Monotonicity properties for the viable control of discrete-time systems (original) (raw)

Monotonic properties for the viable control of discrete time systems

Citeseer

This paper deals with the control of nonlinear systems in the presence of state and control constraints for discrete time dynamics in finite dimensional spaces. The viability kernel is known to play a basic role for the analysis of such problems and the design of viable control feedbacks. Unfortunately, this kernel may display very non regular geometry and its computation is not an easy task in general. In the present paper, we show how monotonic properties of both dynamics and constraints allow for relevant analytical upper and lower approximations of the viability kernel through weakly and strongly invariant sets. An example on fish harvesting management illustrates some of the assertions.

Viable states for monotone harvest models

Systems & Control Letters, 2011

This paper deals with the control of discrete-time dynamical, monotone both in the state and in the control, in the presence of state and control monotone constraints. A state x is said to belong to the viability kernel if there exists a trajectory, of states and controls, starting from x and satisfying the constraints. Under monotonicity assumptions, we present upper

Approximating Viability Kernels of Non-linear Systems

Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics, 2018

A typical concern in Robotics is to assess if it is possible to keep a robot inside a set of safe states, e.g., an autonomous car that must stay on the road. That is closely tied with the problem of computing the viability kernel of the system, i.e., the largest set of initial states for which it is guaranteed that the system has controls that keep maintain the trajectories inside the constraint set. The approach in this paper builds on previous work, on linear sampled-data systems. It is based on sampling the boundary of the constraint set, finding the states inside the viability kernel using finite-horizon forward simulation. Our adaptation extends the original algorithm, approximating the viability kernel for some non-linear systems through linearization methods. The non-linear systems here approached are the ones described by first order differential equations with continuous derivatives and convex with respect to the inputs. Existence and uniqueness conditions are also established to ensure adequate results for the whole algorithm. A practical example, with a simple non-linear system, to illustrate the proposed algorithm is also presented.

APPROXIMATE CONTROLLABILITY WITH POSITIVE CONTROLS

1997

In this paper, controllability of the linear discrete-time systems (A,B,Ω):x k+1 =Ax k +Bu k , x k ∈X, u k ∈Ω, is studied, where X is a Banach space and the control set Ω is assumed to be a cone in a Banach space U . Some criteria for approximate controllability are given. The case where the operator A is compact is examined in detail by using the spectral decomposition of the state space X. As a result, a criterion for approximate controllability of (A,B,Ω) is obtained without imposing a restrictive condition that the system with no control constraints (A,B,U ) is exactly controllable. The obtained results are then applied to consider the problem of controllability for linear functional differential equations with positive controls. Some necessary and sufficient conditions of approximate controllability to the state space R n ×L p are presented and some illustrating examples are given.

Further results on controllability properties of discrete-time nonlinear systems

Dynamics and Control, 1994

Controllability questions for discrete-time nonlinear systems are addressed in this paper. In particular, we continue the search for conditions under which the group-like notion of transitivity implies the stronger and semigroup-like property of forward accessibility. We show that this implication holds, pointwise, for states which have a weak Poisson stability property, and globally, if there exists a global \attractor" for the system.

Feedback control of linear discrete-time systems under state and control constraints

International Journal of Control, 1988

In this paper the problem of stabilizing linear discrete-time systems under state and control linear constraints is studied. Based on the concept of positive invariance, existence conditions of linear state feedback control laws respecting both the constraints are established. These conditions are then translated into an algorithm of linear programming.

Positive output controllability of linear discrete–time invariant systems

Control and Cybernetics

This paper studies the output controllability of discrete linear time invariant systems (LTI) with non-negative input constraints. Some geometrical arguments and positive invariance concepts are used to derive the necessary and/or sufficient conditions for the positive output controllability of discrete LTI systems. The paper also provides several academic examples, which support the theoretical results.

The stabilizability of a controlled system describing the dynamics of a fishery

Comptes Rendus Biologies, 2005

This work presents two stock-effort dynamical models describing the evolution of a fish population growing and moving between two fishing zones, on which it is harvested by a fishing fleet, distributed on the two zones. The first model corresponds to the case of constant displacement rates of the fishing effort, and the second one to fish stock-dependent displacement rates. In equations of the fishing efforts, a control function is introduced as the proportion of the revenue to be invested, for each fleet. The stabilizability analysis of the aggregated model, in the neighborhood of the equilibrium point, enables the determination of a Lyapunov function, which ensures the existence of a stabilizing discontinuous feedback for this model. This enables us to control the system and to lead, in an uniform way, any solution of this system towards this desired equilibrium point. To cite this article: R. Mchich et al., C. R. Biologies 328 (2005).  2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Robustness of viability controllers under small perturbations

Journal of Optimization Theory and Applications, 1994

In this note, a perturbed control problem with state constralnts depending on a parameter u is considered. Assuming that, for a certain value of u, there exists a viability controller, we explicitly estimate the range of variations of u for which the same controller gives viable solutions.