A time-scaling technique for time-delay switched systems

Linna Li; Changjun Yu; Ning Zhang; Yanqin Bai; Zhiyuan Gao

doi:10.3934/dcdss.2020108

Article Contents

2020, Volume 13, Issue 6: 1825-1843. Doi: 10.3934/dcdss.2020108

This issue Previous Article An optimal pid tuning method for a single-link manipulator based on the control parametrization technique Next Article An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints

A time-scaling technique for time-delay switched systems

1.
Department of Mathematics, Shanghai University, Shanghai 200444, China
2.
Mechatronics Engineering and Automation, Shanghai University, Shanghai 200444, China

^* Corresponding author: Changjun Yu
^* Corresponding author: Changjun Yu

Received: April 30, 2018

Revised: July 31, 2018

Published: April 2020

This work is supported by National Natural Science Foundation of China (NSFC grant number 11871039, 11771275)

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Abstract

In this paper, we consider a class of optimal control problems governed by nonlinear time-delay switched systems, in which the system parameters and switching times between different subsystems are decision variables to be optimized. We propose a new computational approach to deal with the computational difficulties caused by variable switching times. The original time-delay switched system is firstly transformed into an equivalent switched system defined on a new time horizon where the switching times are fixed, but each of the subsystems contain a variable time-delay that depends on the durations of each sub-system in the original system. By deriving the analytical form for the variable time-delay in the new time horizon, we can solve the new time-delay switched system. Then, gradient-based optimization algorithm can be applied to solve the equivalent problem efficiently. Numerical results show that this new approach is effective.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. The relationship between $ (s, \mu(s|\mathit{\boldsymbol{\sigma}})) $ and $ (\omega(s|\mathit{\boldsymbol{\sigma}}), \mu(s|\mathit{\boldsymbol{\sigma}})-h) $. Note that $ \mu(s|\mathit{\boldsymbol{\sigma}}) $ is a piecewise linear function

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Figure 2. The relationship between $ (s, t) $ and $ (\omega(s), \mu(\omega(s))) $

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Figure 3. A demonstration for two scenarios of $ \sigma_{\lfloor s\rfloor +1} $

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Figure 4. Optimal state trajectory for Problem 1 for $ \sigma_i\in[0, 1) $

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Figure 5. Optimal state trajectory for Problem 2

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Figure 6. Optimal control $ \xi_1, \xi_2, \xi_3, \xi_4 $ for Problem 2

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Table 1. Optimal cost if the interval of $ \sigma_i $ is different

the interval of $ \sigma_i $	$ \tilde{{G}}^*_{0}(s, \mathit{\boldsymbol{\sigma}}) $	the optimal $ \mathit{\boldsymbol{\sigma}} $
$ \sigma_i\in[0, 1] $	$ 3.9855\times 10^{-3} $	$ \mathit{\boldsymbol{\sigma}}^\star=(0.000, 0.46836125, 0.53163875) $
$ \sigma_i\in[0.001, 1) $	$ 4.0432\times 10^{-3} $	$ \mathit{\boldsymbol{\sigma}}^\star=(0.001, 0.46779421, 0.53120579) $
$ \sigma_i\in[0.01, 1) $	$ 4.5411\times 10^{-3} $	$ \mathit{\boldsymbol{\sigma}}^\star=(0.010, 0.46257412, 0.52742588) $

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