Using Interval Analysis to Compute the Invariant Set of a Nonlinear Closed-Loop Control System

Abstract

In recent years, many applications, as well as theoretical properties of interval analysis have been investigated. Without any claim for completeness, such applications and methodologies range from enclosing the effect of round-off errors in highly accurate numerical computations over simulating guaranteed enclosures of all reachable states of a dynamic system model with bounded uncertainty in parameters and initial conditions, to the solution of global optimization tasks. By exploiting the fundamental enclosure properties of interval analysis, this paper aims at computing invariant sets of nonlinear closed-loop control systems. For that purpose, Lyapunov-like functions and interval analysis are combined in a novel manner. To demonstrate the proposed techniques for enclosing invariant sets, the systems examined in this paper are controlled via sliding mode techniques with subsequently enclosing the invariant sets by an interval based set inversion technique. The applied methods for the control synthesis make use of a suitably chosen Gröbner basis, which is employed to solve Bézout’s identity. Illustrating simulation results conclude this paper to visualize the novel combination of sliding mode control with an interval based computation of invariant sets.

Experiments-Based Comparison of Different Power Controllers for a Solid Oxide Fuel Cell Against Model Imperfections and Delay Phenomena

“…For the third case, the power control was realized using a sliding-mode controller. In general, the sliding-mode control is split into two phases [19]. First, the trajectory approaches a switching manifold in finite time.…”

“…for the term sign(S(x)) as a regularization to avoid discontinuities in the control signal for operating states crossing the sliding manifold and to minimize chattering [19,22]. For compensating possible gain errors, which result from model uncertainties, the steady-state offset…”

Verifying Provable Stability Domains for Discrete-Time Systems Using Ellipsoidal State Enclosures

“…Note that this interval box needs to be employed for generating the polytopic uncertainty representation (38). Hence, the direct application of a stability contractor to the system models ( 4) and ( 14), making use of an ellipsoid with a shape matrix Q computed by (37) simplifies the evaluation of ( 40) and provides reasonably large provable stability domains as long as the contractor itself can be evaluated with a small amount of overestimation.…”

Linear Matrix Inequalities for an Iterative Solution of Robust Output Feedback Control of Systems with Bounded and Stochastic Uncertainty

“…Future work will aim at validating the proposed design methodology experimentally and at interfacing it with LMI-based design approaches for interval observers [ 35 ] as a technique for the state estimation in a bounded-error framework. In addition, also combinations with sliding mode-type control procedures such as those in [ 36 ] can be investigated. Finally, it should be pointed out that the technique is readily applicable also to higher dimensional system models, such as the interconnection of multiple spring-mass-damper elements in the frame of mechanical vibration control or the interconnection of RLC networks, which may serve either as a representation of long electric transmission lines or as a finite-dimensional approximation of volume flow and pressure variations in fluidic networks [ 10 , 37 , 38 ].…”