A Variable Power Surface Error Function backstepping based Dynamic Surface Control of Non-Lower Triangular Nonlinear Systems
Abdulrazaq Nafiu Abubakar Ali Nasir Md Muzakkir Quamar





























a]Abdulrazaq Nafiu Abubakar 
a,b,c]Ali Nasir*
a]Md Muzakkir Quamar 
[a]Control and Instrumentation Engineering Department, KFUPM, Dhahran 31261, SAUDI Arabia
[b]Interdisciplinary Research Center for Intelligent Manufacturing and Robotics, KFUPM, Dhahran 31261, SAUDI Arabia
[c]Interdisciplinary Research Center for Aviation and Space Exploration (guest affiliate), KFUPM, Dhahran 31261, SAUDI Arabia

*ali.nasir@kfupm.edu.sa

























§ ABSTRACT 
A control design for error reduction in the tracking control for a class of non-lower triangular nonlinear systems is presented by combining techniques of Variable Power Surface Error Function (VPSEF), backstepping, and dynamic surface control. At each step of design, a surface error is obtained, and based on its magnitude, the VPSEF technique decides the surface error to be used. Thus, the backstepping-based virtual and actual control law is designed to stabilize the corresponding subsystem. To address the issue of circular structure, a first-order low-pass filter is used to handle the virtual control signal at each intermediate stage of the recursive design. The stability analysis of the closed-loop system demonstrates that all signals indicate semi-global uniform ultimate boundedness. Moreover, by using the switching strategy of the control input using the VPSEF technique suitably, it is possible to ensure that the steady-state tracking error converges to a neighborhood of zero with an arbitrarily very small size. The effectiveness of the proposed concept has been verified using two different simulated 
demonstrations.




Error correction Surface error Backstepping Dynamic surface control (DSC) Non-lower Triangular Nonlinear Systems Variable Power Surface Error Function (VPSEF)












§ INTRODUCTION

The backstepping method is a frequently used strategy for designing controllers and analyzing stability in nonlinear systems that have a strictly triangular structure <cit.>, <cit.>, <cit.>. Nevertheless, the system architecture of triangular nonlinear systems may be influenced by many uncertainties, such as modeling inaccuracies, unidentified parameters, time delays, disturbances, unknown faults, input and state delays, and so on <cit.>,<cit.>,<cit.>,<cit.>,<cit.>,<cit.>. Several design approaches for adaptive control were developed specifically for lower triangular nonlinear systems with linearly parameterized uncertainty using backstepping techniques <cit.>,<cit.>,<cit.>. while online approximation methods based adaptive backstepping control approaches were presented for specific types of lower triangular systems with nonlinearly specified uncertainty for example adaptive control based neural network <cit.>, Adaptive fuzzy control <cit.>. Traditional backstepping based control design approaches have a limitation related to the complexity of controller design. In order to simplify this control design, several approaches such as dynamic surface control (DSC) are used <cit.>, <cit.>, <cit.>, minimal learning parameter <cit.>, <cit.>, <cit.>, online single neural network approximation <cit.>, <cit.> and so on were integrated into the framework of backstepping-based control design.

Although there has been much research on the management of lower triangular nonlinear systems using backstepping design, there are comparatively few findings in research about the control of nonlinear systems having non-lower triangular structures. A significant challenge in designing controls for nonlinear systems that are not lower triangular is the circular structure of the controller. This circular structure arises from the structural characteristics of the class of nonlinear systems and the differentiation of virtual control laws in the backstepping design stage. Indeed, in certain lower triangular nonlinear systems with pure-feedback structure, the issue of circular structure also arises in backstepping based control design. Most recently relevant research to the backstepping approach as the circularity problem solver include <cit.> which introduces a tracking control design technique suited for nonlinear systems with good synthesis and filtering to deal with slow rates and circular structure drawbacks. The authors in <cit.> enhance the capacity to control trajectories for spherical rolling robots by using the backstepping approach on a ball-pendulum system. The performance of the controller is shown via simulations. the study in <cit.> examine output feedback approaches for integrators that are globally stable and systems that include uncertainties and also consider the use of delayed static observers and adding backstepping techniques to expand the range of control applications. A control design strategy based on adaptive backstepping was presented in <cit.> for a specific kind of uncertain affine pure-feedback nonlinear systems. In the study, a DSC approach was used to resolve the issue of circular construction in the design. Nevertheless, the dynamic surface control reduces the circular structure, but there are tracking errors that tends to distabilize the system at initial stage with high overshoot which is due to the initial magnitude of the error, thus, reducing the performance of the controller. These errors tends to reduce as the controller becomes more adaptive. In this situation, the control performance is good but the error changes with time. Therefore, there is a need to control  the magnitude of the error. 

In this work, a problem of tracking error reduction for non lower triangular nonlinear class of system is considered. A Variable Power Surface Error Function (VPSEF) backstepping based DSC design method will be developed for an nth-order non lower triangular nonlinear system. At each step of the control design, say the intermediate step i of the design being recursive, a switching full state feedback virtual control law is firstly designed to stabilize the corresponding switched subsystem i=1,2,3,...n-1. This control law will switched based on the magnitude of the error i.e. being higher or lower error at a time t. Therefore, this entails that, the surface error of the control law will be different in respect of the magnitude of the error. By using a first-order low-pass filter to process the switching virtual control signal and allowing the filtered output to be used in the next design stage, the issue of circular construction may be efficiently resolved. Finally, in the last step n, a complete state feedback real switching control is provided to stabilize the final subsystem. To demonstrate reliability of the suggested method, the stability analysis of the system under study is shown in a subsection below. The closed-loop system signals are uniformly finally bounded, indicating that the system tracking error may converge to a small neighborhood of zero by carefully reducing the error through the switched strategy and selecting parameters.

Motivated by the literature and the discussion above, this article studies the error reduction in trajectory tracking of a special class of nonlinear system. First, a switching virtual controller is designed for the ith subsystem considering the surface error of the backstepping technique to switched based on the magnitude of the error at an instant time t. We then apply the first order low pass filter to process the virtual switching control signal and finally the full state feedback actual switching control is designed to stabilize the last subsystem.  To the best of our knowledge, this is the first time the Switching control of DSC based backstepping, is proposed for the tracking control of a non lower triangular nonlinear system. The main contributions of this paper are enumerated as follows:


    
  * Contrary to the the backstepping methods in  <cit.>, <cit.>, <cit.>, <cit.>, we used the switching control strategy to eliminate the effect of high and low surface errors related to the tracking performance using the VPSEF technique.
    
    
  * Compared with the control methods in <cit.>, <cit.> and <cit.> the switched VPSEF  backstepping based DSC show a superior trajectory tracking in respect of the tracking error and change in reference signals.
    
    
  * The Control strategy proposed guarantees stability, fast and finite time convergence to the desired signals.


The paper is outlined as follows. The nonlinear dynamic model of the non lower triangular nonlinear system is given in Section <ref>. The proposed switching VPSEF backstepping based DSC are designed in Section <ref>. Stability analysis is proved in <ref>. Simulation results are presented in Section <ref>. Some concluding remarks are provided in Section <ref>.



§ MATHEMATICAL MODELLING
 
The description as well as the mathematical model of a class of non lower triangular nonlinear systems as follow has been extensively studied in the literature <cit.>,  <cit.>, <cit.>. The state space dynamic equations describing such class of nonlinear systems are 


    dx_i    =f_i^*(x_1, x_2, …, x_n),    i=1,2, …, n-1  1
    
    dx_n    =g_n(x_1, x_2, …, x_n) u+f_n(x_1, x_2, …, x_n) 
    
    y     =x_1


in the above equation x_i∈ R are the state variables of the system, i=1,2, …, n; u ∈ R and y ∈ R are input control signal and the output to the system respectively; f_i^*(·) and f_n(·) are nonlinear functions that are known to be smooth, i=1,2, …, n ; g_n(·) is a function being continuous that satisfies g_n(·) ≠ 0.
The objective of the control to be applied in (1) is, to design a controller for the system such that, the output y will track a reference signal given as  y_r, and therefore all signals in the closed-loop system are evenly distributed and eventually bounded.  

  
The system (1) has a non-lower triangular shape, which means that the usual design approaches based on backstepping are not applicable. This study aims to design a switching control approach for the class of systems using an error reduction DSC based methodology.  



  
    The signs of the term ∂ f_i^*(·) / ∂ x_i+1 are well known, i=1,2, …, n-1. For simplicity, it is considered that ∂ f_i^*(·) / ∂ x_i+1≥ 0.


  
    The reference signal y_r is substantially continuous, and y_r, ẏ_r and ÿ_r are sufficiently bounded.




§ SWITCHING BACKSTEPPING BASED DSC CONTROL DESIGN
 
 The objective of our control is to provide a switching control scheme that minimizes errors using backstepping based DSC approaches, ensuring the semi-globally uniformly stable behavior of the closed loop system. Like the conventional backstepping-based DSC approaches, the recursive design process consists of n procedures and two different controllers depending on the magnitude of the error. At each intermediate step i, a virtual switching control law β_i+1 is implemented to stabilize the i th subsystem, where i ranges from 1 to n-1. In order to avoid cyclic dependence of control laws in the following procedures, a first-order low-pass filter is used to process the virtual control signal β_i+1. The resulting signal from the filter is then fed into the subsequent subsystem. In the last stage, an actual control is formulated to stabilize the nth subsystem. The aforementioned process results in the formation of a controller that consists of n-1 virtual control laws, n-1 first-order filters, and an actual control law. In order to implement the design of the switching control technique, the following theorem are presented.  
   Variable Power Surface Error Function (VPSEF) 

Within the framework of backstepping-based dynamic surface control, we present the notion of a Variable Power Surface Error Function (VPSEF), which is denoted as ψ_i, that is designed to adjust the influence of surface error on the effectiveness of control, depending on their magnitude. The VPSEF is represented by the following equation:


    ψ_1=(x_1-yr)^p/q
    2ψ_i=(x_i-a_i)^p/q


where x_1 represent the current state variable, yr denotes the reference signal, the parameters p and q determine the behavior of the function,  with p>q for higher errors and p<q for smaller errors. 

This formulation enables the control response to be adjusted adaptively based on the size of the error. This allows for precise control actions that are customized to different levels of deviation from the reference trajectory. The choice of p and q variables allows for adjusting the VPSEF to the unique dynamics and performance needs of a system. 

To enhance the clarity and conciseness of the design process, system (1) is first reformulated in the following form:


    dx_i    =x_i+1+f_i(x_1, x_2, …, x_n), i=1,2, …, n-1  3
    
        dx_n    =g_n(x_1, x_2, …, x_n) u+f_n(x_1, x_2, …, x_n) 
    
    y     =x_1


where f_i(x_1, x_2, …, x_n)=f_i^*(x_1, x_2, …, x_n)-x_i+1 are well-known nonlinear smooth function, i=1,2, …, n-1.

The detailed methodology for controller design is provided in the following.

Step 1: Given the surface tracking error ψ_1=x_1-y_r (i.e. the initial surface error). By applying the VPSEF at each iteration step, by checking the switching condition below:

    ψ_i=(x_i-y_r)^p/q  p>q 
     for- ψ_i > Threshold,  4
    ψ_i=(x_i-y_r)^p/q  p<q  
     for- ψ_i < Threshold,   
    
           Threshold>0





We can obtain the derivative of ψ_i as 




    ψ̇_1   = ẋ_1-ẏ_r5
       =x_2+f_1(x_1, x_2, …, x_n)-ẏ_r

 If we choose x_2 as the control signal for subsystem (5), we can design a virtual control that will stabilize the subsystem in the following manner.    

    β_2=-k_1ψ_1-f_1(x_1, x_2, …, x_n)+ẏ_r6

where k_1>0 being a control parameter and  ψ_1 is the switching parameter and is obtained as in equation (4). Therefore, we obtained different values of β_2 depending on the magnitude of the errors and thus, dual virtual control.

To address the issue of circular structure, a DSC approach is used to manipulate the virtual control signal. This involves the implementation of a first-order low-pass filter, as shown below.


    σ_2ȧ_2+a_2=β_27
  
The variables β_2 and a_2 represent the input and output of the filter, respectively. Additionally, σ_2 is a positive filtering parameter.


  
 The primary focus of the control is to ensure that the tracking error ψ_1 converges to a limited neighborhood around zero. Hence, if x_2 is derived as -k_1ψ_1-f_1(x_1, x_2, …, x_n)+ẏ_r, Subsequently, the subsystem (5) may be mathematically converted into the equation ψ̇_1=-k_1ψ_1, where ψ_1 represents the tracking error. It is important to note that the tracking error ψ_1 gradually approaches zero in an asymptotic manner. Nevertheless, x_2 is considered a system state variable and is not subject to design. Thus, a switching virtual control law β_2 is implemented, with the aim of ensuring the convergence of x_2-β_2 to neighborhood of zero.   


  
Since the switching virtual control law β_2 incorporates x_n, the derivative of β_2 will therefore include the real control input u. If β_2 is immediately sent to the x_2-subsystem, the subsequent design of the switching virtual control law β_3 will be depending upon the real control u in the next stage of the design. Consequently, the issue of cyclical reliance of control laws would emerge, making it challenging to practically apply the controller. Hence, the use of DSC methodology is employed to resolve the issue. By allowing the output a_2 of the filter to reach the following step (step 2), the intended switching virtual control law β_3 will not have a direct dependence on u.


Furthermore, the states x_1, x_2, …, x_n are included in β_2, which implies that the derivative of β_2 will include nonlinear functions f_i(x_1, x_2, …, x_n) for i=1,2, …, n. If β_2 is immediately included into the next subsystem, it will result in a complicated design of the switching virtual control rule. The DSC approach is used to send the filter output a_2 to the next subsystem. The derivative of a_2 may be derived by the algebraic operation ȧ_2=(β_2-a_2) / σ_2. Consequently, the complicated nature of the design is decreased.

Step 2: Let's examine the second error surface, denoted as ψ_2=x_2-a_2 by applying the VPSEF as in equation (4). The derivative of ψ_2 may be calculated as:


    ψ̇_2    =ẋ_2-ȧ_2
        =x_3+f_2(x_1, x_2, …, x_n)-ȧ_28

If we choose x_3 as the control input for subsystem (8), we can create a switching virtual control law to stabilize the subsystem in the following manner:


    β_3=-k_2ψ_2-f_2(x_1, x_2, …, x_n)+ȧ_29


where k_2>0 being a control parameter and  ψ_2 is the switching parameter and is obtained as in equation (4). Therefore, we obtained different values of β_3 depending on the magnitude of the errors and thus, dual virtual control. A first-order low-pass filter is introduced to process β_3 as follows,


    σ_3ȧ_3+a_3=β_310

The filtering parameter is denoted as σ_3 and it must be greater than zero. Similar to step 1, the result of the filter a_3 will be used in the subsequent stage of the design.

For each step i where i(i=3, …, n-1) : Let's examine the error surface ψ_i, which is defined as the difference between x_i and a_i. The derivative of ψ_i may be calculated as


    ψ̇_i    =ẋ_i-ȧ_i
        =x_i+1+f_i(x_1, x_2, …, x_n)-ȧ_i11

If we use x_i+1 as the control input for subsystem (11), we can build a switching virtual control law to stabilize the subsystem in the following manner:


    β_i+1=-k_iψ_i-f_i(x_1, x_2, …, x_n)+ȧ_i12


where k_i>0 being a control parameter and  ψ_i is the switching parameter and is obtained as in equation (4). Therefore, we obtained different values of β_i+1 depending on the magnitude of the errors and thus, dual virtual control. Implementing a first-order low-pass filter to modify β_i+1 in the following equation:


    σ_i+1ȧ_i+1+a_i+1=β_i+113

The variable σ_i+1 represents a filtering parameter with a positive value. Similar to step 1, the result of the filter a_i+1 will be used in the subsequent stage of the design.

Step n: The final error surface is represented by the equation ψ_n=x_n-a_n. The derivative of ψ_n may be calculated as


    ψ̇_n    =ẋ_n-ȧ_n
        =g_n(x_1, x_2, …, x_n) u+f_n(x_1, x_2, …, x_n)-ȧ_n14

In order to provide stability for subsystem (14) above, an actual switching control is formulated below:

    u=1/g_n(x_1, x_2, …, x_n)(-k_nψ_n-f_n(x_1, x_2, …, x_n)+ȧ_n) 15

where k_n is a positive control parameter.

The aforementioned approach yields the designed controller explained by the following







§ STABILITY ANALYSIS
 
Given the nonlinear system in (1), with the controller designed as in equation (15). If we consider the error of the dynamics corresponding to the final state of the system to be:

    ψ_n=(x_n-a_n)^p/q17


Where a_n is the desired trajectory for x_n.

let the error be,  e=x_n-a_n, where the error dynamics is given as:


    ẋ_n-ȧ_n=-k_n(x_n-a_n)

 and this simplifies to:
 
 
    ė=-k_n e

The given equation is a linear differential equation with a solution

    e(t)=e(0)e^-k_nt

As t →∞ e(t) → 0 if k_n > 0, this shows that, x_n→ a_n. Since, x_n converges to a_n it is important that we understand the impact of this on the whole system, particularly x_n-1. 

Consider the dynamics for the equation of x_n-1 as, 


    ẋ_n-1    =f^*_n-1(x_1, x_2, …, x_n)


As x_n→ a_n we will substitute x_n=a_n into f^*_n-1


    ẋ_n-1    =f^*_n-1(x_1, x_2, …, a_n)


f^*_n-1 is designed such that if x_n = a_n, x_n-1→ a_n-1 and will continue throughout the system. Therefore, It is possible to show recursively that each state x_i converges towards its desired trajectory a_i for i=n-1, n-2,..., 1

Overall, the switching VPSEF  backstepping based DSC approach developed in the previous section ensures the stability of each subsystem step by step as the controller ensures that x_n→ a_n. Furthermore, in the n-1 stage, given that x_n→ a_n, the designed controller ensures that a_n-1 is designed such that x_n-1→ a_n-1. Finally, x_i is assured to converge to a_i, thus, stabilizing the entire system.  





§ SIMULATION RESULTS
 
In this section, We present the aforementioned theoretical result, to validate the efficacy of the proposed VPSEF Back-stepping Based DSC Control method. Let us consider a nonlinear system that is non-lower triangular and has a third-order as shown below.



    ẋ_1    =x_1^2+x_2^3+x_3
    ẋ_2    =x_1^2 x_2+x_3^5
    ẋ_3    =u+x_1 x_2 x_3^2
    
    y     =x_1

    
The control goal for the system above is to make x(t) to track y_r=sin(t). 
Following Algorithm 1, a VPSEF Back-stepping Based DSC Control is designed with threshold of 0.1. Based on the control design approach outlined in Algorithm 1, the controller for the system may be created as follows:

  
    β_2    =-k_1(ψ_1)-(x_1^2+x_2^3+x_3-x_2)+cos t  16
    β_3    =-k_2(ψ_2)-(x_1^2 x_2+x_3^5-x_3)+ȧ_2
    
    u     =-k_3(ψ_3)-x_1 x_2 x_3^2+ȧ_3
   

where the variables a_2 and a_3 represent the outputs of first-order filters. β_2=σ_2ȧ_2+ a_2 and β_3=σ_3ȧ_3+a_3 respectively. The selected control parameters for simulation are k_1=3, k_2=3, and k_3=3. The initial states of the system are x_1(0)=0, x_2(0)=-1, x_3(0)=1. The initial states of the filters are a_2(0)=0, a_3(0)=0.

In the process of implementing β_3, ȧ_2 is obtained by ȧ_2=(β_2-a_2) / σ_2. In order to prevent an excessively large signal of β_3 during the early phase of operation of the control system, the filtering parameter σ_2 is selected as σ_2=exp (-t)+0.05. By using this approach, the output of the filter may closely track the input of the filter over time, ensuring that the value of ȧ_2 remains within an acceptable range. We use a similar strategy for the implementation of u, and the selection of the filtering parameter is determined as σ_3=exp (-t)+0.05. 

The figures below illustrate the simulation result of this particular scenario. It is evident that all signals exhibit uniform ultimate boundedness. Figure 1 demonstrates the system's ability to rapidly and accurately track the reference signal y_r with the system output y. The time required for adjustment is less than 1 second, and the maximum steady state tracking error is less than 1 %. This is due to the error correction mechanism, where the initial error magnitudes are being reduced preventing the system from initial overshoot. The control input signal and the system states are shown in figure 2 and 3 respectively. Figure 5 and 6 depict the input and output signals of the two first-order low-pass filters respectively. Figure 4 clearly shows the impact of the proposed technique, where by the tracking error is less than 1 %. The stability of the system and the achievement of perfect tracking performance has being obtained using the suggested controller designed in this work. Figure 7 depict the high error reduction in traccking the reference y_r resulting from the proposed control algorithm compared to the control technique presented in <cit.>.

Likewise, when the reference signal is taken as y_r=1-e^-t. To further validate the effectiveness of the suggested control mechanism given in (16) another reference signal y_r=1-e^-t and the simulation result is given in the figures 8 to 13. The effective achievement of stability and tracking performance in a closed-loop system in question is clearly seen. Figure 7-12 shows the system output, control input, system states, tracking error and the first order filter respectively related to the second reference using the proposed control design.  

































§ CONCLUSIONS
 
This article introduces a new technique called Variable Power Surface Error Function backstepping based Dynamic Surface Control for non lower-triangular non linear systems. The approach is supported by strong theoretical proofs. The control law is switching based on the magnitude of error in order to mitigate the high impact of these errors. The designed control technique not only ensures accurate and prescribed settling time tracking of the reference, but also actively reduces oscillations in the tracking error and intermediate errors. Furthermore, the use of the DCS in this approach effectively eliminates the issue associated with the circular design. The stability analysis shows that all signals in the closed-loop control system are uniformly ultimately bounded, and the tracking error converges to a small vicinity of zero. Two simulations were conducted using various reference signals to illustrate the enhanced trajectory tracking achieved by the suggested technique. The simulations showed a reduction in tracking error and changes in reference signals, highlighting the effectiveness of the proposed approach.













Declarations

    Ethical Approval

Not applicable

    Funding

No funding was received for this work.

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