Global Asymptotic Stabilization Control for a Class of Nonlinear Systems with Dynamic Uncertainties

Abstract

This paper is concerned with the global asymptotic stabilization control problem for a class of nonlinear systems with input-to-state stable (ISS) dynamic uncertainties and uncertain time-varying control coefficients. Unlike the existing works, the ISS dynamic uncertainty is characterized by the uncertain supply rates. By using the backstepping control approach, a systematic controller design procedure is developed. The designed control law can guarantee that the system states are asymptotically regulated to the origin from any initial conditions and the other signals are bounded in closed-loop systems. Moreover, it is shown that, under some additional conditions, a linear control law can be designed by the proposed methodology. The simulation example demonstrates its effectiveness.

1. Introduction

The nonlinear control theory is an active research direction in the control field because of its widespread applications in the real world. During the past two decades, various novel methodologies have been generated for the nonlinear feedback control; see the recent survey [1] and references therein for an interesting introduction to this area. One of the influential notions is the input-to-state stability (ISS) and its several variants. Since they are introduced by Sontag in [2, 3], the notion of ISS as well as its integral variant—integral ISS (iISS)—has become a foundational concept upon which much of modern nonlinear feedback analysis and design rest. As noted in [4], ISS provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite gains, and it plays a central role in recursive design, coprime factorizations, controllers for nonminimum phase systems, and many other areas. Based on the series of works on ISS, the nonlinear small-gain theorem was proposed in the state-space setting and is widely used in the stability analysis and control design for complex interconnected systems in [5]. The stochastic results can be found in [6, 7] and the references therein.

It is noted that a unifying framework is presented in [8] for the global output feedback regulation control problem from ISS to iISS. The framework established in [8] extends many known classes of output feedback form systems. However, the system uncertainties investigated there depend only on the system output and the inverse system state. With unmeasured states dependent growth, in [9, 10], the problem of global stabilization by output/state feedback is investigated for a class of nonlinear systems with uncertain control coefficients. However, there is no dynamic uncertainty for the system under consideration. In [11], this work is further studied for a larger class of nonlinear uncertain systems, in which the observer gain is governed by a Riccati differential equation. Moreover, the output regulation problem is also considered in [12] for this class of nonlinear systems with iISS inverse dynamics. Later, in [13, 14], this work is further investigated for the nonlinear systems with uncertain nonlinearities dependent on all unmeasured states. However, the control coefficients in above results are required to be known a priori or unknown nonzero constants. In [15], the global set-point tracking control is investigated for a class of cascaded nonlinear systems with unknown control coefficients. However, a restrictive condition is that the control coefficients are required to have the same signs.

In this paper, we will further investigate this problem for a class of nonlinear systems with more general nonlinear uncertainties. Unlike the existing works such as in [9, 12, 15–17], the studied system is with the uncertain control coefficients, which could be unknown time-varying functions. Another feature of this work is that the dynamic uncertainties are characterized by the uncertain ISS supply rates. This is different from the existing results reported in literatures where the ISS dynamic uncertainty is investigated under the hypothesis that supply rates are known a priori such as in [8, 11, 13, 15]. With the help of the backstepping approach [18], we design a robust adaptive controller which could achieve the system states convergent to the origin while the other signals are bounded. Moreover, it is of interest to note that a linear control law can be designed using the developed scheme if some stronger conditions are imposed on the nonlinear system.

The rest of the paper is organized as follows. In Section 2, we provide some mathematical preliminaries and state the problem. The controller design procedure is developed in Section 3, and the main result is presented in Section 4. Section 5 illustrates the obtained results by a numerical example. Section 6 concludes this paper.

Notation. Let denote the set of all (positive) real numbers and let denote the real -dimensional space. For a given vector or matrix , denotes its transpose. For any column vector , denotes the column vector consisting of the first elements of in the original order; that is, . Specifically, for , , . A continuous function is said to belong to class- if it is strictly increasing and . It is said to belong to class- if and as . The notation means that there exist two positive constants and such that , .

2. Problem Formulation

In this paper, we consider the following class of cascaded nonlinear systems with dynamic uncertainties: where is the control input, is the system output, are the system states, and is referred to as dynamic uncertainties, which is unmeasured and hence is not available for feedback design. The continuous functions called the control coefficients are assumed to be unknown; particularly, ; the unmodeled (or uncertain) dynamics and are locally Lipschitz.

The control objective in this paper is to find a smooth, dynamic, partial-state feedback law of the form where and are smooth functions such that all solutions in closed-loop system are bounded on and specially the system states asymptotically converge to the origin. Toward this end, throughout the paper, we make the following assumptions on system (1).

Assumption 1. The -subsystem is input-to-state stable (ISS) with state and input ; that is, there exists a positive-definite and proper ISS-Lyapunov function , such that where and is an unknown constant.

Remark 2. According to [19], one knows that -subsystem satisfying (3) is ISS, and the function pair is viewed as the supply rates. Since in (3) is unknown, the dynamic uncertainty has uncertain ISS supply rates. This is different from the existing results reported in literatures, where the ISS dynamic uncertainty is investigated with the supply rates assumed to be known a priori, such as [8, 11, 13, 15].

Assumption 3. For the uncertain nonlinearities , there exist unknown positive constants such that where are known smooth functions and ;   .

Assumption 4. There exist known positive constants and , such that To deal with the unmeasured state , we have the following lemma, which plays an important role in the coming feedback design and stability analysis.

Lemma 5. Consider the -subsystem satisfying Assumption 1. Suppose and then we can choose a positive continuous function , such that the function is another candidate ISS-Lyapunov function satisfying where is a small design constant, is an unknown constant, and is a -function with .

Remark 6. If , according to Lemma  2 in [19], there exists a smooth nondecreasing function satisfying

Lemma 7. For any function , there exist continuous functions , such that

Remark 8. According to Lemma 7, from in Assumption 3, it is known that, for each , there exist smooth functions satisfying

3. Controller Design

In this section, we give the controller design procedure using the backstepping design method.

Step 1. Starting with the -subsystem . We consider the variable as the virtual control input. Let and where is the intermediate control input. Considering Lemma 5 and Remark 6, along solutions of (1), the time derivative of the function satisfies According to Assumption 3 and the completion of squares, we have Define , and we get with a new smooth function . As a result, there holds Considering the unknown constant in (16), we use an adaptive signal to estimate . Consequently, we augment with the parameter estimation error , such as where is the design parameter. In view of (16) and , a direct substitution leads to Considering Assumption 4, we take the virtual control where is a design constant to be determined later. Let and then we get

Remark 9. It is noted that, in (19), we assume that . In fact, from the updating law of given later, this property can be guaranteed by choosing the initial condition . Alternatively, using the idea in [20], we also can apply the or instead of .

Step 2. Let , where is the virtual control law. We consider the Lyapunov function In view of (21), we have From (4) and (11), the following calculations hold: Like the calculations in (14), by completing the squares, we have Define , and then we have In the same manner, using the completion of squares again, it can be verified that As a result, Define , and then Let , and then we get From Assumption 4, it is deduced that Define , and there holds Take the following notation: and furthermore, in view of (30) and (32), we obtain Considering is unknown, the term of can not be canceled. In fact, we express (34) as For the term of , according to (33), it can be dealt with as follows: Define the following smooth function: + , and we get Denote , and a direct substitution leads to Take the virtual control which is such that

Step (). Assume that, in Step , we have designed the virtual control and the tuning function , such that the Lyapunov function satisfies In what follows, it will be shown that the property (42) also holds in Step .

Let be the virtual control and . Consider the Lyapunov function To begin with, the dynamics of can be expressed as For notational convenience, denote . From , like the calculations in (26), it can be verified that there exist smooth functions , such that, for , Define ; from (45), it follows that Similar to (31) and (32), there exists a smooth function such that Let and then, from (47) and (48), there holds

Remark 10. In (49), we subtract two terms and in the brackets to generate the term .
However, from (49), it can be seen that, due to the unknown control coefficient , the terms and could not be directly canceled by the coming virtual control . We get around this burden by the following estimates: Let , , and then we get By substituting (51) into (49), it follows that Take the virtual control and then the following holds: In particular, when , the actual control appears, and we choose the controller and updating law for as follows: such that the Lyapunov function satisfies This completes the controller design procedure.

4. Main Results

After the above controller design procedure, we are now ready to state the main results.

Theorem 11. Suppose the investigated system (1) satisfies Assumptions 1, 3, and 4 together with the local conditions Then all the signals of the closed-loop system (1) with the controller (55) and updating law (56) are bounded on . Specifically, the following convergent property holds:

Proof. From the local conditions (59), one can choose the smooth function such that One can choose positive constants satisfying Then, from (58), (61), and (62), it follows that Now, assume that the maximal interval of existence of the solution of the closed-loop system starting from any given initial conditions is for some . In view of , from (63), it can be concluded that and hence the variables are bounded on . In terms of , we obtain the boundedness of . Considering (53), it can be derived that are bounded. In view of , we further obtain that the states are bounded on .
So far all the closed-loop system signals are bounded on . This guarantees that the finite time escape will not happen. Therefore, it is natural that can be maximized to by means of Theorem in [21]. Next we will prove the convergence property of (60).
Again, according to (63), considering and are -functions, a direct application of LaSalle’s invariance principle in [21] guarantees the convergence property of ; that is, As a consequence, from (53) and (55), the following holds: Particularly, In terms of (64), (65), and , we can obtain This completes the proof.

It is noted that, under some stronger conditions, the designed control law can be a linear controller. In fact, we have the following statement.

Theorem 12. Suppose that the conditions for Theorem 11 are satisfied with known a priori and the following additional assumptions hold: (i)The uncertain functions satisfy (ii) There exist known constants , such that (iii) where is a positive constant.
Then, the proposed design method can result in a linear control law where are some sufficiently large positive constants.

Proof. Under the above hypotheses (i)–(iii), it is known that the constant is known, and hence the estimation for is no longer needed. Moreover, since conditions (59) and (68) are satisfied, the function in (7) can be chosen as a constant . For , we consider the following function: where are design constants. In view of (3), we can get with some positive constants . We will prove that if the constants and are chosen suitably, the following inequality holds: In fact, because of , there exist positive constants , satisfying , for . Take , , and then and . In view of , there exist positive constants , satisfying if . Similarly, if we take and , then , . In the finite closed interval , let be the minimum value of and let be the maximum value of , respectively, and if we take , , then , . According to the previous analysis, we choose and , and then (72) holds. Therefore we get In view of , a direct substitution in (74) results in To deal with the unmeasured dynamics in this case, we can choose the candidate Lyapunov function as follows: Consequently, a modified version of the design procedure in Section 3 leads to the linear control law with some sufficiently large positive constants .

5. Simulation Example

In this section, we provide a simulation example to illustrate the proposed method in the paper. Consider the following nonlinear systems: with , , , and . The inverse system is ISS, and is a candidate ISS-Lyapunov function with the supply pair ,

Next, we use the proposed algorithm in Section 3 to design the partial-state feedback controller.

Step 1. We consider the function , whose time derivative satisfies Like the calculations in (14), we have where satisfying .
Define , and we get with a new smooth function .
Similar to (17), we augment as follows: where is the design parameter. In view of (79) and , a direct substitution leads to We take the virtual control and the tuning function and then we get

Step 2. To find the actual control law , we consider the Lyapunov function In view of (78) and (85), we have As in (80), we have Consequently, in view of the definition of , the following holds: with . Define , and one gets As a result, with , we have For the term of , according to (36), it can be verified that Define , and we get Denote , and a direct substitution leads to As in Step , we design the following partial-state feedback controller: with the updating law for unknown parameter which is such that

The Lyapunov function can be made by choosing , , and the stability analysis can be done in the similar way to Theorem 11. The simulation plots shown in Figures 1 and 2 are performed by MATLAB with the following parameters: , , , , , , , , , the derived functions: , , , , , , and the initial conditions: , , , .

Figure 1 

The response of the closed-loop system.

Figure 2 

The control input of the closed-loop system.

According to our results reported in Theorem 11, the states must asymptotically converge to the origin and the parameter estimate is bounded on . This fact can be verified from Figure 1, which plots the trajectories of these dynamic signals . It can be seen that, at about , , and , the states approach the origin, and at , the parameter estimate is bounded near . In addition, according to Theorem 11, the control input is convergent to the origin. Figure 2 demonstrates this result, and it can be shown that, at about , the input signal approaches the origin. As can be seen from Figures 1 and 2, our control scheme provides a fairly good asymptotic stabilization performance.

6. Conclusion

The state feedback stabilization problem is investigated for a class of nonlinear systems with dynamic uncertainties and uncertain control coefficients in this paper. The dynamic uncertainty is characterized by the uncertain ISS supply rates. A global asymptotic stabilization control scheme is proposed using the backstepping design scheme. The tuning function technique is applied in this procedure, which avoids the disadvantage of overparameterization. It is shown that, under some more restrictive conditions, a linear state feedback controller can be designed by the presented algorithm. The simulation example demonstrates the effectiveness of the proposed method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China under Grant no. 61304008, the Shandong Provincial Natural Science Foundation of China under Grant no. ZR2013FQ033, and the Doctor Research Foundation of Shandong Jianzhu University under Grant no. XNBS1272.