State Consistency

LPV systems are time-varying, state-space models of the form:

$$\left[ \begin{array}{c} \dot x (t) \\ y (t)\end{array} \right]
= \left[ \begin{array}{cc} A(\rho(t)) & B(\rho(t)) \\ C(\rho(t)) & D(\rho(t))
\end{array} \right] \left[ \begin{array}{c} x (t) \\ u (t)\end{array} \right]
\ \ \ \ \ \ \ (1)$$

where $\rho \in \mathcal{R}^{n_\rho}$ is a vector of measurable parameters, $y \in \mathcal{R}^{n_y}$ is a vector of outputs, $x \in \mathcal{R}^{n_x}$ is the state vector, $u \in \mathcal{R}^{n_u}$ is a vector of inputs, and $A\in \mathcal{R}^{n_x \times n_x}$, $B\in \mathcal{R}^{n_x \times n_u}$, $C\in \mathcal{R}^{n_y \times n_x}$ and $D\in \mathcal{R}^{n_y \times n_u}$ are parameter dependent matrices.

Note that the state-vector of the system in Equation (1) remains the same for all values of the parameter, i.e. the states in $x$ are ordered the same way, and their interpretation remains the same, irrespective of the value of $\rho$. This property is referred to as state consistency, and it must be kept in mind when working with LPV models.

State Consistency in LPV Model Construction

A common approach to constructing LPV models is to use Jacobian linearization along a grid of parameter values (e.g. batch linearization of Simulink models) to construct a grid-based LPV system. In this case, the user must ensure that the models generated by the linearization all share the same state-vector.

Figure 1 illustrates the concept. A nonlinear model is linearized along a grid of Mach and altitude values, resulting in an array of linearized systems. State consistency requires the state vectors ($x$) of all the individual linearizations to be identical if these models are to be used to contruct a grid-based LPV system.

Figure 1: A grid-based LPV system.

Maintaining State Consistency

There are some operations that are commonly applied to Linear Time-Invariant (LTI) systems, that can result in loss of state-consistency of a LPV model. A good example is balreal, which performs a Gramian-based balancing of a LTI state-space realization. If balreal is applied to a grid-based LPV system it will balance each of the LTI models, which the grid-based LPV system is comprised of, and the resulting systems will no longer have state consistency. An alternative function that will maintain state consistency is lpvbalreal which computes a balancing realization for the LPV system as a whole, yielding a balanced LPV system with state consistency.