National Instruments Xmath

Chapter 2 Additive Error Reduction

of the balanced system occurs, (assuming nsr is less than the number of

states). Thus, if the state-space representation of the balanced system is

with A11 possessing dimension nsr × nsr, B1 possessing nsr rows and C1

possessing nsr columns, the reduced order system SysR is:

The following error formula is relevant:

It is this error bound which is the basis of the determination of the order

ofthe reduced system when the keyword bound is specified. If the error

bound sought is smaller than , then no reduction is possible which is

consistent with the error bound. If it is larger than , then th e constant

transfer function matrix D achieves the bound.

For continuous systems, the actual approximation error depends on

frequency, but is always zero at ω = ∞. In practice it is often greatest at

ω=0 ; if the reduction of state dimension is 1, the error bound is exact, with

the maximum error occurring at DC. The bound also is exact in the special

case of a single-input, single-output transfer function which has poles and

zero alternating along the negative real axis. It is far from exact when the

poles and zeros approximately alternate along the imaginary axis (with the

poles stable).

AA11 A12

A21 A22

=BB1

=CC1C2

·1A11x1B1u+=

1xDu+=

(continuous) (discrete)

x1k1+()A11x1k() B1uk()+=

yk() C1x1k() Du k()+=

CjωIA–()

1–

[]C1jωIA

–()

1–B1D+()[]–∞

2σnsr 1+σnsr 2+... σns

+++[]≤

(continuous)

jωIA–()

1–BD+[]C1ejωIA

–()

1–B1D+[]–∞

2σnsr 1+σnsr 2+... σns

+++[]≤

(discrete)

2σn

2trΣ