CN112015087A - Non-fragile optimal control method of Lipschitz nonlinear system - Google Patents

Non-fragile optimal control method of Lipschitz nonlinear system Download PDF

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CN112015087A
CN112015087A CN202010860864.6A CN202010860864A CN112015087A CN 112015087 A CN112015087 A CN 112015087A CN 202010860864 A CN202010860864 A CN 202010860864A CN 112015087 A CN112015087 A CN 112015087A
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马卫国
许霞
刘羡飞
陈�峰
吴晓新
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Abstract

The invention discloses a non-fragile optimal control method of a Lipschitz nonlinear system, which comprises the following steps: 1) the sensor samples the state of the Lipschitz nonlinear system with model uncertainty, converts the state into digital quantity by adopting a logarithmic quantizer and transmits the digital quantity to a remote controller; 2) describing the data transmission process by adopting a Markov chain, and adopting a corresponding control strategy by a controller according to the condition of receiving data; 3) a closed-loop system is described as a Markov jump system, and a non-fragile infinite time optimal controller is designed by using a linear matrix inequality based on a system model and a performance index of the system. The invention provides a design method of an optimal controller aiming at the design problem of a non-fragile infinite time optimal controller of a Lipschitz nonlinear system with quantitative and incomplete measurement data, so that the system is stable and a performance index functional takes a minimum value.

Description

Non-fragile optimal control method of Lipschitz nonlinear system
Technical Field
The invention relates to the field of optimal control, in particular to a non-fragile optimal control method for a Lipschitz nonlinear system.
Background
The Lipschitz system is a common type of nonlinear system that can describe many physical processes. For example, robotic systems where sinusoidal terms exist, nonlinear systems that satisfy the Lipschitz condition within a given interval, and the like. Information transfer may occur over a network when the sensor, controller and actuator components of the system are distributed in a remote physical space. The control system based on the network has the advantages of low cost, good expandability, convenience in installation and maintenance and the like, and becomes a development trend of the control system. When the sensor measures and samples the state of the controlled system and converts the state into a digital signal, quantization is needed. The quantized data may be lost due to network congestion or interference during transmission to the controller over the network. These problems will affect the analysis and design of the control system.
In the process of establishing the system model, factors which have small influence on the system, such as small change of parameters of system components, friction force, external interference and the like, are usually ignored, and therefore an error exists between the obtained mathematical model and the actual mathematical model of the system. Aiming at a given controlled system, a weighting matrix in a performance index is a constant value, the integral upper limit is infinite, and the designed infinite time optimal controller is an optimal control mode with wide application. Due to the influence of factors such as analog-to-digital conversion precision and word length, uncertainty exists when the controller is realized through digital equipment. However, the existing Lipschitz nonlinear system optimal control has the following problems: firstly, the uncertainty of the system model is not considered, and the control performance of the actual system is influenced. And secondly, the factors such as conversion precision, word length and the like are not considered when the controller is realized, and the non-vulnerability is poor.
Disclosure of Invention
The purpose of the invention is as follows: aiming at a Lipschitz nonlinear system with quantitative and incomplete measurement data, a non-fragile optimal control method of the Lipschitz nonlinear system is provided, so that the system is asymptotically stable, and the performance index obtains a minimum value.
The technical scheme is as follows: a non-fragile optimal control method of a Lipschitz non-linear system comprises the following steps:
1) the sensor samples the state of the Lipschitz nonlinear system with model uncertainty, and sampling data is converted into digital quantity by adopting a logarithmic quantizer and transmitted to a remote controller;
2) in the process of transmitting quantized data to a remote controller, because of network congestion or interference loss, a Markov chain is adopted to describe the data transmission process, and the remote controller adopts a corresponding control strategy according to the condition of receiving data;
3) the closed-loop system is described as a Markov jump system, and a non-fragile infinite time optimal controller is designed by utilizing a linear matrix inequality based on a Markov jump system model and performance indexes of the system.
Further, in the step 1), a state space expression of the Lipschitz nonlinear system is as follows:
Figure BDA0002648030000000021
wherein x (k) e RnIs the system state vector, u (k) e RpIs the system control input, y (k) e RmIs the system output, f (k, x (k)) is a nonlinear vector function that satisfies the Lipschitz condition: f (k,0) ═ 0, | f (k, x (k)) | ≦ | fx (k)) |; g, H, L, C and F are known coefficient matrices of the corresponding dimensions, Δ G, Δ H, Δ L and Δ C are system uncertainties, and the condition is satisfied:
[ΔG ΔH ΔL ΔC]＝DF1(k)[EG EH EL EC]，F1 T(k)F1(k)≤I (2)
D，EG，EH，ELand ECIs a matrix of known corresponding dimensions;
assuming that the system initial state x (0) is a random variable, E { x (0) } is 0, E { x (0) xT(0) I, the performance index of a given system is:
Figure BDA0002648030000000022
where Q and R are given symmetric positive definite matrices.
3. The design of the non-fragile optimal controller of the Lipschitz nonlinear system as recited in claim 2, wherein in the step 1), a logarithmic quantizer is adopted to convert the sampled data into digital quantity, specifically as follows:
the sensor samples the system state and converts the system state into digital quantity, the quantizer adopts a logarithmic quantizer, and then the output of the quantizer is:
Figure BDA0002648030000000023
where ρ is the quantization density.
Further, in the step 2), a markov chain is used to describe the data transmission process, which specifically includes:
loss occurs in the process of transmitting measurement data to a controller through a network due to the influence of network congestion or external interference, the loss process is described as a Markov chain, alpha (k) ═ 0 represents data loss, the controller adopts the value at the previous moment, alpha (k) ═ 1 represents normal data transmission, and the state transition matrix of the Markov chain
Figure BDA0002648030000000031
Further, in step 2), the controller adopts a corresponding control strategy according to the data receiving condition, which is specifically as follows:
the inputs to the controller are:
xc(k)＝[1-α(k)]xc(k-1)+α(k)xq(k)，α(k)＝i＝{0,1} (5)
aiming at a controlled system and performance indexes, a non-fragile controller is designed to minimize the performance index J, and the controller is in the form of:
u(k)＝(Kα(k)+ΔK)xc(k)，ΔK＝DF1(k)EK，F1 T(k)F1(k)≤I (6)
wherein D and EKIs a matrix of known corresponding dimensions.
Further, in step 3), the closed-loop system is described as a markov jump system, specifically as follows:
is provided with
Figure BDA0002648030000000032
The closed loop system is then:
Figure BDA0002648030000000033
wherein the content of the first and second substances,
Figure BDA0002648030000000034
Gcl111＝G+ΔG+(H+ΔH)(K1+ΔK)(I+F2(k))，
Figure BDA0002648030000000035
Ccl0＝Ccl1＝[C+ΔC 0]。
further, in the step 3), a non-fragile infinite time optimal controller is designed by using a linear matrix inequality, specifically as follows:
given a Lipschitz nonlinear system (1) and its performance index (3), if a symmetric positive definite matrix X exists011，X022，X111，X122，W0，W1Matrix M0，M1Scalar quantity1＞0，2＞0，3> 0, the following linear matrix inequality is satisfied:
Figure BDA0002648030000000041
Figure BDA0002648030000000042
wherein omega11＝diag{-X011,-X022}，Ω22＝-1I，
Figure BDA0002648030000000043
Figure BDA0002648030000000044
Figure BDA0002648030000000045
Ω51＝[0M0]，Ω55＝-R-1，
Figure BDA0002648030000000046
Figure BDA0002648030000000047
Ω66＝diag{-W0,-3I,-2I}，
Figure BDA0002648030000000048
Ω77＝diag{-Q-1,-1I}，
Figure BDA0002648030000000049
Ω88＝diag{-W0,-3I,-2I}，Λ11＝diag{-X111,-X122}，Λ22＝-1I，
Figure BDA0002648030000000051
Figure BDA0002648030000000052
Λ51＝M1，Λ55＝-R-1，
Figure BDA0002648030000000053
Λ66＝diag{-W,-3I}，Λ71＝[EGX111+EHM1 0]，Λ72＝1EL，Λ76＝[EHM1 2EHD]，Λ77＝-2I，
Figure BDA0002648030000000054
Λ88＝diag{-Q-1,-1I,-W}，
Figure BDA0002648030000000055
Figure BDA0002648030000000056
Λ99＝diag{-3I,-2I}。
Then the non-fragile infinite time controller
Figure BDA0002648030000000057
The closed loop system (7) is stable, and the performance index is less than
Figure BDA0002648030000000058
For a given Lipschitz nonlinear system (1) and its performance index (3), if the following optimization problem has a solution
Figure BDA0002648030000000059
Figure BDA00026480300000000510
Wherein N is2＝(X011,X022,X111,X122,W0,W1,M0,M1,1,2,3,N1) Then, then
Figure BDA00026480300000000513
Figure BDA00026480300000000514
Is a non-fragile infinite time optimal controller of the system (1).
Has the advantages that: (1) the invention designs the non-fragile infinite time optimal controller based on the Lipschitz nonlinear system model and corresponding performance indexes, and compared with the prior art, the non-fragile infinite time optimal controller takes the factors of conversion precision, word length and the like into consideration when realizing the optimal controller, thereby improving the non-fragility of the optimal controller.
(2) The invention considers the parameter uncertainty, the quantification and the integrity in the data transmission process in the system model, and the designed controller has better practical significance and practical value.
(3) The invention describes the system as a Markov jump system, and provides a solving method of an optimal controller aiming at the design problem of a non-fragile optimal controller of a Lipschitz nonlinear system, so that the system is kept stable and the performance index is the minimum value.
Drawings
FIG. 1 is a flow diagram of the non-fragile optimal control of a Lipschitz nonlinear system;
FIG. 2 is a block diagram of the non-fragile optimal control of the Lipschitz nonlinear system.
Detailed Description
The invention is further explained below with reference to the drawings.
As shown in fig. 1 and fig. 2, which are a flow chart and a structural diagram of the non-fragile optimal control of the Lipschitz non-linear system, respectively, the non-fragile optimal control method of the Lipschitz non-linear system of the present invention includes the following steps:
1) the sensor samples the state of the Lipschitz nonlinear system with model uncertainty, converts the state into digital quantity by adopting a logarithmic quantizer and transmits the digital quantity to a remote controller;
2) during the process of transmitting quantized data to a controller, the data transmission process is described by adopting a Markov chain because network congestion or interference is possible to lose, and the controller adopts a corresponding control strategy according to the condition of receiving the data;
3) a closed-loop system is described as a Markov jump system, and a non-fragile infinite time optimal controller is designed by using a linear matrix inequality based on a system model and a performance index of the system.
Wherein, the Lipschitz nonlinear system state space expression with model uncertainty in the step 1) is shown as formula (1). The sensor samples the system state and converts the system state into digital quantity, and the method comprises the following steps: the quantizer is a logarithmic quantizer, and the output of the quantizer is equation (4).
The measured data may be lost during the transmission process to the controller through the network in step 2), and the loss process is described as a markov chain, and the method is as follows: alpha (k) ═ 0 represents data loss, the controller adopts the value of the previous moment, alpha (k) ═ 1 represents normal data transmission, the state transition matrix of Markov chain
Figure BDA0002648030000000061
The controller adopts a corresponding control strategy according to the condition of receiving data, the input of the controller is formula (5), and the performance index of the given system is formula (3). Aiming at a controlled system and a performance index, a non-fragile controller is designed to minimize the performance index J, and the form of the controller is an equation (6).
Describing the closed-loop system as a Markov jump system in the step 3), and designing a non-fragile infinite time optimal controller by using a linear matrix inequality based on a system model and a performance index of the system, wherein the method comprises the following steps: is provided with
Figure BDA0002648030000000071
The closed loop system is equation (7).
Given a Lipschitz nonlinear system (1) and its performance index (3), if a symmetric positive definite matrix X exists011，X022，X111，X122，W0，W1Matrix M0，M1Scalar quantity1＞0，2＞0，3If the linear matrix inequalities (8) and (9) are satisfied, the non-fragile infinite time controller
Figure BDA0002648030000000072
The closed loop system (7) is stable, and the performance index is less than
Figure BDA0002648030000000073
For a given Lipschitz nonlinear system (1) and its performance index (3), if the optimization problem equation (10) has a solution
Figure BDA0002648030000000074
Then
Figure BDA0002648030000000075
Is a non-fragile infinite time optimal controller of the system (1).
Lemma gives matrices Φ, D and E with corresponding dimensions, where Φ is a symmetric matrix. For all satisfy FT(k) F (k) a matrix F (k) of ≦ I, (+ DF) (k) E + ETFT(k)DTAn essential condition for < 0 to be true is that > 0 is present, such that Φ + DDT+- 1ETE＜0。
The following Lyapunov functional was chosen:
Figure BDA0002648030000000079
Pi＝diag{Pi11,Pi22}
Figure BDA0002648030000000076
wherein the content of the first and second substances,
Figure BDA0002648030000000077
Figure BDA0002648030000000078
Figure BDA0002648030000000081
wherein the content of the first and second substances,
Figure BDA0002648030000000082
Figure BDA0002648030000000083
Figure BDA0002648030000000084
Figure BDA0002648030000000085
because f isT(k,x(k))f(k,x(k))≤xT(k)FTFx (k), so there is > 0, making the following equation hold xT(k)FTFx(k)-fT(k,x(k))f(k,x(k))≥0。
Figure BDA0002648030000000086
Wherein the content of the first and second substances,
Figure BDA0002648030000000087
is provided with
Figure BDA0002648030000000088
Figure BDA0002648030000000089
1＝-1Equation (8) two sides simultaneous left and right multiplication diag { P }011,P022,I,I,I,I,I,I,P011,3I,I,I,P011I, I }, (9) formula two-sided simultaneous left-and right-multiplication diag { P }111,P122,I,I,I,I,I,I,P111,3I,I,I,P111,I,I}。
Formula (8) and formula (9) are equivalent to xi according to Schur's complement and lemma3If less than 0, then
xT(k)Qx(k)+uT(k)Ru(k)+ΔV[xa(k)]＜0，ΔV[xa(k)]＜0。
The closed loop system (7) is thus stable.
Because of the fact that
Figure BDA00026480300000000810
Therefore, it is not only easy to use
Figure BDA00026480300000000811
Because E { x (0) } 0, E { x (0) xT(0) I, so performance index
Figure BDA0002648030000000091
If it is not
Figure BDA0002648030000000092
Is the solution of the optimization problem (10), the constraint conditions (a) and (b) are satisfied, so
Figure BDA0002648030000000093
Is a non-fragile infinite time controller of the system (1). According to the Schur complementary theorem, the constraints (c) and (d) are equivalent to
Figure BDA0002648030000000094
Namely, it is
Figure BDA0002648030000000095
The performance indicator upper bound is minimal.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and decorations can be made without departing from the principle of the present invention, and these modifications and decorations should also be regarded as the protection scope of the present invention.

Claims (7)

1. A non-fragile optimal control method of a Lipschitz nonlinear system is characterized by comprising the following steps:
1) the sensor samples the state of the Lipschitz nonlinear system with model uncertainty, and sampling data is converted into digital quantity by adopting a logarithmic quantizer and transmitted to a remote controller;
2) in the process of transmitting quantized data to a remote controller, because of network congestion or interference loss, a Markov chain is adopted to describe the data transmission process, and the remote controller adopts a corresponding control strategy according to the condition of receiving data;
3) the closed-loop system is described as a Markov jump system, and a non-fragile infinite time optimal controller is designed by utilizing a linear matrix inequality based on a Markov jump system model and performance indexes of the system.
2. The method as claimed in claim 1, wherein in step 1), the expression of the state space of the Lipschitz nonlinear system is as follows:
Figure FDA0002648029990000011
wherein x (k) e RnIs the system state vector, u (k) e RpIs the system control input, y (k) e RmIs the system output, f (k, x (k)) is a nonlinear vector function that satisfies the Lipschitz condition: f (k,0) ═ 0, | f (k, x (k)) | ≦ | fx (k)) |; g, H, L, C and F are known coefficient matrices of the corresponding dimensions, Δ G, Δ H, Δ L and Δ C are system uncertainties, and the condition is satisfied:
[ΔG ΔH ΔL ΔC]＝DF1(k)[EG EH EL EC]，F1 T(k)F1(k)≤I(2)
D，EG，EH，ELand ECIs a matrix of known corresponding dimensions;
assuming that the system initial state x (0) is a random variable, E { x (0) } is 0, E { x (0) xT(0) I, the performance index of a given system is:
Figure FDA0002648029990000012
where Q and R are given symmetric positive definite matrices.
3. The design of the non-fragile optimal controller of the Lipschitz nonlinear system as recited in claim 2, wherein in the step 1), a logarithmic quantizer is adopted to convert the sampled data into digital quantity, specifically as follows:
the sensor samples the system state and converts the system state into digital quantity, the quantizer adopts a logarithmic quantizer, and then the output of the quantizer is:
xq(k)＝[I+F2(k)]x(k)，F2 T(k)F2(k)≤2I，＝(1-ρ)/(1+ρ) (4)
where ρ is the quantization density.
4. The design of the non-fragile optimal controller of the Lipschitz nonlinear system as claimed in claim 3, wherein in the step 2), the data transmission process is described by using Markov chain, and the specific steps are as follows:
loss occurs in the process of transmitting measurement data to a controller through a network due to the influence of network congestion or external interference, the loss process is described as a Markov chain, alpha (k) ═ 0 represents data loss, the controller adopts the value at the previous moment, alpha (k) ═ 1 represents normal data transmission, and the state transition matrix of the Markov chain
Figure FDA0002648029990000021
5. The design of the non-fragile optimal controller of the Lipschitz nonlinear system as recited in claim 4, wherein in the step 2), the controller adopts a corresponding control strategy according to the received data condition, specifically as follows:
the inputs to the controller are:
xc(k)＝[1-α(k)]xc(k-1)+α(k)xq(k)，α(k)＝i＝{0,1} (5)
aiming at a controlled system and performance indexes, a non-fragile controller is designed to minimize the performance index J, and the controller is in the form of:
u(k)＝(Kα(k)+ΔK)xc(k)，ΔK＝DF1(k)EK，F1 T(k)F1(k)≤I (6)
wherein D and EKIs a matrix of known corresponding dimensions.
6. The design of the non-fragile optimal controller of the Lipschitz nonlinear system as claimed in claim 5, wherein in the step 3), the closed loop system is described as a Markov jump system, specifically as follows:
is provided with
Figure FDA0002648029990000022
The closed loop system is then:
Figure FDA0002648029990000023
wherein the content of the first and second substances,
Figure FDA0002648029990000024
Gcl111＝G+ΔG+(H+ΔH)(K1+ΔK)(I+F2(k))，
Figure FDA0002648029990000025
Ccl0＝Ccl1＝[C+ΔC 0]。
7. the design of the non-fragile optimal controller of the Lipschitz non-linear system as claimed in claim 6, wherein in said step 3), the non-fragile infinite time optimal controller is designed by using the linear matrix inequality, specifically as follows:
given a Lipschitz nonlinear system (1) and its performance index (3), if a symmetric positive definite matrix X exists011，X022，X111，X122，W0，W1Matrix M0，M1Scalar quantity1＞0，2＞0，3> 0, the following linear matrix inequality is satisfied:
Figure FDA0002648029990000031
Figure FDA0002648029990000032
wherein omega11＝diag{-X011,-X022}，Ω22＝-1I，
Figure FDA0002648029990000033
Figure FDA0002648029990000034
Figure FDA0002648029990000035
Ω51＝[0 M0]，Ω55＝-R-1，
Figure FDA0002648029990000036
Figure FDA0002648029990000041
Ω66＝diag{-W0,-3I,-2I}，
Figure FDA0002648029990000042
Ω77＝diag{-Q-1,-1I}，
Figure FDA0002648029990000043
Ω88＝diag{-W0,-3I,-2I}，Λ11＝diag{-X111,-X122}，Λ22＝-1I，
Figure FDA0002648029990000044
Figure FDA0002648029990000045
Λ51＝M1，Λ55＝-R-1，
Figure FDA0002648029990000046
Λ66＝diag{-W,-3I}，Λ71＝[EGX111+EHM1 0]，Λ72＝1EL，Λ76＝[EHM1 2EHD]，Λ77＝-2I，
Figure FDA0002648029990000047
Λ88＝diag{-Q-1,-1I,-W}，
Figure FDA0002648029990000048
Figure FDA0002648029990000049
Λ99＝diag{-3I,-2I}。
Then the non-fragile infinite time controller
Figure FDA00026480299900000410
The closed loop system (7) is stable, and the performance index is less than
Figure FDA00026480299900000411
For a given Lipschitz nonlinear system (1) and its performance index (3), if the following optimization problem has a solution
Figure FDA00026480299900000412
Figure FDA00026480299900000413
s.t.(a)Ω＜0
(b)Λ＜0
Figure FDA00026480299900000414
Figure FDA0002648029990000051
Wherein N is2＝(X011,X022,X111,X122,W0,W1,M0,M1,1,2,3,N1) Then, then
Figure FDA0002648029990000052
Figure FDA0002648029990000053
Is a non-fragile infinite time optimal controller of the system (1).
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