xsort

Sort states based on state partition

Since R2020b

Syntax

xsys = xsort(sys)

Description

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

Sparse Second-Order Model in a Feedback Loop

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 degrees of freedom.

Use "spy" and "showStateInfo" to inspect model structure. 
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)

Physical Connections Between Components in Sparse Second-Order Model

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')
model = ...
   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');
sys = model;

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 degree of freedom (DOF) index data from dofData.mat and use interface to create the physical connections between the two plates and the four pillars. dofs is a 6x7 cell array where the first two rows contain DOF index data for the first and second plates while the remaining four rows contain index data for the four pillars. By default, the function uses dual-assembly method of physical coupling.

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

Specify connection between the bottom plate and the ground.

sysConDual = interface(sys,"Plate2",dofs{2,7});

Use showStateInfo to confirm the physical interfaces.

showStateInfo(sysConDual)

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(sysConDual)

Now, specify physical connections using the primal-assembly method.

sys = model;
for i=3:6
   sys = interface(sys,"Plate1",dofs{1,i},"Pillar"+i,dofs{i,1},'primal');
   sys = interface(sys,"Plate2",dofs{2,i},"Pillar"+i,dofs{i,2},'primal');
end
sysConPrimal = interface(sys,"Plate2",dofs{2,7},'primal');

Use showStateInfo to confirm the physical interfaces.

showStateInfo(sysConPrimal)

The state groups are:

    Type        Name      Size
  ----------------------------
  Component    Plate1     2646
  Component    Plate2     2640
  Component    Pillar3     108
  Component    Pillar4     108
  Component    Pillar5     108
  Component    Pillar6     108

Primal assembly eliminates half of the redundant DOFs associated with the shared set of DOFs in the global finite element mesh.

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

spy(sysConPrimal)

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

Input Arguments

sys — Sparse state-space model

sparss model object | mechss model object

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

Output Arguments

xsys — Sparse state-space model with sorted components

sparss model object | mechss model object

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 $$s\text{E}-\text{A}$$ and $$\text{M }{s}^2+\text{C }s+\text{K}$$ 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.

Version History

Introduced in R2020b