Sliding mode expansion state observer and application thereof

Abstract

The invention discloses a sliding-mode expansion state observer. The sliding-mode expansion state observer is obtained by replacing a nonlinear function of a traditional nonlinear expansion state observer with a sliding-mode nonlinear convergence function. The invention further discloses an active disturbance rejection controller which is obtained by replacing a traditional extended state observer with the sliding mode extended state observer. The method has the advantages that compared with a traditional extended state observer, the tracking precision and the tracking speed are improved, and the method can better adapt to occasions with large-amplitude disturbance; the method can provide more accurate estimated values of the state variable and the total disturbance, and provides a reliable basis for the design of the controller. Compared with a controller using a traditional extended state observer, the active-disturbance-rejection controller using the sliding-mode extended state observer has smoother control output under the condition that measurement noise exists, and a more accurate control effect can be achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of industrial control systems, and particularly to a sliding mode extended state observer and its application. Background Art

[0002] The extended state observer is widely used in modern control systems. Its main function is to estimate the system state in real time and compensate for unknown disturbances. The extended state observer estimates the unknown disturbances by constructing a system model with an extended state and regarding the unknown disturbances as part of the system, thereby improving the disturbance rejection performance of the system. However, there are some deficiencies in the practical application of the traditional extended state observer, mainly manifested as phase lag, slow tracking speed, and low tracking accuracy. Since the extended state observer needs to estimate the system state and disturbances, it usually introduces a certain delay, which may cause the system response to be not timely enough and affect the control accuracy. In addition, when dealing with fast-changing dynamic systems, the traditional extended state observer has a slow tracking speed and is difficult to meet the requirements of high-dynamic environments. The lack of tracking accuracy also limits the application of the extended state observer in high-precision control occasions. Summary of the Invention

[0003] The purpose of the present invention is to provide a sliding mode extended state observer and also apply it to an active disturbance rejection controller.

[0004] The technical solution for achieving the purpose of the present invention is as follows:

[0005] A sliding mode extended state observer is obtained by using a sliding mode nonlinear convergence function to replace the nonlinear function of the nonlinear extended state observer;

[0006] The sliding mode nonlinear convergence function is

[0007]

[0008] where, e1 is the error between the estimated value and the true value of the state variable, is the error between the estimated value and the true value of the lumped disturbance; 0 < a1 < 1, a2 > 1, q1 > 1; k1, k2, k3, k4 are reaching law gains, all greater than 0; k5, k6 are sliding mode surface gains, all greater than 0; is the gain of the sliding mode extended state observer;

[0009] s is a non-singular fixed-time sliding mode surface, ω(e1) is a piecewise function, In the formula, 0 < p1 < 1, n1 ≥ 1, n2 ≥ 1, ε is a constant, 0 < ε < 1; k7, k8 satisfy:

[0010] where α s > 0, 0 < β s < 1, λ s > 0;

[0011] where α r > 0, 0 < β r < 1, λ r > 0.

[0012] An active disturbance rejection controller is obtained by using the aforementioned sliding mode extended state observer to replace the extended state observer.

[0013] The beneficial effects of the present invention are as follows:

[0014] 1. Compared with the traditional extended state observer, the tracking accuracy and tracking speed are improved, and it is more adaptable to the occasion with large-amplitude disturbances; it can provide more accurate estimated values of state variables and total disturbances, providing a reliable basis for the design of the controller.

[0015] 2. The active disturbance rejection controller using the sliding mode extended state observer has a smoother control output and can achieve a more accurate control effect compared with the controller using the traditional extended state observer in the presence of measurement noise. Description of the Drawings

[0016] Figure 1 It is the tracking effect diagram of the sliding mode extended state observer for the system state x1.

[0017] Figure 2 It is the tracking effect diagram of the sliding mode extended state observer for the system state x2.

[0018] Figure 3 It is the tracking effect diagram of the sliding mode extended state observer for the system state x3.

[0019] Figure 4 It is the schematic diagram of the active disturbance rejection controller using the sliding mode extended state observer.

[0020] Figure 5 It is the schematic diagram of the initial condition setting value of the intake test device.

[0021] Figure 6 It is the intake pressure control effect diagram of the intake test device using the active disturbance rejection controller of the present invention without measurement noise.

[0022] Figure 7 It is the intake temperature control effect diagram of the intake test device using the active disturbance rejection controller of the present invention without measurement noise.

[0023] Figure 8The effect diagram of intake pressure control of the intake test device using the auto-disturbance rejection controller of the present invention under the condition of measurement noise.

[0024] Figure 9 The effect diagram of intake temperature control of the intake test device using the auto-disturbance rejection controller of the present invention under the condition of measurement noise. Detailed implementation manners

[0025] Sliding mode control is known for its strong robustness, fast response speed and small steady-state error. By combining sliding mode control with an extended state observer, the sliding mode extended state observer can effectively improve the tracking speed and accuracy of the system, and can provide the controller with the observed values of the system state and disturbance with high tracking accuracy and fast convergence speed, ensuring the effectiveness of the controller.

[0026] The present invention provides a novel sliding mode extended state observer. Aiming at the defects of the traditional extended state observer, such as phase lag, slow tracking speed and low tracking accuracy, by designing a novel fast power reaching law and a novel nonsingular fixed-time sliding mode surface, and then designing a sliding mode nonlinear convergence function, on the basis of the traditional non-extended state observer, replacing the traditional nonlinear function to obtain a novel sliding mode extended state observer, realizing the fast and accurate estimation of the system state and unknown total disturbance; and when there is measurement noise, ensuring the smoothness of the control output, and then effectively improving the control accuracy of the system.

[0027] Based on the novel sliding mode extended state observer, the present invention also provides an auto-disturbance rejection controller based on the novel sliding mode extended state observer.

[0028] The present invention mainly includes the following contents:

[0029] First, based on the traditional non-extended state observer, a novel sliding mode extended state observer is designed to obtain an accurate state estimation of the system. By designing a novel reaching law and a sliding mode surface, the convergence speed of the system state converging to the equilibrium point is effectively improved.

[0030] Secondly, the traditional auto-disturbance rejection algorithm uses the traditional extended state observer to observe signals, and uses the output of the traditional extended state observer to obtain the error signal and the error differential signal to implement the error feedback control law, realizing the auto-disturbance rejection control based on the traditional extended state observer. However, under the influence of system internal uncertainties, unmodeled dynamics and external disturbances, the tracking accuracy and speed of the traditional extended state observer are insufficient, and the control effect is poor when encountering large-amplitude disturbances. Therefore, the present invention provides an auto-disturbance rejection controller based on the novel sliding mode extended state observer. Through the tracking of the system state by the novel sliding mode extended state observer, it can provide an accurate state estimation for the auto-disturbance rejection controller in the presence of a large number of disturbances, and then effectively improve the control accuracy in the presence of disturbances.

[0031] The auto-disturbance rejection controller based on a novel sliding mode extended state observer mainly consists of the following modules: a novel sliding mode extended state observer and an error feedback control law. The novel sliding mode extended state observer is used to obtain an accurate estimate of the system state. The error feedback control law obtains the initial control quantity u0 according to the estimated system state, and obtains the final control quantity u through disturbance compensation for the estimated value of the lumped disturbance, and finally inputs it to the actuator of the controlled object for control.

[0032] Specifically, the novel sliding mode extended state observer given by the present invention is obtained through the following steps:

[0033] The first step: Design a novel fast power reaching law that can make the system state converge quickly and without chattering, and its expression is:

[0034]

[0035] where s is the sliding mode switching function, and in addition, 0 < a1 < 1, a2 > 1, and k1, k2, k3, k4 are reaching law gains, all greater than 0;

[0036] p(s) is the designed non-linear function, and its expression is:

[0037]

[0038] where α r > 0, 0 < β r < 1, λ r > 0.

[0039] The second step: Design a novel non-singular fixed-time sliding mode surface (i.e., the sliding mode switching function) that can ensure that the controller is non-singular and at the same time ensure that the tracking error converges quickly when approaching the equilibrium state, and its expression is:

[0040]

[0041] where e1 is the error between the estimated value and the true value of the state variable, q1 > 1, and k5, k6 are sliding mode surface gains, all greater than 0;

[0042] In order to ensure that there is no negative exponent term after the sliding mode surface is differentiated, a piecewise function ω(e1) is designed, and its expression is:

[0043]

[0044] where 0 < p1 < 1, n1, n2 ≥ 1, and 0 < ε < 1 is a very small constant.

[0045] To ensure that ω(e1) is differentiable, the parameters k7, k8 satisfy the relationship

[0046]

[0047] h(e1) can ensure the convergence speed of the tracking error to the equilibrium point, and its expression is:

[0048]

[0049] where, α s > 0, 0 < β s < 1, λ s > 0.

[0050] Step 3: Design a sliding-mode nonlinear convergence function based on the novel fast power reaching law described in the first step and the novel nonsingular fixed-time sliding surface described in the second step. First, differentiate the novel nonsingular fixed-time sliding surface described in the second step, and perform an operation on the differentiated result and the novel fast power reaching law described in the first step to obtain the sliding-mode nonlinear convergence function. The specific derivation process is as follows:

[0051] For a typical single-input single-output second-order system, it can be described by the following state equation through simplification:

[0052]

[0053] where, u represents the control input, y represents the system output, x i represents the system state variable, f(x1, x2, w) represents the lumped disturbance, which includes external disturbances, internal uncertainties, and unmodeled dynamics, and b is the control input gain.

[0054] According to the traditional nonlinear extended state observer, by substituting the nonlinear function therein, the following prototype of the sliding-mode extended state observer can be designed:

[0055]

[0056] where, z i represents the estimated values of the system state variable and the lumped disturbance by the extended state observer.

[0057] From this, the error equation can be obtained:

[0058]

[0059] where, e1 and e2 are the tracking errors between the estimated values of the state variable by the extended state observer and the actual values of the system state variable, and e3 represents the tracking error between the estimated value of the lumped disturbance and the actual value f of the lumped disturbance.

[0060] Derive the error equation and combine it with the system state equation Σ1 and the prototype of the sliding mode extended state observer Σ2 to obtain the differential error equation:

[0061]

[0062] First, derive the new non-singular fixed-time sliding mode surface and combine it with the differential error equation to obtain:

[0063]

[0064] Introduce the new fast power reaching law, and make it equal to the expression after the derivative of the sliding mode surface to obtain the sliding mode non-linear convergence function Its expression is:

[0065]

[0066] Where is the maximum estimated value of the tracking error between the lumped disturbance and its estimated value.

[0067] Step 4: Replace the non-linear function of the non-linear extended state observer with the sliding mode non-linear convergence function.

[0068] The traditional non-linear extended state observer has the following expression:

[0069]

[0070] Where β n =[β1,β2,β3] T is the gain of the non-linear extended state observer, and φ1(e), φ2(e) are both non-linear functions used in the traditional non-linear extended state observer.

[0071] Based on the traditional non-linear extended state observer, replace the traditional non-linear functions φ1(e), φ2(e) with the designed sliding mode non-linear convergence function to obtain a sliding mode extended state observer with fast error convergence, and its expression is as follows:

[0072]

[0073] Where x1 is the expected value of the system output, u is the control input of the system, z n =[z1,z2,z3] T is the estimated value of the expected value of the system output, the derivative of the expected value of the system output, and the lumped disturbance, b0 is the approximate gain of the control input, is the gain of the new sliding mode extended state observer.

[0074] To better illustrate the advantages of the novel sliding mode extended state observer, the traditional linear extended state observer (LESO) and the nonlinear extended state observer (NLESO) are used as comparison objects for performance comparison below.

[0075] The form of LESO is:

[0076]

[0077] where α n = [α1, α2, α3] T is the gain of LESO.

[0078] The form of NLESO is:

[0079]

[0080] where β n = [β1, β2, β3] T is the gain of NLESO, and φ i (e) is a nonlinear function, and its form is:

[0081]

[0082] Set the following second-order system model:

[0083]

[0084] A simulation model is established through this expression, where the known term is 3sin(x1(t)+x2(t)), and the external disturbance is cos(0.6t)+5sin(cos(t)). By introducing the traditional linear extended state observer (LESO) and the nonlinear extended state observer (NLESO) for comparison, the tracking performance of the present invention (SMESO) is analyzed. Figure 1 、 Figure 2 、 Figure 3 are the comparison curves of the extended state observer designed for the present invention and the traditional extended state observer in terms of the tracking effects of the states x1, x2, x3 of the Σ5 system. It can be seen from the figure that after improvement through sliding mode control, the convergence speed is effectively improved, enabling the sliding mode extended state observer to track the system state variables and the total disturbance faster, with smaller tracking errors, and smaller chattering amplitude and frequency.

[0085] The present invention also provides an active disturbance rejection controller, which is obtained by replacing the extended state observer with the aforementioned sliding mode extended state observer. By introducing its estimated value, the error feedback control law can be designed as follows:

[0086] After the new sliding mode extended state observer estimates the system state, the control output u is modified by introducing the control initial quantity u0 through the lumped disturbance estimated value z3:

[0087]

[0088] where b is the control quantity gain. Ignoring the estimation error between z3 and the actual lumped disturbance, the controlled object can be modified into a unit integral series system:

[0089]

[0090] For the above integral series system, the control can be realized by the error feedback control law:

[0091] u0 = k p (r1 - z1) + k d (r2 - z2)

[0092] where r1 is the set value of the system state variable, r2 is the differential of the system state variable set value, and k p , k d are the proportional gain and the differential gain.

[0093] The structure of the active disturbance rejection controller provided by the present invention is as Figure 4 shown. The core link is to replace the traditional linear extended state observer with a new sliding mode extended state observer to achieve accurate estimation of the system state. Specifically, the feedback signal y output of the control system is connected to the new sliding mode extended state observer, thereby observing the system state v n = [v1, v2, v3] T and outputting its estimated value z n = [z1, z2, z3] T to the subtractor and the error feedback control law. z1 and z2 obtain the errors e1 and e2 by subtraction from the given values v1 and v2 as the inputs of the error feedback control law. Finally, the error feedback control law outputs the final control quantity to the controlled object.

[0094] According to the actual control situation of the system, for Figure 4The shown active disturbance rejection controller only needs to tune the parameters of the fast sliding mode extended state observer, following the principle of improving the tracking accuracy and speed while ensuring the non-singularity of the controller. First, adjust the parameters of the fast power reaching law. Here, a1 and a2 are power parameters, and k1, k2, k3, and k4 are reaching law gains. Fine-tune them to ensure that the system state has a fast speed when far from and approaching the sliding surface. Then, adjust the parameters of the non-linear function α r , β r , λ r to further improve the reaching speed and ensure no chattering in the system. Secondly, adjust the parameters of the non-singular fixed-time sliding surface. The adjustment idea is similar to that of the reaching law adjustment. First, to ensure the non-singularity of the controller, adjust n1 and n2 in the non-linear function. Secondly, to ensure that the tracking error quickly converges to the equilibrium point, at this time, fine-tune the power parameters q1, p1 and the sliding surface gains k5, k6. Finally, adjust the parameters of the non-linear function α s , β s to improve the overall convergence speed.

[0095] After completing the parameter tuning of the new sliding mode extended state observer, according to the final effect of the controller, tune the parameters of the error feedback control law. Use the bandwidth method for parameter tuning. Introduce the controller bandwidth w c , and configure the two poles of the closed-loop system at -w c , k p = w c 2 , k d = 2w c , which can simplify the parameter adjustment.

[0096] Apply the present invention to the control system of a certain flight environment simulation intake test device. The intake test device control system can be defined as a second-order system through system identification:

[0097]

[0098] In the formula, the state v is the controlled output pressure (temperature) of the system; f is the total disturbance received by the system; b1 and b2 are the parameters of the system model, b is the control input gain, and u is the control input of the controlled system.

[0099] To obtain the two system model parameters b1 and b2, it is necessary to model the regulating valve. The flow coefficient of the regulating valve is a parameter that determines the flow rate that the regulating valve can pass under specific opening and pressure ratio conditions. Its expression is:

[0100]

[0101] Among them, is the flow coefficient of the regulating valve, V pis the opening degree of the regulating valve, p r is the ratio of the pressure after the regulating valve to the pressure before the regulating valve.

[0102] The opening degree of the regulating valve under the hydraulic system is essentially a third-order system. After system identification and decoupling control of the regulating valve, it can be simplified to a first-order inertial link, that is:

[0103]

[0104] where, K v is the proportional coefficient, T Θ is the time constant. For the control system of the flight environment simulation intake test device:

[0105]

[0106] Simulation and test description

[0107] To verify the tracking ability of the novel sliding mode extended state observer proposed in the present invention, a simulation model was established through the above intake test device control system, and two test conditions were carried out: 1. The pressure set value increased at 200 s and 300 s; the intake temperature set value decreased at 250 s and 350 s and increased at 400 s; the flow disturbance of the engine increased at 100 s, 200 s and 300 s and decreased at 150 s. 2. A measurement noise of 30 dB was added under the condition of Case 1. The detailed change curves are as Figure 5 shown.

[0108] Through the simulation test, it can be obtained that under the condition of no measurement noise, the control effects of the intake pressure and intake temperature are as Figure 6 、 Figure 7 shown, where the curve SET represents the set value of the intake pressure (temperature), and LADRC, NLADRC, and SMADRC respectively represent the control effects of the active disturbance rejection controller based on the linear extended state observer, the active disturbance rejection controller based on the nonlinear extended state observer, and the active disturbance rejection controller based on the novel sliding mode extended state observer. Through the analysis and calculation of the data, compared with the traditional scheme, the novel sliding mode extended state observer designed in the present invention can effectively track the system state and total disturbance, improving the control accuracy of the system. The overshoot of the active disturbance rejection controller based on the novel sliding mode extended state observer decreased by 45.7% and 32.7% respectively compared with the other two methods in pressure control; the overshoot decreased by 60.2% and 50% respectively compared with the other two methods in temperature control. At the same time, after introducing 30 dB measurement noise, the simulation results are as Figure 8 、 Figure 9As shown in the figure, it can be seen that there is a certain degree of chattering in the other two methods, while the active disturbance rejection controller based on the new sliding mode extended state observer can still track the given value well, ensuring the smoothness of the control quantity and improving the control accuracy.

Claims

1. A sliding mode extended state observer, characterized in that: The nonlinear function of the nonlinear extended state observer is replaced by the nonlinear convergence function of the sliding mode; The sliding mode nonlinear convergence function is Among them, e1 is the error between the estimated value and the true value of the state variable, is the error between the estimated value and the true value of the lumped disturbance; 0<a1<1, a2>1, q1>1; k1, k2, k3, k4 are reaching law gains, all greater than 0; k5, k6 are sliding surface gains, all greater than 0; is the gain of the sliding mode extended state observer; s is a non-singular fixed-time sliding surface, ω(e1) is a piecewise function, In the formula, 0<p1<1, n1≥1, n2≥1, ε is a constant, 0<ε<1; k7 and k8 satisfy: where α s > 0, 0 < β s < 1, λ s > 0; where α r > 0, 0 < β r < 1, λ r > 0.

2. An active disturbance rejection controller, characterized in that: The method is obtained by using the sliding mode extended state observer as claimed in claim 1 instead of the extended state observer.

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