CN110376886A - A kind of Model Predictive Control Algorithm based on expansion state Kalman filter - Google Patents

Abstract

The invention discloses a model predictive control algorithm based on an extended state Kalman filter, which includes the following steps: (1) aggregate the system nonlinearity, uncertainty and external disturbance into a new state quantity, and amplify the original The state space model of the system, design the extended state Kalman filter to observe the state quantity and the lumped disturbance quantity of the system; (2) Based on the known state quantity and disturbance quantity, while considering the system input, output and state constraints, design a model predictive controller . Based on the extended state Kalman filter, the present invention proposes a model predictive control algorithm based on the extended state Kalman filter, which can simultaneously solve system process noise, measurement noise, and input and output constraints, and improve the observation of the observer when noise exists performance, improved control performance.

Description

A Model Predictive Control Algorithm Based on Extended State Kalman Filter

technical field

The invention relates to the technical field of industrial process control, in particular to a model predictive control algorithm based on an extended state Kalman filter.

Background technique

Extended state observer (Extended state observer, ESO) expands the internal and external lumped disturbances in the system into a new first-order state of the system, selects appropriate observer parameters, and obtains the observed values of all state quantities of the system including the lumped disturbance. Due to its obvious advantages in dealing with uncertain problems such as unknown system parameters, unmodeled dynamics, and unknown disturbances, it has gradually attracted extensive attention from researchers and has been successfully applied to various systems. Existing studies have shown that the observation performance of ESO is directly related to the bandwidth of the observer. The larger the bandwidth, the higher the observation accuracy of the state quantity, but at the same time, it is more sensitive to noise. However, the existing extended state observers do not consider the process noise and measurement noise in the system when they are designed, and these noises exist widely in reality, which will affect the performance of the observer. In addition, the gain of the general linear extended state observer is generally adjusted by the bandwidth method, and decoupling design is required when dealing with multivariable systems, which is relatively complicated. Therefore, some literature proposes an extended state Kalman filter, which uses Kalman theory to optimize the gain of the observer in real time, and improves the observation accuracy of the observer in the presence of noise.

Contents of the invention

The technical problem to be solved by the present invention is to provide a model predictive control algorithm based on an extended state Kalman filter, which can be applied to the state observation and model predictive control problems of a class of nonlinear uncertain systems.

In order to solve the above technical problems, the present invention provides a model predictive control algorithm based on extended state Kalman filter, comprising the following steps:

(1) Summarize the system nonlinearity, uncertainty and external disturbance into a new state quantity, amplify the state space model of the original system, and design an extended state Kalman filter to observe the system state quantity and aggregate disturbance quantity;

(2) Design a model predictive controller based on known state quantities and disturbances, while considering system input, output, and state constraints.

Preferably, in step (1), consider the discrete nonlinear uncertain system as

Among them, X(k)∈R ^m is the state quantity, k is the current sampling moment, A _d ∈ R ^m×m , B _u ∈ R ^m×p , B _f ∈ R ^m×r , C _d ∈ R ^q×m is the known system matrix, F(k)∈R ^r is the lumped item of nonlinearity, uncertainty and external disturbance in the system, and its nominal model is a known nonlinear function, the process noise W(k) is m-dimensional uncorrelated zero-mean Gaussian random noise, its covariance matrix is Q _w , y(k)∈R ^q is the measurement output vector, n(k)∈R ^q is the measurement noise vector, and its covariance matrix is Q _n .

Preferably, in step (1), the lumped disturbance F _k in the discrete nonlinear uncertain system model is regarded as the augmented state, then the augmented state space model of the system is

Among them, G(k)=F(k+1)-F(k), in order to make better use of model information, consider the nominal model of G(k) as h is the sampling time,

Preferably, in step (1), for the augmented state space system model, an extended state Kalman filter can be designed as follows:

Among them, '∧' represents the observed value, K _k and P _k+1 are the filter gain at time k and the state error covariance estimate at time k+1 respectively, The saturation function sat( ) is defined as sat(f,b)=max{min{f,b},-b}, b>0, the adjustment parameter

Preferably, in step (2), the specific design of the model predictive controller is as follows:

Based on the state value and disturbance value observed by the extended state Kalman filter at the current sampling moment, the prediction model of the system state at the next P sampling moments can be obtained as

in,

Consider the system optimization performance index function as follows

Among them, Q and R are the error weight matrix and control weight matrix respectively, X _s is the state setting value, and the input, output, and state constraints are satisfied at the same time as follows:

U _min ≤ U(k) ≤ U _max

Y _min ≤ Y(k) ≤ Y _max

X _min ≤ X(k) ≤ X _max

Solve the optimization problem at each sampling time to obtain the optimal control law The first term of the optimal control law is applied to the control object, the state observation value is updated at the next sampling time, and the optimal control law is repeatedly calculated.

The beneficial effects of the present invention are: the system nonlinearity, uncertainty and external disturbance are aggregated into a new state quantity, and the extended state Kalman filter is designed to observe the state value and the aggregated disturbance value, avoiding the nonlinear observer design and the resulting stability problems; use Kalman theory to optimize the gain of the observer in real time, which improves the observation accuracy of the observer in the presence of process noise and measurement noise; combine the extended state Kalman filter with the model predictive controller Combined, it can overcome the shortcomings of the system's large inertia and large delay, and improve the system response speed; at the same time, the constraints of the system's input, output, and state quantities are considered, and the control performance degradation caused by the saturation of the actuator is avoided.

Description of drawings

Fig. 1 is a schematic flow chart of the algorithm of the present invention.

Fig. 2 is a schematic diagram of the output curve in the embodiment of the present invention.

Fig. 3 is a schematic diagram of a control amount curve in an embodiment of the present invention.

Fig. 4 is a schematic diagram of an observation error curve of a state quantity x ₁ in an embodiment of the present invention.

Fig. 5 is a schematic diagram of the observation error curve of the state quantity x ₂ in the embodiment of the present invention.

Fig. 6 is a schematic diagram of the observation error curve of the state quantity x ₃ in the embodiment of the present invention.

Fig. 7 is a schematic diagram of the observation error curve of the state quantity x ₄ in the embodiment of the present invention.

FIG. 8 is a schematic diagram of an observation error curve of a lumped disturbance F in an embodiment of the present invention.

Detailed ways

As shown in Figure 1, a model predictive control algorithm based on extended state Kalman filter includes the following steps:

(1) Summarize the system nonlinearity, uncertainty and external disturbance into a new state quantity, amplify the state space model of the original system, and design an extended state Kalman filter to observe the system state quantity and aggregate disturbance quantity;

(2) Design a model predictive controller based on known state quantities and disturbances, while considering system input, output, and state constraints.

In conjunction with the motion control of the underactuated unmanned ship as an embodiment, the model predictive control algorithm based on the extended state Kalman filter (Extended state kalman filter, ESKF) of the present invention is adopted, and at the same time it is combined with the extended state observer (Extended state observer, The model predictive control algorithm of ESO) is compared with the model predictive control algorithm based on extended kalman filter (Extended kalman filter, EKF).

The motion model of the underactuated UAV can be described as:

Among them, x ₁ , x ₂ , x ₃ and x ₄ are the heading, heading speed, heading acceleration and rudder angle of the underactuated UAV respectively, the control variable u is the rudder order, and the controlled variable y is the heading of the UAV. d(t) represents the external environment disturbance, w(t) and n(t) are the system process noise and measurement noise respectively. K, T ₁ , T ₂ , T ₃ , T _c and α are system parameters, K=0.5900, T ₁ =0.9526, T ₂ =0.0247, T ₃ =0.2215, α=0.0001, T _c =0.1000.

The continuous model is discretized, and the discrete time h=0.01s, the discrete state space model of the following form can be obtained:

in C _d =[1 0 0 0].

Assuming that the initial state of the system is x=[0 0 0 0] ^T , the control objective is to keep the heading constant at 20° when the external disturbance d(t) is a fixed value of 10, while the heading speed, heading acceleration, and rudder angle remain is 0, that is, x _s =[20 0 0 0] ^T . The system state and input constraints are: -30°≤x ₄ (k+i)≤30°, -30°≤u(k+i-1)≤30°, -20° h≤Δu(k+i- 1) ≤20°·h. σ _w and σ _n are the standard deviations of process noise w(k) and measurement noise n(k) respectively, σ _w =0.001·h, σ _n =0.001.

The parameters of the extended state Kalman filter are set as follows:

The parameters of the model predictive controller based on extended state Kalman filter are set as follows: prediction time domain P=200s, control time domain M=2s, error weight matrix Q=diag[10 ⁵ ,10 ² ,10 ² ,10 ² ], Control right matrix R=1, simulation time t=20s.

For comparison, the model predictive control algorithm based on the extended state observer and the model predictive control algorithm based on the extended Kalman filter are introduced for comparison. Based on the model predictive controller of the extended state observer and the prediction time domain of the model predictive controller based on the extended Kalman filter, the control time domain, the error weight matrix, the setting of the control weight matrix and the method based on the extended state Kalman in the present invention The parameters of the filter model predictive controller are set the same.

In the model predictive controller based on the extended state observer, the construction and parameter settings of the extended state observer are as follows:

Wherein, w _e is the bandwidth of the extended state observer, which is set to w _e =4.

In the model predictive controller based on the extended Kalman filter, the construction and parameter settings of the extended Kalman filter are as follows:

in

A schematic structural diagram of a model predictive control algorithm based on an extended state Kalman filter proposed in the present invention is shown in FIG. 1 . Fig. 2 and Fig. 3 are respectively the output quantity curve and the control quantity curve in the embodiment of the present invention, it can be seen that when external constant value disturbance, process noise and measurement noise exist, under the condition of satisfying the constraints of state quantity and input quantity , the controlled quantity can quickly and accurately reach the set value, and the controlled quantity changes smoothly without frequent fluctuations. Fig. 4, Fig. 5, Fig. 6, Fig. 7 and Fig. 8 respectively show the state quantities x ₁ , x ₂ , x obtained by the extended state Kalman filter, the extended state observer and the extended Kalman filter in the embodiment of the present invention ₃ , x ₄ and the observation error curves of the lumped disturbance F. It can be seen from the figure that there is a steady-state observation bias in the estimation of the state quantity and disturbance quantity by the extended Kalman filter. Although the steady-state observation deviation of the extended state observer is 0, the observed value is sensitive to noise, which will cause the controller to fluctuate frequently. The extended state Kalman filter combines the advantages of the extended state observer and the Kalman filter, the observation value is not greatly affected by the noise, and the steady-state observation error is 0, so the observation effect is the best.

Based on the extended state Kalman filter, the present invention proposes a model predictive control algorithm based on the extended state Kalman filter, which can simultaneously solve system process noise, measurement noise, and input and output constraints, and improve the observation of the observer when noise exists performance, improved control performance.

Claims (5)

1. A model prediction control algorithm based on an extended state Kalman filter is characterized by comprising the following steps:

(1) integrating system nonlinearity, uncertainty and external disturbance into a new state quantity, amplifying a state space model of an original system, and designing an expanded state Kalman filter observation system state quantity and an integrated disturbance quantity;

(2) and designing a model predictive controller based on the known state quantity and disturbance quantity and considering system input, output and state constraints simultaneously.

2. The extended-state kalman filter-based model predictive control algorithm according to claim 1, wherein in step (1), the discrete nonlinear uncertainty system is considered as

Wherein X (k) e R^mIs the state quantity, k is the current sampling time, A_d∈R^m×m，B_u∈R^m×p，B_f∈R^m×r，C_d∈R^q×mFor a known system matrix, F (k) e R^rIs a nominal model of the lumped terms of nonlinearity, uncertainty and external disturbance in the systemFor a known nonlinear function, the process noise W (k) is m-dimensional uncorrelated zero mean Gaussian random noise with a covariance matrix of Q_w,y(k)∈R^qIs the measurement output vector, n (k) e R^qIs to measure the noise vector with a covariance matrix of Q_n。

3. The extended-state Kalman filter-based model prediction control algorithm of claim 1, characterized in that in step (1), lumped disturbances F in the discrete nonlinear uncertainty system model are treated_kWhen viewed as an amplification state, the system amplifies the state space model as

Wherein g (k) ═ F (k +1) -F (k), in order to better utilize the model information, the nominal model of g (k) is considered ash is the sampling time of the sample, C＝[C_d 0]。

4. the extended state kalman filter-based model predictive control algorithm of claim 3, wherein in step (1), the extended state kalman filter is designed for the extended state space system model as follows:

wherein ` Λ' represents an observed value, K_kAnd P_k+1Respectively the filter gain at time k and the state error covariance estimate at time k +1,the saturation function sat (-) is defined as sat (f, b) ═ max { min { f, b }, -b }, b > 0, and the adjustment is adjustedParameter(s)

5. The extended-state kalman filter-based model predictive control algorithm according to claim 1, wherein in step (2), the model predictive controller is specifically designed as follows:

based on a state value and a disturbance value observed by an extended state Kalman filter at the current sampling moment, a prediction model of the system state at the future P sampling moments can be obtained as

Wherein,

consider the system optimization performance indicator function as follows

Wherein Q and R are an error weight matrix and a control weight matrix, X_sFor the state set value, the input, output and state constraints are satisfied simultaneously as follows:

U_min≤U(k)≤U_max

Y_min≤Y(k)≤Y_max

X_min≤X(k)≤X_max

solving the optimization problem at each sampling time to obtain the optimal control lawAnd applying the first item of the optimal control law to the control object, updating the state observation value at the next sampling moment, and repeatedly calculating the optimal control law.