A Duality Approach for Solving Control-Constrained Linear-Quadratic Optimal Control Problems

Burachik, Regina S.; Kaya, C. Yalçın; Majeed, S. N.

doi:10.1137/130910221

Cited by 33 publications

(53 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This approach is often referred to as the direct method or the (first-)discretize-then-optimize approach. A survey and discussion of Euler discretization of linear-quadratic optimal control problems and convergence of their discretized solutions to their continuous-time solutions can be found in [10,Section 5]. Figure 2 depicts the discrete solution of Problem (P) with the instance where a = 2.5, s 0 = 0, s f = 0, v 0 = 1, v f = 0.…”

Section: Minimum-energy Control Of Double Integratormentioning

confidence: 99%

Constraint Splitting and Projection Methods for Optimal Control of Double Integrator

Bauschke

Burachik

Kaya

2019

Splitting Algorithms, Modern Operator Theory, and Applications

View full text Add to dashboard Cite

We consider the minimum-energy control of a car, which is modelled as a point mass sliding on the ground in a fixed direction, and so it can be mathematically described as the double integrator. The control variable, representing the acceleration or the deceleration, is constrained by simple bounds from above and below. Despite the simplicity of the problem, it is not possible to find an analytical solution to it because of the constrained control variable. To find a numerical solution to this problem we apply three different projection-type methods: (i) Dykstra's algorithm, (ii) the Douglas-Rachford (DR) method and (iii) the Aragón Artacho-Campoy (AAC) algorithm. To the knowledge of the authors, these kinds of (projection) methods have not previously been applied to continuous-time optimal control problems, which are infinite-dimensional optimization problems. The problem we study in this article is posed in infinite-dimensional Hilbert spaces. Behaviour of the DR and AAC algorithms are explored via numerical experiments with respect to their parameters. An error analysis is also carried out numerically for a particular instance of the problem for each of the algorithms.

show abstract

Section: Minimum-energy Control Of Double Integratormentioning

confidence: 99%

Constraint Splitting and Projection Methods for Optimal Control of Double Integrator

Bauschke

Burachik

Kaya

2019

Splitting Algorithms, Modern Operator Theory, and Applications

View full text Add to dashboard Cite

show abstract

“…The construction of the dual problem is based on the unperturbed case in [1] and [6]. The dual problem can be formulated as…”

Section: Formulation Of Dual Problemmentioning

confidence: 99%

“…A tight lower bound, however, has been more difficult to obtain due to a lack of a duality framework in which to formulate the optimal control problem and, moreover, a lack of a strong duality property that ensures that any tight lower bound to a dual problem will also be an tight lower bound to the primal problem. In order to obtain this lower bound, we adapt the duality construction in [1], [2] and [6] to the case of clustered consensus networks and derive a dual problem with the strong duality property. We then use the strong duality result along with the optimal state of the reduced problem to construct an asymptotic expansion of the optimal control to the dual problem.…”

Section: Introductionmentioning

confidence: 99%

Error bounds on the solution to an optimal control problem over clustered consensus networks

Howe¹,

Parpas²

2017

2017 Proceedings of the Conference on Control and Its Applications

View full text Add to dashboard Cite

In this paper, we obtain a bound on the error term of the solution to a control constrained, linear-quadratic optimal control problem over a clustered consensus network. For large scale systems, the solution may be regarded as computationally infeasible due to the increase in dimensionality. However, the two time-scale property of the network that arises from the cluster formation indicates that the optimal control problem can be written in standard singularly perturbed form. Thus, we are able to obtain a reduced dimension optimal control problem over a reduced dimension network where individual nodes within a cluster are collapsed into an aggregate node. The solution to the reduced problem can be shown to be asymptotically equivalent in the singular perturbation parameter δ to the solution of the original problem. However, for many values of δ this asymptotic result may fail to be of practical use. We improve on this result by applying a duality theory to the clustered network and derive an upper bound χu(δ) and lower bound χ l (δ) on the solution to the optimal control problem that holds for arbitrary δ and, furthermore, satisfies the inequality |χu(δ) − χ l (δ)| = O(δ) as δ → 0.

show abstract

“…Accordingly, we define the distance of two spaces and as (5) With Equations (4) and (5), the criterion is suggested that if the condition…”

Section: The Criterion To Judge a Couplingmentioning

confidence: 99%

“…In engineering, a practical optimization problem often involves various disciplines. For instance, the design of aircraft involves disciplines such as aerodynamics, structure analysis, propulsion, and control [1][2][3][4][5] , and hence the issue of multidisciplinary design optimization (MDO) was put forward. In the early design stage, the optimization is individually carried out for each discipline.…”

Section: Introductionmentioning

confidence: 99%

An improved collaborative optimization for multidisciplinary problems with coupled design variables

Jia²,

Wen

et al. 2016

Advances in Engineering Software

View full text Add to dashboard Cite

In this paper, the collaborative optimization is improved to tackle multidisciplinary problems with coupling design variables within disciplines. The design variables are first distinctly classified into private and public ones according to disciplines. A criterion is then proposed to judge the coupling of private variables with the public variables. For the coupling case, a strategy is suggested to decouple the variables before performing the collaborative optimization.To demonstrate the effectiveness of the improved collaborative optimization, two practical optimization problems are comparatively studied for uncoupled design variables as well as coupled ones. It is shown that the present framework can achieve the better optimal results as compared with other methods. Furthermore, the judge criterion and the decoupling strategy are also validated.

show abstract

A Duality Approach for Solving Control-Constrained Linear-Quadratic Optimal Control Problems

Cited by 33 publications

References 30 publications

Constraint Splitting and Projection Methods for Optimal Control of Double Integrator

Constraint Splitting and Projection Methods for Optimal Control of Double Integrator

Error bounds on the solution to an optimal control problem over clustered consensus networks

An improved collaborative optimization for multidisciplinary problems with coupled design variables

Contact Info

Product

Resources

About