Zhang3：创建页面，内容为“{{Control Systems/Page|MIMO Systems|Gain}} == Realization == '''Realization''' is the process of taking a mathematical model of a system (either in the Laplace domain or the State-Space domain), and creating a physical system. Some systems are not realizable. An important point to keep in mind is that the Laplace domain representation, and the state-space representations are equivalent, and both representations describe the same physical systems. We want, th…”

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创建页面，内容为“{{Control Systems/Page|MIMO Systems|Gain}} == Realization == '''Realization''' is the process of taking a mathematical model of a system (either in the Laplace domain or the State-Space domain), and creating a physical system. Some systems are not realizable. An important point to keep in mind is that the Laplace domain representation, and the state-space representations are equivalent, and both representations describe the same physical systems. We want, th…”

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{{Control Systems/Page|MIMO Systems|Gain}}

== Realization ==

'''Realization''' is the process of taking a mathematical model of a system (either in the Laplace domain or the State-Space domain), and creating a physical system. Some systems are not realizable.

An important point to keep in mind is that the Laplace domain representation, and the state-space representations are equivalent, and both representations describe the same physical systems. We want, therefore, a way to convert between the two representations, because each one is well suited for particular methods of analysis.

The state-space representation, for instance, is preferable when it comes time to move the system design from the drawing board to a constructed physical device. For that reason, we call the process of converting a system from the Laplace representation to the state-space representation "realization".

== Realization Conditions ==

{{SideBox|'''Note:''' Discrete systems ''G(z)'' are also realizable if these conditions are satisfied.}}

*A transfer function ''G(s)'' is realizable if and only if the system can be described by a finite-dimensional state-space equation.
*''(A B C D)'', an ordered set of the four system matrices, is called a '''realization''' of the system ''G(s)''. If the system can be expressed as such an ordered quadruple, the system is realizable.
*A system ''G'' is realizable if and only if the transfer matrix '''G'''(s) is a proper rational matrix. In other words, every entry in the matrix '''G'''(s) (only 1 for SISO systems) is a rational polynomial, and if the degree of the denominator is higher or equal to the degree of the numerator.

We've already covered the method for realizing a SISO system, the remainder of this chapter will talk about the general method of realizing a MIMO system.

== Realizing the Transfer Matrix ==

We can decompose a transfer matrix '''G'''(s) into a ''strictly proper'' transfer matrix:

:<math>\mathbf{G}(s) = \mathbf{G}(\infty) + \mathbf{G}_{sp}(s)</math>

Where Gsp(s) is a strictly proper transfer matrix. Also, we can use this to find the value of our ''D'' matrix:

:<math>D = \mathbf{G}(\infty)</math>

We can define ''d(s'') to be the lowest common denominator polynomial of all the entries in '''G'''(s):

{{SideBox|Remember, ''q'' is the number of inputs, ''p'' is the number of internal system states, and ''r'' is the number of outputs.}}

:<math>d(s) = s^r + a_1s^{r-1} + \cdots + a_{r-1}s + a_r</math>

Then we can define '''G'''sp as:

:<math>\mathbf{G}_{sp}(s) = \frac{1}{d(s)}N(s)</math>

Where

:<math>N(s) = N_1s^{r-1} + \cdots + N_{r-1}s + N_r</math>

And the ''Ni'' are ''p × q'' constant matrices.

If we remember our method for converting a transfer function to a state-space equation, we can follow the same general method, except that the new matrix ''A'' will be a block matrix, where each block is the size of the transfer matrix:

:<math>A = \begin{bmatrix}
-a_1I_p & -a_2I_p & \cdots & -a_{r-1}I_p & -a_rI_p \\
I_p & 0 & \cdots & 0 & 0 \\
0 & I_p & \cdots & 0 & 0 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & I_p & 0
\end{bmatrix}</math>

:<math>B = \begin{bmatrix}I_p \\ 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}</math>

:<math>C = \begin{bmatrix}N_1 & N_2 & N_3 & \cdots & Nr\end{bmatrix}</math>

=== Realizing System by Column ===
We can divide the '''G(s)''' into multiple column, realize them individually and join them back together later, for '''G(s)''':

<math>G(s) =\begin{bmatrix} G_1 & G_2 & G_3 &\dots & G_n \end{bmatrix}</math>

where we realize them and yield:

<math>G_i => (A_i,B_i,C_i,D_i)</math>

and the realization of the system will be:

<math>A = \begin{bmatrix}
A_1 & 0 & 0 & \dots &0\\
0 & A_2 & 0&&\vdots\\
0 & 0 & A_3\\
\vdots& & & \ddots &0\\
0&0&0&\dots &A_n
\end{bmatrix}</math>

<math>B = \begin{bmatrix}
B_1 & 0 & 0 & \dots &0\\
0 & B_2 & 0&&\vdots\\
0 & 0 & B_3\\
\vdots& & & \ddots &0\\
0&0&0&\dots &B_n
\end{bmatrix}</math>

<math>C = \begin{bmatrix} C_1 & C_2 & C_3 &\dots& C_n \end{bmatrix}</math>

<math>D = \begin{bmatrix} D_1 & D_2 & D_3 &\dots& D_n \end{bmatrix}</math>{{Control Systems/Nav|MIMO Systems|Gain}}

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