On Approximation of Maximal Admissible Sets for Nonlinear Continuous-Time Systems with Constraints

Abstract

The concept of the maximal admissible set (MAS) is very important in several issues in control theory, such as constrained control and so on. For linear uncertain discrete time systems, a method to compute the MAS was proposed. For linear time invariant systems without uncertainty, a theoretical result on inner and outer approximations of the MAS was derived. On the other hand, for uncertain and/or nonlinear continuous time systems, only a few restrictive results were reported. This paper treats uncertain and/or nonlinear continuous time systems with constraints and derive some theoretical results about inner and outer approximations of the MAS for such a kind of systems. I. IFor almost all practical control systems, we need to take into account the existence of constraints on state and/or control input caused by amplitude limitation of state variables and saturation property of actuators. If we ignore these constraints, the real performance of the system degrade because of the wind-up phenomena, or in worst cases the control system become unstable. In these respect, extensive researches have been done to cope with such constraints: anti-windup control [1], reference management control using reference governor [2]-[6], predictive reference management control [7], [8], reference shaping [9]. The concept of the maximal admissible set (MAS) plays very important role in the approach of reference management control using reference governor. In most of the literature, linear discrete time systems are treated [2] -[6], [10]-[13]. But plants are continuous time systems. Therefore, it is desired to compute MASs for continuous time systems. As long as we can examine, the reference [14]is the first paper considering the computation of MASs for continuous time systems. In this paper, we treat uncertain and/or nonlinear continuous time systems with constraints, and derive some theoretical results about inner and outer approximations of MASs for such a kind of systems. Theorem 1 is the main result of this paper, and it gives inner approximations of MASs for nonlinear continuous time systems. An example is shown to demonstrate the usefulness of the obtained results. Notation. Let N, Z + and R + denote the set of natural numbers, nonnegative integers and nonnegative real numbers. For a set A in R n , int A, bd A, and co A denote interior, boundary, and convex hull of A. For a set X ⊆ R n , X ±x = {x ′ = x ±x, x ∈ X}. For a matrix A and a vector y, [A] i and y i denote the i-th row vector of A and i-the element of y.

Decoupled Reference Governors for Multi-Input Multi-Output Systems

“…The idea of maximal output admissible set was first proposed in 1991 by Gilbert and Tan [48]. After that, more exploration on MAS has been proposed, such as MAS with disturbance inputs [53], MAS for nonlinear systems with constraints [54,55], MAS with time delay in states and inputs [56,57], computation of polytopic MAS [58], and MAS for periodic systems [59]. We will explain the computation of MAS in detail in Chapter 2.…”

Estimate of Attractive Regions for Systems Satisfying Polytopic Uncertainties in Given Regions

“…We denote a PIS and a AR for Σ C (A)inX by Ω C ∞ (A) and D C (A), respectively, Moreover, we define a robust PIS (RPIS)Ω C ∞ and a robust AR (RAR)D C of the system Σ C (A)inX bỹ Finally, we say thatΩ C * ∞ andD C * are the maximal RPIS (MR-PIS) and the maximal RAR (MRAR) for Σ C (A)i nX if for anyΩ C ∞ and anyD C are subsets ofΩ C * ∞ andD C * , respectively. We note that the MRPISΩ C * ∞ is the maximal admissible set (MAS, see Ohta & Tanizawa [2007]) for Σ C (A) since we are considering a very special constraints for the system Σ C , that is, the variables z(t) to be constrained for this system is z(t) = x(t). Let x(t; x 0 ) be a solution of Σ C .…”

Sliding Mode Control under State and Control Constraints

“…We estimate the region of state, in which the system state is satisfied constraints (24), (25), or (26), using the inner approximation of MAS for nonlinear continuous-time system [11].…”

“…Lemma 3 [11] We haveΩ ∞ ⊆ Ω ∞ that is, MASΩ ∞ for the system (29) is the inner approximation of MAS Ω ∞ for the system (27).…”

“…The main issue of this method is take into account constraints. To approach this issue, we employ the inner approximation of maximal output admissible set(MAS) concept for a nonlinear continuoustime system [11] Moreover, we propose a control strategy of switching "sliding" hyperplanes to achieve better performance. Notation Let Z + and R + denote the set of nonnegative integers and nonnegative real numbers, respectively.…”