High order sliding mode observer for linear systems with unbounded unknown inputs

Abstract

A global observer is designed for strongly detectable systems with unbounded unknown inputs. The design of the observer is based on three steps. First, the system is extended taking the unknown inputs (and possibly some of their derivatives) as a new state; then, using a global high-order sliding mode differentiator, a new output of the system is generated in order to fulfil, what we will call, the Hautus condition, which finally allows decomposing the system, in new coordinates, into two subsystems; the first one being unaffected directly by the unknown inputs, and the state vector of the second subsystem is obtained directly from the original system output. Such decomposition permits designing of a Luenberger observer for the first subsystem, which satisfies the Hautus condition, i.e. all the outputs have relative degree one w.r.t. the unknown inputs. This procedure enables one to estimate the state and the unknown inputs using the least number of differentiations possible. Simulations are given in order to show the effectiveness of the proposed observer.

Keywords:

• sliding mode observer
• unknown inputs
• strong detectability

Acknowledgements

Francisco J. Bejarano thanks the support given by the Mexican CONACyT postdoctoral grant CVU 103957. Both authors acknowledge the Mexican CONACyT project 56819 and PAPIIT 111208.

Notes

1. Clearly all results obtained here can be applied to the more general sort of systems of the form , y = Cx + Eu + Fw. In that case it is enough to construct the system ż = Aż + Bu, y _z = Cz + Eu, and define . Thus, we get a new system and , which belongs to the sort of systems considered in (1).

2. Recall that Σ is strongly detectable if y ≡ 0 implies x(t) → 0. Some times it is said that the fourfold (A, C, D, F) is strongly detectable meaning that this property is fulfilled for system Σ, associated to the fourfold. It was proven in Hautus (1983) that Σ is strongly detectable if, and only if, the set of zeros of (A, C, D, F) lies within the interior of the left half side of the complex plane.