Stability Criteria

Why This Matters

In Control Theory, stability isn't just a nice-to-have—it's the fundamental question you must answer before anything else matters. A system that's unstable will diverge, oscillate wildly, or crash, making all your careful design work irrelevant. You're being tested on your ability to predict system behavior using mathematical tools, and stability criteria give you that predictive power. These methods connect directly to core concepts like transfer functions, frequency response, pole placement, and feedback system design.

What makes stability analysis challenging—and testable—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. That's what separates a surface-level answer from one that earns full credit.

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Algebraic Methods: Testing Stability Without Solving

These criteria let you determine stability directly from the characteristic polynomial—no need to actually find the roots. The key insight: the coefficients of a polynomial contain hidden information about where its roots lie.

Routh-Hurwitz Stability Criterion

• Analyzes the characteristic polynomial of an LTI system to determine stability without computing roots directly
• Routh array construction—arrange coefficients systematically, then compute subsequent rows using a specific formula to reveal root locations
• First-column sign test—a system is stable if and only if all elements in the first column are positive; sign changes indicate right half-plane poles

Jury Stability Test (for Discrete-Time Systems)

• Discrete-time equivalent of Routh-Hurwitz, designed for systems described by z-domain transfer functions
• Jury array construction—similar tabular approach but with different computational rules suited to the unit circle criterion
• Unit circle requirement—stability demands all roots of the characteristic polynomial lie inside the unit circle in the z-plane, not the left half-plane

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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

• Maps the open-loop frequency response onto the complex plane, creating a polar plot as frequency sweeps from $-\infty$ to $+\infty$
• Encirclement counting—the number of clockwise encirclements of the critical point $(-1, 0)$ reveals the difference between closed-loop right half-plane poles and open-loop right half-plane poles
• 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

Bode Stability Criterion

• Uses magnitude and phase plots versus frequency to assess stability through crossover analysis
• Gain crossover frequency—where magnitude equals 0 dB; phase margin is measured here as the distance from $-180°$
• Assumes open-loop stability—this method works cleanly only when the open-loop transfer function has no right half-plane poles

Phase Margin and Gain Margin

• Phase margin (PM)—additional phase lag the system can tolerate at gain crossover before hitting $-180°$ total phase; typical design targets are $45°$ to $60°$
• Gain margin (GM)—factor by which gain can increase at phase crossover (where phase equals $-180°$) before the system becomes unstable
• Robustness indicators—larger margins mean the system can handle more uncertainty in plant parameters; both must be positive for stability

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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

• Traces pole trajectories as a single parameter (typically feedback gain $K$) varies from 0 to $\infty$
• Design by pole placement—choose gain values that position closed-loop poles in desirable regions for stability and transient response
• Stability boundaries visible—the imaginary axis crossing points tell you exactly what gain values cause instability

Pole-Zero Analysis

• Pole locations determine stability—all poles must have negative real parts (left half-plane) for a continuous-time system to be stable
• Zeros affect response shape—while zeros don't directly determine stability, they influence overshoot, settling time, and can cause non-minimum phase behavior
• Right half-plane zeros—create inverse response (initial movement opposite to final value), complicating controller design

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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

• Eigenvalue criterion—stability requires all eigenvalues of the system matrix $A$ to have negative real parts
• Direct connection to poles—for LTI systems, eigenvalues of $A$ equal the poles of the transfer function, unifying time-domain and frequency-domain views
• MIMO capability—handles multi-input multi-output systems naturally, where transfer function methods become unwieldy

Lyapunov Stability Theory

• Energy-based reasoning—if you can find a positive definite function $V(x)$ that always decreases along system trajectories, the equilibrium is stable
• No solution required—proves stability without solving the differential equations, which may be impossible for nonlinear systems
• Nonlinear system applicability—extends stability analysis beyond linear systems where eigenvalue methods don't apply

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Nonlinear System Methods: Beyond Linearization

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

Circle Criterion

• Graphical Nyquist extension—draws a circle (or disk) in the complex plane based on the nonlinearity's sector bounds
• Sector-bounded nonlinearities—applies when the nonlinear element's input-output relationship stays within a cone defined by two slopes
• Sufficient condition—if the Nyquist plot avoids the critical circle, stability is guaranteed; conservative but broadly applicable

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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. An FRQ asks you 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?