Ch 5: Closed Loop Systems

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Ch 5 Closed Loop Systems

• Learning Objectives
• extend techniques of open loop systems
• understand impact of unsampled inputs
• analyze systems with plant, ZOH, and digital
  controller
• Deriving Transfer Function for Closed Loop System
• with sampled inputs (set point)
• with unsampled inputs (disturbance)
• Deriving State Variable Models
• derived from Transfer Function
• converted from Continuous System
• developint state equations from block diagram

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Ch 5 Closed Loop Systems extend techniques of
open loop systems (Ch4)
5.2 Prelimnary Concepts
(2) 2 plants, 1st with data hold
C(s) G1(s) G2(s)E(s)
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(3) 2 plants with intermediate data hold
A(s) G1(s)E(s) ? from convolution
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(4) Plant with ZOH Digital Controller
C(z) G(z)D(z)E(z)
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Deriving Transfer Function Closed Loop Sampled
Data System
C(s) G(s)E(s) E(s) R(s)-H(s)C(s) R(s)
H(s)G(s)E(s)
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• Deriving Transfer Functions when only System
  Response is sampled
• System Input isnt sampled before analog plant ?
  cant derive transfer function

C(s) G(s)E(s) E(s) R(s) H(s)C(s) C(s)
G(s)R(s) G(s)H(s)C(s)
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• e.g. digital autopilot on an aircraft
• control commands (elevation, azimuth, speed, )
• (i) sampled input ? can generate transfer
  function for either or both
• inputs from external sources via A/D device
• inputs generated by autopilots CPU

• (ii) wind affects aircraft as an unsampled input
  that affects altitude, speed,
• transfer function cannot be derived for speed
  control for wind input

• equation should not be if a signal is lost as
  a factor
• for more complex systems, derivation can be more
  complex
• Section 5.3 develops a simpler method

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• 5.3 TF Derivation Procedure for Closed Loop
  Discrete Time Systems
• TF for ideal sampler doesnt exist ? complicates
  determination of TF for sampled data system
• Alternate Procedure construct original signal
  flow graph to find sampled output, C(s)
• omit sampler from signal flow graph
• include affects of sampler treat sampler ouput,
  E1, as an input

(1) assign a variable to each sampler input,
Ei (2) assign starred variable to sampler output,
Ei (3) treat each sampler output, Ei, as an
input (4) express Ei s C(s) in terms of Eis
R(s)
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(5) take starred transform and solve equations

• Solution for System of Equations
• (i) for more than 1 sampler or many loops -
  solve simoultanous linear eqns
• Cramers Rule
• Matrix Methods
• (ii) construct sampled signal flow graph from
  eqns in (5) use Masons Gain Formula
• works well for simple signal flow graph where
  loops are easily identified

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• e.g 5.1 Find Transfer Function between C(z)
  R(z)
• derive model from block diagram
• derive flow graph from model

E R HGDE
C GDE
C GDE
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e.g 5.2 Find Transfer Function between C(z)
R(z)
(1) derive signal flow graph from model
(2) derive equations from signal flow graph
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e.g 5.2 (continued)
(3) Derive equations

(4) Draw sampled flow graphs from equations
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e.g 5.2 (continued) (5) Solve using Masons Gain
Formula or as a System of Linear Equations
(5a) use Masons Gain Fomula
M1 G1G2, ?1 1
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e.g 5.2 (continued)
(5b) Solve as linear equation
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e.g. 5.3 No Transfer Function input R(s)
reaches C(s) without being sampled
(1a) Derive Signal Flow graph from simulation
model, solve for C and E1
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e.g. 5.3 (continued)
(1b) use Masons Gain Formula, solve for C and E1

? 1 (-1 G2) 2 G2
M1 RE1C ? 1, ?1 1
M2 E1G1G2C ? G1G2, ?2 1
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e.g. 5.3 (continued)
(2) Take transform
(3a) use Masons Gain Formula to solve for C(s)
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(3b) or solve for C(s) using linear equations
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• e.g. 5.4 Find Transfer Function CPU with
  compute time t0
• model as delay

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C(z)
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5.4 State Variable Models obtain discrete state
variable model
(1) derived fromTF

• analog physical variables generally are not
  discrete state variables

Y(z) (bn-1z -1 bn-2 z -2 b0 z n) E(z)
R(z) ( 1 an-1 z -1 a1z 1-n a0 z
n)E(z) E(z) R(z) - ( an-1 z -1E(z) a1z
1-nE(z) a0 z nE(z) )
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e(k) r(k) an-1e(k-1) - an-2e(k-2) - .
-a0e(k-n)
y(k) bn-1xn(k) b2 x2(k) b0x1(k)
y(k) b0 b1 bn-1 x(k)
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(2) Conversion of Continuous State Equations
example from 5.2

• (i) redraw system so that
• ZOH outputs are shown as inputs
• sampler inputs system outputs are shown as
  outputs
• if Ei(s) is determined direcly from Y(s) ? Ei(s)
  not shown as an output

• (ii) Develop Continuous State Equations for
  Analog Part of System
• draw flow graph for analog part of plant
• write continiuous state equations
• System Inputs are ZOH outputs with star transform
  inputs ei(t)

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• (iii) Derive Discrete State Equations for Analog
  Part of System
• obtain discrete matrices from 4.10

• (iv) Construct Discrete Simulation Diagram or
  Flow Graph
• include all connecting paths of closed loop
  system external to diagram for (5.28)
• (v) Develop System Discrete State Equations

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e.g. 5.5 Derive Discrete State Equations from
Continuous Plants
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Discrete Simulation Model

• (i) Redraw Discrete Simulation Model so that
• ZOH outputs are shown as inputs
• outpus are sampler inputs system outputs
  E1(s), E2(s), Y(s)
• E1(s) determined direcly from Y(s) ? not shown as
  an output

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• (ii) Develop Continuous State Equations for
  Analog Part of System of form
• draw flow graph for analog part of plant
• write continuous state equations
• System Inputs are ZOH outputs with star transform
  inputs ei(t)

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• write continuous state equations

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• write continuous state equations

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(iii) Derive Discrete State Equations for Analog
Part of System (use CPU)
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(iv) Construct Simulation Diagram with all
external paths
Discrete Flow Graph for Analog Part
Discrete Flow Graph for External Paths
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Complete Discrete Flow Graph
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(v) Write Discrete State Equations for entire
closed loop system by including external paths
input e1(k) becomes e1(k) r(k) - x4(k)
x1(k1) x1(k) 0.095 x2(k) 0.005e1(k)
x1(k) 0.095 x2(k) 0.005( r(k) - x4(k) )
x2(k1) 0.905 x2(k) 0.095e1(k) 0.905
x2(k) 0.095( r(k) - x4(k))
input e2(k) becomes e2(k) x1(k) x3(k)
x4(k1) 0.819 x4(k) 0.181e2(k)
0.819x4(k) 0.181(x1(k) x3(k))
x3(k1) ) 0.069e2(k) 0.069(
x1(k) x3(k))
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Discrete State Equations for entire closed loop
system
y(k) 0 0 0 1x(k)

• Technique above is Similar to Technique for
  finding closed loop TF
• system is opened at each sampler
• each ZOH output is assumed to be an input
• each sampler input is assumed to be an output
• discrete state equations are written for
  specified inputs/outputs
• equations are manipulated to obtain state
  equations of closed loop system
• ok for low order systems, for complex systems ?
  use computer and matrix procedure on next slide

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combined equations to yield x(k1) (A1 B1
C1)x(k) B1D1 r(k) solve for A1 B1 C1 and
B1D1

• less prone to errors, can be implemented using a
  computer
• for (5-30) ? each system must sample input prior
  to application to analog part of system
• if unsampled input varies slowly over sample
  period ? can approximate sampled signal
• state equations are more complex if system
  contains digital controller

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e.g. 5.6 (continuing example 5.5 )
then write e(k) as
e(k) C1 x(k) D1r(k)
prevously,
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• State Equations for a system that contains a
  Digital Controller
• assign states v1(k)..vi(k) to plant (ith order
  plant)
• assign states vi1(k)..vn(k) to filter (n-i
  order filter)

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obtain system state equations eliminate e(k)
m(k) from (5.33) (5.35) m(k) C1v(k) D1
u(k)-Cv(k) C1 D1Cv(k) D1u(k)
then
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e.g. 5.7 Develop State Equations for figure
from example 4.13, plant state equations are
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since e(t) u(t) - y(t) ? e(k) u(k) y(k) and
e(k) u(k) 1 0 0 v(k)
since m(k) 0.9v3(k1) 0.8v3(k) and v3(k1)
e(k) 0.9v3(k), we have m(k) 0.9(e(k)
0.9v3(k)) 0.8v3(k) 0 0 0.01 v(k)
0.9e(k)
recall
from (5.33) C1 0 0 0.01 and D1 0.9 from
(5.35) C 1 0 0 solve for (A1 B1C1), (B2C
B1 D1 C), and (B1 D1 B2)
then substitute into v(k1) (A1 B1C1 -
B2C B1 D1 C) v(k) (B1 D1 B2) u(k)
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5.5 Summary closed loop discrete time systems
2 techniques were discussed.

• method 1 determine Laplace transform and
  z-transforms of system output
• assumes we have block diagram or signal flow
  graph ? used to develop sampled flow graph
• sampled flow graph used to determine Laplace
  transform and z-transforms of system output

method 2 develop state variable model, provided
all inputs are sampled