Resource-aware control for cyber-physical systems

An efficient usage of available resources is a substantial requirement for the successful control design in cyber-physical systems. Recent results indicate major benefits of event-based control compared to conventional designs, when resources such as communication, energy, and/or computation, are scarce. In this work we consider multiple control loops which share the communication resource. We propose a novel approach for a distributed solution for the resource sharing. In particular, the adaptation ability of event-triggered control in terms of communication traffic elasticity is exploited. This property is used to implement a distributed price exchange mechanism, where event-triggers adapt their thresholds according to the resource constraints. I. MOTIVATION The recent increased interest in systems has led to various paradigm shifts in the digital control design. The systems under consideration usually consist of a multitude of smallscale integrated entities coupled through common computational and communication resources. The efficient usage of available resources is a prerequisite for the successful operation of such control systems. This fact has stimulated researchers to look for advanced sampling schemes beyond the conventional periodic sampling scheme to reduce resource consumption. It is well known that event-triggered control schemes achieve the same control performance as time-triggered control schemes, while reducing the number of samples significantly. Besides the necessity for an efficient use of resources, other non-functional requirements like self-configurability and adaptability need to be addressed within the design of cyber-physical systems. In the envisioned system, local entities, such as sensor nodes, are aware that a common resource is shared among them. Such awareness is reflected in the capability of adjusting the sampling rate adaptively to reduce resource needs, while maintaining a certain amount of performance. To be more specific, multiple independent control systems are considered here whose control loops are sharing a common resource. Because of the property of adaptation, a distributed algorithm for an event-triggered control system can be developed, where each subsystem adjusts its event-triggering mechanism to optimally meet a global resource constraint. The problem is formulated in the framework of Markov decision processes (MDPs) with an average cost criterion. Each subsystem is modelled as a discrete-time stochastic linear system with a quadratic control cost. The constraint is given by limiting the total average number of requests of all subsystems. Despite of the relaxation to an average resource constraint which ignores that more requests may occur than available, it turns out that the approximative approach serves as a good approximation for the problem with hard resource constraints. Inspired by distributed optimization and adaptive MDPs, a distributed sample-path based algorithm is proposed. In order to tackle the underlying optimization problem, a dual pricing mechanism forces each subsystem to adjust their eventtriggering thresholds accordingly. Therefore, the subsystems are able to set the optimal rate at which the resource is acquired by only having local information. Apart from the fact that the adaptation mechanism enables the distributed architecture, the local event-triggers are capable to adjust their thresholds according to runtime changes that are often found in real applications. These are for example given by adding or removing control loops during runtime, changes in the resource constraint, or changes in the local system parameters. II. PROBLEM STATEMENT Figure 1 depicts the cyber-physical system under consideration. It shows N independent control subsystems whose feedback loops are connected through a shared communication network. A control subsystem i consists of a process P , a controller C that is implemented at the actuator and a sensor S. The process P i is given by a controlled timehomogeneous Markov chain with state xk taking values in Ri and evolving by the following difference equation xik+1 = A ixik +B iuik + w i k, (1) where system parameters A ∈ Rii , B ∈ Rii ,control input uk and system noise w i k. At each time step k, the scheduler situated at the sensor station S decides, whether the controller C shall update its state. The ith scheduler is described by the variable δ k ∈ {0, 1}, where δ k = { 1 update xik is sent 0 otherwise The received signal at the ith controller, z k, is given by z k = { xik δ i k = 1 ∅ otherwise (2)