Controlling a Class of Nonlinear Systems on Rectangles

Departments of Manuf. & Aerosp. & Mech. Eng., Boston Univ., Brookline, MA
IEEE Transactions on Automatic Control (Impact Factor: 2.78). 12/2006; 51(11):1749 - 1759. DOI: 10.1109/TAC.2006.884957
Source: IEEE Xplore


In this paper, we focus on a particular class of nonlinear affine control systems of the form xdot=f(x)+Bu, where the drift f is a multi-affine vector field (i.e., affine in each state component), the control distribution B is constant, and the control u is constrained to a convex set. For such a system, we first derive necessary and sufficient conditions for the existence of a multiaffine feedback control law keeping the system in a rectangular invariant. We then derive sufficient conditions for driving all initial states in a rectangle through a desired facet in finite time. If the control constraints are polyhedral, we show that all these conditions translate to checking the feasibility of systems of linear inequalities to be satisfied by the control at the vertices of the state rectangle. This work is motivated by the need to construct discrete abstractions for continuous and hybrid systems, in which analysis and control tasks specified in terms of reachability of sets of states can be reduced to searches on finite graphs. We show the application of our results to the problem of controlling the angular velocity of an aircraft with gas jet actuators

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    • "Our work has some relations with theoretical qualitative control techniques used for piecewise linear systems in the field of genetic regulatory networks (Chaves and Gouzé (2011)). The approach is also similar to the domain approaches used in hybrid systems theory, where there are some (controlled) transitions between regions, forming a transition graph (Belta and Habets, 2006; Habets and van Schuppen, 2004). "
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    DESCRIPTION: Abstract The oscillator made of a negative loop of two genes is one of the most classical motifs of genetic networks. In this paper we give two solutions to control such an oscillator by modifying the synthesis rates. Our models are given by Piecewise Affine systems, and the control is very qualitative, taking only two values. The necessary measurements for implementing this control only depend on the fact that some gene is expressed or not. Our goal is to obtain sustained oscillations.
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    • "This approach is similar to those deducing the global dynamics from a " transition graph " of possible transitions between regions [5]. "
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    ABSTRACT: We consider the problem of global stabilization of an unstable bioreactor model (e.g. for anaerobic digestion), when the measurements are discrete and in finite number ("quantized"), with control of the dilution rate. The model is a differential system with two variables, and the output is the biomass growth. The measurements define regions in the state space, and they can be perfect or uncertain (i.e. without or with overlaps). We show that, under appropriate assumptions, a quantized control may lead to global stabilization: trajectories have to follow some transitions between the regions, until the final region where they converge toward the reference equilibrium. On the boundary between regions, the solutions are defined as a Filippov differential inclusion. If the assumptions are not fulfilled, sliding modes may appear, and the transition graphs are not deterministic.
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    • "The Reach Control Problem (RCP), first introduced in [4] and given a modern formulation in [5] [9], is a fundamental problem in piecewise affine and hybrid system theory. A reach control approach has been shown to be useful in a number of applications, including aircraft and underwater vehicles [1], genetic networks [2], and aggressive maneuvers of mechanical systems [12]. Nevertheless, for a given system, it is still not known in general whether the RCP is solvable by either affine or continuous state feedback. "
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    ABSTRACT: This ArXiv paper is a supplement to [7] and contains proofs of preliminary claims omitted in [7] for lack of space. The paper deals with exploring a necessary condition for solvability of the Reach Control Problem (RCP) using affine feedback. The goal of the RCP is to drive the states of an affine control system to a given facet of a simplex without first exiting the simplex through other facets. In analogy to the problem of a topological obstruction to the RCP for continuous state feedback studied in [7], this paper formulates the problem of an affine obstruction and solves it in the case of two- and three-dimensional systems. An appealing geometric cone condition is identified as the new necessary condition.
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