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Discrete (Digital) Feedback Control:Direct Design and Stability, etc.

ELEC 3004: Systems: Signals & ControlsDr. Surya Singh (with material from Dr. Paul Pounds and from Prof. Mark Cannon (Oxford Discrete Systems))

Lecture 21

[email protected]://robotics.itee.uq.edu.au/~elec3004/

© 2013 School of Information Technology and Electrical Engineering at The University of Queensland

May 17, 2013

Next Time in Linear Systems ….

ELEC 3004: Systems 17 May 2013 - 2

Week Date Lecture Title

127-FebIntroduction1-MarSystems Overview

26-MarSignals & Signal Models8-MarSystem Models

313-MarLinear Dynamical Systems15-MarSampling & Data Acquisition

420-MarTime Domain Analysis of Continuous Time Systems22-MarSystem Behaviour & Stability

527-MarSignal Representation29-MarHoliday

610-AprFrequency Response12-Aprz-Transform

717-AprNoise & Filtering19-AprAnalog Filters

824-AprDiscrete-Time Signals26-AprDiscrete-Time Systems

91-MayDigital Filters & IIR/FIR Systems3-MayFourier Transform & DTFT

108-MayIntroduction to Digital Control

10-MayStability of Digital Systems

11 15-MayPID & Computer Control17-MayApplications in Industry

1222-MayState-Space24-MayControllability & Observability

1329-MayInformation Theory/Communications & Review31-MaySummary and Course Review

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• Stability of Digital Systems

• Lead/Lag Compensators

• ZOH and Discretisation

Today:

• Stability

• Design tricks


Goals for the Week

(ZOH)=??? : What is it?

• Lathi

• Franklin, Powell, Workman

• Franklin, Powell, Emani-Naeini

• Dorf & Bishop

• Oxford Discrete Systems: (Mark Cannon)

• MIT 6.002 (Russ Tedrake)

• Matlab

Proof!


• Wikipedia

3

• Assume that the signal x(t) is zero for t<0, then the outputh(t) is related to x(t) as follows:

Zero-order-hold (ZOH)x(t) x(kT) h(t)Zero-order

HoldSampler

17 May 2013 -ELEC 3004: Systems 5

• Recall the Laplace Transforms ( ) of:

• Thus the of h(t) becomes:


Transfer function of Zero-order-hold (ZOH)

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… Continuing the of h(t) …

Thus, giving the transfer function as:


Transfer function of Zero-order-hold (ZOH)

Lead/lag compensation• Serve different purposes, but have a similar dynamic structure:

Note:Lead-lag compensators come from the days when control engineers cared about constructing controllers from networks of op amps using frequency-phase methods. These days pretty much everybody uses PID, but you should at least know what the heck they are in case someone asks.


5

Lead compensation: a < b

• Acts to decrease rise-time and overshoot– Zero draws poles to the left; adds phase-lead– Pole decreases noise

• Set a near desired ; set b at ~3 to 20x a

Img(s)

Re(s)

Faster than system dynamics

Slow open-loop plant dynamics

-a-b


Lag compensation: a > b

• Improves steady-state tracking– Near pole-zero cancellation; adds phase-lag– Doesn’t break dynamic response (too much)

• Set b near origin; set a at ~3 to 10x b

Img(s)

Re(s)

Very slow

plant dynamics

-a -b

Close to pole


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1. Derive the dynamic system model ODE2. Convert it to a continuous transfer function3. Design a continuous controller4. Convert the controller to the z-domain5. Implement difference equations in software

Emulation design process

Img(s)

Re(s)

Img(s)

Re(s)

Img(z)

Re(z)


Recap: Emulation Design Method 1: Tustin’s method• Tustin uses a trapezoidal integration approximation (compare

Euler’s rectangles)• Integral between two samples treated as a straight line:

1Taking the derivative, then z-transform yields:

which can be substituted into continuous models

1

x(tk)

x(tk+1)


7

Recap: Emulation Design Method 2: Matched pole-zero• If , why can’t we just make a direct substitution and go

home?

• Kind of!– Still an approximation– Produces quasi-causal system (hard to compute)– Fortunately, also very easy to calculate.


Matched pole-zeroThe process:

1. Replace continuous poles and zeros with discrete equivalents:

2. Scale the discrete system DC gain to match the continuous system DC gain

3. If the order of the denominator is higher than the enumerator, multiply the numerator by 1 until they are of equal order*

* This introduces an averaging effect like Tustin’s method


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Modified matched pole-zero• We’re prefer it if we didn’t require instant calculations to

produce timely outputs• Modify step 2 to leave the dynamic order of the numerator one

less than the denominator– Can work with slower sample times, and at higher frequencies


• Handy rules of thumb:– Sample rates can be as low as twice the system bandwidth

• but 5 to 10× for “stability”• 20 to 30 × for better performance

– A zero at 1 makes the discrete root locus pole behaviour more closely match the s-plane

– Beware “dirty derivatives”• ⁄ terms derived from sequential digital values are called ‘dirty

derivatives’ – these are especially sensitive to noise!• Employ actual velocity measurements when possible

Discrete design process


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Designing in the Purely Discrete…


Direct Design:Second Order Digital Systems

10


Response of 2nd order system [1/3]



11



• Response of a 2nd order system to increasing levels of damping:


2nd Order System Response

12

Damping and natural frequency

[Adapted from Franklin, Powell and Emami-Naeini]-1.0 -0.8 -0.6 -0.4 0-0.2 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

Re(z)

Img(z)

where 1

0.1

0.2

0.3

0.4

0.50.6

0.7

0.8

0.9

235

710

910

25

1

25

0

310

5

10

20


• Poles inside the unit circleare stable

• Poles outside the unit circleunstable

• Poles on the unit circleare oscillatory

• Real poles at 0 < z < 1give exponential response

• Higher frequency ofoscillation for larger

• Lower apparent dampingfor larer and r


Pole positions in the z-plane

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Characterizing the step response:


2nd Order System Specifications

• Rise time (10% 90%):

• Overshoot:

• Settling time (to 1%):

• Steady state error to unit step: ess

• Phase margin:

Characterizing the step response:


2nd Order System Specifications

• Rise time (10% 90%) & Overshoot: tr, Mp ζ, ω0 : Locations of dominant poles

• Settling time (to 1%): ts radius of poles:

• Steady state error to unit step: ess final value theorem

14

Design a controller for a system with:• A continuous transfer function:• A discrete ZOH sampler • Sampling time (Ts): Ts= 1s• Controller:

The closed loop system is required to have:• Mp < 16%• ts < 10 s• ess < 1


Ex: System Specifications Control Design [1/4]



15





16


LTID Stability


Characteristic roots location and the corresponding characteristic modes [1/2]

17


Characteristic roots location and the corresponding characteristic modes [2/2]


S-Plane to z-Plane [1/2]

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S-Plane to z-Plane [2/2]


Relationship with s-plane poles and z-plane transforms

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• For small T:

• Hence, the unit circle under the map from z to s-plane becomes:


Fast sampling revisited

Specification bounds• Recall in the continuous domain, response performance

metrics map to the s-plane:

Img(s)

Re(s)

4.6

Img(s)

Re(s)

sin

Img(s)

Re(s)

1.8


20

• These map to the discrete domain:

In practice, you’d use Matlab to plot these, and check that the spec is satisfied

Discrete bounds

Img(z)

Re(z)

Img(z)

Re(z)

Img(z)

Re(z)


• This code was shown “by accident” in class during a MATLAB demo. No promises are made for it. It alone is not a solution to Question 7.

%% Input System Model G

numg=5; deng=[1 20 0]; sysg=tf(numg, deng);

%% Approximate the ZOH (1-e^-sT)/(s)

[nd, dd]=pade(1,2); %pade gives us the "hold" or -e^-sT of a ZOH

sysp=tf(nd, dd); sysi=tf([1],[1,0]); %Now we need the "1/s" portion

sys1=series(1-sysp, sysi); % Approximation as a series

%% Open loop response

syso=series(sys1, sysg); % computer the open loop G with the ZOH

sys=feedback(syso,1); % Computer the unity feedback response

step(sys) % Display the step response


Example Code:

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• Final Exam:– 15 Questions (60% Short Answer, 40% Regular Problems)– 3 Hours– Closed-book– Took tutor ~90min to complete– Equation sheet will be provided (in addition to your own)

[See Prac Final – Coming out next week]– Yes, it has an unexpected twist at the end, but you’ll like it.

Saturday, June 15 at 9:30 AM (sorry!)


Announcements:

!

Next Time in Linear Systems ….


Week Date Lecture Title

127-FebIntroduction1-MarSystems Overview

26-MarSignals & Signal Models8-MarSystem Models

313-MarLinear Dynamical Systems15-MarSampling & Data Acquisition

420-MarTime Domain Analysis of Continuous Time Systems22-MarSystem Behaviour & Stability

527-MarSignal Representation29-MarHoliday

610-AprFrequency Response12-Aprz-Transform

717-AprNoise & Filtering19-AprAnalog Filters

824-AprDiscrete-Time Signals26-AprDiscrete-Time Systems

91-MayDigital Filters & IIR/FIR Systems3-MayFourier Transform & DTFT

108-MayIntroduction to Digital Control

10-MayStability of Digital Systems

1115-MayPID & Computer Control17-MayApplications in Industry

12 22-MayState-Space24-MayControllability & Observability

1329-MayInformation Theory/Communications & Review31-MaySummary and Course Review

Next Time in Linear Systemsrobotics.itee.uq.edu.au/~elec3004/2013/lectures/... · L A æ Í, why...

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