tzero

Invariant zeros of linear system

Syntax

z = tzero(sys)

z = tzero(A,B,C,D,E)

z = tzero(___,tol)

[z,nrank] = tzero(___)

Description

z = tzero(sys) returns the invariant zeros of MIMO dynamic system, sys. If sys is a minimal realization, the invariant zeros coincide with the transmission zeros of sys.

z = tzero(A,B,C,D,E) returns the invariant zeros of the state-space model described by matrices A, B, C, D, and E.

z = tzero(___,tol) specifies the relative tolerance controlling rank decisions for any of the previous syntaxes.

[z,nrank] = tzero(___) returns the normal rank of the transfer function of sys or of the transfer function H(s) = D + C(sE – A)^–1B.

Examples

Find Transmission Zeros of MIMO Transfer Function

Create a MIMO transfer function, and locate its invariant zeros.

s = tf('s'); 
H = [1/(s+1) 1/(s+2);1/(s+3) 2/(s+4)];
z = tzero(H)

z = 2×1 complex

  -2.5000 + 1.3229i
  -2.5000 - 1.3229i

The output is a column vector listing the locations of the invariant zeros of H. This output shows that H has a complex pair of invariant zeros. Confirm that the invariant zeros coincide with the transmission zeros.

Check whether the first invariant zero is a transmission zero of H.

If z(1) is a transmission zero of H, then H drops rank at s = z(1).

H1 = evalfr(H,z(1));
svd(H1)

ans = 2×1

    1.5000
    0.0000

H1 is the transfer function, H, evaluated at s = z(1). H1 has a zero singular value, indicating that H drops rank at that value of s. Therefore, z(1) is a transmission zero of H.

A similar analysis shows that z(2) is also a transmission zero.

Identify Unobservable and Uncontrollable Modes of MIMO Model

Obtain a MIMO model.

load ltiexamples gasf
size(gasf)

State-space model with 4 outputs, 6 inputs, and 25 states.

gasf is a MIMO model that might contain uncontrollable or unobservable states.

To identify the unobservable and uncontrollable modes of gasf, you need the state-space matrices A, B, C, and D of the model. tzero does not scale state-space matrices. Therefore, use prescale with ssdata to scale the state-space matrices of gasf.

[A,B,C,D] = ssdata(prescale(gasf));

Identify the uncontrollable states of gasf.

 uncon = tzero(A,B,[],[])

uncon = 6×1

   -0.0568
   -0.0568
   -0.0568
   -0.0568
   -0.0568
   -0.0568

When you provide A and B matrices to tzero, but no C and D matrices, the command returns the eigenvalues of the uncontrollable modes of gasf. The output shows that there are six degenerate uncontrollable modes.

Identify the unobservable states of gasf.

unobs = tzero(A,[],C,[])

unobs =

  0x1 empty double column vector

When you provide A and C matrices, but no B and D matrices, the command returns the eigenvalues of the unobservable modes. The empty result shows that gasf contains no unobservable states.

Input Arguments

sys — MIMO dynamic system

dynamic system model

MIMO dynamic system, specified as one of the following dynamic system models.

• Continuous-time or discrete-time numeric LTI models, such as tf, zpk, or ss models.

• Generalized or uncertain LTI models such as genss or uss (Robust Control Toolbox) models. (Using uncertain models requires Robust Control Toolbox™ software.)

• Frequency-response data models, such as frd models.

• Identified LTI models, such as idtf (System Identification Toolbox), idss (System Identification Toolbox), or idproc (System Identification Toolbox) models. (Using identified models requires System Identification Toolbox™ software.)

If sys is not a state-space model, then tzero computes tzero(ss(sys)).

A,B,C,D,E — State-space matrices

matrices

State-space matrices describing the following linear system.

$$\begin{array}{c}E\frac{dx}{dt}=Ax+Bu\\ y=Cx+Du.\end{array}$$

tzero does not scale the state-space matrices when you use the syntax. To scale the matrices before using tzero, prescale.

Omit E for an explicit state-space model, where E = I.

tol — Relative tolerance controlling rank decisions

eps^(3/4) (default) | positive scalar

Relative tolerance controlling rank decisions, specified as a positive scalar. Increasing tolerance helps detect nonminimal modes and eliminate very large zeros (near infinity). However, increased tolerance might artificially increase the number of transmission zeros.

Output Arguments

z — Invariant zeros

column vector

Column vector containing the invariant zeros of sys or the state-space model described by A,B,C,D,E.

nrank — Normal rank

positive integer

Normal rank of the transfer function of sys or of the transfer function H(s) = D + C(sE – A)^–1B. The normal rank is the rank for values of s other than the transmission zeros.

To obtain a meaningful result for nrank, the matrix s*E-A must be regular (invertible for most values of s). In other words, sys or the system described by A,B,C,D,E must have a finite number of poles.

More About

Invariant zeros

For a MIMO state-space model

$$\begin{array}{c}E\frac{dx}{dt}=Ax+Bu\\ y=Cx+Du,\end{array}$$

the invariant zeros are the complex values of s for which the rank of the system matrix

$$\left[\begin{array}{cc}A-sE& B\\ C& D\end{array}\right]$$

drops from its normal value. (For explicit state-space models, E = I).

Transmission zeros

For a MIMO state-space model

$$\begin{array}{c}E\frac{dx}{dt}=Ax+Bu\\ y=Cx+Du,\end{array}$$

the transmission zeros are the complex values of s for which the rank of the equivalent transfer function H(s) = D + C(sE – A)^–1B drops from its normal value. (For explicit state-space models, E = I.)

Transmission zeros are a subset of the invariant zeros. For minimal realizations, the transmission zeros and invariant zeros are identical.

Tips

• You can use the syntax z = tzero(A,B,C,D,E) to find the uncontrollable or unobservable modes of a state-space model. When C and D are empty or zero, tzero returns the uncontrollable modes of (A-sE,B). Similarly, when B and D are empty or zero, tzero returns the unobservable modes of (C,A-sE). For an example, see Identify Unobservable and Uncontrollable Modes of MIMO Model.

Algorithms

tzero is based on SLICOT routines AB08ND, AB08NZ, AG08BD, and AG08BZ. tzero implements the algorithms in [1] and [2].

References

[1] Emami-Naeini, A. and P. Van Dooren, "Computation of Zeros of Linear Multivariable Systems," Automatica, 18 (1982), pp. 415–430.

[2] Misra, P, P. Van Dooren, and A. Varga, "Computation of Structural Invariants of Generalized State-Space Systems," Automatica, 30 (1994), pp. 1921-1936.

Version History

Introduced in R2012a