Finite time continuous full-order terminal sliding mode variable structure controller design method

Abstract

The invention discloses a finite time continuous full-order terminal sliding mode variable structure controller design method, and belongs to the technical field of automation. The method comprises the following steps of: 1, designing a sliding mode variable structure controller for a general second-order continuous time system; 2, designing a full-order continuous terminal sliding mode variable structure controller; step 2.1, designing a novel full-order terminal sliding mode surface; 2.2, designing a finite time continuous terminal sliding mode variable structure controller; 3, performing simulation and contrast experiments; 3.1, performing numerical value comparison examples; and step 3.2, designing a controller of the rotary motion servo device. The invention provides a novel design method of a finite-time full-order continuous terminal sliding mode variable structure controller, and provides a novel full-order terminal sliding mode surface based on a double-limit homogeneity principle. The sliding mode surface can ensure that the system state can reach the original point of the system along the sliding mode surface within limited time when the system is in a sliding mode.

Description

Technical Field

[0001] The present invention belongs to the technical field of automation, and particularly relates to a design method for a finite-time continuous full-order terminal sliding mode variable structure controller. Background Art

[0002] As a kind of extensive dynamic system, the second-order nonlinear control system has long been the focus of numerous research works. In real life, many actual systems can be described by the second-order nonlinear control system model, and then the system can be analyzed and designed. However, in actual control systems, there are often adverse factors such as model uncertainty and external disturbances. The existence of these factors will deteriorate the control quality of the system, resulting in a large error in the closed-loop control system, and even leading to the instability of the control system. Therefore, designing a suitable controller to enable the closed-loop system to still have good control quality in the presence of adverse factors such as system uncertainty and external disturbances is a research problem that urgently needs to be solved.

[0003] In recent years, for the control problem of uncertain nonlinear control systems, many controller solutions have been proposed by researchers, such as adaptive controllers, active disturbance rejection controllers, controllers based on disturbance observers, and sliding mode variable structure controllers. Although the above controller solutions can solve the negative impacts brought by system uncertainty and external disturbances of the system, however, the above controllers can only ensure that the state of the closed-loop system asymptotically stabilizes to the origin of the system. In actual applications, however, the asymptotic stability of the system is not the desired effect, because the state of the closed-loop system can only converge to the origin of the system when time approaches infinity. Therefore, researchers have begun to shift the focus of research to finite-time stability control methods. Compared with traditional methods, this control method can not only make the closed-loop system have a faster convergence speed, but also make the closed-loop system have better robustness.

[0004] The terminal sliding mode variable structure control method is an ideal finite-time robust control method. By introducing a switching term, this control method suppresses the influence of uncertain terms and disturbances in the system. Although it has a good suppression effect, the introduction of the switching term causes high-frequency dynamics, i.e., chattering, to appear in the closed-loop system. The occurrence of chattering will cause harm to the system and increase the energy loss of the system. In addition, the existence of chattering will also reduce the control accuracy of the system. To suppress the negative impact of chattering on the system, researchers have proposed a series of solutions, such as the boundary layer method, the high-order sliding mode variable structure control method, etc. However, the boundary layer-based method approximates the switching term in the controller, which reduces the robustness of the system while suppressing the chattering phenomenon. The high-order sliding mode variable structure control method hides the discontinuous terms in the system into the high-order derivative information. Therefore, the chattering phenomenon no longer appears in the system. However, the structure of this control method is relatively complex. Summary of the Invention

[0005] Technical Objective: In view of the above problems, the present invention proposes a design method for a finite-time continuous full-order terminal sliding mode variable structure controller, which designs a continuous sliding mode variable structure controller to make the state of the closed-loop system reach its desired state within a finite time. At the same time, the traditional discontinuous sliding mode variable structure controller is replaced by a continuous sliding mode variable structure controller, so as to avoid the generation of chattering in the system.

[0006] Technical Solution: To solve the above technical problems, the technical solution adopted by the present invention is as follows:

[0007] A design method for a finite-time continuous full-order terminal sliding mode variable structure controller includes the following steps:

[0008] Step 1: Design a sliding mode variable structure controller for a general second-order continuous-time system;

[0009] Step 2: Design a full-order continuous terminal sliding mode variable structure controller;

[0010] Step 2.1: Design a new full-order terminal sliding mode surface;

[0011] Step 2.2: Design a finite-time continuous terminal sliding mode variable structure controller;

[0012] Step 3: Simulation and comparative experiments;

[0013] Step 3.1: Numerical comparison examples;

[0014] Step 3.2: Design a controller for a rotational motion servo device.

[0015] Preferably, in Step 1, for the general second-order continuous-time system, its expression is:

[0016]

[0017]

[0018] where x 1 (t) and x 2 (t) are the states of the system, f(x, t) is a smooth nonlinear function related to the state x(t) and time t; b is the input vector of the system, u(t) ∈ R is the control input to be designed for the system; d(x, t) ∈ R represents the model uncertainty, parameter perturbation and external disturbance information existing in the system, which satisfies the following conditions: |d(x, t)| ≤ l d and where l d > 0, k d > 0 are two constants;

[0019] Continuous-time nonlinear dynamic system:

[0020]

[0021] where, x = [x 1 ,..., x n T ∈ R n represents the system state in system (3.2), f(x(t)) ∈ R n is a vector of nonlinear functions, and its internal elements may be discontinuous functions.

[0022] Preferably, in step 2.1, the design of the novel full-order terminal sliding mode surface is as follows:

[0023] First, design the following novel full-order terminal sliding mode surface, whose expression is:

[0024]

[0025] where, ci, ki, αi, βi (i = 1, 2, 3) are adjustable parameters and are all constants, and satisfy ci > 0, ki > 0. The function x → 「x」α in this sliding mode surface is specifically defined as 「x」α = |x| α sgn(x);

[0026] According to the homogeneity principle, the parameter αi in the sliding mode surface can be designed as:

[0027]

[0028] And the parameter β i can be designed as where ν ∈ (0, 1);

[0029] ​When the system state trajectory reaches the sliding surface, we can obtain:

[0030]

[0031] Equation (3.6) can also be written as:

[0032]

[0033] Let represent the state vector corresponding to the sliding mode, and the following definition can be obtained:

[0034] f s0 (x) = [x 2 , -k 1 「x 1 」 β1 , -c 1 「x 2 」 α1 T , (3.8)

[0035] f s∞ (x) = [x 2 , -k 3 「x 1 」 β3 , -c 3 「x 2 」 α3 T (3.9)

[0036] For any ν ∈ (0, 1), the parameter α i and the parameter β i satisfy 0 < α 1 < 1 < α 3 and 0 < β 1 < 1 < β 3 ; the vector f s0 (x) and the vector f s∞ (x) are taken as the approximate functions of f s (x) at the 0 limit and the ∞ limit;

[0037] Let the homogeneity weight vector be:

[0038]

[0039]

[0040] It can be obtained that the vector f s0 (x) is a homogeneous function vector of degree k s0 = -1 of r s0 homogeneity, while f s∞ (x) is a homogeneous function vector of degree k s∞ ​​r = 1 s∞ Homogeneous function vector; the sliding mode (3.6) is a two - bounded homogeneous function vector that depends on (r s0 , k s0 , f s0 (x)) and (r s∞ , k ∞ , f s∞ (x));

[0041] Theorem 3.1 Consider the sliding mode in the form of equation (3.6). Design the parameters c i , k i , α i , β i (i = 1, 2, 3) according to the homogeneity principle. Then the sliding mode can reach the origin in finite time, and the upper bound of this finite time is independent of the state of the system at the initial moment.

[0042] Preferably, in step 2.2, the design of the finite - time continuous terminal sliding - mode variable - structure controller is specifically as follows:

[0043] Theorem 3.2 According to system (3.1), design the continuous terminal sliding - mode variable - structure controller as shown in the following equation:

[0044] u = b -1 (t, x)(u eq + u n ), (3.17)

[0045]

[0046]

[0047] v = -(k d + η)sgn(s)-λ「s」 ξ - μ[s」 ε , (3.20)

[0048] where η > 0 is an adjustable parameter variable, and λ, μ, ξ, ε are constants greater than zero; the system state can reach the sliding - mode surface in finite time and converge to the origin of the system along the sliding - mode surface in finite time; and the upper bounds of the finite times required for these two processes are both independent of the state of the system at the initial moment.

[0049] Preferably, in step 2.3, the design of the continuous terminal sliding - mode variable - structure controller based on the observed state is specifically as follows: Assume that the state x 1 (t) is measurable, and assume that the function f(x) is bounded. For system (3.1), define the observation error of the observer as Then the corresponding finite - time state observer can be designed as:

[0050]

[0051] Define the error state and design the corresponding observer control law as follows:

[0052]

[0053] The expression of the observer error dynamic system can be obtained as follows:

[0054]

[0055] where Δ(t) = f(x) - f(^x) + d(t), and here δ and δd are known constants, and the parameters and the parameters can be selected from the following region:

[0056]

[0057] Based on the state observed by the observer and the state x of the system itself 1 (t), design the corresponding continuous terminal sliding mode variable structure controller, and its specific form is as follows:

[0058]

[0059] Preferably, in step 3.1, the specific numerical comparison example is:

[0060] To verify the effectiveness of the method designed by the present invention, the controller method designed by the present invention is used to control the following system, and the expression form of the system is:

[0061]

[0062] According to formula (3.5), the specific form of the full-order terminal sliding mode surface to be designed can be obtained as:

[0063]

[0064] where The remaining parameters are designed as: k 1 = c 1 = 2, k 2 = c 2 = 35, k 3 = c 3 = 3. Based on Theorem (3.2), the specific form of the corresponding continuous terminal sliding mode variable structure controller can be obtained as u = u eq + u n , where:

[0065]

[0066]

[0067] When controlling the system, four different initial states are selected respectively for verification.

[0068] Preferably, the design of the rotational motion servo device controller in step 3.2 is specifically as follows:

[0069] According to the classical Kirchhoff voltage law, we can obtain a relationship about the armature voltage V m (t):

[0070]

[0071] where V m (t) represents the armature circuit voltage, I m (t) represents the armature current, R m represents the armature resistance, L m represents the armature inductance, θ m (t) represents the angular position of the motor shaft, T m (t) represents the torque generated by the motor, E emf (t) represents the back electromotive force of the motor;

[0072] According to formula (3.37), the calculation expression for the armature current is obtained as: I m (t) = V m (t) - E emf (t) / R m ;

[0073] According to Newton's second law, the equivalent load of the motor can be calculated by the following formula:

[0074]

[0075] where T l (t) / η g k g represents the load torque transmitted through the gear, η g represents the gear efficiency, and further the load balance equation is obtained as:

[0076]

[0077] where T l (t) is the load torque, B eq is the viscous damping ratio at the output end. Substituting formula (3.38) into (3.39) gives:

[0078]

[0079] Because ω m (t) = k g ω l (t), and T m (t) = η m k t I m (t), equation (3.40) can be rewritten as:

[0080]

[0081] Combining formula (3.38) with formula (3.41) and performing Laplace transform, we can get the transfer function between DC motor voltage and motor speed:

[0082]

[0083] Among them J eq =J l +η g J m k g 2 , the transfer function between the DC motor voltage and the motor shaft position is:

[0084]

[0085] Among them, θ l Represents the shaft position of the DC motor, select x 1 (t) = θ l (t), As the state variable of the system, and taking the matching disturbance d(x, t) in the system into account in the system model, the state equation of the DC motor can be obtained as follows:

[0086]

[0087] Beneficial effects: Compared with the prior art, the present invention has the following advantages:

[0088] (1) This invention proposes a new finite-time full-order continuous terminal sliding mode variable structure controller design method for a general second-order nonlinear system. Based on the principle of double-limit homogeneity, a new full-order terminal sliding surface is proposed. The sliding surface can ensure that when the system is in sliding mode, the system state can reach the origin of the system along the sliding surface within a finite time. Moreover, the upper bound of the finite time is independent of the state value of the system when it first reaches the sliding surface.

[0089] (2) The present invention also proposes a continuous sliding mode variable structure controller based on the novel full-order terminal sliding mode surface. The control input signal generated by this controller is a continuous signal, thus avoiding the high-frequency chattering phenomenon in the system. At the same time, this controller can ensure that the system state reaches the sliding mode surface within a finite time, and the upper bound of this finite time is independent of the state of the system at the initial moment. Description of the Drawings

[0090] Figure 1 is the response curve of the state x 1 (t) of the closed-loop system of the present invention under different initial values;

[0091] Figure 2 is the response curve of the state x 2 (t) of the closed-loop system of the present invention under different initial values;

[0092] Figure 3 is u of the present invention eq (t) and u n (t) under different initial states;

[0093] Figure 4 is the corresponding maximum convergence time under different initial states of the present invention;

[0094] Figure 5 is the basic structure diagram of the DC servo motor of the present invention;

[0095] Figure 6 is the rotary servo motion system (SRV02) of the present invention;

[0096] Figure 7 is the tracking effect diagram of the motor shaft position of the closed-loop system of the present invention with respect to the reference tracking signal;

[0097] Figure 8 is the diagram of the motor shaft position tracking error eθ(t) of the present invention;

[0098] Figure 9 is the tracking diagram of the motor shaft speed with respect to the reference speed (without additional load) of the present invention;

[0099] Figure 10 is the diagram of the motor shaft speed tracking error eω(t) (without additional load) of the present invention;

[0100] Figure 11 is the tracking diagram of the motor shaft position with respect to the reference signal (with additional load) of the present invention;

[0101] Figure 12 is the response diagram of the motor shaft position tracking error eθ(t) (with additional load) of the present invention;

[0102] Figure 13It is the diagram of the tracking of the motor shaft speed of the present invention with respect to the reference speed (with additional load);

[0103] Figure 14 It is the diagram of the tracking error eω(t) of the motor shaft speed of the present invention (with additional load). Specific embodiments

[0104] The following further clarifies the present invention in conjunction with specific embodiments. The embodiments are implemented on the premise of the technical solution of the present invention. It should be understood that these embodiments are only used to illustrate the present invention and not to limit the scope of the present invention.

[0105] As Figures 1-14 shown, the present invention proposes a design method for a finite-time continuous full-order terminal sliding mode variable structure controller, including:

[0106] Step 1: Design a sliding mode variable structure controller for a general second-order continuous-time system, and the system description is;

[0107] Consider a general class of second-order continuous-time systems, and its expression is:

[0108]

[0109]

[0110] where, x 1 (t) and x 2 (t) are the states of the system, f(x, t) is a smooth non-linear function related to the state x(t) and time t. b is the input vector of the system, and u(t) ∈ R is the control input to be designed for the system. d(x, t) ∈ R represents the model uncertainty, parameter perturbation and external disturbance information existing in the system, and it satisfies the following conditions: |d(x, t)| ≤ l d and where l d > 0, k d > 0 are two constants.

[0111] The present invention mainly aims at system (3.1) to design a continuous sliding mode variable structure controller, so that the state of the closed-loop system reaches its desired state within a finite time. At the same time, the continuous sliding mode variable structure controller is used to replace the traditional discontinuous sliding mode variable structure controller, so as to avoid the generation of chattering phenomenon in the system.

[0112] Consider the following continuous-time non-linear dynamic system:

[0113]

[0114] where x = [x 1 ,..., x n ​T ∈R n represents the system state in system (3.2), and f(x(t)) ∈ R n is a non-linear function vector, and its internal elements may be discontinuous functions. Therefore, the solution of system (3.2) exists in the sense of Filippov.

[0115] Let r = [r 1 ,..., r n ∈ R n be a weight vector, where the element r i > 0, (i = 1,..., n). For any λ > 0, the dilation mapping of the system is defined as Λ r λ (x) = [λ r1 x1,..., λ rn x n T, so we get the following definition:

[0116] Definition and If the condition f(Λ r (x)) = λ k g(x) is satisfied, then the function f(x) ∈ R is called an r-homogeneous function with homogeneous degree k ∈ R. For the function vector f(x) ∈ R n , if each component f i (x), i ∈ {1,..., n}, and satisfies the condition f i (Λ r (x)) = λ k+r if i (x), then the component f i (x) is an r-homogeneous function with homogeneous degree k + r i , and at the same time the function vector f(x) ∈ R n is called an r-homogeneous function with homogeneous degree k.

[0117] Definition 3.2 If the function g(x) ∈ R is a continuous function, the function g p is a non-constant continuous function, and for the set C ∈ R n \{0}, the condition:

[0118]

[0119] is satisfied, then the function g(x) ∈ R is called a p-limit homogeneous function dependent on (r p , k p , g p ). Where rp = [r p,1 ,..., r p,n ∈ R n is the weight vector, k p is the homogeneous degree, g p is the approximation function.

[0120] Definition 3.3 If a function satisfies both 0-limit homogeneity and ∞-limit homogeneity, then this function is called a double-limit homogeneous function.

[0121] Definition 3.4 If the origin of the system (3.2) satisfies Lyapunov stability, and for any R > 0, there exists T > 0 such that the state trajectory of the system starting from the spherical domain ||x|| ≤ R can be stabilized at this origin within time T, then we call this origin globally finite-time stable.

[0122] Definition 3.5 If the origin of the system (3.2) satisfies global finite-time stability, and there exists a fixed positive constant T max such that for any initial time x 0 ∈ R n , the condition T < T max is satisfied, then this origin is called fixed-time stable.

[0123] Lemma 3.1 For the system (3.2), let the function vector f(x) be a double-limit homogeneous function vector depending on (r 0 , k 0 , f 0 ) and (r ∞ , k ∞ , f ∞ ). If the origins of the systems and the system are globally asymptotically stable, then there are the following conclusions:

[0124] 1. If the condition k ∞>0> k 0 is satisfied, then the origin of the system (3.2) is fixed-time stable;

[0125] 2. Let d V0 and d V∞ be two real numbers, and satisfy d V0 > max 1≤i≤n r 0,i and d V∞ > max 1≤i≤n r ∞,i . Then there exists a continuous positive definite function V(x) such that the function is a double-limit homogeneous function depending on and and the function is a negative definite function.

[0126] Lemma 3.2 Consider the following scalar differential equation:

[0127]

[0128] where λ, μ > 0, α > 1, and γ < 1, then there exists a finite time T 0 such that the equation reaches the origin in finite time, and the upper bound of this time can be expressed as:

[0129]

[0130] Step 2: Design of the full-order continuous terminal sliding mode variable structure controller;

[0131] Step 2.1: Design of a new type of full-order terminal sliding mode surface;

[0132] For system (3.1), first, design the following new type of full-order terminal sliding mode surface, whose expression is:

[0133]

[0134] where ci, ki, αi, βi (i = 1, 2, 3) are adjustable parameters and are all constants, and satisfy ci > 0, ki > 0. The function x → 「x」α in this sliding mode surface is specifically defined as 「x」α = |x| α sgn(x).

[0135] According to the homogeneity principle, the parameter αi in the sliding mode surface can be designed as:

[0136]

[0137] And the parameter β i can be designed as where ν ∈ (0, 1).

[0138] When the system state trajectory reaches the sliding mode surface, that is, the condition s = 0 is satisfied, we can obtain:

[0139]

[0140] Equation (3.6) can also be written as:

[0141]

[0142] Let represent the state vector corresponding to the sliding mode, and the following definition can be obtained:

[0143] f s0 (x) = [x 2, -k 1 「x 1 」β1 - c 1 「x 2 」α1] T , (3.8)

[0144] f s∞ (x) = [x 2 , -k 3 「x 1 」β3 - c 3 「x 2 」α3] T (3.9)

[0145] Obviously, for any ν ∈ (0, 1), the parameter α i and the parameter β i satisfy 0 < α 1 < 1 < α 3 and 0 < β 1 < 1 < β 3 . The vector f s0 (x) and the vector f s∞ (x) can be regarded as the approximate functions of f s (x) at the 0 - limit and ∞ - limit.

[0146] Let the homogeneity weight vectors be:

[0147] and

[0148] It can be obtained that the vector f s0 (x) is a vector of r s0 -homogeneity functions with homogeneity degree k s0 = - 1, while f s∞ (x) is a vector of r s∞ -homogeneity functions with homogeneity degree k s∞ = 1. Therefore, it can be obtained that the sliding mode (3.6) is a two - limit homogeneity function vector depending on (r s0 , k s0 , f s0 (x)) and (r s∞ , k ∞ , f s∞ (x)). Thus, the following theorem can be obtained:

[0149] Theorem 3.1 Consider the sliding mode shown in Equation (3.6). If the parameters c i , k i , α i , β iIf (i = 1, 2, 3), then the sliding mode can reach the origin within a finite time, and the upper bound of this finite time is independent of the state of the system at the initial moment.

[0150] Proof: First, select the following Lyapunov function

[0151]

[0152] Taking the derivative of the above Lyapunov function (3.10), we can get:

[0153]

[0154] where, Θ = (β 1 + 1)(β 2 + 1)(β 3 + 1).

[0155] Substituting formula (3.7) into formula (3.11), we can get:

[0156]

[0157] From the above formula (3.12), it can be seen that the derivative of the Lyapunov function satisfies everywhere except at the point x 2 = 0 And when x 2 = 0, it satisfies

[0158] Therefore, it can be obtained that the sliding mode dynamics (3.6) is asymptotically stable to the origin of the system. For the approximate function f s0 (χ), design the corresponding Lyapunov function as:

[0159]

[0160] It can be concluded that:

[0161]

[0162] Therefore, for the approximate system Its global asymptotic stability is guaranteed. In addition, for the approximate system Select the corresponding Lyapunov function as:

[0163]

[0164] Taking the derivative of the Lyapunov function (3.15), we can get:

[0165]

[0166] It holds for any system state χ. Therefore, the approximate system It is globally asymptotically stable at the origin. Based on the above analysis, it can be obtained that the system and the system are both globally asymptotically stable. According to the properties of the two-sided homogeneous system, it can be obtained that the sliding mode (3.6) can converge to the origin of the system in finite time.

[0167] Let \(T_s\) represent the time required for the state of the sliding mode system (3.6) to converge to the origin of the system, and it can be obtained that:

[0168]

[0169] where, is a real number greater than 0,

[0170] Step 2.2: Design of the finite-time continuous terminal sliding mode variable structure controller;

[0171] Theorem 3.2 For the system (3.1), if a continuous terminal sliding mode variable structure controller is designed as shown in the following formula:

[0172] \(u = b\) -1 (t, x)(u eq + u n ), (3.17)

[0173]

[0174]

[0175] \(v = -(k\) d + η)sgn(s) - λ「s」 ξ - μ「s」 ε , (3.20)

[0176] where \(k_i\), \(c_i\), \(α_i\), \(β_i\) (\(i = 1, 2, 3\)) and \(k_d\) have been given above, \(η>0\) is an adjustable parameter variable, and \(λ\), \(μ\), \(ξ\), \(ε\) are constants greater than zero. Then, the system state can reach the sliding surface in finite time and converge to the origin of the system along the sliding surface in finite time. And the upper bounds of the finite time required for these two processes are both independent of the state of the initial moment of the system.

[0177] Proof: For the system (3.1), we can rewrite the sliding surface (3.5) designed in this chapter as:

[0178]

[0179] Substituting the continuous terminal sliding mode variable structure controller (3.17) into formula (3.21) gives:

[0180]

[0181] Substitute the controller (3.18) into formula (3.22), and after rearrangement, we can get:

[0182] s = d(x, t) + u n . (3.23)

[0183] Next, for formula (3.23), after taking its derivative, we can obtain:

[0184]

[0185] Substitute the controller (3.20) into formula (3.24), and we have the following formula:

[0186]

[0187] Select the Lyapunov function V s (s) = 1 / 2s 2 , take the derivative of this Lyapunov function, and substitute formula (3.25) into this derivative, then we can get the derivative of this Lyapunov function as:

[0188]

[0189] Combined with s = 「s」, 「s」「s」 ξ = |s||s| ξ , we can further derive the following formula:

[0190]

[0191] Let ξ > 1, ε ∈ (0, 1), then we can get and Therefore, it can be deduced that the conditions in Lemma (3.2) are satisfied. The system state can reach the sliding mode surface within a finite time. Let the reaching time be T r , and we can obtain the upper bound of this finite time T r as:

[0192]

[0193] Obviously, the upper bound of this reaching time T r has nothing to do with the initial state of the system.

[0194] In addition, the method proposed by the present invention avoids 「xi」 during the design process of the controller αiDerivatives are taken for such power terms to avoid the singularity problem in the system. The controller design method proposed by the present invention can generate continuous control signals, thereby enabling the system to avoid chattering. This makes the controller design method proposed by the present invention more suitable for use in actual control systems.

[0195] So far, by synthesizing the results of Theorem (3.1) and Theorem (3.2), we can obtain the following theorem:

[0196] Theorem 3.3 For system (3.1), designing a full-order terminal sliding mode surface as shown in Equation (3.5) and the corresponding continuous terminal sliding mode variable structure controller (3.17)-(3.20) can make the system state reach the origin of the system within a finite time, and the upper bound of this finite time can be expressed as:

[0197]

[0198] Step 2.3: Design of a continuous terminal sliding mode variable structure controller based on the observed state;

[0199] In the above controller design problem, it is assumed that all state information in the system is available. However, in actual systems, it is very difficult to obtain all the information of the system. For example, in a motor control system, the position of the motor rotor is relatively easy to obtain, but the rotational speed of the motor rotor is difficult to obtain. In such a case, we need to design an observer to observe the state information of the system, and then design a controller based on the observed quantities of the observer for the system.

[0200] Considering the system (3.1) again, assume that only the state x 1 (t) is measurable, and assume that the function f(x) is bounded. In this case, for the system (3.1), define the observation error of the observer as ζ 1 (t) = x 1 (t) - ^x 1 (t), then the corresponding finite-time state observer can be designed as:

[0201]

[0202] Define the error state ζ2(t) = x2(t) - ^x2(t), and design the corresponding observer control law as:

[0203]

[0204] The expression of the observer error dynamic system can be obtained as:

[0205]

[0206] where Δ(t) = f(x) - f(^x) + d(t), and δ and δd here are known constants. The parameters and the parameters can be selected from the following regions:

[0207]

[0208] so as to ensure the finite-time stability of the system.

[0209] Based on the state ^x 2 (t) observed by the observer and the state x 1 (t) of the system itself, a corresponding continuous terminal sliding mode variable structure controller can be designed, and its specific form is as follows:

[0210]

[0211] Substitute the controller (3.32) into the system (3.1), and using a homogeneous Lyapunov function similar to Equation (3.10), it can be proved that the closed-loop system is finite-time stable.

[0212] Step 3: Simulation and comparative experiments

[0213] Step 3.1: Numerical comparison examples;

[0214] To verify the effectiveness of the method designed by the present invention, the controller method designed by the present invention will be used to control the following system. The expression form of this system is:

[0215]

[0216] According to formula (3.5), the specific form of the full-order terminal sliding mode surface to be designed can be obtained as:

[0217]

[0218] where The remaining parameters can be designed as k 1 = c 1 = 2, k 2 = c 2 = 35, k 3 = c 3 = 3. Based on Theorem (3.2), the specific form of the corresponding continuous terminal sliding mode variable structure controller can be obtained as u = u eq + u n , where:

[0219]

[0220]

[0221] When controlling the system, four different groups of initial states are selected respectively to verify the effectiveness and characteristics of the proposed method. The simulation results are as Figures 1 to 4 shown.

[0222] Figure 1 , Figure 2 are the response curves of the closed-loop system states x 1 (t) and x 2 (t). From the simulation results, it can be seen that the state of the system can always converge to the origin of the system within a finite time. Moreover, the upper bound of this finite time is independent of the state of the system at the initial moment. At the same time, it can be seen from the simulation that the adverse effects brought by the uncertain term d(x,t) to the system are also suppressed by the controller.

[0223] Figure 3 is the control input response of the closed-loop control system. The control input signal u(t) is continuous, and there is no chattering phenomenon in the system. At the same time, the amplitude of the equivalent control input ueq(t) changes with the change of the state of the system at the initial moment. When the state of the system at the initial moment is far from the origin of the system, the equivalent control input signal needs to generate a large control force to make the closed-loop system achieve finite-time stability.

[0224] For further illustration, 1000 different initial value conditions are selected for the upper bound of the finite convergence time of the closed-loop system, which is independent of the state of the system at the initial moment. The range of the initial values is from (1, -1) to (104, -104). The 1000 different initial value conditions are divided into four groups. Within each group, the upper bound of the maximum finite convergence time of the group is selected. The results are as Figure 4 shown. Although the range of the initial value conditions of the system varies greatly, there is always an upper bound for the convergence time of the closed-loop system, and this upper bound is independent of the state of the system at the initial moment. In this application, a full-order terminal sliding mode variable structure controller, a high-order super-twisting algorithm controller, and a homogeneous sliding mode variable structure controller are selected to control the system, and the time required for the closed-loop system state to converge to the origin under different initial states for various different types of controllers is compared. As shown in Table 1.

[0225] Table 1 Comparison of the convergence time of the closed-loop system with different controllers

[0226]

[0227] It can be obtained from Table 1 that except for the method proposed in this application, the convergence time of the closed-loop systems corresponding to the other methods all changes with the change of the initial state of the system. By using the method proposed in this application, it can be obtained that the upper bound of the convergence time of the closed-loop system is 6.8 seconds, and this upper bound does not change with the change of the initial state of the system.

[0228] Step 3.2: Design of the controller for the rotational motion servo device;

[0229] Design a finite-time continuous terminal sliding mode variable structure controller for the rotational motion servo device so that the device can achieve the desired motion effect. Controlling the servo device means controlling the DC motor inside the device to drive the load to the desired position.

[0230] As Figure 5 shown, the basic structure of the DC motor is given in Figure 5 , where V m (t) represents the armature circuit voltage, I m (t) represents the armature current, R m represents the armature resistance, L m represents the armature inductance, θ m (t) represents the angular position of the motor shaft, T m (t) represents the torque generated by the motor. E emf (t) represents the back electromotive force of the motor.

[0231] According to the classical Kirchhoff's voltage law, a relation about the armature voltage V m (t) is obtained:

[0232]

[0233] In a traditional DC motor, Lm << Rm. Therefore, the inductance term can be ignored. Then, according to formula (3.37), the calculation expression for the armature current can be obtained as: I m (t) = V m (t) - E emf (t) / R m .

[0234] By further considering the relationship between the motor back electromotive force E emf (t) and the angular velocity ω m (t) of the motor shaft, the circuit current can be obtained as I m (t) = (V m (t) - k m ω m (t)) / R m , where k m is the back electromotive force constant. According to Newton's second law, the equivalent load of the motor can be calculated by the following formula:

[0235]

[0236] where, T l (t) / η g kg represents the load torque transmitted through the gear, η g represents the gear efficiency. Thus, the load balance equation can be further obtained as:

[0237]

[0238] where T l (t) is the load torque, and B eq is the viscous damping ratio at the output end. Substituting Equation (3.38) into (3.39) gives:

[0239] Since ω m (t) = k g ω l (t), and at the same time T m (t) = η m k t I m (t), Equation (3.40) can be rewritten as:

[0240]

[0241] Combining Equation (3.38) and Equation (3.41), and taking the Laplace transform, the transfer function between the DC motor voltage and the motor speed can be obtained as:

[0242]

[0243] where J eq = J l + η g J m k g 2 . At the same time, the transfer function between the DC motor voltage and the motor shaft position can be obtained as:

[0244]

[0245] where θ l represents the shaft position of the DC motor. Selecting x 1 (t) = θ l (t), as the state variable of the system, and considering the matching disturbance d(x,t) in the system into the system model, the state equation of this DC motor can be obtained as:

[0246]

[0247] Next, for the DC motor model (3.44), a finite-time continuous terminal sliding mode variable structure controller will be designed for it. The control objective is to make the shaft position and speed of the motor track the reference target signal. The rotating motion servo system SRV02 produced by Quanser is used to verify the effectiveness of the designed controller in this application.

[0248] Table 3.2 Explanation of Symbol Parameters of the Rotary Servo System

[0249]

[0250] This experimental system is as Figure 6 shown. Its main components are: a Faulhaber coreless DC motor, model 2338S006. This motor is wrapped by an aluminum outer package and contains an internal gearbox. The DC motor provides driving force for the external gearbox through the internal gearbox. The SRV02 system includes three parts of sensors, namely a potentiometer with technical parameters of 10k / ω per revolution and an electric range of 352deg. Secondly, its sensors also include an American Digital E2 single-ended optical shaft encoder, which can provide a resolution of up to 4096 counts per revolution. These two parts of sensors are mainly used to measure the angular position state of the load shaft. Secondly, a tachometer directly connected to the DC motor is also included in this set of systems, and its function is to measure the rotational speed of the load shaft. In the problem of DC motor control, if the control input signal contains high-frequency signals, it will cause serious damage to the brushes and gears of the DC motor. Therefore, the traditional sliding mode variable structure controller cannot be used to control the DC motor, while the continuous terminal sliding mode controller proposed in this application can avoid high-frequency chattering in the system while maintaining the original advantages of the sliding mode variable structure control, so it is more suitable for application in this system.

[0251] The specific meanings and specific values of the parameters in this simulation system are given in Table 2. To more clearly reflect the control effect of the controller, the experiment is divided into two parts, namely the case without an external load and the case with an external load. First, for the case without an external load, let the reference tracking signal be θ d (t) = 30°sin(t), and define the tracking error of the system as e θ (t) = x 1 (t) - θ d , e ω (t) = x 2 (t) - ω d (t). According to the results in Theorem 3.2, the controller can be designed as u(t) = 0.0164(u eq (t) + u n (t)), where:

[0252]

[0253] The tracking effect of the closed-loop system is as Figures 7 to 10 shown. From Figure 7 , 8 the experimental results, it can be obtained that the upper bound of the steady-state tracking error of the closed-loop system is 0.03 rad. Figure 9 Shown in d is the tracking effect of the motor speed on the reference tracking speed ω Figure 10 (t) = -30cos(t) deg / second. The tracking error of its speed tracking is as

[0254] In this set of test systems, additional load counterweights are added to test the robustness of the control system against external disturbances, uncertainties and other factors. The SRV02 system provides an additional disk load, and the load can be connected to the SRV02 load gear to change the moment of inertia. The mass of the disk load is md = 0.04 kg, and the radius of the disk load is rd = 0.05 m. This increase in additional load will be regarded as an external disturbance, and the load counterweight will also generate a displacement force or vibration that can be regarded as a disturbance. For this case, the controller (3.45) is also used to control the system. The control results of the closed-loop system are as Figures 11 to 14 shown.

[0255] The above are only the preferred embodiments of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements should also be regarded as the protection scope of the present invention.

Claims

1. A design method for a finite-time continuous full-order terminal sliding mode variable structure controller, characterized in that, it includes the following steps: Step 1: Design a sliding mode variable structure controller for a general second-order continuous-time system; Step 2: Design a full-order continuous terminal sliding mode variable structure controller; Step 2.1: Design a new full-order terminal sliding mode surface; Step 2.2: Design a finite-time continuous terminal sliding mode variable structure controller; Step 3: Simulation and comparative experiments; Step 3.1: Numerical comparison examples; Step 3.2: Design a controller for a rotary motion servo device.

2. A design method for a finite-time continuous full-order terminal sliding mode variable structure controller according to claim 1, characterized in that, in Step 1, for a general second-order continuous-time system, its expression is: where x 1 (t) and x 2 (t) are the states of the system, f(x, t) is a smooth nonlinear function related to the state x(t) and time t; b is the input vector of the system, u(t) ∈ R is the control input to be designed for the system; d(x, t) ∈ R represents the model uncertainty, parameter perturbation and external disturbance information existing in the system, which satisfies the following conditions: |d(x, t)| ≤ l d and where l d > 0, k d > 0 are two constants; Continuous-time nonlinear dynamic system: where \(x = [x 1 ,\cdots,x n \ T \in\mathbb{R} n represents the system state in system (3.2), and \(f(x(t))\in\mathbb{R} n is a vector of non-linear functions, and its internal elements may be discontinuous functions.

3. A design method for a finite-time continuous full-order terminal sliding mode variable structure controller according to claim 1, characterized in that, in Step 2.1, the design of the new full-order terminal sliding mode surface is specifically as follows: First, design the following new full-order terminal sliding mode surface, and its expression form is: wherein, ci, ki, αi, βi (i = 1, 2, 3) are adjustable parameters and are all constants, and satisfy ci > 0, ki > 0, and the function in this sliding mode surface is specifically defined as According to the homogeneity principle, the parameter αi in the sliding mode surface can be designed as: α 1 = ν, α 2 = 1, while the parameter β i can be designed as β 2 = 1, where ν ∈ (0, 1); When the system state trajectory reaches the sliding mode surface, it can be obtained that: Formula (3.6) can also be written as: Let represent the state vector corresponding to the sliding mode, and the following definition can be obtained: For any ν ∈ (0, 1), the parameter α i and the parameter β i satisfy 0 < α 1 < 1 < α 3 and 0 < β 1 < 1 < β 3 ; the vectors f s0 (x) and the vector f s∞ (x) are taken as the approximate functions of f s (x) at the 0 limit and the ∞ limit; Let the homogeneity weight vector be: The vector f can be obtained s0 (x) is a homogeneous function vector of degree k s0 = -1 for r s0 homogeneous function vector, and f s∞ (x) is a homogeneous function vector of degree k s∞ = 1 for r s∞ homogeneous function vector; The sliding mode (3.6) is a double-limited homogeneous function vector depending on (r s0 , k s0 , f s0 (x)) and (r s∞ , k ∞ , f s∞ (x)); Theorem 3.1 Consider the sliding mode in the form of equation (3.6), and design the parameters c, k, α, β (i = 1, 2, 3) according to the homogeneity principle. Then the sliding mode can reach the origin in finite time, and the upper bound of this finite time is independent of the state of the system at the initial moment. i , k i , α i , β i (i = 1, 2, 3). Then the sliding mode can reach the origin in finite time, and the upper bound of this finite time is independent of the state of the system at the initial moment.

4. A design method for a finite-time continuous full-order terminal sliding mode variable structure controller according to claim 1, characterized in that, in Step 2.2, the design of the finite-time continuous terminal sliding mode variable structure controller is specifically as follows: Theorem 3.2 According to the system (3.1), design a continuous terminal sliding mode variable structure controller as shown in the following formula: u = b -1 (t, x)(u eq + u n ), (3.17) where η>0 is an adjustable parameter variable, and λ, μ, ξ, ε are constants greater than zero; the system state can reach the sliding mode surface within a finite time, and within a finite time, it converges to the origin of the system along the sliding mode surface; and the upper bounds of the finite times required for these two processes are both independent of the state of the initial moment of the system.

5. A design method for a finite-time continuous full-order terminal sliding mode variable structure controller according to claim 1 or 2, characterized in that, in Step 2.3, the design of the continuous terminal sliding mode variable structure controller based on the observed state is specifically as follows: Assume the state \(x\) 1 (t) is measurable, and assume that the function \(f(x)\) is bounded. For the system (3.1), define the observation error of the observer as Then the corresponding finite-time state observer can be designed as: Define the error state and design the corresponding observer control law as follows: The expression of the observer error dynamic system can be obtained as: where Δ(t) = f(x) - f(^x) + d(t), and |Δ(t)| ≤ δ, where δ and δd are known constants, parameters and parameter can be selected from the following region: The state observed by the observer and the state x of the system itself 1 (t), a corresponding continuous terminal sliding mode variable structure controller is designed, and its specific form is as follows:

6. A design method for a finite-time continuous full-order terminal sliding mode variable structure controller according to claim 1 or 2, characterized in that, in Step 3.1, the numerical comparison examples are specifically as follows: In order to verify the effectiveness of the design method of the present invention, the controller method designed by the present invention is used to control the following system, and the expression form of this system is: According to formula (3.5), the specific form of the full-order terminal sliding mode surface to be designed can be obtained as: where β 2 = 1, α 2 = 1, The remaining parameters are designed as: k 1 = c 1 = 2, k 2 = c 2 = 35, k 3 = c 3 = 3. Based on Theorem (3.2), the specific form of the corresponding continuous terminal sliding mode variable structure controller is u = u eq + u n , where: When controlling this system, four different initial states are respectively selected for verification.

7. A design method for a finite-time continuous full-order terminal sliding mode variable structure controller according to claim 1, characterized in that, in Step 3.2, the design of the controller for the rotary motion servo device is specifically as follows: According to the classical Kirchhoff voltage law, we can obtain a relationship regarding the armature voltage V m (t): Among them, V m (t) represents the armature circuit voltage, I m (t) represents the armature current, R m represents the armature resistance, L m represents the armature inductance, θ m (t) represents the angular position of the motor shaft, T m (t) represents the torque generated by the motor, E emf (t) represents the back electromotive force of the motor; According to formula (3.37), the calculation expression of the armature current is obtained as: I m (t) = V m (t) - E emf (t) / R m ; According to Newton's second law, the equivalent load of the motor can be calculated by the following formula: Among them, T l (t) / η g k g represents the load torque transmitted through the gear, and η g represents the gear efficiency. Further, the load balance equation is obtained as follows: where, T l (t) is the load torque, B eq is the viscous damping ratio at the output end. Substituting Equation (3.38) into (3.39) gives: Since ω m (t) = k g ω l (t), and at the same time T m (t) = η m k t I m (t), Equation (3.40) can be rewritten as: By simultaneously considering Equation (3.38) and Equation (3.41) and performing Laplace transform, the transfer function between the DC motor voltage and the motor speed can be obtained as follows: Among which J eq = J l + η g J m k g 2 , the transfer function between the DC motor voltage and the position of the motor shaft is: where, θ l represents the shaft position of the DC motor, and x 1 (t) = θ l (t), is taken as the state variable of the system, and the matching disturbance d(x, t) in the system is considered in the system model. The state equation of the DC motor can be obtained as follows:

Images