Design method of discrete terminal sliding mode variable structure controller based on novel reaching law

Abstract

The invention discloses a discrete terminal sliding mode variable structure controller design method based on a novel reaching law, and belongs to the technical field of automation. Comprising the following steps: step 1, designing a sliding mode variable structure controller for a general discrete time control system; step 2, designing a discrete terminal sliding mode controller, specifically comprising the following steps: step 2.1, designing a discrete terminal sliding mode surface; 2.2, designing a hyperbolic function type reaching law; 2.3, analyzing the tracking error of the closed-loop system; 3, experimental verification; 3.1, tracking a reference signal when the system does not contain uncertainty; and step 3.2, tracking the reference signal when the system comprises uncertainty. The invention provides a novel discrete sliding mode reaching law, and a discrete terminal sliding mode variable structure controller based on the reaching law is designed on the basis of the novel reaching law, so that the chattering phenomenon of a closed-loop system is inhibited, the tracking control precision of the closed-loop system is improved, and the control performance of the closed-loop control system is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of automation, and particularly relates to a design method of a discrete terminal sliding mode variable structure controller based on a novel reaching law. Background Art

[0002] With the improvement of computer processing power and the continuous improvement of the functions of digital controllers, the research on the design of sliding mode variable structure controllers for discrete-time control systems has become a hot topic among many scholars in recent years. In a discrete-time sliding mode variable structure control system, the input signal of the controller is calculated and maintained as a constant value within each sampling period until the next sampling moment. Since the sampling frequency for sampling the system in a discrete-time control system is a finite value, the ideal sliding mode that appears in a continuous-time sliding mode variable structure control system will not appear in a discrete-time sliding mode variable structure control system. Instead, a quasi-sliding mode occurs, which means that the system state does not move along the sliding surface but makes a sawtooth-like movement within the domain of the sliding surface. The region where such a quasi-sliding mode occurs is called the quasi-sliding mode region of the system. Because of the existence of such a quasi-sliding mode in the system, there is high-frequency dynamics in the closed-loop control system, that is, the chattering phenomenon. The appearance of this phenomenon also greatly restricts the application of the sliding mode variable structure control method.

[0003] The appearance of the chattering phenomenon will have an adverse impact on the control effect of the closed-loop control system and will also introduce high-frequency dynamics into the system. Therefore, in the design problem of a sliding mode variable structure control system, how to reduce the adverse effects brought by the chattering phenomenon to the system has become a key issue of concern to researchers. In response to this problem, many researchers have proposed numerous solutions. Among them, the design method of a discrete sliding mode variable structure controller based on a reaching law has a good effect in solving the chattering problem in a discrete sliding mode variable structure control system. The design of a sliding mode variable structure controller based on a reaching law is mainly divided into two steps. First, the reaching law of the system is designed according to the desired sliding mode dynamics. Second, according to the designed reaching law, the discrete sliding mode variable structure controller of the system is obtained. It should be noted that in the previous design problems of sliding mode variable structure controllers based on a reaching law, suppressing the chattering phenomenon of the system and accelerating the convergence speed of the system are the main goals of concern to researchers. However, there is a mutually restrictive relationship between these two goals. Specifically, suppressing the chattering phenomenon of the system can be achieved by reducing the gain of the controller. However, reducing the gain of the controller will slow down the process of the system state converging to the quasi-sliding mode region. Therefore, how to eliminate the mutually restrictive relationship between these two goals is a problem worthy of consideration. Summary of the Invention

[0004] Technical objective: In view of the above problems, the present invention proposes a design method for a discrete terminal sliding mode variable structure controller based on a novel reaching law.

[0005] Technical solution: To solve the above technical problems, the technical solution adopted by the present invention is as follows:

[0006] A design method for a discrete terminal sliding mode variable structure controller based on a novel reaching law, comprising the following steps:

[0007] Step 1: Design a sliding mode variable structure controller for a general discrete-time control system;

[0008] Step 2: Design a discrete terminal sliding mode controller, specifically including:

[0009] Step 2.1: Design a discrete terminal sliding mode surface;

[0010] Step 2.2: Design a hyperbolic function type reaching law;

[0011] Step 2.3: Analyze the tracking error of the closed-loop system;

[0012] Step 3: Experimental verification;

[0013] Step 3.1: Tracking of the reference signal when the system does not contain uncertainties;

[0014] Step 3.2: Tracking of the reference signal when the system contains uncertainties.

[0015] Preferably, in Step 1, for a general discrete-time control system, its system model is expressed as:

[0016]

[0017] where x(k) ∈ R n is the system state of the discrete system, u(k) ∈ R is the discrete sliding mode variable structure control input to be designed, d(k) ∈ R represents the uncertainties existing in the system, and it is assumed that d(k) is smooth and bounded; let y(k) ∈ R represent the output state of the discrete system, and the matrix A ∈ R n×n represents the state matrix of the discrete system, and the matrix C ∈ R 1×n is the output matrix of the discrete system, and b ∈ R n is the input vector of the system.

[0018] Preferably, in Step 2.1, the design of the discrete terminal sliding mode surface includes:

[0019] Define the tracking error of the system as E(k) = y(k) - R(k), where R(k) represents the reference signal of the system. According to the tracking error state of the system, the discrete terminal sliding mode surface is:

[0020] s(k) = λ 1 E(k) + λ 2 E α (k - 1) - λ 1 (1 - β)E(k - 1), (5.2)

[0021] where, the two parameters λ 1 > 0, λ 2 > 0, and the parameter α ∈ (0, 1) is the ratio of two odd numbers, the parameter β ∈ (0, 1) is an arbitrary adjustable parameter. Assume that when k = 0, the error state E(k - 1) = 0.

[0022] Preferably, in step 2.2, the design of the hyperbolic function - type reaching law includes:

[0023] The new - type sliding - mode reaching law, specifically:

[0024] s(k + 1) = φ 1 (k)s(k) - φ 2 ((k)sgn[s(k)] + ξ(k), (5.3)

[0025]

[0026]

[0027] where, q > 0 is the convergence coefficient of this reaching law, and its value satisfies the condition 0 < 1 - qT < 1, δ is an adjustable parameter to control the change rate of the exponential term, λ > 0 is the gain coefficient. The specific form of ξ(k) in the reaching law is ξ(k) = λ 1 Cb(d(k) - 2d(k - 1) + d(k - 2)), assume ||ξ(k)|| < η, where η is a positive constant;

[0028] Let s(k) in formula (5.2) be s(k + 1), substitute it into the reaching law (5.3), and use the definition of the error state and formula (5.1) to obtain the specific form of the discrete - time terminal - sliding - mode variable - structure controller based on the new - type reaching law (5.3) as:

[0029]

[0030] Preferably, in step 2.3, the analysis of the tracking error of the closed - loop system includes:

[0031] Analyze the tracking error of the closed - loop error system:

[0032] Theorem 5.3 For the discrete - time control system (5.1), under the action of the discrete - time terminal - sliding - mode variable - structure controller (5.4), the tracking error E(k) of the closed - loop system satisfies:

[0033]

[0034] Among them, where λ 1 , λ 2 and β are adjustable parameters, and γ is a bounded constant.

[0035] Preferably, in step 3.1, the tracking of the reference signal when the system has no uncertainty includes:

[0036] Taking a triangular wave signal with a frequency of 0.25 Hz and a phase of 180 degrees as the reference tracking signal of the system, the specific form of the discrete sliding mode variable structure controller is obtained as:

[0037]

[0038] Let the discrete first-order linear sliding mode surface be s(k) = Clx(k). According to Gao's reaching law, the specific form of the discrete sliding mode controller can be obtained as:

[0039]

[0040] Beneficial effects: Compared with the prior art, the present invention has the following advantages:

[0041] (1) Aiming at the system chattering problem caused by system sampling in the discrete-time sliding mode control system, the present invention designs a new discrete sliding mode reaching law, and designs a discrete terminal sliding mode variable structure controller based on this reaching law; two nonlinear functions are introduced, and the value ranges of these two functions change with the change of the distance between the system state and the sliding mode surface, so that the system state can reach the sliding mode surface within a finite number of sampling steps, and at the same time, the chattering phenomenon of the closed-loop system can be suppressed, thereby improving the tracking control accuracy of the closed-loop system and improving the control performance of the closed-loop control system. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] Figure 1 is the industrial servo simulator system ECP220 of the present invention;

[0043] Figure 2 is the reference signal tracking effect diagram under the action of three different controllers when there is no uncertainty in the present invention;

[0044] Figure 3 is the tracking error diagram under the action of three different controllers when there is no uncertainty in the present invention;

[0045] Figure 4 is the response curve diagram of three different sliding mode surfaces when there is no uncertainty in the present invention;

[0046] Figure 5It is the reference signal tracking effect diagram under the action of three different controllers when there is uncertainty in the present invention;

[0047] Figure 6 It is the tracking error diagram under the action of three different controllers when there is uncertainty in the present invention;

[0048] Figure 7 It is the response curve diagram of three different types of sliding surfaces when there is uncertainty in the present invention. Specific implementation manner

[0049] 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.

[0050] As Figure 1-7 shown, the present invention proposes a design method for a discrete terminal sliding mode variable structure controller based on a new reaching law, including:

[0051] Step 1: Design a sliding mode variable structure controller for a general continuous-time multi-input multi-output control system;

[0052] For a general discrete-time control system, its system model is expressed as:

[0053]

[0054] where, x(k) ∈ R n is the system state of the discrete system, u(k) ∈ R is the discrete sliding mode variable structure control input to be designed, d(k) ∈ R represents the uncertainty existing in the system, and it is assumed that d(k) is smooth and bounded. Let y(k) ∈ R represent the output state of the discrete system, and the matrix A ∈ R n×n represents the state matrix of the discrete system, and the matrix C ∈ R 1×n is the output matrix of the discrete system, and b ∈ R n is the input vector of the system.

[0055] Lemma 5.1 The uncertainty d(k) in the discrete-time system (5.1) has the following properties:

[0056] d(k) = O(T);

[0057] d(k) - d(k - 1) = O(T 2 );

[0058] d(k) - 2d(k - 1) + d(k - 2) = O(T 3 );

[0059] where, T is the sampling time of the system.

[0060] Definition 5.1 For a discrete-time sliding mode variable structure control system, if the sliding mode dynamics s(k) of the system satisfies then we call the sliding mode of the system a quasi-sliding mode. The region where the quasi-sliding mode occurs in the system is called the quasi-sliding mode region, and Ω represents the width of the quasi-sliding mode region of the system.

[0061] For a discrete-time sliding mode control system, if for all k ≥ 0, each element si(k) in the sliding mode vector s(k) ∈ Rm satisfies the following conditions:

[0062] · When si(k) > Ω, -Ω ≤ <si(k + 1) < s i (k);

[0063] · When si(k) < -Ω, si(k) < si(k + 1) ≤ Ω;

[0064] · When |si(k)| ≤ Ω, |si(k + 1)| ≤ Ω;

[0065] Step 2: Design of the discrete terminal sliding mode controller, specifically including:

[0066] Step 2.1: Design of the discrete terminal sliding mode surface;

[0067] Define the tracking error of the system as E(k) = y(k) - R(k), where R(k) represents the reference signal of the system. According to the tracking error state of the system, design the discrete terminal sliding mode surface as follows:

[0068] s(k) = λ 1 E(k) + λ 2 E α (k - 1) - λ 1 (1 - β)E(k - 1), (5.2)

[0069] where the two parameters λ 1 > 0, λ 2 > 0, and the parameter α ∈ (0, 1) is the ratio of two odd numbers. Since the parameter α < 1 is introduced, s(k) can generate a larger slope near the origin of the phase plane, so this parameter can accelerate the convergence speed of the system. The parameter β ∈ (0, 1) is an arbitrary adjustable parameter, and the adjustment of this parameter will affect the error of the closed-loop system. In addition, it is assumed that when k = 0, the error state E(k - 1) = 0.

[0070] Step 2.2: Design of the hyperbolic function type reaching law;

[0071] The new sliding mode reaching law, its specific form is as follows:

[0072] s(k + 1) = φ 1(k)s(k)-φ 2 ((k)sgn[s((k)]+ξ(k), (5.3)

[0073]

[0074]

[0075] where \(q > 0\) is the convergence coefficient of this reaching law, and its value satisfies the condition \(0 < 1 - qT < 1\), \(\delta\) is an adjustable parameter, whose main purpose is to control the change rate of the exponential term, \(\lambda>0\) is the gain coefficient, and the specific form of \(\xi(k)\) in the reaching law is \(\xi(k)=\lambda\) 1 \(C_b(d(k)-2d(k - 1)+d(k - 2))\). Assume that the change of the system uncertainty term is bounded, that is, \(\|\xi(k)\|<\eta\), where \(\eta\) is a positive constant. Let \(s(k)\) in formula (5.2) be \(s(k + 1)\), substitute it into the reaching law (5.3), and use the definition of the error state and formula (5.1) to obtain the specific form of the discrete terminal sliding mode controller based on the new reaching law (5.3) as follows:

[0076]

[0077] Based on the discrete terminal sliding mode surface (5.2) and the reaching law (5.3), the following conclusions are obtained:

[0078] Theorem 5.1 For the discrete-time control system (5.1), design the discrete terminal sliding mode controller as shown in formula (5.4). If the parameters \(\eta\) and \(\lambda\) in the controller satisfy the following conditions:

[0079] \(\eta\leq\lambda\), (5.6)

[0080] Then, for any initial state of the system, the state motion trajectory of the closed-loop system will be restricted to the following quasi-sliding mode region \(\Omega=\{s(k)|\vert s(k)\vert<\varphi\) 2 (k)+\eta\}\). (5.7).

[0081] Theorem 5.2 For the discrete-time control system (5.1), design the discrete terminal sliding mode surface as shown in (5.2) and the reaching law as shown in (5.3). The system state will reach the sliding mode surface within a finite number of sampling steps. The number of steps to reach the sliding mode surface is specifically: (5.18), where,

[0082]

[0083] where represents the largest integer not greater than , M * , N* and The expression form of

[0084]

[0085] Step 2.3: Analysis of the tracking error of the closed-loop system;

[0086] Analyze the tracking error of the closed-loop error system, and the following results are obtained:

[0087] Theorem 5.3 For the discrete-time control system (5.1), under the action of the discrete terminal sliding mode variable structure controller (5.4), the tracking error E(k) of the closed-loop system satisfies:

[0088]

[0089] where λ 1 , λ 2 and β are adjustable parameters, and γ is a bounded constant.

[0090] Step 3: Experimental verification;

[0091] As Figure 1 shown, the simulation experimental system mainly consists of three parts: a system electronic emulator, a system actuator, and a sensor. In addition, the system also includes a high-resolution encoder. The driving force of the system is generated by a brushless servo DC motor. However, while generating the driving force, this DC motor will also generate a disturbance signal in the system input. At the same time, this set of systems provides adjustable inertia and variable gear ratios. The driving motor is connected to the reduction mechanism through a synchronous belt, and the load disk is connected to the driving disk through a synchronous belt. By adjusting the mass of the load brass block, the load disk and the driving disk can present different system inertias. Let the gear ratio of the driving disk to the load disk in this experimental system be defined as:

[0092]

[0093] where n = 24 represents the number of teeth on the driving disk gear; N = 36 represents the number of teeth on the load disk gear; npl represents the number of teeth on the bottom gear of the system reduction mechanism; npd represents the number of teeth on the top gear of the system reduction mechanism.

[0094] This system provides two signal conversion channels, which are connected to two 16-bit digital-to-analog converters established in the real-time controller. The output range of the analog signal is ±10V (-32768 to +32768). The priorities of the output tasks of these digital-to-analog converters are updated by the real-time controller.

[0095] The continuous-time dynamic model of this experimental system is:

[0096]

[0097] wherein represents the state of the system. Specifically, represents the position state of the load disk, represents the speed state of the system. Sampling the continuous system (5.42) with a sampling period T = 0.001 s, the discrete-time system expression of (5.42) can be obtained as:

[0098]

[0099] wherein,

[0100] Step 3.1: Tracking of the reference signal when the system has no uncertainty;

[0101] Taking a triangular wave signal with a frequency of 0.25 Hz and a phase of 180 degrees as the reference tracking signal of the system, based on the linear discrete sliding mode control scheme of Gao's reaching law and the high-rate output feedback discrete sliding mode control scheme, the controller of the system is designed so that the output signal of the system tracks the reference tracking signal. The specific form of the obtained discrete sliding mode variable structure controller is:

[0102]

[0103] Let the discrete first-order linear sliding mode surface be s(k) = C1x(k). According to Gao's reaching law, the specific form of the discrete sliding mode controller can be obtained as:

[0104]

[0105] From the system (5.43), it can be obtained that the observability index of the system is N = 2. Based on the high-rate output feedback technology, the specific form of the high-rate output feedback discrete sliding mode variable structure controller is:

[0106]

[0107] where T s = 0.0005 sec, F y = -(C 1 b Ts ) -1 (C1A Ts - C 1 + q 1 T s C 1 )L y , F u = -(C 1 b Ts )-1 (C 1 A Ts -C 1 +q 1 T s C 1 )L u For more information about other parameters in the controller, please refer to . The specific parameters of the three different types of discrete sliding mode variable structure controllers are shown in Table 1.

[0108] Table 1 Controller parameters

[0109]

[0110]

[0111] MAXE=mar|e(k)|,

[0112]

[0113] The results after quantitative processing are presented in Table 2. The results in the table show that the control method proposed in this application can improve the quality of tracking control to a great extent.

[0114] Table 2 Quantitative comparison of closed-loop system tracking errors of three different controllers

[0115]

[0116] Step 3.2: The system includes tracking of the reference signal when there is uncertainty;

[0117] Add the uncertainty term d(k) to the system (5.43), let d(k) = 0.2sin(1.5t), and this uncertainty appears in the system at the experimental time t = 10 seconds.

[0118] The triangular wave signal is also selected as the reference tracking signal. By designing three different controllers with the system, the output of the system can track the reference signal in the presence of uncertainty. Figure 5 shown.

[0119] The closed-loop system tracking errors generated by the three different controllers are as follows: Figure 6 In order to further and more clearly reflect the tracking effect of the closed-loop system on the reference signal under the action of three different controllers, the tracking error is also quantified. The results of the three quantitative indicators and the specific error quantification processing are shown in Table 3.

[0120] Table 3 Quantitative comparison of closed-loop tracking errors of three different controllers (with uncertainty)

[0121]

[0122] From the data in the table, it can be obtained that in the presence of uncertainties, by adopting the controller design method proposed in this chapter, the output signal of the system can track the reference tracking signal more accurately, and the system has better robustness to uncertainties.

[0123] In Figure 7 , the response curves of the sliding mode surfaces corresponding to three different types of sliding mode controllers. From Figure 7 The results shown, it can be concluded that the controller design scheme proposed in this application can still greatly suppress the chattering problem in the discrete sliding mode variable structure control system even in the presence of uncertainties. Thereby further improving the tracking control accuracy and improving the performance of the closed-loop control system.

[0124] 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. Design method of discrete terminal sliding mode variable structure controller based on new reaching law, characterized in that, it includes the following steps: Step 1: Design a sliding mode variable structure controller for a general discrete-time control system; Step 2: Design a discrete terminal sliding mode controller, specifically including: Step 2.1: Design of discrete terminal sliding mode surface; Step 2.2: Design of hyperbolic function type reaching law; Step 2.3: Analysis of tracking error of closed-loop system; Step 3: Experimental verification; Step 3.1: Tracking of reference signal when the system has no uncertainty; Step 3.2: Tracking of reference signal when the system contains uncertainty.

2. The design method of discrete terminal sliding mode variable structure controller based on new reaching law according to claim 1, characterized in that, in Step 1, for a general discrete-time control system, its system model is expressed as: where \(x(k)\in\mathbb{R}\) n is the system state of the discrete system, \(u(k)\in\mathbb{R}\) is the discrete sliding mode variable structure control input to be designed, \(d(k)\in\mathbb{R}\) represents the uncertainty existing in the system. It is assumed that \(d(k)\) is smooth and bounded. Let \(y(k)\in\mathbb{R}\) represent the output state of the discrete system. Matrix \(A\in\mathbb{R}\) n×n represents the state matrix of the discrete system, matrix \(C\in\mathbb{R}\) 1×n is the output matrix of the discrete system, \(b\in\mathbb{R}\) n is the input vector of the system.

3. The design method of discrete terminal sliding mode variable structure controller based on new reaching law according to claim 1, characterized in that, in Step 2.1, the design of discrete terminal sliding mode surface includes: Define the tracking error of the system as E(k) = y(k) - R(k), where R(k) represents the reference signal of the system. According to the tracking error state of the system, the discrete terminal sliding mode surface is: s(k) = λ 1 E(k) + λ 2 E α (k - 1) - λ 1 (1 - β)E(k - 1), (5.2) Among them, two parameters λ 1 > 0, λ 2 > 0, and the parameter α ∈ (0, 1) is the ratio of two odd numbers, the parameter β ∈ (0, 1) is an arbitrarily adjustable parameter. Assume that when k = 0, the error state E(k - 1) = 0.

4. The design method of discrete terminal sliding mode variable structure controller based on new reaching law according to claim 1 or 3, characterized in that, in Step 2.2, the design of hyperbolic function type reaching law includes: New sliding mode reaching law, specifically: s(k + 1) = φ 1 (k)s(k) - φ 2 (k)sgn[s(k)] + ξ(k), (5.3) Among them, q>0 is the convergence coefficient of this reaching law, and its value satisfies the condition 0<1-qT<1. δ is an adjustable parameter used to control the change rate of the exponential term. λ>0 is the gain coefficient. The specific form of ξ(k) in the reaching law is ξ(k)=λ 1 Cb(d(k)-2d(k - 1)+d(k - 2)). Assume ||ξ(k)||<η, where η is a positive constant; Let s(k) in formula (5.2) be s(k + 1), substitute it into the reaching law (5.3), and use the definition of error state and formula (5.1) to obtain the specific form of the discrete terminal sliding mode variable structure controller based on the new reaching law (5.3):

5. The design method of discrete terminal sliding mode variable structure controller based on new reaching law according to claim 1, characterized in that, in Step 2.3, the analysis of tracking error of closed-loop system includes: Analyze the tracking error of the closed-loop error system: Theorem 5.3 For the discrete-time control system (5.1), under the action of the discrete terminal sliding mode variable structure controller (5.4), the tracking error E(k) of the closed-loop system satisfies: where λ 1 , λ 2 and β are adjustable parameters, and γ is a bounded constant.

6. The design method of discrete terminal sliding mode variable structure controller based on new reaching law according to claim 1, characterized in that, in Step 3.1, the tracking of reference signal when the system has no uncertainty includes: Take a triangular wave signal with a frequency of 0.25Hz and a phase of 180 degrees as the reference tracking signal of the system, and obtain the specific form of the discrete sliding mode variable structure controller: Let the discrete first-order linear sliding mode surface be s(k) = C1x(k). According to Gao's reaching law, the specific form of the discrete sliding mode controller can be obtained:

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