A design method of nonlinear disturbance observer based on adaptive gain

Abstract

The application relates to a nonlinear disturbance observer design method based on adaptive gain. For a nonlinear system containing time-varying disturbance and measurement noise: firstly, the mathematical representation of the nonlinear dynamic behavior of the system is carried out, and a virtual system is constructed, and the difference between the outputs of the real system and the virtual system is used to correct the estimation effect of the disturbance observer; secondly, the constant gain of the disturbance observer is designed to positively change with the increase of the absolute value of the approximation error of the virtual system and reversely change with the increase of the output variance of the system; finally, the constant gain and the adaptive gain are combined to obtain the nonlinear disturbance observer containing the adaptive gain. The application has the advantages of less measurement information demand, self-adjustment of the observation gain, etc., can realize the compromise between the estimation accuracy and the measurement noise suppression capability, and can improve the adaptive capacity of the nonlinear disturbance observer to the change information.

Description

Technical Field

[0001] This invention belongs to the field of anti-interference control for nonlinear systems, and specifically relates to a design method for an adaptive gain nonlinear interference observer. Background Technology

[0002] Interference generally originates from three aspects: internal noise, external disturbances, and model errors. It has the characteristics of multiple sources, multiple types, and multiple channels. It is a common physical phenomenon and an important factor causing system instability, which brings great difficulties to system analysis and observation design.

[0003] Methods for handling multi-source disturbances can be mainly divided into two categories. The first category is robust control, which uses feedback principles to suppress disturbances. However, this type of method has difficulties in estimating disturbances and tends to overestimate the upper bound of the disturbance. Furthermore, to cope with worst-case scenarios, a large gain is required to generate sufficient control force. Its high conservatism makes it difficult for the closed-loop system to achieve satisfactory transient and steady-state performance. The second category is composite disturbance suppression control, which generally consists of two steps: First, designing a nominal controller for the nonlinear system without considering disturbances to meet the expected performance indicators; second, achieving disturbance attenuation by estimating the disturbance using a disturbance observer and canceling it out through a control law. The nominal controller plus disturbance cancellation constitutes composite control.

[0004] To improve the anti-interference capability of composite control and overcome the limitations of using constant fixed gain, the estimation capability of interference observers needs further development. First, for nonlinear systems, using linearized models to design interference observers and treating nonlinear dynamics as part of the interference results in unsatisfactory interference estimation. In fact, the nonlinear dynamics of many systems are partially known; applying them to interference observer design can significantly improve the estimation performance. Second, nonlinear observers constructed using system state measurement information cannot be used when the number of sensors is insufficient or measurements are difficult. Summary of the Invention

[0005] To address the aforementioned technical problems, this invention provides a design method for a nonlinear disturbance observer based on adaptive gain. This method utilizes system output measurement information to construct an observer with better adaptability. Furthermore, to enhance the adaptability of the disturbance observer parameters, an adaptive rate of change is designed, enabling the gain to vary based on estimation error and measurement noise: the gain increases when the error increases, and decreases when the noise increases. This achieves real-time and accurate estimation of system disturbances by the adaptive gain nonlinear disturbance observer, providing a theoretical basis and technical support for chemical process observation and control.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] A design method for a nonlinear disturbance observer based on adaptive gain includes the following steps:

[0008] The first step is to mathematically characterize the nonlinear dynamic behavior of the system using input-output differential equations;

[0009] The second step is to construct a virtual system based on mathematical representation, and multiply the difference between the output of the real system and the output of the virtual system by a constant gain to approximate the system input, thereby realizing real-time estimation of external disturbances of the nonlinear system.

[0010] The third step is to add an adaptive term proportional to the absolute value of the approximation error between the real system output and the virtual system output, based on the constant gain designed in the second step. The adaptive term only works when the observation error exists, effectively reducing the estimation error of the nonlinear interference observer for time-varying interference.

[0011] The fourth step involves adding an adaptive term inversely proportional to the output measurement variance, based on the constant gain and adaptive term designed in the third step. This adaptive term effectively reduces the gain of the interference observer and improves robustness to measurement noise when the system output jitter occurs.

[0012] Furthermore, the first step is specifically implemented as follows:

[0013] Consider the following nonlinear system described by input-output differential equations:

[0014]

[0015] Where y1 represents the system output variable, f(Y1) and g i (Y1), i = 0, ..., m, represent known smooth nonlinear functions, and the state set Y1 is:

[0016]

[0017] v represents the system input variable: v = u + d(t), where u represents the control input, d(t) represents the external disturbance received by the system, and y1 (n) and v (m) These represent the nth derivative of the system output and the mth derivative of the system input, respectively.

[0018] Furthermore, the second step is implemented as follows:

[0019] Using the nonlinear dynamic model established in the first step, construct a virtual system:

[0020]

[0021] In the formula, f(Y2) and g i(Y2), i = 0, ..., m, represents the known nonlinear dynamics of the system, and the state settling Y2 is:

[0022]

[0023] Where y2 and h are the output and input of the virtual system, respectively, y2 (n) and h (m) These represent the nth derivative of the output and the mth derivative of the input of the virtual system, respectively.

[0024] The input to the virtual system is approximated by the difference between the real system and the virtual system:

[0025] h = K(y1 - y2)

[0026] In the formula, K is a sufficiently large adjustable constant gain; based on this, the interference observation output device is designed as follows:

[0027]

[0028] This indicates an estimate of the interference.

[0029] Furthermore, the third step is specifically implemented as follows:

[0030] Based on the adjustable constant gain K of the disturbance observer, an adaptive variable term is introduced:

[0031] K a =K+σK

[0032] In the formula, K a The gain is variable, and σK is an adaptive gain that varies with the absolute value of the system output approach error.

[0033] σK=K0|y1-y2|

[0034] Where K0 is the set ratio; when K0 = 0, σK loses its adaptive capability, and the variable gain K a It degenerates into an adjustable constant gain K; when the virtual system output is consistent with the real system output, the adaptive term σK is zero and has no effect; for convenient parameter tuning, K0 = K is chosen.

[0035] Furthermore, the fourth step is specifically implemented as follows:

[0036] An adaptive mechanism is introduced to reduce the gain of the interference observer as noise increases:

[0037]

[0038] Where K1 is the set ratio, D(·) represents the system output approach variance, and the latest 100 data points in each observation period are taken as the variance calculation sample:

[0039]

[0040] In the formula, y 1k ,y 2k These represent the outputs of the real system and the virtual system at time k, respectively. This represents the average of these 100 sample data points:

[0041]

[0042] Construct a variable-gain nonlinear disturbance observer that is adaptive to both observation errors and measurement noise:

[0043]

[0044] The advantages of this invention compared to existing technologies are as follows: This invention considers a class of nonlinear systems described by input-output differential equations. Unlike existing nonlinear observer designs, the designed observer only requires system output measurement information and does not need measurement information for each state. It fully utilizes the information from the nonlinear model to construct a virtual "digital twin" system for designing an interference observer. It overcomes the limitations of fixed gain, enabling the observer gain to adaptively change with estimation error and measurement noise, making it suitable for fine estimation of interference in chemical process control. Attached Figure Description

[0045] Figure 1 This is a flowchart of a nonlinear disturbance observer design method based on adaptive gain according to the present invention.

[0046] Figure 2 A structural diagram for disturbance estimation in a continuous stirred tank reactor;

[0047] Figure 3 A comparison chart showing the interference estimation effect of a continuously stirred tank reactor without considering measurement noise;

[0048] Figure 4 A comparison chart showing the interference estimation error of a continuously stirred tank reactor without considering measurement noise;

[0049] Figure 5 A comparison chart showing the effect of interference estimation for continuously stirred tank reactors when considering measurement noise;

[0050] Figure 6 A comparison chart showing the error estimation of interference in a continuously stirred tank reactor when considering measurement noise. Detailed Implementation

[0051] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0052] This invention addresses nonlinear systems described by input-output differential equations. First, the nonlinear dynamic behavior of the system is mathematically represented using these equations. Second, based on this mathematical representation, a virtual "digital twin" system is constructed to approximate the real system and achieve real-time estimation of disturbances. Third, at a constant baseline, the gain of the disturbance observer is adaptively adjusted based on the estimation error. Finally, under the matched coefficients, the gain of the observer is adaptively adjusted based on the measurement noise. The specific implementation steps are as follows:

[0053] The first step is to mathematically represent the nonlinear dynamic behavior of the system using input-output differential equations:

[0054]

[0055] Where y1 represents the system output variable, f(Y1) and g i (Y1), i = 0, ..., m, represent known smooth nonlinear functions, and the state set Y1 is:

[0056]

[0057] v represents the system input variable: v = u + d(t), where u represents the control input, d(t) represents the external disturbance received by the system, and y1 (n) and v (m) These represent the nth derivative of the system output and the mth derivative of the system input, respectively.

[0058] The second step, based on the mathematical representation, is to construct a virtual system and multiply the difference between the output of the real system and the output of the virtual system by a constant gain to approximate the system input, thereby achieving real-time estimation of external disturbances to the nonlinear system.

[0059] Using the nonlinear dynamic model established in the first step, construct a virtual system:

[0060]

[0061] In the formula, f(Y2) and g i (Y2), i = 0, ..., m, represents the known nonlinear dynamics of the system, and the state settling Y2 is:

[0062]

[0063] Where y2 and h are the output and input of the virtual system, respectively, y2 (n) and h (m) These represent the nth derivative of the output and the mth derivative of the input of the virtual system, respectively.

[0064] The input to the virtual system is approximated by the difference between the real system and the virtual system:

[0065] h = K(y1 - y2)

[0066] In the formula, K is a sufficiently large adjustable constant gain; based on this, the interference observation output device is designed as follows:

[0067]

[0068] This indicates an estimate of the interference.

[0069] The third step involves adding an adaptive term proportional to the absolute value of the approximation error between the real system output and the virtual system output, based on the constant gain designed in the second step. This adaptive term only takes effect when observation error exists, effectively reducing the estimation error of the nonlinear disturbance observer for time-varying disturbances.

[0070] In the second step, the virtual system output is driven to follow the actual system by a fixed constant gain, which lacks adaptability. Therefore, based on the adjustable constant gain K of the disturbance observer, a variable term with adaptive capability is introduced:

[0071] K a =K+σK

[0072] In the formula, K a The gain is variable, and σK is an adaptive gain that varies with the absolute value of the system output approach error.

[0073] σK=K0|y1-y2|

[0074] Where K0 is the set ratio; when K0 = 0, σK loses its adaptive capability, and the variable gain K a It degenerates into an adjustable constant gain K; when the virtual system output is consistent with the real system output, the adaptive term σK is zero and has no effect; for convenient parameter tuning, K0 = K is chosen.

[0075] The fourth step involves using the constant gain and adaptive term designed in the third step, along with K. a Using this as a base point, an adaptive term inversely proportional to the output measurement variance is added. This adaptive term effectively reduces the interference observer gain and improves robustness to measurement noise under system output jitter.

[0076] In the second step, the interference estimation error monotonically changes with the unique adjustable gain of the interference observer. The larger the value of K, the closer the virtual system is to the real system, and the more accurate the interference estimation. However, the actual system is subject to physical constraints, and the gain K cannot approach infinity. Moreover, the system output is affected by measurement noise, and an excessively large gain will cause significant jitter in the observer output, or even instability. Therefore, an adaptive mechanism is introduced in which the interference observer gain decreases as noise increases:

[0077]

[0078] Where K1 is the set ratio, D(·) represents the system output approach variance, and the latest 100 data points in each observation period are taken as the variance calculation sample:

[0079]

[0080] In the formula, y 1k ,y 2k These represent the outputs of the real system and the virtual system at time k, respectively. This represents the average of these 100 sample data points:

[0081]

[0082] When the initial data is insufficient, the variance and mean are padded with zeros.

[0083] In summary, the following variable-gain nonlinear disturbance observer is constructed to adapt to both observation errors and measurement noise:

[0084]

[0085] Specifically, such as Figure 1 As shown, the specific implementation of this invention takes a continuous stirred tank reactor chemical process as an example, and the steps are as follows:

[0086] The first step is to establish a nonlinear dynamic model for a continuous stirred tank reactor that couples mass balance, energy balance, and cooling balance:

[0087] A nonlinear kinetic model is established to address the coupling relationship between reactant mass balance, reactant energy balance, and cooling energy balance within the jacket in a jacketed continuous stirred tank reactor, as follows:

[0088]

[0089] In the formula, x1, x2, and x3 represent reactant concentration, reactant temperature, and cooling jacket temperature, respectively; u1, u2, and u3 represent reactant input concentration, reactant input temperature, and coolant input temperature in the jacket, respectively; d(t) represents disturbances including external temperature, chemical feed mass, particle level, parameter perturbations, and cross-coupling factors that vary with time; and q is the reciprocal of the reactant residence time. c δ1 is the reciprocal of the coolant residence time in the jacket, and represents system parameters. Temperature values ​​x2 and x3 are typically measurable, while the real-time reactant concentration information x1 is often difficult to measure due to process or cost limitations.

[0090] The nonlinear dynamic constants are:

[0091]

[0092] The second step involves constructing a virtual system based on the nonlinear dynamic model established in the first step. The difference between the real and virtual systems is multiplied by a constant gain to approximate the system's input state, thus achieving real-time estimation of external disturbances to the nonlinear system.

[0093] Taking the system state as y1 = x2, we use the measurable states y1 and x3 to estimate the disturbance d(t). We then rearrange the second fraction of Σ1:

[0094]

[0095] Among them, nonlinear functions Differentiating with respect to x1, we get:

[0096]

[0097] Rearranging the first fraction of Σ1 according to whether it contains x1, we have:

[0098]

[0099] Substituting the unmeasurable state x1 and the derivative in the above equation with other variables, we get:

[0100]

[0101] After simplification and organization, we have:

[0102]

[0103] For the nonlinear reactor dynamics system characterized by Σ2, a nonlinear disturbance observer is designed to estimate unknown disturbances, based on full utilization of model parameters and structure:

[0104]

[0105] Where y2 represents the output state of the virtual system, and h represents the difference state between the actual system and the virtual system after gain. This represents the nonlinear observer's estimate of the disturbance, where K is the observer's adjustable constant gain.

[0106] The third step involves adding an adaptive term proportional to the absolute value of the output observation error, based on the adjustable constant gain designed in the second step. This adaptive term only takes effect when the observation error exists, effectively reducing the estimation error of the nonlinear observer to time-varying disturbances.

[0107] Σ3 drives the virtual system's output state to follow the actual system through a fixed constant gain, lacking adaptability. Therefore, the observer gain, based on a constant, introduces a variable term with adaptive capabilities:

[0108] K a =K+σK

[0109] In the formula, K a The gain is variable, and σK is an adaptive gain that varies with the absolute value of the system output approach error.

[0110] σK=K0|y1-y2|

[0111] Where K0 is the set ratio. When K0 = 0, σK loses its adaptive capability, and the variable gain K... a The gain degenerates to a constant value K; furthermore, when the virtual system output matches the real system output, the adaptive term σK is zero and has no effect. Generally, for ease of parameter tuning, K0 can be chosen to be K.

[0112] The fourth step involves adding an adaptive term inversely proportional to the output measurement variance, based on the adaptive gain for observation error designed in the third step. This adaptive term effectively reduces the observer gain and improves robustness to measurement noise under system output jitter.

[0113] The estimation error of interference by Σ3 monotonically changes with the unique adjustable constant gain of the interference observer. The larger the value of K, the closer the virtual system is to the real system, and the more accurate the interference estimation. However, the real system is subject to physical constraints, and the adjustable constant gain K cannot approach infinity. Moreover, the system output is affected by measurement noise, and excessive gain will cause significant jitter in the observer output, or even instability. Therefore, an adaptive mechanism is introduced in which the observer gain decreases as the noise increases:

[0114]

[0115] Where K1 is the set ratio. D(·) represents the system output approach variance. To reduce the computational burden, the latest 100 data points are taken as the variance calculation sample in each observation period:

[0116]

[0117] In the formula, y 1k ,y 2k These represent the outputs of the real system and the virtual system at time k, respectively. This represents the average of these 100 sample data points:

[0118]

[0119] When the initial data is insufficient, the variance and mean are padded with zeros.

[0120] In summary, a variable-gain nonlinear disturbance observer with adaptive capabilities to both observation errors and measurement noise is constructed:

[0121]

[0122] like Figure 2 The diagram shows the structure for disturbance estimation in a continuous stirred tank reactor. The adaptive gain disturbance observer adjusts its gain based on the real-time changes in the virtual system output approximation error and measurement noise, using an adaptive mechanism.

[0123] like Figure 3 As shown, the estimation performance of the two observers is compared when external disturbances occur. In the case of mismatched initial value estimations for the nonlinear system, the disturbance observer using adaptive gain provides a more accurate estimate than the observer using fixed gain under the same conditions.

[0124] like Figure 4 As shown, this compares the estimation errors of the two observers when external disturbances occur. Compared to... Figure 3 , Figure 4 The shorter ordinate makes the comparison more obvious. The observer with the adaptive mechanism has a greater gain when the estimation error increases, thus reducing the estimation error of disturbances; in addition, when the estimation error is small, the gain change is not significant and does not increase the burden on the system.

[0125] like Figure 5 As shown, the estimation performance of Σ3 and Σ4 for interference is compared when interference and noise act together. Under the influence of measurement noise at the output end, the interference observer using adaptive gain provides a smoother estimation than the observer using fixed gain under the same conditions. The less estimation jitter is more friendly to the observation mechanism, and it also consumes less energy.

[0126] like Figure 6As shown, the estimation errors of Σ3 and Σ4 for interference are compared when interference and noise act together. It is clear that the interference observer using adaptive gain has a smaller estimation error amplitude. When the output is affected by noise, the Σ4 adaptive mechanism reduces the observer gain, thus reducing the jitter amplitude of the estimation error. However, the smaller observation gain has adverse effects; the amplitude estimation of the sinusoidal signal is biased within 45-55 seconds. Therefore, a compromise is needed to set parameter K1 based on the magnitude of the noise.

[0127] comprehensive Figure 3-6 As can be seen, the nonlinear interference observer based on adaptive gain proposed in this invention has good interference estimation and noise suppression capabilities.

[0128] The contents not described in detail in this specification are prior art known to those skilled in the art. Those skilled in the art will readily understand that the above descriptions are merely preferred embodiments of the present invention and are not intended to limit the invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A design method for a nonlinear disturbance observer based on adaptive gain, characterized in that, Includes the following steps: The first step is to mathematically characterize the nonlinear dynamic behavior of the system using input-output differential equations; The second step is to construct a virtual system based on mathematical representation, and multiply the difference between the output of the real system and the output of the virtual system by a constant gain to approximate the system input, thereby realizing real-time estimation of external disturbances of the nonlinear system. The third step is to add an adaptive gain proportional to the absolute value of the approximation error between the real system output and the virtual system output, based on the constant gain designed in the second step. The adaptive gain only works when the observation error exists, effectively reducing the estimation error of the nonlinear interference observer for time-varying interference. The fourth step is to introduce an adaptive mechanism where the gain of the interference observer decreases as noise increases: ； Where K is the constant gain, K1 is the set ratio, and D(·) represents the output measurement variance. The latest 100 data points in each observation period are used as the variance calculation sample. ； In the formula, y 1k , y 2k These represent the outputs of the real system and the virtual system at time k, respectively. This represents the average of these 100 sample data points: ； Construct a variable-gain nonlinear disturbance observer that is adaptive to both observation errors and measurement noise: 。 2. The design method for a nonlinear disturbance observer based on adaptive gain according to claim 1, characterized in that: The first step is implemented as follows: Consider the following nonlinear system described by input-output differential equations: ； Where y1 represents the system output variable, f(Y1) and g i (Y1), i=0,…,m, represent known smooth nonlinear functions, and the state set Y1 is: ； v represents the system input variable: Where u represents the control input, d(t) represents the external disturbance experienced by the system, and y1 (n) and v (m) These represent the nth derivative of the system output and the mth derivative of the system input, respectively.

3. The design method for a nonlinear disturbance observer based on adaptive gain according to claim 2, characterized in that: The second step is implemented as follows: Using the nonlinear dynamic model established in the first step, construct a virtual system: ； In the formula, f(Y2) and g i (Y2), i=0,…,m, represents the known nonlinear dynamics of the system, and the state ensemble Y2 is: ； Where y2 and h are the output and input of the virtual system, respectively, y2 (n) and h (m) These represent the nth derivative of the output and the mth derivative of the input of the virtual system, respectively. The input to the virtual system is approximated by the difference between the real system and the virtual system: ； In the formula, K is a sufficiently large adjustable constant gain; based on this, the interference observation output device is designed as follows: ； in, This indicates an estimate of the interference.

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