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Introduction to Control Theory COMP417 Instructor: Philippe Giguère

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Actuators Earlier lecture presented actuators –Electrical motor Now how do we use them in an intelligent manner?

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Open-loop? We want to spin a motor at a given angular velocity. We can apply a fixed voltage to it, and never check to see if it is rotating properly. Called open loop.

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Open-loop? What if there is a changing load on the motor? –Our output velocity will change! torque speed no torque at max speed stall torque

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Closing the loop Lets measure the actual angular velocities. Now we can compensate for changes in load by feeding back some information.

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Control Theory Roots in another science: Cybernetics Cybernetics is the study of feedback and derived concepts such as communication and control in living organisms, machines and organisations. Expression was coined by Norbert Weiner in 1948.

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Early Example of Feedback System James Watts Centrifugal Governor in 1788. Regulates the steam engine speed.

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Other Simple Examples Body temperature regulation –If cold, shiver (muscles produce heat) –If hot, sweat (evaporation takes away heat) Maintaining social peace –If a crime is found (sensor), the guilty party is punished (actuator). Cruise control in cars –You set a speed, CC will increase fuel intake uphill, and decrease it downhill. Etc…

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Why Control Theory Systematic approach to analysis and design –Transient response –Consider sampling times, control frequency –Taxonomy of basic controllers (PID, open-loop, Model-based, Feedforward…) –Select controller based on desired characteristics Predict system response to some input –Speed of response (e.g., adjust to workload changes) –Oscillations (variability) Assessing stability of system

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Characteristics of Feedback System Power amplification –Compare neural signal power ( W) vs muscle power output (tens of W) –Means it is an active system, as opposed to passive. Actuator Feedback –measurement (sensor) Error signal Controller

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Classic Feedback Diagram

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

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

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Approaches to System Modeling First Principles –Based on known laws. –Physics, Queueing theory. –Difficult to do for complex systems. Experimental (System Identification) –Requires collecting data –Is there a good training set?

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Common Laplace Transfom Name f(t)f(t)F(s)F(s) Impulse Step Ramp Exponential Sine 1 Damped Sine

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Transfer Function Definition: H(s) = Y(s) / X(s) Relates the output of a linear system to its input. Describes how a linear system responds to an impulse… called impulse response –(remember! Convolving with impulse yields the same thing. Hence probing a system with and impulse is finding its transfer function…) All linear operations allowed –Scaling, addition, multiplication

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Block Diagram Expresses flows and relationships between elements in system. Blocks may recursively be systems. Rules –Cascaded elements: convolution. –Summation and difference elements

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Laplace Transform of Classic Feedback Sysem

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Key Transfer Functions

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First Order System

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Response of System Impulse Step Exponential

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Steady-State vs. Transient Step Response illustrates how a system response can be decomposed into two components: –Steady-state part: –Transient

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Second Order System

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Second Order Response Typical response to step input is: overshoot -- % of final value exceeded at first oscillation rise time -- time to span from 10% to 90% of the final value settling time -- time to reach within 2% or 5% of the final value

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Basic Controller Function

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Effect of Controller Functions Proportional Action –Simplest Controller Function Integral Action –Eliminates steady-state error –Can cause oscillations Derivative Action (rate control) –Effective in transient periods –Provides faster response (higher sensitivity) –Never used alone

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Control Performance (2 nd O), P-type K p = 20 K p = 200 K p = 50 K p = 500

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Steady-state Errors, P-type K p = 200K p = 50

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Control Performance, PI - type K p = 100 K i = 50K i = 200

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Youve been integrated... K p = 100 instability & oscillation

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Control Performance, PID-type K p = 100 K i = 200K d = 2 K d = 10K d = 20 K d = 5

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PID final control

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PID Tuning How to get the PID parameter values ? (1) If we know the transfer function, analytical methods can be used (e.g., root-locus method) to meet the transient and steady-state specs. (2) When the system dynamics are not precisely known, we must resort to experimental approaches. Using only Proportional control, turn up the gain until the system oscillates w/o dying down, i.e., is marginally stable. Assume that K and P are the resulting gain and oscillation period, respectively. Then, use Ziegler-Nichols Tuning for second or higher order systems for P controlfor PI controlfor PID control K p = 0.6 K K i = 2.0 / P K d = P / 8.0 K p = 0.45 K K i = 1.2 / P K p = 0.5 K Ziegler-Nichols Rules for Tuning PID Controller:

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Layered Approach to Control Robotic systems often have a layered approach to control: movie

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Multiple Loops Inner loops are generally faster that outer loop.

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Advanced Control Topics Adaptive Control –Controller changes over time (adapts). MIMO Control –Multiple inputs and/or outputs. Predictive Control –You measure disturbance and react before measuring change in system output. Optimal Control –Controller minimizes a cost function of error and control energy. Nonlinear systems –Neuro-fuzzy control. –Challenging to derive analytic results.

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RC Servo Is a controller + actuator in a small package. Not expensive 10$-100$. Those in 417 will use some of these.

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PWM PWM -- pulse width modulation Duty cycle: –The ratio of the On time and the Off time in one cycle. –Determines the fractional amount of full power delivered to the motor. Why: –Transistor acts as a resistor when partially on. –Leads to energy being wasted heat.

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