### Matrix Diagonal Stability in Systems and Computation

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For example, if sys is a two-input model, enter:. You can use the shorthand notation u to refer to the InputName property. For example, sys. Character vector — For single-input models, for example, 'seconds'. Use InputUnit to keep track of input signal units. InputUnit has no effect on system behavior. Input channel groups.

## Convex Design of Structured Controllers using Block-Diagonal Lyapunov Functions

The InputGroup property lets you assign the input channels of MIMO systems into groups and refer to each group by name. Specify input groups as a structure. In this structure, field names are the group names, and field values are the input channels belonging to each group. For example:. You can then extract the subsystem from the controls inputs to all outputs using:. Character vector — For single-output models. For example, 'measurements'.

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Alternatively, use automatic vector expansion to assign output names for multi-output models. For example, if sys is a two-output model, enter:. You can use the shorthand notation y to refer to the OutputName property.

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For example, 'seconds'. Use OutputUnit to keep track of output signal units. OutputUnit has no effect on system behavior. Output channel groups. The OutputGroup property lets you assign the output channels of MIMO systems into groups and refer to each group by name. Specify output groups as a structure. In this structure, field names are the group names, and field values are the output channels belonging to each group. You can then extract the subsystem from all inputs to the measurement outputs using:.

## Matrix Diagonal Stability in Systems and Computation

System name, specified as a character vector. Any text that you want to associate with the system, stored as a string or a cell array of character vectors. The property stores whichever data type you provide. For instance, if sys1 and sys2 are dynamic system models, you can set their Notes properties as follows:. For model arrays that are derived by sampling one or more independent variables, this property tracks the variable values associated with each model in the array.

This information appears when you display or plot the model array. Use this information to trace results back to the independent variables. Set the field names of the data 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 should be numeric and scalar valued, and all arrays of sampled values should match the dimensions of the model array. The following code stores the time samples with the linear models. Similarly, suppose you create a 6-by-9 model array, M , by independently sampling two variables, zeta and w. The following code attaches the zeta,w values to M. When you display M , each entry in the array includes the corresponding zeta and w values. Create a state-space model with a sample time of 0. The number of state and input names must be consistent with the dimensions of A , B , C , and D.

To obtain a minimal realization of H s , enter. The resulting model has an order of three, which is the minimum number of states needed to represent H s.

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To see this number of states, refactor H s as the product of a first-order system and a second-order system. This example shows how to create a state-space genss model having both fixed and tunable parameters. Create the tunable parameters using realp. Define a generalized matrix using algebraic expressions of a and b. A is a generalized matrix whose Blocks property contains a and b.

The initial value of A is [1 2;0 -3] , from the initial values of a and b. Use ss to create the state-space model. Confirm that the A property of sys is stored as a generalized matrix. Extract the measured and noise components of an identified polynomial model into two separate state-space models. The former measured component can serve as a plant model while the latter can serve as a disturbance model for control system design. It uses an algorithm similar to tf2ss , but further rescales the state vector to compress the numerical range in state matrix A and to improve numerics in subsequent computations.

For example, in the following code, A and sys. A differ by a diagonal state transformation:. For details, see balance. A modified version of this example exists on your system. Do you want to open this version instead? The data shown below were generated using the model:. The generalized inverse is calculated using the command linalg.

These two commands differ in how they compute the generalized inverse. The first uses the linalg. In many applications it is useful to decompose a matrix using other representations. There are several decompositions supported by SciPy. The eigenvalue-eigenvector problem is one of the most commonly employed linear algebra operations. In addition, linalg. By definition, eigenvectors are only defined up to a constant scale factor. The eigenvectors corresponding to each eigenvalue can be found using the original equation.

The eigenvectors associated with these eigenvalues can then be found. Singular Value Decomposition SVD can be thought of as an extension of the eigenvalue problem to matrices that are not square. It is known that the eigenvalues of square hermitian matrices are real and non-negative. The following example illustrates the use of linalg. The SciPy command for this decomposition is linalg.

Such a decomposition is often useful for solving many simultaneous equations where the left-hand-side does not change but the right hand side does. For example, suppose we are going to solve.