Abstract
This paper is concerned with disturbance observer-based input-output finite-time control of a class of nonlinear systems with one-sided Lipschitz condition, as well as multiple disturbances. Firstly, a disturbance observer is constructed to estimate the disturbance generated by an exogenous system. Secondly, by integrating the estimation of disturbance with a classical state feedback control law, a composite control law is designed and sufficient conditions for input-output finite-time stability (IO-FTS) of the closed-loop system are attained. Such conditions can be converted into linear matrix inequalities (LMIs). Finally, two examples are given to show the effectiveness of the proposed method.
1. Introduction
The robust Lyapunov stability reflects the asymptotic behavior; that is, the result only is achieved in an infinite-time interval. However, in many practical applications (for example, in biochemical reaction systems, communication network systems, or robot control systems), one is more interested in what happens over a finite-time interval rather than the asymptotical property. To discuss this transient performance, Dorato [1] firstly defined finite-time stability (FTS) for linear deterministic systems. A system is said to be FTS if, given a bound on the initial condition, its state does not exceed a certain threshold during a specified time interval. Up until now, much work has been done in this field [2–4]. Recently, the definition of input-output finite-time stability (IO-FTS) has been firstly introduced in [5], which is a more practical concept and means that, given a class of norm-bounded input signals over a specified time interval of length , the outputs of the system do not exceed an assigned threshold during such time interval. This definition of IO-FTS is fully consistent with the definition of FTS. IO-FTS involves signals defined over a finite-time interval and does not necessarily require the inputs and outputs to belong to the same class, and IO-FTS constraints permit specifying quantitative bounds on the controlled variables to be fulfilled during the transient response [6, 7]. Some related results are also presented, such as linear systems [8], hybrid systems via static output feedback [9], nonlinear systems via sliding mode control [10], discrete-time impulsive switch systems [11], nonlinear stochastic systems [12], and Markovian jump systems [13, 14].
On the other hand, the complex systems include multiple disturbances, such as unknown frictions or loads, harmonic disturbances, modeling uncertainties, and stochastic noises. The presence of different types of disturbances will seriously affect control accuracy. Therefore, how to design a controller to suppress disturbances is a hot topic. So disturbance observer-based control technique is proposed as an effective approach, and many related meaningful results are presented [15–19]. It is worth noting that the most existing results involve the asymptotic stability and system nonlinearity functions are assumed to satisfy the Lipschitz condition. As we know, one-sided Lipschitz condition is shown to be an extension of the Lipschitz condition and is less conservative, and the one-sided Lipschitz constant is significantly smaller than the Lipschitz constant, which makes it much more suitable for estimating the influence of nonlinear part [20–24]. Recently, [25] considers the finite-time control of nonlinear systems with one-sided Lipschitz condition. However, there are few results about disturbance observer-based input-output finite-time control of nonlinear systems with one-sided Lipschitz condition, which motivates our study.
This paper considers disturbance observer-based input-output finite-time control of a class of nonlinear systems with one-sided Lipschitz condition, as well as disturbances. The system model includes two parts of disturbances. One part is a norm-bounded disturbance. The other part is supposed by an exogenous system, which is supposed to have a modeling perturbation. Firstly, a reduced-order disturbance observer is designed to estimate the disturbance generated by this exogenous system. Secondly, a composite control law is designed, which includes the estimation of disturbance and the state feedback control law. Moreover, sufficient conditions are derived to guarantee that the closed-loop system is IO-FTS. Such conditions can be converted into linear matrix inequalities (LMIs). Finally, two examples are given to show the effectiveness of the proposed method.
Notations. In this paper, and denote, respectively, the spaces of -dimensional real numbers and real matrices. Let be a real symmetric matrix; means is positive definite. stands for the space of square integrable vector functions. refers to the Euclidean vector norm. represents the omitted symmetric element of a matrix. is the inner product in ; that is, given , then , where is the transpose of the column vector .
2. Problem Formulation
Consider the following nonlinear system:where is the state vector and is the control input. can represent the constant and the harmonic noises, which is described by an exogenous system in Assumption 4. is the external disturbance, which is assumed to be an arbitrary signal in . represents a nonlinear function that is continuous with respect to and . are matrices with compatible dimensions and is controllable.
The following concepts about Lipschitz property, the one-sided Lipschitz property, and quadratic inner-boundedness property for the nonlinear function are introduced to further our study.
Definition 1. The nonlinear function is said to be locally Lipschitz in a region including the origin with respect to , if there exists a constant satisfying
Definition 2. The nonlinear function is said to be one-sided Lipschitz, if there exists a constant such thatwhere is called the one-sided Lipschitz constant.
From Definitions 1 and 2, Lipschitz constant must be positive; however, one-sided Lipschitz constant can be positive, zero, or even negative. It is true that any Lipschitz function is also one-sided Lipschitz, not vice versa [24].
Definition 3. The nonlinear function is called quadratic inner-bounded in the region , if there exist constants such thatwith .
From the definition, any Lipschitz function is quadratically inner-bounded with and , but the converse is not true. Note that is not necessarily positive. In fact, if is restricted to be positive, then it can be shown that must be Lipschitz.
Assumption 4. The disturbance in (1) can be described bywhere , , and are matrices with compatible dimensions. is the addition disturbance in , which results from the perturbations and uncertainties in the exogenous system.
Remark 5. In (5), if with is held, then represents the harmonic disturbance and denotes the frequency of the harmonic disturbance [19].
The disturbance observer is constructed asand a feedback controller is designed aswhere the observer gain and the controller gain will be designed later, respectively.
Let the estimation error be . From (1) and (5)–(7), the error equation isDenote and . From (1) and (6)–(8), the resulting closed-loop system can be written in the form as follows:The reference output is set aswhere , , and .
In this work, a class of norm-bounded square integrable signals over is defined as follows:where and .
Definition 6 (IO-FTS). Given a time interval , disturbance signals defined by (11) and a weighted matrix . The closed-loop system (9) is said to be IO-FTS with respect to , if for ,
Remark 7. In [5, 7], the authors have proposed two definitions of IO-FTS for two different classes of disturbance signals, respectively, that is, the norm-bounded square integrable signals ( satisfies ) and the uniformly bounded signals ( satisfies ). Because of similar approach, we only focus on the former in Definition 6.
Remark 8. In (10), the reference output includes the estimation error . From Definition 6, our goal is that the weighted system output does not exceed threshold 1 in a given time interval ; then the estimation error might not converge to zero in a given time interval . If a smaller threshold is chosen, then the estimation error will become very small.
3. IO-FTS Analysis
In this section, we will give some sufficient conditions for IO-FTS of the closed-loop system (9).
Theorem 9. Given a scalar . Suppose the function satisfies conditions (3) and (4) with constants , , and . If there exist matrices , , and the scalars and such that the following nonlinear matrix inequalities are true:where , , , and , then the closed-loop system (9) is IO-FTS with respect to .
Proof. Consider the following Lyapunov functional candidate:where .
The time derivative of along the trajectories of system (10) is given byFrom (3), for any positive scalar , we haveFrom (4), similarly, for any positive scalar , we haveFrom (16) to (18), we havewhere .
If (13) is held, then we haveIntegrating (20) from 0 to , with , and using (11), we obtainNoting that , from (14) and (21), we haveThe proof is completed.
Theorem 10. Given a scalar . Suppose the function satisfies conditions (3) and (4) with constants , , and . If there exist matrices , , , and the scalars, , such that the following nonlinear matrix inequalities are true: where and , then the closed-loop system (9) is IO-FTS with respect to . Furthermore, and .
Proof. For (13), denote that . Left- and right-multiplying both sides of (13) by , left- and right-multiplying both sides of (14) by , respectively, and using Shur’s complement, we easily obtain (23) and (24).
For (23), it is a nonlinear matrix inequality, and there are no effective algorithms for solving , , , , , and simultaneously. If the parameters and are given in advance, then (23) is converted to an LMI. So we easily solve it by MATLAB LMI toolbox. The procedure for constructing the gains and is summarized as follows.
Step 1. For a given scalar , choose the parameters and .
Step 2. Calculate the matrices , , , and by solving LMIs (23) and (24). If there exists no solution, then the procedure returns to Step .
Step 3. Obtain the gains and .
4. The Examples
In this section, two examples are given to illustrate the effectiveness of the proposed scheme.
Example 1. Consider system (1) with (5) and (10); the system parameters are given as follows:It is shown that satisfies conditions (3) and (4) with , , and .
The matrix is selected; condition (11) can be satisfied.
Choose , , , , and . Solving (23) and (24) yieldsTherefore, we have The initial values , , and are set as 0. The simulation results are shown in Figures 1 and 2.
Figures 1 and 2 show the responses for weighted system output and disturbance estimation error , respectively. From the simulation results, we know ; this implies the effectiveness of our proposed method.
Example 2. Consider system (1) with (5) and (10); the system parameters are given as follows:The above system model (1) can be used to describe the motion of a moving object [24]. It is shown that does not satisfy condition (2). But it is one-sided Lipschitz with .
Let According to [16], the quadratically inner-bounded property of is verified in , . As the system is globally one-sided Lipschitz, that is, , . Note that the region can be made arbitrarily large by choosing appropriate values for and . If the parameters and are chosen, then .
The matrix is selected; condition (11) can be satisfied.
Choose , , , , and . Solving (23) and (24) yieldsTherefore, we have The simulation results are shown in Figures 3 and 4.
Figures 3 and 4 show the responses for weighted system output and disturbance estimation error , respectively. From the simulation results, is held, so it is concluded that the proposed method is effective.
Remark 3. In Example 1, the nonlinear term also satisfies the Lipschitz condition (2) with Lipschitz constant . On the other hand, it also satisfies conditions (3) and (4) with , , and . Obviously, the one-sided Lipschitz constant is smaller than the Lipschitz constant. In Example 2, only satisfies the one-sided Lipschitz condition. From Examples 1 and 2, can be held; that is, the weighted system output does not exceed an assigned threshold 1 during given time intervals, respectively.
5. Conclusion
This paper investigates the problem of disturbance observer-based input-output finite-time control of a class of nonlinear systems with one-sided Lipschitz condition, as well as multiple disturbances. Firstly, a reduced-order disturbance observer is designed to estimate the disturbance generated by this exogenous system. Secondly, by integrating the estimation of disturbance with the classical state feedback control law, a composite control law is designed to guarantee that the closed-loop system is IO-FTS. The obtained sufficient conditions can be converted into linear matrix inequalities (LMIs). Finally, two examples are given to show the effectiveness of the proposed method.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors are grateful for the support of the National Natural Science Foundation of China under Grants U1404610, 61473115, and 61374077.