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xsort

Sort states based on state partition

    Description

    example

    xsys = xsort(sys) sorts the x or q vector based on the state partition. Signal-based connections and physical interfaces between model components gives rise to differential algebraic equation (DAE) models where some internal signals and forces become extra states. The StateInfo property of sparss and mechss model objects keeps track of the state partition into sub-components, interface variables, and signal variables.

    Examples

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    For this example, consider sparseSOSignal.mat that contains a sparse second-order model. Define an actuator, sensor, and controller and connect them together with the plant in a feedback loop.

    Load the sparse matrices and create the mechss object.

    load sparseSOSignal.mat
    plant = mechss(M,C,K,B,F,[],[],'Name','Plant');

    Next, create an actuator and sensor using transfer functions.

    act = tf(1,[1 0.5 3],'Name','Actuator');
    sen = tf(1,[0.02 7],'Name','Sensor');

    Create a PID controller object for the plant.

    con = pid(1,1,0.1,0.01,'Name','Controller');

    Use the feedback command to connect the plant, sensor, actuator, and controller in a feedback loop.

    sys = feedback(sen*plant*act*con,1)
    Sparse continuous-time second-order model with 1 outputs, 1 inputs, and 7111 nodes.
    
    Use "spy" and "showStateInfo" to inspect model structure. 
    Type "properties('mechss')" for a list of model properties. 
    Type "help mechssOptions" for available solver options for this model.
    

    The resultant system sys is a mechss object since mechss objects take precedence over all other model object types.

    Use showStateInfo to view the component and signal groups.

    showStateInfo(sys)
    The state groups are:
    
        Type          Name       Size
      -------------------------------
      Component      Sensor         1
      Component      Plant       7102
      Signal                        1
      Component     Actuator        2
      Signal                        1
      Component    Controller       2
      Signal                        1
      Signal                        1
    

    Use xsort to sort the components and signals, and then view the component and signal groups.

    sysSort = xsort(sys);
    showStateInfo(sysSort)
    The state groups are:
    
        Type          Name       Size
      -------------------------------
      Component      Sensor         1
      Component      Plant       7102
      Component     Actuator        2
      Component    Controller       2
      Signal                        4
    

    Observe that the components are now ordered before the signal partition. The signals are now sorted and grouped together in a single partition.

    You can also visualize the sparsity pattern of the resultant system using spy.

    spy(sysSort)

    For this example, consider a structural model that consists of two square plates connected with pillars at each vertex as depicted in the figure below. The lower plate is attached rigidly to the ground while the pillars are attached rigidly to each vertex of the square plate.

    Load the finite element model matrices contained in platePillarModel.mat and create the sparse second-order model representing the above system.

    load('platePillarModel.mat')
    sys = ...
       mechss(M1,[],K1,B1,F1,'Name','Plate1') + ...
       mechss(M2,[],K2,B2,F2,'Name','Plate2') + ...
       mechss(Mp,[],Kp,Bp,Fp,'Name','Pillar3') + ...
       mechss(Mp,[],Kp,Bp,Fp,'Name','Pillar4') + ...
       mechss(Mp,[],Kp,Bp,Fp,'Name','Pillar5') + ...
       mechss(Mp,[],Kp,Bp,Fp,'Name','Pillar6');

    Use showStateInfo to examine the components of the mechss model object.

    showStateInfo(sys)
    The state groups are:
    
        Type        Name      Size
      ----------------------------
      Component    Plate1     2646
      Component    Plate2     2646
      Component    Pillar3     132
      Component    Pillar4     132
      Component    Pillar5     132
      Component    Pillar6     132
    

    Now, load the interfaced node index data from nodeData.mat and use interface to create the physical connections between the two plates and the four pillars. nodes is a 6x7 cell array where the first two rows contain node index data for the first and second plates while the remaining four rows contain index data for the four pillars.

    load('nodeData.mat','nodes')
    for i=3:6
       sys = interface(sys,"Plate1",nodes{1,i},"Pillar"+i,nodes{i,1});
       sys = interface(sys,"Plate2",nodes{2,i},"Pillar"+i,nodes{i,2});
    end

    Specify connection between the bottom plate and the ground.

    sysCon = interface(sys,"Plate2",nodes{2,7});

    Use showStateInfo to confirm the physical interfaces.

    showStateInfo(sysCon)
    The state groups are:
    
        Type            Name         Size
      -----------------------------------
      Component        Plate1        2646
      Component        Plate2        2646
      Component       Pillar3         132
      Component       Pillar4         132
      Component       Pillar5         132
      Component       Pillar6         132
      Interface    Plate1-Pillar3      12
      Interface    Plate2-Pillar3      12
      Interface    Plate1-Pillar4      12
      Interface    Plate2-Pillar4      12
      Interface    Plate1-Pillar5      12
      Interface    Plate2-Pillar5      12
      Interface    Plate1-Pillar6      12
      Interface    Plate2-Pillar6      12
      Interface    Plate2-Ground        6
    

    You can use spy to visualize the sparse matrices in the final model.

    spy(sysCon)

    The data set for this example was provided by Victor Dolk from ASML.

    Input Arguments

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    Sparse state-space model, specified as a sparss or mechss model object.

    Output Arguments

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    Sparse state-space model with sorted components, returned as a sparss or mechss model object. In the sorted xsys, all components appear first, followed by the interfaces, and then followed by a single group of all internal signals. The matrices sEA and s2+s+K have the following block arrow structure:

    Here, each diagonal block is a sub-component of sys. The last row and column combines the Interface and Signal groups to capture all couplings and connections between components.

    Introduced in R2020b