CH 4: Open Loop Discrete Time Systems

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• CH 4 Open Loop Discrete Time Systems
• covered to this point
• review of continuous systems
• z-transforms and system modeling
• recovery of sampled data
• discrete systems
• ? this chapter derives analysis methods for open
  loop discrete sysems

• Key Learning Points of this section
• E(z) vs E(s)
• Pulse Transfer Function
• Open Loop Systems with Digital Filters
• Modified Z-transform
• Systems with Time Delays
• State Variable Models
• Discrete State Equations

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4.2 Relationshiip between E(z) E(s)

• in this case, z-transform can be considered
  special case of Laplace transform
• for analyzing discrete systems ? use z-transform
  instead of transform

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e.g. 4.1

• E(s) has infinite number of poles zeros in
  the s-plane
• E(z) has a 1 zero at z0 and 2 poles at ? -2T and
  ? -T
• ? analysis using pole-zero locations greatly
  simplified using z-transform

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4.3 Pulse Transfer Function this section develops
z-transform for the output of open-loop sampled
data systems expression used to form closed loop
system

• plant transfer function Gp(s)

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assume c(t) is continuous at all sampling
instants
then if e(t) is continuous at all sampling
instants
then C(s)
G(s) E(s)
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Pulse Transfer Function
G(z) C(z)/E(z)

• TF between sampled input, e(t) and output c(t)
  at sampling instants
• doesnt indicate nature of output c(t) between
  sampling instants ? continuous output is not
  captured
• generally choose sampling frequency, ws such
  that response between sampling instants
  approximates response at sampling instants

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Equations (4.8) (4.9) are general derivations
for any F(s) f0 f1?-Ts f2?-2Ts if
A(s) B(s)F(s) then
A(s) B(s)F(s) A(z) B(z)F(z)
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e.g.
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e.g. if input, e(t) unit step ? determine
output C(z)
and c(nT) 1-?-nT
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• (i) z-transform analysis yields response only at
  sampling instants
• output, c(kT) rises to final value of unity at
  sampling instants
• nothing known about response between sampling
  instants
• normally this information is needed ? find the
  complete response by simulation
• (ii) if input, e(t) to sampler/ZOH unit step,
  u(t), ? output of sampler/ZOH is also u(t)
• ZOH reconstructs sampled unit-step exactly
• response of c(kT) is step response of continuous
  time system with
• G(s) 1/(s1)
• c(t) 1-?-nT
• z-transform analysis in example is correct
• given c(t) ? c(nT) c(t)tnT
• converse is not true c(t) ? c(nT)nTt

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Effect of ZOH on step function
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(iii) often, dc gain is important

• dc Gains Steady State (SS) Gain with constant
  input

• dc gain G(z)z1 G(1)
• with constant input ? gain of sampler/ZOH 1

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dc gain can be used to check calculation of G(z)
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Open Loop Transfer Functions with 2 Plants (3
cases)
(1) 2 plants, both with data holds
C(s) G2(s) A(s) C(z) G2(z)A(z)
A(s) G1(s) E(s) A(z) G1(z)E(z)

• ? C(z) G2(z)G1(z)E(z)
• total transfer function is product of pulse
  transfer functions

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(2) 2 plants, 1st with data hold
C(s) G1(s) G2(s)E(s) C(z) ZG1(s)G2(s)E(z)
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(3) 2 plants with intermediate data hold
C(s) G2(s)A(s)
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• 4.4 Open Loop System with Digital Filters
• extend 4.3 for open loop systems that contain
  digital filters

(i) A/D converts continuous signal e(t) into
discrete sequence e(kT)
(ii) digital filter processes each e(kT)
generates output sequence m(kT)

• solves linear difference equation with constant
  coefficients

• transfer function represented by

• M(z) D(z)E(z) and M(s) D(s)E(s)
  (substitute z ?sT)

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• Complete Model combines with ideal-sampler/ZOH
  for accurate model
• CPU (digital filter) processes each e(kT)
• A/D - digital filter - D/A model processes
  impulse function of weight e(kT)

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e.g.
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e.g. (continued)
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e.g. (continued)
final value, c(?)

• if steady state input constant value of unity
• steady state output, c(?) dc gain? input 1

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4.5 Modified z-transform consider systems
containing ideal time delays let a time function
e(t) be delayed by ?T, 0 lt ? ? 1

e(t-?T)u(t-?T)
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Modified Z-transform E(z,m) is defined from
delayed z-transform replace ? by 1-m
m1 ? no delay E(z,1) E(z) e(0) m0 ? delay
T E(z,0) z-1E(z) special tables used for
modified z-transform

• property of modified z-transform
• Zme(t) ? z-transform of e(t) time shifted by
  1-m
• theorms of E(z) from chapter 2 apply to E(z,m)
• shifting theorm ZmE(s) Z?-?TsE(s) ?
  1-m
• for integer, k Zm?-kTs E(s) z-k Zm E(s)
  z-kE(z,m)

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e.g.
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4.6 Systems with time delays Use Zm (1)
determine pulse transfer function of discrete
time systems with ideal time delays
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• (2) determine pulse transfer function where
  computation time cant be neglected
• nth order digital controller solves difference
  equation every T seconds

• model as digital controller with no delay
  followed by ideal time delay t0

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e.g
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e.g.

• t0 1ms
• T 50ms
• ? m 1- ? ? mT T-?T (50ms-1ms ) 49ms

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4.8 State Variable Models (as opposed to
transfer functions)
x(k1) Ax(k) Bu(k) y(k) Cx(k) Du(k)

• state vector x(k)
• input vector u(k)
• output vector y(k)

Transfer Function approach to obtain state
variable model of open loop sampled data
systems (1) draw simulation diagram from
z-transform TF (2) assign state variables to
output of each time delay (3) write state
equations from simulation diagram
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e.g.
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4.9 Review of Continuous State Variables main
disadvantages of using transfer function to
obtain state equations for discrete systems (1)
difficult to derive pulse transfer function for
higher order systems (2) lose natural system
states as state variables
e.g. motion of mass without friction
state variables (natural, physical variables)
state equations are
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if system were part of sampled data system (i)
choose position as a state in discrete-state
model by letting position be system output (ii)
if TF approach used for discrete-state modelling
choosing velocity as 2nd state would be
difficult state variables are outputs of delays
Alternate Approach for Obtaining Discrete State
Model using continuous state variables

• system states v(t)
• system inputs u(t)
• system outputs y(t)
• c used to indicate continuous matrices

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e.g. mass, damper, spring system
M mass B damping coefficient K spring
constant
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e.g. (continued) state equations
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State Equations for Continuous Single
Input-Output Systems
(1) Use Laplace Transform ? solve for V(s)
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(2) Alternate derivation of ?c(t) - useful when
deriving discrete state model if ?c(t) is
assumed to be an infinite series
Ki are constant, nonsingular matrices
then by substitution into 4.57 we have
thus (K1 2K2 t3K3 t2 ) (AcK0t0 Ac K1t1
Ac K2 t2 Ac K3 t3 )
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thus for i ? 1 we have iKi t(i-1) Ac
Ki-1t(i-1)
without loss of generality, let K0 I, then we
have K1 AcK0 Ac K2 Ac K1/2 Ac2 K0/2
Ac2 /2 K3 Ac K2/3 Ac3 K0/3! Ac3 /3! Ki
Ac Ki-1/i Aci K0/i! Aci /i!
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alternate method for obtaining state equations in
standard format for algebraic loops algebraic
loop loop without integrator in figure
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• 4.10 Discrete State Equations
• determine discrete state equations of sampled
  data system from continuous state equations
• preserve natural system states

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• Discrete State Model
• obtained by evaluating (4.65) at t0 kT and t
  kTT
• during interval kT ? t lt kTT, u(t) m(kT) ?
  replace u(t) by m(kT)
• valid only if u(t) is output of ZOH

compare with discrete equations in (4-52) x(k1)
A x(k) Bm(k) y(k) C x(k) Du(k)
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(4) evaluate A and B either by (i) Laplace
Transform (difficult) (ii) convergent power
series
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• for multiple input outputs
• D, Dc are matrices
• each input must be output of ZOH or errors can
  result, especially if inputs change rapidly

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e.g 4.13, T 0.1s
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e.g 4.13 (continued)

• alternately, use power series to solve for A B
• 3 significant figures of accuracy requires 3
  terms

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e.g 4.13 (continued)
x(k1) Ax(k) Bu(k) y(k) 1 0x(k)
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• 4.11 Practical Calculations by Computer
• required for higher order systems.
• preferred for lower order systems

(2) If Transfer Function of Analog Part of System
Required, then determine
sV(s) Ac V(s) Bc U(s) V(s) sI-Ac-1 BU(s)
Y(s) Cc V(s) Dc U(s) Cc sI-Ac-1 BU(s)
Dc U(s) Cc sI-Ac-1 B Dc U(s)
then Gp(s) CcsI-Ac-1Bc Dc
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(3) Calculate Discrete Matrices of Analog Part of
System from Ac, Bc, Cc ,Dc
C Cc D Dc