Read Microsoft PowerPoint - DCS lecture 02 Introduction to Digital Control System ktsai v1.6 20070921.pptx text version

Digital Control Systems g y Introduction

https://ceiba.ntu.edu.tw/961_dcs_ee Professor Kuen-Yu Tsai Yu-Chen K Y Ch Kung Dept. of Electrical Engineering, National Taiwan University Autumn 2007

Outline

· Introduction to Control Engineering · Why Digital Control? · Major Differences in DCS versus CCS

­ Sampler ­ D/A and Hold ­ Difference Equations

DCS aut '07 20070921

Prof. KY Tsai/YCKung/NTUEE

2

What is Control Engineering?

· "Control is the process of causing a system p g y variable to conform to some desired value, called a reference value ."1 · "A control system is a system in which a desired effect (or objective) is achieved by operating on the various manipulable inputs to the plant until the output which is a measure of the desired effect output, effect, falls within in an acceptable range of values." 2

1. G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamics Systems, 3rd ed., Addison Wesley, 1994 2. F. Jay, IEEE Standard Dictionary of Electrical and Electronics Terms. IEEE, 1977

DCS aut '07 20070921

Prof. KY Tsai/YCKung/NTUEE

3

What is Control Engineering? (cont.) (cont )

· Any engineering/math. problem associated with constraining or minimizing the difference (error) between desired output yd and system output y w.r.t. some decision ( (control/input) variable(s) p ) () · e.g. Least-square problems: minU ||Yd-AU||2, where Y=AU is a vector of system output, U is a vector of decision (input) variables. It can be a rocket propulsion problem, or variables problem DVD read-head positioning prob.; an estimation prob. for wireless communication, or wafer alignment purposes... · With this general sense, most of system engineering sense problems fall within the scope of control engineering!!

­ e.g. signal processing, communication, circuit/photomask design... ­ There are always jobs for control engineers to do!! ­ I spent 9 years to realize this.

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE 4

Two major approaches in control

· Feedforward (FF) ( ) · Feedback (FB)

­ UFF is not a direct function of system output (y) ­ UFB is a function of system output (y)

· Most control textbooks only talk about FB and the associated stability properties. However both FF i t d t bilit ti H b th and FB should be considered.

­ FF is not effective if there is un-measurable un measurable disturbances or if the system needs stabilization. ­ However, sometimes FF is more effective in performance enhancement. enhancement ­ Try to apply both FF and FB in this DCS course!!

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE 5

FF/FB

· The feedback method corrects the current or next control input based on the current and past measurement of the output. · The feedforward method does not use the output measurement directly. Instead, it predicts the best control input based on some assumptions, e.g. the prior knowledge of reference input and deterministic disturbances. disturbances

r : reference input y : output G : model uncertainty G : plant K : robust feedback controllers F : feedforward filter dm : measurable disturbance dum : unmeasurable disturbances

Prof. Kuen-Yu Tsai/NTUEE 6

Prof. Tsai PhD Dissertation, 2003, , , Stanford DCS aut '07 20070921

Outline

· Introduction to Control Engineering · Why Digital Control? · Major Differences in DCS and CCS

­ Sampler ­ D/A and Hold ­ Difference Equations

DCS aut '07 20070921

Prof. KY Tsai/YCKung/NTUEE

7

Continuous Control Systems

· The continuous controllers are built using analog electronics such as resistor, capacitors, and operational amplifiers. amplifiers

Continuous controller

r (t )

e(t )

+

-

y (t )

D( s)

u (t )

Plant

G(s)

Sensor

y (t )

S (s)

[FPW] pp.58

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE 8

Digital Control Systems

· The digital controllers are built using digital computers (or microprocessors) with the necessary input/output hardware to implement the controllers. controllers

Digital controller

r (t )

T

+

r (kT )

e(kT )

-

Difference equations

u (kT )

D/A and hold

u (t )

Plant

G(s)

y (t )

Clock

A/D

y (t )

Sensor

T

S (s)

[ [FPW] pp ] pp.58

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE 9

Notation

· Continous signal: r(t) or r(kT) when t=kT · Discrete sequence: r[k] = r(k·T)

DCS aut '07 20070921

Prof. KY Tsai/YCKung/NTUEE

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Why use digital computers for control?

· Flexibility

­ Easy to implement complex/nonlinear algorithms

· Versus analog computer/analog OPAs

­ Easy to change controllers without rewiring/hardware redesign ­ Possible to change plants without rewiring/hardware redesign

· Robustness

­ Digital circuits are less sensitive to EMI and noise ­ Digital filter coefficients are more accurate

· How accurate are the R, L, C in OPA circuits?

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE 11

Must we use digital computers for control?

· NO! · Cost:

­ Components of analog OPA circuits are often cheaper than digital controllers (DSP+AD/DA)

· Speed:

­ Some systems such as MEMS or optical communications need very high (e.g. >10MHz) sampling rate ­ High speed DSP+AD/DA are usually quite expensive

· Size:

­ Circuit area with analog controllers can be smaller when there are many (e.g. >1000) input/output channels

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE 12

Outline

· Introduction to Control Engineering · Why Digital Control? · Major Differences in DCS and CCS

­ Sampler ­ D/A and Hold ­ Difference Equations

DCS aut '07 20070921

Prof. KY Tsai/YCKung/NTUEE

13

Difference Between DCS and CCS

discrete D( s ) Difference equations approximation

Digital Di i l controller ll

r (t )

D/ A u (kT ) u (t ) hold

T

+

r (kT )

e(kT )

Difference equations

u (kT )

-

D/A and hold

u (t )

Plant

G (s)

y (t )

Clock

A/D

y (t )

Sensor

T

S ( s)

A/ D y (t ) y (kT ) p g sampling

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE 14

Sampler

· A sampler is basically a switch that closes every T seconds for one instant of time.

r * (t ) = k =- r (t ) (t - k T )

+

· Define [k] as discrete unit impulse [ ] p

1 , k = 0 [k ] = 0 , k 0

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE 15

Sampled Data (lecture ?)

· The series for r*(t) is a string of impulses starting at t=0, spaced at T seconds, and of amplitude r(kT).

r (t )

sampler l

r * (t )

sampling period : T

T 2T 3T 4T 5T 6T 7T

r (t ) =

*

k =-

r (kT ) (t - kT )

16

DCS aut '07 20070921

Prof. KY Tsai/YCKung/NTUEE

Sample and hold Sample-and-hold circuit

· In order to give the computer an accurate representation of the signal exactly at the vin sampling instant kT kT, the A/D converter is v typically presented by a sample-and-hold circuit.

R

C

R

S

vout

T

H

H

T

T

H

H = Hold; S in position2 T = Track; S in position1 p

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE

t

17

[FPW] pp.157

Difference Between DCS and CCS

discrete D( s ) Difference equations approximation

Digital controller

r (t )

D/ A u (kT ) u (t ) hold

T

+

r (kT )

e(kT )

Difference equations

u (kT )

-

D/A and hold

u (t )

Plant

G (s)

y (t )

Clock

A/D

y (t )

Sensor

T

S ( s)

A/ D y (t ) y (kT ) sampling li

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE 18

D/A and Hold

· A D/A converter serves as a device that converts the discrete signal p*(t) to a continuous signal p(t). Usually be represented by a zero-order hold (ZOH) y p y ( ) circuit.

p* (t )

p (t )

ZOH

T 2T 3T 4T 5T 6T 7T

sampling period : T

T 2T 3T 4T 5T 6T 7T

· The ZOH takes the value r(kT) and holds it constant for kT t < (k+1)T.

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE 19

D/A and Hold (cont.) (cont )

· A sampler and ZOH can accurately follow the input signal of T is small compared to the transient change in the signal. signal · Usually, the choose of sampling rate (T = 2/T) will be T 30 BW , where BW i the closedill b h is th l d loop bandwidth.

DCS aut '07 20070921

Prof. KY Tsai/YCKung/NTUEE

20

Difference Between DCS and CCS

discrete D( s ) Difference equations approximation

Digital controller

r (t )

D/ A u (kT ) u (t ) hold

T

+

r (kT )

e(kT )

Difference equations

u (kT )

-

D/A and hold

u (t )

Plant

G (s)

y (t )

Clock

A/D

y (t )

Sensor

T

S ( s)

A/ D y (t ) y (kT ) sampling li

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE 21

Difference Equations

· The differential equation of the continuous compensation is approximated by a difference equation which is the discrete approximation to the differential equation and can be made to duplicate the dynamic behavior of a D(s) if the sample period is short enough. · Th result of the difference equation is a discrete The lt f th diff ti i di t u(kT) at each sample instant.

DCS aut '07 20070921

Prof. KY Tsai/YCKung/NTUEE

22

Discrete Approximation

· O particular simple way to make a digital One ti l i l t k di it l computer approximate the real time solution of differential equations is to use E l ' method. diff ti l ti i t Euler's th d · Definition of a derivative :

x & x = lim t 0 t

where x is the change in x over a time internal t.

DCS aut '07 20070921

Prof. KY Tsai/YCKung/NTUEE

23

Discrete Approximation (cont.) (cont )

· If t is not quite equal to zero zero,

& x[k ] x[k + 1] - x[k ] T

where

T = tk+1 ­ tk (the sample interval in seconds), tk = kT (for a constant sample interval), ( p ), k is an integer, x[k ] s e v ue o x[k+1] is the value of x at tk+1 (new v ue)., and ( ew value)., d x[k] is the value of x at tk (past value).

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE

& x( k ) x(k + 1) - x(k ) T

24

Discrete Approximation (cont.) (cont )

· · · This approximation can be used in place of all derivatives that appear in pp p pp the controller differential equations that can be solved by a digital computer. These equations are called difference equations and are solved repetitively with time steps of length T. Ex. Find the difference equation that corresponds to D(s).

D( s) =

approximating difference equation

U ( s) s+a = K0 E (s) s+b

corresponding differential equation

& & u + bu = K 0 (e + ae)

u[k + 1] - u[k ] e[k + 1] - e[k ] + b u[k ] K 0 + a e[k ] T T

u[k + 1] (1 - bT ) u[k ] + K 0 (aT - 1) e[k ] + K 0 e[ K + 1]

DCS aut '07 20070921

Prof. KY Tsai/YCKung/NTUEE

25

Real Time Controller Implementation

u[k + 1] (1 - bT ) u[k ] + K 0 (aT - 1) e[k ] + K 0 e[ K + 1]

Up = 0, ep = 0 (initialization of past values for first loop) Define constants:

1 = 1 bT 1-bT 2 = K0(aT-1) Read A/D to obtain y and r e=r-y u = 1up + 2ep + k0e Output u to D/A and ZOH now set up and ep for the next loop ep = e up = u Go back to Read when T seconds have elapsed

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE

[FPW] pp.61

26

HW1 Progress Check

· Dose everybody get access to [FPW] and [FPE] now? · Any clarification is needed for HW1?

­ Format for writing up? writing-up? ­ Control related problems?

· For people who wonna present on LW3.5, please send your presentation draft to TA by 10/11 for me to comment

­ Chance for you to revise and get a better grade

DCS aut '07 20070921 Prof. KY Tsai/YCKung/NTUEE 27

LW/WW/LD 1/38/0921 2/39/0928 3/40/1005 4/41/1012 5/42/1019 6/43/1026 7/44/1102 8/45/1109 9/46/1116

DCS aut '07 20070920

Topic(s)

Course Information Introduction to digital control g systems Review of continuous-time control Discrete systems analysis (I) Discrete systems analysis (II) Di t t l i Sample-data systems (I) Sample-data systems (II) Design using transform techniques Discrete equivalents

Comment(s)

HW1 out (CT-LTI)

HW1 due/present HW2 out t HW3 out (DTLTI) HW2 due/present HW3 due/present

Prof. Kuen-Yu Tsai/NTUEE

28

LW/WW/LD 10/47/1123 11/48/1130 12/49/1207 13/50/1214 14/51/1221 15/52/1228 16/1/0104 17/2/0111 18/3/0118

DCS aut '07 20070920

Topic(s)

MIMO design using state-space methods (I) h d MIMO design using state-space methods (II) Quantization Effects, Sample rate selection Spectrum analysis by FFT Introductory System Identification* Review Session* Final Exam Final project report/presentation 1st version due Final project report/presentation final fi l version due i d

Prof. Kuen-Yu Tsai/NTUEE

Comment(s)

HW4 due; Final project proposal due ld

HW4 present HW5 due HW5 present

Open book, notes, and p laptop. No collaboration 10% penalty for each day; class present Last lecture Final-exam week Late date to b it t submit grades: 01/28 d

29

Abbreviation/terminology

· · · · · · · · · · · · · · ADC - Analog to Digital Converter g g ASAP - As Soon As Possible CCS - Continuous Control Systems Cont. - Continue CMU - Carnegie Mellon University CACSD - Computer Aided Control System Design CSCAD - Control System Computer Aided Design DAC - Digital to Analog Converter DCS - Digital Control Systems HW - H HomeWork W k TBD - To Be Determined LW - Lecture Week per NTU calendar LW5.3 - Lecture Week 5, Wednesday WW - Work Week per Microsoft Outlook calendar

Prof. KY Tsai/YCKung/NTUEE 30

DCS aut '07 20070921

References

1. 1 [FPW98] G F F kli J. D P G. F. Franklin, J D. Powell, and M ll d M. L. Workman, "Digital Control of Dynamic Systems," 3rd ed., 1998 .m S t "3 d d 2. [FPE] G. F. Franklin, J. D. Powell, and A. Emami-Naeini, "Feedback Control of Dynamic Systems," 5th ed., 2006 .m 3. [DB]R. C. Dorf and R. H. Bishop, "Modern Control Systems," 10th ed., 2005 .m y

DCS aut '07 20070921

Prof. KY Tsai/YCKung/NTUEE

31

Information

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