Real-time balance and speed tracking control method of under-actuated self-balancing vehicle

Abstract

The invention discloses a real-time balance and speed tracking control method for an under-actuated self-balancing vehicle, and the method is characterized in that the method comprises the steps: building a UMS model; predefined time stability and related sufficiency conditions are introduced; a PTHSMC is designed; and NDO is designed. According to the method, the requirements for accurate modeling of the system and accurate measurement of parameters are reduced, and the design of the controller is simplified. And meanwhile, the influence of external interference on the system is overcome, and the stability and the control precision of system operation are improved.

Description

Technical Field

[0001] The present invention relates to the field of underactuated system control, and more specifically to a real-time balance and speed tracking control method for an underactuated self-balancing vehicle. Background Art

[0002] The characteristics of an underactuated mechanical system (UMS) are that the number of actuators is less than the number of degrees of freedom. Therefore, such a system can complete complex tasks with fewer actuators, and has the advantages of low energy consumption, low cost, compact structure, flexible movement, etc. Thus, UMS has now been widely applied in the fields of aerospace, maritime transportation, disaster relief, etc. However, the reduction of actuators will also lead to a strong coupling effect between degrees of freedom, resulting in relatively complex control design. Although traditional control algorithms can effectively control UMS, they cannot fundamentally solve the non-linear control problem of the system. In order to improve the static and dynamic performance of UMS, the UMS control technology based on hierarchical sliding mode control (HSMC) has received increasing attention. Sliding mode control is a control method with strong robustness and is widely applied in various fields. When applied to UMS, hierarchical design is required to effectively control the system. Therefore, in recent years, the optimization research on HSMC in UMS has increased.

[0003] Most of the optimizations of HSMC adopt the fuzzy control theory, which can perfectly imitate the actual control experience and methods of experts and technical workers. However, it has the disadvantages of relying on the rule base, high computational complexity, and poor interpretability. Therefore, there are many limitations in its application in actual engineering. In addition, during actual operation, in order to achieve high-quality control, factors such as parameter uncertainty and external disturbance cannot be ignored when designing the controller. Therefore, it is of great significance to propose a composite control scheme with anti-disturbance performance based on the above background research. Summary of the Invention

[0004] In order to overcome the above defects of the prior art, the present invention provides a real-time balance and speed tracking control method for an underactuated self-balancing vehicle.

[0005] The technical solution of the present invention is as follows:

[0006] A real-time balance and speed tracking control method for an underactuated self-balancing vehicle, comprising:

[0007] Establishing a model of UMS;

[0008] Introducing predefined time stability and related sufficiency conditions;

[0009] Designing PTHSMC;

[0010] Designing NDO.

[0011] Furthermore, it includes: First, introduce the principle of predefined-time stability (PTS), and adopt a fast and effective sufficient condition for predefined-time stability. Second, construct a sliding mode controller based on this sufficient condition; introduce the HSMC design method to conduct hierarchical sliding mode controller design; propose a new non-singular scheme during the design process to handle the singularity problem that appears in the design of predefined-time HSMC (PTHSMC). For matched and unmatched disturbances, propose to use a non-linear disturbance observer for disturbance observation and compensation. Finally, propose a sliding mode control scheme that combines predefined-time stability, hierarchical sliding mode control, non-linear disturbance observer, and non-singular scheme.

[0012] The technical effects and advantages of the present invention:

[0013] The present invention not only reduces the requirements for accurate system modeling and parameter measurement, but also simplifies the design of the controller. At the same time, it also overcomes the influence of external disturbances on the system, and provides the stability and control accuracy of the system operation. Description of the Drawings

[0014] Figure 1 It is the force and moment diagram of the structure of TWSBV acting on the TWSBV body and the double wheels. In the figure: (a) Vehicle force and motion diagram (b) Wheel force diagram;

[0015] Figure 2 It is the overall control framework of PTHSMC based on NDO;

[0016] Figure 3 It is the dynamic response curves of the segway under five different control algorithms. In the figure: (a) Position response curve, (b) Velocity response curve, (c) Roll angle response curve, (d) Roll angular velocity response curve;

[0017] Figure 4 It is the comparison response curves of the disturbance rejection performance under three different algorithms. In the figure: (a) Position response curve, (b) Velocity response curve, (c) Roll angle response curve, (d) Roll angular velocity response curve;

[0018] Figure 5 It is the segway experimental platform, where: (a) Physical diagram of the segway, (b) 500g load diagram;

[0019] Figure 6 It is the hardware block diagram of the segway experimental platform;

[0020] Figure 7 It is the comparison of the roll angle response curves of four different algorithms under the experimental platform;

[0021] Figure 8Comparison of the control response curves of three algorithms under disturbance conditions for TWSBV, where: (a) roll angle, (b) speed;

[0022] Figure 9 Control input comparison of three algorithms for TWSBV under disturbance conditions. Specific implementation mode

[0023] The technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0024] Embodiment 1

[0025] A real-time balance and speed tracking control method for an underactuated self-balancing vehicle, comprising the following steps:

[0026] Step 1: Taking the two-wheeled self-balancing vehicle (TWSBV) as an example, establish the model of the UMS.

[0027] The structure of the TWSBV mainly consists of two parts: the vehicle body and the two wheels, and can be regarded as a moving inverted pendulum. The force and moment diagrams acting on the TWSBV body and the two wheels are respectively shown in Figure 1 (a) and (b) below. Among them, (a) vehicle force and motion diagram (b) wheel force diagram

[0028] The movement of the TWSBV is realized by the rotation of the wheels. Therefore, first construct the wheel mathematical model:

[0029]

[0030] In the formula, m i is the wheel mass, x i represents the horizontal displacement of the wheel, H Ti is the friction force of the wheel on the ground, H i is the horizontal component of the force exerted by the vehicle body on the wheel, f di is the interference force applied to the center of the wheel, y i is the vertical displacement of the wheel, V Ti is the reaction force between the wheel and the ground, V i is the vertical component of the force exerted by the vehicle body on the wheel, J i is the moment of inertia of the wheel, θ i is the rotation angle of the wheel, H Ti is the friction force of the wheel on the ground, r is the radius of the wheel, and i = R, L respectively represent the right wheel and the left wheel.

[0031] Then, the motion mathematical model of the vehicle body is constructed as follows:

[0032]

[0033] In the formula, M is the total mass of the chassis and the driver, x P is the horizontal displacement of the center of mass, f dP is the disturbing force applied to the center of gravity of the vehicle, y P is the vertical displacement of the center of mass, J θP is the moment of inertia when the vehicle body rotates around the center of mass, θ P is the angle between the vehicle body and the vertical direction, L is the distance from the center of mass to the center of the chassis, T R 、T L are the output torques of the right and left wheel motors.

[0034] To simplify the model and construct the state - space expression, linearization and equivalent substitution are required, as follows:

[0035]

[0036] The final state - space expression is:

[0037]

[0038] The parameters in the formula are:

[0039]

[0040] B 22 = B 21

[0041]

[0042] B 42 = B 41

[0043]

[0044] x 1 = x, x 3 = θ P , u L = u R = T L = T R = u

[0045] d 1 、d 2 is the sum of the internal and external disturbances of the system.

[0046] For the d 1 、d given in the system (4) of TWSBV2 is the total internal and external disturbance of the system. The main reasons for its formation are the unmodeled dynamics in the process of modeling linearization, system parameters, and the disturbing forces applied to the wheels and chassis.

[0047] Step 2: Introduce the predefined time stability and related sufficiency conditions.

[0048] First, there exists a system as follows:

[0049]

[0050] where is the system state and f: is a nonlinear function, is the initial state. To illustrate PTS, first introduce the definition of fixed-time stability.

[0051] Definition 1: When the system (5) is Lyapunov stable and there exists a stable time function Τ(x 0 ) bounded by a constant, the system (5) can be called fixed-time stable.

[0052] Definition 2: When the system (5) is fixed-time stable and the upper-bound time constant Τ max has an obvious functional relationship with the controller parameters. Then the system (5) is called predefined-time stable.

[0053] For the design of the auxiliary controller, introduce a lemma.

[0054] Theorem 1: For the system (5), if there exists a positive definite function V(x) that satisfies:

[0055]

[0056] Then the system (5) is PTS. Where T>0 is the preset time and α∈(0,1).

[0057] The control objective of TWSBV is to make the moving distance track the reference value x f and the inclination angle remain at the mechanical median θ f , that is, x→x 1d , θ P →x 2d . Therefore, define the distance error and inclination angle error to facilitate the design of the controller:

[0058] e 1 =x - x 1d , e 2 =θ P -x 2d (7)

[0059] Hypothesis 1: For the convenience of subsequent controller design, the system (4) is decomposed and simplified:

[0060]

[0061] where f 1 (x) = A 23 x 3 , f 2 (x) = A 43 x 3 , g 1 (x) = B 21 , g 2 (x) = B 41 .

[0062] Step 3: Design process of PTHSMC

[0063] To implement the PTS of the sliding mode controller, it is first necessary to design the sliding mode surface (SMS) as follows:

[0064]

[0065] where α i ∈(0, 1), T i > 0, i = 1, 2.

[0066] When the state error of the system reaches the SMS, s i = 0, and then:

[0067]

[0068] The final input of the SMC is:

[0069] u = u eq + u sw (11)

[0070] where u eq equivalent control law and u sw switching control law together constitute the SMC control output.

[0071] First, find u eq , and take the derivative of Equation (9):

[0072]

[0073] Substitute Equation (8) into it:

[0074]

[0075]

[0076] According to Equation (4), there is coupling between speed control and inclination control, and it is a typical underactuated system. Equation (11) is the control output of a fully actuated control system, but it is impossible to find a suitable u for the underactuated system using this equation. sw . Therefore, a hierarchical control scheme is introduced. That is, (9) is used as the first layer of SMS, and the second layer of SMS is constructed as follows:

[0077] S = as 1 + bs 2 (14)

[0078] Then the final input of HSMC is:

[0079] u = u eq1 + u eq2 + u sw (15)

[0080] Next, it is necessary to determine the reaching law. Similarly, to ensure the PTS of the second layer of SMS, the designed reaching law is:

[0081]

[0082] where T 3 > 0, α 3 ∈(0, 1).

[0083] To obtain u sw , take the derivative of (14) and then substitute it into (13):

[0084]

[0085] Substitute into (16), u sw , u is as follows:

[0086]

[0087] There are two aspects of singularity in the designed controller. One is that ag 1 (x) + bg 2 (x) = 0, which will cause singularity in the overall control output and can be solved directly by tuning a and b. That is:

[0088] aB 21 + bB 41 ≠ 0

[0089] aB 21 ≠ -bB 41

[0090] The other is the negative power term that appears when taking the derivative of SMS in Equation (12) When and e iWhen = 0, a singular phenomenon occurs. In this case, many literatures have carried out relevant work, which is mainly divided into two categories. One is to limit the output of the negative power term by adopting the saturation function class. The other is by means of piecewise functions. However, using the saturation class function will not only lead to a weakening of the convergence speed, but also the selection of the threshold will affect the stability of the system. For the selection of piecewise functions, first, the function used for the non-singular part has only two parameters itself, which increases the difficulty of adjusting the parameters of the algorithm. At the same time, according to the analysis of Equation (24), the use of this method often makes the control input too large due to inappropriate parameter selection, and subsequent experiments have also verified this phenomenon.

[0091] To simplify the tuning of the controller, this paper comprehensively uses the method of piecewise functions and adopts the parameter piecewise method. That is:

[0092]

[0093] By using Equation (19), the problem of the limitation of the saturation function on the control performance is solved. At the same time, except for the necessary tuning based on the predefined time, there are no extra parameters to be tuned.

[0094] Step 4: Design of NDO (Nonlinear Disturbance Observer, NDO)

[0095] To cope with the interference caused by mismatched disturbances, NDO is introduced. The design process is as follows.

[0096] To reduce the influence of the observation noise on the observation effect, an auxiliary variable Z i is introduced. The specific form is:

[0097]

[0098] where NDO only needs to design a suitable p i (x) to complete.

[0099] The final u eqi is:

[0100]

[0101] The final control input is Equation (18). The overall design is completed, and the specific control design framework is as Figure 2 .

[0102] Example 2

[0103] The feasibility of the proposed scheme is verified through numerical simulation and experiments.

[0104] To verify the feasibility of the theoretical design of the PTHSMC-NDO algorithm, a mathematical model of the TWSBV was first established using MATLAB / Simulink for simulation verification. Then, to verify whether the algorithm can be applied in practice, a self-balancing vehicle experimental platform was designed and a series of experimental studies were carried out. At the same time, simulation parameters were set based on the experimental platform, and the specific parameters are shown in Table 1.

[0105]

[0106] Comparison Simulation of the Dynamic Response of the Self-Balancing Vehicle Based on PTHSMC

[0107] Based on the controller and mathematical theory derived in 2.1, simulation verification was carried out in this section, and the results are shown in Fig 3. As shown in the figure, the dynamic response curves of the self-balancing vehicle under five different algorithms, namely LQR, HSMC, finite-time-HSMC, fixed-time-HSMC, and PTHSMC, are presented. It should be noted that the parameter selection of the LQR algorithm adopted in this paper needs to be designed separately for each system configuration. Its parameters need to correspond one by one to the control quantities, specifically [-22.3607 -45.2048 -206.1926 -26.5134]. At the same time, to make the comparative experiment effective, the parameters were kept as consistent as possible. As can be seen from Equation (18), six parameters need to be determined and can be adjusted arbitrarily according to actual requirements. The final parameters are α 1 = α 2 = α 3 = 0.5, T 1 = 10, T 2 = 0.5, T 3 = 0.8.

[0108] At the same time, to conduct a comparative experiment, there are the following three methods.

[0109] HSMC:

[0110] finite-time-HSMC:

[0111] fixed-time-HSMC:

[0112] The above controller parameters are

[0113] Figure 3 (a), (b), (c), and (d) respectively show the position, speed, roll angle, and roll angular velocity response curves under different control algorithms. First, by Figure 3(a) and (b) show that all control algorithms can make the speed converge finally, but there is a certain static error in the position response of HSMC. At the same time, the overshoot and response speed of the proposed algorithm are the best on the position curve. Although on the speed response curve, the overshoot is slightly higher than that of fixed-time-HSMC, the convergence speed is still the optimal. Further from Figure 3 (c) and (d) show that the overshoot and response speed of the proposed algorithm are similar to those of fixed-time-HSMC and finite-time-HSMC, but it is significantly better than HSMC. It is worth mentioning that in the response curves of roll angle and angular velocity, the overshoot of LQR simulation is significantly better than other algorithms. The fundamental reason is that LQR is a controller established based on an accurate model, and it will present better simulation results for the corresponding model.

[0114] Robustness simulation test

[0115] To further illustrate the advantages of the algorithm proposed in this paper, a robustness experiment is carried out to further verify the proposed method. For simplicity, at 10 s, a step signal with an amplitude of 10 is added, and the final results are as Figure 4 shown. The anti-disturbance algorithms for comparison are HSMC-PE and LQR. From Figure 4 (a) and (b), it can be seen that after the step signal is added, the speed response curves can all converge, but the dynamic performance of the proposed control algorithm is the best. For the position response curve, except for the proposed algorithm, the other two algorithms both have static errors. Further from Figure 4 (c) and (d), for the roll angle and angular velocity response curves, when the step signal is not added, the overshoot of LQR is the smallest, but its convergence speed is slower than that of the proposed algorithm. When the disturbance is added, the overshoot of LQR is the largest and the convergence speed is the slowest. Generally speaking, the proposed algorithm has the fastest convergence speed and the best anti-disturbance performance.

[0116] Experimental platform construction

[0117] Previously, the effectiveness and robustness of the algorithm proposed in this paper were verified by mathematical model simulation, but whether the algorithm can be used in an actual platform has not been verified yet. In this section, a self-balancing vehicle experimental platform is designed, as Figure 5 shown. Figure 5 (a) is a physical picture of the self-balancing vehicle, Figure 5 (b) is the load test with a 500 g weight. The self-balancing vehicle consists of an STM32 single-chip microcomputer, an MPU6050 gyroscope, a TB6612FNG motor drive IC, a power battery, a DC / DC buck voltage stabilizing circuit, a DC brush motor, a Hall encoder, etc. A PCB circuit board is designed independently, and program development is carried out based on STM32CubeMX and Keil. Figure 6 The hardware block diagram of the self-balancing vehicle experimental platform is presented. Based on this, subsequent experimental verification is carried out.

[0118] Algorithm verification based on the experimental platform

[0119] First, the simplest control experiment is carried out, that is, to ensure that the TWSBV stays at the equilibrium position. Figure 7 . shows the comparison of the response curves of the roll angle of the TWSBV under four different algorithms. It can be seen from the figure that the fluctuation ranges of the roll angle under the control of LQR, HSMC, fixed-time-HSMC and the proposed algorithm are -7.6~5.7deg, -5.4~6.2deg, -4.1~4.4deg, and ±2.8deg respectively. That is, the proposed algorithm has the smallest fluctuation range of the roll angle, and its control performance is better than the other three compared algorithms.

[0120] In practice, matching or mismatching disturbances are inevitable, so Figure 5 . the 500g weight in (b) is used as an external disturbance to compare the algorithms. The final response curves of the roll angle and speed of the TWSBV are as Figure 8 . shown. From Figure 8 . (a), it can be seen that an external disturbance is added around 8s, and the TWSBV under the control of the LQR algorithm falls down around 10s. The fluctuation range of the proposed algorithm after adding the disturbance is -4.3~3.4deg, and the fluctuation range of HSMC-PE after adding the disturbance is -11.1~9.7deg. Figure 8 . (b), it can be seen that after adding the disturbance, the fluctuation range of the speed of the proposed algorithm is ±0.27m / s, and the fluctuation range of the speed of HSMC-PE is -0.29~0.35m / s. That is, the proposed algorithm is robust to disturbances, and its anti-disturbance performance is better than that of LQR and HSMC-PE. The control input voltage is as Figure 9 . shown. It can be seen from the figure that the proposed algorithm in this paper has the smallest fluctuation range of the control input. In summary, the proposed algorithm in this paper has the advantages of superior anti-disturbance performance and strong robustness compared with the compared algorithms.

[0121] The above are only the preferred embodiments of the present invention and are not used to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A real-time balancing and speed tracking control method for an underactuated self-balancing vehicle, characterized in that: include: Modeling of underactuated systems (UMS); Introducing stability and related sufficiency conditions for predefined time; Design of predefined time hierarchical sliding mode controller (PTHSMC); Design a nonlinear disturbance observer (NDO).

2. The real-time balancing and speed tracking control method of an underactuated self-balancing vehicle according to claim 1, characterized in that: The steps of establishing the UMS model are as follows: taking the two-wheeled self-balancing vehicle (TWSBV) as an example, establishing the UMS model; The structure of TWSBV mainly consists of two parts: the body and the two wheels, which can be regarded as a moving inverted pendulum; The movement of TWSBV is achieved through the rotation of the wheels, so the mathematical model of the wheels is first constructed: In the formula, m i is the wheel mass, x i Represents the horizontal displacement of the wheel, H Ti is the friction force of the wheel on the ground, H i is the horizontal component of the force exerted on the wheel by the vehicle body, f di is the disturbance force applied to the wheel center, y i is the vertical displacement of the wheel, V Ti is the reaction force between the wheel and the ground, V i is the vertical component of the force exerted on the wheel by the vehicle body, J i is the moment of inertia of the wheel, θ i is the rotation angle of the wheel, H Ti is the friction force exerted on the wheel by the ground, r is the radius of the wheel, i=R, L represents the right wheel and the left wheel respectively; Then the motion mathematical model of the vehicle body is constructed as follows: Where M is the total mass of the chassis and the driver, x P is the horizontal displacement of the center of mass, f dP is the disturbance force applied to the center of gravity of the vehicle, y P is the vertical displacement of the center of mass, J θP is the moment of inertia of the vehicle body when it rotates around the center of mass, θ P is the angle between the vehicle body and the vertical direction, L is the distance between the center of mass and the center of the chassis, T R , T L Output torque for the right and left wheel motors; In order to simplify the model and construct the state space expression, linearization and equivalent replacement are required, as follows: The final state space expression is: The parameters in the formula are: B 22 =B 21 B 42 =B 41 x1=x, x3=θ P , u L =u R =T L =T R =u d1 and d2 are the sum of the internal and external disturbances of the system; For the TWSBV system, d1 and d2 given in (4) are the total internal and external disturbances of the system; the main reasons for their formation are the unmodeled dynamics, system parameters and disturbance forces applied to the wheels and chassis during the modeling linearization process.

3. The real-time balancing and speed tracking control method of an underactuated self-balancing vehicle according to claim 2, characterized in that: Introducing predefined temporal stability and related sufficiency conditions, including: First, there is a system: in, is the system status and f: is a nonlinear function, is the initial state; to illustrate the predefined time stability (PTS), the definition of fixed time stability is first introduced; Definition 1: When the system (5) is Lyapunov stable and there exists a stable time function Τ(x0) bounded by a constant, the system (5) is called fixed-time stable; Definition 2: When the system (5) is fixed-time stable and the upper bound time constant Τ max There is an obvious functional relationship with the controller parameters; then the system (5) is called PTS; To assist controller design, a lemma is introduced; Theorem 1: For system (5), if there exists a positive definite function V(x) that satisfies: Then the system (5) is PTS; where T>0 is the preset time and α∈(0,1); The control goal of TWSBV is to make the moving distance track the reference value x f and the inclination angle is maintained at the mechanical midpoint θ f , that is, x→x 1d ,θ P →x 2d ; Therefore, the distance error and inclination error are defined to facilitate the design of the controller: e1=x-x 1d ,e2=θ P -x 2d (7) Assumption 1: For the convenience of subsequent controller design, system (4) is decomposed and simplified: where \(f_1(x)=A 23 x^3\), \(f_2(x)=A 43 x^3\), \(g_1(x)=B 21 , \(g_2(x)=B 41 .\) 4. The real-time balancing and speed tracking control method of an underactuated self-balancing vehicle according to claim 3, characterized in that: The design process of PTHSMC is: To implement the PTS of the sliding mode controller, we first need to design the sliding mode surface (SMS) as follows: where α i ∈(0,1),T i >0,i=1,2; When the system state error reaches SMS, there is s i =0, and then: The final input to the SMC is: in=in eq +in sw (11) where u eq Equivalent control law and u sw The switching control laws together constitute the SMC control output; First, find u eq , and derive formula (9): Substituting formula (8) into: According to formula (4), there is coupling between speed control and tilt angle control, and it is a typical under-actuated system; formula (11) is the control output of the full-drive control system, and the under-actuated system cannot find a suitable u using this formula. sw ; Therefore, a hierarchical control scheme is introduced; that is, (9) is used as the first layer SMS, and the second layer SMS is constructed: S=as1+bs2 (14) Then the final input of HSMC is: in=in eq1 +in eq2 +in sw (15) Next, we need to determine the reaching law. Similarly, to ensure the PTS of the second SMS, the designed reaching law is: Where T3>0,α3∈(0,1); To obtain u sw , derive (14) and substitute it into (13): Substituting into (16), u sw ,u is as follows: The designed controller has two singular situations. One is ag1(x)+bg2(x)=0, which will cause the overall control output to be singular. This can be solved by directly adjusting a and b; that is: aB 21 +bB 41 ≠0 aB 21 ≠-bB 41 The other is the negative power term that appears in the derivative of SMS in equation (12): when And e i =0, a singular phenomenon occurs; this situation can be divided into two categories; one is to limit the output of negative power terms by using a saturated function; the other is to use a piecewise function; but the use of a saturated function will not only reduce the convergence speed, but also affect the stability of the system due to the selection of the threshold; and the selection of a piecewise function, first of all, the function used for the non-singular part has only two parameters, which increases the difficulty of adjusting the algorithm parameters; In order to simplify the controller parameter adjustment, this paper integrates the piecewise function method and adopts the parameter piecewise method; That is: By adopting formula (19), the limitation of saturation function on control performance is solved. At the same time, except for the necessary parameter adjustment based on predefined time, no extra parameters need to be adjusted.

5. The real-time balancing and speed tracking control method of an underactuated self-balancing vehicle according to claim 4, characterized in that: The design of NDO includes: In order to deal with the interference caused by mismatched disturbances, NDO is introduced; the design process is as follows; In order to reduce the impact of observation noise on the observation effect, an auxiliary variable Z is introduced i ; The specific form is: in NDO only needs to design appropriate i (x) to complete; The final u eqi for: The final control input is formula (18), and the overall design is completed.

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