H-infinity Methods in Control Theory - Problem Formulation

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.