Let's break down the bouncing ball hybrid process shown in Fig. 2 and **formally specify it** as a hybrid automaton, listing all its components: --- # 1. **States (Modes)** - **Single mode:** `Fall` --- # 2. **Continuous Variables** - **Height** \(h\) - **Velocity** \(v\) --- # 3. **Inputs** - No external inputs in this model (the system is autonomous). --- # 4. **Outputs** - **Discrete:** `bump!` (event when the ball hits the ground) - **Continuous:** Height \(h\) (continuously updated) --- # 5. **Mode Invariants** - In `Fall` mode: \( h \geq \) (the ball cannot go below the ground) --- # 6. **Mode Dynamics (Flows)** - In `Fall`: \[ \dot{h} = v \] \[ \dot{v} = -g \] where \(g\) is gravitational acceleration. --- # 7. **Transitions** - **Guard:** \( h = \) (when the ball hits the ground) - **Reset:** - Issue the discrete event `bump!` - Update velocity: \( v \coloneqq -a v \), where \( < a < 1 \) (coefficient of restitution; reverses direction and reduces speed) --- # 8. **Initial Conditions** - At \( t = \): \[ h := h_, \quad v := v_ \] --- # 9. **Formal Hybrid Automaton Specification** | Component | Description | |---------------------|------------------------------------------------------------------------------------------------------| | **Modes** | `Fall` | | **Variables** | \( h \) (height), \( v \) (velocity) | | **Input** | None | | **Output** | Discrete: `bump!`, Continuous: \( h \) | | **Invariant** | \( h \geq \) | | **Flow** | \( \dot{h} = v \), \( \dot{v} = -g \) | | **Guard** | \( h = \) | | **Reset** | `bump!`; \( v := -a v \) | | **Init** | \( h := h_ \), \( v := v_ \) | --- ## **Step-by-Step Explanation** 1. **Initialization**: Set initial height \(h_\) and velocity \(v_\). 2. **Continuous Evolution**: While \(h > \), evolve according to \(\dot{h} = v\), \(\dot{v} = -g\). 3. **Guard Check**: When \(h = \), the guard triggers. 4. **Transition**: Issue the `bump!` event and reset velocity: \( v := -a v \). 5. **Repeat**: The process continues in the `Fall` mode with updated velocity. --- ## **Summary Table** | Step | Description | |--------------------------|--------------------------------------------------------------------| | Initial state | \( h = h_, v = v_ \) | | Mode (Fall) | \( \dot{h} = v, \dot{v} = -g, \ h \geq \) | | Guard (Transition) | \( h = \) | | Reset (on bump) | Issue `bump!`, set \( v := -a v \) | | Output | Discrete: `bump!`, Continuous: \( h \) | --- ## **Final Answer** The hybrid model consists of a single mode (`Fall`) with continuous variables \(h\) and \(v\), evolving according to gravity, constrained by \(h \geq \). When the ball hits the ground (\(h = \)), a transition is triggered: the `bump!` event is output, and velocity is updated to \(v := -a v\). The outputs of the process are the sequence of `bump!` events (discrete) and the continuously updated height \(h\) (continuous).