sdhinfsyn

Compute H_∞ controller for sampled-data system

Syntax

[K,GAM]=sdhinfsyn(P,NMEAS,NCON)
[K,GAM]=sdhinfsyn(P,NMEAS,NCON, KEY1,VALUE1,KEY2,VALUE2,...)

Description

sdhinfsyn controls a continuous-time LTI system P with a discrete-time controller K. The continuous-time LTI plant P has a state-space realization partitioned as follows:

$P = [\begin{matrix} A & B_{1} & B_{2} \\ C_{1} & 0 & 0 \\ C_{2} & 0 & 0 \end{matrix}]$

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 must be zero.

sdhinfsyn synthesizes a discrete-time LTI controller K to achieve a given norm (if possible) or find the minimum possible norm to within tolerance TOLGAM.

Similar to hinfsyn, the function sdhinfsyn employs a γ iteration. Given a high and low value of γ, GMAX and GMIN, the bisection method is used to iterate on the value of γ in an effort to approach the optimal H_∞ control design. If GMAX = GMIN, only one γ value is tested. The stopping criterion for the bisection algorithm requires that the relative difference between the last γ value that failed and the last γ value that passed be less than TOLGAM.

Input arguments

`P`	LTI plant
`NMEAS`	Number of measurements output to controller
`NCON`	Number of control inputs

Optional input arguments (KEY, VALUE) pairs are similar to hinfsyn, but with additional KEY values 'Ts' and 'DELAY'.

KEY	VALUE	Meaning
`'GMAX'`	real	Initial upper bound on GAM (default=`Inf`)
`'GMIN'`	real	Initial lower bound on GAM (default=0)
`'TOLGAM'`	real	Relative error tolerance for GAM (default=.01)
`'Ts'`	real	(Default=1) sample time of the controller to be designed
`'DELAY'`	integer	(Default=0) a nonnegative integer giving the number of sample periods delay for the control computation
`'DISPLAY'`	`'off'` `'on'`	(Default) no command window display, or the command window displays synthesis progress information

Output arguments

`K`	H_∞ controller
`GAM`	Final γ value of H_∞ cost achieved

Algorithms

sdhinfsyn uses a variation of the formulas described in the Bamieh and Pearson paper [1] to obtain an equivalent discrete-time system. (This is 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 γ-value. This requires a search and is computationally a relatively expensive step.

References

[1] 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, 1992, pp. 418-435.

Version History

Introduced before R2006a