CN110007602B - Low-complexity self-adaptive saturation control method for nonlinear system - Google Patents

Low-complexity self-adaptive saturation control method for nonlinear system Download PDF

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CN110007602B
CN110007602B CN201910374143.1A CN201910374143A CN110007602B CN 110007602 B CN110007602 B CN 110007602B CN 201910374143 A CN201910374143 A CN 201910374143A CN 110007602 B CN110007602 B CN 110007602B
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adaptive
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saturation
complexity
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CN110007602A (en
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张刚
刘志坚
李德路
侯文宝
沈永跃
吴玮
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Dragon Totem Technology Hefei Co ltd
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Jiangsu Institute of Architectural Technology
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Abstract

The invention discloses a low-complexity self-adaptive saturation control method of a nonlinear system, which comprises the following steps: firstly, converting control saturation nonlinearity into a linear model relative to an actual input signal by adopting a dead zone model; then, converting an original system with time-varying output constraint into unconstrained output constraint, and designing a new self-adaptive saturation control law along the filtered error manifold based on the system; by adopting the minimum learning parameter technology and the virtual error concept, only two self-adaptive parameters need to be updated on line, and the calculation burden is greatly reduced. Compared with the existing work, the calculation amount of the self-adaptive scheme is small, and the asymptotic stability of the tracking error system can be ensured, but not the consistent final bounded stability. The simulations used show that an improvement of about 4 times can be obtained in terms of tracking accuracy, which demonstrates the effectiveness of the proposed control law.

Description

Low-complexity self-adaptive saturation control method for nonlinear system
Technical Field
The invention relates to a control algorithm, in particular to a low-complexity self-adaptive saturation control method for a nonlinear system.
Background
In the past decades, there has been an increasing interest in research on adaptive control system methods for nonlinear systems, such as Duffing-Holmes chaotic systems and reluctance motor systems, and significant research efforts have been made, such as adaptive Neural Network (NN) -based control, perturbed strict feedback nonlinear systems, and Barrier Lyapunov Function (BLF) -based output control, all for strict feedback systems with unknown nonlinearity.
In existing adaptive control schemes, the back-stepping technique has been developed as an effective tool to handle nonlinear cascade systems (also called strict or pure feedback systems). For example, an adaptive neural control protocol based on input-to-state stability (ISS) was developed for unknown pure feedback systems using backstepping techniques, where the unknown smoothing function is approximated with a radial basis function nn (rbfnn). Similar to NN, due to its wide approximation capacity, fuzzy logic systems have been applied to approximate unknown non-linearities. An effective self-adaptive fuzzy control scheme is developed for a strict feedback system by utilizing a fuzzy logic system and an inversion technology. However, the complex and cumbersome computation of the virtual controllers involved in the recursive design step easily causes the problem of "complexity explosion" of the backstepping technique, which makes it very challenging to guarantee the utility of the developed control law, and to alleviate this computational complexity, a Dynamic Surface Control (DSC) method is proposed by introducing a first-order low-pass filter in each recursive step in the conventional backstepping design process. The DSC method was further explored in the nonlinear rigorous feedback system and the interconnected nonlinear pure feedback system of the previous studies, due to its outstanding advantages in solving the "complex explosion" problem. While the computational complexity can be greatly reduced, updating NN or fuzzy logic system elements is still computationally expensive, especially for higher order nonlinear systems. To overcome this problem, the Minimum Learning Parameter (MLP) technique with fewer adaptation parameters for a strict feedback system provides an alternative approach to reduce the complexity of the adaptation scheme. Therefore, how to reduce the computational complexity of the adaptive control law is worth further research.
Disclosure of Invention
In light of the deficiencies of the prior art, the present invention provides a low complexity adaptive saturation control method for a nonlinear system.
The invention is realized according to the following technical scheme:
a low-complexity self-adaptive saturation control method for a nonlinear system comprises the following steps: firstly, converting control saturation nonlinearity into a linear model relative to an actual input signal by adopting a dead zone model; then, converting an original system with time-varying output constraint into unconstrained output constraint, and designing a new self-adaptive saturation control law along the filtered error manifold based on the system; by adopting the minimum learning parameter technology and the virtual error concept, only two self-adaptive parameters need to be updated on line, and the calculation burden is greatly reduced.
Further, the model is represented as:
Figure BDA0002050522780000021
Figure BDA0002050522780000022
respectively representing the state vector and the system output;
Figure BDA0002050522780000023
is an unknown continuous non-linear function with a minimum order derivative;
Figure BDA0002050522780000024
is a known control gain, which is not equal to zero, considering the controllability of the system;
Figure BDA0002050522780000025
representing control input and unknown external disturbances, respectively.
Further, in practical engineering applications, actuator saturation is often encountered, considering symmetric control constraints, i.e., and u | ≦ u |0Thus, the output of the actuator is given by
u(t)＝sat(v(t))＝sign(v(t))min{u0,|v(t)|} (2)
Wherein u is0Is a control inputV (t) is the true input to the actuator as will be determined in the following section, and for the sake of brevity (t) is omitted hereafter without any ambiguity, and for ease of controller design the saturation non-linearity in equation (2) can be converted to a relatively linear form, denoted as
Figure BDA0002050522780000026
In the formula (3)
Figure BDA0002050522780000027
Is a normal number, phi (tau), density function, satisfies
Figure BDA0002050522780000028
Is a dead zone operator.
Further, for input-constrained equation (1), there is a feasible actual control input v, i.e., v, bounded, such that the desired control objective can be achieved;
thus substituting equation (3) into equation (1)
Figure BDA0002050522780000031
Figure BDA0002050522780000032
Is a compound disturbance, and therefore the control target for this work is to track the output reference command y in the expectation equationrWhile ensuring asymptotic stability of the tracking error system. In this case, the parameter y is passedrThe output y is derived within the pre-planned envelope. To obtain
Figure BDA0002050522780000033
In the formula
Figure BDA0002050522780000034
Upper and lower limits.
Further, as shown in equation (5), the output is constrained, which increases the difficulty and complexity of designing an efficient controller directly; to overcome this problem, an output transformation is used, i.e.
Figure BDA0002050522780000035
Or
Figure BDA0002050522780000036
Wherein z is1Is the converted output variable;
function(s)
Figure BDA0002050522780000037
The following conditions should be satisfied
Figure BDA0002050522780000038
Function(s)
Figure BDA0002050522780000039
Is composed of
Figure BDA00020505227800000310
Then, based on equation (11), the transformed output is equal to
Figure BDA00020505227800000311
Thus, let
Figure BDA00020505227800000312
Then can obtain
Figure BDA0002050522780000041
Wherein
Figure BDA0002050522780000042
In the formula
Figure BDA0002050522780000043
Wherein
Figure BDA0002050522780000044
Are binomial coefficients.
Further, based on the transformed system (13), the following filtered state variables are defined as
s＝c1z1+c2z2+…+cn-1zn-1+zn (15)
In which a normal vector is present
Figure BDA0002050522780000045
Thus, polynomial
λn+cn-1λn-1+…+c2λ2+c 10 is Hurwitz, and λ is laplace operator.
Further, the adaptive saturation controller is designed to
Figure BDA0002050522780000046
Where is eta, mu1,μ2Is a normal number.
Figure BDA0002050522780000047
Are respectively unknown constants
Figure BDA0002050522780000048
An estimated value of (d);
the corresponding adaptive scheme is
Figure BDA0002050522780000049
Further, under equations (24) and (25), the required output reference commands that evolve within the envelope pre-programmed by the time-varying constraint are tracked with asymptotic stability.
The invention has the beneficial effects that:
compared with the existing work, the calculation amount of the self-adaptive scheme is small, and the asymptotic stability of the tracking error system can be ensured, but not the consistent final bounded stability. The simulations used show that an improvement of about 4 times can be obtained in terms of tracking accuracy, which demonstrates the effectiveness of the proposed control law.
Drawings
FIG. 1 is a schematic phase image of a system without a control input;
FIG. 2 is a schematic diagram of output tracking trajectories under two control laws;
FIG. 3 is a schematic diagram of output tracking error under two control laws;
FIG. 4 is a schematic diagram of control inputs according to two control rules;
FIG. 5 is a diagram illustrating adaptive parameters under two control laws;
FIG. 6 shows the weight W of RBFNN under the conventional back-stepping control lawbSchematic representation.
Detailed Description
In order to make the implementation objects, technical solutions and advantages of the present invention clearer, the technical solutions in the embodiments of the present invention will be described in more detail below with reference to the accompanying drawings in the embodiments of the present invention. In the drawings, the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The described embodiments are only some, but not all embodiments of the invention. The embodiments described below with reference to the drawings are illustrative and intended to be illustrative of the invention and are not to be construed as limiting the invention. 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. Embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
The invention researches a novel low-complexity self-adaptive control law for an uncertain nonlinear system which considers time-varying output constraint and controls saturation and external disturbance by adopting an MLP (maximum likelihood ratio) and RBFNN (radial basis function network) method. Using MLP techniques and virtual error concepts, little work has been done on adaptive saturation control studies for nonlinear systems with time-varying output constraints. Compared to the existing work, our main contributions are two: 1) under the condition that time-varying output constraint, control saturation and external disturbance exist, a new low-complexity adaptive neural saturation control law is firstly proposed, wherein control saturation nonlinearity is similar to a dead zone model. 2) The number of adaptation parameters is reduced to only two, which is independent of the system order, the hidden nodes of the NN and the number of rules of the fuzzy logic system. In this case, the computational load is significantly reduced and in practice the corresponding control laws are easier to implement.
Note that:
Figure BDA0002050522780000051
a set of non-negative integers, positive integers, n-dimensional real numbers and n-dimensional positive real numbers, respectively. T, | | |, respectively, represents the absolute value of the vector transpose, euclidean norm, and scalar.
1. Problem description and preliminary measures
2.1 model description
The system considered herein is denoted as
Figure BDA0002050522780000061
Figure BDA0002050522780000062
Representing the state vector and the system output, respectively. In the formula (1)
Figure BDA0002050522780000063
Is an unknown continuous non-linear function with a minimum derivative of order.
Figure BDA0002050522780000064
Is a known control gain which is not equal to zero in view of the controllability of the system.
Figure BDA0002050522780000065
Representing control input and unknown external disturbances, respectively.
In practical engineering applications including robotic systems, power systems, etc., actuator saturation is often encountered. In this work, the symmetric control constraints are considered, i.e., u ≦ u |0Thus, the output of the actuator is given by
u(t)＝sat(v(t))＝sign(v(t))min{u0,|v(t)|} (2)
Wherein u is0Is the upper limit of the control input. v (t) is the true input to the actuator as will be determined in the following section, and for the sake of brevity (t) is omitted hereafter without any ambiguity. For the convenience of controller design, the saturation nonlinearity in (2) can be converted to a relatively linear form, which is denoted as
Figure BDA0002050522780000066
In the formula (3)
Figure BDA0002050522780000067
Is a normal number, phi (tau), density function, satisfies
Figure BDA0002050522780000068
Is a dead zone operator.
Assume that 1: for an input-constrained system (1), there is a feasible actual control input v, i.e., v, bounded so that the desired control objective can be achieved.
Note that 1: the actual control input v is bounded due to the controllability of the actual system, which isIn addition, many functions may be used as the density function including a gaussian function. Therefore, the approximation error under the premise of assumption 1
Figure BDA0002050522780000069
Is bounded.
Thus bringing (3) into (1)
Figure BDA0002050522780000071
Figure BDA0002050522780000072
Is a compound disturbance, and therefore the control objective of the operation is to track the expectation
Output reference command yrWhile ensuring asymptotic stability of the tracking error system. In this case, the parameter y is passedrThe output y is derived within the pre-planned envelope. To obtain
Figure BDA0002050522780000073
In the formula
Figure BDA0002050522780000074
Upper and lower limits, e.g.
Figure BDA0002050522780000075
Suppose that
Figure BDA0002050522780000076
Is a continuous function with at least an (n-1) order derivative, which is known and bounded. It is to be noted that the initial output y (0) or x in the above expression (5)1(0) Are also assumed.
2.2 radial basis function NN approximation
Radial basis function NN (RBFNN) techniques have been widely used to approximate unknown continuous functions with arbitrary precision on a compact set. For unknown continuous values
Figure BDA0002050522780000077
The output of RBFNN has an approximation error. The output of RBFNN with approximation error is
Figure BDA0002050522780000078
In the formula
Figure BDA0002050522780000079
Is an input vector in which
Figure BDA00020505227800000710
Figure BDA00020505227800000711
Is the best weight vector and M is the number of nodes of the hidden layer. O (X) is the error of approximation,
Figure BDA00020505227800000712
wherein
Figure BDA00020505227800000713
Represented by a gaussian kernel function.
Figure BDA00020505227800000714
In the formula
Figure BDA00020505227800000715
Representing the respective center and width of the gaussian function.
For the approximation function (6), there is an ideal weight W*Vector such that O (X) is arbitrarily small. To facilitate subsequent research of adaptive saturation controllers, certain assumptions have been added.
Hypothesis 2. there is an unknown normal number θ where | | W*||≤θ.
Assumption 3. unknown external disturbance d is boundedI.e. d is less than or equal to d0D in (1)0Is an unknown normal number.
Note 2. based on hypothesis 2, the adaptive scheme to be determined can be studied to estimate the unknown constant θ instead of the vector W*This greatly reduces the amount of computation. This approach is also referred to as the Minimum Learning Parameter (MLP) scheme [12]。
Note 3, assume 3 is reasonable because if the external disturbance is unbounded, the system (1) will runaway with bounded control input.
Based on the above analysis, the following work is mainly focused on adaptive saturation controller design considering unknown non-linearity, external disturbances and time-varying constraints.
3. Main results
3.1 output conversion
As shown in (5), the output is constrained, which increases the difficulty and complexity of designing an efficient controller directly. To overcome this problem, an output transformation is used, i.e.
Figure BDA0002050522780000081
Or
Figure BDA0002050522780000082
Wherein z is1Is the converted output variable. Function(s)
Figure BDA0002050522780000083
The following conditions should be satisfied
Figure BDA0002050522780000084
Function(s)
Figure BDA0002050522780000085
Is composed of
Figure BDA0002050522780000086
Then, based on (11), the transformed output is equal to
Figure BDA0002050522780000087
Thus, let
Figure BDA0002050522780000088
Then can obtain
Figure BDA0002050522780000091
Wherein
Figure BDA0002050522780000092
In the formula
Figure BDA0002050522780000093
Wherein
Figure BDA0002050522780000094
Are binomial coefficients.
3.2 adaptive saturation controller design
To continue, based on the transformed system (13), the following filtered state variables are defined as
s＝c1z1+c2z2+…+cn-1zn-1+zn (15)
In which a normal vector is present
Figure BDA0002050522780000095
Thus, the polynomial λn+cn-1λn-1+…+c2λ2+c 10 is Hurwitz, and λ is laplace operator.
Theorem 1. for the filter state variable in (15), if s converges to zero, it is t → ∞
Then the newly defined tracking error z1And its derivative zk(k → 2.., n) converges to zero at the same convergence rate as t → ∞.
And (3) proving that: (15) is expressed as
Figure BDA0002050522780000096
Wherein parameter A is associated with B in (15). Then solving the first equation in (16) to obtain
Figure BDA0002050522780000097
Equation (17) implies that if the condition is
Figure BDA0002050522780000098
Is bounded, then xn-1(t) → 0 if t → ∞ if conditions
Figure BDA0002050522780000099
Is borderless, for (17), a rule using L' Hopital can be generated
Figure BDA0002050522780000101
Therefore, if t → ∞ then s (t) → 0, and if t → ∞ then χn-1(t) → 0, and s (t) have the same convergence rate. Similarly, for the second equation in (16), based on (18), a rule of L' Hopital can be used to generate
Figure BDA0002050522780000102
As shown in (19), when t → ∞, s (t) → 0as, when t → ∞ χn-2Convergence rate of (t) → 0 and sThe same is true. For the remaining equations in (16), the same conclusions can be drawn by employing the same procedures as (17) - (19). Thus, the certification of lemma 1 is completed.
Taking the time derivative of s to obtain
Figure BDA0002050522780000103
Figure BDA0002050522780000104
Wherein
Figure BDA0002050522780000105
Due to the fact that
Figure BDA0002050522780000106
Is also unknown. Thus, based on the knowledge in subsection 2.2, RBFNN is used to approximate the lumped unknown function f*Given by:
Figure BDA0002050522780000107
wherein
Figure BDA0002050522780000108
From assumption 2 above, one can derive
Figure BDA0002050522780000109
Wherein
Figure BDA00020505227800001010
For complex disturbances d*Under remark 1 and hypothesis 3, one can obtain
Figure BDA0002050522780000111
In the formula psi2Beta wherein beta > 0(14)Time-varying positive parameters.
Figure BDA0002050522780000112
Is an unknown normal number. Based on the foregoing analysis, the adaptive saturation controller is designed to
Figure BDA0002050522780000113
Where is eta, mu1,μ2Is a normal number.
Figure BDA0002050522780000114
Are respectively unknown constants
Figure BDA0002050522780000115
An estimate of (d). The corresponding adaptive scheme is
Figure BDA0002050522780000116
Remark 4. the controller has two parts. One is the nominal part:
Figure BDA0002050522780000117
for stabilizing the tracking error system.
And
the other being a stabilising part, e.g.
Figure BDA0002050522780000118
Figure BDA0002050522780000119
The method is used for further compensating the influence caused by unknown nonlinearity, external interference and control saturation. Only two parameters are needed to estimate the 2m parameters involved in the RBFNN approximation weight vector. In this sense, the computational load drops sharply.
3.3 stability analysis
Under the designed controller (24), the following important results are given.
Theorem 1 under a designed controller (24) and adaptive scheme (25), a desired output reference command that evolves within an envelope preplanned by a time-varying constraint can be tracked with asymptotic stability.
And (5) proving. First, the following Lyapunov function is constructed
V＝V1+V2 (26)
Wherein
Figure BDA0002050522780000121
Virtual error in the equation
Figure BDA0002050522780000128
Is defined as
Figure BDA0002050522780000123
By using V in (27)1Is derived from the time derivative of
Figure BDA0002050522780000124
Substituting (20) and (24) into (28) to obtain
Figure BDA0002050522780000125
In view of (22) and (23), (29) equals
Figure BDA0002050522780000126
Taking V in (27)2Is derived from the time derivative of
Figure BDA0002050522780000127
Substituting (30) and (31) into the time derivative of V in (26) to obtain
Figure BDA0002050522780000131
Based on virtual errors in (27)
Figure BDA0002050522780000132
Then (32) becomes
Figure BDA0002050522780000133
Substituting (25) into (33) to obtain
Figure BDA0002050522780000134
Only when s is 0
Figure BDA0002050522780000135
Therefore, the tracking error system is asymptotically stable, i.e., s → 0 time corresponds to t → ∞, and therefore, based on (15), by using the theorem 1, all newly defined tracking errors z can be obtained1,z2,...,znConvergence to zero is equivalent to b.
4. Description case
In order to verify the effectiveness of the proposed control law, a simulation of a second-order Duffing Holmes chaotic system is adopted. In this application example, the chaotic system is expressed as follows
Figure BDA0002050522780000141
In the formula of omega1＝0.3+0.2sin(10t),ω2＝0.2+0.2cos(5t),ω3＝1,q＝5+0.1cos(t),ξ＝0.5+0.1sin(t),|u|≤10,d＝0.4sin(0.2πt)+0.3sin(x1x2) The phase diagram of the Duffing Holmes chaotic system is shown in fig. 1. According to remark 1, a gaussian kernel function of the detailed form given in (7) can be used as part of the density function. When selecting the density function
Figure BDA0002050522780000142
Then, ρ can be obtained0The 12.5 simulation parameters are set as: y in (6) and (7)r＝sin(0.5t)+cos(0.25t),
Figure BDA0002050522780000143
y＝yr-exp(-0.3t)-0.45in(5)；m＝50,εi＝0.5+2rand(·)，γi＝[2,-8,6,2]T(i ═ 1,2,. ·, m); in (15) and (24) c1＝2,η＝100,μ1＝μ2＝3,
Figure BDA0002050522780000144
To illustrate the effectiveness of the proposed control law, a conventional back-stepping control method was used as a comparative test, with RBFNN node number of 50, centered at [ -1,3 ] and]×[-17,1]×[-5,17]×[-2,6]×[-5,7]×[-4,10]the initial weight of RBFNN in the traditional backstepping control technology is Wb-3+6rand (·), initial simulation condition x1＝1.7,x2The relevant simulation results are shown in fig. 2-6, at 0. Note that 'ASC' and 'TBC' refer to a proposed control method and a conventional control method, respectively.
From the simulation results of fig. 2-6, the following conclusions can be drawn: (i) after 30 seconds, the output tracking error under the proposed control law is much smaller than under the conventional backstepping control law (the tracking accuracy depicted in fig. 3 is improved by about 4 times). (ii) from FIGS. 5 and 6, it can be seen that only two adaptive parameters are needed for updating, whereas in the conventional backstepping control law, W isbOf which 50 elements need to be updated. In this sense, the computational load drops sharply.
Finally, it should be noted that the above examples are only used to illustrate the technical solutions of the present invention and not to limit the same; although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art will understand that: modifications to the specific embodiments of the invention or equivalent substitutions for parts of the technical features may be made; without departing from the spirit of the present invention, it is intended to cover all aspects of the invention as defined by the appended claims.

Claims (6)

1. A low-complexity self-adaptive saturation control method for a nonlinear system is characterized by comprising the following steps: the control method comprises the following steps: firstly, converting control saturation nonlinearity into a linear model relative to an actual input signal by adopting a dead zone model;
then, converting an original system with time-varying output constraint into unconstrained output constraint, and designing a new self-adaptive saturation control law along the filtered error manifold based on the system;
updating two self-adaptive parameters on line by adopting a minimum learning parameter technology and a virtual error concept;
the model is represented as:
Figure FDA0003405002280000011
Figure FDA0003405002280000012
respectively representing the state vector and the system output;
Figure FDA0003405002280000013
Figure FDA0003405002280000014
is an unknown continuous non-linear function with a minimum order derivative;
Figure FDA0003405002280000015
is a known control gain, which is not equal to zero, considering the controllability of the system;
Figure FDA0003405002280000016
respectively representing control inputs and unknown external disturbances;
in practical engineering applications, actuator saturation is often encountered, taking into account symmetric control constraints, i.e., and u | ≦ u |0Thus, the output of the actuator is given by
u(t)＝sat(v(t))＝sign(v(t))min{u0,|v(t)|} (2)
Wherein u is0Is the upper limit of the control input, v (t) is the actual input to the actuator as will be determined in the following section, and for the sake of brevity, (t) is omitted hereafter without any ambiguity, and for the convenience of controller design, the saturation non-linearity in equation (2) can be converted to a relatively linear form, expressed as
Figure FDA0003405002280000017
In the formula (3)
Figure FDA0003405002280000018
Is a normal number, phi (tau) is a density function, satisfies
Figure FDA0003405002280000019
Figure FDA00034050022800000110
dzτ(v) Max { v- τ, min {0, v + τ } } is the dead zone operator.
2. A low complexity adaptive saturation control method for nonlinear system according to claim 1, characterized in that, for formula (1) constrained by input, there is feasible practical control input v, v bounded, so that the desired control target can be achieved;
thus substituting equation (3) into equation (1)
Figure FDA0003405002280000021
In the formula
Figure FDA0003405002280000022
Is a compound disturbance, and therefore the control objective of the job is to track the desired output reference command yrWhile ensuring asymptotic stability of the tracking error system, in this case by means of the parameter yrIs derived within a pre-planned envelope, derived
Figure FDA0003405002280000023
In the formulay(t),
Figure FDA0003405002280000024
Upper and lower limits.
3. The low-complexity adaptive saturation control method for the nonlinear system according to claim 2, wherein: as shown in equation (5), the output is constrained, which increases the difficulty and complexity of designing an efficient controller directly; to overcome this problem, an output transformation is used, i.e.
Figure FDA0003405002280000025
Or
Figure FDA0003405002280000026
Wherein z is1Is the converted output variable;
function(s)
Figure FDA0003405002280000027
The following conditions should be satisfied
Figure FDA0003405002280000028
Function(s)
Figure FDA0003405002280000029
Is composed of
Figure FDA00034050022800000210
Then, based on equation (11), the transformed output is equal to
Figure FDA00034050022800000211
Thus, let
Figure FDA0003405002280000031
Then can obtain
Figure FDA0003405002280000032
Wherein
Figure FDA0003405002280000033
In the formula
Figure FDA0003405002280000034
Wherein
Figure FDA0003405002280000035
Figure FDA0003405002280000036
Are binomial coefficients.
4. The low-complexity adaptive saturation control method for the nonlinear system according to claim 3, wherein: based on the transformed system (13), the following filtered state variables are defined as
s＝c1z1+c2z2+…+cn-1zn-1+zn (15)
In which a normal vector is present
Figure FDA0003405002280000037
Thus, the polynomial λn+cn-1λn-1+…+c2λ2+c10 is Hurwitz, and λ is laplace operator.
5. The low-complexity adaptive saturation control method for the nonlinear system according to claim 1, wherein: the adaptive saturation controller is designed to
Figure FDA0003405002280000038
Where is eta, mu1,μ2Is a normal number which is a positive number,
Figure FDA0003405002280000039
are respectively an unknown constant theta1,θ2Is determined by the estimated value of (c),
Figure FDA00034050022800000310
ψ2β wherein β > 0;
the corresponding adaptive scheme is
Figure FDA00034050022800000311
Figure FDA00034050022800000312
6. The low-complexity adaptive saturation control method for the nonlinear system according to claim 5, wherein: under equations (24) and (25), the required output reference commands that evolve within the envelope preplanned by the time-varying constraint are tracked with asymptotic stability.
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