High-speed train fixed time tracking control method of distributed state observer

Abstract

According to the fixed-time nonsingular terminal sliding-mode control strategy based on the distributed observer, compared with a common sliding-mode consistency control method, a high-speed train fixed-time nonsingular terminal sliding-mode control method based on the distributed fixed-time observer is adopted; the method can weaken the influence that the speeds of all carriages are not synchronous due to external disturbance when a high-speed train runs, is more suitable for a nonlinear and strong-coupling high-speed train system, and meanwhile, adopts a distributed fixed time state observer to estimate the speed and position information of a navigator carriage, so that the robustness of the method is improved; the control method disclosed by the invention has quick responsiveness and relatively high control precision, and can enable each compartment to quickly and accurately track a target speed curve, so that the high-speed train can efficiently, stably and reliably run under the conditions of uncertain factors and external interference.

Description

Technical Field

[0001] The present invention relates to the technical field of high-speed train control systems, and more specifically, to a high-speed train fixed-time non-singular terminal sliding mode tracking control method based on a distributed state observer. Background Art

[0002] As an important part of modern transportation, high-speed trains play an increasingly important role in promoting national construction, economic development and improving people's livelihood. With the advantages of high speed, strong transportation capacity, high safety and large passenger capacity, high-speed trains have become an indispensable means of transportation in my country. However, when high-speed trains are running under actual working conditions, they are affected by the complex and changeable operating environment, such as temperature, running time and external interference. The speeds of the carriages are not synchronized, causing problems such as longitudinal impulse of the train, which makes the train operation unstable and even leads to safety accidents. Therefore, the safety and stability of high-speed train operation are gradually becoming prominent.

[0003] The coordinated speed control of high-speed train carriages is the key to train operation control. Research on the coordinated speed control algorithm of high-speed train carriages can not only improve the synchronization accuracy and operation efficiency, but also have a significant impact on the safe operation of high-speed trains and the development of China's high-speed railways. Therefore, in order to ensure the stable operation of high-speed train systems under parameter perturbations and external disturbances, it is necessary to seek new control methods to achieve efficient, safe and reliable operation of high-speed trains under actual operating conditions. Summary of the invention

[0004] The technical problem to be solved by the present invention is to provide a high-speed train fixed-time non-singular terminal sliding mode tracking control method based on a distributed state observer in view of the shortcomings and defects of the existing technology.

[0005] To achieve the above object, the present invention adopts the following technical solutions:

[0006] A fixed-time non-singular terminal sliding mode control method based on a distributed state observer, characterized in that it comprises the following steps:

[0007] Step 1: Establish a high-speed train multi-agent mathematical model:

[0008] Step 1.1, the high-speed train system is a multi-agent system consisting of a pilot car and i (i = 1, ..., n) follower cars. The dynamic equation of the pilot car is Among them, v 0 , x 0 ,u 0 are the speed, displacement, and control input of the pilot car, respectively;

[0009] Step 1.2, the dynamic equation of the i-th (i=1,…,n) follower car of the high-speed train is:

[0010] Where: x i , v i (i=1,2,…,n) is the displacement and running speed of the i-th carriage; m i ,u i , f i (i=1,2,…,n) is divided into the mass of the i-th car, control input, and coupler force; a ij is the connection weight between adjacent follower carriages, a ij =1 means there is information transmission between carriage i and carriage j, a ij = 0 means there is no information transmission between carriage i and carriage j; D fi (i=1,2,…,n) is the total resistance of the i-th carriage;

[0011] Step 1.3, the dynamic equation of the i-th (i=1,…,n) follower car can be simplified to Among them, b i =1 / m i ; is the unknown composite disturbance term of the i-th carriage;

[0012] Step 2: Design a fixed-time terminal sliding mode controller for the pilot car

[0013] Step 2.1, define the speed error e of the pilot car 0 =v 0 -v * , where v 0 is the actual speed of the pilot car, v * is the target speed of the pilot car;

[0014] Step 2.2: Construct terminal sliding surface where 0<κ<1, λ>1, α 0 , β 0 , κ, λ are the parameters to be designed; sig λ (·)=|·| λ sign(·), sign(·) is the sign function; is the integral term;

[0015] Step 2.3, select the double power reaching law where r 1 >1,0 <r 2 <1, l 1 , l 2 、r 1 、r 2are the parameters to be designed;

[0016] Step 2.4, design the control law of the pilot car as:

[0017] Step 3: Design a fixed-time distributed state observer for the i-th (i=1,…,n) follower car to observe the leader's state information:

[0018] Step 3.1, for the i-th (i=1,…,n) follower car, define the observer error equation as: represents the displacement x of the i-th carriage relative to the navigator i With speed v i An estimated value of is the displacement x of the ith carriage relative to the pilot i With speed v i The observer error of

[0019] Step 3.2: In the high-speed train system, since some follower cars do not have direct information exchange with the leader car and cannot obtain the position and speed of the leader, a fixed-time distributed state observer is designed to estimate the state information of the leader. For the i-th follower car, the fixed-time distributed state observer is designed as

[0020]

[0021] Among them, m 1 >n 1 、m 2 <n 2 ;m 1 , n 1 , m 2 , n 2 , p x1 ,q x2 , p v1 ,q v2 , η are all positive real numbers and are the parameters to be designed;

[0022] Step 4: Design a fixed-time non-singular terminal sliding mode controller for the i-th (i=1,…,n) follower car:

[0023] Step 4.1, define the position tracking error of the follower car in section i as The velocity tracking error of the i-th follower car is defined as Among them, e xi , e vi are the position x of the i-th carriage respectively i The tracking error and velocity v iThe tracking error;

[0024] Step 4.2, when t ≥ T L , The position tracking error in step 4.1 can be written as e xi =x i -x 0 ; The speed tracking error in step 4.1 can be written as e vi =v i -v 0 ; T L is the convergence time of the fixed-time distributed observer;

[0025] Step 4.3, design a new non-singular terminal sliding surface Among them, α 1 , β 1 , k 1 , ε is a positive real number, k 1 >0,α 1 , β 1 , k 1 , ε is the parameter to be designed;

[0026] Step 4.4, design the fixed-time non-singular terminal sliding mode controller of the i-th (i=1,…,n) follower car as:

[0027] Among them, 0<ρ<1, is the parameter to be designed; the disturbance term d i is bounded, that is, |d i | <D 1 , D 1 >0.

[0028] Furthermore, the high-speed train pilot fixed-time terminal sliding mode controller selects the gain α 0 >0,β 0 >0,l 1 >0,l 2 >0, the pilot car can be in a fixed time T 0 The target speed characteristic curve is tracked internally, and the convergence time is T 0 satisfy

[0029] Furthermore, the fixed-time distributed state observer selects the gain m 1 >n 1 、m 2 <n 2 , when t ≥ T 1 hour, The observer velocity error equation converges, with a fixed convergence time T 1 satisfy When t ≥ T2 , The observer position error equation converges, with a fixed convergence time T 2 satisfy Therefore, the distributed state observer can be L Internal estimated navigator displacement x i With speed v i Information, and the convergence time T L Satisfaction: T L ≤T 1 +T 2 ; Among them, m 1 , n 1 , m 2 , n 2 , p x1 ,q x2 , p v1 ,q v2 Represents the parameters to be designed for the i-th follower car.

[0030] Furthermore, the follower fixed-time non-singular terminal sliding mode controller selects the gain α 1 >0,β 1 >0,|d i | <D 1 ;

[0031] 1) When |e vi |>εThe follower will reach the corresponding sliding surface within a fixed time, and the convergence time is T 3 for After the follower car reaches the sliding surface, according to the sliding mode equivalence principle, s1 = 0, and the follower car tracks the leader car within a fixed time and converges to the set A = {e xi :e xi ≤ε}, convergence time T 4 satisfy The follower will reach the corresponding sliding surface within a fixed time, and the convergence time is T 5 satisfy When the follower car reaches the sliding surface s 1 After that, we get 1 = 0, the speed and displacement error of the follower car converge to 0, at a fixed time T 6 Internal tracking navigator, stable time meets:

[0032] 2) When |e vi |≤ε, since the sign function sign(·) is not differentiable, we consider two cases; before the follower follows the leader, if e xi = 0 and e vi ≠0, then e xiwill leave the origin immediately, so e xi = 0 can be ignored, and we can get After the follower follows the leader, there is a sign (e xi )=0; so we can also get In summary, the speed and displacement of the i-th (i=1,…,n) follower car are synchronized with the leader car.

[0033] In order to effectively solve the speed asynchrony problem of each carriage caused by unknown composite disturbances during the operation of high-speed trains, the present invention adopts a fixed time sliding mode control strategy for high-speed trains based on a fixed time distributed state observer (FTDSO), so that each carriage of the high-speed train can be consistent with the speed of the pilot carriage. First, a multi-agent model of high-speed trains is constructed. Secondly, the terminal sliding surface is selected, and combined with the bi-power reaching law, the fixed time terminal sliding mode controller (FTTSMC) of the pilot carriage is designed. Since some follower cars in a high-speed train cannot communicate directly with the leader car, the position and speed of the leader car cannot be obtained. Therefore, FTDSO is designed to observe the state information of the leader car. At the same time, a fixed time non-singular terminal sliding mode controller (FTNSTSMC) of the follower car is designed using a non-singular terminal sliding surface to make the system state converge within a fixed time, ensuring the speed synchronization of each car, which effectively improves the robustness of the high-speed train system. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 This is an embodiment of the present invention: a structural diagram of a “coupler” model.

[0035] Figure 2 This is a system structure block diagram of an embodiment of the present invention;

[0036] In the figure, v * is the target speed of the pilot car; v 0 、x 0 Respectively represent the speed and displacement of the pilot car; x i 、v i 、u idenote the displacement, velocity, and control input of the ith (i=1, 2, …, n) follower car, respectively; FTFSO1, FTFSO1, …, FTFSOn denote the fixed-time distributed state observer (FTFSO) of the ith (i=1, 2, …, n) follower car, d i are the unknown composite disturbances of the i-th (i=1,2,…,n) follower car, They represent the observed values ​​of the velocity and displacement of the leader car by the i-th (i=1,2,…,n) follower car respectively;

[0037] Figure 3 This is an embodiment of the present invention: a communication topology diagram between high-speed train carriages.

[0038] Figure 4 This is an embodiment of the present invention: a target speed-displacement curve diagram.

[0039] Figure 5 This is an embodiment of the present invention: a comparison diagram of PID and FTTSMC speed tracking.

[0040] Figure 6 This is an embodiment of the present invention: a comparison diagram of PID and FTTSMC speed tracking errors.

[0041] Figure 7 An embodiment of the present invention: FTDSO and v 0 Tracking effect diagram.

[0042] Figure 8 An embodiment of the present invention: FTDSO and x 0 Tracking effect diagram.

[0043] Fig. 9 An embodiment of the present invention: FTDSO velocity estimation error picture.

[0044] Fig.10 An embodiment of the present invention: FTDSO displacement estimation error picture.

[0045] Fig.11 This is an embodiment of the present invention: FTNSTSMC two-dimensional displacement tracking effect diagram.

[0046] Fig.12 This is an embodiment of the present invention: FTNSTSM three-dimensional displacement tracking effect diagram.

[0047] Fig.13 This is an embodiment of the present invention: FTNSTSMC two-dimensional displacement tracking effect.

[0048] Fig.14 This is an embodiment of the present invention: FTNSTSM three-dimensional displacement tracking effect.

[0049] Fig.15 This is an embodiment of the present invention: FTNSTSMC follower velocity error diagram.

[0050] Fig.16 This is an embodiment of the present invention: SMCC follower speed error diagram. DETAILED DESCRIPTION

[0051] The present invention is further described in detail below in conjunction with the accompanying drawings and embodiments.

[0052] Figure 1 This is a structural diagram of the "coupler" model proposed in the present invention. Figure 2 It is a block diagram of an embodiment of the fixed-time non-singular terminal sliding mode control method for high-speed trains based on FTDSO proposed by the present invention. Figure 3 This is the communication topology diagram between high-speed train carriages proposed by the present invention.

[0053] The traditional single-mass point model cannot reflect the forces between high-speed train carriages. Therefore, each carriage of the train is regarded as a mass point, and the coupling relationship between the carriages is simplified to a "coupler" model. A multi-mass point model that reflects the actual operating state of the high-speed train is established. The resistance encountered by the high-speed train during actual operation is as follows:

[0054] Basic resistance: During the operation of high-speed trains, the basic resistance generated by the rolling of wheels and rails can be expressed as

[0055]

[0056] Where: v i is the running speed of the i-th carriage (m / s); a 1i ,a 2i ,a 3i It is the basic running resistance coefficient of the carriage.

[0057] Additional resistance: When a high-speed train is running, it will generate additional resistance when passing through ramps and tunnels, which mainly include the following three types:

[0058] 1) Additional resistance on the slope: When a high-speed train passes through an uphill or downhill slope, the gravity of the carriage itself will generate additional resistance on the slope, which can be expressed as

[0059]

[0060] in: are the slopes of the two sections where the carriage is located; l 1 , l2 They are the lengths of the two sections of the car respectively; L is the total length of the car body.

[0061] 2) Additional resistance of curve: When a high-speed train passes through a curve, the resistance caused by the bending of the wheel and rail is the additional resistance of the curve, which can be expressed as

[0062]

[0063] Where: R 1 , R 2 is the curve radius of the carriage at two locations, l 1 , l 2 It is the length of the two curved sections where the carriage is located.

[0064] 3) Tunnel additional resistance: When a high-speed train passes through a tunnel, air turbulence will generate resistance on its surface, which is called tunnel additional air resistance and can be expressed as

[0065] D f4i =0.00013·L s (4)

[0066] Where: f s Additional air resistance per unit tunnel, L s is the tunnel length.

[0067] Therefore, the total resistance of the high-speed EMU when running on the line is

[0068] D fi =D f1i +D f2i +D f3i +D f4i (5)

[0069] Train coupler force: When analyzing the coupling relationship between high-speed train carriages, the coupling relationship between each carriage is simplified to a "coupler" model to describe the change process of the coupler force between the carriages, so as to more accurately simulate the actual operation of the entire high-speed train system. The train coupler force is a linear function of the relative displacement and relative speed of the carriages, and its expression is

[0070] f i =k i (x i+1 -x i )+c i (v i+1 -v i ) (6)

[0071] Where: k i , c i are the spring constant and damping coefficient of the “coupler” model respectively; x iis the displacement of the i-th carriage, v i is the running speed of the i-th carriage, f i Indicates the coupling force between adjacent carriages.

[0072] In summary, each carriage of the high-speed train is regarded as a mass point, and the force analysis of each carriage of the train is carried out during operation. The dynamic equation of the multi-mass point model of the high-speed train is established as follows:

[0073]

[0074] Where: m i ,u i , f i (i=1,2,…,n) is divided into the mass of the i-th car, control input, and coupler force; x i is the displacement of the i-th carriage, v i is the running speed of the i-th carriage; D fi is the total resistance of the i-th carriage.

[0075] The high-speed train system can be regarded as consisting of a leader car and multiple follower cars i (i = 1, 2, ..., n). Each car is regarded as a particle, and the entire train forms a multi-agent system of a multi-particle model. Assume that the follower car is a node in the multi-agent system, and the leader car is the root node in the graph theory. The information transmission between the cars can be described by the communication topology through graph theory.

[0076] Definition: The information interaction between multiple agents can be represented by the node set V n , edge set E n and the weighted adjacency matrix A n The undirected graph G = {V n ,E n ,A n}describe.

[0077] 1) Node set V n = {v 1 ,v 2 ,...,v n Each node in} represents a follower car.

[0078] 2) E n ={(v i ,v j ):v i ,v j ∈V n ,i,j=1,2,...n,i≠j} edge set, if the follower car v i Can be v j Send information, then (v i ,vj ) is the edge set between these two followers, and

[0079] 3) Weight connection matrix A n =(a ij ∈R n×n ), if there is information exchange between follower cars i and j, then a i =a j =1, otherwise a ij =0, so the matrix A n is a symmetric matrix.

[0080] Laplace matrix L n =D n -A n , where D n =[d ij ]∈R n×n is the degree matrix. When i=1,2,…,n, For node v i The degree of ij =0, i≠j. If there is information exchange between any two nodes, then the undirected graph G is called connected, and its corresponding L n is a symmetric semi-positive definite matrix whose eigenvalues ​​satisfy 0 = λ 1 (L n )≤λ 1 (L n )≤λ 2 (L n )≤…≤λ n (L n ). If we define the matrix M = L + B, B = diag (a 10 ,a 20 ,…,a n0 ), then M is a symmetric positive definite matrix.

[0081] Definition: Define sig γ (x)=|x γ sign(x), where γ>0, x∈R, sign(x) is the standard sign function.

[0082] Assumption 1: The disturbance term d of the i-th carriage of a high-speed train i is bounded, that is, there exists D 1 >0, so that |d i | <D 1 .

[0083] Assumption 2: The displacement x of the i-th carriage of the high-speed train i and its first-order derivative and second-order derivative exist and are bounded.

[0084] Lemma 1: Consider a nonlinear system x(0)=x 0 , if there exists a continuous positive definite function V(x(t)) that satisfies:

[0085]

[0086] Where α>0, β>0, n, m, p, q are all odd numbers and satisfy m>n>0, q>p>0. Then the system is stable in a fixed time, and the stable time is

[0087]

[0088] Lemma 2: Consider a nonlinear system x(0)=x 0 , if there exists a continuous positive definite function V(x(t)) that satisfies:

[0089]

[0090] Where μ>0, τ>0.

[0091] When τ = 1, the nonlinear system is stable at the origin;

[0092] When 0<τ<1, the system is stable in a finite time, and the convergence time satisfies:

[0093]

[0094] If τ ≥ 1, for any ε > 0, V(x) < ε is stable in a fixed time, and its convergence time satisfies:

[0095]

[0096] Lemma 3: Assume σ 1 ,σ 2 ,...,σ n ≥0, ω>1, then

[0097]

[0098] The traditional high-speed train controller is a PI controller. When faced with complex working conditions, the high-speed train control system cannot adapt well, especially when there are unknown compound disturbances such as parameter perturbations, external disturbances, and the dynamic model is not established. This embodiment proposes a fixed-time sliding mode control strategy for high-speed trains based on FTDSO. The leader car transmits the target speed-displacement information to the controller, and then transmits the speed and displacement information to each follower car through a fixed-time non-singular terminal sliding mode controller. Secondly, since some follower cars cannot communicate directly with the leader car and cannot obtain the position and speed of the leader car, the FTDSO is designed to observe the state information of the leader car so that each follower car of the high-speed train can be consistent with the speed and displacement of the leader car.

[0099] S1: Establishing a multi-agent mathematical model for high-speed trains

[0100] The high-speed train system consists of a leader car and i (i = 1, ..., n) follower cars. From equation (7), the multi-agent model of the high-speed train is:

[0101]

[0102] Where: m i ,u i , f i (i=1,2,…,n) is divided into the mass of the i-th car, control input, and coupler force; x i , v i (i=1,2,…,n) is the displacement and running speed of the i-th carriage; D fi (i=1,2,…,n) is the unknown composite disturbance of the i-th carriage; a ij is the connection weight between adjacent follower carriages, a ij =1 means there is information transmission between carriage i and carriage j, a ij =0 means there is no information transmission between carriage i and carriage j;

[0103] Formula (14) can be further simplified as:

[0104]

[0105] Where b i =1 / m i , It can be considered as an unknown composite disturbance term.

[0106] S2: Design of fixed-time terminal sliding mode controller (FTTSMC) for pilot

[0107] Since the speed convergence time of the follower car is directly related to the speed convergence time of the leader car, the traditional leader car generally uses a PID controller. In order to improve the rapidity and robustness of the leader controller, the FTTSMC is designed to make the speed of the leader and the target speed curve v * Quick and consistent.

[0108] The dynamic equation of the pilot car is

[0109]

[0110] Among them, v 0 , x 0 ,u 0 are the speed, displacement, and control input of the pilot car respectively.

[0111] Define the speed error e 0 is the actual speed v of the navigator 0 With target speed v * difference

[0112] e 0 =v 0 -v * (17)

[0113] From formula (17), we can get

[0114]

[0115] Construct terminal sliding surface

[0116]

[0117] In the formula, α 0 >0,β 0 >0, 0<κ<1, λ>1.

[0118] Select the bi-power reaching law

[0119]

[0120] In the formula, l 1 >0,l 2 >0,r 1 , r 2 is a positive number greater than 0, r 1 >1,0 <r 2 <1.

[0121] Therefore, the pilot control law can be obtained as

[0122]

[0123] When the terminal sliding surface s is selected 0, the speed of the pilot car can be controlled within a fixed time T under the control input formula (21). 0 Internal tracking target speed characteristic curve.

[0124] Proof: Constructing the Lyapunov function

[0125]

[0126] V 0 The derivative is

[0127]

[0128] From equation (23), we can see that the system satisfies the Lyapunov stability criterion, the designed controller is stable, and the system state converges within a fixed time. From Lemma 1, we can see that the time T for the leader to track the target speed curve is 0 satisfy

[0129]

[0130] S3: Design the distributed state observer of the i-th (i=1,...,n) follower car to observe the state information of the leader

[0131] In the high-speed train system, since some follower cars do not have direct information exchange with the leader car and cannot obtain the position and speed of the leader, the FTDSO is designed to estimate the state information of the leader. For the i-th follower car, the FTDSO is designed as

[0132]

[0133] Among them, m 1 , n 1 , m 2 , n 2 are all positive real numbers and satisfy m 1 >n 1 , m 2 <n 2 ;p x1 ,q x2 , p v1 ,q v2 is a positive real number. represents the estimated displacement and speed of the i-th car relative to the navigator.

[0134] For the i-th carriage, the observer error equation is defined as

[0135]

[0136] Taking the derivative of equation (25) we can get

[0137]

[0138] make

[0139]

[0140] definition ε x =[ε x,1 ,...,ε x,n ] T , ε v =[ε v,1 ,...,ε v,n ] T , we can get

[0141]

[0142] Among them, M=L+B is a symmetric positive definite matrix.

[0143] Design the observer formula (25), select the appropriate parameter η>0, the observer error dynamic equation (26) converges to 0 in a fixed time, and FTDSO can be L The displacement and velocity of the navigator are estimated within , and the convergence time is T L satisfy

[0144] T L ≤T 1 +T 2 (30)

[0145] Construct the following Lyapunov function

[0146]

[0147] By V 1 The derivative is

[0148]

[0149] According to Lemma 3 and Lemma 4, we can further obtain

[0150]

[0151] Also because Can get

[0152]

[0153] According to Lemma 4, we can get V 1 (t)≡0, And fixed time T 1 satisfy

[0154]

[0155] When t ≥ T 1 hour, Therefore, the first equation in (27) can be simplified to

[0156]

[0157] Similar to the above steps, construct the Lyapunov function

[0158]

[0159] V 2 The derivative is

[0160]

[0161] According to Lemma 1, we can get V 2 (t)≡0, And fixed time T 2 satisfy

[0162]

[0163] Therefore, in T L =T 1 +T 2 Afterwards, you can get Therefore, the distributed fixed-time observer is L The status information of the pilot car can be estimated.

[0164] S4: Design a fixed-time non-singular terminal sliding mode controller for the i-th (i=1,...,n) follower car

[0165] By designing the above FTDSO, the follower car can make an accurate estimate of the state of the leader car in a fixed time. In order to make the follower car track the speed of the leader car in a fixed time, FTNSTSMC is designed.

[0166] The tracking error of the follower car is defined as

[0167]

[0168] Among them, e xi , e vi are the position error and speed error of the i-th carriage respectively. L , Formula (40) can be written as

[0169]

[0170] For the follower car, a new non-singular terminal sliding surface is constructed

[0171]

[0172] Among them, α 1 , β 1 , k 1 , ε is a positive real number; α 1 >0,β 1 >0,k 1 >0.

[0173] Design the follower car i controller as:

[0174]

[0175] Where 0<ρ<1.

[0176] Assuming that Assumption 1 holds, the multi-agent system (14) will be in a state of constant time T under the control input (43). f =max(T 3 +T 4 ,T 5 +T 6 ) reaches the sliding surface (42), and the states of the leader car and the follower car finally reach the same state.

[0177] Proof: 1) When |e vi When |>ε, verify the sliding surface s 1 Convergence characteristics, constructing Lyapunov function

[0178]

[0179] By V 3 The derivative is:

[0180]

[0181] According to Lemma 1, the follower will reach the corresponding sliding surface in a fixed time, and the convergence time T 3 for

[0182]

[0183] After the follower car reaches the sliding surface, according to the sliding mode equivalence principle, s 1 =0, from formula (42) we can get

[0184]

[0185] According to Lemma 2, if s 1 = 0, the follower car follows the leader car within a fixed time and converges to the set A = {e xi :e xi≤ε}, convergence time T 4 satisfy

[0186]

[0187] 2) When |e vi |≤ε, since the sign function sign(·) is not differentiable, we consider two cases. Before the follower follows the leader, if e xi = 0 and e vi ≠0, then e xi will leave the origin immediately, so e xi = 0 can be ignored, and we can get After the follower follows the leader, there is a sign (e xi )=0. So it is also possible In summary. In either case, the follower car will reach the corresponding sliding surface at a fixed time, and the proof process is similar to |e vi |>ε.

[0188] To analyze the sliding surface s 1 Convergence characteristics, select Lyapunov function

[0189]

[0190] V 5 The derivative is

[0191]

[0192] According to Lemma 1, the follower will reach the corresponding sliding surface in a fixed time, and the convergence time T 5 satisfy

[0193]

[0194] According to the sliding mode equivalence principle, we can get s 1 =0, from formula (42), we can get:

[0195] e vi =-α 1 sign(e xi ) (52)

[0196] When the follower car reaches the sliding surface s 1 Finally, in order to verify that it can accurately and stably track the pilot car, the Lyapunov function is selected

[0197]

[0198] V 6 The derivative is:

[0199]

[0200] According to Lemma 2, in s 1 = 0, the velocity and displacement errors of the follower car converge to 0, and it tracks the leader within a fixed time, and the stabilization time satisfies: In summary, the designed controller is stable.

[0201] Next, the fixed-time non-singular terminal sliding mode control system of high-speed train based on distributed state observer is modeled and simulated. The system model is as follows: Figure 2 In order to verify the rationality and effectiveness of the proposed algorithm, it is assumed that the high-speed train has 6 carriages, each carriage is regarded as an agent, a multi-agent model is constructed, and simulation analysis is carried out in MATLAB software.

[0202] The leader car obtains the speed target curve signal through FTTSMC, designs FTDSO for each follower car to estimate the state information of the leader, and designs FTNSTSMC so that each follower car accurately tracks the leader car, thereby achieving consistent speed of each car in the high-speed train.

[0203] Taking the initial position of the pilot car as the coordinate origin and the train running direction as the positive direction, the high-speed train multi-agent communication topology is as follows: Figure 3 .Depend on Figure 3 The adjacency matrix A and degree matrix D are obtained as

[0204]

[0205] Considering that the actual operation process of high-speed trains includes acceleration, cruising, and braking stages, the speed-displacement target curve of the designed train is as follows: Figure 4 The entire operation process includes two acceleration stages, two deceleration stages and three cruising stages. The high-speed train operation time is 1000s and the maximum speed is 80m / s.

[0206] Assuming that the model parameters of the six carriages are the same, as shown in Table 1, the above parameters meet the parameters of the CRH type train. The length of each carriage is 20m, the original length of the spring of the "coupler" model is 1m, and the elastic deformation range is set to [-0.5, 0.5]m. The safety distance between adjacent carriages is [19.5, 20.5]m. The initial position and initial speed of each carriage are shown in Table 2.

[0207] Table 1 High-speed train parameters

[0208]

[0209] Table 2 Initial position and speed of each carriage

[0210]

[0211] The traditional PID control algorithm has a slow convergence speed and a lower control accuracy than the sliding mode control. However, the navigator car of the present invention adopts FTTSMC control. The system convergence speed is only related to the designed controller parameters and has nothing to do with external disturbances. The system convergence time is fixed, achieving a better control effect.

[0212] Figure 5 This is a comparison chart of the speed tracking effect of the pilot car using PID and FTTSMC. The PID parameter is k p =200, k i =500, k d =20; FTTSMC parameter is α 0 =3,β 0 =5, κ=1 / 3, λ=5 / 3, l 1 =15,l 2 =10. Figure 5 It can be seen that when the target speed curve has a working condition switch, the convergence time of FTTSMC is significantly shorter than that of PID control, and the jitter amplitude is smaller, which reflects the characteristics of FTTSMC with faster convergence speed and stronger robustness. Figure 6 It can be seen that when FTTSMC faces working condition switching, the tracking error is significantly smaller than that of PID, and the convergence time is a fixed time of 0.5s. The PID speed tracking error is between -0.006 and 0.012m / s, and the FTTSMC speed tracking error is between 0.004 and 0.008m / s, which shows that FTTSMC converges faster, can converge within a fixed time, has stronger anti-disturbance ability, and achieves better control effect.

[0213] In order to better test the tracking accuracy of FTDSO and enable the follower car to better estimate the state information of the leader car, it is assumed that the initial speed of the follower car is [v 1 (0),…,v 6 (0)]=[1,1.5,2,2.5,3,3.5], the designed FTDSO parameters are η=0.1, p x1 =8,q x2 =8, p v1 =8,q v2 =8,m 1 =3,m 2 =5,n 1 =5,n 2 =3.

[0214] Depend on Figure 7 It can be seen that the speed of the follower car tracks the speed information of the leader at 1.7s, given by Figure 8It can be seen that the follower car displacement tracks the leader's position information in 3.8s, and FTDSO enables all follower cars to estimate the leader's state information within a fixed time. FTDSO has fast observation and tracking speed and strong stability.

[0215] The state estimation error of the distributed fixed-time state observer is Fig. 9 As shown. Fig.10 It can be seen that the state errors between all follower cars and the leader car are small, among which the speed error is within [-0.0002, 0.0002] m / s, the displacement error is within [-0.0001, 0.0002] m / s, the error fluctuation range is small, and the convergence speed is fast, which meets the performance requirements of the observer design.

[0216] In order to illustrate the fast convergence speed and strong robustness of FTNSTSMC, an experimental comparison is conducted with the sliding mode consensus controllor (SMCC). To reflect the complexity of the actual operation of high-speed trains, this paper selects six signals, namely step signal, mutation signal, pulse signal, slowly changing signal, sinusoidal signal, and high-frequency signal, to act on each carriage to simulate the unknown composite disturbances suffered by high-speed trains during operation. The designed FTNSTSMC parameters are α, 1 =3,β 1 =5, k 1 =5 / 3,ρ=1 / 3,D 1 =0.5,ε=2.

[0217] Fig.11 is a three-dimensional graph of the speed curve of the follower car. Fig.12 It can be seen that there is no obvious fluctuation in the speed of each carriage during the operation of the high-speed train, and from the partial enlarged figure, it can be seen that the speed tracking effect is good, the speed error between each follower and the leader does not exceed 0.0001m / s, and the curve has a good tracking effect.

[0218] from Fig.13 It can be seen that there is no obvious fluctuation in the displacement of each carriage during the operation of the high-speed train, and from the local enlarged figure, it can be seen that the running spacing of each carriage is 20m. Fig.14 is the three-dimensional displacement curve of the follower car,

[0219] Depend on Fig.15 It can be seen that the speed error of adjacent carriages in FTNSTSMC converges quickly, and the speed error of adjacent carriages varies within a range of -2×10 -6 ~2×10 -6 m / s, which meets the requirements of the actual operation of high-speed trains and has a smaller speed error than SMC. Fig.16It can be obtained that when the train just starts, the maximum speed error between adjacent carriages of SMCC is 0.005m / s. During the operation of the train, the speed error between the carriages is stable between 0.004 and 0.005m / s. In summary, FTNSTSMC has a smaller speed error, stronger anti-interference ability, and better robustness.

[0220] In summary, the fixed-time non-singular terminal sliding mode control of high-speed trains based on distributed state observers can effectively solve the problem of speed asynchrony between train carriages in high-speed train systems caused by the inability of some followers to obtain the signals of the leader. The state information of the high-speed train leader is estimated in real time using FTDSO, so that all follower carriages can obtain the information of the leader. At the same time, the unknown disturbance term is introduced to effectively eliminate the impact of unknown disturbances on the high-speed train, so that the speed and displacement errors of each carriage can converge at a fixed time, improving the robustness of the system. The traditional PID controller of the leader carriage is replaced by a non-singular fixed-time sliding mode controller, and the follower carriage adopts a fixed-time non-singular terminal sliding mode control algorithm, which improves the stability of train operation.

[0221] The above embodiments are only preferred embodiments for fully illustrating the present invention, and the protection scope of the present invention is not limited thereto. Any equivalent substitution or change made by a person skilled in the art based on the present invention is within the protection scope of the present invention.

Claims

1. A fixed-time non-singular terminal sliding mode control method based on a distributed state observer, characterized in that: The following steps are involved: Step 1: Establish a high-speed train multi-agent mathematical model: Step 1.1, the high-speed train system is a multi-agent system consisting of a pilot car and i (i = 1, ..., n) follower cars. The dynamic equation of the pilot car is Among them, v0, x0, and u0 are the velocity, displacement, and control input of the pilot car, respectively; Step 1.2, the dynamic equation of the i-th (i=1,…,n) follower car of the high-speed train is: Among them, x i , v i (i=1,2,…,n) is the displacement and running speed of the i-th carriage; m i ,u i , f i (i=1,2,…,n) is divided into the mass of the i-th car, control input, and coupler force; a ij is the connection weight between adjacent follower carriages, a ij =1 means there is information transmission between carriage i and carriage j, a ij = 0 means there is no information transmission between carriage i and carriage j; D fi (i=1,2,…,n) is the total resistance of the i-th carriage; Step 1.3, the dynamic equation of the i-th (i=1,…,n) follower car can be simplified to Among them, b i =1 / m i ; is the unknown composite disturbance term of the i-th carriage; Step 2: Design a fixed-time terminal sliding mode controller for the pilot car Step 2.1, define the speed error of the pilot car as e0 = v0-v * , where v0 is the actual speed of the pilot car, v * is the target speed of the pilot car; Step 2.2: Construct terminal sliding surface Where 0<κ<1, λ>1, α0, β0, κ, λ are the parameters to be designed; sig λ (·)=|·| λ sign(·), sign(·) is the sign function; is the integral term; Step 2.3, select the double power reaching law where r1 > 1, 0 < r2 < 1, and l1, l2, r1, r2 are parameters to be designed; Step 2.4, design the control law of the pilot car as: Step 3: Design a fixed-time distributed state observer for the i-th (i=1,…,n) follower car to observe the leader's state information: Step 3.1, for the i-th (i=1,…,n) follower car, define the observer error equation as: represents the displacement x of the i-th carriage relative to the navigator i With speed v i An estimated value of is the displacement x of the ith carriage relative to the pilot i With speed v i The observer error of Step 3.2: In the high-speed train system, since some follower cars do not have direct information exchange with the leader car and cannot obtain the position and speed of the leader, a fixed-time distributed state observer is designed to estimate the state information of the leader. For the i-th follower car, the fixed-time distributed state observer is designed as Among them, m1>n1, m2 <n2;m1,n1,m2,n2,p x1 ,q x2 , p v1 ,q v2 , η are all positive real numbers and are the parameters to be designed; Step 4: Design a fixed-time non-singular terminal sliding mode controller for the i-th (i=1,…,n) follower car: Step 4.1, define the position tracking error of the follower car in section i as The velocity tracking error of the i-th follower car is defined as Among them, e xi , e vi are the position x of the i-th carriage respectively i The tracking error and velocity v i The tracking error; Step 4.2, when t ≥ T L , The position tracking error in step 4.1 can be written as e xi =x i -x0; the velocity tracking error in step 4.1 can be written as e vi =v i -v0;T L is the convergence time of the fixed-time distributed observer; Step 4.3, design a new non-singular terminal sliding surface Among them, α1, β1, k1, ε are positive real numbers, k1>0, α1, β1, k1, ε are parameters to be designed; Step 4.4, design the fixed-time non-singular terminal sliding mode controller of the i-th (i=1,…,n) follower car as: Among them, 0<ρ<1, is the parameter to be designed; the disturbance term d i is bounded, that is, |d i |<D1,D1> 0.

2. According to claim 1, a high-speed train fixed-time non-singular terminal sliding mode control method based on a distributed state observer is characterized in that: The high-speed train pilot fixed-time terminal sliding mode controller selects gains α0>0, β0>0, l1>0, l2>0, and the pilot car can track the target speed characteristic curve within a fixed time T0, and the convergence time T0 satisfies 3. The method for high-speed train fixed-time non-singular terminal sliding mode control based on distributed state observer according to claim 1 is characterized in that: For the fixed-time distributed state observer, gains m1 > n1 and m2 < n2 are selected. When t ≥ T1, the observer speed error equation converges, and the fixed convergence time T1 satisfies When t ≥ T2, the observer position error equation converges, and the fixed convergence time T2 satisfies Therefore, the distributed state observer can estimate the displacement x L and velocity v i of the leader within a fixed time T i , and the convergence time T L satisfies: T L ≤ T1 + T2; Among them, m1, n1, m2, n2, p x1 ,q x2 , p v1 ,q v2 Represents the parameters to be designed for the i-th follower car.

4. The method for high-speed train fixed-time non-singular terminal sliding mode control based on distributed state observer according to claim 1 is characterized in that: The follower fixed-time non-singular terminal sliding mode controller selects gains α1>0, β1>0, |d i | <D1; 1) When |e vi |>εThe follower will reach the corresponding sliding surface within a fixed time, and the convergence time T3 is After the follower car reaches the sliding surface, according to the sliding mode equivalence principle, s1 = 0, and the follower car tracks the leader car within a fixed time and converges to the set A = {e xi :e xi ≤ε}, the convergence time T4 satisfies The follower will reach the corresponding sliding surface within a fixed time, and the convergence time T5 satisfies When the follower car reaches the sliding surface s1, s1 = 0, the speed and displacement errors of the follower car converge to 0, and it tracks the leader within a fixed time T6. The stable time satisfies: 2) When |e vi |≤ε, since the sign function sign(·) is not differentiable, we consider two cases; before the follower follows the leader, if e xi = 0 and e vi ≠0, then e xi will leave the origin immediately, so e xi = 0 can be ignored, and we can get After the follower follows the leader, there is a sign (e xi )=0; so we can also get In summary, the speed and displacement of the i-th (i=1,…,n) follower car are synchronized with the leader car.

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