A method for tuning PID controller parameters in a time delay system

Abstract

The application discloses a PID controller parameter setting method in a time-delay system. In view of modeling error and uncertain disturbance problems existing in the time-delay system, the application combines a linear quadratic optimal control algorithm, a linear extended state observer method after modification and a dominant pole placement technology to design a PID controller in the time-delay system. In the application, a selection criterion of a weighting matrix in the linear quadratic optimal control method is also given, so that the closed-loop system response has expected performance indexes. Compared with other time-domain optimization methods (performance index functions are square error integral (ISE), time square error integral (ITSE), absolute error integral (IAE) and time absolute error integral (ITAE)), dynamic response performance indexes such as system rise time and regulation time and the anti-interference ability of the system are obviously improved under the design method of the application.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of disturbance estimation and suppression, and in particular to a PID controller parameter tuning method in a time-delay system, which further improves dynamic response performance indicators such as system rise time, regulation time, and the anti-interference ability of the system. BACKGROUND

[0002] In engineering control applications, systems are often affected by external disturbances and internal uncertainties, which seriously affect the stability performance and control effect of the system, and even may cause the closed-loop system to be unstable. Taking the inertially stabilized system as an example, in the literature [3] (Tian J, Yang W, Peng Z, et al. Application of MEMS accelerometers and gyroscopes in fast steering mirror control systems [J]. Sensors, 2016, 16(4): 440.), a multi-loop feedback control system is composed of fiber-optic gyroscopes, accelerometers and high-resolution position detectors. The total disturbance suppression capability of the system is the superposition of the effects of each loop, but this method requires the installation of additional inertial sensors on the inertially stabilized platform, which is not conducive to the realization of the requirements of small inertia and rapidity of the inertially stabilized platform, and also increases the experimental space and economic cost. In the literature [4] (Glück M, Pott J U, Sawodny O. Piezo-actuated vibration disturbance mirror for investigating accelerometer-based tip-tilt reconstruction in large telescopes [J]. IFAC-PapersOnLine, 2016, 49(21): 361-366.), the external vibration measured by the base sensor is suppressed by using a direct feedforward method based on measurement, but it is necessary to accurately identify the disturbance transfer characteristics from the base to the tilt mirror. In the literature [5] (Tang T, Niu S, Chen X, et al. Disturbance observer-based control of tip-tilt mirror for mitigating telescope vibrations [J]. IEEE Transactions on Instrumentation and Measurement, 2018, 68(8): 2785-2791.), DOB is introduced into the inertially stabilized system to enhance the anti-disturbance ability of the system. However, the characteristics of the controlled object model are often not accurately identified, and the design of the compensator is limited.

[0003] In order to further improve the disturbance suppression capability of the system, an extended state observer can be used to observe the external disturbance and the system internal uncertainty. The traditional linear extended state observer design method classifies the external disturbance and the system internal uncertainty as a total disturbance, and expands it into a state variable for observation and compensation. The system under this design method will be simplified as a double integral series standard type, which reduces the difficulty of control. However, the design of the traditional extended state observer has nothing to do with the known model information, which will reduce the observation accuracy of the observer and lead to the decline of the anti-interference ability of the system. The present application improves the design of the traditional extended state observer, so as to improve the observation accuracy of the observer.

[0004] With the continuous development of classical control theory, linear quadratic optimal control has been widely used in modern control theory. The optimal control law obtained by the linear quadratic optimal control method has many excellent characteristics, including closed-loop stability and when the system process is single-input single-output, the phase margin of the system under linear quadratic optimal control is at least 60°, and the system amplitude margin is infinite. In addition, in the linear quadratic optimal control method, by selecting the weighting matrix Q and R, the trade-off between state regulation requirements and control energy consumption can be controlled. This excellent property prompts the control designer to use it for PID controller parameter tuning. In the present application, the linear quadratic optimal control is combined with the dominant pole placement technology to tune the PID controller parameters of the time delay system with the improved extended state observer. SUMMARY

[0005] The present application aims at: since the influence of system time delay factor on system stability cannot be ignored in engineering control application, aiming at the modeling error and uncertain disturbance problem existing in time delay system, the present application provides a PID controller parameter tuning method in time delay system.

[0006] The technical scheme adopted by the present application is: a PID controller parameter tuning method in time delay system, the specific implementation steps are as follows:

[0007] For a second-order model system G(s):

[0008]

[0009] Wherein, a1=2ζ ol ω ol , ζ ol , ω ol are the damping ratio and natural frequency of the open-loop system respectively; b is the gain of the open-loop system; L is the time delay coefficient of the system.

[0010] Step 1: establish the system differential equation:

[0011]

[0012] where y, u are system output and input, w is external disturbance, is partial model information, a0, a1, b are known parameters of system; since a0, a1, b are usually not accurately identified in practice, f x represents part of model inaccuracy and part of internal dynamics change; f w represents external disturbance; represents the comprehensive effect of known model dynamics and unknown disturbance.

[0013] The differential equation is converted into the form of extended state space equation, and the system extended state space equation is as follows:

[0014]

[0015] where, respectively represent the position, velocity and total disturbance of the time-delay system.

[0016] Step 2: According to the design of state observer in linear system theory, the continuous extended state observer ESO is as follows:

[0017]

[0018] where z = [z1 z2 z3] T is the observer state vector, L = [β1 β2 β3] T is the observer gain matrix to be determined, u c = [u(t-L) y(t-L)] T is the observer input combination, y c is the extended state observer output.

[0019] Step 3: The extended state observer can accurately estimate the uncertain disturbance f′ in a certain frequency range and compensate for the extended state z3, as shown in Figure 1 . Figure 1 The control signal u(t-L) in the middle is:

[0020] u(t-L) = u0(t-L) - z3 / b = u0(t-L) - (f x +f w ) / b (1.49)

[0021] Bringing formula (1.49) into formula (1.46), we can get:

[0022]

[0023] Will Substituting the available dynamic model into equation (1.50), after eliminating unnecessary disturbances, the system becomes:

[0024]

[0025] Equation (1.51) can be rewritten as a transfer function as follows:

[0026]

[0027] By comparing with Equation (1.45), the system transfer function in Equation (1.51) is consistent with the model in Equation (1.45). This is an improvement on the linear extended state observer in this invention. The traditional linear extended state observer design method is independent of known model information, and the system will be simplified to a double-integral cascaded canonical form.

[0028] Step 4: Tuning is performed using a combination of linear quadratic optimal control (LQR) and dominant pole techniques. Figure 1 The PID controller parameters in the control framework enable control of the system after disturbance compensation. Figure 1 In this context, u0(t) is a PID controller, as shown below:

[0029] u0(t)=K p x2(t)+K i x1(t)+K d x3(t) (1.53)

[0030] in, K p ,K i ,K d These are the coefficients corresponding to the proportional element x2(t), the integral element x1(t), and the differential element x3(t), respectively; e(t) is the tracking error signal.

[0031] Under the influence of external disturbance w(t) and system reference signal r(t), the position output signal of the controlled object is y(t). Assuming the reference signal r(t) = 0, the tracking error e(t) = r(t) - y(t) = -y(t). Under this condition, the controlled system formula (1.52) can be expressed as:

[0032]

[0033] Based on the form of the state space, the derivatives of the state variables in formula (1.52) of the controlled system can be written as:

[0034]

[0035] in,

[0036]

[0037] A, B, X, L are state transition matrix, control matrix, state matrix and time delay term, respectively. As can be seen from equation (1.55), when t < L, the control signal is invalid, and only when t >= L, the control signal is valid. Therefore, equation (1.55) is divided into two parts: as follows:

[0038]

[0039]

[0040] u m (t) in equation (1.58) is:

[0041] u m (t) = u0(t - L) (1.59)

[0042] By changing in equation (1.59), that is, u m (t) acts as an intermediate variable. From a mathematical point of view, equation (1.57), equation (1.58) is now delay-free, and the standard LQR method for delay-free process can be applied to find the optimal control vector u m (t).

[0043] In order to make the system of equation (1.56) have the performance of LQR, it is necessary to minimize the following quadratic cost function:

[0044]

[0045] Where Q is a semi-positive definite state weight matrix, and R is a positive definite control weight matrix. The standard LQR method gives the optimal control vector u m (t) as:

[0046] u m (t) = -R -1 B T Px(t) (1.61)

[0047] Where P is a symmetric positive definite Riccati coefficient matrix, which can be obtained by solving the following continuous algebraic Riccati equation:

[0048] A T P + PA + Q - PBR -1 B T P = 0 (1.62)

[0049] From equation (1.59), we have:

[0050] u0(t) = u m(t + L) = -R -1 B T Px(t + L) (1.63)

[0051] Equation (1.63) gives the control signal u0(t) for the whole time range t >= 0, however the value of x(t + L) at time t is unknown. According to the conclusion in reference [1] (He J B, Wang Q G, Lee T H. PI / PID controller tuning via LQR approach [J]. Chemical Engineering Science, 2000, 55(13): 2429-2439.), the optimal control vector u0(t) at this time is:

[0052]

[0053]

[0054] where A c is:

[0055] A c = A - BR -1 B T P(1.66)

[0056] Since the system matrix obtained in equation (1.66) does not contain any time delay, the method of pole placement is directly applied to obtain the desired closed-loop time performance. Equation (1.61) is brought into equation (1.58) to obtain:

[0057]

[0058] Then by establishing the characteristic equation of the closed-loop system, △(s) = |sI - A c | is equal to the required closed-loop equation.

[0059] When A c is a 2x2 matrix, the characteristic equation △(s) is as follows:

[0060]

[0061] where ζ cl , ω cl are the damping ratio and natural frequency.

[0062] When A c is a 3x3 matrix, using the dominant pole placement method, the characteristic equation △(s) is as follows:

[0063]

[0064] where non-dominant pole p3 = mζ cl ω cl m times the real part of dominant closed-loop poles p1, p2, the value of m should be chosen as 3 or more.

[0065] Step 5: In linear quadratic optimal control, the standard practice is to further design the controller parameters by changing the weighting matrix Q and keeping the weighting matrix R unchanged.

[0066] Assume,

[0067]

[0068] Through equations (1.56), (1.66) and (1.70), the corresponding closed-loop system characteristic equation can be obtained as follows:

[0069] Δ(s) = | sI - A c | = s 3 + (a1+ηp 33 )s 2 + (a0+ηp 23 )s + ηp 13 (1.71)

[0070] where η = r -1 b 2 .

[0071] By comparing the coefficients of the same state variables on the right side of equation (1.69) and equation (1.73), we can get:

[0072]

[0073] The remaining elements of matrix P and matrix Q can be obtained by solving the Riccati equation (1.62) as follows:

[0074]

[0075] Step 6: In order to obtain the optimal control signal u0(t), we need to calculate e A(L-t) .

[0076]

[0077] where p 01 and p 02 are the poles of the open-loop system in equation (1.54), as follows:

[0078]

[0079] By using the partial fraction method, f1'1, f1'2, f1'3, f2'1, f2'2, f2'3, f3'1, f3'2, f3'3are calculated as follows:

[0080]

[0081]

[0082]

[0083] Step 7: To get the optimal control signal u0(t), one needs to calculate

[0084]

[0085] where γ = ηp 13 , α = a1+ ηp 33 , β = a0+ ηp 23 .

[0086] are calculated as follows:

[0087]

[0088] By using the partial fraction method, f 11 ,f 12 ,f 13 ,f 21 ,f 22 ,f 23 ,f 31 ,f 32 ,f 33 are calculated as follows:

[0089]

[0090] where

[0091]

[0092] p1, p2, p3 are the same as p1, p2, p3 in equation (1.25).

[0093] Step 8: By solving equations (1.79), (1.76) and (1.64) simultaneously, one can calculate the PID parameters for 0 ≤ t < L.

[0094]

[0095] By comparing equation (1.53) with equation (1.81), one can get the PID controller parameters:

[0096]

[0097] Step 9: Calculate the PID parameters for t≥L by simultaneous equations (1.79) and (1.65).

[0098]

[0099] Comparing equation (1.53) with equation (1.83), we can get the PID controller parameters:

[0100]

[0101] where K p ,K i ,K d are the coefficients corresponding to the proportional element x2(t), the integral element x1(t) and the derivative element x3(t) respectively.

[0102] Step 10: Design the extended state observer gain matrix. Assuming the error state variable is e(t) = x(t) - z(t), subtracting equation (1.47) from equation (1.48), we get the observer error matrix equation:

[0103]

[0104] From the above equation, we can see that in the observer error matrix equation, (A1-LC) determines the characteristic values of the closed-loop system. By ensuring that the characteristic values of (A1-LC) are less than zero, the observer equation converges. The characteristic equation corresponding to the observer error matrix equation is as follows:

[0105] |sI-(A1-LC)| = s 3 +(β1+a1)s 2 +(a0+a1β1+β2)s+β3 (1.86)

[0106] According to the conclusion in the relevant literature [2] (Herbst G. A simulative study on active disturbance rejection control (ADRC) as a control tool for practitioners [J]. Electronics, 2013, 2(3): 246-279.), the extended state observer can be parameterized to place the poles of the corresponding characteristic equation at the same position (-w0, w0 is the observer bandwidth), as follows:

[0107] |sI-(A1-LC)| = (s+ω o ) 3 =s 3 +3ω o s2 + 3ω o 2 s + ω o 3 (1.87)

[0108] Comparing the coefficients of the same variables on the right side of formula (1.86) and formula (1.87), the following can be obtained:

[0109]

[0110] According to the above technical scheme, the following beneficial effects can be achieved:

[0111] 1. Compared with the method without the improved extended state observer, the anti-interference ability of the system under the design method of the application is further improved, that is, the improved extended state observer method in the application is effective.

[0112] 2. Compared with other time domain optimization methods (performance index function is square error integral (ISE), time square error integral (ITSE), absolute error integral (IAE), time absolute error integral (ITAE)), the dynamic response performance indicators such as the rise time and the regulation time of the system under the design method of the application and the anti-interference ability of the system are significantly improved. BRIEF DESCRIPTION OF DRAWINGS

[0113] Figure 1 is a control block diagram based on linear quadratic optimal control and improved extended state observer in the application;

[0114] Figure 2 is a step response comparison chart of the system under different methods under the action of step disturbance;

[0115] Figure 3 is a step response comparison chart of the system with the improved extended state observer and the system without the improved extended state observer under the method of the application under the action of step disturbance. DETAILED DESCRIPTION

[0116] The specific implementation steps of the application will be described in detail below in combination with the figures and a non-minimum phase system as an example:

[0117] The PID controller parameter tuning method of a time delay system of the application comprises the following steps:

[0118] Step 1: for a non-minimum phase system:

[0119]

[0120] The above non-minimum phase controlled object is converted into an extended state space equation form, and the system extended state space equation is as follows:

[0121]

[0122] where, x1, x2, x3 represent the position, velocity and total disturbance of the system respectively.

[0123] Step 2: Design the modified linear extended state observer according to the system extended state space equation:

[0124]

[0125] where, z = [z1 z2 z3] T is the observer state vector, L = [β1 β2 β3] T is the observer gain matrix to be determined, u c = [u(t-L) y(t-L)] T is the observer input combination, y c is the extended state observer output.

[0126] According to formula (1.88), the gain matrix L of the modified extended state observer is:

[0127]

[0128] Step 3: On the basis of the modified extended state observer, design a PID controller to realize the control of the system after disturbance compensation. As shown in Figure 1 , LQR method and dominant pole placement technology are used to realize the tuning of PID controller parameters. Usually, the weighting matrix R = 1. Take the system expected damping ratio ζ cl , natural frequency ω cl and relative dominance m as ζ cl = 0.8, ω cl = 0.793, m = 6, which satisfies m >> ζ cl ω cl , so as to ensure that the closed-loop poles are at the dominant pole position.

[0129] Through the use of formula (1.72), (1.73), the LQR method tuned weighting matrix P, Q parameters are as follows:

[0130] Q = diag(5.7158 1.4889 9.829) (1.89)

[0131]

[0132] The eigenvalues of formula (1.89), (1.90) are:

[0133] eig(P) = [1.405 3.6579 29.3282] T eig(Q) = [5.7158 1.4889 9.829] T (1.91)

[0134] From equation (1.91), it can be seen that the eigenvalues of the weighting matrix P, Q are greater than zero, which meets the use condition of the LQR method.

[0135] Step 4: The parameters of the PID controller for 0≤t

[0136] The parameters of the PID controller for t>L are:

[0137] [K p K i K d ] = [0.6984 0.4602 0.1543] (1.92)

[0138] Step 5: Compare the method in the present application with other time domain optimization methods (performance index functions are integral square error (ISE), time integral square error (ITSE), absolute error integral (IAE), time absolute error integral (ITAE)). In the simulation, the fmincon() function of the MATLAB optimization toolbox is used to find the optimized PID controller parameter set based on the given time domain performance index of the performance index function, and all optimization methods start from the same initial value of the PID parameters, i.e.

[0139] [K p K i K d ] = [0.3 0.3 0.3] (1.93)

[0140] Compare the method in the present application with the method applied to the system with improved extended state observer and the system without improved extended state observer. In the simulation, the bandwidth of the extended state observer is taken as 40HZ.

[0141] Figure 2 The step response comparison chart of the system under the action of the disturbance with a given step amplitude of 1 at 40s under different methods. From Figure 2It can be seen from the above that compared with other time domain optimization methods (performance index functions are integral square error (ISE), integral time square error (ITSE), integral absolute error (IAE) and integral time absolute error (ITAE)), the dynamic response performance indexes such as system rising time and regulation time and the anti-interference ability of the system are significantly improved under the design method of the application.

[0142] Figure 3 A comparison chart of step responses of the system with the improved extended state observer under the method of the application and the system without the improved extended state observer under the method of the application under the disturbance of a given step amplitude of 1 at 40 s is shown in FIG. 4. Figure 3 It can be seen from the above that the anti-interference ability of the system is further improved under the design method of the application.

[0143] The above detailed description of the specific implementation, process and effect of the application is combined with the drawings and examples, but the above description is only one embodiment of the method and cannot limit the implementation range of the method.

Claims

1. A method for tuning PID controller parameters in a time delay system, characterized in that: The following steps are implemented in detail: For the second-order model system G(s): wherein a1= 2ζ ol ω ol , ζ ol , ω ol are the damping ratio and the natural frequency of the open-loop system, respectively; b is the gain of the open-loop system; L is the time delay coefficient of the system; Step 1: Establish the system differential equation: where y, u are system output and input, w is external disturbance, is the part of model information, a0, a1, b are known parameters of the system; since a0, a1, b are usually not accurately identified in practice, f x represents the part of model inaccuracy and the part of internal dynamics variation; f w represents the external disturbance; represents the combined effect of known model dynamics and unknown disturbance; Convert the differential equation into the extended state space equation form, and the system extended state space equation is as follows: wherein C = [1 0 0], y, f' denotes the position, velocity and total disturbance in the system, respectively; Step 2: According to the design of the state observer in linear system theory, the continuous extended state observer ESO is as follows: where z = [z1 z2 z3] T is the observer state vector, L = [β1 β2 β3] T is the observer gain matrix to be determined, u c = [u(t-L) y(t-L)] T is the observer input combination, y c is the extended state observer output; Step 3: The extended state observer can accurately estimate the uncertain disturbance f' within a certain frequency range and compensate for the extended state z3, and the control signal u(t-L) is: u(t - L) = u0(t - L) - z3 / b = u0(t - L) - (f x +f w ) / b (1.5) Bring formula (1.5) into formula (1.2) to get: The By introducing the available kinetic model into equation (1.6), and after the system is excluded from unnecessary interference, the system becomes: Rewrite formula (1.7) as a transfer function: By comparing formula (1.1) with formula (1.8), the system transfer function in formula (1.8) is consistent with the model in formula (1.1), which is an improvement of the linear extended state observer. The traditional linear extended state observer design method is independent of the known model information, and the system will be simplified to a double-integral series standard type; Step 4: Use the linear quadratic optimal control (LQR) combined with the dominant pole technique to tune the PID controller parameters in the control framework, realize the control of the system after disturbance compensation, and u0(t) is the PID controller, as shown below: u0(t) = K p x2(t) + K i x1(t) + K d x3(t) (1.9) wherein x1(t) =∫e(t)dt, x2(t) = e(t), K p ,K i ,K d are coefficients corresponding to the proportional element x2(t), the integral element x1(t) and the differential element x3(t) respectively; e(t) is a tracking error signal; Under the action of external disturbance w(t) and system reference signal r(t), the controlled object position output signal is y(t), assuming that the reference signal r(t)=0, then the tracking error e(t)=r(t)-y(t)=-y(t), under this condition, the controlled system formula (1.8) can be expressed as: According to the state space form, the derivative of the state variable in the controlled system formula (1.8) can be written as: Where, A, B, X, and L are state transition matrix, control matrix, state matrix, and time delay term, respectively. From formula (1.11), it can be seen that when t<L, the control signal is invalid, and only when t>=L, the control signal is effective, therefore, formula (1.11) is divided into two parts as follows: In equation (1.14) u m (t) is: u m (t) = u0(t - L) (1.15) By the change in equation (1.15), i.e. u m (t) acts as an intermediate variable, mathematically, equations (1.13), (1.14) are now delay-free, and the standard LQR method for delay-free processes can be applied to find the optimal control vector u m (t); In order to make the formula (1.12) system have the performance of LQR, it is necessary to minimize the following quadratic cost function: where Q is a semi-positive definite state weight matrix, R is a positive definite control weight matrix, the standard LQR method gives the optimal control vector u m (t) is: u m (t) = -R -1 B T Px(t) (1.17) Where P is a symmetric positive definite Riccati coefficient matrix, which can be obtained by solving the following continuous algebraic Riccati equation: A T P+PA+Q-PBR -1 B T P=0 (1.18) From formula (1.15), we get: u0(t) = u m (t+L) = -R -1 B T Px(t+L) (1.19) Formula (1.19) gives the control signal of u0(t) in the entire time range t>=0, but the value of x(t+L) at time t is unknown, at this time the optimal control vector u0(t) is: wherein A c is: A c = A - BR -1 B T P (1.22) Since the system matrix obtained in equation (1.22) does not contain any time delay, the pole placement method is directly applied to obtain the desired closed-loop time performance. In order to obtain the optimal control signal u0(t), one needs to calculate and e A(L-t) Substituting equation (1.17) into equation (1.14), one obtains The characteristic equation of the closed loop system is then established as Δ(s) = |sI - A c | equals the required closed loop equation; When A c is a 2x2 matrix, the characteristic equation Δ(s) is as follows: wherein ζ cl , ω cl is the damping ratio and the natural frequency; When A c is a 3x3 matrix, the characteristic equation Δ(s) is as follows using the dominant pole placement method: where the non-dominant pole p3 = mζ cl ω cl m times the real part of the dominant closed loop poles p1, p2, the value of m should be chosen to be 3 or more; In the linear quadratic optimal control, the standard method is to further design the controller parameters by changing the weighting matrix Q and keeping the weighting matrix R unchanged; Assume that Through formula (1.12), (1.22) and (1.26), the corresponding closed-loop system characteristic equation is as follows: Δ(s) = |sI - A c = s 3 + (a1+ ηp 33 )s 2 + (a0+ ηp 23 )s + ηp 13 (1.27) wherein η = r -1 b 2 ; By comparing the coefficients of the same state variables on the right side of formula (1.25) and formula (1.29), we get: The remaining elements of matrix P and matrix Q can be obtained by solving the Riccati equation (1.18) as follows: Step 6: Calculate e A(L-t) , where p 01 and p 02 are the poles of the open-loop system of equation (1.10), as follows: By using the partial fraction method, f1'1, f1'2, f1'3, f2'1, f2'2, f2'3, f3'1, f3'2, f3'3 are calculated as follows: Step 7: Calculation where γ = ηp 13 , α = a1+ ηp 33 , β = a0+ ηp 23 ; The calculation is as follows: By using the partial fraction method, f 11 ,f 12 ,f 13 ,f 21 ,f 22 ,f 23 ,f 31 ,f 32 ,f 33 is calculated as follows: where, p1, p2, p3 are consistent with p1, p2, p3 in formula (1.25); Step 8: by formula (1.35), (1.32) and (1.20), the PID parameters of 0≤t By comparing formula (1.9) with formula (1.37), the PID controller parameters can be obtained: where K p ,K i ,K d are the coefficients corresponding to the proportional element x2(t), the integral element x1(t) and the derivative element x3(t) respectively. Step 9: by formula (1.35) and (1.21), the PID parameters of t≥L are calculated, By comparing formula (1.9) with formula (1.39), the PID controller parameters can be obtained: Step 10: design the gain matrix of the extended state observer. Assuming that the error state variable is e(t) = x(t)-z(t), the observer error matrix equation is obtained by subtracting formula (1.3) from formula (1.4): From the above formula, it can be seen that in the observer error matrix equation, (A1-LC) determines the characteristic value of the closed-loop system. By ensuring that the characteristic value of (A1-LC) is less than zero, the observer equation converges, and the corresponding characteristic equation of the observer error matrix equation is as follows: |sI-(A1-LC)| = s 3 +(β1+a1)s 2 +(a0+a1β1+β2)s+β3 (1.42) After parameterization of the extended state observer, the poles of the corresponding characteristic equation can be placed in the same position, as shown below: |sI-(A1-LC)| = (s+ω o ) 3 = s 3 + 3ω o s 2 + 3ω o 2 s+ω o 3 (1.43) Where -w0, w0 are the bandwidth of the observer; By comparing the coefficients of the same variables on the right side of formula (1.42) and formula (1.43), we can get: β1= 3ω o -a1, β2= 3ω o 2 -3a1ω o -a0+ a1 2 , β3= ω o 3 (1.44).

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