BangBangHysteresis.md

Bang-Bang / Hysteresis (Relay) Controller

Overview & Motivation

The bang-bang controller is the simplest closed-loop regulator: its output switches between two discrete levels depending on whether the controlled variable is above or below a threshold. Without hysteresis, a plain threshold comparator chatters — switching at high frequency whenever noise nudges the signal across the boundary.

Adding a dead-band (Schmitt trigger) eliminates chatter by giving the relay memory: once the output goes High it stays High until the measurement falls all the way to a lower threshold, and vice versa. This makes the relay a practical actuator-safe control primitive for thermostats, fridge compressors, tank level switches, and power-stage on/off regulation.

Mathematical Theory

State Transition

The relay holds a binary state $s \in {\text{Low}, \text{High}}$ with the following transition rules:

$$ s[k] = \begin{cases} \text{High} & \text{if } s[k-1] = \text{Low} \text{ and } x[k] \geq \theta_H \\ \text{Low} & \text{if } s[k-1] = \text{High} \text{ and } x[k] \leq \theta_L \\ s[k-1] & \text{otherwise} \end{cases} $$

where $\theta_L < \theta_H$ are the lower and upper switching thresholds (the hysteresis band).

Output Map

$$ u[k] = \begin{cases} u_H & \text{if } s[k] = \text{High} \\ u_L & \text{if } s[k] = \text{Low} \end{cases} $$

The output levels $u_L$ and $u_H$ are arbitrary; common choices are ${0, 1}$ or ${-1, +1}$.

Hysteresis Band Width

The band width $\Delta = \theta_H - \theta_L$ is the key design parameter. It bounds the switching frequency $f_s$ given a signal slope $\dot{x}$:

$$ f_s \leq \frac{|\dot{x}|}{2\Delta} $$

A wider band reduces $f_s$ (protecting relays and power stages) at the cost of a larger steady-state limit cycle amplitude.

Complexity Analysis

Case	Time	Space	Notes
All	$O(1)$	$O(1)$	Two comparisons, one state bit, one select

No arithmetic on the signal path — only comparisons — so the relay introduces no numerical error and is exactly representable in any floating-point format.

Step-by-Step Walkthrough

Setup: band $[\theta_L, \theta_H] = [-0.2, 0.2]$, outputs $u_L = 0$, $u_H = 1$, initial state Low.

Step	$x[k]$	Condition	$s[k]$	$u[k]$
1	0.0	Low, $x < 0.2$	Low	0
2	0.3	Low, $x \geq 0.2$ → switch	High	1
3	0.1	High, $x > -0.2$ → hold	High	1
4	−0.3	High, $x \leq -0.2$ → switch	Low	0
5	0.0	Low, $x < 0.2$ → hold	Low	0

Pitfalls & Edge Cases

• Inverted band. $\theta_H \leq \theta_L$ latches the output in an undefined state; reject this at construction time via a precondition assertion.
• Exactly on threshold. Transitions are inclusive: $x = \theta_H$ triggers Low→High and $x = \theta_L$ triggers High→Low. This avoids a dead-zone at the switching points.
• Zero-width band. $\theta_H = \theta_L$ collapses the relay to a pure comparator (no hysteresis). The logic is still correct but offers no chatter suppression.
• Noise sizing. The band width must exceed the peak-to-peak noise amplitude; otherwise noise alone drives state transitions at the sampling rate.
• Actuator minimum on-time. Size $\Delta$ so that the minimum on-time (derived from $\Delta / |\dot{x}|_\text{max}$) is above the actuator's rated minimum switching period.

Variants & Generalizations

Variant	Key Difference
Plain comparator	$\Delta = 0$; no memory, chatters on noise
Asymmetric band	$\theta_H$ and $\theta_L$ not symmetric around the set-point; biases the duty cycle
Three-state relay	Adds a dead-band output level $u_0$; used in motor direction control
Adaptive hysteresis	Band width tracks signal variance online to maintain a target switching rate
Relay feedback test (Åström–Hägglund)	Deliberate oscillation under relay feedback to identify the ultimate gain/period for automatic PID tuning

Applications

• Thermostats and HVAC — heating/cooling switched on/off around a temperature set-point.
• Tank and vessel level control — pump on/off between high- and low-level floats.
• Power-stage converters — hysteretic current-mode control in DC-DC converters and class-D amplifiers.
• Motor drive enable/disable — protecting power stages with a current-band relay.
• Åström–Hägglund auto-tuning — the relay feedback experiment that drives limit-cycle oscillation for PID parameter identification.

Connections to Other Algorithms

Algorithm	Relationship
PID Controller	The relay's limit cycle can be used to identify PID tuning parameters via the Åström–Hägglund relay-feedback test
Saturation / Rate Limiter	Continuous-output counterpart for actuator constraint; often combined with a relay in cascaded loops

References & Further Reading

• K. J. Åström, R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers, Princeton University Press, 2008 — relay feedback, Chapter 10.
• Ya. Z. Tsypkin, Relay Control Systems, Cambridge University Press, 1984.
• K. J. Åström, T. Hägglund, "Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins," Automatica, 20(5), 1984.