Unknown state and noise estimation method of nonlinear system

Abstract

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

Description

Technical Field

[0001] The invention relates to the technical field of nonlinear system automatic control, in particular to an unknown state and noise estimation method for a nonlinear system. Background Art

[0002] In actual control systems, the outputs of many sensors are processed discretely in a sampling manner, while the actual objects work in a continuous form. Such systems are widely studied and are called continuous-discrete systems. Since the states of many systems cannot be directly measured, the design of state observer algorithms 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 speaking, the shorter the sampling period, the better the performance of the observer. However, shortening the sampling period may increase the computational burden and complexity of the system.

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

[0004] In addition to the unknown measurement noise during system synthesis, the observer gain may be perturbed, which may make the traditional observer fragile or unable to guarantee good tracking performance and stability performance. Therefore, it is very important to design a non-fragile observer algorithm to deal with the perturbation of the observer gain. But so far, there are few results on the observer design that simultaneously considers the effects of measurement noise and gain perturbation. In addition, due to the widespread application of numerical computing, the form of sampled data output is more practical. Therefore, it is very attractive to study non-fragile sampled data state observers that can resist measurement noise. Summary of the invention

[0005] The invention provides an unknown state and noise estimation method for a nonlinear system. In the presence of observer gain disturbance and measurement noise, the observer proposed by the method can still track the unknown state of the extended system well.

[0006] The technical solution adopted by the present invention is:

[0007] A method for estimating unknown states and noise of a nonlinear system comprises 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 non-fragile observer to estimate the unknown states and noises of the extended nonlinear system constructed in step 2.

[0011] Step 4: Verify the non-fragile observer proposed in step 3 and prove that it can still track the unknown nonlinear system state under the conditions of noise and gain disturbance.

[0012] Step 5: Use the single-link manipulator model to verify the proposed method.

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

[0014]

[0015] In formula (1), represents the set of real numbers; Indicates the status of the system; Indicates the status of the system; Indicates the status of the system; Indicates the status 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 are the system input and output respectively; the unknown measurement noise d 0 (t) is a bounded continuous function; η(t) is an interval [t k ,t k+1 ] on the sample output; and d 0 (t k ) are and d 0 (t) at t k The sampling value at the moment;

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

[0018] is a continuous nonlinear function; where Respectively represent the status of the system; Represents a continuous nonlinear function; Represents a continuous nonlinear function; Represents a continuous nonlinear function; Represents a continuous nonlinear function; Represents a continuous nonlinear function.

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

[0020]

[0021] In formula (2), represents the derivative of the state; Ξ≥1, Ξ is a positive constant; k 0 is a positive constant; θ(t) is an unknown disturbance; Indicates status;

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

[0023]

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

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

[0026]

[0027] In formula (4), represents the observer state; represents the observer state; represents the observer state; represents the observer state; represents the observer state; Indicates d 0 (t) estimate; represents a nonlinear function; represents a nonlinear function; Represents a nonlinear function. represents a nonlinear function; ν 1 represents a positive constant; ν 2 represents a positive constant; k 0 represents a positive constant; k 1 represents a positive constant; k n Represents a positive constant.

[0028] Introducing coordinate transformation

[0029]

[0030] According to equations (3) and (4), the observation error system can be obtained as:

[0031]

[0032] In the formula, All states of the extended nonlinear system of formula (3) are unknown, and only the output is known. The extended non-fragile observer of formula (4) is used to estimate the unknown states of the extended nonlinear system of formula (3), and all states of the observer are known.

[0033] The observation error system means the state of the extended nonlinear system in equation (3) minus the state of the extended non-fragile observer in equation (4). If the difference is small, the state of the extended non-fragile 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, that is, the state estimation error system.

[0034] In step 4, through the Lyapunov candidate function, it is proved that all signals of the observation error system of formula (5) are ultimately uniformly bounded, that is, it is proved that even if there are large observer gain disturbances and measurement noise, the proposed extended non-fragile observer can still track the unknown state of the extended system well;

[0035] The Lyapunov candidate function is as follows:

[0036] V(t)=V 1 (t)+Ξ 2 V 2 (t)+V 3 (t)+Ξ 2 V 4 (t) (6);

[0037] In formula (6): V 1 (t) = ζ T (t)Pζ(t),V 1 (t) represents the Lyapunov candidate function;

[0038] V 2 (t) represents the Lyapunov candidate function;

[0039] V 3 (t) represents the Lyapunov candidate function;

[0040] V 4(t) represents the Lyapunov candidate function.

[0041] The present invention provides a method for estimating unknown state and noise of a nonlinear system, and the technical effects are as follows:

[0042] 1) The present invention obtains an extended nonlinear system through filtering transformation, and the output including measurement noise is converted into a state equation. The system is expanded through auxiliary transformation, making the design more standardized and systematic.

[0043] 2) The present invention uses an extended non-fragile observer to estimate the unknown state and unknown measurement noise, thereby improving the robustness.

[0044] 3) Based on a time-varying matrix inequality, the present invention proves that the extended non-fragile observer can track the extended system well even in the presence of large observer gain disturbances and measurement noise. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] Figure 1 The figure is a flow chart of the method of the present invention.

[0046] Figure 2 This is the error trajectory of the method based on the non-fragile observer of the present invention under sampling time T=0.2 and θ(t)=1.

[0047] Figure 3 This is the error trajectory of the method based on the non-fragile observer of the present invention under sampling time T=0.01 and θ(t)=1.

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

[0049] Figure 5 This is the error trajectory based on the method of the non-fragile observer of the present invention under sampling time T=0.01 and θ(t)=1.15+0.2sint.

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

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

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

[0053] Fig. 9It is the error trajectory of the method based on the traditional observer with sampling time T=0.01 and θ(t)=1.15+0.2sint. DETAILED DESCRIPTION

[0054] A method for estimating unknown states and noise of nonlinear systems is proposed. In order to estimate the unmeasurable state, an extended nonlinear system is constructed through output filtering transformation, and the sampled output with unknown measurement noise is converted into the state equation of the extended system. Then, an extended non-fragile observer is designed to estimate the state of the extended system, and the unknown measurement noise is estimated by 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, the following steps are included:

[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 non-fragile observer to estimate the unknown states and noises of the extended nonlinear system constructed in step S2.

[0059] Step S4: Verify the non-fragile observer proposed in step S3 to prove that it can still track the unknown nonlinear system state under the conditions of noise and gain disturbance.

[0060] Step S5: Use a single-link manipulator model to verify the proposed method.

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

[0062]

[0063] In the formula, and are the system state variables, input and output respectively. The unknown measurement noise d 0 (t) is a bounded continuous function, η(t) is the interval [t k ,t k+1 ] on the sample output, and d 0 (t k ) are and d0 (t) at t k The sampling value at time t k is a strictly increasing sequence that satisfies t k+1 =t k +T, T is the sampling period. is a continuous nonlinear function.

[0064] and d 0 (t) The following conditions are met:

[0065] Nonlinear functions The following Lipschitz conditions are met:

[0066]

[0067] Where δ is a positive constant.

[0068] Positive real number and is the unknown measurement noise d 0 (t) and The known upper bound of

[0069]

[0070] In particular, the noise of the measurement 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 the conversion Multiplicative noise can be converted to additive noise. On the other hand, the transformation matrix Additive noise can be converted into multiplicative noise. Similarly, observer gain disturbance can also be divided into additive disturbance and multiplicative disturbance. The present invention studies the observer design problem of a nonlinear system whose sampled output contains additive noise and whose observer gain contains multiplicative disturbance.

[0071] Among them, in step S2, for the system with sampled output and unknown measurement noise, the following output filter strategy is proposed:

[0072]

[0073] In the formula, Ξ≥1, k 0 and θ(t) represent high gain, positive constant gain, and gain perturbation.

[0074] Therefore, according to equations (1) and (2), the following extended nonlinear system can be obtained:

[0075]

[0076] In step S3, the following extended non-fragile 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), ν 1 and ν 2 are the observer state, the observer gain, an estimate of the unknown measurement noise, and two positive numbers.

[0079] According to equations (3) and (4), the estimated error system can be derived as follows:

[0080]

[0081] In the formula, and

[0082] The following coordinate changes are introduced:

[0083]

[0084] According to equations (3) and (4), the observation error system can be obtained as:

[0085]

[0086] In the formula,

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

[0088] V(t)=V 1 (t)+Ξ 2 V 2 (t)+V 3 (t)+Ξ 2 V 4 (t)(6);

[0089] Where:

[0090] V 1 (t) = ζ T (t)Pζ(t),

[0091]

[0092] It is proved that all signals in the observation error system (5) are eventually uniformly bounded, that is, it is proved that even in the presence of 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 the time-varying matrix for:

[0095]

[0096]

[0097] In the formula, ι 0,i (i=1,…,n), 0<ε≤1 and θ 0 (t) The following conditions are met:

[0098]

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

[0100]

[0101] and

[0102]

[0103] Then, the following inequality holds:

[0104]

[0105] In the formula, and I n is the n-th 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] Then, the following inequality holds:

[0110] A T P+PA≤-γ 1 I n (17);

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

[0112] Proof: Since A=ΘA 0 Θ -1 , P = Θ -1 P 1 Θ -1 From equation (17), we can get the following inequality:

[0113] Θ(A T P+PA)Θ≤-I n (18);

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

[0115] According to (17), V 1 The derivative of (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 The following inequality holds:

[0118] According to formula (20), we can get:

[0119]

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

[0121]

[0122] According to formula (5) and formula (20), we can get:

[0123]

[0124] In the formula,

[0125] From formula (24), we can get:

[0126]

[0127] According to formula (7), we can get:

[0128]

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

[0130]

[0131]

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

[0133]

[0134] V 2 (t) and V 4 The derivative of (t) is given by:

[0135]

[0136] because and

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

[0138]

[0139] V 3 The derivative of (t) is as follows:

[0140]

[0141] According to formula (30)-formula (32), and You can get:

[0142]

[0143] make and so:

[0144]

[0145] For the following nonlinear sampling system

[0146]

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

[0148]

[0149] Where: and l are three positive numbers;

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

[0151] According to formula (34), we can get:

[0152]

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

[0154] Through equations (28), (31), (33) and (44), we can find that the observed residual term increases with and Ξ. However, the sampling period T will decrease, which may impose more load on the sensor. Therefore, it is important to choose the appropriate and Ξ to ensure stability while minimizing the observation residuals. In other words, choosing too large values ​​for these parameters should be avoided. It is also worth mentioning that finding a proper balance between these parameters is crucial for the effective implementation of the proposed observer design in practical applications.

[0155] Compared to the previously proposed inventive method, which considers observer gain perturbations and uses the LMI method to handle them, the present invention considers multiplicative observer gain perturbations and utilizes time-varying matrix inequalities to solve them. 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 often encountered in practical systems.

[0156] In step S5, a single-link manipulator model is used to verify the feasibility and effectiveness of the proposed method.

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

[0158]

[0159] In the formula, ζ 1 , 2, B, J, M, G, and q are the angle of the link, 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 given,

[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 that the measurement noise d 0 (t) = sin50t and the initial value is

[0164] Design high gain Ξ=1.1, parameter ν 1 =3,ν 2 =3;

[0165] The observer gain (k 0 ,k 1 ,k 2 )=(-15,-3.006,-0.6012);

[0166] Observer Gain Based on Linear Matrix Inequality

[0167] In order to show the difference in performance between the parameters selected by the proposed non-fragile observer and the parameters selected by the traditional observer method, the sampling times T = 0.2 and T = 0.01, and the observer gain perturbations θ(t) = 1 and θ(t) = 1.15 + 0.2sint are selected. Figure 2 to Figure 5 The estimated error e using the non-fragile observer method is shown i (t)(0≤i≤2) simulation results, Figure 6 to Figure 9 The simulation results using the traditional observer method are shown in Table 1. The corresponding results under different parameters are shown in Table 1. Figure 8 It can be seen that when the traditional observer method is used, the system may become unstable due to the presence of disturbances in the observer gain, which also reflects that the non-fragile observer method has better robust performance. In addition, it can be observed that when the sampling time T is small, the system is more robust.

[0168] Table 1 Comparison between the non-fragile observer method and the traditional observer method under different gain perturbations.

[0169]

Claims

1. A method for estimating unknown states and noise of a nonlinear system, characterized in that The following steps are involved: 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 non-fragile observer to estimate the unknown states and noises of the extended nonlinear system constructed in step 2.

2. The method for estimating unknown states and noise of a nonlinear system according to claim 1, characterized in that: It also includes step 4: verifying the non-fragile observer proposed in step 3 to prove that it can still track the unknown nonlinear system state under the conditions of noise and gain disturbance.

3. The method for estimating unknown states and noise of a nonlinear system according to claim 1, characterized in that: In step 1, the n-order nonlinear sampling system model with unknown measurement noise is as follows: In formula (1), represents the set of real numbers; Indicates the status of the system; represents the state derivative of the system; and are the system input and output respectively; the unknown measurement noise d0(t) is a bounded continuous function; η(t) is an interval [t k ,t k+1 ] on the sample output; and d0(t k ) are and d0(t) at t k The sampling value at time t k is a strictly increasing sequence that satisfies t k+1 =t k +T, T is the sampling period; is a continuous nonlinear function; where Respectively represent the status of the system; Represents a continuous nonlinear function.

4. The method for estimating unknown states and noise of a nonlinear system according to claim 1, characterized in that: In step 2, for a system with sampled output and unknown measurement noise, the following output filter strategy is proposed: In formula (2), represents the derivative of the state; Ξ≥1, Ξ is a positive constant; k0 is a positive constant; θ(t) is an unknown disturbance; Indicates status; Therefore, according to equations (1) and (2), the following extended nonlinear system can be derived: In formula (3), represents the output of the extended nonlinear system; t represents the running time of the system.

5. The method for estimating unknown states and noise of a nonlinear system according to claim 1, characterized in that: In step 3, the following extended non-fragile observer is proposed to estimate the unknown state and noise of the extended nonlinear system (3); In formula (4), represents the estimate of d0(t); represents a nonlinear function; ν1 represents a positive constant; ν2 represents a positive constant; k0 represents a positive constant; k1 represents a positive constant; k n represents a positive constant; Introducing coordinate transformation According to equations (3) and (4), the observation error system is obtained as follows: In the formula, 6. The method for estimating unknown states and noise of a nonlinear system according to claim 2, characterized in that: In step 4, through the Lyapunov candidate function, it is proved that all signals of the observation error system of formula (5) are ultimately uniformly bounded, that is, it is proved that even if there are large observer gain disturbances and measurement noise, the proposed extended non-fragile observer can still track the unknown state of the extended system well; The Lyapunov candidate function is as follows: V(t)=V1(t)+Ξ 2 V2(t)+V3(t)+Ξ 2 V4(t) (6); In formula (6): V1(t) = ζ T (t)Pζ(t),V1(t) represents the Lyapunov candidate function; V2(t) represents the Lyapunov candidate function; V3(t) represents the Lyapunov candidate function; V4(t) represents the Lyapunov candidate function.

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