Continuous state observability and mode reconstructability of switched nonlinear systems with unknown switching function

Abstract

Summary For switched nonlinear systems, the problems of continuous state observability and mode reconstructability are tackled, assuming the switching function is unknown. First, we give a condition under which the continuous state is observable and the active mode is reconstructible. Furthermore, we provide some relaxed conditions that guarantee the observability of the continuous state vector x(t); those conditions allow estimating x(t) even if the active mode cannot be reconstructed. The observability analysis is carried out by the study of an invariant manifold on which the outputs of two systems (fixing each pair of different modes) are equal. Moreover, by assuming that there is a known minimum dwell‐time, we propose a way of reconstructing the state vector and the switching function, provided the respective observability conditions are fulfilled. The reconstruction procedure involves the use of finite‐time homogeneous sliding mode differentiator. In a short section, we show how the obtained observability conditions match with the known structural ones for linear systems. Then, we see that the proposed state reconstruction scheme can also be applied to that particular class of systems. We point out that the original system is not required to be in any specific form, which may be an advantage of the estimation method proposed in this paper. By an academic example with simulations, we illustrate the analysis carried out along the paper and the respective reconstruction method.

Non‐fragile dynamic output‐feedback control for ‐gain performance of positive FM‐II model with PDT switching: An event‐triggered mechanism

“…Denote 𝛽 = (𝜆 2 ∕𝜆 1 )𝜒 2(1+T) and 𝜖 = −[((1 + T) ln 𝜒)∕(𝜏 + T) + ln 𝛼], one can easily obtain 𝛽 > 0 and 𝜖 > 0 based on condition (18). Thus, system (9) is robustly exponentially stable according to Definition 3.…”

“…hold for all p ∈ M, 𝜍, 𝜍 1 ∈ N, then system ( 9) is positive, robustly exponentially stable, and has an l 1 -gain bound 𝛾 with 𝛾 ≜ (𝛾∕𝛼)𝜒 2(1+T) under the PDT switching scheme satisfying (18) and (19), where ⃗ f 6) could be designed as in (20), and the corresponding gain matrices Ĉp , Dp , Ŝp , and Tp are designed as…”

“…The other parameters in this example are given by a 𝜄 = b 𝜄 = c = d = 0.1, 𝛼 = 0.95, 𝜒 = 1.07, 𝛾 = 1.5 and T = 4. By solving the conditions in Theorem 2, the following solutions can be obtained: Then based on ( 20) and ( 41), the controller parameters can be obtained as In addition, it can be calculated that the PDT satisfies 𝜏 > 2.5953 based on (18). Without loss of generality, the PDT is selected as 𝜏 = 3, and the simulation results of the obtained closed-loop system are shown in Figures 2-5 under the switching sequence given in Figure 6, where the nonnegative initial condition employed is as follows:…”

“…In general, this phenomenon could be described by a switched system which consists of several subsystems and a switching law. [18][19][20] In the research of such kind of models, one key problem is to design the appropriate switching scheme because this will determine whether a desired performance could be obtained or not. 21 In the literature, the switching mechanism with average dwell time (ADT) constraint is widely used, where the upper bound of the average switching numbers on a time interval is limited, and this certainly leads to a loss in the design flexibility to some extent.…”