Anti-interference fuzzy prediction control method based on linear matrix inequality

Abstract

The invention discloses an anti-interference fuzzy prediction control method based on a linear matrix inequality. The method comprises the following steps: 1) establishing a system state equation of a cleaning robot with noise; 2) designing a fuzzy controller according to a T-S fuzzy method; 3) converting an anti-interference fuzzy prediction control problem into a solution minimization problem; and 4) solving a minimization problem based on a linear matrix inequality method. According to the invention, the model is reasonable and has certain robustness in switching control of various states; according to the method, the cleaning robot can adapt to various different working states, the system is kept stable, meanwhile, the robustness of the system can be kept under the condition that noise disturbance exists, and the working efficiency of the cleaning robot is improved.

Description

Technical Field

[0001] The invention belongs to the technical field of agricultural intelligent equipment, and in particular relates to an anti-interference fuzzy predictive control method for a cleaning robot based on linear matrix inequality. Background Art

[0002] With the development of society, the scale of animal husbandry has increased rapidly. While large-scale and intensive breeding generates economic benefits and improves production efficiency, the feces produced by livestock and poultry are prone to cause environmental pollution and infectious diseases. In order to reduce the risk of environmental pollution and infectious diseases, animal husbandry cleaning robots are introduced to clean the feces produced by livestock and poultry. Faced with the narrow space in the breeding farm, the cleaning robots need to adapt to the working states of departure, switching, cleaning and recycling.

[0003] Model predictive control uses a rolling finite-horizon optimization strategy and has good dynamic control performance, but it usually relies on accurate system models and parameter determination to predict future behavior, which may be challenging for modeling complex systems. In order to meet the needs of various working states of cleaning robots, TS fuzzy control is introduced.

[0004] TS fuzzy control is based on a set of fuzzy rules that describe the relationship between input variables and output variables. Each fuzzy rule corresponds to a local linear model, which is activated under different input conditions. Therefore, a set of fuzzy rules is defined, each fuzzy rule corresponds to a working state of the cleaning robot, and the output of the fuzzy rules is weighted and summed by linear combination to obtain the final control output, so that the cleaning robot can adapt to all working states during operation. Summary of the invention

[0005] In order to overcome the shortcomings of the prior art, and in response to the various working state requirements of the cleaning robot in the narrow space of the farm, the present invention proposes an anti-interference fuzzy predictive control method for the cleaning robot based on linear matrix inequality. The method is suitable for working scenarios with various motion states. TS fuzzy control is introduced into the model predictive control, and linear matrix inequality is used to solve it, so that the cleaning robot can achieve all working states and has a certain robustness.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] An anti-interference fuzzy predictive control method based on linear matrix inequality comprises the following steps:

[0008] 1) Establish the state equation of the cleaning robot system with noise;

[0009] 2) Design fuzzy controller based on TS fuzzy method;

[0010] 3) Transform the anti-disturbance fuzzy predictive control problem into a minimization problem;

[0011] 4) Solve the minimization problem based on the linear matrix inequality method.

[0012] Furthermore, in 1), the system state equation is established: the cleaning robot system state vector x(k), the control input vector u(k) and the noise w(k) are defined, where k is the sampling time, and the following model is established:

[0013]

[0014] Among them A z is the state matrix of the cleaning robot, B z is the control input matrix, y(k) represents the system output, and C is the output matrix.

[0015] Furthermore, in 2), a TS-type fuzzy controller is designed: the TS-type fuzzy model of the system model of formula (1) is described as the following r fuzzy rules:

[0016] Rule ith:

[0017] Then x(k+1)=A i x(k)+B i u(k),i=1,2,…,r (2)

[0018] Among them, x j is the jth state variable of the system, is the t-th membership function of the ith rule, A i and B i are the system matrix and control input matrix of the ith rule respectively. The total output of the TS-type fuzzy model composed of fuzzy rule formula (2) is:

[0019]

[0020] Among them, w i is the membership function of the ith rule, h i is the proportional weight of the i-th rule, and From equations (1) and (3), we can get Using the state feedback method, we get:

[0021] Rule ith:

[0022] Then u(k)=K i x(k),i=1,2,…,r (4)

[0023] Where K i is the controller gain of the i-th rule, and the TS-type fuzzy controller designed by the parallel distributed compensation method is:

[0024]

[0025] make have to:

[0026] u(k)=K z x(k) (6)

[0027] Furthermore, in 3), the performance index is determined: let the infinite time domain quadratic performance index be:

[0028]

[0029] Where λ is the level of noise suppression; to obtain the final controller gain K z , solve the following minimization performance index J(k):

[0030]

[0031] In order to simplify the solution of problem (8), the upper bound of the performance index is derived. Let β(k) be the upper bound of J(k) at time kT, then:

[0032] J(k)<β(k) (9)

[0033] Convert the problem of minimizing the performance index into the problem of minimizing the upper bound:

[0034]

[0035] J(k) can be rewritten as follows:

[0036]

[0037] From equation (9) and equation (11), we can get:

[0038]

[0039] By using Schur's complement theorem, equation (12) can be rewritten as:

[0040]

[0041] In 4), the stability condition is: define the quadratic function And let the function satisfy:

[0042] V(x(k+i+1|k))-V(x(k+i|k))<-[||y(k+i|k)|| 2 -λ2 ||w(k+i|k)|| 2 ] (14)

[0043] Since the right side of the inequality is always negative definite at non-zero points, according to the Lyapunov stability theorem, if the above inequality holds, the system is asymptotically stable; equation (14) can be written as:

[0044]

[0045] Formula (15) can be rewritten as:

[0046]

[0047] Where Q 0 =C T C, applying Schur complement theorem, formula (16) is rewritten as:

[0048]

[0049] Let K z =Y z G -1 , using the expansion lemma, we can rewrite equation (17) as:

[0050]

[0051] Constraints: Set the control input vector and state vector to:

[0052] ||u(k)||≤θ u (19)

[0053] ||x(k)||≤θ x (20)

[0054] where θ u and θ x are the maximum values ​​of the control input vector and the state vector respectively, which can be obtained from formula (19):

[0055]

[0056] Where k and k+i represent the sampling time, m represents x(k+i|k), and formula (21) yields:

[0057]

[0058] Applying Schur's complement lemma, equation (22) can be rewritten as:

[0059]

[0060] Multiply both sides of the matrix by diag{I,G} and apply the expansion lemma to rewrite equation (23) as follows:

[0061]

[0062] From formula (20), we can get:

[0063]

[0064] Where m represents x(k+i|k), and from formula (25) we get:

[0065]

[0066] Applying Schur's complement lemma, equation (26) can be rewritten as:

[0067]

[0068] Multiply both sides of the matrix by diag{I,G} and apply the expansion lemma to rewrite equation (27) as follows:

[0069]

[0070] The upper bound of the performance index set in step 3) is converted into a minimization problem solved by linear matrix inequality, and combined with the set stability conditions and constraints, the minimization upper bound problem solved by formula (10) is According to the decision variable Q z ,Y z ,G, converted to solve A new anti-interference fuzzy predictive control algorithm is obtained, as shown in formula (29):

[0071]

[0072] According to the above analysis, the anti-interference fuzzy predictive control process based on linear matrix inequality is obtained as follows:

[0073] S1: Initialization: Set the rolling time domain N = 200, k = 0, sampling time T = 0.01, and select λ∈(0,20];

[0074] S2: Solving the minimization problem

[0075] S3: Calculate the feedback controller K z =Y z G -1 ;

[0076] S4: Update system state x(k);

[0077] S5: Set k=k+1 and jump to step S2.

[0078] The effective effects of the present invention are mainly manifested in: enabling the cleaning robot to adapt to a variety of different working conditions and keeping the system stable, and at the same time the control method can maintain the robustness of the system in the presence of noise disturbances, thereby improving the working efficiency of the cleaning robot. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] Figure 1 It is a flow chart for realizing the method of the present invention;

[0080] Figure 2 It is a schematic diagram of the verification effect of the method of the present invention. DETAILED DESCRIPTION

[0081] The present invention will be further described below in conjunction with the accompanying drawings.

[0082] Reference Figure 1 and Figure 2 , an anti-interference fuzzy predictive control method based on linear matrix inequality, comprising the following steps:

[0083] 1) Establish the state equation of the cleaning robot system with noise;

[0084] In 1), the system state equation is established: the cleaning robot system state vector x(k), control input vector u(k) and noise w(k) are defined, where k is the sampling time, and the following model is established:

[0085]

[0086] Among them A z is the state matrix of the cleaning robot, B z is the control input matrix, y(k) represents the system output, and C is the output matrix.

[0087] 2) Design fuzzy controller based on TS fuzzy method;

[0088] In the above 2), a TS-type fuzzy controller is designed: The TS-type fuzzy model of the system model of formula (1) is described as the following r fuzzy rules:

[0089] Rule ith:

[0090] Then x(k+1)=A i x(k)+B i u(k),i=1,2,…,r (2)

[0091] Among them, x j is the jth state variable of the system, is the t-th membership function of the ith rule, A iand B i are the system matrix and control input matrix of the ith rule respectively. The total output of the TS-type fuzzy model composed of fuzzy rule formula (2) is:

[0092]

[0093] Among them, w i is the membership function of the ith rule, h i is the proportional weight of the i-th rule, and From equations (1) and (3), we can get Using the state feedback method, we get:

[0094] Rule ith:

[0095] Then u(k)=K i x(k),i=1,2,…,r (4)

[0096] Where K i is the controller gain of the i-th rule, and the TS-type fuzzy controller designed by the parallel distributed compensation method is:

[0097]

[0098] make have to:

[0099] u(k)=K z x(k) (6)

[0100] 3) Transform the anti-disturbance fuzzy predictive control problem into a minimization problem;

[0101] In the above 3), determine the performance index: let the infinite time domain quadratic performance index be:

[0102]

[0103] Where λ is the level of noise suppression; to obtain the final controller gain K z , solve the following minimization performance index J(k):

[0104]

[0105] In order to simplify the solution of problem (8), the upper bound of the performance index is derived. Let β(k) be the upper bound of J(k) at time kT, then:

[0106] J(k)<β(k) (9)

[0107] Convert the problem of minimizing the performance index into the problem of minimizing the upper bound:

[0108]

[0109] J(k) can be rewritten as follows:

[0110]

[0111] From equation (9) and equation (11), we can get:

[0112]

[0113] By using Schur's complement theorem, equation (12) can be rewritten as:

[0114]

[0115] 4) Solve the minimization problem based on the linear matrix inequality method.

[0116] In 4), the stability condition is: define the quadratic function And let the function satisfy:

[0117] V(x(k+i+1|k))-V(x(k+i|k))<-[||y(k+i|k)|| 2 -λ 2 ||w(k+i|k)|| 2 ] (14)

[0118] Since the right side of the inequality is always negative definite at non-zero points, according to the Lyapunov stability theorem, if the above inequality holds, the system is asymptotically stable; equation (14) can be written as:

[0119]

[0120] Formula (15) can be rewritten as:

[0121]

[0122] Where Q 0 =C T C, applying Schur complement theorem, formula (16) is rewritten as:

[0123]

[0124] Let K z =Y z G -1 , using the expansion lemma, we can rewrite equation (17) as:

[0125]

[0126] 5) Constraints: Set the control input vector and state vector to:

[0127] ||u(k)||≤θ u (19)

[0128] ||x(k)||≤θ x (20)

[0129] where θ u and θ x are the maximum values ​​of the control input vector and the state vector respectively, which can be obtained from formula (19):

[0130]

[0131] Where k and k+i represent the sampling time, m represents x(k+i|k), and formula (21) yields:

[0132]

[0133] Applying Schur's complement lemma, equation (22) can be rewritten as:

[0134]

[0135] Multiply both sides of the matrix by diag{I,G} and apply the expansion lemma to rewrite equation (23) as follows:

[0136]

[0137] From formula (20), we can get:

[0138]

[0139] Where m represents x(k+i|k), and from formula (25) we get:

[0140]

[0141] Applying Schur's complement lemma, equation (26) can be rewritten as:

[0142]

[0143] Multiply both sides of the matrix by diag{I,G} and apply the expansion lemma to rewrite equation (27) as follows:

[0144]

[0145] The upper bound of the performance index set in step 3) is converted into a minimization problem solved by linear matrix inequality, and combined with the stability condition set in step 4) and the constraint condition set in step 5), the minimization upper bound problem solved by formula (10) is According to the decision variable Q z ,Y z ,G, converted to solve A new anti-interference fuzzy predictive control algorithm is obtained, as shown in formula (29):

[0146]

[0147] The anti-interference fuzzy predictive control method based on linear matrix inequality of this embodiment includes the following steps:

[0148] S1. Initialization: set the rolling time domain N = 200, k = 0, sampling time T = 0.01, and select λ∈(0,20];

[0149] S2. Solve the minimization problem

[0150] S3. Calculate the feedback controller K z =Y z G -1 ;

[0151] S4. Update system state x(k);

[0152] S5. Set k=k+1 and jump to step S2.

[0153] In the example of the present invention, the cleaning robot can adapt to a variety of different working states and keep the system stable, presenting the effect as follows Figure 2 As shown in the figure; a fuzzy controller is introduced into the model predictive control, and the final system model is obtained by weighting multiple working states through fuzzy rules. The problem is transformed into a minimum value problem solved by linear matrix inequality. At the same time, this control method can maintain the robustness of the system in the presence of noise disturbances, thereby improving the working efficiency of the cleaning robot.

[0154] The contents described in the embodiments of this specification are merely an enumeration of the implementation forms of the inventive concept and are for illustrative purposes only. The protection scope of the present invention should not be considered to be limited to the specific forms described in this embodiment, and the protection scope of the present invention also extends to equivalent technical means that can be thought of by ordinary technicians in this field based on the inventive concept.

Claims

1. An anti-interference fuzzy predictive control method based on linear matrix inequality, characterized in that: The method comprises the following steps: 1) Establish the state equation of the cleaning robot system with noise; 2) Design fuzzy controller based on TS fuzzy method; 3) Transform the anti-disturbance fuzzy predictive control problem into a minimization problem; 4) Solve the minimization problem based on the linear matrix inequality method.

2. The anti-interference fuzzy predictive control method based on linear matrix inequality according to claim 1, characterized in that: In 1), the system state equation is established: the cleaning robot system state vector x(k), control input vector u(k) and noise w(k) are defined, where k is the sampling time, and the following model is established: Among them A z is the state matrix of the cleaning robot, B z is the control input matrix, y(k) represents the system output, and C is the output matrix.

3. The anti-interference fuzzy predictive control method based on linear matrix inequality according to claim 2, characterized in that: In the above 2), a TS-type fuzzy controller is designed: The TS-type fuzzy model of the system model of formula (1) is described as the following r fuzzy rules: Rule ith: Then x(k+1)=A i x(k)+B i u(k),i=1,2,…,r (2) Among them, x j is the jth state variable of the system, is the t-th membership function of the ith rule, A i and B i are the system matrix and control input matrix of the ith rule respectively. The total output of the TS-type fuzzy model composed of fuzzy rule formula (2) is: Among them, w i is the membership function of the ith rule, h i is the proportional weight of the i-th rule, and From equations (1) and (3), we can get By adopting the state feedback method, we get: Rule ith: Then u(k)=K i x(k),i=1,2,…,r (4) Where K i is the controller gain of the i-th rule; The TS type fuzzy controller is designed using the parallel distributed compensation method as follows: where w i is the membership function of the ith rule, u i is the control input vector of the ith rule, h i is the proportional weight of the i-th rule, K i is the controller gain of the i-th rule, let have to: u(k)=K z x(k) (6)。 4. The anti-interference fuzzy predictive control method based on linear matrix inequality according to claim 3, characterized in that: In said 3), determine the performance index: Where λ is the level of noise suppression; to obtain the final controller gain K z , solve the following minimization performance index J(k): Let β(k) be the upper bound of J(k) at time kT, then: J(k)<β(k) (9) Convert the problem of minimizing the performance index into the problem of minimizing the upper bound: Rewrite it as follows: From equation (9) and equation (11), we can get: By using Schur's complement theorem, equation (12) can be rewritten as:

5. The anti-interference fuzzy predictive control method based on linear matrix inequality according to claim 4, characterized in that: In 4), the stability condition is: let the quadratic function And let the function satisfy: V(x(k+i+1|k))-V(x(k+i|k))<-[||y(k+i|k)|| 2 -λ 2 ||w(k+i|k)|| 2 ] (14) Since the right side of the inequality is always negative definite at non-zero points, according to the Lyapunov stability theorem, if the above inequality holds, the system is asymptotically stable; equation (14) can be written as: Formula (15) can be rewritten as: Where Q0 = C T C, applying Schur complement theorem, formula (16) is rewritten as: Let K z =Y z G -1 , using the expansion lemma, we can rewrite equation (17) as: Constraints: Set the control input vector and state vector to: ||u(k)||≤θ u (19) ||x(k)||≤θ x (20) where θ u and θ x are the maximum values ​​of the control input vector and the state vector respectively, which can be obtained from formula (19): Where k and k+i represent the sampling time, m represents x(k+i|k), and formula (21) yields: Applying Schur's complement lemma, equation (22) can be rewritten as: Multiply both sides of the matrix by diag{I,G} and apply the expansion lemma to rewrite equation (23) as follows: From formula (20), we can get: Where m represents x(k+i|k), and from formula (25) we get: Applying Schur's complement lemma, equation (26) can be rewritten as: Multiply both sides of the matrix by diag{I,G} and apply the expansion lemma to rewrite equation (27) as follows: The upper bound of the performance index set in step 3) is transformed into a minimization problem solved by linear matrix inequality, and combined with the set stability conditions and constraints, the minimization upper bound problem solved by formula (10) is According to the decision variable Q z ,Y z ,G, converted to solve A new anti-interference fuzzy predictive control algorithm is obtained, as shown in formula (29): According to the above analysis, the anti-interference fuzzy predictive control process based on linear matrix inequality is obtained as follows: S1: Initialization: Set the rolling time domain N = 200, k = 0, sampling time T = 0.01, and select λ∈(0,20]; S2: Solving the minimization problem S3: Calculate the feedback controller K z =Y z G -1 ; S4: Update system state x(k); S5: Set k=k+1 and jump to step S2.

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