Counterclockwise hysteresis maps are known to be dissipative in the energy sense as well as in the system-theoretic sense. In a recent paper, Angeli studied feedback interconnections of counterclockwise systems, where counterclockwise refers to the net signed area of the input-output map, which need not be a simple closed curve. In the present paper we apply this notion to the study of hysteretic models. In particular, we give conditions under which the semilinear Duhem model is counterclockwise.
We consider the semilinear Duhem model and develop an identification method for rate-independent and rate-dependent hysteresis. For rate-independent hysteresis, we reparameterize the system in terms of the input signal, so that the system has the form of a switching linear time-invariant system with ramp-plus-step forcing. For rate-dependent hysteresis, the system can be viewed as a switching linear time-invariant system for triangle wave inputs. Least-squares-based methods are developed to identify the rate-independent and rate-dependent semilinear Duhem models.Index Terms-Duhem model, hysteresis, rate dependence.
In this paper we develop ® xed-order (i.e. full-and reduced-order) controllers for continuous-time and discrete-time linear systems with actuator amplitude and rate saturation constraints. The problem is formulated as a multiobjective problem involving a convex combination of an L 1 norm and the H 2 norm to capture actuator saturation constraints and closedloop system performance in the face of exogenous white noise disturbances. The L 1 convolution operator norm considered is induced by bounded amplitude persistent L 1 disturbances and L 1 performance variables involving the actuator amplitude and rate signals. Hence, the peak pointwise-in-time actuator amplitude and actuator rate excursion are guaranteed to be less than the product of the L 1 convolution operator norm and the L 1 disturbance amplitude bound. Application of the proposed framework to the design of multivariable saturation controllers for the control of a bank-to-turn missile and a high-performance ® ghter aircraft is demonstrated.
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The classical Duhem model provides a finite-dimensional differential model of hysteresis. In this paper, we consider rate-independent and rate-dependent semilinear Duhem models with provable properties. The vector field is given by the product of a function of the input rate and linear dynamics. If the input rate function is positively homogeneous, then the resulting input-output map of the model is rate independent, yielding persistent nontrivial input-output closed curve (that is, hysteresis) at arbitrarily low input frequency. If the input rate function is not positively homogeneous, the input-output map is rate dependent and can be approximated by a rate-independent model for low frequency inputs. Sufficient conditions for convergence to a limiting input-output map are developed for rate-independent and rate-dependent models. Finally, the reversal behavior and orientation of the rate-independent model are discussed.Index Terms-Duhem, hysteresis, rate dependence.
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