Embedded and networked control systems often rely on the periodic sampling and transmission of data. This periodic data abstraction is advantageous from the design standpoint. It permits real-time system engineers and control system engineers to pursue their design objectives in relative isolation from each other. While this so-called separation-of-concerns has proven advantageous from a designer's perspective, it does not necessarily lead to cost effective implementations of the control system. By separating the concerns of the control engineer from the real-time system engineer, we force each designer to adopt a conservative viewpoint that may lead to unnecessary over-provisioning in the system implementation and hence to higher system costs. When we apply these traditional design principles to extremely large-scale systems, then the cost of enforcing the periodic data abstraction may become prohibitive.
As a result of these scaling issues, there has been recent interest in developing co-design frameworks where the concerns of real-time systems and control systems engineers are treated in a unified manner. One of the first statements of the co-design problem was given by Seto et al. [11]. This work presented co-design as an optimization problem that sought to minimize a traditional quadratic integral measure of control cost subject to task schedulability constraints. Seto’s problem was an off-line design approach since the optimization problem was solved prior to system deployment. Since that time, a number of other co-design approaches have been suggested. A list of such methods will be found in a paper by K-E Arzen [2]. This list includes feedback modification of task attributes [5] [9] [6] [7] , anytime controllers [4][8] , and event-triggered sampling[1][12][13] .
This project focuses on the event-triggered approach to real-time control system co-design. Event-triggered controllers adapt the real-time system’s task period directly in response to the application performance [1]. Under event-triggering the control task is only executed when the application’s error signal exceeds a specified threshold. Ostensibly, this error provides a measure of how ”valuable” the current state is to the overall system’s closed-loop behavior. In this way the real-time system is only used when it is essential for the system’s performance. Since the system’s state is always changing, this approach generates a sporadic sequence of control task invocations. In general, the hope is that the average rate of this sporadic task will be much lower than the rate of a comparable periodic task.
There is, in fact, ample experimental evidence to support the assertion that event-triggered feedback improves overall control system performance while reducing the real-time system’s use of computational resources. Let's first review a results from [3] shown in the plot below. This paper considers a controlled scalar diffusion process of the form,
where a is real constant and w is a standard Brownian motion. The signal u is the control signal generated by a full-state controller. This control is computed in either a periodic or event-triggered manner. Under event-triggering, the control is updated whenever the state magnitude, |x|, exceeds a specified threshold. The performance of the system is characterized by the steady-state variance of the system state. The variance of the periodically triggered system is denoted as VR whereas the variance of the event-triggered system is denoted as VL. The plot below shows the ratio VR/VL as a function of the mean sampling period. Note that for all choices of the system constant, a, this performance ratio is greater than one, thereby showing that the event-triggered system has better performance than periodically triggered systems operating at the same mean sampling period.
Another example is found in [10] as shown in the plot below. This paper considers the control of a linear plant under a PID controller. The controller is discretized at a specified sampling rate and the results of the system response are shown in the top plot. This trace is then compared against the response obtained by using an event-triggered version of the PID controller. The event-triggered control is computed when the gap between the current system state and the last sampled system state exceeds a specified threshold, eT ,
In the above equation rj denotes the jth consecutive time when the state was sampled. The middle plot is the associated trace when the threshold eT is chosen to match the peak error of the periodically triggered system. This means that these first two plots are comparing behaviors that have similar performance levels. The bottom plot shows the number of samples that were generated by the periodically triggered (time-driven) PID control versus the event-triggered PID control. As can be seen from this plot, the number of event-triggered samples is smaller than the time-driven control. Moreover, as the system approaches its equilibrium point, we see that the number of samples begins to level off, thereby suggesting that as the ”information” content within the error signal decreases, the controller needs to be invoked less often.
The first example shown above suggests an event-triggered system will perform better than a periodically-triggered systems with similar computational usage. The second example suggests an event- triggered system will use fewer computational resources than a periodically triggered system with similar performance levels. These results, unfortunately, are only empirical in nature. One objective of this project is to analytically characterize the relationship between event-triggered control system performance and computational effort in a way that can lead to better embedded and networked control systems.