Control System Toolbox™

Arithmetic Operations on LTI Models

Supported Arithmetic Operations

You can apply almost all arithmetic operations to LTI models, including those shown below.

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Addition and Subtraction

Adding LTI models is equivalent to connecting them in parallel. Specifically, the LTI model

sys = sys1 + sys2

represents the parallel interconnection shown below.

If sys1 and sys2 are two state-space models with data and , the state-space data associated with sys1 + sys2 is

Scalar addition is also supported and behaves as follows: if sys1 is MIMO and sys2 is SISO, sys1 + sys2 produces a system with the same dimensions as sys1 whose ijth entry is sys1(i,j) + sys2.

Similarly, the subtraction of two LTI models

sys = sys1 - sys2

is depicted by the following block diagram.

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Multiplication

Multiplication of two LTI models connects them in series. Specifically,

sys = sys1 * sys2

returns an LTI model sys for the series interconnection shown below.

Notice the reverse orders of sys1 and sys2 in the multiplication and block diagram. This is consistent with the way transfer matrices are combined in a series connection: if sys1 and sys2 have transfer matrices and , then

For state-space models sys1 and sys2 with data and , the state-space data associated with sys1*sys2 is

Finally, if sys1 is MIMO and sys2 is SISO, then sys1*sys2 or sys2*sys1 is interpreted as an entry-by-entry scalar multiplication and produces a system with the same dimensions as sys1, whose ijth entry is sys1(i,j)*sys2.

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Inversion and Related Operations

Inversion of LTI models amounts to inverting the following input/output relationship.

This operation is defined only for square systems (that is, systems with as many inputs as outputs) and is performed using

inv(sys)

The resulting inverse model is of the same type as sys. Related operations include:

• Left division sys1\sys2, which is equivalent to inv(sys1)*sys2

• Right division sys1/sys2, which is equivalent to sys1*inv(sys2)

For a state-space model sys with data , inv(sys) is defined only when is a square invertible matrix, in which case its state-space data is

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Transposition

You can transpose an LTI model sys using

sys.'

This is a literal operation with the following effect:

• For TF models (with input arguments, num and den), the cell arrays num and den are transposed.

• For ZPK models (with input arguments, z, p, and k), the cell arrays, z and p, and the matrix k are transposed.

• For SS models (with model data ), transposition produces the state-space model A^T, C^T, B^T, D^T.

• For FRD models (with complex frequency response matrix Response), the matrix of frequency response data at each frequency is transposed.

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Pertransposition

For a continuous-time system with transfer function , the pertransposed system has the transfer function

The discrete-time counterpart is

Pertransposition of an LTI model sys is performed using

sys'

You can use pertransposition to obtain the Hermitian (conjugate) transpose of the frequency response of a given system. The frequency response of the pertranspose of , , is the Hermitian transpose of the frequency response of : .

To obtain the Hermitian transpose of the frequency response of a system sys over a frequency range specified by the vector w, type

freqresp(sys', w);

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Extracting and Modifying Subsystems

Model Interconnection Functions