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, 1517], 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 formwhere 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 thatwhere 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 thatwhere are known smooth functions and ;   .

Assumption 4. There exist known positive constants and , such thatTo 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. Supposeand then we can choose a positive continuous function , such that the functionis another candidate ISS-Lyapunov function satisfyingwhere 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 functionsatisfiesAccording to Assumption 3 and the completion of squares, we haveDefine , and we get with a new smooth function . As a result, there holdsConsidering the unknown constant in (16), we use an adaptive signal to estimate . Consequently, we augment with the parameter estimation error , such aswhere is the design parameter. In view of (16) and , a direct substitution leads toConsidering Assumption 4, we take the virtual controlwhere is a design constant to be determined later. Letand 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 haveDefine , and then we haveIn the same manner, using the completion of squares again, it can be verified that As a result,Define , and thenLet , and then we getFrom Assumption 4, it is deduced thatDefine , and there holdsTake the following notation:and furthermore, in view of (30) and (32), we obtainConsidering 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 getDenote , and a direct substitution leads toTake the virtual controlwhich is such that

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

Let be the virtual control and . Consider the Lyapunov functionTo 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 thatSimilar to (31) and (32), there exists a smooth function such thatLetand 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 getBy substituting (51) into (49), it follows thatTake the virtual controland 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 functionsatisfiesThis 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 conditionsThen 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 thatOne can choose positive constants satisfyingThen, from (58), (61), and (62), it follows thatNow, 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 obtainThis 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 lawwhere 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 getwith 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 getIn view of , a direct substitution in (74) results inTo 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 satisfiesLike the calculations in (14), we havewhere satisfying .
Define , and we getwith 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 toWe take the virtual control and the tuning functionand then we get

Step 2. To find the actual control law , we consider the Lyapunov functionIn view of (78) and (85), we have