Sampled-Data Systems and the Z-Transform

Every controller designed in this book so far has been an analogue object. The PID controller of Chapter 19 was an arrangement of operational amplifiers; the lead network of Chapter 21 was a resistor and a capacitor; the state feedback of Chapter 26 was a bank of summing amplifiers. Each of them processes a voltage that exists at every instant.

Almost no controller built today works that way. The compensator is a program running on a microcontroller, a DSP, or a programmable logic controller. It reads the plant output through an analogue-to-digital converter at regular instants, computes a number, and writes that number to a digital-to-analogue converter which holds it steady until the next instant. The reasons are practical and overwhelming: gains that do not drift with temperature or age, arbitrarily complicated control laws at no extra hardware cost, gain scheduling, self-tuning, fault logging, and a design that can be changed by editing a line of code rather than unsoldering a capacitor.

The price is that the signals inside the loop are no longer continuous functions of time. Two operations, absent from every earlier chapter, now sit between the plant and the controller.

Neither operation is describable by a transfer function in \(s\), because neither is time-invariant: a sampler applied to \(x(t)\) and to \(x(t-\tau)\) gives results that are not simply shifted versions of one another unless \(\tau\) happens to be a multiple of \(T\). The Laplace transform, which rests on time invariance, therefore cannot handle the loop directly. We need a new transform, and Section 27-2 shows that it is forced upon us rather than invented.

A real A/D converter closes a switch for a very short time \(\tau\), takes a reading, and opens again. Modelling that finite window exactly is awkward and, since \(\tau\ll T\) in every practical converter, unnecessary. The standard idealisation replaces each narrow pulse of area \(x(kT)\tau\) by an impulse of the same strength. The result is the ideal impulse sampler: multiplication of \(x(t)\) by a train of unit impulses spaced \(T\) apart.

The sifting property of the impulse has done the essential work: \(x(t)\delta(t-kT) = x(kT)\delta(t-kT)\), so the starred signal is a train of impulses whose areas are the sample values. All information between the samples has been discarded, which is exactly what the converter does.

Now take the Laplace transform of \(x^*(t)\), using \(\mathcal{L}\{\delta(t-kT)\}=e^{-kTs}\):

Something has gone wrong, and it is instructive to see what. \(X^*(s)\) is not a rational function of \(s\); it is an infinite series in the transcendental quantity \(e^{-Ts}\). Every tool of Parts 3 and 4 — partial fractions, Routh's array, the root locus, all of which assume polynomials — is useless against it. But the difficulty also contains its own cure. The variable \(s\) never appears alone; it appears only inside \(e^{Ts}\). So define a new variable that absorbs the exponential:

That is the whole origin of the z-transform. It is not a separate invention parallel to Laplace but the Laplace transform of an impulse-sampled signal, rewritten in a variable that makes it rational. Everything that follows — stability tests, transfer functions, frequency response — is the Part 3 to Part 5 apparatus transplanted through the map \(z=e^{sT}\).

Before building machinery on the sampled signal, one question must be settled: how much has been lost? The impulse train \(\delta_T(t)\) is periodic, so it has a Fourier series, and with \(\omega_s = 2\pi/T\) that series is remarkably simple — every harmonic has the same amplitude \(1/T\):

Multiplying \(x(t)\) by each exponential shifts its transform along the \(s\)-axis, by the frequency-shift theorem of Chapter 3. Summing the shifts gives the fundamental frequency-domain description of sampling:

The consequence follows at once. If \(X(j\omega)\) is zero above some \(\omega_{\max}\), the copies are separate blocks of width \(2\omega_{\max}\) placed \(\omega_s\) apart. They fail to overlap precisely when \(\omega_s > 2\omega_{\max}\), and in that case an ideal low-pass filter could recover \(x(t)\) exactly from its samples. That is Shannon's sampling theorem. If instead \(\omega_s < 2\omega_{\max}\), neighbouring copies overlap, high frequencies fold down into the baseband, and are indistinguishable from genuine low-frequency content. This is aliasing, and it is irreversible: no filter placed after the sampler can separate a real 1 Hz signal from a 49 Hz signal that has folded onto it.

Two engineering consequences follow, and both are non-negotiable in a real design. First, an anti-aliasing filter — an analogue low-pass filter — must precede the A/D converter, so that whatever reaches the sampler genuinely is band-limited below \(\omega_s/2\). Second, the theoretical bound \(\omega_s > 2\omega_{\max}\) is far too generous for control. It guarantees only that the samples contain the information, ignoring the phase lag the hold introduces and the delay that lag costs in the loop. Practical control rules are much more aggressive.

Take the definition of Section 27-2 as the starting point in its own right. For a sequence \(x(k)\) defined for \(k\ge 0\),

The series converges outside some circle in the complex \(z\)-plane; that region is the region of convergence, and for the one-sided transform it always has the form \(|z| > R\). Almost every pair in the table below comes from one summation, so it is worth doing that summation once. Let \(x(k)=a^k\):

It is the geometric series and nothing more, and it converges exactly when \(|a/z| < 1\). Every other entry follows from special cases and simple operations: \(a=1\) gives the unit step; \(a=e^{-\alpha T}\) gives the sampled exponential, because \(x(kT)=e^{-\alpha kT}=(e^{-\alpha T})^k\); differentiating with respect to \(a\) generates \(ka^k\); and writing \(\cos\omega kT\) as the sum of two complex exponentials and adding the two resulting fractions gives the trigonometric pairs.

One warning about notation. The symbol \(\mathcal{Z}\{G(s)\}\) is used throughout the literature to mean "sample the inverse transform of \(G(s)\) and take its z-transform" — it is a two-step operation, not a transform of \(G(s)\) itself. Because of that, \(\mathcal{Z}\{G_1(s)G_2(s)\}\neq G_1(z)G_2(z)\) in general, a trap Section 27-9 returns to.

The z-transform is linear, which needs no proof beyond the linearity of the defining sum. The properties that matter for control are the shifting theorems, because they are what convert a difference equation into an algebraic one.

Consider a delay. If \(y(k)=x(k-1)\) with \(x\) zero for negative argument, then

which confirms the interpretation of \(z^{-1}\) as the unit delay. Advancing is not quite symmetrical, because the samples that leave the window must be accounted for:

The remaining properties are collected below. The final-value theorem is the one used most often in design, and its validity condition is the exact analogue of the stability caveat that accompanied the continuous version in Chapter 9.

The final-value theorem is worth reading against its continuous cousin. There, the operator was \(s\) and the point of evaluation was \(s\to 0\); here it is \((1-z^{-1})\) and the point is \(z\to 1\). That is not a coincidence: \(s=0\) maps to \(z=e^{0}=1\), so the entire steady-state analysis of Chapter 9 transplants by replacing "at the origin" with "at \(z=1\)". A digital system's type is the number of poles it has at \(z=1\), and the static error constants of Chapter 9 reappear as limits taken there.

The controller's output is a number, and the plant needs a voltage. The zero-order hold supplies one in the simplest possible way: it holds each sample constant until the next arrives, producing the staircase seen in the block diagram of Section 27-1.

Its transfer function follows from its impulse response. Feed the hold a single unit impulse at \(t=0\); it responds with a rectangular pulse of unit height lasting exactly \(T\) seconds. That pulse is the difference of two unit steps, one delayed:

Now put the hold in front of the plant and ask for the transfer function from the input sequence to the output sequence. Because \(e^{-Ts}=z^{-1}\) is a constant as far as the \(\mathcal{Z}\) operation is concerned, it factors out of the transform:

Two structural facts about \(G(z)\) deserve attention, because both surprise students meeting them for the first time.

The poles map, the zeros do not. A plant pole at \(s=p\) becomes a \(z\)-plane pole at \(z=e^{pT}\), which is exactly the map of Section 27-8. The zeros obey no such rule; the discretisation manufactures new zeros with no continuous counterpart, and for fast sampling of a plant with a large pole excess some of them sit outside the unit circle. A minimum-phase continuous plant can therefore have a non-minimum-phase discrete model, which is why the tempting design idea of cancelling all the plant zeros with controller poles is dangerous in digital control.

The order can rise. The example of a pure delay makes this vivid: \(e^{-Ts}\) is not rational in \(s\) at all, yet its discrete equivalent is simply \(z^{-1}\), a single extra pole at the origin. Delays, which are painful in continuous design and needed the Padé approximation, are exact and trivial in \(z\). This is one respect in which the discrete world is genuinely easier than the continuous one.

Having produced \(Y(z)\) we usually want the sequence \(y(k)\). Three methods are standard, and they answer different questions.

Long division divides the numerator by the denominator in ascending powers of \(z^{-1}\). Since \(Y(z)=y(0)+y(1)z^{-1}+y(2)z^{-2}+\cdots\) by definition, the quotient's coefficients are the samples. The method needs no algebra beyond division and gives the first few values immediately, which is exactly what one wants when checking a simulation. It gives no closed form, so it cannot tell you the behaviour as \(k\to\infty\).

Partial fractions gives the closed form. There is one trick that must be learned: expand \(X(z)/z\), not \(X(z)\). The reason is visible in the table of Section 27-4 — every standard pair carries a factor \(z\) in the numerator, so an expansion of \(X(z)\) itself produces terms like \(1/(z-a)\) that are not in the table, whereas expanding \(X(z)/z\) and multiplying back through by \(z\) produces \(z/(z-a)\), which is.

The last step reads off the table, and it displays the central fact about discrete stability: each pole \(p_i\) contributes a geometric sequence \(p_i^k\), which decays if and only if \(|p_i| < 1\).

The residue (inversion integral) method is the formal inverse, \(x(k)=\frac{1}{2\pi j}\oint X(z)z^{k-1}dz\), evaluated as a sum of residues at the poles of \(X(z)z^{k-1}\). It is the tool for repeated poles and for proving general results, and for simple pole patterns it reduces to the partial-fraction answer.

Everything about discrete stability follows from the single relation \(z=e^{sT}\). Write \(s=\sigma+j\omega\) and split it into magnitude and angle:

The real part of \(s\) — which governed stability in every chapter from 10 onwards — becomes the magnitude of \(z\). Three cases exhaust the picture. If \(\sigma < 0\) then \(|z| < 1\): the left half-plane maps inside the unit circle. If \(\sigma=0\) then \(|z|=1\): the imaginary axis maps onto the unit circle itself. If \(\sigma > 0\) then \(|z| > 1\): the right half-plane maps outside.

The angle carries the second half of the story, and it introduces something with no continuous analogue. Since \(\angle z=\omega T\), frequencies differing by \(\omega_s=2\pi/T\) produce the same angle and hence the same point in the \(z\)-plane. The map is many-to-one: the horizontal strip \(-\pi/T < \omega \le \pi/T\), called the primary strip, covers the whole \(z\)-plane by itself, and every other strip lies on top of it. This is aliasing again, now visible as a geometric fact rather than a spectral one.

The loci that mattered in root-locus design map to curves worth recognising. A line of constant \(\sigma\) — constant settling time — becomes a circle of radius \(e^{\sigma T}\). A line of constant \(\omega\) becomes a radial ray at angle \(\omega T\). And the constant-\(\zeta\) ray of Chapter 13, along which \(\sigma=-\zeta\omega_n\) and \(\omega=\omega_n\sqrt{1-\zeta^2}\) with \(\omega_n\) sweeping, becomes a logarithmic spiral winding inward toward the origin, drawn dashed in the figure above.

This is the practical bridge from Chapter 8 to digital design: convert the overshoot and settling-time specification into \(\zeta\) and \(\omega_n\) exactly as before, then map the resulting dominant pair through \(z=e^{sT}\) to get the target \(z\)-plane locations. The design itself — root locus in \(z\), or the discrete equivalents of the compensators of Part 5 — is the subject of Chapter 28.

The block-diagram algebra of Chapter 4 mostly survives the move to \(z\), but with one genuinely new rule that must be respected: a signal can be given a z-transform only where it is sampled. Two continuous blocks in cascade with no sampler between them form a single block, and the pair must be transformed together:

The notation \(G_1G_2(z)\) — one transform of the product — is standard and deliberately different from \(G_1(z)G_2(z)\). For the common configuration of a digital controller followed by a hold and plant, with the error sampled and the output measured continuously, the loop reduces exactly as in Chapter 4:

Two further remarks complete the picture. First, not every sampled-data configuration has a closed-loop pulse transfer function: if the reference enters at a point that is never sampled, \(C(z)\) can be expressed in terms of the starred transform of \(RG\) but not as a ratio involving \(R(z)\) alone. Second, once the characteristic polynomial in \(z\) is in hand, its roots can be tested against the unit circle either directly, or by the Jury test — the discrete counterpart of Routh–Hurwitz from Chapter 11 — or by the bilinear transformation \(z=(1+w)/(1-w)\), which maps the unit circle to the imaginary axis and lets Routh's array be used unchanged. For a second-order polynomial \(P(z)=z^2+a_1z+a_0\) the conditions collapse to three inequalities that are worth memorising:

Example 5 uses them, and reaches a conclusion that should trouble anyone comfortable with continuous design: a plant that is stable in closed loop for every positive gain becomes unstable at a modest gain once it is sampled. Sampling is not a neutral implementation detail. It is a destabilising element in its own right, and the loss must be paid for either with a faster sampling rate or with a compensator designed in \(z\) — which is where Chapter 28 begins.

Unless told otherwise, assume an ideal sampler and a zero-order hold, and take all initial conditions to be zero. Keep four decimal places in the exponentials; rounding early destroys the stability boundaries in the later problems. Difficulty rises down the list.