2.10 Hybrid and Switched Systems

Dynamics plus logic¶

A hybrid system combines continuous state evolution with discrete modes. In mode $q$,

and guards trigger transitions $q^+ = \Delta(q,x,u,d,p)$. State resets may occur at impact, connection, or switching events.

Examples include contact and flight in legged robots, gear shifts in vehicles, below-rated and above-rated wind-turbine operation, valve logic in fluid systems, charging and discharging modes in batteries, and passive/semi-active/active suspension modes.

Why hybrid behavior matters for CCD¶

Physical design changes when transitions occur. Pendulum length affects impact speed; battery size affects when power limits activate; gearing affects shift timing; platform motion affects turbine shutdown events. If an optimizer simulates only one smooth mode, it may exploit a model that never represents the limiting event.

Switching also creates nonsmooth objectives and constraints. Small design changes can add or remove an event, making finite-difference derivatives unreliable. Event detection, consistent reset maps, smoothing, mixed-integer formulations, complementarity methods, or derivative-free searches may be required.

Avoid hidden simulation logic¶

Common hidden logic includes clipping, lookup-table discontinuities, conditional controller gains, solver event resets, and safety shutdowns. These operations must be documented because they shape the design landscape.

Chapter summary¶

Dynamic-system design begins with clear roles for states, inputs, outputs, disturbances, information, and physical variables. Continuous models must be reconciled with sampled implementation. Linear models reveal local structure, while nonlinear and hybrid models protect against invalid extrapolation. Feedback, feedforward, stability, controllability, and observability connect plant choices to achievable closed-loop behavior. Actuator and sensor limitations are not implementation details; they can change the system-optimal physical design.

Exercises¶

1. Derive continuous- and discrete-time state-space models for a base-excited mass-spring-damper. Retain $m$, $k$, and $c$ symbolically.

2. Linearize the adjustable inverted pendulum about upright. Explain how $m$ and $l$ affect instability and torque authority.

3. For a two-mode beam model, compare actuator locations using controllability rank and a Gramian metric.

4. Propose two sensor layouts for the beam and compare observability, noise, and implementation cost.

5. Show how a 10 ms delay changes phase at 5, 20, and 50 Hz. Discuss the resulting architecture implications.

6. Simulate a saturated PI-controlled mass-spring-damper with and without anti-windup.

7. Explain why a controller designed on a continuous model may fail after sampling and zero-order hold.

8. Formulate a hybrid model for a semi-active suspension whose damper switches between two settings.

9. Create a table that separates measured outputs, estimated states, performance outputs, and constrained outputs for the quarter car.

10. Identify one plant change that improves controllability but worsens observability, cost, or robustness.

Open research question¶

How can actuator placement, sensor placement, control architecture, and physical geometry be optimized together while preserving robust closed-loop guarantees for nonlinear and hybrid systems?

Sources and further reading¶

• G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems.

• K. Ogata, Modern Control Engineering.

• J. T. Allison and D. R. Herber, “Multidisciplinary Design Optimization of Dynamic Engineering Systems,” AIAA Journal, 2014.

• M. Garcia-Sanz, “Control Co-Design: An Engineering Game Changer,” Advanced Control for Applications, 2019.