Chapter 1 Introduction Control system engineering is concerned - - PDF document

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Chapter 1 Introduction

Control system engineering is concerned with modifying the behavior

• f

dynamical systems to achieve certain pre-specified goals.

Control design loop:

• Modeling:

Physical plants ⇒ Mathematical models.

• Analysis:

Given performance specifications, check whether the specifications are satisfied; Robustness issues.

• Design:

Design controllers such that the closed-loop system sat- isfies the specifications.

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Control system design

• sensors
• sampling rates
• filter selection
• data storage
• physical
• black box
• system identification
• parameter estimation
• model validation

Data acquisition Modeling (hardware) “closing the loop” Plant -system “definition” information a priori constraints OK! design & analysis Controller implementation spec’s met? specification specification constraints yes no Control system ESAT–SCD–SISTA

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3 Motivating Examples Philips Glass Tube Manufacturing Process

Mandrel Pressure Mandrel Glass Tube Furnace

100 200 300 400 500 600 700 800

• 0.02

0.02 0.04 100 200 300 400 500 600 700 800

• 0.04
• 0.02

0.02 0.04 100 200 300 400 500 600 700 800

• 20
• 10

10 20 100 200 300 400 500 600 700 800

• 0.1
• 0.05

0.05 0.1

Melted Glass Quartz Sand

Input 1 Output 2

Thickness Drawing Speed

Input 2

Diameter

Output 1

Control design specifications: design a controller such that the wall thickness and diameter are as constant as possible.

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Modeling via system identification:

information a priori disturbance model Mathematical Model identification algorithms

• utputs

inputs simulated predicted

• utputs

disturbance model disturbances Plant ESAT–SCD–SISTA

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Plot with simulation and validation results:

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System identification toolbox

• Xmath GUI identification toolbox.
• Matlab Toolbox of system identification.
• RaPID (some algorithms developed at ESAT/SISTA)
• · · · · · ·

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After system identification, a 9th order linear discrete-time model with two inputs and two outputs was obtained: xk+1 = Axk + Buk + wk, yk = Cxk + Duk + vk. E

• wk

vk wT

l

vT

l

• =
• Q S

ST R

• δkl

where A, B, C, D are matrices, uk is the control input and wk and vk are process noise and measurement noise respectively. Control design results:

Filt 1 Filt 2 Setpoint Diameter Setpoint Thickness Static Ffwd

• Decoupling

Static PIID 2 PIID 1

Plant

Thickness Diameter

Controller =            two feedforward filters a static feedforward controller a static decoupling controller two PIID controllers

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The two PIIDs control the decoupled loops. Parameter tuning follows from a multi-objective optimization algo- rithm. The control design loop for this application :

State space model Kalman filter Hardware constraints Hardware implementation PIID/Static decoupling Multi-objective optimization LQG Static feedback Controller design ESAT–SCD–SISTA

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Quality improvement via control: Histogram of the measured diameter and thickness

• 0.2
• 0.1

0.1 0.2 100 200 300 Diameter - No Control

• 0.2
• 0.1

0.1 0.2 100 200 300 Diameter - PIID Controlled

• 0.02
• 0.01

0.01 0.02 100 200 300 400 Thickness - No Control

• 0.02
• 0.01

0.01 0.02 100 200 300 400 Thickness - PIID Controlled

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Boeing 747 aircraft control:

• utput

Velocity vector β x,u α φ,p θ,q y,v Aileron z,w ψ,r Rudder δr Elevator δe input

x, y, z = position coordinates φ = roll angle u, v, w = velocity coordinates θ = pitch angle p = roll rate ψ = yaw angle q = pitch rate β = slide-slip angle r = yaw rate α = angle of attack

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Control specifications for lateral control:

• Stability
• Damping ratio ≃ 0.5

Modeling: A lateral perturbation model in horizontal flight for a nom- inal forward speed of 774 ft/s at 40.000 ft can be de- rived based on physical laws. A 4th order, linear, single- input/single-output system was obtained : ˙ x = Ax + Bu, y = Cx + Du, where

A =       −0.0558 −0.9968 0.0802 0.0415 0.5980 −0.1150 −0.0318 −3.0500 0.3880 −0.4650 0.0805 1.0000       , B =       0.0073 −0.4750 0.1530       , C =

• 0 1 0 0
• , D = 0,

and x =

• β r p φ

T , u = δr, y = r.

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Control design: The controller is a proportional feedback from yaw rate to rudder, designed based on the root-locus method.

δr r e rc − Rudder servo Aircraft K

30

Im Re

stable unstable damping ratio = 0.5

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Analysis:

• For good pilot handling, the damping ratio of the sys-

tem should be around 0.5. The open loop system, with-

• ut control, has a damping ratio of 0.03, far less than

0.5. With the controller, the damping ratio is 0.35, near to 0.5 : a big improvement !

• Consider the initial-condition response for initial slide-

slip angle β0 = 1◦ :

5 10 15 20 25 30 −0.01 −0.005 0.005 0.01 0.015 Time (secs)

Response at r to a small initial β No feedback Yaw rate feedback yaw rate r rad/sec ESAT–SCD–SISTA

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Automobile control: CVT control

psc Output

Rsc

f

channel

• il pump

cylinder secondary valve secondary Cch

c

Csv

k

Csc

k

Rsv

f

Nengine Isc

m

Isv

m

pcl Gsv

r3

qsv

r3

Gsv

r2

qsv

r2

Gsv

r1

qsv

1

psv

1 Input

Csv

c1

Csv

c2

qsv vsv

m

vsc

m

Gsc

r

qsc

r

f sc

1

Csc

c

belt Nengine Nwheel primary pulley set secondary pulley set

Ck = spring constant Im = inertia R = friction resistor Gr = hydraulic impedance p = pressure q = oil flow vm = velocity of a mass f = force

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Control specifications (tracking problem):

• stability,
• no overshoot,
• steady state error on the step response ≤ 2%,
• rise time of the step response is ≤ 50ms.

Modeling: A 6th order single-input/single-output nonlinear system was obtained by physical (hydraulic and mechanical) mod- eling. Controller design: The control problem is basically a tracking problem:

− K P e psv

1

psc psc

ref

A PID controller is designed using optimal and robust con- trol methods.

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Analysis: It is required that there is no overshoot and that the rise time and steady state error of the step time response should be less than 50 ms and 2% respectively. Without feedback control: overshoot 20% with oscillation! With a PID controller: no overshoot, rise time is less than 30ms, no steady-state error. Step (from 40 bar to 20 bar) responses:

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 15 20 25 30 35 40 45 Time:sec. Pressure (bar) Step responses

no feedback control with a PID controller

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General Control Configuration

e-Ts sensor noise sensor uncertainty

• Plant

time delay process noise

Cpre Cff Ccon1 Ccon2 ∆ S G v w y u

• ref. e
• dist. d

e

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Systems and Models

• Linear - Nonlinear systems

A system L is linear ⇔ if input u1 yields output L(u1) and input u2 yields out- put L(u2), then : L(c1u1 + c2u2) = c1L(u1) + c2L(u2), c1, c2 ∈ R In this course, we only consider linear systems.

• Lumped - distributed parameter systems

Many physical phenomena are described mathemati- cally by partial differential equations (PDEs). Such sys- tems are called distributed parameter systems. Lumped parameter systems are systems which can be described by ordinary differential equations (ODEs). Example : – Diffusion equation → discretize in space – Heat equation In this course, we only consider lumped parameter sys- tems.

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• Time invariant - time varying systems

A system is time varying if one or more of the param- eters of the system may vary as a function of time,

• therwise, it is time invariant.

Example: Consider a system M d2y dt + F dy dt + Ky = u(t) If all parameters (M, F, K) are constant, it is time in-

• variant. Otherwise, if any of the parameters is a func-

tion of time, it is time varying. In this course, we only consider time-invariant systems.

• Continuous time - discrete time systems

A continuous time system is a system which describes the relationship between time continuous signals, and can be described by differential equations. A discrete system is a system which describes the relationship be- tween discrete signals, and can be described by differ- ence equations. In this course, we’ll consider systems both in continuous and discrete time.

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• Causal - a-causal systems

A system is called causal, if the output to time T de- pends only on the input up to time T, for every T,

• therwise it is called a-causal.

In this course, we only consider causal systems. This course: Linear Lumped parameters Time-invariant Causal systems in discrete time continuous time Realistic? In many real cases, YES!

• Industrial processes around an equilibrium point: glass
• ven, aircraft and CVT.
• Linearization of a nonlinear system → linear system.
• . . .

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System Modeling

System models are developed in two ways mainly:

• Physical modeling consists of applying various laws of

physics, chemistry, thermodynamics, . . . , to derive ODE or PDE models. It is modeling from “First Prin- ciples”. Example:

k m F = kx F = mg (Newton)

md2x dt2 = mg − kx.

• Empirical modeling or identification consists of devel-
• ping models from observed or collected data.

Experiments: − System identification, e.g.: glass oven − Parameter estimation

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