Control loops that are closed over a communication network get more and more common. A problem with such systems is that the transfer delays will be varying with different characteristics depending on the network hardware and software. The network delays are typically varying due to varying network load, scheduling policies in the network and the nodes, and due to network failures. Two network models of different complexity are studied: Random delays that are independent from transfer to transfer, Random delays with probability distribution functions governed by an underlying Markov chain. The delay models are verified by experimental measurements of network delays. In the thesis it is shown how to analyze stability and expected performance of linear controllers where the network delays are described by one of the two network models above. Methods to evaluate quadratic cost functions are developed. Through the same analysis we find criteria for mean square stability of the closed loop for the different network models. The Linear Quadratic Gaussian (LQG) optimal controller is developed for the two delay models. The derived controller uses knowledge of old time delays. These can be calculated using ``timestamping'' of messages in the network. ``Timestamping'' means that every transfered signal is marked with the time of generation. The receiving node can then calculate how long the transfer delay was by comparing the timestamp with the node's internal clock. (Less)

/pdf/real-time-control-systems-with-delays-aau6neyvii.pdf

Real-Time Control Systems with Delays

Jump linear systems are defined as a family of linear systems with randomly jumping parameters (usually governed by a Markov jump process) and are used to model systems subject to failures or changes in structure. The authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties. It is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability, and a testable necessary and sufficient condition for second moment stability is derived. The Lyapunov exponent method for the study of almost sure sample stability is discussed, and a theorem which characterizes the Lyapunov exponents of jump linear systems is presented. Finally, for one-dimensional jump linear system, it is proved that the region for delta -moment stability is monotonically converging to the region for almost sure stability at delta down arrow 0/sup +/. >

Stochastic stability properties of jump linear systems

Stability of stochastic differential equations with Markovian switching

In this note, we consider the problems of stochastic stability and sliding-mode control for a class of linear continuous-time systems with stochastic jumps, in which the jumping parameters are modeled as a continuous-time, discrete-state homogeneous Markov process with right continuous trajectories taking values in a finite set. By using Linear matrix inequalities (LMIs) approach, sufficient conditions are proposed to guarantee the stochastic stability of the underlying system. Then, a reaching motion controller is designed such that the resulting closed-loop system can be driven onto the desired sliding surface in a limited time. It has been shown that the sliding mode control problem for the Markovian jump systems is solvable if a set of coupled LMIs have solutions. A numerical example is given to show the potential of the proposed techniques.

https://www.researchgate.net/profile/Peng_Shi5/publication/3032315_On_designing_of_sliding-mode_control_for_stochastic_jump_systems/links/0c960519604c828e4a000000.pdf

On designing of sliding-mode control for stochastic jump systems

This note investigates the output feedback stabilization of networked control systems (NCSs). The sensor-to-controller (S-C) and controller-to-actuator (C-A) random network-induced delays are modeled as Markov chains. The focus is on the design of a two-mode-dependent controller that depends on not only the current S-C delay but also the most recent available C-A delay at the controller node. The resulting closed-loop system is transformed to a special discrete-time jump linear system. Then, the sufficient and necessary conditions for the stochastic stability are established. Further, the output feedback controller is designed via the iterative linear matrix inequality (LMI) approach. Simulation examples illustrate the effectiveness of the proposed method.

/pdf/output-feedback-stabilization-of-networked-control-systems-2e4bny39sn.pdf

Output Feedback Stabilization of Networked Control Systems With Random Delays Modeled by Markov Chains

Consideration is given to the control of continuous-time linear systems that possess randomly jumping parameters which can be described by finite-state Markov processes. The relationship between appropriately defined controllability, stabilizability properties, and the solution of the infinite time jump linear quadratic (JLQ) optimal control problems is also examined. Although the solution of the continuous-time Markov JLQ problem with finite or infinite time horizons is known, only sufficient conditions for the existence of finite cost, constant, stabilizing controls for the infinite time problem appear in the literature. In this paper necessary and sufficient conditions are established. These conditions are based on new definitions of controllability, observability, stabilizability, and detectability that are appropriate for continuous-time Markovian jump linear systems. These definitions play the same role for the JLQ problem as the deterministic properties do for the linear quadratic regulator (LQR) problem. >

Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control

The optimal quadratic control of continuous-time linear systems that possess randomly jumping parameters which can be described by finite-state Markov processes is addressed. The systems are also subject to Gaussian input and measurement noise. The optimal solution for the jump linear-quadratic-Gaussian (JLQC) problem is given. This solution is based on a separation theorem. The optimal state estimator is sample-path dependent. If the plant parameters are constant in each value of the underlying jumping process, then the controller portion of the compensator converges to a time-invariant control law. However, the filter portion of the optimal infinite time horizon JLQC compensator is not time invariant. Thus, a suboptimal filter which does converge to a steady-state solution (under certain conditions) is derived, and a time-invariant compensator is obtained. >

Jump linear quadratic Gaussian control in continuous time

This paper is concerned with the optimal quadratic control of continuous-time linear systems that possess randomly jumping parameters which can be described by finite-state Markov processes. The systems are also subject to gaussian input and measurement noises. This Jump Linear Quadratic Gaussian (JLQG) optimal control problem can be used to consider the control of systems which are subject to abrupt changes in their structure and components and also with disturbances on actuators and sensors. The solution of the continuous-time Jump Linear Quadratic (JLQ) problem, (the JLQG problem with complete state information and no input noise) is known. However in many applications, the plant state is available only through noisy observations on the output channel. In this paper, the optimal solution for the JLQG problem in finite time is given. This solution is based on a separation theorem. The optimal state estimator is sample path dependent (it may depend upon past as well as the current values of the jump parameter). For the infinite time JLQG problem, the optimal solution is also obtained. If the plant parameters are constant in each value of the underlying jumping process, then the controller part of the compensator converges to a time invariant control law (which depends on the jump parameter). However the filter portion of the optimal infinite time horizon JLQG compensator is not time invariant. A suboptimal filter which does converge to a steady-state solution (under certain stochastic stabilizabilty and observability conditions) is also derived.

Y. Ji

Papers

Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control

Stochastic stability properties of jump linear systems

Jump linear quadratic Gaussian control in continuous time

Jump Linear Quadratic Gaussian Control in Continuous Time