Stability Criteria

Why This Matters

In Control Theory, stability is the fundamental question you must answer before anything else matters. An unstable system will diverge, oscillate wildly, or crash, making all your careful design work irrelevant. Stability criteria give you the mathematical tools to predict system behavior, and they connect directly to core concepts like transfer functions, frequency response, pole placement, and feedback system design.

What makes stability analysis challenging is that different methods reveal different insights. Some work in the time domain, others in frequency domain; some handle linear systems elegantly while others extend to nonlinear cases. Don't just memorize which criterion uses which plot. Know why you'd choose one method over another, what each reveals about system behavior, and how they connect to design decisions.

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Algebraic Methods: Testing Stability Without Solving

These criteria let you determine stability directly from the characteristic polynomial, with no need to actually find the roots. The coefficients of a polynomial contain information about where its roots lie, and these methods extract it systematically.

Routh-Hurwitz Stability Criterion

This is the workhorse for continuous-time LTI systems. It analyzes the characteristic polynomial to determine stability without computing roots directly.

• Routh array construction: Arrange the polynomial coefficients into the first two rows of a table, then compute each subsequent row using cross-multiplication of the two rows above it. Each new element follows the pattern $b_i = \frac{a_{1} \cdot c_{i} - a_{0} \cdot c_{i+1}}{a_{1}}$, where the specific entries depend on your row positions.
• First-column sign test: The system is stable if and only if every element in the first column of the completed array is positive. Each sign change in the first column corresponds to exactly one root in the right half-plane.
• Special cases to watch for: A zero in the first column (replace with a small $\epsilon$ and continue) or an entire row of zeros (indicates symmetric roots about the origin, such as purely imaginary pairs).

Jury Stability Test (for Discrete-Time Systems)

This is the discrete-time counterpart of Routh-Hurwitz, designed for systems described by z-domain transfer functions.

• Jury array construction: A similar tabular approach, but with different computational rules. You build rows by reversing and cross-multiplying, suited to testing root locations relative to the unit circle.
• Unit circle requirement: Stability demands all roots of the characteristic polynomial lie inside the unit circle in the z-plane ($|z| < 1$), not in the left half-plane. This is the key geometric difference from continuous-time stability.
• Quick necessary conditions: Before building the full array, check that $P(1) > 0$, that $(-1)^n P(-1) > 0$, and that $|a_0| < a_n$. If any of these fail, the system is already unstable.

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Frequency-Domain Methods: Stability from Response Curves

These graphical techniques extract stability information from how a system responds to sinusoidal inputs at different frequencies. The underlying principle: a system's frequency response encodes its pole locations and stability margins.

Nyquist Stability Criterion

The Nyquist criterion is the most general frequency-domain stability test. It maps the open-loop frequency response $G(j\omega)H(j\omega)$ onto the complex plane as frequency sweeps from $-\infty$ to $+\infty$, creating a closed polar plot (the Nyquist contour).

• Encirclement counting: The number of clockwise encirclements $N$ of the critical point $(-1, 0)$ relates to stability through $N = P - Z$, where $P$ is the number of open-loop right half-plane poles and $Z$ is the number of closed-loop right half-plane poles. For closed-loop stability, you need $Z = 0$, so the plot must encircle $(-1, 0)$ exactly $P$ times counter-clockwise.
• Handles open-loop unstable systems: Unlike Bode methods, Nyquist can assess closed-loop stability even when the open-loop system has right half-plane poles. This is its major advantage.
• Detour handling: When the open-loop transfer function has poles on the imaginary axis (like a pure integrator), you indent the Nyquist contour around them with small semicircles, which produce large arcs at infinity on the plot.

Bode Stability Criterion

This uses magnitude and phase plots versus frequency to assess stability through crossover analysis. It's faster and more intuitive for design work than Nyquist, but less general.

• Gain crossover frequency ($\omega_{gc}$): Where the magnitude plot crosses 0 dB. You measure phase margin here.
• Phase crossover frequency ($\omega_{pc}$): Where the phase plot crosses $-180°$. You measure gain margin here.
• Assumes open-loop stability: This method works cleanly only when the open-loop transfer function has no right half-plane poles. If the open-loop system is unstable, you must use Nyquist instead.

Phase Margin and Gain Margin

These are the quantitative measures of how close a stable system is to becoming unstable. They're your primary robustness indicators.

• Phase margin (PM): The additional phase lag the system can tolerate at gain crossover before total phase reaches $-180°$. Calculated as $PM = 180° + \angle G(j\omega_{gc})H(j\omega_{gc})$. Typical design targets are $45°$ to $60°$.
• Gain margin (GM): The factor by which gain can increase at phase crossover before the system becomes unstable. Expressed in dB as $GM = -20\log_{10}|G(j\omega_{pc})H(j\omega_{pc})|$. A GM of at least 6 dB is a common design target.
• Both must be positive for a stable system. If either margin is small, the system is sensitive to parameter variations and modeling errors. A phase margin of $30°$, for instance, is generally considered marginal.

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Graphical Root Methods: Visualizing Pole Movement

These techniques show you where poles are and how they move, giving geometric intuition about stability and transient behavior.

Root Locus Method

The root locus traces closed-loop pole trajectories as a single parameter (typically feedback gain $K$) varies from 0 to $\infty$.

• Starting and ending points: Loci start at open-loop poles (when $K = 0$) and end at open-loop zeros or at infinity (as $K \to \infty$). The number of branches equals the number of open-loop poles.
• Design by pole placement: Choose gain values that position closed-loop poles in desirable regions. Poles further left in the s-plane give faster response; poles closer to the real axis give less oscillation.
• Stability boundaries visible: The points where loci cross the imaginary axis tell you exactly what gain values cause the transition from stability to instability. You can find these crossover gains using Routh-Hurwitz on the closed-loop characteristic polynomial.

Pole-Zero Analysis

This is the most direct stability check: examine where the poles of the closed-loop transfer function sit in the complex plane.

• Pole locations determine stability: All poles must have negative real parts (left half-plane) for a continuous-time system to be BIBO stable. For discrete-time, all poles must be inside the unit circle.
• Zeros affect response shape: While zeros don't directly determine stability, they influence overshoot, settling time, and the overall transient shape.
• Right half-plane zeros create inverse response, where the output initially moves opposite to its final value. This limits achievable bandwidth and complicates controller design significantly.

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State-Space and Energy Methods: Modern Stability Analysis

These approaches work with system matrices and energy-like functions rather than transfer functions, enabling analysis of complex multi-variable and nonlinear systems.

State-Space Stability Analysis

For a state-space model $\dot{x} = Ax + Bu$, stability depends entirely on the system matrix $A$.

• Eigenvalue criterion: Stability requires all eigenvalues of $A$ to have strictly negative real parts. If any eigenvalue has a positive real part, the system is unstable. Purely imaginary eigenvalues indicate marginal stability.
• Direct connection to poles: For LTI systems, the eigenvalues of $A$ are exactly the poles of the transfer function $G(s) = C(sI - A)^{-1}B + D$. This unifies the time-domain and frequency-domain perspectives.
• MIMO capability: State-space methods handle multi-input multi-output systems naturally, where transfer function methods require working with transfer function matrices and become unwieldy.

Lyapunov Stability Theory

Lyapunov's method uses energy-based reasoning to prove stability without ever solving the system's differential equations.

• The core idea: If you can find a scalar function $V(x)$ that is positive definite (like energy: $V(x) > 0$ for all $x \neq 0$, and $V(0) = 0$) and whose time derivative $\dot{V}(x)$ is negative definite along system trajectories, then the equilibrium at the origin is asymptotically stable.
• No solution required: This is powerful because for nonlinear systems, closed-form solutions to the differential equations often don't exist.
• For linear systems: Lyapunov stability reduces to solving the matrix equation $A^T P + PA = -Q$ for a positive definite matrix $P$, given a positive definite $Q$. If such a $P$ exists, the system is stable. This is equivalent to checking eigenvalues of $A$, but the Lyapunov equation becomes useful in robust control and optimization contexts.
• The challenge: For nonlinear systems, there's no systematic way to find a suitable Lyapunov function. Choosing one is often described as an art.

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Nonlinear System Methods: Beyond Linearization

When systems have saturation, dead zones, hysteresis, or other nonlinearities, linear methods fail. These criteria extend stability analysis to real-world nonlinear behavior.

Circle Criterion

The circle criterion is a graphical extension of Nyquist for systems with a nonlinear element in the feedback loop.

• Sector-bounded nonlinearities: The method applies when the nonlinear element's input-output relationship stays within a sector defined by two slopes $\alpha$ and $\beta$, meaning $\alpha \leq \frac{f(x)}{x} \leq \beta$ for all $x \neq 0$.
• Critical disk: Based on the sector bounds, you construct a disk (or circle) in the complex plane. The center and radius of this disk depend on $\alpha$ and $\beta$.
• Stability condition: If the Nyquist plot of the linear part avoids this critical disk entirely, stability is guaranteed. This is a sufficient condition, so it's conservative. The system might actually be stable even if the criterion isn't satisfied.

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Quick Reference Table

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Self-Check Questions

1. You're given a discrete-time system's characteristic polynomial. Which stability criterion would you use, and what geometric region must all roots occupy?

2. Compare Routh-Hurwitz and Nyquist: both assess stability, but what fundamental difference determines when you'd choose one over the other?

3. A system has a gain margin of 6 dB and a phase margin of $30°$. What do these values tell you about robustness, and which margin suggests the system is closer to instability?

4. Why can Lyapunov stability theory analyze systems that eigenvalue methods cannot? Give an example of when you'd need Lyapunov's approach.

5. You need to design a controller for a system with an open-loop unstable pole. Which frequency-domain stability criterion can handle this case, and why does the alternative fail?