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Abstract

This article presents finite-time stabilization methods of switched linear systems with disturbances. After extending finite-time stabilization and finite-time boundedness definitions to switched linear systems, sufficient conditions guaranteeing system finite-time boundedness are proposed, by which the state feedback controller method is obtained. For a class of switched terminal guidance systems, the methods are illustrated by application to guidance design to solve the finite-time stabilization problem considering nonzero initial conditions and state constraints.

Keywords

Linear system finite-time stabilization finite-time boundedness switched dynamics

Introduction

A switched system is a hybrid system that includes various subsystems and a rule that governs the switching among the subsystems. The main efforts typically focus on the analysis of dynamic behaviors, such as stability, controllability, accessibility, observability, and fault diagnosis, and aim to design controllers with guaranteed stability and optimized performance.^1

–6 So far, the most stabilities involved in the existing results basically focus on ‘infinite’ time interval, such as Lyapunov asymptotic stability. Recently, finite-time stabilization (FTS) for some stochastic nonlinear systems was studied in the literature.⁷ Finite-time state feedback controller was constructed for high-order stochastic nonlinear systems in strict-feedback form in the study by Wang and Zhu.⁸ Finite-time output feedback stabilization of a class of high-order stochastic nonlinear systems was investigated in the study by Zhai.⁹ The FTS of a class of switched stochastic nonlinear systems in p-normal form was invested in the study by Huang and Xiang.¹⁰

However, in various practical operations, system stability (usually in the sense of Lyapunov) as well as a bound of system trajectories over a fixed short time are both interested. In other words, the main concern is the behavior of the system within a finite-time interval, such as aerospace control systems and guidance systems, the control processes of which are implemented in a finite-time interval. Moreover, the disturbances in practice should not be ignored. Hence, we aim to investigate the FTS of switched linear systems with disturbances, as dealt in this article.

In the 1960s, Yin and Khoo⁷ initially raised up the concept about “short-term stability” under FTS. Followed by a large amount of research arise after the concept was carried out, the idea of “finite-time contractive stability” was first brought up by Wang and Zhu⁸ and Zhai⁹ in 1965. Finite-time contractive stability is defined as “finite-time bounded-input bounded-output (BIBO) stability”; it is approved to consider nonlinear systems with disturbance and it is now commonly known as “finite-time boundedness (FTB).” Up until now, much work has been done in this field. Given a bound on the initial condition, a system is said to be finite-time stable if the state does not exceed a certain threshold during a specified time interval. It is worth noting that there is a different notion of FTS,¹⁰ which requires the system state to reach the system equilibrium in a finite-time interval, and the property is called finite-time attractiveness in some research.^11,12 In the remainder of this article, FTS that we mentioned refers to the former one.

While external disturbances are considered, FTS is extended to FTB. In the light of results coming from linear matrix inequality (LMI) theory, many optimization and control problems can be formulated and solved using LMIs.^13,14 Considering linear Time-invariant systems, Chen and Jiao¹⁵ proposed the sufficient conditions for FTB and FTS method. Then, the results are extended to time-varying continuous systems and discrete systems.¹⁶ The dynamic output feedback controller design problem was studied in the literature.¹⁷ Considering impulsive dynamical systems, the sufficient conditions of FTS were proposed in the study by He and Wang.¹⁸ FTS of linear discrete-time system with time-varying delay was studied in the literature.¹⁹ Considering linear time-varying systems with jumps and impulsive dynamical linear systems, the conditions of FTS and FTB were proposed.^20,21 Recently, Ambrosino et al.²² also proposed the necessary and sufficient conditions for FTS of impulsive dynamical linear systems.

Therefore, the article aims to redefine the FTS for the switched linear system and investigate the corresponding control methods. The “Problem formulation” section of this article presents the basic descriptions and the problem statement. The “Main result” section proposes the requirements for FTB of a switched linear system. The state feedback controller design methods are then presented. The “Application example” section shows the appropriate applications for this method for the terminal guidance systems with switched dynamics. The last section gives the conclusion.

Problem formulation

Let $M = {1, 2, \dots, m}$ be a finite index set, and $ℜ_{+}$ be the set of nonnegative real numbers. Consider the following system switching among M linear dynamics

\dot{x} (t) = A_{σ} x (t)

where $x (t) \in ℜ^{n}$ is the state, σ(t) ∈ M, $A_{i} \in ℜ^{n \times n}$ , and i ∈ M.

Definition 1 (FTS of switched linear system):

System (1) is the FTS with respect to $(p, q, T, R)$ , with p < q, R > 0, and T > 0, if

x^{T} (0) R x (0) \leq p \Rightarrow x^{T} (t) R x (t) < q, \forall t \in [0, T]

The concept of state boundedness is more general and also concerns the behavior of the state in the attendance of both given initial and external disturbances.

Definition 2 (FTB of switched linear system):

Consider system

\dot{x} (t) = A_{σ} x (t) + G_{σ} w (t)

where $w (t) \in ℜ^{p}$ is the exogenous disturbance, $G_{i} \in ℜ^{n \times p}$ , i ∈ M, and the system is FTB with respect to $(p, q, T, R, S_{w})$ , with p < q, R > 0, and T > 0, if

x^{T} (0) R x (0) \leq p \Rightarrow x^{T} (t) R x (t) < q, \forall t \in [0, T], \forall w : \int_{Δ t} w^{T} (t) S_{w} w (t) d t \leq \frac{1}{Δ t}

Problem 1 (FTS of switched linear systems with disturbances):

Consider the system

\dot{x} (t) = A_{σ} x (t) + B_{σ} u (t) + G_{σ} w (t)

where $u (t) \in ℜ^{r}$ represents the input, $B_{i} \in ℜ^{n \times r}$ , and i ∈ M. Define a state feedback

u (t) = K_{σ} x (t)

where K _σ is a control gain matrix function to be further resolved. Placing this controller into the system (5) results in the closed-loop system

\dot{x} (t) = {\bar{A}}_{σ} x (t) + G_{σ} w (t)

where ${\bar{A}}_{σ} = A_{σ} + B_{σ} K_{σ}$ . Therefore, a state feedback controller in the form of equation (6) needs to be found, such that the closed-loop system (7) is FTB with respect to $(p, q, T, R, S_{w})$ .

Main result

Theorem 1 (FTB sufficient condition of switched linear system):

System (3) is FTB with respect to $(p, q, T, R, S_{w})$ , if, while $σ (t) = i, \forall t \in [t_{i}, t_{i + 1}] \subset [0, T], i \in M$ , letting ${\tilde{Q}}_{i} = R^{- \frac{1}{2}} Q_{i} R^{- \frac{1}{2}}$ , c₁ = p, and c_m = q, there exist Q _i, α_i, c_i, and c_{i + 1} such that

[\begin{matrix} A_{i} {\tilde{Q}}_{i} + {\tilde{Q}}_{i} A_{i}^{T} - α_{i} {\tilde{Q}}_{i} & G_{i} \\ G_{i}^{T} & - S_{w} \end{matrix}] < 0

\frac{e^{α_{i} t_{i}}}{t_{i + 1} - t_{i}} + \frac{c_{i}}{λ_{min} (Q_{i})} < \frac{c_{i + 1} e^{- α_{i} (t_{i + 1} - t_{i})}}{λ_{max} (Q_{i})}

where $α_{i} \in ℜ^{+}$ , p ≤ c_i ≤ q, Q _i ≥ 0, and $Q_{i} \in ℜ^{n \times n}$ .

Proof: Considering time interval $t \in [t_{i}, t_{i + 1}]$ , letting $V_{i} (x (t)) = x^{T} (t) {\tilde{Q}}_{i}^{- 1} (t) x (t)$ , we have

{\dot{V}}_{i} = {[\begin{array}{l} x (t) \\ w (t) \end{array}]}^{T} [\begin{matrix} A_{i}^{T} {\tilde{Q}}_{i}^{- 1} + {\tilde{Q}}_{i}^{- 1} A_{i} & {\tilde{Q}}_{i}^{- 1} G_{i} \\ G_{i}^{T} {\tilde{Q}}_{i}^{- 1} & 0 \end{matrix}] [\begin{array}{l} x (t) \\ w (t) \end{array}]

Pre- and post-multiplying (8) by

[\begin{matrix} {\tilde{Q}}^{- 1} & 0 \\ 0 & I \end{matrix}]

we obtain

[\begin{matrix} A_{i}^{T} {\tilde{Q}}_{i}^{- 1} + {\tilde{Q}}_{i}^{- 1} A_{i} - α_{i} {\tilde{Q}}_{i}^{- 1} & {\tilde{Q}}_{i}^{- 1} G_{i} \\ G_{i}^{T} {\tilde{Q}}_{i}^{- 1} & - S_{w} \end{matrix}] < 0

From equations (10) and (12), we can obtain

{\dot{V}}_{i} - α_{i} V_{i} - w^{T} S_{w} w < 0

multiplying (13) by $e^{- α_{i} t}$ , and integrating from t_i to t, $t \in [t_{i}, t_{i + 1}]$ , we obtain

\int_{t_{i}}^{t} {(e^{- a_{i} τ} V_{i})}^{'} d τ < \int_{t_{i}}^{t} e^{- a_{i} τ} w^{T} S_{w} w d τ

Noting α_i > 0, we have

e^{- α_{i} t} V_{i} (x (t)) - e^{- α_{i} t_{i}} V_{i} (x (t_{i})) < \int_{t_{i}}^{t} w^{T} S_{w} w d τ

Noting $\int_{Δ t} w^{T} (t) S_{w} w (t) d t \leq \frac{1}{Δ t}$ , $t \in [t_{i}, t_{i + 1}] \subset [0, T]$ , we have

\int_{t_{i}}^{t_{i + 1}} w^{T} S_{w} w d τ \leq \frac{1}{t_{i + 1} - t_{i}}

From equations (15) and (16), we can obtain

V_{i} (x (t)) < \frac{e^{α_{i} t}}{t_{i + 1} - t} + e^{α_{i} (t - t_{i})} V_{i} (x (t_{i}))

which can be rewritten as

x^{T} (t) R^{\frac{1}{2}} Q_{i}^{- 1} R^{\frac{1}{2}} x (t) < e^{α_{i} t} (\frac{1}{t_{i + 1} - t_{i}} + e^{- α_{i} t_{i}} x^{T} (t_{i}) R^{\frac{1}{2}} Q_{i}^{- 1} R^{\frac{1}{2}} x (t_{i}))

Noting $t \in [t_{i}, t_{i + 1}]$ , then

λ_{min} (Q_{i}^{- 1}) x^{T} (t) R x (t) < e^{α_{i} t_{i + 1}} (\frac{1}{t_{i + 1} - t_{i}} + e^{- α_{i} t_{i}} λ_{max} (Q_{i}^{- 1}) x^{T} (t_{i}) R x (t_{i}))

If $x^{T} (t_{i}) R x (t_{i}) < c_{i}$ , combining with equation (9), in time interval $t \in [t_{i}, t_{i + 1}]$ , we can obtain

x^{T} (t) R x (t) < c_{i + 1}, \forall t \in (t_{i}, t_{i + 1}]

Then, the proof follows by the recursive way.

Remark 1: From a computational point of view, for fixed α_i, c_i, and c_{i + 1} and a fixed finite-time interval $[t_{i}, t_{i + 1}]$ , the feasibility of the conditions stated in theorem 1 can be turned into the LMIs-based feasibility problem. Conditions (8) and (9) are guaranteed by imposing the conditions

λ_{1} I < Q_{i} < λ_{2} I

\frac{e^{α_{i} t_{i}}}{t_{i + 1} - t_{i}} + \frac{c_{i}}{λ_{1}} < \frac{c_{i + 1} e^{- α_{i} (t_{i + 1} - t_{i})}}{λ_{2}}

For positive numbers λ₁ and λ₂, the inequality can be converted to an LMI using Schur complements.

Theorem 2: While $σ (t) = i, \forall t \in [t_{i}, t_{i + 1}] \subset [0, T], i \in M$ , letting ${\tilde{Q}}_{i} = R^{- \frac{1}{2}} Q_{i} R^{- \frac{1}{2}}$ , c₁ = p, and c_m = q, if there exist Q _i, L _i, α_i, c_i, and c_{i + 1}, such that

[\begin{matrix} A_{i} {\tilde{Q}}_{i} + {\tilde{Q}}_{i} A_{i}^{T} + B L + L^{T} B^{T} - α_{i} {\tilde{Q}}_{i} & G_{i} \\ G_{i}^{T} & - S_{w} \end{matrix}] < 0

\frac{e^{α_{i} t_{i}}}{t_{i + 1} - t_{i}} + \frac{c_{i}}{λ_{min} (Q_{i})} < \frac{c_{i + 1} e^{- α_{i} (t_{i + 1} - t_{i})}}{λ_{max} (Q_{i})}

where $α_{i} \in ℜ^{+}$ , p ≤ c_i ≤ q, Q _i ≥ 0, $Q_{i} \in ℜ^{n \times n}$ , and $L_{i} \in ℜ^{r \times n}$ . Then, problem 1 can be solved by $K = L {\tilde{Q}}^{- 1}$

Proof: Replacing A _i in equation (8) with ${\bar{A}}_{i}$ and ${\bar{A}}_{i} = A_{i} + B_{i} K_{i}$ , letting $L = K \tilde{Q}$ , then equation (23) holds. According to theorem 1, the proof follows.

Application example

A terminal guidance scenario^23

–27 is the typical finite-time process,^28

–32 which is presented in Figure 1.

Figure 1.

Terminal guidance scenario geometry.

The distance between the missile and the target is indicated by r, l is the perpendicular deviation between the missile and the target, and velocities for the missile and the target are expressed as $v_{M} a n d v_{T}$ , respectively. The maneuver accelerations of the missile and the target are shown as $a_{M} a n d a_{T},$ respectively; $ϕ_{M} a n d ϕ_{T}$ in Figure 1 correspond to the velocity direction angles with respect to the reference line ox.

In this situation, ϕ_M and ϕ_T are small, since the missile and the target can be approximately considered to be head-on form, then

\ddot{l} = a_{M} \cos ϕ_{M} - a_{T} \cos ϕ_{T} \approx a_{M} - a_{T}

Define $a_{i C}, i = M, T$ as acceleration commands and $τ_{i}, i = M, T$ as consuming time of acceleration responses of the missile and the target, respectively. For the different types of actuator, τ is always different. For example, τ of the missile with traditional air rudders is always between 300 ms and 800 ms, and, in order to have quick response, some types of agile missile equip lateral jet engine, which cause τ being only between 30 ms and 100 ms. Some missiles have several types of actuator, which work in different time intervals cooperatively or independently. In our scenario, the missile with two types of actuator is considered. Hence, τ_M is different in different time intervals. Then, we have

{\dot{a}}_{M} = \frac{(a_{M C} - a_{M})}{τ_{M (t)}}

which means that the missile needs time τ_M(t) to adjust from current acceleration a_M to expected acceleration a_MC. And the target has a fixed τ_T, then we have

{\dot{a}}_{T} = \frac{(a_{T C} - a_{T})}{τ_{T}}

From equations (22) and (23), a state space description of the system can be obtained

[\begin{array}{l} \dot{l} \\ \ddot{l} \\ {\dot{a}}_{M} \\ {\dot{a}}_{T} \end{array}] = [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & - 1 \\ 0 & 0 & - \frac{1}{τ_{M (t)}} & 0 \\ 0 & 0 & 0 & - \frac{1}{τ_{T}} \end{matrix}] [\begin{array}{l} l \\ \dot{l} \\ a_{M} \\ a_{T} \end{array}] + [\begin{array}{l} 0 \\ 0 \\ \frac{1}{τ_{τ_{M (t)}}} \\ 0 \end{array}] a_{M C} + [\begin{array}{l} 0 \\ 0 \\ 0 \\ \frac{1}{τ_{T}} \end{array}] a_{T C}

y = l

where y is the output, which can be used as the miss distance of the missile to the target. The guidance process completes in a finite-time interval [0, T], and

T = \frac{r (0)}{V_{c}}

where $V_{c} = \dot{r}$ is the approaching speed.

For the missile, acceleration command of the target a_TC can be considered as an external disturbance, the purpose of guidance design is to minimize the miss distance in finite-time interval [0, T], and some indices not exceeding the physical constraints, such as seeker’s view field and missile acceleration. Therefore, the proposed method can be used to guidance designing.

Consider systems (28) and (29), finite-time interval t ∈ [0, 10],

τ_{M (t)} = {\begin{matrix} 0.3, t \in [0, 4) \cup [5, 9) \\ 0.05, t \in [4, 5) \cup [9, 10] \end{matrix}

which means the missile switches between two types of actuator, the suitable α_i, c_i, c_{i + 1}, i = (1, 2, 3, 4), a_TC is shown in Figure 2.

Figure 2.

Target maneuver acceleration command a_TC.

Other simulation parameters are given in Table 1.

Table 1.

Simulation parameters.

Parameter	Value	Unit
τ_T	0.2	s
x(0)	${(\begin{matrix} 50 & - 10 & 0 & 0 \end{matrix})}^{T}$	—
S _w	$diag (0.01, 0.01, 0.01, 0.01)$	—
R	$diag (0.01, 0.01, 1, 0.01)$	—

According to theorem 2, the controller gain K s can be obtained as follows

K_{1} = [\begin{matrix} - 9.23 & - 3.05 & - 0.41 & 0.30 \end{matrix}]

K_{2} = [\begin{matrix} - 5.09 & - 8.42 & - 0.12 & 3.89 \end{matrix}]

K_{3} = [\begin{matrix} - 6.18 & - 2.31 & - 0.33 & 0.54 \end{matrix}]

K_{4} = [\begin{matrix} - 4.83 & - 8.15 & - 0.07 & 1.97 \end{matrix}]

which are added into the original system. Simulation results are shown in Figures 3 to 5. For comparison, the cases using K guaranteed each subsystem FTB are also presented.

Figure 3.

Comparison of the output y.

Figure 4.

Acceleration a comparison of the proposed method and only-subsystem-1-FTB. FTB: finite-time boundedness.

Figure 5.

State measure x ^T Rx comparison of the proposed method and only-subsystem-1-FTB. FTB: finite-time boundedness.

It can be concluded from Figure 3 that all three controllers can drive the output near to zero in the concerned finite-time interval, which means the small miss distance. However, the minimum one is led by the proposed method. Figure 4 shows the acceleration comparison of the proposed method and only-subsystem-1-FTB. In the simulation results, it is shown that the guidance law designed by the proposed method has better performances in acceleration demanding aspect.

Figures 5 and 6 show that state measures x ^T(t) Rx (t) of only-subsystem-1-FTB and only-subsystem-2-FTB are significantly larger than the proposed method, which is unacceptable in practical application, since the missile acceleration demanding may exceed the physical constraints.

Figure 6.

State measure x ^T Rx comparison of the proposed method and only-subsystem-2-FTB. FTB: finite-time boundedness.

Therefore, it can be concluded that, compared with only-subsystem-1-FTB and only-subsystem-2-FTB methods, the proposed method has the minimum miss distance and the acceleration demanding, which means the better system performances in terminal guidance scenario. Meanwhile, the state measure x ^T Rx of lead by the proposed method does not exceed the predetermined boundedness, which also verifies the effectiveness of the methods.

Conclusion

FTS of switched linear systems with disturbances has been investigated in this article. After extending FTS and FTB definitions to switched linear systems, sufficient conditions guaranteeing system FTB are proposed, by which the state feedback controller designing method can be obtained. These circumstances, which contain the feasibility of LMIs, have then been used in the design context for controller synthesis. For a class of switched terminal guidance systems, the methods can be used to guidance design to solve the FTS problem considering nonzero initial conditions and state constraints. Performance comparisons of the proposed method, only-subsystem-1-FTB method and only-subsystem-2-FTB method, are presented. The results that indicate the proposed methods are suitable to the system with switched dynamics in the finite-time interval.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Finite-time stabilization of switched linear systems and its application in terminal guidance

Abstract

Keywords

Introduction

Problem formulation

Definition 1 (FTS of switched linear system):

Definition 2 (FTB of switched linear system):

Problem 1 (FTS of switched linear systems with disturbances):

Main result

Theorem 1 (FTB sufficient condition of switched linear system):

Application example

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References