MOTOR-20 — Cascaded Torque, Speed, and Current Loops

FACT — Opening the throttle of an internal combustion engine requires $100 - 300\text{ ms}$ for intake manifold filling and cylinder combustion pressure build-up. In contrast, an electric traction motor can swing from zero to full peak torque in under $1 - 2\text{ ms}$! While this instantaneous response delivers exhilarating acceleration, it also threatens to wind up the mechanical driveline like a torsional spring, triggering violent vehicle longitudinal bucking (driveline jerk).

In this lesson, we break down the multi-rate cascaded control structure of EV powertrains, voltage saturation Anti-Windup algorithms, and active Anti-Jerk driveline damping:

1. Cascaded Control Hierarchy & Multi-Rate Frequency Separation
2. The Inner Current Loop ($10\text{ kHz}$): Dynamic Torque Engine
3. Voltage Saturation & Back-Calculation Anti-Windup
4. Mechanical Vulnerability: The 8–15 Hz Driveline Torsional Resonance
5. Active Anti-Jerk Filter: Closed-Loop Torsional Damping
6. Brake Blending Coordination (Regen vs. Hydraulic Friction)

---

1. Cascaded Hierarchy & Multi-Rate Separation

In an EV motor controller, algorithms are segmented into hierarchical layers based on physical dynamics. A golden rule of cascaded control dictates: Each inner loop must operate at least 5–10 times faster than the outer loop enclosing it.

Cascaded Powertrain Hierarchy:
  [ Vehicle Supervisory Unit (VCU / ADAS) ] ---> 10 Hz - 100 Hz  (CAN-FD / Ethernet)
               |  (Driver Torque / Target Speed)
               v
  [ Speed / Anti-Jerk Controller ] -------------> 1 kHz - 2 kHz   (Damps Torsional Oscillation)
               |  (I_q_ref, I_d_ref)
               v
  [ FOC Current Regulators ] -------------------> 10 kHz - 20 kHz (Synthesizes Voltage Demands)
               |  (PWM Duty Cycles)
               v
  [ Silicon Power Inverter & Motor ] -----------> 100 kHz+        (Air-Gap Magnetic Field)

Loop Layer	Execution Rate	Hardware Target	Function
Outer (VCU/ESP/ADAS)	$10 - 100\text{ Hz}$	Vehicle Control Unit	Pedal maps, cruise control, traction stability
Middle (Speed / Anti-Jerk)	$1 - 2\text{ kHz}$	MCU Safety Core	Speed error regulation, driveline damping
Inner (Current Loop)	$10 - 20\text{ kHz}$	MCU DSP/Floating-Point Core	Clarke/Park, PI current control, SVPWM

---

2. Inner Current Loop: High-Bandwidth Torque Generation

Passenger electric vehicles operate predominantly in direct torque control mode; the accelerator pedal maps driver position directly into an electromagnetic shaft torque demand ($T_{ref}$).

The torque request indexes the MTPA lookup tables, generating optimal current setpoints:

$T_{ref} \implies I_d^*, I_q^*$

With current loop bandwidth tuned to $\approx 1000\text{ Hz}$, current tracking errors settle within $0.5 - 1.0\text{ ms}$, ensuring near-instantaneous electromagnetic force generation.

---

3. Integrator Saturation & Anti-Windup

Under aggressive tip-ins or high-speed driving, demanded output voltage magnitude ($V_d^{*2} + V_q^{*2}$) can exceed the physical battery ceiling ($V_{max} = V_{dc}/\sqrt{3}$).

The inverter cannot physically output more than $V_{max}$. If the PI regulators are unaware of this ceiling, the integrator term continues accumulating tracking error, ramping its internal state to astronomical values. This phenomenon is termed Integrator Windup.

When the driver lifts off the pedal, the bloated integrator takes hundreds of milliseconds to unwind, causing the vehicle to continue accelerating uncontrollably!

Back-Calculation Anti-Windup:

To neutralize windup, the error difference between saturated output and demanded output ($\Delta V = V_{sat} - V_{demand}$) is fed back to the integrator input via a tracking gain ($K_{aw}$):

Back-Calculation Anti-Windup Schematic:
  Error (e) --(+)---> [ Kp ] -----------------(+)---> [ Voltage Limiter ] ---> V_out
               ^                               ^             |
               |                               |             v
               |--(+)-> [ Ki/s ] --(+)---------+        (V_out - V_in)
                   ^ -              |                        |
                   |                |                        v
                   +-- [ Kaw ] <----+------------------------+

$\frac{d}{dt} x_i(t) = K_i \cdot e(t) + K_{aw} \cdot \left[ V_{out}(t) - V_{in}(t) \right]$

In linear operation ($V_{out} = V_{in}$), the feedback term is zero. The instant saturation occurs ($V_{out} < V_{in}$), the negative correction freezes the integrator, clamping state accumulation instantly.

---

4. Driveline Torsional Resonance (8–15 Hz)

Automotive powertrains are not perfectly rigid. The half-shafts, constant-velocity (CV) joints, and rubber tire compliance behave like a torsional spring.

The heavy vehicle chassis and wheels represent a massive inertia ($J_{veh}$), while the motor rotor represents a smaller inertia ($J_{mot}$). Coupled via compliant driveshafts, they form a classic two-mass spring-damper torsional pendulum.

In modern passenger cars, the natural frequency of this driveline mode falls precisely in the $8\text{ Hz} - 15\text{ Hz}$ band!

Two-Mass Driveline Model:
  [ Motor Rotor: J_m ] <=====( Half-Shaft Spring: K_s, D_s )=====> [ Vehicle Mass: J_v ]
         |                                                                 |
    Fast Torque                                                    Slow Wheel Inertia
    Response (1 ms)                                                Resonance (10 Hz)

If the motor applies a step torque of $400\text{ Nm}$ in $1\text{ ms}$:

1. The half-shaft winds up like a torsion spring before the wheels break static friction.
2. The stored elastic energy uncoils violently, accelerating the wheels past equilibrium.
3. The vehicle oscillates forward and backward at $10\text{ Hz}$ (vehicle bucking / kangaroo effect), causing unacceptable cabin shudder and passenger motion sickness.

---

5. Active Anti-Jerk Damping Control

Artificially slowing down motor torque ramps degrades the crisp throttle response expected of an EV. Instead, automakers implement Active Anti-Jerk Damping.

The MCU computes the differential speed between motor shaft speed ($\omega_m$) and wheel speed ($\omega_w$) transformed across the final drive ratio:

$\Delta \omega = \omega_m - G_{gear} \cdot \omega_{wheel}$

During shaft twist, motor speed oscillates relative to wheel speed. The Anti-Jerk algorithm passes this speed difference through a band-pass or high-pass filter tuned to the $8-15\text{ Hz}$ resonance band, generating a counter-phase damping torque ($\Delta T_{damp}$):

$T_{motor}^* = T_{driver}^* - \underbrace{K_{damp} \cdot \frac{s}{s + \omega_f} \cdot \Delta \omega}_{\text{Active Damping Torque}}$

Anti-Jerk Damping Action:
  Torque
   ^       Raw Driver Torque Step (Triggers severe driveline shudder)
   |       ----------------------------------------
   |
   |       Anti-Jerk Modulated Torque (Counter-phase oscillation suppresses bounce)
   |          /\    /\
   |         /  \  /  \----------------------------
   |        /    \/
   +--------------------------------------------------> Time

By modulating motor air-gap torque in exact opposition to half-shaft springback, torsional bounce is extinguished within one or two cycles ($<150\text{ ms}$), restoring velvety smooth acceleration.

---

6. Brake Blending Coordination

When the driver applies the brake pedal, deceleration is shared across two domains:

1. Regenerative Braking (Electric Motor): Recovers kinetic energy into the battery pack (handling up to $0.3\text{ g}$ of normal deceleration).
2. Hydraulic Friction Brakes: Conventional caliper-and-disc friction braking.

The Brake System Controller (e.g. Bosch iBooster / ESP) evaluates battery charge acceptance limits in real time. If the battery is cold or at 100% SoC, regen is restricted; hydraulic calipers seamlessly clamp down to make up the deficit.

The cascaded torque controller coordinates this Brake Blending, ensuring seamless transitions without pedal modulation or deceleration steps perceptible to the driver.

---

7. Summary

• Cascaded control separates high-speed current dynamics ($10-20\text{ kHz}$) from outer vehicle supervisory loops ($10-100\text{ Hz}$).
• Back-calculation Anti-Windup prevents integrator blow-up during inverter voltage saturation.
• Instantaneous electric torque easily excites the $8-15\text{ Hz}$ mechanical torsional resonance of vehicle half-shafts.
• Active Anti-Jerk filters inject counter-phase damping torque derived from motor-wheel speed differentials, eliminating driveline bucking.
• The torque controller seamlessly orchestrates brake blending between regenerative braking and hydraulic calipers.

Next Lesson

• MOTOR-21 — Observers and Sensorless Control: Redundant limp-home control under resolver failure, Sliding Mode Observers (SMO), Extended Kalman Filters (EKF), and High-Frequency Injection (HFI).