B.E.A. Milani

Generation of optimal schedules for metro lines using model predictive control

This paper presents a new methodology for computation of optimal train schedules in metro lines using a linear-programming-based model predictive control formulation. The train traffic model is comprised of dynamic equations describing the evolution of train headways and train passenger loads along the metro line, considering the time variation of the passenger demand and all relevant safety and operational constraints for practical use of the generated schedule. The performance index is a weighted sum of convex piecewise-linear functions for directly or indirectly modelling the waiting time of passengers at stations, onboard passenger comfort, train trip duration and number of trains in service. The proposed methodology is computationally very efficient and can generate optimal schedules for a whole day operation as well as schedules for transition between two separate time periods with known schedules. The use and performance of the proposed methodology is illustrated by an application to a metro line similar to the North-South line of Sao Paulo Underground.

Brief Piecewise-affine Lyapunov functions for discrete-time linear systems with saturating controls

This paper is concerned with piecewise-affine (PWA) functions as Lyapunov function candidates for stability analysis of time-invariant discrete-time linear systems with saturating closed-loop control inputs. Using a PWA model of saturating closed-loop system, new necessary and sufficient conditions for a PWA function be a Lyapunov function are presented. Based on linear programming formulation of these conditions, an effective algorithm is proposed for construction of such Lyapunov functions for estimation of the region of local asymptotic stability. Compared to piecewise-linear functions, like Minkowski functions, PWA functions are more adequate to capture the dynamical effects of saturation nonlinearities, giving strictly less conservative results. The complexity of the proposed approach is polynomial in state dimension and exponential in saturating control dimension, being hence appropriate for problems with large state dimension but with few saturating inputs.

Design of L-Q regulators for state constrained continuous-time systems

A new methodology to the design of LQ regulators for continuous-time systems subject to linear state constraints is proposed, consisting of two parts. In the first one, the positive invariance of the polyhedron defined by the constraints in the state space is imposed, guaranteeing thereby that the constraints will not be violated. Furthermore, the admissible constrained controllers are parameterized via the determination of the elements which are fixed in the state feedback matrix. This parameterization enables in a second part the L-Q regulator to be obtained from the solution of a parameter optimization problem subject to linear constraints, for which it is proposed a specialized feasible directions method. >

On invariant polyhedra of continuous-time systems subject to additive disturbances

This paper presents new necessary and sufficient algebraic conditions on the existence of positively D-invariant polyhedra of continuous-time linear systems subject to additive disturbances. In particular, for a convex unbounded polyhedron containing the origin in its interior, it is also shown that the positive D-invariance conditions can be split into two lower-dimensional sets of algebraic relations: the first corresponds to disturbance decoupling conditions and the second to positive D-invariance conditions for bounded polyhedra of a reduced-order system. The stability of the overall system is discussed as well. By exploring the results obtained, an LP approach is proposed for solution of a state-constrained regulator problem in the presence of additive disturbances.

Piecewise-affine Lyapunov functions for discrete-time linear systems with saturating controls

This paper deals with piecewise-affine functions as Lyapunov function candidates for stability analysis of time-invariant discrete-time linear systems with saturating closed-loop control inputs. Considering the nonlinear behavior of the closed-loop system, new necessary and sufficient conditions for a piecewise-affine function as a Lyapunov function are presented. Based on linear programming formulation of these conditions, an effective procedure is proposed for determination of such Lyapunov functions and associated polyhedral regions of local asymptotic stability, with reduced conservativeness. Compared to piecewise-linear functions, like the Minkowski functions, piecewise-affine functions are more adequate to capture the dynamical effects of saturation nonlinearities, giving strictly less conservative results.

Robust linear regulator design for discrete-time systems under polyhedral constraints

The robust linear constrained regulation problem for uncertain linear discrete-time systems (A, B) is investigated using the theory of positively invariant polyhedral sets. Closed and bounded polyhedral uncertain domains are considered, involving parameters of both A and B-matrices and polyhedral constraints on both state and control vectors. New sufficient conditions for the solvability of the regulation problem are presented. For computational solution of the regulation problem, a linear programming approach is proposed where the constraints are given by the solvability conditions and the performance index represents a guaranteed trade-off between the closed-loop stability margin and the level of control effort.

A computational method for optimal L-Q regulation with simultaneous disturbance decoupling

The disturbance decoupling problem using state feedback (DDP), with simultaneous infinite-time horizon optimal L-Q regulation (LQR), for continuous time-invariant linear systems, is formulated as a parameter optimization problem in L-Q regulators subjected to control constraints imposed by the solution of DDP. For computational solution of DDP it is proposed an efficient numerical procedure, which gives the solution directly in the form of constraints on some parameters of the control matrix. For computational solution of the optimization problem, it is proposed a specialized hybrid descent method, suitable for problems with severe control structural constraints, composed by a sequence of steps of the following methods: Modified Newton, Newtons and Quasi-Newton. The results are illustrated by a numerical example.

Contractive polyhedra for discrete-time linear systems with saturating controls

This paper presents new necessary and sufficient linear programming conditions for convex closed polyhedra to be /spl lambda/-contractive with respect to discrete-time linear systems with saturating feedback control inputs. Based on linear programming formulation of these conditions, an effective procedure is proposed for construction of /spl lambda/-contractive polyhedra with nonempty intersection with the region of nonlinear behavior of the closed-loop system. The procedure starts with the supremal /spl lambda/-contractive polyhedron contained in the region of linear behavior and progressively expands it over the region of nonlinear behavior.

Robust linear regulator design for continuous-time systems under state and control constraints

Two sufficient solvability conditions, derived from the theory of positively invariant polyhedral sets, are presented for robust linear regulation problems of uncertain continuous-time systems, considering state and control polyhedral constraints and uncertain parameter domains defined by convex compact polyhedral sets. Based on these conditions, an LP approach is proposed.<<ETX>>

Linear Regulator Design for Bounded Uncertain Discrete-Time Systems with Additive Disturbances

Abstract The design of robust linear regulators, lor bounded uncertain linear discrete-time systems with additive disturbances, is treated using the concepts of robust positively invariant and robust control admissible sets. For constraint;, on stale, control and additive disturbance variables defined by convex closed polyhedra and uncertain parameter domains defined by convex compact polyhedra, new necessary and sufficient algebraic conditions for robust positive invariance and robtsr. control admissibility are presented. Based on these results, an LP approach is proposed to the design of robust regulators by static output feedback, for constrained uncertain linear systems with input and measurement additive disturbances.