Problem Formulation
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 variable 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 D11 (when the system P is given in the form (A, B1, B2, C1, C2, D11, D12, D22, D21)). 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 2 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.
Read more about this topic: H-infinity Methods In Control Theory
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