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System Identification as used in control engineering for the purpose of treating stochastic systems
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System Identification
6.435
SET 11
Computation
Levinson Algorithm
Recursive Estimation
Munther A. Dahleh
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
1
Computation
Least Squares: QR factorization
Q is invertible and
error
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Initial Conditions:
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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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).
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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In the 2nd case
only depends on
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Block Toeplitz.
Structure allows for fast computations
AR model of order n
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Levinson Algorithm
definition
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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def. of
flip flip
add 1st + 2nd
Flip:
Clearly
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Initial conditions
good reduction.
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Numerical Methods
Both have no analytical
solutions in general
General Procedure
step size
direction of the search depends on
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Different Methods
f depends on
f depends on
f depends on (Newtons)
Newtons
Quasi Newton: Approximate
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Special Schemes: Nonlinear Least Squares
A family of Algorithms
step size, chosen so that
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Newtons
negligible around min.
Newton-Gauss, Newton-Raphson
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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gradient steepest descent
For instrumental method
Newton-Raphson
Computing the gradient:
ARMAX:
diff. with respect to
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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diff. w. r. to
diff. w. r. to
Recall
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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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
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Recursive Methods
Computational advantages
Carry the covariance matrix in the estimate.
General form:
Specific form:
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Least-square estimate as a recursive estimate
Off line
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Recursive Identification (LS)
Assume
Example exponential weight
means in general
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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A recursive algorithm
Normalized Gain estimate:
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Recursive Algorithm:
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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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)
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Recursive Algorithms withEfficient Matrix Conversion
Define
Inversion formula
Consider the matrix
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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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
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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Recursive Instrumental Variable Method
For a fixed, not model dependent Instrument,
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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You can show:
where satisfies (by assumption)
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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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
Lecture 11 6.435, System Identification Prof. Munther A. Dahleh
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System Identification 6.435Computation