Active-disturbance-rejection control method for hypersonic flight vehicle

Abstract

The invention discloses an active disturbance rejection control method for a hypersonic flight vehicle. In the invention, a new linear and nonlinear switching function is provided, and an extended state observer of a speed subsystem and a height subsystem is designed based on the new linear and nonlinear switching function. In particular, for mismatched interference of the altitude subsystem, it is estimated as a separate state. In addition, when a control law is designed, a system interference value is fused, and the purpose of compensating the system interference value is achieved. Theoretical derivation and result verification show that the designed control strategy fully utilizes the advantages of a linear state observer and a nonlinear state observer, the influence of uncertainty on the control system can be effectively inhibited, and the robustness of the control system is enhanced. The method is oriented to engineering practical problems, the achievement can provide reference for high-performance control of the hypersonic flight vehicle, and the method has important application value.

Description

Technical Field

[0001] The present invention belongs to the field of aerospace control technology, and specifically relates to a hypersonic aircraft self-disturbance rejection control method. Background Art

[0002] Hypersonic vehicles are a new type of aircraft that fly at Mach 5 or above in near space. Their excellent concealment and defense raid capabilities may make existing defense systems powerless. They have extremely important military value and have received widespread attention. However, hypersonic aircraft systems have strong uncertainty, and the flight environment is complex and the tasks are diverse. In order to meet the guidance requirements, the design of high-precision control systems faces huge challenges. Among them, improving anti-interference ability is the key to high-precision control. Control strategies based on interference compensation can greatly enhance the robustness of hypersonic aircraft systems. These control strategies include adaptive control, intelligent control, and observer-based control. Active disturbance rejection control is a type of observer-based control. It regards the total disturbance of the system as a state, and estimates and compensates for the disturbance through an extended state observer and feedback control law. It has also been widely studied and applied in hypersonic aircraft control.

[0003] However, the common aircraft anti-disturbance control methods have the following shortcomings:

[0004] 1) The extended state observer is used too singly, either the linear extended state observer or the linear extended state observer. However, the estimation capabilities of these two observers are different, and they need to be switched reasonably according to the system error situation. 2) The estimation of system interference is too "rough". The observer is used to estimate the total interference, and no targeted estimation strategy is adopted according to the different interference characteristics, such as independent estimation of matched and unmatched interference.

[0005] Therefore, how to select a suitable extended state observer to make targeted estimates of disturbance characteristics is crucial to constructing compensation control laws and helps to enhance the robustness of hypersonic vehicle control systems. Summary of the invention

[0006] The purpose of the present invention is to provide a hypersonic vehicle active disturbance rejection control method in order to solve the above-mentioned problems.

[0007] The technical solution adopted by the present invention is as follows: a hypersonic vehicle anti-disturbance control method, the method comprising the following steps:

[0008] S1: Establish a mathematical model of hypersonic aircraft;

[0009] S2: Design an active disturbance rejection controller, which includes three parts: tracking differentiator design, extended state observer design and control law design;

[0010] Control law design includes the pitch angle loop control law design of the altitude subsystem;

[0011] S3: Design the speed subsystem control law first;

[0012] S4: Designing the altitude loop control law of the altitude subsystem;

[0013] S5: Design the pitch angle loop control law of the altitude subsystem;

[0014] S6: Verify the design results.

[0015] In a preferred embodiment, in step S1, the longitudinal dynamics model of the aircraft is considered, including five rigid states and six elastic states:

[0016]

[0017] Among them V, h, γ, α, Q, η i are speed, altitude, flight path angle (FPA), angle of attack (AOA), pitch rate and elastic mode; m, g, I yy , i (i=1,2,3),ω i (i=1,2,3) are mass, gravitational acceleration, moment of inertia, damping of the first three elastic states and natural vibration frequency respectively; d1, d2, d3, d4, d5 are system uncertainties, including unmodeled dynamics, parameter perturbations and external disturbances; lift L, drag D, thrust T, pitching moment M and generalized elastic force N i The expression is:

[0018]

[0019] in is the dynamic pressure, S and c are the reference area and the average aerodynamic chord length, is the combination vector of elastic mode; in addition, the aerodynamic parameter expression of the aircraft is:

[0020]

[0021] in δ e and δ c are the fuel equivalence ratio, elevator deflection angle, and canard deflection angle, respectively, which are all control inputs of the aircraft; due to δ e and δc Satisfy the relationship Therefore the control input can be simplified to and δ e ;

[0022] Considering the elastic mode as interference, models (1)-(6) can be decoupled into two relatively independent subsystems. The velocity subsystem can be described as:

[0023]

[0024] in The height subsystem can be described as:

[0025]

[0026] in

[0027]

[0028] In a preferred embodiment, in step S2, the speed subsystem is a first-order system, and its tracking differentiator is:

[0029]

[0030] Among them, v0, v1, and v2 are the set value, the tracking value, and the differential of the tracking value, respectively; r0 and h0 are the speed factor and the filter factor, respectively; fhan(v1-v0, v2, r0, h0) is the fast optimal comprehensive function, and the specific form can be referred to the reference (Highperformance control method of electro-mechanical actuator based on activedisturbance rejection control, 2022);

[0031] The height subsystem is a second-order system, and its tracking differentiator is:

[0032] v& 1i =v 2i

[0033] v& 2i =v 3i

[0034] v& 3i =-a i (a i (a i (v 1i -v 0i )+3v 2i )+3v 3i) (13)

[0035] where v 0i (i=H,Q) is the expected signal, v 1i ,v 2i ,v 3i Yes 0i (i=H,Q) tracking, first and second order derivative signals, a i is the speed factor.

[0036] In a preferred embodiment, in step S2, the extended state observer

[0037] The following linear / nonlinear switching function is proposed:

[0038]

[0039] Where St is the switching instruction; when St = 1, bas(e, α1, δ1, St) behaves nonlinearly, while when St = 2, it behaves linearly; since there is no non-matching interference in the speed subsystem, based on the above switching function (14), the extended state observer is designed as follows:

[0040]

[0041] in is the parameter to be designed, z 1V 、z 2V are the estimated values ​​of V and ΔV+d1 respectively; the error e is defined V =V d -V, here V d is the tracking signal obtained by tracking differentiator (12); the switching rule of observer (15) is: when |e V |<δ V (δ V When is a threshold, select the nonlinear coefficient At the same time, St V =1; otherwise select linear coefficient At the same time, St V =2; In addition, g 0V (V) = (0.5ρSC T,Φ (α)cosα / m)V 2 is a time-varying function;

[0042] The two loops of the altitude subsystem are both second-order systems, and there is non-matching interference. Based on the above switching function (14), the extended state observer of the altitude loop is designed as follows:

[0043]

[0044] in is the coefficient to be designed, z 1H 、z 2H 、z 3H 、z 4H They are H, d3, and The estimated value of H =H d -H, here H d is the tracking signal obtained by tracking differentiator (13); the switching rule of observer (16) is: when |e H |<δ H (δ H When is a threshold, select the nonlinear coefficient At the same time, St H =1; otherwise select linear coefficient At the same time, St H =2; In addition, Take it as a function of speed V; similarly, the extended state observer of the pitch angle loop is designed as follows:

[0045]

[0046] in is the coefficient to be designed, z 1Q 、z 2Q 、z 3Q 、z 4Q It's Q, d2-d4, and the estimated value of ΔQ+d5; observer (17) and observer (16) are switched synchronously, that is, when St H =1Select nonlinear coefficient Otherwise, select the linear coefficient β i l Q ;also, It is also a function of velocity V.

[0047] In a preferred embodiment, in step S4, the altitude subsystem altitude loop control law design includes:

[0048] The integral terminal sliding mode function is selected as:

[0049]

[0050] where k 1H >0,k 2H >0,k 3H >0,0<α 2H ≤1,α 1H =α 2H / (2-α 2H ), e H =Hd -H; its first-order derivative with respect to time is:

[0051]

[0052] The virtual control law of the desired pitch angle is designed as:

[0053]

[0054] Where m H >0,n H >0 is the design parameter.

[0055] In a preferred embodiment, in step S4, the altitude subsystem pitch angle loop control law design includes:

[0056] The integral terminal sliding mode function is selected as:

[0057]

[0058] where k 1Q >0,k 2Q >0,k 3Q >0,0<α 2Q ≤1, e Q =Q d -Q, α 1Q =α 2Q / (2-α 2Q ); its first-order derivative with respect to time is:

[0059]

[0060] The rudder angle control law is designed as:

[0061]

[0062] Where m Q >0,n Q >0 is the design parameter.

[0063] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are:

[0064] In the present invention, new linear and nonlinear switching functions are proposed, and based on this, extended state observers of the speed and altitude subsystems are designed. In particular, for the mismatched interference of the altitude subsystem, it is estimated as a separate state. In addition, the system interference value is incorporated into the design of the control law to achieve the purpose of compensating it. Theoretical derivation and result verification show that the designed control strategy makes full use of the advantages of linear and nonlinear state observers, can effectively suppress the influence of uncertainty on the control system, and enhances the robustness of the control system. The present invention is aimed at practical engineering problems, and the results will provide a reference for high-performance control of hypersonic aircraft, and have important application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 Design a control block diagram for the control law of the present invention;

[0066] Figure 2 This is a diagram of the system response results under external interference in the present invention. DETAILED DESCRIPTION

[0067] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0068] Reference Figure 1-2 ,

[0069] A new method of anti-disturbance control aims to further improve the anti-disturbance capability of the system, thereby improving the control performance of the system. First, the mathematical model of the hypersonic vehicle is introduced; then the controller design is discussed in detail, including the estimation of disturbances, the analysis and design of the controller; finally, the theoretical design is simulated and verified.

[0070] S1. Mathematical Model of Hypersonic Vehicle

[0071] Consider the longitudinal dynamics model of the aircraft, which includes five rigid states and six elastic states:

[0072]

[0073] Among them V, h, γ, α, Q, η i are speed, altitude, flight path angle (FPA), angle of attack (AOA), pitch rate and elastic mode; m, g, I yy , i (i=1,2,3),ω i(i=1,2,3) are mass, gravitational acceleration, moment of inertia, damping of the first three elastic states and natural vibration frequency respectively; d1, d2, d3, d4, d5 are system uncertainties, including unmodeled dynamics, parameter perturbations and external disturbances; lift L, drag D, thrust T, pitching moment M and generalized elastic force N i The expression is:

[0074]

[0075] in is the dynamic pressure, S and c are the reference area and the average aerodynamic chord length, is the combination vector of elastic mode. In addition, the aerodynamic parameter expression of the aircraft is:

[0076]

[0077] in δ e and δ c are respectively the fuel equivalence ratio, the elevator deflection angle and the canard deflection angle, which are all control inputs of the aircraft. For more parameter meanings and values, please refer to the literature (Nonlinear adaptive controller design for air-breathing hypersonic vehicles, The Ohio State University, 2010). e and δ c Satisfy the relationship Therefore the control input can be simplified to and δ e .

[0078] By considering the elastic mode as interference, models (1)-(6) can be decoupled into two relatively independent subsystems.

[0079] Among them, the speed subsystem can be described as:

[0080]

[0081] in The height subsystem can be described as:

[0082]

[0083] in

[0084]

[0085] Design of S2 Active Disturbance Rejection Controller

[0086] It includes three parts: tracking differentiator, extended state observer and control law design.

[0087] S2.1 Tracking Differentiator

[0088] The velocity subsystem is a first-order system, and its tracking differentiator is:

[0089]

[0090] Among them, v0, v1, and v2 are the set value, tracking value, and the differential of the tracking value, respectively; r0 and h0 are the speed factor and the filter factor, respectively. fhan(v1-v0, v2, r0, h0) is the fast optimal comprehensive function. The specific form can be found in the reference (Highperformance control method of electro-mechanical actuator based on activedisturbance rejection control, 2022).

[0091] The height subsystem is a second-order system, and its tracking differentiator is:

[0092] v& 1i =v 2i

[0093] v& 2i =v 3i

[0094] v& 3i =-a i (a i (a i (v 1i -v 0i )+3v 2i )+3v 3i ) (13)

[0095] where v 0i (i=H,Q) is the expected signal, v 1i ,v 2i ,v 3i Yes 0i (i=H,Q) tracking, first and second order derivative signals, a i is the speed factor.

[0096] S2.2 Extended State Observer

[0097] The following linear / nonlinear switching function is proposed:

[0098]

[0099] Where St is the switching instruction. When St = 1, bas(e, α1, δ1, St) behaves nonlinearly, while when St = 2, it behaves linearly. Since there is no non-matching interference in the speed subsystem, based on the above switching function (14), the extended state observer is designed as follows:

[0100]

[0101] in is the parameter to be designed, z 1V 、z 2V are the estimated values ​​of V and ΔV+d1 respectively. Define the error e V =V d -V, here V d is the tracking signal obtained by tracking differentiator (12). The switching rule of observer (15) is: when |e V |<δ V (δ V When is a threshold, select the nonlinear coefficient At the same time, St V =1; otherwise select linear coefficient At the same time, St V =2. In addition, g 0V (V) = (0.5ρSC T,Φ (α)cosα / m)V 2 is a time-varying function.

[0102] The two loops of the altitude subsystem are both second-order systems, and there is non-matching interference. Based on the above switching function (14), the extended state observer of the altitude loop is designed as follows:

[0103]

[0104] in is the coefficient to be designed, z 1H 、z 2H 、z 3H 、z 4H They are H, d3, and The estimated value of . Define the error e H =H d -H, here H d is the tracking signal obtained by tracking differentiator (13). The switching rule of observer (16) is: when |e H |<δ H (δ H When is a threshold, select the nonlinear coefficient At the same time, St H =1; otherwise select linear coefficient At the same time, StH =2. In addition, is taken as a function of velocity V. Similarly, the extended state observer of the pitch angle loop is designed as follows:

[0105]

[0106] in is the coefficient to be designed, z 1Q 、z 2Q 、z 3Q 、z 4Q It's Q, d2-d4, and the estimated value of ΔQ+d5. Observer (17) and observer (16) are switched synchronously, that is, when St H =1Select nonlinear coefficient Otherwise, select the linear coefficient also, It is also a function of velocity V.

[0107] S2.3 Control law design

[0108] According to the established speed and altitude subsystems, the control block diagram of the system is as follows Figure 1 As shown:

[0109] S2.3.1 Speed ​​subsystem control law design

[0110] The integral terminal sliding mode function is selected as:

[0111]

[0112] in k 1V >0,0<α 1V <1, e V =V d -V. Its first-order derivative with respect to time is:

[0113]

[0114] The fuel equivalence ratio control law is designed as:

[0115]

[0116] Where m V is the design parameter.

[0117] S2.3.2 Altitude subsystem altitude loop control law design

[0118] The integral terminal sliding mode function is selected as:

[0119]

[0120] where k1H >0,k 2H >0,k 3H >0,0<α 2H ≤1,α 1H =α 2H / (2-α 2H ), e H =H d -H. Its first-order derivative with respect to time is:

[0121]

[0122] The virtual control law of the desired pitch angle is designed as:

[0123]

[0124] Where m H >0,n H >0 is the design parameter.

[0125] S2.3.3 Design of the pitch angle loop control law for the altitude subsystem

[0126] The integral terminal sliding mode function is selected as:

[0127]

[0128] where k 1Q >0,k 2Q >0,k 3Q >0,0<α 2Q ≤1, e Q =Q d -Q, α 1Q =α 2Q / (2-α 2Q ). Its first-order derivative with respect to time is:

[0129]

[0130] The rudder angle control law is designed as:

[0131]

[0132] Where m Q >0,n Q >0 is the design parameter.

[0133] S3. Result verification:

[0134] Assume that the aircraft maneuvers during cruising, the speed increases by 2000ft / s, and the altitude rises by 10000ft. The design parameters of the integral terminal sliding mode control (expressed as ITSMC) are shown in Tables 1 and 2 below. In order to show the advantages of the switching state observer (expressed as SESO) in this paper, it is compared with the nonlinear state observer (expressed as NESO) and the linear state observer (LESO). Assume that the system is subjected to external disturbances d1=0.1sin(0.1πt), d2=0.01sin(0.001πt), d3=2sin(0.01πt), d4=0.02sin(0.001πt) and d5=0.01sin(0.01πt) in the interval [200-300]s. The results are shown in Figure 2 As shown. Figure 2 As shown in (a, b), the tracking trajectories under the control systems based on SESO and NESO are good, while the tracking trajectory under the control system based on LESO diverges. The height tracking trajectory converges faster under the control system based on SESO than that under the control system based on NESO. Under the control system based on LESO, the tracking trajectory diverges at 300s. This is because when the external disturbance disappears, the system needs a lot of control force to return the trajectory to the expected value. However, due to physical constraints (as shown in Table 1), the control force at this time reaches the upper limit, as shown in Table 1. Figure 2 (c) In addition, for the velocity subsystem SESO, it always works under NESO. For the altitude subsystem SESO, it works under LESO during disturbances and under NESO at other times, as shown in Figure 2 (d) As shown. It is precisely because of the adaptive switching of SESO and the rotation of linear and nonlinear states that the system can achieve good control performance. In short, the proposed control strategy is also robust under strong disturbances.

[0135] Table 1 Control parameters

[0136]

[0137] Table 2 Extended state observer parameters

[0138]

[0139] In the present invention, new linear and nonlinear switching functions are proposed, and based on this, extended state observers of the speed and altitude subsystems are designed. In particular, for the mismatched interference of the altitude subsystem, it is estimated as a separate state. In addition, the system interference value is incorporated into the design of the control law to achieve the purpose of compensating it. Theoretical derivation and result verification show that the designed control strategy makes full use of the advantages of linear and nonlinear state observers, can effectively suppress the influence of uncertainty on the control system, and enhances the robustness of the control system. The present invention is aimed at practical engineering problems, and the results will provide a reference for high-performance control of hypersonic aircraft, and have important application value.

[0140] It should be noted that, in this article, relational terms such as first and second, etc. are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Moreover, the term "comprise" or any other variant thereof is intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device. In the absence of further restrictions, the elements defined by the sentence "comprise a ..." do not exclude the existence of other identical elements in the process, method, article or device including the elements.

[0141] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the same. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that the technical solutions described in the aforementioned embodiments may still be modified, or some of the technical features may be replaced by equivalents. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A hypersonic vehicle active disturbance rejection control method, characterized in that: The method comprises the following steps: S1: Establish a mathematical model of hypersonic aircraft; S2: Design an active disturbance rejection controller, which includes three parts: tracking differentiator design, extended state observer design and control law design; The control law design includes the speed subsystem and the pitch angle loop control law design of the altitude subsystem; S3: Design the speed subsystem control law first; S4: Designing the altitude loop control law of the altitude subsystem; S5: Design the pitch angle loop control law of the altitude subsystem; S6: Verify the design results.

2. The hypersonic vehicle active disturbance rejection control method according to claim 1, characterized in that: In step S1, the longitudinal dynamics model of the aircraft is considered, including five rigid states and six elastic states: Among them V, h, γ, α, Q, η i are speed, altitude, flight path angle (FPA), angle of attack (AOA), pitch rate and elastic mode; m, g, I yy , i (i=1,2,3),ω i (i=1,2,3) are mass, gravitational acceleration, moment of inertia, damping of the first three elastic states and natural vibration frequency respectively; d1, d2, d3, d4, d5 are system uncertainties, including unmodeled dynamics, parameter perturbations and external disturbances; lift L, drag D, thrust T, pitching moment M and generalized elastic force N i The expression is: in is the dynamic pressure, S and are the reference area and the average aerodynamic chord length, is the combination vector of elastic mode; in addition, the aerodynamic parameter expression of the aircraft is: in δ e and δ c are the fuel equivalence ratio, elevator deflection angle, and canard deflection angle, respectively, which are all control inputs of the aircraft; due to δ e and δ c Satisfy the relationship Therefore the control input can be simplified to and δ e ; Considering the elastic mode as interference, models (1)-(6) can be decoupled into two relatively independent subsystems. The velocity subsystem can be described as: in The height subsystem can be described as: in 3. The hypersonic vehicle active disturbance rejection control method according to claim 1, characterized in that: In step S2, the speed subsystem is a first-order system, and its tracking differentiator is: Among them, v0, v1, and v2 are the set value, the tracking value, and the differential of the tracking value, respectively; r0 and h0 are the speed factor and the filter factor, respectively; fhan(v1-v0, v2, r0, h0) is the fast optimal comprehensive function, and the specific form can be referred to the reference (Highperformance control method of electro-mechanical actuator based on activedisturbance rejection control, 2022); The height subsystem is a second-order system, and its tracking differentiator is: v& 1i =v 2i v& 2i =v 3i v& 3i =-a i (a i (a i (in 1i -v 0i )+3v 2i )+3v 3i ) (13) where v 0i (i=H,Q) is the expected signal, v 1i ,v 2i ,v 3i Yes 0i (i=H,Q) tracking, first and second order derivative signals, a i is the speed factor.

4. The hypersonic vehicle active disturbance rejection control method according to claim 1, characterized in that: In step S2, the extended state observer The following linear / nonlinear switching function is proposed: Where St is the switching instruction; when St = 1, bas(e, α1, δ1, St) behaves nonlinearly, while when St = 2, it behaves linearly; since there is no non-matching interference in the speed subsystem, based on the above switching function (14), the extended state observer is designed as follows: in is the parameter to be designed, z 1V 、z 2V are the estimated values ​​of V and ΔV+d1 respectively; the error e is defined V =V d -V, here V d is the tracking signal obtained by tracking differentiator (12); the switching rule of observer (15) is: when |e V |<δ V (δ V When is a threshold, select the nonlinear coefficient At the same time, St V =1; otherwise select linear coefficient At the same time, St V =2; In addition, g 0V (V) = (0.5ρSC T,Φ (α)cosα / m)V 2 is a time-varying function; The two loops of the altitude subsystem are both second-order systems, and there is non-matching interference. Based on the above switching function (14), the extended state observer of the altitude loop is designed as follows: in is the coefficient to be designed, z 1H 、z 2H 、z 3H 、z 4H They are H, d3, and The estimated value of H =H d -H, here H d is the tracking signal obtained by tracking differentiator (13); the switching rule of observer (16) is: when |e H |<δ H (δ H When is a threshold, select the nonlinear coefficient At the same time, St H =1; otherwise select linear coefficient At the same time, St H =2; In addition, Take it as a function of speed V; similarly, the extended state observer of the pitch angle loop is designed as follows: in is the coefficient to be designed, z 1Q 、z 2Q 、z 3Q 、z 4Q It's Q, d2-d4, and the estimated value of ΔQ+d5; observer (17) and observer (16) are switched synchronously, that is, when St H =1Select nonlinear coefficient Otherwise, select the linear coefficient also, It is also a function of velocity V.

5. The hypersonic vehicle active disturbance rejection control method according to claim 1, characterized in that: In step S3, the speed subsystem control law design selects the integral terminal sliding mode function as: in k 1V >0,0<α 1V <1, e V =V d -V; its first-order derivative with respect to time is: The fuel equivalence ratio control law is designed as: Where m V is the design parameter.

6. The hypersonic vehicle active disturbance rejection control method according to claim 1, characterized in that: In step S4, the altitude subsystem altitude loop control law design includes: The integral terminal sliding mode function is selected as: where k 1H >0,k 2H >0,k 3H >0,0<α 2H ≤1,α 1H =α 2H / (2-α 2H ), e H =H d -H; its first-order derivative with respect to time is: The virtual control law of the desired pitch angle is designed as: Where m H >0,n H >0 is the design parameter.

7. The hypersonic vehicle active disturbance rejection control method according to claim 1, characterized in that: In step S4, the pitch angle loop control law design of the altitude subsystem includes: The integral terminal sliding mode function is selected as: where k 1Q >0,k 2Q >0,k 3Q >0,0<α 2Q ≤1, e Q =Q d -Q, α 1Q =α 2Q / (2-α 2Q ); its first-order derivative with respect to time is: The rudder angle control law is designed as: Where m Q >0,n Q >0 is the design parameter.

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