Adaptive tracking control for nonlinear systems with uncertain control gains and its application to a TCP/AQM network

This paper is concerned with the adaptive tracking control problem for nonlinear systems with virtual control coefficients including known and unknown items. The known items are employed for controller design directly, such that more information is utilized to achieve better performance. To deal with the unknown items, a novel real control law is firstly constructed by introducing an auxiliary system. The proposed controller is designed and applied to an uncertain TCP/AQM network system, which guarantees the practical boundedness of all the signals in the closed-loop system. Finally, the effectiveness and practicability of the developed control strategy are validated by simulation results.

controller was developed for a TCP/AQM network based on a new finite-time performance function 33 .Considering fuzzy logic and funnel control, an adaptive congestion control approach was given for a TCP/AQM network 34 .In view of barrier Lyapunov and neural network, the adaptive TCP/AQM network congestion control was studied 35 .Furthermore, aiming at a novel system model with the uncertain link bandwidth, which was supposed to be known [33][34][35] , two adaptive TCP network congestion controllers were designed 36,37 .However, it is noted that the link bandwidth C is unknown, but the round-trip time R(t) is taken to be known, which is a function of C in 36,37 .Obviously, this is contradictory.
Encouraged by the previous results, this paper is devoted to the adaptive control problem for nonlinear systems with unknown virtual control coefficients and its application to TCP/AQM network systems.Firstly, both controller design and stability analysis are achieved for a class of uncertain nonlinear systems.Furthermore, the proposed control method is applied to a TCP/AQM network, which guarantees the practical boundness of all the signals in the closed-loop system.Finally, the effectiveness of the developed controller is validated by two simulation examples.
The main contributions of this paper are stated as (1) The virtual control coefficients of nonlinear systems are formulated by known and unknown terms so that the bounds of virtual control coefficients are smaller than those expressed as one unknown item in [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] .In applications, virtual control coefficients of many real systems are partially known.Therefore, It is feasible to separate virtual control coefficients into known constants and unknown functions by selecting suitable nominal values of system states according to practical requirements.This is demonstrated by the application to a TCP/AQM network.(2) By defining the variable ˙ r in (23), a new auxiliary system ( 21) is designed to compensate the items due to the unknown parts of virtual control coefficients, and a novel control law ( 17) is developed, which ensures the practical bounded control of nonlinear systems.(3) Compared to these adaptive TCP/AQM network congestion control schemes [33][34][35][36][37] , not only the link bandwidth C but also the round-trip time R(t) are considered to be unknown.This is an improvement on the methods in [33][34][35] .Simultaneously, the contradiction existing in 36,37 is excluded.
The outline of the paper is organized as: second section presents the problem formulation and preliminaries.
The control design and stability analysis is given in next section, and its application to a TCP/AQM network follows in next section.Finally, the simulation results and conclusion are shown in section V and VI, respectively.
where x 1 and x 2 are the queue length of router and TCP window size, respectively, T p is the propagation delay, N is the TCP network load, both known, C is the link bandwidth, which is an unknown constant, u r is the packet drop probability.
Remark 1 From the actual physical connotations, it is reasonable to assume that , C m and C M being positive constants.Thus, it is known that there is no singularity in the virtual control coefficients.Firstly, C x 1 +CT p is transformed into the following form by introducing x 10 , C 0 , which are the nominal values of x 1 and C. where and Assumption 1 Suppose that the queue length of router x 1 and link bandwidth C are bounded.
According to Assumption 1, it is shown that is bounded, which is expressed as with and Then, the system (1) is changed to be Remark 2 It is easy to check that the range of is much smaller than the bound of C x 1 +CT p .Besides, all the link bandwidth C are unknown, such that the contradiction in 36,37 does not exist in this paper.
Assumption 2 Suppose that the continuity and boundedness of ideal trajectory vector ȳd Different from the existing dynamics of uncertain TCP/AQM networks, the virtual control coefficients in this paper are divided into the known item p 0 and unknown bounded item .There exist ε r1 N�x 2 , ε r2 ∂ρ r1 x 1 N�x 2 , ε r2 �v r and ε r2 � inevitably in the following controller design, which will be explained later.It is a severe chal- lenge to deal with the above items.

Controller design and main results
The following coordinate transformation is employed to design a controller via backstepping.
Step 1: Select the Lyapunov function candidate as where q r > 0 is a constant and C = C − C * with C * being the approximation of C.
Vr1 is deduced as Design the first virtual control signal as where k r1 is a positive constant.Invoking (12) leads to where Step 2: The 2th Lyapunov function candidate is given as By some direct calculations, Vr2 is written as  18) by (19) produces Design the following function where v r max is the practical maximum of v r , and Then, choose the control law as and ( 16) www.nature.com/scientificreports/ where k rf , c r1 and c r2 are positive constants and c r1 < c r2 /k rf , � r (0) is the initial value of r , which is chosen in the interval of c r1 , c r2 /k rf , i = 1, 2 , and Remark 3 Although there is ρr1 in the controller (22), the repeated differentiation will not occur due to the system order only being 2. Besides, the proposed control scheme does not utilize approximators based on fuzzy logics or neural networks, which means that the computational complexity does not increase exponentially as the number of the rules increases.Therefore, the explosion of computation does nor exist.At worst, there have been many methods to deal with the problem 38 .
Substituting (22) in (20) results in Construct the adaptive law as By invoking (26) and considering (5), Vr2 is rewritten as The main results of the paper are stated below.
Theorem 1 Considering the nonlinear system (1) with Assumption 1, the developed control strategy, including the control law (17) and adaptive law (26), guarantees the practical boundedness of all the signals in the resulting closed-loop systems.

Proof Design the 3th Lyapunov synthesis candidate as
Case 1: Then, Vr3 is computed as It is known from ( 29) that V r3 is bounded, which means that ε r1 , ε r2 , C * , and r are all bounded.From ε r1 = x 1 − y * d and the boundedness of y * d , it is obtained that x 1 is bounded.According to (12), the boundedness of ρ r1 is verified, which implies that x 2 is bounded by considering ε r2 = x 2 − ρ r1 .It follows that u r is bounded.Therefore, the conclusion as Theorem 1 is obtained.
Case 2: www.nature.com/scientificreports/It can not be obtained that Vr3 ≤ −κV r3 + γ , which means ε i and C * may not be convergent.However, Case 2 will be switched to Case 1 when |ε ri | ≥ σ ri or C * ≥ Ĉ , such that the practical boundedness of all the signals is obtained.
Remark 4 (1) The range of k ri is wide.The bigger the parameter k ri , the faster the response and smaller the steady state error.However, it cannot be too big.(2) k rf should be small enough and chosen according to the value of c r2 , such that c r2 /k rf is not too big.(3) q r should be big enough, which can guarantee that the steady state error is small enough.(4) c r1 and c r2 should satisfy c r1 < c r2 /k rf , and c r2 can not be selected too big, which may cause a bigger steady state error.

Simulation results
In this section, the proposed control scheme is simulated for an uncertain TCP/AQM network and compared with the controller in 37 , which are abbreviated to be C2 and C1, respectively.
Figures 1, 2, 3, 4, 5 and reff6 show the simulation results.In Fig. 1, the tracking errors under C2 and C1 of the queue length are shown.It is easy to see that the error under C2 is smaller than the error under C1, whose absolute mean values are 0.0824, 1.0564 and root mean square values are 0.5063, 2.5891, respectively.The actual queue lengths y * are displayed in Fig. 2, which can track desired queue length y * d , after transient oscillation.However, the tracking performance under C2 is much better than C1 during the transient state.Figure 3 gives the packet drop probability u r , which varies between 0 and 1 in the transient state and varies around 0.1 in the steady state.The TCP window size x 2 is drawn in Fig. 4, which oscillates before 0.2s and varies according to y * d .Figure 5 presents the adaptive link bandwidth C * , it can be known that it tends to its real value 2000b/s.The auxiliary variable r is presented in Fig. 6, which is bounded and converges to 10.

Conclusion
This paper has been devoted to the study of adaptive tracking control for nonlinear systems in a new form, whose virtual control coefficients consist of known and unknown items.The proposed controller not only utilized known information fully to pursue better control performance, but also handled unknown items by defining a novel auxiliary system.To demonstrate the feasibility of the developed control scheme, it was further applied to the congestion control of an uncertain TCP/AQM network system.In future, we plan to combine the proposed control scheme with fixed-time control 39,40 and apply it to some other real systems, such as robots, quadrotors, and so on.