Balanced model truncation via square root method

Algorithms

Given a state space (A,B,C,D) of a system
and k, the desired reduced order, the following
steps will produce a similarity transformation to truncate the original
state-space system to the kth order
reduced model.

  1. Find the SVD of the controllability
    and observability grammians

    P = Up ΣpVpT

    Q = UqΣqVqT

  2. Find the square root of the grammians
    (left/right eigenvectors)

    Lp = Up Σp½

    Lo =
    U
    q Σq½

  3. Find the SVD of (LoTLp)

    LoT Lp=
    U
    Σ VT

  4. Then the left and right transformation
    for the final kth order
    reduced model is

    SL,BIG =
    Lo
    U(:,1:k)
    Σ(1;k,1:k))–½

    SR,BIG =
    Lp
    V(:,1:k)
    Σ(1;k,1:k))–½

  5. Finally,

The proof of the square root balance truncation algorithm can
be found in [2].

Description

balancmr returns a reduced
order model GRED of G and a
struct array redinfo containing the error bound
of the reduced model and Hankel singular values of the original system.

The error bound is computed based on Hankel singular values
of G. For a stable system these values indicate
the respective state energy of the system. Hence, reduced order can
be directly determined by examining the system Hankel singular values, σι.

With only one input argument G, the function
will show a Hankel singular value plot of the original model and prompt
for model order number to reduce.

This method guarantees an error bound on the infinity norm of
the additive error G-GRED
∞ for well-conditioned model reduced problems [1]:

This table describes input arguments for balancmr.

Argument

Description

G

LTI model to be reduced. Without any other inputs, balancmr will
plot the Hankel singular values of G and prompt
for reduced order

ORDER

(Optional) Integer for the desired order of the reduced
model, or optionally a vector packed with desired orders for batch
runs

A batch run of a serial of different reduced order models can
be generated by specifying order = x:y, or a vector
of positive integers. By default, all the anti-stable part of a system
is kept, because from control stability point of view, getting rid
of unstable state(s) is dangerous to model a system.

'MaxError' can be specified in the
same fashion as an alternative for 'Order'.
In this case, reduced order will be determined when the sum of the
tails of the Hankel singular values reaches the 'MaxError'.

This table lists the input arguments 'key' and
its 'value'.

Argument

Value

Description

'MaxError'

Real number or vector of different errors

Reduce to achieve H error. When present,
'MaxError' overrides
ORDER input.

'Weights'

{Wout,Win} cell array

Optional 1-by-2 cell array of LTI weights Wout (output) and
Win (input). The weights must be stable, minimum phase and invertible.
When you supply these weights, balancmr finds the reduced model that
minimizes the Hankel norm of

You can use weighting functions to make the model reduction algorithm
focus on frequency bands of interest. See:

As an alternative, you can use balred to focus model reduction on a particular frequency band without defining
a weighting function. Using balancmr and providing your own weighting
functions allows more precise control over the error profile.

Default weights are both identity.

'Display'

'on'' or 'off'

Display Hankel singular plots (default 'off').

'Order'

Integer, vector or cell array

Order of reduced model. Use only if not specified as
2nd argument.

This table describes output arguments.

Argument

Description

GRED

LTI reduced order model. Becomes multidimensional array
when input is a serial of different model order array

REDINFO

A STRUCT array with three fields:

  • REDINFO.ErrorBound (bound on ∥ G-GRED ∥∞)

  • REDINFO.StabSV (Hankel SV of stable
    part of G)

  • REDINFO.UnstabSV (Hankel SV of
    unstable part of G)

G can be stable or unstable, continuous or
discrete.

References

[1] Glover, K., “All Optimal Hankel Norm
Approximation of Linear Multivariable Systems, and Their Lµ-error
Bounds,“ Int. J. Control, Vol. 39, No. 6, 1984, p. 1145-1193

[2] Safonov, M.G., and R.Y. Chiang, “A
Schur Method for Balanced Model Reduction,” IEEE
Trans. on Automat. Contr.
, Vol. 34, No. 7, July 1989,
p. 729-733

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