Data-Based System Analysis and Control of Flat Nonlinear Systems

Alsalti, Mohammad; Berberich, Julian; Lopez, Victor G.; Allgöwer, Frank; Müller, Matthias A.

doi:10.48550/arxiv.2103.02892

Cited by 8 publications

(17 citation statements)

References 24 publications

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“…In Part (ii).c, we exploit that the optimal solution of Problem ( 16) depends in a piecewise affine fashion on σ and thus, the difference between the optimal input for σ = 0 (i.e., ǔ * (t)) and σ = σε (i.e., ũ(t)) is bounded in terms of ε, which then implies that ū * (t) and ǔ * (t) are close, cf. (18). In Proposition 2 (see Appendix C), we show that a result analogous to Proposition 1 holds in the presence of additional input disturbances affecting the initial conditions.…”

Section: B Proposed Mpc Schemementioning

confidence: 55%

“…Thus, Problem ( 12) is strongly convex in ū[0,L−n−1] (t) u s (t) and hence, using (18) trivially holds over the first n time steps and therefore, we focus on k ∈ I [0,L] in the following.…”

Section: B Proposed Mpc Schemementioning

confidence: 99%

“…Although different successful applications to complex nonlinear systems have been reported in the literature, see, e.g., [11], [12], providing theoretical guarantees of data-driven MPC for nonlinear systems remains a widely open research problem. The literature contains various extensions and variations of [2] for specific classes of nonlinear systems such as Hammerstein and Wiener systems [13], Volterra systems [14], polynomial systems [15], [16], systems with rational dynamics [17], flat systems [18], and linear parameter-varying systems [19]. However, all of these works assume that the system is linearly parametrized in known basis functions, which restricts their practical applicability.…”

Section: Introductionmentioning

confidence: 99%

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Linear tracking MPC for nonlinear systems Part II: The data-driven case

Berberich,

Köhler,

Müller

et al. 2021

Preprint

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We present a novel data-driven MPC approach to control unknown nonlinear systems using only measured inputoutput data with closed-loop stability guarantees. Our scheme relies on the data-driven system parametrization provided by the Fundamental Lemma of Willems et al. We use new inputoutput measurements online to update the data, exploiting local linear approximations of the underlying system. We prove that our MPC scheme, which only requires solving strictly convex quadratic programs online, ensures that the closed loop (practically) converges to the (unknown) optimal reachable equilibrium that tracks a desired output reference. As intermediate results of independent interest, we extend the Fundamental Lemma to affine systems and we propose a data-driven tracking MPC scheme with guaranteed robustness. The theoretical analysis of this MPC scheme relies on novel robustness bounds w.r.t. noisy data for the open-loop optimal control problem, which are directly transferable to other data-driven MPC schemes in the literature. The applicability of our approach is illustrated with a numerical application to a continuous stirred tank reactor.

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Section: B Proposed Mpc Schemementioning

confidence: 55%

Section: B Proposed Mpc Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Linear tracking MPC for nonlinear systems Part II: The data-driven case

Berberich,

Köhler,

Müller

et al. 2021

Preprint

Self Cite

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“…This means that both the state x k and input u k at time step k can be determined from a finite number of future values of the output y k [18]. The key concept of discrete-time flatness is that the trajectory of the (flat) output in a certain finite window uniquely determines the state and input at any time step [19].…”

Section: B Discrete-time Flatnessmentioning

confidence: 99%

Fly Out The Window: Exploiting Discrete-Time Flatness for Fast Vision-Based Multirotor Flight

Greeff¹,

Zhou²,

Schoellig³

2021

Preprint

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Current control design for fast vision-based flight tends to rely on high-rate, high-dimensional and perfect state estimation. This is challenging in real-world environments due to imperfect sensing and state estimation drift and noise. In this letter, we present an alternative control design that bypasses the need for a state estimate by exploiting discrete-time flatness. To the best of our knowledge, this is the first work to demonstrate that discrete-time flatness holds for the Euler discretization of multirotor dynamics. This allows us to design a controller using only a window of input and output information. We highlight in simulation how exploiting this property in control design can provide robustness to noisy output measurements (where estimating higher-order derivatives and the full state can be challenging). Fast vision-based navigation requires high performance flight despite possibly noisy high-rate real-time position estimation. In outdoor experiments, we show the application of discrete-time flatness to vision-based flight at speeds up to 10 m/s and how it can outperform controllers that hinge on accurate state estimation.

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“…The success of the Fundamental Lemma is evident from the number of data-based applications for LTI systems that followed from it, such as extensions for databased simulation and control [3], data-driven state-feedback control [4], [5], data-based dissipativity analysis [6], [7] and data-driven predictive control [8]. There exists preliminary work that aims to extend the Fundamental Lemma towards nonlinear (NL) [9] and Linear Time-Varying (LTV) [10] systems. However, these results impose heavy restrictions on the systems as they leverage model transformations and linearisations.…”

Section: Introductionmentioning

confidence: 99%

Fundamental Lemma for Data-Driven Analysis of Linear Parameter-Varying Systems

Verhoek,

Tóth,

Haesaert

et al. 2021

Preprint

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In data-driven analysis and control, the so-called Fundamental Lemma by Willems et al. has gained a lot of interest in recent years. Using behavioural system theory, the Fundamental Lemma shows that the full system behaviour of a Linear Time-Invariant (LTI) system can be characterised by a single sequence of data of the system, as long as the input is persistently exciting. In this work, we aim to generalize this LTI result to Linear Parameter-Varying (LPV) systems. Based on the behavioural framework for LPV systems, we prove that one can obtain a result similar to Willems'. This implies that the result is also applicable to nonlinear system behaviour that can be captured with an LPV representation. We show the applicability of our result by connecting it to earlier works on data-driven analysis and control for LPV systems.

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Data-Based System Analysis and Control of Flat Nonlinear Systems

Cited by 8 publications

References 24 publications

Linear tracking MPC for nonlinear systems Part II: The data-driven case

Linear tracking MPC for nonlinear systems Part II: The data-driven case

Fly Out The Window: Exploiting Discrete-Time Flatness for Fast Vision-Based Multirotor Flight

Fundamental Lemma for Data-Driven Analysis of Linear Parameter-Varying Systems

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