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getDelayModel

State-space representation of internal delays

    Description

    example

    [H,tau] = getDelayModel(sys) decomposes a state-space model, sys, with internal delays into a delay-free state-space model, H, and a vector of internal delays, tau. The relationship among sys, H, and tau is shown in the following diagram.

    [A,B1,B2,C1,C2,D11,D12,D21,D22,E,tau] = getDelayModel(sys) returns the set of state-space matrices and internal delay vector that explicitly describe the state-space model sys.

    Examples

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    Decompose the following closed-loop system with internal delay into a delay-free component and a component representing the internal delay.

    Create the closed-loop model sys from r to y.

    G = tf(1,[1 10],'InputDelay',2.1); 
    C = pid(0.5,2.3);
    sys = feedback(C*G,1);

    sys is a state-space (ss) model with an internal delay that arises from closing the feedback loop on a plant with an input delay.

    Decompose sys into a delay-free state-space model and the value of the internal delay.

    [H,tau] = getDelayModel(sys);

    Confirm that the internal delay matches the original input delay on the plant.

    tau
    tau = 2.1000
    

    Input Arguments

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    State-space model, specified as one of the following objects.

    • ss — Numeric state-space model

    • sparss — Sparse first-order state-space model

    • mechss — Sparse second-order state-space model

    Output Arguments

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    Delay-free state-space model, returned as a state-space object of the same type as sys). H results from decomposing sys into a delay-free component and a component exp(-tau*s) that represents all internal delays.

    If sys has no internal delays, H is equal to sys.

    Internal delays of sys, returned as a vector. tau is expressed in the time units of sys. tau results from decomposing sys into a delay-free state-space model H and a component exp(-tau*s) that represents all internal delays.

    If sys has no internal delays, tau is empty.

    State-space matrices that, along with the internal delay vector tau, describe the state-space model sys according to the following equations.

    • Continuous-time sys:

      Edx(t)dt=Ax(t)+B1u(t)+B2w(t)y(t)=C1x(t)+D11u(t)+D12w(t)z(t)=C2x(t)+D21u(t)+D22w(t)w(t)=z(tτ)

    • Discrete-time sys:

      Ex[k+1]=Ax[k]+B1u[k]+B2w[k]y[k]=C1x[k]+D11u[k]+D12w[k]z[k]=C2x[k]+D21u[k]+D22w[k]w[k]=z[kτ]

    For explicit state-space models (E = I or sys.e = []), the output E is [].

    If sys has no internal delays, the outputs B2, C2, D12, D21, and D22 are all empty ([]).

    Version History

    Introduced in R2006a