Observer and control synthesis for positive systems with faults

Abstract

This thesis is concerned with the observer and control synthesis for positive systems subject to actuator/sensor faults. Several approaches are studied to improve the reliability and transient performance of positive systems. Three classes of systems are considered, positive Markov jump systems, positive multi-agent systems, and positive switched systems under dwell-time constraints.

In practice, there is usually limited access to all state variables of a system, and it is difficult to directly measure all of them due to technical limitations and/or sensor costs. Therefore, the use of observers and observer-based controllers are proved to be useful in system monitoring and regulation. When positive systems are considered, new challenges arise in the observation and control synthesis due to nonnegativity. In this thesis, observation and observer-based control problems for positive systems are investigated. Specially, the following three aspects are discussed. (a) The problem of observer design for positive Markov jump systems subject to interval uncertainties and sensor faults is investigated. The main difficulty lies in the inherent constraint of nonnegativity on the state estimate in the presence of sensor faults. To cope with this, an intersectional design of descriptor observer and interval observer is considered. Rigorous theoretical analysis is given with a necessary and sufficient condition on existence of a desired observer, based on which both of the faults and state can be encapsulated at all times. (b) The existence of actuator faults in positive Markov jump systems is considered. Since the occurrence of unexpected faults in actuators may lead to instability or severely degraded performance, a joint design of interval observer and fault-tolerant controller is proposed. By utilizing the obtained interval estimates of the system state and actuator faults, the closed-loop system is guaranteed to be positive, mean exponentially stable, and satisfies a prescribed L1-gain performance with fault tolerance. Generally speaking, this part provides an interval estimation-based fault-tolerant control framework, which can cover a wide spectrum of uncertain positive systems and lead to more convenient implementation in practice. (c) The output consensus problem for positive multi-agent systems is studied. The leader-follower case is considered and a distributed observer-based consensus protocol is developed. By utilizing information exchange among agents and resorting to the positivity of the observer error dynamics, the presented distributed scheme enables each follower obtain an asymptotic state estimation of the leader. When sensor faults exist, by introducing the descriptor observer approach, a distributed state and fault estimation-based control scheme is proposed to solve the output consensus problem.

Apart from designing of observer-based controllers, the issue of bumpless transfer for positive systems is also studied for enhanced transient performance. A general class of positive switched systems with actuator faults is considered in this case. With the known minimum dwell time, a novel characterization of bumpless transfer among a variety of subsystem controllers satisfying some interpolation constraints is introduced. By utilizing adjustable interpolation functions and multiple co-positive Lyapunov functions, a bumpless switching controller is developed to guarantee a desirable closed-loop performance (globally uniformly exponential stability and L1-gain performance), trajectory smoothness, and fault tolerance.