Control method based on specified time nonsingular preset performance

Abstract

The invention provides a control method based on specified time nonsingular preset performance. The control method comprises the following steps: establishing an error dynamic model; establishing a preset performance function of specified time; designing an adaptive item of envelope adjustment based on a preset performance function; designing an error conversion variable based on an envelope-adjusted adaptive item to obtain a constrained error dynamic model; and designing a controller based on the error conversion variable. According to the controller designed by the invention, the error variable of the system is sufficient for performance constraint, and the error can still be converged to a neighborhood near zero without generating singularity under the condition of controller saturation limitation and system disturbance; according to the invention, it can be ensured that the preset performance function converges to the final function value at the specified time, and the system error variable satisfies the performance constraint, that is, the error variable is always located between the upper envelope and the lower envelope; according to the method, the singularity of the controller is avoided by adding the adaptive item, and the controller enables the error conversion variable to converge.

Description

Technical Field

[0001] The present invention relates to the field of control technology, and in particular to a control method based on non-singular preset performance at a specified time. Background Art

[0002] Prescribed Performance Control (PFC) is an advanced control strategy. Its core idea is to limit the convergence trajectory of the tracking error by designing a performance function, thereby realizing the constraint and adjustment of the dynamic performance of the control system. This control method not only ensures the ideal control accuracy, but also takes into account the dynamic performance such as overshoot and adjustment time, which can ensure that the control system has good transition quality. Prescribed performance control is widely used in drone control, train control and unmanned boat control.

[0003] However, the traditional preset performance function based on exponential function cannot establish a direct relationship between parameters, and the time for the preset performance function to reach the final value is often infinite. In order to establish a direct relationship between the convergence time and the parameters, the prior art proposes a preset performance function with a specified time, whose convergence time can be directly given by a parameter, which improves the function performance and makes parameter adjustment simple. However, when the actuator suddenly saturates or is disturbed by external disturbances, even if timely and effective compensation measures are taken, the control error will show a significant instantaneous increase compared to before the saturation and disturbance occurs, which may cause the control error to gradually approach or even exceed the constraint envelope, thereby causing control singularity. Summary of the invention

[0004] In view of the shortcomings of the prior art, the present invention provides a control method based on non-singular preset performance at a specified time. The controller designed by the present invention makes the error variable of the system meet the performance constraints, and in the presence of controller saturation limitations and system disturbances, the error can still converge to a neighborhood near zero without generating singularity.

[0005] The technical solution of the present invention is: a control method based on non-singular preset performance at a specified time, comprising the following steps:

[0006] S1), establish error dynamics model;

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

[0008] S3), designing an adaptive item of envelope adjustment based on a preset performance function;

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

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

[0011] As a preferred embodiment, in step S1), it is assumed that the expected value of the system is x d , the systematic error variable is defined as X 1 =x 1 -x d ; According to the kinetic equation:

[0012]

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

[0014]

[0015] In the formula, and is the error variable X 1 and X 2 The derivative of and is the expected value x d The first and second derivatives of f(x 1 ,x 2 ) is a nonlinear function with f(0,0)=0; g(x 1 ,x 2 ) is the input function, which is the control input of the system; d is the external disturbance, x 1 、x 2 are the system state variables and their derivatives, For x 1 The first derivative of .

[0016] Preferably, in step S2), the expression of the preset performance function at the specified time is:

[0017]

[0018] Where t is time, ρ 0 is the initial value of the preset performance function, ρ ∞ is the final value of the preset performance function, T a is the time parameter to reach the final value.

[0019] Preferably, in step S2), the preset performance function at the specified time satisfies:

[0020] ρ(0)=ρ 0 ;

[0021] ρ(T a )=ρ ∞ ;

[0022] Preferably, in step S3), the preset performance control uses an error conversion function so that the system error satisfies the following inequality:

[0023] P l <X 1 <P u ;

[0024] Where P l is the lower envelope, P u It is the upper envelope.

[0025] As a preferred embodiment, in step S3), the upper and lower envelopes P u , P l The definitions are:

[0026] P u =ρ(t)+ψ;

[0027] P l = -ρ(t)-ψ;

[0028] Where ρ(t) is the preset performance function and ψ is the adaptive term.

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

[0030]

[0031] Where τ = ρ ∞ / 2 is a constant.

[0032] Preferably, in step S4), for the error variable X 1 Satisfy P l <X 1 <P u The preset performance of the design error conversion variable ε is:

[0033]

[0034] In the formula,

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

[0036]

[0037] In the formula, u is the output of the controller, that is, the control input of the system; t is time; k P , k I and k D is the controller parameter; ε is the error conversion variable; is the error conversion variable ε.

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

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

[0040] 2. The present invention can ensure that the preset performance function converges to the function final value within the specified time, and ensure that the system error variable meets the performance constraint, that is, the error variable is always between the upper envelope and the lower envelope;

[0041] 3. The present invention avoids the occurrence of singularity in the controller by adding an adaptive term, and the controller of the present invention makes the error conversion variable converge. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 Schematic diagram of the process of the method of Example 1 of the present invention;

[0043] Figure 2 This is a simulation effect diagram of Example 2 of the present invention;

[0044] Figure 3 This is a diagram of the roll angle simulation result in Example 3 of the present invention;

[0045] Figure 4 This is a diagram of the pitch angle simulation result in Example 3 of the present invention;

[0046] Figure 5 This is a diagram of the yaw angle simulation result in Example 3 of the present invention; DETAILED DESCRIPTION

[0047] The specific implementation of the present invention will be further described below in conjunction with the accompanying drawings:

[0048] Example 1

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

[0050] S1) Establish an error dynamics model; specifically:

[0051] In this embodiment, it is assumed that the expected value of the system is x d , the systematic error variable is defined as X 1 =x 1 -x d ; According to the kinetic equation:

[0052]

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

[0054]

[0055] In the formula, and is the error variable X 1 and X 2 The derivative of and is the expected value x d The first and second derivatives of f(x 1 ,x 2 ) is a nonlinear function with f(0,0)=0; g(x 1 ,x 2 ) is the input function, which is the control input of the system; d is the external disturbance, x 1 、x 2 are the system state variables and their derivatives, For x 1 The first derivative of .

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

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

[0058] ρ(0)=ρ 0 ;

[0059] ρ(T a )=ρ ∞ ;

[0060] In the formula, ρ 0 is the initial value of the preset performance function, ρ ∞ is the final value of the preset performance function, T a is the time parameter to reach the final value.

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

[0062]

[0063] Where t is time.

[0064] S3) Designing an adaptive item for envelope adjustment based on a preset performance function; specifically:

[0065] In this embodiment, the preset performance control uses an error conversion function to make the system error satisfy the following inequality:

[0066] P l <X 1 <P u ;

[0067] Where P l is the lower envelope, P u is the upper envelope;

[0068] The upper and lower envelopes are defined as follows:

[0069] P u =ρ(t)+ψ;

[0070] P l = -ρ(t)-ψ;

[0071] Where ρ(t) is the preset performance function; ψ is the adaptive term;

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

[0073]

[0074] Where τ = ρ ∞ / 2 is a constant.

[0075] S4), designing error conversion variables based on the adaptive term of envelope adjustment to obtain a constrained error dynamics model;

[0076] In this embodiment, the error variable X 1 Satisfy P l <X 1 <P u The preset performance of the design error conversion variable ε is:

[0077]

[0078] In the formula,

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

[0080]

[0081] In the formula, u is the output of the controller, that is, the control input of the system; t is time; k P , k I and k D is the controller parameter; ε is the error conversion variable; is the error conversion variable ε.

[0082] Example 2

[0083] This example provides the following kinetic equation:

[0084]

[0085] Where t is the simulation time, and the initial value of the system is set to x 1 (0) = 2; x 2 (0) = -2;

[0086] The default performance function is defined as:

[0087]

[0088] In the formula, ρ 0 is the initial value of the preset performance function, ρ ∞ is the final value of the preset performance function, T a is the time parameter to reach the terminal value;

[0089] The default performance function parameter is set to ρ 0 =5,ρ ∞ = 0.1 and T a =5;

[0090] PID Controller The parameter is set to k P =-5; k I = -1 and k D =-1; the control input saturation limit is [-10,10]; the result is as follows Figure 2 As shown, from Figure 2 It can be seen from the figure that the method of this embodiment can ensure that the preset performance function converges to the function final value within the specified time, and ensure that the system error variable X 1 Satisfy the performance constraint, that is, the error variable is always within the upper envelope P u and the lower envelope P l Moreover, this embodiment adds the preset performance boundary convergence trend of the adaptive term, and shows the "elasticity" of the preset performance function when the error is close to the boundary, avoiding the error variable X 1 Equal to or exceeding the boundary, the occurrence of singularity is avoided.

[0091] Example 3

[0092] This embodiment applies the control method of Embodiment 1 to the attitude control of the UAV, and specifically includes the following steps:

[0093] Step 1: Build the error dynamics model

[0094] Among them, the attitude dynamics of the drone are as follows

[0095]

[0096] Among them, φ, θ and ψ are the roll angle, pitch angle and yaw angle of the UAV 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 They are the moments of inertia of the three axes of the drone in the body coordinate system; u 1 、u 2 and u 3 It is the three-axis control input defined in the drone body coordinate system;

[0097] The attitude error of the drone is e 1 =Ψ-Ψ d , where Ψ = [φθψ] T is the attitude angle vector; d =[φ d θ d ψ d ] T is the desired attitude angle vector of the drone;

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

[0099]

[0100] In the formula, e 2 is the error variable e 1 The derivative of is the error variable e 1 The derivative of for e 2 The derivative of ; A and B are the system matrix and input matrix respectively; u is the control input;

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

[0102]

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

[0104] Step 2: Design a preset performance function for a specified time

[0105] The preset performance function for a specified time is defined as follows:

[0106]

[0107] Among them, ρ 0 is the initial value of the preset performance function, ρ ∞ is the final value of the preset performance function, T a is the time parameter to reach the final value, and ρ(T a )=ρ ∞ , and there are

[0108] Step 3: Design the adaptive terms for envelope adjustment

[0109] The preset performance control uses the error conversion function to make the system error satisfy the following inequality

[0110] P l1 <e 1 <P u1

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

[0112] Among them, the upper and lower envelopes P u1 , P l1 They are defined as:

[0113] P u1 =ρ(t)+ψ 1

[0114] P l1 =-ρ(t)-ψ 1

[0115] Where ρ(t) is the preset performance function; ψ 1 is an adaptive term;

[0116] Therefore, the adaptive term ψ 1 Defined as:

[0117]

[0118] Where τ = ρ ∞ / 2 is a constant.

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

[0120] In order to make the error variable e 1 Satisfy P l1 <e 1 <P u1 The preset performance, error conversion variable ε 1 Designed for:

[0121]

[0122] in

[0123] Step 5: Design the Controller

[0124] In order to make the error conversion variable ε 1 Converge to zero and design the following PID controller:

[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, is ε 1 The derivative of

[0127] In addition, in this embodiment, the initial value of the simulation posture is set to Ψ = [50 -50 0] T deg, expected postureΨ d =[0 00] T deg. The default performance function parameter is set to ρ 0 =90deg,ρ ∞ =1deg and T a =5s.

[0128] The simulation results are shown in Figure 3-5 As shown, from Figure 3-5 It can be seen that the state variable envelope converges to the final value of the preset performance function at the specified time. In addition, the system error is strictly constrained within the envelope of the preset performance function. In the presence of external disturbances and control input saturation limits, the system error variable is still constrained within the domain of the preset performance function final value, and no singular phenomenon occurs.

[0129] The above embodiments and descriptions are only for illustrating the principles and best embodiments of the present invention. Without departing from the spirit and scope of the present invention, the present invention may be subject to various changes and improvements, all of which fall within the scope of the present invention to be protected.

Claims

1. A control method based on non-singular preset performance at a specified time, characterized in that: The following steps are involved: S1), establish error dynamics model; S2), establishing a preset performance function for a specified time; S3), designing an adaptive item of envelope adjustment based on a preset performance function; S4), designing error conversion variables based on the adaptive term of envelope adjustment to obtain a constrained error dynamics model; S5) Design a controller based on the error conversion variable.

2. A control method based on non-singular preset performance at a specified time according to claim 1, characterized in that: In step S1), assume that the expected value of the system is x d , the system error variable is defined as X1 = x1-x d ; According to the kinetic equation: The expression of the error dynamics model is obtained as follows: In the formula, and are the derivatives of the error variables X1 and X2, and is the expected value x d The first and second order derivatives of ; f(x1, x2) is a nonlinear function and f(0, 0) = 0; g(x1, x2) is the input function, which is the control input of the system; d is the external disturbance, x1 and x2 are the system state variables and their derivatives respectively, is the first-order derivative of x1.

3. The control method based on non-singular preset performance at a specified time according to claim 1, characterized in that: In step S2), the expression of the preset performance function at the specified time is: Among them, t is the time, ρ0 is the initial value of the preset performance function, ρ ∞ is the final value of the preset performance function, T a is the time parameter to reach the final value.

4. The control method based on non-singular preset performance at a specified time according to claim 3, characterized in that: In step S2), the preset performance function at the specified time satisfies: ρ(0)=ρ0; ρ(T a )=ρ ∞。 5. The control method based on non-singular preset performance at a specified time according to claim 3, characterized in that: In step S3), the preset performance control uses the error conversion function to make the system error satisfy the following inequality: P l <X1<P u ; Where Pl is the lower envelope, P u It is the upper envelope.

6. A control method based on non-singular preset performance at a specified time according to claim 5, characterized in that: In step S3), the upper and lower envelopes P u , P l The definitions are: P u =ρ(t)+ψ; P l =-ρ(t)-ψ; Where ρ(t) is the preset performance function and ψ is the adaptive term.

7. A control method based on non-singular preset performance at a specified time according to claim 6, characterized in that: In step S3), the expression of the adaptive term ψ is: Where τ = ρ ∞ / 2 is a constant.

8. The control method based on non-singular preset performance at a specified time according to claim 5, characterized in that: In step S4), in order for the error variable X1 to satisfy P l <X1<P u The preset performance of the design error conversion variable ε is: In the formula, 9. A control method based on non-singular preset performance at a specified time according to claim 8, characterized in that: In step S5), the expression of the controller is: In the formula, u is the output of the controller, that is, the control input of the system; t is time; k P , k I and k D is the controller parameter; ε is the error conversion variable; is the error conversion variable ε.

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