CDS 110b: Receding Horizon Control

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L7-2: Sensitivity

L8-1: Robust Stability

L9-1: Robust Perf

Schedule

This lecture presents an overview of receding horizon control (RHC). In addition to providing a summary of the available theoretical results, we introduce the concept of differential flatness for simplifying RHC problems and provide an example of RHC control on the Caltech ducted fan. .

Lecture Outline

1. Receding Horizon Control
   • Problem Formulation
   • Stability theorems
2. Differential Flatness and Trajectory Generation
   • Definitions
   • Properties
   • Examples
3. Examples: Caltech ducted fan, satellite formation flight, multi-vehicle testbed

Lecture Materials

• Lecture Presentation (no MP3)
• Lecture notes: Online Control Customization via Optimization-Based Control, R. Murray et al, 2003.
• Homework 3

Frequently Asked Questions

Q: How is differential flatness defined?

A system of the form

${\dot {x}}=f(x,u)$

is said to be differentially flat if there exists an integer $p$ and a (smooth) function $h$ of the form

$z=h(x,u,{\dot {u}},{\ddot {u}},\dots ,u^{(p)})$

such that all solutions of the differential equation can be written in terms of $z$ and a finite number of its derivatives with respect to time. In other words, $x$ and $u$ satisfying the dynamics of the system have the form

${\begin{matrix}x&=&\alpha (z,{\dot {z}},{\ddot {z}},\dots ,z^{(q)})\\u&=&\beta (z,{\dot {z}},{\ddot {z}},\dots ,z^{(q)})\end{matrix}}$

for some integer $q$ and smooth functions $\alpha$ and $\beta$.

Checking a system for flatness is difficult, but there are certain classes of systems for which there are necessary and sufficient conditions. Usually you find the flat outputs by a combination of physical insight and trial and error.

References:

• Flat systems, equivalence and trajectory generation, Phillipe Martin, Richard Murray, Pierre Rouchon, CDS Technical Report, 2003.