MEGR 3171 — Digital Control & System Integration

Learning Objectives

Apply Euler forward, Euler backward, and Tustin (bilinear) methods to discretize a continuous PID controller.
Select an appropriate sampling period and explain the sample-and-hold effect on closed-loop stability.
Configure a timer interrupt on a microcontroller for a fixed-rate control loop.
Design a Moore or Mealy finite state machine for a control mode switching application.
Compare I2C, SPI, UART, and CAN bus protocols on throughput, topology, and use case.
Describe the complete mechatronic system design process from requirements to documentation.

1. Discretization of Continuous Controllers

A continuous PID must be converted to a discrete-time difference equation for implementation on a microcontroller. The choice of method affects stability and frequency response of the digital implementation.

Discretization Methods (s to z transform)

Euler Forward:  s ≈ (z-1)/T_s        [simple but unstable for large T_s]

Euler Backward: s ≈ (z-1)/(T_s·z)   [always stable but distorts frequency]

Tustin (Bilinear): s ≈ (2/T_s)(z-1)/(z+1)  [best frequency preservation]

Discrete PID (Tustin, velocity form):
u[k] = u[k-1] + K_p(e[k]-e[k-1])
       + K_i(T_s/2)(e[k]+e[k-1])
       + K_d(2/T_s)(e[k]-2e[k-1]+e[k-2])  [with filter]

Sampling Period Selection Rule of thumb: T_s ≤ T_d/10 (sample at least 10× per derivative time constant). For bandwidth-limited systems: T_s ≤ 0.1/ω_c. Tustin introduces frequency warping; pre-warp critical frequencies if the Bode shape matters.

2. Interrupt-Driven Control Loops

The control loop must execute at a precise, fixed rate. Using delay() in Arduino is not acceptable for real control because it blocks all other code and the rate varies with computation time. Timer interrupts enforce precise execution.

Arduino Timer Interrupt (Conceptual)

// Configure Timer1 for 100 Hz interrupt (10 ms period)
void setup() {
  noInterrupts();
  TCCR1A = 0; TCCR1B = 0;
  OCR1A = 156;         // 16 MHz / 1024 / 100 Hz - 1
  TCCR1B |= (1<




  3. Finite State Machines
  A finite state machine (FSM) models a system with a discrete set of states, transitions triggered by events, and actions associated with states or transitions.
  
    Moore Machine
Output depends only on current state. Simpler to analyze. State transition diagram shows outputs labeled on states.
    Mealy Machine
Output depends on current state AND input. More compact (fewer states). Outputs labeled on transitions.
    Implementation
Use an enum for states, a switch-case for transitions. Update state at fixed intervals or on event. Document all transitions explicitly.
    Application
Control mode switching (idle → startup → run → fault), sequence control, communication protocols, user interface logic.
  



  4. Communication Protocols Comparison
  
    Protocol Topology Speed Wires Best Use
    
      UART Point-to-point Up to 1 Mbps 2 (TX, RX) Debugging, GPS, PC communication
      I2C Multi-drop bus 100 kHz / 400 kHz / 3.4 MHz 2 (SDA, SCL) Short-range sensors, displays (IMUs, EEPROMs)
      SPI Master-slave, full-duplex Up to 50 MHz 4 (MOSI, MISO, SCK, CS) High-speed DACs, ADCs, SD cards
      CAN bus Multi-node bus, differential Up to 1 Mbps 2 (CAN-H, CAN-L) Automotive, industrial, noisy environments
    
  



  5. Complete System Design Process
  
    Requirements Definition: Performance specs, environmental constraints, cost, safety, reliability targets
    System Architecture: Block diagram of complete signal chain from physical quantity to output
    Sensor and Actuator Selection: Range, accuracy, bandwidth, interface compatibility
    Signal Conditioning Design: Amplification, filtering, impedance matching
    Controller Design: Plant model, stability analysis, gain selection, anti-windup
    Embedded Implementation: Interrupt structure, state machine, communication
    Integration and Test: Subsystem testing, then system-level verification against requirements
    Documentation: Calibration records, uncertainty analysis, test data, user guide
  



  Practice Problems
  
    
      Problem 1 — Discrete PID
      A continuous PI controller has K_p = 3.0 and K_i = 0.5 s&sup-¹. Discretize the integral term using the Tustin method with T_s = 0.1 s. Write the discrete difference equation for the integral output.
    
    
    
      Tustin discretization of integrator 1/s → T_s(z+1)/(2(z-1)):
      Integral output: I[k] = I[k-1] + (K_i · T_s/2)(e[k] + e[k-1])
      = I[k-1] + (0.5 × 0.1/2)(e[k] + e[k-1]) = I[k-1] + 0.025(e[k] + e[k-1])
      Total: u[k] = K_p·e[k] + I[k] = 3.0·e[k] + I[k-1] + 0.025(e[k] + e[k-1])
      I[k] = I[k-1] + 0.025(e[k] + e[k-1]);  u[k] = 3.0·e[k] + I[k]
    
  
  
    
      Problem 2 — Protocol Selection
      You are designing an IMU data logger that must read a 6-axis IMU at 1 kHz and transmit data to a laptop for logging. Which protocol connects the IMU to the microcontroller, and which connects the microcontroller to the laptop? Justify.
    
    
    
      IMU to microcontroller: SPI is preferred at 1 kHz because it has higher speed than I2C and full-duplex operation. Most modern IMU ICs (e.g., ICM-42688) support SPI at speeds up to 24 MHz, easily handling 1 kHz reads of 12 bytes.
      Microcontroller to laptop: UART (USB-UART bridge). Standard for debug and logging. At 115200 baud this transmits ~11,000 bytes/s, which comfortably handles 12 bytes per read × 1000 Hz = 12,000 bytes/s — marginal. Use 921600 baud or pack multiple reads per transmission.
      IMU to MCU: SPI (high speed, low latency). MCU to laptop: UART at high baud rate (921600) or USB CDC.

Protocol	Topology	Speed	Wires	Best Use
UART	Point-to-point	Up to 1 Mbps	2 (TX, RX)	Debugging, GPS, PC communication
I2C	Multi-drop bus	100 kHz / 400 kHz / 3.4 MHz	2 (SDA, SCL)	Short-range sensors, displays (IMUs, EEPROMs)
SPI	Master-slave, full-duplex	Up to 50 MHz	4 (MOSI, MISO, SCK, CS)	High-speed DACs, ADCs, SD cards
CAN bus	Multi-node bus, differential	Up to 1 Mbps	2 (CAN-H, CAN-L)	Automotive, industrial, noisy environments