An adaptive tracking problem for a family of retarded time‐delay plants

Abstract

Most of the existing switching control techniques are developed specifically for finite-dimensional linear time-invariant (LTI) systems. In many practical applications, however, it is essential to take time delay into consideration in the modelling as the control system can be highly sensitive to delay. In this paper, a multi-model switching control algorithm is proposed for retarded time-delay systems. It is assumed that the plant is represented by a family of known multi-input multi-output, observable, LTI models with multiple delays in the states, and that corresponding to each model in the known family, there exists a high-performance finite-dimensional LTI controller. In addition, it is supposed that a bound on the magnitude of the external inputs and disturbances is available. It is then shown that the proposed switching controller can stabilize the uncertain system, and that under some mild conditions, output tracking can be achieved in the given problem setting.

On observer design for a class of impulsive switched systems

“…replacing the second term in the above inequality by 0 P i H i, j T P −1 i 0 P i H i, j and using the Schur-complement formula one can verify that the above inequality becomes the same as (17). Thus all the conditions of Lemma 1 are satisfied by this choice of V .…”

“…Otherwise (0 < µ * ≤ 1), the error dynamic in the switching observer O is globally uniformly exponentially stable for arbitrary switching signals. Proof: Proof is similar to the proof of the previous theorem, in fact LMIs (23) and (24) guarantees that convergence rate of each Luenberger observer is in the desired region, also by substituting H i j = G i j , one can verify that LMI's (25) and (17) are the same. Thus if the same Lyapunov function V as in the previous theorem is considered all the conditions of Lemma 1 are satisfied by this choice of V and error in the observer is asymptotically stable for any impulsive switched system with the average dwell time greater than (27).…”

“…Switching can also be applied to control in order to cope with highly uncertain systems [1], [2], [17]. There are many examples of switched systems in power electronics [12], process control, biomedical and biochemical processes [9] and aerospace, to name only a few.…”

An adaptive switching scheme for uncertain discrete time‐delay systems

“…, and i(k) ∈ P (P is a finite set), it follows from Lemma 3 that there exist finite positive constants M 0 and¯ satisfying (16) and since (15) behaves like an exponentially stable system for any k ∈[k s , k s+1 ), it follows from Lemmas 1 and 2 that there exist constants M i(k s ) , i(k s ) , and a…”

“…For uncertain time-delay systems there are only a few references that can handle large uncertainties in both system parameters and delay in the system dynamics [15]. In [15], a pre-routed switching approach is developed to stabilize a class of uncertain continuous-time systems with known delay.…”