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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! speed w no torque at max speed stall torque torque t

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**Closing the loop Let’s 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 Watt’s “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 (mW) 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**

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

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**Control System Goals Regulation Tracking Optimization thermostat**

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(s) Impulse d 1 Step Ramp Exponential Sine 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 (2nd O), P-type**

Kp = 20 Kp = 50 Kp = 500 Kp = 200

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**Steady-state Errors, P-type**

Kp = 50 Kp = 200

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**Control Performance, PI - type**

Kp = 100 Ki = 50 Ki = 200

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**You’ve been integrated...**

Kp = 100 instability & oscillation

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**Control Performance, PID-type**

Kp = 100 Kd = 5 Kd = 2 Ki = 200 Kd = 10 Kd = 20

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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. Ziegler-Nichols Rules for Tuning PID Controller: 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 for P control for PI control for PID control Kp = 0.5 K Kp = K Kp = 0.6 K Ziegler-Nichols Tuning for second or higher order systems Ki = 1.2 / P Ki = 2.0 / P Kd = P / 8.0

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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: Why:**

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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