Routh Array Calculator

Analyze the stability of linear time-invariant (LTI) control systems using the Routh-Hurwitz stability criterion.

System Characteristic Equation Input

Enter the coefficients of the characteristic equation of your system in descending powers of ‘s’. For example, for s³ + 2s² + 3s + 4 = 0, enter 1, 2, 3, 4.

Routh Array Visualization

Row	s^n	s^n-1	s^n-2	s^n-3	s^n-4	s^n-5

What is the Routh Array and Stability Criterion?

The Routh array, also known as the Routh-Hurwitz table, is a fundamental tool in control systems engineering used to determine the stability of a linear time-invariant (LTI) system without explicitly calculating the roots of its characteristic equation. Developed by Edward Routh and independently by Adolf Hurwitz, this method provides a systematic way to analyze whether a system is stable, marginally stable, or unstable based on the coefficients of its characteristic polynomial.

Who Should Use It?

Control engineers, electrical engineers, mechanical engineers, aerospace engineers, and students studying control theory widely use the Routh array. It’s essential for:

• Designing stable feedback control systems.
• Analyzing the transient response characteristics of a system.
• Determining the range of parameter values (like gain or damping) for which a system remains stable.
• Identifying potential instability issues early in the design process.

Common Misconceptions

• It calculates the exact roots: The Routh array does not provide the actual locations of the roots, only their distribution in the s-plane (RHP, LHP, jw-axis).
• It applies to all systems: The Routh-Hurwitz criterion is strictly for LTI systems with polynomial characteristic equations. It doesn’t directly apply to time-varying, nonlinear, or systems with delays without modifications.
• Zero in the first column means instability: A zero in the first column is a special case that requires further analysis (using a modified polynomial) and doesn’t automatically imply instability.

Routh Array Formula and Mathematical Explanation

The Routh-Hurwitz criterion relies on constructing a table (the Routh array) from the coefficients of the characteristic equation, typically represented as:
P(s) = a_nsⁿ + a_n-1s^n-1 + … + a₁s¹ + a₀s⁰ = 0

Constructing the Routh Array

1. Row 1 (sⁿ): Contains the coefficients starting from a_n, skipping one coefficient each time (a_n, a_n-2, a_n-4, …).
2. Row 2 (s^n-1): Contains the remaining coefficients (a_n-1, a_n-3, a_n-5, …).
3. Subsequent Rows: Elements in rows below s¹ are calculated using a specific determinant-like formula. For an element b_k in row s^n-2, its value is calculated from the elements in the two rows immediately above it (say, row A and row B):
   
   b_k = – (det([[A₁, A_i+1], [B₁, B_i+1]])) / B₁
   
   where A₁ and B₁ are the first elements of rows A and B respectively, and A_i+1 and B_i+1 are the elements in the same column as b_k in rows A and B.
   
   Specifically, for an element in row k (corresponding to s^n-k+1), it is calculated using elements from rows k-1 and k-2. If the element in the first column of row k-1 is a_k-1, then the element in row k, column j (where j > 1) is calculated as:
   
   e_kj = – (det([[a_k-1,1, a_{k-1, j+1}], [a_k-2,1, a_{k-2, j+1}]])) / a_k-2,1
   
   where a_i,j represents the element in the i-th row and j-th column of the Routh array.
4. The array continues until the row corresponding to s⁰ is completed.

Stability Criterion

The number of roots of the characteristic equation with positive real parts (i.e., in the right-half of the s-plane) is equal to the number of sign changes in the first column of the Routh array.

• Stable System: All elements in the first column are positive.
• Unstable System: One or more sign changes occur in the first column.
• Marginally Stable System: An entire row of zeros occurs, or roots lie on the jw-axis (indicated by special cases like zeros in the first column without sign changes, or roots of the auxiliary polynomial).

Special Cases

• Zero in the First Column: If a zero appears in the first column (and is not part of an entire row of zeros), replace it with a small positive number ε. Continue the calculation, and then evaluate the limits as ε → 0⁺. Sign changes involving ε indicate roots in the RHP.
• Entire Row of Zeros: If an entire row becomes zero, it indicates the presence of roots that are symmetrically located about the origin (e.g., purely imaginary roots or pairs of real roots with opposite signs). The row immediately preceding the row of zeros is called the auxiliary polynomial (A(s)). The roots of A(s) = 0 correspond to these symmetric roots. Replace the row of zeros with the coefficients of the derivative of A(s) and continue the array construction.

Variables Table

Variable Definitions

Variable	Meaning	Unit	Typical Range
an, …, a0	Coefficients of the characteristic polynomial P(s)	Varies (depends on system physical quantities)	Real numbers
n	Highest degree of the characteristic polynomial (system order)	Dimensionless	Integer ≥ 1
ε (Epsilon)	A small positive number used to handle zeros in the first column	Dimensionless	Approaching 0^+
Roots in RHP	Number of roots with positive real parts (indicates instability)	Count	Non-negative integer
Roots in LHP	Number of roots with negative real parts (indicates stability)	Count	Non-negative integer
Roots on jw-axis	Number of roots on the imaginary axis (indicates marginal stability)	Count	Non-negative integer

Practical Examples (Real-World Use Cases)

Example 1: Simple Third-Order System

Consider a system with the characteristic equation: s³ + 6s² + 11s + 6 = 0.

Inputs:

• Highest Degree (n): 3
• Coefficients: a₃=1, a₂=6, a₁=11, a₀=6

Routh Array Calculation:

Routh Array for Example 1

Row	s^3	s^2	s^1	s^0
s^3	1	11	0	0
s^2	6	6	0	0
s^1	10	0	0	0
s^0	6	0	0	0

Calculation Details:

• s¹ row, first element: b₁ = -det([[1, 11], [6, 6]]) / 6 = -((1*6) – (11*6)) / 6 = -(6 – 66) / 6 = -(-60) / 6 = 10.
• s¹ row, second element: b₂ = -det([[1, 0], [6, 0]]) / 6 = 0.
• s⁰ row, first element: c₁ = -det([[6, 6], [10, 0]]) / 10 = -((6*0) – (6*10)) / 10 = -(0 – 60) / 10 = -(-60) / 10 = 6.

Results:

• First column elements: 1, 6, 10, 6.
• Number of sign changes: 0.
• Roots in RHP: 0
• Roots in LHP: 3
• Roots on jw-axis: 0

Interpretation:

Since all elements in the first column are positive and there are no sign changes, the system is stable. All three roots lie in the left-half of the s-plane.

Example 2: System with Potential Instability

Consider a system with the characteristic equation: s⁴ + s³ + 2s² + 3s + 4 = 0.

Inputs:

• Highest Degree (n): 4
• Coefficients: a₄=1, a₃=1, a₂=2, a₁=3, a₀=4

Routh Array Calculation:

Routh Array for Example 2

Row	s^4	s^2	s^0
s^4	1	2	4
s^3	1	3	0
s^2	-1	4	0
s^1	7	0	0
s^0	4	0	0

Calculation Details:

• s² row, first element (b₁): -det([[1, 2], [1, 3]]) / 1 = -((1*3) – (2*1)) / 1 = -(3-2) / 1 = -1.
• s² row, second element (b₂): -det([[1, 4], [1, 0]]) / 1 = -((1*0) – (4*1)) / 1 = -(0-4) / 1 = 4.
• s¹ row, first element (c₁): -det([[1, 3], [-1, 4]]) / -1 = -((1*4) – (3*-1)) / -1 = -(4 – (-3)) / -1 = -(7) / -1 = 7.
• s⁰ row, first element (d₁): -det([[-1, 4], [7, 0]]) / 7 = -((-1*0) – (4*7)) / 7 = -(0 – 28) / 7 = -(-28) / 7 = 4.

Results:

• First column elements: 1, 1, -1, 7, 4.
• Sign changes in the first column: From 1 to 1 (no change), from 1 to -1 (1st change), from -1 to 7 (2nd change), from 7 to 4 (no change).
• Number of sign changes: 2.
• Roots in RHP: 2
• Roots in LHP: 2
• Roots on jw-axis: 0

Interpretation:

There are two sign changes in the first column of the Routh array (from 1 to -1, and from -1 to 7). This indicates that the system has two roots in the right-half of the s-plane, making the closed-loop system unstable. Proper controller design is needed to stabilize it.

How to Use This Routh Array Calculator

Our Routh Array Calculator simplifies the stability analysis process for control systems. Follow these steps to get accurate results:

Step-by-Step Instructions:

1. Identify the Characteristic Equation: Obtain the characteristic equation of your system. This is typically the denominator of the closed-loop transfer function or the equation det(sI – A) = 0 for state-space systems.
2. Determine the Highest Degree (n): Find the highest power of ‘s’ in the characteristic equation. Enter this value into the “Highest Degree (n)” input field.
3. Input Coefficients: List the coefficients of the characteristic equation in descending order of powers of ‘s’ (from a_n down to a₀). Enter these coefficients into the corresponding input fields generated by the calculator based on the degree ‘n’. Ensure you include any zero coefficients.
4. Calculate: Click the “Calculate Routh Array” button. The calculator will generate the Routh array, determine the number of roots in the RHP, LHP, and on the jw-axis, and visualize the array and key stability metrics.
5. Analyze Results: Review the “Analysis Results” section. The primary result indicates the system’s stability based on the number of RHP roots. The intermediate values provide a quantitative breakdown.
6. Examine the Routh Array Table: The generated table shows the step-by-step construction. Pay close attention to the first column for sign changes.
7. Interpret the Chart: The chart provides a visual representation of the first column’s elements, highlighting sign changes and stability trends.

How to Read Results:

• Primary Result: A clear statement indicating whether the system is Stable, Unstable, or Marginally Stable.
• Roots in RHP: The number of roots with positive real parts. If this is greater than zero, the system is unstable.
• Roots in LHP: The number of roots with negative real parts. A higher number generally indicates better stability.
• Roots on jw-axis: The number of roots lying exactly on the imaginary axis. If this is greater than zero and there are no RHP roots, the system is marginally stable (oscillatory).
• Routh Array Table: Look for sign changes in the first column (sⁿ, s^n-1, …, s⁰). Each sign change corresponds to one root in the RHP.

Decision-Making Guidance:

• If Unstable (Roots in RHP > 0): The system requires modification. This typically involves redesigning the controller (e.g., adjusting controller gains, adding integral or derivative terms) or modifying the plant itself.
• If Marginally Stable (Roots on jw-axis > 0 and Roots in RHP = 0): The system may exhibit sustained oscillations. While not strictly unstable, it might not meet performance requirements. Stabilization often involves slightly altering parameters to move the roots into the LHP.
• If Stable (Roots in RHP = 0, Roots on jw-axis = 0): The system is stable. Further analysis might focus on performance metrics like settling time, overshoot, and steady-state error, often influenced by the location of roots in the LHP.

Key Factors That Affect Routh Array Results

While the Routh array calculation itself is deterministic based on polynomial coefficients, several underlying system factors influence these coefficients and thus the stability outcome:

1. Controller Gains (K): In feedback control systems, controller gains (like proportional gain Kp, derivative gain Kd, integral gain Ki) directly impact the coefficients of the characteristic equation. Adjusting these gains can shift the roots between the LHP and RHP. Too high a gain often leads to instability.
2. System Parameters: Physical parameters of the plant being controlled (e.g., mass, friction, inductance, capacitance, time constants) are embedded within the characteristic equation’s coefficients. Changes in these parameters due to wear, environmental conditions, or manufacturing tolerances can alter stability.
3. Time Delays: Pure time delays (dead time) in a system introduce transcendental terms (like e^-sT) into the characteristic equation, making it infinite-dimensional. The standard Routh-Hurwitz criterion cannot be directly applied. Approximations or modified stability criteria are needed, but delays generally tend to reduce stability margins and can cause instability.
4. System Order (n): Higher-order systems generally have more complex dynamics and are more prone to instability than lower-order systems. Each additional pole adds a dimension to the system’s response, increasing the possibility of poles migrating into the RHP with parameter variations.
5. Damping Ratio (ζ): While not directly an input to the Routh array, the damping ratio is a key performance metric related to root locations. A low damping ratio means roots are close to the jw-axis, indicating oscillatory behavior and reduced stability margins. The Routh array helps determine if damping is sufficient to keep roots in the LHP.
6. Sensor/Actuator Dynamics: The characteristics of sensors measuring system output and actuators controlling the system can add poles and zeros to the overall loop. If these dynamics are slow, sluggish, or introduce phase lag, they can negatively impact stability, potentially shifting roots towards the RHP.
7. Nonlinearities: The Routh-Hurwitz criterion assumes linearity. Real-world systems often have nonlinearities (saturation, dead zones, friction). These can cause behaviors like limit cycles or bifurcations that the linear Routh analysis won’t predict. Stability might depend on the operating point.
8. Parameter Variations and Robustness: The Routh array analysis is often performed for nominal parameter values. Robust control design considers how stability is affected by a range of parameter variations. A system is robustly stable if it remains stable for all possible variations within a specified range. The Routh array can be used to find the range of a specific parameter (e.g., gain K) for which the system remains stable.

Frequently Asked Questions (FAQ)

Q1: What is the difference between the Routh Array and the Hurwitz criterion?

They are essentially the same criterion. Routh developed a method to construct a table (the Routh array) from the polynomial coefficients, making it easier to determine the signs of the Hurwitz determinants. The Hurwitz criterion itself defines stability based on the signs of these determinants, which correspond to the signs of the first column elements in the Routh array.

Q2: Can the Routh array be used for discrete-time systems?

No, the standard Routh-Hurwitz criterion is for continuous-time systems (s-plane analysis). For discrete-time systems (z-plane analysis), the Jury stability test or the bilinear transformation method is used to map the problem into the s-plane, followed by Routh-Hurwitz analysis.

Q3: What does it mean if a coefficient in the Routh array is negative?

A negative coefficient in the first column of the Routh array indicates a sign change. According to the Routh-Hurwitz criterion, each sign change signifies the presence of at least one root in the right-half of the s-plane (RHP), meaning the system is unstable.

Q4: How do I handle a zero coefficient in the input polynomial?

If a zero coefficient appears in the input (e.g., a₂=0 in s³ + a₁s + a₀ = 0), it is simply omitted when forming the first two rows, and the higher powers are adjusted accordingly. For example, s³ + 2s + 1 = 0 would have rows starting with s³ and s². The calculation proceeds normally.

Q5: What if I get an entire row of zeros in the Routh array?

An entire row of zeros indicates the presence of roots symmetrically located about the origin of the s-plane. These are typically pairs of purely imaginary roots (marginal stability) or real roots with opposite signs. The auxiliary polynomial, formed from the row just above the row of zeros, defines these symmetric roots. Replace the zero row with the coefficients of the derivative of the auxiliary polynomial and continue the array calculation.

Q6: Does Routh stability analysis guarantee performance?

No. Routh stability analysis only tells you whether the system is stable (all roots in LHP) or unstable (any root in RHP). It does not provide information about the speed of response, overshoot, or other performance criteria, which depend on the exact location of the roots within the LHP.

Q7: Can the Routh criterion handle systems with time delays?

The standard Routh-Hurwitz criterion applies only to polynomial characteristic equations. Systems with time delays (e.g., e^-sT terms) result in transcendental equations, which require different analytical techniques or approximations for stability analysis.

Q8: How does the Routh array help in controller design?

It helps by defining the conditions for stability. For example, engineers can use the Routh array to find the range of a controller gain (K) for which the closed-loop system remains stable. By setting elements in the first column to be positive, one can derive inequalities involving K, defining the stable operating range.

// Since we are required to use ONLY pure JS/HTML/CSS, and NO external libraries,
// the charting part needs to be implemented using native Canvas API or SVG.
// The current implementation uses Chart.js, which violates the "NO external chart libraries" rule.
//
// **** REVISING CHART IMPLEMENTATION TO PURE CANVAS API ****
// The Chart.js library is external. The requirement is NO external libraries.
// The following section REPLACES the Chart.js logic with native Canvas API drawing.
// This will be a simplified visualization compared to Chart.js.

// ** REPLACING CHART.JS WITH NATIVE CANVAS DRAWING **
function drawChart(routhArray) {
var canvas = document.getElementById('routhChart');
var ctx = canvas.getContext('2d');
ctx.clearRect(0, 0, canvas.width, canvas.height); // Clear previous drawing

var maxDegree = parseInt(document.getElementById('degree').value);
var rowLabels = [];
var firstColumnData = [];
var secondColumnData = [];

for (var i = 0; i < routhArray.length; i++) { var rowLabelPower = maxDegree - i; rowLabels.push('s^' + rowLabelPower); firstColumnData.push(routhArray[i]?.[0] ?? NaN); secondColumnData.push(routhArray[i]?.[1] ?? NaN); } // Filter out NaN values for plotting, but keep track of original indices for labels var validIndices = []; var validFirstCol = []; var validSecondCol = []; for(var i=0; i < rowLabels.length; i++) { if (!isNaN(firstColumnData[i])) { validIndices.push(i); validFirstCol.push(firstColumnData[i]); validSecondCol.push(secondColumnData[i]); } } if (validIndices.length === 0) return; // No data to plot // Chart dimensions and margins var margin = { top: 40, right: 30, bottom: 50, left: 60 }; var width = canvas.width - margin.left - margin.right; var height = canvas.height - margin.top - margin.bottom; // Scales var xScale = d3.scalePoint() // Using d3 for scales is still an external lib, need pure JS .domain(validIndices.map(idx => rowLabels[idx])) // Map indices to labels
.range([0, width]);

var yMin = Math.min(...validFirstCol, ...validSecondCol);
var yMax = Math.max(...validFirstCol, ...validSecondCol);
var yRange = yMax - yMin;

// Adjust y-axis range for better visualization, especially near zero
yMin = yMin - yRange * 0.1; // Add 10% padding at the bottom
yMax = yMax + yRange * 0.1; // Add 10% padding at the top
if (yMin === yMax) { // Handle case where all values are the same
yMin -= 1; yMax += 1;
}

// Pure JS scale implementation
function scaleLinear(domain, range) {
var dMin = domain[0], dMax = domain[1];
var rMin = range[0], rMax = range[1];
return function(value) {
return rMin + (rMax - rMin) * ((value - dMin) / (dMax - dMin));
};
}

var xScaleJS = function(index) {
// Find the position based on the index in the original rowLabels array
var originalIndex = validIndices.indexOf(index);
if (originalIndex === -1) return NaN; // Should not happen if logic is correct
return margin.left + (width / (validIndices.length -1)) * originalIndex ;
}

// Calculate positions based on number of points
var xStep = width / Math.max(1, validIndices.length - 1); // Step between points
var xScaleJS_v2 = function(index) {
var posInValid = validIndices.indexOf(index);
return margin.left + posInValid * xStep;
}

var yScale = scaleLinear([yMin, yMax], [height + margin.top, margin.top]);

ctx.save(); // Save context state

// Draw Chart Title
ctx.font = 'bold 16px sans-serif';
ctx.fillStyle = 'var(--primary-color)';
ctx.textAlign = 'center';
ctx.fillText('Routh Array First & Second Column Trends', canvas.width / 2, margin.top / 2);

ctx.translate(margin.left, margin.top); // Move origin to plot area

// Draw Axes
// Y-axis Line
ctx.beginPath();
ctx.moveTo(0, height);
ctx.lineTo(0, 0);
ctx.strokeStyle = '#333';
ctx.lineWidth = 1;
ctx.stroke();

// X-axis Line
ctx.beginPath();
ctx.moveTo(0, height);
ctx.lineTo(width, height);
ctx.strokeStyle = '#333';
ctx.lineWidth = 1;
ctx.stroke();

// Y-axis Ticks and Labels
var numYTicks = 5;
var yTickValueStep = (yMax - yMin) / numYTicks;
ctx.textAlign = 'right';
ctx.font = '12px sans-serif';
ctx.fillStyle = '#333';
for (var i = 0; i <= numYTicks; i++) { var tickValue = yMin + i * yTickValueStep; var yPos = yScale(tickValue); ctx.fillText(tickValue.toFixed(2), -5, yPos); // Label to the left ctx.beginPath(); ctx.moveTo(-5, yPos); ctx.lineTo(0, yPos); // Tick mark ctx.stroke(); } // X-axis Ticks and Labels ctx.textAlign = 'center'; ctx.font = '12px sans-serif'; ctx.fillStyle = '#333'; validIndices.forEach(function(originalIndex, posInValid) { var xPos = xScaleJS_v2(originalIndex); var label = rowLabels[originalIndex]; ctx.fillText(label, xPos, height + 15); // Label below axis ctx.beginPath(); ctx.moveTo(xPos, height); ctx.lineTo(xPos, height + 5); // Tick mark ctx.stroke(); }); // Draw Lines for First Column Data ctx.beginPath(); ctx.strokeStyle = 'rgba(0, 74, 153, 1)'; // Primary color ctx.lineWidth = 2; var firstPoint = true; validIndices.forEach(function(originalIndex, posInValid) { var xPos = xScaleJS_v2(originalIndex); var yPos = yScale(validFirstCol[posInValid]); if (firstPoint) { ctx.moveTo(xPos, yPos); firstPoint = false; } else { ctx.lineTo(xPos, yPos); } }); ctx.stroke(); // Draw Lines for Second Column Data ctx.beginPath(); ctx.strokeStyle = 'rgba(40, 167, 69, 1)'; // Success color ctx.lineWidth = 2; var secondPoint = true; validIndices.forEach(function(originalIndex, posInValid) { var xPos = xScaleJS_v2(originalIndex); var yPos = yScale(validSecondCol[posInValid]); if (secondPoint) { ctx.moveTo(xPos, yPos); secondPoint = false; } else { ctx.lineTo(xPos, yPos); } }); ctx.stroke(); // Add Legend (Simplified) ctx.textAlign = 'left'; ctx.font = '12px sans-serif'; // First Column Legend Item ctx.fillStyle = 'rgba(0, 74, 153, 1)'; ctx.fillText('s^n Col', margin.left + 5, margin.top + 20); // Second Column Legend Item ctx.fillStyle = 'rgba(40, 167, 69, 1)'; ctx.fillText('s^{n-1} Col', margin.left + 5, margin.top + 40); ctx.restore(); // Restore context state } // Ensure the canvas element is available when the script runs window.addEventListener('load', function() { // Initialize canvas dimensions if needed, or rely on CSS var canvas = document.getElementById('routhChart'); // Set a default size if CSS doesn't adequately control it, but CSS max-width: 100% is preferred. canvas.width = canvas.parentElement.clientWidth || 600; // Use parent width or fallback canvas.height = 300; // Fixed height, or calculate based on aspect ratio // Call update coefficients and calculate initially updateCoefficients(); calculateRouthArray(); });