A method for unknown state and noise estimation of a nonlinear system

Abstract

The application discloses a method for estimating unknown state and noise of a nonlinear system, which comprises the following steps: constructing a general nonlinear system model with measurement noise and sampling output; constructing an extended nonlinear system through filtering transformation; and proposing an extended non-fragile observer to estimate unknown state and noise of the constructed extended nonlinear system. The proposed non-fragile observer is verified, and it is proved that the non-fragile observer can still track unknown state of the nonlinear system under the conditions of noise and gain disturbance. The proposed observer can still well track unknown state of the extended system.

Description

Technical Field

[0001] This invention relates to the field of automatic control technology for nonlinear systems, specifically to a method for estimating the unknown state and noise of a nonlinear system. Background Technology

[0002] In practical control systems, the outputs of many sensors are processed discretely through sampling, while the actual object operates continuously. Such systems are extensively studied and are called continuous-discrete systems. Since the state of many systems cannot be directly measured, state observer algorithm design has become an important research area in control theory. When designing an observer, the dynamic characteristics of the system and the sampling period need to be considered. Generally, the shorter the sampling period, the better the observer's performance. However, shortening the sampling period may increase the computational burden and complexity of the system.

[0003] Due to faults, nonlinearities, drift, and other influencing factors in sensors, amplifiers, circuits, and other electronic components, the output of a system often contains uncertain random noise. To estimate and compensate for disturbances and noise, two basic methods are proposed: filters and observers. The filters include low-pass and high-pass filters, while the observers include extended observers and reduced-order observers.

[0004] Besides unknown measurement noise during system synthesis, observer gain can be perturbed, potentially making traditional observers vulnerable or unable to guarantee good tracking performance and stability. Therefore, designing a non-vulnerable observer algorithm to handle observer gain perturbations is crucial. However, to date, there are few results on observer designs that simultaneously consider the effects of measurement noise and gain perturbations. Furthermore, due to the widespread application of numerical computation, the form of sampled data output is more practically meaningful. Therefore, researching non-vulnerable sampled data state observers that are robust to measurement noise is extremely attractive. Summary of the Invention

[0005] This invention provides a method for estimating the unknown state and noise of a nonlinear system. Even with observer gain perturbation and measurement noise, the observer proposed in this method can still track the unknown state of the extended system very well.

[0006] The technical solution adopted in this invention is as follows:

[0007] A method for estimating the unknown state and noise of a nonlinear system includes the following steps:

[0008] Step 1: Construct a general nonlinear system model with measurement noise and sampled output;

[0009] Step 2: Construct an extended nonlinear system through filtering transformation;

[0010] Step 3: Propose an extended nonfragile observer to estimate the unknown state and noise of the extended nonlinear system constructed in Step 2.

[0011] Step 4: Verify the nonfragile observer proposed in Step 3 to prove that it can still track the state of the unknown nonlinear system under noise and gain perturbation conditions.

[0012] Step 5: Validate the proposed method using a single-link robot model.

[0013] In step 1, the model of the nth-order general nonlinear sampling system with unknown measurement noise is as follows:

[0014]

[0015] In equation (1), Represents the set of real numbers; Indicates the state of the system; Indicates the state of the system; Indicates the state of the system; Indicates the state of the system; Represents the state derivative of the system; Represents the state derivative of the system; Represents the state derivative of the system; Represents the state derivative of the system;

[0016] and These are the system input and output, respectively; the unknown measurement noise d0(t) is a bounded continuous function; η(t) is the interval [t...]. k ,t k+1 The sampling output on ]; and d0(t) k They are respectively and d0(t) at t k The sampled value at time;

[0017] t k It is a strictly increasing sequence and satisfies t k+1 =t k +T, where T is the sampling period;

[0018] It is a continuous nonlinear function; where, These represent the system states respectively; Represents a continuous nonlinear function; Represents a continuous nonlinear function; Represents a continuous nonlinear function; Represents a continuous nonlinear function; It represents a continuous nonlinear function.

[0019] In step 2, for systems with sampled output and unknown measurement noise, the following output filter strategy is proposed:

[0020]

[0021] In equation (2), The derivative of the state is represented by Ξ ≥ 1, where Ξ is a positive constant; k0 is a positive constant; θ(t) is an unknown perturbation. Indicates state;

[0022] Therefore, based on equations (1) and (2), the following extended nonlinear system can be derived:

[0023]

[0024] In equation (3), Represents a continuous nonlinear function; Represents a continuous nonlinear function; Represents a continuous nonlinear function; t represents the output of system (3); t represents the system running time.

[0025] In step 3, the following extended nonfragile observer is proposed to estimate the unknown state and noise of the extended nonlinear system (3);

[0026]

[0027] In equation (4), Indicates the observer state; Indicates the observer state; Indicates the observer state; Indicates the observer state; Indicates the observer state; This represents an estimate of d0(t); Represents a nonlinear function; Represents a nonlinear function; This represents a nonlinear function. ν1 represents a nonlinear function; ν2 represents a positive constant; k0 represents a positive constant; k1 represents a positive constant; k n Represents a positive integer.

[0028] Introducing coordinate transformation

[0029]

[0030] Based on equations (3) and (4), the observation error system can be obtained as follows:

[0031]

[0032] In the formula, In the extended nonlinear system of Equation (3), all states are unknown, only the output is known. The extended nonfragile observer of Equation (4) serves to estimate the unknown states of the extended nonlinear system of Equation (3), and all states of the observer are known.

[0033] The observation error system is the state of the extended nonlinear system in equation (3) minus the state of the extended nonfragile observer in equation (4). If the difference is small, the state of the extended nonfragile observer in equation (4) can be approximated as the state of the extended nonlinear system in equation (3). Therefore, equation (5) is called the observation error system, which is also the state estimation error system.

[0034] In step 4, by using the Lyapunov candidate function, it is proved that all signals of the observation error system in equation (5) are eventually uniformly bounded, that is, it is proved that even if there are large observer gain perturbations and measurement noise, the proposed extended nonfragile observer can still track the unknown state of the extended system well.

[0035] The Lyapunov candidate functions are as follows:

[0036] V(t)=V1(t)+Ξ 2 V2(t)+V3(t)+Ξ 2 V4(t) (6);

[0037] In equation (6): V1(t)=ζ T (t)Pζ(t),V1(t) denote Lyapunov candidate functions;

[0038] V2(t) represents the Lyapunov candidate function;

[0039] V3(t) represents the Lyapunov candidate function;

[0040] V4(t) represents the Lyapunov candidate function.

[0041] This invention provides a method for estimating the unknown state and noise of a nonlinear system, with the following technical advantages:

[0042] 1) This invention obtains an extended nonlinear system through filtering transformation, where the output containing measurement noise is transformed into state equations. The auxiliary transformation extends the system, making the design more standardized and systematic.

[0043] 2) This invention employs an extended nonfragile observer to estimate unknown states and unknown measurement noise, thereby improving robustness.

[0044] 3) Based on a time-varying matrix inequality, this invention proves that the extended nonfragile observer can track the extended system well, even with large observer gain perturbations and measurement noise. Attached Figure Description

[0045] Figure 1 This is a flowchart of the method of the present invention.

[0046] Figure 2 The error trajectory is given for sampling time T = 0.2 and θ(t) = 1, based on the nonfragile observer method of this invention.

[0047] Figure 3 The error trajectory is given for sampling time T = 0.01 and θ(t) = 1, based on the non-fragile observer method of this invention.

[0048] Figure 4 The error trajectory is given for sampling time T = 0.2 and θ(t) = 1.15 + 0.2sint, based on the non-fragile observer method of this invention.

[0049] Figure 5 The error trajectory is given for sampling time T = 0.01 and θ(t) = 1.15 + 0.2sint, based on the nonfragile observer method of this invention.

[0050] Figure 6 The error trajectory is based on the traditional observer method, with sampling time T = 0.2 and θ(t) = 1.

[0051] Figure 7 The error trajectory is based on the traditional observer method, with sampling time T = 0.01 and θ(t) = 1.

[0052] Figure 8 The error trajectory is based on the traditional observer method, with sampling time T = 0.2 and θ(t) = 1.15 + 0.2sint.

[0053] Figure 9 The error trajectory is based on the traditional observer method, with sampling time T = 0.01 and θ(t) = 1.15 + 0.2sint. Detailed Implementation

[0054] A method for estimating the unknown state and noise of a nonlinear system is proposed. To estimate the unmeasurable state, an extended nonlinear system is constructed through output filtering transformation, converting the sampled output containing unknown measurement noise into the state equation of the extended system. Then, an extended nonfragile observer is designed to estimate the state of the extended system, while the unknown measurement noise is estimated using a first-order estimator. Based on a time-varying matrix inequality, it is proved that the proposed observer algorithm can still track the unknown state of the extended system well even in the presence of observer gain perturbations and measurement noise. Finally, a nonlinear dynamic model is used to verify the feasibility and effectiveness of the proposed method.

[0055] like Figure 1 As shown, it includes the following steps:

[0056] Step S1: Construct a general nonlinear system model with measurement noise and sampled output;

[0057] Step S2: Construct an extended nonlinear system through filtering transformation;

[0058] Step S3: Propose an extended nonfragile observer to estimate the unknown state and noise of the extended nonlinear system constructed in step S2.

[0059] Step S4: Verify the nonfragile observer proposed in Step S3, proving that it can still track the state of the unknown nonlinear system under noise and gain perturbation conditions.

[0060] Step S5: Validate the proposed method using a single-link robot model.

[0061] In step S1, consider the following n-order general nonlinear sampling system model with unknown measurement noise:

[0062]

[0063] In the formula, and These are the system state variables, input, and output, respectively. The unknown measurement noise d0(t) is a bounded continuous function, and η(t) is a function of the interval [t]. k ,t k+1 The sampling output on ] and d0(t) k They are respectively and d0(t) at t k The sampled value at time t. k It is a strictly increasing sequence and satisfies t k+1 =t k +T, where T is the sampling period. It is a continuous nonlinear function.

[0064] And d0(t) satisfy the following conditions:

[0065] nonlinear functions The following Lipschitz conditions must be met:

[0066]

[0067] In the formula, δ is a positive constant.

[0068] positive real numbers and The unknown measurement noise d0(t) and The known upper bound, that is to say

[0069]

[0070] Specifically, the noise measured can be divided into additive noise and multiplicative noise. Under certain conditions, these two types of noise can be converted into each other. For example, by using a converter... Multiplicative noise can be converted into additive noise. On the other hand, the transformation matrix... Additive noise can be converted into multiplicative noise. Similarly, observer gain perturbations can be classified as additive or multiplicative perturbations. This invention studies the observer design problem for nonlinear systems where the sampled output contains additive noise and the observer gain contains multiplicative perturbations.

[0071] In step S2, for systems with sampled output and unknown measurement noise, the following output filter strategy is proposed:

[0072]

[0073] In the formula, Ξ≥1, k0 and θ(t) represent high gain, normal gain and gain perturbation, respectively.

[0074] Therefore, based on equations (1) and (2), the following extended nonlinear system can be derived:

[0075]

[0076] In step S3, the following extended nonfragile observer is proposed to estimate the unknown state and noise of the extended nonlinear system:

[0077]

[0078] In the formula, ki(i=0,1,…,n), ν1 and ν2 are the observer state, the observer gain, the estimate of the unknown measurement noise, and two positive numbers, respectively.

[0079] Based on equations (3) and (4), the estimation error system can be derived as follows:

[0080]

[0081] In the formula, and

[0082] Introduce the following coordinate transformations:

[0083]

[0084] Based on equations (3) and (4), the observation error system can be obtained as follows:

[0085]

[0086] In the formula,

[0087] In step S4, a Lyapunov candidate function is considered:

[0088] V(t)=V1(t)+Ξ 2 V2(t)+V3(t)+Ξ 2 V4(t)(6);

[0089] In the formula:

[0090] V1(t)=ζ T (t)Pζ(t),

[0091]

[0092] It is proven that all signals in the observation error system (5) are eventually uniformly bounded, that is, it is proven that even with large observer gain perturbations and measurement noise, the proposed observer algorithm can still track the unknown state of the extended system well.

[0093] The specific proof is as follows:

[0094] Define a positive definite matrix and time-varying matrix for:

[0095]

[0096]

[0097] In the formula, ι 0,i(i = 1, ..., n), 0 < ε ≤ 1 and θ0(t) satisfy the following conditions:

[0098]

[0099] In the formula, υ>0, 0<θ min ≤1, 1≤θ max <+∞ and satisfy:

[0100]

[0101] and

[0102]

[0103] Therefore, the following inequality holds:

[0104]

[0105] In the formula, and I n It is an n-order identity matrix.

[0106] Let a matrix In the formula, ι i (i=1,…,n) and θ(t) satisfy the following conditions:

[0107] ι i =ι 0,i ε 1+…+i-1 (15);

[0108] θ min ≤θ(t)≤θ max (16);

[0109] Therefore, the following inequality holds:

[0110] A T P+PA≤-γ1I n (17);

[0111] In the formula, γ1≥1, P=Θ -1 P1Θ -1 and Θ=diag(1,ε,ε 1+2 ,…,ε 1+2+…+n-1 ).

[0112] Proof: Since A = ΘA0Θ -1 P = Θ -1 P1Θ -1 From equation (17), we can obtain the following inequality:

[0113] Θ(A T P+PA)Θ≤-In (18);

[0114] Since Θ>0 and 0<ε≤1, the conclusion holds.

[0115] According to (17), the derivative of V1(t) is as follows:

[0116]

[0117] For any α>β≥0, the continuous function Π(t) is integrable on [β,α], and the positive definite matrix is... The following inequalities hold:

[0118] According to equation (20), we can obtain:

[0119]

[0120] According to Young's inequality and equation (21), we can obtain:

[0121]

[0122] According to equations (5) and (20), we can obtain:

[0123]

[0124] In the formula,

[0125] From equation (24), we can obtain:

[0126]

[0127] According to equation (7), we can conclude that:

[0128]

[0129] Combining equations (19) and (22) to (26), we can obtain:

[0130]

[0131]

[0132] because and The following inequality can be derived from equation (27).

[0133]

[0134] The derivatives of V2(t) and V4(t) are given below:

[0135]

[0136] because and

[0137] because From equations (28) and (29), we can obtain:

[0138]

[0139] The derivative of V3(t) is as follows:

[0140]

[0141] According to equations (30)-(32), and We can obtain:

[0142]

[0143] make and so:

[0144]

[0145] For the following nonlinear sampling system

[0146]

[0147] In the formula, and They are at time t respectively k The sampled state vector and continuous function, if there exists a Lyapunov function V(t) such that:

[0148]

[0149] In the formula: l and are three positive numbers;

[0150] Then the nonlinear system is eventually uniformly bounded and and

[0151] According to equation (34), we can obtain:

[0152]

[0153] In the formula, Therefore, by It can be concluded that all signals in the error system are eventually uniformly bounded, thus the proof is complete.

[0154] By using equations (28), (31), (33), and (44), it can be found that the observed residual term changes with... The value of Ξ decreases as the value of Ξ increases. However, the sampling period T will decrease, which may place a greater load on the sensor. Therefore, it is important to choose an appropriate Ξ value. And Ξ, to ensure stability while minimizing observation residuals. In other words, excessively large values ​​for these parameters should be avoided. It is also worth mentioning that finding an appropriate balance among these parameters is crucial for the effective implementation of the proposed observer design in practical applications.

[0155] Compared to previously proposed methods that consider observer gain perturbations and handle them using the LMI method, this invention considers multiplicative observer gain perturbations and addresses these issues using time-varying matrix inequalities. The proposed method not only provides a more practical and convenient design process but also allows for easier selection of appropriate parameters. This method is expected to be more effective in real-world applications because it takes into account measurement noise and gain perturbations, which are frequently encountered in practical systems.

[0156] In step S5, the feasibility and effectiveness of the proposed method are verified using a single-link manipulator model.

[0157] Considering the mathematical model of a single-link manipulator, its state space can be given as follows:

[0158]

[0159] In the formula, ζ1, ζ2, B, J, M, G, and q represent the angle, angular velocity, total damping coefficient, total moment of inertia of the motor, total mass of the link, gravitational acceleration, and distance from the joint axis to the center of mass, respectively. Consider the output filter strategy:

[0160]

[0161] Next, an extended non-fragile sampled data observer is presented.

[0162]

[0163] Where: input u(t) = 20cos(20t), J = 1kg·m 2 ,B=2kg·m / s,M=1kg,G=10m / s 2 , q = 1m. Assume the measured noise d0(t) = sin50t and the initial value is

[0164] Design a high-gain system with Ξ = 1.1 and parameters ν1 = 3 and ν2 = 3.

[0165] The observer gain based on the time-varying matrix inequality is (k0,k1,k2)=(-15,-3.006,-0.6012);

[0166] Observer gain based on linear matrix inequalities

[0167] To demonstrate the performance difference between the parameters selected by the proposed nonfragile observer and those selected by traditional observer methods, sampling times T = 0.2 and T = 0.01, and observer gain perturbations θ(t) = 1 and θ(t) = 1.15 + 0.2sint were chosen. Figures 2-5 The estimation error e using the nonfragile observer method is shown. i Simulation results for (t)(0≤i≤2), Figures 6-9 This shows the simulation results using the traditional observer method. Table 1 presents the corresponding results under different parameters. Figure 8 It can be seen that when using the traditional observer method, the system may become unstable due to perturbations in the observer gain, which reflects the better robustness of the non-fragile observer method. Furthermore, it can be observed that the system is more robust when the sampling time T is small.

[0168] Table 1 Comparison of nonfragile observer methods and traditional observer methods under different gain perturbations.

[0169]

Claims

1. A method for estimating the unknown state and noise of a nonlinear system, characterized in that... Includes the following steps: Step 1: Construct a general nonlinear system model with measurement noise and sampled output; Step 2: Construct an extended nonlinear system through filtering transformation; Step 3: Propose an extended nonfragile observer to estimate the unknown state and noise of the extended nonlinear system constructed in Step 2; In step 1, there is unknown measurement noise. The model of the order nonlinear sampling system is shown below: （1）； In equation (1), , ; Represents the set of real numbers; Indicates the state of the system; Represents the state derivative of the system; and These are the system input and output; unknown measurement noise. It is a bounded continuous function; It is an interval The sampling output on; and They are and exist The sampled value at time; It is a strictly increasing sequence and satisfies , , It is the sampling period; It is a continuous nonlinear function; where, These represent the system states respectively; Represents a continuous nonlinear function; In step 2, for systems with sampled output and unknown measurement noise, the following output filter strategy is proposed: （2）； In equation (2), The derivative representing the state; , It is a positive number; It is a positive number; It is an unknown disturbance; Indicates state; Therefore, based on equations (1) and (2), the following extended nonlinear system can be derived: （3）； In equation (3), Represents the output of an extended nonlinear system; Indicates the system uptime; In step 3, the following extended nonfragile observer is proposed to estimate the unknown state and noise of the extended nonlinear system (3); （4）； In equation (4), express The estimate; Represents a nonlinear function; Represents positive integers; Represents positive integers; Represents positive integers; Represents positive integers; Represents positive integers; Introducing coordinate transformation Based on equations (3) and (4), the observation error system is obtained as follows: （5）； In the formula, , .

2. The method for estimating the unknown state and noise of a nonlinear system according to claim 1, characterized in that: Also includes Step 4: Verify the nonfragile observer proposed in Step 3 to prove that it can still track the state of the unknown nonlinear system under noise and gain perturbation conditions.

3. The method for estimating the unknown state and noise of a nonlinear system according to claim 2, characterized in that: In step 4, by using the Lyapunov candidate function, it is proved that all signals of the observation error system in equation (5) are eventually uniformly bounded, that is, it is proved that even if there are large observer gain perturbations and measurement noise, the proposed extended nonfragile observer can still track the unknown state of the extended system well. The Lyapunov candidate functions are as follows: （6）； In formula (6): Represents Lyapunov candidate functions; Represents Lyapunov candidate functions; ; Represents Lyapunov candidate functions; ; This represents a Lyapunov candidate function.

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