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Introduction to Control Theory COMP417 Instructor: Philippe Giguère.

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1 Introduction to Control Theory COMP417 Instructor: Philippe Giguère

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

3 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.

4 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

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

6 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.

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

8 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…

9 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

10 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

11 Classic Feedback Diagram

12 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

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

14 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?

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

16 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

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

18 Laplace Transform of Classic Feedback Sysem

19 Key Transfer Functions

20 First Order System

21 Response of System Impulse Step Exponential

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

23 Second Order System

24 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

25 Basic Controller Function

26 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

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

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

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

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

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

32 PID final control

33 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:

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

35 Multiple Loops Inner loops are generally faster that outer loop.

36 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.

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

38 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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