Fast finite-time control method for non-strict feedback systems under quantized input conditions

Abstract

The present invention relates to a fast finite-time control method for a non-strict feedback system under quantized inputs, belonging to the technical field of control engineering. The control method includes: S1, determining a non-strict feedback control system and selecting a suitable quantizer Q(·); S2, designing a suitable fuzzy system and extending the backstepping method to the non-strict feedback system by using the characteristics of the fuzzy system kernel function; S3, selecting a suitable Lyapunov function and determining the conditions that need to be satisfied to achieve fast finite-time stability; S4, based on steps S1, S2, and S3, designing a controller by combining the backstepping method, fuzzy control technology, and nonlinear decomposition method; S5, stability analysis to prove that the system satisfies the fast finite-time stability conditions. The present invention simultaneously considers the quantized inputs and non-strict feedback structure of the system. The experimental results show that the present invention can effectively extend the backstepping method to the non-strict feedback system, and has the characteristics of high control accuracy and fast response speed.

Description

Technical Field

[0001] The present invention relates to a fast finite-time control method for a non-strict feedback system under quantized input conditions, and belongs to the field of control engineering technology. Background Art

[0002] Due to the complexity of actual control systems, system models often involve uncertain functions. Approximation-based control methods such as fuzzy control and neural network control have received extensive attention due to their strong ability to approximate unknown functions. In order to overcome the difficulties brought by unknown functions in control systems, many studies use neural networks to approximate uncertain functions or adopt adaptive fuzzy control methods to achieve control objectives.

[0003] In addition, with the continuous development of control technology, digital control technology plays a crucial role. When information is transmitted, it needs to be quantized first. Therefore, in most control systems, it is an essential process to use the quantized input signal generated by a quantizer to replace the continuous form of the control rate. It is undeniable that the backstepping method makes the design of the controller more systematic and institutionalized, and has unique advantages in solving problems of uncertain systems. However, most of the methods based on backstepping solve strict feedback systems with a triangular structure, while actual systems often cannot be modeled as such systems due to their structural complexity. For example, the single-link manipulator control system in the field of robot control needs to be modeled as a non-strict feedback system, which leaves room for further research.

[0004] For actual engineering systems, control methods are usually required to achieve corresponding performance indicators in terms of system stability or convergence, which makes asymptotic stability unable to meet actual needs. In recent years, the method of finite-time stability has successfully solved some constraint problems, time-delay problems, etc. The fast finite-time stability control scheme has good transient performance and can meet more requirements of control systems in practical applications.

[0005] However, there is still no good solution to how to design a control strategy for a non-strict feedback system with quantized input conditions that can achieve fast finite-time stability, and while ensuring good dynamic performance of the system, can minimize the number of control parameters to reduce the computational burden. Summary of the Invention

[0006] Technical Problem:

[0007] The main technical problem to be solved by the present invention is: for a control system with a quantized input signal, how to use adaptive control technology to design a quantized input signal and an adaptation rate that can meet the system performance index. It should be noted that when using the backstepping method to design the controller, the present invention not only considers the case where the system is a non-strict feedback system rather than a strict feedback system, but also considers how to satisfy the fast stability of the system.

[0008] Technical solution:

[0009] To solve the above technical problem, the present invention provides a fast finite-time control method for a non-strict feedback system under quantized input conditions, and the method includes the following steps:

[0010] 1. A fast finite-time control method for a non-strict feedback system under quantized input conditions, and the method includes the following steps:

[0011] S1. Determine the non-strict feedback control system and select a suitable quantizer Q(·);

[0012] S2. Design a suitable fuzzy system, and use the characteristics of the fuzzy system kernel function to extend the backstepping method to the design of the non-strict feedback system controller;

[0013] S3. Select a suitable Lyapunov function and judge the conditions that need to be met to achieve fast finite-time stability;

[0014] S4. Based on steps S1, S2 and S3, combine the backstepping method, fuzzy control technology and nonlinear decomposition method to design the controller;

[0015] S5. Stability analysis, and prove that the system meets the fast finite-time stability conditions.

[0016] Further, the step S1 includes the following content:

[0017] For the single-link manipulator control system in the field of robot control, under the condition of considering the application of quantized input, modeling is carried out, and the obtained dynamic model is:

[0018]

[0019] Among them, x1, x2, x3 are the state variables of the system, y is the system output, N is the inertia coefficient, G is the friction coefficient, F is the viscous friction coefficient at the joint, L h is the armature inductance, K m is the back electromotive force coefficient, R m is the armature resistance, and Q(u) is the quantized input signal.

[0020] Further, the step S2 specifically includes the following content:

[0021] (1) Use the approximation ability of the fuzzy system to handle the non - linear terms in the control system, and select the Gaussian function as the basis function of the fuzzy system:

[0022]

[0023] where χ is the basis vector, c j is the center of the receptive field, and w j is the width of the basis function.

[0024] (2) Since the non - strict feedback system does not have a triangular structure, the traditional backstepping method cannot be directly applied to such systems. To solve this problem, this method discovers and utilizes the characteristics of the Gaussian basis function, and the constructed fuzzy system has the following properties:

[0025] For the designed fuzzy system, select Choose the mid - point of the basis function as c i = [c i1 ,..., c im T , i = [1, 2,..., M].

[0026] From the above conditions, it can be known that:

[0027]

[0028]

[0029] where both p and q are positive constants. When they satisfy the condition p < q, the following conclusion can be obtained:

[0030]

[0031] Furthermore, the specific content of step S3 is as follows:

[0032] (1) Considering that this method uses the backstepping method for the design of the controller, it is necessary to design the Lyapunov function for each step in the backstepping method. Here, the strategy of adaptive control is adopted. For an n - order system, the final Lyapunov function is shown in formula (1.7):

[0033]

[0034]

[0035] where V i (x) is the Lyapunov function of the i - th step, V(x) is the final Lyapunov function, is the estimation error, σ1 and μ​i are all normal constants, z i is an error variable and can be expressed as:

[0036]

[0037] (2) For a nonlinear system, define the Lyapunov function as V(x), and select some appropriate scalars as m1 > 0, m2 > 0, and 1 > τ > 0. If the system satisfies the conditions of fast finite-time stability, then it is necessary to satisfy:

[0038]

[0039] At this time, for a constant ∈ in the open interval (0, m2), there exists a settling time T. When the time t satisfies t ≥ T, there is:

[0040]

[0041] When the initial time is t0, the time T can be calculated by the following formula:

[0042]

[0043] Furthermore, the step S4 specifically includes the following content:

[0044] (1) For the nth-order nonlinear control system shown in formula (1.12), use the backstepping method to design the control law and the adaptation law,

[0045]

[0046] where i = 2, 3,..., n - 1, function and are two unknown smooth functions, y ∈ R is the system output, and Q(u) is the quantized input signal.

[0047] Considering the influence of the quantized input signal, when designing the controller, first complete the first n - 1 steps in the backstepping method, and then decompose the quantized input signal to complete the design of the nth step.

[0048] (2) Design of the virtual control law and the adaptation law (the first n - 1 steps):

[0049] For the general model (1.12) discussed in this method, considering the fuzzy system involved in step S2 and the stability criterion involved in step S3, design the virtual control law and the adaptation law.

[0050] Using the fuzzy system to process the nonlinear terms, we can get:

[0051]

[0052] Among them, is the output of the fuzzy system, Further calculation gives:

[0053]

[0054] Virtual controller and adaptation rate are:

[0055]

[0056] Among them, the main parameters respectively satisfy g i > 0, k i > 0, 1 > γ > 0 and δ i > 0.

[0057] (3) Design of actual control rate and adaptation rate (step n):

[0058] Adopt steps similar to the first n - 1 steps of design in S4, and consider using the method of nonlinear decomposition to process the quantized input signal while performing corresponding calculations, and it can be calculated that:

[0059]

[0060] Actual controller u(t) and adaptation rate are:

[0061]

[0062] Among them, g n > 0, k n > 0, 1 > γ > 0 and δ n > 0.

[0063] Furthermore, the step S5 specifically includes the following content:

[0064] (1) According to the content involved in step S3 in claim 1 and combining the content in step S4 in claim 1, it can be calculated that:

[0065]

[0066] Among them, A = min{A i}, B = min{B i}, -A i = max{-2k i ψ0, -μ i (1 - γ)},

[0067] (2) The set time can be calculated by formula (1.11). When the time satisfies t≥T, the tracking error of this method can converge to an interval approaching 0 by adjusting the parameters. The calculation formula of the tracking error is as follows:

[0068]

[0069] where 0<β<1.

[0070] Beneficial effects:

[0071] Different from the system with a triangular structure in the strict feedback control system, the research object of the present invention is a non-strict feedback system, and such a system can more specifically describe the actual application system. However, due to the complexity of the structure of this type of system, the prior knowledge cannot be fully utilized, so the backstepping method cannot be directly used for controller design. The present invention cleverly uses a fuzzy system to approximate the nonlinear terms, enabling the backstepping method to solve the controller design problem of non-strict feedback systems. In addition, the present invention considers the quantization input of the actual system, decomposes the input signal, and designs an adaptive fast finite-time control strategy with a simple form, easy to implement, and insensitive to parameter changes. This method can ensure the boundedness of all variables in the system and ensure that the tracking error of the system is within an interval approaching 0. Description of the drawings

[0072] To more clearly illustrate the technical solution of the present invention and the effects achieved in the implementation examples, the technical solution and implementation examples of the present invention will be introduced in the form of drawings below. Figure 2 and Figure 3 are all the results obtained on system (2.1); Figures 4 - 8 is the result obtained on system (2.2). The drawings shown here include:

[0073] Figure 1 is the system structure diagram of the present invention;

[0074] Figure 2 is the comparison chart of the tracking effects between the present invention and the traditional finite-time method (in the legend, output 1 is the output result of the present invention, and output 2 is the output result of the traditional finite-time method);

[0075] Figure 3 is the comparison chart of the tracking errors between the present invention and the traditional finite-time method (in the legend, error 1 is the tracking error of the present invention, and error 2 is the tracking error of the traditional finite-time method);

[0076] Figure 4 is the output effect diagram of system (2.2);

[0077] Figure 5 The tracking error of system (2.2);

[0078] Figure 6 The trajectories of the states x2 and x3 of system (2.2);

[0079] Figure 7 The variations of the adaptive rates θ1, θ2, and θ3 of system (2.2);

[0080] Figure 8 The graphs of the control rate u and the quantized input signal Q(u) of system (2.2). Detailed implementation manners

[0081] In order to more clearly illustrate the technical solutions in the present invention, the following content will describe the present invention in conjunction with the attached Figure 1 The examples cited are only used to explain the present invention and are not used to limit the scope of the present invention. Based on the embodiments in the present invention, other cases obtained by other persons without creative efforts fall within the scope of protection of the present invention.

[0082] According to Figure 1 , the steps for implementing a fast finite-time control method for a non-strict feedback system under quantized input conditions proposed by the present invention mainly include the following steps:

[0083] S1. Determine the non-strict feedback control system and select a suitable quantizer Q(·);

[0084] S2. Design a suitable fuzzy system and use the characteristics of the fuzzy system kernel function to extend the backstepping method to the design of the non-strict feedback system controller;

[0085] S3. Select a suitable Lyapunov function and determine the conditions that need to be met to achieve fast finite-time stability;

[0086] S4. Based on steps S1, S2, and S3, combine the backstepping method, fuzzy control technology, and nonlinear decomposition method to design the controller;

[0087] S5. Conduct stability analysis to prove that the system meets the fast finite-time stability conditions.

[0088] Specifically, the content of step S1 includes the following:

[0089] For a class of non-strict feedback systems under quantized input conditions, its model can be:

[0090]

[0091] For the single-link manipulator control system in the field of robot control, a dynamic model is obtained by modeling under the condition of considering the application of a quantized input:

[0092]

[0093] where x1, x2, x3 are the state variables of the system, y is the system output, N = 1 kg·m 2 is the inertia coefficient, G = 2 is the friction coefficient, F = 1 N·m·s / rad is the viscous friction coefficient at the joint, L h = 1 H is the armature inductance, K m = 2 N·m / A is the back electromotive force coefficient, R m = 1 Ω is the armature resistance, and Q(u) is the quantized input signal.

[0094] The selected quantizer Q(·) is:

[0095]

[0096] where u min is a value set according to the requirements of the system, u i = ω (1-i) u min , u min > 0 (i = 1, 2,...), 1 > ω > 0 and ε = (1 - ω) / (1 + ω).

[0097] Specifically, the step S2 specifically includes the following contents:

[0098] (1) Use the approximation ability of the fuzzy system to process the non-linear terms in the control system, and select the Gaussian function as the basis function of the fuzzy system:

[0099]

[0100] where χ is the basis vector, c j is the center of the receptive field, and w j is the width of the basis function.

[0101] (2) Since the non-strict feedback system is not a system with a triangular structure, the traditional backstepping method cannot be directly used in such systems. To solve this problem, this method discovers and utilizes the characteristics of the Gaussian basis function, and the constructed fuzzy system has the following characteristics:

[0102] For the designed fuzzy system, select Choose the midpoint of the basis function as c i = [c i1 ,..., c im T , i = [1, 2,..., M].​

[0103] It can be known from the above conditions that:

[0104]

[0105]

[0106] Where both p and q are positive constants. When they satisfy the condition p < q, the following conclusions can be obtained:

[0107]

[0108] Specifically, the step S3 specifically includes the following content:

[0109] (1) Considering that this method uses the backstepping method for the design of the controller, it is necessary to design a Lyapunov function for each step in the backstepping method. Here, an adaptive control strategy is adopted. For an n-order system, the final Lyapunov function is shown in formula (2.9):

[0110]

[0111]

[0112] Where V i (x) is the Lyapunov function of the i-th (0 ≤ i ≤ n - 1) step, and V(x) is the final Lyapunov function. is the estimation error, and both σ1 and μ i are positive constants. z i is the error variable and can be expressed as:

[0113]

[0114] (2) For a nonlinear system, define the Lyapunov function as V(x), and select some appropriate scalars m1 > 0, m2 > 0, and 1 > τ > 0. If the system satisfies the conditions of fast finite-time stability, it is necessary to satisfy:

[0115]

[0116] At this time, for a constant ∈ in the open interval (0, m2), there exists a setting time T. When the time t satisfies t ≥ T, there is:

[0117]

[0118] When the initial time is t0, the time T can be calculated by the following formula:

[0119]

[0120] Specifically, step S4 specifically includes the following content:

[0121] (1) For the nth-order nonlinear control system shown in formula (2.14), the backstepping method is used to design the control law and the adaptation law,

[0122]

[0123] where i = 2, 3,..., n - 1, The function and are two unknown smooth functions, y ∈ R is the system output, and Q(u) is the quantized input signal.

[0124] Considering the influence of the quantized input signal, when designing the controller, the first n - 1 steps in the backstepping method are completed first, and then the quantized input signal is decomposed and processed to complete the design of the nth step.

[0125] (2) Design of the virtual control law and the adaptation law (the first n - 1 steps):

[0126] For the general model (2.14) discussed in this method, considering the fuzzy system involved in step S2 and the stability criterion involved in step S3, the virtual control law and the adaptation law are designed.

[0127] According to the design process of the backstepping method, through calculation, it can be obtained that:

[0128]

[0129] It should be noted that at the ith step, the derivative of the virtual control law can be expressed as:

[0130]

[0131] Furthermore, through calculation, it can be obtained that:

[0132]

[0133] where

[0134] Using the fuzzy system to handle the nonlinear term It can be obtained that:

[0135]

[0136] where is the output of the fuzzy system, Further calculation can obtain:

[0137]

[0138] Virtual controller and adaptation rate are as follows:

[0139]

[0140] where the main parameters respectively satisfy g i > 0, k i > 0, 1 > γ > 0 and δ i > 0.

[0141] (3) Design of the actual control rate and adaptation rate (step n):

[0142] For the controller adopted in the present invention, the selected non - linear decomposition is:

[0143] Q(u(t)) = β(u)u(t) + λ(t), (2.21)

[0144] where 1 - ε ≤ β(u) ≤ 1 + ε, |λ(t)| ≤ u min .

[0145] Adopting steps similar to those in the first n - 1 steps of S4, and considering using the non - linear decomposition method to process the quantized input signal while performing corresponding calculations, we can obtain:

[0146]

[0147] The actual controller u(t) and adaptation rate are as follows:

[0148]

[0149] where g n > 0, k n > 0, 1 > γ > 0 and δ n > 0.

[0150] Specifically, step S5 specifically includes the following content:

[0151] (1) According to the content involved in step S3 of claim 1 and combining the content in step S4 of claim 1, we can calculate:

[0152]

[0153] where A = min{A i}, B = min{B i}, -A i = max{-2k iψ0, -μ i (1 - γ)},

[0154] (2) The set time can be calculated by formula (2.13). When the time satisfies t ≥ T, the tracking error of this method can converge to an interval approaching 0 by adjusting the parameters. The calculation formula of the tracking error is as follows:

[0155]

[0156] where 0 < β < 1.

[0157] Simulation experiment:

[0158] To verify the effectiveness of the control scheme proposed in this invention, simulation experiments are respectively carried out for systems (2.1) and (2.2). This part of the simulation experiment mainly consists of two parts.

[0159] 1. Comparative experiment.

[0160] For system (2.1), controllers and adaptation rates are respectively designed according to formula (2.20) and (2.22) in this invention and the finite-time control method, and the simulation results are compared. In the experiment, the main parameters of the quantizer are set as ε = 0.1, u min = 0.1 and ω = 9 / 11; the main parameters of the controller in the fast finite-time control method are k1 = 20, k2 = 12, g1 = 8, g2 = 10, μ1 = 12, μ2 = 10, γ = 79 / 101; the main parameters of the controller in the finite-time control method are k1 = k2 = 10, g1 = g2 = 1, μ1 = μ2 = 7.5, σ1 = σ2 = 0.05, γ = 79 / 101; the initial conditions of the system are all set as [x1(0), x2(0), θ1(0), θ2(0), u(0)] T = [0.5, -0.3, 0.5, -0.5, 0] T ; the desired output is y r = exp(sin(t)); the fuzzy sets of the fuzzy system basis functions are distributed in the interval [-5, 5], and the demarcation points are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4 and 5 respectively. The comparative experiment results of the two methods are as Figure 2 and Figure 3 shown.

[0161] 2. Implementation of actual cases.

[0162] For the system (2.2), the control rate and the adaptation rate are designed according to the method of the present invention. In the experiment, the main parameters of the controller are k1 = 10, k2 = k3 = 20, g1 = g2 = g3 = 10, μ1 = μ2 = μ3 = 12, γ = 99 / 101; the desired output is y r = 0.5sin(0.6t); the initial conditions of the system are set as x1(0) = x2(0) = x3(0) = 0.15, θ1(0) = θ2(0) = θ3(0) = 0.3, u(0) = 0; the main parameters of the quantizer and the fuzzy system are set the same as those in the comparative experiment. The experimental results for this single-link manipulator control system are as Figures 4 - 8 shown.

[0163] The present invention not only solves the problem of quantized input and the problem of how to design a controller for a non-strict feedback system using the backstepping method, but also enables the system to have good dynamic performance, with a faster convergence speed and smaller tracking error than finite-time control.

[0164] As described above, it is only the specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any other changes or substitutions without adding substantial innovations should be covered by the protection scope of the present invention. Therefore, the specific protection scope of the present invention should be subject to the protection scope of the claims.

Claims

1. A fast finite-time control method for a non-strict feedback system under quantized input conditions, characterized in that, It includes the following steps: S1. Determine a non-strict feedback control system and select a quantizer Q(·); S2. Design a fuzzy system, and use the characteristics of the fuzzy system basis functions to extend the backstepping method to the design of the controller for non-strict feedback systems; S3. Select an appropriate Lyapunov function and determine the conditions that need to be satisfied to achieve fast finite-time stability; S4. Based on steps S1, S2, and S3, combine the backstepping method, fuzzy control technology, and nonlinear decomposition method to design the controller; S5. Stability analysis to prove that the system satisfies the fast finite-time stability conditions; The processing method included in step S4 when designing the control law and adaptation rate is as follows: (1) For an nth-order nonlinear control system as shown in formula (1), use the backstepping method to design the control law and adaptation rate. where \(i = 2, 3, \ldots, n - 1\) function and are two unknown smooth functions, \(y\in\mathbb{R}\) is the system output, and \(Q(u)\) is the quantized input signal; Considering the influence of the quantized input signal, when designing the controller, first complete the first n - 1 steps in the backstepping method, and then decompose the quantized input signal to complete the design of the nth step; (2) Design of the virtual control law and adaptation rate, the first n - 1 steps: For formula (1) discussed in this method, considering the fuzzy system involved in step S2 and the stability criterion involved in step S3, design the virtual control law and adaptation rate; Using the fuzzy system to process the nonlinear terms, we can obtain: Among them, is the output of the fuzzy system, Further calculation gives: Virtual controller and adaptive rate are as follows: where the parameters respectively satisfy g i > 0, k i > 0, 1 > γ > 0 and δ i > 0; (3) Design of the actual control law and adaptation rate, the nth step: Adopt steps similar to the first n - 1 steps in S4, and consider using the nonlinear decomposition method to process the quantized input signal while performing the corresponding calculations, and we can calculate and obtain: The actual controller u(t) and the adaptation rate are as follows: where g n > 0, k n > 0, 1 > γ > 0 and δ n > 0.

2. The fast finite-time control method for a non-strict feedback system under quantized input conditions according to claim 1, characterized in that, The content included in step S1 is as follows: For the single-link manipulator control system in the field of robot control, under the condition of considering the application of a quantized input, model it, and the obtained dynamic model is: Among them, x1, x2, x3 are the state variables of the system, y is the system output, N is the inertia coefficient, G is the friction coefficient, F is the viscous friction coefficient at the joint, L h is the armature inductance, K m is the back electromotive force coefficient, R m is the armature resistance, and Q(u) is the quantized input signal.

3. The fast finite-time control method for a non-strict feedback system under quantized input conditions according to claim 1, characterized in that The specific content included in step S2 is as follows: (1) Use the approximation ability of the fuzzy system to process the nonlinear terms in the control system, and select the Gaussian function as the basis function of the fuzzy system; where χ is the basis vector, c j is the center of the receptive field, and w j is the width of the basis function; (2) Since the non-strict feedback system is not a system with a triangular structure and it is difficult to make full use of the system information, the traditional backstepping method cannot be directly used in such systems. To solve this problem, this method discovers and uses the characteristics of the Gaussian basis function, and the constructed fuzzy system has the following properties: For the designed fuzzy system, select Select the midpoint of the basis function as c i = [c i1 , …, c im T , i = [1, 2, …, M];​ From the above conditions, we can know: After further calculation, we can obtain the following conclusion: where both p and q are positive constants, and they satisfy the condition p < q.

4. The fast finite-time control method for a non-strict feedback system under quantified input conditions according to claim 1, characterized in that, The content included in step S3 is as follows: (1) Considering that this method uses the backstepping method to design the controller, it is necessary to design a Lyapunov function for each step in the backstepping method. Here, an adaptive control strategy is adopted. For an nth-order system, the final Lyapunov function is as shown in formula (13): where V i (x) is the Lyapunov function at the i-th step, and V(x) is the final Lyapunov function. is the estimation error, and σ1 and μ i are both positive constants. z i is the error variable and can be expressed as: (2) For a non - linear system, define the Lyapunov function as V(x), and select some appropriate scalars m1 > 0, m2 > 0, and 1 > τ > 0; if the system satisfies the conditions of fast finite - time stability, then it is required to satisfy: At this time, for a constant ∈ in the open interval (0, m2), there exists a set time T. When the time t satisfies t ≥ T, there is: The time T can be calculated by the following formula: where t0 is the initial time.

5. The fast finite-time control method for a non-strict feedback system under quantized input conditions according to claim 4, characterized in that, The content included in step S5 is as follows: (1) According to the content involved in step S3 and combined with the content in step S4, we can calculate and obtain: where A = min{A i}, B = min{B i}, -A i = max{-2k i ψ0, -μ i (1 - γ)}, (2) The set time can be calculated by formula (11), and when the time satisfies t≥T, the tracking error of this method can converge to an interval approaching 0 by adjusting the parameters; the calculation formula of the tracking error is as follows: Where 0 < β < 1.

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