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centrifugal (fly-ball) governor 1788 Picture shows an operation principle of the fly-ball (centrifugal) speed 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 _ Qin thermostat switch air con simple feedback system heat transfer Qout desired temperature _ Qin thermostat switch air con + S room temperature office room

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

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

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

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

Example 1: Liquid Level System Goal: Design the input valve control to maintain a Example 1: Liquid Level System Goal: Design the input valve control to maintain a constant height regardless of the setting of the output valve (input flow) Input valve control float (resistance) (height) (output flow) (volume) Output valve

Example 2: Admission Control Users Goal: Design the controller to maintain a constant queue Example 2: Admission Control Users Goal: Design the controller to maintain a constant queue length regardless of the workload RPCs Reference value Administrator Controller Tuning control Sensor Server Queue Length Server Log

Why Control Theory n Systematic approach to analysis and design • • n Transient Why Control Theory n Systematic approach to analysis and design • • n 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) n Approaches to assessing stability and limit cycles

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

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

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

Laplace transforms n n n The Laplace transform of a signal f(t) is defined Laplace transforms n n n 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 ) Unit impulse d (t ) Unit Laplace transforms of basic functions f (t ) 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 F (s )

Example f (t) Laplace transform: signal 1 t 0 w s Example f (t) Laplace transform: signal 1 t 0 w s

Properties of Laplace transforms n Linear operator: if and then for any two signals Properties of Laplace transforms n Linear operator: if and then for any two signals n f 1(t) and f 2(t) and any two constants a 1 and a 2 Time delay:

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

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

Laplace Transform Properties Laplace Transform Properties

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

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

n Rearrange: External force n Initial conditions The system can be in motion if n Rearrange: External force n Initial conditions The system can be in motion if 1. An external force is applied 2. The initial conditions are not an equilibrium state (not zero) n Apply initial conditions:

n n n Partial fraction expansion: Use tables to find inverse Laplace transform System n n n Partial fraction expansion: Use tables to find inverse Laplace transform System response (in time domain) is x(t) x 0 0 t

Partial fraction expansion n n A partial fraction expansion can be used to find Partial fraction expansion n n 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 n Find the partial fraction expansion of n This Example - Partial fraction expansion n Find the partial fraction expansion of n This can be expanded as n So

Other examples 1. 2. 3. Other examples 1. 2. 3.

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

Transfer Function n Definition • H(s) = Y(s) / X(s) n n n X(s) Transfer Function n Definition • H(s) = Y(s) / X(s) n n n X(s) H(s) Y(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

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

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

Combining Blocks Reference Value + S Combined Block – Transducer Combining Blocks Reference Value + S Combined Block – Transducer

Key Transfer Functions Reference S + Plant Controller – Transducer Key Transfer Functions Reference S + Plant Controller – Transducer

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