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Controls and Systems · Chapter 6 of 10 · Intermediate

Stability and the Routh-Hurwitz Criterion

A control loop is useless if it oscillates or runs away. The Routh test answers whether every pole sits in the left half-plane without ever finding the poles themselves.

Readiness check

This chapter tests stability from a polynomial. Tick only what you can do closed-notes.

Write the characteristic equation 1 + GH = 0.
Recall that stability means all poles in the left half-plane.
Compute a 2-by-2 determinant.
Solve a linear inequality for a parameter.
Identify the coefficients of a polynomial.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsRevisit poles and stability in Chapter 2.

3 or more weak itemsReview polynomials and inequalities before continuing.

The core idea

A system is stable when every root of its characteristic equation lies in the left half-plane. The Routh array reveals how many roots are not, from the coefficients alone.

1 + G(s)H(s) = 0stable ⇔ all poles in left half-planesign changes in column 1 = unstable roots

The closed-loop poles are the roots of the characteristic equation 1 + GH = 0. Factoring a high-order polynomial is hard, but stability only asks whether all roots have negative real parts. The Routh-Hurwitz criterion builds a triangular array from the coefficients; the number of sign changes in its first column equals the number of roots in the right half-plane. Zero sign changes means stable. The same array, with a parameter left symbolic, gives the exact range of gain for which the loop stays stable.

The skill works when: you form the characteristic polynomial, build the array, and read the first column.

The skill breaks down when: the array is built from the forward path instead of 1 + GH, or a sign change is missed.

The concept. As loop gain increases, a pole pair migrates toward the imaginary axis. When it crosses, the system loses stability; Routh finds the exact gain at the boundary without plotting the poles.

The skills, taught in order

Five skills define stability, build the Routh array, and use it to bound a gain.

6.1 Stability and pole location

A linear system is bounded-input bounded-output stable exactly when every pole has a negative real part. A single pole in the right half-plane makes the response grow without bound; a pole on the imaginary axis gives a sustained oscillation, the marginal case.

6.2 The characteristic equation

The closed-loop poles are the roots of 1 + G(s)H(s) = 0. Clearing fractions turns this into a polynomial, the characteristic polynomial, whose coefficients feed the Routh test. A necessary first check is that all coefficients are present and of the same sign.

6.3 Building the Routh array

List the coefficients in two rows by descending power, then compute each later row from the two above it with a cross-multiplication divided by the pivot. The array narrows to a single entry, and only the first column matters for stability.

Row	Entries	Rule
s³	a₃, a₁	odd coefficients
s²	a₂, a₀	even coefficients
s¹	(a₂a₁ − a₃a₀)/a₂	cross-multiply
s⁰	a₀	last term

6.4 Reading the first column

Count the sign changes down the first column: each one is a root in the right half-plane. No sign changes means the system is stable. A zero in the first column signals a root on or near the imaginary axis and needs a special technique.

6.5 Range of gain for stability

Leaving a gain K symbolic in the characteristic polynomial, the Routh conditions become inequalities in K. Solving them gives the exact range of gain for stability, and the boundary value, where a first-column entry hits zero, gives the frequency of the marginal oscillation.

Engineering connection: the gain limit from Routh is the same crossing the root locus reaches in Chapter 7 and the gain margin measures in Chapter 9.

Worked example 1: testing stability with a Routh array

Test the stability of a system whose characteristic equation is s³ + 6s² + 11s + 6 = 0.

Figure 1. The Routh array for the cubic. The first column (green box) is all positive, so there are no right half-plane roots and the system is stable.

ProblemDetermine whether the system in Figure 1 is stable.
Given / findCharacteristic equation s³ + 6s² + 11s + 6 = 0. Find the stability from the Routh array.
AssumptionsLinear time-invariant system; the polynomial is the full characteristic equation.
ModelBuild the Routh array and count sign changes in the first column.
Equationss¹ entry = (6·11 − 1·6)/6
SolveRows: s³ → 1, 11; s² → 6, 6; s¹ → (66 − 6)/6 = 10; s⁰ → 6. The first column is 1, 6, 10, 6, with no sign changes, so the system is stable.
CheckThe polynomial factors as (s + 1)(s + 2)(s + 3), with roots −1, −2, −3, all in the left half-plane, confirming the Routh result without ever factoring during the test.
ConclusionRouth confirmed stability from the coefficients alone. For this cubic the roots happened to be clean, but the array works the same for any order.

Result. First column 1, 6, 10, 6: no sign changes, so the system is stable.

Worked example 2: the range of gain for stability

A unity-feedback loop has the characteristic equation s³ + 3s² + 2s + K = 0. Find the range of K for stability and the frequency of oscillation at the stability boundary.

Figure 2. The Routh condition (6 − K)/3 > 0 bounds stability at K < 6. At the boundary K = 6, the loop sustains an oscillation at ω = √2 rad/s.

ProblemFind the stable range of K and the boundary frequency for the loop in Figure 2.
Given / findCharacteristic equation s³ + 3s² + 2s + K = 0. Find the range of K and ω at the boundary.
AssumptionsK is a positive gain; the polynomial is the full characteristic equation.
ModelBuild the Routh array with K symbolic and require every first-column entry to be positive.
Equationss¹ entry = (3·2 − 1·K)/3 = (6 − K)/3auxiliary: 3s² + K = 0 at the boundary
SolveFirst column: 1, 3, (6 − K)/3, K. Stability needs (6 − K)/3 > 0 and K > 0, so 0 < K < 6. At K = 6 the s¹ row vanishes; the auxiliary 3s² + 6 = 0 gives s² = −2, so ω = √2 ≈ 1.41 rad/s.
CheckSubstituting s = j√2 and K = 6 into the polynomial gives zero, confirming the marginal poles sit exactly on the imaginary axis at that gain.
ConclusionRouth gave both the gain limit and the oscillation frequency at the edge of stability, the information a designer needs to stay safely below it.

Result. Stable for 0 < K < 6; at K = 6 the loop oscillates at ω = √2 rad/s.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Array from the forward path	Wrong stability conclusion	"Did I use 1 + GH?"	Build the array from the characteristic polynomial, not G alone.
Missing-coefficient oversight	Calling it stable when a term is absent	"Are all powers of s present and same-signed?"	A missing or sign-changed coefficient already implies instability.
Ignoring a first-column zero	Division by zero in the array	"Did a pivot become zero?"	Use the epsilon or auxiliary-polynomial method for the special case.
Forgetting the boundary frequency	Gain limit found but no ω	"What does the auxiliary polynomial give?"	Solve the auxiliary equation at the boundary for the oscillation frequency.

Practice ladder

Level 1 · Direct skill

Is s² + 5s + 6 = 0 stable? Use the coefficients.

Show answer

All coefficients are positive and present, and for a second-order polynomial that is sufficient: stable, with roots −2 and −3.

Level 2 · Mixed concept

Build the Routh array for s³ + 2s² + 4s + 8 = 0 and judge stability.

Show answer

s³: 1, 4; s²: 2, 8; s¹: (2·4 − 1·8)/2 = 0; this zero row signals roots on the imaginary axis. The auxiliary 2s² + 8 = 0 gives s = ±2j: marginally stable.

Level 3 · Independent problem

Find the range of K for stability of s³ + 4s² + 5s + K = 0.

Show answer

s¹ entry = (4·5 − K)/4 = (20 − K)/4 > 0 needs K < 20, and K > 0. Stable for 0 < K < 20; at K = 20 the loop oscillates at ω = √5.

Level 4 · Transfer to real engineering

Take a real feedback loop with an adjustable gain and explain how you would find the gain at which it starts to oscillate, and why staying below it matters.

What good work looks like

A characteristic equation in K, a Routh condition that gives the critical gain, and a margin argument for operating well below it.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I built the array from 1 + GH and counted sign changes correctly."

"Give me four characteristic polynomials; I will judge stability for each."

"Is this stable?" Building the array and reading the first column is the skill.

"What gain makes it oscillate?" Solving the Routh inequality is the point.

Portfolio task

Take a real loop with a tunable gain, form its characteristic equation, use Routh to find the stable gain range, and identify the oscillation frequency at the boundary.

Must include: a characteristic polynomial, a Routh array, a stable gain range, and a boundary frequency.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. When is a system stable?

When every root of the characteristic equation has a negative real part.

2. What does the Routh first column tell you?

The number of sign changes equals the number of right half-plane roots.

3. What polynomial feeds the array?

The characteristic polynomial from 1 + GH = 0.

4. How do you find the gain limit?

Keep K symbolic, require all first-column entries positive, and solve the inequalities.

5. What gives the oscillation frequency at the boundary?

The auxiliary polynomial from the row above the zero row.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive a Routh array and a gain range from a blank page.

+3 daysTest two new polynomials for stability.

+7 daysCarry the gain limit into the root locus, Chapter 7.

+30 daysCompare the Routh gain limit with the gain margin from Bode.

Textbook mapping

Item	Mapping
Primary source	Ogata, Modern Control Engineering, Chapter 5 (Stability, Routh-Hurwitz Criterion)
Cross-reference	Nise, Ch. 6 · Dorf and Bishop, Ch. 6
Core topics	6.1 Stability and poles · 6.2 Characteristic equation · 6.3 Building the array · 6.4 Reading column 1 · 6.5 Gain range
Engineering connection	The Routh gain limit reappears in root locus and gain margin.
Read next	Chapter 7: The Root-Locus Method.