An algorithm for multi-parametric quadratic programming and explicit MPC solutions

Tøndel, P.; Johansen, Tor Arne; Bemporad, Alberto

doi:10.1016/s0005-1098(02)00250-9

Cited by 499 publications

(179 citation statements)

References 14 publications

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“…DMAPS, diffusion maps; MPC, model predictive control rescaled from 0 to 1, and scaled back to their original ranges for presentation. Equation (11) was built using pyTorch and trained using the Adam optimizer (a similar algorithm to stochastic gradient descent) with learning rate of 1 × 10 −3.40,41 Alternatively, we predict ϕ from x * α using GP regression. Using the Matérn covariance function in Equation (5), we optimize hyperparameters for prediction over our training data set by minimizing negative log marginal likelihood 38 to obtainφ GP .…”

Section: Constrained Mpc For Linear Systemsmentioning

confidence: 99%

“…For constrained linear systems, an explicit solution can be found by solving a single multiparametric quadratic programming problem off-line. [10][11][12][13] The "parameters" are the system states and the solution to this optimization problem is piecewise affine as long as the constraints are linear; with this solution in hand, the controller only needs to first follow a look-up table to determine the relevant polytopic region of state space in which the system currently lies and then perform an affine computation. An advantage of the multiparametric programming approach to finding an explicit controller is that the resulting control law is identical to the implicit MPC controller, and thus assured to preserve all closed-loop properties including stability and constraint satisfaction when it is solved exactly.…”

Section: Introductionmentioning

confidence: 99%

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Some manifold learning considerations toward explicit model predictive control

et al. 2020

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Model predictive control (MPC) is a de facto standard control algorithm across the process industries. There remain, however, applications where MPC is impractical because an optimization problem is solved at each time step. We present a link between explicit MPC formulations and manifold learning to enable facilitated prediction of the MPC policy. Our method uses a similarity measure informed by control policies and system state variables, to “learn” an intrinsic parametrization of the MPC controller using a diffusion maps algorithm, which will also discover a low‐dimensional control law when it exists as a smooth, nonlinear combination of the state variables. We use function approximation algorithms to project points from state space to the intrinsic space, and from the intrinsic space to policy space. The approach is illustrated first by “learning” the intrinsic variables for MPC control of constrained linear systems, and then by designing controllers for an unstable nonlinear reactor.

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Section: Constrained Mpc For Linear Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Some manifold learning considerations toward explicit model predictive control

et al. 2020

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show abstract

“…The main limitation of MPC is its requirement of high computational power which needs to be available on board the controlled vehicle. However, in the recent work, it was shown that, for constrained linear systems, most of the computation can be moved offline using, eg, multiparametric quadratic programming to solve the MPC problem, [26][27][28][29][30][31] thus reducing the online computation to an evaluation of an analytical function, which yields the optimal control law. Such techniques have been implemented, eg, for the altitude control of a microsatellite.…”

Section: Related Workmentioning

confidence: 99%

Adaptive‐optimal control under time‐varying stochastic uncertainty using past learning

Abdollahi

Chowdhary²

2019

Adaptive Control & Signal

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An adaptive-optimal control architecture is presented for adaptive control of constrained aerospace systems with matched uncertainties that are subject to dynamic stochastic change. The architecture brings together three key elements, ie, model predictive control-based reference command shaping, Gaussian process (GP)-based Bayesian nonparametric model reference adaptive control (MRAC), and online GP clustering over nonstationary GPs. Model predictive control optimizes reference model and its shaped output is passed into GP-based MRAC, which is used to learn the model in presence of significant time-varying stochastic uncertainty while maintaining stability. Based on a likelihood ratio test, the changepoints are detected and learned. Lastly, the models are created and clustered by non-Bayesian clustering algorithm. The key salient feature of our architecture is that not only can it detect changes but also it uses online GP clustering to enable the controller to utilize past learning of similar models to significantly reduce learning transients. Furthermore, persistence of excitation conditions are significantly relaxed due to the use of GP-MRAC. Stability of the architecture is argued theoretically and performance is validated empirically on different scenarios for wing rock dynamics.

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“…is a function of the line load q(x), the QP has to be solved at every time step. By introducing multi-parametric quadratic programming (MP-QP) [4], the computational burden of reoccurring optimization is reduced. MP-QP distinguishes between active and inactive constraints.…”

Section: Application Of Multi-parametric Quadratic Programmingmentioning

confidence: 99%

Application of Multi‐Parametric Quadratic Programming to Simulation of Deflected Wires in a Wire Saw

Treyer

Gaulocher

Niederberger

et al. 2017

Proc Appl Math and Mech

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In the wire sawing process, a silicon brick is fed into a moving wire web, thus deflecting the wires. By considering static deflections only, the wire displacement and the contact forces between wire and brick may be computed by minimizing the potential energy of the wire, which introduces a constrained quadratic optimization problem. In a time-domain simulation, the continuously changing contour inside the kerf requires the optimization problem to be solved recurringly. This work aims at reducing the optimization-related computational effort by applying multi-parametric quadratic programming.

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An algorithm for multi-parametric quadratic programming and explicit MPC solutions

Cited by 499 publications

References 14 publications

Some manifold learning considerations toward explicit model predictive control

Some manifold learning considerations toward explicit model predictive control

Adaptive‐optimal control under time‐varying stochastic uncertainty using past learning

Application of Multi‐Parametric Quadratic Programming to Simulation of Deflected Wires in a Wire Saw

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