Review last lectures

centrifugal (fly-ball) governor 1788 Picture shows an operation principle of the fly-ball (centrifugal) speed governor developed by James Watt.

simple feedback system heat transfer Qout desired temperature _ room temperature Qin S thermostat switch air con office room +

closed-loop (feedback) system error or actuating signal input or reference disturbance summing junction or comparator plant output or controlled variable control signal + S input filter (transducer) controller actuator process _ sensor or output transducer sensor noise

closed-loop system advantages disadvantages high accuracy not sensitive on disturbance controllable transient response controllable steady state error more complex more expensive possibility of instability recalibration needed need for output measurement

open-loop system input or reference disturbance plant output or controlled variable control signal input filter (transducer) controller actuator process

open-loop system advantages disadvantages simple construction ease of maintenance less expensive no stability problem no need for output measurement disturbances cause errors changes in calibration cause errors output may differ from what is desired recalibration needed

Example 1: Liquid Level System (input flow) Goal: Design the input valve control to maintain a constant height regardless of the setting of the output valve Input valve control float This can be applied to computer systems as a fluid model of queues: inflow is in work/sec; volume represents work in system; R relates to service rate. (resistance) (height) (output flow) (volume) Output valve

Example 2: Admission Control Goal: Design the controller to maintain a constant queue length regardless of the workload Users Administrator Controller RPCs Sensor Server Reference value Queue Length Tuning control Log Jlh: May want a slide that explains how this controller works?

Why Control Theory Systematic approach to analysis and design Transient response Consider sampling times, control frequency Taxonomy of basic controls Select controller based on desired characteristics Predict system response to some input Speed of response (e.g., adjust to workload changes) Oscillations (variability) Approaches to assessing stability and limit cycles Jlh: Some edits on this slide

Controller Design Methodology Start System Modeling Controller Design Block diagram construction Controller Evaluation Transfer function formulation and validation Objective achieved? Y Stop Model Ok? N Y N

Control System Goals Regulation Tracking Optimization thermostat, target service levels Tracking robot movement, adjust TCP window to network bandwidth Optimization best mix of chemicals, minimize response times

Approaches to System Modelling First Principles Based on known laws Physics, Queuing theory Difficult to do for complex systems Experimental (System ID) Statistical/data-driven models Requires data Is there a good “training set”?

Laplace transforms The Laplace transform of a signal f(t) is defined as The Laplace transform is an integral transform that changes a function of t to a function of a complex variable s = s + jw The inverse Laplace transform changes the function of s back to a function of t

Laplace transforms of basic functions f(t) F(s) Unit impulse d (t) Unit step u (t) Exponential e−at Sine wave sin(t) Cosine wave cos(t) Polynomial tn e−at x(t) Note: f(t) = 0, t < 0

Example f (t) Laplace transform: signal 1 t w w s s

Properties of Laplace transforms Linear operator: if and then for any two signals f1(t) and f2(t) and any two constants a1 and a2 Time delay:

Properties of Laplace transforms cont. Laplace transforms of derivatives: if then

Properties of Laplace transforms cont. Laplace transform of integrals: The Laplace transform changes differential equations in t into arithmetic equations in s

Laplace Transform Properties

Using Laplace transforms to solve ODEs The Laplace transform can be used to solve differential equations Method: Transform the differential equation into the ‘Laplace domain’ (equation in t → equation in s) Rearrange to get the solution Transform the solution back from the Laplace domain to the time domain (signal in s → signal in t) Usually the Laplace transform (step 1) and the inverse transform (step 3) are done using a Table of Laplace transforms

Example m = 1 kg k = 2 N/m b = 3 Ns/m Use Laplace transforms to find the unforced response of a spring-mass-damper with initial conditions x m = 1 kg k = 2 N/m b = 3 Ns/m k m f b x Equation of motion m f Free body diagram

Example – solution Take Laplace transform of both sides of equation of motion: Equation of motion in Laplace domain is

The system can be in motion if Rearrange: External force Initial conditions The system can be in motion if An external force is applied The initial conditions are not an equilibrium state (not zero) Apply initial conditions:

Partial fraction expansion: Use tables to find inverse Laplace transform System response (in time domain) is x(t) x0 t

Partial fraction expansion A partial fraction expansion can be used to find the inverse transform of This can be expanded as Note that So

Example - Partial fraction expansion Find the partial fraction expansion of This can be expanded as So

Other examples 1. 2. 3.

Insights from Laplace Transforms What the Laplace Transform says about f(t) Value of f(0) Initial value theorem Does f(t) converge to a finite value? Poles of F(s) Does f(t) oscillate? Value of f(t) at steady state (if it converges) Limiting value of F(s) as s ---> 0

Transfer Function Definition H(s) = Y(s) / X(s) Relates the output of a linear system (or component) to its input Describes how a linear system responds to an impulse All linear operations allowed Scaling, addition, multiplication X(s) H(s) Y(s)

Block Diagrams Pictorially expresses flows and relationships between elements in system Blocks may recursively be systems Rules Cascaded (non-loading) elements: convolution Summation and difference elements Can simplify

Block Diagram of System Disturbance Reference Value + S Controller S Plant – Transducer

Combining Blocks Reference Value + S Combined Block – Transducer

Key Transfer Functions Reference + S Controller Plant – Jlh: Can we make the diagram bigger and the title smaller? Transducer

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