Article Contents
Article Contents

# A time-scaling technique for time-delay switched systems

• * Corresponding author: Changjun Yu

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

• 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.

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

 Citation:

• 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

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

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

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

Figure 5.  Optimal state trajectory for Problem 2

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

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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