Article

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: 3.17). 12/2006; DOI: 10.1109/TAC.2006.884957
Source: IEEE Xplore

ABSTRACT 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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    • "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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    ABSTRACT: Networked Control Systems (NCS) are distributed control systems where the communication among plants, sensors, actuators and controllers occurs in a shared communication network. NCS have been the subject of intensive study in the last few years. Important results have been obtained, mainly on stability and stabilizability of these systems. While stability is a basic property and as such very important, the design of these systems requires considering other properties that can be captured by logical requirements or automata. In this paper, we propose an approach based on symbolic models, which are abstract descriptions of continuous systems where one symbol corresponds to an "aggregate" of continuous states. We use symbolic models for NCS control design where specifications are expressed in terms of non-deterministic automata.
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    • "Early results for classes of control systems were based on dynamical consistency properties [7], natural invariants of the control system [18], l-complete approximations [21], and quantized inputs and states [11] [5]. Recent results include work on controllable discrete-time linear systems [27], piecewise-affine and multi-affine systems [15] [4], set-oriented discretization approach for discrete-time nonlinear optimal control problems [17] and abstractions based on convexity of reachable sets [24]. The notion of approximate bisimulation [12] provided an important tool to identify other classes of control systems admitting symbolic models, examples of which are stabilizable linear control systems [28], incrementally stable nonlinear control systems [22], switched systems [14], time-delay systems [23] and incrementally forward complete nonlinear control systems [29]. "
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    ABSTRACT: Symbolic models have recently spurred the interest of the research community because they offer a correct-by-design approach to the control of embedded and cyber-physical systems. In this paper we address construction of symbolic models for networks of discrete-time nonlinear control systems. The main result of the paper shows that under some small gain theorem-type conditions, a network of symbolic models can be constructed which approximates a network of incrementally stable control systems in the sense of approximate bisimulation with any desired accuracy. Compositional design of quantization parameters of the symbolic models is also derived and based on the topological properties of the network.
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    • "We follow the hierarchical approach to robot controller synthesis as outlined above and we narrow our attention to the second step of the approach, i.e., to generating discrete plans. The application of the algorithm that we propose is, however, not restricted to discrete systems: For the first step of the hierarchical approach, methods for discrete modeling of robotic systems can be used (e.g., [12], [14], [15], [21] and the references therein); for the third step, low-level controllers exist that can drive a robot from any position within a region to a goal region (e.g., [2]). As a mission specification language, we use Linear Temporal Logic (LTL), for its resemblance to natural language [10], and expressive power. "
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