CN111694277A - Nonlinear stochastic model prediction control method based on multi-step state feedback - Google Patents

Nonlinear stochastic model prediction control method based on multi-step state feedback Download PDF

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CN111694277A
CN111694277A CN202010536914.5A CN202010536914A CN111694277A CN 111694277 A CN111694277 A CN 111694277A CN 202010536914 A CN202010536914 A CN 202010536914A CN 111694277 A CN111694277 A CN 111694277A
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CN111694277B (en
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孔小兵
冯乐
刘向杰
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North China Electric Power University
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Abstract

The invention relates to a nonlinear stochastic model predictive control method based on multi-step state feedback, which comprises the steps of establishing a system stochastic interference model by researching the characteristics of a nonlinear system with stochastic interference and a nonlinear discrete system stochastic model with probability constraint by utilizing a statistical method, providing a stochastic optimization control problem aiming at the model by combining with a control target design method in the traditional predictive control, and then solving the bounded model mismatch problem between the nonlinear model and a linear nominal model by applying a robust tube invariant set idea. By designing a multi-step state feedback control law, optimizing statistical performance indexes in an infinite time domain, constructing a multi-step probability Tube invariant set with higher degree of freedom, and ensuring that probability constraint recursion is satisfied, thereby controlling future random states of the system and effectively solving the problems of traceability, economy and stability of the random system under random interference.

Description

Nonlinear stochastic model prediction control method based on multi-step state feedback
Technical Field
The invention relates to a nonlinear system which is not limited in the field of chemical engineering, in particular to a nonlinear stochastic model prediction control method based on multi-step state feedback.
Background
With the development of control technology, the requirements for the model are gradually increased due to the improvement of the control requirements of industrial processes, aviation fields and the like. The systems which are more common in the practical process have strong nonlinear characteristics and even random uncertainties. Poor robustness, poor stability and low conservation become difficulties in the control process of such systems. Therefore, with the nonlinearity of the system, the randomness is increasingly highlighted, and due to simplified processing when the model is built, neglecting unmodeled dynamics can have serious influence on the performance of the controller.
The traditional model prediction control adopts deviation to carry out online correction, but if the actual system has strong randomness, the state can be seriously deviated from the predicted value, and the feasibility and the stability of the closed-loop system can be influenced. Therefore, in order to control the nonlinear system with random interference uncertainty more accurately, the random characteristics of the nonlinear system need to be accurately characterized. Although the robust model predictive control can solve the problem of random uncertainty, probability distribution of uncertainty is ignored, and the objective function with the maximum disturbance influence is minimized, so that the robust model predictive control has large conservative property. Thereby promoting the development of stochastic predictive control. The probability invariant set in the random predictive control uses the idea of parametric disturbance feedback for reference to obtain an easily-processed convex predictive control optimization problem. However, the control capability provided by the fixed feedback law is limited, so that the calculation result of the feasible domain of the probability constraint is conservative, and the design and implementation are difficult. Meanwhile, a learner combines the discrete Markov chain with the invariant probability set to process probability constraint, but different invariant sets only meet one-step forward probability transition constraint, the degree of freedom of the invariant set is limited, and system parameters have sensitive influence on the existence and the size of the invariant set. Therefore, how to improve the degree of freedom of calculation and provide stronger control capability, and the expansion of the application range of the algorithm is a challenge in the development process of the stochastic model predictive control.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a nonlinear stochastic model prediction control method based on multi-step state feedback, which aims to solve the problem of low degree of freedom of probability invariant set calculation.
The invention provides a nonlinear stochastic model predictive control method based on multi-step state feedback, which comprises the following steps:
step 1: and analyzing the random process by utilizing statistics aiming at the nonlinear discrete random system, and establishing a corresponding model.
Step 2: designing a control target with economy, traceability and stability, introducing probability constraint and hard constraint, and proposing a random infinite time domain optimization control problem aiming at the system;
and step 3: linearizing the state space model of the nonlinear discrete random system to obtain a nominal discrete linear random model, and calculating a robust Tube invariant set in an off-line manner by combining the model mismatch between the nominal model and the nonlinear model and the characteristics of a random interference process to ensure the effectiveness of the nominal model on replacing the nonlinear model;
and 4, step 4: aiming at a nominal discrete linear random model, introducing a dual-mode prediction paradigm with a prediction sequence, designing a multi-step state feedback control law meeting the autonomy of a system, and obtaining a corresponding prediction model to increase the design freedom of a subsequent random prediction controller;
and 5: combining the characteristics of multi-step feedback, converting the statistical performance index in the infinite time domain of the original random optimization problem into a finite value with the certainty related to the initial state;
step 6: combining the characteristics of multi-step feedback, constructing a multi-step probability Tube invariant set to force a future prediction state to enter the probability invariant set, and ensuring that a nominal state meets tightening constraints by using the characteristics of a robust invariant set to ensure that an original probability constraint is met;
and 7: and introducing dual-mode control to ensure that the state is always in a probability invariant set or gradually shrinks into the probability invariant set, and tightening each constraint condition in combination with the probability invariant set to ensure that the precondition for carrying out optimization control is obtained.
Further, the nonlinear discrete random system is expressed as follows:
Figure BDA0002537325300000021
in the formula: x is the number ofkIs a state quantity, ukAs an input quantity, wkIs random interference.
Further, random interference of the system
Figure BDA0002537325300000031
Is the 0 mean value, and for any time, { w0,w1,. } are independent, identically distributed seed states whose random probability distribution is expressed as follows:
Figure BDA0002537325300000032
in the formula: pr [ ] represents the probability of a random event, the scalar γ > 0, the function F is continuous;
as can be obtained from equation (2), wkDefining multiple cells with orthogonal edges
Figure BDA0002537325300000033
And (4) the following steps.
Further, the system is represented by the probability constraint of the states and the hard constraint of the input as follows:
Pr[G1xk≤g1]≥p (3a)
G2uk≤g2(3b)
in the formula: p ∈ [0,1] is the probability of allowing a constraint violation.
Further, the control objective with economy, tracking and stability is usually expressed as a least square function of the control quantity and the output quantity, and is expressed as follows:
Figure BDA0002537325300000034
in the formula: weight coefficient r1And r2A non-negative constant, u*As input quantity reference value, y*Is an output reference value related to economic benefit;
by mixing y*Set as the maximum set value of the output quantity
Figure BDA0002537325300000036
To obtain better economic efficiency, the set objective function (4) is rewrittenComprises the following steps:
Figure BDA0002537325300000035
further, when random interference is included in the controller design, the objective function is set as an expected performance index based on random variable probability distribution, and in combination with the constraint (3), the random infinite time domain optimization control problem for the nonlinear random system is expressed as follows:
Figure BDA0002537325300000041
the system is highly nonlinear, and the predicted output power depends on future control movement, so the output power in the objective function is implicit, and a deterministic quadratic programming method cannot be applied to solve a stochastic optimization problem.
Further, taylor expansion of the non-linear terms in the system is as follows:
Figure BDA0002537325300000042
in the formula:
Figure BDA0002537325300000043
Figure BDA0002537325300000044
Figure BDA0002537325300000045
omitting higher order terms
Figure BDA0002537325300000046
And
Figure BDA0002537325300000047
to obtain a nominally linearized stochastic model:
Figure BDA0002537325300000048
in the formula:
Figure BDA0002537325300000049
and
Figure BDA00025373253000000410
the model mismatch between the nonlinear system and the nominal linear stochastic model is high-order term
Figure BDA00025373253000000411
And
Figure BDA00025373253000000412
it can be limited by the following formula:
Figure BDA00025373253000000413
Figure BDA00025373253000000414
in the formula: | | represents a 2 norm of the variable,
Figure BDA00025373253000000415
is composed of
Figure BDA00025373253000000416
The upper bound of (a) is,
Figure BDA00025373253000000417
and
Figure BDA00025373253000000418
are all non-linear functions f1(xk,uk,wk) The constant of the relative amount of the acid to the base,
Figure BDA00025373253000000419
is composed of
Figure BDA00025373253000000420
The upper bound of (a) is,
Figure BDA00025373253000000421
and
Figure BDA00025373253000000422
are all non-linear functions f2(xk,uk,wk) T represents the measurement interval,
Figure BDA0002537325300000051
Figure BDA0002537325300000052
further, a linear feedback control law K is obtained by constructing linear matrix inequalities LMIs(1)And a robust Tube invariant set solved in an off-line manner is used for overcoming the state deviation problem caused by model mismatch, and a linear matrix inequality is expressed as follows:
Figure BDA0002537325300000053
Figure BDA0002537325300000054
in the formula:
Figure BDA0002537325300000055
is a positive definite matrix and the matrix is a negative definite matrix,
Figure BDA0002537325300000056
is a non-square matrix. Scalar lambda0＞0,μ2Is greater than 0. Thus, the gain K is fed back(1)And a symmetric positive definite matrix P1Given by:
K(1):＝YX-1(11)
P1:＝X-1(12)
the robust Tube invariant set can be expressed as:
Figure BDA0002537325300000057
in the formula: max (M)e,2γ1) Wherein the neutralization can be by
Figure BDA0002537325300000058
Finding a scalar λ1，αmin(P1) Representative matrix P1The smallest real part of the eigenvalues.
Further, by using the further invariance to compute an implicit constraint on the initial state; constructing an autonomous system to eliminate propagation uncertainty effects within a prediction horizon; defining a dual-mode predictive paradigm with multi-step state feedback to increase the freedom of the controller design:
Figure BDA0002537325300000061
in the formula: kk+iIs a state feedback gain matrix, ci|kIs a decision variable; the control input of the first step is free, and a prescribed state feedback law is adopted in subsequent infinite prediction; suppose that
Figure BDA0002537325300000062
Information with perturbation sequence
Figure BDA0002537325300000063
Wherein N iscIs the control time domain; at each time k, assume a prediction state of
Figure BDA0002537325300000064
An equivalent augmentation prediction model is obtained by recursion of the prediction dynamics to reduce steady-state errors and improve the steady-state performance of the system, and the equivalent augmentation prediction model is expressed as follows:
Figure BDA0002537325300000065
in the formula:
Figure BDA0002537325300000067
thereby to obtain
Figure BDA0002537325300000068
Further, the future predicted state trajectories in the model are random variable sequences; an objective function with a stochastic process is processed using statistical methods to transform a desired performance indicator function that is contained within an infinite time domain into a finite value associated with an initial state, as follows:
Figure BDA0002537325300000069
in the formula:
Figure BDA0002537325300000071
Figure BDA0002537325300000072
further, under the condition of giving an initial state and a multi-step state feedback strategy, a sub-optimization problem needs to be satisfied to obtain a cluster of probability invariant set to ensure that the probability constraint is satisfied, and the optimization problem is expressed as follows:
Figure BDA0002537325300000073
Figure BDA0002537325300000074
Figure BDA0002537325300000075
Figure BDA0002537325300000076
in the formula:
Figure BDA0002537325300000077
Figure BDA0002537325300000078
is a probability invariant set.
Further, a dual-mode control online optimization predictive control algorithm is adopted to ensure closed-loop trajectory convergence and system stability, and the method comprises the following steps:
minimizing the performance indicator function when the nominal random state is within the probability invariant set;
when the nominal random state is not in the probability invariant set, rapidly returning the nominal random state to the probability invariant set;
the dual-mode control online optimization problem is expressed as follows:
if it is not
Figure BDA0002537325300000079
Figure BDA00025373253000000710
Figure BDA0002537325300000081
If it is not
Figure BDA0002537325300000082
Figure BDA0002537325300000083
Figure BDA0002537325300000084
In the formula:
Figure BDA0002537325300000085
can be obtained in advance by the optimization problem (17). And tightening each constraint condition in combination with the probability invariant set to ensure that a precondition for performing optimal control is obtained.
Compared with the prior art, the random model prediction control method based on the multi-step state feedback accurately models a large number of nonlinear random processes existing in the actual process by introducing the random characteristic of interference, and establishes the random optimization problem with probability constraint in the infinite time domain. A nominal linear random model is obtained through linearization processing, and the state deviation is limited by applying a robust Tube invariant set, so that the problem of model mismatch is effectively solved. Based on a multi-step state feedback strategy, a cluster of ellipse probability invariant set with higher degree of freedom is designed by using random interference information. And (3) utilizing a dual-mode control to limit the future nominal random state of the system to be in a probability invariant set, and combining the property tightening constraint of the Luban Tube invariant set to finally ensure the satisfaction of the probability constraint. Compared with a random model predictive control strategy of a fixed feedback control law, the random model predictive control method based on multi-step state feedback has better optimization capability on random characteristics and better control performance.
Drawings
Fig. 1 is a schematic diagram of a stochastic model predictive control method based on multi-step state feedback according to an embodiment of the present invention.
FIG. 2 is a graph illustrating the relationship between controller quality and economic performance according to an embodiment of the present invention.
Detailed Description
Preferred embodiments of the present invention are described below with reference to the accompanying drawings. It should be understood by those skilled in the art that these embodiments are only for explaining the technical principle of the present invention, and do not limit the scope of the present invention.
Referring to fig. 1, a stochastic model predictive control method based on multi-step state feedback according to an embodiment of the present invention includes the following steps:
step 1, aiming at a nonlinear discrete random system, analyzing a random process by using statistical knowledge, and establishing a corresponding model;
step 2, designing a control target for ensuring economy, traceability and stability, introducing probability constraint and hard constraint, and proposing a random infinite time domain optimization control problem for the system;
and 3, linearizing the state space model of the nonlinear discrete random system to obtain a nominal discrete linear random model. Combining the model mismatch between the nominal model and the nonlinear model and the characteristics of the random interference process, and calculating a robust Tube invariant set in an off-line manner;
step 4, aiming at a nominal discrete linear random model, introducing a dual-mode prediction paradigm with a prediction sequence, designing a multi-step state feedback control law meeting the autonomy of the system, and obtaining a corresponding prediction model;
step 5, combining the characteristics of multi-step feedback, converting the statistical performance index in the infinite time domain of the original random optimization problem into a finite value with the certainty related to the initial state;
step 6, combining the characteristics of multi-step feedback, constructing a multi-step probability Tube invariant set with larger degree of freedom to force a future prediction state to enter the probability invariant set, and then utilizing the characteristics of the robust invariant set to ensure that a nominal state meets tightening constraints, thereby ensuring that the original probability constraints are met;
and 7, introducing dual-mode control to ensure that the state is always in the probability invariant set or is gradually contracted into the probability invariant set, and tightening each constraint condition in combination with the probability invariant set to ensure that the precondition for optimizing control is obtained.
Wherein, the step 1 specifically comprises the following steps:
step 1.1, the nonlinear discrete random system is:
Figure BDA0002537325300000091
in the formula: x is the number ofkIs a state quantity, ukAs an input quantity, wkIs random interference.
Step 1.2, the probability distribution characteristic of the random process is as follows:
Figure BDA0002537325300000101
in the formula: pr [. X]Representing the probability of a random event, the scalar γ > 0, the function F is continuous. In the case of equation (2), wkDefined as a plurality of cells with orthogonal edges
Figure BDA0002537325300000102
And (4) the following steps. Because the perturbations from physical processes are finite, the finite support uncertainty assumption matches the real world more closely than the unrealistic gaussian assumption. And this assumption is necessary to demonstrate the recursive feasibility and stability of the control strategy.
The step 2 specifically comprises the following steps:
step 2.1, the system (1) is subjected to the probability constraint of the state and the hard constraint of the input:
Pr[G1xk≤g1]≥p (3a)
G2uk≤g2(3b)
in the formula: p ∈ [0,1] is the probability of allowing a constraint violation.
Step 2.2, the control targets for ensuring the economy, the tracking performance and the stability are as follows:
Figure BDA0002537325300000103
in the formula: weight coefficient r1And r2A non-negative constant. u. of*Is an input quantity reference value. y is*Is an output reference value related to economic benefits. FIG. 2 depicts different control strategiesa and b, the probability density function of the output quantity in a steady state. Set value y in preferred controller a*Set value y in controller b for comparison*Larger, but violating the upper constraint limit ymaxIs smaller, a better controller reduces the dispersion of the output, and therefore guarantees that the upper limit y is violatedmaxOn the premise of the same risk, a higher set value y is set*And the economic benefit is improved. Thus y*Can be set to the maximum setting of the output quantity
Figure BDA0002537325300000104
To obtain better economic benefits, the objective function (8) can be rewritten as:
Figure BDA0002537325300000105
step 2.3, the general nonlinear stochastic predictive control optimization problem is expressed as follows:
Figure BDA0002537325300000111
the output power in the objective function (5) is implicit, since the system (1) has a high degree of non-linearity, whereas the predicted output power depends on future control movements. Therefore, the deterministic quadratic programming method cannot be applied to solve the stochastic optimization problem (6).
The step 3 specifically comprises the following steps:
step 3.1, linearizing to obtain the following model:
Figure BDA0002537325300000112
in the formula:
Figure BDA0002537325300000113
Figure BDA00025373253000001120
Figure BDA0002537325300000114
ignoring high order terms, the nominal linearized stochastic model can be obtained as:
Figure BDA0002537325300000115
in the formula:
Figure BDA0002537325300000116
and
Figure BDA0002537325300000117
is a nominal random state, a nominal input quantity and a nominal random output quantity. The model mismatch between the nonlinear system and the nominal linear stochastic model is the high-order term
Figure BDA0002537325300000118
And
Figure BDA0002537325300000119
it can be limited by the following formula:
Figure BDA00025373253000001110
Figure BDA00025373253000001111
in the formula: | | represents a 2 norm of the variable,
Figure BDA00025373253000001112
is that
Figure BDA00025373253000001113
The upper bound of (a) is,
Figure BDA00025373253000001114
and
Figure BDA00025373253000001115
is a non-linear function f1(xk,uk,wk) The constant of the relative amount of the acid to the base,
Figure BDA00025373253000001116
is that
Figure BDA00025373253000001117
The upper bound of (a) is,
Figure BDA00025373253000001118
and
Figure BDA00025373253000001119
is a non-linear function f2(xk,uk,wk) T represents the measurement interval,
Figure BDA0002537325300000121
Figure BDA0002537325300000122
and 3.2, solving a linear matrix inequality group in the offline robust Tube invariant set as follows:
Figure BDA0002537325300000123
Figure BDA0002537325300000124
in the formula:
Figure BDA0002537325300000125
is a positive definite matrix and the matrix is a negative definite matrix,
Figure BDA0002537325300000126
is a non-square matrix. Scalar lambda0＞0,μ2Is greater than 0. Thereby to obtainFeedback gain K(1)And a symmetric positive definite matrix P1Given by:
K(1):＝YX-1(11)
P1:＝X-1(12)
thus, the robust Tube invariant set can be expressed as:
Figure BDA0002537325300000127
in the formula: max (M)e,2γ1) Wherein the neutralization can be by
Figure BDA0002537325300000128
Finding a scalar λ1，αmin(P1) Representative matrix P1The smallest real part of the eigenvalues.
The step 4 specifically comprises the following steps:
the dual-mode predictive paradigm with multi-step state feedback is:
Figure BDA0002537325300000129
in the formula: kk+iIs a state feedback gain matrix, ci|kIs a decision variable. The control input for the first step is free and a prescribed state feedback law is adopted in the subsequent infinite prediction. Suppose that
Figure BDA0002537325300000131
Information with perturbation sequence
Figure BDA0002537325300000132
Wherein N iscIs the control time domain. In order to reduce the steady-state error and improve the steady-state performance of the system, at each time k, the predicted state is assumed to be
Figure BDA0002537325300000133
By recursion of the prediction dynamics, the following equivalent augmentation prediction model can be obtained:
Figure BDA0002537325300000134
in the formula:
Figure BDA0002537325300000135
Figure BDA0002537325300000136
thereby to obtain
Figure BDA0002537325300000137
The step 5 specifically comprises the following steps:
further, the desired performance indicator function within the infinite domain may be translated into a finite value associated with the initial state, which is the following equation:
Figure BDA0002537325300000138
in the formula:
Figure BDA0002537325300000139
Figure BDA00025373253000001310
Figure BDA0002537325300000141
the step 6 specifically comprises the following steps:
the solving optimization problem of a cluster of probability invariant sets based on the multi-step state feedback control law is as follows:
Figure BDA0002537325300000142
Figure BDA0002537325300000143
Figure BDA0002537325300000144
Figure BDA0002537325300000145
in the formula:
Figure BDA0002537325300000146
Figure BDA0002537325300000147
is a probability invariant set.
The step 7 specifically comprises the following steps:
the problems of dual-mode control online optimization are as follows:
if it is not
Figure BDA0002537325300000148
Figure BDA0002537325300000149
Figure BDA00025373253000001410
If it is not
Figure BDA00025373253000001411
Figure BDA00025373253000001412
Figure BDA00025373253000001413
In the formula:
Figure BDA0002537325300000151
can be obtained in advance by the optimization problem (17). And tightening each constraint condition in combination with the probability invariant set to ensure that a precondition for performing optimal control is obtained.
Compared with the prior art, the random model prediction control method based on the multi-step state feedback accurately models a large number of nonlinear random processes existing in the actual process by introducing the random characteristic of interference, and establishes the random optimization problem with probability constraint in the infinite time domain. A nominal linear random model is obtained through linearization processing, and the state deviation is limited by applying a robust Tube invariant set, so that the problem of model mismatch is effectively solved. Based on a multi-step state feedback strategy, a cluster of ellipse probability invariant set with higher degree of freedom is designed by using random interference information. And (3) utilizing a dual-mode control to limit the future nominal random state of the system to be in a probability invariant set, and combining the property tightening constraint of the Luban Tube invariant set to finally ensure the satisfaction of the probability constraint. Compared with a random model predictive control strategy of a fixed feedback control law, the random model predictive control method based on multi-step state feedback has better optimization capability on random characteristics and better control performance.
So far, the technical solutions of the present invention have been described in connection with the preferred embodiments shown in the drawings, but it is easily understood by those skilled in the art that the scope of the present invention is obviously not limited to these specific embodiments. Equivalent changes or substitutions of related technical features can be made by those skilled in the art without departing from the principle of the invention, and the technical scheme after the changes or substitutions can fall into the protection scope of the invention.

Claims (10)

1. A nonlinear stochastic model predictive control method based on multi-step state feedback is characterized by comprising the following steps:
step 1: and analyzing the random process by utilizing statistics aiming at the nonlinear discrete random system, and establishing a corresponding model.
Step 2: designing a control target with economy, traceability and stability, introducing probability constraint and hard constraint, and proposing a random infinite time domain optimization control problem aiming at the system;
and step 3: linearizing the state space model of the nonlinear discrete random system to obtain a nominal discrete linear random model, and calculating a robust Tube invariant set in an off-line manner by combining the model mismatch between the nominal model and the nonlinear model and the characteristics of a random interference process to ensure the effectiveness of the nominal model on replacing the nonlinear model;
and 4, step 4: aiming at a nominal discrete linear random model, introducing a dual-mode prediction paradigm with a prediction sequence, designing a multi-step state feedback control law meeting the autonomy of a system, and obtaining a corresponding prediction model to increase the design freedom of a subsequent random prediction controller;
and 5: combining the characteristics of multi-step feedback, converting the statistical performance index in the infinite time domain of the original random optimization problem into a finite value with the certainty related to the initial state;
step 6: combining the characteristics of multi-step feedback, constructing a multi-step probability Tube invariant set to force a future prediction state to enter the probability invariant set, and ensuring that a nominal state meets tightening constraints by using the characteristics of a robust invariant set to ensure that an original probability constraint is met;
and 7: and introducing dual-mode control to ensure that the state is always in a probability invariant set or gradually shrinks into the probability invariant set, and tightening each constraint condition in combination with the probability invariant set to ensure that the precondition for carrying out optimization control is obtained.
2. The nonlinear stochastic model predictive control method based on multi-step state feedback according to claim 1, wherein the nonlinear discrete stochastic system is expressed as follows:
Figure FDA0002537325290000011
in the formula: x is the number ofkIs a state quantity, ukAs an input quantity, wkIs random interference.
3. The nonlinear stochastic model predictive control method based on multi-step state feedback of claim 1, wherein stochastic disturbance of the system
Figure FDA0002537325290000021
Is the 0 mean value, and for any time, { w0,w1,. } is independent co-distributed random interference, and its random probability distribution is expressed as follows:
Figure FDA0002537325290000022
in the formula: pr [ ] represents the probability of a random event, the scalar γ > 0, the function F is continuous;
as can be obtained from equation (2), wkDefining multiple cells with orthogonal edges
Figure FDA0002537325290000023
And (4) the following steps.
4. The nonlinear stochastic model predictive control method based on multi-step state feedback according to claim 1, wherein the probability constraint of the system subjected to the state and the input hard constraint are expressed as follows:
Pr[G1xk≤g1]≥p (3a)
G2uk≤g2(3b)
in the formula: p ∈ [0,1] is the probability of allowing a constraint violation.
The control objectives for economy, tracking and stability are usually expressed as a least squares function of the control and output quantities:
Figure FDA0002537325290000024
in the formula: weight coefficient r1And r2A non-negative constant, u*As input quantity reference value, y*Is related to economic benefitsThe output quantity reference value of (1);
by mixing y*Set as the maximum set value of the output quantity
Figure FDA0002537325290000025
In order to obtain better economic benefit, the set objective function (4) is rewritten as follows:
Figure FDA0002537325290000026
when random interference is brought into the design of a controller, an objective function is set as an expected performance index based on random variable probability distribution, and in combination with constraint (3), the random infinite time domain optimization control problem aiming at a nonlinear random system is as follows:
Figure FDA0002537325290000031
the system is highly nonlinear, and the predicted output power depends on future control movement, so the output power in the objective function is implicit, and a deterministic quadratic programming method cannot be applied to solve a stochastic optimization problem.
5. The nonlinear stochastic model predictive control method based on multi-step state feedback according to claim 1, wherein Taylor expansion of nonlinear terms in the system is as follows:
Figure FDA0002537325290000032
in the formula:
Figure FDA0002537325290000033
Figure FDA0002537325290000034
Figure FDA0002537325290000035
Figure FDA0002537325290000036
omitting higher order terms
Figure FDA0002537325290000037
And
Figure FDA0002537325290000038
to obtain a nominally linearized stochastic model:
Figure FDA0002537325290000039
in the formula:
Figure FDA00025373252900000310
and
Figure FDA00025373252900000311
the model mismatch between the nonlinear system and the nominal linear stochastic model is high-order term
Figure FDA00025373252900000312
And
Figure FDA00025373252900000313
it can be limited by the following formula:
Figure FDA00025373252900000314
Figure FDA00025373252900000315
in the formula: | | represents a 2 norm of the variable,
Figure FDA00025373252900000316
is composed of
Figure FDA00025373252900000317
The upper bound of (a) is,
Figure FDA00025373252900000318
and
Figure FDA00025373252900000319
are all non-linear functions f1(xk,uk,wk) The constant of the relative amount of the acid to the base,
Figure FDA00025373252900000320
is composed of
Figure FDA00025373252900000321
The upper bound of (a) is,
Figure FDA00025373252900000322
and
Figure FDA00025373252900000323
are all non-linear functions f2(xk,uk,wk) T represents the measurement interval,
Figure FDA0002537325290000041
Figure FDA0002537325290000042
6. the nonlinear stochastic model predictive control method based on multi-step state feedback of claim 1, characterized in that the nonlinear stochastic model predictive control method is realized by constructing a linear matrix inequality LMIs obtaining linear feedback control law K(1)And a robust Tube invariant set solved in an off-line manner is used for overcoming the state deviation problem caused by model mismatch, and a linear matrix inequality is expressed as follows:
Figure FDA0002537325290000043
Figure FDA0002537325290000044
in the formula:
Figure FDA0002537325290000045
is a positive definite matrix and the matrix is a negative definite matrix,
Figure FDA0002537325290000046
is a non-square matrix. Scalar lambda0＞0,μ2Is greater than 0. Thus, the gain K is fed back(1)And a symmetric positive definite matrix P1Given by:
K(1):＝YX-1(11)
P1:＝X-1(12)
the robust Tube invariant set can be expressed as:
Figure FDA0002537325290000047
in the formula: max (M)e,2γ1) Wherein the neutralization can be by
Figure FDA0002537325290000048
Finding a scalar λ1，αmin(P1) Representative matrix P1The smallest real part of the eigenvalues.
7. The nonlinear stochastic model predictive control method based on multi-step state feedback of claim 1, characterized in that implicit constraints on initial states are calculated by using one-step invariance; constructing an autonomous system to eliminate propagation uncertainty effects within a prediction horizon; defining a dual-mode predictive paradigm with multi-step state feedback to increase the freedom of the controller design:
Figure FDA0002537325290000051
in the formula: kk+iIs a state feedback gain matrix, ci|kIs a decision variable; the control input of the first step is free, and a prescribed state feedback law is adopted in subsequent infinite prediction; suppose that
Figure FDA0002537325290000052
Information with perturbation sequence
Figure FDA0002537325290000053
Wherein N iscIs the control time domain; at each time k, assume a prediction state of
Figure FDA0002537325290000054
An equivalent augmentation prediction model is obtained by recursion of the prediction dynamics to reduce steady-state errors and improve the steady-state performance of the system, and the equivalent augmentation prediction model is expressed as follows:
Figure FDA0002537325290000055
in the formula:
Figure FDA0002537325290000056
Figure FDA0002537325290000057
E＝[I 0 … 0],
Figure FDA0002537325290000058
thereby to obtain
Figure FDA0002537325290000059
8. The nonlinear stochastic model predictive control method based on multi-step state feedback of claim 1, wherein a future predicted state trajectory in the model is a random variable sequence; an objective function with a stochastic process is processed using statistical methods to transform a desired performance indicator function that is contained within an infinite time domain into a finite value associated with an initial state, as follows:
Figure FDA0002537325290000061
in the formula:
Figure FDA0002537325290000062
Figure FDA0002537325290000063
P1,k+i＝-tr(Θk+iPz,k+i)，
Figure FDA0002537325290000064
Figure FDA0002537325290000065
9. the nonlinear stochastic model predictive control method based on multi-step state feedback according to claim 1, wherein given an initial state and a multi-step state feedback strategy, a sub-optimization problem is required to be satisfied to find a cluster of probability invariant set to ensure that probability constraints are satisfied, and the optimization problem is expressed as follows:
Figure FDA0002537325290000066
Figure FDA0002537325290000067
Figure FDA0002537325290000068
Figure FDA0002537325290000069
in the formula:
Figure FDA00025373252900000610
Figure FDA00025373252900000611
Figure FDA00025373252900000612
is a probability invariant set.
10. The nonlinear stochastic model predictive control method based on multi-step state feedback according to claim 1, wherein a dual-mode control online optimization predictive control algorithm is adopted to ensure closed-loop state trajectory convergence and system stability, and the method comprises the following steps:
minimizing the performance indicator function when the nominal random state is within the probability invariant set;
when the nominal random state is not in the probability invariant set, rapidly returning the nominal random state to the probability invariant set;
the dual-mode control online optimization problem is expressed as follows:
if xk∈Ξx,k
Figure FDA0002537325290000071
Figure FDA0002537325290000072
If it is not
Figure FDA0002537325290000073
Figure FDA0002537325290000074
Figure FDA0002537325290000075
In the formula:
Figure FDA0002537325290000076
Figure FDA0002537325290000077
can be obtained by an optimization problem (17) in advance, and each constraint condition is tightened in combination with the probability invariant set to ensure that a precondition for optimization control is obtained.
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Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113267997A (en) * 2020-10-22 2021-08-17 华北电力大学 Nonlinear stochastic model prediction control simplification method based on multi-step feedback control law
CN113673103A (en) * 2021-08-19 2021-11-19 杭州电子科技大学 Control method for load capacity saturation of automatic logistics sorting equipment
CN114167732A (en) * 2021-12-13 2022-03-11 西北工业大学 Coupling constraint multi-agent distributed robust nonlinear model prediction control method

Citations (14)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20010021900A1 (en) * 1998-09-28 2001-09-13 Aspen Technology, Inc. Robust steady-state target calculation for model predictive control
US20050107895A1 (en) * 2001-05-25 2005-05-19 Efstratios Pistikopoulos Process control
EP1538319A1 (en) * 2003-12-05 2005-06-08 General Electric Company Apparatus for model predictive control of aircraft gas turbine engines
JP2006172364A (en) * 2004-12-20 2006-06-29 Fujitsu Ten Ltd Model predictive control device
US20090319059A1 (en) * 2008-06-20 2009-12-24 Honeywell International Inc. Apparatus and method for model predictive control (mpc) of a nonlinear process
CN105893654A (en) * 2016-03-11 2016-08-24 中南大学 Robust predictive control method for first-order continuous stirred tank reactor (CSTR)
CN106094524A (en) * 2016-07-07 2016-11-09 西北工业大学 The rapid model prediction control method compensated based on input trend
CN108873698A (en) * 2018-07-07 2018-11-23 福州大学 A kind of disturbance rejection two stages fixed point method of servo-controlling
CN108897217A (en) * 2018-07-04 2018-11-27 西北工业大学 A kind of drive lacking waterborne vessel Trajectory Tracking Control method based on Model Predictive Control
CN109613830A (en) * 2019-01-31 2019-04-12 江南大学 Model predictive control method based on prediction step of successively decreasing
CN110145436A (en) * 2019-04-28 2019-08-20 华北电力大学 Non-linear economic model forecast control method applied to blower
CN110994582A (en) * 2019-11-19 2020-04-10 南京邮电大学 Uncertain direct-current micro-grid output feedback fuzzy model prediction control method
CN110985294A (en) * 2019-12-10 2020-04-10 华北电力大学 Stochastic model prediction control method for joint use of robust probability tube
CN111259525A (en) * 2020-01-09 2020-06-09 曲阜师范大学 Model prediction control method for nonlinear unstable wind power engine room suspension system

Patent Citations (14)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US20010021900A1 (en) * 1998-09-28 2001-09-13 Aspen Technology, Inc. Robust steady-state target calculation for model predictive control
US20050107895A1 (en) * 2001-05-25 2005-05-19 Efstratios Pistikopoulos Process control
EP1538319A1 (en) * 2003-12-05 2005-06-08 General Electric Company Apparatus for model predictive control of aircraft gas turbine engines
JP2006172364A (en) * 2004-12-20 2006-06-29 Fujitsu Ten Ltd Model predictive control device
US20090319059A1 (en) * 2008-06-20 2009-12-24 Honeywell International Inc. Apparatus and method for model predictive control (mpc) of a nonlinear process
CN105893654A (en) * 2016-03-11 2016-08-24 中南大学 Robust predictive control method for first-order continuous stirred tank reactor (CSTR)
CN106094524A (en) * 2016-07-07 2016-11-09 西北工业大学 The rapid model prediction control method compensated based on input trend
CN108897217A (en) * 2018-07-04 2018-11-27 西北工业大学 A kind of drive lacking waterborne vessel Trajectory Tracking Control method based on Model Predictive Control
CN108873698A (en) * 2018-07-07 2018-11-23 福州大学 A kind of disturbance rejection two stages fixed point method of servo-controlling
CN109613830A (en) * 2019-01-31 2019-04-12 江南大学 Model predictive control method based on prediction step of successively decreasing
CN110145436A (en) * 2019-04-28 2019-08-20 华北电力大学 Non-linear economic model forecast control method applied to blower
CN110994582A (en) * 2019-11-19 2020-04-10 南京邮电大学 Uncertain direct-current micro-grid output feedback fuzzy model prediction control method
CN110985294A (en) * 2019-12-10 2020-04-10 华北电力大学 Stochastic model prediction control method for joint use of robust probability tube
CN111259525A (en) * 2020-01-09 2020-06-09 曲阜师范大学 Model prediction control method for nonlinear unstable wind power engine room suspension system

Non-Patent Citations (6)

* Cited by examiner, † Cited by third party
Title
XIANGJIE LIU 等: "Nonlinear Model Predictive Control for DFIG-Based Wind Power Generation", 《IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING》 *
YANYAN YIN 等: "CONSTRAINED MODEL PREDICTIVE CONTROL ON CONVEX POLYHEDRON STOCHASTIC LINEAR PARAMETER VARYING SYSTEMS", 《INTERNATIONAL JOURNAL OF INNOVATIVE COMPUTING, INFORMATION AND CONTROL》 *
YANYAN YIN 等: "Constrained predictive control of nonlinear stochastic systems", 《JOURNAL OF SYSTEMS ENGINEERING AND ELECTRONICS》 *
刘向杰 等: "基于多步反馈的变桨距风力发电系统随机模型预测控制", 《2017中国自动化大会(CAC 2017 )》 *
孔小兵: "连续非线性模型预测控制的研究与应用", 《中国优秀硕士学位论文全文数据库信息科技辑》 *
秦伟伟 等: "基于鲁棒一步集的Tube不变集鲁棒模型预测控制", 《自动化学报》 *

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113267997A (en) * 2020-10-22 2021-08-17 华北电力大学 Nonlinear stochastic model prediction control simplification method based on multi-step feedback control law
CN113673103A (en) * 2021-08-19 2021-11-19 杭州电子科技大学 Control method for load capacity saturation of automatic logistics sorting equipment
CN114167732A (en) * 2021-12-13 2022-03-11 西北工业大学 Coupling constraint multi-agent distributed robust nonlinear model prediction control method

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