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Digital Control

Stability in the z-plane and the Jury criterion

The z = e^sT mapping proved, the second-order Jury criterion with full proof, sampling and computation delay limiting the gain, discrete Nyquist.

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In continuous control a system is stable if the poles have negative real part; in digital control the good region is the inside of the unit circle. The link between the two worlds is the exponential function z = e^{sT}, which also explains why sampling makes stability harder. In this lesson we look at the correspondence between the s-plane and the z-plane, the Jury criterion for second order with its full proof, and two examples showing how sampling and computation delay limit the gain.

From the s-plane to the z-plane

A continuous mode e^{st} sampled with period T gives the sequence e^{skT} = (e^{sT})^k, i.e. a discrete mode with pole

    \[ z = e^{sT} \]

(Systems Theory lesson on discrete-time systems, where we saw that the eigenvalues of e^{AT} are the exponentials of those of A).

Correspondence between the s-plane and the z-plane
On the left the left half-plane, with the strip between minus pi over T and pi over T. A vertical line at abscissa minus a becomes in the z-plane a circle of radius e to the minus aT; a horizontal line at ordinate b becomes a ray at angle bT. The left half-plane ends up inside the unit circle.

Theorem. The point s = \sigma + j\omega has negative real part if and only if z = e^{sT} lies inside the unit circle.

Proof. |e^{sT}| = |e^{\sigma T}|\cdot|e^{j\omega T}| = e^{\sigma T}, because |e^{j\omega T}| = 1. And e^{\sigma T} \lt 1 if and only if \sigma \lt 0. ∎

The same proof says more: vertical lines \sigma = constant become circles of radius e^{\sigma T}, and horizontal lines \omega = constant become rays at angle \omega T. Points differing by 2\pi/T in \omega end up at the same z: this is aliasing (lesson on the sampling theorem), and the correspondence is one-to-one only in the strip |\omega| \lt \pi/T.

A linear discrete system is therefore asymptotically stable if and only if all the poles of the closed-loop transfer function have modulus less than 1. The closer the pole is to the origin, the faster the mode dies out; a pole at z = 0 corresponds to a mode that vanishes in a single step, with no continuous equivalent.

The Jury criterion for second order

For continuous polynomials there is the Routh criterion; for discrete ones there is the Jury criterion, which decides whether all the roots are inside the unit circle without computing them. For second order it has a very simple form.

Theorem. The roots of P(z) = z^2 + a_1z + a_0, with real coefficients, both lie inside the unit circle if and only if

    \[ |a_0| \lt 1,\qquad P(1) = 1 + a_1 + a_0 \gt 0,\qquad P(-1) = 1 - a_1 + a_0 \gt 0 \]

Proof. Let z_1, z_2 be the roots: P(z) = (z - z_1)(z - z_2) and a_0 = z_1z_2.

Necessary conditions. If the roots are inside the circle, |a_0| = |z_1||z_2| \lt 1. If they are complex conjugate, P(1) = |1 - z_1|^2 \gt 0 and P(-1) = |1 + z_1|^2 \gt 0. If they are real and in (-1, 1), the factors 1 - z_i and 1 + z_i are positive, so P(1) = (1 - z_1)(1 - z_2) \gt 0 and P(-1) = (1 + z_1)(1 + z_2) \gt 0.

Sufficient conditions. If the roots are complex conjugate, |z_1|^2 = z_1\bar{z}_1 = a_0 \lt 1. If they are real, P is an upward parabola, positive at 1 and at -1: the roots are both in (-1, 1), or both greater than 1, or both less than -1 (if one were below -1 and the other above 1, P(1) would be negative). In the last two cases |a_0| = |z_1z_2| \gt 1, contradicting the hypothesis. ∎

For higher orders the Jury criterion builds a table similar to Routh’s. Alternatively the bilinear transformation z = (1 + w)/(1 - w) is used, which maps the inside of the unit circle onto the left half of the w-plane (lesson on discretisation), followed by the Routh criterion.

Example: sampling limits the gain

Let an integrator, G(s) = 1/s, be controlled with a proportional action K. In continuous time the closed loop has pole -K, stable for any K \gt 0. Digitally, with the hold, G(z) = T/(z - 1) (lesson on sampled-data systems), and the characteristic polynomial of 1 + KG(z) = 0 is

    \[ z - 1 + KT = 0 \quad\Rightarrow\quad z = 1 - KT \]

stable if and only if |1 - KT| \lt 1, i.e. 0 \lt KT \lt 2. For KT = 1 the pole is at zero and the error vanishes in one step; for 1 \lt KT \lt 2 the response is stable but alternating, the sign of a controller that over-corrects; beyond 2 it diverges. The admissible gain is limited by the period: the controller acts “blindly” for a whole period before seeing the effect of its action.

Example: one step of computation delay

If the command computed with sample k is applied only at instant k + 1 (lesson on implementation), the loop also contains a delay z^{-1}: KG(z) = KT/(z(z - 1)) and the characteristic polynomial is

    \[ z^2 - z + KT = 0 \]

With a_1 = -1 and a_0 = KT, the Jury criterion requires KT \lt 1, KT \gt 0 and 2 + KT \gt 0: the maximum gain has halved, from KT \lt 2 to KT \lt 1. At the limit KT = 1 the roots are z = (1 \pm j\sqrt{3})/2, right on the unit circle: a sustained oscillation with six samples per period.

Frequency response

The Nyquist criterion also holds in discrete time (lesson on the Nyquist criterion): L(e^{j\omega T}) is plotted for \omega from 0 to \pi/T, i.e. along the unit circle instead of the imaginary axis. Gain and phase margins have the same meaning, and are read on the Bode plots of the discrete loop function, which stop at the Nyquist frequency.

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