Question

For a system with the transfer function H(s) = 3(s-2)/s^3+4s^2-2s+1 , the matrix A in the state space form is equal to:

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Query is from State Variable Analysis in chapter Time Response Analysis, Design Specifications and Performance Indices of Control Systems

Answer 1

To find the matrix $A$ in the state-space form for the transfer function H(s)=3(s−2)s3+4s2−2s+1H(s) = \frac{3(s-2)}{s^3+4s^2-2s+1}H(s)=s3+4s2−2s+13(s−2)​, we first need to write the transfer function in a form that can be converted to state-space representation.

Since the denominator is a third-degree polynomial, we can use the following canonical form for state-space:

1. Rewrite the denominator as a differential equation, expressing the output $y$ in terms of the input $u$.
2. Construct the state-space matrices, where $A$ will capture the system's internal dynamics.

For a transfer function with a third-degree denominator, $A$ typically takes the form of a 3x3 matrix representing the dynamics of a third-order system. Calculations will yield:

A=[010001−12−4]A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & 2 & -4 \end{bmatrix}A=​00−1​102​01−4​​

This matrix $A$ represents the state transition matrix in the state-space representation of the given system.