1,881 research outputs found
Sensitivity analysis of hybrid systems with state jumps with application to trajectory tracking
This paper addresses the sensitivity analysis for hybrid systems with
discontinuous (jumping) state trajectories. We consider state-triggered jumps
in the state evolution, potentially accompanied by mode switching in the
control vector field as well. For a given trajectory with state jumps, we show
how to construct an approximation of a nearby perturbed trajectory
corresponding to a small variation of the initial condition and input. A major
complication in the construction of such an approximation is that, in general,
the jump times corresponding to a nearby perturbed trajectory are not equal to
those of the nominal one. The main contribution of this work is the development
of a notion of error to clarify in which sense the approximate trajectory is,
at each instant of time, a firstorder approximation of the perturbed
trajectory. This notion of error naturally finds application in the (local)
tracking problem of a time-varying reference trajectory of a hybrid system. To
illustrate the possible use of this new error definition in the context of
trajectory tracking, we outline how the standard linear trajectory tracking
control for nonlinear systems -based on linear quadratic regulator (LQR) theory
to compute the optimal feedback gain- could be generalized for hybrid systems
Backstepping controller synthesis and characterizations of incremental stability
Incremental stability is a property of dynamical and control systems,
requiring the uniform asymptotic stability of every trajectory, rather than
that of an equilibrium point or a particular time-varying trajectory. Similarly
to stability, Lyapunov functions and contraction metrics play important roles
in the study of incremental stability. In this paper, we provide
characterizations and descriptions of incremental stability in terms of
existence of coordinate-invariant notions of incremental Lyapunov functions and
contraction metrics, respectively. Most design techniques providing controllers
rendering control systems incrementally stable have two main drawbacks: they
can only be applied to control systems in either parametric-strict-feedback or
strict-feedback form, and they require these control systems to be smooth. In
this paper, we propose a design technique that is applicable to larger classes
of (not necessarily smooth) control systems. Moreover, we propose a recursive
way of constructing contraction metrics (for smooth control systems) and
incremental Lyapunov functions which have been identified as a key tool
enabling the construction of finite abstractions of nonlinear control systems,
the approximation of stochastic hybrid systems, source-code model checking for
nonlinear dynamical systems and so on. The effectiveness of the proposed
results in this paper is illustrated by synthesizing a controller rendering a
non-smooth control system incrementally stable as well as constructing its
finite abstraction, using the computed incremental Lyapunov function.Comment: 23 pages, 2 figure
Exploring Strategies for Teaching Creatively Online
The way we learn and teach is changing. There is more emphasis on collaboration and personalization of the learning. Teaching online is becoming common. For my Master’s project I have developed a product that will help designing and delivering teaching programs creatively through the use of online learning opportunities. The product discusses the opportunities and challenges of a creative climate when teaching online and provides strategies to develop creativity during the online learning process. It is developed with the use of the Torrance Incubation Model of Teaching and Learning (TIM) and Ekvall’s ten dimensions of a creative climate. This paper describes the process of developing the product. Relevant literature and resources about creative climate, TIM and teaching in an online environment are identified. A first version of the product is received as comprehensive and useful. It needs further development to make the product appropriate for a broader audience. The process of developing the product has shown the importance of experimenting and of keeping your goal in mind and acting deliberately towards your goal
Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact
In this paper, we will give conditions under which the equilibrium set of multi-degree-of-freedom non-linear mechanical systems with an arbitrary number of frictional unilateral constraints is attractive. The theorems for attractivity are proved by using the framework of measure differential inclusions together with a Lyapunov-type stability analysis and a generalisation of LaSalle's invariance principle for non-smooth systems. The special structure of mechanical multi-body systems allows for a natural Lyapunov function and an elegant derivation of the proof. Moreover, an instability theorem for assessing the instability of equilibrium sets of non-linear mechanical systems with frictional bilateral constraints is formulated. These results are illustrated by means of examples with both unilateral and bilateral frictional constraint
Stability Properties of Equilibrium Sets of Controlled Linear Mechanical Systems with Dry Friction
The dynamics of mechanical systems with dry friction elements, modeled by set-valued force laws, can be described by differential inclusions. The switching and set-valued nature of the friction force law is responsible for the hybrid character of such models. An equilibrium set of such a differential inclusion corresponds to a stationary mode for which the friction elements are sticking. The attractivity properties of the equilibrium set are of major importance for the overall dynamic behavior of this type of systems. Conditions for the attractivity of the equilibrium set of linear MDOF mechanical systems with multiple friction elements are presented. These results are obtained by application of a generalization of LaSalle’s principle for differential inclusions of Filippov-type. Besides passive systems, also systems with negative viscous damping are considered. For such systems, only local attractivity of the equilibrium set can be assured under certain conditions. Moreover, an estimate for the region of attraction is given for these cases. The results are illustrated by means of a 2DOF example. Moreover, the value of the attractivity results in the context of the control of mechanical systems with friction is illuminated
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