xelim

Eliminate states from state-space models

Since R2023b

Syntax

rsys = xelim(sys,elim)

rsys = xelim(sys,elim,method)

Description

rsys = xelim(sys,elim) simplifies the state-space model sys by eliminating the states specified in the vector elim. The full state vector x is partitioned as x = [x₁;x₂] where x₁ is the reduced state vector and x₂ is eliminated.

This function is useful to eliminate states known to settle quickly (fast modes) or contribute little to the input/output map. When you don’t know which states to eliminate, use reducespec and the model-order reduction workflow.

rsys = xelim(sys,elim,method) also specifies the elimination method. Set method to either "MatchDC" or "Truncate".

Examples

Order Reduction by Matched-DC-Gain and Direct-Deletion Methods

Consider the following continuous fourth-order model.

$$h(s)=\frac{s^3+11s^2+36s+26}{s^4+14.6s^3+74.96s^2+153.7s+99.65}.$$

To reduce its order, first compute a balanced state-space realization with balreal.

h = tf([1 11 36 26],[1 14.6 74.96 153.7 99.65]);
[hb,g] = balreal(h);

Examine the Gramians.

g'

ans = 1×4

    0.1394    0.0095    0.0006    0.0000

The last three diagonal entries of the balanced Gramians are relatively small. Eliminate these three least-contributing states with xelim, using both matched DC gain and direct-deletion methods.

hmdc = xelim(hb,2:4,"MatchDC");
hdel = xelim(hb,2:4,"Truncate");

Both hmdc and hdel are first-order models. Compare their Bode responses against that of the original model.

op = bodeoptions;
op.PhaseMatching = "on";
bodeplot(h,hmdc,'r--',hdel,'k-.',op)
legend("Original","State elimination (match DC)",...
    "State elimination (truncate)")

The reduced-order model hdel is clearly a better frequency-domain approximation of h. Now compare the step responses.

stepplot(h,hmdc,'r--',hdel,'k-.')
legend("Original","State elimination (match DC)",...
    "State elimination (truncate)",Location="southeast")

While hdel accurately reflects the transient behavior, only hmdc gives the true steady-state response.

For faster and more accurate results, use reducespec for model reduction workflows.

Input Arguments

sys — Dynamic system model

dynamic system model

Dynamic system model, specified as an ordinary or sparse LTI model.

The input model must have a valid state-space representation, such as tf, ss, sparss, mechss models. For generalized or uncertain state-space models (genss, uss), the function uses the current value of the model. For identified models (idss), the function uses the identified value.

elim — State elimination vector

vector

State elimination vector, specified as one of these.

• A vector containing index values of states you want to discard.

• A vector of logical values of the same size as the number of states, where the true (1) values specifies the states you want to discard.

method — State elimination method

"MatchDC" (default) | "Truncate"

State elimination method, specified as "MatchDC" or "Truncate". This argument specifies how the function eliminates the states with weak contribution.

• "MatchDC" — Enforce matching DC gains. To do so, the algorithm treats x₂ as infinitely fast and sets its derivative to zero. The resulting algebraic equation is used to express x₂ in terms of x₁ and eliminate it. For details, see Algorithms

• "Truncate" — Simply drop x₂, and use x₁ as reduced state.

The "Truncate" option tends to produce a better approximation in the frequency domain, but the DC gains are not guaranteed to match.

Output Arguments

rsys — Reduced-order model

state-space model

Reduced-order model, returned as a state-space model.

Algorithms

For a state-space model

$$\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du\end{array}$$

the function partitions the state vector into x₁ (to keep) and x₂ (to eliminate).

$$\begin{array}{l}\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right]u\\ y=\left[\begin{array}{cc}{C}_1& C_2\end{array}\right]x+Du\end{array}$$

"MatchDC" Method

For continuous-time models, this method sets the derivative of x₂ to zero and solves the resulting equation for x₁. The reduced-order model is given by

$$\begin{array}{l}{\dot{x}}_1=\left[A_{11}-A_{12}{A}_{22}{}^{-1}{A}_{21}\right]x_1+\left[B_1-A_{12}{A}_{22}{}^{-1}{B}_2\right]u\\ y=\left[C_1-C_2{A}_{22}{}^{-1}{A}_{21}\right]x+\left[D-C_2{A}_{22}{}^{-1}{B}_2\right]u\end{array}$$

Similarly, for discrete-time models, the algorithm sets $$x_2[n+1]=x_2[n]$$ to recompute the matrices.

xelim returns a scaled version of this realization. To disable this scaling, set sys.Scaled to true before eliminating the states.

"Truncate" Method

For this method, the algorithm simply drops x₂, and uses x₁ as reduced state. The reduced-order model is given by

$$\begin{array}{l}{\dot{x}}_1=A_{11}{x}_1+B_{1}u\\ y=C_1{x}_1\end{array}$$

xelim returns a scaled version of this realization. To disable this scaling, set sys.Scaled to true before eliminating the states.

Version History

Introduced in R2023b