1 A canonical form for the design of unknown input sliding mode observers

The concept of sliding mode control [11, 27, 32] has been extended to the problem of the state estimation by an observer, both for linear [10, 32] and nonlinear systems [1, 8, 28, 34]. Using the same design theory as variable structure control laws, the observer trajectories are constrained to evolve after a finite time on a suitable sliding manifold, by the use of a discontinuous output injection (the sliding manifold is usually given by the difference between the observer and the system output). The sliding motion provides an estimation (asymptotically or in finite time) of the system state. Sliding mode observers have been shown to be efficient in many applications, such as in robotics [3, 21], electrical engineering [6, 33], chemical reactors [25] or fault detection [14, 18]. The problem of designing an observer for a multivariable linear system partially driven by unknown inputs is of great interest. Such a problem arises in systems subject to disturbances or with inaccessible inputs and in many applications such as fault detection and isolation, parameter identification or cryptography. Since, in the sliding mode, the resulting dynamics are insensitive to a class of perturbations and parametric uncertainties, variable structure techniques for robust state reconstruction have been developed in many papers. Observers are designed using canonical forms which are derived under the assumption that some matching conditions are fulfilled by the system. The main contribution of this chapter is to introduce a constructive algorithm that transforms the system into a new canonical form suitable for the design of finite time sliding mode observers 3. An interesting property of the algorithm is that it can be performed to estimate the state of some systems 3 Note that the system has to be fully observable for a finite time convergence property. that do not necessarily satisfy the classical matching conditions involved in the design of sliding mode observers. This work is organized as follows. Section 1.2 gives a brief overview of robust state reconstruction based on sliding mode observers and the topic of this work. In Section 1.3 the algorithm is developed, which results in a canonical form made of block triangular observable forms. Then in Section 1.4, the state and the unknown inputs are estimated by means of a finite time observer designed under second order sliding mode considerations. An example, that points out the feasibility and the advantages of the proposed method, is given …