A factorization with update procedures for a KKT matrix arising in direct optimal control

Kirches, Christian; Bock, Hans Georg; Schlöder, Johannes P.; Säger, Sebastian

doi:10.1007/s12532-011-0030-z

Cited by 28 publications

(36 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Due to the required factorizations (or re-factorizations, respectively) of the large matrices involved, a typical runtime between O(n 2 ) and O(n 3 ) results [37]. The second-order derivatives were approximated by an iterative update using first-order derivative information.…”

Section: Numerical Solution Of the Optimal-control Problemmentioning

confidence: 99%

Numerical Optimal Control of Turbo Dynamic Ventricular Assist Devices

et al. 2013

View full text Add to dashboard Cite

Abstract:The current paper presents a methodology for the derivation of optimal operating strategies for turbo dynamic ventricular assist devices (tVADs). In current clinical practice, tVADs are typically operated at a constant rotational speed, resulting in a blood flow with a low pulsatility. Recent research in the field has aimed at optimizing the interaction between the tVAD and the cardiovascular system by using predefined periodic speed profiles. In the current paper, we avoid the limitation of using predefined profiles by formulating an optimal-control problem based on a mathematical model of the cardiovascular system and the tVAD. The optimal-control problem is solved numerically, leading to cycle-synchronized speed profiles, which are optimal with respect to an arbitrary objective. Here, an adjustable trade-off between the maximization of the flow through the aortic valve and the minimization of the left-ventricular stroke work is chosen. The optimal solutions perform better than constant-speed or sinusoidal-speed profiles for all cases studied. The analysis of optimized solutions provides insight into the optimized Bioengineering 2014, 1 23 interaction between the tVAD and the cardiovascular system. The numerical approach to the optimization of this interaction represents a powerful tool with applications in research related to tVAD control. Furthermore, patient-specific, optimized VAD actuation strategies can potentially be derived from this approach.

show abstract

Section: Numerical Solution Of the Optimal-control Problemmentioning

confidence: 99%

Numerical Optimal Control of Turbo Dynamic Ventricular Assist Devices

et al. 2013

View full text Add to dashboard Cite

show abstract

Section: Related Workmentioning

confidence: 99%

Structure-Exploiting Numerical Algorithms for Optimal Control

Nielsen¹

View full text Add to dashboard Cite

Cover illustration: An illustration of the structure of the Karush-Kuhn-Tucker matrix for the unconstrained finite-time optimal control problems considered in this thesis. Thank you Jonas Linder for helping me with the design.Linköping studies in science and technology. Dissertations.No. 1848 Tillägnas hela min underbara familj Structure-Exploiting Numerical Algorithms for Optimal Control AbstractNumerical algorithms for efficiently solving optimal control problems are important for commonly used advanced control strategies, such as model predictive control (mpc), but can also be useful for advanced estimation techniques, such as moving horizon estimation (mhe). In mpc, the control input is computed by solving a constrained finite-time optimal control (cftoc) problem on-line, and in mhe the estimated states are obtained by solving an optimization problem that often can be formulated as a cftoc problem. Common types of optimization methods for solving cftoc problems are interior-point (ip) methods, sequential quadratic programming (sqp) methods and active-set (as) methods. In these types of methods, the main computational effort is often the computation of the second-order search directions. This boils down to solving a sequence of systems of equations that correspond to unconstrained finite-time optimal control (uftoc) problems. Hence, high-performing second-order methods for cftoc problems rely on efficient numerical algorithms for solving uftoc problems. Developing such algorithms is one of the main focuses in this thesis. When the solution to a cftoc problem is computed using an as type method, the aforementioned system of equations is only changed by a low-rank modification between two as iterations. In this thesis, it is shown how to exploit these structured modifications while still exploiting structure in the uftoc problem using the Riccati recursion. Furthermore, direct (non-iterative) parallel algorithms for computing the search directions in ip, sqp and as methods are proposed in the thesis. These algorithms exploit, and retain, the sparse structure of the uftoc problem such that no dense system of equations needs to be solved serially as in many other algorithms. The proposed algorithms can be applied recursively to obtain logarithmic computational complexity growth in the prediction horizon length. For the case with linear mpc problems, an alternative approach to solving the cftoc problem on-line is to use multiparametric quadratic programming (mp-qp), where the corresponding cftoc problem can be solved explicitly off-line. This is referred to as explicit mpc. One of the main limitations with mp-qp is the amount of memory that is required to store the parametric solution. In this thesis, an algorithm for decreasing the required amount of memory is proposed. The aim is to make mp-qp and explicit mpc more useful in practical applications, such as embedded systems with limited memory resources. The proposed algorithm exploits the structure from the qp problem in the parametric solution in order to reduc...

show abstract

“…The computational complexity of this algorithm grows quadratically in the number of active inequality constraints. An alternative sparse non-Riccati factorization is used in Kirches et al (2011), where the factorization is updated after changes in the as iterations. In Nielsen et al (2013), an algorithm for performing low-rank modifications of the Riccati factorization by exploiting the structured changes in an as iteration is presented.…”

Section: Low-rank Modifications Of Riccati Factorizationsmentioning

confidence: 99%

On Structure Exploiting Numerical Algorithms for Model Predictive Control

Nielsen

2015

View full text Add to dashboard Cite

One of the most common advanced control strategies used in industry today is Model Predictive Control (mpc), and some reasons for its success are that it can handle multivariable systems and constraints on states and control inputs in a structured way. At each time-step in the mpc control loop the control input is computed by solving a constrained finite-time optimal control (cftoc) problem on-line. There exist several optimization methods to solve the cftoc problem, where two common types are interior-point (ip) methods and active-set (as) methods. In both these types of methods, the main computational effort is known to be the computation of the search directions, which boils down to solving a sequence of Newton-system-like equations. These systems of equations correspond to unconstrained finite-time optimal control (uftoc) problems. Hence, high-performance ip and as methods for cftoc problems rely on efficient algorithms for solving the uftoc problems.The solution to a uftoc problem is computed by solving the corresponding Karush-Kuhn-Tucker (kkt) system, which is often done using generic sparsity exploiting algorithms or Riccati recursions. When an as method is used to compute the solution to the cftoc problem, the system of equations that is solved to obtain the solution to a uftoc problem is only changed by a low-rank modification of the system of equations in the previous iteration. This structured change is often exploited in as methods to improve performance in terms of computation time. Traditionally, this has not been possible to exploit when Riccati recursions are used to solve the uftoc problems, but in this thesis, an algorithm for performing low-rank modifications of the Riccati recursion is presented.In recent years, parallel hardware has become more commonly available, and the use of parallel algorithms for solving the cftoc problem and the underlying uftoc problem has increased. Some existing parallel algorithms for computing the solution to this type of problems obtain the solution iteratively, and these methods may require many iterations to converge. Some other parallel algorithms compute the solution directly (non-iteratively) by solving parts of the system of equations in parallel, followed by a serial solution of a dense system of equations without the sparse structure of the mpc problem. In this thesis, two parallel algorithms that compute the solution directly (non-iteratively) in parallel are presented. These algorithms can be used in both ip and as methods, and they exploit the sparse structure of the mpc problem such that no dense system of equations needs to be solved serially. Furthermore, one of the proposed parallel algorithms exploits the special structure of the mpc problem even in the parallel computations, which improves performance in terms of computation time even more. By using these algorithms, it is possible to obtain logarithmic complexity growth in the prediction horizon length.v Populärvetenskaplig sammanfattningModellprediktiv reglering (eng. Model Predictive...

show abstract

A factorization with update procedures for a KKT matrix arising in direct optimal control

Cited by 28 publications

References 35 publications

Numerical Optimal Control of Turbo Dynamic Ventricular Assist Devices

Numerical Optimal Control of Turbo Dynamic Ventricular Assist Devices

Structure-Exploiting Numerical Algorithms for Optimal Control

On Structure Exploiting Numerical Algorithms for Model Predictive Control

Contact Info

Product

Resources

About