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 closed-loop 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 worst-case 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 exponential-of-quadratic 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: -20-0 > K (I -3IJ 0 .1 3 O z -4o.u (0 -5lJ-0 -50-fl 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+y-2Q1€m)TG>+<P(A+y-2QI€,.)+>\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+@(£+7-2Qfim)+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 (A-7) Expanding (2.14) and (A.4) yields 0=AQ1+QIAT+BCcQS+Q120337+7'2Q1R1mQ1 +BZ’Y_2Q12CZR2CCQ1T2+Vh (A-s) 0=AQ12+QizAZ+BCcQ2+QICTBZ+7_2Q1R1uQI2 +327—2Q12C3R2CcQ2, (A-9) 0=AcQz+ Q2A6T+BcCQ12+ QITZCTBCT‘F‘Y _2Q{2RlooQ12 +fi2‘y'2Q2CcTRzCL-Q2+BCVZBCT, (A.10) 0=ATP. +P1A+CTBCTPTZ+PIZBCC +7’2R1w(P1Q1+P12Q.T2)T +7_2(P1Q1+P12Q1T2)R1oo+R1. (A-ll) 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—PuPz-IPITZ, 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 0-70 0.00 [SD -00 303 IC'UIL L2 6031' (I 10'-! 1 L2 VEIFBIII‘OCI V. M-INFIII'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 left-hand 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 right-hand coefficient of C, in (A.16) is given by s a 1,c+327-2eror. 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 + {*Iz-y’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. +7-2(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 +(A-Q§+7‘2QR1w)TP+P(A-Q§+7'ZQR1.»), (A26) 0=[(A_Q2+7_ZQle)Tp+p(A_Q:+7-2QR100) +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 discussions, and D. Mustafa for providing a preprint of [45]. REFERENCES [1] H. Kwakemaak and R. Sivan, Linear Optimal Control Systems. New York: Wiley, 1972. [2] G. Zames, “Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate in- tltggsps,” IEEE Trans. Automat. Contr., vol. AC—26, pp. 301-320, [3] B. A. Francis and J. C. Doyle, “Linear control theory with an Ho, optimality criterion,” SIAM J. Contr. Optimiz., vol. 25, pp. 8157 844, 1987. [4] B. A. Francis, A Course in H, Control Theory. New York: Springer-Verlag, 1987. [5] J. C. Doyle and G. Stein, “Multivariable feedback design: Concepts for a classical modern synthesis,” IEEE Trans. Automat. Contr., vol. AC-26, pp. 4—6, 1981. [6] B. A. 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Haddad, “LQG control with an H... [13] [14] [15] [16] [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. AC-l7, pp. 79-86, 1972. K. Glover and .l. C. Doyle, “State—space formulae for all stabilizing controllers that satisfy an Hog-norm bound and relations to risk sensitivity," Syst. Comr. Lett.. vol. 11, pp. 167—172, 1988. D. Mustafa and K. Glover, “Controllers which satisfy a closed-loop 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’S7-M’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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