Efficient Solution of Second Order Cone Program for Model Predictive Control

Åkerblad, Magnus; Hansson, Anders

doi:10.3182/20020721-6-es-1901.00298

Cited by 8 publications

(12 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The two real valued control signals were constrained by a random upper and random lower bound, chosen such that the problems are feasible and such that the constraints are "reasonable active" along the prediction horizon. The result from the experiment clearly confirms the theoretical result in (10). It should be mentioned that the tightness is problem dependent, and the quality of the bounds may vary.…”

Section: Numerical Experimentssupporting

confidence: 82%

See 1 more Smart Citation

Relaxations applicable to mixed integer predictive control comparisons and efficient computations

Axehill

Hansson

Vandenberghe

2007

2007 46th IEEE Conference on Decision and Control

Self Cite

View full text Add to dashboard Cite

In this work, different relaxations applicable to an MPC problem with a mix of real valued and binary valued control signals are compared. In the problem description considered, there are linear inequality constraints on states and control signals. The relaxations are related theoretically and both the tightness of the bounds and the computational complexities are compared in numerical experiments. The relaxations considered are the Quadratic Programming (QP) relaxation, the standard Semidefinite Programming (SDP) relaxation and an equality constrained SDP relaxation. The result is that the standard SDP relaxation is the one that usually gives the best bound and is most computationally demanding, while the QP relaxation is the one that gives the worst bound and is least computationally demanding. The equality constrained relaxation presented in this paper often gives a better bound than the QP relaxation and is less computationally demanding compared to the standard SDP relaxation. Furthermore, it is also shown how the equality constrained SDP relaxation can be efficiently computed by solving the Newton system in an Interior Point algorithm using a Riccati recursion. This makes it possible to compute the equality constrained relaxation with approximately linear computational complexity in the prediction horizon.Keywords: Predictive control, Hybrid systems, Binary control, Integer programming, Semidefinite programming Relaxations Applicable to Mixed Integer Predictive Control -Comparisons and Efficient Computations Daniel Axehill, Anders Hansson and Lieven VandenbergheAbstract-In this work, different relaxations applicable to an MPC problem with a mix of real valued and binary valued control signals are compared. In the problem description considered, there are linear inequality constraints on states and control signals. The relaxations are related theoretically and both the tightness of the bounds and the computational complexities are compared in numerical experiments. The relaxations considered are the Quadratic Programming (QP) relaxation, the standard Semidefinite Programming (SDP) relaxation and an equality constrained SDP relaxation. The result is that the standard SDP relaxation is the one that usually gives the best bound and is most computationally demanding, while the QP relaxation is the one that gives the worst bound and is least computationally demanding. The equality constrained relaxation presented in this paper often gives a better bound than the QP relaxation and is less computationally demanding compared to the standard SDP relaxation. Furthermore, it is also shown how the equality constrained SDP relaxation can be efficiently computed by solving the Newton system in an Interior Point algorithm using a Riccati recursion. This makes it possible to compute the equality constrained relaxation with approximately linear computational complexity in the prediction horizon.

show abstract

Section: Numerical Experimentssupporting

confidence: 82%

“…A derivation of the Riccati recursions can be found in, e.g., [9], [10], and the resulting recursions can be found in Algorithm 1. Finally, by using (28), the linearized version of the equation in (13f) is found to be…”

Section: E(t) = Cδx(t) + Dδu(t) − R(t)mentioning

confidence: 99%

Relaxations applicable to mixed integer predictive control comparisons and efficient computations

Axehill

Hansson

Vandenberghe

2007

2007 46th IEEE Conference on Decision and Control

Self Cite

View full text Add to dashboard Cite

show abstract

“…It can be factored efficiently using a specialized method described below for block tridiagonal matrices (which is related to the Riccati recursion in control theory) [27]- [29], [55], [56], or by treating it as a banded matrix, with bandwidth . Both of these methods require order flops ( [3], [4], [29]- [32], [55], [56] flops. The cost of step 3 is dominated by the other steps, since the Cholesky factorization of was already computed in step 1.…”

Section: Fast Computation Of the Newton Stepmentioning

confidence: 99%

Fast Model Predictive Control Using Online Optimization

Wang

Boyd

2008

IFAC Proceedings Volumes

487

765

View full text Add to dashboard Cite

Abstract-A widely recognized shortcoming of model predictive control (MPC) is that it can usually only be used in applications with slow dynamics, where the sample time is measured in seconds or minutes. A well-known technique for implementing fast MPC is to compute the entire control law offline, in which case the online controller can be implemented as a lookup table. This method works well for systems with small state and input dimensions (say, no more than five), few constraints, and short time horizons. In this paper, we describe a collection of methods for improving the speed of MPC, using online optimization. These custom methods, which exploit the particular structure of the MPC problem, can compute the control action on the order of 100 times faster than a method that uses a generic optimizer. As an example, our method computes the control actions for a problem with 12 states, 3 controls, and horizon of 30 time steps (which entails solving a quadratic program with 450 variables and 1284 constraints) in around 5 ms, allowing MPC to be carried out at 200 Hz.Index Terms-Model predictive control (MPC), real-time convex optimization.

show abstract

“…A similar approach was taken in [6]. For work detailing efficient primal-dual interior-point methods to solve the quadratic programs (QPs) that arise in optimal control, see [7]- [9].…”

Section: ) Interior-point Methodsmentioning

confidence: 99%

A Splitting Method for Optimal Control

O’Donoghue

Stathopoulos

Boyd

2013

IEEE Trans. Contr. Syst. Technol.

160

136

View full text Add to dashboard Cite

Abstract-We apply an operator splitting technique to a generic linear-convex optimal control problem, which results in an algorithm that alternates between solving a quadratic control problem, for which there are efficient methods, and solving a set of single-period optimization problems, which can be done in parallel, and often have analytical solutions. In many cases, the resulting algorithm is division-free (after some off-line precomputations) and can be implemented in fixed-point arithmetic, for example on a field-programmable gate array (FPGA). We demonstrate the method on several examples from different application areas.Index Terms-Alternating directions method of multipliers (ADMM), convex optimization, embedded control, fixed point algorithms for control, model predictive control (MPC), operator splitting, optimal control.

show abstract

Efficient Solution of Second Order Cone Program for Model Predictive Control

Cited by 8 publications

References 22 publications

Relaxations applicable to mixed integer predictive control comparisons and efficient computations

Relaxations applicable to mixed integer predictive control comparisons and efficient computations

Fast Model Predictive Control Using Online Optimization

A Splitting Method for Optimal Control

Contact Info

Product

Resources

About