CDS 110b: Receding Horizon Control

CDS 110b

Schedule

Project

Course Text

This set of lectures builds on the previous three weeks and explores the use of online optimization as a tool for control of nonlinear control. We being with an high-level discussion of optimization-based control, refining some of the concepts initially introduced in Week 1. We then describe the technique of receding horizon control (RHC), including a proof of stability for a particular form of receding horizon control that makes use of a control Lyapunov function as a terminal cost. A detailed implementation example, the Caltech ducted fan, is used to explore some of the computational tradeoffs in optimization-based control.

Prerequisites: Students should be familiar with the concepts of trajectory generation and optimal control as described in Weeks 1-3. For the proof of stability for the receding horizon controller that we use, familiarity with Lyapunov stability analysis at the level given in AM08, Chapter 4 (Dynamic Behavior) is required.

• Lecture slides on RHC overview (Mon)
• Lecture notes on RHC analysis (Wed) - the notation in these notes is slightly different than the text. In lecture, I will use the textbook notation.
• HW #4 (due 6 Feb 08): problems 3.1, 3.3 and 3.2. Students working on the course project should do problems 3.1 and 3.3 only.

References and Further Reading

• R. M. Murray, Optimization-Based Control. Preprint, 2008: Chapter 3 - Receding Horizon Control

• Online Control Customization via Optimization-Based Control, R. M. Murray et al. In Software-Enabled Control: Information Technology for Dynamical Systems, T. Samad and G. Balas (eds.), IEEE Press, 2001. This paper talks about the CLF-based nonlinear RHC approach and its application on the Caltech ducted fan using NTG.

• Constrained model predictive control: Stability and optimality, D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scokaert. Automatica, 2000, Vol. 36, No. 6, pp. 789-814. This is one of the most referenced comprehensive survey papers on MPC. Gives a nice overview about its history and explains the most important issues and various approaches.

Frequently Asked Questions

Q: How do you do trajectory optimization using differential flatness

The basic idea in using flatness for optimal trajectory generation is to rewrite the cost function and constraints in terms of the flat outputs and then parameterize the flat outputs in terms of a set of basis functions:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z(t) = \sum_i \alpha_i \psi_i(t) }

Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_i} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 1, \dots, N} are the basis functions (eg, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_i(t) = t^i} ) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_i} are constant coefficients.

Once you have parameterized the flat outputs by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} , you can convert all expressions involving Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} into functions involving Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} . This process is described in a more detail in the lectures notes (Section 4).

Q: Is the condition given by Jadbabaei and Hauser and example of a CLF or the definition of a CLF?

I was a bit sloppy defining CLFs in lecture. The formal definition is given in the lectures notes (Section 2.2, Defn 1). Briefly, given a system

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot x = f(x,u), }

we say that a (smooth) function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x)} is a control Lyapunov function (CLF) if

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x) > 0} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \neq 0}
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x) = 0} if and only if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 0}
• The derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} along trajectories of the system satisfies

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \min_u \dot V(x) |_{\dot x = f(x,u)} = \min_u \frac{\partial V}{\partial x} f(x, u) < 0 }

for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}

The condition for stability given in lecture is that there exists a CLF for the system that in addition satisfies the relationship

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \min_u (\dot V(x) + L(x, u)) < 0 }

along the trajectories of the system. Thus we have to have the derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} be sufficiently negative definite in order to insure that the terminal cost Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x)} provides stability.

Q: Why do receeding horizon trajectories need to go to zero (on slide 4)?

It is common in control problems to assume that the desired trajectory goes to zero as its desired end state. This is implicitly the case whenever you see an integral cost of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^T Q x} or a terminal cost Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^T(T) P_1 x(T)} , both of which are minimized when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is zero. There are two ways to think about this:

• If we wish to move to a different (equilibrium) point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_d} , we can always change the state to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{new} = x - x_d} and then the new state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{new}} has zero as the desired equilibrium point.
• If we want to track a trajectory Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_d(t)} (not constant), then we can solve the problem for the error system given by substrating the desired state.

This is explained in more dtail in the lecture notes on LQR control (Section 3).