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

Part of the SpringerBriefs in Electrical and Computer Engineering book series (BRIEFSELECTRIC)


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.


Robust Stability Hard Constraint Performance Constraint Output Feedback Controller Actuator Signal 
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© The Author(s) 2013

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringState University of MichiganEast LansingUSA
  2. 2.Department of Mechanical EngineeringMichigan State UniversityEast LansingUSA
  3. 3.Department of Mechanical EngineeringMichigan State UniversityEast LansingUSA

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