The synthesis of externally positive filters that can be used to modulate control signals of output-feedback sliding mode controllers is presented. These filters are useful to estimate the norm of unmeasured signals in systems with parametric uncertainties. These norm observers are a robust alternative to state observers, that may be difficult to design for uncertain systems. The first step of the synthesis is the search of the envelope function, which is an upper bound for the norm of each admissible impulse response of the uncertain system. After this step, the approximation of a positive filter with an impulse response that is an upper bound for the envelope function is carried out. The search of the envelope function and the synthesis of the approximation filters are formulated as non-convex optimisation problems that can be solved by genetic algorithms. It is important to design norm observers that are not too conservative, because the modulation signals should be kept small in order to reduce the amplitude of the control signals, power losses, and undesirable chattering effects that may occur in sliding-mode control. The proposed methods are illustrated by numerical examples.

This study introduces a problem of positive filter design for positive piecewise homogeneous Markovian jump Takagi–Sugeno (T–S) fuzzy systems. The difference between the existing achievements is that the considered transition rate of the positive Markovian jump system is time-varying. This time-varying nature is finite piecewise homogeneous. The time variation of the transition rate is characterised by two cases: stochastic variation and arbitrary variation. First of all, sufficient conditions are obtained for stochastic stability and performance under the transition rate in a manner of stochastic variation and arbitrary variation by means of choosing a linear co-positive Lyapunov function. Then, based on the obtained achievements, the problem of positive filter design for the positive piecewise homogeneous Markovian jump T–S fuzzy system is studied. All the mentioned problems in this study can be solved by linear programming. Finally, a biological model is proposed to illustrate the effectiveness of theoretical results.

In this study, the problem of positive filtering is investigated for positive two-dimensional (2D) Roesser T–S fuzzy systems under -induced performance. Error-bounding positive filters are constructed to estimate the output of positive 2D T–S fuzzy systems. Firstly, a novel characterisation for asymptotic stability of the filtering error system with -induced performance is derived. Then, based on that characterisation, sufficient conditions are derived to design the required filters expressed by linear programming problems. Finally, the theoretical results are illustrated through a numerical example.

This study addresses the observability of Boolean networks (BNs), using semi-tensor product (STP) of matrices. First, unobservable states can be divided into two types, and the first type of unobservable states can be easily determined by blocking idea. Second, it is found that all states reaching to observable states are observable. Based on subgraph of transition matrix and blocking idea, the second type of unobservable states can be also determined. Approaches are obtained to directly determine some observable or unobservable states. An algorithm is designed for determining the observability of BNs as well. Examples are given to illustrate the effectiveness of the given results.