H∞

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The correct title of this article is H_∞. It features superscript or subscript characters that are substituted or omitted because of technical limitations.

H infinity or $\mathcal{H}_\infty$ is a method in control theory for the design of optimal controllers. Essentially it is an optimization method, that takes into consideration a strong mathematical definition of the restrictions on the expected behaviour of the closed loop and the strict stability of it. It is noted for its strong mathematical foundation, the mathematical way to express engineering restrictions, its optimization/optimality aspect, and the ability to include both classic and robust control concepts within a single design framework.

Operators sometimes object that controllers synthesised with the $\mathcal{H}_\infty$ method are not "optimal," and can instead be somewhat sluggish. The word "optimal" is used strictly in the mathematical sense, as the synthesised controller will be one that minimizes the effect of system inputs on outputs. This may not be seen as "optimal" in the sense meant by operators.

The $\mathcal{H}$ stands for Hardy space. "Infinity" implies that it is designed to accomplish minimax restrictions in the frequency domain. The $\mathcal{H}_\infty$ norm of a dynamic system is the maximum amplification the system can make to the energy of the input signal. In the MIMO case, it is equal to the system's maximum singular value, reducing to the maximum value of its frequency response magnitude in the SISO case.

[edit] Problem formulation

First, the process has to be represented according to the following standard configuration:

Plant P has two inputs, the exogenous input w, that includes reference signal and disturbances, and the manipulated variables u. There are two outputs, the error signals z that we want to minimize, and the measured variables v, that we use to control the system. v is used in K to calculate the manipulated variable u. Remark that all these are generally vectors, whereas P and K are matrices.

In formulae, the system is:

$\begin{bmatrix} z\\ v \end{bmatrix} = P(s)\, \begin{bmatrix} w\\ u\end{bmatrix} = \begin{bmatrix}P_{11}(s) & P_{12}(s)\\P_{21}(s) & P_{22}(s)\end{bmatrix} \, \begin{bmatrix} w\\ u\end{bmatrix}$

$u = K(s) \, v$

It is therefore possible to express the dependency of z on w as:

$z=F_l(P,K)\,w$

Having defined:

$F_l(P,K) = P_{11} + P_{12}\,K\,(I-P_{22}\,K)^{-1}\,P_{21}$

Therefore, the objective of $\mathcal{H}_\infty$ control design is to find a controller K such that $F l (P, K)$ is minimised according to the $\mathcal{H}_\infty$ norm. The same definition applies to $\mathcal{H}_2$ control design. The infinity norm of the transfer function matrix $F l (P, K)$ is defined as:

$||F_l(P,K)||_\infty = \sup_\omega \bar{\sigma}(F_l(P,K)(j\omega))$

where $\bar{\sigma}$ is the maximum singular value of the matrix $F l (P, K)(j ω)$ .

[edit] See also

Hardy space
H square
Linear quadratic gaussian control (LQG)

[edit] References

Sigurd Skogestad and Ian Postlethwaite: Multivariable Feedback Control - Analysis and Design, Wiley (1996), ISBN 0-471-94277-4 (hardback), ISBN 0-471-94330-4 (paperback). 2nd. ed. (2005) ISBN-13 978-0-470-01167-6, ISBN 0-470-01167-X (HB); ISBN-13 978-0-470-01168-3, ISBN 0-470-01168-8 (PBK).

Dan Simon: Optimal State Estimation: Kalman, H-infinity, and Nonlinear Approaches, Wiley (2006). Book web site http://academic.csuohio.edu/simond/estimation/.