Reachable set bounding for linear systems with mixed delays and state constraints

doi:10.1016/j.amc.2022.127085

Applied Mathematics and Computation

Volume 425, 15 July 2022, 127085

https://doi.org/10.1016/j.amc.2022.127085 Get rights and content

Highlights

•
This paper considers the state constraint problem in reachable set estimation.
•
Instead of using the commonly used reciprocally convex combination method, this paper uses a less conservative method, which reduced the related decision variables.
•
Parameter-dependent L-K functional are constructed for the condition of uncertain differentiable parameters.

Abstract

This paper studies the reachable set estimation for systems with mixed delays and state constraints. Firstly, by constructing an appropriate maximal Lyapunov–Krasovskii functional, combined with the wirtinger-based integral inequality and the extended reciprocally convex combination method, we derive a new criterion about reachable set estimation which reduces the related decision variables without increasing the computational burden. Then, the results are extended to polytopic uncertainties systems and we consider the condition of uncertain differentiable parameters. Three examples are presented to demonstrate the validity of our theorems.

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: $\begin{matrix} \dot{x} (t) = A x (t) + A_{b} x (t - τ (t)) + A_{h} \int_{t - h}^{t} x (s) d s + B w (t), \\ x (t) = 0, \forall t \in [- max {h, \bar{τ}}, 0], \\ | F_{j} x (t) | \leq 1, j = 1, 2, \dots, M, \end{matrix}$ where $x (t) \in R^{n}$ is the state vector; $A$ , $A_{b}$ , $A_{h}$ , $B$ are system matrices; $F_{j}$ are real matrices and $| F_{j} x (t) | \leq 1$ represents the state constraints on the system; $τ (t)$ is a time-varying delay and $h$ is a distributed delay; external disturbance $w (t)$ satisfies $w^{T} (t) w (t) \leq w_{m},$ where $w_{m}$ is a constant.

We denote the reachable set of system (1) by $R_{x} ≜ {x (t) \in R^{n} | x (t), ω (t)$

Reachable set estimation for mixed delays systems with state constraints

For $0 \leq τ (t) \leq \bar{τ}$ , $| \dot{τ} (t) | \leq μ \leq 1$ , the reachable set of system (1) is considered in this part, where $\bar{τ}$ is the upper bounds of the time-varying delay and $μ$ is its derivative.

Theorem 1

Given scalars $β_{j k} > 0$ , $α > 0$ , $(j, k = 0, 1, 2, \dots, M)$ , if there exist positive definite symmetric matrix $P_{0}$ , semi-positive definite symmetric matrix $P_{j} = F_{j}^{T} F_{j} \geq 0, j \in [1, M]$ and compatible matrix $Z$ such that $\begin{matrix} [\begin{matrix} Γ | τ (t) = 0 & E_{1}^{T} Z \\ * & - \tilde{Q} \end{matrix}] < 0, \end{matrix}$ $\begin{matrix} [\begin{matrix} Γ | τ (t) = \bar{τ} & E_{2}^{T} Z \\ * & - \tilde{Q} \end{matrix}] < 0, \end{matrix}$ where $\begin{matrix} Γ = Γ_{1} + Γ_{2} + Γ_{3} + Γ_{4} + Γ_{5} + α e_{1} P_{j} e_{1}^{T} + \sum_{k = 0}^{M} β_{j k} e_{1} (P_{j} - P_{k}) e_{1}^{T} - \frac{α}{ω_{M}} e_{8}^{T} e_{8}, \\ Γ_{1} = sym (e_{1} P_{j} e_{9}^{T}), \\ Γ_{2} = h e_{1} e_{1}^{T} - \frac{e^{- α h}}{h} e_{4} e_{4}^{T}, \\ Γ_{3} = \end{matrix}$

Reachable set estimation of uncertain mixed delays system

System (1) with polytopic uncertainties is expressed as: $\begin{matrix} \dot{x} (t) = (A + Δ A (t)) x (t) + (A_{b} + Δ A_{b} (t)) x (t - τ (t)) + (A_{h} + Δ A_{h} (t)) \int_{t - h}^{t} X (s) d s + (B + Δ B (t)) w (t), \\ x (t) = 0, \forall t \in [- max {h, \bar{τ}}, 0], \\ | F_{j} x (t) | \leq 1, j = 1, 2, \dots M, \end{matrix}$ where $\begin{matrix} Δ A (t) = \sum_{i = 0}^{M} θ_{i} (t) A_{i}, Δ A_{b} (t) = \sum_{i = 0}^{M} θ_{i} (t) A_{b i}, \\ Δ A_{h} (t) = \sum_{i = 0}^{M} θ_{i} (t) A_{h i}, Δ B (t) = \sum_{i = 0}^{M} θ_{i} (t) B_{i} . \end{matrix}$

We consider $0 ⩽ θ_{i} (t) ⩽ 1, \sum_{i = 0}^{M} θ_{i} (t) = 1$ first.

Theorem 2

Given scalars $α > 0$ , $β_{j k} > 0$ , $(j, k = 0, 1, 2, \dots, M)$ , existing positive definite symmetric matrix $P_{0}$ , semi-positive definite symmetric matrix $P_{j} = F_{j}^{T} F_{j} \geq 0, j \in [1, M]$ and compatible matrix $Z$ such that $\begin{matrix} [\begin{matrix} \tilde{Γ} | τ (t) = 0 & E_{1} \end{matrix} \end{matrix}$

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] $\begin{matrix} A = [\begin{matrix} 0 & 1 \\ - 2 & - 3 \end{matrix}], A_{b} = [\begin{matrix} 0 & - 0.1 \\ - 0.2 & 0.3 \end{matrix}], \\ A_{h} = [\begin{matrix} 0.2 & 0.2 \\ 0 & - 0.2 \end{matrix}], B = [\begin{matrix} 1 \\ 1 \end{matrix}], F = [\begin{matrix} 1 & 1 \end{matrix}] . \end{matrix}$

The state constraint on the system is $| F x (t) | \leq 1$ and external disturbance $w_{m} = 1$ . We select parameter $β = 1$ , $μ = 0.5$ , $α = 0.1$ , $\bar{τ} = 0.2$ with different values of $h$ to solve the optimization problem (14) and the results are listed in Table 1. Tables 2 and 3 also list the computed $\bar{δ}$ 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.

References (33)

L. Zhang et al.
Improved results on reachable set estimation of singular systems
Appl. Math. Comput.
(2020)
Y. Sheng et al.
Improved reachable set bounding for linear time-delay systems with disturbances
J. Frankl. Inst.
(2016)
X. Wang et al.
Reachable set estimation for Markov jump LPV systems with time delays
Appl. Math. Comput.
(2020)
Z. Feng et al.
On reachable set estimation of singular systems
Automatica
(2015)
G. Liu et al.
New insight into reachable set estimation for uncertain singular time-delay systems
Appl. Math. Comput.
(2018)
Y. Chen et al.
Estimation and synthesis of reachable set for switched linear systems
Automatica
(2016)
E. Fridman et al.
On reachable sets for linear systems with delay and bounded peak inputs
Automatica
(2003)
O. Kwon et al.
On the reachable set bounding of uncertain dynamic systems with time-varying delays and disturbances
Inf. Sci.
(2011)
B. Zhang et al.
Reachable set estimation and controller design for distributed delay systems with bounded disturbances
J. Frankl. Inst.
(2014)
T. Hu et al.
An analysis and design method for linear systems subject to actuator saturation and disturbance
Automatica
(2002)

W. Tang et al.

Interval estimation for discrete-time linear systems: a two-step method

Syst. Control Lett.

(2019)

W. Wu et al.

Fuzzy adaptive tracking control for switched nonlinear systems with full time-varying state constraints

Neurocomputing

(2019)

Z. Zuo et al.

Reachable set bounding for delayed systems with polytopic uncertainties: the maximal Lyapunov–Krasovskii functional approach

Automatica

(2010)

P. Park et al.

Reciprocally convex approach to stability of systems with time-varying delays

Automatica

(2011)

C. Zhang et al.

An extended reciprocally convex matrix inequality for stability analysis of systems with time-varying delay

Automatica

(2017)

Z. Feng et al.

An improved result on reachable set estimation and synthesis of time-delay systems

Appl. Math. Comput.

(2014)

Cited by (3)

Estimation of reachable set for switched singular systems with time-varying delay and state jump
2023, Applied Mathematics and Computation
This paper addresses the estimation problem of the reachable set for switched singular systems with state jump under initial condition and bounded peak disturbance. The aim is to determine a bounded set that contains all reachable states. Initially, a state jump lemma is presented for switched singular systems. Then, a new bounding lemma is shown by analyzing the variation of piecewise continuous functional in subintervals and referencing the definition of the average dwell-time. Afterwards, using lemmas and linear matrix inequalities, sufficient conditions are derived for switched singular systems with and without time-varying delay, which ensure that the state trajectory of the system remains within a closed bounded set. Finally, the obtained results are validated through numerical examples.
A dimensionality reduction method for computing reachable tubes based on piecewise pseudo-time dependent Hamilton–Jacobi equation
2023, Applied Mathematics and Computation
Citation 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.
Reachability analysis is a powerful tool for studying the safety of nonlinear systems, in which one of the key points is the computation of reachable tubes. As a common method in engineering, the Hamiltonian Jacobi technique often faces the “curse of dimensionality”. Its computational complexity grows exponentially with the dimensionality of the system state space. This paper proposes a dimensionality reduction method for the computation of reachable tubes that can be used for problems with dynamical systems of a particular form and a columnar target set. In the proposed method, one state variable is considered a pseudo-time variable, and the remaining state variables are contained within a low-dimensional dynamical system. Multiple slices of the original reachable tube are obtained by solving the Hamilton–Jacobi equation constructed based on this low-dimensional dynamical system and then stacking these slices to reconstruct the original reachable tube. Since the solved Hamilton–Jacobi equation is one dimension lower than the Hamilton–Jacobi equation in the original problem, the complexity of the computation is significantly reduced. Furthermore, the proposed method can be combined with existing methods to further reduce the dimensionality of the reachability problem. The computational accuracy and efficiency of the proposed method are demonstrated by some examples.
Reachable set estimation for neutral semi-Markovian jump systems with time-varying delay
2024, AIMS Mathematics

View full text

Reachable set bounding for linear systems with mixed delays and state constraints

Highlights

Abstract

Introduction

Section snippets

Problem statement

Reachable set estimation for mixed delays systems with state constraints

Reachable set estimation of uncertain mixed delays system

Illustrative examples

Conclusion

Acknowledgments

Appl. Math. Comput.

J. Frankl. Inst.

Appl. Math. Comput.

Automatica

Appl. Math. Comput.

Automatica

Automatica

Inf. Sci.

J. Frankl. Inst.

Automatica

Syst. Control Lett.

Neurocomputing

Automatica

Automatica

Automatica

Appl. Math. Comput.