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

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.


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  2 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 2 Mathematical Problems in Engineering uncertainties.Unlike the existing works such as in [9,12,[15][16][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.

Problem Formulation
In this paper, we consider the following class of cascaded nonlinear systems with dynamic uncertainties: where  ∈ R is the control input,  ∈ R is the system output,  = ( 1 , . . .,   ) ∈ R n are the system states, and  ∈ R r is referred to as dynamic uncertainties, which is unmeasured and hence is not available for feedback design.The continuous functions   () ( = 1, . . ., ) called the control coefficients are assumed to be unknown; particularly,   () ̸ = 0; the unmodeled (or uncertain) dynamics (⋅) and   (⋅) ( = 1, . . ., ) are locally Lipschitz.
The control objective in this paper is to find a smooth, dynamic, partial-state feedback law of the form ξ =   (, ) ,  =   (, ) , (2) where   and   are smooth functions such that all solutions ((), (), ()) in closed-loop system are bounded on [0, ∞) and specially the system states ((), ()) asymptotically converge to the origin.Toward this end, throughout the paper, we make the following assumptions on system (1).
Remark 2. According to [19], one knows that -subsystem satisfying (3) is ISS, and the function pair ( 0 ,  0  0 ) is viewed as the supply rates.Since  0 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 4.There exist known positive constants   and   ( = 1, . . ., ), 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    2 ( 1 , . . . ,  ) ,  = 1, . . . ,. (11)

Controller Design
In this section, we give the controller design procedure using the backstepping design method.
Step 1. Starting with the  1 -subsystem ẋ 1 =  1 () 2 +  1 (, , ).We consider the variable  2 as the virtual control input.Let  1 =  1 and  2 =  2 −  1 where  1 is the intermediate control input.Considering Lemma 5 and Remark 6, along solutions of (1), the time derivative of the function satisfies with a new smooth function φ1 ( As a result, there holds Considering the unknown constant  * in ( 16), we use an adaptive signal p to estimate  * .Consequently, we augment  1 with the parameter estimation error p =  * − p, such as where Υ > 0 is the design parameter.In view of ( 16) and  * = p + p, a direct substitution leads to Considering Assumption 4, we take the virtual control where ] 1 > 0 is a design constant to be determined later.Let and then we get Remark 9.It is noted that, in (19), we assume that p() ≥ 0.
In fact, from the updating law of ṗ given later, this property can be guaranteed by choosing the initial condition p(0) ≥ 0.
Step 2. Let  3 =  3 −  2 , where  2 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 )), and then we have In the same manner, using the completion of squares again, it can be verified that From Assumption 4, it is deduced that 2 ), and there holds Take the following notation: and furthermore, in view of (30) and (32), we obtain For the term of − 2 ( 1 / p) 2 , according to (33), it can be dealt with as follows: )) . ( Define the following smooth function: 2 ), and we get Take the virtual control which is such that Step  (3 ≤  ≤ ).Assume that, in Step −1, we have designed the virtual control   ( = 1, . . ., −1) and the tuning function   ( = 1, . . .,  − 1), such that the Lyapunov function In what follows, it will be shown that the property (42) also holds in Step .

Main Results
After the above controller design procedure, we are now ready to state the main results.
In terms of p() =  * − p(), we obtain the boundedness of p().Considering (53), it can be derived that   ( = 1, . . ., ) are bounded.In view of   =   −  −1 ( 0 = 0), we further obtain that the states   ( = 1, . . ., ) are bounded on [0,   ).So far all the closed-loop system signals are bounded on [0,   ).This guarantees that the finite time escape will not happen.Therefore, it is natural that   can be maximized to +∞ by means of Theorem 3.3 in [21].Next we will prove the convergence property of (60).
It is noted that, under some stronger conditions, the designed control law can be a linear controller.In fact, we have the following statement.
Then, the proposed design method can result in a linear control law where ]  ( = 1, . . ., ) 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 p for  * is no longer needed.Moreover, since conditions (59) and (68) are satisfied, the function (⋅) in ( 7) can be chosen as a constant  > 0. For  = 1, . . ., , we consider the following function: where   ( = 1, . . ., ) are design constants.In view of ( with some positive constants ρ ( = 1, . . ., ).We will prove that if the constants   and ρ are chosen suitably, the following inequality holds: 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 ]  ( = 1, . . ., ).

Simulation Example
In this section, we provide a simulation example to illustrate the proposed method in the paper.Consider the following nonlinear systems:  4  1 .Next, we use the proposed algorithm in Section 3 to design the partial-state feedback controller.
Step 1.We consider the function  1 = (1/2) 2  1 + (1/2) 2 , whose time derivative satisfies Like the calculations in ( 14), we have where  0 > 0 satisfying  2 () ≤  0 .Define  * = max{ 0 , |  |,  2  |  = 1, 2, 3}, and we get with a new smooth function φ1 ( 1 ) ≥ 1 + 1/2 +  2 1 > 0. Similar to (17), we augment  1 as follows: where Υ > 0 is the design parameter.In view of (79) and  * = p + p, 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 φ2 ( , and one gets As a result, with  2 =  1 + Υ 2 2 φ2 ( 1 , p), we have For the term of − which is such that The Lyapunov function  2 can be made V 2 ≤ 0 by choosing ] 1 > 5, ] 2 > 0, and the stability analysis can be done in the similar way to Theorem 11.The simulation plots shown in Figures 1 and 2  According to our results reported in Theorem 11, the states (,  1 ,  2 ) must asymptotically converge to the origin and the parameter estimate p is bounded on [0, ∞).This fact can be verified from Figure 1, which plots the trajectories of these dynamic signals ((),  1 (),  2 (), p()).It can be seen that, at about  = 4.17 s,  = 0.68 s, and  = 0.59 s, the states approach the origin, and at  = 0.36 s, the parameter estimate p() is bounded near  * = 1.1.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  = 0.51 s, 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.

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.

Figure 1 :
Figure 1: The response of the closed-loop system.

20 Figure 2 :
Figure 2: The control input of the closed-loop system.