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: 2.72). 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

  • [Show abstract] [Hide abstract]
    ABSTRACT: This paper addresses the symbolic motion planning and control of robots to meet high level specifications through hybrid supervisory control. The basic idea is to partition the motion space of robots into logically equivalent regions, based on which a bisimulation quotient transition system is derived and supervisor is synthesized. The bisimulation relation between the abstracted model and the original continuous dynamics is formally proved, which guarantees the existence of feasible continuous control signals and closed-loop trajectories for robots to satisfy the high level specifications as well. The main contribution of the paper lies in the development of a unified hybrid hierarchical control framework whose top layer is a discrete supervisor that is responsible for decision making to satisfy the assigned specification. This discrete supervisor is connected to the low level continuous dynamics of the system via an interface layer. The interface layer is responsible for translating discrete commands of the supervisor to a continuous control signals implementable by the continuous plant and vice versa.
    Control and Automation (ICCA), 2013 10th IEEE International Conference on; 01/2013
  • [Show abstract] [Hide abstract]
    ABSTRACT: Symbolic approaches to the control design over complex systems employ the construction of finite-state models that are related to the original control systems, then use techniques from finite-state synthesis to compute controllers satisfying specifications given in a temporal logic, and finally translate the synthesized schemes back as controllers for the concrete complex systems. Such approaches have been successfully developed and implemented for the synthesis of controllers over non-probabilistic control systems. In this paper, we extend the technique to probabilistic control systems modeled by controlled stochastic differential equations. We show that for every stochastic control system satisfying a probabilistic variant of incremental input-to-state stability, and for every given precision $\varepsilon>0$, a finite-state transition system can be constructed, which is $\varepsilon$-approximately bisimilar (in the sense of moments) to the original stochastic control system. Moreover, we provide results relating stochastic control systems to their corresponding finite-state transition systems in terms of probabilistic bisimulation relations known in the literature. We demonstrate the effectiveness of the construction by synthesizing controllers for stochastic control systems over rich specifications expressed in linear temporal logic. The discussed technique enables a new, automated, correct-by-construction controller synthesis approach for stochastic control systems, which are common mathematical models employed in many safety critical systems subject to structured uncertainty and are thus relevant for cyber-physical applications.
  • [Show abstract] [Hide abstract]
    ABSTRACT: The technical note studies the problem of making the trajectories of an affine system defined on a polytopic state space reach a prescribed facet of the polytope in finite time without first leaving the polytope. The focus is on solvability by continuous piecewise affine feedback, and we formulate a variant of the problem in which trajectories exit in a monotonic sense. This allows to obtain necessary and sufficient conditions for solvability in certain geometric situations.
    IEEE Transactions on Automatic Control 01/2013; 58(10):2704-2709. · 2.72 Impact Factor


Available from