Robust data-driven model-reference control of linear perturbed systems via sliding mode generation

TL;DR

The paper addresses robust data-driven control for linear, model-unknown plants subject to matched disturbances and aims to enforce a prescribed reference model in closed-loop. It introduces a data-driven integral sliding mode control (DD-ISMC) where the integral sliding variable depends only on the reference model, enabling exact reference-model enforcement without full plant modeling. Theoretical contributions include conditions for integral sliding mode generation, stability via invariant zeros, and residual disturbance characterization, together with a VRFT-based design path and data-driven tuning of the discontinuous component guided by Petersen’s lemma. Numerical and experimental validations on a triple-tank benchmark and a Quanser Aero helicopter demonstrate robust reference tracking despite disturbances and model uncertainty, highlighting the method’s practicality and potential for real-world deployments.

Abstract

This paper introduces a data-based integral sliding mode control scheme for robustification of model-reference controllers, accommodating generic multivariable linear systems with unknown dynamics and affected by matched disturbances. Specifically, an integral sliding mode control (ISMC) law is recast into a data-based framework relying on an integral sliding variable depending only on the reference model, without the need of modeling the plant. The main strength of the proposed approach is the enforcement of the desired reference model in closed-loop under sliding mode conditions, despite the lack of knowledge of the model dynamics and the presence of the matched disturbances. Moreover, the conditions required to guarantee an integral sliding mode generation and the closed-loop stability are formally analyzed in the paper, remarking the generality of the proposed data-driven integral sliding mode control (DD-ISMC) with respect to the related model-based counterpart. Finally, the main practices for the data-based design of the proposed control scheme are deeply discussed in the paper, and the proposed method is tested in simulation on a benchmark example, and experimentally on a real laboratory setup. Simulation and experimental evidence fully corroborates the theoretical analysis, thus motivating further research in this direction.

Figures (11)

• Figure 1: DD-ISMC scheme. The control signal $u(t)$ is given by the sum of the output of the ideal (data-driven) controller $\hat{\mathcal{R}}_0$ and the output of the sliding mode controller $\mathcal{R}_1$ (receiving as inputs the reference signal $r(t)$ and the controlled output $\hat{y}_\text{o}(t)$), aimed at compensating the mismatches with respect to the reference model $\mathcal{M}$, including also the matched disturbance $d$ affecting the plant $\mathcal{P}$.
• Figure 2: Schematic rendering of the triple tanks system.
• Figure 3: Open-loop input-output data collection. The PRBS input signals (plots on the left) $q_1$ (black line) and $q_2$ (blue line) shifted by the corresponding equilibrium values $\bar{q}_1$ and $\bar{q}_2$ are fed into the plant affected by the matched disturbances $d_1$ and $d_2$ (gray lines). The corresponding output levels (plot on the right) are $h_1$ (black line) and $h_2$ (blue line).
• Figure 4: Bode diagrams of the magnitude associated with the frequency response of the ideal regulator $R_0(s)$ (black line) and the VRFT based controller $\hat{R}_0(s)$ (gray line) with all their components $R_{0,ij}(s)$ and $\hat{R}_{0,ij}(s)$, $i,j=1,2$, respectively.
• Figure 5: Estimation of the bound of the residual disturbance $d_0$. The time evolution of the norm associated with the real residual disturbance (gray line), $\norm{d_0(t)}$, is compared with the one (black line) related to the estimation in \ref{['eq:d0_est']}, $\norm{\hat{d}_0(t)}$, so that the available bound is $\bar{d}_0$ in \ref{['eq:d0_approx']} (dashed black line), close to the real one (dashed gray line) which is not available in practice (see Remark \ref{['rem:d0']}).

Theorems & Definitions (25)

• remark 1: Reference model design and $R_0(s)$ existence
• Lemma 1
• proof
• Proposition 2
• proof
• Lemma 3
• proof
• Proposition 4
• proof
• remark 2: Equivalence with model-based ISMC