Robust stability/stabilization with variable convergence rate for uncertain impulsive stochastic systems

This article researches the interval‐driven variable convergence rate robust stability/stabilization of uncertain impulsive stochastic systems. First, a new conception of variable convergence rate stability for uncertain impulsive stochastic systems is presented in conjunction with the idea of generalized pole configurations. Also, a new criterion for variable convergence rate robust stability/stabilization of uncertain impulsive stochastic systems is given, which can reflect and regulate the convergence rate of the system state. In addition, a new algorithm is designed by means of an interval‐driven stabilization method. This algorithm, combined with the variable convergence rate criterion, achieves accurate control of the convergence rate of the state at the operational level. Double instances are proposed to verify the superiority of the method.


INTRODUCTION
2][3][4] Impulsive behavior is a transient and abrupt phenomenon, which is prevalent in the real world.For example, biological neural networks, 5 the flow of funds in financial markets 6 and so forth.][18] In fact, it is inevitable that disturbances will be encountered in actual systems due to uncertainties in the inputs and outputs of the system or in the system itself.For example, pharmacology has the problem of changes in drug standards in an organ after drug uptake (or intramuscular injection), 19 the effects of heat coefficients, unknown flow coefficients and temperature variations on pneumatic cylinders in pneumatic servo-driven systems, 20 and the problem of state changes in biological populations due to abrupt environmental influences. 21,22These disturbances may lead to worse system performance and even oscillations or instabilities.Therefore, in order to better analyze the effects of disturbances on system performance, the framework of impulsive system analysis is extended to the performance analysis of uncertain impulsive stochastic systems (UISSs).][25][26] Numerous scholars around the world have devoted to the study of various properties of systems, and stability is a prerequisite for the proper functioning of all control systems.8][29][30] Specifically, 27 investigated the input-to-state stability of impulsive reaction-diffusion neural networks with infinite distributed time delays, while 30 studied the input-state stability of impulsive delay systems with multiple impulses.Stability analysis of a system contributes to recognize and understand the internal and external structure of the system, thus providing direction to other performance targets such as controller design.However, stochastic terms as well as impulsive discrete moments 11,29 pose various challenges to the stability analysis of UISSs.Currently, the stability analysis of UISSs is limited to relatively coarse stability conditions such as asymptotic stability and input-output stability as well as input-state stability, which cannot reflect more complex stability characteristics.][33][34][35][36] Motivated by the results of the discussion above, the robust stability of UISSs driven by interval with variable convergence rates is analyzed.The focus is on the stability problem driven by intervals, which improves the performance of controller design.Meanwhile, sufficient conditions for robust stability of interval-driven variable convergence rate for this class of systems and the design of interval-driven state feedback controllers are given.The innovations of this article are as follows: 1.A new concept of variable convergence rate stability for UISSs is presented in conjunction with the idea of generalized pole configurations.Also, a new criterion for variable convergence rate robust stability/stabilization of UISSs is given, which describes the performance of the system more accurately than the existing stability criterion.2. A more precise controller design approach is presented according to the variable convergence rate robust stability, which not only ensures the stability of the uncertain impulse stochastic system, but also accurately controls the dynamic performance of the system.3. A novel algorithm is designed by means of an interval-driven stability approach.The algorithm combines the variable convergence rate criterion to achieve accurate control of the system state convergence rate at the operational level.
The remaining of this article is arranged as follows.To prepare the main results, some preliminaries and definitions are put forward in Section 2. In Section 3, sufficient conditions for robust stability of UISSs with variable convergence rates are introduced.In addition, a state feedback controller for UISSs is designed to achieve interval-driven stabilization.The practicality of the main results is verified by numerical arithmetic examples in Section 4. Section 5 concludes the article.
Notation: ℑ min () denotes the impulse time sequence meeting inf k {t k − t k−1 } ≥ ; ℑ max () denotes the impulse time sequence meeting sup k {t k − t k−1 } ≤ . 1, s = {1, 2, … , s}.  denotes the average time separation between continuous impulses.Given (Ω,  , ), a complete probability space with Ω the sample space and  the -algebra of subsets of the sample space, {• • • } denotes the expectation operator with respect to the given probability measure . * is the symmetric terms in a symmetric matrix.|| ⋅ || is the Euclidean norm for vectors or the spectral norm of matrices.dV(x(t)) denotes the differential term of the function V(x(t)), LV (x(t)) is said to be a differential operator.

PRELIMINARIES
Consider UISSs: where x(t) ∈  n is the state vector, u a (t) ∈  p1 is a continuously controlled input, u b (t) ∈  p2 is an input for impulse control. i (t), i ∈ 1, s are a set of mutually uncorrelated standard Brown motion defined on a complete probability space (Ω,  , ).A(t), H(t), G a (t), G b (t), M i (t), i = 1, s are matrix functions with time-varying uncertainties, that is where A, H, G a , G b , M i ∈ R n×n are known real constant matrices and ΔA(t), ΔH(t), ΔG a (t), ΔG b (t), ΔM i (t), i ∈ 1, s are unknown matrices representing time-varying parameter uncertainties.We assume that uncertains are norm bounded and can be described as ( Definition 1 (37).If there are constants  > 0 and  > 0 satisfying the following condition (3) at t ≥ 0, then the trivial solution of ( 1) is robustly exponentially stable.
Definition 2 (38).Let  A,M i be a operator associated with system (1) as follows: where X ∈  n .According to operator theory, the existence of a nonzero matrix X ∈  n and a number  will give  A,M i X = X.Where  is the eigenvalue, and X is the eigenvector of  A,M i corresponding to the eigenvalue .
A new definition of variable convergence rate robust stability is given by combining the idea of generalized pole configurations.Definition 3.There exists real numbers p > 0, q > 0 (q > p) and we transform system (1) into two systems as follows If both of the above systems satisfy definition 1, then system (1) is described as variable convergence rate robustly stable driven by p, q.Remark 1.Based on the theory of pole placement, the eigenvalues of a system are closely related to the steady-state and dynamic performance of the system.Therefore, the eigenvalues of the target system can be used to study the convergence rate problem.For UISSs, the uncertainties present in the system and the nonlinear terms make it harder to find the corresponding eigenvalues of the system matrix, and therefore have to be combined with the relevant definitions, lemmas or inequality scaling as an auxiliary tool presented during the proof of subsequent theorems.The two transformed systems are stable, and using the eigenvalue criterion, the eigenvalues of each transformed system can be limited to specific intervals, which in turn ensures that the eigenvalues of the original system are in the desired intervals.The values p, q are the endpoint values of the intervals, and it is in fact by adjusting the endpoint values that the eigenvalues of the systems are restricted to be in the desired interval.We introduce this new concept in order to reflect the rate-adjustability of the target system states, the idea of which can be found in the literature. 34,36,39mma 1 (38).Let x(t) ∈  n be an Itô process on t > 0 with a stochastic differential equation (1) , and V is a one-dimensional Itô process satisfying the conditions below. where Therefore, dV can also be expressed as Lemma 2 (37).
≤ I and scalar  > 0, the following inequalities hold:

STABILITY WITH VARIABLE CONVERGENCE RATE
This section considers the stability with variable convergence rate driven by p, q of system (1).For this purpose, we restrict our study to the case of u a (⋅) = 0, u b (⋅) = 0, that is, 4) is robustly stable with variable convergence rate driven by p, q over ℑ , if for the prescribed real numbers p > 0, q > 0 (q > p) and given a positive scalar , there exist scalars  > 0,  > 0,  1 > 0,  2 > 0,  3i > 0, i ∈ 1, s, and a matrix P > 0 that satisfy the following inequalities: where Proof.To begin with, it is proved that system (1) is robustly stable with variable convergence rate if (6)-( 8) hold.Consider systems: and Using Schur complement, ( 6)-( 8) can be introduced respectively ln h() ln h() using (iii) of Lemma 2 for any positive scalar  2 satisfying P −1 −  2 EE T > 0. For (10), choosing the stochastic Lyapunov function V(x(t)) = x T (t)Px(t) leads to Substituting ( 12) into (15) yields To complete the proof of Theorem 1, two cases are discussed:  ≥ 1 and  < 1. Case 1:  ≥ 1.
By (iii) of Lemma 2, the following inequality can be introduced for any given  3i > 0 (i ∈ 1, s).

V(x(t)) = x T (t)[P(A(t) + pI) + (A(t) + pI)
Accordingly, Integrating (26) from t k to t and computing the expectation leads to In accordance with ( 9), it can be obtained V(x(t)) < e  e −(t−t 0 ) V(x(t 0 )). Thus, It can be shown that ( 10) is robustly stable.Similarly, it can be introduced that system (11) is also robustly stable.Thus system (4) is robustly stable with variable convergence rate driven by p, q over ℑ , .The proof is completed.▪ Remark 2. The impulse appears as the form of disturbance when  ≥ 1.In this case, the impulse time series is required to satisfy inf k {t k − t k−1 } ≥  and  ≥ ln ∕(ln ∕ − ).
Remark 3. The impulse appears as the form of stabilization when  < 1, whereas ( 7) and (8) may cause (4) to be unstable.In this case, (6) can be run frequently to suppress the continuous dynamics that may be unstable.It is required that the impulse time series satisfy sup k {t k − t k−1 } ≤  and  < ln ∕(ln ∕ − ).
On the basis of Theorem 1, the common stability conditions of UISSs can be acquired as follows.

STABILIZATION WITH VARIABLE CONVERGENCE RATE
Next, the variable convergence speed stabilization problem for uncertain impulse stochastic closed-loop systems driven by p, q will be explored.Consider the following closed-loop system (1): Designing the state feedback controllers where K a and K b are gain matrices.
Theorem 2. Given a positive scalar  and real numbers p > 0, q > 0 (q > p), there exists a matrix X > 0 and positive scalars , ,  1 ,  2 ,  3i , i ∈ 1, s, the matrices inequalities satisfy: where 1) robust stabilization with variable convergence rate driven by p, q over ℑ min (); when  < 1, the controller (29) with K a = Y a X −1 and K b = Y b X −1 makes system (1) robust stabilization with variable convergence rate driven by p, q over ℑ max ().
Proof.Pre-and post-multiplying diag {P, P, I} on both sides of (31), and then (34) can be introduced by Schur complement theorem.
where X = P −1 .Meanwhile, multiplying diag {P, I, s ⏞⏞⏞ I, … , I, s ⏞⏞⏞ I, … , I, I} on both sides of ( 32)-( 33), and then ( 35)-( 36) can be deduced from the Schur complement theorem.ln h() ln h() Combining the above obtained inequalities, the following proof procedure is analogous to Theorem 1, thus Theorem 2 holds.▪ Remark 4. As far as the authors know, the existing stability methods for UISSs, such as asymptotic stability, do not take into account the convergence rate of the system state approaching the equilibrium point.The robust stability and stabilization with variable convergence rate driven by p, q method in this article solves this limitation.Compared with the existing stabilization methods, the system states' convergence rate can be more precisely modulated by Theorem 2. This method primary thought is evolved from pole assignment.At present, no such conclusion had been obtained before for UISSs.By adjusting the values of p and q, the system state's convergence rate of UISSs can be adjusted to the desired level.
Under Theorem 2, if only the stabilization of UISSs is controlled, a more particular corollary can be obtained as follows.
Corollary 2. Given a positive scalar , there exists a matrix X > 0 and positive scalars , , such that (30), (31)  and the matrix inequality satisfy: where When  ≥ 1, the controller (29) with K a = Y a X −1 and K b = Y b X −1 makes the system (1) robust stabilization over ℑ min (); when  < 1, the controller (29) with K a = Y a X −1 and K b = Y b X −1 makes system (1) robust stabilization over ℑ max ().
Theorem 2 is founded on the thought of pole configuration, which regulates the convergence rate of system by restricting the generalized eigenvalues of UISSs to a suitable interval.The relevant algorithm is devised by combining this idea.As for the interval H = (−q, −p), Algorithm 1 follows: Algorithm 1. Eigenvalue adjustment algorithm of system matrix Require: p > 0 ∨ q > 0; Ensure: adjusts system state convergence speed; H > −q&&H < −p; use [Theorem2] to solve corresponding controller; if (−q < [the real parts of eigenvalues] &&[the real parts of eigenvalues] < −p)==1 then determine p,q; end if while p,q does not meet the requirements do if the system convergence slowly then p++; q++; else the system convergence rapidly p−−; q−−; end if end while Remark 5. Arbitrary 0 < p < q cannot guarantee the existence of the solution of Theorem 1 or Theorem 2, but Theorem 1 or Theorem 2 can be solved by increasing the value of p or decreasing the value of q.Therefore, in practice, the appropriate p and q can always be found to control the convergence rate of the system.

SIMULATION RESULTS
In this component, two examples will be put forward to examine the effectiveness and applicability of the above conclusions.
Example 1.When the impulse is in the form of disturbance, consider the data of UISSs (1) with u a (t) = K a x(t) and u b (t) = K b x(t) as As a common method of stabilization, the system gain matrices are solved via Corollary 2 as follows ] .
The system state trajectories are displayed in Figure 2. When t > 10.5(s), |x i (t)| < 0.02.F I G U R E 1 Open-loop system state trajectories.Remark 6.So far, the usual stabilization methods can only render certain the asymptotic convergence of UISSs, not to exert influence on the convergence speed.Whereas, Theorem 2 can regulate the system convergence speed through regulating the interval (−p, −q), as stated below.
In accordance with Theorem 2, the interval is adjusted to take the generalized eigenvalues of the system, which in turn adjusts the convergence rate.Execute the Matlab toolbox with  = 0.3 and set (−p, −q) = (−2, −1).Solving in ( 30)-( 33) yields the respective gain matrices as which can exhibit a slower convergence rate.
The system state trajectories are illustrated in Figure 3.When t > 26(s), |x i (t)| < 0.02.Meanwhile, to make the system converge faster, the interval (−p, −q) is set to (−8, −5).Combining with Theorem 2, the gain matrix can be obtained as follows The system state trajectories are given in Figure 4.When t > 1.9(s), |x i (t)| < 0.02.Remark 7. As shown in Figure 2, when stabilizing UISSs using Corollary 2, it is not possible to adjust the convergence speed of the system.Theorem 2 addresses this limitation by providing a remedy.It emphasizes the control of convergence speed while ensuring robustness, enabling precise control.As depicted in Figures 3  and 4, the convergence speed can be accelerated or decelerated.These improvements are advantageous for exploring the dynamic performance indicators of UISSs.x 1 (t) x 2 (t)  Figure 7 shows that the convergence speed will become slower.When t > 9(s), |x i (t)| < 0.02.In addition, the interval can be adjusted to (−8, −6) in order to speed up the convergence.Solve for the correlation gain matrices as  The system state trajectories are given in Figure 8.When t > 1.9(s), |x i (t)| < 0.02.Remark 8. Subject to the stabilization impulses, the interval-driven stabilization approach can be seen in Figures 6 and 7 to be more flexible in practice to adjust the convergence rate of the states as required.
Remark 9.The adjustment of convergence speed holds paramount practical significance in practical applications.For instance, when operating a self-balancing vehicle, it becomes imperative to modulate the convergence speed in order to effectively control acceleration and deceleration, taking into account the ever-changing road conditions.This fine-tuning optimizes both the safety and comfort aspects of the vehicle's operation.

CONCLUSION
The interval-driven variable convergence rate robust stability/stabilization problem for UISSs has been studied, and a new concept of variable convergence rate stability has been presented.Also, we proposed a more precise controller design method based on interval-driven stability and given a new robust stabilization criterion for variable convergence rate.On this basis, a novel algorithm has been designed by means of an interval-driven stability approach.The algorithm combines the variable convergence rate criterion to achieve accurate control of the system state convergence rate at the operational level.Finally, double instances have been proposed to verify the superiority of the the method.

2
State trajectories under the stabilization controller.

3
State trajectories under the stabilization controller.

4
State trajectories under the stabilization controller.

Example 2 .E 1
When the impulse is in the form of stabilization, consider the data of UISSs (1) with u a (t) = K a x(t) and u b (t) = K b x(t) as = 0.2I, E 2 = 0.4I, E 3 = 0.2I, D a = D b = 0.1I, D g1 = 0.01I,

5
Open-loop system state trajectories.

F I G U R E 6
State trajectories under the stabilization controller. = 0.9,  = 0.25.Let the initial states be x 1 (t 0 ) = 10, x 2 (t 0 ) = −10.The open-loop system state response is shown in Figure5.The state feedback controllers K a1 , K a2 of the system are obtained from Corollary 2The system state trajectories under feedback control are depicted in Figure6.When t > 4.5(s), |x i (t)| < 0.02.According to Theorem 2, it can be set to (−p, −q) = (−3, −0.5), and the respective gain matrices as

7
State trajectories under the stabilization controller.

8
State trajectories under the stabilization controller.