Design of interval observers and controls for PDEs using finite-element approximations

Kharkovskaya, Tatiana; Efimov, Denis; Polyakov, Andrey; Richard, Jean‐Pierre

doi:10.1016/j.automatica.2018.03.016

Cited by 21 publications

(19 citation statements)

References 34 publications

(63 reference statements)

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“…hold. During the observer parameterization, which ensures ISS due to the inclusion of (20) in the design LMIs, the parameter µ ∞ is minimized. Cooperativity of the error dynamics is obtained by the element-wise defined inequality constraint…”

Section: Between the Open-loop Statementioning

confidence: 99%

“…If simplified sparse solutions for H according to (11) shall be optimized by a minimization of µ ∞ ≥ 0 according to the constraint (21) with the help of the H ∞ design described above, the LMIs M (Σ) ≺ 0 in (19) with (20)…”

Section: Between the Open-loop Statementioning

confidence: 99%

“…Although the summary above was mainly focused on finite-dimensional system models, also sets of PDEs such as the parabolic differential equation of heat conduction were investigated. There, two fundamentally different approaches could be distinguished, namely, techniques relying on a replacement of the infinite-dimensional dynamics by a finite-dimensional system model before the observer design (early lumping) and the design of observers and its stability proof on the basis of the PDE model (late lumping) [18,20].…”

Section: Introductionmentioning

confidence: 99%

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From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers

Rauh

Kersten

2020

Acta Cybern

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One of the most important advantages of interval observers and the associated trajectory computation is their capability to provide estimates for a given dynamic system model in terms of guaranteed state bounds which are compatible with measured data subject to bounded uncertainty. However, the inevitable requirement for being able to produce such verified bounds is the knowledge about a dynamic system model in which possible uncertainties and inaccuracies are themselves represented by guaranteed bounds. For that reason, classical point-valued parameter identification schemes are often not sufficient or should, at least, be handled with sufficient care if safety critical applications are of interest. This paper provides an application-oriented description of the major steps leading from a control-oriented system model with an associated interval-valued parameter and disturbance identification to a verified design of interval observers which provide the basis for the development and implementation of cooperativity-preserving feedback controllers. Such combined control and observer structures allow for forecasting guaranteed lower and upper state bounds that can be determined by solving initial value problems for crisp-parameter models. As such, they replace the significantly more demanding task of computing tubes of reachable states by means of general-purpose interval methods. The corresponding computational steps for the cooperativity-preserving control and observer synthesis are described and visualized for the temperature control of a laboratory-scale test rig available at the Chair of Mechatronics at the University of Rostock.

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Section: Between the Open-loop Statementioning

confidence: 99%

Section: Between the Open-loop Statementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers

Rauh

Kersten

2020

Acta Cybern

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“…The idea of interval observer construction has been proposed rather recently in [26], but it has already received numerous extensions for various classes of dynamical models. Interval observers for systems described by PDEs have been proposed in [27,28,29,31]. The finite-dimensional approximation approach was applied in [29] using the discretization error estimates from [30], and in [31] for temperature estimation in fuel cells.…”

Section: Introductionmentioning

confidence: 99%

Interval observer design and control of uncertain non-homogeneous heat equations

et al. 2020

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The problems of state estimation and observer-based control for heat non-homogeneous equations under distributed in space point measurements are considered. First, an interval observer is designed in the form of Partial Differential Equations (PDEs), without Galerkin projection. Second, conditions of boundedness of the interval observer solutions with non-zero boundary conditions and measurement noise are proposed. Third, the obtained interval estimates are used to design a dynamic outputfeedback stabilizing controller. The advantages of the PDE-based interval observer over the approximation-based one are clearly shown in the numerical example.

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“…Further, approaches for the control of systems with input saturations that do not rely on interval analysis were handled, for example, in [26] and the references therein. In addition to the implementation of robust controllers, also the dual task of interval based state estimation and observer design can be solved; see, for example, [27][28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Using Interval Analysis to Compute the Invariant Set of a Nonlinear Closed-Loop Control System

2019

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In recent years, many applications, as well as theoretical properties of interval analysis have been investigated. Without any claim for completeness, such applications and methodologies range from enclosing the effect of round-off errors in highly accurate numerical computations over simulating guaranteed enclosures of all reachable states of a dynamic system model with bounded uncertainty in parameters and initial conditions, to the solution of global optimization tasks. By exploiting the fundamental enclosure properties of interval analysis, this paper aims at computing invariant sets of nonlinear closed-loop control systems. For that purpose, Lyapunov-like functions and interval analysis are combined in a novel manner. To demonstrate the proposed techniques for enclosing invariant sets, the systems examined in this paper are controlled via sliding mode techniques with subsequently enclosing the invariant sets by an interval based set inversion technique. The applied methods for the control synthesis make use of a suitably chosen Gröbner basis, which is employed to solve Bézout’s identity. Illustrating simulation results conclude this paper to visualize the novel combination of sliding mode control with an interval based computation of invariant sets.

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Design of interval observers and controls for PDEs using finite-element approximations

Cited by 21 publications

References 34 publications

From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers

From Verified Parameter Identification to the Design of Interval Observers and Cooperativity-Preserving Controllers

Interval observer design and control of uncertain non-homogeneous heat equations

Using Interval Analysis to Compute the Invariant Set of a Nonlinear Closed-Loop Control System

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