13.2 Basic Mechatronics, Sensors & Closed-Loop Controls

13.2 Basic Mechatronics, Sensors & Closed-Loop Controls

Modern mechanical systems are rarely purely mechanical; they are integrated mechatronic systems that combine precision mechanisms, sensors, electronic signal conditioning, embedded digital processors, and electromechanical actuators. For machine designers, understanding how physical quantities (force, strain, displacement, temperature, and speed) are measured, converted into electrical signals, and regulated by feedback controllers is essential for designing automated machinery, robotic arms, precision positioning stages, and dynamic test equipment.

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Mechatronic System Architecture & Signal Conditioning

A complete mechatronic control system functions as a closed information loop:

                      Mechatronic Control Loop
                      
         Reference  +    Error     +------------+  Control  +----------+
  r(t) ------------>(+)----------> | Controller | --------> | Actuator |
         (Setpoint)  ^ -  e(t)     | (DSP/PLC)  |   u(t)    | Driver   |
                     |             +------------+           +----------+
                     |                                            |
                     |           +-----------------------+        v
                     |           |   Mechanical Plant    | <--------------+
                     |           | (Mechanism / Load)    |   Driven       |
                     |           +-----------------------+   Motion       |
                     |                       |                            |
                     |                       v                            |
                     |             +-------------------+                  |
                     +------------ | Sensor / Signal   | <----------------+
                      Measurement  | Conditioning      |   Process Feedback
                          y(t)     +-------------------+

Signal Conditioning & Interfacing Fundamentals

1. Amplification & Common-Mode Rejection: Sensors frequently output low-level differential voltages ($1\,\text{mV} - 50\,\text{mV}$). Differential instrumentation amplifiers provide high input impedance and a high Common-Mode Rejection Ratio (CMRR) to amplify microvolt sensor signals while rejecting 60 Hz electrical mains noise and electromagnetic interference (EMI).

2. Anti-Aliasing Filtering & Sampling (Nyquist Criterion): Analog signals must be converted to digital numbers via an Analog-to-Digital Converter (ADC). To prevent aliasing (where high-frequency noise folds back into the measurement band as false low-frequency signals), an analog low-pass anti-aliasing filter must precede the ADC. By the Nyquist-Shannon sampling theorem, the sampling frequency $f_s$ must exceed at least twice the maximum frequency component present in the analog signal:

   $$f_s \ge 2 f_{\max}$$

   In practical machine control systems, $f_s$ is commonly set to $10 \times$ to $20 \times$ the closed-loop system bandwidth.

3. ADC Resolution & Quantization Error: An $N$-bit ADC with an input voltage range $V_{\text{ref}}$ divides the signal into $2^N$ discrete digital levels. The voltage resolution (quantization step size $Q$) is:

   $$Q = \frac{V_{\text{ref}}}{2^N}$$

   The maximum quantization error is $\pm \frac{Q}{2}$. For example, a 16-bit ADC operating over a $0-10\,\text{V}$ range has $Q = \frac{10}{65,\!536} \approx 0.153\,\text{mV}$.

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Sensors & Transducers

1. Resistive Strain Gages & Wheatstone Bridges

A resistive foil strain gage consists of a thin metallic grid bonded to a flexible backing. When subjected to mechanical strain $\epsilon$, its electrical resistance changes proportionally due to physical elongation and piezoresistive effects:

$$\frac{\Delta R}{R} = GF \cdot \epsilon$$

where:

• $R$ = nominal gage resistance (commonly $120\,\Omega, 350\,\Omega, \text{or } 1,\!000\,\Omega$).
• $\epsilon = \frac{\Delta L}{L}$ = engineering strain (dimensionless, commonly expressed in microstrain $\mu\varepsilon = 10^{-6}$).
• $GF$ = Gage Factor (dimensionless sensitivity factor; typically $GF \approx 2.0 - 2.1$ for metallic constantan foil gages, and $GF \approx 100 - 150$ for semiconductor silicon gages).

                     Wheatstone Bridge Circuit
                                (+) V_s (Excitation)
                                    *
                                   / \
                       R_1 (Arm 1)/   \ R_2 (Arm 2)
                                 /     \
                                *-------* 
                             (-)  V_out  (+)
                                 \     /
                       R_4 (Arm 4)\   / R_3 (Arm 3)
                                   \ /
                                    *
                                   GND

Wheatstone Bridge Circuit Formulations

Because $\Delta R / R$ is tiny (typically $< 0.1\%$), a Wheatstone bridge converts small fractional resistance changes into measurable differential voltages $V_{\text{out}}$. The general balanced bridge relation is $R_1 R_3 = R_2 R_4$. For small variations, the bridge output is:

$$\frac{V_{\text{out}}}{V_s} \approx \frac{1}{4} \left( \frac{\Delta R_1}{R_1} - \frac{\Delta R_2}{R_2} + \frac{\Delta R_3}{R_3} - \frac{\Delta R_4}{R_4} \right) = \frac{GF}{4} (\epsilon_1 - \epsilon_2 + \epsilon_3 - \epsilon_4)$$

• Quarter-Bridge (1 active gage in arm 1, 3 fixed resistors):

  $$V_{\text{out}} = V_s \frac{GF \cdot \epsilon}{4}$$

  Susceptible to thermal drift because temperature changes cause thermal resistance changes that mimic mechanical strain. A dummy unstrained gage in arm 2 exposed to the same ambient temperature cancels temperature drift.

• Half-Bridge (2 active gages):

  • Bending Application: Gage 1 mounted on top surface in tension ($+\epsilon$), Gage 2 on bottom surface in compression ($-\epsilon$): $$V_{\text{out}} = V_s \frac{GF}{4} [(\epsilon) - (-\epsilon)] = V_s \frac{GF \cdot \epsilon}{2}$$ Doubles sensitivity, and inherently cancels both axial tension/compression and thermal expansion.

• Full-Bridge (4 active gages):

  • Bending Beam: Gages 1 and 3 in tension ($+\epsilon$), Gages 2 and 4 in compression ($-\epsilon$): $$V_{\text{out}} = V_s \frac{GF}{4} [(\epsilon) - (-\epsilon) + (\epsilon) - (-\epsilon)] = V_s \cdot GF \cdot \epsilon$$ Quadruples output sensitivity, completely rejects temperature drift, cancels axial loads, and provides maximum signal-to-noise ratio.

2. Linear Variable Differential Transformers (LVDTs)

An LVDT is an electromechanical transducer that produces an electrical output proportional to linear displacement:

• Construction: One primary coil centered between two identical secondary coils wound in series-opposition around a cylindrical hollow bobbin. A moveable high-permeability ferromagnetic core attaches to the moving machine member.
• Operation: The primary coil is excited with an AC carrier voltage ($1-10\,\text{kHz}$, $3-15\,\text{V}_{\text{rms}}$). When the core is centered at the null position, equal magnetic flux links both secondary coils; because they are wired in series-opposition, the net output voltage is zero ($V_{\text{out}} = V_{s1} - V_{s2} = 0$).
• Displacement: Moving the core from null increases flux in one secondary while decreasing flux in the other. The amplitude of $V_{\text{out}}$ scales linearly with core displacement, while its phase ($0^\circ$ vs $180^\circ$ relative to primary excitation) indicates direction.
• Features: Frictionless non-contact operation, infinite theoretical resolution, high linearity ($< 0.25\%$), and rugged resistance to shock, vibration, and contaminated environments.

3. Optical Encoders (Incremental vs. Absolute)

Optical encoders convert rotational displacement into digital pulses using a patterned glass or polymer disc and photodetectors.

• Incremental Encoders:

  • Utilize two optical tracks: Channel A and Channel B, offset by $90^\circ$ electrical phase (quadrature).
  • Direction Sensing: If Channel A leads Channel B, shaft rotation is clockwise; if B leads A, rotation is counterclockwise.
  • Resolution Multiplication ($4\times$ Decoding): By counting both rising and falling edges of channels A and B, the effective resolution is quadrupled ($4 \times \text{PPR}$ pulses per revolution).
  • Index Channel (Z): Produces one pulse per complete revolution, serving as a physical reference marker for homing.
  • Limitation: Loses absolute shaft position upon loss of power; requires a homing routine on power-up.

• Absolute Encoders:

  • Disc features multiple concentric tracks, where each angular sector corresponds to a unique digital binary word.
  • Retains absolute position even during power outages; no homing sequence required.
  • Gray Code vs. Binary: In natural binary code, transitioning between consecutive numbers often requires multiple bits to flip simultaneously (e.g., $011_2 \to 100_2$ flips all 3 bits). Slight mechanical disc misalignment causes intermediate read glitches (e.g., momentarily reading $111_2$). Gray code is a reflected binary code designed so that only one bit changes between any two consecutive values, eliminating transition glitches.

4. Temperature Sensors

                               Temperature Sensor Comparison
  +-----------------------------------------------------------------------------------------+
  | Sensor Type  | Transduction Principle          | Operating Range     | Key Characteristic |
  |--------------+---------------------------------+---------------------+--------------------|
  | Thermocouple | Seebeck thermoelectric effect   | -200C to +1800C     | Rugged, self-power |
  | RTD (Pt100)  | Positive temp-coefficient (PTC) | -200C to +650C      | High accuracy/line |
  | Thermistor   | Negative temp-coefficient (NTC) | -50C to +150C       | High sensitivity   |
  +-----------------------------------------------------------------------------------------+

• Thermocouples:

  • Two dissimilar metal wires joined at one end form a measuring (hot) junction. The Seebeck effect generates an open-circuit voltage proportional to the temperature difference between hot junction $T_{\text{hot}}$ and reference (cold) junction $T_{\text{ref}}$: $V = S_{AB}(T_{\text{hot}} - T_{\text{ref}})$.
  • Requires Cold-Junction Compensation (CJC) using a local IC temperature sensor on the isothermal terminal block.
  • Common types: Type K (Chromel-Alumel, wide industrial range $-200^\circ\text{C}$ to $+1250^\circ\text{C}$, $\approx 41\,\mu\text{V}/^\circ\text{C}$), Type J (Iron-Constantan), Type T (Copper-Constantan).

• Resistance Temperature Detectors (RTDs):

  • Operating principle: Pure metals exhibit a linear increase in electrical resistance with temperature: $$R(T) = R_0 [1 + \alpha(T - T_0)]$$
  • Industrial standard: Pt100 ($R_0 = 100.0\,\Omega$ at $0^\circ\text{C}$, $\alpha = 0.00385\,\Omega/\Omega/^\circ\text{C}$) and Pt1000 ($1,\!000\,\Omega$ at $0^\circ\text{C}$).
  • Highly linear, stable, and accurate ($\pm 0.1^\circ\text{C}$). 3-wire and 4-wire lead configurations cancel lead-wire resistance in remote installations.

• Thermistors:

  • Semiconductor metal oxides with a Negative Temperature Coefficient (NTC): $$R(T) = R_0 \exp\left[ \beta \left(\frac{1}{T} - \frac{1}{T_0}\right) \right]$$
  • Extremely sensitive to small temperature changes over narrow ranges, but highly non-linear.

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Basic Circuit Relations for Mechatronics

Sensor, actuator and controller hardware rests on a few circuit relations:

• Ohm's law and power: $V = IR$; $P = VI = I^2 R = V^2/R$.
• Series and parallel resistors: $R_{eq} = R_1 + R_2 + \dots$ in series; $1/R_{eq} = 1/R_1 + 1/R_2 + \dots$ in parallel.
• Voltage divider: $V_{\text{out}} = V_{\text{in}} R_2/(R_1 + R_2)$, used to scale signals and in potentiometer position sensors.
• RC circuits: time constant $\tau = RC$; a capacitor charges to $63.2\%$ of its final voltage in one $\tau$ and about $99\%$ in $5\tau$. A first-order RC low-pass filter has cutoff frequency $f_c = 1/(2\pi R C)$ and is the simplest anti-aliasing filter.
• RL circuits: $\tau = L/R$. Current in solenoids and motor windings rises with this time constant, which is why stepper torque falls at high step rates.
• Inductive loads (relays, solenoids, motor windings) produce a large voltage spike ($V = L\,di/dt$) when switched off, so a flyback (freewheeling) diode is placed across the coil to protect the switching transistor.
• Switching and drive circuits: transistors and MOSFETs let low-power logic switch loads; an H-bridge of four switches reverses a DC motor; pulse-width modulation (PWM) sets the average voltage $V_{\text{avg}} = D\,V_{\text{supply}}$ for duty cycle $D$. Relays, contactors and solid-state relays interface controller outputs with motors and heaters.

Mini-example: A $10\,\text{k}\Omega$ resistor and a $0.1\,\mu\text{F}$ capacitor give $\tau = RC = 1\,\text{ms}$ and $f_c = 1/(2\pi \times 10{,}000 \times 0.1 \times 10^{-6}) = 159\,\text{Hz}$. A $24\,\text{V}$ supply switched at $40\%$ duty cycle delivers an average of $9.6\,\text{V}$ to a motor.

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Electromechanical Actuators

1. DC Motors (Brushed vs. Brushless)

In a direct current (DC) motor, magnetic fields interact to produce torque:

$$V = I_a R_a + E_b = I_a R_a + K_e \omega$$ $$T = K_t I_a$$

where:

• $V$ = applied armature terminal voltage (V)
• $I_a$ = armature current (A)
• $R_a$ = armature winding resistance ($\Omega$)
• $E_b = K_e \omega$ = counter-electromotive force / back-EMF (V)
• $\omega$ = rotor angular velocity (rad/s)
• $K_e$ = back-EMF constant ($\text{V}\cdot\text{s/rad}$)
• $K_t$ = torque sensitivity constant ($\text{N}\cdot\text{m/A}$)

(In consistent SI units, $K_e = K_t$ numerically).

Torque-Speed Characteristic

Combining the electrical and torque equations yields the linear torque-speed curve for a permanent magnet DC motor:

$$\omega(T) = \frac{V}{K_e} - \frac{R_a}{K_t K_e} T = \omega_{nl} \left(1 - \frac{T}{T_{\text{stall}}}\right)$$

• No-Load Speed ($\omega_{nl}$ at $T = 0$): $$\omega_{nl} = \frac{V}{K_e}$$
• Stall Torque ($T_{\text{stall}}$ at $\omega = 0$): $$T_{\text{stall}} = \frac{K_t V}{R_a}$$

                  DC Motor Torque-Speed Curve
       Speed w ^
          w_nl +\ 
               | \ 
               |  \   Operating Characteristic Curve
               |   \ 
        w_nl/2 +----+----\  <-- Maximum Power Point: P_max
               |    |     \ 
               +----+------+---------> Torque T
               0  T_stall/2 T_stall

Mechanical Output Power

The mechanical output power is $P(T) = T \omega$:

$$P(T) = T \omega_{nl} \left(1 - \frac{T}{T_{\text{stall}}}\right)$$

Differentiating with respect to $T$ and setting to zero demonstrates that maximum mechanical power output occurs at exactly half the stall torque and half the no-load speed:

$$T_{P\max} = \frac{1}{2} T_{\text{stall}}, \qquad \omega_{P\max} = \frac{1}{2} \omega_{nl}$$ $$P_{\max} = T_{P\max} \omega_{P\max} = \frac{1}{4} T_{\text{stall}} \omega_{nl} = \frac{V^2}{4 R_a}$$

Brushed vs. Brushless DC Motors (BLDC)

• Brushed DC: Mechanical commutator rings and graphite brushes continuously reverse armature coil currents. Simple two-lead drive, but subject to brush arcing, wear, acoustic/RF noise, and limited operational life.
• Brushless DC (BLDC): Permanent magnets on rotor, multiphase electromagnetic windings on stator. Electronic inverter commutates currents based on rotor position (Hall effect sensors or sensorless back-EMF sensing). Offers higher efficiency, superior thermal dissipation, higher continuous torque, and virtually maintenance-free operation.

2. Stepper Motors

Stepper motors convert discrete digital pulses into precise incremental shaft rotations without closed-loop feedback.

• Step Angle ($\theta_s$): $$\theta_s = \frac{360^\circ}{2\,m\,Z}$$ for a hybrid stepper driven in full steps, where $m$ = number of phases (typically 2) and $Z$ = number of rotor teeth (commonly 50). Each tooth pitch is crossed in $2m$ full steps, giving the standard hybrid stepper angle: $$\theta_s = \frac{360^\circ}{2 \times 2 \times 50} = 1.8^\circ \quad (\implies 200\,\text{steps per revolution})$$
• Microstepping: Modulating phase currents using Pulse-Width Modulation (PWM) into sinusoidal proportions divides each fundamental full step into fractions ($1/4, 1/16, 1/64, 1/256$), smoothing shaft torque and suppressing mid-frequency mechanical resonance.
• Torque Behavior: Stepper motors generate high holding torque at zero speed, but usable torque drops rapidly at higher stepping frequencies due to winding inductance ($L/R$ time constant) limiting current rise.

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Closed-Loop Feedback Control Systems

Open-Loop vs. Closed-Loop Systems

• Open-Loop: Control input is applied without measuring system output. Simple, low cost, unconditionally stable, but unable to compensate for external disturbances (load changes, friction variations) or plant parameter drift.
• Closed-Loop (Feedback): Sensor monitors the output $y(t)$, subtracts it from reference setpoint $r(t)$ to generate error signal $e(t) = r(t) - y(t)$. Provides high disturbance rejection, reduced sensitivity to plant parameter changes, and stabilization of unstable systems.

Block Diagram Algebra & Transfer Functions

Consider a negative feedback system with forward controller $C(s)$, plant $G(s)$, and unity feedback ($H(s) = 1$):

$$T(s) = \frac{Y(s)}{R(s)} = \frac{C(s) G(s)}{1 + C(s) G(s)}$$

The open-loop loop transfer function is $L(s) = C(s) G(s)$. The characteristic equation governing system stability is:

$$1 + L(s) = 1 + C(s) G(s) = 0$$

PID Control Fundamentals

The Proportional-Integral-Derivative (PID) controller is the most widely deployed feedback algorithm in machine design:

$$u(t) = K_p e(t) + K_i \int_0^t e(\tau) d\tau + K_d \frac{de(t)}{dt}$$

In the Laplace domain:

$$C(s) = K_p + \frac{K_i}{s} + K_d s = \frac{K_d s^2 + K_p s + K_i}{s}$$

                                PID Tuning Influences
  +----------------------------------------------------------------------------------------+
  | Parameter | Rise Time     | Overshoot     | Settling Time | Steady-State Error | Stability |
  |-----------+---------------+---------------+---------------+--------------------+-----------|
  | Increase K_p | Decreases  | Increases     | Minor Effect  | Decreases (offset) | Degrades  |
  | Increase K_i | Decreases  | Increases     | Increases     | Eliminates to ZERO | Degrades  |
  | Increase K_d | Minor Effect| Decreases     | Decreases     | Minor Effect       | Improves  |
  +----------------------------------------------------------------------------------------+

• Proportional Gain ($K_p$): Provides immediate restoring action proportional to current error. Accelerates response and reduces steady-state error, but high $K_p$ induces excessive overshoot and oscillation.
• Integral Gain ($K_i$): Integrates past error over time. Adds a free integrator (pole at $s = 0$), raising system type by 1 and forcing steady-state error to zero ($e_{ss} = 0$) under step commands and constant disturbance loads. However, introduces $-90^\circ$ phase lag, which reduces damping and can cause integral windup during actuator saturation.
• Derivative Gain ($K_d$): Anticipates future error based on rate of change ($de/dt$). Adds phase lead, increasing effective damping, suppressing overshoot, and accelerating settling time. However, amplifies high-frequency measurement noise; practical implementations require a first-order low-pass filter on the derivative term: $C_d(s) = \frac{K_d s}{\tau_f s + 1}$.

Stability Analysis & Routh-Hurwitz Stability Criterion

A closed-loop linear system is Bounded-Input Bounded-Output (BIBO) stable if and only if all roots of the characteristic equation lie strictly in the open left-half complex plane ($\text{Re}(s) < 0$).

For a general characteristic polynomial:

$$A(s) = a_n s^n + a_{n-1} s^{n-1} + a_{n-2} s^{n-2} + \dots + a_1 s + a_0 = 0$$

1. Necessary Condition (Stodola): All coefficients $a_i$ must be strictly positive ($a_i > 0$) and non-zero.
2. Routh-Hurwitz Array: Form the array from polynomial coefficients: $$\begin{array}{c|cccc} s^n & a_n & a_{n-2} & a_{n-4} & \dots \\ s^{n-1} & a_{n-1} & a_{n-3} & a_{n-5} & \dots \\ s^{n-2} & b_1 & b_2 & b_3 & \dots \\ s^{n-3} & c_1 & c_2 & c_3 & \dots \end{array}$$ where: $$b_1 = \frac{a_{n-1} a_{n-2} - a_n a_{n-3}}{a_{n-1}}, \qquad b_2 = \frac{a_{n-1} a_{n-4} - a_n a_{n-5}}{a_{n-1}}$$ $$c_1 = \frac{b_1 a_{n-3} - a_{n-1} b_2}{b_1}$$ Criterion: The number of roots with positive real parts (unstable poles) equals the number of sign changes in the first column of the Routh array. For absolute stability, there must be zero sign changes in the first column.

For a Third-Order System ($s^3 + a_2 s^2 + a_1 s + a_0 = 0$):

$$\begin{array}{c|cc} s^3 & 1 & a_1 \\ s^2 & a_2 & a_0 \\ s^1 & \frac{a_2 a_1 - a_0}{a_2} & 0 \\ s^0 & a_0 & 0 \end{array}$$

The system is strictly stable if and only if:

$$a_2 > 0, \quad a_1 > 0, \quad a_0 > 0, \quad \text{and} \quad a_2 a_1 > a_0$$

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Step-by-Step Worked Sensor & Control Scenario

Problem Statement

An automated machine spindle drive requires precision torque measurement and closed-loop speed control.

Part A: Wheatstone Bridge Strain Gage Transducer

A solid steel shaft of diameter $d = 30.0\,\text{mm}$ ($0.030\,\text{m}$) transmits a steady working torque $T = 250.0\,\text{N}\cdot\text{m}$. The shear modulus of elasticity of the steel is $G = 79.3\,\text{GPa}$. Four identical metallic foil strain gages with nominal resistance $R = 350.0\,\Omega$ and Gage Factor $GF = 2.10$ are bonded to the shaft surface at $\pm 45^\circ$ helix angles to measure the principal tensile and compressive strains induced by torsion in a full Wheatstone bridge configuration. The bridge excitation voltage is $V_s = 10.0\,\text{V}$.

Calculate:

1. The maximum torsional surface shear stress $\tau_{\max}$ and the resulting surface principal strain $\epsilon$.
2. The differential output voltage $V_{\text{out}}$ produced by the full Wheatstone bridge in millivolts.

Part B: Closed-Loop Speed Control & Stability

The DC spindle drive and load have the plant transfer function:

$$G(s) = \frac{25}{s(s + 5)}$$

A proportional-derivative (PD) controller $C(s) = K_p + K_d s$ is placed in the forward path with unity negative feedback ($H(s) = 1$).

Calculate:

1. The closed-loop characteristic equation in terms of $K_p$ and $K_d$.
2. The required gains $K_p$ and $K_d$ to achieve a closed-loop natural frequency $\omega_n = 10.0\,\text{rad/s}$ and a damping ratio $\zeta = 0.707$.
3. If an integral gain $K_i$ is added to form a full PID controller ($C(s) = \frac{K_d s^2 + K_p s + K_i}{s}$), apply the Routh-Hurwitz criterion to determine the maximum upper bound on $K_i$ for which the closed-loop system remains strictly stable.

Step-by-Step Solution

Step 1: Torsional Stress & Principal Strain

Compute the polar moment of inertia $J$ of the solid circular shaft:

$$J = \frac{\pi d^4}{32} = \frac{\pi (0.030\,\text{m})^4}{32} = \frac{\pi \times 8.1 \times 10^{-7}}{32} = 7.9522 \times 10^{-8}\,\text{m}^4$$

Compute maximum surface shear stress $\tau_{\max}$:

$$\tau_{\max} = \frac{T (d/2)}{J} = \frac{16 T}{\pi d^3} = \frac{16 \times 250.0}{\pi (0.030)^3} = \frac{4,\!000}{8.4823 \times 10^{-5}} = 4.7157 \times 10^7\,\text{Pa} = 47.16\,\text{MPa}$$

Compute the surface engineering shear strain $\gamma$:

$$\gamma = \frac{\tau_{\max}}{G} = \frac{47.157 \times 10^6\,\text{Pa}}{79.3 \times 10^9\,\text{Pa}} = 5.9467 \times 10^{-4}\,\text{rad}$$

In pure torsion, the principal normal stresses occur at $\pm 45^\circ$ to the shaft axis with magnitudes $\sigma_1 = +\tau_{\max}$ and $\sigma_2 = -\tau_{\max}$. The corresponding principal normal strains are:

$$\epsilon_1 = +\frac{\gamma}{2} = +2.9734 \times 10^{-4} = +297.3\,\mu\varepsilon$$ $$\epsilon_2 = -\frac{\gamma}{2} = -2.9734 \times 10^{-4} = -297.3\,\mu\varepsilon$$

Step 2: Full-Bridge Output Voltage

In a full-bridge torsion setup, gages 1 and 3 experience tensile principal strain $\epsilon_1 = \epsilon_3 = +\epsilon$, and gages 2 and 4 experience compressive principal strain $\epsilon_2 = \epsilon_4 = -\epsilon$:

$$\frac{V_{\text{out}}}{V_s} = \frac{GF}{4} (\epsilon_1 - \epsilon_2 + \epsilon_3 - \epsilon_4) = \frac{GF}{4} [\epsilon - (-\epsilon) + \epsilon - (-\epsilon)] = \frac{GF}{4} (4\epsilon) = GF \cdot \epsilon$$ $$V_{\text{out}} = V_s \cdot GF \cdot \epsilon = (10.0\,\text{V}) \times (2.10) \times (2.9734 \times 10^{-4}) = 6.244 \times 10^{-3}\,\text{V} = 6.24\,\text{mV}$$

The full-bridge circuit outputs $6.24\,\text{mV}$, which can be directly fed into an instrumentation amplifier.

Step 3: PD Control Parameter Tuning

The closed-loop transfer function is:

$$T(s) = \frac{C(s) G(s)}{1 + C(s) G(s)} = \frac{(K_p + K_d s) \frac{25}{s(s + 5)}}{1 + (K_p + K_d s) \frac{25}{s(s + 5)}} = \frac{25 K_d s + 25 K_p}{s(s + 5) + 25(K_d s + K_p)}$$

The characteristic equation is:

$$s^2 + (5 + 25 K_d) s + 25 K_p = 0$$

Compare to the standard second-order canonical form $s^2 + 2 \zeta \omega_n s + \omega_n^2 = 0$:

1. Match the constant term: $$\omega_n^2 = 25 K_p \implies (10.0)^2 = 25 K_p \implies 100 = 25 K_p \implies K_p = 4.0$$
2. Match the linear damping coefficient: $$2 \zeta \omega_n = 5 + 25 K_d \implies 2 (0.707)(10.0) = 5 + 25 K_d \implies 14.14 = 5 + 25 K_d$$ $$25 K_d = 14.14 - 5 = 9.14 \implies K_d = \frac{9.14}{25} = 0.3656\,\text{s}$$

Step 4: PID Stability Boundary via Routh-Hurwitz

With PID controller $C(s) = \frac{K_d s^2 + K_p s + K_i}{s}$, the loop transfer function is:

$$L(s) = C(s) G(s) = \frac{K_d s^2 + K_p s + K_i}{s} \cdot \frac{25}{s(s + 5)} = \frac{25(K_d s^2 + K_p s + K_i)}{s^2(s + 5)}$$

The new third-order closed-loop characteristic equation is:

$$s^2(s + 5) + 25(K_d s^2 + K_p s + K_i) = 0 \implies s^3 + (5 + 25 K_d) s^2 + 25 K_p s + 25 K_i = 0$$

Substitute $K_p = 4.0$ and $K_d = 0.3656$:

• $a_2 = 5 + 25(0.3656) = 5 + 9.14 = 14.14$
• $a_1 = 25 K_p = 25(4.0) = 100.0$
• $a_0 = 25 K_i$

The characteristic equation is:

$$s^3 + 14.14 s^2 + 100.0 s + 25 K_i = 0$$

Applying the Routh-Hurwitz stability criterion for a 3rd-order system ($a_2 a_1 > a_0$ with all $a_i > 0$):

$$25 K_i > 0 \implies K_i > 0$$ $$a_2 a_1 > a_0 \implies (14.14) \times (100.0) > 25 K_i \implies 1,\!414 > 25 K_i$$ $$K_i < \frac{1,\!414}{25} = 56.56\,\text{s}^{-1}$$

Therefore, for the system to remain strictly stable, the integral gain must satisfy:

$$0 < K_i < 56.56\,\text{s}^{-1}$$

Exceeding $K_i = 56.56$ will drive the closed-loop poles across the imaginary axis into the right-half plane, causing sustained oscillations and unstable runaway.