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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    • "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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    • "Early results for classes of control systems were based on dynamical consistency properties [26], natural invariants of the control system [27], l-complete approximations [28], and quantized inputs and states [29] [30]. Recent results include work on controllable discrete-time linear systems [31], piecewise-affine and multi-affine systems [32] [33], set-oriented discretization approach for discrete-time nonlinear optimal control problems [34], abstractions based on convexity of reachable sets [35], incrementally stable and incrementally forward complete nonlinear control systems with and without disturbances [36] [37] [38] [39], switched systems [40] and time-delay systems [41] [42]. "
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