Control Systems · 96 h · Topic 5 of 8

Stability of control systems

In 1877 Routh found how to tell whether a polynomial has unstable roots without computing them. For a third-order loop, the same table gives the gain at which it starts to oscillate and the frequency.

Characteristic equation Routh–Hurwitz Critical gain Gain margin Relative stability

01What a stable system is

A linear system is stable if every bounded input produces a bounded output. For a system described by a rational transfer function, that happens if and only if all its poles have negative real part: each pole contributes to the response an exponential \( e^{p t} \), which decays only if \( \mathrm{Re}(p) < 0 \). The general notion was covered in stability, in Signals and Systems. In control, the problem is that feedback moves the poles: a stable plant can become unstable when the loop is closed with too much gain, and an unstable one can be stabilized.

\[ T(s) = \frac{G(s)}{1 + G(s)} \] \[ 1 + G(s) = 0 \] The closed-loop poles are the roots of the characteristic equation \( 1 + G(s) = 0 \). With \( G = N/D \), they are the roots of \( D(s) + N(s) \).

02The Routh–Hurwitz criterion

Computing the roots of a high-degree polynomial was impractical by hand. In 1877, Edward Routh found a procedure to tell how many of them have positive real part without computing them. With the polynomial \( a_n s^n + a_{n-1} s^{n-1} + \dots + a_0 \), a table is built: the first two rows with the alternating coefficients, and each following row from the two before it:

\[ b_1 = \frac{a_{n-1} a_{n-2} - a_n a_{n-3}}{a_{n-1}} \] The number of sign changes in the first column is the number of roots with positive real part. The system is stable if all the elements of the first column have the same sign. A necessary, but not sufficient, condition is that all the coefficients exist and have the same sign.

For a loop with \( G(s) = K/(s(s+a)(s+b)) \), the characteristic equation is \( s^3 + (a+b)s^2 + ab\,s + K = 0 \), and the table gives the gain limit:

\[ K_{crit} = ab\,(a + b) \] \[ \omega_{osc} = \sqrt{ab} \] With \( K = K_{crit} \) the third row vanishes: there is a pair of roots on the imaginary axis, and the system oscillates at \( \omega_{osc} \), which comes from the previous row. It is the starting point of the Ziegler–Nichols tuning method.

Lab · Routh table and roots

The loop \( K/(s(s+a)(s+b)) \) with unity feedback. On the left, the first column of the Routh table; on the right, the roots of the characteristic equation in the s-plane, computed numerically.

a =

b =

Gain K =

There are two special cases: if an element of the first column vanishes and the rest of the row does not, it is replaced by a small number ε and the table continues; if an entire row vanishes, there are roots symmetric about the origin, and the table continues with the derivative of the auxiliary polynomial formed from the previous row.

03Relative stability

Knowing that a system is stable is not enough: a loop with poles barely to the left of the imaginary axis oscillates heavily and any change in parameters makes it unstable. Relative stability measures how far away it is: with the damping of the dominant poles, with the gain margin (how much K can be multiplied before reaching K_crit) or with the margins of the frequency response. Routh's criterion can be used for this by shifting the axis: replacing \( s \) with \( s - \sigma \) tells whether all the roots are to the left of \( -\sigma \).

04In the lab

Activity 1 · Critical gain

Build a third-order loop with op amps and adjustable gain, raise the gain until it oscillates and compare the gain and the oscillation frequency with those from Routh.

Activity 2 · Routh in Python

Program the Routh table for any polynomial, with the special cases, and check it against the roots computed numerically.

Activity 3 · Gain margin

With the lab loop, measure the gain margin as the ratio between the critical gain and the working gain.

05Common mistakes

Looking at the stability of the plant instead of the closed loop. Feedback moves the poles.
Settling for the positive-coefficients condition. It is necessary but not sufficient from third order up.
Forgetting the special cases of the table. A zero in the first column does not mean stability.
Designing just below K_crit. The system is stable, but it oscillates heavily and does not tolerate changes.

06Self-assessment

What is the critical gain of K/(s(s+2)(s+3))?

\( 6 \cdot 5 = 30 \), with oscillation at \( \sqrt 6 \approx 2.45 \) rad/s.

How many roots with positive real part are there if the first column is 1, 4, −2, 5?

Two: there are two sign changes.

Is s³ + s² + 2s + 8 stable?

No: the third row gives \( (2 - 8)/1 = -6 \), two sign changes.

What gain margin does the loop of the first question have with K = 10?

3, that is, 9.5 dB.

07Further reading

Katsuhiko Ogata. Modern Control Engineering. 5th ed., Prentice Hall, 2010 (Spanish edition: “Ingeniería de control moderna”, Pearson). Routh's criterion with its special cases and relative stability.
Richard C. Dorf and Robert H. Bishop. Modern Control Systems. 10th ed., Pearson, 2005 (Spanish edition: “Sistemas de control moderno”). The stability of feedback systems with design examples.
Edward J. Routh. A Treatise on the Stability of a Given State of Motion. Macmillan, 1877. The original essay, awarded a prize by the University of Cambridge.

← Previous topic

Steady state and system type

Next topic →

The root locus