Abstract
A nonlinear backstepping control is proposed for the coupled normal form of nonlinear systems. The proposed method is designed by combining the sliding-mode control and backstepping control with a disturbance observer (DOB). The key idea behind the proposed method is that the linear terms of state variables of the second subsystem are lumped into the virtual input in the first subsystem. A DOB is developed to estimate the external disturbances. Auxiliary state variables are used to avoid amplification of the measurement noise in the DOB. For output tracking and unmatched disturbance cancellation in the first subsystem, the desired virtual input is derived via the backstepping procedure. The actual input in the second subsystem is developed to guarantee the convergence of the virtual input to the desired virtual input by using a sliding-mode control. The stability of the closed-loop is verified by using the input-to-state stable (ISS) property. The performance of the proposed method is validated via numerical simulations and an application to a vehicle system based on CarSim platform.
1. Introduction
Control for nonlinear systems has attracted considerable attention, and, therefore, various control methods have been investigated for nonlinear systems. Control methods using the Lyapunov function were proposed for nonlinear systems [1–4]. Input-output linearization and feedback linearization were proposed to transform nonlinear systems to the normal form and to control the nonlinear systems [5, 6]. Sliding-mode control techniques were developed for nonlinear systems owing to the decoupling and invariance properties [7, 8]. Control algorithms based on the singular perturbation theory were developed for nonlinear systems involving fast and slow dynamics [9, 10]. Backstepping method is one of the breakthroughs for the control of nonlinear systems. This method is a recursive procedure using a Lyapunov function and a systematic design approach for special forms of the nonlinear systems (the strict feedback form or the normal form or both) [11]. It can guarantee global stability and improvement of tracking and transient performances. In the past decades, various backstepping methods were widely used to solve the control problems of nonlinear systems. A backstepping control method was developed to improve the force control performance for an electro-hydraulic actuator [12]. An adaptive control technique was implemented to the backstepping controller for unknown disturbance or parameters [13]. An adaptive backstepping sliding-mode controller was designed to improve the tracking performance in the sliding and presliding phases [14]. In [15], an output feedback nonlinear control was proposed for a hydraulic system with mismatched modeling uncertainties; in this control, an extended state observer (ESO) and a nonlinear robust controller are synthesized via the backstepping method. A recurrent fuzzy neural network backstepping control was proposed for the prescribed output tracking performance of nonlinear dynamic systems [16]. An adaptive robust control using ESO was developed for the DC motor control [17]. An ESO-based backstepping was proposed to improve the output-tracking performance with external disturbance using only output feedback [18]. Active disturbance rejection adaptive control schemes were proposed for both parametric uncertainties and uncertain nonlinearities of the nonlinear systems [19, 20]. An observer-based backstepping control method using reduced lateral dynamics was developed for autonomous lane-keeping system [21].
Let us consider a class of nonlinear systems coupled with two normal form subsystems as follows: where is the state variable vector, , , denotes the output, is nonzero constant, the input function is always positive or negative for all , and and are disturbances. In this paper, this class of systems (1) is called the coupled normal form in nonlinear systems. This nonlinear system (1) is not in the general normal form because cannot be regarded as the virtual input in dynamics. Furthermore, the disturbance is in dynamics. Thus, previous backstepping control methods cannot be used directly for tracking control of these systems. Several methods were presented to solve the control problem of nonlinear systems [22, 23]. In [22], two nonlinear control techniques using backstepping and sliding mode techniques were applied to an autonomous microhelicopter. Recently, a robust nonlinear control was developed for the synchronization control of cross-strict feedback hyperchaotic systems [23]. Unfortunately, because systems in [22, 23] have multi-inputs, these techniques cannot solve the control problems for the coupled normal form in a nonlinear system (1).
In this paper, we propose a nonlinear backstepping control for the coupled normal form of nonlinear systems. The proposed method combines a sliding mode technique and backstepping control with the disturbance observer (DOB). The key idea of the proposed method is that the terms are lumped into the virtual input in the first subsystem. A DOB is developed to estimate the external disturbances. Auxiliary state variables are used to avoid amplification of the measurement noise in the DOB. For output tracking and unmatched disturbance cancellation in the first subsystem, the desired virtual input is derived via the backstepping procedure. The actual input in the second subsystem is developed to guarantee the convergence of the virtual input to the desired virtual input by using a super twisting algorithm (STA). The stability of the closed-loop is proven by using the input-to-state stable (ISS) property. The performance of the proposed method is validated via numerical simulations and an application to a vehicle system based on CarSim platform.
2. Disturbance Observer Design
In this paper, we describe the situation that satisfies the following Assumptions 1 and 2.
Assumption 1. , and their derivatives are also bounded.
In general, prior information about the derivative of the disturbances is unknown but bounded, at least locally [24]. The unknown constants and exist such that
Assumption 2. The polynomial is Hurwitz.
Most real systems that are in the form of (1) satisfy Assumption 2. For example, the lateral dynamics of a vehicle with differential braking force input satisfy Assumption 2 [25]. Thus, this assumption is not strict for actual physical systems.
From (1), the disturbances, and , can be rewritten as We define the estimations of the disturbances, and . The dynamics of and are designed as where and are the observer gains. The estimation errors are defined as From, (3), (4), and (5), the estimation error dynamics can be derived as To suppress the bounded derivatives of the disturbances, the high gains, i.e., the low values of and , are required. In practice, measurement noises do appear in sensors. The dynamics of and (4) employ the derivative of the state. If high observer gains are used, the noise is amplified by the high gains. Thus, the observer is not practical for implementation. To avoid the use of the derivative of the state, we use the auxiliary state variables, , .
Theorem 3. With Assumption 1, given the auxiliary state variables, , such as the dynamics of the auxiliary state variables are Then, for .
Proof. Differentiating the auxiliary state variables with respect to time gives for and . From (1), (5), (7), and (8), the disturbance estimation error dynamics are obtained as In (10), the dynamics of are The following result is thus derived from (11), using Lemma 6.20 and Theorem C.2 in [11]: The upper bound of thus decreases as gets smaller.
Remark 4. The proposed DOB (8) with the auxiliary state variable (7) does not require the derivatives of states, i.e., and , to obtain and . Thus, if (7) and (8) are used to estimate the disturbances instead of (4), amplification of the measurement noise by the high gain can be reduced, such that it is negligible in practice.
3. Sliding Mode Backstepping Controller Design
In this paper, the control goal is to determine that makes the output track the desired reference trajectory , which is assumed to have continuous derivatives up to the th order. The tracking error is defined as To eliminate the steady-state error, the integral error is defined as The error dynamics from to can be written as The linear combination of the tracking errors, , is designed in terms of the error where the coefficients are chosen such that the polynomial is Hurwitz. From (15) and (16), we obtain as We define the terms, , as the virtual input of the first subsystem in (1): Equation (17) then becomes where is the desired virtual input and the sliding surface . The desired virtual input, , is designed as where is positive and constant. The derivative of with respect to time is The input is designed using STA as where , , and and are positive constants. In order to avoid the chattering problem, STA [26, 27] is applied to the controller (22).
Theorem 5. With Assumption 2, suppose that the control law, (20) and (22), is applied to system (1). Then the output tracking error converges to zero and is ultimately bounded.
Proof. Step 1. From (19) and (20), we have By defining the positive-definite Lyapunov function as we obtain Using as the input and as the output in (23) gives Then (26) shows that the relationship between and is strictly output passive [9] and is zero-state observable. Therefore, system is ISS. With control law (22), the dynamics of and become We define the vector . The derivative of with respect to time is where and . Because and , is Hurwitz. We define the Lyapunov candidate function as where is positive definite. The derivative of with respect to time is given by where is positive definite such that . From [27], the origin is finite-time stable. Consequently is equal to zero, identically, after a finite time interval.
Step 2. With , Then, (16) can be rewritten as Equation (33) is simplified as where , Because is Hurwitz, is bounded-input bounded output (BIBO) stable. With the convergence of to zero, , , , converge to zeros. Consequently, also converges to zero.
Step 3. With , Because , , , converge to zeros and has continuous derivatives up to the th order, is bounded. Thus, a positive constant exists such that . From (1), (18), and (36), we obtain Equation (37) is simplified as where , We define the positive-definite Lyapunov function as Because is Hurwitz, a positive definite matrix exists such that The derivative of is where , and and are the minimum and maximum eigenvalues of the matrix , respectively. Then, for all where . There exists such that for all . Consequently, is also ultimately bounded with the ultimate bound of .
Remark 6. Owing to in (38), only the boundedness of is guaranteed. Furthermore, the convergence rate of is fixed by the system parameters. Only the convergence of to is sufficient to make converge to zero, regardless of the convergence rate of .
Remark 7. When converges to zero and and are zero, becomes zero. Consequently, also converges to zero.
Remark 8. In [22, 23], the coupled systems with multi-inputs were dealt; these techniques cannot solve the control problems for the coupled nonlinear system with single input (1). On the other hand, the proposed method (20) and (22) can solve the tracking control problems for the coupled nonlinear system with one input (1).
4. Closed-Loop Stability Analysis
Actually, controller (20) and (22) uses the estimated disturbances and instead of the disturbances and . The controller becomes where , , and , , and are positive constants. The closed-loop system including controller (45) and observer (4) is given as follows:
Theorem 9. With Assumptions 1 and 2, suppose that controller (45) and disturbance observer (7) and (8) are used in (1). Further, and exist such that for all . The overall tracking error system (46) is the serial interconnected system of the ISS system. As , where . Consequently, and are ultimately bounded.
Proof. Step 1. In (46), the dynamics of and are We define the vector . The derivative of with respect to time is where and . Because and , is Hurwitz. We define the Lyapunov candidate function as where is positive definite. The derivative of with respect to time is given by where is positive definite such that . From (10) and the definition of , is given by where . Then, for all where . There exists such that for all When , (53) becomes where . Then, for all where . There exists such that for all . Consequently, and exist such that for all . In (46), dynamics can be obtained as Then we obtain the dynamics of as From (62), the following result is derived using Lemma 6.20 and Theorem C.2 in [11]: Equation (63) shows that the relationship between and , and has ISS property. The overall tracking error system (46) is the serial interconnected system of the ISS system. As , where .
Step 2. Equation (16) can be rewritten as Equation (65) is simplified as where , Because is Hurwitz, is BIBO stable. In (63) it was shown that is ultimately bounded and that as . Thus, as , (66) can be rewritten as follows: We define the positive-definite Lyapunov function as Because is Hurwitz, a positive definite matrix exists such that The derivative of is Then, for all where . Consequently, converges to the bounded ball, as as .
Step 3. With , we obtain Because , , , converge to the bounded ball and because has continuous derivatives up to the th order, is bounded. Equation (74) is simplified as We define the positive-definite Lyapunov function as Because is Hurwitz, a positive definite matrix exists such that The derivative of is where . Then, for all where . From Theorem 4.18 of [9], there exists such that for all . Consequently, is also ultimately bounded with the ultimate bound of .
Remark 10. As the controller gains and , and the observer gains and increase, becomes smaller. If the disturbances and are constant, we see that the disturbance estimation errors and converge to zeros from (12). Then, converges to zero. Consequently, the output tracking error converges to zero.
5. Performance Analysis
5.1. Numerical Simulation Study
Simulations were performed to analyze the performance of the proposed method. In these simulations, we used the system where and . The desired reference trajectory was used. The controller was designed as where and . In controller (82), the following parameters were used: , , , , , , and .
The estimation performance of the DOB is shown in Figure 1. The disturbances were well estimated by the DOB. The tracking performances of and are shown in Figure 2. and converged to the neighborhood of zero and neighborhood of , respectively, because the controller and observer gains were sufficiently high to suppress the effect of the estimation error. The tracking performance and state variables are shown in Figures 3 and 4. We see that both tracking errors and converged to almost zero because of the proposed controller (82). Because of the disturbance and nonzero trajectory, state variables and did not converge to zeros, but were bounded. Figure 5 shows the control input. Because the control method was designed using STA, there was no chattering problem.
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5.2. Application to Differential Braking Control in Vehicle Lateral Dynamics
To evaluate the performance of the proposed method in a practical system, the proposed method was applied to the differential braking control system in a vehicle. In the vehicular control system, the lateral position is controlled for avoiding collisions using differential brake forces when the driver changes the lane under collision risk or with a vehicle in the blind spot [25]. The lateral control system with the differential braking is where is the lateral offset, is the derivative of the lateral offset, is the yaw error, is the yaw error rate, is the differential braking force control input, is the steering wheel angle, is the desired yaw rate, is the disturbance including the driver torque and modeling error, and is the disturbance including self-aligned torque, modeling error, etc. The detailed definitions of the parameters can be found in [28]. The aim of controller design is to determine the brake steer force that makes when the driver attempts the lane change under a collision risk or with a vehicle in the blind spot. This system (83) satisfies Assumption 2. The controller is designed as where and , , , , , , , , , and . The velocity of the vehicle is 80 km/h on a straight road. The test scenario is as follows: ) at 5 sec., the driver attempts lane change under collision risk with the object vehicle in the target lane; ) as soon as the driver attempts lane change, the differential braking control (DBC) system is activated with a warning against the collision risk; ) the driver attempts to keep the original lane with the help of the DBC; ) the DBC system operates to move the vehicle to the center of the original lane.
Simulations were performed using the vehicle dynamic software CarSim and Matlab/Simulink as shown in Figure 6. The S-function coded in C language was used for implementing the proposed sliding mode backstepping control method. The output of CarSim consists of vehicle motion data such as steer angle, lateral velocity, and brake force. We also modeled the lane camera sensor to obtain the lane coefficients [29]; denotes the lateral lane center offset at c.g., denotes the in-lane heading slop, the heading angle error at c.g., denotes curvature/2 at , and denotes the curvature-rate/6. The steering wheel angle used in the simulations is shown in Figure 7.
(a) Overall simulation structure that consists of CarSim vehicle model
(b) Vehicle part. The output of lane camera is lane coefficients; denotes the lateral lane center offset at c.g., denotes the in-lane heading slop, the heading angle error at c.g., denotes curvature/2 at , and denotes the curvature-rate/6
Two cases were tested: ) driving with proposed controller (85); ) driving without proposed controller (85). The simulation results are shown in Figure 8. The lateral offset errors of the two cases are depicted in Figure 8(a). Figure 8(b) shows the control input. Because the control method was designed using STA, there was no chattering problem. In case 2 (without DBC), for the given steering wheel angle, the lateral offset error becomes 1.1 m owing to the steering wheel angle. On the other hand, in case 1 (with DBC), the lateral offset error was maintained to nearly zero because the brake steer force compensated for the steering wheel angle using the proposed method (85). The yaw rate error was also kept to nearly zero. Figure 9 shows the estimated disturbance. External disturbances appeared owing to the bank angle, road reaction force, the assumptions for this modeling, etc. The external disturbances were compensated by using utilizing the proposed method. Consequently, the lateral offset converged to zero despite the steer angle and the disturbances.
(a) Lateral offset error
(b) Brake steer force input
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6. Conclusions
In this paper, we proposed a sliding-mode backstepping control for the coupled normal form of nonlinear systems. The proposed method was developed by combining backstepping and sliding-mode control. The key idea of the proposed method is that the linear terms of the state variables of the second subsystem are lumped into the virtual input in the first subsystem. To compensate for the disturbances, a DOB was developed. The stability of the closed-loop is validated by using the ISS property. Through numerical simulations and application to a vehicle system, the proposed method was observed to lead to convergence of the output to the desired output trajectory under the described disturbances.
The main drawback is the use of the derivative of the measured signal in the controller. It may result in the amplification of the measurement noise. Generally, the filter technique is widely used to obtain the derivatives of the measured signals without the amplification of the measurement noise [30, 31]. However, the use of the filter may cause the phase lag in the feedback loop. In future works, we will develop the control method with the consideration of the amplification of the measurement noise in the derivatives of the measured signals.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education under Grant NRF-2016R1C1B1014831 and the Research Program, Development of High Voltage Brake System for Response to Safety Regulations of Micro eMobility (20003066), funded by the Ministry of Trade, Industry and Energy (MOTIE, Korea).