A robust tracking control method for helicopter based on LPV model

Abstract

The invention discloses a helicopter robust tracking control method based on an LPV model, comprising step 1: selecting a state variable characterizing the nonlinear characteristics of a helicopter as a dispatching parameter, and establishing a longitudinal LPV mathematical model for the helicopter; step 2: performing an augmented transformation on the LPV mathematical model to obtain an affine parameter-dependent LPV model; step 3: for the model obtained in step 2, considering the rate of change of the dispatching parameter, applying a parameter-dependent Lyapunov function method to obtain a parameterized linear matrix inequality for solving the control law; step 4: introducing a parameter slack variable, decoupling the parameterized linear matrix inequality, and obtaining a linear matrix inequality that is easy to solve; step 5: solving the linear matrix inequality, and obtaining an LPV controller at the corresponding equilibrium point. The invention can effectively solve the problem that the helicopter flight control system is difficult to design because the helicopter is a statically unstable complex system with serious coupling between channels.

Description

Technical Field

[0001] The invention belongs to the technical field of aircraft control, and in particular relates to a helicopter robust tracking control method based on an LPV model. Background Art

[0002] A helicopter is an aircraft that is mainly composed of three major systems: lift, power, and transmission, as well as onboard flight equipment. With the continuous improvement of aircraft performance requirements in modern military defense, aircraft must not only meet the basic requirements of daily flight, but also be able to cope with various interferences and adapt to complex environmental changes. At this time, the traditional local linearization method can no longer meet the increasingly complex and changing requirements. In this case, the linear variable parameter (LPV) model has become a hot research topic in recent years due to its unique advantages.

[0003] The LPV system is a special type of nonlinear dynamic system. In this type of system, only the parameters are variable. Its control method belongs to a type of variable gain control. Its dynamic characteristics depend on external parameters that can be measured in real time. By modeling the complex aircraft object with LPV, the purpose of describing the aircraft motion can be achieved, so that it is convenient to use linear control theories such as robust control to design variable gain controllers. Compared with the traditional linear time-invariant system (LTI) system, the LPV system has the advantages of being more convenient and closer to the nonlinear characteristics of the controlled object.

[0004] At present, many scholars at home and abroad have focused on the establishment of LPV models and the research of related control issues. Since the 1990s, variable gain control technology has gradually matured, but the variable gain control at that time was limited to the open-loop change of controller parameters, without the feedback of closed-loop performance. The stability of its controller lacked rigorous theoretical verification, and the relevant performance of the controller could not be guaranteed. In order to solve the shortcomings of traditional variable gain technology, Shamma proposed the LPV system and effectively verified the stability of the system theoretically. Marcos et al. analyzed three methods of converting linear models into LPV models, namely Jacobi linearization method, state transformation method and function substitution method. Apkrian et al. proposed a robust variable gain controller for a class of affine dependent parameter systems to ensure global robust performance.

[0005] In addition, the single quadratic Lyapunov function method generally assumes that the Lyapunov function matrix is ​​a constant matrix and does not consider the rate of change of the scheduling parameters, which brings conservatism to the design of the system. Therefore, researchers proposed a method of affine parameter-dependent Lyapunov function. However, when these literatures use the method of affine parameter-dependent Lyapunov function to design control systems, they all target LPV systems. When solving the parameterized linear matrix, the LPV system cannot be converted into a polyhedral structure, and then the vertices of the system are designed through convex theory, which leads to the need to solve infinite linear matrix inequalities. When solving the control gain of the parameterized linear matrix, the grid method must be used, but the size and density of the grid have a great influence on the accuracy of the solution. Therefore, the controller design method based on affine parameter-dependent Lyapunov function also needs further study. Summary of the invention

[0006] The technical problem to be solved by the present invention is to provide a helicopter robust tracking control method based on the LPV model in view of the deficiencies of the above-mentioned prior art.

[0007] In order to achieve the above technical objectives, the technical solution adopted by the present invention is:

[0008] A helicopter robust tracking control method based on LPV model, comprising:

[0009] Step 1: Select the state variables that characterize the nonlinear characteristics of the helicopter as the dispatching parameters and establish a longitudinal LPV mathematical model for the helicopter;

[0010] Step 2: Perform augmented transformation on the LPV mathematical model to obtain the affine parameter-dependent LPV model;

[0011] Step 3: For the model obtained in step 2, considering the rate of change of the dispatching parameters, the parameter-dependent Lyapunov function method is applied to obtain the parameterized linear matrix inequality for solving the control law;

[0012] Step 4: Introduce parametric slack variables to decouple the parameterized linear matrix inequality to obtain a linear matrix inequality that is easy to solve;

[0013] Step 5: Solve the linear matrix inequality obtained in step 4 to obtain the LPV controller at the corresponding equilibrium point.

[0014] To optimize the above technical solutions, the specific measures taken also include:

[0015] In step 1 above, the method for establishing the longitudinal LPV mathematical model is as follows:

[0016] Step 1.1: The dynamic equation of the helicopter longitudinal motion mode is established as follows:

[0017]

[0018]

[0019]

[0020] Where m is the total mass of the helicopter, u, v, and w are the linear velocities of the helicopter along the body axis, θ is the pitch angle, φ is the roll angle, p is the roll rate, q is the pitch rate, r is the yaw rate, and I x , I y , I z is the inertia moment of the helicopter mass about the axes of the body coordinate system, X, Z are the vector sum of the aerodynamic forces of the helicopter components under the body axis, and M is the pitch moment;

[0021] Step 1.2: In the longitudinal motion mode, the influence of the roll angle φ, the roll angular velocity p, and the yaw angular rate r are not considered, that is, these quantities are all 0, then the dynamic equation in step 1.1 is simplified to the following differential equation:

[0022]

[0023]

[0024]

[0025]

[0026] Step 1.3: Use the Jacobi linearization method to obtain the LPV model of the helicopter, that is, select the state variables that characterize the flight characteristics of the helicopter as scheduling parameters within the full flight envelope of the helicopter, and numerically fit the linearized model at a series of equilibrium points to obtain the LPV model of the helicopter, which is the longitudinal LPV mathematical model.

[0027] The above step 1.3 specifically includes:

[0028] The helicopter dynamics equation is expressed as follows:

[0029]

[0030] x=[u,w,q,θ] T

[0031] Performing a first-order Taylor expansion at any equilibrium point gives:

[0032]

[0033] In the formula, x trim , δtrim They represent the values ​​of the state quantity x and the input quantity δ at a certain equilibrium point, respectively. represents the partial derivative of the function f with respect to the state quantity x, represents the partial derivative of the function f with respect to the input δ;

[0034] Keeping the linear part in the formula, we get:

[0035]

[0036] Then we get the linearized model at a certain equilibrium point:

[0037]

[0038] By selecting a series of equilibrium points within the flight envelope and performing Taylor expansion, a series of LTI models that can locally approximate the nonlinear dynamic characteristics are obtained. These linearized models are then numerically fitted to obtain the LPV model of the helicopter, which is the longitudinal LPV mathematical model.

[0039] The specific process of step 2 above is:

[0040] Step 2.1: For the tracking control problem of helicopter maneuvering flight, define the reference input signal r w =[u r ,θ r ] T , the actual output of the system y=[u,θ] T , after the above definition, the tracking control problem of helicopter maneuvering flight is solved by designing an LPV state feedback controller and the designed controller makes the following equation hold:

[0041]

[0042] In addition, in the feedback control strategy, the steady-state error is eliminated by defining the following tracking error integral term;

[0043]

[0044] Step 2.2: According to the tracking error integral term, the helicopter longitudinal LPV mathematical model is augmented and transformed to obtain the following augmented system, which is the affine parameter-dependent LPV model:

[0045]

[0046] In the formula, A(ρ), B(ρ) are both affine functions of ρ, C and D are constant matrices,

[0047] The specific process of step 3 above is:

[0048] Step 3.1: Design an affine parameter-dependent state feedback controller of the form δ(t) = K(ρ)x(t) such that the LPV system obtained in step 2 is asymptotically stable and H ∞ The performance index is equal to or less than γ;

[0049] In the formula, ρ=[u,u 2 ,u 3 ,u 4 ,w,w 2 ,w 3 ,w 4 ,θ,θ 2 ,θ 3 ];

[0050] Therefore, the closed-loop system is obtained:

[0051]

[0052] In the formula,

[0053] Step 3.2: Determine the range of scheduling parameters and their rate of change:

[0054]

[0055]

[0056]

[0057] Step 3.3: Convert the feedback controller problem into solving the control law under the constraint of parameterized linear matrix inequality and given performance index γ. The parameterized linear matrix inequality is as follows:

[0058]

[0059] In the formula,

[0060] Step 4 above: introduce parametric slack variables, decouple the parameterized linear matrix inequality, and obtain a linear matrix inequality that is easy to solve;

[0061] Step 4.1: Introduce parameter relaxation and transform the range constraints of the scheduling parameter ρ and its change rate in step 3.2 into:

[0062]

[0063] Among them, the matrix satisfy

[0064] (Λ+Λ T )≥0,

[0065] Step 4.2: Based on step 4.1, the parameterized linear matrix inequality is decoupled to obtain a linear matrix inequality that is easy to solve:

[0066] Given a scalar γ>0, if there exist n+1 symmetric positive definite matrices P0,…,P n satisfy And there are n+1 real matrices satisfy Then the closed-loop system is asymptotically stable and the L2 gain of the closed-loop system is equal to or less than γ. The specific linear matrix inequality is as follows:

[0067]

[0068]

[0069] P0>0,P i >0

[0070] In the formula

[0071]

[0072]

[0073]

[0074]

[0075]

[0076]

[0077]

[0078]

[0079]

[0080] The above step 5 includes the following steps:

[0081] Step 5.1: Set the range of scheduling parameters:

[0082] u min =0.5≤u≤72=u max

[0083] w min =-0.03≤w≤0.18=w max

[0084] θ min =-0.005≤θ≤0.102=θ max

[0085] Step 5.2: Set the range of the conversion rate:

[0086]

[0087]

[0088]

[0089] Step 5.3: Based on the ranges set in Step 5.1 and Step 5.2, solve the linear matrix inequality obtained in Step 4 to obtain the control law of the designed tracking controller, that is, the LPV controller at the corresponding equilibrium point.

[0090] In the above step 5.3, the linear matrix inequality obtained in step 4 is solved by using the LMI toolbox in Matlab.

[0091] The present invention has the following beneficial effects:

[0092] The present invention can effectively solve the problem that the helicopter flight control system is difficult to design because the helicopter is a statically unstable complex system with serious coupling between channels.

[0093] 1. A longitudinal LPV model of the helicopter was established. For linear control methods, the problem of only being able to design locally and not having a theoretical guarantee for global stability was avoided; for nonlinear control methods, the problem of large workload and difficulty in actual engineering due to the need to calculate high-order derivatives of certain state quantities in the aircraft was avoided; that is, in order to ensure that the helicopter control system meets certain performance indicators, a linear variable parameter model that can fit the nonlinear characteristics of the helicopter under a large flight envelope was established, and relevant theoretical research was carried out on this basis.

[0094] 2. Introduce parameter-dependent Lyapunov function and consider the rate of change of state quantity when the helicopter is flying forward. Avoid the problem of conservativeness caused by the rate of change of scheduling parameters, which is generally assumed by the single quadratic Lyapunov function method. That is, for the tracking control problem of the helicopter flying forward, introduce the error integral term of the scheduling parameter and augment the LPV model of the helicopter. Considering the rate of change of scheduling parameters, the parameter-dependent Lyapunov function method is adopted to reduce the conservatism brought by the single quadratic robust controller.

[0095] 3. When the traditional grid method is used to solve the control law, the size and density of the grid have a great influence on the accuracy of the solution. Therefore, the present invention introduces slack variables to decouple the parameterized linear matrix inequality, avoiding the need to solve the control gain by the grid method. That is, by introducing slack variables, the parameterized linear matrix inequality constraints for solving the control law are decoupled, and LPV robust variable gain tracking control is realized. BRIEF DESCRIPTION OF THE DRAWINGS

[0096] Figure 1 Tracking curve for velocity component u;

[0097] Figure 2 is the tracking curve of the pitch angle θ;

[0098] Figure 3 is the velocity component w response curve;

[0099] Figure 4 is the pitch angle rate q response curve;

[0100] Figure 5 Comparison diagram of velocity tracking based on single quadratic form and affine parameter-dependent Lyapunov function method;

[0101] Figure 6 It is a comparison diagram of velocity tracking within a large flight envelope based on the affine parameter-dependent Lyapunov function method;

[0102] Figure 7 This is a schematic diagram of the helicopter LPV model fitting;

[0103] Figure 8 The figure is a flow chart of the method of the present invention. DETAILED DESCRIPTION

[0104] The embodiments of the present invention are further described in detail below with reference to the accompanying drawings.

[0105] See also Figure 8 The present invention provides a helicopter robust tracking control method based on the LPV model, comprising the following steps:

[0106] Step 1: Select the state variables that characterize the nonlinear characteristics of the helicopter as the dispatching parameters and establish a longitudinal LPV mathematical model for the helicopter;

[0107] In the embodiment, firstly, based on the original nonlinear flight dynamics model of the helicopter, the Jacobi linearization method is used to perform linearization processing at the equilibrium point of the nonlinear model, and a data fitting method is used to theoretically establish an LPV model that is dynamically close to the nonlinear model. Step 1 includes:

[0108] Step 1.1: Consider that when the helicopter is in a steady linear motion with constant speed and direction, it is in a state of equilibrium. At this time, the resultant force and torque acting on the body axis are both zero. According to Newton's laws of motion, six equilibrium dynamic equations can be obtained. In order to establish the dynamic model of the helicopter's longitudinal motion mode, its longitudinal equilibrium equation in level flight is as follows:

[0109]

[0110]

[0111]

[0112] Where m is the total mass of the helicopter, u, v, and w are the linear velocities of the helicopter along the body axis, θ is the pitch angle, φ is the roll angle, p is the roll rate, q is the pitch rate, r is the yaw rate, and I x , I y , I z is the inertia moment of the helicopter mass about the axes of the body coordinate system, X, Z are the vector sum of the aerodynamic forces of the helicopter components under the body axis, and M is the pitch moment.

[0113] Step 1.2: In the longitudinal motion mode, the influence of the roll angle φ, the roll angular velocity p, and the yaw angular rate r are not considered, that is, these quantities are all 0, then the dynamic equation in step 1.1 is simplified to the following differential equation:

[0114]

[0115]

[0116]

[0117]

[0118] Step 1.3: There are three commonly used methods to express nonlinear models as LPV models, namely Jacobi linearization method, state transformation method and function substitution method. Among them, Jacobi linearization method is the most commonly used method to obtain LPV model, which is applicable to any nonlinear system that can be linearized at the equilibrium point. Its theoretical basis is to perform a first-order Taylor expansion of the nonlinear system at the equilibrium point. Therefore, this method is used to obtain the LPV model of the helicopter, that is, the state value that can characterize the flight characteristics of the helicopter is selected as the gain scheduling variable within the full flight envelope of the helicopter, and the linearized model at a series of equilibrium points is numerically fitted to obtain the helicopter LPV model.

[0119] Specific:

[0120] The above helicopter dynamics differential equation can be expressed as follows:

[0121]

[0122] x=[u,w,q,θ] T

[0123] Performing a first-order Taylor expansion at any equilibrium point gives:

[0124]

[0125] In the formula, x trim , δ trim They represent the values ​​of the state quantity x and the input quantity δ at a certain equilibrium point, respectively. represents the partial derivative of the function f with respect to the state quantity x, Represents the partial derivative of the function f with respect to the input δ.

[0126] Keeping the linear part in the formula, we can get:

[0127]

[0128] make Then we can get the linearized model at a certain equilibrium point:

[0129]

[0130] By selecting a series of equilibrium points within the flight envelope and performing Taylor expansion, a series of LTI models that can locally approximate the nonlinear dynamic characteristics can be obtained. Then, these linearized models are numerically fitted to obtain the LPV model of the helicopter.

[0131] Here are some examples:

[0132] Since the pitch angle rate q can be expressed by the derivative of the pitch angle θ among the four state variables of the forward flight state, u, w, θ are selected as the scheduling state when selecting the gain scheduling variable ρ. The helicopter LPV model fitting diagram is shown in the figure. Figure 7 As shown:

[0133] The forward flight speed u range of the LPV model is 0m / s~72m / s, the w range is -0.03m / s~0.18m / s, and the pitch angle range is -0.005rad~0.102rad. Six balance points in the forward flight range are selected for balancing calculation to obtain six LTI models in the balancing mode. Then, the six models are fitted using the multivariate linear regression fitting method to obtain the LPV model of the helicopter nonlinear model. The scheduling parameters are selected as high-order powers of u, w, and θ, that is,

[0134] ρ=[u,u2 ,u 3 ,u 4 ,w,w 2 ,w 3 ,w 4 ,θ,θ 2 ,θ 3 ]

[0135] The LTI model at a series of equilibrium points obtained above is numerically fitted using the Cftool toolbox in Matlab, and the LPV model of the helicopter can be obtained as follows:

[0136]

[0137] Step 2: Perform augmented transformation on the LPV mathematical model to obtain the affine parameter-dependent LPV model;

[0138] The step 2 applies the quadratic stability theory of robust variable gain control to the full envelope flight mode of the helicopter to conduct research and construct an affine parameter-dependent robust controller based on the LPV model. Then, based on the different requirements of the flight mode and the different characteristics of each control variable, a strategy of real-time dynamic adjustment of the controller gain is adopted. The specific process is as follows:

[0139] Step 2.1: For the tracking control problem of helicopter maneuvering flight, define the reference input signal r w =[u r ,θ r ] T , the actual output of the system y=[u,θ] T , after the above definition, the tracking control problem of helicopter maneuvering flight can be solved by designing an LPV state feedback controller and the designed controller makes the following equation hold:

[0140]

[0141] In addition, in the feedback control strategy, the steady-state error is eliminated by defining the following tracking error integral term.

[0142]

[0143] Step 2.2: Based on the tracking error integral term and combined with the helicopter longitudinal LPV mathematical model, the following augmented system can be obtained, which is the affine parameter-dependent LPV model:

[0144]

[0145] In the formula, A(ρ), B(ρ) are both affine functions of ρ, C and D are constant matrices,

[0146] Step 3: For the model obtained in step 2, considering the rate of change of the dispatching parameters, the parameter-dependent Lyapunov function method is applied to obtain the parameterized linear matrix inequality for solving the control law;

[0147] In step 3, in order to reduce the conservatism brought by a single Lyapunov function, a parameter-dependent Lyapunov function method is used to replace a single quadratic Lyapunov function to constrain the change rate of the scheduling parameters in the LPV system. The specific process is as follows:

[0148] Step 3.1: Design an affine parameter-dependent state feedback controller of the form δ(t) = K(ρ)x(t) such that the LPV system is asymptotically stable and H ∞ The performance index is equal to or less than γ, where ρ=[u,u 2 ,u 3 ,u 4 ,w,w 2 ,w 3 ,w 4 ,θ,θ 2 ,θ 3 ].

[0149] Therefore, the closed-loop system can be obtained:

[0150]

[0151] In the formula,

[0152] Step 3.2: Consider the range of scheduling parameters and their change rates

[0153]

[0154]

[0155]

[0156] Step 3.3: The control objective of the robust tracking control of the helicopter is to make the above closed-loop system asymptotically stable, that is, to have a certain robust performance in tracking the state quantity. Based on this, the control system synthesis conditions based on linear matrix inequalities are given. The controller synthesis problem can be transformed into solving the control law under the constraints of parameterized linear matrix inequalities and given performance indicators γ. The constraints with parameterized linear matrix inequalities are as follows:

[0157]

[0158] In the formula,

[0159] Step 4: Introduce parametric slack variables to decouple the parameterized linear matrix inequality to obtain a linear matrix inequality that is easy to solve;

[0160] In the embodiment, step 4 introduces slack variables to decouple the scheduling parameters from the parameterized linear matrix inequality, thereby ensuring that the closed-loop system is stable and has certain robust performance. Compared with the controller design by grid division and interpolation method in the traditional affine parameter-dependent Lyapunov function method, the solution complexity is effectively reduced. Step 4 includes:

[0161] Step 4.1: By introducing parameter relaxation, the range constraints on the scheduling parameter ρ and its change rate can be converted to:

[0162]

[0163] Among them, the matrix satisfy

[0164] (Λ+Λ T )≥0,

[0165] Step 4.2: Since the solution of parameterized linear matrix inequalities is complex, decoupling is required. Therefore, the controller synthesis problem can be transformed from parameterized linear matrix inequalities to solving the control law under the constraints of linear matrix inequalities. Given a scalar γ>0, if there are n+1 symmetric positive definite matrices P0,…,P n satisfy And there are n+1 real matrices satisfy Then the closed-loop system is asymptotically stable and the L2 gain of the closed-loop system is equal to or less than γ. The specific linear matrix inequality constraints are as follows:

[0166]

[0167] (Λ+Λ T )≥0,

[0168] P0>0,P i >0

[0169] In the formula

[0170]

[0171]

[0172]

[0173]

[0174]

[0175]

[0176]

[0177]

[0178]

[0179] Step 5: Solve the linear matrix inequality obtained in step 4 to obtain the LPV controller at the corresponding equilibrium point.

[0180] In an embodiment, step 5 comprises:

[0181] Step 5.1: Set the range of scheduling parameters:

[0182] u min =0.5≤u≤72=u max

[0183] w min =-0.03≤w≤0.18=w max

[0184] θ min =-0.005≤θ≤0.102=θ max

[0185] Step 5.2: Set the range of the conversion rate:

[0186]

[0187]

[0188]

[0189] Step 5.3: Solve the linear matrix inequality obtained in step 4 by using the LMI toolbox in Matlab to obtain the control law of the designed tracking controller, that is, the LPV controller at the corresponding equilibrium point.

[0190] Example

[0191] The established longitudinal LPV mathematical model is specifically expressed as follows:

[0192]

[0193] In the formula, A(ρ), B(ρ) are both affine functions of ρ, C and D are constant matrices,

[0194] According to the helicopter's own performance, the amplitude of the scheduling parameters and their rate range are selected as shown in Table 1.

[0195] Table 1

[0196] Scheduling parameters Amplitude range Rate range u [0.5,72] [-10,10] w [-0.03,0.18] [-10,10] θ [-0.005,0.102] [-10,10]

[0197] Given the reference input state u r =20m / s,θ r =0.068rad, given γ=15.6.

[0198] The response tracking curves of the helicopter forward flight speed u and its pitch angle θ obtained by simulation in Simulink are as follows: Figure 1 and Figure 2 The response curves of velocity component w and pitch angular rate q are shown as Figure 3 and Figure 4 shown.

[0199] Can be obtained from Figure 1 and 2 It can be seen that the tracking error between the helicopter's forward flight speed u and its pitch angle θ tends to zero, the overshoot is small, and it has a good tracking effect.

[0200] from Figure 3 and 4 It can be seen that the response curves of the two state quantities of the helicopter also tend to be stable, which verifies the effectiveness of the proposed control scheme.

[0201] from Figure 5 It can be seen that the control methods based on single quadratic form and parameter-dependent Lyapunov method can track the target speed and have certain tracking performance, but it can be clearly seen that the response time of parameter-dependent Lyapunov function is shorter and there is almost no overshoot. The comparative simulation shows that the controller designed based on parameter-dependent Lyapunov function has better control effect, which reflects the superiority of the method.

[0202] Next, the tracking performance of the designed controller in tracking the state quantity within the full flight envelope of the helicopter is further analyzed, and the command signal of the helicopter forward flight speed u is processed in sections. At t = 0 ~ 30s, the command signal u is given. r =20m / s, at t=30~60s, give command signal u r =35m / s, at t=60~90s, the command signal u is given r =55m / s, the forward flight speed response curve in the whole time period is as follows Figure 6 shown.

[0203] from Figure 6 It can be seen that in any time period, the forward flight speed reference command signal ur , can quickly reduce the tracking error of the forward flight speed u, with basically no overshoot. This shows that the controller can track the state quantity well and has good tracking performance, verifying the effectiveness of the designed controller.

[0204] In summary, the present invention firstly takes the original nonlinear flight dynamics model of the helicopter as the basis, uses the Jacobi linearization method to perform linearization processing at the equilibrium point of the nonlinear model, and uses the data fitting method to theoretically establish an LPV model that is dynamically close to the nonlinear model;

[0205] Then the quadratic stability theory of robust variable gain control is applied to the full envelope flight mode of helicopters to construct an affine parameter-dependent robust controller based on the LPV model.

[0206] Then, based on the different requirements of flight modes and the different characteristics of each control variable, a strategy of real-time dynamic adjustment of controller gain is adopted;

[0207] In addition, in order to reduce the conservatism brought by a single Lyapunov function, a parameter-dependent Lyapunov function method is used to replace a single quadratic Lyapunov function to constrain the rate of change of the scheduling parameters in the LPV system;

[0208] At the same time, slack variables are introduced to decouple the scheduling parameters from the parameterized linear matrix inequality, and a design method that can ensure the stability of the closed-loop system and has certain robust performance is proposed. Compared with the traditional affine parameter-dependent Lyapunov function method that designs the controller through grid division and interpolation, the solution complexity is effectively reduced.

[0209] The above are only preferred embodiments of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions under the concept of the present invention belong to the protection scope of the present invention. It should be pointed out that for ordinary technicians in this technical field, some improvements and modifications without departing from the principle of the present invention should be regarded as the protection scope of the present invention.

Claims

1. A helicopter robust tracking control method based on LPV model, characterized in that: include: Step 1: Select the state variables that characterize the nonlinear characteristics of the helicopter as the dispatching parameters and establish a longitudinal LPV mathematical model for the helicopter; Step 2: Perform augmented transformation on the LPV mathematical model to obtain the affine parameter-dependent LPV model; Step 3: For the model obtained in step 2, considering the rate of change of the dispatching parameters, the parameter-dependent Lyapunov function method is applied to obtain the parameterized linear matrix inequality for solving the control law; Step 4: Introduce parametric slack variables to decouple the parameterized linear matrix inequality to obtain a linear matrix inequality that is easy to solve; Step 5: Solve the linear matrix inequality obtained in step 4 to obtain the LPV controller at the corresponding equilibrium point; In step 1, the method for establishing the longitudinal LPV mathematical model is as follows: Step 1.1: The dynamic equation of the helicopter longitudinal motion mode is established as follows: Where m is the total mass of the helicopter, u, v, and w are the linear velocities of the helicopter along the body axis, θ is the pitch angle, φ is the roll angle, p is the roll rate, q is the pitch rate, r is the yaw rate, and I x , I y , I z is the inertia moment of the helicopter mass about the axes of the body coordinate system, X, Z are the vector sum of the aerodynamic forces of the helicopter components under the body axis, and M is the pitch moment; Step 1.2: In the longitudinal motion mode, the influence of the roll angle φ, the roll angular velocity p, and the yaw angular rate r are not considered, that is, these quantities are all 0, then the dynamic equation in step 1.1 is simplified to the following differential equation: Step 1.3: Use the Jacobi linearization method to obtain the LPV model of the helicopter, that is, select the state variables that characterize the flight characteristics of the helicopter as the scheduling parameters within the full flight envelope of the helicopter, and numerically fit the linearized models at a series of equilibrium points to obtain the LPV model of the helicopter, which is the longitudinal LPV mathematical model; The step 1.3 specifically includes: The helicopter dynamics equation is expressed as follows: x=[u,w,q,θ] T Performing a first-order Taylor expansion at any equilibrium point gives: In the formula, x trim , δ trim They represent the values ​​of the state quantity x and the input quantity δ at a certain equilibrium point, represents the partial derivative of the function f with respect to the state quantity x, represents the partial derivative of the function f with respect to the input δ; Keeping the linear part in the formula, we get: make Then we get the linearized model at a certain equilibrium point: By selecting a series of equilibrium points in the flight envelope and performing Taylor expansion, a series of LTI models that can locally approximate nonlinear dynamic characteristics are obtained. Then, these linearized models are numerically fitted to obtain the helicopter's LPV model, which is the longitudinal LPV mathematical model. The specific process of step 2 is: Step 2.1: For the tracking control problem of helicopter maneuvering flight, define the reference input signal r w =[u r ,θ r ] T , the actual output of the system y=[u,θ] T , after the above definition, the tracking control problem of helicopter maneuvering flight is solved by designing an LPV state feedback controller and the designed controller makes the following equation hold: In addition, in the feedback control strategy, the steady-state error is eliminated by defining the following tracking error integral term; Step 2.2: According to the tracking error integral term, the helicopter longitudinal LPV mathematical model is augmented and transformed to obtain the following augmented system, which is the affine parameter-dependent LPV model: In the formula, A(ρ), B(ρ) are both affine functions of ρ, C and D are constant matrices, The specific process of step 3 is as follows: Step 3.1: Design an affine parameter-dependent state feedback controller of the form δ(t) = K(ρ)x(t) such that the LPV system obtained in step 2 is asymptotically stable and H ∞ The performance index is equal to or less than γ; In the formula, ρ = [u, u 2 , u 3 , u 4 , w, w 2 , w 3 , w 4 , θ, θ 2 , θ 3 ; Therefore, the closed-loop system is obtained: In the formula, Step 3.2: Determine the range of scheduling parameters and their rate of change: Step 3.3: Convert the feedback controller problem into solving the control law under the constraint of parameterized linear matrix inequality and given performance index γ. The parameterized linear matrix inequality is as follows: In the formula, Step 4: Introduce parametric slack variables to decouple the parameterized linear matrix inequality to obtain a linear matrix inequality that is easy to solve; Step 4.1: Introduce parameter relaxation and transform the range constraints of the scheduling parameter ρ and its change rate in step 3.2 into: Among them, the matrix satisfy Step 4.2: Based on step 4.1, the parameterized linear matrix inequality is decoupled to obtain a linear matrix inequality that is easy to solve: Given a scalar γ>0, if there exist n+1 symmetric positive definite matrices P0,...,P n satisfy And there are n+1 real matrices satisfy Then the closed-loop system is asymptotically stable and the L2 gain of the closed-loop system is equal to or less than γ. The specific linear matrix inequality is as follows: P0>0,P i >0 In the formula The step 5 comprises the following steps: Step 5.1: Set the range of scheduling parameters: in min =0.5≤u≤72=u max In min =-0.03≤w≤0.18=w max i min =-0.005≤θ≤0.102=θ max Step 5.2: Set the range of the conversion rate: Step 5.3: Based on the ranges set in Step 5.1 and Step 5.2, solve the linear matrix inequality obtained in Step 4 to obtain the control law of the designed tracking controller, that is, the LPV controller at the corresponding equilibrium point; In step 5.3, the linear matrix inequality obtained in step 4 is solved by using the LMI toolbox in Matlab.

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