CN114326372B - Non-smooth feedback optimal tracking control method of position servo system - Google Patents

Non-smooth feedback optimal tracking control method of position servo system Download PDF

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CN114326372B
CN114326372B CN202111488936.XA CN202111488936A CN114326372B CN 114326372 B CN114326372 B CN 114326372B CN 202111488936 A CN202111488936 A CN 202111488936A CN 114326372 B CN114326372 B CN 114326372B
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position servo
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CN114326372A (en
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任雪梅
韩宁
成云
郑冬冬
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Beijing Institute of Technology BIT
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Abstract

The invention discloses a non-smooth feedback optimal tracking control method of a position servo system, which relates to the technical field of electromechanical control and is used for inhibiting buffeting of the system while improving transient and steady-state performances of tracking errors of the position servo system. The scheme comprises the following steps: setting a reference signal of the position servo system. And obtaining model parameters of the position servo system and constructing a discrete model of the position servo system. And setting one-time simulation time length according to the discrete model and model parameters of the position servo system. The non-smoothing controller is designed according to the discrete values of the reference signal and the discrete model of the position servo system. And designing discrete ITAE performance indexes according to the discrete values of the reference signals, the discrete model of the position servo system and the primary simulation time length. And (3) adopting a particle swarm optimization algorithm and combining with discrete ITAE performance indexes to perform off-line optimization on parameters of the non-smooth controller. And obtaining the optimal parameters of the non-smooth controller according to the optimization result.

Description

Non-smooth feedback optimal tracking control method of position servo system
Technical Field
The invention relates to the technical field of electromechanical control, in particular to a non-smooth feedback optimal tracking control method of a position servo system.
Background
The position servo system can be used for tracking and controlling the track of mechanical and electrical equipment such as a mechanical arm, and in the practical application of the position servo system, in order to obtain the expected control performance, the parameters in the controller need to be set off-line or on-line. On the other hand, for various control performances and complex system working environments, it is necessary to design a control method which gives consideration to different control performances and disturbance rejection capabilities.
In the research of high-performance tracking control of a position servo system, a typical method comprises the following steps: the self-adaptive control method based on the neural network on-line estimation and compensation has a certain compensation effect on regular interference, but has high calculation complexity and poor compensation effect on white noise signals; the dynamic surface control method based on the preset performance can arbitrarily set the control performance of the system in theory, but is difficult to treat the escape problem caused by random interference; the sliding mode control method has a simple control structure and can completely inhibit system model errors, external disturbance and the like, but the inherent buffeting problem makes the sliding mode control method difficult to apply to engineering practice; the non-smooth feedback control method has the capabilities of finite time convergence, small overshoot and disturbance inhibition, and is widely focused and applied in engineering. However, the non-smooth feedback control method still has the problem that the optimal control parameters are difficult to directly calculate in theory and buffeting can exist in the system.
Disclosure of Invention
In view of the above, the invention provides a non-smooth feedback optimal tracking control method of a position servo system, which can inhibit buffeting of the system while improving transient and steady state performances of tracking errors of the position servo system.
In order to achieve the above purpose, the technical scheme of the invention comprises the following steps:
s1: setting reference signals of the position servo system, wherein the reference signals comprise a reference position, a reference speed and a reference acceleration.
S2: and obtaining model parameters of the position servo system and constructing a discrete model of the position servo system.
S3: and setting one-time simulation time length according to the discrete model and model parameters of the position servo system.
S4: the non-smoothing controller is designed according to the discrete values of the reference signal and the discrete model of the position servo system.
S5: and designing discrete ITAE performance indexes according to the discrete values of the reference signals, the discrete model of the position servo system and the primary simulation time length.
S6: and (3) adopting a particle swarm optimization algorithm and combining with discrete ITAE performance indexes to perform off-line optimization on parameters of the non-smooth controller.
S7: and (3) obtaining the optimal parameters of the non-smooth controller according to the optimization result of the step S6.
Further, in the second step, the model parameters of the position servo system are obtained and a discrete model of the position servo system is constructed, specifically:
the open loop transfer function of the position servo system is:
wherein G(s) is a system model, s is a Laplacian, k 0 Input magnification, ω, to the system k Is the natural frequency of the system ρ k Is a system damping coefficient;
the dynamics model of the position servo system obtained by the open loop transfer function is as follows:
wherein [ x ] 1 ,x 2 ,x 3 ] T Is a system state vector, x 1 、x 2 And x 3 Respectively representing the position, the speed and the acceleration of the position servo system, u is the input of the position servo system, b 2 、b 3 And k is three intermediate variables, whereinb 3 ＝2ρ k ω k And->
Constructing a discrete model of the position servo system according to the dynamics model of the position servo system, wherein the discrete model comprises the following steps:
wherein t is s For differential step size, n=0, 1,2,..n, N is the total number of simulated steps, x j (n) and u (n) represent the system state and input, x, respectively, at the current time j (n+1) represents the system state at the next time, j=1, 2 or 3; at the current time, n.t s Time of day.
Further, setting a model according to the discrete model and model parameters of the position servo systemThe secondary simulation duration is specifically as follows: setting a primary simulation time length to be T=N.t according to the discrete model and model parameters of the position servo system s N is the total number of simulation steps.
In the fourth step, a non-smoothing controller is designed according to the discrete value of the reference signal and the discrete model of the position servo system, specifically:
wherein u (n) is the system input at the current time,j=1, 2,3, e j (n) tracking errors, x, of the position, velocity and acceleration at the current time, respectively j (n) is the system state at the current time, < >>For the discrete value of the reference signal at the current moment, the control parameters of the non-smooth controller comprise six parameters, namely: k (k) 1 ,k 2 ,k 3 ，α 1 ,α 2 ,α 3 Wherein k is 1 ,k 2 ,k 3 > 0 and alpha 1 ,α 2 ,α 3 E (0, 1) is the control parameter of the non-smooth controller.
Further, according to the discrete value of the reference signal, the discrete model of the position servo system and the primary simulation time length, designing a discrete ITAE performance index, wherein the discrete ITAE performance index is specifically:
where i is an integer, and the value is from 0 to T/T s 。
Further, the parameters of the non-smooth controller are optimized offline by adopting a particle swarm optimization algorithm and combining with discrete ITAE performance indexes, and the optimization process specifically comprises the following steps:
s601: parameter k for non-smooth controller according to initialization procedure 1 、k 2 、k 3 、α 3 Setting and initializing the range of random positions and speeds of the particle swarm, and assigning the inertia weight and acceleration of the particle swarm.
S602: the adaptive value of each particle is calculated, i.e. the discrete ITAE performance index is calculated from the results of a single simulation of the discrete system at discrete values of the non-smooth controller and the reference signal.
S603: for each particle, its fitness value is compared with the fitness value of the best location experienced, and if so, it is taken as the current best location.
S604: for each particle, its fitness value is compared with the fitness value of the global best location, and if it is better, it is taken as the current global best location.
S605: the speed and the position of the particles are evolved, and an evolution equation is as follows;
wherein v and x represent the position and velocity of the particle, respectively, the subscript i represents the ith particle, the subscript j represents the jth dimension of the particle, z represents the zth generation, P ij (z) and P gj (z) represents the optimal positions that the individual and population have undergone in that dimension, c 1 、c 2 Is acceleration constant, r 1 、r 2 Two mutually independent random numbers meeting standard normal distribution are adopted, ω is inertia weight, and the function of maintaining the balance of global and local searching capability is achieved.
S606: if the preset iteration number M is reached or the preset optimization performance index is met, the optimization process is ended, otherwise, the process returns to S602.
Further, parameters of the non-smooth controller need to satisfy the following relationship:
α 4 ＝1,α 3 ＝α∈(0,1)。
the beneficial effects are that:
1. the invention introduces a punishment term e of speed buffeting in the original ITAE performance index 2 To improve the traditional ITAE performance index, thereby avoiding the problem of high-frequency oscillation of the traditional non-smooth controller under the action of a sign function. The invention optimizes the control parameters of the non-smooth controller by using the particle swarm algorithm and the improved ITAE performance index, and the obtained optimal control parameters not only improve the transient state and steady state performance of the position servo system, but also avoid the buffeting problem.
2. The embodiment of the invention constructs the optimized discrete ITAE index and the discrete form of the original ITAE indexCompared with the optimized discrete ITAE index, a buffeting penalty term is addedHelping to obtain non-smooth controller parameters that keep the system from jittering. It can be seen that when the system tracking error is not converged, the punishment force on the speed tracking error is smaller; and after the system tracking error converges, the punishment force on the speed tracking error becomes larger. Therefore, the optimized system can avoid jitter while maintaining a fast convergence characteristic.
Drawings
FIG. 1 is a flow chart of the steps of a non-smooth feedback optimal tracking control method of a position servo system;
FIG. 2 is an optimized block diagram of a non-smooth feedback optimal tracking control method for a position servo system;
FIG. 3 is a position tracking curve of a third-order hydraulic servo system under original ITAE performance index optimization;
FIG. 4 is a graph of velocity tracking for a third-order hydraulic servo system under original ITAE performance index optimization;
FIG. 5 is a position tracking curve of a third-order hydraulic servo system under new ITAE performance index optimization;
FIG. 6 is a speed tracking curve of a third-order hydraulic servo system under new ITAE performance index optimization.
Detailed Description
The invention will now be described in detail by way of example with reference to the accompanying drawings.
The step flow chart of the non-smooth feedback optimal tracking control method of the position servo system shown in fig. 1 specifically comprises the following steps:
s1: setting a reference signal of a position servo system, wherein the reference signal comprises a reference position, a reference speed and a reference acceleration; in the embodiment of the invention, the following steps are included: setting the reference signal asWherein y is d Is the reference position->For reference speed +.>Is the reference acceleration.
S2: obtaining model parameters of a position servo system and constructing a discrete model of the position servo system; the embodiment of the invention is realized by the following steps:
the open loop transfer function of the hydraulic servo system is as follows:
wherein G(s) is a system model, s is a Laplacian, k 0 Input magnification, ω, to the system k Is the natural frequency of the system ρ k Is the damping coefficient of the system.
The kinetic model of the system (1) is described as:
wherein [ x ] 1 ,x 2 ,x 3 ] T Is a system state vector, x 1 、x 2 And x 3 Respectively representing the position, the speed and the acceleration of the position servo system, u is the input of the position servo system, b 2 、b 3 And k is three intermediate variables, whereinb 3 ＝2ρ k ω k And->
Discretizing the dynamics model (2) in a differential mode to obtain a discrete dynamics model:
wherein t is s For differential step size, n=0, 1,2,..n, N is the total number of simulated steps, x j (n) and u (n) represent the system state and input, x, respectively, at the current time j (n+1) represents the system state at the next time, j=1, 2 or 3; at the current time, n.t s Time of day.
S3: setting one-time simulation time length according to a discrete model and model parameters of the position servo system; according to the above model (3) and model parameters thereofb 3 ＝2ρ k ω k ，/>Setting the primary simulation time period t=n·t in S3 s 。
S4: designing a non-smoothing controller according to the discrete value of the reference signal and the discrete model of the position servo system; the non-smoothing controller in the embodiment of the invention is designed as follows:
wherein u (n) is the system input at the current time,j=1, 2,3, e j (n) tracking errors, x, of the position, velocity and acceleration at the current time, respectively j (n) is the system state at the current time, < >>For the discrete value of the reference signal at the current moment, the control parameters of the non-smooth controller comprise six parameters, namely: k (k) 1 ,k 2 ,k 3 ，α 1 ,α 2 ,α 3 Wherein k is 1 ,k 2 ,k 3 > 0 and alpha 1 ,α 2 ,α 3 E (0, 1) is the control parameter of the non-smooth controller.
Meanwhile, in order to ensure the limited time convergence of the system tracking error, the parameters of the non-smooth controller need to satisfy the following relationship:
s5: designing discrete ITAE performance indexes according to discrete values of reference signals, a discrete model of a position servo system and one-time simulation time length; according to the discrete value y of the reference signal d (n),Discrete model (3) of position servo system and primary simulation time period t=n·t s The discrete ITAE performance index in the above S5 is designed as follows:
wherein i is the wholeNumber, value from 0 to T/T s 。
Discrete form with original ITAE indexCompared with the optimized discrete ITAE index, the optimized discrete ITAE index is added with a buffeting penalty term +.>Helping to obtain non-smooth controller parameters that keep the system from jittering. It can be seen that when the system tracking error is not converged, the punishment force on the speed tracking error is smaller; and after the system tracking error converges, the punishment force on the speed tracking error becomes larger. Therefore, the optimized system can avoid jitter while maintaining a fast convergence characteristic.
S6: and designing a particle swarm optimization algorithm and carrying out off-line optimization on parameters of the non-smooth controller by combining with discrete ITAE performance indexes. The optimization process of the particle swarm algorithm is specifically as follows:
s601: parameter k for non-smooth controller according to initialization procedure 1 、k 2 、k 3 、α 3 Setting and initializing the range of random positions and speeds of the formed particle groups;
s602: calculating an adaptive value of each particle, namely a discrete ITAE performance index (6) calculated according to a simulation result of a discrete system under a discrete value of a non-smooth controller and a reference signal;
s603: for each particle, comparing the adaptation value with the adaptation value of the best position experienced, and if so, taking the best position as the current best position;
s604: comparing the adaptive value of each particle with the adaptive value of the global best position, and taking the particle as the current global best position if the adaptive value of the global best position is better;
s605: the speed and the position of the particles are evolved, and an evolution equation is as follows;
wherein v and x represent the position and velocity of the particle, respectively, the subscript i represents the ith particle, the subscript j represents the jth dimension of the particle, z represents the zth generation, P ij (z) and P gj (z) represents the optimal positions that the individual and population have undergone in that dimension, c 1 、c 2 Is acceleration constant, r 1 、r 2 Two mutually independent random numbers meeting standard normal distribution are adopted, ω is inertia weight, and the function of maintaining the balance of global and local searching capability is achieved.
S606: ending if the preset iteration times M are reached or the preset optimization performance index is met, otherwise returning to the step 2).
S7: and obtaining the optimal parameters of the non-smooth controller according to the optimization result.
FIG. 2 is an optimized block diagram of a non-smooth feedback optimal tracking control method of a position servo system, which consists of a control block diagram under a system discrete model and a particle swarm optimization algorithm with discrete ITAE performance indexes. And combining the improved ITAE performance indexes, and ensuring the transient state and steady state performance of the position servo system by the obtained optimal control parameters, and simultaneously avoiding the buffeting problem in the system.
The technical scheme disclosed by the invention is verified in a simulation way, and the simulation method is as follows:
step 1: super-parameter design of hydraulic servo system model parameters and particle swarm algorithm
According to the natural frequency omega of the hydraulic servo system k =10rad/s, damping coefficient ρ k =0.2, amplification factor k 0 =5, each parameter in the calculated hydraulic servo system dynamics equation (2) is b 2 ＝100，b 3 =4, k=500. The sampling step length in the discrete dynamics model (3) is set as t s =0.01 s, while selecting 10 times the upper bound of the system adjustment time obtained by simulation as the upper bound of the integral, i.e. t=4s=400T, in order to ensure rapid convergence of the system and to take into account possible buffeting conditions of the system s 。
Is made possible by a discrete dynamics model (3), a non-smooth controller (4) and a parameter relationship (5)It is known that only the parameter k is required 1 、k 2 、k 3 、α 3 And (5) optimizing. Therefore, taking the dimension of the particles as 4, the number of particles as 500, and the learning factor c 1 、c 2 And respectively taking 1.2 and 2, taking 0.6 of inertia weight, wherein the maximum iteration number is 1000, and the condition of stopping the iteration is that the performance index is smaller than 0.5. At the same time, define parameter k 1 ,k 2 ,k 3 ∈[0,50]And alpha 3 ∈(0,1)。
Step 2: based on traditional discrete ITAE performance indexOptimizing parameters in the non-smooth controller (4) by using a particle swarm algorithm to obtain an optimal parameter (k) of the non-smooth controller under the current index 1 ＝50，k 2 ＝12.05，k 3 ＝0.61，α 3 =0.76) and tracking results under optimal parameters (fig. 3-4).
Step 3: based on the improved discrete ITAE performance index (6), the parameters in the non-smooth controller (4) are optimized by using a particle swarm algorithm to obtain the optimal parameters (k) of the non-smooth controller under the current index 1 ＝50，k 2 ＝8.58，k 3 ＝0.27，α 3 =0.86) and tracking results under optimal parameters (fig. 5-6).
Fig. 3-4 are tracking curves and tracking error curves of the third-order hydraulic servo system under the original ITAE performance index optimization. After the original ITAE performance index is adopted for optimization, the system obtains a faster convergence speed, the position and the speed signals are tracked quickly within 0.25 seconds, but the speed signals of the system have obvious high-frequency buffeting in certain time periods.
Fig. 5-6 are tracking curves and tracking error curves of the third-order hydraulic servo system under the optimization of the new ITAE performance index. After the improved ITAE performance index is adopted for optimization, the system eliminates buffeting under the condition that the convergence rate is not reduced.
Based on the technical content, a particle swarm optimization algorithm of the non-smooth controller can be obtained, and the obtained parameter of the non-smooth controller can eliminate system buffeting while maintaining the system convergence speed by using an improved ITAE index as a particle swarm fitness function.
In summary, the above embodiments are only preferred embodiments of the present invention, and are not intended to limit the scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (2)

1. The non-smooth feedback optimal tracking control method of the position servo system is characterized by comprising the following steps of:
s1: setting reference signals of a position servo system, wherein the reference signals comprise a reference position, a reference speed and a reference acceleration;
s2: obtaining model parameters of the position servo system and constructing a discrete model of the position servo system;
the open loop transfer function of the position servo system is:
wherein G(s) is a system model, s is a Laplacian, k 0 Input magnification, ω, to the system k Is the natural frequency of the system ρ k Is a system damping coefficient;
the dynamics model of the position servo system obtained by the open loop transfer function is as follows:
wherein [ x ] 1 ,x 2 ,x 3 ] T Is a system state vector, x 1 、x 2 And x 3 Respectively representing the position, the speed and the acceleration of the position servo system, u is the input of the position servo system, b 2 、b 3 And k is three intermediate variables, whereinb 3 ＝2ρ k ω k And (b)
Constructing a discrete model of the position servo system according to the dynamics model of the position servo system, wherein the discrete model comprises the following steps:
wherein t is s For differential step size, n=0, 1,2,..n, N is the total number of simulated steps, x j (n) and u (n) represent the system state and input, x, respectively, at the current time j (n+1) represents the system state at the next time, j=1, 2 or 3; at the current time, n.t s Time;
s3: setting a primary simulation time length according to a discrete model and model parameters of the position servo system;
setting a primary simulation time length to be T=N.t according to the discrete model and model parameters of the position servo system s N is the total simulation step number;
s4: designing a non-smooth controller according to the discrete value of the reference signal and the discrete model of the position servo system;
where u (n) is the system input at the current time,j=1, 2,3, e j (n) tracking errors, x, of the position, velocity and acceleration at the current time, respectively j (n) is the system state at the current time, < >>For the discrete value of the reference signal at the current moment, the control parameters of the non-smooth controller comprise six parameters, namely: k (k) 1 ,k 2 ,k 3 ，α 1 ,α 2 ,α 3 Wherein k is 1 ,k 2 ,k 3 >0 and alpha 1 ,α 2 ,α 3 E (0, 1) is the control parameter of the non-smooth controller;
s5: designing discrete ITAE performance indexes according to the discrete value of the reference signal, the discrete model of the position servo system and the primary simulation time length;
the discrete ITAE performance index is specifically:
where i is an integer, and the value is from 0 to T/T s ；
S6: adopting a particle swarm optimization algorithm and combining the discrete ITAE performance indexes to perform off-line optimization on parameters of the non-smooth controller;
the optimization process specifically comprises the following steps:
s601: parameter k for the non-smooth controller 1 、k 2 、k 3 、α 3 Setting and initializing a range of random positions and speeds of the particle swarm, and assigning inertia weight and acceleration of the particle swarm;
s602: calculating an adaptive value of each particle, namely calculating the discrete ITAE performance index according to a simulation result of a discrete system under the discrete values of the non-smooth controller and the reference signal;
s603: comparing the current adaptive value with the adaptive value of the best position experienced by each particle, and taking the best adaptive value as the current best position if the best adaptive value is good;
s604: comparing the adaptive value of each particle with the adaptive value of the global best position, and taking the particle as the current global best position if the adaptive value of the global best position is better;
s605: the speed and the position of the particles are evolved, and an evolution equation is as follows;
wherein v and x represent the position and velocity of the particle, respectively, the subscript i represents the ith particle, the subscript j represents the jth dimension of the particle, z represents the zth generation, P ij (z) and P gj (z) represents the optimal positions that the individual and population have undergone in that dimension, c 1 、c 2 Is acceleration constant, r 1 、r 2 Two mutually independent random numbers meeting standard normal distribution, wherein omega is inertia weight, and has the function of maintaining the balance of global and local searching capability;
s606: if the preset iteration times M are reached or the preset optimization performance index is met, ending the optimization process, otherwise returning to S602;
s7: and (3) obtaining the optimal parameters of the non-smooth controller according to the optimization result of the step S6.
2. A non-smooth feedback optimal tracking control method of a position servo system as claimed in claim 1, wherein parameters of the non-smooth controller satisfy the following relationship:
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