Analysis and Design of Digital Control Systems

Chapter 27 built the description of a sampled-data loop: a sampler that reads the error every \(T\) seconds, a zero-order hold that freezes the computed command between samples, and the z-transform that turns the resulting difference equations into algebra. What it did not do was design anything. That is the work of this chapter.

The layout is fixed by the hardware. The plant \(G_p(s)\) is continuous — a motor, a heater, an aircraft. The controller is a program running on a processor: it wakes on a clock tick, reads the sampled error \(e(kT)\), computes a number \(u(kT)\), writes it to a converter, and sleeps. Everything the designer chooses lives inside that program, and everything the program can do is captured by one discrete transfer function \(D(z)\).

Chapter 27 showed that the hold and the plant together form a single pulse transfer function, and it is that object — not \(G_p(s)\) — that the controller sees:

From here the argument writes itself. Every question the continuous course asked — is the loop stable, how fast does it respond, what error remains, how much gain can I afford — has a z-domain counterpart obtained by asking the same question of \(1+D(z)G(z)=0\). Two things change, and only two. First, the stability boundary is the unit circle rather than the imaginary axis. Second, the sampling period \(T\) is now a design variable in its own right: choose it badly and no \(D(z)\) will rescue the loop.

Chapter 10 established that a continuous system is stable when every characteristic root has a negative real part. The discrete criterion follows from that statement and from the defining relation \(z=e^{sT}\), with no new physics at all.

Write a characteristic root of the sampled system as \(s=\sigma+j\omega\). The corresponding point in the z-plane is \(z=e^{\sigma T}e^{j\omega T}\), so \(|z|=e^{\sigma T}\). The condition \(\sigma<0\) is therefore the condition \(|z|<1\): the left half of the s-plane maps onto the interior of the unit circle, the imaginary axis maps onto the circle itself, and the right half maps outside. A discrete mode \(z_i^{\,k}\) decays if and only if \(|z_i|<1\), which is the same statement read directly in the time domain.

Chapter 11 gave us Routh–Hurwitz so that stability could be settled without solving for roots. The same convenience is wanted here, but Routh cannot be applied directly: it tests for roots to the left of a line, and we need a test for roots inside a circle. Two responses are possible. One is to map the unit circle onto the imaginary axis with a bilinear substitution and then run Routh — that is the \(w\)-transformation of Section 28-6. The other is a test built for the circle from the start: Jury's stability test.

Jury's test operates on the characteristic polynomial \(P(z)=a_nz^n+a_{n-1}z^{n-1}+\dots+a_1z+a_0\) with \(a_n>0\). Three of its conditions are immediate and are worth understanding before the array. The requirement \(P(1)>0\) says the polynomial has not changed sign between the stability region and the point \(z=1\); since \(P(z)\to+\infty\) as \(z\to+\infty\), a value \(P(1)\le 0\) forces a real root at or beyond \(z=1\), which is outside the circle. The requirement \((-1)^nP(-1)>0\) is the same argument applied on the negative real axis, the sign factor accounting for the behaviour of \(P\) as \(z\to-\infty\). The requirement \(|a_0|<a_n\) comes from the fact that \(|a_0/a_n|\) is the product of the magnitudes of all the roots: if that product already exceeds one, at least one root must lie outside the circle.

For the second-order case the three conditions are not merely necessary but sufficient, and they are the ones a GATE problem will almost always want. The last of them, \(|a_0|<1\), is worth remembering in words: for a monic quadratic, \(a_0\) is the product of the two roots, so it is the squared magnitude of a complex pair. Requiring \(|a_0|<1\) is requiring that pair to sit inside the unit circle.

Specifications arrive in the language of Chapter 8: a damping ratio \(\zeta\), a natural frequency \(\omega_n\), an overshoot, a settling time. Those quantities describe a complex pole pair in the s-plane. To design in \(z\) we need to know where that pair lands.

The dominant pair is \(s=-\zeta\omega_n\pm j\omega_d\) with \(\omega_d=\omega_n\sqrt{1-\zeta^{2}}\). Applying \(z=e^{sT}\) term by term separates the magnitude from the angle.

Two families of curves follow at once. Fixing \(\omega_n\) and sweeping \(\zeta\) traces a curve of constant natural frequency; fixing \(\zeta\) and sweeping \(\omega_n\) traces a curve of constant damping. The latter is the more useful, and it is a logarithmic spiral: eliminating \(\omega_n\) between \(|z|=e^{-\zeta\omega_nT}\) and \(\theta=\omega_nT\sqrt{1-\zeta^2}\) gives

The radius shrinks as the angle grows, so a constant-\(\zeta\) locus spirals inward from \(z=1\) toward the origin. Poles near \(z=1\) are slow; poles near \(z=0\) are extremely fast — a pole at the origin corresponds to a mode that vanishes in a single sample. That observation, followed to its conclusion, is deadbeat control.

The shaded band in the left panel is the primary strip, \(-\omega_s/2<\omega<\omega_s/2\) where \(\omega_s=2\pi/T\). That strip alone covers the whole z-plane. Every strip above and below it maps onto exactly the same points, which is aliasing seen geometrically: a continuous mode at \(\omega\) and one at \(\omega+\omega_s\) produce identical sample sequences and are indistinguishable to the controller. Chapter 27 stated Shannon's condition; here we see why violating it is not merely inconvenient but genuinely irreversible.

Two transient specifications translate cleanly. Overshoot depends only on \(\zeta\), so the formula \(M_p=e^{-\pi\zeta/\sqrt{1-\zeta^2}}\) of Chapter 8 carries over unchanged. Settling time is more instructive when counted in samples: since \(|z|=e^{-\zeta\omega_nT}\), the 2% settling time \(t_s\approx4/(\zeta\omega_n)\) corresponds to

A pole at \(|z|=0.9\) needs about \(38\) samples to settle; one at \(|z|=0.5\) needs about six. The number of samples, not the number of seconds, is what determines whether the controller has enough resolution to shape the transient.

Chapter 9 built the whole steady-state theory on two ingredients: the error expression \(E=R/(1+G)\) and the final-value theorem. Both have exact discrete counterparts, so the theory rebuilds itself with almost no new work.

For a unity-feedback sampled loop the error transform is \(E(z)=R(z)/[1+D(z)G(z)]\). Writing \(G_{\!L}(z)=D(z)G(z)\) for the open-loop pulse transfer function, the discrete final-value theorem gives

The type of a discrete system is the number of poles of \(G_{\!L}(z)\) at \(z=1\), because \(z=1\) is the image of \(s=0\) and \(1/(1-z^{-1})\) is the discrete integrator. Feeding in the three standard inputs produces three constants. The step \(r(kT)=A\) has transform \(Az/(z-1)\); substituting and cancelling gives \(e_{ss}=A/[1+G_{\!L}(1)]\), so the position constant is simply the DC gain of the loop.

The ramp \(r(kT)=AkT\) has transform \(ATz/(z-1)^2\). Substituting,

The bracket in the denominator becomes \((z-1)G_{\!L}(z)\) in the limit because a type-1 loop has \(G_{\!L}\to\infty\) at \(z=1\), which swallows the \(1\). The quantity that emerges is defined as \(K_v\). The parabola \(r(kT)=A(kT)^2/2\), with transform \(AT^2z(z+1)/[2(z-1)^3]\), yields \(K_a\) by the same route.

The diagonal rule of Chapter 9 survives intact: a type-\(N\) discrete loop tracks inputs of order below \(N\) exactly, the order-\(N\) input with finite error, and higher inputs not at all. The one structural difference is the explicit \(T\) in \(K_v\) and \(K_a\). It is there because a discrete ramp advances by \(AT\) per sample rather than by \(A\) per second; the factor makes the discrete constants numerically comparable with their continuous originals, and in fact for a plant preceded by a zero-order hold the two agree exactly, as Example 3 confirms.

The root-locus construction of Chapter 12 was never about the s-plane. It was about the equation \(1+KP(z)=0\), which splits into an angle condition \(\angle P=180^\circ(2q+1)\) and a magnitude condition \(K=1/|P|\). Nothing in the derivation of the rules — real-axis segments, asymptotes, breakaway points, departure angles — mentions the imaginary axis. Every rule therefore transfers to the z-plane verbatim.

What changes is the interpretation. In the s-plane we watched for the locus crossing the imaginary axis; in the z-plane we watch for it crossing the unit circle. The gain at that crossing is the limit of stability, and the angle of the crossing point gives the frequency of the sustained oscillation through \(\omega=\theta/T\).

The figure is the locus of the running example of this chapter, the plant \(1/[s(s+1)]\) sampled at \(T=1\,\text{s}\). Two branches leave the poles at \(z=1\) and \(z=0.368\), meet at the breakaway point \(z=0.648\), travel round a circle centred on the finite zero at \(z=-0.718\), and return to the real axis at \(z=-2.084\). One branch then runs into the zero and the other to infinity. The circle leaves the unit circle at \(z=0.244\pm j0.970\), an angle of \(75.9^\circ\); at that point \(K=2.39\), the same number Jury's test produces in Example 2.

Read the frequency off the crossing: \(\theta=75.9^\circ=1.325\) rad and \(T=1\,\text{s}\), so the loop would oscillate at \(\omega=1.325\) rad/s at the critical gain. The continuous loop, by contrast, has no critical gain at all. The whole of that difference is contained in the finite zero at \(z=-0.718\) and the pole at \(z=0.368\), which are the algebraic fingerprints of the hold.

Chapters 14 to 22 built an entire design methodology on the frequency response: Bode asymptotes, gain and phase margins, lead and lag compensation. All of it rests on being able to sweep a frequency along a straight boundary and plot straight-line asymptotes against \(\log\omega\). In the z-plane the boundary is a circle and the substitution \(z=e^{j\omega T}\) produces transcendental expressions with no asymptotes at all. The frequency-response toolkit appears to be lost.

It is recovered by one substitution. Define the bilinear or \(w\)-transformation

The mapping is easy to verify on the boundary. Put \(z=e^{j\omega T}\) and factor out a half-angle:

So the unit circle does become the imaginary axis, but the frequency scale is stretched. The fictitious frequency \(\nu=(2/T)\tan(\omega T/2)\) agrees with \(\omega\) at low frequency — \(\tan x\approx x\) — and then runs away, reaching infinity at \(\omega=\pi/T\), the Nyquist frequency. The entire physical frequency range \(0\le\omega<\omega_s/2\) is spread across the whole positive \(\nu\) axis. This frequency warping is the price of recovering the Bode machinery, and it must be undone at the end: a crossover placed at \(\nu_c\) in the \(w\)-plane corresponds to the physical frequency \(\omega_c=(2/T)\arctan(\nu_cT/2)\).

Transformed into \(w\), the running example becomes something a Bode plot can handle:

Three features of this expression carry the whole message of digital control. There is a pole at \(w=0\): the loop is still type 1, and its \(w\)-plane velocity constant is \(1\), matching the continuous plant exactly. There is a fast, harmless zero at \(w=-12.19\). And there is a zero at \(w=+2\) — in the right half-plane. That non-minimum-phase zero is not an artefact of the algebra: \(2/T\) is precisely where the zero-order hold's half-sample delay makes itself felt, and it contributes phase lag while contributing gain, capping the achievable bandwidth at a fraction of the sampling rate no matter how clever the controller. Design in the \(w\)-plane by the lead–lag procedures of Chapter 21, then map \(D(w)\) back through \(w=2(z-1)/[T(z+1)]\) to obtain \(D(z)\).

The quickest route to a digital controller is to pretend the sampling is not there. Design \(D(s)\) by any continuous method — root locus from Chapter 13, lead–lag from Chapter 21, PID from Chapters 19 and 20 — and then find a discrete transfer function that behaves like it. This is emulation, or redesign, and it is what most industrial controllers actually are.

Every emulation rule is a rule for approximating the derivative or the integral. Consider the simplest possible continuous controller, the integrator \(U(s)/E(s)=1/s\), whose time-domain statement is \(\dot u=e\). Approximate the derivative by a difference over one sample and three natural choices appear.

Taking z-transforms of each and solving for \(U(z)/E(z)\) gives \(T/(z-1)\), \(Tz/(z-1)\) and \((T/2)(z+1)/(z-1)\). Comparing each with \(1/s\) identifies the substitution to make in any \(D(s)\):

The stability column deserves a moment. Forward Euler sets \(z=1+sT\), so a continuous pole at \(s=-a\) is placed at \(z=1-aT\); if \(aT>2\) that lands outside the unit circle and a perfectly good compensator has been turned into an unstable one. Tustin is the inverse of the \(w\)-transformation of Section 28-6, and Section 28-6 already established that the \(w\)-map carries the left half-plane onto the unit disc exactly. Tustin can therefore never destabilise a stable design — its only sin is the same frequency warping, which is why pre-warping exists: choosing \(\omega_1\) as the crossover frequency forces the discrete controller to be exact where the loop is most sensitive.

Matched pole–zero takes a different view. Rather than approximating an operator, it applies \(z=e^{sT}\) to each pole and zero of \(D(s)\) individually — the exact mapping, applied root by root. Zeros at infinity are conventionally mapped to \(z=-1\) (or one of them dropped, to leave the controller one sample of computational headroom), and an overall constant is chosen to make \(D(z)\big|_{z=1}=D(s)\big|_{s=0}\).

Sampling faster is not free. Every doubling of the rate doubles the processor load, halves the time available per computation, and — because differences of nearly equal numbers appear in the derivative term — amplifies quantisation noise. There is a genuine optimum, and it usually sits well below the fastest rate the hardware could manage.

Direct design abandons the continuous intermediary. The plant is described by \(G(z)\), which already contains the hold and the sampling delay exactly, and the controller is chosen in \(z\) to place the closed-loop roots or to realise a specified closed-loop response. Nothing is approximated, and the design remains correct even at sampling rates where emulation would fail outright.

The most common direct design is simply the PID controller of Chapter 19, written as a difference equation. Start from the continuous law and discretise the two dynamic terms with a rectangular sum and a backward difference:

The form on the second line is not what one programs. Writing \(u(k-1)\) from the same expression and subtracting removes the running sum entirely, leaving the velocity or incremental form:

A more ambitious direct method specifies the closed-loop pulse transfer function outright. Let \(M(z)=C(z)/R(z)\) be the response we want. Solving \(M=DG/(1+DG)\) for the controller gives the direct synthesis or Ragazzini formula

Choosing \(M(z)=z^{-N}\) — the output equal to the input, delayed by \(N\) samples and thereafter exact — is deadbeat control. The error becomes zero in a finite number of samples, something no continuous controller can achieve. Example 6 designs one and then shows its characteristic weakness: the response is exact at the sampling instants only, and between them the output can ripple substantially while the control signal rings.

The same idea appears in state-space clothing. Chapter 24 discretised \(\dot{\mathbf{x}}=A\mathbf{x}+B u\) into \(\mathbf{x}(k+1)=\Phi\,\mathbf{x}(k)+\Gamma u(k)\) with \(\Phi=e^{AT}\) and \(\Gamma=\left(\int_0^Te^{A\lambda}\,d\lambda\right)B\). Chapter 26's pole-placement argument then applies unchanged to \(\Phi-\Gamma K\), the only difference being that the target eigenvalues must lie inside the unit circle instead of in the left half-plane. Ackermann's formula computes \(K\) exactly as before. Placing all \(n\) eigenvalues at \(z=0\) makes \(\Phi-\Gamma K\) nilpotent, so any initial state is driven to zero in at most \(n\) steps: deadbeat control recovered from the state-space side, and a clean illustration that the two descriptions are one subject.

Assume a zero-order hold in front of every continuous plant and unity feedback throughout. Where a numerical answer is wanted, carry four significant figures through the working. Difficulty rises down the list.