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Robust Design of an Anticipative Feedforward-feedback Controller for MIMO Systems

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  • Control Theory and Applications
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Abstract

In this paper, multi-input multi-output preview systems, for which future values of the reference and disturbance signals are known a number of time-steps ahead, are considered. The paper focuses on designing control algorithms that improve the performance of the closed-loop system taking into account the preview information. A robust anticipative controller described through an \({{\cal L}_\infty }/{{\cal L}_2}\) optimisation problem is proposed. Its design focuses on the minimization of the tracking error in the presence of disturbances while at the same time limiting the control input amplitude. The proposed solution combines a robust feedback controller with a preview feedforward filter. The feedback controller’s purpose is to ensure the robustness of the closed-loop system to model uncertainties. The proposed solution is assessed by using the model of a MIMO system with uncertain parameters.

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Abbreviations

t :

Continuous-time.

t k :

Discrete-time.

T s :

Sampling period.

f s :

Sampling frequency.

s :

Laplace variable.

q −1 :

Unit delay operator used in the time domain

z −1 :

Unit delay operator used in the frequency domain.

q :

Unit advance operator used in the feedforward. filters to represent anticipative action.

x :

Scalar.

G(s):

Transfer function or polynomial.

G(s):

Transfer function matrix.

x :

Vector signal.

x T :

Transpose of the x vector.

G T :

Transpose of the G(s) transfer function matrix.

I :

Matrix Identity.

ℕ:

Set of natural numbers.

ℤ:

Set of integers.

ℝ:

Set of real numbers.

ℂ:

Set of complex numbers.

\({{\cal M}_G}\) :

Model-set.

j :

Complex operator such as j2 = −1

ω :

Frequency (rad/s)

v :

Pseudo-frequency such as w = jv and \(v = \tan \left( {{{\omega {T_s}} \over 2}} \right)\).

w :

Operational variable in the pseudo-continuous domain.

G(jω):

Frequency response of G(s).

ω l :

Low frequency.

ω h :

High frequency.

ℜe(.):

Real part of complex number.

ℐm(.):

Imaginary part of a complex number.

u :

Control input vector.

y :

System output vector.

r :

Reference input vector.

dy :

Disturbance vector on system output.

\({{\bf{u}}_{{\bf{f}}{{\bf{f}}_{\bf{F}}}}}\) :

Feedforward control vector for reference tracking.

\({{\bf{u}}_{{\bf{f}}{{\bf{f}}_{\bf{P}}}}}\) :

Feedforward control vector for disturbance rejection.

u ff :

Total feedforward control vector.

u fb :

Feedback control vector.

F :

Preview feedforward filter for reference tracking.

P :

Preview feedforward filter for disturbance rejection.

ε y = r − y :

Error.

E c :

Control energy.

\({\cal R}{\cal H}_\infty ^{m \times m}\) :

Set of proper real-rational transfer matrices analytic in ∣z∣ ≥ 1 with m-inputs and m-outputs.

H yu :

Matrix transfer function linking the MIMO plant input to its output.

Y/U = H yu :

Right division of vector values signals to indicate the transfer function matrix.

β :

Open loop transfer matrix.

κ :

Exponent to mean “inverse”.

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Author information

Authors and Affiliations

  1. CNRS, Grenoble INP (Institute of Engineering), GIPSA-Lab, University of Grenoble Alpes, 38000, Grenoble, France

    Asma Achnib

  2. Bordeaux INP, CNRS, IMS, UMR 5218, University of Bordeaux, 33405, Talence, France

    Tudor-Bogdan Airimitoaie & Patrick Lanusse

Authors
  1. Asma Achnib
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  2. Tudor-Bogdan Airimitoaie
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  3. Patrick Lanusse
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Corresponding author

Correspondence to Asma Achnib.

Additional information

Asma Achnib was born in Gabes, Tunisia, on June 5, 1991. She received her Ph.D. degree in automatic control in 2019 from the University of Bordeaux, France, and National Engineering School of Gabes (ENIG), Tunisia. Her Ph.D. research focused on the design of CRONE anticipative controllers for SISO and MIMO systems. Currently, she is with the Automatic Control Department, GIPSA-lab, Grenoble, where she is an ATER at Grenoble Institute of Engineering. Her research interests are preview control and robust control for uncertain systems, with applications to automotive systems.

Tudor-Bogdan Airimitoaie received an Automatic Control Engineering degree from the University Politehnica of Bucharest, Romania, in 2008 and a Ph.D. degree from the University of Grenoble, France, and the University Politehnica of Bucharest, Romania, in 2012. He is currently an Associate Professor with the University of Bordeaux, France and the Automatic Control Department of IMS-lab, Bordeaux. He is a coauthor of Adaptive and Robust Active Vibration Control (Springer, 2017) (with I. Landau, A. Castellanos, and A. Constantinescu). His main research interests are in the fields of adaptive and robust digital control and flat dynamic systems characterization; the main applications are oriented toward active noise and vibration suppression, control of hybrid unmanned aerial vehicles, and building energy management. Recently, he has been interested also in predictive maintenance with neural networks-based models.

Patrick Lanusse was born in Bordeaux, France, on January 12, 1966. Since 1990 he has been with the CRONE team of the IMS laboratory where he has been working on Robust Control, Fractional Order Controllers and more particularly on CRONE Control. He received his Ph.D. degree in 1994 on the use of a complex order fractional operator to design robust control-systems. Since 2019 he has been an Associate Professor, Habilitation of Control Theory at the Bordeaux Institute of Technology. His research concerns CRONE control to increase performance without decreasing robustness. He developed the control-system design CRONE Toolbox for Matlab which can be downloaded freely since 2010. He is the author or co-author of more than 25 journal papers, 5 international patents, about 100 international conference papers, he has collaborated in 15 books, and has also organized several special sessions on fractional systems in international conferences.

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Achnib, A., Airimitoaie, TB. & Lanusse, P. Robust Design of an Anticipative Feedforward-feedback Controller for MIMO Systems. Int. J. Control Autom. Syst. 20, 924–940 (2022). https://doi.org/10.1007/s12555-021-0034-z

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