5.6 Stability Analysis

5.6 Stability Analysis

Introduction to System Stability

Stability is the most critical requirement for any control system. An unstable system is useless and potentially dangerous, as its output may diverge to infinity or oscillate uncontrollably in response to a bounded input. Stability analysis provides mathematical tools to determine whether a system will remain well-behaved over time. This unit explores stability from multiple perspectives: algebraic criteria like Routh-Hurwitz that work directly on the characteristic equation, graphical methods like Root Locus that show how closed-loop poles migrate with gain, and frequency-domain techniques like Bode and Nyquist that assess stability from open-loop data. These methods allow engineers to predict, analyze, and ensure stable operation.

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1. Definitions of Stability

1.1 Bounded-Input Bounded-Output (BIBO) Stability

• Definition: A system is BIBO stable if, for every bounded input, the output is also bounded.

• Mathematical Condition: For a linear time-invariant (LTI) system with impulse response $h(t)$, BIBO stability requires: $\int_{0}^{\infty} |h(t)| dt < \infty$

• Practical Implication: The system's natural response must decay to zero as time goes to infinity.

1.2 Asymptotic Stability (Internal Stability)

• Definition: A system is asymptotically stable if its natural response (response due to initial conditions) approaches zero as time approaches infinity.

• Condition for LTI Systems: All poles of the closed-loop transfer function must have negative real parts (i.e., lie in the left half of the s-plane).

• Key Point: For LTI systems, asymptotic stability implies BIBO stability, but the converse is not always true (e.g., systems with pole-zero cancellation in the right-half plane).

1.3 Marginal Stability

• Definition: A system is marginally stable if its natural response neither decays to zero nor grows unbounded, but remains bounded or oscillates at a constant amplitude.

• Condition: The system has non-repeated poles on the imaginary axis (e.g., $s = \pm j\omega$) and all other poles in the left half-plane.

• Example: An undamped second-order system ($\zeta = 0$) is marginally stable.

1.4 Instability

• Definition: A system is unstable if its natural response grows without bound as time increases.

• Condition: At least one pole lies in the right half of the s-plane (positive real part), OR there are repeated poles on the imaginary axis.

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2. Routh-Hurwitz Stability Criterion

2.1 Purpose

An algebraic method to determine the number of closed-loop poles in the right-half s-plane without factoring the characteristic polynomial.

2.2 Procedure

Given the characteristic equation: $a_n s^n + a_{n-1} s^{n-1} + \cdots + a_1 s + a_0 = 0$

1. Arrange the Routh Array:

   • Row 1: $a_n, a_{n-2}, a_{n-4}, \dots$

   • Row 2: $a_{n-1}, a_{n-3}, a_{n-5}, \dots$

   • Row 3 onwards: $b_1, b_2, \dots$ where: $b_1 = \frac{-1}{a_{n-1}} \begin{vmatrix} a_n & a_{n-2} \\ a_{n-1} & a_{n-3} \end{vmatrix}, \quad b_2 = \frac{-1}{a_{n-1}} \begin{vmatrix} a_n & a_{n-4} \\ a_{n-1} & a_{n-5} \end{vmatrix}, \dots$

   • Continue until the $s^0$ row is completed.

2. Stability Criterion: The system is stable if and only if: a) All coefficients $a_i$ of the characteristic equation are positive (necessary condition). b) All elements in the first column of the Routh array are positive.

3. Number of RHP Poles: The number of sign changes in the first column equals the number of poles in the right-half plane.

2.3 Special Cases

1. Zero in the first column (but entire row is not zero):

   • Replace the zero with a small positive number $\epsilon$.

   • Complete the array.

   • Examine the signs in the first column as $\epsilon \to 0$.

2. Entire row is zero:

   • Indicates symmetrical roots (pairs of roots symmetric about the origin).

   • Form an auxiliary polynomial from the coefficients of the row above.

   • Take the derivative of the auxiliary polynomial.

   • Replace the zero row with coefficients of this derivative.

   • Continue. The roots of the auxiliary polynomial are the symmetrical roots.

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3. Root Locus Plot Construction Rules

3.1 Definition

A root locus is a graphical plot showing how the closed-loop poles move in the s-plane as a single parameter (typically gain $K$) varies from $0$ to $\infty$.

3.2 Basic Construction Rules

Given the open-loop transfer function $KG(s)H(s) = K \frac{(s+z_1)(s+z_2)\dots}{(s+p_1)(s+p_2)\dots}$:

1. Start/End Points:

   • Loci start (K=0) at open-loop poles.

   • Loci end (K→∞) at open-loop zeros (finite zeros + $(n-m)$ zeros at infinity).

2. Number of Branches: Equal to $n$ (number of poles).

3. Symmetry: Symmetrical about the real axis.

4. Real Axis Segment: A point on the real axis lies on the root locus if the number of poles + zeros to its right is odd.

5. Asymptotes (for large K):

   • Number of asymptotes = $n - m$.

   • Centroid (σ): $\sigma = \frac{\sum \text{Poles} - \sum \text{Zeros}}{n - m}$.

   • Angles: $\theta = \frac{(2q+1)180^\circ}{n-m}$, $q = 0, 1, 2, \dots$.

6. Breakaway/Break-in Points:

   • Occur on the real axis between two adjacent poles or zeros.

   • Found by solving $\frac{dK}{ds} = 0$ from the characteristic equation $1+KG(s)H(s)=0$.

7. Departure/Arrival Angles:

   • Angle of departure from a complex pole: $\theta_{dep} = 180^\circ - \sum \angle \text{poles} + \sum \angle \text{zeros}$.

   • Angle of arrival at a complex zero: $\theta_{arr} = 180^\circ + \sum \angle \text{poles} - \sum \angle \text{zeros}$.

8. Imaginary Axis Crossings:

   • Use the Routh array.

   • Find the gain $K$ that makes a row zero (marginal stability).

   • Solve the auxiliary polynomial to find the crossing points $s = \pm j\omega$.

3.3 Using Root Locus for Stability

• The system is stable for gains K where all branches lie in the left-half plane.

• The gain at the imaginary axis crossing gives the maximum gain for stability.

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4. Frequency Response Analysis

Analyzes system stability and performance using sinusoidal inputs. Based on the open-loop transfer function $G(j\omega)H(j\omega)$.

4.1 Bode Plots and Stability Assessment

• Bode Plot: Consists of two graphs:

  1. Magnitude Plot: $20 \log_{10} |G(j\omega)H(j\omega)|$ vs. $\log \omega$.

  2. Phase Plot: $\angle G(j\omega)H(j\omega)$ vs. $\log \omega$.

• Stability Criterion (Bode): For a stable open-loop system, the closed-loop system is stable if the phase is greater than -180° at the gain crossover frequency (where magnitude = 0 dB).

• Key Frequencies:

  • Gain Crossover Frequency ($\omega_{gc}$): Frequency where magnitude = 0 dB.

  • Phase Crossover Frequency ($\omega_{pc}$): Frequency where phase = -180°.

4.2 Polar Plots and Nyquist Stability Criterion

• Polar (Nyquist) Plot: Plot of $G(j\omega)H(j\omega)$ in the complex plane as $\omega$ varies from $0$ to $\infty$.

• Nyquist Stability Criterion:

  • Let $P$ = number of open-loop poles in the RHP.

  • Let $N$ = net number of clockwise encirclements of the point $(-1, j0)$ by the Nyquist plot.

  • Then, the number of closed-loop poles in the RHP, $Z$, is: $Z = N + P$

  • For closed-loop stability, we require $Z = 0$, i.e., $N = -P$.

  • Simplified Rule (for stable open-loop, P=0): The closed-loop system is stable if and only if the Nyquist plot does NOT encircle the $(-1, j0)$ point.

4.3 Gain Margin and Phase Margin

These are quantitative measures of relative stability (how far the system is from instability).

1. Gain Margin (GM):

   • Definition: The amount of additional gain (in dB) that can be added to the system before it becomes unstable.

   • Calculation from Bode: $GM_{dB} = 0 - |G(j\omega_{pc})|_{dB} = -|G(j\omega_{pc})|_{dB}$

   • Calculation from Nyquist: Reciprocal of the distance from the plot to the -1 point along the real axis.

   • Interpretation: A positive GM (in dB) indicates stability. Larger GM = more stable.

2. Phase Margin (PM):

   • Definition: The amount of additional phase lag (in degrees) that can be added to the system before it becomes unstable.

   • Calculation from Bode: $PM = \angle G(j\omega_{gc}) - (-180^\circ) = 180^\circ + \angle G(j\omega_{gc})$

   • Interpretation: A positive PM indicates stability. Typical desired PM: 30° to 60°.

3. Relationship to Time Response:

   • Higher Gain Margin → Less oscillatory, more sluggish response.

   • Higher Phase Margin → Less overshoot, better damping.

   • Both margins must be positive for stability. One margin alone is insufficient.

4.4 Stability Assessment Summary

• Routh-Hurwitz: Algebraic, gives absolute stability.

• Root Locus: Graphical, shows pole movement with gain.

• Bode Plot: Frequency domain, gives GM and PM easily.

• Nyquist Plot: Frequency domain, works for open-loop unstable systems and gives absolute stability.

Conclusion: Stability analysis is a multi-faceted discipline. The Routh-Hurwitz criterion provides a quick algebraic check. The root locus offers deep insight into how closed-loop dynamics change with a parameter. Frequency response methods (Bode and Nyquist) are exceptionally powerful as they can use experimentally obtained open-loop data to predict closed-loop stability and provide direct measures (Gain Margin and Phase Margin) for robust design. Together, these tools form the essential toolkit for ensuring that control systems perform predictably and safely.