CHEN90032 · Process Simulation and Control
Control-System Stability & Routh Analysis
Control-System Stability & Routh Analysis is the pivot of University of Melbourne CHEN90032 Process Simulation and Control, where the closed-loop transfer function built earlier in the subject is turned into a stability verdict. You learn the core theorem — a feedback loop is stable if and only if every root of the characteristic equation 1 + GOL = 0 lies in the left-half plane (all poles have negative real part) — and the Routh–Hurwitz criterion that tests it from the coefficients alone, without factoring the polynomial. From the same array you extract the ultimate gain Kcu and ultimate period Pu that seed controller tuning, and you read the identical story off a root-locus diagram — the analytical spine of the transfer-function and tuning question on the written final.
What this chapter covers
- 01The stability theorem: a loop is stable if and only if every closed-loop pole has a negative real part (poles in the open left-half plane)
- 02Forming the characteristic equation 1 + G_OL = 0 with G_OL = GcGvGpGm (all loop elements included)
- 03Reading poles in the s-plane: real part sets decay rate, imaginary part sets oscillation, imaginary axis = stability boundary
- 04The necessary coefficient test: all coefficients present and positive (necessary but not sufficient from third order up)
- 05Building the Routh array and applying the criterion (stable iff the first column is all positive; sign changes = right-half-plane poles)
- 06The collapsed 3rd-order condition a2a1 > a3a0 and the 4th-order condition a1(a2a3 - a1a4) > a3^2 a0
- 07Finding the ultimate gain Kcu (the first-column entry driven to zero) and the ultimate frequency/period from the auxiliary equation
- 08Root-locus diagrams: closed-loop poles moving with Kc and crossing the imaginary axis at Kcu
- 09Why Routh cannot handle a pure time delay (transcendental equation) and when to switch to a Pade approximation or the Bode method
- 10How Kcu and Pu feed the Ziegler-Nichols and Tyreus-Luyben tuning rules on the provided formula sheet
Stability limit and ultimate gain of a three-lag loop
- +1Characteristic equation, 1 + GOL = 0, i.e. (s+1)(s+2)(s+3) + 6Kc = 0. Expanding the product gives s3 + 6s2 + 11s + (6 + 6Kc) = 0, so a3 = 1, a2 = 6, a1 = 11, a0 = 6 + 6Kc.
- +1Necessary check: all coefficients positive needs 6 + 6Kc > 0, i.e. Kc > −1. This is necessary but not sufficient for a cubic, so build the array.
- +1Routh array. s3 row: 1, 11. s2 row: 6, (6 + 6Kc). The only entry that can change sign is b1 = [a2a1 − a3a0]/a2 = [6(11) − 1(6 + 6Kc)]/6 = (60 − 6Kc)/6 = 10 − Kc. The s0 row is a0 = 6 + 6Kc.
- +1Criterion: the first column (1, 6, 10 − Kc, 6 + 6Kc) must be all positive. b1 > 0 gives Kc < 10; a0 > 0 gives Kc > −1. So the stable range is −1 < Kc < 10 (physically 0 < Kc < 10).
- +1Ultimate gain: the binding upper limit is Kcu = 10 (dimensionless). At this gain the s1 first-column entry b1 falls to zero, i.e. a pair of poles reaches the imaginary axis.
- +1At Kc = Kcu = 10, a0 = 6 + 60 = 66. The auxiliary equation from the s2 row is 6s2 + 66 = 0, so s2 = −11 and the ultimate frequency is ωu = √11 = 3.32 rad/min. The ultimate period is Pu = 2π/ωu = 6.283/3.32 = 1.89 min.
Key terms
- Characteristic equation
- The equation 1 + G_OL = 0, where G_OL = GcGvGpGm is the product of every element around the loop. Setting the closed-loop denominator to zero, its roots are the closed-loop poles that decide stability.
- Pole / left-half plane (LHP)
- A pole is a root of the characteristic equation, a value of s where the closed-loop transfer function blows up. The left-half plane is the region Re(s) < 0; a loop is stable exactly when all poles lie there.
- Stability (BIBO)
- A loop is bounded-input bounded-output stable when a bounded input can never produce an unbounded output. This holds if and only if every closed-loop pole has a negative real part; a single pole with positive real part makes the response grow without bound.
- Routh array
- A table generated from the characteristic-polynomial coefficients by successive determinant-style cross-multiplications. It reveals how many roots lie in the right-half plane without factoring the polynomial.
- Routh-Hurwitz criterion
- The rule that a loop is stable if and only if every entry in the first column of the Routh array is positive. The number of sign changes down that column equals the number of right-half-plane (unstable) poles.
- Ultimate gain Kcu
- The largest proportional-controller gain for which the loop is still stable. At Kcu a binding first-column Routh entry falls to zero and a pair of poles sits on the imaginary axis, giving sustained (marginal) oscillation.
- Ultimate frequency and period, wu and Pu
- At Kc = Kcu the loop oscillates at the ultimate frequency wu (rad per time), found from the auxiliary equation of the row above the vanished row; the ultimate period is Pu = 2*pi/wu. Both feed the Ziegler-Nichols and Tyreus-Luyben tuning rules.
- Root locus
- A plot of the closed-loop poles in the s-plane as the controller gain is swept from zero upward. Branches start at the open-loop poles; the gain at which a branch crosses the imaginary axis is Kcu, and the crossing height is wu.
Control-System Stability & Routh Analysis FAQ
If every coefficient of the characteristic polynomial is positive, is the loop stable?
Not necessarily. All coefficients being present and positive is a necessary condition for stability, but from third order upward it is not sufficient — a cubic or quartic with every coefficient positive can still have roots in the right-half plane. For a first- or second-order polynomial the positive-coefficient test alone does guarantee stability, but for anything higher you must build the Routh array (or apply the collapsed 3rd- and 4th-order inequalities) and confirm the whole first column is positive. If a coefficient is missing or negative you can stop immediately: the loop is already unstable.
Why can't the Routh array be used when the process has a time delay?
A pure time delay contributes a factor e^(-theta*s) to the loop transfer function, which makes the characteristic equation transcendental rather than a finite polynomial — and the Routh array only works on polynomials. To keep using Routh you first replace the delay with a Pade approximation, which turns e^(-theta*s) back into a ratio of polynomials so the equation becomes polynomial again. Otherwise you switch to a frequency-response (Bode) analysis, which handles dead time directly and, for a delay-free loop, returns exactly the same ultimate gain and period that Routh gives.
Can AI help me with control-system stability and Routh analysis in CHEN90032?
Yes, as a study aid. Sia can walk you through the method step by step — how to assemble 1 + G_OL = 0, expand it into a characteristic polynomial, build the Routh array, apply the first-column criterion, and read off the ultimate gain and period — and it can check your algebra and unit work on practice problems. Treat it as a tutor that explains the reasoning, not a source of ready-made assessment answers: it does not sit your exam or guarantee a grade, and re-deriving each array yourself is what builds the speed and accuracy you need to clear the exam hurdle.
Studying with AI? Sia — free AI chemical engineering tutor works through CHEN90032 step by step.
Exam move
Drill the sequence until it is automatic: write 1 + G_OL = 0 with every loop element included, expand to a characteristic polynomial with a positive leading coefficient, run the necessary positive-coefficient check, then build the Routh array and test the first column. Practise pushing Kc into the constant term and solving the binding first-column entry equal to zero for Kcu, then use the auxiliary equation of the row above for wu and Pu — those two numbers are what the tuning rules need, so stability and tuning are always examined together. The written final is 4 questions and 100 marks over 3 hours, about 1.8 minutes per mark, so a 35-mark transfer-function question is roughly 63 minutes; budget the stability sub-part by its printed marks and use the reading time to locate the Routh and tuning rows on the provided formula sheet. On every practice loop, state Kcu as dimensionless, wu in rad per time and Pu in time, and check the direction: stable below Kcu, marginal at it, unstable above it.
