FPIO zyxw = zyxwvu 4:45 Proceedings of 23rd Conference on Decision and Control Las Vegas, NV, December 1984 zy ON zyxwvutsrq INNER-OUTER AND SPECI'RAL FACTORIZATIONS Cheng-Chih Chu John C. Doyle Honeywell Inc., Systems and Research Center, Minneapolis, Minnesota Abstract This paper outlines methods for computing the key factori- zations necessary to solve general zyxwvutsrqp Hz and H, linear optimal control problems. Notation Lebesque space 1 I Hardy space 1 zyxwvutsrqponmlkj t Proper, red-rational 1 zyxwvutsrqponmlkjihgf Ipxm matrices in R{ (similarly for zyxwvuts H and L) zyxwvuts D t C(sI -A)-'B right coprime factorization over RH, Throughout this paper, a will be used whenever either zyxwvut a=2 M E RH, such that K 1 E RH,. When R is used as a prefix, it or a=- would apply equally. The term unit in RH, refers to any denotes real-rational. 1. Introduction : The importance of inner-outer (IOF'), spectral and coprime factorizations in obtaining solutions to certain Hz and H, optimal control problems has been hown for some time. The solutiontothegeneral Ha (a=Z,=) optimal control problems [1],[2] uses these factorizations and,in addition, the "comple- mentary inner factor" (CIF'), to reduce the general problem to that of approximating an La rational matrix by one in Ha. This paper focuses on the factorizations used in [Z] and, in particular, on explicit formulas and methods for computation. We show that all thefactorizationsneeded in the Hz and H, optimal synthesis problem can be obtained using standard real matrix operations on state-space representations. The Algebraic Riccati Equation (ARE) playsacentralrole in computingthe desired factorizations. Because of space limitations, the "proofs" of the results in this paper are extremely sketchy. 2. Background : lowing figure. The general H, optimal control problem is shown in the fol- R The objective is to find a stabilizing K E Rmpxpz which solves min I !Fl(P;K)j i, where Fi(P;K) 4 PI1 tP12K(I -PZK)-'P21 . For nontriviality, assume that p1 > mz and mi > pz. K * This work has been supported by Honeywell Internal Research and Development Funding, the Office of Naval Research under Oh'R Research Grant KOOO14-82-C-0157, and the U.S. Air Force of Scientific Research Grant F49620-82- c-0090. ~~ stable and affine for any Q E RHEzxp'. This is the Youla parametrization of all stabilizing controllers and is obtained by finding coprime factorizations of P over the ring of stable ration- als and solving a double Bezout identity to obtain the coefficients of KC. le are interested in a particular KO xk1ch results in both 1% ' and N being inner. That is, N*N=I and .hJN*=I. This requires a coprime factorization with inner numerat r, In addition, we require .VI and .nilinner so that N IV I] and llvd are square and inner. A iand 51 are called complementary inner factors (m. With these we have that N 7, 1 since both the a = 2 and = norms are unitary invariant. The He case immediately reQ:es to a best approximation problem with QOpt = PH,(N* [TI]] N ), where PX* denotes projec- tion onto Hz. The H, case is somewhat more complicated and requires an additional spectral factorization._To see how this arises, consider t e spe ial case when TllNl* = 0 and (2.1) reduces to ~iR EQ] ~~, with R = N* [Tll] Nh* and G = zyxwvu .vi [ T ~ ~ ] E*. It is easily verified that for any y > / / G 11- where (H)g denotes the unit spectral factor of the para- Hermitianmatrix H. Thus, the H, best approximation problem since can be absorbed into Q. The general case similarly involves both inner-outer and spectral factorizations [Z]. The remainder of the paper outlines methods for computing these factoriza- tions. 3. Algebraic Riccati Equation : Consider the Algebraic Riccati Equation, FTX + XF - XWX t Q = 0 where P, W, Q E IRnxn, W = WT 2 0 and Q = QT with the associated Hamiltonian matrix Ir -wl Our main interest is to find the unique real symmetric stabiliz- ing solutionsuchthatthematrix (F - WX) is asymptotically stable. For simplicity we will use "solution" of the to mean a real symmetric one. The ARE considered here is more general than the ARE which arises in linear quadratic optimal control and Kalman-Bucy Altering theory in that there is no assumption on the definiteness of the matrix Q. The following theorem gives thenecessaryandsufficient conditions for the existence of auniquestabilizingsolution of (AIEE). Without loss of generality, we will assume that W = GGT. This is a slight generalization of a theorem of Kucera [3]. CH2093-3/84/0000-1764 $1.00 0 1984 IEEE 1764