Digital and Discrete-Time Control

Introduction

Digital controllers are central to modern communication equipment, robotics, power electronics, process automation, autonomous systems and embedded instrumentation. Unlike an analog controller, which operates continuously, a digital controller measures signals at discrete instants, executes numerical algorithms and updates an actuator through a digital-to-analog interface or a digitally controlled power stage.

A complete sampled-data system is therefore hybrid: the plant is usually continuous-time, while sensing, computation and command generation are discrete-time. Its performance depends not only on control laws but also on sampling frequency, computational delay, quantization, numerical precision, actuator saturation and communication timing.

This chapter develops difference-equation and Z-transform models, explains sampling and discretization, establishes discrete stability criteria, and presents digital PID and state-space controller design. Implementation, safety and contemporary intelligent-control considerations are also examined.

Learning Objectives

After studying this chapter, the reader should be able to:

• represent discrete-time systems using difference equations, pulse transfer functions and state-space models;
• apply the Z-transform to solve and interpret discrete-time models;
• select sampling rates while accounting for aliasing and dynamic performance;
• evaluate quantization and finite-word-length effects;
• discretize continuous-time models using appropriate methods;
• assess stability in the Z-plane;
• formulate and implement digital PID algorithms;
• design state-feedback, observer and optimal digital controllers; and
• evaluate practical timing, saturation, safety and cybersecurity constraints.

1. Architecture of a Digital Control System

A typical digital feedback loop contains:

1. a continuous-time plant;

2. sensors and signal-conditioning circuits;

3. an anti-aliasing filter;

4. a sample-and-hold and analog-to-digital converter (ADC);

5. a processor executing the control algorithm;

6. a digital-to-analog converter (DAC), pulse-width modulator or digital actuator interface; and

7. a zero-order hold or equivalent command-holding mechanism.

Let the sampling period be \(T_s\) seconds and the sampling frequency be

\[

f_s=\frac{1}{T_s}.

\]

A continuous signal \(x(t)\) generates the sequence

\[

x[k]=x(kT_s),\qquad k=0,1,2,\ldots.

\]

The controller typically computes \(u[k]\) from current and previous samples. Under zero-order holding, \(u(t)=u[k]\) for \(kT_s\leq t<(k+1)T_s\).

The sampling period is a design variable. A very large \(T_s\) can produce poor damping, aliasing and instability. An unnecessarily small \(T_s\) increases processor load, communication traffic and sensitivity to measurement noise.

2. Difference-Equation Models

A linear time-invariant discrete system can be represented by

\[

y[k]+a_1y[k-1]+\cdots+a_ny[k-n]

=b_0u[k]+b_1u[k-1]+\cdots+b_mu[k-m].

\]

Here, \(u[k]\) is the input, \(y[k]\) is the output, and the coefficients describe the system dynamics. This equation is causal when the current output depends only on present and past inputs and past outputs.

For example,

\[

y[k]=0.8y[k-1]+0.2u[k-1]

\]

describes a first-order system with a one-sample input delay. If \(y[0]=0\) and \(u[k]=1\), then

\[

y[1]=0.2,\quad y[2]=0.36,\quad y[3]=0.488.

\]

The response approaches unity because the homogeneous component \(0.8^k\) tends to zero.

Difference equations are directly implementable, but their realization matters. Direct-form, cascade and parallel implementations can exhibit different finite-precision behavior even when they represent the same ideal transfer function.

3. The Z-Transform

3.1 Definition

For a sequence \(x[k]\), the bilateral Z-transform is

\[

X(z)=\sum_{k=-\infty}^{\infty}x[k]z^{-k}.

\]

For causal engineering systems, the unilateral transform is often convenient:

\[

X^+(z)=\sum_{k=0}^{\infty}x[k]z^{-k}.

\]

The region of convergence must accompany a bilateral transform because identical algebraic expressions can represent different sequences. For a causal rational sequence, the region of convergence lies outside the outermost pole.

Important properties include:

• linearity: \(\mathcal{Z}\{ax[k]+by[k]\}=aX(z)+bY(z)\);
• delay: \(\mathcal{Z}\{x[k-r]\}=z^{-r}X(z)\) under zero initial conditions;
• convolution: \(\mathcal{Z}\{x[k]*h[k]\}=X(z)H(z)\);
• final value theorem:

\[

\lim_{k\to\infty}x[k]=\lim_{z\to1}(1-z^{-1})X(z),

\]

provided all poles of \((1-z^{-1})X(z)\), except a possible pole at \(z=1\), lie strictly inside the unit circle.

3.2 Pulse Transfer Function

Applying the Z-transform to a difference equation under zero initial conditions gives

\[

G(z)=\frac{Y(z)}{U(z)}

=\frac{b_0+b_1z^{-1}+\cdots+b_mz^{-m}}

{1+a_1z^{-1}+\cdots+a_nz^{-n}}.

\]

For the earlier first-order system,

\[

G(z)=\frac{0.2z^{-1}}{1-0.8z^{-1}}=\frac{0.2}{z-0.8}.

\]

Its pole is at \(z=0.8\), inside the unit circle.

For a continuous plant \(G(s)\) preceded by a zero-order hold, the sampled pulse transfer function may be calculated from

\[

G_d(z)=(1-z^{-1})\mathcal{Z}\left\{g_s(kT_s)\right\},

\]

where

\[

g_s(t)=\mathcal{L}^{-1}\left\{\frac{G(s)}{s}\right\}

\]

is the continuous-time step response. This formula applies specifically to zero-order-held inputs and sampled outputs.

4. Sampling and Aliasing

4.1 Sampling Theorem

If a continuous signal is strictly band-limited to \(B\) Hz, exact reconstruction is theoretically possible when

\[

f_s>2B.

\]

The value \(2B\) is the Nyquist rate. Equality is normally avoided because practical filters do not have infinitely sharp transitions.

Sampling replicates the continuous spectrum at integer multiples of \(f_s\). If the replicas overlap, high-frequency components appear as lower-frequency components. This irreversible phenomenon is aliasing.

A sinusoid of frequency \(f\) sampled at \(f_s\) is indistinguishable from sinusoids at \(|f-nf_s|\) for appropriate integers \(n\). For example, a 900 Hz signal sampled at 1 kHz aliases to

\[

|900-1000|=100\text{ Hz}.

\]

4.2 Sampling-Rate Selection for Control

The sampling theorem prevents spectral ambiguity but does not by itself guarantee satisfactory feedback performance. A common engineering starting point is to select a sampling frequency approximately 10 to 20 times the desired closed-loop bandwidth. Faster sampling may be necessary for lightly damped modes, switching converters or systems with strict phase-margin requirements.

An anti-aliasing low-pass filter should attenuate frequencies above the Nyquist frequency before conversion. Its phase lag and tolerances must be included in high-performance designs.

4.3 Delay and Jitter

A computation delay of \(d\) complete samples introduces the factor \(z^{-d}\). Fractional-sample delays require more detailed models. Delays reduce phase margin and can destabilize fast loops.

Sampling jitter is variation in the actual sampling instant. For a signal with slope \(dx/dt\), a small timing error \(\delta t\) produces approximately

\[

\delta x\approx \frac{dx}{dt}\delta t.

\]

Thus, jitter is especially important for high-frequency or rapidly changing signals.

5. Quantization and Numerical Effects

An ideal uniform \(N\)-bit converter spanning \([V_{\min},V_{\max})\) has step size

\[

\Delta=\frac{V_{\max}-V_{\min}}{2^N}.

\]

When the converter is not overloaded and rounding is used, the quantization error is bounded by approximately

\[

|e_q|\leq\frac{\Delta}{2}.

\]

For a 12-bit sensor covering 0–100 °C,

\[

\Delta=\frac{100}{4096}=0.024414\ ^\circ\text{C},

\]

so the maximum ideal rounding error is approximately \(0.012207\ ^\circ\text{C}\).

For an ideal converter driven by a full-scale sinusoid, the approximate signal-to-quantization-noise ratio is

\[

\mathrm{SQNR}\approx6.02N+1.76\text{ dB}.

\]

This formula does not include thermal noise, nonlinearity, clipping or non-full-scale operation.

Finite-word-length implementation can also cause coefficient quantization, round-off accumulation, overflow and limit cycles. Limit cycles are persistent oscillations that can remain even with zero input because nonlinear rounding interacts with feedback. Mitigation measures include increased precision, saturation arithmetic, scaling, dithering and numerically robust realizations.

6. Discretization of Continuous-Time Models

6.1 Exact Zero-Order-Hold Discretization

Consider

\[

\dot{x}(t)=Ax(t)+Bu(t),\qquad y(t)=Cx(t)+Du(t).

\]

If \(u(t)\) is constant over each sampling interval, exact integration gives

\[

x[k+1]=A_dx[k]+B_du[k],

\]

where

\[

A_d=e^{AT_s},

\]

\[

B_d=\int_0^{T_s}e^{A\tau}B\,d\tau,

\]

and \(C_d=C\), \(D_d=D\). If \(A\) is nonsingular, one may use

\[

B_d=A^{-1}(A_d-I)B,

\]

but the integral or augmented-matrix exponential is preferable when \(A\) is singular or poorly conditioned.

For the first-order plant

\[

G(s)=\frac{1}{\tau s+1},

\]

zero-order-hold discretization gives

\[

x[k+1]=e^{-T_s/\tau}x[k]+\left(1-e^{-T_s/\tau}\right)u[k].

\]

With \(\tau=10\) s and \(T_s=2\) s,

\[

A_d=e^{-0.2}=0.818731,

\]

\[

B_d=1-e^{-0.2}=0.181269.

\]

The corresponding transfer function is

\[

G_d(z)=\frac{0.181269z^{-1}}{1-0.818731z^{-1}}.

\]

6.2 Approximate Discretization Methods

The forward Euler substitution is

\[

s\approx\frac{z-1}{T_s}.

\]

It is simple but can map stable continuous poles outside the unit circle if the sampling period is too large.

Backward Euler uses

\[

s\approx\frac{z-1}{T_sz},

\]

and has stronger numerical stability properties, although it introduces frequency distortion.

The bilinear or Tustin transformation is

\[

s\approx\frac{2}{T_s}\frac{z-1}{z+1},

\]

or equivalently

\[

z=\frac{1+sT_s/2}{1-sT_s/2}.

\]

It maps the open left half of the \(s\)-plane into the interior of the unit circle, but warps frequencies according to

\[

\Omega=\frac{2}{T_s}\tan\left(\frac{\omega T_s}{2}\right),

\]

where \(\omega\) is digital frequency in rad/s and \(\Omega\) is the corresponding analog frequency. Prewarping can preserve one selected critical frequency.

Matched pole-zero and impulse-invariant methods are useful in specific applications. Impulse invariance can introduce aliasing and is inappropriate when significant continuous-time dynamics exist above the Nyquist frequency.

7. Discrete-Time Stability and Performance

7.1 Pole Criterion

A causal rational discrete-time transfer function is bounded-input bounded-output stable if all poles of its reduced transfer function lie strictly inside the unit circle:

\[

|z_i|<1.

\]

Internal stability additionally requires all internal modes to be stable; unstable pole-zero cancellation can conceal an unstable internal mode.

Under exact sampling, a continuous pole \(s_i\) maps to

\[

z_i=e^{s_iT_s}.

\]

Thus, the open left half-plane maps inside the unit circle, the imaginary axis maps onto the unit circle, and the right half-plane maps outside it.

For a discrete pole \(z=re^{j\theta}\), the associated continuous modal values are

\[

s=\frac{1}{T_s}\ln z

=\frac{\ln r}{T_s}+j\frac{\theta+2\pi n}{T_s}.

\]

The frequency interpretation is unique only after selecting an appropriate logarithm branch.

7.2 Jury Stability Test

The Jury test determines whether a polynomial has all roots inside the unit circle without explicitly calculating them. For the monic second-order polynomial

\[

P(z)=z^2+a_1z+a_2,

\]

strict Schur stability is equivalent to

\[

1+a_1+a_2>0,

\]

\[

1-a_1+a_2>0,

\]

\[

1-a_2>0.

\]

For \(P(z)=z^2-1.2z+0.32\), the three quantities are \(0.12\), \(2.52\) and \(0.68\), all positive. Its roots, 0.8 and 0.4, are consequently inside the unit circle.

7.3 Steady-State Accuracy

Discrete system type is determined by the number of uncancelled open-loop poles at \(z=1\). Under closed-loop stability, increasing the system type improves tracking of polynomial reference inputs but may reduce robustness. Integral action introduces a pole at \(z=1\) in the controller and can eliminate steady-state error to a constant reference or disturbance when the resulting closed loop remains stable.

8. Digital PID Control

8.1 Positional and Incremental Forms

A basic sampled PID controller can be written as

\[

u[k]=K_pe[k]+K_iT_s\sum_{i=0}^{k}e[i]

+\frac{K_d}{T_s}\big(e[k]-e[k-1]\big),

\]

where \(e[k]=r[k]-y[k]\). Subtracting the expression for \(u[k-1]\) produces the incremental or velocity form:

\[

\begin{aligned}

u[k]=u[k-1]&+K_p(e[k]-e[k-1])+K_iT_se[k]\\

&+\frac{K_d}{T_s}\big(e[k]-2e[k-1]+e[k-2]\big).

\end{aligned}

\]

Equivalently,

\[

u[k]=u[k-1]+q_0e[k]+q_1e[k-1]+q_2e[k-2],

\]

with

\[

q_0=K_p+K_iT_s+\frac{K_d}{T_s},

\]

\[

q_1=-K_p-2\frac{K_d}{T_s},\qquad

q_2=\frac{K_d}{T_s}.

\]

These expressions correspond to rectangular integration and backward-difference differentiation. Tustin-based or filtered implementations produce different coefficients.

8.2 Verified Numerical Calculation

Suppose the implemented law is

\[

u[k]=u[k-1]+4.1e[k]-6e[k-1]+2e[k-2].

\]

For \(u[k-1]=0.8\), \(e[k]=0.5\), \(e[k-1]=0.4\) and \(e[k-2]=0.3\),

\[

\begin{aligned}

u[k]&=0.8+4.1(0.5)-6(0.4)+2(0.3)\\

&=0.8+2.05-2.4+0.6\\

&=1.05.

\end{aligned}

\]

8.3 Practical PID Features

An ideal derivative amplifies measurement noise. Practical implementations therefore use a filtered derivative and often apply differentiation to the measured output rather than the error to avoid derivative kick after a set-point step.

When the actuator saturates, the integrator can continue accumulating error, causing windup. Common anti-windup methods include conditional integration and back-calculation. Bumpless transfer is also needed when changing between manual and automatic modes.

PID tuning must be validated on the actual sampled model. Simply copying continuous gains into software may be unacceptable when the sampling period or delay is significant.

9. Digital State-Space Control

Consider

\[

x[k+1]=A_dx[k]+B_du[k],\qquad y[k]=C_dx[k]+D_du[k].

\]

9.1 Controllability and State Feedback

The system is controllable if

\[

\mathcal{C}=\begin{bmatrix}B_d&A_dB_d&\cdots&A_d^{n-1}B_d\end{bmatrix}

\]

has rank \(n\). With state feedback

\[

u[k]=-Kx[k]+N_rr[k],

\]

the closed-loop matrix is

\[

A_{cl}=A_d-B_dK.

\]

For a controllable system, \(K\) may be selected to place the closed-loop eigenvalues at desired locations inside the unit circle. Pole locations must also respect control effort, uncertainty and unmodelled dynamics.

9.2 Observers

If all states are not measured, a full-order observer is

\[

\hat{x}[k+1]=A_d\hat{x}[k]+B_du[k]

+L\big(y[k]-C_d\hat{x}[k]-D_du[k]\big).

\]

The estimation error evolves according to

\[

\tilde{x}[k+1]=(A_d-LC_d)\tilde{x}[k].

\]

Observability requires

\[

\operatorname{rank}\begin{bmatrix}

C_d\\C_dA_d\\\vdots\\C_dA_d^{n-1}

\end{bmatrix}=n.

\]

Very fast observer poles can increase noise sensitivity and numerical demands.

9.3 Linear Quadratic Regulation

The infinite-horizon discrete LQR minimizes

\[

J=\sum_{k=0}^{\infty}\left(x[k]^TQx[k]+u[k]^TRu[k]\right),

\]

where \(Q\succeq0\) and \(R\succ0\). The optimal law is \(u[k]=-Kx[k]\), with

\[

K=(R+B_d^TPB_d)^{-1}B_d^TPA_d,

\]

where \(P\) is the stabilizing solution of the discrete algebraic Riccati equation

\[

P=A_d^TPA_d-A_d^TPB_d(R+B_d^TPB_d)^{-1}B_d^TPA_d+Q.

\]

Standard existence results require stabilizability of \((A_d,B_d)\) and appropriate detectability conditions involving \(Q\).

Integral states may be added to remove constant tracking error. However, feasibility and actuator constraints must still be considered. Model predictive control extends this approach by explicitly handling constraints over a finite prediction horizon.

10. Illustrative Application: Digitally Controlled Thermal Stage

Background

Consider an illustrative electronics manufacturing station whose thermal stage must regulate normalized temperature deviation. The plant is approximated by

\[

\dot{x}(t)=-0.1x(t)+0.1u(t),\qquad y(t)=x(t).

\]

The continuous time constant is 10 s and the steady-state gain is one.

Problem

A controller operating every 2 s must produce a faster response while maintaining stability. The sensor covers 0–100 °C with 12-bit resolution. The actuator has a normalized upper limit of 2.

Analysis

Exact zero-order-hold discretization gives

\[

x[k+1]=0.818731x[k]+0.181269u[k].

\]

The open-loop sampled pole is 0.818731. For a response dominated by this pole, reaching 2% of the initial error requires approximately

\[

k>\frac{\ln(0.02)}{\ln(0.818731)}=19.56,

\]

or about 20 samples, corresponding to 40 s.

The sensor resolution is

\[

\Delta=100/4096=0.024414\ ^\circ\text{C}.

\]

Approach and Solution

Choose a closed-loop pole at 0.5 and apply

\[

u[k]=-Kx[k]+N_rr[k].

\]

Pole placement requires

\[

0.818731-0.181269K=0.5,

\]

so

\[

K=1.7583.

\]

For unit steady-state tracking,

\[

N_r=\frac{1-0.5}{0.181269}=2.7583.

\]

The closed-loop system becomes

\[

x[k+1]=0.5x[k]+0.5r[k].

\]

The 2% settling condition is reached in approximately six samples, or 12 s.

For a unit step from zero, the unconstrained initial command is \(u[0]=2.7583\), which exceeds the actuator limit of 2. The nominal pole-placement response is therefore not immediately achievable. A reference governor, trajectory ramp or constrained predictive controller should be used. Ignoring this limit would make simulation results misleading and could stress the heater or power electronics.

Outcome

The design reduces the nominal settling time from roughly 40 s to 12 s. Sensor quantization is small relative to ordinary thermal tolerances, but actuator saturation is a material design constraint. Timing tests must confirm that sensing and computation complete reliably within the 2 s interval.

Learning Points

• Exact zero-order-hold models preserve intersample input behavior.
• Pole placement alone does not guarantee constraint feasibility.
• Quantization, delay and actuator limits must be included in validation.
• A controller should be tested under nominal, worst-case and fault conditions.

11. Implementation, Governance and Contemporary Practice

Real-time control software should have deterministic scheduling, bounded execution time and documented overrun behavior. Best practices include separating sampling from noncritical communication tasks, using monotonic timing sources, logging overruns, validating numerical ranges and testing saturation logic.

Safety-related applications may require development processes aligned with standards such as IEC 61508, ISO 26262 or domain-specific aerospace requirements. Applicable standards depend on the industry and assurance level. Cybersecurity controls should protect firmware, calibration parameters and communication channels; IEC 62443 provides relevant industrial-security guidance.

AI-assisted tuning, learned models and reinforcement-learning controllers are increasingly researched. They can improve adaptation in complex systems but may lack predictable behavior outside the training distribution. Safety-critical deployment therefore requires constraint enforcement, independent monitoring, traceability, fallback control and rigorous verification. A learned controller should not bypass physical limits or established protective interlocks.

12. Challenges and Limitations

Discrete models can fail when sampling is too slow to capture neglected dynamics. Other recurring difficulties include:

• aliasing from inadequate analog filtering;
• uncertain or time-varying computation and network delays;
• saturation, dead zones and rate limits;
• coefficient sensitivity and finite-word-length limit cycles;
• model mismatch and unmodelled resonances;
• asynchronous sensors and packet loss;
• noise amplification by derivative or observer action; and
• insufficient verification of abnormal operating conditions.

Robust design requires sensitivity analysis, hardware-in-the-loop testing, timing measurements and explicit fault-response strategies rather than nominal simulation alone.

Chapter Summary

Digital control systems combine continuous plants with sampled measurements and numerically generated commands. Difference equations and pulse transfer functions provide input-output descriptions, while discrete state-space models support multivariable control, estimation and optimization. The Z-transform exposes poles, zeros, delays and steady-state behavior.

Sampling must satisfy both signal-reconstruction and control-performance requirements. Anti-aliasing filters, quantization, timing jitter and computational delay affect achievable performance. Exact zero-order-hold discretization is preferred when the plant model and hold mechanism are known, while Euler and Tustin methods provide useful approximations under stated conditions.

Discrete stability requires poles or closed-loop eigenvalues to lie strictly inside the unit circle. Digital PID control can be implemented in positional or incremental form, with derivative filtering, anti-windup and bumpless transfer. State feedback, observers and LQR provide systematic methods for higher-order and multivariable systems. Practical success ultimately depends on constraints, numerical reliability, real-time execution, safety and cybersecurity governance.

Key Terms

Aliasing: Irreversible overlap of spectral replicas caused by inadequate sampling.

Anti-aliasing filter: Analog filter used to attenuate frequencies above the usable sampling band.

Controllability: Ability to transfer a system state using admissible inputs.

Difference equation: Recurrence relation describing a discrete-time dynamic system.

Discrete LQR: Optimal state-feedback method based on a quadratic infinite-horizon cost.

Finite-word length: Restriction caused by representing numbers with limited precision.

Jury test: Algebraic test for determining whether polynomial roots lie inside the unit circle.

Limit cycle: Persistent finite-amplitude oscillation caused by nonlinear numerical effects.

Observability: Ability to infer the internal state from input-output measurements.

Pulse transfer function: Ratio of Z-transformed sampled output to input under zero initial conditions.

Quantization: Mapping of continuous amplitudes to a finite set of digital levels.

Sampling jitter: Variation of sampling instants from their intended times.

Sampling period: Time interval between successive samples.

Schur stability: Property that all polynomial roots lie strictly inside the unit circle.

State observer: Dynamic estimator that reconstructs unmeasured state variables.

Tustin transformation: Bilinear mapping used to convert continuous transfer functions to discrete form.

Z-transform: Transform that represents discrete sequences as functions of the complex variable \(z\).

Zero-order hold: Device or model that maintains each command constant until the next update.

Review Questions

Short Answer Questions

1. Distinguish the sampling frequency from the Nyquist rate.

2. What is the stability region for poles of a causal discrete-time system?

3. Why is an anti-aliasing filter placed before an ADC?

4. Define quantization step size for an ideal uniform converter.

5. What is integral windup in a digital PID controller?

Long Answer Questions

1. Derive the pulse transfer function of a linear difference equation using the Z-transform.

2. Compare exact zero-order-hold, forward Euler, backward Euler and Tustin discretization.

3. Explain how sampling delay, jitter and quantization influence closed-loop performance.

4. Derive the incremental digital PID algorithm and discuss practical implementation modifications.

5. Explain discrete state-feedback, observer and LQR design, including their applicability conditions.

Analytical / Application Questions

1. A plant \(G(s)=1/(5s+1)\) is sampled every second using a zero-order hold. Derive its discrete state equation and pulse transfer function.

2. Apply the second-order Jury conditions to determine the stability of \(P(z)=z^2-0.7z+0.1\).

3. Develop a digital-control validation plan for a networked motor drive, addressing delay, packet loss, saturation, quantization, cybersecurity and safe fallback behavior.

Multiple Choice Questions with Answers

Question 1

A 900 Hz sinusoid is sampled at 1 kHz. What is its apparent aliased frequency in the baseband?

A. 50 Hz

B. 100 Hz

C. 450 Hz

D. 900 Hz

Correct Answer: B. 100 Hz

Question 2

Which discrete pole corresponds to an asymptotically stable mode?

A. z = 1.1

B. z = -1

C. z = 0.8

D. z = 1.02j

Correct Answer: C. z = 0.8

Question 3

For G(s) = 1/(10s+1) sampled with a zero-order hold at Ts = 2 s, what is the discrete state coefficient Ad?

A. 0.181269

B. 0.500000

C. 0.818731

D. 1.221403

Correct Answer: C. 0.818731

Question 4

Which expression is the bilinear or Tustin transformation?

A. s = (z-1)/Ts

B. s = (z-1)/(Tsz)

C. s = (2/Ts)(z-1)/(z+1)

D. s = ln(z)Ts

Correct Answer: C. s = (2/Ts)(z-1)/(z+1)

Question 5

What is the ideal step size of a 12-bit converter covering 0–100 °C?

A. 0.012207 °C

B. 0.024414 °C

C. 0.048828 °C

D. 0.409600 °C

Correct Answer: B. 0.024414 °C

Question 6

Which set of inequalities gives strict Schur stability for P(z) = z² + a1z + a2?

A. 1+a1+a2>0, 1-a1+a2>0, 1-a2>0

B. 1+a1+a2<0, 1-a1+a2<0, 1-a2<0

C. a1>1, a2>1, a1+a2>1

D. a1<0, a2<0, a1a2<1

Correct Answer: A. 1+a1+a2>0, 1-a1+a2>0, 1-a2>0

Question 7

For u[k] = u[k-1] + 4.1e[k] - 6e[k-1] + 2e[k-2], what is u[k] when u[k-1] = 0.8 and the three error samples are 0.5, 0.4 and 0.3, respectively?

A. 0.85

B. 1.05

C. 1.45

D. 2.25

Correct Answer: B. 1.05

Question 8

For exact zero-order-hold discretization of ẋ = Ax + Bu, which expression gives Bd?

A. Bd = e^(BTs)

B. Bd = ATsB

C. Bd = ∫₀^Ts e^(Aτ)B dτ

D. Bd = A + TsB

Correct Answer: C. Bd = ∫₀^Ts e^(Aτ)B dτ

Question 9

For xd[k+1] = 0.818731x[k] + 0.181269u[k], what feedback gain places the closed-loop pole at 0.5 using u[k] = -Kx[k]?

A. 0.3187

B. 1.0000

C. 1.7583

D. 2.7583

Correct Answer: C. 1.7583

Question 10

Which technique directly addresses integrator accumulation during actuator saturation?

A. Prewarping

B. Anti-windup

C. Pole-zero cancellation

D. Impulse invariance

Correct Answer: B. Anti-windup