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LPV systems, robustness and gain-scheduling

Linear Parameter Varying (LPV) systems are a class of linear systems whose internal description depends on some time-varying parameters as

\[ \dot{x}(t)=A(\rho(t))x(t)+B(\rho(t))u(t) \]

where \(\rho\) is the vector of time-varying parameters, x is the state of the system and u is the control input.

Tools for their analysis are the same as the ones used in robust analysis and robust control. In this regard, the considered LPV systems usually take the form of polytopic LPV systems, generic parameter dependent systems and systems in Linear Fractional Representation. LPV systems can model a wide range of real world processes where time-varying parameters are involved. As an example, a vehicle trajectory and its response to a change in the angle of the steering wheel can be described as a parameter-dependent system with car speed as parameter. This follows from the fact that the behavior of the car highly depends on its speed.

The actual power of the LPV framework is only revealed when control enters the picture. In this case, and under the assumption that the parameters can be measured or estimated in real-time, the parameters can be used to adapt the controller expression according to the values taken by the parameters. This gives rise to the concept of LPV-based gain-scheduled controllers. For instance, gain-scheduled state-feedback controllers take the form

\[ u(t)=K(\rho(t))x(t). \]

This framework then offers an elegant way for deriving (conservative) nonlinear controllers when the parameters depend on the state of the system. It is also possible to compute, in a single design, different controllers optimized according to different performance measures, and accordingly scheduling between them. In our car example, we may design for instance two controllers: one optimizing road holding, the other one optimizing comfort. Scheduling between them can be then performed according to the driver's choice or using automatic heuristics.

I am personally interested in the development of new theoretical results for the analysis and control of LPV systems. I have been recently interested in deriving theoretical results that capture more information about the delay trajectories. Indeed, it turns out that the considered Lyapunov functions implies some condition on the trajectories of the parameters: parameter-independent Lyapunov functions correspond to arbitrarily fast parameter variations whereas parameter-dependent ones are associated with bounded rates of variations for the parameter trajectories. This has recently motivated me to consider LPV systems with piecewise constant and piecewise differentiable parameters that can be analyzed using recent hybrid systems methods.

References:

• J. S. Shamma, “Gain scheduling: potential hazards and possible remedies”, IEEE Control Systems Magazine, Vol. 12(3), 2002.

• P. Apkarian, P. Gahinet and G. Becker, “Self-scheduled \(H_\infty\) control of linear parameter-varying systems: A design example”, Automatica, Vol. 31(9), pp. 1251-1261, 1995.

• P. Akarian and P. Gahinet, A convex characterization of gain-scheduled \(H_\infty\) controllers, IEEE Transactions on Automatic Control, Vol. 40(5), pp. 853-864, 1995.

• C. W. Scherer, “LPV control and full-block multipliers”, Automatica, Vol. 37(3), pp. 361-375, 2001.

• J. Mohammadpour and C. W. Scherer (editors), “Control of linear parameter varying systems with applications”, Springer, 2012.

• C. Briat, “LPV & Time-Delay Systems - Analysis, Observation, Filtering & Control”, Advances in Delay and Dynamics, Vol. 3, Springer-Heidelberg, 2015.

• A. P. White, G. Zhu and J. Choi, “Linear Parameter-Varying Control for Engineering Applications”, Springer, 2013.

• C. Briat, “Stability analysis and control of a class of LPV systems with piecewise constant parameters”, Systems & Control Letters, Vol. 82, pp. 10-17, 2015.

• C. Briat, “Stability analysis and state-feedback control of LPV systems with piecewise constant parameters subject to spontaneous Poissonian jumps”, IEEE Control Systems Letters, Vol. 2(2), pp. 230-235, 2018.

• C. Briat, “Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers”, submitted to Nonlinear Analysis: Hybrid Systems