A hybrid adaptive inverse for uncertain SISO linear plants with full relative degree

Matteo Cocetti, Matteo Ragni, Sophie Tarbouriech, Luca Zaccarian

Abstract

We propose a hybrid adaptive feed-forward regulator for single input single output linear plants with a full relative degree. The scheme includes an adaptive law that estimates the inverse of the plant and provides a feed-forward control calculated on the basis of the desired output and its derivatives.

The adaptation is performed during discrete time events, called jumps, while the feed-forward action is continuous. This combination leads to a full hybrid system.

The advantage of this framework is a conceptual separation between the adaptation dynamics, which is discrete, and the plant dynamics, which is continuous.

Under an assumption of a persistence of excitation, we show through examples that the output asymptotically tracks the desired reference and that the estimate of the parameters of the inverse converges.

Example Repository

The repository contains the source code and the simulink model for the example presented in the article.

Requirements:

Matlab/Simulink (R2018a)
Matlab Control Systems Toolbox
Matlab Symbolic Toolbox

Usage:

Open the file example.slx and run the simulation. Data are prepared through initSim, called automatically at simulation start.

Edit initSim in order to test a different plant / initial conditions / filters.

Proposed Example

Plant:

x' = A x + B u
y  = C x

[ A | B ]      [ 0       1 | 0 ]
[ --+-- ] =  = [ -w²  -2zw | 1 ]
[ C |   ]      [ ----------+-- ]
               [ k       0 |   ]

w = 3
k = 3
z = 0.2

x0 = [ 3, 4]'

Filters:

eig(Λ) = (-50 -75 -100)
Φ      = [ 0 0 1 ]'

Adaptation:

γ    = 0.5
τmin = 0.25
τmax = 3.00

θ0 = [ 1, 0.2, 0.1 ]'

Reference:

r(t) := 5 cos(t/5) + cos(t + ?/3) + sin(2t/5)

(derivative are computed symbolically using the Symbolic Toolbox)

Results

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