A set-membership state estimation algorithm based on DC programming

Álamo, T.; Bravo, José Manuel; Redondo, Manuel Jiménez; Camacho, Eduardo F.

doi:10.1016/j.automatica.2007.05.008

Cited by 79 publications

(51 citation statements)

References 23 publications

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“…Theorem 13.6 is the main result of [3] and shows how to bound the image of a zonotope under a nonlinear DC function. A comparison betweeen DC programming (cyan) and first-order Taylor zonotope extension (yellow, computed as in [11]) is provided in Fig.…”

Section: Lemma 131 ([3]) Let Z = P ⊕ Hb M Be a Zonotope And F (X) :mentioning

confidence: 99%

“…In [3] the authors propose a DC programming-based method for constructing a tight outer approximation of f (Z) (where f is a nonlinear C 2 function and Z a zonotope). First they linearize f and compute the image of Z under this linearization.…”

Section: Every Continuous Function Can Be Approximated By a Differencmentioning

confidence: 99%

“…We approximate the optimal control law by adaptive interpolation using second order interpolets, while concurrently verifying feasibility and stability of the resulting feedback system via the computation of an inner approximation of the capture basin (see, e.g., [12]). In contrast to the capture basin computational method considered in [12,31], we develop a mechanism for computing the capture basin using zonotopes [22,11,33] and DC programming [3] that significantly reduces the complexity of the combined function approximation and verification procedure. Using zonotopes and DC programming rather than interval analysis [25,6] additionally leads to an approximate control law with less storage requirements and a larger verifiable region of attraction.…”

Section: Introductionmentioning

confidence: 99%

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A Set-Theoretic Method for Verifying Feasibility of a Fast Explicit Nonlinear Model Predictive Controller

Raimondo

Riverso

Summers

et al. 2012

Distributed Decision Making and Control

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In this chapter an algorithm for nonlinear explicit model predictive control is presented. A low complexity receding horizon control law is obtained by approximating the optimal control law using multiscale basis function approximation. Simultaneously, feasibility and stability of the approximate control law is ensured through the computation of a capture basin (region of attraction) for the closed-loop system. In a previous work, interval methods were used to construct the capture basin (feasible region), yet this approach suffered due to slow computation times and high grid complexity.In this chapter, we suggest an alternative to interval analysis based on zonotopes. The suggested method significantly reduces the complexity of the combined function approximation and verification procedure through the use of DC (difference of convex) programming, and recursive splitting. The result is a multiscale function approximation method with improved computational efficiency for fast nonlinear explicit model predictive control with guaranteed stability and constraint satisfaction. IntroductionThis chapter proposes a method of approximate explicit model predictive control (MPC) for nonlinear systems. While it is possible to compute the optimal control law offline for a limited number of cases (e.g., affine or piecewise affine dynamics [5,20,29]), it is in general necessary to approximate, and therefore validation techniques are required for the resulting approximate closed-loop system. In this chapter, we present a new technique for approximation and certification of stability and recursive feasibility for explicit NMPC controllers, in which the control law is precomputed and verified offline in order to speed online computation. The control law is approximated via an adaptive interpolation using second order interpolets, which results in an extremely fast online computation time and low data storage. The resulting suboptimal closed-loop system is verified by computing an inner approximation of the capture basin and an algorithm is proposed that iteratively improves the approximation where needed in order to maximize the size of the capture basin. The key novelty of this chapter is the use of difference of convex (DC) programming and zonotope approximation in order to significantly improve both the computational performance and efficacy of the calculation of the capture basin.Methods for the approximation of explicit solutions of nonlinear model predictive control (NMPC) problems have been addressed recently by various authors (e.g., see [8,19]). In [8], the authors compute an approximate control lawũ(x) with a bound on the controller approximation error ( u * (x) −ũ(x) ), from which performance and stability properties are derived using set membership (SM) function approximation theory. In [19] the authors use multiparametric nonlinear programming to compute an explicit approximate solution of the NMPC problem defined on an orthogonal structure of the state-space partition. An additional example of the approximat...

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Section: Lemma 131 ([3]) Let Z = P ⊕ Hb M Be a Zonotope And F (X) :mentioning

confidence: 99%

Section: Every Continuous Function Can Be Approximated By a Differencmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Set-Theoretic Method for Verifying Feasibility of a Fast Explicit Nonlinear Model Predictive Controller

Raimondo

Riverso

Summers

et al. 2012

Distributed Decision Making and Control

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“…The zonotopic observer in combination with Kalman filtering is addressed in [6], [7]. Moreover, a set-membership approach based on zonotopes is proposed for dynamical systems in [1], [2].…”

Section: Introductionmentioning

confidence: 99%

Set-membership approach and Kalman observer based on zonotopes for discrete-time descriptor systems

2018

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This paper proposes a set-membership state estimator and a zonotopic Kalman observer for discrete-time descriptor systems. Both approaches are developed in a set-based context considering system disturbances, measurement noise, and unknown inputs. This set-membership state estimation approach determines the set of consistent states with the model and measurements by constructing a parameterized intersection zonotope. Two methods to minimize the size of this intersection zonotope are provided: one inspired by Kalman filtering and the other based on solving an optimization problem involving a series of linear matrix inequalities. Additionally, we propose a zonotopic Kalman observer for discrete-time descriptor systems. Moreover, the relationship between both approaches is discussed. In particular, it is proved that the zonotopic Kalman observer in the current estimation type is equivalent to the set-membership approach. Finally, a numerical example is used to illustrate and compare the effectiveness of the proposed approaches.

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“…In recent years, zonotopes (a particular class of polytopes) have received more attention because of their advantages of having better precision in comparison with ellipsoidal sets and less complexity compared to polyhedral sets. Hence, several setmembership estimation methods based on zonotopes [22][23][24][25] have been proposed.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Output Feedback Robust Model Predictive Control via Zonotopic Set-Membership Estimation for Constrained Quasi-LPV Systems

Ping

Sun

2015

Journal of Applied Mathematics

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For the quasi-linear parameter varying (quasi-LPV) system with bounded disturbance, a synthesis approach of dynamic output feedback robust model predictive control (OFRMPC) is investigated. The estimation error set is represented by a zonotope and refreshed by the zonotopic set-membership estimation method. By properly refreshing the estimation error set online, the bounds of true state at the next sampling time can be obtained. Furthermore, the feasibility of the main optimization problem at the next sampling time can be determined at the current time. A numerical example is given to illustrate the effectiveness of the approach.

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A set-membership state estimation algorithm based on DC programming

Cited by 79 publications

References 23 publications

A Set-Theoretic Method for Verifying Feasibility of a Fast Explicit Nonlinear Model Predictive Controller

A Set-Theoretic Method for Verifying Feasibility of a Fast Explicit Nonlinear Model Predictive Controller

Set-membership approach and Kalman observer based on zonotopes for discrete-time descriptor systems

Dynamic Output Feedback Robust Model Predictive Control via Zonotopic Set-Membership Estimation for Constrained Quasi-LPV Systems

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