Manufacturing Controls

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Manufacturing Controls

• FALL 2001
• Lecture 18 

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Syllabus

• DATE TOPIC NOTES
• 1. Sep. 20 Mechatronics Design Process Ch. 1
• 2. Sep. 25 System Modeling and Simulation Ch. 2
• 3. Sep. 27 Laplace Transforms and Transfer
  Functions Ch. 2
• 4. Oct. 2 Electrical Examples Ch.2, Notes
• 5. Oct. 4 Mechanical Examples Ch.2, Notes
• 6. Oct. 9 More Examples, Thermal and Fluid
  Examples, QUIZ 1 (Take Home)
• 7. Oct. 11 Sensors and Transducers Ch. 3
• 8. Oct. 16 Digital control, Advanced MATLAB
• 9. Oct. 18 Analog and Digital Sensing Ch. 3,
  Notes
• 10. Oct. 23 Actuating Devices, time and frequency
  response Ch. 4
• 11. Oct. 25 DC Motor Model, Ch. 4,
  Notes
• 12. Oct. 30 Examples Ch. 5
• 13. Nov. 1 Boolean Logic ,Programmable Logic
  Controllers Ch. 5, Notes
• 14. Nov. 6 Stability and Compensators, P, PI and
  PD Ch. 6
• 15. Nov. 8 PID Controllers - Review Ch. 7
• 16. Nov. 13 QUIZ 2 (In Class - Open Book)
• 17. Nov. 15 Practical and Optimal Compensator
  Design Ch. 8
• 18. Nov. 20 Frequency Response Methods Ch. 9,
  Notes

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Todays objective

• To test the accomplishment of our objective of
  understanding systems theory by solving problems
  related to the concepts of optimal control
  systems for feedback control.

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Comparison of input and actual response leads to
a natural criteria to optimize both the transient
and steady state responses.
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Simulation and optimization

• Model of the non-linear or linear system using
  Matlab's Simulink toolbox
• Optimization using the Optimization toolbox
• Matlab file that has the models transfer
  function
• Matlab file converting the digital gains to
  analog signals
• Simulink graphics model which takes the analog
  gains and simulates the step response

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Optimization Toolbox

• Is a collection of functions that extend the
  capability of Matlab. The toolbox includes
  routines for
• Unconstrained optimization
• Constrained nonlinear optimization, including
  goal attainment problems, minimax problems, and
  semi-infinite minimization problems
• Quadratic and linear programming
• Nonlinear least squares and curve fitting
• Nonlinear systems of equations solving
• Constrained linear least squares
• Specialized algorithms for large scale problems
• Reference Thomas Coleman, Mary Ann Branch and
  Andrew Grace, Optimization Toolbox, The Math
  Works,1999.

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Optimization

• Optimization concerns the minimization or
  maximization of functions. For example

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To use the optimization toolbox

• Most of these optimization routines require the
  definition of an M-file containing the function,
  f, to be minimized.
• Maximization is achieved by supplying the
  routines with f.
• Optimization options passed to the routines
  change optimization parameters.
• Default optimization parameters can be changed
  through an options structure.

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Medium-Scale Example

• Unconstrained example
• Consider the problem of finding a set of
  valuesx1,x2 that solves

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Solution

• Write an M-file that returns the function value
• Call it objfun.m
• Then invoke the unconstrained minimization
  routine
• Use fminunc

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Function

• function f objfun(x)
• fexp(x(1))(4x(1)22x(2)24x(1)x(2)2x(2)
  1)

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help function FUNCTION Add new function.
New functions may be added to MATLAB's vocabulary
if they are expressed in terms of other
existing functions. The commands and
functions that comprise the new function must
be put in a file whose name defines the name of
the new function, with a filename extension
of '.m'. At the top of the file must be a
line that contains the syntax definition for
the new function. For example, the existence of a
file on disk called STAT.M with
function mean,stdev stat(x)
STAT Interesting statistics. n
length(x) mean sum(x) / n
stdev sqrt(sum((x - mean).2)/n)
defines a new function called STAT that
calculates the mean and standard deviation
of a vector. The variables within the body of
the function are all local variables.
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A subfunction that is visible to the other
functions in the same file is created by
defining a new function with the FUNCTION
keyword after the body of the preceding function
or subfunction. For example, avg is a
subfunction within the file STAT.M
function mean,stdev stat(x) STAT
Interesting statistics. n
length(x) mean avg(x,n)
stdev sqrt(sum((x-avg(x,n)).2)/n)
------------------------- function
mean avg(x,n) MEAN subfunction
mean sum(x)/n Subfunctions are not
visible outside the file where they are defined.
Normally functions return when the end of the
function is reached. A RETURN statement can
be used to force an early return. See also
SCRIPT, RETURN, VARARGIN, VARARGOUT, NARGIN,
NARGOUT, INPUTNAME, MFILENAME.
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Step 2 Invoke one of the optimization routines

• x0 -1,1 Starting guess
• options optimset(LargeScale,off)
• x,fval,exitflag,outputfminunc(objfun,x0,optio
  ns)
• Try this. After 40 iterations, it should produce
  the solution
• X
• 0.50000 -1.0000

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And

• The function at the solution x is returned in
  fval
• fval
• 1.3030e-10
• The exitflag tells if the algorithm converged. An
  exitflag gt 0 means than a local minimum was
  found.
• exitflag
• 1

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Output structure gives more details

• For fminunc, it includes the number of
  iterations, the number of function elaluations,
  the final step size, a measure of first-order
  optimality, and the type algorithm used.
• output
• iterations 7
• funcCount 40
• stepsize 1
• firstorderopt 9.280e-004
• algorithm medium-scale Quasi-Newton line search

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When more than one minimum

• When there exists more than one local minimum,
  the initial guess for the vector x1,x2 affects
  both the number of function evaluations and the
  value of the solution point. In the example, x0
  is initialized to -1,1.
• Try it again with another x0.

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Variable options can be changed

• The variable options can be passed to fminunc to
  change the characteristics of the optimization
  algorithm.
• Xfminunc(objfun,x0,options)
• Options is a structure that contains values for
  termination tolerances and algorithm choices.
• An options structure can be created using the
  optimset function
• optionsoptimset(LargeScale,off) Turning off
  the LargeScale option gives the medium-scale
  algorithm.

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Other options

• Other options include
• controlling the amount of command line display
  during the optimization iteration,
• the tolerances for the termination criteria,
• if a user-supplied gradient or Jacobian is to be
  used,
• and the maximum number of iterations or function
  evaluations.

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Nonlinear Inequality Constrained Example
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Preliminaries

• Since neither of the constraints if linear, you
  cannot pass the constraints to fmincon at the
  command line
• Instead, you can create a second M-file confun.m
  that returns the value at both constraints at the
  current x in a vector, c.
• The constrained optimizer, fmincon, is then
  invoked.
• Because fmincon expects the constraints to be
  written in the form c(x)lt0, the constraints must
  be rewritten in the form

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Process

• Step 1 Write an M-file confun.m for the
  constraints
• Step 2 Invoke the constrained optimization
  routine
• Look at the results

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Step 1

• function c,ceqconfun(x)
• nonlinear inequality constraints
• c1.5x(1)x(2)-x(1)-x(2)
• -x(1)x(2)-10
• nonlinear equality constraints - none
• ceq

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Step 2 Invoke constrained optimization routine

• x0-1,1 Starting guess at the solution
• optionsoptimset(LargeScale,off)
• x,fvalfmincon(objfun,x0,,,,,,,con
  fun,options)
• After 38 function calls, the solution x produced
  with function value fval is
• x
• -9.5474 1.0474
• fval
• 0.0236

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Constraints

• The values of the constraints at the solution can
  be evaluated
• c,ceqconfunx
• c
• 1.0e-15
• -0.8882
• 0
• ceq

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Multiobjective Examples

• The previous examples involved problems with a
  single objective function. Let us now consider
  solving problems with multiobjective functions
  using
• lsqnonlin
• fminmax
• fgoalattain

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Simulink Example

• Suppose we want to optimize the control
  parameters in the Simulink model
• optsim.mdl
• This model may be found in the Optimization
  Toolbox directory.
• Lets look for it. Try it. This will also check
  that Simulink is installed.

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Simulink mdl file

• The model is a nonlinear process plant
• Limit amplifier
• Rate amplifier
• Under-damped third-order system
• The actuator limits are a saturation limit and a
  slew rate limit.
• The saturation limit cuts off input values
  greater than 2 units and less than -2 units
• The slew rate limit is 0.8 units/sec.

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Examine the response

• At the Matlab command prompt, type
• optsim
• Open the Scope block by double clicking on it.
• Run the simulation.
• The output is oscillatory.
• The problem is to design a feedback control law
  that tracks a unit step input function.

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Closed loop plant

• The closed-loop plant is entered in terms of the
  blocks where the plant and actuator have been
  placed in a hierarchical Subsystem block. A Scope
  block displays output trajectories during the
  design process.
• A PID controller is used.

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Optimization approach

• One way to solve the problem is to minimize the
  error between the output and input signals.
• The variables are the parameters of the PID
  controller.
• If you need to minimize the error at one time
  value, the problem would require a single
  objective function. However, let us set the goal
  to minimize the error for all time steps from 0
  to 100. This produces a multiobjective function
  with one function per time step.

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Use the routine lsqnonlin

• lsqnonlin is used to perform a least squares fit
  on tracking the output.
• This is defined by a Matlab function in the file
• tracklsq.m
• It defines the error signal, yout. The output is
  computed by calling sim, minus the input signal 1
• Error yout-1

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tracklsq

• tracklsq must run the simulation.
• The simulation may be run either in the base
  workspace or the current workspace, i.e, the
  workspace of the function calling sim which in
  this case is the tracklsqs workspace.
• In this example, the simset command is used to
  tell sim to run the simulation in the current
  workspace by setting SrcWorkspace to current.

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Running the simulation

• To run the simulation in optsim, the variables
• Kp, Kd and Ki
• And a1 and a2 that are variables in the plant
  block
• Must all be defined.
• You can initialize a1 and a2 before calling
  lsqnonlin and then pass these two variables as
  additional arguments.
• lsqnonlin will then pass a1 and a2 to tracklsq
  each time it is called so you do not have to use
  global variables.

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Choosing a solver

• After choosing a solver using the simset
  function, the simulation is run using sim.
• The simulation is performed using a fixed-step
  fifth-order method to 100 seconds.
• When the simulation completes, the variables
  tout, xout and your are now in the current
  workspace (the workspace of tracklsq).
• The Outport block is used in the block diagram
  model to put yout into the current workspace at
  the end of the simulation.

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Procedure

• Step 1- Write an M-file
• tracklsq.m
• Step 2 Invoke optimization routine

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tracklsq.m

• Function F tracklsq(pid,a1,a2)
• Kp pid(1) Move variables into model
  parameters
• Ki pid(2)
• Kd pid(3)
• choose solver and set model workspace to this
  function
• opt simset('solver', 'ode5', 'SrcWorkspace',
  'Current')
• tout, xout, yout sim('optsim', 0 100, opt)
• F yout-1 compute error signal

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optsim

• Step 2 The second file invokes the model and the
  optimizing routine
• optsim load the model
•   pid0 0.63 0.0504 1.9688 setting initial
  values
• a1 3 a2 43 initialize Plant variables in
  model
• options optimset('LargeScale', 'off',
  'Display', 'iter', 'TolX', 0.001, 'TolFun',
  0.001)
• pid lsqnonlin(tracklsq', pid0, , ,
  options, a1, a2)
• put variables back into the base workspace
• Kp pid(1) Ki pid(2) Kd pid(3)

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Options

• The variable options passed to lsqnonlin define
  the criteria and display characteristics. In this
  case you ask for output, use the medium-scale
  algorithm, and give termination tolerances for
  the step and objective function on the order of
  0.001.

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Try it!

• This should converge in 10 iterations and give
  final values of
• pid 2.9108 0.1443 12.8161

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Homework-Given the Simulink model of the motor
system- select the optimum values of the digital
PID controller (a,b,f,g).
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Model explanation

• Step input signal fed to a summation block
• Constant values to be used are calculated using a
  separate M-file
• Analog values in the Mat lab kernel
• Zero order hold
• DAC
• Amplifier
• Encoder feedback fed to the summation block for
  correction

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Optimization of the parameters

• Non-linear model
• Enables the system to track a unit step input
  into the system
• Two methods are used
• lsqnonlin is the non-linear least squares
  method. It minimizes the error between the output
  and input
• fminmax minimizes the maximum value of the
  output at anytime.
• The first method is chosen
• It is a multi-objective function because the
  error needs to be minimized for all time steps

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lsqnonlin method

• Least square fit on the tracking of output
• E input-output
• Objective is
• Minimize

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Mat lab optimizing files

• Step 0 Creat the Simulink file
• Step 1 the first file defines the function

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Mat lab optimizing files contd.

• Step 2 The second file invokes the model and the
  optimizing routine
• Initial values of the parameters are defined

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Optimization results
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Results

• Turn in a copy of your Simulink mdl file, the
  other Matlab m-files and your iteration results
  with the final values of a,b,f,g.

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Lets try it again with the motor model we have
developed.

• Due to difficulty of using a specific model a
  generic model was chosen. Future team members
  could try to work on the specific model
• Thought the iterative steps were eliminated,
  still the initial value had to be chosen.
• Only one method was dealt with. More methods can
  be used and compared for the best results

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Any questions?