Energy-to-peak control for switched systems with PDT switching

: This paper investigates the issue of energy-to-peak control for continuous-time switched systems. A generalized switching signal, known as persistent dwell-time switching, is considered. Two di ﬀ erent strategies for state-feedback controller design are proposed, using distinct Lyapunov functions and a few decoupling techniques. The critical distinction between these two strategies lies in their temporal characteristics: one is time-independent, while the other is quasi-time-dependent. Compared to the former, the latter has the potential to be less conservative. The validity of the proposed design strategies is demonstrated through an example.


Introduction
Switched systems (SSs) are a significant subclass of hybrid systems consisting of a series of subsystems and a signal for controlling the switch among them [1]. Over the last few decades, SSs have been applied in various domains, such as DC-DC power converters [2], inverter circuits [3], unmanned vehicles [4], secure communication [5], fault estimation [6] and image encryption [7]. At the same time, stability analysis and controller design for SSs, as fundamental issues in the control area, have been intensively studied and a significant amount of results have been proposed in the international literature; see, e.g., [8][9][10][11][12][13][14][15][16].
The stability of an SS is dependent on each subsystem and highly affected by the switching frequency. Although all subsystems are asymptotically stable (AS), the entire SS may have a nonconvergent solution trajectory caused by fast switching [17]. However, the stability can be maintained when the switching is sufficiently slow in the sense that the running time of each active subsystem is not less than a specified threshold called the dwell time (DT) [18]. Actually, this is also true even when fast switching occurs occasionally, as long as the average dwell time (ADT) is long enough [19].
In 2004, a type of switching signal, called the persistent DT (PDT) switching signal, was introduced in [20] to represent the switching signal that has infinitely many intervals of length no less than a given positive constant ε (i.e., the PDT) during which no switch occurs, and distance between two consecutive intervals with this property does not surpass a period of persistence δ. As explained in [20], the PDT switching is more common and suitable than DT and ADT switching for describing switching phenomena associated with hybrid systems.
There have been various control strategies, including uniform tube-based control [21], fault-tolerant control [22], quasi-synchronization control [23], quantized fuzzy control [24], L 1 finite-time control [25], dynamic output-feedback control [26], sliding mode control [27] and model predictive control [28], that have been introduced for discrete-time SSs with PDT switching over the past few years. To the best of our knowledge, however, there are no available reports on the energy-to-peak control of continuous-time SSs (CTSSs) with PDT switching. Energy-to-peak control can guarantee that the infinity norm of the controlled output is less than a certain disturbance attenuation level [29]. In many practical situations, such a control approach serves as an appropriate selection for system design because it is insensitive to specific statistical characteristics of the noise signals and exhibits good robustness [30].
In this paper, we are interested in the energy-to-peak control for CTSSs with PDT switching. Our objective is to ensure that the CTSS is AS with a certain energy-to-peak disturbance-attenuation performance (EPDAP) level [31]. We first introduce a lemma regarding the asymptotic stability and EPDAP analysis of the PDT-based CTSS. Then, we propose a time-independent state-feedback controller design approach using a Lyapunov function (LF) and decoupling techniques. To reduce conservatism, we further present a quasi-time-dependent (QTD) controller design method. The required gains of these two types of controllers can be obtained by means of feasible solutions of linear matrix inequalities (LMIs), which are known to be easily solved with available tools in MATLAB [32]. Finally, we give an example to illustrate the effectiveness of our controller design strategies.
Notation: We denote by R n the n-dimensional Euclidean space, by Z + the set of non-negative integers, by R the set of real numbers, by · the 2-norm, and by · the round-down operator. We apply an asterisk " * " to represent a symmetric term in a matrix and the superscript "T " to stand for the transpose operator. For any square matrix X, we utilize X > 0 (< 0) to imply that the matrix is symmetric positive-definite (negative-definite) and define S (X) as X + X T . Moreover, we let K ∞ be a class of continuous and strictly increasing functions β(·) that satisfy β(0) = 0.

Preliminaries
In the field of control for SSs with PDT switching, the majority of published work primarily focuses on the discrete-time setting; see, e.g., [21][22][23][24][25][26][27][28]. In this paper, we will consider CTSSs with PDT switching as in [33,34]. The system model is described bẏ within which ψ(t) ∈ R n , φ(t) ∈ R p and u(t) ∈ R q stand for the state, controlled output and control input, respectively; (t) ∈ R r denotes the exterior disturbance that belongs to L 2 [0, ∞] [35]; A ς(t) , B ς(t) , E ς(t) and G ς(t) are the given system matrices; ς(t) denotes the switching rule, which is a right-continuous The sequence formed by all switching moments is given as t 0 = 0, . . . , t l , . . .. The minimum time interval between any two switching moments is defined by h t = min{t l+1 − t l } (l ∈ Z + ). To avoid Zeno behavior, we set h t ≥ C 0 , where C 0 is a positive constant.
As described in Figure 1, the whole time axis is divided into infinitely many stages, where each stage consists of two parts, namely the slow switching part (i.e., ε-part) and the frequent switching part (i.e., δ-part). We denote by ε r and δ r the duration of each of these two parts in the r-th stage, respectively. Additionally, we use t m(r) to represent the initial moment of the r-th stage and n r to stand for the number of switches within the period of (t m(r) , t m(r+1) ). Evidently, ε r and δ r satisfy ε r ≥ ε and δ r ≤ δ, respectively, and the switching moments within the interval (t m(r) , t m(r+1) ) can be shown as t m(r)+1 , . . . , t m(r)+κ , . . . , t m(r)+n r . For all κ ∈ Z + , we denote further that δ r, κ = t m(r)+κ+1 − t m(r)+κ . Then, according to the meaning of the symbols introduced, it can be seen that δ r, κ ≤ δ and t m(r+1) = t m(r)+n r +1 . Remark 1. According to the PDT switching scheme, in the r-th stage, the duration of the slow switching part is at least ε, and the duration of the frequent switching part does not surpass δ. As compared to the DT and ADT switching schemes, the PDT switching scheme is more general. To be more specific, when δ takes values of zero and infinity, the PDT switching scheme degenerates into the DT and weak DT switching schemes, respectively [20]. Furthermore, unlike the ADT switching scheme, the PDT switching scheme does not impose any restrictions on the switching frequency of the frequent switching part [36].
Let us now introduce the concepts concerning the asymptotic stability and EPDAP: holds in the case of (t) = 0.
Definition 2. Given a scalar γ > 0, we say that CTSS (2.1) has the EPDAP level γ if Before ending the section, we state three preliminary propositions that we are going to use to prove our main results. Proposition 1.
[37] Given a scalar ρ > 0 and two locally integrable functions V (t) and
The controller designed in (3.13) is time-independent. Next, we focus on the time-dependent design. The controller to be determined takes the form of where q t is a time scheduler that takes values in N = {0, . . . , ε/h t }, described by with t η max t l ∈[0, t] {t l }.
The following result can be deduced from Lemma 1.
Thus, (3.24) also holds when t is a switching instant. The proof is finished.
Then, based on Lemma 2, the desired QTD controller can be constructed according to the following theorem.

Remark 2.
As commonly reported in the literature on SSs with PDT switching (see, e.g., [43][44][45], the design strategy proposed in Theorem 1 is time-independent. In order to reduce conservativeness, Theorem 2 presents a QTD design strategy by incorporating the time scheduler q t effectively. The benefits of this approach will be demonstrated in Section 4. However, it may be worth noting that this improvement comes at the cost of increased computational complexity.

Example
Consider CTSS (2.1) subject to the following parameters: We set δ = 2, h t = 0.2 and θ = 0.1. Then, by solving the LMIs of Theorems 1 and 2, respectively, we can get the comparison outcomes of the optimal EPDAP levelγ min for different values of ρ and µ, as described in Tables 1 and 2, respectively. From these two tables, we have two observations. First, when one of the values of ρ and µ is fixed, the optimal EPDAP levelγ min increases as the value of the other parameter increases. Second, when comparing the controller design method given in Theorem 1 with the one in Theorem 2, it is evident that the latter always yields better performance levelsγ min . This improvement can be attributed to the fact that the design method in Theorem 2 is QTD.
Next, we set ρ = 0.4 and µ = 1.1. By solving the LMIs of Theorem 2, we get the EPDAP level γ min = 0.0866 and the controller gains as follows: In the simulation, we take the exterior disturbance as (t) = e −0.2t sin(2t) and set the initial value as ψ(0) = [5 − 2] T . Figures 2-4 show the trajectories of PDT switching mode ς(t), time scheduler q t and states of the closed-loop CTSS, respectively. It is evident from Figure 4 that the open-loop CTSS is unstable. The trajectories of the states and control input of the closed-loop CTSS are depicted in Figure 5. It is apparent that the curves tend to zero as time t → ∞, indicating that QTD controller (3.18) can ensure that the closed-loop CTSS is AS. At last, we introduce γ(t) = sup s≥0 φ(s) 2 t 0 (s) 2 ds . Figure 6 further describes the curve of γ(t) given that ψ(0) = [0 0] T . Apparently, γ(t) progressively converges to 0.0077, which is smaller than the optimal EPDAP levelγ min = 0.0866. This shows the effectiveness of controller (3.18) in ensuring the EPDAP of the closed-loop CTSS.

Conclusions
This work investigated the issue of energy-to-peak control for CTSSs with PDT switching. With the aid of an LF and a few decoupling techniques, a time-independent controller design approach was proposed in Theorem 1. To reduce conservatism, a QTD controller design method was further presented in Theorem 2. The required gains of these two types of controllers can be acquired by solving LMIs. Finally, an example was utilized to illustrate the validity of our controller design approaches.
The controllers under consideration are based on full-state feedback, which utilizes state variables as feedback signals to generate control inputs. However, there are certain scenarios in which implementing such controllers becomes challenging because directly measuring all of the state variables is often difficult [46]. The issue of energy-to-peak control for CTSSs with PDT switching based on output feedback will be explored as an extension of the current work.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.