Some applications in control require the state vector of a system to be expressed in the appropriate coordinates so as to satisfy to some mathematical properties which constrain the studied system dynamics. This is the case with the theory of linear interval observers which are trivial to implement on cooperative systems, a rather limited class of control systems. The available literature shows how to enforce this limiting cooperativity condition for any considered system through a statecoordinate transformation. This article proposes an overview of the existing numerical techniques to determine such a transformation. It is shown that in spite of being practical, these techniques have some limitations. Consequently, a reformulation of the problem is proposed so as to apply non-smooth control design techniques. A solution is obtained in both the continuous-and discretetime frameworks. Interestingly, the new method allows to formulate additional control constraints. Simulations are performed on three examples.
The problem of state observation, based on spatiallysampled output measurements, is addressed for a class of infinite dimensional systems, modelled by a semi-linear heat equation augmented with a structured uncertain part involving a set of unknown parameters. An adaptive observer is designed that provides online estimates of the system (spatially distributed) state and unknown parameters based on sampled data (in space). Sufficient conditions for the observer to be exponentially convergent are established. These include an ad-hoc persistent excitation condition as well as a condition on how the observer gain must be selected in relation with the space sampling interval.
The problem of observer design is addressed for outputinjection nonlinear systems. A major difficulty with this class of systems is that the state equation involves an output-dependent term that is explicitly dependent on unknown parameters. As the output is only accessible to measurement at sampling times, the outputdependent term turns out to be (almost all time) subject to a double uncertainty, making previous adaptive observers inappropriate. Presently, a new hybrid adaptive observer is designed and shown to be exponentially convergent under ad-hoc conditions.
We provide new bounded backstepping results that ensure global asymptotic stability for a large class of partially linear systems with an arbitrarily large number of integrators. We use a dynamic extension that contains one artificial delay, and a converging-input-converging-state assumption. When the nonlinear subsystem is control affine, we provide sufficient conditions for our converging-input-converging-state assumption to hold. We also show input-to-state stability with respect to a large class of model uncertainties, and robustness to delays in the measurements of the state of the nonlinear subsystem. We illustrate our result using an example that is beyond the scope of classical backstepping.
In this paper, we design a novel observer for a class of semilinear heat 1D equations under the delayed and sampled point measurements. The main novelty is that the delay is arbitrary. To handle any arbitrary delay, the observer is constituted of a chain of sub-observers. Each sub-observer handles a fraction of the considered delay. The resulting estimation error system is shown to be exponentially stable under a sufficient number of sub-observers is used. The stability analysis is based on a specific Lyapunov-Krasovskii functional and the stability conditions are expressed in terms of LMIs.
We are considering the problem of state observation for a class of infinite dimensional systems modeled by parabolic type PDEs. The model is subject to parametric uncertainty entering in both the domain equation and the boundary condition. An adaptive boundary observer, providing online estimates of the system state and parameters, is designed using finite-and infinite-dimensional backsteppinglike transformations. The observer is exponentially convergent under an ad hoc persistent excitation condition.
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