minimum norm state-space identification for discrete-time systems

Minimum Norm State-Space Identification for Discrete-Time Systems

Zachary FeinsteinAdvisor: Professor Jr-Shin Li

Agenda

•Goals•Motivation•Procedure•Application•Future Work

Goals

•Find a linear realization of the form:

•To solve:

Goals

•In the case of output-data only, create realization of the form:

•This is called historically-driven system

Motivating Problem

•Wanted to find a constant linear realization to approximate financial data

•Use for 1-step Kalman Predictor on historically-driven system:

Motivating Problem

•The specific problem being addressed initially was analysis of the credit market

•Try to do noise reduction and prediction of default rates

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)

Procedure: Agenda

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

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)

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}

Procedure: Find Weighting Pattern•Discrete time weighting pattern:

•We can write:

Procedure: Find Weighting Pattern•Our minimization problem can now be

rewritten as:

•Want to solve for optimal Fk for all k

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:

Procedure: Find Realization

•Given that we have weighting pattern

•Now we have an objective function:

•Again want an iterative approach to solve


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


•For the kth update solving for minimum norm:

•These values solve:


•Choose to update the matrices in the order:


•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

Procedure: Find Optimal Delta

•Want to solve for the optimal delta values such that:

Procedure: Find Optimal Delta

•First discuss how to solve for δB

•Then discuss δC since it is similar to δB

•Finally, discuss δA because this case has higher order terms

Procedure: Find Optimal δB

•For simplification rewrite optimization problem to be:

•Through use of counterexample, it can be seen that δB ≥ 0


•Using property of norms, mainly the triangle inequality


•Using these inequalities it can be seen that:


•Therefore we can find upper and lower bounds for δB:


•Using these bounds use a search algorithm to find optimal δB

•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

Procedure: Find Optimal δC

•Analogous to δB

▫Rewrite the objective function as:

▫Can use same properties to find an upper-bound on this objective function

Procedure: Find Optimal δC

•We can use same properties as before to find bounds on δC:

•Therefore we can use the same search algorithm as in the δB case to find the optimal δC

Procedure: Find Optimal δA

•To simplify we first want to find a linear approximation in δA for:

•Using knowledge of exponentials, we can say:


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


•Given the linearization in δA we can use the same properties as with δB to find bounds on δA

•Using these bounds, we can run the same search algorithm as given for δB

•This search will run on the full objective function, not the linearized version

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

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

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 δ drops below some threshold, stop updating the

corresponding matrix▫ After calculation, if A is an unstable matrix (i.e. |λmax| >

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 ε, iterate through again If better by less than ε, stop and choose new realization If worse by any amount, stop and choose old realization

Application: Input-Output Test

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


•The resulting calculated realization was:


•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)

Application: Objective Value Plot

Application: Actual Output Plot

Application: Calculated Output

Application: Discussion• As shown, these results show this technique to

be unsuccessful, this can be due to:▫ It is assumed that the δ 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

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

Future Work: Gradient Descent

•Advantage:▫Mathematically Robust▫Proven that it will find a local minimum

•Disadvantage:▫Given m*n+n2+n*p 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

Future Work: Heuristic Approach•Example: Genetic Algorithm, Simulated

Annealing•Advantage:

▫Can somewhat control level of computational complexity

•Disadvantage▫Only finds a “good” solution

Thank you

Questions?

minimum norm state-space identification for discrete-time systems

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