**Purpose**

To reduce the matrices D12 and D21 of the linear time-invariant system | A | B1 B2 | | A | B | P = |----|---------| = |---|---| | C1 | D11 D12 | | C | D | | C2 | D21 D22 | to unit diagonal form, to transform the matrices B, C, and D11 to satisfy the formulas in the computation of an H2 and H-infinity (sub)optimal controllers and to check the rank conditions.

SUBROUTINE SB10PD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC, $ D, LDD, TU, LDTU, TY, LDTY, RCOND, TOL, DWORK, $ LDWORK, INFO ) C .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LDC, LDD, LDTU, LDTY, LDWORK, $ M, N, NCON, NMEAS, NP DOUBLE PRECISION TOL C .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ), $ D( LDD, * ), DWORK( * ), RCOND( 2 ), $ TU( LDTU, * ), TY( LDTY, * )

**Input/Output Parameters**

N (input) INTEGER The order of the system. N >= 0. M (input) INTEGER The column size of the matrix B. M >= 0. NP (input) INTEGER The row size of the matrix C. NP >= 0. NCON (input) INTEGER The number of control inputs (M2). M >= NCON >= 0, NP-NMEAS >= NCON. NMEAS (input) INTEGER The number of measurements (NP2). NP >= NMEAS >= 0, M-NCON >= NMEAS. A (input) DOUBLE PRECISION array, dimension (LDA,N) The leading N-by-N part of this array must contain the system state matrix A. LDA INTEGER The leading dimension of the array A. LDA >= max(1,N). B (input/output) DOUBLE PRECISION array, dimension (LDB,M) On entry, the leading N-by-M part of this array must contain the system input matrix B. On exit, the leading N-by-M part of this array contains the transformed system input matrix B. LDB INTEGER The leading dimension of the array B. LDB >= max(1,N). C (input/output) DOUBLE PRECISION array, dimension (LDC,N) On entry, the leading NP-by-N part of this array must contain the system output matrix C. On exit, the leading NP-by-N part of this array contains the transformed system output matrix C. LDC INTEGER The leading dimension of the array C. LDC >= max(1,NP). D (input/output) DOUBLE PRECISION array, dimension (LDD,M) On entry, the leading NP-by-M part of this array must contain the system input/output matrix D. The NMEAS-by-NCON trailing submatrix D22 is not referenced. On exit, the leading (NP-NMEAS)-by-(M-NCON) part of this array contains the transformed submatrix D11. The transformed submatrices D12 = [ 0 Im2 ]' and D21 = [ 0 Inp2 ] are not stored. The corresponding part of this array contains no useful information. LDD INTEGER The leading dimension of the array D. LDD >= max(1,NP). TU (output) DOUBLE PRECISION array, dimension (LDTU,M2) The leading M2-by-M2 part of this array contains the control transformation matrix TU. LDTU INTEGER The leading dimension of the array TU. LDTU >= max(1,M2). TY (output) DOUBLE PRECISION array, dimension (LDTY,NP2) The leading NP2-by-NP2 part of this array contains the measurement transformation matrix TY. LDTY INTEGER The leading dimension of the array TY. LDTY >= max(1,NP2). RCOND (output) DOUBLE PRECISION array, dimension (2) RCOND(1) contains the reciprocal condition number of the control transformation matrix TU; RCOND(2) contains the reciprocal condition number of the measurement transformation matrix TY. RCOND is set even if INFO = 3 or INFO = 4; if INFO = 3, then RCOND(2) was not computed, but it is set to 0.

TOL DOUBLE PRECISION Tolerance used for controlling the accuracy of the applied transformations. Transformation matrices TU and TY whose reciprocal condition numbers are less than TOL are not allowed. If TOL <= 0, then a default value equal to sqrt(EPS) is used, where EPS is the relative machine precision.

DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) contains the optimal LDWORK. LDWORK INTEGER The dimension of the array DWORK. LDWORK >= MAX(1,LW1,LW2,LW3,LW4), where LW1 = (N+NP1+1)*(N+M2) + MAX(3*(N+M2)+N+NP1,5*(N+M2)), LW2 = (N+NP2)*(N+M1+1) + MAX(3*(N+NP2)+N+M1,5*(N+NP2)), LW3 = M2 + NP1*NP1 + MAX(NP1*MAX(N,M1),3*M2+NP1,5*M2), LW4 = NP2 + M1*M1 + MAX(MAX(N,NP1)*M1,3*NP2+M1,5*NP2), with M1 = M - M2 and NP1 = NP - NP2. For good performance, LDWORK must generally be larger. Denoting Q = MAX(M1,M2,NP1,NP2), an upper bound is MAX(1,(N+Q)*(N+Q+6),Q*(Q+MAX(N,Q,5)+1).

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: if the matrix | A B2 | had not full column rank | C1 D12 | in respect to the tolerance EPS; = 2: if the matrix | A B1 | had not full row rank in | C2 D21 | respect to the tolerance EPS; = 3: if the matrix D12 had not full column rank in respect to the tolerance TOL; = 4: if the matrix D21 had not full row rank in respect to the tolerance TOL; = 5: if the singular value decomposition (SVD) algorithm did not converge (when computing the SVD of one of the matrices |A B2 |, |A B1 |, D12 or D21). |C1 D12| |C2 D21|

The routine performs the transformations described in [2].

[1] Glover, K. and Doyle, J.C. State-space formulae for all stabilizing controllers that satisfy an Hinf norm bound and relations to risk sensitivity. Systems and Control Letters, vol. 11, pp. 167-172, 1988. [2] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and Smith, R. mu-Analysis and Synthesis Toolbox. The MathWorks Inc., Natick, Mass., 1995.

The precision of the transformations can be controlled by the condition numbers of the matrices TU and TY as given by the values of RCOND(1) and RCOND(2), respectively. An error return with INFO = 3 or INFO = 4 will be obtained if the condition number of TU or TY, respectively, would exceed 1/TOL.

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**Program Text**

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