10 research outputs found
Switching and stability properties of conewise linear systems
Being a unique phenomenon in hybrid systems, mode switch
is of fundamental importance in dynamic and control analysis. In
this paper, we focus on global long-time switching and stability
properties of conewise linear systems (CLSs), which are a class of
linear hybrid systems subject to state-triggered switchings
recently introduced for modeling piecewise linear systems. By
exploiting the conic subdivision structure, the “simple switching
behavior” of the CLSs is proved. The infinite-time mode switching
behavior of the CLSs is shown to be critically dependent on two
attracting cones associated with each mode; fundamental properties
of such cones are investigated. Verifiable necessary and
sufficient conditions are derived for the CLSs with infinite mode
switches. Switch-free CLSs are also characterized by exploring
the polyhedral structure and the global dynamical properties. The
equivalence of asymptotic and exponential stability of the CLSs is
established via the uniform asymptotic stability of the CLSs that
in turn is proved by the continuous solution dependence on initial
conditions. Finally, necessary and sufficient stability conditions
are obtained for switch-free CLSs
Estimating Linear Mixed Effects Models with Truncated Normally Distributed Random Effects
Linear Mixed Effects (LME) models have been widely applied in clustered data
analysis in many areas including marketing research, clinical trials, and
biomedical studies. Inference can be conducted using maximum likelihood
approach if assuming Normal distributions on the random effects. However, in
many applications of economy, business and medicine, it is often essential to
impose constraints on the regression parameters after taking their real-world
interpretations into account. Therefore, in this paper we extend the classical
(unconstrained) LME models to allow for sign constraints on its overall
coefficients. We propose to assume a symmetric doubly truncated Normal (SDTN)
distribution on the random effects instead of the unconstrained Normal
distribution which is often found in classical literature. With the
aforementioned change, difficulty has dramatically increased as the exact
distribution of the dependent variable becomes analytically intractable. We
then develop likelihood-based approaches to estimate the unknown model
parameters utilizing the approximation of its exact distribution. Simulation
studies have shown that the proposed constrained model not only improves
real-world interpretations of results, but also achieves satisfactory
performance on model fits as compared to the existing model
Convergence of time - stepping schemes for passıie and extended linear complementarity systems
Generalizing recent results in [M. K. Camlibel, Complementarity Methods in the Analysis of Piecewise Linear Dynamical Systems, Ph.D. thesis, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, 2001], [M. K. Camlibel, W. P. M. H. Heemels, and J. M. Schumacher, IEEE Trans. Circuits Systems I: Fund. Theory Appl., 49 (2002), pp. 349-357], and [J.-S. Pang and D. Stewart, Math. Program. Ser. A, 113 (2008), pp. 345-424], this paper provides an in-depth analysis of time-stepping methods for solving initial-value and boundary-value, non-Lipschitz linear complementarity systems (LCSs) under passivity and broader assumptions. The novelty of the methods and their analysis lies in the use of "least-norm solutions" in the discrete-time linear complementarity subproblems arising from the numerical scheme; these subproblems are not necessarily monotone and are not guaranteed to have convex solution sets. Among the principal results, it is shown that, using such least-norm solutions of the discrete-time subproblems, an implicit Euler scheme is convergent for passive initial-value LCSs; generalizations under a strict copositivity assumption and for boundary-value LCSs are also established
Convergence of Time-Stepping Schemes for Passive and Extended Linear Complementarity Systems
Generalizing recent results in [M. K. Camlibel, Complementarity Methods in the Analysis of Piecewise Linear Dynamical Systems, Ph.D. thesis, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, 2001], [M. K. Camlibel, W. P. M. H. Heemels, and J. M. Schumacher, IEEE Trans. Circuits Systems I: Fund. Theory Appl., 49 (2002), pp. 349-357], and [J.-S. Pang and D. Stewart, Math. Program. Ser. A, 113 (2008), pp. 345-424], this paper provides an in-depth analysis of time-stepping methods for solving initial-value and boundary-value, non-Lipschitz linear complementarity systems (LCSs) under passivity and broader assumptions. The novelty of the methods and their analysis lies in the use of "least-norm solutions" in the discrete-time linear complementarity subproblems arising from the numerical scheme; these subproblems are not necessarily monotone and are not guaranteed to have convex solution sets. Among the principal results, it is shown that, using such least-norm solutions of the discrete-time subproblems, an implicit Euler scheme is convergent for passive initial-value LCSs; generalizations under a strict copositivity assumption and for boundary-value LCSs are also established.</p
A unified numerical scheme for linear-quadratic optimal control problems with joint control and state constraints
This paper presents a numerical scheme for solving the continuous-time convex linear-quadratic (LQ) optimal control problem with mixed polyhedral state and control constraints. Unifying a discretization of this optimal control problem as often employed in model predictive control and that obtained through time-stepping methods based on the differential variational inequality reformulation, the scheme solves a sequence of finite-dimensional convex quadratic programs (QPs) whose optimal solutions are employed to construct a sequence of discrete-time trajectories dependent on the time step. Under certain technical primal–dual assumptions primarily to deal with the algebraic constraints involving the state variable, we prove that such a numerical trajectory converges to an optimal trajectory of the continuous-time control problem as the time step goes to zero, with both the limiting optimal state and costate trajectories being absolutely continuous. This provides a constructive proof of the existence of a solution to the optimal control problem with such regularity properties. Additional properties of the optimal solutions to the LQ problem are also established that are analogous to those of the finite-dimensional convex QP. Our results are applicable to problems with convex but not necessarily strictly convex objective functions and with possibly unbounded mixed state–control constraints.