CN104698847B - Nonsingular terminal sliding mode (NTSM) designated performance control method of turntable servo system - Google Patents

Abstract

A non-singular terminal sliding mode specified performance control method for a turntable servo system, comprising: establishing a dynamic model of the turntable servo system, initializing the system state, sampling time and related control parameters; using a smooth affine function for the input saturation function in the system Approximation, deduce the model of the turntable servo system with saturation; calculate the tracking error of the control system, FC (funnel control) error variable, its first derivative and second order derivative; based on the turntable servo model with input saturation function, according to the non-singularity Terminal sliding mode theory, choose the neural network to approach the unknown dynamics, design a non-singular terminal sliding mode specified performance controller, and update the weight matrix of the neural network. This method can effectively avoid the influence of saturation function input on the system, and realize the specified performance control effect.

Description

A Nonsingular Terminal Sliding Mode Specified Performance Control Method for Turntable Servo System

technical field

The invention relates to a non-singular terminal sliding mode designated performance control method of a turntable servo system, in particular to a designated performance control method of a turntable servo system with input and output constraints.

Background technique

Turntable servo system is widely used in various fields such as flight control and fire control. For the output-constrained control problem of the turntable servo system, there are many control methods, such as adaptive control, sliding mode control, terminal sliding mode control and so on. The traditional linear sliding mode control method can guarantee the asymptotic convergence of the tracking error of the controlled system, that is: theoretically, the tracking error can converge to the equilibrium point in infinite time. In order to improve the convergence speed of the system tracking error, a nonlinear term is generally added on the basis of the linear sliding mode, and the fast tracking control of the system is realized by designing a dynamic nonlinear sliding mode surface, which is called the terminal sliding mode control method. This method can ensure that the system state can fully track the desired state within a specified time. However, due to the introduction of fractional exponential terms in the design of the sliding surface, the terminal sliding mode method may have problems with singular values and slow convergence when the initial state is far from the equilibrium point. The singular value problem can be avoided by designing the controller with non-singular terminal sliding mode.

There are many methods to realize specified performance control, such as BLF (barrier Lyapunov function) control, PPC (prescribed performance control) control, and FC (funnel control) method. The BLF method can constrain the system state variables to indirectly limit the system tracking error, but the expression of the Lyapunov function in the method is relatively complicated, and it is necessary to ensure that the function is differentiable. PPC uses a new error variable to guarantee the steady-state error specified by the system, but there is a singular value problem. FC proposes a virtual control variable related to the tracking error, and applies the variable to the non-singular terminal sliding mode control.

Saturated nonlinear links widely exist in turntable servo systems, servo motor systems and other industrial engineering fields. The existence of saturation will often lead to reduced system performance, and even cause system instability. Therefore, to improve the control performance, compensation and control methods against saturation are essential. The traditional saturation compensation method is generally to establish an inverse model or an approximate inverse model of saturation, and design an adaptive controller to compensate the influence of saturation by estimating the upper and lower bound parameters of the saturation function. However, in nonlinear systems such as turntable servo systems, the inverse model of saturation is often not easy to obtain accurately. For the saturated input in the system, first use a smooth function to approximate, and then optimize it based on the differential median value theorem to make it a simple affine form, avoiding additional compensation, so that a simple neural network can be used to approximate the unknown function and Unknown parameter.

Contents of the invention

In order to overcome the deficiencies of the existing turntable servo system that cannot avoid additional dead zone compensation and does not allow the system to have unknown parameters, the present invention provides a non-singular terminal sliding mode specified performance control method for the turntable servo system, combined with output restrictions, non-singular terminal Sliding mode and neural network ideas, design controllers, and realize system specified performance tracking.

The technical scheme proposed in order to solve the above technical problems is as follows:

A non-singular terminal sliding mode specified performance control method for a turntable servo system, the control method comprising the following steps:

Step 1, establish the dynamic model of the turntable servo system, initialize the system state, sampling time and control parameters;

1.1 The mechanical dynamic model of the turntable servo system can be described as

in, y=x(t)∈R, u(t)∈R represent system state, control input voltage and motor output respectively; x represents position, m represents load mass, k ₀ represents control gain, f(x,t) is friction force, d(x,t) is a bounded disturbance including measurement noise, electromagnetic interference, and other unknowns, and v(u)∈R represents the input saturation function, expressed as

1.2 Define x ₁ =x, Then formula (1) can be rewritten as

Step 2, the input saturation function in the system is approximated by a smooth affine function, and the model of the turntable servo system with saturation is derived;

2.1 Design a smooth function as follows

Therefore, the expression (2) is approximated as

v(u)=sat(u)=g(u)+d ₁ (u) (5)

Among them, d ₁ (u)=sat(u)-g(u) is a bounded function;

|d ₁ (u)|＝|sat(u)-g(u)|≤v _max (1-tanh(1)) (6)

2.2 According to the differential median value theorem, there is a constant 0<ξ<1, and the formula (4) is transformed into a smooth affine function

in u _ξ = ξu;

2.3 From formula (5) and formula (7), formula (3) is rewritten into the following equivalent form:

Step 3, calculate the control system tracking error, FC error variable, its first order derivative and second order derivative;

3.1 Define the tracking error of the control system as

e(t)=x _d -x (9)

Among them, x _d is the second-order derivable expected trajectory;

3.2 Define the FC error variable as:

in,

F _φ (t)＝δ ₀ exp(a ₀ t)+δ _∞ (11)

Among them, δ ₀ ≥ δ _∞ > 0, |e(0)|<F _φ (0);

3.3 Deriving formula (10), we get

in,

3.4 Deriving formula (12), we get

in,

Step 4, based on the turntable servo model with input saturation function, according to the non-singular terminal sliding mode theory, select the neural network to approximate the unknown dynamics, design the non-singular terminal sliding mode specified performance controller, and update the neural network weight matrix;

4.1 Select the sliding mode flow pattern as

Among them, α>0;

4.2 Differentiate the formula (14) to get

4.3 Substitute formula (8) and formula (13) into formula (15) to get

Among them, the nonlinear function κ is

4.4 In order to approximate the non-linear function κ which cannot be obtained directly, the following neural network is defined

κ＝W ^*T φ(X)+ε (18)

Among them, W ^* is the ideal weight, φ(X) is usually taken as the following Gaussian function

Among them, c=[c ₁ ,c ₂ ,...,c _n ] ^T is the kernel parameter of the Gaussian function, b is the width of the Gaussian function, 0<φ(X)≤1;

4.5 According to the non-singular terminal sliding mode theory, the second sliding mode flow pattern is designed as

4.6 Substitute formula (16) and formula (18) into formula (20) to get

Among them, p, q are positive odd numbers and p<q, is an estimate of W ^* , μ is ε and the estimated value of is the weight estimation error;

4.7 Designing Neural Network Weights regulation of

Step 5, design Lyapunov function

V＝V ₀ +V ₁ +V ₂ (23)

in, with

Deriving formula (23) gives:

Substitute formula (21) and formula (22) into formula (24), if Then the system is judged to be stable.

The invention combines output limitation, non-singular terminal sliding mode and neural network ideas, designs a controller, and realizes specified performance tracking of the system.

The technical idea of the present invention is: aiming at the turntable servo system with unmeasurable state and with saturation function input, a smooth affine function is used to approximate the saturation function, combined with FC method. Nonsingular terminal sliding mode and neural network, design a nonsingular terminal sliding mode specified performance control method for turntable servo system. For the saturated input in the system, a smooth function is firstly used for approximation, and then optimized based on the differential median value theorem to make it a simple affine form, avoiding additional compensation. For the output limit, a new error variable is designed using the FC method, so that the non-singular terminal sliding mode theory is used to design a specified performance tracking control to ensure that the system converges in a finite time. The invention provides a specified performance control method capable of realizing specified performance control and effectively avoiding the influence of saturation function input on the system.

The invention has the advantages of avoiding additional compensation in the dead zone, allowing the system to have unknown parameters, and realizing specified performance control.

Description of drawings

Fig. 1 is a schematic diagram of the saturation function model of the present invention.

Fig. 2 is a schematic diagram of the tracking effect of the present invention.

FIG. 3 is a schematic diagram of the tracking error of the present invention.

Fig. 4 is a control flow chart of the present invention.

detailed description

The present invention will be further described below in conjunction with the accompanying drawings.

Referring to Fig. 1-Fig. 4, a non-singular terminal sliding mode specified performance control method of a turntable servo system comprises the following steps:

Step 1, establish the dynamic model of the turntable servo system, initialize the system state, sampling time and control parameters;

1.1 The mechanical dynamic model of the turntable servo system can be described as

in, y=x(t)∈R, u(t)∈R represent system state, control input voltage and motor output respectively; x represents position, m represents load mass, k ₀ represents control gain, f(x,t) is friction force, d(x,t) is a bounded disturbance including measurement noise, electromagnetic interference, and other unknowns, and v(u)∈R represents the input saturation function, expressed as

1.2 In order to facilitate the design of the controller, define x ₁ = x, Then formula (1) can be rewritten as

Step 2, the input saturation function in the system is approximated by a smooth affine function, and the model of the turntable servo system with saturation is derived;

2.1 Design a smooth function as follows

Therefore, the expression (2) is approximated as

v=sat(u)=g(u)+d ₁ (u) (5)

Among them, d ₁ (u)=sat(u)-g(u) is a bounded function

|d ₁ (u)|＝|sat(u)-g(u)|≤v _max (1-tanh(1)) (6)

2.2 According to the differential median value theorem, there is a constant 0<ξ<1, and the formula (4) is transformed into a smooth affine function

in u _ξ = ξu;

2.3 From formula (3), formula (5) and formula (7), formula (2) is rewritten into the following equivalent form:

Step 3, calculate the control system tracking error, FC error variable, its first order derivative and second order derivative;

3.1 Define the tracking error of the control system as

e(t)=x _d -x (9)

Among them, x _d is the second-order derivable expected trajectory;

3.2 Define the FC error variable as:

in,

F _φ (t)＝δ ₀ exp(a ₀ t)+δ _∞ (11)

Among them, δ ₀ ≥ δ _∞ > 0, |e(0)|<F _φ (0);

3.3 Deriving formula (10), we get

in,

3.4 Deriving formula (12), we get

in,

Step 4, based on the turntable servo model with input saturation function, according to the non-singular terminal sliding mode theory, select the neural network to approximate the unknown dynamics, design the non-singular terminal sliding mode specified performance controller, and update the neural network weight matrix;

4.1 Select the sliding mode flow pattern as

Among them, α>0;

4.2 Differentiate the formula (14) to get

4.3 Substitute formula (8) and formula (13) into formula (15) to get

Among them, the nonlinear function κ is

4.4 In order to approximate the non-linear function κ which cannot be obtained directly, the following neural network is defined

κ＝W ^*T φ(X)+ε (18)

Among them, W ^* is the ideal weight, φ(X) is usually taken as the following Gaussian function

Among them, c=[c ₁ ,c ₂ ,...,c _n ] ^T is the kernel parameter of the Gaussian function, b is the width of the Gaussian function, 0<φ(X)≤1;

4.5 According to the non-singular terminal sliding mode theory, the second sliding mode flow pattern is designed as

4.6 Substitute formula (16) and formula (18) into formula (20) to get

Among them, p, q are positive odd numbers and p<q, is an estimate of W ^* , μ is ε and the estimated value of is the weight estimation error;

4.7 Designing Neural Network Weights regulation of

Step 5, design Lyapunov function

V＝V ₀ +V ₁ +V ₂ (23)

in, with

Deriving formula (23) gives:

Substitute formula (21) and formula (22) into formula (24), if Then the system is judged to be stable.

In order to verify the effectiveness of the proposed method, the following experiments were carried out:

The present invention provides PID method, ordinary sliding mode control (sliding mode control, SMC), nonsingular terminal sliding mode control (NTSMC), nonsingular terminal sliding mode specified performance method (nonsingular terminal sliding mode funnel control, NTSMFC) comparison:

For more effective comparison, all control signal parameters are consistent, x ₁ (0)=0, x ₂ (0)=0, K=0.1, a=2, b=10, c=1, d= -10, δ ₀ =100, δ _∞ =3, a ₀ =0.3, α=2, β=0.2, p=5, q=7, b ₀ =6, μ=0.1. The system model is chosen as h=.2x ₂ sin(x ₂ ), b=6. The limit of the saturation function is chosen as v _max =1.

Tracking the signal of y _d =0.5(sin(t)+sin(0.5t)), it can be seen from Figure 2 that the tracking effect of NTSMFC is better than other methods; it can be seen from Figure 3 that the tracking steady-state error of NTSMFC is the smallest , and the overshoot is also the smallest, and can reach stability within 1 second. Therefore, the present invention provides a specified performance control method capable of realizing specified performance control and effectively avoiding the influence of saturation function input on the system.

The above set forth is the excellent optimization effect shown by an embodiment of the present invention. Obviously, the present invention is not limited to the above-mentioned embodiment. It can be implemented in various modifications.

Claims (1)

1. A non-singular terminal sliding mode specified performance control method of a turntable servo system, characterized in that: the control method may further comprise the steps: Step 1, establish the dynamic model of the turntable servo system, initialize the system state, sampling time and control parameters; 1.1 The mechanical dynamic model of the turntable servo system can be described as in, y=x(t)∈R, u(t)∈R represent system state, control input voltage and motor output respectively; x represents position, m represents load mass, k ₀ represents control gain, f(x,t) is friction force, d(x,t) is a bounded disturbance including measurement noise, electromagnetic interference, and other unknowns, and v(u)∈R represents the input saturation function, expressed as 1.2 Define x ₁ =x, Then formula (1) can be rewritten as Step 2, the input saturation function in the system is approximated by a smooth affine function, and the model of the turntable servo system with saturation is derived; 2.1 Design a smooth function as follows Therefore, the expression (2) is approximated as v(u)=sat(u)=g(u)+d ₁ (u) (5) Among them, d ₁ (u)=sat(u)-g(u) is a bounded function; |d ₁ (u)|＝|sat(u)-g(u)|≤v _max (1-tanh(1)) (6) 2.2 According to the differential median value theorem, there is a constant 0<ξ<1, and the formula (4) is transformed into a smooth affine function in u _ξ = ξu; 2.3 From formula (5) and formula (7), formula (3) is rewritten into the following equivalent form: Step 3, calculate the control system tracking error, FC error variable, its first order derivative and second order derivative; 3.1 Define the tracking error of the control system as e(t)=x _d -x (9) Among them, x _d is the second-order derivable expected trajectory; 3.2 Define the FC error variable as: in, F _φ (t)＝δ ₀ exp(a ₀ t)+δ _∞ (11) Among them, δ ₀ ≥ δ _∞ > 0, |e(0)|<F _φ (0); 3.3 Deriving formula (10), we get in, 3.4 Deriving formula (12), we get in, Step 4, based on the turntable servo model with input saturation function, according to the non-singular terminal sliding mode theory, select the neural network to approximate the unknown dynamics, design the non-singular terminal sliding mode specified performance controller, and update the neural network weight matrix; 4.1 Select the sliding mode flow pattern as Among them, α>0; 4.2 Differentiate the formula (14) to get 4.3 Substitute formula (8) and formula (13) into formula (15) to get Among them, the nonlinear function κ is 4.4 In order to approximate the non-linear function κ which cannot be obtained directly, the following neural network is defined Among them, W ^* is the ideal weight, φ(X) is taken as the following Gaussian function Among them, c=[c ₁ ,c ₂ ,...,c _n ] ^T is the kernel parameter of the Gaussian function, b is the width of the Gaussian function, 0<φ(X)≤1; 4.5 According to the non-singular terminal sliding mode theory, the second sliding mode flow pattern is designed as 4.6 Substitute formula (16) and formula (18) into formula (20) to get Among them, p, q are positive odd numbers and p<q, is the estimated value of W ^* , μ is the estimated value of ε, is the weight estimation error; 4.7 Designing Neural Network Weights regulation of Step 5, design Lyapunov function V＝V ₀ +V ₁ +V ₂ (23) in, with Deriving formula (23) gives: Substitute formula (21) and formula (22) into formula (24), if Then the system is judged to be stable.