CN114839882A

CN114839882A - Nonlinear system composite self-adaptive control method under input constraint

Info

Publication number: CN114839882A
Application number: CN202210723139.3A
Authority: CN
Inventors: 祝汝松; 王平
Original assignee: Equipment Design and Testing Technology Research Institute of China Aerodynamics Research and Development Center
Current assignee: Equipment Design and Testing Technology Research Institute of China Aerodynamics Research and Development Center
Priority date: 2022-06-24
Filing date: 2022-06-24
Publication date: 2022-08-02

Abstract

The invention relates to the field of adaptive control, and discloses a nonlinear system composite adaptive control method under input constraint, which is applied to a system with input constraint; a dynamic equation is reconstructed, an auxiliary signal is generated, and the effect on the system in the actuator saturation state is expressed by the auxiliary signal; and designing a composite adaptive control law. The invention combines the prediction error estimation with the traditional tracking error estimation by combining the uncertainty in the prediction error estimation system, is used for enhancing the estimation of the uncertainty in the system and brings extra stability characteristics to the error dynamic state of the parameter estimation; and (3) performing adaptive control on a nonlinear uncertain system with input constraint by adopting a composite adaptive method.

Description

Nonlinear system composite self-adaptive control method under input constraint

Technical Field

The invention relates to the field of self-adaptive control, in particular to a nonlinear system composite self-adaptive control method under input constraint.

Background

In practical control applications, uncertainties and disturbances, such as unknown parameters, noise, perturbations, and unmodeled higher order dynamics, are common in controlled objects and systems, such as flexible structures, fluid flows, combustion processes, aircraft, and bioengineering [1-3 ]. Designing control systems using high-confidence system models rarely occurs in practice. Adaptive control methods developed in the past decades are powerful tools to solve the problem of parameter uncertainty in control systems [3-6 ].

The basic idea of adaptive control is to estimate uncertain system parameters on-line or equivalently corresponding controller parameters from the measured signals of the system and to use the estimated parameters in the control law. This gives the system some ability to self-learn and self-improve as this adaptation process proceeds. By adopting the self-adaptive control, the control system can keep relatively consistent control performance under the condition that the equipment parameters have uncertainty or unknown changes. The stability and performance of adaptive control is often analyzed using Lyapunov theory and other tools in nonlinear control theory [3 ].

However, when applying adaptive control in real systems, robustness issues related to non-parametric uncertainties, such as external disturbances, time-varying parameters, unmodeled dynamics, time lags and other non-structural disturbances, need to be addressed. Otherwise, parameter drift and instability of the system may result. Some developed techniques, such as the improved robust adaptation law [2,6], may partially solve this problem.

Another disadvantage of conventional adaptive control is that the dynamics of parameter estimation using conventional adaptive laws do not have strong stability due to the purely integral form of the adaptive laws. In some cases, this may be unacceptable in terms of transient response of the closed-loop system, such as standard model reference adaptive control. Currently, several techniques have been developed to address this problem. One of them is adaptive control using the fusion and invariance (I & I) method [7,8], which can impart tailorable (consistently stable) dynamics to the estimation error of a parameter by constructing a zero estimation error manifold with progressive stability. But designing I & I adaptive control usually requires finding a nonlinear filter function, sometimes by solving partial differential equations, which makes designing more difficult than conventional adaptive control. Another technique is the use of composite adaptation [1,9-11] which combines tracking errors and parameter prediction errors in parameter adaptation, which is not new and is intensively studied in the literature [3,10,11 ]. It is easier to design and may bring additional stability to the adaptation law, similar to I & I adaptive control.

When the self-adaptive control method is applied to an actual engineering physical system, another important problem to be solved is that: control input constraints due to limitations of system actuators (actuators). Typically, these constraints include amplitude and rate saturation of actuators, such as control valve speed and amplitude limits in process control, control surface yaw angle limits in aircraft control, and the like. This can lead to reduced control system performance and even instability of the closed loop system if this problem is not properly designed and accounted for in the adaptive control design [12-17 ].

Some studies on adaptive control with input constraint systems using composite adaptation can be found in document [18 ]. In [18], actuator amplitude saturation is taken into account in the composite adaptive output feedback control of the nonlinear system, but actuator velocity saturation and gain uncertainty are not taken into account. Preliminary literature studies have shown that there is little comprehensive research on adaptive control of nonlinear uncertain systems with input constraints using a composite adaptive approach.

The references cited in the present invention are as follows:

[1] J.-J. E. Slotine, and W. Li, Applied nonlinear control: Prentice hall Englewood Cliffs, NJ, 1991.

[2] E. Lavretsky, and K. Wise, “Robust and adaptive control: With aerospace applications, ser,” Advanced textbooks in control and signal processing. London and New York: Springer, 2013.

[3] K. S. Narendra, and A. M. Annaswamy, Stable adaptive systems: Courier Corporation, 2012.

[4] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and adaptive control design: Wiley New York, 1995.

[5] S. Sastry, Nonlinear systems: analysis, stability, and control: Springer Science & Business Media, 2013.

[6] P. A. Ioannou, and J. Sun, Robust adaptive control: PTR Prentice-Hall Upper Saddle River, NJ, 1996.

[7] A. Astolfi, D. Karagiannis, and R. Ortega, Nonlinear and adaptive control with applications: Springer Science & Business Media, 2007.

[8] A. Astolfi, and R. Ortega, “Immersion and invariance: A new tool for stabilization and adaptive control of nonlinear systems,” IEEE Transactions on Automatic control, vol. 48, no. 4, pp. 590-606, 2003.

[9] J. Nakanishi, J. A. Farrell, and S. Schaal, “Composite adaptive control with locally weighted statistical learning,” Neural Networks, vol. 18, no. 1, pp. 71-90, 2005.

[10] E. Lavretsky, “Combined/composite model reference adaptive control,” IEEE Transactions on Automatic Control, vol. 54, no. 11, pp. 2692-2697, 2009.

[11] M. A. Duarte, and K. S. Narendra, “Combined direct and indirect approach to adaptive control,” IEEE Transactions on Automatic Control, vol. 34, no. 10, pp. 1071-1075, 1989.

[12] J. Zhou, and C. Wen, Adaptive Backstepping Control of Uncertain Systems: Nonsmooth Nonlinearities, Interactions or Time-Variations, 2008.

[13] J. Farrell, M. Polycarpou, and M. Sharma, "Adaptive backstepping with magnitude, rate, and bandwidth constraints: aircraft longitude control." pp. 3898-3904 vol.5.

[14] S. P. Karason, and A. M. Annaswamy, “Adaptive control in the presence of input constraints,” IEEE Transactions on Automatic Control, vol. 39, no. 11, pp. 2325-2330, 1994.

[15] A. M. Annaswamy, S. Evesque, S.-I. Niculescu, and A. Dowling, "Adaptive control of a class of time-delay systems in the presence of saturation," Adaptive Control of Nonsmooth Dynamic Systems, pp. 289-310: Springer, 2001.

[16] J. E. Gaudio, A. M. Annaswamy, and E. Lavretsky, "Adaptive control of hypersonic vehicles in the presence of rate limits." p. 0846.

[17] J. Yang, A. Calise, and J. I. Craig, Adaptive output feedback control with input saturation, 2003.

[18] Y. Li, S. Tong, and T. Li, “Composite adaptive fuzzy output feedback control design for uncertain nonlinear strict-feedback systems with input saturation,” IEEE Transactions on Cybernetics, vol. 45, no. 10, pp. 2299-2308, 2014.

[19] A. Isidori, Nonlinear control systems: Springer Science & Business Media, 2013.

[20] S. Gao, H. Dong, B. Ning, and L. Chen, “Neural adaptive control for uncertain nonlinear system with input saturation: State transformation based output feedback,” Neurocomputing, vol. 159, pp. 117-125, 2015/07/02/, 2015.

[21] H. K. Khalil, “Nonlinear systems third edition,” Patience Hall, vol. 115, 2002。

disclosure of Invention

Therefore, to overcome the above-mentioned deficiencies, the present invention herein provides an adaptive control method with input constrained nonlinear system that incorporates uncertainty in the prediction error estimation system; the invention uses the prediction error estimation and the traditional tracking error estimation together to enhance the estimation of uncertainty in the system and bring extra stability characteristics to the parameter estimation error dynamic state; and (3) adopting a composite self-adaptive method to carry out self-adaptive control on a nonlinear uncertain system with input constraint.

Specifically, a nonlinear system composite adaptive control method under input constraint is applied to a nonlinear single-input single-output system with input constraint, and the method specifically comprises the following steps:

considering a class of nonlinear single-input single-output systems with actuator constraints, and writing a state space form with uncertain control gain and unknown constant parameters;

constructing a dynamic equation, generating an auxiliary signal, and expressing the effect of the actuator on the nonlinear single-input single-output system in a saturated state by the auxiliary signal;

defining an augmented tracking error vector, defining a scalar tracking error as a linear combination of elements of the augmented tracking error vector;

calculating a prediction error;

constructing a composite adaptive law for uncertain control gain estimated values and unknown estimated values of constant parameters;

and designing a composite adaptive control law according to the composite adaptive law.

The invention has the following beneficial effects:

the invention relates to an adaptive control method which is applied to a nonlinear system with input constraint by combining uncertainty in a prediction error estimation system. The prediction error estimate is used together with the conventional tracking error estimate to enhance the estimation of uncertainty in the system and to bring additional stability properties to the parameter estimation error dynamics, which is not present in the conventional adaptive control of non-linear systems with input constraints. An auxiliary dynamic system driven by the actuator saturation error is constructed and forms an augmented tracking error used in a composite adaptive law to ensure the stability of a closed-loop system under the condition of actuator saturation. Model uncertainties such as unknown parameters and interference are taken into account in the system, including unknown control input gain.

Drawings

FIG. 1 is a schematic flow diagram of the process of the present invention;

FIG. 2 is a schematic view of a delta wing aircraft (100, fuselage; 200, delta wing; 300, engine; 400, left aileron; 500, right aileron; 600, roll angle dynamics, FIG. 2);

FIG. 3 is a roll angle

A control response to the step change command;

FIG. 4 is a graph of roll angle control

Control signal after time-saturation output

；

FIG. 5 is a graph of roll angle control

A parameter estimation process.

Detailed Description

The present invention will be described in detail with reference to fig. 1, and the technical solutions in the embodiments of the present invention will be clearly and completely described, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all 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.

As shown in fig. 1, the present invention is a nonlinear system composite adaptive control method under input constraint, and the specific method is as follows:

the first step is as follows: consider a class of non-linear single-input single-output systems where there are constraints on the actuators to perform, the equation for which is as follows:

wherein

And its derivatives are the states of the nonlinear single-input single-output system where there are constraints on the actuation of the actuators,

is the derivative of X and is the sum of the,

is an unknown constant parameter and is an unknown vector

The (i) th element of (a),

is a known non-linear smooth function and is a vector

0 < i < r, i, r and n are integers greater than 0, and the nonlinear smoothing function satisfies the consistent Lipschitz (Lipschitz) condition for each parameter in the function, b _c Is an indeterminate control gain, v is the output of the control actuator, d is the time-varying bounded disturbance, and u is the control command input to the system actuator or the output of the controller; wherein

And

are the actuator amplitude and rate saturation functions, respectively, defined as

Wherein

And

are the known amplitude saturation upper and lower limits of the actuator,

and

are the known rate saturation upper and lower limits of the actuator.

The second step is that: equation (1) is rewritten as the following state space form,

wherein the content of the first and second substances,

,

,… ,

，

is the output of a non-linear single-input single-output system with actuator constraints, where the vector is unknown

Is an unknown vector

The transposing of (1).

The control objective of the system is to synthesize a composite adaptive control law with composite adaptation, so that the system output X tracks a given smooth reference trajectory signal X asymptotically or within a limited error _r And all other signals of the closed loop system remain bounded.

Equation (4) is a canonical form of nonlinear system, called Brunovsky form. Although the range of the system covered by equation (4) seems limited compared to some other forms of nonlinear systems, many strict feedback nonlinear systems can be converted into this canonical form (as described in document [5 ]), and many physical systems can also be directly represented as dynamic models of this form.

Equation (4) contains uncertain control gain and control input rate constraints. This is different from the process in document [20], although in the prior art, the uncertainty of the system is handled by an adaptive fuzzy method. Since uncertainty in control gain and input rate constraints are very common in practice, this will make the adaptive control method of the nonlinear system under the input constraints combined with uncertainty in prediction error estimation according to the present invention more applicable.

In order to have a proper solution to the above control problem, the following assumption is made for equation (4).

Assume that 1: there is a compact set of system states

The system of equation (4) is controllable within the state space in the presence of control input constraints.

Assume 2: the order n of the system is known, the state vector of equation (4)

Is measurable.

Assume 3: sign of high frequency control gain

Are known, and

wherein b is _o Is a known lower limit.

Assume 4: reference track x _r And before it

Derivative of order

Given and bounded, the output x of the system with control input constraints ₁ Can achieve the purpose of

。

Where for a system with control input constraints, assumptions 1 and 4 are reasonable. Since it is not practical for a system with amplitude and velocity constraints on the control inputs to be expected to control the system to track unreachable trajectories outside the operating range.

Where hypothesis 2 denotes that the state feedback control of the system is considered herein. When only the output x of equation (4) is available ₁ When applicable to feedback control, i.e. output feedback control, the control method studied herein is also applicable with the help of a suitable observer design, which will not be described in detail.

The third step: self-adaptive control design;

the auxiliary dynamics generated by actuator saturation, before designing the composite adaptive control law, the following dynamic equations are first constructed to generate the auxiliary signal, which is the augmentation error

And the auxiliary signal is used to represent the effect on the system in the event of actuator saturation.

Wherein the content of the first and second substances,

is a normal number that can be designed in the design,

；

is b in equation (4) _c And is provided by the subsequent complex adaptive law equation (16); wherein

The error signal is input for control of the actuator in saturation, where v, u are as shown in equation (4),

is to increase the error

The ith element of (1).

Here, it is assumed that the output v of the actuator is measurable or available. If v is not measurable or accurate in a real system, v can be obtained by numerical calculation using the definitions of equation (2) and equation (3), and the known upper and lower amplitude limits, rate saturation limits.

In equation (5), there is an indeterminate control gain b _c Is estimated value of

，

Representing errors caused by actuator saturation or other input constraints. In said document [18]]These factors are not taken into account.

Defining an augmented tracking error vector as

Wherein:

is a measurable state of the system that is,

is a known reference trajectory vector and is,

is saturated by the actuator

Resulting in an error of amplification. When there is no actuator saturation, i.e.

Then, then

Equation (6) is a normal tracking error vector. It is clear that the signal e is computable.

If the control law of the system can be used

（

Is a small normal number) and after some transient process the saturation of the actuator disappears, i.e. the actuator is saturated

This means that

The control objective is reached.

Defining scalar tracking error

Linear combination of elements of tracking error vector e for augmentation

In the formula (I), the compound is shown in the specification,

elements of

Is selected as a polynomial

Is a value stable to Herwitz (Hurwitz). Thus, it is clear that if the control law is such that

Occur, then

And

the same is true.

Scalar tracking error

The kinetic equations (c) can be derived from equations (4) to (7) as follows:

wherein the content of the first and second substances,

,

is uncertain control gain

An estimate of (c), which will be given later,

is a diagonal matrix whose elements b _i Defined in equation (5).

According to the scalar tracking error

The compound adaptive control law is designed as follows:

wherein

Is a designable normal number which is,

is a vector of unknown parameters

An estimate of (d).

Substituting the composite adaptive control law (9) into a tracking error dynamics equation (8),

becomes a closed loop kinetic equation

Wherein, the first and the second end of the pipe are connected with each other,

is the error of the parameter estimation.

In the process of providing

And

before the complex adaptation law of the above unknown parameters, let us first introduce the concept of prediction error in parameter estimation, which will be used for the complex adaptation law of the above unknown parameters.

In conventional adaptive control, scalar tracking error

Is used in a complex adaptation law to update parameters unknown to the system

Is estimated. However, scalar tracking error

Is not the only source of parameter information (as in said document [ 1]]And document [10 ]]). In many techniques of online parameter estimation, the prediction error, rather than the tracking error, is used to extract the parameter information in the estimation.

In the form of a linear parameterization, a general parameter estimation model is

In the formula

Is the output of the model and is,

is the unknown parameter vector to be estimated,

is a known signal in the model.

If it is not

Is an estimate of a at time t by some updating law or estimator, then at time t, the use is made of

To the output

Prediction of (2)

Can be expressed as

Then, the output is predicted

And measuring the output

The difference between them is called the prediction error, using

To represent

Wherein

(ii) a The estimation based on the prediction error is based on the use of a different signal than the tracking error

To design a parameter estimator; one commonly used online estimation method is a gradient estimator (e.g., document [1 ])]As recited in (1) above),

wherein

Is the estimated gain; utilizing Lyapunov candidate function

Can be easily proven

Stability of the gradient estimator (14).

To design the unknown parameters in equation (4) using the above prediction error

And

using a first order exponentially stabilized filter

For the equation (4)

Is filtered on both sides of the kinetic equation of (a), where s is the laplace operator,

is a known normal number. The arrangement of the expressions can result in:

in the formula (I), the compound is shown in the specification,

,

definition of

,

And

then equation (15) is exactly the form of equation (11). Determining a prediction error according to equation (13)

The gradient estimator (14) may then also be used to estimate

。

Combining the prediction error estimation with the conventional tracking error based adaptation, we construct the algorithm for equation (10)

And

(corresponding to

And

) The adaptation law of (2):

in the formula (I), the compound is shown in the specification,

and

is dimension of

Of symmetric positive definite matrix gain, wherein

As defined in equation (4) above,

is a vector of unknown parameters

Is estimated value of，

For the purpose of error in the estimation of the parameters,

。

the adaptive control method provided by the embodiment combines a prediction error estimation method and a tracking error estimation method of parameters, is used for constructing a composite adaptive law, and is used for processing the control problem of a Brunovsky nonlinear uncertain system with control input constraints. The prediction error estimation method brings more information for parameter self-adaptation, improves the convergence speed and robustness of uncertainty estimation, and accordingly obtains better system response in closed-loop control. Meanwhile, under the condition of properly selecting parameters, the auxiliary dynamic signal driven by the actuator saturation error can effectively deal with and process the actuator constraint problem in control. The two technologies are combined to solve the problem of self-adaptive control of the nonlinear uncertain system under the control input constraint. By adopting the self-adaptive control method provided by the invention, all signals in a closed-loop system are consistent and finally bounded, and the composite self-adaptive law of the invention has better dynamic performance than the conventional self-adaptive law.

The stability analysis for the method specifically comprises the following steps:

theorem 1: considering the system described by equation (4) subject to input constraints, the composite adaptive control law (9) and adaptive law (16) ensure that the following conclusions hold:

1) all signals in a closed loop system are ultimately consistently bounded;

2) system output

The smooth reference trajectory can be tracked with bounded error

. When the actuator saturation disappears, the system outputs

Asymptotic tracking

。

The demonstration process is as follows:

consider a candidate Lyapunov (Lyapunov) function,

calculating the derivative of the equation, and substituting the equation (10) to obtain:

then, using the complex adaptation law of equation (16), we get:

note that according to the formula (13)

In the definition of (a) is,

then, the user can use the device to perform the operation,

it is easy to see

Is a symmetric semi-positive definite matrix. Because of the fact that

Is symmetricalPositive definite matrix, therefore

Also a symmetric semi-positive definite matrix. Then obtain

Direct method based on Lyapunov stability (see e.g. document [21 ]]Khalil, "Nonlinear systems third edition," Patience Hall, vol. 115, 2002), the inequality being the scalar tracking error

And error of parameter

(corresponding to

And

) Are consistent and ultimately bounded. As can be seen from equation (7), the tracking error vector is enlarged

Is also bounded.

According to hypothesis 4, the reference trajectory

And a first

Derivative of order

Is bounded and the system inputs

Is amplitude limited and therefore safe and rationalConcluding that a control input signal for the actuator is achievable

And is also bounded as will be explained further below.

Therefore, saturation error of actuator

And subsequently the auxiliary signal in equation (5) are also bounded. Then, according to the definition of the formula (6), it is apparent that the system state

Is bounded. Due to the fact that

For each of which the condition of uniform Lipschitz (Lipschitz) is satisfied, and therefore,

is also bounded. Then, the input is controlled according to the definition of the composite adaptive control law (9)

Is truly bounded. Thereby proving the first conclusion.

As is apparent from the formula (18), if

Then, then

. Due to the fact that

And

is also uniformly bounded, using equation (18), it is easy to see the second derivative of the Lyapunov function

Is consistently bounded. Therefore, the temperature of the molten metal is controlled,

are consistently continuous. It has furthermore been shown that the flow rate, as a function of time,

tending to a limit. The Barbalt theorem is applied (see, for example, document [2]]Description) of the invention can be obtained

This means that

. Therefore, with the composite adaptive control law,

asymptotic convergence to 0. When the actuator saturation disappears, i.e.

And then, according to equation (5),

this will result in

This means that the system outputs

Asymptotic tracking

(ii) a Thus proving a second conclusion.

Note that in equation (18)

Including additional parameter estimation errors

The second order term, which is not present in conventional adaptive control. By this design, it will be possible to give the estimation error

The process of variation sets a consistent and stable dynamic characteristic, which brings some additional stability to the adaptation law. One of the essential features of such composite adaptation is the ability to achieve rapid adaptation (e.g. document [1 ])]Description of the document). Thus, the performance of the adaptive system can be significantly improved. This outstanding property and I&I adaptive control method (see, e.g., document [ 7]]And literature description [8]) Very similar but implemented in a different way. This makes the composite adaptation law (16) more robust to noise and unmodeled dynamics in the system.

If there is no prediction term in the complex adaptation law (16)

It becomes the conventional adaptation law and this additional, semi-negatively-determined quadratic term of the parameter estimation error

Will not appear in equation (18)

In (1). This will make the adaptive control more susceptible to some of the disadvantages associated with conventional adaptive methods, such as parameter drift, oscillation behavior when using high adaptive gain, etc. [3, 6]]. The complex adaptation law (16) is more like a time-varying low-pass filter, which can be compared to the adaptation law of the traditional pure integrator form (see document [1 ])]And document [ 7]]Documentation), preserving scalar tracking error

More low frequency components.

From the formula (18),can easily deduce

Wherein

，

Is composed of

Is determined by the minimum characteristic value of (c),

is that

The maximum eigenvalue of (c). This indicates scalar tracking error

In effect, the exponent converges to zero. In addition, if in the complex adaptation law (16)

Satisfying the condition of persistent excitation, that is, in (18)

Is a symmetric positive definite matrix, then the parameter estimation error

And also the exponential converges to zero, i.e. the adaptive controller has the exponential convergence property in both tracking error and parameter estimation. This is a strong stability that cannot be achieved explicitly in conventional adaptive control.

For the simulation of the present embodiment, the following is specific:

this simulation example, from example 9.3 in document [2], controls the roll motion of a delta-wing aircraft at large angles of attack, without taking into account the saturation of the input actuators. Here, the above proposed adaptive control method is used for delta-roll motion control while considering the constraints of the input actuators. A schematic view of a delta wing aircraft is shown in figure 2.

In a delta-wing aircraft flying at a large angle of attack, the zero point of balance is unstable in an open loop during rolling motion, and there is a limit cycle (e.g., wing flapping) phenomenon (as described in document [2 ]). This local roll angle oscillation about the roll origin is caused by asymmetric unsteady aerodynamic forces acting on the delta wing. Therefore, there is a need to actively control such roll dynamics of an aircraft.

In this case, the rolling motion is accommodated by the ailerons of the aircraft, which are movable control surfaces located symmetrically behind the left and right wings of the aircraft. Moving the left aileron downward (positive yaw) and moving the right aileron upward (negative yaw) results in a downward rolling motion (positive roll rate) of the aircraft right wing. The difference in yaw between the left and right ailerons is referred to as the "differential aileron" and is the primary control input for adjusting the roll angle dynamics.

A commonly used delta wing roll dynamics model is,

wherein

Is the roll angle (rad) of the aircraft,

is the roll rate (rad/s),

is the actual flap deflection difference (rad), is the actual control input to the aircraft,

see equation (4).

Is the differential aileron control signal (rad) from the control law. In equation (19)

And

is an unknown constant parameter with a true value of

，

，

，

，

，

，

。

The magnitude and rate saturation of differential aileron deflection is increased in this model compared to the original model (using

Representation), and constant perturbation

。

Note that the angular unit in equation (19) is rad, and the angular rate is in rad/s. For convenience, in the following simulation results, degrees are used to represent angles and degrees/seconds are used to represent angular rates.

The purpose of the simulation is that, with the adaptive controller designed in this embodiment,actively controlling roll angle of delta-wing aircraft in the presence of uncertainty and control input constraints

To accurately follow the specified roll angle command

。

In this simulation, the adaptive controller is implemented as follows:

for the amplitude and rate saturation of differential aileron deflection in equation (19), the following values were used in the simulation:

，

，

，

(these values may be less than actual values and are used here for illustrative purposes only).

Roll angle command

Passing through a second order filter

Wherein

，

. The second order filter is used for generating a tracking reference track

And derivatives thereof

、

And is used in the design of composite adaptive control laws. Roll angle of aircraft

The need to track the reference trajectory within the differential aileron deflection constraints

。

The composite adaptive control law is designed by adopting equation (9) and composite adaptive law (16). The design parameters are as follows:

，

，

，

，

，

，

. Adaptive parameters in a complex adaptation law (16)

And

is initially of

，

. These values mean that at the start of the closed-loop control of the scrolling movement, the parameters of the scrolling dynamics are very poorly understood, only the control gain

There is an initial estimate of 1.6 that is more than twice its true value of 0.75.

In this embodiment, the results of the simulation are as follows:

step change command for roll angle

The control simulation results for a series of step changes are shown in fig. 3-5. For comparison, the results of conventional adaptive control without prediction error estimation are also given in fig. 3, where the design parameters are the same as above, except that

。

Simulation results show that the self-adaptive control of the two self-adaptive methods can ensure the roll angle of the airplane

Tracking a reference track within a certain error range

And following the final step change command

. They all exhibit reasonable self-learning and adaptive capabilities and the response of the system control improves as the control process and adaptation proceeds. As can be seen from FIG. 4, the actual control signal Vdynamic aileron deflection of the system has its amplitude and rate limited rangeAnd (4) the following steps.

As can be seen from fig. 3-5, the performance of adaptive control in combination with prediction error estimation is significantly better than conventional adaptive control, especially in terms of initial control response, control signal oscillation. The control signal of conventional adaptive control is jittered more during the first step change of the command, at which point the uncertainty of the system is not well estimated at the beginning. As can be seen from fig. 5, the parameter estimation of the composite adaptation is generally smoother, less oscillatory, and has a better convergence speed to reach its true value, compared to the conventional adaptive control. These advantages mainly come from the extra prediction error estimation in the complex adaptation law, which means that more information in the system is utilized and the estimation of the system uncertainty is improved. And prediction error estimation is introduced into the composite adaptive law, so that the robustness of a closed-loop system is improved.

The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A nonlinear system composite self-adaptive control method under input constraint is characterized in that,

the method is applied to a nonlinear single-input single-output system with input constraint, and comprises the following steps:

s1, considering a nonlinear single-input single-output system with actuator constraint, and writing a state space form with uncertain control gain and unknown constant parameters;

s2, constructing a kinetic equation, and generating an auxiliary signal to represent the effect of the actuator on the nonlinear single-input single-output system in a saturated state;

s3, defining an augmented tracking error vector, and defining a scalar tracking error as a linear combination of elements of the augmented tracking error vector;

s4, calculating a prediction error;

s5, constructing a composite adaptive law aiming at uncertain control gains and unknown constant parameters;

and S6, designing a composite adaptive control law according to the composite adaptive law in the step S5.

2. The method for compound adaptive control of nonlinear systems under input constraints as claimed in claim 1, wherein the specific method of step S1 is as follows:

s11, considering a nonlinear single-input single-output system with actuator constraint, the equation is as follows:

wherein

is the derivative of X and is the sum of,

is an unknown constant parameter and is an unknown vector

The (i) th element of (a),

is a known non-linear smooth function and is a vector

0 < i < r, i, r and n are integers greater than 0, b _c It is the control gain that is not deterministic,

is the output of the control actuator, d is the time-varying bounded disturbance, u is the control command input to the system actuator or the output of the controller;

and

are the actuator amplitude and rate saturation functions, respectively, and are defined as follows:

wherein

And

are the known amplitude saturation upper and lower limits of the actuator,

and

are the known rate saturation upper and lower limits of the actuator;

s12, rewriting the equation of the non-linear single-input single-output system with the constraint of executing the actuator in the step S11 into a state space form,

wherein the content of the first and second substances,

,

,… ,

，

Is an unknown vector

The transposing of (1).

3. The method for compound adaptive control of nonlinear systems under input constraints as claimed in claim 2, wherein the specific method of step S3 is as follows:

s31, constructing the following dynamical equation to generate an auxiliary signal, wherein the auxiliary signal is an amplification error driven by a saturation error signal of a control actuator

And the auxiliary signal is used to represent the effect on the system in the event of actuator saturation;

wherein the content of the first and second substances,

is a designable normal number that is,

and is provided by the composite adaptive law in step S5,

is that

Wherein, in

For the control input error signal for the actuator in the saturated condition, v, u are represented in the form of a state space rewritten as described in step S12,

is to increase the error

The ith element of (1);

s32, defining the augmented tracking error vector as

And is and

，

defining scalar tracking error

Tracking error vector for augmentation

A linear combination of elements, and

wherein:

is a measurable state of the system that is,

is a known reference trajectory vector and is,

elements of

Is selected as a polynomial

Is a herwitz stable value.

4. The adaptive control method according to claim 3, wherein the method for calculating the prediction error in step S4 is as follows:

filter using first order exponential stabilization

To the state space form rewritten in step S12

Is filtered on both sides of the kinetic equation of (1), wherein

Is the laplacian operator, and is,

is a known normal number; after filtering, the following can be obtainedAn equation;

in the formula (I), the compound is shown in the specification,

,

；

definition of

,

And

,

,

then the prediction error

Is defined as:

wherein;

is an unknown vector

Is determined by the estimated value of (c),

is unknown control gain

An estimate of (d).

5. The method for complex adaptive control of nonlinear system under input constraint according to claim 4, wherein the method for constructing complex adaptive law for uncertain control gain and unknown constant parameters in step S5 is as follows:

combining the prediction error estimation with the traditional tracking error-based adaptation, a composite adaptation law for uncertain control gain and unknown constant parameters is constructed by the following formula,