Minimum Norm State-Space Identification for Discrete-Time Systems

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Minimum Norm State-Space Identification for
Discrete-Time Systems

• Zachary Feinstein
• Advisor Professor Jr-Shin Li

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Agenda

• Goals
• Motivation
• Procedure
• Application
• Future Work

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Goals

• Find a linear realization of the form
• To solve

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Goals

• In the case of output-data only, create
  realization of the form
• This is called historically-driven system

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Motivating Problem

• Wanted to find a constant linear realization to
  approximate financial data
• Use for 1-step Kalman Predictor on
  historically-driven system

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Motivating Problem

• The specific problem being addressed initially
  was analysis of the credit market
• Try to do noise reduction and prediction of
  default rates

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Motivation

• Why do we need a new technique?
• Financial Data does not follow any clear
  frequency response
• Cannot use any identification technique that
  finds peaks of transfer function (e.g. ERA or FDD)

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Procedure Agenda

• Background
• Find Weighting Pattern
• Find Updated Realization
• Find Optimal Delta Value
• Discussion of Output-Only Case

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Procedure Background

• Let A, B, C have elements which lie in the
  complex plane.
• Let p length of output vector y(k)
• Let n length of state vector x(k)
• Let m length of input vector u(k)

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Procedure Background

• For simplification assume x0 0
• Want to solve
• Remove all points at the beginning such that u(k)
  0 for all k 0,,M

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Procedure Find Weighting Pattern

• Discrete time weighting pattern
• We can write

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Procedure Find Weighting Pattern

• Our minimization problem can now be rewritten as
• Want to solve for optimal Fk for all k

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Procedure Find Weighting Pattern

• Want an iterative approach
• Since each norm in the sum only has Fl for l k
  we can solve find such a formula
• Solving each as a minimum norm problem

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Procedure Find Realization

• Given that we have weighting pattern
• Now we have an objective function
• Again want an iterative approach to solve

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Procedure Find Realization

• Would use Convex Combination of previous best
  solution and optimal case for next norm

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Procedure Find Realization

• For the kth update solving for minimum norm
• These values solve

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Procedure Find Realization

• Choose to update the matrices in the order

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Procedure Find Realization

• This update order was chosen since
• Let C be a constant then from F0 we can find
  optimal B
• Using this optimal B and C then use F1 we can
  find optimal A
• Logical to update C next

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Procedure Find Optimal Delta

• Want to solve for the optimal delta values such
  that

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Procedure Find Optimal Delta

• First discuss how to solve for dB
• Then discuss dC since it is similar to dB
• Finally, discuss dA because this case has higher
  order terms

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Procedure Find Optimal dB

• For simplification rewrite optimization problem
  to be
• Through use of counterexample, it can be seen
  that dB 0

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Procedure Find Optimal dB

• Using property of norms, mainly the triangle
  inequality

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Procedure Find Optimal dB

• Using these inequalities it can be seen that

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Procedure Find Optimal dB

• Therefore we can find upper and lower bounds for
  dB

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Procedure Find Optimal dB

• Using these bounds use a search algorithm to find
  optimal dB
• Evaluate at 2 endpoints and 2 interior points
• If value at endpoint is smallest recursively call
  again with new endpoints of that endpoint and the
  nearest interior point
• Otherwise choose the 2 points surrounding that
  minimum as the new endpoints and call recursively
• Terminate if interval is below some threshold

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Procedure Find Optimal dC

• Analogous to dB
• Rewrite the objective function as
• Can use same properties to find an upper-bound on
  this objective function

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Procedure Find Optimal dC

• We can use same properties as before to find
  bounds on dC
• Therefore we can use the same search algorithm as
  in the dB case to find the optimal dC

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Procedure Find Optimal dA

• To simplify we first want to find a linear
  approximation in dA for
• Using knowledge of exponentials, we can say

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Procedure Find Optimal dA

• Using this linear approximation, we can rewrite
  the minimization problem to be

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Procedure Find Optimal dA

• Given the linearization in dA we can use the same
  properties as with dB to find bounds on dA
• Using these bounds, we can run the same search
  algorithm as given for dB
• This search will run on the full objective
  function, not the linearized version

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Procedure Output-Only Case

• More important case for us given the motivating
  problem of financial data
• Input for financial markets is unknown
• Same procedure as given before
• In finding the optimal weighting pattern
• let u(k) yact(k) for all k

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Application

• Implemented in MATLAB with a few additions to the
  Procedure
• Tried on test input-output system
• Discussion of the unsuccessful results for the
  test case

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Application Implementation

• MATLAB chosen due to native handling of matrix
  operations
• Few differences in implementation and procedure
  given before
• Initial choice of C matrix is chosen as a random
  matrix with elements between 0 and 1
• If d drops below some threshold, stop updating
  the corresponding matrix
• After calculation, if A is an unstable matrix
  (i.e. ?max gt 1) then restart with new initial C
  matrix
• At end of implementation compare new value of
  objective function to previous one
• If better by more than e, iterate through again
• If better by less than e, stop and choose new
  realization
• If worse by any amount, stop and choose old
  realization

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Application Input-Output Test

• Run MATLAB code on well-defined state-space
  system

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Application Input-Output Test

• The resulting calculated realization was

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Application Input-Output Test

• The objective function had a
  value of 37.7 for this calculated realization
• Easier to see in plots on next 3 slides.
• Value of with
  x-axis of k
• Output of the test system (first output only)
• Output of the calculated system (first output
  only)

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Application Objective Value Plot
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Application Actual Output Plot
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Application Calculated Output
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Application Discussion

• As shown, these results show this technique to be
  unsuccessful, this can be due to
• It is assumed that the d values are small, which
  is not necessarily true
• It is assumed that the convex combination will
  bring us towards a better solution, which is seen
  to not be the case
• Changing from the initial minimization problem to
  finding the best approximation for the weighting
  pattern means that some of the relationships
  between the elements of A,B,C could be lost

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Future Work

• There are 2 types of techniques that may be
  useful to solve this problem and find a better
  solution than the shown solution
• Gradient Descent Method
• Heuristic Approach

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Future Work Gradient Descent

• Advantage
• Mathematically Robust
• Proven that it will find a local minimum
• Disadvantage
• Given mnn2np variables, this will take a long
  time to solve
• The objective function (as a sum of norms) is
  large, therefore the gradient may take an
  incredible amount of computational power and
  memory to compute and store

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Future Work Heuristic Approach

• Example Genetic Algorithm, Simulated Annealing
• Advantage
• Can somewhat control level of computational
  complexity
• Disadvantage
• Only finds a good solution

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• Thank you
• Questions?