B.2 LPV Systems

An LPV system is described by a state-space realisation of the form

where ( ), ( ), ( ), ( ) are continuous functions of some time-varying parameter vector ? = [ ? ₁ , , ] ^T.

It is assumed that the time-varying parameter ?( t) and its rate of variation ( t) are bounded as follows:

1. the parameter ? satisfies

   
   

   where ? is a compact set, and

2. the rate of variation satisfies

   
   

   The inequalities (B.3) define a hypercube

   
   

   with vertices in

   
   

The set of parameter trajectories satisfying Assumptions B.2 and B.3 will be denoted by .

Note that an LPV system is reduced to an LTI system for a constant parameter trajectory ? = ? ₀ ? t ? 0. This property allows us to analyse the local behaviour of the LPV plant from the underlying LTI systems. However, it must be pointed out that the time-varying properties of an LPV system cannot be inferred from the underlying LTI systems. Besides, an LPV system differs from an LTV system in the fact that it is not necessary to know beforehand the matrices ( t), ( t), ( t) and ( t). They are completely specified when the LPV plant works over a particular trajectory ?( t).

Note also that (B.1) describes a family of linear models. Therefore, in the context of gain scheduling, an LPV system may be...