Feedback Control Models

In this thesis we use principles of the branch of Control Theory concerned with Feedback control systems to reason quantitatively about a system’s ability to meet the operational goals and environmental constraints (policies), which govern its operation.

We use feedback control models as one of our evaluation tools because of the framework it provides for realizing predictable systems – systems where the expected response of the system to changes (in the system and/or in its environment) can be characterized and/or evaluated quantitatively. Further, feedback control can also be used to realize robust adaptive systems – systems that automatically adjust to reject disturbances and accommodate noise while continuing to meet their operational goals.

We posit that predictable systems are easier to manage than unpredictable systems, and as a result predictability affects the Serviceability characteristics of a system concerned with meeting objectives in the presence of failures and/or remediations. In this section we identify the properties of feedback control systems that can be used to quantify facets of system Serviceability in the development of our RAS models.

Control Theory and feedback control has been widely studied and employed in other engineering disciplines including, but not limited to: mechanical engineering and electrical engineering. Further, despite the stochastic nature of computing systems, feedback control has been applied to their analysis and design with encouraging results [90,145,45,196]2. We now provide some background on Feedback control, highlighting the properties that make it suitable for use in constructing RAS models.

Control Theory is concerned with the study of dynamical systems and is commonly used to achieve one or more of the following objectives: regulatory control, disturbance rejection and optimization.

2_{One approach for dealing with system stochastics involves building on results from Queuing Theory[90]}

Regulatory control ensures that some measurable characteristic (measured output) of the (target) system is equal to or near a desired/specified reference value (reference input). Disturbance rejection ensures that disturbances acting on the system do not significantly affect its measured output. And finally, optimization is concerned with obtaining the best value of the measured output of the system. In our development of RAS models we are interested primarily in regulatory control.

There are two main classes of control systems, open-loop (feedforward) control systems (Figure4.2) and closed-loop (feedback) control systems (Figure 4.3). Before discussing the differences between feedforward and feedback control systems we first describe the elements typically found in control systems:

Figure 4.2: Block diagram of feedforward control [90]

Figure 4.3: Block diagram of a feedback control system [90]

• The Target System – the system to be controlled.

• Measured output – one or more measurable characteristics of the target system. • Reference input – desired value(s) of the measured output.

• Control input – one or more (dynamically adjustable) parameters that affect the behavior of the target system.

• The Controller – manipulates/determines the setting of the control input to achieve the reference input.

• The Transducer – transforms the measured output such that it can be compared with the reference input and/or used by the Controller. Examples include moving-average filters and unit conversions.

• Control error – difference between the measured output and the reference input. • Disturbance input – changes that affect the way the control input influences the

measured output.

• Noise input – any effect that changes the measured output of the target system. The main difference between feedforward control and feedback control is the role of the measured output in each of these control systems and its implications for controller design. Feedforward controllers use the reference input (and sometimes the disturbance input) to determine the setting of the control input needed to achieve the desired measured output. Unlike feedback controllers, they do not use the measured output to adjust the control input. As a result, feedforward control is more suitable for systems where the control input is a deterministic function of the reference and/or disturbance input and an accurate model of the system that is robust to changes in the system and its operating environment is available or can be constructed [90]. These properties of feedforward control systems, while making them less complex to design than feedback control systems, also make them less flexible/adaptive.

In addition to being more flexible than feedforward control systems, feedback control systems and the design principles used to realize them can be used to develop systems that exhibit four desirable properties – referred to as SASO properties – and analyze whether

systems exhibit any or all of these properties:

1. Stability – a stable system produces bounded output for any bounded input (these con- trol systems are sometimes referred to as being Bounded Input Bounded Output/BIBO stable).

2. Accuracy – the measured output of an accurate control system converges or becomes sufficiently close to the reference input (small steady-state control error).

3. Short settling times – a control system with short settling times quickly converges to its steady state value.

4. Avoids overshoot – a control system that avoids overshoot allows changes to the control input to made while maintaining its measured output.

The flexible/adaptive nature of feedback control systems and the ability to analyze the SASO properties of such systems provides a framework for codifying operational policies (internally and externally visible service level objectives, environmental constraints, SLAs etc.) for a computing system and reasoning about the ability of the system to meet these goals in the presence of failures and/or remediation activities.