Mu Analysis and Synthesis Toolbox

sdhfsyn computes an H controller for a sampled-data SYSTEM interconnection matrix

Syntax

• [k,gfin] = sdhfsyn(p,nmeas,ncon,gmin,gmax,tol,h,... 
  delay,ricmethd,epr,epp)
  

Description
sdhfsyn is concerned with the control of a continuous-time system by a discrete-time controller. The continuous-time interconnection structure structure, p of type SYSTEM, has state-space realization partitioned as follows

where the continuous-time disturbance inputs enter through B₁, the outputs from the controller are held constant between sampling instants and enter through B₂, the continuous-time errors to be kept small correspond to the C₁ partition, and the output measurements that are sampled by the controller correspond to the C₂ partition. B₂ has column size (ncon) and C₂ has row size (nmeas). Note that the D matrix is assumed to be zero.

sdhfsyn synthesizes a discrete-time controller to achieve a given norm (if possible) or find the minimum possible norm to within some tolerance.

sdhfsyn provides a iteration using the bisection method. Given a high and low value of , gmax and gmin, the bisection method is used to iterate on the value of in an effort to approach the optimal H control design. If gmax = gmin, only one value is tested. The stopping criteria for the bisection algorithm requires the relative difference between the last value that failed and the last value that passed be less than tol. You can select either the eigenvalue or Schur method for solution of the Riccati equations with and without balancing. The eigenvalue method is faster but can have numerical problems, while the Schur method is slower but generally more reliable.

The algorithm employed calculates an equivalent purely discrete-time problem for each value of and then calls dhfsyn with = 1. The screen printing is then derived from the tests performed by dhfsyn.

Input arguments

p	SYSTEM interconnection structure matrix
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
delay	number of samples computational delay (default = 0)
h	time between samples
ricmethod	1 Eigenvalue decomposition with balancing -1 Eigenvalue decomposition with no balancing  2 Schur decomposition with balancing (default) -2 Schur decomposition with no 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
gfin	final value achieved

You might design a first controller using the dhfsyn function on the SYSTEM (samhld(p,h)), followed by sdhfnorm to determine an upper bound gmax to use for the start of this sampled data control design iterative process.

The sdhfsyn 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 H 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.

Examples
An illustrative example is given in the "Discrete-time and Sampled-data H· Control" section in Chapter 3.

Algorithm
sdhfsyn uses variations of the formulae described in the Bamieh and Pearson paper to obtain an equivalent discrete-time system. (These variations are done to improve the numerical conditioning of the algorithms.) A preliminary step is to determine whether the norm of the continuous-time system over one sampling period without control is less than the given -value, this requires a search and is computationally a relatively expensive step.

Reference
Bamieh, B.A., and J.B. Pearson, "A General Framework for Linear Periodic Systems with Applications to Sampled-Data Control," IEEE Transactions on Automatic Control, vol. AC-37, pp. 418-435, 1992.

See Also
dhfsyn, hinfsyne, hinffi, hinfnorm, hinfsyn, h2syn, h2norm, ric_eig, ric_schr

sdhfnorm

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