Reachable set bounding for linear systems with mixed delays and state constraints
Introduction
Reachable set is a set bounding all possible states of system under the condition of zero initial state. There are lots of results [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] about reachable set for different systems have been studied. The S-procedure method was used by Boyd to obtain an ellipsoid estimation of the reachable set for linear systems in Boyd et al. [11], Fridman and Shaked [12]. Kwon studied reachable set for time-varying delay systems with convex polytopic uncertainties in Kwon et al. [13]. However, the range of time delays derived by method used in this paper expanded the estimation condition can withstand. Therefore, there is still a lot of room to improve the existing results.
It is noted that the above research all focuses on discrete time-delay systems, while distributed time-delay also exists in systems. The estimation for linear systems with distributed delays are studied based on the delay-partitioning technique in Zhang et al. [14]. So far, there are few studies on reachable set of mixed time-delay systems. It is more difficult to deal with mixed time-delay systems because of their complex representation. In [15], Zuo employed the appropriate lyapunov-krasovskii functions investigated reachable set for mixed time-delay systems, which have less computational burden. To find an ellipsoid to bound the reachable sets, improved delay-dependent linear matrix inequalities criteria are derived by using free-weighting matrix approach, reciprocally convex combination lemma and convex analysis technique in Zhao and Hu [16].
On the other hand, state constraints inevitably exist and closely relate to the dynamic performance of the systems in practical engineering. Exceeding the state constraints of the systems is likely to result in the instability of the systems and even cause serious damage to the systems [17], [18]. In general, the state constraints of the systems are intricately related, and the constraint range of one state variable is likely to change with the change of another state variable. Therefore, the problem of state limitation of control systems has become a valuable research field in recent years, which drives a large number of scholars to think and explore this problem and obtain fruitful research results [19], [20], [21], [22], [23], [24], [25].
However, to the best of our knowledge, there are no results on the problem of reachable set estimation for linear systems with distributed delays subject to state constraints, which mainly motivates this paper. The contribution of this paper are as follows: (1) The existing articles on reachable sets estimation do not take state constraints into account, hence we consider mixed delay systems with state constraints in this paper. (2) Instead of using the commonly used reciprocally convex combination method [27], we used a less conservative method in this paper, which reduced the related decision variables. (3) Parameter-dependent L-K functionals are constructed for the condition of uncertain differentiable parameters.
Section snippets
Problem statement
Consider system (1) with mixed delays and state constraints:where is the state vector; , , , are system matrices; are real matrices and represents the state constraints on the system; is a time-varying delay and is a distributed delay; external disturbance satisfieswhere is a constant.
We denote the reachable set of system (1) by
Reachable set estimation for mixed delays systems with state constraints
For , , the reachable set of system (1) is considered in this part, where is the upper bounds of the time-varying delay and is its derivative. Theorem 1 Given scalars , , , if there exist positive definite symmetric matrix , semi-positive definite symmetric matrix and compatible matrix such thatwhere
Reachable set estimation of uncertain mixed delays system
System (1) with polytopic uncertainties is expressed as:where
We consider first. Theorem 2 Given scalars , , , existing positive definite symmetric matrix , semi-positive definite symmetric matrix and compatible matrix such that
Illustrative examples
We present three examples to demonstrate the values of our method in this paper. Example 1 Consider system (1) with the same parameters as Zuo et al. [15], Zhao and Hu [16] The state constraint on the system is and external disturbance . We select parameter , , , with different values of to solve the optimization problem (14) and the results are listed in Table 1. Tables 2 and 3 also list the computed of
Conclusion
The reachable set estimation for linear systems with mixed delays and state constraints are studied in this paper. A new criterion about reachable set is established based on maximal Lyapunov–Krasovskii functional. Then, our results extend to the polytopic uncertainties systems. Three examples are given to demonstrate the effectiveness of our method. In the future, we will study the reachable set problem of a system with stochastic items.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 61873271 and Grant 61873272; in part by the Fundamental Research Funds for the Central Universities under Grant 2018XKQYMS15; and in part by the Double-First-Rate Special Fund for Construction of China University of Mining and Technology under Grant 2018ZZCX14; and Assistance Program for Future Outstanding Talents of China University of Mining and Technology 2020WLJCRCZL075.
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2023, Applied Mathematics and ComputationCitation Excerpt :These methods can be classified as Lyapunov function-based methods and grid-based methods [1,2,12,15–17]. The former often uses ellipsoid [18–22], polyhedron [23,24], Taylor models [25], and semi-algebraic sets [26] to approximate the reachable tube, and can solve problems with high dimensional state space, but are usually limited to linear systems. The latter requires meshing the state space and can be used for nonlinear systems.
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