The characteristic equation \({s^3} + 8{s^2} + 14s + 24 = 0\) represents ______
stable system
Understanding System Stability from Characteristic Equation
The stability of a linear time-invariant (LTI) system is fundamentally linked to the roots of its characteristic equation. The characteristic equation is typically a polynomial in the complex variable 's', derived from the system's differential equations or transfer function. For a system to be considered stable, all the roots of its characteristic equation must lie in the left half of the complex s-plane (i.e., have negative real parts).
The given characteristic equation is:
$$s^3 + 8s^2 + 14s + 24 = 0$$
To determine the stability, we can use the Routh-Hurwitz stability criterion. This method checks the location of the roots without explicitly calculating them.
Applying Routh-Hurwitz Criterion
The Routh-Hurwitz criterion involves constructing a Routh array (or Routh table) from the coefficients of the characteristic equation.
Step 1: Check Necessary Conditions
First, we check if all coefficients of the characteristic polynomial are present and positive. In the equation \(s^3 + 8s^2 + 14s + 24 = 0\), the coefficients are 1, 8, 14, and 24. All coefficients are positive and present for all powers of 's' from \(s^3\) down to \(s^0\). This condition is necessary but not sufficient for stability.
Step 2: Construct the Routh Array
The Routh array is constructed as follows:
| Power of s | 1 | 14 |
|---|---|---|
| Power of s | 8 | 24 |
| \(s^3\) | 1 | 14 |
| \(s^2\) | 8 | 24 |
| \(s^1\) | ||
| \(s^0\) |
Now, we calculate the elements for the \(s^1\) and \(s^0\) rows.
Calculation for \(s^1\) row:
The first element (\(b_1\)) is calculated as:
$$b_1 = \frac{(8 \times 14) - (1 \times 24)}{8} = \frac{112 - 24}{8} = \frac{88}{8} = 11$$
The next element in the \(s^1\) row is calculated similarly:
$$b_2 = \frac{(8 \times 0) - (1 \times 0)}{8} = 0$$
Calculation for \(s^0\) row:
The first element (\(c_1\)) is calculated using the elements from the \(s^2\) and \(s^1\) rows:
$$c_1 = \frac{(11 \times 24) - (8 \times 0)}{11} = \frac{264 - 0}{11} = 24$$
Step 3: Complete the Routh Array and Interpret Results
The completed Routh array is:
| Power of s | 1 | 14 |
|---|---|---|
| Coefficient Value | 8 | 24 |
| \(s^3\) | 1 | 14 |
| \(s^2\) | 8 | 24 |
| \(s^1\) | 11 | 0 |
| \(s^0\) | 24 | 0 |
Stability Interpretation:
The Routh-Hurwitz stability criterion states that a system is stable if and only if all the elements in the first column of the Routh array have the same sign (and are non-zero). In this case, the elements in the first column are 1, 8, 11, and 24.
- \(s^3\) row, first column: 1 (Positive)
- \(s^2\) row, first column: 8 (Positive)
- \(s^1\) row, first column: 11 (Positive)
- \(s^0\) row, first column: 24 (Positive)
Since all the elements in the first column are positive, there are no sign changes. This indicates that all the roots of the characteristic equation have negative real parts.
Conclusion on System Stability
Because all the roots of the characteristic equation \(s^3 + 8s^2 + 14s + 24 = 0\) lie in the left half of the s-plane (as confirmed by the Routh-Hurwitz criterion), the system represented by this equation is stable.
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