CN116125818B - Finite-time cabin suspension control method with closed-loop information feedback dynamic assignment performance - Google Patents
Finite-time cabin suspension control method with closed-loop information feedback dynamic assignment performance Download PDFInfo
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Abstract
The invention relates to a limited-time cabin suspension control method with closed-loop information feedback dynamic specified performance, and belongs to the technical field of automation. The method obtains system input through an error transformation function and designs a finite time suspension sliding mode controller: establishing a two-degree-of-freedom suspension linearization model; introducing an s-shaped function into a performance function, designing a boundary expansion smooth transition function and a state feedback transition function, transforming a state variable into an s-shaped function form on the basis, introducing the performance function, constructing a second-order microscopic bounded state feedback function, and designing a dynamic appointed performance function containing closed-loop information feedback; based on the method, a limited-time sliding mode main controller is designed, and a self-adaptive method based on a reconstructed sliding mode surface is adopted to estimate lumped disturbance. The invention can effectively inhibit the influence of external disturbance and parameter uncertainty on the system operation, has good transient performance, and ensures that the magnetic suspension system operates stably and reliably under the influence of external disturbance.
Description
Technical Field
The invention relates to a control method, in particular to a limited-time sliding mode control method for improving transient performance of a nonlinear system and simultaneously restraining system output or tracking performance and dynamically specifying performance by closed-loop information feedback, and belongs to the technical field of automation.
Background
The key of ensuring the good robustness of the system is that the magnetic levitation system of the wind turbine cabin is stably levitated so as to realize yaw on wind. The wind power cabin magnetic suspension system works on a tower with the height of 80 meters, and the working conditions are bad. The pitching moment causes the nacelle to pitch and axial disturbances induce axial vibrations. In order to solve the pitching problem, two-degree-of-freedom motions are carried out by adopting two-end suspension control of two rotor coils and a converter, the two sides are in resultant force to suspend the engine room, and the differential force is used for restraining pitching. Compared with the PID control with self-adaptive compensation, the self-adaptive robust control strategy with synchronous compensation has faster dynamic response speed, smaller steady-state error and synchronous error. However, the control strategy does not effectively restrict the system state, and still has the problems of poor interference suppression capability, low dynamic response speed and the like, and cannot effectively cope with high-frequency disturbance and external interference existing in an actual system. To limit the output or tracking error performance of the controlled system, prescribed performance control techniques are currently commonly employed. The scholars have further proposed a dynamic prescribed performance control to avoid the singular problem caused by overrun.
However, it should be noted that, the conventional method for controlling the specified performance sets a static error range, and a smaller boundary needs to be set to ensure transient performance, which will cause that the system may also operate in a sensitive interval in a steady state, so that the parameters of the main controller cannot be too large, which results in that the method has weak constraint in an actual system, and the actual system is inevitably affected by external interference and uncertainty in the operation process, so that the singular problem is very easy to occur. For the research on the aspect of the existing dynamic regulation performance control, the discontinuous problem exists when the performance function is reset, so that the singular problem of the transformation error higher-order term appears, and the spike problem is brought in an actual system. And because the error boundary is enlarged after reset, the control force is weakened, so that the problem of backlash is caused after the problem of sharp-rushing occurs. Therefore, the improvement of the transient performance of the system by the method mainly depends on the main controller, the introduction of the performance function does not effectively improve the transient performance of the system, and the expanded boundary may cause the tracking error to exceed the maximum boundary allowed by the system.
Disclosure of Invention
The main purpose of the invention is that: aiming at the defects and the blank of the prior art, the invention provides a limited-time cabin suspension control method with closed-loop information feedback dynamic assignment performance. The s-shaped function is initially introduced into the performance function to construct a smooth transition function, so that smooth transition from starting to steady-state process is realized, and after the system enters steady state, a large constraint boundary base value is generated after the transition of the performance function is completed, so that the system operates in an insensitive area of reconstruction error. On the basis, the system state quantity is introduced into the performance function, and a state feedback transition function is designed for ensuring the continuity of the performance function when state feedback information is introduced. In order to make the performance function have the capability of identifying the working condition of the system and realize dynamic adjustment in a bounded range, a second-order differentiable bounded state feedback function is designed, wherein the bounded state feedback function is an s-shaped function related to the state variable of the system, and based on the s-shaped function, a dynamic designated performance function containing state closed-loop feedback is designed. When the system tracking error is interfered to punch out an allowable steady-state error band, firstly, a damping item reconstructed from the gradient of the performance function relative to the actual error and the gradient of the actual error relative to time can quickly inhibit the interference; secondly, along with the increase of errors, the constraint boundary is reduced, so that the reconstruction errors are increased rapidly, and the limited-time sliding mode main controller is ensured to act rapidly and forcefully; and finally, rapidly estimating interference based on a self-adaptive method for reconstructing the sliding mode surface, so that the sliding mode surface is converged in a limited time.
In order to achieve the above purpose, the invention provides a limited time cabin suspension control method with closed loop information feedback dynamic specified performance, which comprises the following steps:
step 1, establishing a linearization model of a wind turbine cabin with two degrees of freedom suspension height considering axial direction and pitching;
step 2, introducing an s-shaped function into a traditional preset performance function, constructing a boundary expansion smooth transition function, a state feedback transition function and a second-order microscopic bounded state feedback function, and designing a dynamic appointed performance function containing closed-loop information feedback based on the s-shaped function;
step 3, obtaining virtual control input of a system through an error transformation function, and designing a limited-time cabin suspension controller with dynamic specified performance and closed-loop information feedback;
the step of designing the linearization model of the two-degree-of-freedom suspension height of the wind turbine in the step 1 is as follows:
a) Establishing a model of a two-degree-of-freedom suspension height taking into account axial height and pitch angle:
wherein H is the suspension height of the center point of the engine room, theta is the pitching angle and mu 0 Is vacuum magnetic conductivity, N is number of turns of suspension windings on two sides, S is magnetic pole area, i A And i B Exciting currents on the blade side and the tail wing side respectively, H A And H B The front side suspension height and the rear side suspension height are respectively, J is cabin pitching moment of inertia, and m is the mass of the wind power cabin; (L-H) A ) And (L-H) B ) The lengths of the air gaps at the front side and the rear side are respectively; g is gravity acceleration; f (f) d Is a nacelle axial disturbance; t (T) r For nacelle overturning moment, r is nacelle turning radius, and L is the sum of air gap length and levitation height.
B) Obtaining a linearization model of the two-degree-of-freedom suspension height:
wherein DeltaH x ＝H x -H 0 ，Δi x ＝i x -i 0 Wherein x is A or B, i 0 And H 0 Target current and target air gap, (L-H) 0 ) Is the center air gap length.
The step of designing the dynamic designated performance function with closed-loop information feedback in the step 2 is as follows:
a) The smooth transition function without singular problem is designed to be F (t) = (1+exp (-a (t-t) i ))) -1 Wherein a, t i Is positive constant, the value of a affects the transition speed, t i The value of (2) may locate the transition time.
B) The concept of the transition function is initially introduced into a traditional preset performance function, and the preset performance function comprising the boundary expansion transition function is designed as follows:
wherein ρ is i-0 、ρ i-∞ And K i Is a positive scalar, ρ i-0 Maximum bound, ρ representing overshoot and undershoot of tracking error i-∞ Representing a preset maximum limit of tracking error at steady state, K i Characterizing the gradient of the performance function with respect to time, i being 1 or 2, N designating the expansion boundary, said N and ρ i-∞ The sum represents the outer layer boundary, t 1 Determining transition time to satisfy t 1 >t st Wherein t is st A, the time required for the system to complete the start-up n And the transition time is influenced, the smooth transition from the realization performance function to the steady-state process is started, the constraint boundary is further enlarged, and conditions are created for realizing dynamic adjustment of the performance function according to the change of the working condition of the system.
C) In order to realize the dynamic adjustment of the performance function along with the change of the working condition of the system, the tracking error of the system is introduced into the performance function, and in order to ensure the continuity of the performance function when the state feedback information is introduced, the state feedback transition function is constructed, and the dynamic designated performance function containing the state feedback transition function is designed as follows:
wherein D designates a contribution degree, e i (t) is the system tracking error, a d Influence the transition time t 1 <t 2 ，t 2 And determining the time for introducing the system state quantity, wherein the performance function is dynamically adjusted according to the working condition of the system. Setting the tracking error of the axial suspension air gap as e 1 ＝H ref H, synchronous tracking error e 2 ＝θ ref - θ, the axial air gap and pitch angle are denoted h= (H) respectively A +H B )/2，θ＝(H A -H B ) 2r, where H ref Is the reference height of the center point of the engine room, theta ref Is the nacelle pitch reference angle.
D) In order to enable a dynamically designated performance function containing a state feedback transition function to have the capability of identifying the working condition of a system and realize dynamic adjustment in a bounded range, a bounded state feedback function is designed, wherein the bounded state feedback function is an s-shaped function related to a state variable of the system. From this, D can define the dynamic range of the performance function, and D<N, introducing the dynamic appointed performance function containing the state feedback transition function, and simultaneously introducing parameters for ensuring the continuity of the dynamic appointed performance function and higher-order terms thereofDesign->The order-differentiable bounded state feedback function is:
wherein a is i Influencing the gradient of the performance function with respect to the tracking error of the system, thereby improving the response speed by adjusting the value thereof, c i Steady state error interval affecting performance function, its value selection can be referred to c i ＝2|e iss I, wherein e iss Representing steady state error effective values. To avoid singular problems of the performance function in the process of changing tracking errorObtaining a second order microscopic bounded state feedback function. The dynamically specified performance function containing state closed loop feedback is designed to be:
e) Building dynamic specified performance functions for axial and pitch sides, respectively
The dynamic specified performance function containing the axial air gap tracking error is constructed as follows:
because the initial value of the system synchronization error is set to 0, a starting process is not required to be set, and a dynamic designated performance function containing the pitching angle tracking error is constructed as follows:
the controller design in the step 3 comprises the following steps:
a) The design error transformation function is:
the virtual control variables are introduced as follows:
the control law for differentiating the expression (10) and setting the sliding mode is as follows:
wherein lambda is 1 ,λ 1 ,k，k 1 ，k 2 ，k 3 Gamma is a non-zero positive constant.
Will beThe unfolding can be obtained:
will beSubstitution into formula (11) with formula (12) can be achieved:
b) Designing a finite time sliding mode controller based on a dynamically specified performance function containing an axial air gap tracking error: expanding the first derivative of the performance function with respect to time may result in:
the second derivative is developed to obtain:
wherein the method comprises the steps of
The second derivative is expressed as:
wherein the method comprises the steps of
The preliminary design axial control rate is
Pair DFe 1 ρ 1 -1 To discuss the limitations of (a):
wherein the method comprises the steps of
Order theThen when |e 1 When the I is less than or equal to 1, I is bounded; when |e 1 When the I is greater than 1, there are
I.e. l is bounded, thus DFe 1 ρ 1 -1 Is always bounded. Based on this, the design axial control law is:
the external disturbance term and the uncertainty term are expressed as:
wherein,Δu 1 * ＝(Δi A +Δi B )/2，Θ＝(1+e 1 /ρ 1 )(1-e 1 /ρ1)ρ 1 ，a10＝μ0N 2 Si 0 2 /(2m(L-H 0 ) 3 ),b 10 ＝μ0N 2 Si 0 /(2m(L-H 0 ) 2 )。
c) Designing a finite time sliding mode controller based on a dynamically specified performance function containing pitch angle tracking errors:
expanding the first derivative of the performance function with respect to time may result in:
the second derivative is developed to obtain:
wherein,the second derivative is expressed as:
wherein the method comprises the steps of
It can be seen that D 2 F 2 e 2 ρ 2 -1 Always bounded, based on this, the synchronous control law is designed as
The external disturbance term and the uncertainty term are expressed as:
wherein,Δu 2 * ＝(Δi A -Δi B )/2，Θ 2 ＝(1+e 2 /ρ 2 )(1-e 2 /ρ 2 )ρ 2 ，a 20 ＝rμ 0 N 2 Si 0 2 /(2J(L-H 0 ) 3 ),b 20 ＝rμ 0 N 2 Si 0 /(2J(L-H 0 ) 2 )。
the main control current of the blade side and the tail wing side can be obtained by the methods (23), (29)
The beneficial effects of the invention are as follows:
1) The dynamic specified performance function and the higher-order term thereof designed in the control method are continuous, and no singular problem exists in dynamic adjustment;
2) A large constraint boundary is set, so that the singular problem caused by overrun is further avoided;
3) The method solves the problem that the reconstruction quantity is insensitive to the system state variable under the condition of a large constraint boundary, and when the system tracking error is interfered to punch out an allowable steady-state error band, firstly, the interference is rapidly inhibited by a damping item reconstructed by the gradient of the performance function relative to the actual error and the gradient of the actual error relative to the time;
4) Secondly, along with the increase of errors, the constraint boundary is reduced, so that the reconstruction errors are increased rapidly, and the limited-time sliding mode main controller is ensured to act rapidly and forcefully;
5) And finally, rapidly estimating interference based on a self-adaptive method for reconstructing the sliding mode surface, so that the sliding mode surface is converged in a limited time. The method has excellent robustness and provides powerful guarantee for the magnetic suspension system to cope with external high-frequency interference and uncertain items.
Drawings
FIG. 1 is a schematic diagram of a nacelle suspension structure of a horizontal axis wind yaw system of the present invention.
FIG. 2 is a diagram of a nacelle levitation control architecture for a horizontal axis wind yaw system of the present invention.
FIG. 3 is a graph of the experimental application of axial disturbance force to the nacelle under the control of the present invention and other algorithms.
FIG. 4 is a graph of the pitch disturbance applied to the nacelle under the control of the present invention and other algorithms.
FIG. 5 is a graph of the synchronization error of the nacelle applied pitch disturbance force under the control of the present invention and other algorithms.
Fig. 6 is a graph of an experimental cabin applied pitch disturbance force under PPSMC control.
FIG. 7 is a graph of an experimental application of pitch disturbance force to a nacelle under the control of the present invention.
In the figure: 1-fan blade, 2-fan nacelle, 3-yaw stator, 4-front side winding, 5-back side winding, 6-front side air gap sensor, 7-back side air gap sensor, 8-tower, 9-axial dynamic specified performance function, 10-axial conversion error, 11-axial reconstruction error-based slip mode face, 12-pitch dynamic specified performance function, 13-pitch conversion error, 14-pitch reconstruction error-based slip mode face, 15-axial air gap, 16-pitch angle, 17-axial controller, 18-pitch controller, 19-axial finite time state observer, 20-pitch finite time state observer, 21-blade side current tracking controller, 22-tail side current tracking controller, 23-blade side suspension winding, 24-tail side suspension winding, 25-wind nacelle.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings.
As shown in fig. 1, the air gap length is between the air sensors 6 and 7 and the yaw stator 3, and the yaw stator 3 is fixed. The suspension electromagnet consists of windings 4 and 5 and an iron core. When a voltage u (t) is applied to the windings 4 and 5, a current i flows through the windings 4 and 5 1 (t)、i 2 (t) the levitation electromagnet will generate an electromagnetic attraction force and the yaw stator 3 will be attracted. In the floating process, after the windings 4 and 5 are electrified, the suspension electromagnet will move upwards under the action of electromagnetic attractionWhen disturbance is applied, u (t) is adjusted along with the change of the suspension air gap, so that i (t) is tracked and changed until the stable suspension balance point is reached, and a control block diagram is shown in figure 2.
The invention relates to a limited-time cabin suspension control method with closed-loop information feedback and dynamic specified performance, which aims to restrict the output and tracking error performance of a cabin suspension system so as to realize good transient performance, and specifically comprises the following steps:
step 1, establishing a linearization model of a wind turbine nacelle with two degrees of freedom in suspension height considering axial direction and pitching:
the modeling process is as follows:
according to fig. 1, the windings 4, 5 will generate an upward axial levitation force F (i x (t),δ x ) The method comprises the following steps:
F(i x (t),δ x )＝ki x 2 (t)/δ x 2
according to FIG. 1, the magnetic levitation system is axially subjected to a levitation suction force F with both ends upward A 、F B Downward gravity mg and axial disturbance force f d The method comprises the steps of carrying out a first treatment on the surface of the In the process of floating, the rising acceleration is-During landing, the landing acceleration is +.>The mechanical equation of the magnetic suspension system in the axial direction is as follows:
the length of the air gap is converted into the axial height:
the mechanical equation of the magnetic suspension system about the pitching angle is as follows:
in summary, a two-degree-of-freedom suspension height model is available that takes into account the axial height and pitch angle:
wherein H is the suspension height of the center point of the engine room, delta x ＝L-H x X epsilon { A, B } is the air gap length, θ is the pitch angle, μ 0 Is vacuum magnetic conductivity, N is number of turns of suspension windings on two sides, S is magnetic pole area, i A And i B Exciting currents on the blade side and the tail wing side respectively, H A And H B The front side suspension height and the rear side suspension height are respectively, J is cabin pitching moment of inertia, and m is the mass of the wind power cabin; (L-H) A ) And (L-H) B ) The lengths of the air gaps at the front side and the rear side are respectively; g is gravity acceleration; f (f) d Is a nacelle axial disturbance; t (T) r For nacelle overturning moment, r is nacelle turning radius, and L is the sum of air gap length and levitation height.
From the linearization model, a two-degree-of-freedom suspension height is obtained:
wherein DeltaH x ＝H x -H 0 ，Δi x ＝i x -i 0 Wherein x is A or B, i 0 And H 0 Target current and target air gap, (L-H) 0 ) Is the center air gap length.
Step 2, the step of dynamic appointed performance function design with state closed loop feedback is as follows:
a) The smooth transition function without singular problem is designed to be F (t) = (1+exp (-a (t-t) i ))) -1 Wherein a, t i Is positive constant, the value of a affects the transition speed, t i Can be fixed in valueBit transition time.
B) The concept of the transition function is initially introduced into a traditional preset performance function, and the preset performance function comprising the boundary expansion transition function is designed as follows:
wherein ρ is i -0、ρ i-∞ And K i Is a positive scalar, ρ i-0 Maximum bound, ρ representing overshoot and undershoot of tracking error i-∞ Representing a preset maximum limit of tracking error at steady state, K i Characterizing the gradient of the performance function with respect to time, i being 1 or 2, N designating the expansion boundary, said N and ρ i-∞ The sum represents the outer layer boundary, t 1 Determining transition time to satisfy t 1 >t st Wherein t is st A, the time required for the system to complete the start-up n And the transition time is influenced, the smooth transition from the realization performance function to the steady-state process is started, the constraint boundary is further enlarged, and conditions are created for realizing dynamic adjustment of the performance function according to the change of the working condition of the system.
C) In order to realize the dynamic adjustment of the performance function along with the change of the working condition of the system, the tracking error of the system is introduced into the performance function, and in order to ensure the continuity of the performance function when the state feedback information is introduced, the state feedback transition function is constructed, and the dynamic designated performance function containing the state feedback transition function is designed as follows:
wherein D designates a contribution degree, e i (t) is the system tracking error, a d Influence the transition time t 1 <t 2 ，t 2 And determining the time for introducing the system state quantity, wherein the performance function is dynamically adjusted according to the working condition of the system. Setting the tracking error of the axial suspension air gap as e 1 ＝H ref H, synchronous tracking error e 2 ＝θ ref - θ, axial air gap and pitch angle respectivelyExpressed as h= (H) A +H B )/2，θ＝(H A -H B ) 2r, where H ref Is the reference height of the center point of the engine room, theta ref Is the nacelle pitch reference angle.
D) In order to enable a dynamically designated performance function containing a state feedback transition function to have the capability of identifying the working condition of a system and realize dynamic adjustment in a bounded range, a bounded state feedback function is designed, wherein the bounded state feedback function is an s-shaped function related to a state variable of the system. From this, D can define the dynamic range of the performance function, and D<N, introducing the dynamic appointed performance function containing the state feedback transition function, and simultaneously introducing parameters for ensuring the continuity of the dynamic appointed performance function and higher-order terms thereofDesign->The order-differentiable bounded state feedback function is:
wherein a is i Influencing the gradient of the performance function with respect to the tracking error of the system, thereby improving the response speed by adjusting the value thereof, c i Steady state error interval affecting performance function, its value selection can be referred to c i ＝2|e iss I, wherein e iss Representing steady state error effective values. To avoid singular problems of the performance function in the process of changing tracking errorObtaining a second order microscopic bounded state feedback function. The dynamically specified performance function containing state closed loop feedback is designed to be:
the dynamic specified performance function containing state closed-loop feedback is a dynamic specified performance function comprehensively considering mechanical constraint, steady-state performance and anti-interference capability of the system. N (N) max For the system mechanical constraint boundary ρ ∞ Specifying steady-state constraint boundaries, N 1 Specifying a performance function dynamic range, N 0 An outer layer boundary is specified.
E) Building dynamic specified performance functions for axial and pitch sides, respectively
The dynamic specified performance function containing the axial air gap tracking error is constructed as follows:
because the initial value of the system synchronization error is set to 0, a starting process is not required to be set, and a dynamic designated performance function containing the pitching angle tracking error is constructed as follows:
and 3, adopting the dynamic designated performance functions of the axial side and the pitching side of the design, and carrying out controller design based on the transformation error function, wherein the controller design comprises the following steps:
a) The design error transformation function is:
the virtual control variables are introduced as follows:
the control law for differentiating the expression (10) and setting the sliding mode is as follows:
wherein lambda is 1 ,λ 1 ,k，k 1 ，k 2 ，k 3 Gamma is a non-zero positive constant.
Will beThe unfolding can be obtained:
/>
will beSubstitution into formula (11) with formula (12) can be achieved:
b) Designing a finite time sliding mode controller based on a dynamically specified performance function containing an axial air gap tracking error: expanding the first derivative of the performance function with respect to time may result in:
the second derivative is developed to obtain:
wherein the method comprises the steps of
The second derivative is expressed as:
wherein the method comprises the steps of
The preliminary design axial control rate is
Pair DFe 1 ρ 1 -1 To discuss the limitations of (a):
wherein the method comprises the steps of
Order theThen when |e 1 When the I is less than or equal to 1, I is bounded; when |e 1 When the I is greater than 1, there are
I.e. l is bounded, thus DFe 1 ρ 1 -1 Is always bounded. Based on this, the design axial control law is:
the external disturbance term and the uncertainty term are expressed as:
wherein,Δu 1 * ＝(Δi A +Δi B )/2，Θ＝(1+e 1 /ρ 1 )(1-e 1 /ρ 1 )ρ 1 ，a 10 ＝μ 0 N 2 Si 0 2 /(2m(L-H 0 ) 3 ),b 10 ＝μ 0 N 2 Si 0 /(2m(L-H 0 ) 2 )。
c) Designing a finite time sliding mode controller based on a dynamically specified performance function containing pitch angle tracking errors:
expanding the first derivative of the performance function with respect to time may result in:
the second derivative is developed to obtain:
wherein,the second derivative is expressed as:
wherein the method comprises the steps of
It can be seen that D 2 F 2 e 2 ρ 2 -1 Always bounded, based on this, the synchronous control law is designed as
The external disturbance term and the uncertainty term are expressed as:
wherein,Δu 2 * ＝(Δi A -Δi B )/2，Θ 2 ＝(1+e 2 /ρ 2 )(1-e 2 /ρ 2 )ρ 2 ，a 20 ＝rμ 0 N 2 Si 0 2 /(2J(L-H 0 ) 3 ),b 20 ＝rμ 0 N 2 Si 0 /(2J(L-H 0 ) 2 )。
the main control current of the blade side and the tail wing side can be obtained by the methods (23), (29)
The invention is further illustrated by the following preferred embodiment.
Example 1:
the system parameters of the magnetic suspension system are as follows: effective area s= 235050mm of pole surface of suspension electromagnet 2 The total mass m of the suspension body is=500 kg, the number of turns of the exciting coil of the suspension electromagnet is n=6400 turns, the resistance R of the exciting coil is 4.4Ω, and the vacuum magnetic permeability is mu 0 ＝4π×10 -7 H/m; suspension electromagnet height H when stabilizing suspension equilibrium point 0 =0.013 m, at rest positionHeight H of suspension electromagnet 1 ＝0.009m。
Based on the above system parameters, system simulation conditions: and (I) an axial interference resistance simulation experiment: the running time is t=0-20 s, and the expected track tracking function of the process is selected as H ref (t) =0.013, and axial interference was added at t=4 s, and interference was removed at t=15 s; (II) anti-pitching interference simulation experiment: the running time is t=0-20 s, and the expected track tracking function of the process is selected as H ref (t) =0.013, pitch disturbance was added at t=4 s, and disturbance was removed at t=15 s.
And simulating the system according to the simulation conditions, so as to verify the anti-interference capability of the system to external disturbance at the moment of operation. As shown in fig. 3, 4, 5, 6 and 7.
FIG. 3 shows a simulation curve of a track tracking suspension air gap in an anti-axial interference simulation experiment, wherein the simulation effect of five types of algorithms is distinguished by a line type. As can be seen from the graph, when axial interference is applied, the maximum drop value of the air gap is only 0.0015mm, and the recovery time is only 0.008s, so that the method has stronger anti-interference capability and quicker response speed compared with other algorithms.
Fig. 4 shows a synchronous error simulation curve in a simulation experiment of anti-pitch interference, in which simulation effects of five types of algorithms are distinguished linearly. As can be seen from the graph, the maximum synchronization error is only 0.02mm and the recovery time is only 0.01s when the pitching interference is applied, so that the method has a better synchronization effect compared with other algorithms.
FIG. 5 shows a trace tracking suspension air gap simulation curve in a pitching interference resistance simulation experiment, wherein the simulation effect of five types of algorithms is distinguished by the line type. As can be seen from the graph, when pitch interference is applied, the maximum drop value of the air gap is only 0.007mm, and the recovery time is only 0.022s, so that compared with other algorithms, the method has stronger anti-interference capability and quicker response speed.
FIG. 6 shows a simulation curve of the PPSMC algorithm tracking the suspension air gap at two ends in a pitching interference resistance simulation experiment, wherein the line type distinguishes the blade side and the tail wing side. As can be seen from the graph, the method has an air gap maximum drop value of 0.032mm and a recovery time of 0.03s when pitch disturbance is applied.
FIG. 7 shows a suspension air gap simulation curve of the proposed algorithm with both end trajectory tracking in anti-pitch disturbance simulation experiments, in which the line type distinguishes between blade side and tail wing side. As can be seen from the graph, the method has an air gap maximum drop value of 0.007mm and a recovery time of 0.022s when pitch disturbance is applied. Compared with the PPSMC algorithm, the method has stronger anti-interference capability and quicker response speed.
The results show that the control method of the invention can effectively inhibit the influence of external disturbance and parameter uncertainty on the system operation while ensuring the rapid tracking capability and stability of the system, has good robustness and ensures the stable and reliable operation of the magnetic suspension system.
Claims (2)
1. A limited time cabin suspension control method with closed loop information feedback dynamic specified performance is characterized in that: dynamic performance function formulation comprising cabin suspension height dynamic modeling, closed-loop information feedback and finite time suspension sliding mode controller design; the cabin suspension height dynamic modeling is a two-degree-of-freedom suspension height model based on cabin center suspension height and pitching angle; the dynamic performance function of the closed-loop information feedback is a dynamic appointed performance function of state-containing closed-loop feedback, which comprises a starting constraint, a smooth transition and an anti-interference performance constraint, wherein the starting constraint is constrained by a static boundary which gradually converges with respect to time, the convergence target of the static boundary is a steady-state constraint boundary, the smooth transition is realized by the boundary expansion smooth transition function and the state feedback transition function in a coordinated manner, the boundary expansion smooth transition function is an s-shaped function with respect to starting time and comprises an appointed transition speed, a transition time and fixed parameters of an expansion boundary, the sum of the steady-state constraint boundary and the expansion boundary is an outer boundary, the outer boundary is positioned between the maximum mechanical boundary of the system and the steady-state constraint boundary, the state feedback transition function is an s-shaped function with respect to starting time and comprises fixed parameters which are appointed transition speed, transition time and contribution degree, the contribution degree is appointed to a dynamic adjustment range based on state feedback, the value of the anti-interference performance constraint is smaller than the expansion boundary, the anti-interference performance constraint is realized by introducing a micro-bounded state feedback function, and the second-micro-order micro-bounded state feedback function is an steady-state variable with respect to the state variable; the finite time suspension sliding mode controller takes an axial height conversion error and a pitching angle conversion error of a system as control inputs, and is divided into a finite time sliding mode controller containing an axial air gap tracking error and a finite time sliding mode controller based on a dynamic specified performance function containing a pitching angle tracking error, and a self-adaptive method based on a reconstruction sliding mode surface is adopted to estimate lumped disturbance; the method comprises the following steps:
step 1, establishing a linearization model of a cabin two-degree-of-freedom suspension height considering an axial height and a pitching angle;
a) Establishing a model of a two-degree-of-freedom suspension height taking into account axial height and pitch angle:
wherein H is the suspension height of the center point of the engine room, theta is the pitching angle and mu 0 Is vacuum magnetic conductivity, N is number of turns of suspension windings on two sides, S is magnetic pole area, i A And i B Exciting currents on the blade side and the tail wing side respectively, H A And H B Respectively the front side suspension height and the rear side suspension height, wherein J is the pitching moment of inertia of the nacelle, m is the mass of the wind power nacelle, (L-H) A ) And (L-H) B ) Respectively, the front side and the rear side are respectively provided with a suspension air gap, g is gravity acceleration, f d For axial disturbance of nacelle, T r The nacelle overturning moment is represented by r, the nacelle rotating radius is represented by L, and the sum of the suspension air gap and the height is represented by L;
b) Obtaining a linearization model of the two-degree-of-freedom suspension height:
wherein DeltaH x ＝H x -H 0 ，Δi x ＝i x -i 0 Wherein x is A or B, i 0 And H 0 Target current and target air gap, (L-H) 0 ) Is an axial suspension air gap;
step 2, constructing a boundary expansion smooth transition function, a state feedback transition function and a second-order differentiable bounded state feedback function, and designing a dynamic appointed performance function containing closed-loop information feedback;
a) The smooth transition function without singular problem is designed to be F (t) = (1+exp (-a (t-t) i ))) -1 Wherein a, t i Is positive constant, the value of a affects the transition speed, t i The value of (2) may locate the transition time;
b) The smooth transition function is initially introduced into a preset performance function, and the preset performance function containing the boundary expansion transition function is designed as follows:
wherein ρ is i-0 、ρ i-∞ And K i Is a positive scalar, ρ i-0 Maximum bound, ρ representing overshoot and undershoot of tracking error i-∞ Representing a preset maximum limit of tracking error at steady state, K i Characterizing the gradient of the performance function with respect to time, i being 1 or 2, N designating the expansion boundary, said N and ρ i-∞ The sum represents the outer layer boundary, t 1 Determining transition time to satisfy t 1 >t st Wherein t is st A, the time required for the system to complete the start-up n Setting parameters for the transition time;
c) Introducing a system tracking error into a performance function, constructing a state feedback transition function, and designing a dynamic designated performance function containing the state feedback transition function as follows:
wherein D designates a contribution degree, e i (t) is the system tracking error, a d Influence the transition time t 1 <t 2 ，t 2 Determining the time for introducing the system state quantity, dynamically adjusting the performance function according to the system working condition, and setting the axial suspension air gap tracking error as e 1 ＝H ref H, synchronous tracking error e 2 ＝θ ref - θ, where H ref Is the reference height of the center point of the engine room, theta ref Is the cabin pitching reference angle, and the axial air gap and the pitching angle are respectively expressed as H= (H) A +H B )/2，θ＝(H A -H B )/2r；
D) Designing a bounded state feedback function that is an s-shaped function with respect to a system state variable, from which D can define a dynamic range of the performance function, and D<N, introducing the dynamic appointed performance function containing the state feedback transition function, and simultaneously introducing parameters for ensuring the continuity of the dynamic appointed performance function and higher-order terms thereofDesign->The order-differentiable bounded state feedback function is:
wherein a is i Influencing the gradient of the performance function with respect to the tracking error of the system, thereby improving the response speed by adjusting the value thereof, c i Steady state error interval affecting performance function, its value selection can be referred to c i ＝2|e iss I, wherein e iss To represent the steady-state error effective value and avoid singular problems of the performance function in the process of tracking error variation, the method comprises the following steps ofAnd obtaining a second-order microscopic bounded state feedback function, and designing a dynamic appointed performance function containing closed-loop information feedback as follows:
e) Building dynamic specified performance functions for axial and pitch sides, respectively
The dynamic specified performance function containing the axial air gap tracking error is constructed as follows:
because the initial value of the system synchronization error is set to 0, the starting process is not required to be set, and therefore, the dynamic appointed performance function containing the pitching angle tracking error is constructed as follows:
and step 3, obtaining virtual control input of the system through an error transformation function, and designing a limited-time cabin suspension controller with dynamic specified performance and closed-loop information feedback.
2. A limited time nacelle suspension control method with closed loop information feedback dynamic specified performance according to claim 1, wherein: the step of designing the limited-time cabin suspension controller in the step 3 is as follows:
a) The design error transformation function is:
the virtual control variables are introduced as follows:
the control law for differentiating the expression (10) and setting the sliding mode is as follows:
wherein lambda is 1 ,λ 2 ,k，k 1 ，k 2 ，k 3 Gamma is a non-zero positive constant;
will beThe unfolding can be obtained:
will beSubstitution into formula (11) with formula (12) can be achieved:
b) Designing a finite time sliding mode controller based on a dynamically specified performance function containing an axial air gap tracking error: expanding the first derivative of the performance function with respect to time may result in:
the second derivative is developed to obtain:
wherein the method comprises the steps of
The second derivative is expressed as:
wherein the method comprises the steps of
The preliminary design axial control rate is
Pair DFe 1 ρ 1 -1 To discuss the limitations of (a):
wherein the method comprises the steps of
Order theThen when |e 1 When the I is less than or equal to 1, I is bounded; when |e 1 When the I is greater than 1, there are
I.e. l is bounded, thus DFe 1 ρ 1 -1 Always bounded, based on which the design axial control law is:
the axial external interference and uncertainty term is expressed as:
wherein,Δu 1 * ＝(Δi A +Δi B )/2，Θ＝(1+e 1 /ρ 1 )(1-e 1 /ρ 1 )ρ1 1 ，a 10 ＝μ 0 N 2 Si 0 2 /(2m(L-H 0 ) 3 ),b 10 ＝μ 0 N 2 Si 0 /(2m(L-H 0 ) 2 )；
c) Designing a finite time sliding mode controller based on a dynamically specified performance function containing pitch angle tracking errors:
expanding the first derivative of the performance function with respect to time may result in:
the second derivative is developed to obtain:
wherein,the second derivative is expressed as:
wherein the method comprises the steps of
It can be seen that D 2 F 2 e 2 ρ 2 -1 Always bounded, based on this, the synchronous control law is designed as
The pitch ambient interference and uncertainty term is expressed as:
wherein,Δu 2 * ＝(Δi A -Δi B )/2，Θ 2 ＝(1+e 2 /ρ 2 )(1-e 2 /ρ 2 )ρ 2 ，a 20 ＝rμ 0 N 2 Si 0 2 /(2J(L-H 0 ) 3 )，b 20 ＝rμ 0 N 2 Si 0 /(2J(L-H 0 ) 2 )；
the main control current of the blade side and the tail wing side can be obtained by the methods (23), (29)
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