# Research

### Network systems, Switched systems, Control applications

### The reserach activities are mainly focused on three different areas:

Network systems

Switched systems

Control applications

### Network systems

**Bounded confidence opinion dynamics**

**Opinion dynamics for modeling Paris agreement process**

### Switched systems

**Linear c****omplementarity systems: limit cycles and power converters modeling**

The classic approaches for modeling and simulating power converters (averaged, switched and hybrid models) and the commercial tools available (e.g. PSpice, Power System Blockset in the Matlab environment, PLECS) often do not allow the analysis of non-standard innovative converter topologies or operating conditions. The problem is tackled by using the complementarity systems theory. The complementarity approach, coming from the nonsmooth mechanics, allows to model the electronic devices by using piecewise affine characteristics, and then integrating them with detailed models of the converter and the network. The complementarity framework allows the formal and simulation analysis of innovative power converters topologies and electrical networks in terms of dynamic performance, failure and fault conditions, high frequency phenomena, switching behavior of the converters, frequency domain response. Further, the complementarity approach with respect to simulation tools based on standard hybrid models, it also allows to gain in terms of simulation time and numerical convergence.

A similar approach to that used for the computation of steady state solutions for switched electronic systems can be used for the computation of limit cycles in Lur'e systems. It can be shown that the complementarity representation of the feedback characteristic allows to represent the discretized closed loop system as a linear complementarity system. A static linear complementarity problem, whose solutions correspond to periodic solutions of the discrete–time system, can then be formulated. The proposed technique is able to compute steady state oscillations with known period for continuous–time systems, so as demonstrated by simulation results on Lur’e systems which exhibit asymmetric unstable and sliding limit cycles.

**Averaging and dithering**

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. On the other hand only few formal approaches have been proposed in the literature to deal with the averaging of nonsmooth systems. The dithering, the phasor dynamics and the hybrid framework techniques can be recast and compared by considering pulse-modulated switched linear systems as the common modeling platform.

In particular, dither signals provide an effective way to compensate for nonlinearities in control systems. Previous studies present tools for systematic design of dithered systems, but the results rely only on a Lipschitz assumption on the nonlinearity, thus they don’t cover important applications with discontinuities. We obtained initial results on how to analyze and design dither in nonsmooth systems. In particular, we showed how a dither relay feedback system can be approximated by a smoothed system, and we gave guidelines for tuning the amplitude and the period time of the dither signal, in order to stabilize one class of particularly interesting nonsmooth systems, namely relay feedback systems with triangular dither. Then the study of dither signals is enlarged by considering square and trapezoidal dither; we have shown how the dither shape affects the behavior of nonsmooth feedback system, differently from the case of dither in Lipschitz continuous systems. A process used to show the results obtained is the DC/DC power converter.

**Picewise Quadratic Lyapunov functions and absolute stability**

Piecewise linear systems represent a class of hybrid systems characterized by a partition of the state-space into regions where system dynamics can be described by linear models. Unfortunately, mathematical models of practical systems are always affected by uncertainties of various kind. We consider uncertain autonomous piecewise linear systems where the state partition consists of convex polyhedral cones and in each cone the uncertain dynamic matrix can be expressed as a convex hull of known matrices. Such class of systems can be viewed as piecewise linear differential inclusions. Among others, examples of nonlinear systems which can be embedded in this framework are Lur’e systems with possibly asymmetric domain of the feedback characteristic and power electronics converters.

We tackle the absolute stability issue for of piecewise linear differential inclusions by using a cone-copositive approach. The stability problem is formulated as a set of linear matrix inequalities each constrained into a specific cone, i.e. a set of cone-copositive programming problems. A procedure for solving the set of constrained inequalities is under study. The proposed approach allows less conservative estimation of the robust stability region with respect to the classical Circle criterion and to other approaches based on piecewise quadratic Lyapunov function.**State jumps and bifurcations**

For discontinuities in the state variables (typically voltages across capacitors and currents through inductors), a good deal of literature has been devoted to the problem of state reinitialization, i.e., determining the state after a discontinuity. We study the state reinitilization problem for electrical networks consisting of linear passive elements, independent voltage/current sources, and ideal switches. We employ the so-called linear switched systems framework in which these circuits can be analyzed for any given switch configuration. After providing a complete characterization of admissible inputs and consistent initial states with respect to a switch configuration, we introduce a new state reinitialization rule that is based on energy minimization at the time of switching.

Inspired by nonlinear phenomena exhibited by power electronics converters, we study the classification of non-standard bifurcations in piecewise smooth feedback systems. Bifurcations involving fixed points in maps, equilibria and limit cycles in flows are considered with particular attention to: border collisions of fixed points in maps; non-smooth bifurcations of equilibria, grazing bifurcations and sliding bifurcations of limit cycles in flows. The aim is to describe existing and novel results to form the basis of a consistent theory of bifurcations in such systems. In so doing, a novel approach to classify non-smooth bifurcations of equilibria in flows can be proposed.

### Control applications

**Dry clutch transmissibility: modeling, control and real time hardware in the loop**

Dry clutches are widely used in conventional and innovative automotive drivelines and represent a key element for automated manual transmissions (AMTs). In practical applications, it is fundamental to model the clutch behavior through its torque transmissibility characteristic, i.e., the relationship between the throwout bearing position (or the pressure applied by the clutch actuator) and the torque transmitted through the clutch during the engagement phase. In this research activity we study new models for the torque transmissibility of dry clutches. We analyze how the transmissibility characteristic depends on: friction pads geometry, cushion spring compression, cushion spring load, and slip-speed-dependent friction. Corresponding functions are suitably composed determining the torque transmissibility expression. An experimental procedure for tuning the characteristic parameters can be defined and suitable engagement control algorithms can be designed.**Battery modeling and energy management****D****roop control****Modeling and control for****LEDs****R****eal time hardware in the loop applications**