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The major motivation of the averaging technique for switched systems is the construction of a smooth average system whose state trajectory approximates in some sense the state trajectory of the switched system. Averaging of dynamic systems represented by switched ordinary differential equations (ODEs) has been widely analyzed in the literature. The averaging approach can be useful also for the analysis of switched differential algebraic equations (DAEs). Indeed by analyzing the evolution of the switched DAEs state it is possible to conjecture the existence of an average model. However a trivial generalization of the ODE case is not possible due to the presence of state jumps. In this paper we discuss the averaging approach for switched DAEs and an approximation result is derived for homogenous switched linear DAE with periodic switching signals commuting among several modes. This approximation result extends a recent averaging result for switched DAEs with only two modes. Numerical simulations confirm the validity of the averaging approach for switched DAEs.

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Abstract-Averaging of fast switching systems is an effective technique used in many engineering applications. Practical stability and control design for a nonsmooth switched system can be inferred by analyzing the smooth averaged system. In this paper we overview the few formal approaches proposed in the literature to deal with the averaging of nonsmooth systems. The dithering, the phasor dynamics and the hybrid framework techniques are recast and compared by considering pulsemodulated switched linear systems as the common modeling platform.

Emission requirements for diesel engines are becoming increasingly strict, leading to the increase of engine architecture complexity. This evolution requires a more systematic approach in the development of control systems than presently adopted, in order to achieve improved performances and reduction of times and costs in design, implementation and calibration. To this end, large efforts have been devoted in recent years to the application of advanced Model-Based MIMO control systems. In the present paper a new MIMO nonlinear feedback control is proposed, based on an innovative data-driven method, which allows to design the control directly from the experimental data acquired on the plant to be controlled. Thus, the proposed control design does not need the intermediate step of a reliable plant model identification, as required by Model-Based methods. In this way, significant advantages over Model-Based methods can be achieved in terms of times and costs in design and deployment as well as in terms of control performances. The method is applied to the control design for the air and charging systems, using experimental data measured on a four cylinder diesel engine with single stage turbocharger. The performances of the designed controller are evaluated on an accurate nonlinear engine model, showing significant reductions of up to 2.7 times for the intake manifold pressure, up to 2.7 times for the oxygen concentration tracking errors and about 4 times in controller design and calibration efforts with respect to a decoupled-gain-scheduled PID controller typically applied for the air charging system control of diesel engines.