How to choose state variables in a sytem?
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As systems become more complex, representing them with differential equations or transfer functions becomes cumbersome. This is even more true if the system has multiple inputs and outputs. This document introduces the state space method which largely alleviates this problem. The state space representation of a system replaces an nth order differential equation with a single first order matrix differential equation. The state space representation of a system is given by two equations :

Note: Bold face characters denote a vector or matrix.The variable x is more commonly used in textbooks and other references than is the variable qwhen state variables are discussed. The variable qwill be used here since we will often use x to represent position.
The first equation is called the state equation, the second equation is called the output equation. For an nth order system (i.e., it can be represented by an nth order differential equation) with r inputs and moutputs the size of each of the matrices is as follows:
q is nx1 (n rows by 1 column); q is called the state vector, it is a function of timeA is nxn; A is the state matrix, a constantB is nxr; B is the input matrix, a constantu is rx1; u is the input, a function of timeC is mxn; C is the output matrix, a constantD is mxr; D is the direct transition (or feedthrough) matrix, a constanty is mx1; y is the output, a function of time
Note several features:
The state equation has a single first order derivative of the state vector on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.The output equation has the output on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.
For systems with a single input and single output (i.e., most of the systems we will consider) these variables become (with r=1 and m=1):

Note: Bold face characters denote a vector or matrix.The variable x is more commonly used in textbooks and other references than is the variable qwhen state variables are discussed. The variable qwill be used here since we will often use x to represent position.
The first equation is called the state equation, the second equation is called the output equation. For an nth order system (i.e., it can be represented by an nth order differential equation) with r inputs and moutputs the size of each of the matrices is as follows:
q is nx1 (n rows by 1 column); q is called the state vector, it is a function of timeA is nxn; A is the state matrix, a constantB is nxr; B is the input matrix, a constantu is rx1; u is the input, a function of timeC is mxn; C is the output matrix, a constantD is mxr; D is the direct transition (or feedthrough) matrix, a constanty is mx1; y is the output, a function of time
Note several features:
The state equation has a single first order derivative of the state vector on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.The output equation has the output on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.
For systems with a single input and single output (i.e., most of the systems we will consider) these variables become (with r=1 and m=1):
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