A fixed-wing unmanned aerial vehicle fault-tolerant optimization control method based on hidden Markov chain

Abstract

The application discloses a fixed-wing unmanned aerial vehicle fault-tolerant optimization control method based on a hidden Markov chain, and relates to the technical field of unmanned aerial vehicle control.The technical points of the application include: selecting multiple initial states of unmanned aerial vehicle systems, calculating optimal solutions for each initial state of the system based on a segmented homogeneous emission probability hidden Markov jump model, obtaining feedback control gains and feasible regions that satisfy the stability of a closed-loop system, recursive feasibility and system constraints, and thus obtaining the corresponding relationship among system states, feasible regions and feedback control gains; for the actual state of the unmanned aerial vehicle system, the feasible region where the current system state is located is determined at each time; according to the corresponding relationship, the corresponding feedback control gain is obtained, so that the controller controls the unmanned aerial vehicle to fly according to the corresponding feedback control gain at each time.The application effectively solves the problem that the observation mode does not match the actual mode of the system and the emission probability is time-varying, and improves the safety and reliability of a faulty unmanned aerial vehicle during flight.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of unmanned aerial vehicle control, and particularly relates to a fixed-wing unmanned aerial vehicle fault-tolerant optimization control method based on a hidden Markov chain. BACKGROUND

[0002] Fixed-wing unmanned aerial vehicles have the advantages of fast flight speed, long endurance time and strong carrying capacity, and are widely used in fields such as terrain exploration, cargo transportation and military reconnaissance. With the diversification of flight tasks performed by fixed-wing unmanned aerial vehicles, the flight environment they face is becoming increasingly complex, and the safety and reliability requirements for unmanned aerial vehicles are continuously increasing. On the one hand, strong gust disturbances inevitably cause wear and tear to the control mechanisms such as the rudder and the control surface of the unmanned aerial vehicle; on the other hand, the long-term operation of the unmanned aerial vehicle also causes stress fatigue to the main force components such as the propeller and the wing. The above problems may all cause the unmanned aerial vehicle to randomly fail during operation. In this context, it is of great significance to study the fault-tolerant control technology of fixed-wing unmanned aerial vehicles when random failures occur, so as to prolong the service life of the unmanned aerial vehicle, improve the stable operation ability of the system and improve the success rate of autonomous operation tasks.

[0003] In actual tasks, under random disturbances such as gust disturbance, irregular airflow changes and system actuator failure, the system parameters of the unmanned aerial vehicle will change to varying degrees. This random change can be modeled by a Markov jump process. It is worth noting that in previous research on Markov jump systems, most of them assume that the switching mode is completely known, which does not fully match the phenomenon that the fault observation mode does not match the actual system mode due to reasons such as sensor failure or observation delay. This asynchronous problem can be effectively solved by modeling into a more general hidden Markov model [1] . The hidden Markov model can be regarded as a two-layer Markov process, with the upper layer representing the actual system mode and the lower layer representing the observed system mode. The information of the lower layer is determined by the information of the upper layer and the emission probability. In addition, to improve the generality of the hidden Markov model, a class of piecewise homogeneous Markov processes can be used to model the time-varying emission probability [2] . In this case, the time-invariant emission probability can be regarded as a special case.

[0004] In the design of controllers for hidden Markov models, the model predictive control method has attracted widespread attention from domestic and foreign scholars due to its advantages of considering model uncertainty, multiple input-output coupling, system constraints and good dynamic control performance [3][4]The explicit model predictive control is divided into two steps of offline calculation and online searching, the heavy calculation burden is performed in the offline process, and therefore, compared with the traditional model predictive control, the online calculation efficiency is greatly improved; in addition, since the state feedback control law is selected for the optimized control law corresponding to each feasible region partition, the robustness is stronger, the software implementation of the control law is simpler and more reliable, and the method is suitable for relatively fast dynamic systems or large-scale control processes [5] .

[0005] In summary, the existing model predictive control method does not consider the case that the fault observation mode of the unmanned aerial vehicle system does not match the actual mode of the system, and the online solving calculation cost is too high and the control frequency is low. SUMMARY

[0006] To this end, the application provides a fixed-wing unmanned aerial vehicle fault-tolerant optimization control method based on a hidden Markov chain, in an attempt to solve or at least alleviate at least one of the problems existing above.

[0007] A fixed-wing unmanned aerial vehicle fault-tolerant optimization control method based on a hidden Markov chain, comprising the following steps:

[0008] Step one, selecting a plurality of initial states of the unmanned aerial vehicle system, and obtaining an optimal solution for each initial state of the system based on a piecewise homogeneous emission probability hidden Markov jump model, to obtain a set of feedback control gains and a feasible region that satisfy the stability of the closed-loop system, the recursive feasibility and the system constraints, so as to obtain the corresponding relationship among the system state, the feasible region and the feedback control gain;

[0009] Step two, for the actual state of the unmanned aerial vehicle system, determining the feasible region in which the current system state is located at each time; according to the corresponding relationship, the corresponding feedback control gain is obtained, so that the controller controls the unmanned aerial vehicle to fly according to the corresponding feedback control gain at each time.

[0010] Further, the specific process of obtaining a set of feedback control gains and a feasible region that satisfy the stability of the closed-loop system, the recursive feasibility and the system constraints for each initial state of the system based on a piecewise homogeneous emission probability hidden Markov jump model in step one comprises:

[0011] Step one, based on the random fault of the unmanned aerial vehicle, a piecewise homogeneous emission probability hidden Markov jump model is established; based on the performance of the unmanned aerial vehicle, system state constraints and control input constraints are established;

[0012] Step two, an observation mode dependent state feedback controller is established, the observation mode includes a system mode and an emission probability mode;

[0013] Step three, an infinite time domain cost function and a rolling horizon optimization control problem are established;

[0014] Step 14: Establish a state feedback controller that satisfies the recursive feasibility and closed-loop system mean square stability conditions;

[0015] Step 15: Based on the recursive feasibility, closed-loop system mean square stability and constraints that the state feedback controller must satisfy, design a solution method for the receding horizon optimization control problem based on linear matrix inequalities to obtain the feedback control gain and feasible region.

[0016] Furthermore, the specific process of step one includes:

[0017] 1) Combine the uncertainty of the UAV model and the fault type to establish a continuous-time Markov jump model of the UAV:

[0018]

[0019] Where t represents time; x(t) represents the state vector of the UAV system; u(t) represents the control input; represents the UAV system matrix under fault; represents the input matrix of the UAV under fault conditions; {r k} represents a random process that obeys the Markov characteristic, and the random process has a finite set of M switching modes. and obey the transition probability:

[0020] Perform Euler iteration to obtain the UAV Markov jump model in discrete time:

[0021]

[0022] Where k represents the time; The positive constant s is the sampling period of the system, and I represents the identity matrix;

[0023] 2) Considering the mismatch between the observed mode and the true mode of the system, it is assumed that the true mode and the observed mode sequence of the system obey two independent random processes, where the random process {φ k} is a Markov chain, representing the real mode of the system {r k} observation mode and from a finite set of N modes The emission probability mode is used to describe the random process {φ k} and {r k}, that is, for The emission probability is expressed as:

[0024]

[0025] Among them, λ m,n(k) e [0, 1],

[0026] The piecewise homogeneous emission probability is introduced to simplify the time-varying emission probability into a homogeneous one, which is denoted as The piecewise homogeneous emission probability is denoted as

[0027]

[0028] where,

[0029] The variation of the piecewise homogeneous emission probability is determined by a higher-level Markov chain {ψ k} and {ψ k} takes values from a finite set of H modes; the transition probability of the random process {ψ k} is defined as

[0030]

[0031] where,

[0032] 3) The hard constraints of the system state and control quantity are established according to the actual performance of the UAV as

[0033] |[Ex(k) + Fu(k)] q |≤[μ] q

[0034] where, E represents the state constraint matrix; F represents the input constraint matrix; μ represents the constraint boundary value; D represents the number of system hard constraints, and [·] q represents the qth element of the vector.

[0035] Further, the expression of the state feedback controller in step one is

[0036]

[0037] where, represents the feedback control gain of the state feedback controller.

[0038] Further, the expression of the infinite time domain cost function in step one is

[0039]

[0040] where, Q and R are known weight matrices; x(i; k) and u(i; k) represent the predicted system state and system input at k+i time at k time, respectively.

[0041] Further, the receding horizon optimal control problem in step 113 is established as follows:

[0042]

[0043] wherein, denotes the closed-loop system matrix at k+i predicted at k, A m and B m denote the system matrix and input matrix at k+i predicted at k, respectively.

[0044] Further, the condition in step 114 is:

[0045] If there exists a series of positive definite matrices {P1, P2…P M} such that the system has an observation modal dependent Lyapunov function V(x(i;k), r i；k )=x T (i;k)P i；k x(i;k) at initial time k=0 and satisfies:

[0046]

[0047] and the state initial feasible region is a generalized positive invariant set, and the receding horizon optimal control problem has a solution, then the state feedback controller satisfies the recursive feasibility and the closed-loop system mean square stability; wherein {P1, P2…P M} is a series of Lyapunov function solution arrays, γ is the feasible region boundary value, which is also the upper bound of the cost function; P rk denotes the Lyapunov function solution array under r k modal; r i；k denotes the system modal at k+i predicted at k.

[0048] Further, the specific process of solving the receding horizon optimal control problem to obtain the feedback control gain and the feasible region based on the linear matrix inequality solvable in step 115 is:

[0049] Given positive definite matrices Q, R; given matrices E, F; vector μ; for the system with a piecewise homogeneous emission probability hidden Markov jump model, the receding horizon optimal control problem is transformed into an optimization problem solvable based on a linear matrix inequality by using the schur complement lemma and matrix congruence transformation, and satisfies the recursive feasibility, the closed-loop system mean square stability, the system state and input control quantity constraint conditions, and the optimization problem is expressed as follows:

[0050]

[0051]

[0052] wherein, denotes that the matrix is negative definite; x denotes a given initial state; * denotes an omitted term due to a symmetric matrix; s.t. denotes for positive definite matrix T m ,P m , Θ, the invertible matrix and the matrix satisfy the above inequality condition; then for the initial state x, the observation mode-dependent state feedback controller is its feasible region is

[0053] The beneficial technical effects of the present application are:

[0054] The present application aims at the random fault problem of unmanned aerial vehicle, models it as a piecewise homogeneous emission probability hidden Markov jump model, and designs an unmanned aerial vehicle fault-tolerant optimization control method based on offline model predictive control. For an unmanned aerial vehicle whose fault change obeys a Markov model and has model uncertainty, a mode-dependent controller design method is proposed, which effectively solves the problem of mismatch between observation mode and actual mode and time-varying emission probability, and improves the safety and reliability of the fault unmanned aerial vehicle during flight. The controller is also applicable to the cases of complete matching between observation mode and actual mode and time-invariant emission probability, thereby increasing the application range of the controller. In view of the problem that the calculation time of the conventional model predictive control is too long to meet the control requirement due to the high control frequency required by the actual control system, the offline control algorithm used in the present application performs a large amount of optimal control problem solving in the offline process, greatly reduces the online calculation burden under the premise of ensuring the same control performance, and has high engineering application value. BRIEF DESCRIPTION OF DRAWINGS

[0055] The present application can be better understood by reference to the description given in the following text in conjunction with the drawings, which are included in the present specification and form part of the present specification, and are used to further illustrate the preferred embodiments of the present application and explain the principles and advantages of the present application.

[0056] Figure 1 It is a flow chart of a fixed-wing unmanned aerial vehicle fault-tolerant optimization control method based on hidden Markov chain in an embodiment of the present application.

[0057] Figure 2 It is a set of open-loop state response curves in 15 groups in an embodiment of the present application.

[0058] Figure 3 It is a set of random processes {r k}, {ψk},{φ k} random switching sequence.

[0059] Figure 4 These are 15 groups of closed-loop system state response curves in an embodiment of the present invention.

[0060] Figure 5 These are 15 sets of closed-loop system control input curves in an embodiment of the present invention.

[0061] Figure 6 These are 15 groups of closed-loop system state response curves based on the online model predictive control method in an embodiment of the present invention. DETAILED DESCRIPTION

[0062] In order to enable those skilled in the art to better understand the present invention, exemplary embodiments or examples of the present invention will be described below with reference to the accompanying drawings. Obviously, the described embodiments or examples are only some of the embodiments or examples of the present invention, and not all of them. Based on the embodiments or examples of the present invention, all other embodiments or examples obtained by those skilled in the art without creative work should fall within the scope of protection of the present invention.

[0063] The embodiment of the present invention provides a fault-tolerant optimization control method for a fixed-wing UAV based on a hidden Markov chain, the method comprising the following steps:

[0064] Step 1: Select multiple UAV system initial states and find the optimal solution for each system initial state based on the piecewise homogeneous emission probability hidden Markov jump model to obtain a set of feedback control gains and a feasible region that meet the closed-loop system stability, recursive feasibility, and system constraints, thereby obtaining the corresponding relationship between system state, feasible region, and feedback control gain; specifically:

[0065] Step 1: Based on the random failure of the UAV, a segmented homogeneous emission probability hidden Markov jump model is established; based on the performance of the UAV, system state constraints and control variable input constraints are established;

[0066] Step 12: Establishing a state feedback controller that depends on the observed mode, wherein the observed mode includes the system mode and the emission probability mode;

[0067] Step 13: Establish infinite horizon cost function and rolling horizon optimization control problem;

[0068] Step 14: Establish a state feedback controller that satisfies the recursive feasibility and closed-loop system mean square stability conditions;

[0069] Step one five, based on the state feedback controller must meet the recursive feasibility, closed-loop system mean square stability and constraint conditions, design the solution method based on linear matrix inequality solvable for the rolling horizon optimization control problem, so as to obtain the feedback control gain and feasible region;

[0070] Step two, for the actual state of the unmanned aerial vehicle system, judge the feasible region where the current system state is at each time; According to the corresponding relationship, the corresponding feedback control gain is obtained, so that the controller controls the unmanned aerial vehicle flight according to the corresponding feedback control gain at each time.

[0071] In this embodiment, preferably, the specific process of step one one includes:

[0072] 1) Combine the uncertainty of the unmanned aerial vehicle model and the fault type to establish the continuous time Markov jump model of the unmanned aerial vehicle:

[0073]

[0074] Wherein, t represents time; x(t) represents the state vector of the unmanned aerial vehicle system; u(t) represents the control input; represents the unmanned aerial vehicle system matrix under fault; represents the unmanned aerial vehicle input matrix under fault;{r k} represents a random process subject to Markov property, and the random process takes values in a finite set containing M switching modes And subject to transition probability:

[0075] Euler iteration is performed on it to obtain the discrete time unmanned aerial vehicle Markov jump model:

[0076]

[0077] Wherein, k represents time; Normal number s is the sampling period of the system, and I represents the unit matrix;

[0078] 2) Considering the mismatch between the system observation mode and the system real mode, it is assumed that the system real mode and the observation mode sequence respectively subject to two independent random processes, wherein, random process{φ k} is a Markov chain, representing the observation mode of the real mode{r k} of the system and taking values from a finite set N modes; The emission probability mode is used to describe the transition probability between the random processes{φ k} and{r k}, that is, for The emission probability is expressed as:

[0079]

[0080] where λ m,n (k)∈[0,1],

[0081] The piecewise homogeneous emission probability is introduced to simplify the time-varying emission probability into the emission probability with homogeneous characteristics, and at this time, for The piecewise homogeneous emission probability is expressed as:

[0082]

[0083] where

[0084] The change of the piecewise homogeneous emission probability is determined by the higher layer Markov chain {ψ k}, and {ψ k} is valued from the finite set of H modes; the transition probability of the random process {ψ k} is defined as:

[0085]

[0086] where

[0087] 3) The hard constraints of the system state and the control quantity according to the actual performance of the unmanned aerial vehicle are:

[0088] |[Ex(k)+Fu(k)] q |≤[μ] q

[0089] where E represents the state constraint matrix; F represents the input constraint matrix; μ represents the constraint boundary value; D represents the number of system hard constraints, and [·] q represents the qth element of the vector.

[0090] In this embodiment, preferably, the expression of the state feedback controller in step one two is:

[0091]

[0092] In the formula, represents the feedback control gain of the state feedback controller.

[0093] In this embodiment, preferably, the expression of the infinite time domain cost function in step one three is:

[0094]

[0095] Where Q and R are known weight matrices; x(i; k) and u(i; k) represent the system state and system input at time k+i predicted at time k, respectively.

[0096] In this embodiment, preferably, the rolling horizon optimization control problem in steps 1 and 3 is established as follows:

[0097]

[0098] Where, A represents the closed-loop system matrix at time k+i predicted at time k, m and B m They represent the system matrix and input matrix at time k+i predicted at time k respectively.

[0099] In this embodiment, preferably, the conditions in step 1 to 4 are:

[0100] If there exists a series of positive definite matrices {P1,P2…P M} makes the system have an observation mode dependent Lyapunov function at the initial time k = 0 And obey:

[0101]

[0102] And the initial feasible region of the state is a generalized positive invariant set, and the reciprocal horizon optimization control problem has a solution, then the state feedback controller satisfies the recursive feasibility and the mean square stability of the closed-loop system; where {P1,P2…P M} is a series of Lyapunov function solution matrices, γ is the feasible region boundary value, and is also the upper bound of the cost function; Represents r k The Lyapunov function matrix under the mode, r i；k Represents the system mode at time k+i predicted at time k.

[0103] In this embodiment, preferably, the specific process of solving the rolling horizon optimization control problem based on the linear matrix inequality in step 15 to obtain the feedback control gain and the feasible region is as follows: given positive definite matrices Q and R; given matrices E and F; vector μ; for a system with a piecewise homogeneous emission probability hidden Markov jump model, the rolling horizon optimization control problem is converted into an optimization problem based on a linear matrix inequality through Schur's complement lemma and matrix congruence transformation, and the constraints of recursive feasibility, closed-loop system mean square stability, system state and input control quantity are satisfied. The optimization problem is expressed as follows:

[0104]

[0105]

[0106] wherein, denotes that the matrix is negative definite; x denotes a given initial state; * denotes an item omitted for abbreviation due to a symmetric matrix; s.t. denotes for Both of them need to exist a positive definite matrix T m ,P m , Θ, an invertible matrix and a matrix satisfy the above inequality condition; then for the initial state x, the observation modal dependent state feedback controller is The feasible region thereof is

[0107] Another embodiment of the present application provides a fixed-wing unmanned aerial vehicle fault-tolerant optimization control method based on a hidden Markov chain. As shown in the figure, the method is implemented according to the following steps: Figure 1

[0108] Step 1: According to the unmanned aerial vehicle fault type and model uncertainty, a segmented homogeneous emission probability hidden Markov jump model is established.

[0109] The specific steps are as follows: first, a continuous-time Markov jump model is established according to the unmanned aerial vehicle fault type and model uncertainty, the continuous-time state space equation is discretized through Euler iteration to obtain the unmanned aerial vehicle Markov jump model; second, a segmented homogeneous emission probability is introduced to represent the relationship between fault detection information and actual information, and a segmented homogeneous emission probability hidden Markov jump model is obtained; finally, the system state and control constraints are given.

[0110] According to the embodiment of the present application, 1) first consider the linear dynamics and kinematics nominal model of the unmanned aerial vehicle

[0111]

[0112] Wherein x(t) and u(t) represent the unmanned aerial vehicle state and control input respectively; and represent the unmanned aerial vehicle system matrix and input matrix under standard state respectively.

[0113] On the basis of formula (1) combined with the unmanned aerial vehicle model uncertainty and fault type, the unmanned aerial vehicle continuous-time Markov jump model is established:

[0114]

[0115] Wherein, represents the unmanned aerial vehicle system matrix under fault; is an unmanned aerial vehicle input matrix under fault.​ represents the uncertainty coefficient of the UAV model; represents the UAV actuator fault gain matrix. k} represents a random process that obeys the Markov characteristic, and the random process has a finite set of M switching modes. and obey the transition probability:

[0116]

[0117] By performing Euler iteration on it, we can obtain the Markov jump model of the UAV in discrete time:

[0118]

[0119] in The positive constant s is the sampling period of the system, and I represents the identity matrix.

[0120] 2) Secondly, considering the mismatch between the observed mode and the true mode of the system, it is assumed that the true mode and the observed mode sequence of the system obey two independent random processes. k} is a Markov chain, representing the real mode of the system {r k} observation mode and from a finite set of N modes The emission probability is used to describe the random process {φ k} and {r k}, that is, for The emission probability is expressed as

[0121]

[0122] where λ m,n (k)∈[0,1],

[0123] Introducing the segmented homogeneous emission probability, the time-varying emission probability is simplified to the emission probability of homogeneous characteristics. The emission probability of the piecewise homogeneous characteristic is expressed as:

[0124]

[0125] in

[0126] The change of the piecewise homogeneous emission probability is determined by the higher-level Markov chain {ψ k}determines, and {ψ k}From a finite set of H modes Define the random process {ψ k The transition probability of} is:

[0127]

[0128] where

[0129] In summary, the system model and the emission probability of the compliance attribute together constitute the piecewise homogeneous emission probability hidden Markov jump model of the failed UAV.

[0130] 3) Finally, the hard constraints of the system state and control quantity are established according to the actual performance of the UAV, as follows:

[0131] |[Ex(k)+Fu(k)] q |≤[μ] q (7)

[0132] where D represents the number of system hard constraints, and [·] q represents the qth element of the vector.

[0133] Step 2: For the system with the piecewise homogeneous emission probability hidden Markov jump model, a model predictive controller is designed to meet the recursive feasibility and closed-loop system stability.

[0134] The specific steps are as follows: first, the structure of the observation mode-dependent state feedback controller is established; second, the cost function is designed according to the model predictive control form, and the rolling horizon optimization control problem is established; then, the mode-dependent Lyapunov function and the generalized positive invariant set are constructed, and the conditions for ensuring the recursive feasibility and the mean square stability of the closed-loop system of the model predictive controller are proposed; finally, a solution method for the controller is designed, which can be solved by using the linear matrix inequality method.

[0135] According to the embodiment of the application, 1) first, the observation mode-dependent state feedback controller is designed for the system, which is in the form of:

[0136]

[0137] wherein, is the state feedback controller gain designed by the model predictive control method. By substituting into the closed-loop system is obtained as follows:

[0138]

[0139] wherein represents the matrix of the closed-loop system.

[0140] 2) secondly, for the infinite-time domain cost function is designed in the form of:

[0141]

[0142] where Q, R are known weight matrices; x(i; k) and u(i; k) are the predicted system state and system input at time k+i, respectively.

[0143] At each time instant k, the following optimization problem is formulated:

[0144]

[0145] The control sequence {u(k), u(i; k),...} is obtained by solving the above optimization problem, and the first control action u(k) is applied to the system, the next time instant continues to solve the problem to carry out the rolling horizon optimization control.

[0146] 3) Then construct the following modal dependent Lyapunov function:

[0147] V(x(i; k), r i；k = m) = x T (i; k) P m x(i; k) (12)

[0148] And it is subject to

[0149]

[0150] Construct the generalized positive invariant set Φ k , which is expressed as:

[0151]

[0152] That is, at the current time instant k, the system state belongs to Φ k , then at the future time instant k+i, the system state belongs to the following set:

[0153]

[0154] Therefore, the set Φ k is the feasible region of the system state at time instant k. Where the upper index T represents the transpose of the matrix.

[0155] 4) The conditions to guarantee the recursive feasibility of the model predictive controller and the mean square stability of the closed-loop system are proposed:

[0156] Theorem 1: If there exists a series of matrices {P1, P2... P M} such that the system at the initial time instant k = 0 exists the above modal dependent Lyapunov function and the generalized positive invariant set, and the optimization problem has a solution, then the recursive feasibility of the model predictive controller and the mean square stability of the closed-loop system.

[0157] Proof: The proof is divided into two steps to prove the recursive feasibility and then prove the mean square stability.

[0158] Recursive feasibility: If there is a solution to the optimization problem for state x(k) at time k, then there exists a control gain matrix The control sequence {u(k),u(1;k),…} is obtained, and the first control action u(k) is applied to the system, which results in the following equation:

[0159]

[0160] So easy to get It is a feasible control sequence for the optimization problem of state x(k+1) at time k+1. Its recursive feasibility can be easily proved by mathematical induction.

[0161] Mean square stability: If there is a solution to the optimization problem for the state x(k) at time k, then there exists a set of matrices Based on available:

[0162]

[0163] According to the recursive feasibility, there must be a set of matrices at time k+1 According to optimality, we can get:

[0164]

[0165] Substituting in:

[0166]

[0167] Therefore V(x(i;k),r i；k ) is a strictly decreasing quadratic function, as k→∞, x(k)→∞.

[0168] 5) Finally, based on the conditions proposed in Theorem 1 above, a solution method for the model predictive controller with observation mode dependence is designed.

[0169] Theorem 2: The positive definite matrices Q, R, E, F, and μ are given. For a hidden Markov system with piecewise homogeneous emission probabilities, if There exists a positive definite matrix T m ,P m , Θ, reversible matrix and matrix The following conditions are met:

[0170]

[0171]

[0172]

[0173]

[0174]

[0175]

[0176] where Then, for the state x(k) at time k, the model predictive controller satisfies the recursive feasibility and closed-loop system mean square stability conditions and the system and input control constraints, and the observation mode dependent state feedback controller is

[0177]

[0178] Proof: According to the left side of the condition, it can be equivalently transformed as

[0179]

[0180] where The further condition can be equivalently transformed as

[0181]

[0182] Further transformed as

[0183]

[0184]

[0185]

[0186] By the Schur complement lemma, and introducing variables Then, inequalities (20)-(22) can be proved.

[0187] Combining conditions (13) and (14), and using mathematical induction, it is easy to get that the condition inequality is established as long as inequality is satisfied.

[0188] Considering the system constraints, the following derivation can be made:

[0189]

[0190] According to the Schur complement lemma, inequality and is proved.

[0191] Taking the expectation of inequality from i = 0→∞ and accumulating, according to V(x(∞;k)) = 0, we have:

[0192]

[0193] In summary, the linear matrix inequality problem to be solved is:

[0194]

[0195] Step three: design the offline part of the controller.

[0196] The specific steps are as follows: according to the state constraint range, select a limited number of initial states for the controller; according to step two, solve the linear matrix inequality for each state to obtain an observation mode dependent controller and related parameters that satisfy stability and recursive feasibility, and store these parameters and the controller in a sequence list. In the implementation process, it is required that each corresponding elliptical region obtained should be gradually expanded in a step-by-step manner, the first elliptical region should be very close to the origin, and the last elliptical region should contain the actual initial state of the aircraft.

[0197] According to an embodiment of the present application, a set of state sequences wherein and for each ω, solve the optimization problem to obtain the feedback control gain and the feasible region Φ ω wherein the feasible region and it is required to ensure that for any ω≠1. In this way, Ω gradually expanded feasible regions Φ ω corresponding to a set of control gains

[0198] Step four: online controller selection.

[0199] The specific steps are as follows: obtain the actual initial state of the fixed-wing unmanned aerial vehicle system, search the sequence list stored in step three online at each time, determine the smallest elliptical region in which the current state of the unmanned aerial vehicle is located at each time, thereby obtaining the controller and related parameters under the state. Calculate the control input required by the observation mode, and apply the control input to the next time until the state converges, and finally realize the stable flight of the unmanned aerial vehicle.

[0200] According to an embodiment of the present application, at each time k, determine the region in which the current system state x(k) is located, if x(k) is in Φ 1 , then use the corresponding controller If not in Φ 1 , then further determine whether it is in Φ 2 , and so on. In this way, the controller gain is obtained at each time k and is applied to the next time k+1 until the system state converges.

[0201] The technical effects of the present application are further verified through experiments.

[0202] For the fault-tolerant optimal control of the unmanned aerial vehicle, the simulation example and algorithm parameters are as follows: the system matrix of the lateral nominal model of the unmanned aerial vehicle refers to the literature [6]:

[0203]

[0204]

[0205] The sampling time is 0.4s. Assume that the UAV system has three working modes, corresponding to the rudder efficiency reduction Reduced aileron efficiency Aileron and rudder efficiency are reduced simultaneously On the other hand, the uncertainty coefficients of the system model ξ1=1,ξ2=0.95,ξ3=1.05. Assume that the UAV fault mode switching obeys a Markov random process, and the process {r k The transition probability matrix of} is:

[0206]

[0207] The three modes of emission probability are:

[0208]

[0209] And its mode switching obeys the Markov process {ψ k}, process {ψ k The transition probability matrix of} is:

[0210]

[0211] Consider the UAV performance state and control constraints as |x1|≤20,|x2|≤20,|x3|≤20,|x4|≤20,|u1|≤45,|u2|≤45. Assume that the weight matrix in the infinite time domain cost function is Q=I4,R=I2. Six initial states are selected in the offline part, namely And obtain 6 groups of controllers and their corresponding feasible regions. In the online part, the initial condition of the system is x(0)=[3 3 3 3] T , select the appropriate controller at each moment and simulate 30 steps.

[0212] Due to the randomness of drone failures, 15 different experiments were conducted to verify the effectiveness of the algorithm. The simulation running device and environment is Intel(R) Core(TM) i7-8700K CPU@3.70GHz. Figure 2 , Figure 3 Given 15 groups of open-loop state response curves and one group of random processes {r k},{ψ k},{φ k}, which indicates that all states of the open-loop system cannot converge. Figure 4 , Figure 5It is shown that all state response curves of the method converge to the origin before k=30 under the premise of meeting system state and control constraints, thereby indicating the random stability of the fault unmanned aerial vehicle system. Figure 6 It can be seen that the offline method proposed in the application has similar control performance compared with the online method. The online running time of a single experiment of the online algorithm is 330.32s, and the online running time of a single experiment of the method of the application is 0.02s. By comparing the online running time, it can be seen that the method of the application greatly reduces the online calculation burden.

[0213] In summary, the application models the unmanned aerial vehicle fault by using the piecewise homogeneous emission probability hidden Markov jump model, and provides an unmanned aerial vehicle fault-tolerant optimization control method based on the explicit model predictive control. Compared with the traditional method, on the one hand, the application considers the case that the fault observation mode does not match the actual mode of the system and the case that the emission probability is time-varying, designs an observation mode dependent controller, and greatly improves the application range of the controller; on the other hand, the application considers the problem that the online solving calculation cost of the existing model predictive control technology is too high, designs an offline algorithm, guarantees the recursive feasibility, closed-loop system stability and system constraint control performance, and greatly improves the online calculation efficiency and control frequency.

[0214] Although the application is described according to a limited number of embodiments, those skilled in the art, with the benefit of the above description, understand that other embodiments can be conceived within the scope of the application described herein. The disclosure of the application is illustrative rather than restrictive, and the scope of the application is defined by the appended claims.

[0215] The application cites the following documents:

[0216] [1] Cai B. Analysis and synthesis of hidden semi-Markov jump systems based on semi-Markov kernel [D]. Harbin Institute of Technology, 2019.

[0217] [2] S Fang, H Li, D Pei, et al. Stabilization of Discrete-Time Hidden Semi-Markov Jump Systems with Time-varying Emission Probability [C] / / 2021 11th International Conference on Intelligent Control and Information Processing (ICICIP). IEEE, 2021: 202-208.

[0218] [3] T. Shi, P. Shi and Z. G. Wu. Dynamic Event-Triggered Asynchronous MPC of Markovian Jump Systems With Disturbances[J]. IEEE Transactions on Cybernetics, 2021.

[0219] [4] B Zhang, Y Song. Asynchronous Constrained Resilient Robust Model Predictive Control for Markovian Jump Systems[J]. IEEE Transactions on Industrial Informatics, 2019, 16(11): 7025-7034.

[0220] [5] Luo Jia, Zhao Haoran, Gao Zhuning, et al. Low Voltage Ride Through Strategy for Doubly-fed Wind Turbines Based on Explicit Model Predictive Control and Improved Virtual Impedance[J]. Power System Technology, 2021, 45(5): 1716-1723.

[0221] [6] G. J. J. Ducard. Fault-tolerant Flight Control and Guidance Systems: Practical Methods for Small Unmanned Aerial Vehicles[M]. Springer Science & Business Media, 2009.

Claims

1. A fixed-wing unmanned aerial vehicle fault-tolerant optimization control method based on hidden Markov chain, characterized in that, The method comprises the following steps: Step one, selecting multiple initial states of unmanned aerial vehicle systems, and solving the optimal solution for each initial state of the system based on a piecewise homogeneous emission probability hidden Markov jump model to obtain a set of feedback control gains and a feasible region that satisfy the stability of a closed-loop system, the feasibility of recursion, and system constraints, thereby obtaining the correspondence between system states, feasible regions, and feedback control gains; The specific process comprises: Step one, establishing a piecewise homogeneous emission probability hidden Markov jump model based on random faults of unmanned aerial vehicles; and establishing system state constraints and control input constraints based on the performance of unmanned aerial vehicles; Step two, establishing an observation mode-dependent state feedback controller, wherein the observation mode comprises a system mode and an emission probability mode; Step three, establishing an infinite-time-domain cost function and a receding horizon optimization control problem; the expression of the infinite-time-domain cost function is as follows: In the formula, Q and R are known weight matrices; x(i; k) and u(i; k) represent the predicted system state and system input at the k+i time point at the k time point; and Ξ represents a mathematical expectation; Step four, establishing a condition under which the state feedback controller satisfies the feasibility of recursion and the mean square stability of a closed-loop system; Step five, based on the conditions under which the state feedback controller satisfies the feasibility of recursion, the mean square stability of a closed-loop system, and constraints, designing a solution method based on the solvability of a linear matrix inequality for the receding horizon optimization control problem, thereby obtaining feedback control gains and a feasible region; Step two, for the actual state of the unmanned aerial vehicle system, determining the feasible region in which the current system state is located at each time point; and according to the correspondence, obtaining the corresponding feedback control gains, so that the controller controls the unmanned aerial vehicle to fly according to the corresponding feedback control gains at each time point. 2.The method of claim 1, wherein, The specific process of step one comprises: 1) establishing a continuous-time Markov jump model of an unmanned aerial vehicle by combining the model uncertainty and fault types of the unmanned aerial vehicle: where t denotes time; x(t) denotes the UAV system state vector; u(t) denotes the control input; denotes the UAV system matrix under fault; denotes the UAV input matrix under fault; k denotes a random process with Markov property, and the random process takes values in a finite set containing M switching modes and subject to transition probabilities: Euler iteration is performed on the model to obtain a discrete-time Markov jump model of the unmanned aerial vehicle: where k represents the time; The normal number s is the sampling period of the system, and I represents the unit matrix. 2) Considering the mismatch between the system observation modalities and the system true modalities, it is assumed that the system true modalities and the observation modality sequences follow two independent stochastic processes, where the stochastic process {φ k} is a Markov chain, representing the observation modalities of the system true modalities {r k} and taking values from a finite set of N modalities ; the emission probability modality is used to describe the transition probability between the stochastic processes {φ k} and {r k}, i.e., for the emission probability is represented as: wherein π m,n (k) e [0, 1], The piecewise homogeneous emission probability is introduced to simplify the time-varying emission probability into the emission probability with homogeneous characteristics, and at this time, the emission probability is expressed as The piecewise homogeneous emission probability is expressed as: wherein The variation of the piecewise homogeneous emission probability is determined by a higher level Markov chain {ψ k} and {ψ k} takes values from a finite set of H modalities ; the transition probabilities of the stochastic process {ψ k} are defined as: wherein 3) establishing the hard constraints of system states and control inputs according to the actual performance of the unmanned aerial vehicle: |[Ex(k) + Fu(k)] q |≤[μ] q Wherein, E represents state constraint matrix; F represents input constraint matrix; μ represents constraint boundary value; D represents the number of system hard constraints, [·] q represents the qth element of the vector.

3. The fault-tolerant optimization control method for fixed-wing UAV based on hidden Markov chain according to claim 2, characterized in that, The expression of the state feedback controller in step two is as follows: In the formula, represents the feedback control gain of the state feedback controller.

4. The fault-tolerant optimal control method for fixed-wing UAV based on hidden Markov chain according to claim 3, characterized in that, The receding horizon optimization control problem in step three is established as follows: wherein denotes the closed loop system matrix at time k+i predicted at time k, A m and B m denote the system matrix and input matrix at time k+i predicted at time k, respectively.

5. The fault-tolerant optimization control method for fixed-wing UAV based on hidden Markov chain according to claim 4, characterized in that, The condition in step four is as follows: If there exists a sequence of positive definite matrices {P1, P2... P M} such that the system has an observation modality dependent Lyapunov function V(x(i; k), r i；k = m) = x T (i; k)P m x(i; k) and obeys: and the state initial feasible region is a generalized positive invariant set, and the receding horizon optimal control problem has a solution, then the state feedback controller satisfies the recursive feasibility and the closed-loop system is mean-square stable; where 1, P2…P M} is a series of Lyapunov function solution matrix, γ is the feasible region boundary value, and is also the upper bound of the cost function; represents r k Lyapunov function solution matrix under the modal; r i；k represents the system modal at time k+i predicted at time k.

6. The fault-tolerant optimization control method for fixed-wing UAV based on hidden Markov chain according to claim 5, characterized in that, The specific process of obtaining feedback control gains and a feasible region by solving the receding horizon optimization control problem based on the solvability of a linear matrix inequality in step five is as follows: Given positive definite matrices Q and R; given constraint matrices E and F; and a vector μ; for a system with a piecewise homogeneous emission probability hidden Markov jump model, the receding horizon optimization control problem is converted into an optimization problem based on the solvability of a linear matrix inequality by using the Schur complement lemma and matrix congruence transformation, and the optimization problem satisfies the conditions of the feasibility of recursion, the mean square stability of a closed-loop system, and the constraints of system states and input control, and the optimization problem is expressed as follows: wherein, [·] <0 means the matrix is negative definite; x means the given initial state; * means the terms omitted due to the symmetry of the matrix; s.t. means subject to There exists a positive definite matrix T m ,P m , Θ, an invertible matrix and a matrix satisfying the above inequality conditions; then for the initial state x, the observer mode dependent state feedback controller is whose feasible region is

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