Simulink Blocks in LPVTools

LPVTools provides Simulink blocks to interface to the state-space LPV objects: pss, upss and plftss. The Simulink blocks enable users to include LPV systems in Simulink simulation models. One Simlink block is for systems that depend on a time-varying parameter and its derivative, as seen in Equation 1, while the other is for systems that do not depend explicitly on the derivative, as seen in Equation 2.

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

$$\left[ \begin{array}{c} \dot x (t) \\ y (t)\end{array} \right]
= \left[ \begin{array}{cc} A(\rho(t)) & B(\rho(t)) \\ C(\rho(t)) & D(\rho(t))
\end{array} \right] \left[ \begin{array}{c} x (t) \\ u (t)\end{array} \right]
\ \ \ \ \ \ \ (2)$$

The Simulink block for the system shown in Equation 2 is shown in Figure 1.

Figure 1: Simulink LPV block and block mask.

The block in Figure 1 has inputs for the system input $u(t)$ and the parameter vector $\rho(t)$, and an output for $y(t)$. The block mask contains entries for the user to specify the system variable name, the order of the input parameter vectors $\rho$, and the state initial condition $x(0)$. The block is implemented as a Simulink S-function under the block mask. The block currently performs a multidimensional linear interpolation to evaluate the state-space matrices at the specified parameter vector. An efficient implementation of this linear interpolation has been coded to reduce computation and speed up the simulation time.

LPVTools also includes a block for systems that depend explicitly on both the time-varying parameter and its derivative, as seen in Equation 1. This block is shown in Figure 2.

Figure 2: Rate-dependent Simulink LPV block and block mask.