State estimation for linear impulsive systems

Abstract

In this paper, we treat the fundamental problem of state estimation for a class of linear impulsive systems with time-driven impulsive effects. We show that a strong observability property enables an impulsive observer to be constructed that yields uniformly exponentially stable estimation error dynamics. This approach accommodates impulsive systems with arbitrarily-spaced impulse times and singular state transition matrices in a manner reminiscent of well-known results for time-varying discrete-time linear systems. As an example, an observer is constructed for an impulsive system that produces general cubic spline signals.

Duality Properties of Linear Impulsive Systems

“…The notion of strong observability shown in [7] to be sufficient for the construction of an impulsive observer for (1) that produces an asymptotically convergent state estimate involves the existence a positive integer ᐉ* such that Qfixed(t0, tf, T) = 0 holds for all time intervals [t0, tf] impulse time sets T for which card{T പ (t0, tf)} Ն ᐉ*. As a special case of Theorem V.3, we have the following condition that is necessary and almost sufficient for strong observability.…”

Observability criteria for impulsive control systems with applications to biomedical engineering processes

“…Conversely, if there exists an integer l such that the observability Gramian is positive definite for any impulse time set and any finite interval containing at least l impulse times, then the system is observable (Medina & Lawrence, 2009). Note that this criterion meets Definition 2.…”

“…The input u(τ k ) = 0, ∀k is considered without loss of generality. Proof of (1) ⇒ (2) is obtained from Theorem 2, but considering linear ICS and the proof of (1) ⇔ (3) can be found in Medina and Lawrence (2009), so it will be omitted here.…”

“…The linear case: A comparable definition of observability over a finite interval along the lines of the observability Gramian can be found in Medina and Lawrence (2009), in which it is adapted to linear ICS with discrete measurements of the output. It reads as: given the impulsive system (8) on [0, t f ], and the time instants…”

“…Discontinuities in the state of the form x(τ + k ) = A I x(τ k ) are allowed, where A I defines a diagonal matrix. A different class of impulsive control systems is considered in Medina and Lawrence (2009), for which the states evolve in continuous form but the output is available for measurement at discrete times only. Suitable criteria based on geometric properties of the invariant observable space and the observability Gramian were worked out for this case.…”