Abstract
The output feedback controllers of stochastic nonholonomic systems under arbitrary switching are discussed. We adopt an observer which can simplify the design process. The designed control laws cause the calculation of the gain parameter to be very convenient since the denominator of virtual controllers does not contain the gain parameter. Finally, an example is given to show the effectiveness of controllers.
1. Introduction
In recent years, switched system’s control, especially under arbitrary switching, has become an active field [1–3]. The global stabilization of switched systems based on arbitrary switching was given [4–6]. The stabilizing controllers of switched systems with arbitrary switching were given [7–10]. The output feedback controllers of nonlinear systems with arbitrary switching were designed [11, 12].
In the past ten years, the problem of stabilization for stochastic nonholonomic systems (SNSs) received much attention. They mainly can be classified into two types. The first is state-feedback control: stabilization [13, 14], adaptive stabilization [15–18], finite-time stabilization [19], stabilization with time-varying delays [20], and stabilization of mobile robots [21, 22]. The second is the output feedback stabilization [23–25].
Since sometimes part of the system states are unmeasurable, output feedback controllers are needed. Zhang et al. discussed the output feedback stabilizing controllers of SNSs whose virtual control contains gain parameter [25]. This will lead to a problem where the calculation of is very difficult, especially for , since the inequalities about were quintic. In addition, to the authors’ knowledge, there are some results about state-feedback stabilization of SNSs with Markovian switching [13, 14], with few available results for the output stabilization of SNSs under arbitrary switching. Based on the above analysis, there exists a problem, that is, how to choose a proper observer under arbitrary switching where the virtual control in controllers does not contain gain parameter , which causes the calculation of to be easier.
Notations. denotes the set of all real numbers. denotes the real -dimensional space. For a vector or matrix , denotes its transpose, denotes the Euclidean norm, is its trace when is square, and is a stochastic differential operator [26].
2. Problem Formulation
The stochastic nonholonomic nonlinear systems are given bywhere and are inputs, and are system states, , are smooth functions named as nonlinear drifts with , and and are smooth functions with and , . is a piecewise constant switching signal, and is an m-dimensional independent standard Wiener process.
Assumption 1. If there exist smooth function and positive constant , thenwhere is a known bounded nonnegative function.
Assumption 2. For , one has known constants and satisfying
Remark 3. There are two main differences between systems (1a) and (1b) and those in [24, 25]. The first is the arbitrary switching mentioned in this paper. The second will be illustrated in Remark 7. In addition, Assumptions 1 and 2 are similar to those in [24, 25], but in fact they hold under arbitrary switching; for example, when .
3. Output Feedback Stabilization
The controller design in Sections 3.1 and 3.2 is under . The other one is discussed in Section 3.3.
3.1. Controller Design
For subsystem (1a), one obtainswith Lyapunov function and controller as follows:where is a design real number.
Theorem 4. For system (1a), the closed-loop system with controller (4) is asymptotically stable in probability.
By (4) and (1a), one has
Remark 5. For any , one has in (6) at the time interval a.s. with a similar proof of Proposition in [25].
3.2. Controller Design
In order to design controller , let
Remark 6. For any , from Remark 5, we have that (7) is meaningful a.s.
By (1b) and (7), one haswhere and
We adopt the following observer [27]:where is a gain parameter and real numbers , , such that is Hurwitz.
Remark 7. The second main difference between this manuscript and [25] is that the observer (10) we choose is the same as that in [27], but different from that in [25]. This observer has two advantages. The first is that it can simplify the process of designing controllers. The second is that in virtual control that we design in the following does not contain the gain parameter .
Denoting one has the error systemswhereTherefore, for positive definite matrix , is a Hurwitz matrix; that is, . Now, one has
Proposition 8. By Assumptions 1 and 2, there exist constants and a.s., such that where .
Remark 9. The proof of the above proposition is similar to that of Proposition 2 in [25]. In fact, we only need to let .
One can define variables as follows in order to utilize the backstepping method: with virtual smooth controllers .
Step 1. For positive parameter , letting , with a similar method in [24], one has where . Defining , we have
The following inequalities hold with Lemma in [28]:where, together with (18), . Choosingone has
Step . From (16), we havewhere Now, we have finished the design step , and is chosen as follows:Let , such that
Defining , we have
By Lemma 2.1 in [28], one gets Choosing as one has
Step . Letting and choosingone getswhere and
Remark 10. From (20), one has that do not contain under arbitrary switching. From (28), one has that all do not contain under the designed controllers.
Remark 11. For example, with , by (41) in [25] and (32), we haverespectively. It is easy to see that and in (33) are all quintic functions about , but they are linear functions about in (34). So, the calculation of and will be more simple with the method in this manuscript.
Choose , which together with (5) and (31) result in where .
Theorem 12. For system (1b), the closed-loop system with controller (30) is asymptotically stable in probability.
3.3. Switching Control
In the above two subsections, we give the controllers and with as (4) and (30). Now, we turn to the case of . If , we firstly choose constant control . Secondly, there will exist a time such that . After that, controllers and as (4) and (30) can be applied.
Theorem 13. If we apply the above switching procedure, systems (1a) and (1b) will be asymptotically stabilized in probability.
4. A Simulation Example
Consider systems (1a) and (1b) with and , , , , , , , , , and .
From (8), we have , , , , , , , and .
Letting , by Proposition 8, we have and . Choosing and , then and .
From (20) and (30), one gets where . From (32) and (35), we have Solving the above inequalities, one has and , which means , , and .
If we choose initial values , , , , and , responses of systems are as in Figures 1, 2, and 3.
Remark 14. From the above example, the observer we adopted is the same as that in [27], but it can simplify the calculation of and compared with the observer in [25].
5. Conclusions
The output feedback stabilization for SNSs under arbitrary switching is discussed. We proposed an observer which is different from that in [25]. of the designed output feedback stabilizing control laws do not contain the gain parameter. We will give some new results, for example, how to design an adaptive controller with the method LMI based on results in [29, 30].
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was partially supported by the National Natural Science Foundation of China (nos. 61503262 and 61374040), the Foundation for High-Level Talents of Hebei Province of China (nos. A2016001137 and A2016001144), and the Natural Science Foundation of Henan Educational Committee of China (no. 17A110027).