Mu Analysis and Synthesis Toolbox

dhfsyn computes an H controller for a discrete-time SYSTEM interconnection matrix

Syntax

• [k,g,gfin,ax,ay,hamx,hamy] = dhfsyn(p,nmeas,ncon,...
                         gmin,gmax,tol,h,z0,quiet,ricmethd,epr,epp)
  

Description
dhfsyn calculates a discrete-time H controller that achieves the infinity norm gfin for the interconnection structure p. The controller, k, stablizes the discrete-time SYSTEM matrix p and has the same number of states as p. The SYSTEM p is partitioned

where B₁ are the disturbance inputs, B₂ are the control inputs, C₁ are the errors to be kept small, and C₂ are the output measurements provided to the controller. B₂ has column size (ncon) and C₂ has row size (nmeas).

The closed-loop system is returned in g. The same bilinear transformation method described for dhfnorm is used. The controller k is returned that minimizes the entropy integral,

The program calls the continuous-time routine hinfsyne and the corresponding conditions and tests need to be satisfied.

Input arguments

p	SYSTEM interconnection structure matrix, (stable, discrete time)
nmeas	number of measurements output to controller
ncon	number of control inputs
gmin	lower bound on
gmax	upper bound on
tol	relative difference between final values
h	time between samples (optional)
z0	point at which entropy is evaluated (default )
quiet	controls printing on the screen 1. no printing 0. header not printed -1. full printing (default)
ricmethod	1.Eigenvalue decomposition (with balancing) -1. Eigenvalue decomposition (without balancing) 2. Schur decomposition (with balancing, default) -2. Schur decomposition (without balancing)
epr	measure of when a real part of an eigenvalue of the Hamiltonian matrix is zero (default epr = 1e-10)
epp	positive definite determination of the X and Y solution (default epp = 1e-6)

Output arguments

k	H (sub) optimal controller (discrete time)
g	closed-loop system with H controller (discrete time)
gfin	final value associated with k and g
ax	X Riccati solution as a VARYING matrix with independent variable
ay	Y Riccati solution as a VARYING matrix with independent variable
hamx	X Hamiltonian matrix as a VARYING matrix with independent variable
hamy	Y Hamiltonian matrix as a VARYING matrix with independent variable

:

Note that the outputs ax, ay, hamx, and hamy correspond to the equivalent continuous-time problems and can also be scaled and/or balanced.

The dhfsyn program outputs several variables, which can be checked to ensure that the above conditions are being met. For each value the minimum magnitude, real part of the eigenvalues of the X Hamiltonian matrices is displayed along with the minimum eigenvalue of X, which is the solution to the X Riccati equation. A # sign is placed to the right of the condition that failed in the printout. This additional information can aid you in the control design process.

Algorithm
dhfsyn uses the above bilinear transformation to continuous-time and then the formulae described in the Glover and Doyle paper for solution to the optimal H control design problem.

Reference
Doyle, J.C., K. Glover, P. Khargonekar, and B. Francis, "State-space solutions to standard H₂ and H control problems," IEEE Transactions on Automatic Control, vol. 34, no. 8, pp. 831-847, August 1989.

Glover, K., and J.C. Doyle, "State-space formulae for all stabilizing controllers that satisfy an H norm bound and relations to risk sensitivity," Systems and Control Letters, vol. 11, pp. 167-172, 1988.

See Also
hinfsyne, hinffi, hinfnorm, hinfsyn, h2syn, h2norm, ric_eig, ric_schr, sdhfnorm, sdhfsyn

dhfnorm

dkit