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

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# 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 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 ]'
``````

``````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)

### Cite

The paper is currently under review for ACC 2019. Citation instruction will be provided as soon as we receive an acceptance notification.

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

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