Multiparametric Linear Programming with Applications to Control

Jones, Colin N.; Barić, Miroslav; Morari, Manfred

doi:10.3166/ejc.13.152-170

Cited by 53 publications

(51 citation statements)

References 48 publications

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“…Remark IV.3. As was shown in [12] there exists an analytical relation between the approximation error ǫ and the complexity N P of a PWA offline approximation. Requirements on the approximation error can therefore also be imposed using the parameter N P .…”

Section: Lemma Iv2 (Convergence Of ν(· ·))mentioning

confidence: 97%

“…e.g. [15], [6], [12], [16]. All of the cited explicit approximation methods provide a feasible control law and allow verification of closed-loop stability by means of Theorem III.11 for some minimally required complexity.…”

Section: Proposed Control Lawmentioning

confidence: 99%

“…We take x =x fixed and show by contradiction that equation (12) computes the distance to the optimal cost at x. Assume z * (x) is the optimal solution to (5) …”

Section: Analysis Of the Proposed Control Lawmentioning

confidence: 99%

See 2 more Smart Citations

Real-time suboptimal model predictive control using a combination of explicit MPC and online optimization

Zeilinger

Jones

Morari

2008

2008 47th IEEE Conference on Decision and Control

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Abstract-Limits on the storage space or the computation time restrict the applicability of model predictive controllers (MPC) in many real problems. Currently available methods either compute the optimal controller online or derive an explicit control law. In this paper we introduce a new approach combining the two paradigms of explicit and online MPC to overcome their individual limitations. The algorithm computes a piecewise affine approximation of the optimal solution that is used to warm-start an active set linear programming procedure. A preprocessing method is introduced that provides hard real-time execution, stability and performance guarantees for the proposed controller. By choosing a combination of the quality of the approximation and the number of online active set iterations the presented procedure offers a tradeoff between the warm-start and online computational effort. We show how the problem of identifying the optimal combination for a given set of requirements on online computation time, storage and performance can be solved. Finally, we demonstrate the potential of the proposed warm-start procedure on numerical examples.

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Section: Lemma Iv2 (Convergence Of ν(· ·))mentioning

confidence: 97%

Section: Proposed Control Lawmentioning

confidence: 99%

See 1 more Smart Citation

Real-time suboptimal model predictive control using a combination of explicit MPC and online optimization

Zeilinger

Jones

Morari

2008

2008 47th IEEE Conference on Decision and Control

View full text Add to dashboard Cite

show abstract

“…Based on this, the critical region can be formulated based on Equation (D.11). However, this might lead to lower-dimensional regions due to underlying degeneracies (see [144,168] for excellent treatments on degeneracy in multi-parametric programming). In order to identify these cases, we solve the following 3 Within our numerical studies, we successfully utilized M = 10 5 .…”

Section: Appendix Amentioning

confidence: 99%

Model-based multi-parametric programming strategies towards the integration of design, control and operational optimization

Diangelakis

Pistikopoulos

2017

Computer Aided Chemical Engineering

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“…Equation (5.1) is, in fact, a bilinear program which can be reformulated using its dual to a multiparamteric linear optimization problem [30,33]. However, the direct resolution of a multiparametric program is very expensive since it requires to find exponentially many critical regions, and for each region we have to find our optimal value which will be an affine function depending on the parameter vector c. Therefore, rather than solve the optimization problem (5.1), we simply seek values of c such that…”

Section: Encoding Positivity Of Parametric Polynomialmentioning

confidence: 99%

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

Sassi

Sankaranarayanan

Chen

et al. 2015

IMA J Math Control Info

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We examine linear programming (LP) based relaxations for synthesizing polynomial Lyapunov functions to prove the stability of polynomial ODEs. Our approach starts from a desired parametric polynomial form of the polynomial Lyapunov function. Subsequently, we encode the positive-definiteness of the function, and the negation of its derivative, over the domain of interest. We first compare two classes of relaxations for encoding polynomial positivity: relaxations by sum-of-squares (SOS) programs, against relaxations based on Handelman representations and Bernstein polynomials, that produce linear programs. Next, we present a series of increasingly powerful LP relaxations based on expressing the given polynomial in its Bernstein form, as a linear combination of Bernstein polynomials. Subsequently, we show how these LP relaxations can be used to search for Lyapunov functions for polynomial ODEs by formulating LP instances. We compare our techniques with approaches based on SOS on a suite of automatically synthesized benchmarks. Posi-

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Multiparametric Linear Programming with Applications to Control

Cited by 53 publications

References 48 publications

Real-time suboptimal model predictive control using a combination of explicit MPC and online optimization

Real-time suboptimal model predictive control using a combination of explicit MPC and online optimization

Model-based multi-parametric programming strategies towards the integration of design, control and operational optimization

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

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