Sparse first-order state-space model

## Description

Use sparss to represent sparse descriptor state-space models using matrices obtained from your finite element analysis (FEA) package. FEA involves the concept of dynamic substructuring where a mechanical system is partitioned into components that are modeled separately. These components are then coupled using rigid or semi-rigid physical interfaces that express consistency of displacements and equilibrium of internal forces. The resultant matrices from this type of modeling are quite large with a sparse pattern. Hence, using sparss is an efficient way to represent such large sparse state-space models in MATLAB® to perform linear analysis. You can also use sparss to convert a second-order mechss model object to a sparss object.

You can use sparss model objects to represent SISO or MIMO state-space models in continuous time or discrete time. In continuous time, a first-order sparse state-space model is represented in the following form:

Here, x, u and y represent the states, inputs and outputs respectively, while A, B, C, D and E are the state-space matrices. The sparss object represents a state-space model in MATLAB storing sparse matrices A, B, C, D and E along with other information such as sample time, names and delays specific to the inputs and outputs.

You can use a sparss object to:

• Perform time-domain and frequency-domain response analysis.

• Specify signal-based connections with other LTI models.

• Transform models between continuous-time and discrete-time representations.

## Creation

### Description

example

sys = sparss(A,B,C,D,E) creates a continuous-time first-order sparse state-space model object of the following form:

For instance, consider a plant with Nx states, Ny outputs, and Nu inputs. The first-order state-space matrices are:

• A is the sparse state matrix with Nx-by-Nx real- or complex-values.

• B is the sparse input-to-state matrix with Nx-by-Nu real- or complex-values.

• is the sparse state-to-output matrix with Ny-by-Nx real- or complex-values.

• D is the sparse gain or input-to-output matrix with Ny-by-Nu real- or complex-values.

• E is the sparse mass matrix with the same size as matrix A. When E is omitted, sparss populates E with an identity matrix.

example

sys = sparss(A,B,C,D,E,ts) creates a discrete-time sparse state-space model with sample time ts with the form:

To leave the sample time unspecified, set ts to -1. When E is an identity matrix, you can set E as [] or omit E as long as A is not a scalar.

example

sys = sparss(D) creates a sparse state-space model that represents the static gain, D. The output state-space model is equivalent to sparss([],[],[],D,[]).

example

sys = sparss(___,Name,Value) sets properties of the first-order sparse state-space model using one or more name-value pair arguments. Use this syntax with any of the previous input-argument combinations.

example

sys = sparss(ltiSys) converts the dynamic system model ltiSys to a first-order sparse state-space model.

### Input Arguments

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State matrix, specified as an Nx-by-Nx sparse matrix, where Nx is the number of states. This input sets the value of property A.

Input-to-state matrix, specified as an Nx-by-Nu sparse matrix, where Nx is the number of states and Nu is the number of inputs. This input sets the value of property B.

State-to-output matrix, specified as an Ny-by-Nx sparse matrix, where Nx is the number of states and Ny is the number of outputs. This input sets the value of property C.

Input-to-output matrix, specified as an Ny-by-Nu sparse matrix, where Ny is the number of outputs and Nu is the number of inputs. This input sets the value of property D.

Mass matrix, specified as an Nx-by-Nx sparse matrix, where Nx is the number of states. This input sets the value of property E.

Sample time, specified as a scalar. For more information see the Ts property.

Dynamic system to convert to first-order sparse state-space form, specified as a SISO or MIMO dynamic system model or array of dynamic system models. Dynamic systems that you can convert include:

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

### Output Arguments

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Output system model, returned as a sparss model object.

## Properties

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State matrix, specified as an Nx-by-Nx sparse matrix, where Nx is the number of states.

Input-to-state matrix, specified as an Nx-by-Nu sparse matrix, where Nx is the number of states and Nu is the number of inputs.

State-to-output matrix, specified as an Ny-by-Nx sparse matrix, where Nx is the number of states and Ny is the number of outputs.

Input-to-output matrix, specified as an Ny-by-Nu sparse matrix, where Ny is the number of outputs and Nu is the number of inputs. D is also called as the static gain matrix which represents the ratio of the output to the input under steady state condition.

Mass matrix, specified as a Nx-by-Nx sparse matrix. E is the same size as A.

Differential algebraic equations (DAEs) are characterized by their differential index, which is a measure of their singularity. A linear DAE is of index ≤1 if it can be transformed by congruence to the following form with E11 and A22 being invertible matrices.

$\begin{array}{r}{E}_{11}{\stackrel{˙}{x}}_{1}={A}_{11}{x}_{1}+{A}_{12}{x}_{2}+{B}_{1}u\\ 0={A}_{21}{x}_{1}+{A}_{22}{x}_{2}+{B}_{2}u\end{array}$

The index of the DAE is 0 if x2 is empty. The index is 1 if x2 is not empty. In other words, a linear DAE has structural index ≤1 if it can be brought to the above form by row and column permutations of A and E. Some functionality, such as computing the impulse response of the system, is limited to DAEs with structural index less than 1.

For more information on DAE index, see Solve Differential Algebraic Equations (DAEs).

State partition information containing state vector components, interfaces between components and internal signal connecting components, specified as a structure array with the following fields:

• Type — Type includes component, signal or physical interface

• Name — Name of the component, signal or physical interface

• Size — Number of states or degrees of freedom in the partition

You can view the partition information of the sparse state-space model using showStateInfo. You can also sort and order the partitions in your sparse model using xsort.

Options for model analysis, specified as a structure with the following fields:

• UseParallel — Set this option true to enable parallel computing and false to disable it. Parallel computing is disabled by default. The UseParallel option requires a Parallel Computing Toolbox™ license.

• DAESolver — Use this option to select the type of Differential Algebraic Equation (DAE) solver. The available DAE solvers are:

• 'trbdf2' — Fixed-step solver with an accuracy of o(h^2), where h is the step size.[1]

• 'trbdf3' — Fixed-step solver with an accuracy of o(h^3), where h is the step size.

Reducing the step size increases accuracy and extends the frequency range where numerical damping is negligible. 'trbdf3' is about 50% more computationally intensive than 'trbdf2'

Internal delays in the model, specified as a vector. Internal delays arise, for example, when closing feedback loops on systems with delays, or when connecting delayed systems in series or parallel. For more information about internal delays, see Closing Feedback Loops with Time Delays.

For continuous-time models, internal delays are expressed in the time unit specified by the TimeUnit property of the model. For discrete-time models, internal delays are expressed as integer multiples of the sample time Ts. For example, InternalDelay = 3 means a delay of three sampling periods.

You can modify the values of internal delays using the property InternalDelay. However, the number of entries in sys.InternalDelay cannot change, because it is a structural property of the model.

Input delay for each input channel, specified as one of the following:

• Scalar — Specify the input delay for a SISO system or the same delay for all inputs of a multi-input system.

• Nu-by-1 vector — Specify separate input delays for input of a multi-input system, where Nu is the number of inputs.

For continuous-time systems, specify input delays in the time unit specified by the TimeUnit property. For discrete-time systems, specify input delays in integer multiples of the sample time, Ts.

Output delay for each output channel, specified as one of the following:

• Scalar — Specify the output delay for a SISO system or the same delay for all outputs of a multi-output system.

• Ny-by-1 vector — Specify separate output delays for output of a multi-output system, where Ny is the number of outputs.

For continuous-time systems, specify output delays in the time unit specified by the TimeUnit property. For discrete-time systems, specify output delays in integer multiples of the sample time, Ts.

Sample time, specified as:

• 0 for continuous-time systems.

• A positive scalar representing the sampling period of a discrete-time system. Specify Ts in the time unit specified by the TimeUnit property.

• -1 for a discrete-time system with an unspecified sample time.

Note

Changing Ts does not discretize or resample the model. To convert between continuous-time and discrete-time representations, use c2d and d2c. To change the sample time of a discrete-time system, use d2d.

Time variable units, specified as one of the following:

• 'nanoseconds'

• 'microseconds'

• 'milliseconds'

• 'seconds'

• 'minutes'

• 'hours'

• 'days'

• 'weeks'

• 'months'

• 'years'

Changing TimeUnit has no effect on other properties, but changes the overall system behavior. Use chgTimeUnit to convert between time units without modifying system behavior.

Input channel names, specified as one of the following:

• A character vector, for single-input models.

• A cell array of character vectors, for multi-input models.

• '', no names specified, for any input channels.

Alternatively, you can assign input names for multi-input models using automatic vector expansion. For example, if sys is a two-input model, enter the following:

sys.InputName = 'controls';

The input names automatically expand to {'controls(1)';'controls(2)'}.

You can use the shorthand notation u to refer to the InputName property. For example, sys.u is equivalent to sys.InputName.

Use InputName to:

• Identify channels on model display and plots.

• Extract subsystems of MIMO systems.

• Specify connection points when interconnecting models.

Input channel units, specified as one of the following:

• A character vector, for single-input models.

• A cell array of character vectors, for multi-input models.

• '', no units specified, for any input channels.

Use InputUnit to specify input signal units. InputUnit has no effect on system behavior.

Input channel groups, specified as a structure. Use InputGroup to assign the input channels of MIMO systems into groups and refer to each group by name. The field names of InputGroup are the group names and the field values are the input channels of each group. For example, enter the following to create input groups named controls and noise that include input channels 1 and 2, and 3 and 5, respectively.

sys.InputGroup.controls = [1 2];
sys.InputGroup.noise = [3 5];

You can then extract the subsystem from the controls inputs to all outputs using the following.

sys(:,'controls')

By default, InputGroup is a structure with no fields.

Output channel names, specified as one of the following:

• A character vector, for single-output models.

• A cell array of character vectors, for multi-output models.

• '', no names specified, for any output channels.

Alternatively, you can assign output names for multi-output models using automatic vector expansion. For example, if sys is a two-output model, enter the following.

sys.OutputName = 'measurements';

The output names automatically expand to {'measurements(1)';'measurements(2)'}.

You can also use the shorthand notation y to refer to the OutputName property. For example, sys.y is equivalent to sys.OutputName.

Use OutputName to:

• Identify channels on model display and plots.

• Extract subsystems of MIMO systems.

• Specify connection points when interconnecting models.

Output channel units, specified as one of the following:

• A character vector, for single-output models.

• A cell array of character vectors, for multi-output models.

• '', no units specified, for any output channels.

Use OutputUnit to specify output signal units. OutputUnit has no effect on system behavior.

Output channel groups, specified as a structure. Use OutputGroupto assign the output channels of MIMO systems into groups and refer to each group by name. The field names of OutputGroup are the group names and the field values are the output channels of each group. For example, create output groups named temperature and measurement that include output channels 1, and 3 and 5, respectively.

sys.OutputGroup.temperature = [1];
sys.InputGroup.measurement = [3 5];

You can then extract the subsystem from all inputs to the measurement outputs using the following.

sys('measurement',:)

By default, OutputGroup is a structure with no fields.

User-specified text that you want to associate with the system, specified as a character vector or cell array of character vectors. For example, 'System is MIMO'.

User-specified data that you want to associate with the system, specified as any MATLAB data type.

System name, specified as a character vector. For example, 'system_1'.

Sampling grid for model arrays, specified as a structure array.

Use SamplingGrid to track the variable values associated with each model in a model array.

Set the field names of the structure to the names of the sampling variables. Set the field values to the sampled variable values associated with each model in the array. All sampling variables must be numeric scalars, and all arrays of sampled values must match the dimensions of the model array.

For example, you can create an 11-by-1 array of linear models, sysarr, by taking snapshots of a linear time-varying system at times t = 0:10. The following code stores the time samples with the linear models.

sysarr.SamplingGrid = struct('time',0:10)

Similarly, you can create a 6-by-9 model array, M, by independently sampling two variables, zeta and w. The following code maps the (zeta,w) values to M.

[zeta,w] = ndgrid(<6 values of zeta>,<9 values of w>)
M.SamplingGrid = struct('zeta',zeta,'w',w)

By default, SamplingGrid is a structure with no fields.

## Object Functions

The following lists show functions you can use with sparss model objects.

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 mechss Sparse second-order state-space model getx0 Map initial conditions from a mechss object to a sparss object full Convert sparse models to dense storage imp2exp Convert implicit linear relationship to explicit input-output relation inv Invert models getDelayModel State-space representation of internal delays
 sparssdata Access first-order sparse state-space model data mechssdata Access second-order sparse state-space model data showStateInfo State vector map for sparse model spy Visualize sparsity pattern of a sparse model
 c2d Convert model from continuous to discrete time d2c Convert model from discrete to continuous time d2d Resample discrete-time model
 step Step response plot of dynamic system; step response data impulse Impulse response plot of dynamic system; impulse response data initial Initial condition response of state-space model lsim Plot simulated time response of dynamic system to arbitrary inputs; simulated response data bode Bode plot of frequency response, or magnitude and phase data nyquist Nyquist plot of frequency response nichols Nichols chart of frequency response sigma Singular value plot of dynamic system passiveplot Compute or plot passivity index as function of frequency dcgain Low-frequency (DC) gain of LTI system evalfr Evaluate frequency response at given frequency freqresp Frequency response over grid
 interface Specify physical connections between components of mechss model xsort Sort states based on state partition feedback Feedback connection of multiple models parallel Parallel connection of two models append Group models by appending their inputs and outputs connect Block diagram interconnections of dynamic systems lft Generalized feedback interconnection of two models (Redheffer star product) series Series connection of two models

## Examples

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For this example, consider sparseFOContinuous.mat which contains sparse matrices for a continuous-time sparse first-order state-space model.

Extract the sparse matrices from sparseFOContinuous.mat.

Sparse continuous-time state-space model with 1 outputs, 1 inputs, and 199 states.

Use "spy" and "showStateInfo" to inspect model structure.
Type "properties('sparss')" for a list of model properties.
Type "help sparssOptions" for available solver options for this model.

The output sys is a continuous-time sparss model object containing with 199 states, 1 input and 1 output.

You can use the spy command to visualize the sparsity of the sparss model object.

spy(sys)

For this example, consider sparseFODiscrete.mat which contains sparse matrices for a discrete-time sparse first-order state-space model.

Extract the sparse matrices from sparseFODiscrete.mat.

Sparse discrete-time state-space model with 1 outputs, 1 inputs, and 398 states.

Use "spy" and "showStateInfo" to inspect model structure.
Type "properties('sparss')" for a list of model properties.
Type "help sparssOptions" for available solver options for this model.

The output sys is a discrete-time sparss model object containing with 398 states, 1 input and 1 output.

You can use the spy command to visualize the sparsity of the sparss model object.

spy(sys)

You can also view model properties of the sparss model object.

A
B
C
D
E
Scaled
StateInfo
SolverOptions
InternalDelay
InputDelay
OutputDelay
Ts
TimeUnit
InputName
InputUnit
InputGroup
OutputName
OutputUnit
OutputGroup
Notes
UserData
Name
SamplingGrid

Create a static gain MIMO sparse first-order state-space model.

Consider the following three-input, two-output static gain matrix:

$D=\left[\begin{array}{ccc}1& 5& 7\\ 6& 3& 9\end{array}\right]$

Specify the gain matrix and create the static gain sparse first-order state-space model.

D = [1,5,7;6,3,9];
size(sys)
Sparse state-space model with 2 outputs, 3 inputs, and 0 states.

For this example, consider mechssModel.mat that contains a mechss model object ltiSys.

Load the mechss model object from mechssModel.mat.

ltiSys
Sparse continuous-time second-order model with 1 outputs, 1 inputs, and 872 degrees of freedom.

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.

Use the sparss command to convert to first-order sparse representation.

Sparse continuous-time state-space model with 1 outputs, 1 inputs, and 1744 states.

Use "spy" and "showStateInfo" to inspect model structure.
Type "properties('sparss')" for a list of model properties.
Type "help sparssOptions" for available solver options for this model.

The resultant sparss model object sys has exactly double the number of states than the mechss object ltisys since the mass matrix M is full rank. If the mass matrix is not full rank then the number of states in the resultant sparss model when converting from a mechss model is between n and 2n. Here, n is the number of nodes in the mechss model object.

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

Next, create an actuator and sensor using transfer functions.

act = tf(1,[1 2 3],'Name','Actuator');
sen = tf(1,[6 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 state-space model with 1 outputs, 1 inputs, and 29 states.

Use "spy" and "showStateInfo" to inspect model structure.
Type "properties('sparss')" for a list of model properties.
Type "help sparssOptions" for available solver options for this model.

The resultant system sys is a sparss object since sparss objects take precedence over tf and PID 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         20
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         20
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)

## References

[1] M. Hosea and L. Shampine. "Analysis and implementation of TR-BDF2." Applied Numerical Mathematics, vol. 20, no. 1-2, pp. 21-37, 1996.