Question

Find the state equations and output equation for the phase-variable representation of the transfer function. Show all steps in your working. Attach your answer in any suitable format. G(s) =2s + 1 /s² + 7s + 9

Answer 1

To find the state equations and output equation for the state-space representation in phase-variable (canonical) form, assign the coefficients of the transfer function denominator to the system matrices and the numerator to the output matrix. Compile the state matrix (A), input matrix (B), output matrix (C), and feedforward matrix (D) appropriately, and then confirm by ensuring that the Laplace transform of the state equations corresponds to the original transfer function G(s).

Step-by-step explanation:

To find the state equations and output equation for the phase-variable representation of the transfer function G(s) = (2s + 1) / (s² + 7s + 9), we first need to express this transfer function in terms of state-space variables. State-space representation is a mathematical model of a physical system as a set of input, output, and state variables related by first-order differential equations.

For the given transfer function:

• We first assign the denominator polynomial coefficients as the system matrices.
• Then, we assign the numerator coefficients appropriately in the output matrix.
• Lastly, we ensure that the state matrix (A), input matrix (B), output matrix (C), and feedforward matrix (D) represent the system dynamics correctly.

The state-space representation in phase-variable (canonical) form is:

1. A = [[-7, -9], [1, 0]]
2. B = [[1], [0]]
3. C = [2, 1]
4. D = [0]

The state equations are:

• ˙x(t) = Ax(t) + Bu(t)
• x(t) = [x1(t) x2(t)]T (state vector)

And the output equation is:

• y(t) = Cx(t) + Du(t)

where x(t) is the state vector, u(t) is the input, and y(t) is the output of the system. To verify that these equations represent the original transfer function, we can take the Laplace transform of the state equations and solve for Y(s)/U(s) to obtain the original G(s).

Always remember to check that your resulting state-space model is controllable and observable, which are necessary conditions for a good system representation.