To determine the stability of a given system, we utilize the Routh-Hurwitz stability criterion. This powerful mathematical tool helps us identify if any roots of the system's characteristic equation lie in the right half of the S-plane, which would indicate an unstable system. The presence of roots with positive real parts signifies that the system's output will grow unbounded over time, leading to instability.

System Stability Fundamentals

In the field of control systems, stability is a fundamental property. A stable system is one that returns to or stays near its equilibrium state after being subjected to a disturbance. Conversely, an unstable system will diverge from its equilibrium state following a disturbance, leading to an uncontrolled output. The stability of a linear time-invariant (LTI) system is directly determined by the locations of the roots of its characteristic equation. Specifically, if even one root of the characteristic equation has a positive real part (i.e., lies in the right half of the S-plane), the system is considered unstable.

Routh-Hurwitz Criterion Application

The Routh-Hurwitz criterion offers a systematic method to ascertain the number of roots of a polynomial that possess positive real parts without the need to explicitly calculate these roots. This is achieved through the construction and analysis of a specific tabular arrangement known as the Routh array.

The given characteristic equation of the system is:

This is a third-order polynomial. When compared to the general form of a polynomial $a_n S^n + a_{n-1} S^{n-1} + \dots + a_1 S + a_0 = 0$, the coefficients for our equation are:

• $a_3 = 1$ (coefficient of $S^3$)
• $a_2 = 1$ (coefficient of $S^2$)
• $a_1 = 1$ (coefficient of $S^1$)
• $a_0 = 4$ (constant term, coefficient of $S^0$)

Constructing the Routh Array for Stability Analysis

The Routh array is constructed using the coefficients of the characteristic equation. The first two rows of the array are populated directly from these coefficients, alternating between terms.

Next, we compute the elements for the subsequent rows ($S^1$ and $S^0$) using specific determinant-like formulas based on the elements in the two rows directly above them.

For the $S^1$ row, the first element ($b_1$) is calculated as:

The second element ($b_2$) for the $S^1$ row would involve coefficients from further columns, but since there are no more, $b_2 = 0$.

For the $S^0$ row, the first element ($c_1$) is calculated using the elements from the $S^2$ and $S^1$ rows:

Now, we can present the complete Routh array:

Analyzing the Routh Array's First Column

The stability of the system is solely determined by examining the signs of the elements in the first column of the Routh array. The elements in the first column are:

Let's trace the sign changes as we move from top to bottom:

• From the first element ($1$, positive) to the second element ($1$, positive): No sign change.
• From the second element ($1$, positive) to the third element ($-3$, negative): One sign change.
• From the third element ($-3$, negative) to the fourth element ($4$, positive): One sign change.

In total, there are two sign changes observed in the first column of the Routh array.

Conclusion on System Stability

Based on the Routh-Hurwitz criterion, the interpretation of the sign changes in the first column is straightforward:

• If there are no sign changes in the first column of the Routh array, the system is considered stable, meaning all roots lie in the left half of the S-plane.
• If there are sign changes in the first column, the system is inherently unstable. The number of sign changes directly corresponds to the number of roots of the characteristic equation that are located in the right half of the S-plane.
• If an entire row of zeros appears in the Routh array, it indicates the presence of roots located symmetrically with respect to the origin (e.g., on the imaginary axis or as conjugate pairs). This scenario requires further investigation to determine if the system is marginally stable or unstable.

Since our analysis revealed two sign changes in the first column of the Routh array, it signifies that two roots of the characteristic equation $S^3 + S^2 + S + 4 = 0$ lie in the right half of the S-plane. The presence of any root in the right half of the S-plane causes the system to be unstable, as it leads to an unbounded response over time.

Therefore, the system described by the characteristic equation $S^3 + S^2 + S + 4 = 0$ is definitively unstable.