Lecture Notes- MIT- System Identification

System Identification

6.435

SET 11

Computation

Levinson Algorithm

Recursive Estimation

Munther A. Dahleh

Lecture 11 6.435, System Identification Prof. Munther A. Dahleh

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Computation

Least Squares: QR factorization

Q is invertible and

error


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Initial Conditions:


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What about initial conditions?

Solution 1: sum starts

appropriately shifted, assume data is available at

(Covariance method)

Solution 2:

assume

and

augment the sum to

(autocorrelation method).


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In the 2nd case

only depends on


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Block Toeplitz.

Structure allows for fast computations

AR model of order n


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Levinson Algorithm

definition


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def. of

flip flip

add 1st + 2nd

Flip:

Clearly


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Initial conditions

good reduction.


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Numerical Methods

Both have no analytical

solutions in general

General Procedure

step size

direction of the search depends on


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Different Methods

f depends on

f depends on

f depends on (Newtons)

Newtons

Quasi Newton: Approximate


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Special Schemes: Nonlinear Least Squares

A family of Algorithms

step size, chosen so that


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Newtons

negligible around min.

Newton-Gauss, Newton-Raphson


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gradient steepest descent

For instrumental method

Newton-Raphson

Computing the gradient:

ARMAX:

diff. with respect to


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diff. w. r. to

diff. w. r. to

Recall


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Re-arrange

dep. on , however assumed stable

Of course a special case for ARX:

Exercise

Jenkins Black-Box model

State-space model


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Recursive Methods

Computational advantages

Carry the covariance matrix in the estimate.

General form:

Specific form:


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Least-square estimate as a recursive estimate

Off line


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Recursive Identification (LS)

Assume

Example exponential weight

means in general


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A recursive algorithm

Normalized Gain estimate:


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Recursive Algorithm:


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Properties of Recursive Algorithms

Need to store and to compute the next estimate

is the covariance (an estimate) of and hence gives an

estimate of the accuracy of [Recall that ]

is symmetric, so you only need to store the lower part of it.

(Save on memory)


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Recursive Algorithms withEfficient Matrix Conversion

Define

Inversion formula

Consider the matrix


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Recursive Algorithm:

Advantage: no need to compute at each iteration

is iterated directly!

Kalman filter interpretation

is the gain

is the solution of the Riccati equation


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Recursive Instrumental Variable Method

For a fixed, not model dependent Instrument,


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You can show:

where satisfies (by assumption)


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Adaptive Control

P

Estimator (recursive)

Controller

r is a bdd input.

Estimator: Std least squares

Controller : Condition is a stable time-varying system.

is bold for any bdd


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System Identification 6.435Computation

Lecture Notes- MIT- System Identification

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