CN114527660B - Self-adaptive tracking control method and system for unmanned helicopter yaw channel dynamics - Google Patents

Self-adaptive tracking control method and system for unmanned helicopter yaw channel dynamics Download PDF

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CN114527660B
CN114527660B CN202210164561.XA CN202210164561A CN114527660B CN 114527660 B CN114527660 B CN 114527660B CN 202210164561 A CN202210164561 A CN 202210164561A CN 114527660 B CN114527660 B CN 114527660B
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张纪峰
张言军
孙健
赵延龙
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Beijing Institute of Technology BIT
Academy of Mathematics and Systems Science of CAS
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Abstract

The invention relates to a self-adaptive tracking control method and a self-adaptive tracking control system for the dynamics of a yaw channel of an unmanned helicopter, wherein the method comprises the following steps: determining a parameterized model of the discrete non-affine nonlinear system according to a dynamic equation of a target; updating a parameter of the parameterized model, and estimating unknown parameters according to the parameter updating law to obtain estimated parameters; constructing an estimate of the signal at a future time based on the parameter update law to obtain an estimated signal; determining an adaptive implicit function equation according to the estimation parameters and the estimation signals; designing an adaptive control law with an iterative form based on the adaptive implicit function equation; the solution of the adaptive control law with iterative form is the control signal of the discrete non-affine nonlinear system. The invention constructs an analytic self-adaptive control law, realizes self-adaptive output tracking performance, and ensures that the closed-loop signal of a controlled system is bounded and practical output tracking is realized.

Description

Self-adaptive tracking control method and system for unmanned helicopter yaw channel dynamics
Technical Field
The invention relates to the technical field of self-adaptive control, in particular to a self-adaptive tracking control method and a self-adaptive tracking control system for the dynamics of a yaw channel of an unmanned helicopter.
Background
The self-adaptive control of the non-affine non-linear uncertain system is always the hot spot direction of the control community. The prior literature proposes a plurality of methods to solve the problem that the control signal of the non-affine nonlinear system is difficult to solve. Many achievements have been achieved in adaptive control of a non-affine non-linear uncertain system, but how to solve the adaptive control problem of the non-affine non-linear uncertain system in a parameterized adaptive control framework is not found in relevant research. Therefore, the research on solving the adaptive control problem of the non-affine non-linear uncertain system in the parameterized adaptive control framework has important theoretical and practical significance.
In addition to the non-affine non-linearity problem, the input-output delay of a non-linear discrete system is crucial for the adaptive control law construction. When the input-output time delay d of the system is greater than 1, the causal contradiction of the control law is easily caused, and how to design the adaptive control law which accords with the causal property is the first technical problem. When the non-affine non-linear dependence bounded set condition is processed based on the neural network or the fuzzy approximation, the optimal control method generally has difficulty in having an analytic solution for processing the non-affine non-linear dependence. The dynamic nonlinearity of the system depends on the input signal, and how to design the self-adaptive control law of an analytic form is the second technical problem.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention aims to provide a self-adaptive tracking control method and a self-adaptive tracking control system for the dynamics of a yaw channel of an unmanned helicopter.
In order to achieve the purpose, the invention provides the following scheme:
an adaptive tracking control method for unmanned helicopter yaw channel dynamics comprises the following steps:
determining a parameterized model of the discrete non-affine nonlinear system according to a dynamic equation of a target;
updating a parameter of the parameterized model, and estimating unknown parameters according to the parameter updating law to obtain estimated parameters;
constructing an estimate of the signal at a future time based on the parameter update law to obtain an estimated signal;
determining an adaptive implicit function equation according to the estimation parameters and the estimation signals;
designing an adaptive control law with an iterative form based on the adaptive implicit function equation; the solution of the adaptive control law with iterative form is the control signal of the discrete non-affine nonlinear system.
Preferably, the designing a parameter updating law for the parameterized model and estimating unknown parameters according to the parameter updating law to obtain estimated parameters includes:
order to
Figure GDA0003764216800000021
Denotes theta * Is estimated, and
Figure GDA0003764216800000022
Figure GDA0003764216800000023
wherein, theta * In order for the parameters to be unknown,
Figure GDA0003764216800000024
and
Figure GDA0003764216800000025
respectively an output signal and an input signal of the discrete non-affine nonlinear system;
Figure GDA0003764216800000026
is a set Lipschitz nonlinear micro-mappable; d is system input-output time delay and satisfies d is more than or equal to 1 and less than or equal to n;
defining the error as ∈ (t) ═ θ T (t)φ f (t-1)-y(t)；
Designing the parameter updating law according to the errors
Figure GDA0003764216800000027
Figure GDA0003764216800000028
Wherein the content of the first and second substances,
Figure GDA0003764216800000029
Γ＝diag{γ 1 ,…,γ p is the gain matrix; g (θ (t), h (t)) [ g [ 1 (θ 1 (t),h 1 (t)),…,g p (θ p (t),h p (t))] T Is a correction term based on a parametric projection technique; gamma ray i ∈(0,2)；
There are known groups of bounded intervals
Figure GDA00037642168000000210
Satisfy the requirement of
Figure GDA00037642168000000211
And is
Figure GDA00037642168000000212
Wherein
Figure GDA00037642168000000213
α 0 Is an unknown normal number, X n+1 Is the n +1 variable of f;
design g i (θ i (t),h i (t)) is of the form:
Figure GDA00037642168000000214
wherein h is i (t) is the ith component of h (t), and h (t) is of the form
Figure GDA00037642168000000215
Determining based on the parameter update law
Figure GDA00037642168000000216
θ(t+i 0 )-θ(t)∈L 2 Wherein i 0 Is any positive integer.
Preferably, the constructing an estimate of the signal at a future time based on the parameter update law to obtain an estimated signal comprises:
updating law design based on the kinetic equation and the parameters
Figure GDA0003764216800000031
The method specifically comprises the following steps:
Figure GDA0003764216800000032
Figure GDA0003764216800000033
wherein the content of the first and second substances,
Figure GDA0003764216800000034
for the estimation of the output signal y (t +1),
Figure GDA0003764216800000035
is an estimate of the output signal y (t + j);
Figure GDA0003764216800000036
is phi f (ii) an estimate of (d); wherein the content of the first and second substances,
Figure GDA0003764216800000037
based on obtained
Figure GDA0003764216800000038
As an estimation signal; the estimated signal is known at the current time and is satisfied
Figure GDA0003764216800000039
Wherein μ (t) ∈ L 2 And μ (t) is the general attenuated signal boundary.
Preferably, the determining an adaptive implicit function equation according to the estimation parameter and the estimation signal includes:
the estimation based on unknown parameters and the estimation structure of future output signals are consistent with the structure of the kinetic equationThe adaptive implicit function equation of (2); the adaptive implicit function equation is
Figure GDA00037642168000000310
Figure GDA00037642168000000311
Wherein, y * (t + d) ═ r (t), r (t) is a reference input signal known at the current time; at each sampling instant t, the adaptive implicit function equation has a unique solution with respect to u (t), which is noted
Figure GDA00037642168000000312
And satisfy
Figure GDA00037642168000000313
Figure GDA00037642168000000314
Wherein, X n+1 Is phi f (t + j) th variable.
Preferably, the designing an adaptive control law having an iterative form based on the adaptive implicit function equation includes:
designing an adaptive control law of an iterative form into
Figure GDA00037642168000000315
Where u0(t) is taken as the control signal value at time t-1, and γ (t) is a given time-varying parameter dependent on time t and satisfies
Figure GDA00037642168000000316
Figure GDA00037642168000000317
Figure GDA00037642168000000318
An adaptive tracking control system for unmanned helicopter yaw channel dynamics, comprising:
the model construction unit is used for determining a parameterized model of the discrete non-affine nonlinear system according to a dynamic equation of the target;
the parameter estimation unit is used for updating the parameter of the parameterized model and estimating unknown parameters according to the parameter updating law to obtain estimated parameters;
a signal estimation unit, configured to construct an estimation of a signal at a future time based on the parameter update law to obtain an estimation signal;
an equation determining unit, configured to determine an adaptive implicit function equation according to the estimation parameter and the estimation signal;
the control law construction unit is used for designing the self-adaptive control law with an iteration form based on the self-adaptive implicit function equation; the solution of the adaptive control law with iterative form is the control signal of the discrete non-affine nonlinear system.
According to the specific embodiment provided by the invention, the invention discloses the following technical effects:
the invention provides a self-adaptive tracking control method and a self-adaptive tracking control system for the dynamics of a yaw channel of an unmanned helicopter, wherein the method comprises the following steps: determining a parameterized model of the discrete non-affine nonlinear system according to a dynamic equation of a target; updating a parameter of the parameterized model, and estimating unknown parameters according to the parameter updating law to obtain estimated parameters; constructing an estimate of the signal at a future time based on the parameter update law to obtain an estimated signal; determining an adaptive implicit function equation according to the estimation parameters and the estimation signals; designing an adaptive control law with an iterative form based on the adaptive implicit function equation; the solution of the adaptive control law with iterative form is the control signal of the discrete non-affine nonlinear system. The invention constructs an analytic self-adaptive control law, realizes self-adaptive output tracking performance, and ensures that the closed-loop signal of a controlled system is bounded and practical output tracking is realized.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings without inventive exercise.
FIG. 1 is a flow chart of a method in an embodiment provided by the present invention;
FIG. 2 is a trace diagram of an output signal and a reference output signal according to an embodiment of the present invention;
FIG. 3 is a schematic trace diagram of a tracking error signal in an embodiment provided by the present invention;
FIG. 4 is a schematic trace diagram of a system input signal in an embodiment of the present invention;
fig. 5 is a diagram illustrating an adaptive response of an estimated parameter in an embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Reference herein to "an embodiment" means that a particular feature, structure, or characteristic described in connection with the embodiment can be included in at least one embodiment of the application. The appearances of the phrase in various places in the specification are not necessarily all referring to the same embodiment, nor are separate or alternative embodiments mutually exclusive of other embodiments. It is explicitly and implicitly understood by one skilled in the art that the embodiments described herein may be combined with other embodiments.
The terms "first," "second," "third," and "fourth," etc. in the description and claims of this application and in the accompanying drawings are used for distinguishing between different objects and not for describing a particular order. Furthermore, the terms "include" and "have," as well as any variations thereof, are intended to cover non-exclusive inclusions. For example, the inclusion of a list of steps, processes, methods, etc. is not limited to only those steps recited, but may alternatively include additional steps not recited, or may alternatively include additional steps inherent to such processes, methods, articles, or devices.
The invention aims to provide a self-adaptive tracking control method and a self-adaptive tracking control system for the dynamics of a yaw channel of an unmanned helicopter, which construct an analytic self-adaptive control law, realize self-adaptive output tracking performance and ensure that a closed-loop signal of a controlled system is bounded and practical to output and track.
In order to make the aforementioned objects, features and advantages of the present invention comprehensible, embodiments accompanied with figures are described in further detail below.
Fig. 1 is a flowchart of a method in an embodiment provided by the present invention, and as shown in fig. 1, the present invention provides an adaptive tracking control method for the dynamics of a yaw channel of an unmanned helicopter, including:
step 100: determining a parameterized model of the discrete non-affine nonlinear system according to a dynamic equation of a target;
step 200: updating a parameter of the parameterized model, and estimating unknown parameters according to the parameter updating law to obtain estimated parameters;
step 300: constructing an estimate of the signal at a future time based on the parameter update law to obtain an estimated signal;
step 400: determining an adaptive implicit function equation according to the estimation parameters and the estimation signals;
step 500: designing an adaptive control law with an iterative form based on the adaptive implicit function equation; the solution of the adaptive control law with iterative form is the control signal of the discrete non-affine nonlinear system.
In particular, the present invention contemplates a non-linear uncertainty system of the input-output type, where y (t) θ *T f (y (T-T), y (T-2T), …, y (T-nT), u (T-dT), u (T-dT-T), …, u (T-mT)), m is larger than or equal to d, and for the sake of simplicity, the adoption is omittedThe sample period T can be:
y(t)＝θ *T f(y(t-1)，y(t-2)，…，y(t-n)，u(t-d)，u(t-d-1)，…，u(t-m))，m≥d …(1)。
wherein the content of the first and second substances,
Figure GDA0003764216800000061
and
Figure GDA0003764216800000062
respectively the output signal and the input signal of the system,
Figure GDA0003764216800000063
Figure GDA0003764216800000064
is a vector of unknown constant values and is,
Figure GDA0003764216800000065
is known Lipschitz nonlinear micro-mappable, d is system input-output time delay and satisfies that d is more than or equal to 1 and less than or equal to n. The present invention assumes that the output y (t) of the system (1) is measurable.
Further, the control targets of the present invention are: for any given reference model y * (t)＝z -d r (t) wherein
Figure GDA0003764216800000066
Is a bounded reference input signal, an output feedback self-adaptive control law u (t) is designed, all closed-loop signals of the system (1) are bounded, and asymptotic output tracking lim (y (t) -y is realized * (t)) -0 or utility output tracking, i.e., y (t)) -y * (t) any set of small residues that converge to the origin.
The adaptive design process proposed by the present invention comprises 5 steps: firstly, designing a parameter updating law and estimating an unknown parameter theta * (ii) a Secondly, based on theta * (ii) an estimate of (d); constructing an estimate of the signal y (t + j) at the end time; wherein j is 1, 2.. d; thirdly, based on the estimation parameters and the estimation signals obtained in the first two steps, an equation theta is derived *T f(y(t-1+d)，y(t-2+d)，…，y(t-n+d)，u(t)，u(t-1)，…，u(t-m+d))＝y * (t + d) … · (2) an adaptive version; replacing the unknown signal and the unknown parameter in (2) with their estimates; obtaining an adaptive implicit function equation related to u (t), and proving that a unique solution exists in the equation related to u (t); fourthly, signal bounded and output tracking performance analysis of the self-adaptive closed-loop system is completed; fifthly, aiming at the situation that the adaptive implicit function equation is difficult to solve; giving an analysis self-adaptive control law in an iterative form; and the control law is proved to realize the bounded closed-loop signal and practical output tracking performance of the controlled system.
Preferably, the designing a parameter updating law for the parameterized model and estimating unknown parameters according to the parameter updating law to obtain estimated parameters includes:
order to
Figure GDA0003764216800000071
Denotes θ * Is estimated, and
Figure GDA0003764216800000072
Figure GDA0003764216800000073
wherein, theta * In order for the parameters to be unknown,
Figure GDA0003764216800000074
and
Figure GDA0003764216800000075
respectively an output signal and an input signal of the discrete non-affine nonlinear system;
Figure GDA0003764216800000076
is a set Lipschitz nonlinear micro-mappable; d is system input-output time delay and satisfies d is more than or equal to 1 and less than or equal to n;
defining an error e (t) as θ T (t)φ f (t-1)-y(t)；
Designing the parameter updating law according to the errors
Figure GDA0003764216800000077
Figure GDA0003764216800000078
Wherein the content of the first and second substances,
Figure GDA0003764216800000079
Γ＝diag{γ 1 ，...，γ p is the gain matrix; g (θ (t), h (t)) [ g [ 1 (θ 1 (t)，h 1 (t))，...，g p (θ p (t)，h p (t))] T Is a correction term based on a parametric projection technique; gamma ray i ∈(0，2)；
There are known groups of bounded intervals
Figure GDA00037642168000000710
Satisfy the requirement of
Figure GDA00037642168000000711
And is
Figure GDA00037642168000000712
Wherein
Figure GDA00037642168000000713
α 0 Is an unknown normal number, X n+1 Is the n +1 variable of f;
design g i (θ i (t)，h i (t)) is of the form:
Figure GDA00037642168000000714
wherein h is i (t) is the ith component of h (t), and h (t) is of the form
Figure GDA00037642168000000715
Determining based on the parameter update law
Figure GDA00037642168000000716
θ(t+i 0 )-θ(t)∈L 2 Wherein i 0 Is any positive integer.
Specifically, make
Figure GDA00037642168000000717
Denotes theta * Is estimated, and
Figure GDA00037642168000000718
Figure GDA00037642168000000719
Figure GDA0003764216800000081
then the error is defined as:
∈(t)＝θ T (t)φ f (t-1)-y(t)…·(4)
by using phi f (t-1) and estimation error, we design the update law of θ (t) as:
Figure GDA0003764216800000082
wherein the content of the first and second substances,
Figure GDA0003764216800000083
Γ＝diag{γ 1 ，...，γ p is the gain matrix, g (correction term for the theta parameter projection technique, gamma) i E (0, 2). There are known groups of bounded intervals
Figure GDA0003764216800000084
Satisfy the requirement of
Figure GDA0003764216800000085
And is
Figure GDA0003764216800000086
Figure GDA0003764216800000087
Wherein
Figure GDA0003764216800000088
α 0 Is an unknown normal number, X n+1 Is the n +1 th variable of f.
Design g based on the above formula i (θ i (t)，h i (t)) is of the form:
Figure GDA0003764216800000089
wherein h is i (t) is the ith component of h (t) * And h (t) is of the form:
Figure GDA00037642168000000810
based on the parameter update law (5), the following reasoning can be derived:
introduction 1: the parameter update law (5) guarantees the following properties,
Figure GDA00037642168000000811
θ(t+i 0 )-θ(t)∈L 2 wherein i 0 Is any positive integer.
Preferably, the constructing an estimate of the signal at a future time based on the parameter update law to obtain an estimated signal comprises:
updating law design based on the kinetic equation and the parameters
Figure GDA00037642168000000812
The method specifically comprises the following steps:
Figure GDA00037642168000000813
Figure GDA00037642168000000814
wherein the content of the first and second substances,
Figure GDA00037642168000000815
for the estimation of the output signal y (t +1),
Figure GDA00037642168000000816
is an estimate of the output signal y (t + j);
Figure GDA00037642168000000817
is phi f (ii) an estimate of (d); wherein the content of the first and second substances,
Figure GDA0003764216800000091
based on obtained
Figure GDA0003764216800000092
As an estimation signal; the estimated signal is known at the current time and is satisfied
Figure GDA0003764216800000093
Wherein μ (t) ∈ L 2 And μ (t) is the general attenuated signal boundary.
Specifically, the second step of the present embodiment is the estimation of the future time signal.
The signal of the last moment is estimated. The estimation signals are designed according to the sequence of y (t +1), y (t +2), … and y (t + d-1). The reason for this order of estimation is that the design of the latter will depend on the former, as detailed below.
The invention uses symbols
Figure GDA0003764216800000094
An estimate of any unknown signal X is indicated. Based on (1), (3) and (5), the design,
Figure GDA0003764216800000095
comprises the following steps:
Figure GDA0003764216800000096
Figure GDA0003764216800000097
wherein the content of the first and second substances,
Figure GDA0003764216800000098
Figure GDA0003764216800000099
the following reasoning is given to reveal one basic property of the estimated signal.
Based on the estimated signals obtained in (9) and (10),
Figure GDA00037642168000000910
is known at the present moment and satisfies
Figure GDA00037642168000000911
Where μ (t) ∈ L 2 Is a general attenuated signal boundary.
Preferably, the determining an adaptive implicit function equation according to the estimation parameter and the estimation signal includes:
constructing an adaptive implicit function equation consistent with the structure of the kinetic equation based on the estimation of the unknown parameters and the estimation of the future output signals; the adaptive implicit function equation is
Figure GDA00037642168000000912
Figure GDA00037642168000000913
Wherein, y * (t + d) ═ r (t), r (t) is a reference input signal known at the current time; at each sampling instant t, the adaptive implicit function equation has a unique solution with respect to u (t), which is noted
Figure GDA00037642168000000914
And satisfy
Figure GDA00037642168000000915
Figure GDA00037642168000000916
Wherein, X n+1 Is phi f (t + j) th variable.
Specifically, the third step in this embodiment is to construct an adaptive control law implicit function equation.
Based on the estimation of the unknown parameters and the estimation of the future output signal, we construct an adaptive implicit function equation that is structurally consistent with the nominal implicit function equation (2):
Figure GDA0003764216800000101
Figure GDA0003764216800000102
notice that y * (t + d) ═ r (t), and r (t) is the reference input signal known at the current time, so that all parameters and signals in equation (12) are known at the current time. To simplify the notation, define:
Figure GDA0003764216800000103
Figure GDA0003764216800000104
Figure GDA0003764216800000105
one key conclusion about equation (12) is derived below:
leading: 3 at each sampling instant t, equation (12) has a unique solution for u (t), noted
Figure GDA0003764216800000106
And satisfies the following conditions:
Figure GDA0003764216800000107
wherein X n+1 Is phi f N +1 th variable of (t + j).
Further, the fourth step in this embodiment is a closed-loop system performance analysis, and based on theorem 2 and theorem 3, we give one of the main results of the present invention:
theorem 1: the system (1) takes the solution of equation (12) as an adaptive control law, and takes (5) as a parameter updating law, so that all signals of the closed-loop system are bounded and asymptotic output tracking lim can be realized t→∞ (y(t)-y * (t))＝0。
Preferably, the designing an adaptive control law with an iterative form based on the adaptive implicit function equation includes:
designing an adaptive control law of an iterative form into
Figure GDA0003764216800000108
Where u0(t) is taken as the control signal value at time t-1, and γ (t) is a given time-varying parameter dependent on time t and satisfies
Figure GDA0003764216800000109
Figure GDA00037642168000001010
Figure GDA00037642168000001011
Specifically, in this embodiment, the fifth step is adaptive control law design in an iterative form, and for the situation where the implicit function equation (12) of the adaptive control law is difficult to solve, an analytic adaptive control law in the iterative form will be given below.
The analytic structure of the iterative adaptive control law is based on an implicit function equation (12) of the adaptive control law, and the adaptive control law in an iterative form is designed at each sampling time t as follows:
Figure GDA0003764216800000111
wherein u is 0 (t) is taken as the control signal value at time t-1, and gamma (t) is a given time-varying parameter dependent on time t and satisfies
Figure GDA0003764216800000112
Figure GDA0003764216800000113
About signal u i (t), the following theorem is given.
Theorem 2: for all t-1, { u i (t) } convergence on i, i.e.
Figure GDA0003764216800000114
Wherein
Figure GDA0003764216800000115
Is an implicit function of
Figure GDA0003764216800000116
A unique solution for u.
Theorem 3: the control law of the closed loop system (1) is u p And (t) the parameter updating law is (5). If u is 1 (t)＝u 0 (t) taking u alone p (t)＝u 0 (t); if u is 1 (t)≠u 0 (t) as long as the number of iterations p satisfies:
Figure GDA0003764216800000117
it can be guaranteed that all closed-loop signals are bounded and the tracking error is satisfied
Figure GDA0003764216800000118
Where e is the given tolerance error and δ is the signal that decays to zero asymptotically depending on the initial value of the system.
Theorem 3: provides a specific method for constructing an adaptive control law, elucidating the input signal u applied at each moment p And (t), all closed-loop signals are guaranteed to be bounded and practical output tracking can be realized only if the iteration number p is satisfied (14).
The invention has established an output feedback self-adaptive output tracking control framework based on implicit functions aiming at a non-affine nonlinear uncertain system (1), and mainly comprises two specific methods: the adaptive control method based on implicit function equation solution can realize asymptotic tracking performance; and secondly, an analytic self-adaptive control method based on iteration can realize practical output tracking performance. The two methods each have advantages: the former can realize the convergence of the tracking error to zero; the latter does not need to solve an implicit function equation, and is convenient for practical application.
Example 2:
the unmanned helicopter yaw channel dynamics has the characteristics of high relative order, high uncertainty and the like. With the increasing performance requirements of modern aircrafts for high maneuverability, it is necessary to improve the yaw control performance of the unmanned helicopter. The applicability of the proposed method in a practical system is shown below by taking an unmanned helicopter yaw channel dynamic model as an example. (unmanned helicopter in hover and low speed flight conditions, the moment is mainly derived from the main rotor and tail rotor; simplified fuselage and vertical tail damping, the unmanned helicopter yaw path dynamics equation can be described as follows) consider the following unmanned helicopter yaw path dynamics equation:
Figure GDA0003764216800000121
wherein
Figure GDA0003764216800000122
And r is the yaw angle and yaw rate of the helicopter, I zz Is the inertia of the helicopter about an axis, Q mr 、T tr And l tr The main rotor moment, the thrust of the tail rotor and the distance from the tail rotor to the shaft of the helicopter, b 1 And b 2 Is the damping constant.
Using blade unit method, main rotor moment Q mr Comprises the following steps:
Figure GDA0003764216800000123
wherein phi is v 1 /(Ωr),C l ＝aα,C d ≈C d0 +C d1 α+C d2 α 2 。
Where ρ, a, r, α, c, v 1 Phi and omega are the air density, lift curve slope, speed radial distance, blade element angle of attack, blade chord, induced speed, inflow angle and main rotor speed, respectively.
According to the prior art results, a main rotor torque Q is obtained mr Comprises the following steps:
Figure GDA0003764216800000131
Figure GDA0003764216800000132
Figure GDA0003764216800000133
wherein Q mr R and b are the pitch angle of the main rotor, the rotor radius and the number, respectively.
Similar to (16) and (17), the thrust of the tail rotor can be obtained as follows:
Figure GDA0003764216800000134
Figure GDA0003764216800000135
Figure GDA0003764216800000136
Figure GDA0003764216800000137
wherein a is tr ,c tr ,b tr ,Ω tr ,θ tr ,r tr And A tr The slope of the lift curve of the tail rotor, the chord of the blade, the number of rotors, the rotating speed of the tail rotor, the pitch angle, the radial distance and the area of a tail rotor disc are respectively shown.
By plotting the moment omega of the main rotor separately mr To the pitch angle theta mr And force T of tail rotor tr To the pitch angle theta tr Can find omega mr And theta mr And T tr And theta tr The relationship between can be approximated by a quadratic polynomial:
Figure GDA0003764216800000138
Figure GDA0003764216800000139
wherein
Figure GDA00037642168000001310
And
Figure GDA00037642168000001311
depending on the blade shape of the main rotor and the rotational speed omega.
Figure GDA00037642168000001312
And
Figure GDA00037642168000001313
blade shape and rotational speed omega dependent on the tail rotor tr 。
Substituting equations (19,20) for equation (15) yields the following system model:
Figure GDA0003764216800000141
generally, consider θ tr ＝θ mr For the pitch angle of the helicopter, the system (21) can be rewritten as:
Figure GDA0003764216800000142
wherein k is i I is 1, …,4 is a coefficient
Figure GDA0003764216800000147
Figure GDA0003764216800000148
θ tr Is the pitch angle of the helicopter, represents the control input to the system,
Figure GDA0003764216800000149
is the yaw angle of the helicopter, representing the output signal of the system. Fully considering the saturated physical property of the pitch angle of the helicopter, the pitch angle with saturation constraint is modeled as follows:
Figure GDA0003764216800000143
wherein theta is trmax >0 is the maximum helicopter pitch yaw. By using h (theta) trs ) De-approximating a saturated input θ tr ，
Figure GDA0003764216800000144
Can easily obtain:
θ tr ＝sat(θ trs )＝d(θ trs )+h(θ trs ) (23)
|d(θ trs )|＝|sat(θ trs )-h(θ trs )|≤θ trmax (1-tanh(1))＝D (24)
due to the fact that d (theta) trs ) Far greater than h (theta) trs ) So the error can be ignored and the unmanned helicopter yaw channel dynamics system is expressed as:
Figure GDA0003764216800000145
and setting the sampling step length as h, and obtaining a first-order Euler discretization model of the unmanned helicopter yawing channel dynamics as follows:
Figure GDA0003764216800000146
Figure GDA0003764216800000151
thus, it is easy to obtain:
Figure GDA0003764216800000152
order:
θ * ＝[1,h 2 k 4 ,h+h 2 k 1 ,h 2 k 2 ,h 2 k 3 ] T (26)
Figure GDA0003764216800000153
the parameterized model from which the system model (25) is derived is then:
Figure GDA0003764216800000154
as can be seen by the form (27) of f, f is non-linear and, in particular, the control input θ trs (t) is non-affine. When the system parameter theta * In the presence of large uncertainties, the applicability of the proposed adaptive control scheme to unmanned helicopter yaw path dynamics discrete systems (28) is demonstrated below.
The control targets for this example are: for any given reference model
Figure GDA0003764216800000155
Wherein
Figure GDA0003764216800000156
Is a bounded reference input signal, and designs an output feedback self-adaptive control law theta trs (t) all closed loop signals of the system (15) are bounded and asymptotic output tracking is achieved
Figure GDA0003764216800000157
Or utility output tracking
Figure GDA0003764216800000158
Any set of small residues that converge to the origin.
Step 1: design of parameter updating law
Order to
Figure GDA0003764216800000159
Denotes theta * Is estimated, and
Figure GDA00037642168000001510
Figure GDA00037642168000001511
defining the estimation error as
Figure GDA00037642168000001512
Using phi f (t-1) and estimating errors, and designing an updating law of theta (t) as follows:
Figure GDA0003764216800000161
wherein the content of the first and second substances,
Figure GDA0003764216800000162
Γ＝diag{γ 1 ，...，γ 4 is the gain matrix, g (θ (t), h (t)) [ g (t) } 1 (θ 1 (t)，h 1 (t))，...，g 4 (θ 4 (t)，h 4 (t))] T Is based on ginsengCorrection term, gamma, for digital projection techniques i E (0, 2). Design g i (θ i (t)，h i (t)), i ═ 1, ·,4, of the form:
Figure GDA0003764216800000163
wherein
Figure GDA0003764216800000164
Given in hypothesis 1, h i (t) is the ith component of h (t), and h (t) is of the form:
Figure GDA0003764216800000165
step 2: estimation of signals at future time instants
When the relative order d is greater than 1, in order to design a causality-compliant adaptive control law, d is greater than 1 output signals at the last moment, and the details are as follows.
Based on (15), (29), (31), the design
Figure GDA0003764216800000166
Is composed of
Figure GDA0003764216800000167
Wherein the content of the first and second substances,
Figure GDA0003764216800000168
and step 3: construction of adaptive control law implicit function equation
Constructing and normalizing implicit function equations based on the estimate of the unknown parameters and the estimate of the future output signal
Figure GDA0003764216800000169
Structurally consistent adaptive implicit function equations:
Figure GDA00037642168000001610
note that
Figure GDA00037642168000001611
And r (t) is the reference input signal for which the current time instant is known. Thus all parameters and signals in equation (37) are known at the current time and are defined
Figure GDA0003764216800000171
Figure GDA0003764216800000172
According to the global implicit theorem, to ensure that (39) is about θ trs (t) well-defined implicit function equations, for
Figure GDA0003764216800000173
Reference output signal
Figure GDA0003764216800000174
Need to be in
Figure GDA0003764216800000175
Within the range of (1).
And 4, step 4: closed loop system performance analysis
The system (15) solves for theta in equation (37) trs And (t) is an adaptive control law, and by taking (31) as a parameter updating law, the closed-loop system has bounded signals and can realize asymptotic output tracking:
Figure GDA0003764216800000176
Figure GDA0003764216800000177
based on an implicit function equation (37) of the adaptive control law, at each sampling time t, the adaptive control law in an iterative form is designed as follows:
Figure GDA0003764216800000178
wherein
Figure GDA0003764216800000179
Taking the control signal value at the time of t-1, the selection of gamma (t) meets the requirement,
Figure GDA00037642168000001710
Figure GDA00037642168000001711
wherein
Figure GDA00037642168000001712
At omega t Above with respect to theta trs And (4) continuous.
About the signal theta trs (t), for all t ≧ 1, { θ trs (t) converges with respect to i, i.e.
Figure GDA00037642168000001713
Figure GDA00037642168000001714
Wherein
Figure GDA00037642168000001715
Is an implicit function of equation
Figure GDA00037642168000001716
About theta trs Is determined.
{θ trs Convergence value of (t) } of
Figure GDA00037642168000001717
Is a limit value that is difficult to obtain in practice. To solve this problem, p iterations are obtained at each sampling instant based on the iteration formula (40)
Figure GDA00037642168000001718
As control signals and expected to achieve practical output tracking performance, i.e.
Figure GDA00037642168000001719
Where e is a given tolerance and δ is a signal that decays asymptotically to zero depending on the initial value of the system, for which the following results are given.
The control law of the closed loop system (15) is
Figure GDA00037642168000001720
The parameter update law is (31) if
Figure GDA00037642168000001721
Only need to take
Figure GDA00037642168000001722
If it is not
Figure GDA00037642168000001723
As long as the number of iterations p satisfies:
Figure GDA0003764216800000181
wherein:
Figure GDA0003764216800000182
it can be guaranteed that all closed loop signals are bounded and the tracking error is met (42).
Example 3:
consider the following system model:
y(t)＝θ *T f(y(t-1),y(t-2),y(t-3),u(t-3)), (45)
wherein
Figure GDA0003764216800000183
Is the output of the system and is,
Figure GDA0003764216800000184
is the input of the system and is,
Figure GDA0003764216800000185
wherein:
Figure GDA0003764216800000186
Figure GDA0003764216800000187
this means that the system order n is 3 and the delay d is 3. Is provided with
Figure GDA0003764216800000188
Denotes theta * The fourth element in (1), then:
Figure GDA0003764216800000189
based on (46) and (48), it can be verified that the input-output delay of the simulation example (45) is 3 and depends on the unknown parameter
Figure GDA00037642168000001810
Based on this, for the closed interval group in (7)
Figure GDA00037642168000001811
Only need to ensure
Figure GDA00037642168000001812
In the interval (0, 1.6)]The above selection ensures that the inequality (7) is always true. In this example, we choose
Figure GDA00037642168000001813
Selecting a parameter updating law (5) based on a closed interval group; the specific analytic form of the projection correction term g (θ (t), h (t)) can be determined, and the simulation example (31) is verified to be a minimum phase system.
Designing a self-adaptive control law: from (46) and (47), the simulation example (45) is non-affine and it is difficult to find an analytical solution of the control law, so that the reference output signal is determined to be y according to theorem 3 * (t) ═ 0.5sin (t) +1, this embodiment uses an iterative adaptive control law of the form:
Figure GDA0003764216800000191
wherein
Figure GDA0003764216800000192
，y * (t+3)＝0.5sin(t+3)+1
Based on (9) and (10), it is possible to obtain
Figure GDA0003764216800000193
And
Figure GDA0003764216800000194
the analytic form of (2) can be used to obtain the analytic form of the update law of theta (t) based on (5); and will be omitted here.
According to theorem 3, the present embodiment uses an iterative solution u p (t) as an adaptation law. The allowable error e in equation (14) is selected to be 0.001 and the reference output signal is selected to be y * 0.5sin (t) + 1. Fig. 1 shows the trajectories of the output signal and the reference output signal, and fig. 2 plots the trajectories of the tracking error signals.
As can be shown from fig. 1 and fig. 2, the system output can track the reference output signal well, fig. 3 shows the trajectory of the system input signal, fig. 4 shows the adaptive response of the estimated parameter, and the above simulation results confirm the effectiveness of the adaptive control algorithm proposed in this embodiment.
Example 4:
the embodiment also provides an adaptive tracking control system for the dynamics of the yaw channel of the unmanned helicopter, which comprises:
the model construction unit is used for determining a parameterized model of the discrete non-affine nonlinear system according to a dynamic equation of the target;
the parameter estimation unit is used for updating the parameter of the parameterized model and estimating unknown parameters according to the parameter updating law to obtain estimated parameters;
a signal estimation unit, configured to construct an estimation of a signal at a future time based on the parameter update law to obtain an estimation signal;
an equation determining unit, configured to determine an adaptive implicit function equation according to the estimation parameter and the estimation signal;
the control law construction unit is used for designing the self-adaptive control law with an iteration form based on the self-adaptive implicit function equation; the solution of the adaptive control law with iterative form is the control signal of the discrete non-affine nonlinear system.
In the present specification, the embodiments are described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same and similar parts among the embodiments are referred to each other. For the system disclosed by the embodiment, the description is relatively simple because the system corresponds to the method disclosed by the embodiment, and the relevant points can be referred to the method part for description.
The principles and embodiments of the present invention have been described herein using specific examples, which are provided only to help understand the method and the core concept of the present invention; meanwhile, for a person skilled in the art, according to the idea of the present invention, the specific embodiments and the application range may be changed. In view of the above, the present disclosure should not be construed as limiting the invention.

Claims (5)

1. An adaptive tracking control method for the dynamics of a yaw channel of an unmanned helicopter is characterized by comprising the following steps:
determining a parameterized model of the discrete non-affine nonlinear system according to a dynamic equation of a target;
the method for determining the parameterized model of the discrete non-affine nonlinear system according to the kinetic equation of the target comprises the following steps:
the formula is y (t) ═ theta *T f (y (T-T), y (T-2T), …, y (T-nT), u (T-dT), u (T-dT-T), …, u (T-mT)), m is larger than or equal to d, and the sampling period T is omitted for simplicity and convenience;
y(t)＝θ *T f(y(t-1),y(t-2),…,y(t-n),u(t-d),u(t-d-1),…,u(t-m)),m≥d ····(1)；
wherein the content of the first and second substances,
Figure FDA0003764216790000011
and
Figure FDA0003764216790000012
respectively the output signal and the input signal of the system,
Figure FDA0003764216790000013
is a vector of unknown constant values and is,
Figure FDA0003764216790000014
the method is known Lipschitz nonlinear micro-mapping, wherein d is system input-output time delay and satisfies that d is more than or equal to 1 and less than or equal to n; assuming that the output y (t) of system (1) is measurable;
updating a parameter of the parameterized model, and estimating unknown parameters according to the parameter updating law to obtain estimated parameters;
the designing a parameter updating law for the parameterized model and estimating unknown parameters according to the parameter updating law to obtain estimated parameters comprises the following steps:
order to
Figure FDA0003764216790000015
Denotes theta * Is estimated, and
Figure FDA0003764216790000016
Figure FDA0003764216790000017
Figure FDA0003764216790000018
wherein, theta * In order for the parameters to be unknown,
Figure FDA0003764216790000019
and
Figure FDA00037642167900000110
respectively an output signal and an input signal of the discrete non-affine nonlinear system;
Figure FDA00037642167900000111
is a set Lipschitz nonlinear micro-mappable; d is system input-output time delay and satisfies d is more than or equal to 1 and less than or equal to n;
defining an error e (t) as θ T (t)φ f (t-1)-y(t)；
Designing the parameter updating law according to the errors
Figure FDA0003764216790000021
Figure FDA0003764216790000022
Wherein the content of the first and second substances,
Figure FDA0003764216790000023
Γ＝diag{γ 1 ,…,γ p is the gain matrix; g (θ (t), h (t)) [ g [ 1 (θ 1 (t),h 1 (t)),…,g p (θ p (t),h p (t))] T Is a correction term based on a parametric projection technique; gamma ray i ∈(0,2)；
There are known groups of bounded intervals
Figure FDA0003764216790000024
Satisfy the requirement of
Figure FDA0003764216790000025
And is provided with
Figure FDA0003764216790000026
Wherein
Figure FDA0003764216790000027
Figure FDA0003764216790000028
α 0 Is an unknown normal number, X n+1 Is the n +1 variable of f;
design g i (θ i (t),h i (t)) is of the form:
Figure FDA0003764216790000029
wherein h is i (t) is the ith component of h (t), and h (t) is of the form
Figure FDA00037642167900000210
Determining based on the parameter update law
Figure FDA00037642167900000211
θ(t+i 0 )-θ(t)∈L 2 Wherein i 0 Is any positive integer;
constructing an estimate of the signal at a future time based on the parameter update law to obtain an estimated signal;
determining an adaptive implicit function equation according to the estimation parameters and the estimation signals;
designing an adaptive control law with an iterative form based on the adaptive implicit function equation; the solution of the adaptive control law with iterative form is the control signal of the discrete non-affine nonlinear system.
2. The method of claim 1, wherein said constructing an estimate of a signal at a future time based on said parameter update law to obtain an estimated signal comprises:
updating the law design based on the kinetic equation and the parameters
Figure FDA00037642167900000212
The method specifically comprises the following steps:
Figure FDA0003764216790000031
wherein the content of the first and second substances,
Figure FDA0003764216790000032
for the estimation of the output signal y (t +1),
Figure FDA0003764216790000033
is an estimate of the output signal y (t + j);
Figure FDA0003764216790000034
is phi f (ii) an estimate of (d); wherein the content of the first and second substances,
Figure FDA0003764216790000035
Figure FDA0003764216790000036
based on obtained
Figure FDA0003764216790000037
As an estimation signal; the estimated signal is known at the current time and is satisfied
Figure FDA0003764216790000038
Figure FDA0003764216790000039
Wherein μ (t) ∈ L 2 Mu (t) is a general attenuation signalThe number bound.
3. The method of claim 2, wherein said determining an adaptive implicit function equation from the estimated parameters and the estimated signals comprises:
constructing an adaptive implicit function equation consistent with the structure of the kinetic equation based on the estimation of the unknown parameters and the estimation of the future output signals; the adaptive implicit function equation is
Figure FDA00037642167900000310
Figure FDA00037642167900000311
Wherein, y * (t + d) ═ r (t), r (t) is a reference input signal known at the current time; at each sampling time t, the adaptive implicit function equation has a unique solution to u (t), which is recorded as
Figure FDA00037642167900000312
And satisfy
Figure FDA00037642167900000313
Figure FDA00037642167900000314
Wherein, X n+1 Is phi f (t + j) th variable.
4. The method of claim 3, wherein said designing an adaptive control law with an iterative form based on said adaptive implicit function equation comprises:
designing an adaptive control law of an iterative form into
Figure FDA00037642167900000315
Wherein u is 0 (t) is taken as the control signal value at time t-1, and gamma (t) is a given time-varying parameter dependent on time t and satisfies
Figure FDA00037642167900000316
Figure FDA0003764216790000041
Figure FDA0003764216790000042
5. An adaptive tracking control system for unmanned helicopter yaw channel dynamics, comprising:
the model construction unit is used for determining a parameterized model of the discrete non-affine nonlinear system according to a dynamic equation of the target;
the model construction unit is used for determining a parameterized model of the discrete non-affine nonlinear system according to a dynamic equation of a target, and comprises the following steps:
the formula is y (t) ═ theta *T f (y (T-T), y (T-2T), …, y (T-nT), u (T-dT), u (T-dT-T), …, u (T-mT)), m is larger than or equal to d, and the sampling period T is omitted for simplicity and convenience;
y(t)＝θ *T f(y(t-1),y(t-2),…,y(t-n),u(t-d),u(t-d-1),…,u(t-m)),m≥d ····(1)；
wherein the content of the first and second substances,
Figure FDA0003764216790000043
and
Figure FDA0003764216790000044
respectively the output signal and the input signal of the system,
Figure FDA0003764216790000045
is a vector of unknown constant values and is,
Figure FDA0003764216790000046
the method is known Lipschitz nonlinear micro-mapping, wherein d is system input-output time delay and satisfies that d is more than or equal to 1 and less than or equal to n; assuming that the output y (t) of system (1) is measurable;
the parameter estimation unit is used for updating the parameter of the parameterized model and estimating unknown parameters according to the parameter updating law to obtain estimated parameters;
the designing a parameter updating law for the parameterized model and estimating unknown parameters according to the parameter updating law to obtain estimated parameters comprises the following steps:
order to
Figure FDA0003764216790000047
Denotes θ * Is estimated, and
Figure FDA0003764216790000048
Figure FDA0003764216790000049
Figure FDA00037642167900000410
wherein, theta * In order for the parameters to be unknown,
Figure FDA00037642167900000411
and
Figure FDA00037642167900000412
respectively an output signal and an input signal of the discrete non-affine nonlinear system;
Figure FDA00037642167900000413
is a set Lipschitz nonlinear micro-mappable; d is the system input-output time delay and satisfies that d is more than or equal to 1 and less than or equal to n;
defining an error e (t) as θ T (t)φ f (t-1)-y(t)；
Designing the parameter updating law according to the errors
Figure FDA0003764216790000051
Figure FDA0003764216790000052
Wherein the content of the first and second substances,
Figure FDA0003764216790000053
Γ＝diag{γ 1 ,…,γ p is the gain matrix; g (θ (t), h (t)) [ g [ [ g ] 1 (θ 1 (t),h 1 (t)),…,g p (θ p (t),h p (t))] T Is a correction term based on a parametric projection technique; gamma ray i ∈(0,2)；
There are known groups of bounded intervals
Figure FDA0003764216790000054
Satisfy the requirement of
Figure FDA0003764216790000055
And is provided with
Figure FDA0003764216790000056
Wherein
Figure FDA0003764216790000057
Figure FDA0003764216790000058
α 0 Is an unknown normal number, X n+1 Is the n +1 variable of f;
design g i (θ i (t),h i (t)) is of the form:
Figure FDA0003764216790000059
wherein h is i (t) is the ith component of h (t), and the form of h (t)Is of the formula
Figure FDA00037642167900000510
Determining based on the parameter update law
Figure FDA00037642167900000511
θ(t+i 0 )-θ(t)∈L 2 Wherein i 0 Is any positive integer;
a signal estimation unit, configured to construct an estimation of a signal at a future time based on the parameter update law to obtain an estimation signal;
an equation determining unit, configured to determine an adaptive implicit function equation according to the estimation parameter and the estimation signal;
the control law construction unit is used for designing the self-adaptive control law with an iteration form based on the self-adaptive implicit function equation; the solution of the adaptive control law with iterative form is the control signal of the discrete non-affine nonlinear system.
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