Brief Review of Control Theory

1. Brief Review of Control Theory E0397 Lecture. Some PPT slides based on Columbia Course by authors of the book Feedback Control of Computing Systems Hellerstein, Diao, Parekh, Tilbury

2. A Feedback Control System Error • Data • Reference input: objective • Control input: manipulated to affect output • Disturbance input: other factors that affect the target system • Transduced output: result of manipulation • Components • Target system: what is controlled • Controller: exercises control • Transducer: translates measured outputs Given target system, transducer, Control theory finds controller that adjusts control input to achieve given measured output in the presence of disturbances.

3. Control System Examples • AC temperature control • Controlled Variable: temperature • Control variables: fan speed, time for which compressor is on? • Car cruise control • Controlled variable: Car speed • Control variable: Angle of pushing of accelerator and brake

4. A Feedback Control System Road surface, gradient Random variation Embedded System that does cruise control Desired Speed Distance Travelled, Time taken Car Pedal Angle Setting Error Speed calculator Speed Actuator: The entity that implements the control Sensor: the entity that measures the output Odometer, Car clock Mechanical Device that Changes pedal angle Feedback!

5. A virtual machine as a FBCS Other Virtual Machines Doing some interfering work Random variation Controller: Process in Domain 0 that periodically changes Cap value of Guest Domain Desired Response time Response Time (of Applications) Xen VM (Guest Domain) With applications running CPU Cap Value Error E.g. Smoothing “filter” Smoothed Response time Actuator: The entity that implements the control Sensor: the entity that measures the output E.g. a “proxy” that Can measure Time taken Xen hypervisor scheduler Feedback!

6. A Feedback Control System Error • Data • Reference input: objective • Control input: manipulated to affect output • Disturbance input: other factors that affect the target system • Transduced output: result of manipulation • Components • Target system: what is controlled • Controller: exercises control • Transducer: translates measured outputs Given target system, transducer, Control theory finds controller that adjusts control input to achieve given measured output in the presence of disturbances.

7. Control System Goals • Reference Tracking • Ensure that measured output “follows” (tracks) a target desired level • Disturbance Rejection • Maintain measured output at a given stable level even in presence of “disturbances” • Optimization • No reference input may be given. Maintain measured output and control input at “optimal” levels • Minimize petrol consumption, maximize speed (not available in reality!!) • Minimize power consumed by CPU, maximize performance

8. Control System: Basic Working • “Control Loop” executed every T time units. • y(k): Value of measure at time instant k • u(k): Value of control input at time instant k • r(k): Value of “reference” at time instant k • e(k): Error between reference and measured value • Next value of control input, i.e. u(k+1) to be computed based on feedback: e(k) r(k) u(k) x(k) e(k) y(k)

9. Determining Change in Control Input • Change in u should depend on relationship between y and u • Relationship should be known • If known, then why feedback? Why not: • y = f(u) • (Car speed as function of accelerator pedal angle) •  u = f-1(y). For a desired reference, just calculate the control input required by using inverse. This is called feed forward Feed-forward

10. Feedforward (model based) • Problems: • Need accurate model • If model wrong, control input can be totally wrong • Imagine wrong model b/w accel. pedal and car speed • Does not take into account time-dependent behaviour • E.g. how long will system require to attain desired value • If car was at speed S1, we set cruise control to speed S2, if we set pedal angle to f-1(S2), will it immediately go to speed S2? • Cannot take care of “disturbances” • E.g. if environmental conditions change, model may not be applicable. • What if car was climbing a slope? (And angle was measured on flat ground?) Feedback addresses many of these problems.

11. Feedback Control • Rather than use only some offline model, also take into account the immediate measured effect of the value of your control variable • If car at S1, want to go to S2, S1 < S2 (positive error), pedal must be pushed further down. • We still need some idea of the relationship between “input” and “output” • Pedal should be pushed? Or released? If pushed – by what angle? • (Intuitively, for larger S2 - S1, angle of pushing in must be larger) • Another e.g. if e(k) = r(k) – y(k) is positive • Should u(k) increase or decrease? • If u is CPU frequency, y is response time. If e(k) is positive, u should? • Increase/Decrease by how much? • If increase/decrease too much: • Measured output will keep on oscillating above and below reference: Unstable behavior

12. …Feedback control • Since we want control to work in a timely manner (not only in “steady-state”), relationship should be time dependent • New value of y: y(k+1) will generally depend on y(k) and u(k) • Speed of car at this instant, will depend on what speed was at the beginning of previous interval, and the pedal angle in the previous interval • Similarly, queuing delay at this intv’l will depend on queuing delay of previous instant, and CPU frequency setting of previous interval

13. System Modeling • Need a model to capture this relationship over time • Can be first order (depending on 1 previous value), or higher order (more than 1 previous values). • First Order Linear model: y(k+1) = a y(k) + b u(k) • Can be done by “first principles” or “analytical modeling” • E.g. If y is response time of queueing system, u is service time, it may be possible to find an equation to relate y(k+1) to y(k) and u(k)

14. System Modeling • If “first principles” difficult: empirical model • Run controlled experiments on target system, record values of y(k), u(k), run regression models to find a and b. • How to do this correctly is a field called “system identification” • Main issues: • Relationship may not be actually be linear

15. m l { Number in System Non-linear relationships Response Time Utilization • E.g. number in queueing system vs buffer size • Relationship can be “linearized” in certain regions. Centre of such a region: Operating Point Operating point: Operating range • Values of N,K for which model applies. Offset value • Difference from operating point Linear Relationship is made between y and u defined as “offsets”

16. Control “Laws” • Once system model is made, need to design “control laws” • Control Laws relate error to new value of control input: • From e(k) determine value of u(k) • Controllers should have good SASO properties: • STABILITY - ACCURACY - SETTLING TIME - OVERSHOOT • Deeply mathematical theory for deriving these laws • Transfer functions, Poles • Estimating SASO properties of control laws • Using this understanding to design good control laws which have good SASO properties Cannot go into details

17. Unstable System Stability Accuracy Short settling Small overshoot Properties of Control Systems

18. Types of Controllers (Control Laws) • Proportional Control Law • u(k) = Kpe(k) (Kp is a constant, called controller gain) • Can be made stable, short settling time, less overshoot – but may be inaccurate • Integral Control Law • u(k) = u(k-1) + KI e(k) (KI is controller gain) • =u(k-2) + KI e(k-1) + KI e(k) • = u(0) +…+ KI * [ e(1) + e(2) + … +e(k) ] • Keeps reacting to previous errors also • Can show that this law will lead to zero error • Has longer settling time

19. Conclusion • Padala paper uses integral control law in most places. • It uses empirical methods for generating system model • Slides gave just enough required background. • Control theory is a huge field with lots of books (mainly used in EE, MECH, CHEM) • Read books/papers/if further interested