An internal model approach to estimation of systems with arbitrary unknown inputs

Abstract Estimation of systems with arbitrary unknown inputs, for which no models or statistical properties are assumed to be known, has received much attention. Given that no prior knowledge is assumed for the unknown inputs, the estimator has to be designed to be dissociated from their effects, i.e., the problem at hand has also been called unknown input decoupled estimation. The most significant results in the literature, pioneered by Hautus and others, show that the strong ∗ detectability requirements, namely, (1) a rank matching condition; (2) the system is minimum phase, are necessary and sufficient for stable estimation of the state/unknown input. Recent research has sought alternative methods, when the above conditions do not hold. However, existing results have only addressed a few specific scenarios. For the most general case with the unknown inputs affecting both the state dynamics and the output, the essential question of whether and under what conditions it is still possible to obtain instantaneous and asymptotically stable estimation of the full state, when the strong ∗ detectability conditions do not hold, has remained open for a long time. Answering such a question is critical for real-time feedback control with closed-loop stability and offset free regulation/tracking performance guarantees. The current paper will fill this gap via an internal model approach. We make a crucial observation that albeit without any model or statistical properties for the unknown inputs, one does have a model for their sum over time (to illustrate, here we consider the discrete-time case), which is an integrator driven by the unknown inputs. By incorporating this model into the original system model to form an augmented system, we establish conditions under which the augmented system is strong ∗ detectable so that a strong observer exists and can be constructed to estimate the augmented state variable, comprised of the original system state and the unknown input sum, with asymptotically stable error. Moreover, the former conditions prove to encompass well-known results in the existing literature on offset free tracking with known disturbance models as special cases. Hence, the proposed results generalize the classical internal model principle from the conventional case when models of the disturbances are assumed to be known to the case with arbitrary unknown disturbances.