# mechss

Sparse second-order state-space model

## Description

Use `mechss`

to represent second-order sparse models using
matrices obtained from your finite element analysis (FEA) package. Such sparse models arise
from finite element analysis (FEA) and are useful in fields like structural analysis, fluid
flow, heat transfer, and electromagnetics. The resultant matrices from this type of modeling
are quite large with a sparse pattern. Hence, using `mechss`

is an efficient
way to represent such large sparse state-space models in MATLAB^{®} to perform linear analysis. You can also use `mechss`

to
convert a first-order `sparss`

model object or other dynamic system models to a `mechss`

object.

You can use `mechss`

model objects to represent SISO or MIMO state-space
models in continuous time or discrete time. In continuous time, a second-order sparse
mass-spring-damper model is represented in the following form:

$$\begin{array}{r}\text{M}\ddot{q}(t)+\text{C}\dot{q}(t)+\text{K}q(t)\text{=B}u(t)\\ y(t)\text{=F}q(t)+\text{G}\dot{q}(t)+\text{D}u(t)\end{array}$$

Here, the full state vector is given by $$\left[q,\dot{q}\right]$$ where $$q$$ and $$\dot{q}$$ are the displacement and velocity vectors. `u`

and
`y`

represent the inputs and outputs, respectively. `M`

,
`C`

and `K`

represent the mass, damping and stiffness
matrices, respectively. `B`

is the input matrix while `F`

and `G`

are the output matrices for displacement and velocity, respectively.
`D`

is the input-to-output matrix.

You can use a `mechss`

object to:

Perform time-domain and frequency-domain response analysis.

Specify signal-based connections with other LTI models.

Specify physical interfaces between components using the

`interface`

command.

For more information, see Sparse Model Basics.

## Creation

### Syntax

### Description

creates a `sys`

= mechss(`M`

,`C`

,`K`

,`B`

,`F`

,`G`

,`D`

)`mechss`

object representing this continuous-time
second-order mass-spring-damper model:

$$\begin{array}{r}\text{M}\ddot{q}(t)+\text{C}\dot{q}(t)+\text{K}q(t)\text{=B}u(t)\\ y(t)\text{=F}q(t)+\text{G}\dot{q}(t)+\text{D}u(t)\end{array}$$

Here, `M`

, `C`

, and
`K`

represent the mass, damping, and stiffness matrices,
respectively. `B`

is the input-to-state matrix while
`F`

and `G`

are the displacement-to-output and
velocity-to-output matrices resulting from the two components of the state
`x`

. `D`

is the input-to-output matrix. You can set
`M`

to `[]`

when the mass matrix is an identity
matrix. Set `G`

and `D`

to `[]`

or omit them when they are empty.

uses the sample time `sys`

= mechss(`M`

,`C`

,`K`

,`B`

,`F`

,`G`

,`D`

,`ts`

)`ts`

to create a `mechss`

object representing this discrete-time second-order mass-spring-damper model:

$$\begin{array}{r}\text{M}q\left[k+2\right]+\text{C}q\left[k+1\right]+\text{K}q\left[k\right]\text{=B}u\left[k\right]\\ y\left[k\right]\text{=F}q\left[k\right]+\text{G}q\left[k+1\right]+\text{D}u\left[k\right]\end{array}$$

To leave the sample time unspecified, set `ts`

to
`-1`

.

### Input Arguments

`M`

— Mass matrix

`Nq`

-by-`Nq`

sparse matrix

Mass matrix, specified as an `Nq`

-by-`Nq`

sparse
matrix, where `Nq`

is the number of degrees of freedom. This input
sets the value of property M.

`C`

— Damping matrix

`Nq`

-by-`Nq`

sparse matrix

Damping matrix, specified as an `Nq`

-by-`Nq`

sparse matrix, where `Nq`

is the number of degrees of freedom. You
can also set `C=[]`

to specify zero damping. This input sets the
value of property C.

`K`

— Stiffness matrix

`Nq`

-by-`Nq`

sparse matrix

Stiffness matrix, specified as an `Nq`

-by-`Nq`

sparse matrix, where `Nq`

is the number of degrees of freedom. This
input sets the value of property K.

`B`

— Input-to-state matrix

`Nq`

-by-`Nu`

sparse matrix

Input-to-state matrix, specified as an
`Nq`

-by-`Nu`

sparse matrix, where
`Nq`

is the number of degrees of freedom and `Nu`

is the number of inputs. This input sets the value of property B.

`F`

— Displacement-to-output matrix

`Ny`

-by-`Nq`

sparse matrix

Displacement-to-output matrix, specified as an
`Ny`

-by-`Nq`

sparse matrix, where
`Nq`

is the number of degrees of freedom and `Ny`

is the number of outputs. This input sets the value of property F.

`G`

— Velocity-to-output matrix

`Ny`

-by-`Nq`

sparse matrix

Velocity-to-output matrix, specified as an
`Ny`

-by-`Nq`

sparse matrix, where
`Nq`

is the number of degrees of freedom and `Ny`

is the number of outputs. This input sets the value of property G.

`D`

— Input-to-output matrix

`Ny`

-by-`Nu`

sparse matrix

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.

`ts`

— Sample time

scalar

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

`ltiSys`

— Dynamic system to convert to second-order sparse state-space form

dynamic system model | model array

### Output Arguments

`sys`

— Output system model

`mechss`

model object

Output system model, returned as a `mechss`

model object.

## Properties

`M`

— Mass matrix

`Nq`

-by-`Nq`

sparse matrix

Mass matrix, specified as an `Nq`

-by-`Nq`

sparse
matrix where, `Nq`

is the number of degrees of freedom.

`C`

— Damping matrix

`Nq`

-by-`Nq`

sparse matrix

Damping matrix, specified as an `Nq`

-by-`Nq`

sparse matrix where, `Nq`

is the number of degrees of freedom.

`K`

— Stiffness matrix

`Nq`

-by-`Nq`

sparse matrix

Damping matrix, specified as an `Nq`

-by-`Nq`

sparse matrix where, `Nq`

is the number of degrees of freedom.

`B`

— Input-to-state matrix

`Nq`

-by-`Nu`

sparse matrix

Input-to-state matrix, specified as an
`Nq`

-by-`Nu`

sparse matrix where,
`Nq`

is the number of degrees of freedom and `Nu`

is
the number of inputs.

`F`

— Displacement-to-output matrix

`Ny`

-by-`Nq`

sparse matrix

Displacement-to-output matrix, specified as an
`Ny`

-by-`Nq`

sparse matrix where,
`Nq`

is the number of degrees of freedom and `Ny`

is
the number of outputs.

`G`

— Velocity-to-output matrix

`Ny`

-by-`Nq`

sparse matrix

Velocity-to-output matrix, specified as an
`Ny`

-by-`Nq`

sparse matrix where,
`Nq`

is the number of degrees of freedom and `Ny`

is
the number of outputs.

`D`

— Input-to-output matrix

`Ny`

-by-`Nu`

sparse matrix

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 the static gain matrix, and represents
the ratio of the output to the input in steady state condition.

`StateInfo`

— State partition information

structure array

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`

.

`SolverOptions`

— Options for model analysis

structure

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

`UseParallel`

— Set this option to`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 following DAE solvers are available:`'trbdf2'`

— Fixed-step solver with an accuracy of`o(h^2)`

, where`h`

is the step size.[2] This is the default solver for the`mechss`

model object.`'trbdf3'`

— Fixed-step solver with an accuracy of`o(h^3)`

, where`h`

is the step size.`'hht'`

— Fixed-step solver with an accuracy of`o(h^2)`

, where`h`

is the step size.[1]

Reducing the step size increases accuracy and extends the frequency range where numerical damping is negligible.

`'hht'`

is the fastest but can run into difficulties with high initial acceleration (for example, an impulse response with initial jerk).`'trbdf2'`

requires about twice as many computations as`'hht'`

and`'trbdf3'`

requires another 50% more computations than`'trbdf2'`

.For an example, see Time and Frequency Response of Sparse Second-Order Model.

`InternalDelay`

— Internal delays in the model

vector

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.

`InputDelay`

— Input delay

`0`

(default) | scalar | `Nu`

-by-1 vector

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`

.

For more information, see Time Delays in Linear Systems.

`OutputDelay`

— Output delay

`0`

(default) | scalar | `Ny`

-by-1 vector

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`

.

For more information, see Time Delays in Linear Systems.

`Ts`

— Sample time

`0`

(default) | positive scalar | `-1`

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.

`TimeUnit`

— Time variable units

`'seconds'`

(default) | `'nanoseconds'`

| `'microseconds'`

| `'milliseconds'`

| `'minutes'`

| `'hours'`

| `'days'`

| `'weeks'`

| `'months'`

| `'years'`

| ...

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.

`InputName`

— Input channel names

`''`

(default) | character vector | cell array of character vectors

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.

`InputUnit`

— Input channel units

`''`

(default) | character vector | cell array of character vectors

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.

`InputGroup`

— Input channel groups

structure

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.

`OutputName`

— Output channel names

`''`

(default) | character vector | cell array of character vectors

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.

`OutputUnit`

— Output channel units

`''`

(default) | character vector | cell array of character vectors

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.

`OutputGroup`

— Output channel groups

structure

Output channel groups, specified as a structure. Use `OutputGroup`

to 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.OutputGroup.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.

`Notes`

— User-specified text

`{}`

(default) | character vector | cell array of character vectors

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'`

.

`UserData`

— User-specified data

`[]`

(default) | any MATLAB data type

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

`Name`

— System name

`''`

(default) | character vector

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

.

`SamplingGrid`

— Sampling grid for model arrays

structure array

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 `mechss`

model
objects.

### Modeling

`sparss` | Sparse first-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 |

### Data Access

`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 |

### Time and Frequency Response

`step` | Step response plot of dynamic system; step response data |

`impulse` | Impulse response plot of dynamic system; impulse response data |

`initial` | System response to initial states 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 system response at specific frequency |

`freqresp` | Evaluate system response over a grid of frequencies |

### Model Interconnection

`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

### Continuous-Time Sparse Second-Order Model

For this example, consider the sparse matrices for the 3-D beam model subjected to an impulsive point load at its tip in the file `sparseBeam.mat`

.

Extract the sparse matrices from `sparseBeam.mat`

.

load('sparseBeam.mat','M','K','B','F','G','D');

Create the `mechss`

model object by specifying `[]`

for matrix `C`

, since there is no damping.

sys = mechss(M,[],K,B,F,G,D)

Sparse continuous-time second-order model with 3 outputs, 1 inputs, and 3408 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.

The output `sys`

is a `mechss`

model object containing a 3-by-1 array of sparse models with 3408 degrees of freedom, 1 input, and 3 outputs.

You can use the `spy`

command to visualize the sparsity of the `mechss`

model object.

spy(sys)

### Discrete-Time Sparse Second-Order Model

For this example, consider the sparse matrices of the discrete system in the file `discreteSOSparse.mat`

.

Load the sparse matrices from `discreteSOSparse.mat`

.

load('discreteSOSparse.mat','M','C','K','B','F','G','D','ts');

Create the discrete-time `mechss`

model object by specifying the sample time `ts`

.

sys = mechss(M,C,K,B,F,G,D,ts)

Sparse discrete-time second-order model with 1 outputs, 1 inputs, and 28408 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.

The output `sys`

is a discrete-time `mechss`

model object with 28408 degrees of freedom, 1 input, and 1 output.

You can use the `spy`

command to visualize the sparsity pattern of the `mechss`

model object. You can right-click on the plot to select matrices to be displayed.

spy(sys)

### Array of Sparse Second-Order Models

For this example, consider `sparseSOArray.mat`

which contains three sets of sparse matrices that define multiple sparse second-order state-space models.

Extract the data from `sparseSOArray.mat`

.

`load('sparseSOArray.mat');`

Preallocate a 3-by-1 array of `mechss`

models.

sys = mechss(zeros(1,1,3));

Next, use indexed assignment to populate the 3-by-1 array with sparse second-order models.

sys(:,:,1) = mechss(M1,[],K1,B1,F1,G1,[]); sys(:,:,2) = mechss(M2,[],K2,B2,F2,G2,[]); sys(:,:,3) = mechss(M3,[],K3,B3,F3,G3,[]); size(sys)

3x1 array of sparse second-order models. Each model has 1 outputs, 1 inputs, and between 385 and 738 degrees of freedom.

Alternatively, you can also create an array of sparse second-order models using the `stack`

command when you have models with the same I/O sizes.

*Copyright 2020 The MathWorks, Inc*

### MIMO Static Gain Sparse Second-Order Model

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

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

$$D=\left[\begin{array}{cc}1& 5\\ 2& 3\\ 5& 9\end{array}\right]$$

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

D = [1,5;2,3;5,9]; sys = mechss(D); size(sys)

Sparse second-order model with 3 outputs, 2 inputs, and 0 degrees of freedom.

### Mass-Spring-Damper Sparse Second-Order Model

For this example, consider `sparseSOSignal.mat`

which contains the mass, stiffness, and damping sparse matrices.

Load the sparse matrices from `sparseSOSignal.mat`

and create the sparse second-order model object.

load('sparseSOModel.mat','M','C','K'); sys = mechss(M,C,K);

`mechss`

creates the model object `sys`

with the following assumptions:

Identity matrices for

`B`

and`F`

with the same size as mass matrix`M`

.Zero matrices for

`G`

and`D`

.

### Convert First-Order Sparse Model to Second-Order Sparse Model Representation

For this example, consider `sparssModel.mat`

that contains a `sparss`

model object `ltiSys`

.

Load the `sparss`

model object from `sparssModel.mat`

.

load('sparssModel.mat','ltiSys'); ltiSys

Sparse continuous-time state-space model with 1 outputs, 1 inputs, and 354 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.

Use the `mechss`

command to convert to `mechss`

model object representation.

sys = mechss(ltiSys)

Sparse continuous-time second-order model with 1 outputs, 1 inputs, and 354 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.

### Time and Frequency Response of Sparse Second-Order Model

For this example, consider `tuningForkData.mat`

that contains the sparse second-order model of a tuning fork being struck gently but quickly on one of its tines. The system has one input, the pressure applied on one of its tines, which results in two outputs - the displacements at the tip and base of the tuning fork.

Load the sparse matrices from `tuningForkData.mat`

into the workspace and create the `mechss`

model object.

load('tuningForkData.mat','M','K','B','F'); sys = mechss(M,[],K,B,F,'InputName','pressure','Outputname',{'y tip','x base'})

Next, set solver options for the model by setting the `UseParallel`

parameter to `true`

and the `DAESolver`

to use `trbdf3`

. Use `spy`

to inspect the model structure. Enabling parallel computing requires a Parallel Computing Toolbox™ license.

```
sys.SolverOptions.UseParallel = true;
sys.SolverOptions.DAESolver = 'trbdf3';
spy(sys)
```

You can also use `showStateInfo`

to examine the components.

showStateInfo(sys)

Use `step`

to obtain the step response plot of the system. You need to provide the time vector or final time for sparse models.

t = linspace(0,0.5,1000); step(sys,t)

Next, obtain the Bode plot to examine the frequency response. You need to provide the frequency vector for sparse models.

w = logspace(1,5,1000); bode(sys,w), grid

### 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 "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)

### 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.

## References

[1] H. Hilber, T. Hughes & R.
Taylor. " Improved numerical dissipation for time integration algorithms in structural
dynamics." *Earthquake Engineering and Structural Dynamics*, vol. 5, no. 3,
pp. 283-292, 1977.

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

## Version History

**Introduced in R2020b**

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