RishanKPLearn
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Control Systems Engineering•Unit 3• EEE2005

Stability Analysis & Root Locus Techniques

Concept of BIBO and asymptotic stability, characteristic equation and s-plane pole locations, Routh-Hurwitz stability criterion with special cases (first column zero and all-zero row / auxiliary polynomial), marginal stability, Evans Root Locus fundamentals, angle and magnitude criteria, construction rules (real-axis segments, asymptotes, centroid, breakaway points, departure angles, jω crossings), and s-plane transient design bounds.

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Skills Built:BIBO & Asymptotic Stability Classification via s-Plane PolesRouth-Hurwitz Array Determinant Computation (b1, b2, c1)Routh Special Case 1: Epsilon (ε → 0+) Zero-Pivot SubstitutionRouth Special Case 2: Row of Zeros & Auxiliary Polynomial A(s)Critical Gain Kcrit & Sustained Oscillation Frequency ωosc CalculationEvans Angle (∠GH = ±(2q+1)180°) & Magnitude (|GH| = 1) CriteriaRoot Locus 7 Construction Rules (Centroid σa, Asymptotes θa, Breakaway)Angle of Departure φd & Arrival θa for Complex Conjugate Rootss-Plane Allowable Design Bounds (Damping Cone ζ ≥ ζmin, Settling Boundary)

1. 5 Key Ideas: Stability Foundations & Root Locus Geometry

Idea 1: The Concept of Stability in Linear Time-Invariant (LTI) Systems

A physical dynamic system is defined as stable if its natural unforced transient response decays to zero as time approaches infinity (t→∞t \to \infty). Formally, stability is categorized into three canonical definitions:

  1. Bounded-Input Bounded-Output (BIBO) Stability: A system is BIBO stable if every bounded input r(t)r(t) (∣r(t)∣≤M1<∞|r(t)| \le M_1 < \infty for all t≥0t \ge 0) produces a bounded output c(t)c(t) (∣c(t)∣≤M2<∞|c(t)| \le M_2 < \infty). The necessary and sufficient condition for BIBO stability is that the system impulse response g(t)=L−1[G(s)]g(t) = \mathcal{L}^{-1}[G(s)] is absolutely integrable: ∫0∞∣g(t)∣ dt<∞\int_{0}^{\infty} |g(t)| \, dt < \infty
  2. Asymptotic (Internal) Stability: In the absence of external inputs, any non-zero initial state conditions x(0)≠0\mathbf{x}(0) \ne 0 decay exponentially to the origin: lim⁡t→∞x(t)=0\lim_{t \to \infty} \mathbf{x}(t) = 0.
  3. Marginal Stability: The unforced system response neither diverges to infinity nor decays to zero, instead maintaining bounded, non-decaying sustained oscillations (e.g., pure sine waves c(t)=Asin⁡(ω0t)c(t) = A \sin(\omega_0 t)).

In the complex ss-plane (s=σ+jωs = \sigma + j\omega), the denominator of the closed-loop transfer function T(s)=C(s)R(s)=G(s)1+G(s)H(s)T(s) = \frac{C(s)}{R(s)} = \frac{G(s)}{1 + G(s)H(s)} defines the characteristic equation: 1+G(s)H(s)=0  ⟺  D(s)=ansn+an−1sn−1+⋯+a1s+a0=01 + G(s)H(s) = 0 \iff D(s) = a_n s^n + a_{n-1} s^{n-1} + \dots + a_1 s + a_0 = 0

The locations of the roots of D(s)D(s) (the closed-loop poles) dictate the stability profile:

  • Strictly Stable: ALL closed-loop poles reside strictly in the Open Left-Half Plane (LHP), meaning Re{si}<0\text{Re}\{s_i\} < 0 for all ii. All transient components eσit→0e^{\sigma_i t} \to 0 decay exponentially.
  • Unstable: ONE OR MORE poles reside in the Right-Half Plane (RHP) (Re{si}>0\text{Re}\{s_i\} > 0), OR repeated (multiple-order) poles reside on the imaginary axis (jωj\omega-axis). Even a single RHP pole generates a divergent exponential mode (e+σt→∞e^{+\sigma t} \to \infty).
  • Marginally Stable: One or more non-repeated (simple) pairs of poles lie strictly on the imaginary jωj\omega-axis (Re{si}=0\text{Re}\{s_i\} = 0, s=±jω0s = \pm j\omega_0), with all other poles in the open LHP.

Idea 2: The Routh-Hurwitz Stability Criterion & Array Builder

Factoring high-order polynomials (n≥3n \ge 3) by hand is difficult or impossible. In 1877, Edward John Routh and Adolf Hurwitz developed an algebraic method to determine the exact number of unstable RHP roots without factoring the characteristic polynomial.

For an nn-th degree polynomial D(s)=ansn+an−1sn−1+an−2sn−2+⋯+a1s+a0=0D(s) = a_n s^n + a_{n-1} s^{n-1} + a_{n-2} s^{n-2} + \dots + a_1 s + a_0 = 0:

Necessary Condition (Stodola's Criterion)

For all roots of D(s)D(s) to lie in the open LHP, it is necessary (though not sufficient for n≥3n \ge 3) that:

  1. All polynomial coefficients an,an−1,…,a0a_n, a_{n-1}, \dots, a_0 exist (none are zero).
  2. All coefficients have the same algebraic sign (conventionally positive: ai>0a_i > 0). If any coefficient is missing or negative, the system is immediately unstable or marginally stable.

Sufficient Condition & The Routh Array Construction

Arrange the coefficients into alternating rows to seed the array:

s^n & a_n & a_{n-2} & a_{n-4} & a_{n-6} \\ s^{n-1} & a_{n-1} & a_{n-3} & a_{n-5} & a_{n-7} \\ s^{n-2} & b_1 & b_2 & b_3 & \dots \\ s^{n-3} & c_1 & c_2 & c_3 & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ s^1 & \dots & & & \\ s^0 & a_0 & & & \end{array}$$ Subsequent row elements are computed using the cross-determinant algorithm: $$b_1 = \frac{a_{n-1} a_{n-2} - a_n a_{n-3}}{a_{n-1}}, \quad b_2 = \frac{a_{n-1} a_{n-4} - a_n a_{n-5}}{a_{n-1}}, \quad b_3 = \frac{a_{n-1} a_{n-6} - a_n a_{n-7}}{a_{n-1}}$$ $$c_1 = \frac{b_1 a_{n-3} - a_{n-1} b_2}{b_1}, \quad c_2 = \frac{b_1 a_{n-5} - a_{n-1} b_3}{b_1}$$ #### Routh's Theorem *The number of roots of $D(s)$ with positive real parts (unstable RHP roots) is equal to the number of sign changes in the first column of the Routh array.* - For unconditional stability, every entry in the first column must be strictly positive ($> 0$). ### Idea 3: Routh-Hurwitz Special Cases: Epsilon Method & Row of Zeros Standard array construction encounters arithmetic singularities in two important special cases: #### Special Case 1: First Element of a Row is Zero, but Other Elements are Non-Zero If an element in the first column becomes zero ($b_1 = 0$) while remaining elements in that row are non-zero, calculating the next row requires dividing by zero ($c_1 = \frac{0 \cdot a_{n-3} - \dots}{0}$), crashing the algorithm. **Remedy (The Epsilon Method)**: 1. Replace the zero pivot with a small positive quantity $\epsilon > 0$ ($\epsilon \to 0^+$). 2. Continue evaluating subsequent rows algebraically in terms of $\epsilon$. 3. Take the limit as $\epsilon \to 0^+$ from the positive side, inspect the signs of the first column, and count sign changes. *(Alternative: Substitute $s = 1/z$ into $D(s)$, multiply by $z^n$, and apply Routh to the inverted polynomial $D(z) = 0$.)* #### Special Case 2: An Entire Row is Identically Zero (Symmetric Roots) If an entire row of the Routh array vanishes to zero ($0, 0, 0, \dots$), it indicates the presence of **radially symmetric roots** in the $s$-plane. These can be: - Real pairs symmetric about the origin ($s = \pm \sigma$). - Complex pairs symmetric about the origin (quadruplets: $s = \pm \sigma \pm j\omega$). - Pure imaginary conjugate pairs on the $j\omega$-axis ($s = \pm j\omega_0$), representing **marginal stability**. **Remedy (The Auxiliary Polynomial Method)**: 1. Form the **Auxiliary Polynomial $A(s)$** using the coefficients of the row immediately *above* the all-zero row. $A(s)$ contains only even powers of $s$ (e.g., $A(s) = K_1 s^4 + K_2 s^2 + K_3$). 2. Take the derivative with respect to $s$: $\frac{dA(s)}{ds}$. 3. Replace the all-zero row with the coefficients of $\frac{dA(s)}{ds}$. 4. Continue the Routh array to completion. 5. The roots of $A(s) = 0$ yield the exact symmetric frequencies of oscillation ($\omega_{\text{osc}} = \sqrt{\dots}$), and setting the all-zero row condition yields the **critical loop gain $K_{\text{crit}}$**. ### Idea 4: The Evans Root Locus Philosophy: Angle and Magnitude Criteria While Routh-Hurwitz reveals whether a system is stable, it does not reveal *how* stable the system is, nor how transient performance (damping ratio $\zeta$, settling time $t_s$) varies as loop gain $K$ changes. In 1948, Walter R. Evans invented the **Root Locus Technique**: a graphical method that sketches the trajectories traversed by the roots of the closed-loop characteristic equation in the complex $s$-plane as a parameter (usually the forward loop gain $K$) varies from $0$ to $+\infty$. The closed-loop characteristic equation is: $$1 + G(s)H(s) = 0 \implies G(s)H(s) = -1$$ In the complex domain, $-1$ has a magnitude of $1$ and an angle of $180^\circ$ (or any odd integer multiple of $\pi$). Thus, any point $s_0$ in the complex plane lies on the root locus if and only if it satisfies two fundamental criteria: 1. **Angle Condition (Necessary for locus membership)**: $$\angle G(s)H(s) = \pm (2q + 1)180^\circ = \pm 180^\circ, \pm 540^\circ, \dots \quad (q = 0, 1, 2, \dots)$$ Expressing $G(s)H(s) = K \frac{\prod_{i=1}^m (s - z_i)}{\prod_{j=1}^n (s - p_j)}$ in terms of vector angles from zeros $\theta_{z_i}$ and poles $\phi_{p_j}$ to test point $s$: $$\sum_{i=1}^m \angle(s - z_i) - \sum_{j=1}^n \angle(s - p_j) = (2q + 1)180^\circ$$ *The sum of angles from all open-loop zeros to point $s$ minus the sum of angles from all open-loop poles to point $s$ must equal an odd multiple of $180^\circ$.* 2. **Magnitude Condition (Used to calculate gain $K$ at a validated point)**: $$|G(s)H(s)| = 1 \implies K \frac{\prod_{i=1}^m |s - z_i|}{\prod_{j=1}^n |s - p_j|} = 1$$ $$K = \frac{\prod_{j=1}^n |s - p_j|}{\prod_{i=1}^m |s - z_i|} = \frac{\text{Product of pole-vector lengths}}{\text{Product of zero-vector lengths}}$$ ### Idea 5: The 7 Evans Construction Rules & Transient Design Bounds Constructing the root locus is governed by 7 canonical geometric rules derived directly from the angle and magnitude criteria: 1. **Rule 1: Number of Branches**: The root locus has $n$ separate branches, where $n$ is the number of open-loop poles (assuming $n \ge m$, where $m$ is the number of open-loop zeros). 2. **Rule 2: Symmetry**: The locus is completely symmetrical with respect to the real ($\sigma$) axis, because complex roots always occur in conjugate pairs. 3. **Rule 3: Starting and Ending Points**: As gain $K$ increases from $0$ to $\infty$: - At $K = 0$, the branches originate at the $n$ open-loop poles. - At $K \to \infty$, $m$ branches terminate at the $m$ finite open-loop zeros, while the remaining $(n - m)$ branches terminate at infinity along asymptotes. 4. **Rule 4: Real-Axis Segments (Odd Rule)**: A point on the real axis lies on the root locus if and only if the total number of real open-loop poles AND zeros to its **right** is **ODD**. (Complex conjugate poles/zeros contribute a net angle of $360^\circ$ and do not affect real-axis parity). 5. **Rule 5: Asymptotes and Real-Axis Centroid**: The $(n - m)$ branches heading to infinity follow straight-line asymptotes radiating from a single real-axis centroid: $$\sigma_a = \frac{\sum \text{Re}(\text{poles}) - \sum \text{Re}(\text{zeros})}{n - m} = \frac{\sum_{j=1}^n p_j - \sum_{i=1}^m z_i}{n - m}$$ The angles of the asymptotes relative to the positive real axis are: $$\theta_a = \frac{(2q + 1)180^\circ}{n - m}, \quad q = 0, 1, 2, \dots, (n - m - 1)$$ 6. **Rule 6: Breakaway and Break-in Points**: Points where locus branches depart from or arrive at the real axis occur at coincident roots, found by solving: $$\frac{dK}{ds} = 0 \quad \text{or equivalently} \quad \sum_{j=1}^n \frac{1}{\sigma - p_j} = \sum_{i=1}^m \frac{1}{\sigma - z_i}$$ *Only solutions that lie on valid real-axis segments (Rule 4) and correspond to real positive gain $K > 0$ are true breakaway/break-in points.* 7. **Rule 7: Angles of Departure and Arrival**: - For a complex open-loop pole $p_k$, the angle of departure $\phi_d$ is: $$\phi_d = 180^\circ - \sum \angle(p_k - p_j) + \sum \angle(p_k - z_i)$$ - For a complex open-loop zero $z_k$, the angle of arrival $\theta_a$ is: $$\theta_a = 180^\circ - \sum \angle(z_k - z_i) + \sum \angle(z_k - p_j)$$ #### s-Plane Allowable Design Bounds By superimposing performance criteria onto the $s$-plane, we obtain an allowable region for closed-loop poles: - **Damping Ratio $\zeta \ge \zeta_{\min}$**: Restricts poles to a wedge subtending half-angle $\theta = \cos^{-1}(\zeta)$ from the negative real axis. - **Settling Time $t_s \le t_{s,\max}$**: Requires $\sigma = \zeta\omega_n \ge \frac{4}{t_{s,\max}}$, placing a vertical boundary line at $s = -\frac{4}{t_{s,\max}}$ (poles must lie to the left of this vertical line). - **Peak Time $t_p \le t_{p,\max}$**: Requires $\omega_d \ge \frac{\pi}{t_{p,\max}}$, bounding poles above and below horizontal lines $j\omega = \pm \frac{\pi}{t_{p,\max}}$. --- ## 2. Key Formula Box <FormulaStrip title="Fundamental Stability & Root Locus Formulas" formulas={[ { name: "Characteristic Equation", symbol: "1 + G(s)H(s) = 0", description: "Fundamental polynomial D(s) whose roots are the closed-loop poles of the feedback system." }, { name: "Routh Array Element", symbol: "b_1 = \\frac{a_{n-1} a_{n-2} - a_n a_{n-3}}{a_{n-1}}", description: "Cross-determinant algorithm evaluating subsequent row coefficients in the Routh array." }, { name: "Evans Angle Condition", symbol: "\\sum \\theta_{z_i} - \\sum \\phi_{p_j} = \\pm (2q+1)180^\\circ", description: "Geometric angle summation required for a point s in the complex plane to belong to the root locus." }, { name: "Asymptote Centroid", symbol: "\\sigma_a = \\frac{\\sum p_j - \\sum z_i}{n - m}", description: "Center of gravity on the real axis where all (n - m) infinite asymptote branches intersect." }, { name: "Asymptote Angles", symbol: "\\theta_a = \\frac{(2q + 1)180^\\circ}{n - m}", description: "Radiating directional angles of the infinite root locus branches (q = 0, 1, ..., n-m-1)." }, { name: "Breakaway Derivative", symbol: "\\frac{dK}{ds} = 0 \\iff \\sum \\frac{1}{s - p_j} = \\sum \\frac{1}{s - z_i}", description: "Condition for multiple real roots where branches collide and break away into the complex plane." }, { name: "Departure Angle", symbol: "\\phi_d = 180^\\circ - \\sum \\phi_p + \\sum \\theta_z", description: "Initial angle at which a root locus branch departs from a complex open-loop pole." } ]} /> --- ## 3. In-Depth Mathematical Derivations ### Derivation 1: Algebraic Equivalence of Routh Array and Continued Fraction Expansion The Routh array is not a heuristic; it is mathematically identical to performing a **Stieltjes Continued Fraction Expansion** on the ratio of the even and odd parts of the characteristic polynomial. <Derivation title="Stieltjes Continued Fraction and Routh Array Equivalence" steps={[ { title: "Step 1: Partition polynomial into even and odd parts", formula: "D(s) = P_{\\text{even}}(s) + P_{\\text{odd}}(s)", explanation: "Let D(s) = a_n s^n + a_{n-1} s^{n-1} + a_{n-2} s^{n-2} + ... For n even, P_even contains s^n, s^{n-2}, ... while P_odd contains s^{n-1}, s^{n-3}, ... For n odd, reverse the roles." }, { title: "Step 2: Form the rational reactance-like function Q(s)", formula: "Q(s) = \\frac{P_{\\text{even}}(s)}{P_{\\text{odd}}(s)} = \\frac{a_n s^n + a_{n-2} s^{n-2} + \\dots}{a_{n-1} s^{n-1} + a_{n-3} s^{n-3} + \\dots}", explanation: "By the Hermite-Biehler theorem, D(s) is a Hurwitz polynomial (all roots in open LHP) if and only if P_even and P_odd have simple, interlaced roots on the imaginary axis." }, { title: "Step 3: Perform Euclidean polynomial division", formula: "\\frac{a_n s^n + a_{n-2} s^{n-2} + \\dots}{a_{n-1} s^{n-1} + a_{n-3} s^{n-3} + \\dots} = c_1 s + \\frac{R_1(s)}{a_{n-1} s^{n-1} + \\dots}", explanation: "Dividing highest power gives quotient c_1 = a_n / a_{n-1} with remainder R_1(s) = (a_{n-2} - c_1 a_{n-3}) s^{n-2} + ... Notice that b_1 = a_{n-2} - (a_n / a_{n-1}) a_{n-3} = (a_{n-1} a_{n-2} - a_n a_{n-3}) / a_{n-1}!" }, { title: "Step 4: Expand Q(s) as a continued fraction", formula: "Q(s) = c_1 s + \\cfrac{1}{c_2 s + \\cfrac{1}{c_3 s + \\cfrac{1}{\\ddots + \\cfrac{1}{c_n s}}}}", explanation: "Each quotient coefficient c_k = Row_k(1) / Row_{k+1}(1) is the ratio of consecutive first-column elements of the Routh array. All roots lie strictly in the LHP if and only if all n continued fraction quotients c_1, c_2, ..., c_n are strictly positive (c_k > 0)." } ]} /> ### Derivation 2: Evans Root Locus Angle Condition and Asymptote Geometry Why do asymptotes radiate from $\sigma_a = \frac{\sum p - \sum z}{n-m}$ at angles $\theta_a = \frac{(2q+1)180^\circ}{n-m}$? <Derivation title="Asymptote Center of Gravity (Centroid) and Radiating Angles" steps={[ { title: "Step 1: Express open-loop transfer function in factored form", formula: "G(s)H(s) = K \\frac{(s - z_1)(s - z_2)\\cdots(s - z_m)}{(s - p_1)(s - p_2)\\cdots(s - p_n)} = K \\frac{s^m - (\\sum z_i) s^{m-1} + \\dots}{s^n - (\\sum p_j) s^{n-1} + \\dots}", explanation: "Using polynomial expansion (Vieta's formulas), the coefficient of the second-highest power is the negative sum of the roots." }, { title: "Step 2: Approximate for very large s (|s| >> 1)", formula: "G(s)H(s) \\approx \\frac{K}{s^{n-m} - [\\sum p_j - \\sum z_i] s^{n-m-1}} \\approx \\frac{K}{(s - \\sigma_a)^{n-m}}", explanation: "Recall the binomial expansion: (s - sigma_a)^(n-m) = s^(n-m) - (n-m) sigma_a s^(n-m-1) + ... Equating the second terms gives (n-m) sigma_a = sum p_j - sum z_i." }, { title: "Step 3: Solve for the centroid coordinate σa", formula: "\\sigma_a = \\frac{\\sum_{j=1}^n p_j - \\sum_{i=1}^m z_i}{n - m}", explanation: "Far from the origin, the n poles and m zeros appear clustered at a single equivalent point mass on the real axis: the centroid sigma_a." }, { title: "Step 4: Apply the Evans Angle Condition to the asymptotic model", formula: "\\angle \\frac{K}{(s - \\sigma_a)^{n-m}} = -(n - m) \\angle(s - \\sigma_a) = -(n - m) \\theta_a = -(2q + 1)180^\\circ", explanation: "Solving for theta_a yields the canonical asymptote angle formula: theta_a = (2q + 1) 180° / (n - m) for q = 0, 1, ..., n-m-1." } ]} /> ### Derivation 3: Breakaway and Break-in Points Derivative Formula Why does $\frac{dK}{ds} = 0$ yield the breakaway and break-in points? <Derivation title="Derivation of dK/ds = 0 and Reciprocal Distance Summation" steps={[ { title: "Step 1: Express gain K explicitly along the locus", formula: "1 + K \\frac{N(s)}{D(s)} = 0 \\implies K = -\\frac{D(s)}{N(s)}", explanation: "Along the root locus, the characteristic equation must be satisfied for real s = sigma on the real axis." }, { title: "Step 2: Differentiate with respect to s using the quotient rule", formula: "\\frac{dK}{ds} = -\\frac{D'(s)N(s) - D(s)N'(s)}{[N(s)]^2} = 0", explanation: "At a multiple root s_b (breakaway/break-in point), two branches meet and depart tangentially, meaning the characteristic equation has a repeated root: D'(s_b) N(s_b) - D(s_b) N'(s_b) = 0." }, { title: "Step 3: Divide by D(s) N(s)", formula: "\\frac{D'(s)}{D(s)} - \\frac{N'(s)}{N(s)} = 0 \\implies \\frac{D'(s)}{D(s)} = \\frac{N'(s)}{N(s)}", explanation: "Notice that D(s) = prod(s - p_j) and N(s) = prod(s - z_i). The logarithmic derivative of a product is the sum of reciprocals!" }, { title: "Step 4: Arrive at the reciprocal distance formula", formula: "\\sum_{j=1}^n \\frac{1}{s - p_j} = \\sum_{i=1}^m \\frac{1}{s - z_i}", explanation: "This elegant electrostatic analog states that breakaway points occur where the repulsive electric fields from all open-loop poles balance the attractive fields from all zeros on the real axis." } ]} /> --- ## 4. Typical Laboratory Hardware Values & Engineering Benchmarks <TypicalValues values={[ { parameter: "Gain Margin (GM)", typical: "GM ≥ 6 dB (2.0×)", range: "6 - 12 dB", notes: "Minimum gain buffer required to absorb component tolerances and temperature drift without instability." }, { parameter: "Phase Margin (PM)", typical: "PM ≈ 45° - 60°", range: "30° - 65°", notes: "Guarantees damping ratio ζ ≈ 0.45 - 0.707, preventing excessive transient ringing." }, { parameter: "Commercial Flight Control Margin", typical: "GM ≥ 8 dB, PM ≥ 45°", range: "6 - 12 dB, 40° - 60°", notes: "FAA / MIL-F-8785C standards for civil and military fly-by-wire flight surfaces." }, { parameter: "Power Grid Frequency Droop (AGC)", typical: "Kcrit / K ≈ 3.5×", range: "2.5× - 5.0×", notes: "Generators operate far below critical gain to prevent inter-area power oscillations (0.2 - 2 Hz)." }, { parameter: "Disk Drive Track Following Servos", typical: "ts ≤ 1.5 ms", range: "0.8 - 3.0 ms", notes: "Requires closed-loop poles placed deep in LHP with σ = ζωn ≥ 2700 rad/s." } ]} /> --- ## 5. Worked Examples: Step-by-Step Analytical Solutions <WorkedSolution problem="[Presidency University Exam Problem] A feedback control system has open-loop transfer function G(s)H(s) = K / [s(s + 3)(s^2 + 2s + 2)]. (1) Formulate the Routh array in terms of loop gain K, (2) Determine the range of K for closed-loop stability, (3) Determine the critical gain Kcrit for marginal stability, and (4) Find the frequency of sustained oscillation at Kcrit." steps={[ { title: "Step 1: Obtain the closed-loop characteristic equation", detail: "1 + G(s)H(s) = 0 => s(s + 3)(s^2 + 2s + 2) + K = 0.\nExpand the polynomial:\ns(s + 3) = s^2 + 3s.\n(s^2 + 3s)(s^2 + 2s + 2) = s^4 + 2s^3 + 2s^2 + 3s^3 + 6s^2 + 6s\n= s^4 + 5s^3 + 8s^2 + 6s + K = 0.\nCoefficients: a4 = 1, a3 = 5, a2 = 8, a1 = 6, a0 = K." }, { title: "Step 2: Construct the Routh array", detail: "Row s^4: 1, 8, K\nRow s^3: 5, 6, 0\nRow s^2:\nb1 = (5*8 - 1*6)/5 = (40 - 6)/5 = 34/5 = 6.8\nb2 = (5*K - 1*0)/5 = K\nRow s^1:\nc1 = (b1*6 - 5*K)/b1 = (6.8*6 - 5K)/6.8 = (40.8 - 5K)/6.8\nRow s^0:\nd1 = (c1*K - b1*0)/c1 = K." }, { title: "Step 3: Apply the Routh stability conditions to Column 1", detail: "For closed-loop stability, all first-column elements must be strictly positive:\n1. From row s^0: K > 0.\n2. From row s^1: (40.8 - 5K)/6.8 > 0 => 40.8 - 5K > 0 => 5K < 40.8 => K < 8.16.\nCombining inequalities: 0 < K < 8.16." }, { title: "Step 4: Find critical gain Kcrit and frequency of oscillation ωosc", detail: "At K = Kcrit = 8.16, row s^1 becomes identically zero (c1 = 0), indicating marginal stability.\nForm the auxiliary polynomial A(s) from row s^2:\nA(s) = b1 s^2 + K = 6.8 s^2 + 8.16 = 0.\ns^2 = -8.16 / 6.8 = -1.20 => s = ± j sqrt(1.20) = ± j 1.095 rad/s.\nTherefore, the sustained oscillation frequency is omega_osc = 1.095 rad/s." } ]} answer="Stable for 0 < K < 8.16; Critical Gain Kcrit = 8.16; Sustained Oscillation Frequency ωosc = 1.095 rad/s." /> <WorkedSolution problem="[Routh Special Case 2 Problem] A system has characteristic equation s^6 + 2s^5 + 8s^4 + 12s^3 + 20s^2 + 16s + 16 = 0. Determine the complete stability profile, identifying the number of roots in the Left-Half Plane, Right-Half Plane, and on the imaginary axis." steps={[ { title: "Step 1: Construct the initial Routh array rows", detail: "Row s^6: 1, 8, 20, 16\nRow s^5: 2, 12, 16, 0 (divide row by 2 to simplify: 1, 6, 8, 0)\nRow s^4:\nb1 = (1*8 - 1*6)/1 = 2\nb2 = (1*20 - 1*8)/1 = 12\nb3 = (1*16 - 1*0)/1 = 16\n(Divide row by 2: 1, 6, 8, 0)\nRow s^3:\nc1 = (1*6 - 1*6)/1 = 0\nc2 = (1*8 - 1*8)/1 = 0\nc3 = 0. ROW OF ZEROS!" }, { title: "Step 2: Form Auxiliary Polynomial A(s) and take derivative", detail: "Using row s^4 (with coefficients 1, 6, 8):\nA(s) = s^4 + 6s^2 + 8 = 0.\nDerivative with respect to s:\ndA/ds = 4s^3 + 12s = 0 (coefficients 4, 12; divide by 4 => 1, 3)." }, { title: "Step 3: Complete the Routh array using dA/ds", detail: "Row s^3: 1, 3, 0\nRow s^2:\nd1 = (1*6 - 1*3)/1 = 3\nd2 = (1*8 - 1*0)/1 = 8\nRow s^1:\ne1 = (3*3 - 1*8)/3 = (9 - 8)/3 = 1/3\nRow s^0:\nf1 = 8." }, { title: "Step 4: Analyze roots and sign changes", detail: "First column entries: s^6: +1, s^5: +1, s^4: +1, s^3: +1, s^2: +3, s^1: +1/3, s^0: +8.\nSign changes in first column: ZERO sign changes => ZERO roots in the Right-Half Plane.\nSolve the auxiliary polynomial A(s) = 0:\ns^4 + 6s^2 + 8 = (s^2 + 2)(s^2 + 4) = 0.\ns^2 = -2 => s = ± j sqrt(2) ≈ ± j 1.414\ns^2 = -4 => s = ± j 2.000.\nAll 4 roots of A(s) are simple pairs on the jw-axis!" } ]} answer="4 roots on the jω-axis (at s = ±j√2 and s = ±j2), 2 roots in the open Left-Half Plane, 0 roots in the Right-Half Plane. System is Marginally Stable." /> <WorkedSolution problem="[Comprehensive Root Locus Construction] A unity negative feedback control system has open-loop transfer function G(s)H(s) = K / [s(s + 2)(s + 4)]. Construct the complete root locus by computing: (1) Asymptote centroid σa and angles θa, (2) Real-axis segments, (3) Breakaway point sb, (4) Imaginary axis crossing frequency ωosc and critical gain Kcrit, and (5) The loop gain K that yields a damping ratio of ζ = 0.5 for the dominant complex closed-loop poles." steps={[ { title: "Step 1: Identify poles, zeros, and real-axis segments", detail: "Poles: n = 3 at s = 0, s = -2, s = -4. Zeros: m = 0.\nNumber of branches = n = 3.\nReal-axis segments (Rule 4: odd number of poles/zeros to the right):\n- Between s = 0 and s = -2: 1 pole to right (ODD) => ON ROOT LOCUS.\n- Between s = -2 and s = -4: 2 poles to right (EVEN) => NOT on root locus.\n- From s = -4 to -∞: 3 poles to right (ODD) => ON ROOT LOCUS." }, { title: "Step 2: Calculate Asymptote Centroid σa and Angles θa", detail: "Number of asymptotes = n - m = 3 - 0 = 3.\nCentroid: sigma_a = [sum(poles) - sum(zeros)] / (n - m) = [0 + (-2) + (-4) - 0] / 3 = -6 / 3 = -2.0.\nAngles: theta_a = (2q + 1)*180° / 3 for q = 0, 1, 2:\nFor q = 0: theta_a0 = 180° / 3 = 60°\nFor q = 1: theta_a1 = 3 * 180° / 3 = 180°\nFor q = 2: theta_a2 = 5 * 180° / 3 = 300° (or -60°)." }, { title: "Step 3: Determine the Breakaway Point sb", detail: "Characteristic equation: s(s + 2)(s + 4) + K = 0 => s^3 + 6s^2 + 8s + K = 0.\nK = -(s^3 + 6s^2 + 8s).\ndK/ds = -(3s^2 + 12s + 8) = 0 => 3s^2 + 12s + 8 = 0.\ns = [-12 ± sqrt(144 - 4*3*8)] / (2*3) = [-12 ± sqrt(144 - 96)] / 6 = [-12 ± sqrt(48)] / 6 = [-12 ± 6.928] / 6.\ns_b1 = (-12 + 6.928) / 6 = -5.072 / 6 = -0.845.\ns_b2 = (-12 - 6.928) / 6 = -18.928 / 6 = -3.155.\nCheck with Rule 4: s = -0.845 lies on the valid segment [0, -2]. s = -3.155 lies on [-2, -4] (invalid).\nTherefore, the true breakaway point is s_b = -0.845 (with gain K = 3.079)." }, { title: "Step 4: Find jω-Axis Crossing via Routh Array", detail: "Polynomial: s^3 + 6s^2 + 8s + K = 0.\nRow s^3: 1, 8\nRow s^2: 6, K\nRow s^1: (48 - K)/6, 0\nRow s^0: K.\nFor marginal stability: 48 - K = 0 => Kcrit = 48.\nAuxiliary equation from row s^2: 6s^2 + 48 = 0 => s^2 = -8 => s = ± j sqrt(8) = ± j 2.828 rad/s.\nLocus crosses the jw-axis at s = ± j 2.828 rad/s with Kcrit = 48.0." }, { title: "Step 5: Determine Gain K for ζ = 0.50", detail: "For zeta = 0.5, the pole line angle from negative real axis is theta = acos(0.5) = 60°.\nPoles lie on ray s = -sigma0 + j sqrt(3) sigma0.\nIntersecting with root locus branch yields dominant pole at s ≈ -0.667 ± j 1.155.\nUsing the Evans magnitude condition K = |s| * |s + 2| * |s + 4|:\n|s| = sqrt(0.667^2 + 1.155^2) = 1.333\n|s + 2| = sqrt(1.333^2 + 1.155^2) = 1.764\n|s + 4| = sqrt(3.333^2 + 1.155^2) = 3.528\nK = (1.333) * (1.764) * (3.528) ≈ 8.29." } ]} answer="Centroid σa = -2.0, Asymptote Angles = ±60°, 180°; Breakaway Point sb = -0.845 (K = 3.08); jω-Crossing at s = ±j2.828 rad/s with Kcrit = 48; Operating Gain for ζ = 0.5 is K ≈ 8.29." /> --- ## 6. Common Exam Traps & Where Marks Are Lost <ExamTraps traps={[ "Routh Array Row Multiplication Pitfall: You may multiply or divide any entire row of the Routh array by a positive constant to simplify arithmetic without changing signs. However, NEVER multiply a row by a negative number or by a variable involving K whose sign is unknown, as this reverses column signs and invalidates the stability count.", "Real-Axis Odd Rule Direction Confusion: The real-axis segment rule strictly counts poles and zeros to the RIGHT of the test point. Counting to the left produces a reversed locus, resulting in an immediate zero on exams.", "Discarding Invalid Breakaway Roots: Solving dK/ds = 0 often produces multiple roots (e.g., quadratic or cubic). You must verify each candidate root against Rule 4 (real-axis segment validation). Any root residing in an 'off-locus' region or producing a negative gain K < 0 must be discarded as non-physical.", "Row of Zeros Auxiliary Derivative Variable: When an all-zero row occurs (e.g., at row s^3), form the auxiliary polynomial A(s) from the row above (row s^4). Differentiate with respect to s: dA/ds. Students frequently make careless arithmetic errors differentiating powers of s (e.g., forgetting that d/ds(8s^2) = 16s).", "Angle of Departure Reference Direction: The departure angle φd is measured relative to the POSITIVE real axis. Remember the formula φd = 180° - sum(pole angles) + sum(zero angles). If your computed angle is negative, add 360° to express it as a standard positive counterclockwise angle." ]} /> --- ## 7. Interactive Circuit Verification & Laboratory Workbench Explore dynamic root locus pole migration vs loop gain $K$, compute live Routh arrays with special cases ($\epsilon$ and $A(s)$), and visualize allowable $s$-plane design regions in the live engineering simulator below: <Simulator name="root-locus-stability" /> <BreakIt prompt="What happens to the root locus if you add an open-loop zero to a stable plant? How does it affect the asymptote centroid σa and transient speed?" hint="Adding an open-loop zero pulls the root locus branches toward the LEFT! Notice from the centroid formula sigma_a = (sum p - sum z) / (n - m): adding a zero decreases (n - m) and shifts sigma_a farther into the left-half plane. This bends the locus away from the jw-axis, drastically increasing the critical gain Kcrit, improving stability margins, and speeding up transient settling time!" /> --- ## 8. Parametric Practice Problem <ParametricProblem type="thevenin-divider" /> --- ## 9. Unit Mastery Quiz export const quizQuestions = [ { question: "A closed-loop system has characteristic equation s^3 + 2s^2 + 4s + 10 = 0. According to the Routh-Hurwitz criterion, is the system stable?", options: [ "Stable (all roots in LHP)", "Unstable (2 roots in RHP)", "Marginally stable (roots on jω axis)", "Cannot be determined without factoring" ], correctAnswer: 1, explanation: "Constructing the Routh array: Row s^3: 1, 4; Row s^2: 2, 10; Row s^1: (2*4 - 1*10)/2 = (8 - 10)/2 = -1; Row s^0: 10. The first column entries are 1, 2, -1, 10. There are 2 sign changes (from +2 to -1, and -1 to +10). Therefore, the system has 2 unstable roots in the RHP.", }, { question: "When does an all-zero row appear in the Routh array, and what does it physically signify?", options: [ "When the system is critically damped with duplicate real roots", "When the system contains radially symmetric roots about the origin (such as jω-axis pairs)", "When loop gain K is set to zero", "When open-loop zeros exceed open-loop poles" ], correctAnswer: 1, explanation: "A row of zeros occurs when the characteristic polynomial contains roots with quadrantal or radial symmetry about the s-plane origin, most commonly a pair of imaginary conjugate roots s = ±jω0 indicating sustained oscillations.", }, { question: "How many asymptote branches radiate to infinity for an open-loop system with 4 poles and 1 zero?", options: [ "1 branch", "3 branches", "4 branches", "5 branches" ], correctAnswer: 1, explanation: "The number of infinite asymptote branches is given by (n - m), where n = 4 poles and m = 1 zero: n - m = 4 - 1 = 3 branches.", }, { question: "For open-loop transfer function G(s)H(s) = K / [s(s + 3)(s + 6)], what is the real-axis centroid σa of the asymptotes?", options: [ "σa = -3.0", "σa = -4.5", "σa = -2.0", "σa = -9.0" ], correctAnswer: 0, explanation: "Poles are at s = 0, -3, -6 (sum = -9). Zeros: m = 0. Centroid sigma_a = (sum p - sum z)/(n - m) = (-9 - 0)/3 = -3.0.", }, { question: "Which of the following points on the real axis lies on the root locus for G(s)H(s) = K(s + 1) / [s(s + 2)(s + 5)]?", options: [ "s = +1.0 (to the right of origin)", "s = -0.5 (between 0 and -1)", "s = -1.5 (between -1 and -2)", "s = -3.5 (between -2 and -5)" ], correctAnswer: 1, explanation: "Using the odd rule: at s = -0.5, there is exactly 1 pole at s = 0 to its right (ODD number) => lies on root locus. At s = -1.5, there are 2 critical points to its right (pole at 0 and zero at -1, sum = 2, EVEN) => NOT on locus.", }, { question: "Why does adding a lead compensator (open-loop zero in LHP) improve closed-loop system stability?", options: [ "It pulls the root locus asymptotes and branches toward the left, increasing gain margin and speed", "It eliminates all closed-loop poles completely", "It restricts the root locus strictly to the real axis", "It prevents loop gain K from exceeding 1.0" ], correctAnswer: 0, explanation: "Open-loop zeros act as attractive sinks in the s-plane, pulling locus branches leftward away from the jw-axis. This shifts the asymptote centroid sigma_a deeper into the LHP and expands the stable gain range.", }, ]; <Quiz questions={quizQuestions} unitTitle="Stability Analysis & Root Locus Techniques" /> --- ## 10. References & Further Reading 1. **Ogata, Katsuhiko**: *Modern Control Engineering*, 5th Edition, Prentice Hall (Chapters 6 & 7: Root-Locus Analysis and Control Systems Design). 2. **Nise, Norman S.**: *Control Systems Engineering*, 8th Edition, John Wiley & Sons (Chapter 6: Stability, and Chapter 8: Root Locus Techniques). 3. **Kuo, Benjamin C., and Farid Golnaraghi**: *Automatic Control Systems*, 9th Edition, Wiley. 4. **Evans, Walter R.**: *Graphical Analysis of Control Systems*, Transactions of the American Institute of Electrical Engineers, Vol. 67, 1948 (The seminal paper establishing Root Locus). 5. **Presidency University Courseware**: *EEE2005 - Control Systems Engineering*, Department of Electrical & Electronics Engineering, Bengaluru.