**Purpose**

To transform a SISO (single-input single-output) system [A,B;C,D] by mirroring its unstable poles and zeros in the boundary of the stability domain, thus preserving the frequency response of the system, but making it stable and minimum phase. Specifically, for a continuous-time system, the positive real parts of its poles and zeros are exchanged with their negatives. Discrete-time systems are first converted to continuous-time systems using a bilinear transformation, and finally converted back.

SUBROUTINE SB10ZP( DISCFL, N, A, LDA, B, C, D, IWORK, DWORK, $ LDWORK, INFO ) C .. Scalar Arguments .. INTEGER DISCFL, INFO, LDA, LDWORK, N C .. Array Arguments .. INTEGER IWORK( * ) DOUBLE PRECISION A( LDA, * ), B( * ), C( * ), D( * ), DWORK( * )

**Input/Output Parameters**

DISCFL (input) INTEGER Indicates the type of the system, as follows: = 0: continuous-time system; = 1: discrete-time system. N (input/output) INTEGER On entry, the order of the original system. N >= 0. On exit, the order of the transformed, minimal system. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the leading N-by-N part of this array must contain the original system matrix A. On exit, the leading N-by-N part of this array contains the transformed matrix A, in an upper Hessenberg form. LDA INTEGER The leading dimension of the array A. LDA >= MAX(1,N). B (input/output) DOUBLE PRECISION array, dimension (N) On entry, this array must contain the original system vector B. On exit, this array contains the transformed vector B. C (input/output) DOUBLE PRECISION array, dimension (N) On entry, this array must contain the original system vector C. On exit, this array contains the transformed vector C. The first N-1 elements are zero (for the exit value of N). D (input/output) DOUBLE PRECISION array, dimension (1) On entry, this array must contain the original system scalar D. On exit, this array contains the transformed scalar D.

IWORK INTEGER array, dimension (max(2,N+1)) DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, if INFO = 0, DWORK(1) returns the optimal value of LDWORK. LDWORK INTEGER The length of the array DWORK. LDWORK >= max(N*N + 5*N, 6*N + 1 + min(1,N)). For optimum performance LDWORK should be larger.

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value; = 1: if the discrete --> continuous transformation cannot be made; = 2: if the system poles cannot be found; = 3: if the inverse system cannot be found, i.e., D is (close to) zero; = 4: if the system zeros cannot be found; = 5: if the state-space representation of the new transfer function T(s) cannot be found; = 6: if the continuous --> discrete transformation cannot be made.

First, if the system is discrete-time, it is transformed to continuous-time using alpha = beta = 1 in the bilinear transformation implemented in the SLICOT routine AB04MD. Then the eigenvalues of A, i.e., the system poles, are found. Then, the inverse of the original system is found and its poles, i.e., the system zeros, are evaluated. The obtained system poles Pi and zeros Zi are checked and if a positive real part is detected, it is exchanged by -Pi or -Zi. Then the polynomial coefficients of the transfer function T(s) = Q(s)/P(s) are found. The state-space representation of T(s) is then obtained. The system matrices B, C, D are scaled so that the transformed system has the same system gain as the original system. If the original system is discrete-time, then the result (which is continuous-time) is converted back to discrete-time.

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**Program Text**

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