# Guaranteed \(\ell _2-\ell _{\infty }\) Gain Control for LPV Systems

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## Abstract

This chapter considers the optimal control of polytopic, discrete-time LPV systems with a guaranteed \(\ell _2\) to \(\ell _{\infty }\) gain. Additionally, to guarantee robust stability of the closed-loop system under parameter variations, \(\fancyscript{H}_{\infty }\) performance criterion is also considered as well. Controllers with a guaranteed \(\ell _2\) to \(\ell _{\infty }\) gain and a guaranteed \(\fancyscript{H}_{\infty }\) performance (\(\ell _2\) to \(\ell _2\) gain) are mixed \(\fancyscript{H}_2/\fancyscript{H}_{\infty }\) controllers. Normally, \(\fancyscript{H}_2\) controllers are obtained by considering a quadratic cost function that balances the output performance with the control input needed to achieve that performance. However, to obtain a controller with a guaranteed \(\ell _2\) to \(\ell _{\infty }\) gain (closely related to the physical performance constraint), the cost function used in the \(\fancyscript{H}_2\) control synthesis minimizes the control input subject to maximal singular-value performance constraints on the output. This problem can be efficiently solved by a convex optimization with LMI constraints. The contribution of this chapter is the characterization of the control synthesis LMIs used to obtain an LPV controller with a guaranteed \(\ell _2\) to \(\ell _{\infty }\) gain and \(\fancyscript{H}_{\infty }\) performance while the control \(\ell _2\) to \(\ell _{\infty }\) gain is minimized. A numerical example is presented to demonstrate the effectiveness of the convex optimization.