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** H_{∞}** (i.e. "

The phrase *H*_{∞} *control* comes from the name of the mathematical space over which the optimization takes place: *H*_{∞} is the *Hardy space* of matrix-valued functions that are analytic and bounded in the open right-half of the complex plane defined by Re(*s*) > 0; the *H*_{∞} norm is the supremum singular value of the matrix over that space. In the case of a scalar-valued function, the elements of the Hardy space that extend continuously to the boundary and are continuous at infinity is the disk algebra. For a matrix-valued function, the norm can be interpreted as a maximum gain in any direction and at any frequency; for SISO systems, this is effectively the maximum magnitude of the frequency response.

*H*_{∞} techniques can be used to minimize the closed loop impact of a perturbation: depending on the problem formulation, the impact will either be measured in terms of stabilization or performance. Simultaneously optimizing robust performance and robust stabilization is difficult. One method that comes close to achieving this is *H*_{∞} loop-shaping, which allows the control designer to apply classical loop-shaping concepts to the multivariable frequency response to get good robust performance, and then optimizes the response near the system bandwidth to achieve good robust stabilization.

Commercial software is available to support *H*_{∞} controller synthesis.

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

The 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 variables *u*. Notice that all these are generally vectors, whereas **P** and **K** are matrices.

In formulae, the system is:

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

Called the *lower linear fractional transformation*, is defined (the subscript comes from *lower*):

Therefore, the objective of control design is to find a controller such that is minimised according to the norm. The same definition applies to control design. The infinity norm of the transfer function matrix is defined as:

where is the maximum singular value of the matrix .

The achievable *H*_{∞} norm of the closed loop system is mainly given through the matrix *D*_{11} (when the system *P* is given in the form (*A*, *B*_{1}, *B*_{2}, *C*_{1}, *C*_{2}, *D*_{11}, *D*_{12}, *D*_{22}, *D*_{21})). There are several ways to come to an *H*_{∞} controller:

- A Youla-Kucera parametrization of the closed loop often leads to very high-order controller.
- Riccati-based approaches solve two Riccati equations to find the controller, but require several simplifying assumptions.
- An optimization-based reformulation of the Riccati equation uses linear matrix inequalities and requires fewer assumptions.

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