CN112286054B - Prediction control method based on magnetic suspension damping device - Google Patents
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- G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
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Abstract
The invention relates to the field of protection of precise instruments, in particular to a prediction control method based on a magnetic suspension damping device. The method utilizes the magnetic suspension technology to actively lift high-precision equipment, so that active shock absorption and noise reduction are realized, and the method specifically comprises the following steps: analyzing the working principle of the magnetic suspension lifting vibration reduction protection device, and writing an offset equation and a rotation equation which are nonlinear equations; linearizing the listed nonlinear equations, and writing a state equation by using current as a control quantity; discretizing a state equation of the magnetic suspension lifting vibration reduction protection device by adopting a front Euler method, writing a predictive control equation in sequence, and designing a predictive controller; aiming at the constraints of the offset position and the rotation angle of the object to be suspended, lifted and protected, an optimization algorithm is designed; and further optimizing the algorithm, giving an iterative solution step, and finally iteratively solving the optimal solution.
Description
Technical Field
The invention relates to the field of protection of precision instruments, in particular to a prediction control method based on a magnetic suspension damping device, which is a control method for lifting precision equipment by using a magnetic suspension technology to realize active shock absorption.
Background
In the design of ships, particularly underwater special ships such as submarines, underwater robots and underwater manned detectors, strict shock-absorbing requirements are imposed on precise instruments. The ship shock absorption is different from the land vehicle, the land vehicle shock absorption only needs to consider the oscillation force in the direction vertical to the ground, and the ship, particularly a special ship sailing underwater needs to consider the omnibearing oscillation interference force. The omnibearing oscillation interference force can cause the structural fatigue of important equipment, and influences the service life of the equipment, and the active shock absorption technology of the important equipment can greatly reduce the running noise of the equipment and enhance the concealment of the running of the ship. At present, the damping measures aiming at the important equipment of the ship mainly comprise:
(1) Increase of stand size and increase of stand rigidity
The method can reduce the amplitude of the equipment to a certain extent and eliminate external interference. But the size and rigidity of the housing cannot be increased without limit, which is also subject to layout and economic requirements.
(2) With elastic support and connection
The method is generally adopted for the shock absorption of important equipment of ships, and the elastic materials mainly comprise rubber vibration isolators, metal vibration isolators and the like. The rubber vibration isolator is low in price and not easy to deform, but is easy to age at high temperature, poor in elasticity and capable of burning; although the metal vibration isolator is not influenced by temperature, the metal vibration isolator cannot be aged and is easy to produce. But since the damping itself is small, the transmission ratio is very large at resonance, and in addition, the metal vibration isolator transmits vibration and the like at high frequencies.
(3) Hydraulic frame
The hydraulic frame body is an important device for vibration reduction and noise elimination of important devices of underwater ships, particularly submarines with extremely high requirements on quietness, the hydraulic rod is utilized to form the deformable frame body, the important devices can be installed in the deformable frame body, the devices and the shell frame body are only connected by the hydraulic rod, and when the devices vibrate during operation or are interfered by external vibration, the hydraulic rod can contract to absorb vibration and damp, so that the effects of noise reduction and vibration elimination are achieved. The hydraulic frame body can realize omnibearing shock absorption, but has larger volume and considerable manufacturing cost.
Disclosure of Invention
Aiming at the defects of the vibration damping method, the invention provides the magnetic suspension vibration damping device, and a control algorithm of equipment is designed, the magnetic suspension lifting vibration damping protection device is utilized to suspend important equipment in the air, and the contact between the important equipment and a ship body is effectively isolated, so that the aims of vibration damping and noise damping are fulfilled. The device adopts a modular design idea, a single module has a simple structure, the whole system is easy to quickly overhaul, and the size is small. In addition, aiming at the defects that an active magnetic suspension PID control device is poor in robustness and sensitive to external disturbance, the invention provides a control algorithm for predicting and adjusting the movement track of the lifted weight.
The invention relates to a magnetic suspension active vibration reduction control technology based on a repulsion type magnetic suspension model, which calculates the optimal input of a system in a dynamic stabilization process according to the repulsion type magnetic suspension model. Aiming at the phenomena that the lifted object is easily interfered by the outside, such as the ship sways due to the influence of water flow when a ship sails, or the phenomena of position deviation, rotation and the like caused by the vibration of the controller, the invention designs the prediction controller, and the controller can enable the system to enter a dynamic stable state more quickly to realize the control of the posture of the object.
The invention discloses a prediction control method based on a magnetic suspension damping device, which comprises the following steps:
the method comprises the following steps: the magnetic suspension damping device model is provided, so that the propagation of vibration and noise can be effectively limited;
step two: based on a magnetic suspension damping device, the working principle of the magnetic suspension damping device is analyzed, the deviation and rotation of an object to be lifted caused by the disturbance are considered, and a kinematic equation and a rotation equation which are nonlinear equations are established;
step three: linearizing the listed nonlinear equations, and writing a state equation by using current as a control quantity;
step four: discretizing a state equation of the magnetic suspension lifting vibration reduction protection device by adopting a front Euler method, writing a predictive control equation in a row, and designing a predictive controller;
step five: aiming at the constraints of the offset position and the rotation angle of the object to be suspended, lifted and protected, designing an optimization algorithm;
step six: and further optimizing the algorithm, giving an iterative solution step, and finally iteratively solving the optimal solution.
In the third step, after the given nonlinear equation is approximately linearized based on the magnetic suspension lifting device, the displacement and the rotation angle of the floater on the x, y and z axes are used as state variables and output, and the current of the x, y and z axes is used as input, so as to establish the state equation, thereby realizing the control of the system on the attitude of the lifted object and maintaining the balance of the lifted object.
In the fourth step, based on the control system of the magnetic suspension damping device, the listed state equations are discretized by adopting a front Euler method, and a prediction controller is designed according to a prediction control method.
In the fifth step, based on the magnetic suspension damping device control system, an optimized objective function of the prediction controller is given:
and then, constraining the amplitude of the state variable, and further giving out an optimized target function with constraint:
in the sixth step, based on the magnetic suspension damping device control system, a relaxed KKT condition is given for a target function with inequality constraint:
based on a magnetic suspension damping device control system, an optimization algorithm is designed, iterative solving steps are given, and the optimal solution meeting the conditions is solved.
Compared with the prior art, the invention has the advantages that:
1. the magnetic suspension lifting vibration reduction protection device is suitable for vibration reduction and noise reduction of important equipment of ships, particularly underwater vehicles, and is different from the traditional method, the contact between the equipment and a ship body is completely isolated by adopting the magnetic suspension lifting vibration reduction protection device, so that the vibration reduction and noise reduction effects are realized;
2. the magnetic suspension vibration damping protection device designed by the invention has the advantages of small volume, modular design and simple structure, and the whole system is easy to quickly overhaul, so that the efficiency is improved, and the economy is improved;
3. aiming at the defects of the traditional PID magnetic suspension control, the invention aims at the problems of the deviation, the rotation and the like of the protected object caused by water flow, course changing and the like of the ship body. The method has the advantages that an object mathematical model is established, a predictive control algorithm is designed, dynamic adjustment capacity is enhanced, the anti-interference capacity of the system is greatly improved, and stability is improved, so that all-round protection is provided for important equipment, and the vibration and noise reduction capacity of the ship is further improved.
Drawings
FIG. 1 is a schematic diagram of an external structure of a magnetic levitation lifting vibration damping protection device in an embodiment of the invention.
Fig. 2 is a schematic diagram of the internal structure of the magnetic suspension lifting vibration-damping protection device in the embodiment of the invention.
FIG. 3 is a front view of a unit module of a magnetic levitation lifting vibration damping protection device in an embodiment of the invention.
FIG. 4 is a top view of a unit module of the magnetic levitation lifting vibration damping protection device in the embodiment of the invention.
Fig. 5 is an equivalent schematic diagram of the force applied to the floating floater in the embodiment of the invention.
Fig. 6 is an equivalent schematic diagram of the x-axis stress of the floating floater in the embodiment of the invention.
FIG. 7 is a block diagram of predictive control in an embodiment of the invention.
FIG. 8 is a flow chart of an iterative solution in an embodiment of the present invention.
Detailed Description
For the purpose of enhancing the understanding of the present invention, the present invention will be described in further detail with reference to the accompanying drawings and examples, which are provided for the purpose of illustration only and are not intended to limit the scope of the present invention.
Example (b): a vibration damping protection device for ships, in particular underwater vehicles, comprises a magnetic suspension lifting vibration damping protection device and a suspension prediction controller; the method comprises the following specific steps:
the method comprises the following steps: the working principle of the magnetic suspension lifting vibration reduction protection device is analyzed, and an offset equation and a rotation equation are written.
Fig. 1 and 2 are schematic diagrams of a magnetic levitation lifting vibration damping protection device, which is in a hexahedral structure and is similar to a box body. When the ship body or the underwater vehicle runs normally, the main magnetic field provided by the coil 5 lifts the protected object, and the balance of the coils 1,2, 3 and 4 is maintained. The other coils are not energized. When the ship body rotates 90 degrees, the coil 10 or the coil 20 is electrified to provide a main magnetic field, and the other coils have the same working state. We analyze one of the states as follows:
as shown in fig. 3 and 4, the lifted object is equivalent to a float, and a magnetic suspension lifting unit module is provided with a main magnetic field by a coil 5, wherein the main magnetic field enables the float to be suspended, and the main magnetic field can also be provided by an electromagnet. Also designed on top of the main magnetic field are 4 main coils, coil 1 and coil 2 to control the X-axis balance, coil 3 and coil 4 to control the Y-axis balance, and coil 5 to control the Z-axis balance. Setting the space coordinates (0, 0) of the balance position of the floater, establishing a space coordinate system, as shown in FIG. 5:
in FIG. 5, f y Is the resolution force in the Y-axis direction of the external disturbance force to which the float is subjected, f x Is the resolving force in the X-axis direction, f z The same is true. mg is the weight force exerted on the float, and F is the repulsion force of the coil 5 to the float.
In a stable magnetic field, the upward electromagnetic force experienced by the float is:
in formula (1), S-coil core area (m) 2 )；
N-number of turns (N) of electromagnet coil;
μ -air permeability (H/m);
i z -z winding coil 5 current (a);
z-the distance (m) of the float from the coil;
let the mass of the float be m, the acceleration of gravity be G, and the force of gravity be G (obviously G = mg). According to newton's second law, the kinetic equation in the z-axis direction is:
when the acceleration of the floater is 0 when the floater is in a stable state in the z-axis direction, the balance equation of the balance point position in the z-axis direction is as follows:
in formula (3), z 0 Is the distance from the coil 5 when the float is in the equilibrium position;
the nonlinear equation of the formula (1) is at an equilibrium point z 0 After approximate linearization by a Taylor formula, neglecting high-order terms, and taking the first two terms to obtain:
order to
Combining equations (2), (3) and (4) can derive the kinematic equation of the device at the equilibrium point as follows:
wherein
The same as in formula (4).
When the float shifts out of equilibrium in the x-axis due to a force, as shown in fig. 6:
when the offset occurs, coil 1 and coil 2 on the x-axis apply a magnitude of 2f to the float x The resultant force of (a) and (b) thereby returning the float to the equilibrium position, the translation equation in the x-axis is:
at the equilibrium point:
the approximation linearization neglects high-order terms to obtain:
order to
Obtaining:
the equation of rotation is:
in the formula (10), J-moment of inertia, J = mR 2 /4；
l-arm of force of electromagnetic force relative to float;
substituting formula (9) into (10) to obtain:
similarly, coil 3 and coil 4 apply 2f to the float when the y-axis is deflected y The force of (a) returns the float to the equilibrium position, the translation equation in the y-direction having:
is simple and easy to obtain
Because the rotation of the object in the z-axis does not affect the object balance, the z-axis rotational equation is not considered.
Step two: the listed non-linear equations are linearized and the state equations are written with the current as the control quantity.
Defining a state variable x 1 ＝[x,y,z,θ x ,θ y ,.θ z ] T ,
Control amount u = [ i = [ i ] x ，i y ，i z ] T 。x 1 The values of the terms in (1) are offset values and rotation angle angles in the x, y, z-axis directions, x 2 Each value of (A) is x 1 The rate of change of each value. And (3) listing a state equation according to the kinematic equation and the rotation equation obtained in the step 1, wherein the state equation is shown as the formula (13):
wherein:
step three: the state equation of the magnetic suspension lifting vibration reduction protection device is discretized by adopting a front Euler method, a predictive control equation is written in a row, and a predictive controller is designed.
FIG. 7 is a schematic block diagram of predictive control, Y r (k) Is the desired system output, Y P (k) Is the predicted output sequence given by the prediction model, and E (k) is the difference between the two. And E (k) generates U (k) through optimization calculation, wherein the U (k) is the future optimal control strategy. Then inputting the actual output y (k) and the predicted output y to the controlled object to obtain the actual output y (k) and the predicted output y of the system p (k) After on-line calibration, this is a cyclic process, and the final goal is to calibrate the model to make it closer to the actual system.
Firstly, a forepart Euler method is adopted, the formula (13) obtained in the step 2 is discretized, and a model is established for prediction control. Assuming that the model performs sampling once every T time, at a certain time k (k is an arbitrary value), there are:
x(k+1)＝A (k) x(k)+B (k) u(k) (14)
in formula (14): a. The (k) ＝I+AT，B (k) ＝BT
So the discretized linear time-varying equation is:
for the following formula (16):
u (k) is a controlled variable at the moment k, u (k-1) is a known quantity, and the value of Δ u (k) is a parameter of the optimal solution and feedback correction which we need to find.
According to equation (15), the prediction model equation is listed. It is assumed that all variables are measurable and that all states before time k are known quantities. With the latest measured value (i.e. the current state value x (k)) and the system input u (k-1) = [ i ] at the last instant (i.e. instant k-1) x (k-1),i y (k-1),i z (k-1)] T Is the initial condition. For this purpose, let the prediction time domain be n and the control time domain be m (n)>m)。
Writing equation (15) in matrix-vector form:
X(k)＝Φx(k)+B Δ ΔU(k)+B u u(k-1) (17)
in formula (17):
the predicted controlled output of the system at the time points (k + 1) to (k + n) can be obtained from equation (15) and equation (17):
wherein
The outputs of the predictions (k + 1) to (k + n) at time k, as found by equation (18), are the displacements and deflection angles that need to be applied to the system to adjust it at each of the times (k + 1) to (k + n). And provides parameters for the correlation process of step 4.
Step four: and designing a predictive controller aiming at the restraint of the offset position and the rotation angle of the object protected by the suspended lifting.
In this step, an optimization objective function for constrained predictive control is first given, and in the case of constraints, a suitable constraint object is first selected for the optimization function. In a complex system, it is the state variable of the system that is often chosen as the constraint object. This patent takes this idea to build an optimal controller. And (4) solving a predictive control model formula (18) by adopting the step (3), and setting an optimization objective function of constrained predictive control as J. J is the predicted value y and the reference track y of the output r The deviation between them, let the objective function be:
in formula (19), Q is an error weighting matrix; r is an input weighting matrix;
a future reference trajectory Y of the system may be defined r (k) The difference value between the system free response track and the system free response track is E (k), and is recorded as E (k) = psiX (k) + eta u (k-1) -Y r (k) Then the objective function (19) can be further represented by E (k):
let H =2 (Θ) T QΘ+R)，f＝2E(k) T QΘ，const＝E(k) T QE(k)；
The objective function is then:
in this case, H, f, and const in equation (21) are constant matrices. Therefore, the optimization problem becomes a problem of solving the standard Quadratic Programming (QP).
Selecting the state variable as a constraint object, the constrained optimization objective function can be further written as:
x imin ，x imax a limiting amplitude value of the state variable;
let S = [ x ] imin ]，X(k)＝[Φx(k)+B Δ ΔU(k)+B u u(k-1)]，L＝[x imax ]
Equation (22) can be reduced to the standard form of the optimization equation with constraints:
in the formula
This is a standard Quadratic Programming (QP) problem and the variable is Δ U (k). The solution of the optimal solution of the magnetic suspension lifting vibration reduction system is to solve the Quadratic Programming (QP) problem.
Step five: and further designing an optimization algorithm aiming at the designed predictive controller.
Solving the quadratic programming problem in step four we adopt the following way.
(1) Setting relaxed KKT condition
As can be seen from step 4, the prediction control for considering the magnetic levitation lifting vibration damping protection device with constraints in this patent can be converted into finding out the optimal solution of the constraint objective function shown in the formula (23), and at this time, the problem is converted into solving a quadratic programming problem with inequality constraints.
Let Δ U (k) = x in equation (23), a general form of the quadratic programming problem can be obtained:
the number of variables p =3m, the number of constraints q =6n, and the sufficient requirement for Δ U (k) to be an optimal solution is the presence of lagrange multiplier γ = (γ =) 1 ,γ 2 ,…,γ q ) The KKT condition is satisfied, that is, equation (25) is satisfied:
now adding the variable w, and
then gamma is added i w i Where =0 (i =1,2, \8230;, p) is denoted as YWe = μ e, the relaxation condition can be obtained, and the inequality constraint problem shown in equation (25) is converted into the equality constraint problem equation (26).
Wherein Y = diag (γ) 1 ,γ 2 ,…γ p ),W＝diag(w 1 ,w 2 ,…w p ),e＝(1,1,…1) T For each real number μ > 0, equation (26) has a unique solution x (μ), γ (μ), w (μ), the set of names { x (μ), γ (μ), w (μ) | μ > 0} is the center path.
(2) Iterative solution step:
for the relaxed KKT condition given in formula (26) in step (1) of this section, the patent adopts the following method to find the optimal solution. The solving step flow chart is shown in fig. 8.
The method comprises the following specific steps:
(1) setting an initial position
Taking an initial point z (0) ＝(x (0) ,γ (0) ,w (0) ) Wherein gamma is (0) ＞0,w (0) Is greater than 0, and the accuracy requirement tau is set.
(2) The equation for KKT is defined as
(3) Solving for the search direction (Δ x) (k) ,Δγ (k) ,Δw (k) )
Take any point (Δ x) in the iterative process (k) ,Δγ (k) ,Δw (k) ) Wherein gamma is (k) ＞0,w (k) > 0, solving for the search direction Δ x (k) ,Δγ (k) ,Δw (k) Such that (x) (k) +Δx (k) ,γ (k) +Δγ (k) ,w (k) +Δw (k) ) KKT conditional expression (26) is satisfied, that is:
wherein Δ Y = diag (Δ γ) 1 ,Δγ 2 ,…Δγ p ),W＝diag(Δw 1 ,Δw 2 ,…Δw p ). After finishing the formula (27), neglecting quadratic terms to obtain
A I Δx (k) -Δw (k) )＝b I -A I x (k) +w (k) (29)
W (k) Y (k) +Y (k) ΔW (k) ＝μe-ΔY (k) W (k) e (30)
From the formula (30)
ΔW (k) ＝(Y (k) ) -1 (μe-Y (k) W (k) e-W (k) ΔY (k) ) (31)
When the formula (31) is substituted into (28) and (29) and expressed by a matrix, there are
Solving equation (32) yields:
will be Δ x (k) Δ γ can be solved by substituting equations (45) and (46) (k) ,Δw (k) 。
(4) Let an integer p (p is smaller than one and infinitely close to one), calculate the along direction (Δ x) (k) ,Δγ (k) ,Δw (k) ) Step size parameter lambda of search (k)
(5) Calculating the next iteration point z (k+1) ＝(x (k+1) ,γ (k+1) ,w (k+1) )
Replace point k with point k +1 and continue the cycle as above. When the temperature is higher than the set temperature
And stopping calculation to obtain the optimal solution.
Claims (4)
1. A prediction control method based on a magnetic suspension damping device is characterized by comprising the following steps:
the method comprises the following steps: providing a magnetic suspension lifting shock-absorbing protection device;
step two: based on the magnetic suspension lifting shock absorption protection device, the working principle of the magnetic suspension lifting shock absorption protection device is analyzed, the deviation and rotation of the lifted object caused by the disturbance are considered, and a kinematic equation and a rotation equation which are nonlinear equations are established;
step three: linearizing the listed nonlinear equations, and writing a state equation by using current as a control quantity;
step four: discretizing a state equation of the magnetic suspension lifting damping protection device by adopting a front Euler method, writing a predictive control equation in sequence, and designing a predictive controller;
step five: aiming at the constraints of the offset position and the rotation angle of the suspended and lifted object, an optimization algorithm is designed;
step six: further optimizing the algorithm, giving an iteration solving step, and finally iterating to obtain an optimal solution;
in the fifth step, based on the control system of the magnetic suspension lifting damping protection device, an optimized objective function of the prediction controller is given:
wherein, delta U (k) is a control quantity difference value obtained by discretization calculation according to k and k-1 time, H is a constant matrix, f is a constant matrix T Is a transposed matrix of the constant matrix f,
and then, constraining the amplitude of the state variable, and further giving out an optimization objective function with constraint:
wherein, delta U (k) is a control quantity difference value obtained by discretization calculation according to k and k-1 time, H is a constant matrix, f is a constant matrix T A transposed matrix which is a constant matrix f;
in the sixth step, based on a magnetic suspension lifting damping protection device control system, a relaxation KKT condition is given for a target function with inequality constraint:
wherein H is a constant matrix, x = Δ U (k), γ is a Lagrange multiplier, and w is a variable; y is a diagonal matrix composed of Lagrange multipliers; w is a diagonal matrix of variables W, e is a unit vector, e = (1, \82301; 1) T And mu is a real number and is greater than 0, an optimization algorithm is designed based on a control system of the magnetic suspension lifting damping protection device, iterative solving steps are given, and the optimal solution meeting the conditions is solved.
2. The predictive control method based on the magnetic suspension damping device according to claim 1, characterized in that in the third step, after approximately linearizing the given nonlinear equation based on the magnetic suspension lifting damping protection device, the displacement and rotation angle of the float on the x, y and z axes are used as the state variable and output, and the current on the x, y and z axes is used as the input, so as to establish the state equation, thereby realizing the control of the attitude of the lifted object by the system and maintaining the balance thereof.
3. The predictive control method based on the magnetic suspension damping device according to claim 1, wherein in the fourth step, based on the magnetic suspension lifting damping protection device, the state equation listed is discretized by adopting the proud euler method, and the predictive controller is designed according to the predictive control method.
4. The prediction control method based on the magnetic suspension damping device according to any one of claims 1 to 3, wherein the magnetic suspension lifting damping protection device in the first step is a hexahedral box, each surface of the box is provided with a unit module, each unit module is composed of a plurality of coils and Hall elements, and the magnetic suspension damping is realized by controlling a single unit module or a plurality of unit modules in combination.
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