ME375 Dynamic System Modeling and Control

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MESB374 System Modeling and AnalysisFeedback
Control Design Process
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Classical Feedback Control Structure
Disturbance D(s)
Reference Input R(s)
Control Input U(s)
E(s)
Output Y(s)

Gf(s)
Gc(s)
-
Plant
filter
Controller
H(s)
Ym(s)
Sensor Dyamics
Plant Equation (Transfer function model that we
all know how to obtain ?!) Control Law
(Algorithm) (we will try to learn how to design)
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Closed-Loop Transfer Function
Can you obtain it by block algebra?
What is Y(s) if disturbance dynamics and/or noise
are taken into account?
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Closed-Loop Transfer Function

• The closed-loop transfer functions relating the
  output y(t) (or Y(s)) to the reference input r(t)
  (or R(s)) and the disturbance d(t) (or D(s)) are
• The objective of control system design is to
  design a controller GC (s) and Gf (s), such that
  certain performance (design) specifications are
  met. For example
• we want the output y(t) to follow the reference
  input r(t), i.e., for certain frequency range.
  This is equivalent to specifying that
• we want the disturbance d(t) to have very little
  effect on the output y(t) within the frequency
  range where disturbances are most likely to
  occur. This is equivalent to specifying that
  

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Performance Specifications

• Given an input/output representation, GYR (s),
  for which the output of the system should follow
  the reference input, what specifications should
  you make to guarantee that the system will behave
  in a manner that will satisfy its functional
  requirements?

Input R(s)
Output Y(s)
GYR (s)
rss
yssGYR(0)rss
t0
t0
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Unit Step Response
yMAX
OS
X
Unit Step Response
tP
tS
Time
tr
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Performance Specifications

• Steady State Performance ? Steady State Gain of
  the Transfer Function
• Specifies the tracking performance of the system
  at steady state. Often it is specified as the
  steady state response, y() (or ySS(t)), to be
  within an X bound of the reference input r(t),
  i.e. the steady state error eSS(t) ySS(t) -
  r(t) should be within a certain percent. For
  example, for step reference input r(t)rss
• To find the steady state value of the output,
  ySS(t)
• Sinusoidal references use frequency response,
  i.e.
• General references use FVT, provided that
  sY(s) is stable, ...

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Performance Specifications

• Transient Performance (Transient Response)
• Transient performance of a system is usually
  specified using the unit step response of the
  system. Some typical transient response
  specifications are
• Settling Time (tS)
• Specifies the time required for the response to
  reach and stay within a specific percent of the
  final (steady-state) value. Some typical
  settling time specifications are 5, 2 and 1.
  For 2nd order systems, the specification is
  usually
• 
  ?
• Overshoot (OS)(2nd order systems)

Recall step response of second order system
Q How can we link this performance specification
to the closed-loop transfer function? (Hint)
What system characteristics affect the system
performance ?
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Performance Specifications

• Transient Performance Specifications and CLTF
  Poles
• Recall that the locations of TF poles directly
  affect the system output. For example, assume
  that the closed-loop transfer function of a
  feedback control system is
• The characteristic poles are
• Settling Time (2)
• ? Puts constraint on the real part of the
  dominating closed-loop poles
• OS
• ? Puts constraint on the damping ratio ? or the
  angle ? of the dominating closed-loop poles.

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Performance Specification CL Pole Locations

• Transient Performance Specifications and CLTF
  Pole Locations
• Transient performance specifications can be
  interpreted as constraints on the positions of
  the poles of the closed-loop transfer function.
  Let a pair of closed-loop poles be represented
  as
• Transient Performance Specifications
• Settling Time (2 ) TS
• OS X

Img.
jwd
Real
-jwd
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Example

• A DC motor driven positioning system can be
  modeled by a second order transfer function
• A proportional feedback control is proposed and
  the proportional feedback gain is chosen to be
  16/3. Find the closed-loop transfer function, as
  well as the 2 settling time and the percent
  overshoot of the closed loop system when given a
  step input.
• Draw block diagram

• Find closed-loop transfer function

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Example

• Find closed-loop poles

• 2 settling time
• OS

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Example

• Find closed-loop transfer function
• Write down the performance specifications

• A DC motor driven positioning system can be
  modeled by a second order transfer function
• A proportional feedback control is proposed. It
  is desired that
• for a unit step response, the steady state
  position should be within 2 of the desired
  position,
• the 2 settling time should be less than 2 sec,
  and
• the percent overshoot should be less than 10.
• Find
• (1) the condition on the proportional gain such
  that the steady state performance is satisfied
• (2) the allowable region in the complex plane for
  the closed-loop poles.

Assume that closed-loop is an under-damped second
order system
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Example

• Percent Overshoot (OS)

Steady state performance constraint Transi
ent performance constraint 2 Settling Time
Assume that closed-loop is an under-damped second
order system
Img.
Real
-2
How to discuss the case of over-damped system?
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Example

• Over-damped case

Steady state performance constraint
Stability Requirement
Transient performance constraint 2 Settling Time
No Overshoot!!
In all, we have
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Review of Structure of Closed-loop System
Disturbance D(s)
Disturbance Channel
GD (s)
Reference Input R(s)
Control Input U(s)

Error E(s)
Output Y(s)

Gf (s)
GC (s)
GA (s)

-
Actuator
Plant
Control algorithm
filter

H(s)

Sensor
GN (s)
Noise Channel
Noise N(s)
How many transfer functions do we need to
determine the output?
Do you know how to find them?
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Feedback Control Design Process

• A typical feedback controller design process
  involves the following steps
• Model the physical system (plant) that we want to
  control and obtain its I/O transfer function
  G(s). (Sometimes, certain model simplification
  should be performed)
• Determine sensor dynamics (transfer function of
  the measurement system) H(s) and actuator
  dynamics (if necessary).
• Draw the closed-loop block diagram, which
  includes the plant, sensor, actuator and
  controller transfer functions GC (s) and Gf (s).
• Obtain the closed-loop transfer functions GYR (s)
  and GYT (s) .
• Based on the performance specifications, find the
  conditions that the CLTFs, GYR (s) and GYT (s),
  have to satisfy.
• Choose controller structure GC (s) and Gf (s) and
  substitute it into the CLTFs GYR (s) and GYT (s).
• Select the controller parameters (e.g. the
  proportional feedback gain of a proportional
  control law) so that the design constraints
  established in (5) are satisfied.
• (8) Verify your design via computer simulation
  (MATLAB) and actual implementation.

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In Class Exercise

• You are the young engineer that is in charge of
  designing the control system for the next
  generation inkjet printer (refer the example
  discussed in previous lecture notes). During the
  latest design review, the following plant
  parameters are obtained
• LA 10 mH
• RA 10 W
• KT 0.06 Nm/A
• JE 6.5 10-6 Kg m2
• BE 1.4 10-5 Nm/(rad/sec)
• The drive roller angular position is sensed by a
  rotational potentiometer with a static
  sensitivity of KS 0.03 V/deg.

• The design specifications for the paper
  positioning system are
• The steady state position for a step input should
  be within 5 of the desired position.
• The 2 settling time should be less than 200
  msec, and
• the percent overshoot should be less than 5.
• You are to design a controller that satisfies the
  above specifications.

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In Class Exercise
(1) Model the physical system (plant) that we
want to control and obtain its I/O transfer
function G(s). (Sometimes, certain model
simplification should be performed.) From
previous example, the DC motor driven paper
positioning system can be modeled by
Ei(s)

IA(s)
Tm(s)
KT
-
Kb
Read textbook about the equivalent moment of
inertia and equivalent damping constant
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In Class Exercise

• The plant transfer function G(s) can be derived
  to be
• As discussed in the previous example, we can
  further simplify the plant model by neglecting
  the electrical subsystem dynamics (i.e., by
  letting LA 0 )
• Substituting in the numerical values, we have our
  plant transfer function

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In Class Exercise

• (2) Determine sensor dynamics (transfer function
  of the measurement system) H(s) and actuator
  dynamics (if necessary).
• (3) Draw the closed-loop block diagram, which
  includes the plant, sensor, actuator and
  controller GC (s) transfer functions.

Reference Input

Error E(s)
Gf (s)
GC (s)
-
H(s)
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In Class Exercise

• (4) Obtain the closed-loop transfer function GYR
  (s).

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In Class Exercise

• (5) Based on the performance specifications, find
  the conditions that GYR (s) has to satisfy.
• Steady State specification
• For step reference input,
• Transient Specifications
• Settling Time Constraint
• Overshoot Constraint

-20
Transient performance Region
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In Class Exercise

• (6) Choose controller structure GC (s) and Gf (s)
  substitute it into the CLTF GCL (s).
• Lets try a simple proportional control, where
  the control input to the plant is proportional to
  the current position error
• In s-domain (Laplace domain), this control law
  can be written as
• Substitute the controller transfer function into
  GCL (s)

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In Class Exercise

• (7) Select the controller parameters (e.g., the
  proportional feedback gain of a proportional
  control law) so that the design constraints
  established in (5) are satisfied.
• Steady State Constraint
• Want
• Transient Constraints
• To satisfy transient performance specifications,
  we need to choose KP such that the closed-loop
  poles are within the allowable region on the
  complex plane. To do this, we first need to find
  an expression for the closed-loop poles

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In Class Exercise

• For every KP , there will be two closed-loop
  poles (closed-loop characteristic roots). Its
  obvious that the two closed-loop poles change
  with the selection of different KP . For
  example
• KP 0 p1,2 0, -57.47
• KP 0.26 p1,2 -8.4, -50
• KP 0.475 p1,2 -20.1, -37.4
• KP 0.52 p1,2 -28.7,-28.7
• KP 0.7 p1,2 -28.717j
• KP 1.08 p1,2 -28.729.7j
• KP 1.75 p1,2 -28.744j
• By inspecting the root-locus, we can find
• that if
• then the closed-loop poles will be in the
• allowable region and the performance
• specifications will be satisfied.

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In Class Exercise

• (8) Verify your design via computer simulation
  (MATLAB) and actual implementation.
• gtgt num 16KsKp
• gtgt den tauM 1 16KsKp
• gtgt T (00.00020.25)
• gtgt y step(num,den,T)
• gtgt plot(T,y)

KpKp180/pi
(Kp0.7)
(Kp1.75)
(Kp0.52)
(Kp0.26)
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In Class Exercise

• (9) Check the Bode Plots of the open loop and
  closed loop systems