Optimal Control of Hybrid Systems with Sliding Modes

Pytlak, R.; Suski, Damian; Tarnawski, Tomasz

doi:10.1007/978-3-319-96598-7_23

Cited by 4 publications

(9 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It can be seen after 120 min of haemodialysis. From that time the value of does not change and the system exhibits a ’sliding mode’ during which it evolves – the details of the system behaviour in this mode are stated in [21].…”

Section: Resultsmentioning

confidence: 99%

“…The numerical procedure which we used to solve the problem (9), (1)–(8),(10)–(15) is described, to much extent, in [20] and [21]. The main features of the procedure are: it is based on the Radau IIa version of a Runge–Kutta method for integrating differential equations,it uses adjoint equations to evaluate gradients of functions defining the optimization problem.…”

Section: Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Application of dynamic optimisation for planning a haemodialysis process

et al. 2019

View full text Add to dashboard Cite

Background The aim of the study is to show that optimization tools can be used in planning the haemodialysis process in order to obtain the most effective treatment aimed at removing both urea and phosphorus. To this end we use the IV–compartment model of phosphorus kinetics. Methods The use of the IV–compartment model of phosphorus kinetics forces us to apply new numerical tools which cope with a rebound phenomenon that can occur during haemodialysis. The proposed algorithm solves optimization problems with various constraints imposed on concentrations of urea and phosphorus. Results We show that the optimization tools are effective in planning haemodialysis processes aimed at achieving desired levels of urea and phosphorus concentrations at the end of these processes. One of the numerical experiments reported in the paper concerns patients data who experienced a rebound phenomenon during haemodialysis due to a low level of phosphorus concentration. Conclusion In order to plan haemodialysis processes one should take into account the fact that these processes, in general, are described by different equations in different regions determined by phosphorus concentrations. This follows from the fact that mechanisms modelled by IV–compartment model are activated during dialysis. Therefore, advanced numerical tools have to be used in order to simulate and optimize these processes. The paper shows that these tools can be constructed and effectively applied in planning haemodialysis processes.

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Application of dynamic optimisation for planning a haemodialysis process

et al. 2019

View full text Add to dashboard Cite

show abstract

“…The computational approach to hybrid systems we advocate in this paper (and in [16], [14], [15]) assumes that for a given control function we attempt to follow true systems trajectories as close as possible by using integration procedures with a high order of convergence. It is in contrast to the approach which uses approximations to the discontinuous right-hand side for the differential equations or inclusions by a smooth right-hand side ( [24], [22], [9]).…”

mentioning

confidence: 99%

“…In the first part of the paper we give an estimate for the order of convergence of discrete adjoint equations assuming that a hybrid system is described by ODEs. In the second part of the paper ( [15]) we carry out analogical analysis for the case when system equations correspond to index 2 DAEs. The second part of the paper complements the result of the first part by paying attention to the jumps of adjoint variables which occur when the hybrid system changes its discrete state.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Numerical procedure for optimal control of hybrid systems with sliding modes, Part I

Pytlak,

Suski

2021

Preprint

Self Cite

View full text Add to dashboard Cite

This paper concerns the numerical procedure for solving hybrid optimal control problems with sliding modes. The proposed procedure has several features which distinguishes it from the other procedures for the problem. First of all a sliding mode is coped with differentialalgebraic equations (DAEs) and that guarantees accurate tracking of the sliding motion surface. The second important feature is the calculation of cost and constraints functions gradients with the help of adjoint equations. The adjoint equations presented in the paper take into account sliding motion and exhibit jump conditions at transition instants. The procedure uses the discretization of system equations by Radau IIA Runge-Kutta scheme and the evaluation of optimization functions gradients with the help of the adjoint equations stated for discretized system equations. In the first part of the paper we demonstrate the correspondence between the discrete adjoint equations and the discretized version of the continuous adjoint equations in the case of system equations described by ODEs. We show that the discrete adjoint state trajectories converge to their continuous counterparts in the case of ODEs.

show abstract