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gram

Controllability and observability Gramians

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Syntax

Wc = gram(sys,'c')
Wo = gram(sys,'o')
Rc = gram(sys,'cf')
Ro = gram(sys,'of')
W = gram(___,opt)

Description

Use gram to construct the controllability and observability Gramians. You can use Gramians to study the controllability and observability properties of state-space models and for model reduction [1]. They have better numerical properties than the controllability and observability matrices formed by ctrb and obsv.

Wc = gram(sys,'c') calculates the controllability Gramian of the state-space model sys.

Wo = gram(sys,'o') calculates the observability Gramian of the state-space model sys.

Rc = gram(sys,'cf') returns the Cholesky factor of the controllability Gramian.

Ro = gram(sys,'of') returns the Cholesky factor of the observability Gramian.

W = gram(___,opt) calculates time-limited or frequency-limited Gramians. opt is an option set that specifies time or frequency intervals for the computation. Create opt using the gramOptions command.

example

Examples

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Open Live Script

Compute the controllability Gramian of the following state-space model. Focus the computation on the frequency interval with the most energy.

sys = ss([-.1 -1;1 0],[1;0],[0 1],0);

The model contains a peak at 1 rad/s. Use gramOptions to specify an interval around that frequency.

opt = gramOptions('FreqIntervals',[0.8 1.2]);
gc = gram(sys,'c',opt)
gc = 2×2

    4.2132   -0.0000
   -0.0000    4.2433

Input Arguments

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Input model, specified as a state-space model or an array of state-space models.

The input model must be stable and without any internal delays. For descriptor state-space models, the matrix E must be nonsingular.

Option set for computing time-limited or frequency-limited Gramians, specified as a gramOptions object.

Output Arguments

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Controllability or observability Gramian, returned as a matrix.

Cholesky factor of controllability or observability Gramian, returned as a matrix.

The Cholesky factors of Gramians are defined as follows:

  • Controllability Gramian — Wc=RcTRc

  • Observability Gramian — Wo=RoTRo

Limitations

The A matrix must be stable (all eigenvalues have negative real part in continuous time, and magnitude strictly less than one in discrete time).

More About

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Controllability and Observability Gramians

Given the continuous-time state-space model

x˙=Ax+Buy=Cx+Du

the controllability Gramian is defined by

Wc=∫0∞eAτBBTeATτdτ

The controllability Gramian is positive definite if and only if (A, B) is controllable.

The observability Gramian is defined by

Wo=∫0∞eATτCTCeAτdτ

The observability Gramian is positive definite if and only if (A, C) is observable.

The discrete-time counterparts of the controllability and observability Gramians are:

Wc=∑k=0∞AkBBT(AT)k,Wo=∑k=0∞(AT)kCTCAk

Use time-limited or frequency-limited Gramians to examine the controllability or observability of states within particular time or frequency intervals. The definition of these Gramians is as described in [2].

Algorithms

The controllability Gramian Wc is obtained by solving the continuous-time Lyapunov equation

AWc+WcAT+BBT=0

or its discrete-time counterpart

AWcAT−Wc+BBT=0

Similarly, the observability Gramian Wo solves the Lyapunov equation

ATWo+WoA+CTC=0

in continuous time, and the Lyapunov equation

ATWoA−Wo+CTC=0

in discrete time.

The computation of time-limited and frequency-limited Gramians is as described in [2].

References

[1] Kailath, Thomas. Linear Systems. Prentice-Hall Information and System Science Series. Englewood Cliffs, N.J: Prentice-Hall, 1980.

[2] Gawronski, Wodek, and Jer-Nan Juang. “Model Reduction in Limited Time and Frequency Intervals.” International Journal of Systems Science 21, no. 2 (February 1990): 349–76. https://doi.org/10.1080/00207729008910366.

Version History

Introduced before R2006a

See Also

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