A control method based on a specified time non-singular preset performance

Abstract

The application provides a control method based on non-singular preset performance at a specified time, comprising: establishing an error dynamics model; establishing a preset performance function at a specified time; designing an envelope adjusted adaptive term based on the preset performance function; designing an error conversion variable based on the envelope adjusted adaptive term to obtain a constrained error dynamics model; and designing a controller based on the error conversion variable. The controller designed by the application can make the error variable of the system meet the performance constraint, and in the case of controller saturation limitation and system disturbance, the error can still converge to the neighborhood of zero and no singularity is generated. The application can guarantee that the preset performance function converges to the final value of the function at the specified time, and guarantee that the error variable of the system meets the performance constraint, that is, the error variable is always located between the upper envelope and the lower envelope. The application avoids singularity of the controller by adding an adaptive term, and the controller of the application makes the error conversion variable converge.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of control, in particular to a control method based on specified time non-singular prescribed performance. BACKGROUND

[0002] Prescribed performance control (PFC) is an advanced control strategy, the core idea of which is to limit the convergence trajectory of tracking error by designing a performance function, so as to realize the constraint and adjustment of the dynamic performance of the control system. This control method can guarantee ideal control precision, and also takes into account the dynamic performance such as overshoot and regulation time, so as to ensure that the control system has good transition quality. Prescribed performance control is widely used in unmanned aerial vehicle control, train control and unmanned ship control.

[0003] However, the traditional prescribed performance function based on exponential function cannot establish a direct relationship between parameters and parameters, and the time for the prescribed performance function to reach the final value is often infinite. In order to establish a direct relationship between the convergence time and the parameter, the prior art proposes a specified time prescribed performance function, the convergence time of which can be directly given by a parameter, which improves the function performance and makes the parameter adjustment simple. However, this fixed boundary prescribed performance control method will cause a significant instantaneous increase in the control error relative to the saturation and disturbance before it occurs, even if timely and effective compensation measures are taken, which may cause the control error to gradually approach or even exceed the constraint envelope, thereby causing control singularity. SUMMARY

[0004] In view of the deficiencies of the prior art, the present application provides a control method based on specified time non-singular prescribed performance. The controller designed by the present application can make the error variable of the system satisfy the performance constraint, and in the case of controller saturation limit and system disturbance, the error can still converge to the neighborhood of zero without singularity.

[0005] The technical scheme of the present application is as follows: a control method based on specified time non-singular prescribed performance, comprising the following steps:

[0006] S1), establishing an error dynamics model;

[0007] S2), establishing a specified time prescribed performance function;

[0008] S3), designing an envelope adjustment adaptive term based on the prescribed performance function;

[0009] S4), designing an error conversion variable based on the envelope adjustment adaptive term to obtain a constrained error dynamics model;

[0010] S5), designing a controller based on the error conversion variable.

[0011] As preferred, in step S1), the desired value of the system is assumed to be x d , and the system error variable is defined as X1=x1-x d ; according to the dynamic equation:

[0012]

[0013] The expression of the error dynamic model is obtained as:

[0014]

[0015] wherein, and are the derivatives of the error variables X1 and X2, and are the first and second derivatives of the desired value x d ; f(x1,x2) is a nonlinear function and has f(0,0)=0; g(x1,x2) is an input function, which is the control input of the system; d is an external disturbance, x1 and x2 are the state variables of the system and their derivatives, is the first derivative of x1.

[0016] As preferred, in step S2), the expression of the specified time pre-set performance function is:

[0017]

[0018] wherein, t is time, ρ0 is the initial value of the pre-set performance function, ρ ∞ is the final value of the pre-set performance function, and T a is the time parameter to reach the final value.

[0019] As preferred, in step S2), the specified time pre-set performance function satisfies:

[0020] ρ(0)=ρ0;

[0021] ρ(T a )=ρ ∞ ;

[0022] As preferred, in step S3), the pre-set performance control makes the system error satisfy the following inequality through the error conversion function:

[0023] P l <X1<P u ;

[0024] wherein, P l is the lower envelope, and P u is the upper envelope.

[0025] As preferred, in step S3), the upper and lower envelopes P u , P l are defined as:

[0026] P u = p(t) + ψ;

[0027] P l = -p(t) - ψ;

[0028] where p(t) is a predetermined performance function; and ψ is an adaptive term.

[0029] As preferred, in step S3), the expression of the adaptive term ψ is:

[0030]

[0031] where τ = p ∞ / 2 is a constant.

[0032] As preferred, in step S4), in order to make the error variable X1 satisfy the predetermined performance P l < X1 < P u , the error conversion variable ε is designed as:

[0033]

[0034] where

[0035] As preferred, in step S5), the expression of the controller is:

[0036]

[0037] where u is the output of the controller, i.e. the control input of the system; t is time; k P , k I and k D are controller parameters; and ε is the error conversion variable.

[0038] The beneficial effects of the present application are:

[0039] 1. The controller designed in the present application makes the error variable of the system satisfy the performance constraint, which is determined by a smooth curve with specified convergence time, and in the presence of controller saturation limit and system disturbance, the error still converges to the neighborhood of zero without singularity;

[0040] ​2、The application can guarantee that the preset performance function converges to the function final value at the specified time, and guarantee that the system error variable meets the performance constraint, that is, the error variable is always located between the upper envelope and the lower envelope;

[0041] 3、The application avoids singularity of the controller by increasing the adaptive term, and the controller converges the error conversion variable. BRIEF DESCRIPTION OF DRAWINGS

[0042] Figure 1 It is a flowchart of the method of the embodiment 1 of the application;

[0043] Figure 2 It is an effect diagram of the simulation of the embodiment 2 of the application;

[0044] Figure 3 It is a roll angle simulation result diagram in the embodiment 3 of the application;

[0045] Figure 4 It is a pitch angle simulation result diagram in the embodiment 3 of the application;

[0046] Figure 5 It is a yaw angle simulation result diagram in the embodiment 3 of the application; DETAILED DESCRIPTION

[0047] The specific embodiment of the application will be further described below in combination with the drawings:

[0048] Embodiment 1

[0049] As shown in the figure, the embodiment provides a control method based on specified time non-singular preset performance, comprising the following steps: Figure 1

[0050] S1), an error dynamics model is established; specifically:

[0051] The embodiment assumes that the expected value of the system is x d , and the system error variable is defined as X1=x1-x d ; according to the dynamics equation:

[0052]

[0053] The expression of the error dynamics model is obtained as:

[0054]

[0055] In the formula, and are the derivatives of the error variables X1 and X2, and are the expected values x d ​first and second derivatives; f(x1, x2) is a nonlinear function and has f(0, 0) = 0; g(x1, x2) is an input function, which is a control input of the system; d is an external disturbance, and x1 and x2 are respectively a state variable of the system and a derivative thereof, is a first derivative of x1.

[0056] S2), establishing a specified time preset performance function;

[0057] In the embodiment, the specified time preset performance function is:

[0058] ρ(0) = ρ0;

[0059] ρ(T a ) = ρ ∞ ;

[0060] In the formula, ρ0 is an initial value of the preset performance function, ρ ∞ is a final value of the preset performance function, T a is a time parameter for reaching the final value.

[0061] The expression of the preset performance function is:

[0062]

[0063] In the formula, t is time.

[0064] S3), designing an adaptive term of envelope adjustment based on the preset performance function; specifically,

[0065] In the embodiment, the preset performance control makes the system error satisfy the following inequality through an error conversion function:

[0066] P l <X1<P u ;

[0067] In the formula, P l is a lower envelope, and P u is an upper envelope.

[0068] In the formula, the definitions of the upper and lower envelopes are respectively:

[0069] P u = ρ(t) + ψ;

[0070] P l = -ρ(t) - ψ;

[0071] In the formula, ρ(t) is the preset performance function, and ψ is the adaptive term.

[0072] Therefore, the expression of the adaptive term ψ is:

[0073] Therefore, the expression of the adaptive term ψ is:

[0074] where τ = p ∞ / 2 is a constant.

[0075] S4), an adaptive term design error conversion variable based on envelope adjustment, to obtain a constrained error dynamics model;

[0076] In this embodiment, in order to make the error variable X1 satisfy P l X1 < P u , the error conversion variable ε is designed as:

[0077]

[0078] where,

[0079] S5), a controller based on the error conversion variable, the expression of the controller being:

[0080]

[0081] where u is the output of the controller, i.e. the control input of the system; t is time; k P , k I and k D are controller parameters; and ε is the error conversion variable.

[0082] Embodiment 2

[0083] This embodiment gives the following dynamics equation:

[0084]

[0085] where t is simulation time, and the initial values of the system are set as x1(0) = 2 and x2(0) = -2.

[0086] The preset performance function is defined as:

[0087]

[0088] where p0 is the initial value of the preset performance function, p ∞ is the final value of the preset performance function, and T a is the time parameter for reaching the final value.

[0089] The preset performance function parameters are set as p0 = 5, p ∞ = 0.1 and T a = 5.

[0090] The parameters of the PID controller are set as k​P = -5; k I = -1 and k D = -1; control input saturation limited to [-10, 10]; results as shown in Figure 2 Figure 2 It can be seen from the above that the method of the embodiment can guarantee that the preset performance function converges to the function terminal value at the specified time, and guarantee that the system error variable X1 satisfies the performance constraint, i.e., the error variable is always located between the upper envelope P u and the lower envelope P l Moreover, the preset performance boundary convergence trend of the embodiment is unchanged by increasing the adaptive term, and presents "elasticity" of the preset performance function when the error is close to the boundary, avoiding the error variable X1 being equal to or exceeding the boundary, and thus avoiding singularity.

[0091] Embodiment 3

[0092] The control method of embodiment 1 is used for unmanned aerial vehicle attitude control in the embodiment, and specifically includes the following steps:

[0093] Step 1: Establishing an error dynamics model

[0094] Wherein, the unmanned aerial vehicle attitude dynamics is as follows

[0095]

[0096] Wherein, φ, θ and ψ are the roll angle, pitch angle and yaw angle of the unmanned aerial vehicle, respectively; are the first and second order derivatives of the roll angle φ, respectively; are the first and second order derivatives of the pitch angle θ, respectively; are the first and second order derivatives of the yaw angle ψ, respectively; I xx , I yy and I zz are the moments of inertia of the three axes of the unmanned aerial vehicle in the body coordinate system; u1, u2 and u3 are three-axis control inputs defined in the body coordinate system of the unmanned aerial vehicle;

[0097] The unmanned aerial vehicle attitude error is e1 = Ψ - Ψ d , wherein Ψ = [φ θ ψ] T is the attitude angle vector; Ψ d = [φ d θ d ψ d ] T is the expected attitude angle vector of the unmanned aerial vehicle;

[0098] Therefore, the dynamics model in the state space form is as follows:

[0099]

[0100] ​where e2 is the derivative of the error variable e1; is the derivative of e2; is the derivative of e2; A, B are system matrix, input matrix respectively; u is control input;

[0101] where the system matrix A and input matrix B are defined as follows:

[0102]

[0103] where diag(·) represents a diagonal matrix.

[0104] Step 2: Design the specified time pre-specified performance function

[0105] The specified time pre-specified performance function is defined as follows:

[0106]

[0107] where ρ0 is the initial value of the pre-specified performance function, ρ ∞ is the final value of the pre-specified performance function, T a is the time parameter to reach the final value, and ρ(T a ) = ρ ∞ , and has

[0108] Step 3: Design the adaptive term of envelope adjustment

[0109] The pre-specified performance control makes the system error satisfy the following inequality through the error conversion function

[0110] P l1 < e1 < P u1

[0111] where P l1 is the lower envelope, and P u1 is the upper envelope;

[0112] where the upper and lower envelopes P u1 , P l1 are defined as follows:

[0113] P u1 = ρ(t) + ψ1

[0114] P l1 = - ρ(t) - ψ1

[0115] where ρ(t) is the pre-specified performance function; ψ1 is the adaptive term;

[0116] Therefore, the adaptive term ψ1 is defined as:

[0117]

[0118] In the formula, τ=ρ ∞ / 2 is a constant.

[0119] Step 4: Obtain the constrained error dynamics model through error transformation.

[0120] To make the error variable e1 satisfy P l1 <e1<P u1 The preset performance, the error transformation variable ε1 is designed as follows:

[0121]

[0122] in

[0123] Step 5: Design the controller

[0124] To make the error transformation variable ε1 converge to zero, the following PID controller is designed:

[0125]

[0126] In the formula, k p,i k I,i k D,i (i = 1, 2, 3) are the control parameters of the PID controller. It is the derivative of ε1;

[0127] Furthermore, in this embodiment, the initial simulated attitude value is set to Ψ = [50 - 50 0]. T deg, desired posture Ψ d =[0 00] T deg. The default performance function parameters are set to ρ0 = 90deg, ρ ∞ =1deg and T a =5s.

[0128] The simulation results can be found in [link to simulation results]. Figures 3-5 As shown, from Figures 3-5 As can be seen, the envelope of the state variables converges to the final value of the preset performance function within a specified time. Furthermore, the system error is strictly constrained within the envelope formed by the preset performance function. Even under external disturbances and control input saturation constraints, the system error variable remains constrained within the domain of the final value of the preset performance function, and no singular phenomena occur.

[0129] The embodiments and descriptions above are merely illustrative of the principles and preferred embodiments of the present invention. Various changes and modifications may be made to the present invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed.

Claims

1. A control method based on a specified time non-singular preset performance, characterized by, The method comprises the following steps: S1), establishing an error dynamics model; S2), establishing a preset performance function at a specified time; the expression of the preset performance function at the specified time is: wherein t is time, is a preliminary value of the preset performance function, is a final value of the preset performance function, is a time parameter for reaching the final value; S3), an adaptive term based on a preset performance function to design envelope adjustment; the adaptive term The expression is: ； wherein is a constant; S4), designing an error conversion variable based on an envelope adjustment adaptive term to obtain a constrained error dynamics model; S5), designing a controller based on the error conversion variable.

2. The control method based on the specified time non-singular preset performance according to claim 1, characterized in that: In step S1), assuming the desired value of the system is , the system error variable is defined as ; according to the kinetic equation: The expression of the error dynamics model is: ； where and are error variables and are derivatives of and are first and second derivatives of ; is a nonlinear function and has ; is an input function, is a control input to the system; is an external disturbance, , are system state variables and their derivatives, respectively, is a first derivative of .

3. The control method based on the specified time non-singular preset performance according to claim 1, characterized in that: In step S2), the preset performance function at the specified time satisfies: ； 。 4. The control method based on the specified time non-singular preset performance according to claim 1, characterized in that: In step S3), the preset performance control makes the system error satisfy the following inequality through an error conversion function: ； wherein is a lower envelope, is an upper envelope.

5. The control method based on a specified time non-singular preset performance according to claim 4, characterized in that: In step S3), the upper and lower envelopes , are defined as: ； ； In the formula, is a preset performance function; is an adaptive term.

6. The control method based on the specified time non-singular preset performance according to claim 4, characterized in that: In step S4, in order to the error variable satisfying the preset performance, the error conversion variable is designed as: In the formulae, .

7. The control method based on a specified time non-singular preset performance according to claim 6, characterized in that: In step S5), the expression of the controller is: wherein is the output of the controller, i.e. the control input to the system; t is time; , and are controller parameters; is the error conversion variable; is the error conversion variable .

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