CN110442020B - Novel fault-tolerant control method based on whale optimization algorithm - Google Patents
Novel fault-tolerant control method based on whale optimization algorithm Download PDFInfo
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Abstract
The invention discloses a novel sliding mode prediction fault-tolerant control method for faults of an actuator of a variable time-lag control system. Aiming at the problem of fault-tolerant control of a quadrotor system with uncertain discrete time lag, a sliding-mode prediction fault-tolerant control method based on a whale optimization algorithm is designed. The global robustness is guaranteed by using a whole-course sliding mode surface as a prediction model, a power function reference track with fault compensation is designed, and the suppression effect is achieved for uncertainty and faults while buffeting influence is weakened. In the rolling optimization process, considering that the optimization process needs high-precision and fast response, a whale optimization algorithm is adopted, and the algorithm is strong in optimization performance, few in parameter setting, fast in convergence and high in precision. The invention is used for robust fault-tolerant control of an uncertain discrete system containing time-varying time lag.
Description
Technical Field
The invention relates to a sliding mode prediction fault-tolerant control method based on a whale optimization algorithm and designed for faults of an actuator of a variable-time-lag control system, and belongs to the technical field of robust fault-tolerant control of an uncertain discrete control system.
Background
With the development of science and technology, the society nowadays becomes a society driven by various intelligent instruments to a great extent. The intelligent instrument is operated for a long time, and a series of faults often occur under the influence of external factors. In order to enable an intelligent agent to safely operate under a fault condition, a fault-tolerant control technology for processing a fault problem is being developed vigorously, and some achievements are made in processing a practical problem. Fault tolerant control has evolved roughly in both the active and passive directions, presenting a number of promising strategies in dealing with actuator and sensor failures.
The quad-rotor unmanned aerial vehicle has advanced daily life, and shows indispensable value in the aspects of agriculture, military industry, transportation, tracing, and the like. In recent years, a great number of control strategies with practical utility, such as sliding mode control, predictive control, adaptive control, sliding mode prediction and the like, are proposed for the aspects of unmanned aerial vehicle formation flight, tracking and obstacle avoidance, fault handling and the like. When a four-rotor discrete system is researched, due to the fact that the system is prone to interference during flying, certain errors exist during system modeling, and the design of the complexity of a control system and the design of a control strategy are greatly influenced. In fact, the control is made more difficult by the presence of time lags in the system due to faults present during flight.
The research of the discrete control system becomes an important component in the control field, and has great exploration value for fault diagnosis and fault-tolerant control of the discrete system. In a discrete system, the sliding mode control can well process uncertain factors such as parameter perturbation and external disturbance in the system and has good robustness. The sliding mode variable structure control is essentially a special nonlinear control, and the control strategy is different from other control methods in that a system can be purposefully and continuously changed according to the current state in a dynamic process, the sliding mode of the system can be designed and is irrelevant to the disturbance of the system, so that the SMC has the advantages of quick response, insensitivity to parameter change and disturbance, simple physical implementation and the like. Therefore, the method is widely researched and applied to uncertain discrete system control at present. However, in an actual control system, a time lag phenomenon is ubiquitous. When a time lag term occurs in the system, the simple sliding mode control is difficult to obtain a good control effect, and particularly when the time lag is large, the sliding mode control is difficult to meet the requirement of the system on rapidity and the instability phenomenon may occur. However, predictive control works well in eliminating the effects of time lag on discrete systems. In the rolling optimization process of the predictive control, the solution of the control sequence is carried out on line at any time, and the optimization problem is continuously solved, which is also the meaning of the rolling optimization. Therefore, the rolling optimization can keep the actual control optimal, and further, the time lag problem can be well treated, and the influence of the time lag on the system is reduced. Therefore, the advantages of sliding mode control and prediction control are combined, a sliding mode prediction algorithm is designed for a time-lag discrete uncertainty system, parameter perturbation and external interference of the system are solved by fully utilizing sliding mode control processing, the influence of time lag is avoided by utilizing model prediction control, and the control effect is further optimized.
At present, sliding mode prediction algorithms are researched more and more, various novel control strategies are proposed continuously, but intensive research and analysis are rarely carried out on the problem of dealing with time-varying time lag.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the existing processing scheme, a control problem of a quadrotor system with uncertain discrete time lag containing fault items is provided, and a sliding mode prediction fault-tolerant control method based on a whale optimization algorithm is designed. The global robustness is guaranteed by using a whole-course sliding mode surface as a prediction model, a power function reference track with fault compensation is designed, and the suppression effect is achieved for uncertainty and faults while buffeting influence is weakened. In the rolling optimization process, considering that the optimization process needs high-precision and fast response, a whale optimization algorithm is adopted, and the algorithm is strong in optimization performance, few in parameter setting, fast in convergence and high in precision. Therefore, the discrete system with uncertain time-varying time lag under the condition of actuator failure keeps robust and stable, and obtains good effects on rapidity and accuracy.
The technical scheme is as follows: a sliding mode prediction fault-tolerant control method for a variable-time-lag control system actuator fault is characterized in that a sliding mode prediction model is designed according to the state of a system, the model is a whole-course sliding mode switching function, instability in an approaching process is avoided, and overall robustness is guaranteed; considering that a time-lag system is simultaneously influenced by internal parameter perturbation and external disturbance, a power function reference track with fault and uncertainty compensation is designed, buffeting is weakened to a greater extent, and meanwhile, the convergence speed is guaranteed; a whale optimization algorithm is designed on the basis of the rolling optimization problem for optimization, and compared with a particle swarm optimization algorithm, the whale optimization algorithm has the advantages of higher convergence speed, more accurate solving precision and less and easier parameter setting. The robust fault-tolerant control method for an uncertain discrete system containing time-varying time lag comprises the following specific steps:
step 1) establishing a discrete system model:
step 1.1) Δ A, Δ B, Δ AdRespectively, the perturbation of the parameters of the system, x (k) e Rn，u(k)∈Rp，y(k)∈RqThe state, input, output, w (k) e R of the systemnFor external disturbances, f (k) is a fault function, τ (k)In order to not determine the time-varying time-lag, but have its upper and lower bounds [ tau ]l，τu]，A，B，C，E，AdBeing a matrix of appropriate dimensions
Step 1.2) rewriting system (1) into formula (2), wherein d (k) ═ Δ ax (k) + Δ bu (k) + Δ adx (k- τ (k)) + v (k)) + ef (k), and d (k) satisfies | d (k) < d (k) -d (k-1) | ≦ d0And dL≤|d(k)|≤dU；
Step 2), designing a sliding mode prediction model:
step 2.1) designing a whole-course sliding mode switching function, so that the initial state of the state track of the system is positioned on a switching surface, the approach process of a linear sliding mode surface is eliminated, and the overall robustness of the system is guaranteed; where y (k) is the actual output of the system, σ can be solved by the pole placement rule, x0Is the initial state of the system, y0The output in the initial state is s (0) ═ 0, the system state track is positioned on the switching surface at the initial moment, and the approach process is omitted;
s(k)＝σy(k)-αkσy0＝σCx(k)-αkσCx0 (3)
step 2.2) the k +1 moment sliding mode prediction model is (4);
s(k+1)＝σCx(k+1)-αk+1σCx0 (4)
step 2.3) according to the nominal system x (k +1) ═ ax (k) + bu (k) + adx (k-tau (k)) can obtain the prediction output (5) of the sliding mode prediction model at the (k + P) moment and the vector representation (6) of the sliding mode prediction model;
SPM(k)＝ΩX(k)+ΞU(k)+ΨXd(k)-ГX0 (6)
wherein P is a prediction time domain, M is a control time domain, and M is less than or equal to P, the control quantity u (k + j) keeps u (k + M-1) unchanged when M-1 is less than or equal to j and less than or equal to P,
SPM(k)＝[s(k+1)，...，s(k+p)]T
X(k)＝[x(k+1)，...，x(k+p)]T
X0＝[x0，...，x0]T
U(k)＝[u(k)，u(k+1)，...，u(k+M-1)]T
Ω＝[(σCA)T，...，(σCAp)T]T
Γ＝[αk，αk+1，...，αk+P]T
step 3), designing a reference track:
step 3.1) in the sliding mode prediction control, the selection of the reference track can be constructed according to a sliding mode approach law, so that how to reduce the influence of buffeting is avoided, and the problem that the buffeting needs to be carefully considered during selection is solved; in view of the great effect of the power function in reducing the trembling, the power function is used as a reference track, meanwhile, the influence of faults and uncertainty is considered, an interference suppression means is embedded in the reference track, the faults and the uncertainty are made up to the maximum extent, and the reference track is designed as shown in the formula (7):
wherein
sgn () is expressed as a sign function, the value range of each parameter is as follows, beta is more than 0 and less than 1, delta is more than 0 and less than 1,
the compensation function is expressed as:
xi (k) is defined as a function of system faults and uncertainty, xiUIs an upper bound of xi (k), xiLA lower bound of ξ (k);
step 3.2) equation (8) is expressed as approximated by a one-step delay estimation method
Pairing s can be done without d (k) being knownrefSolution of (k +1), srefThe vector form of (k +1) satisfies (9);
Sref(k)＝[sref(k+1)，sref(k+2)，...，sref(k+P)]T (9)
step 4), feedback correction design:
step 4.1), the prediction model (10) represents the prediction output of p steps to k time before k time, and the formula (11) represents the error between the actual output and the prediction output of the k time;
e(k)＝s(k)-s(k|k-P) (11)
step 4.2) adding the error represented by the formula (11) into a sliding mode prediction model as correction to obtain P-step prediction output and vector forms thereof which are (12) and (13) respectively;
wherein the content of the first and second substances,
jpas the correction coefficient, the correction coefficient is sequentially decreased as the prediction step is increased, j1＝1，j1＞j2＞...＞jp＞0；
Step 5), optimizing performance index design:
step 5.1) design optimization performance index is as shown in formula (14), wherein lambdaiThe sampling time error is a non-negative weight coefficient and represents the proportion of the sampling time error in the performance index; gamma raylA positive weight coefficient for constraining the control input;
step 5.2) expressing the optimized performance index in a vector form (15);
wherein the content of the first and second substances,
step 6) solving control law of whale optimization algorithm
Step 6.1) taking the optimized performance index J (k) as an adaptive value function psi, initializing whale populations and initializing all parameters
l, ρ. Wherein the content of the first and second substances,
and
are coefficient vectors, representing the wobble factor and the convergence factor respectively,
linearly decreasing from 2 to 0 as the number of iterations increases, l is [ -1, 1 [ ]]A random number therebetween, constant rho ∈ [0, 1 ]]And are uniformly distributed random numbers generated, an
And
can be calculated by the following formula;
wherein the content of the first and second substances,
is [0, 1 ]]A random number in between;
step 6.2) when the parameter rho is less than 0.5, and
then, calculating an optimal value by a whale optimization algorithm by adopting the following iterative formula;
wherein the content of the first and second substances,
is the distance between the individual and the target prey, the current number of iterations is t,
to the position of the optimal solution at iteration t,
and (4) carrying out t iterations of whale individual position vectors.
Step 6.3) when the parameter rho is less than 0.5, and
in time, the whale optimization algorithm searches for the optimal solution by adopting a random searching method, and randomly selects an individual whale position
Optimizing according to a formula (18);
step 6.4) when the parameter rho is larger than 0.5, optimizing by adopting a bubble-net optimizing mode through a whale optimizing algorithm, and iterating according to the following formula;
step 6.5), when the maximum iteration number is reached, optimizing is finished, the current control quantity is implemented, and k +1 → k returns to the step 2).
Has the advantages that: aiming at the control problem of a quadrotor system with uncertain discrete time lag and containing fault items, a sliding mode prediction fault-tolerant control method based on a whale optimization algorithm is designed. The global robustness is guaranteed by using a whole-course sliding mode surface as a prediction model, a power function reference track with fault compensation is designed, and the suppression effect is achieved for uncertainty and faults while buffeting influence is weakened. In the rolling optimization process, considering that the optimization process needs high-precision and fast response, a whale optimization algorithm is adopted, and the algorithm is strong in optimization performance, few in parameter setting, fast in convergence and high in precision. Therefore, the discrete system with uncertain time-varying time lag under the condition of actuator failure keeps robust and stable, and obtains good effects on rapidity and accuracy. Has the following specific advantages:
firstly, according to the state of a system, designing global sliding mode switching as a sliding mode prediction model of the system, wherein the model has time-varying characteristics, avoids instability in an approaching process, ensures global robustness and can dynamically improve the motion quality of the system;
considering the time lag problem and uncertainty of a discrete time lag system, a power function reference track with fault compensation is designed, and the suppression effect is achieved for the uncertainty and the fault while the buffeting influence is weakened;
compared with the traditional derivation method and general optimization, the rolling optimization process improved by the whale algorithm has the advantages of high solving speed and convergence accuracy, few parameter designs and simplicity in operation.
The robust fault-tolerant control method for the discrete system containing time-varying state time lag, actuator fault, system parameter perturbation and disturbance has certain practical value, is easy to implement, good in real-time performance and high in accuracy, can effectively improve the safety of the control system, is strong in operability, saves time, is higher in efficiency, and can be widely applied to actuator fault-tolerant control of an uncertain discrete control system.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is an experimental setup Qball-X4 quad-rotor helicopter developed by Quanser to study control of a quad-rotor helicopter;
FIG. 3 is a graph of the X-axis position of a Qball-X4 quad-rotor helicopter;
FIG. 4 is a Qball-X4 quad-rotor helicopter actuator dynamic graph;
FIG. 5 is a control law graph;
FIG. 6 is a partially enlarged control law graph.
Detailed Description
The invention is further explained below with reference to the drawings.
As shown in fig. 1, considering that an approaching process exists in a linear sliding mode surface designed by a general sliding mode algorithm, and a destabilization phenomenon easily occurs in the approaching process, a full-course sliding mode switching function is designed in the aspect of a sliding mode prediction model for the situation, so that the destabilization in the approaching process is avoided, and the overall robustness is ensured. Considering a plurality of factors which influence the control performance of the system, such as faults, interference, time lag and the like, of the system, a power function reference track with faults and uncertainty compensation is designed, and the power function reference track is designed in consideration of weakening the buffeting problem of the sliding mode self-band to a greater extent. The whale optimization algorithm is designed for optimizing on the basis of the rolling optimization problem, parameters of the algorithm are less in setting, the solution is convenient, the control law can be solved quickly and accurately, and compared with the particle swarm optimization algorithm, the algorithm has the advantages of higher convergence speed and more accurate solution precision. The robust fault-tolerant control method for an uncertain discrete system containing time-varying time lag comprises the following specific steps:
step 1) establishing a discrete system model:
step 1.1) Δ A, Δ B, Δ AdRespectively, the perturbation of the parameters of the system, x (k) e Rn，u(k)∈Rp，y(k)∈RqThe state, input, output, w (k) e R of the systemnFor external interference, f (k) is a fault function, τ (k) is an indeterminate time-varying time lag, but has its upper and lower bounds [ τ [, ]l，τu]，A，B，C，E，AdBeing a matrix of appropriate dimensions
Step 1.2) rewriting system (1) into formula (2), wherein d (k) ═ Δ ax (k) + Δ bu (k) + Δ adx (k- τ (k)) + v (k)) + ef (k), and d (k) satisfies | d (k) < d (k) -d (k-1) | ≦ d0And dL≤|d(k)|≤dU；
Step 2), designing a sliding mode prediction model:
step 2.1) is to provideA whole-course sliding mode switching function is calculated, so that the initial state of the system state track is positioned on a switching surface, the approach process of a linear sliding mode surface is eliminated, and the overall robustness of the system is guaranteed; where y (k) is the actual output of the system, σ can be solved by the pole placement rule, x0Is the initial state of the system, y0Is the output in the initial state. s (0) is 0, the system state track is positioned on the switching surface at the initial moment, and the approach process is omitted;
s(k)＝σy(k)-αkσy0＝σCx(k)-αkσCx0 (3)
step 2.2) the k +1 moment sliding mode prediction model is (4);
s(k+1)＝σCx(k+1)-αk+1σCx0 (4)
step 2.3) according to the nominal system x (k +1) ═ ax (k) + bu (k) + adx (k-tau (k)) can obtain the prediction output (5) of the sliding mode prediction model at the (k + P) moment and the vector representation (6) of the sliding mode prediction model;
SPM(k)＝ΩX(k)+ΞU(k)+ΨXd(k)-ΓX0 (6)
wherein P is a prediction time domain, M is a control time domain, and M is less than or equal to P, the control quantity u (k + j) keeps u (k + M-1) unchanged when M-1 is less than or equal to j and less than or equal to P,
SPM(k)＝[s(k+1)，...，s(k+p)]T
X(k)＝[x(k+1)，...，x(k+p)]T
X0＝[x0，...，x0]T
U(k)＝[u(k)，u(k+1)，...，u(k+M-1)]T
Ω＝[(σCA)T，...，(σCAP)T]T
Γ＝[αk，αk+1，...，αk+P]T
step 3), designing a reference track:
and 3.1) in sliding mode prediction control, the selection of the reference track can be constructed according to the approach rate of the sliding mode, so that how to reduce the influence of buffeting is avoided, and the problem that the buffeting needs to be carefully considered during selection is solved. In view of the great effect of the power function in reducing the trembling, the power function is adopted as a reference track, meanwhile, the influence of faults and uncertainty is considered, interference suppression means are embedded in the reference track, and the faults and the uncertainty are compensated to the maximum extent, and the reference track is designed according to the formula (7):
wherein
sgn () is represented as a sign function. The value ranges of the parameters are as follows, beta is more than 0 and less than 1, delta is more than 0 and less than 1,
the compensation function is expressed as:
xi (k) is defined as a function of system faults and uncertainty, xiUIs an upper bound of xi (k), xiLA lower bound of ξ (k);
step 3.2) equation (8) is expressed as approximated by a one-step delay estimation method
Pairing s can be done without d (k) being knownrefSolution of (k +1), srefThe vector form of (k +1) satisfies (9);
Sref(k)＝[sref(k+1)，sref(k+2)，...，sref(k+P)]T (9)
step 4), feedback correction design:
step 4.1), the prediction model (10) represents the prediction output of p steps to k time before k time, and the formula (11) represents the error between the actual output and the prediction output of the k time;
e(k)＝s(k)-s(k|k-P) (11)
step 4.2) adding the error represented by the formula (11) into a sliding mode prediction model as correction to obtain P-step prediction output and vector forms thereof which are (12) and (13) respectively;
wherein the content of the first and second substances,
jpas the correction coefficient, the correction coefficient is sequentially decreased as the prediction step is increased, j1＝1，j1＞j2＞...＞jp＞0；
Step 5), optimizing performance index design:
step 5.1) design optimization performance index is as shown in formula (14), wherein lambdaiThe sampling time error is a non-negative weight coefficient and represents the proportion of the sampling time error in the performance index; gamma raylIs a positive weight coefficient, and is used for constraint controlMaking and inputting;
step 5.2) expressing the optimized performance index in a vector form (15);
wherein the content of the first and second substances,
step 6) solving control law of whale optimization algorithm
Step 6.1) taking the optimized performance index J (k) as an adaptive value function psi, initializing whale populations and initializing all parameters
l, ρ. Wherein the content of the first and second substances,
and
are coefficient vectors, representing the wobble factor and the convergence factor respectively,
linearly decreasing from 2 to 0 as the number of iterations increases, l is [ -1, 1 [ ]]A random number therebetween, constant rho ∈ [0, 1 ]]And are uniformly distributed random numbers generated, an
And
can be calculated by the following formula;
wherein the content of the first and second substances,
is [0, 1 ]]A random number in between.
Step 6.2) when the parameter rho is less than 0.5, and
then, calculating an optimal value by a whale optimization algorithm by adopting the following iterative formula;
wherein the content of the first and second substances,
is the distance between the individual and the target prey, the current number of iterations is t,
to the position of the optimal solution at iteration t,
the position vector of the whale individual for t iterations;
step 6.3) when the parameter rho is less than 0.5, and
in time, the whale optimization algorithm searches for the optimal solution by adopting a random searching method, and randomly selects an individual whale position
Optimizing according to a formula (18);
step 6.4) when the parameter rho is larger than 0.5, optimizing by adopting a bubble-net optimizing mode through a whale optimizing algorithm, and iterating according to the following formula;
step 6.5), when the maximum iteration number is reached, optimizing is finished, the current control quantity is implemented, and k +1 → k returns to the step 2).
The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and decorations can be made without departing from the principle of the present invention, and these modifications and decorations should also be regarded as the protection scope of the present invention.
The effectiveness of the implementation is illustrated in the following by a practical case simulation.
The Qball-X4 quad-rotor helicopter flight control system actuators, developed by Quanser, Canada, were used as the subject of the application. The Qball-X4 experimental subject is shown in FIG. 2. The Qball-X4 four-rotor helicopter has six-dimensional variables (X, Y, Z, psi, theta, phi), wherein X, Y and Z are position variables, psi is yaw angle, theta is pitch angle and phi is roll angle. The simulation of the case selects the channel signal in the forward direction of the X axis as a research object.
First, for the convenience of modeling, the following assumptions are made for Q-ball:
(1) the whole four-rotor aircraft is regarded as a rigid body which cannot generate elastic deformation, and a Newton-Euler formula can be used;
(2) the machine body has symmetrical structure and uniform mass distribution, and the mass center of the machine body is exactly positioned on the origin of the machine body coordinate system and is superposed with the mass center;
(3) in the whole flight experiment, because the influence of the ground on the four rotors is very little, air friction, gyroscopic effect, air resistance torque and the like are ignored;
(4) neglecting the influence of the curvature of the earth on the flight motion of the four rotors, assuming that the gravity acceleration keeps unchanged, and regarding the ground coordinate system as an inertial coordinate system;
(5) when the four rotors do linear motion, the change of the attitude angles (yaw psi, pitch theta and roll phi) of the body does not exceed +/-5 degrees.
The four-rotor helicopter power is derived from the thrust generated by the rotation of the four rotors, the rotating speeds of the four rotors are changed, namely the flight states of the four rotors are changed, and the thrust model is as follows:
wherein FiFor rotor thrust, K is a positive gain, ω is actuator bandwidth, uiIs an actuator input.
When the aircraft flies in the X direction, the yaw angle psi is 0, the roll angle phi is very small and can be approximated to 0, and then the dynamic model of the X direction position can be simplified as follows:
wherein m is the total mass of the machine body, theta is a pitch angle,
acceleration in the X direction and lift force in the F direction.
Actuator dynamics vi：
In the X-axis position control model, a pitch angle theta is coupled with the X-axis position control model, the integral control can be divided into two stages, one stage is a pitch angle control stage, and the second stage, namely the position control stage, is started after the pitch angle is controlled to a preset value. And when the position reaches the set position, the pitch angle theta is reset to zero through the pitch angle control channel. Under the condition that theta is smaller, a model of an X-axis direction under an ideal condition without external disturbance, parameter perturbation and time-varying time lag is obtained through linearization, and the model is as follows:
supposing that in the X-axis position control stage, the pitch angle is already set to be approximately equal to 0.035rad at 2 degrees, the disturbance, the perturbation of parameters, the network delay and the actuator fault which are dynamically related to the actuator are considered, and the values of each matrix in the system (1) are as follows:
The case simulation result shows that the sliding mode prediction fault-tolerant control algorithm based on the whale optimization algorithm has strong robustness in processing a discrete uncertain system with time lag and actuator faults, and has rapidity and convergence accuracy. Compared with the general traditional algorithm for processing the time-lag discrete uncertainty system, the four-rotor helicopter body can clearly obtain a position curve and an actuator dynamic curve more smoothly under the action of the control method designed by the simulation of the embodiment and by using the graphs in fig. 3-4, and the convergence speed is obviously improved, so that the flight process is smoother and the preset position is reached more quickly. Meanwhile, after the control law is converged, although a certain amount of buffeting and buffeting are still obviously reduced, the amplitude of the buffeting is obviously reduced, as shown in fig. 6. In general, the simulated control method of the embodiment is effective for an actuator fault system containing parameter perturbation, external disturbance and time-varying time lag.
Claims (1)
1. A novel sliding mode prediction fault-tolerant control method for actuator faults of a variable time-lag control system is characterized by comprising the following steps: considering that a linear sliding mode surface designed by a general sliding mode algorithm has an approaching process which is easy to have instability, designing a whole-course switching function on the aspect of a sliding mode prediction model aiming at the condition, avoiding instability of the approaching process, ensuring global robustness, considering that a system has faults, interference, time lag and other factors which influence the control performance of the system, designing a power function reference track with fault and uncertainty compensation, considering the buffeting problem of weakening the sliding mode self-carrying to a greater extent, designing a whale optimization algorithm for optimization on a rolling optimization problem, setting parameters of the algorithm is few, solving is convenient, and a control law can be solved quickly and accurately, the method comprises the following specific steps:
step 1) establishing a discrete system model:
step 1.1) Δ A, Δ B, Δ AdRespectively, the perturbation of the parameters of the system, x (k) e Rn，u(k)∈Rp，y(k)∈RqThe state, input and output of the system are respectively; w (k) ε RnFor external interference, f (k) is a fault function, τ (k) is an indeterminate time-varying time lag, but has its upper and lower bounds [ τ [, ]l，τu]。A，B，C，E，AdBeing a matrix of appropriate dimensions
Step 1.2) rewriting system (1) into formula (2), wherein d (k) ═ Δ ax (k) + Δ bu (k) + Δ adx (k- τ (k)) + v (k)) + ef (k), and d (k) satisfies | d (k) < d (k) -d (k-1) | ≦ d0And dL≤|d(k)|≤dU；
Step 2), designing a sliding mode prediction model:
step 2.1) designing a whole-course sliding mode switching function, so that the initial state of the state track of the system is positioned on a switching surface, the approach process of a linear sliding mode surface is eliminated, and the overall robustness of the system is guaranteed; where y (k) is the actual output of the system, σ can be solved by the pole placement rule, x0Is the initial state of the system, y0The output in the initial state is s (0) is 0, the system state track is positioned in the switching surface at the initial moment, the approach process is omitted,
s(k)＝σy(k)-αkσy0＝σCx(k)-αkσCx0 (3)
step 2.2) the k +1 moment sliding mode prediction model is (4);
s(k+1)＝σCx(k+1)-αk+1σCx0 (4)
step 2.3) according to the nominal system x (k +1) ═ ax (k) + bu (k) + adx (k-tau (k)) can obtain the prediction output (5) of the sliding mode prediction model at the (k + P) moment and the vector representation (6) of the sliding mode prediction model;
SPM(k)＝ΩX(k)+ΞU(k)+ΨXd(k)-ΓX0 (6)
wherein P is a prediction time domain, M is a control time domain, and M is less than or equal to P, the control quantity u (k + j) keeps u (k + M-1) unchanged when M-1 is less than or equal to j and less than or equal to P,
SPM(k)＝[s(k+1)，...，s(k+p)]T
X(k)＝[x(k+1)，...，x(k+p)]T
X0＝[x0，...，x0]T
U(k)＝[u(k)，u(k+1)，...，u(k+M-1)]T
Ω＝[(σCA)T，...，(σCAP)T]T
Γ＝[αk，αk+1，...，αk+P]T
step 3), designing a reference track:
step 3.1) in the sliding mode prediction control, the selection of the reference track can be constructed according to the approach rate of the sliding mode, so that how to reduce the influence of buffeting is avoided, and the problem that the buffeting needs to be carefully considered during selection is solved; in view of the great effect of the power function in reducing the trembling, the power function is adopted as a reference track, meanwhile, the influence of faults and uncertainty is considered, interference suppression means are embedded in the reference track, and the faults and the uncertainty are compensated to the maximum extent, and the reference track is designed according to the formula (7):
wherein
sgn () is expressed as a sign function, the value range of each parameter is as follows, beta is more than 0 and less than 1, delta is more than 0 and less than 1,
the compensation function is expressed as:
xi (k) is defined as a function of system faults and uncertainty, xiUIs an upper bound of xi (k), xiLA lower bound of ξ (k);
step 3.2) equation (8) is expressed as approximated by a one-step delay estimation method
Pairing s can be done without d (k) being knownrefSolution of (k +1), srefThe vector form of (k +1) satisfies (9);
Sref(k)＝[sref(k+1)，sref(k+2)，...，sref(k+P)]T (9)
step 4), feedback correction design:
step 4.1), the prediction model (10) represents the prediction output of p steps to k time before k time, and the formula (11) represents the error between the actual output and the prediction output of the k time;
e(k)＝s(k)-s(k|k-P) (11)
step 4.2) adding the error represented by the formula (11) into a sliding mode prediction model as correction to obtain P-step prediction output and vector forms thereof which are (12) and (13) respectively;
wherein the content of the first and second substances,
jpas the correction coefficient, the correction coefficient is sequentially decreased as the prediction step is increased, j1＝1，j1＞j2＞…＞jp＞0；
Step 5), optimizing performance index design:
step 5.1) design optimization performance index is as shown in formula (14), wherein lambda1The sampling time error is a non-negative weight coefficient and represents the proportion of the sampling time error in the performance index; gamma raylA positive weight coefficient for constraining the control input;
step 5.2) expressing the optimized performance index in a vector form (15);
wherein the content of the first and second substances,
step 6) solving a control law by a whale optimization algorithm:
step 6.1) taking the optimized performance index J (k) as an adaptive value function psi, initializing whale populations and initializing all parameters
l, rho; wherein，
And
are coefficient vectors, representing the wobble factor and the convergence factor respectively,
linearly decreasing from 2 to 0 as the number of iterations increases, l is [ -1, 1 [ ]]A random number therebetween, constant rho ∈ [0, 1 ]]And are uniformly distributed random numbers generated, an
And
can be calculated by the following formula;
wherein the content of the first and second substances,
is [0, 1 ]]A random number in between;
step 6.2) when the parameter rho is less than 0.5, and
then, calculating an optimal value by a whale optimization algorithm by adopting the following iterative formula;
wherein the content of the first and second substances,
is the distance between the individual and the target prey, the current number of iterations is t,
to the position of the optimal solution at iteration t,
the position vector of the whale individual for t iterations;
step 6.3) when the parameter rho is less than 0.5, and
in time, the whale optimization algorithm searches for the optimal solution by adopting a random searching method, and randomly selects an individual whale position
Optimizing according to a formula (18);
step 6.4) when the parameter rho is larger than 0.5, optimizing by adopting a bubble-net optimizing mode through a whale optimizing algorithm, and iterating according to the following formula;
step 6.5), when the maximum iteration number is reached, optimizing is finished, the current control quantity is implemented, and k +1 → k returns to the step 2).
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