# dhinf

H_infinity design of discrete-time systems

### Syntax

[AK, BK, CK, DK] = dhinf(A,B,C,D, ncon, nmeas, gamma) [AK, BK, CK, DK, RCOND] = dhinf(A,B,C,D, ncon, nmeas, gamma)

### Arguments

- A
the n-by-n system state matrix A.

- B
the n-by-m system input matrix B.

- C
the p-by-n system output matrix C.

- D
the p-by-m system matrix D.

- ncon
the number of control inputs. m >= ncon >= 0, p-nmeas >= ncon.

- nmeas
the number of measurements. p >= nmeas >= 0, m-ncon >= nmeas.

- gamma
the parameter gamma used in

`H_infinity`

design. It is assumed that gamma is sufficiently large so that the controller is admissible. gamma >= 0.- AK
the n-by-n controller state matrix AK.

- BK
the n-by-nmeas controller input matrix BK.

- CK
the ncon-by-n controller output matrix CK.

- DK
the ncon-by-nmeas controller matrix DK.

- RCOND
a vector containing estimates of the reciprocal condition numbers of the matrices which are to be inverted and estimates of the reciprocal condition numbers of the Riccati equations which have to be solved during the computation of the controller. (See the description of the algorithm in [1].)

- RCOND
(1) contains the reciprocal condition number of the matrix R3,

- RCOND
(2) contains the reciprocal condition number of the matrix R1 - R2'*inv(R3)*R2

- RCOND
(3) contains the reciprocal condition number of the matrix V21,

- RCOND
(4) contains the reciprocal condition number of the matrix St3,

- RCOND
(5) contains the reciprocal condition number of the matrix V12,

- RCOND
(6) contains the reciprocal condition number of the matrix Im2 + DKHAT*D22,

- RCOND
(7) contains the reciprocal condition number of the X-Riccati equation,

- RCOND
(8) contains the reciprocal condition number of the Z-Riccati equation.

### Description

`[AK,BK,CK,DK,(RCOND)] = dhinf(A,B,C,D,ncon,nmeas, gamma)`

To compute the matrices of an H-infinity (sub)optimal n-state
controller

| AK | BK | K = |----|----|, | CK | DK |

for the discrete-time system

| A | B1 B2 | | A | B | P = |----|---------| = |---|---|, | C1 | D11 D12 | | C | D | | C2 | D21 D22 |

and for a given value of gamma, where B2 has column size of the number of control inputs (ncon) and C2 has row size of the number of measurements (nmeas) being provided to the controller.

### References

[1] P.Hr. Petkov, D.W. Gu and M.M. Konstantinov. Fortran 77 routines for Hinf and H2 design of linear discrete-time control systems. Report99-8, Department of Engineering, Leicester University, April 1999.

### Examples

//example from Niconet report SLWN1999-12 //Hinf A=[-0.7 0 0.3 0 -0.5 -0.1 -0.6 0.2 -0.4 -0.3 0 0 -0.5 0.7 -0.1 0 0 -0.8 -0.7 0 0 -0.5 -1 0 0 0.3 0.6 -0.9 0.1 -0.4 0.5 -0.8 0 0 0.2 -0.9]; B=[-1 -2 -2 1 0 1 0 1 -2 1 -3 -4 0 2 -2 1 -2 1 0 -1 0 1 -2 0 3 1 0 3 -1 -2]; C=[ 1 -1 2 -2 0 -3 -3 0 1 -1 1 0 0 2 0 -4 0 -2 1 -3 0 0 3 1 0 1 -2 1 0 -2]; D=[1 -1 -2 0 0 0 1 0 1 0 2 -1 -3 0 1 0 1 0 1 -1 0 0 1 2 1]; ncon=2 nmeas=2 gam=111.30; [AK,BK,CK,DK] = dhinf(A,B,C,D,ncon,nmeas,gam)

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