Research · 01 / 06

LPV systems, robustness & gain-scheduling

Linear parameter-varying systems form a class of linear systems whose internal description depends on time-varying parameters. They offer an elegant pathway to nonlinear control.

Linear parameter-varying (LPV) systems take the form

$$ \dot{x}(t) = A(\rho(t))\, x(t) + B(\rho(t))\, u(t) $$

where $\rho$ is the vector of time-varying parameters, $x$ is the state, and $u$ is the control input. The mathematical tools for their analysis are shared with robust analysis and robust control, and the systems typically appear as polytopic LPV systems, generic parameter-dependent systems, or systems in linear fractional representation.

LPV systems can model a wide range of real-world processes. A vehicle's response to a change in the steering wheel angle can be described as a parameter-dependent system with the car speed as the parameter, since the behavior of the car depends strongly on its speed.

Why gain-scheduling matters

The real power of the LPV framework appears with control. Assuming the parameters can be measured or estimated in real time, they can be used to adapt the controller as the parameters change. This yields LPV-based gain-scheduled controllers. A gain-scheduled state-feedback controller takes the form

$$ u(t) = K(\rho(t))\, x(t). $$

This is an elegant way of deriving conservative nonlinear controllers when the parameters depend on the state. It also makes it possible to compute, in a single design, different controllers optimized for different performance measures and to schedule between them. In a car, for instance, two controllers can be designed, one optimizing road holding and the other comfort, with scheduling driven either by the driver's choice or by automatic heuristics.

What I work on

I am interested in the development of new theoretical results for the analysis and control of LPV systems, with particular attention to how Lyapunov functions implicitly impose conditions on the trajectories of the parameters. Capturing more information about those parameter trajectories tightens stability and performance results in ways that classical methods miss.

Related publications

For the full list, see the publications page (filter by "LPV" or "gain-scheduled" in the search box).