CN111290274B - H-infinity control method of network control system with data packet loss - Google Patents

H-infinity control method of network control system with data packet loss Download PDF

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CN111290274B
CN111290274B CN202010101007.8A CN202010101007A CN111290274B CN 111290274 B CN111290274 B CN 111290274B CN 202010101007 A CN202010101007 A CN 202010101007A CN 111290274 B CN111290274 B CN 111290274B
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CN111290274A (en
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王燕锋
夏卫锋
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Foshan Haixie Technology Co ltd
Weishitong (Guangzhou) Information Security Technology Co.,Ltd.
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Suqian College
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Abstract

The invention discloses an H-infinity control method of se:Sub>A network control system with datse:Sub>A packet loss, which comprises the steps of firstly constructing an observer at se:Sub>A controller end, modeling the network control system with S-C and C-A packet loss into se:Sub>A Markov jump system with four modes, secondly providing sufficient and necessary conditions for random stability of se:Sub>A closed-loop system in se:Sub>A matrix inequality form, and thirdly providing se:Sub>A solving method of an observer gain matrix, se:Sub>A controller gain matrix and se:Sub>A minimum disturbance suppression performance index under the condition that the system mode transition probability is partially unknown, and obtaining the relation between the information quantity of the transition probability and the H-infinity performance index of the system.

Description

H-infinity control method of network control system with data packet loss
Technical Field
The invention belongs to the technical field of network control, and particularly relates to an H-infinity control method of a network control system with data packet loss.
Background
A feedback control system (NCS) that forms a closed loop by a network is called a Network Control System (NCS). The network control system has the advantages of low cost, easy expansion, easy maintenance and the like, and is widely applied to the fields of aerospace, industrial control and the like. Due to the introduction of the network, the control system may exhibit some new characteristics, such as transmission of signals in the network with limited bandwidth, and inevitably generate packet loss and the like. Packet loss exists not only from Sensor to Controller (S-C) but also from Controller to Actuator (C-A).
Existing methods for handling packet loss can be divided into three categories: the first is to treat packet loss as an event, modeling NCS as a control system with event rate constraints; the second type is that random variables obeying Bernoulli distribution are used for describing the phenomenon of data packet loss, and NCS is modeled as a control system containing the random variables; neither the first-type nor the second-type methods can describe the relationship between packet loss at the current time and packet loss at the previous time. The third method is to model packet loss as a finite-mode Markov chain and NCS as a Markov hopping system. At present, an H-infinity control method aiming at NCS is still incomplete, and further analysis is needed for S-C packet loss and C-A packet loss. The problems existing in the prior art are as follows:
1) Most technologies only consider S-C packet loss or C-A packet loss, assume that the state of the system can be measured, and lack se:Sub>A controller design method which considers S-C packet loss and C-A packet loss based on an observer;
2) Most of the technologies assume that the transition probability of the system mode is known, and a collaborative design method of an observer gain matrix and a controller gain matrix under the condition that the transition probability of the system mode is partially unknown is lacked;
3) Under the condition that the transition probability is partially unknown, the relationship between the minimum disturbance rejection performance index and the transition probability information amount needs to be further researched.
The significance of solving the technical problems is as follows:
simultaneously considering S-C packet loss and C-A packet loss, and designing se:Sub>A controller based on an observer has important practical significance for promoting the application of NCS theory; the relation between the minimum disturbance suppression performance index and the transfer probability information quantity is disclosed, and a compromise scheme can be selected between the system performance index and the transfer probability information quantity, so that all transfer probabilities do not need to be obtained on the premise of meeting the system performance requirements.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides an H-infinity control method for a network control system with data packet loss, which specifically comprises the following steps:
step 1: establishing a closed-loop system mathematical model:
using a random variable alpha having a value of 0 or 1 k And beta k Respectively describing S-C packet loss and C-A packet loss: alpha is alpha k =0 denotes that S-C packet loss has occurred, α k =1 indicates that no S-C packet loss has occurred; beta is a k =0 denotes that C-se:Sub>A packet loss occurred, β k =1 means that C-se:Sub>A packet loss has not occurred;
the controlled object is a linear system, and the state space is expressed as:
Figure BDA0002386868800000011
wherein x is k Is the system state vector, u k Is the system control input vector, ω k Is the input vector that is external to the system,y k is the system output vector; a, B ω ,C,D ω Is a known real matrix;
constructing an observer at a controller end:
Figure BDA0002386868800000021
wherein
Figure BDA0002386868800000022
Is the state of the observer and,
Figure BDA0002386868800000023
is the output of the observer and,
Figure BDA0002386868800000024
is the system output received by the observer,
Figure BDA0002386868800000025
is the control input of the observer, L is the observer gain matrix to be determined;
adopting a state feedback control law based on an observer:
Figure BDA0002386868800000026
where K is the controller gain matrix to be determined;
due to S-C packet loss, the system output obtained by the controller at time k is:
Figure BDA0002386868800000027
because of the packet loss of C-se:Sub>A, the control amount acting on the controlled object at time k is:
Figure BDA0002386868800000028
defining a state estimation error e k And an augmented vector ζ k ：
Figure BDA0002386868800000029
The closed-loop system expression is obtained from equations (1) to (6):
Figure BDA00023868688000000210
wherein
Figure BDA00023868688000000211
I is an identity matrix;
step 2: the closed-loop system is modeled into se:Sub>A Markov jump system with 4 modes through the influence of S-C packet loss and C-A packet loss on parameters of the closed-loop system:
1) When alpha is k ＝β k =0, i.e. S-C packet loss and C-se:Sub>A packet loss occur simultaneously, the closed-loop system (7) can be represented as:
Figure BDA00023868688000000212
wherein
Figure BDA00023868688000000213
N 1 ＝[-I I]；
2) When alpha is k ＝0,β k =1, i.e. S-C packet loss has occurred, the closed-loop system (7) can be represented as:
Figure BDA00023868688000000214
wherein
Figure BDA00023868688000000215
N 2 ＝[I-I]；
3) When alpha is k ＝1,β k =0, i.e. C-se:Sub>A packet loss has occurred, the closed-loop system (7) can be represented as:
Figure BDA00023868688000000216
wherein
Figure BDA0002386868800000031
N 3 ＝[-I I]，F 3 ＝[0 -C]，
Figure BDA0002386868800000032
4) When alpha is k ＝1,β k =1, i.e. no packet loss has occurred, the closed-loop system (7) can be represented as:
Figure BDA0002386868800000033
wherein
Figure BDA0002386868800000034
N 4 ＝[-I I]，F 4 ＝[0 -C]，
Figure BDA0002386868800000035
With the packet loss under different conditions, the closed-loop system (7) jumps among the 4 subsystems (8) - (11); the closed loop system (7) was modeled as a Markov hopping system with 4 subsystems:
Figure BDA0002386868800000036
wherein r k K ∈ Z } is a discrete Markov chain, takes values from the set M = {1,2,3,4}, and
Figure BDA0002386868800000037
Figure BDA0002386868800000038
r k the transition probability matrix is pi = [ pi = pq ]，π pq ＝Pr{r k+1 ＝q|r k ＝p}，π pq ≥0，
Figure BDA0002386868800000039
p,q∈M；
And step 3: the condition that the closed loop system (12) is randomly stable and has H ∞ performance as shown in formula (13) is given:
Figure BDA00023868688000000310
wherein γ is a disturbance rejection performance indicator;
if a positive definite matrix P exists p ＞0，Y p > 0 and the matrix K, L is such that
Figure BDA00023868688000000311
Figure BDA00023868688000000312
Figure BDA00023868688000000316
Wherein
Figure BDA00023868688000000313
For p , q ∈ M all holds, then the closed-loop system (12) is randomly stable and has H ∞ performance (13);
and 4, step 4: giving out a controller gain matrix K, an observer gain matrix L and a minimum disturbance rejection performance index gamma min The solving algorithm of (1):
the first step is as follows: given γ = γ 0 And maximum number of iterations R max ；
The second step is that: solving the equations (23), (24) and
Figure BDA00023868688000000314
p belongs to M, and a set of feasible solutions is obtained
Figure BDA00023868688000000315
Let k =0;
the third step: solving the non-linear minimization problem:
Figure BDA0002386868800000041
constrained by the formulae (23), (24) and
Figure BDA0002386868800000042
p is an element of M, such that
Figure BDA0002386868800000043
K k ＝K，L k ＝L；
The fourth step: checking whether the formulas (23) to (25) are satisfied, if so, reducing gamma appropriately, namely, making gamma = gamma-tau and tau be a positive real number, and making k = k +1, and going to a third step; if not, directly turning to the third step;
the fifth step: if the number of iterations is greater than R max The iteration is terminated; check γ after iteration terminates: if γ = γ 0 Then the optimization problem has no solution within a given number of iterations; if gamma is to be<γ 0 Then γ min ＝γ+τ。
The beneficial effects of the invention are as follows: designing an observer for NCS with S-C packet loss and C-A packet loss at the same time, and modeling se:Sub>A closed-loop system as se:Sub>A Markov jump system; a collaborative design method of an observer and a controller gain matrix is provided, and a relation between the information quantity of the system modal transition probability and the H-infinity performance of the system is obtained, so that on the premise of meeting the H-infinity performance of the system, all transition probabilities are not required to be obtained.
Drawings
Fig. 1 is a block diagram of an NCS with packet loss according to an embodiment of the present invention.
Fig. 2 is a diagram of jump values of a system mode according to an embodiment of the present invention.
FIG. 3 shows a closed loop system state x provided by an embodiment of the invention 1 And its estimated value
Figure BDA00023868688000000412
Drawing.
FIG. 4 shows a closed-loop system state x according to an embodiment of the present invention 2 And its estimated value
Figure BDA00023868688000000413
Figure (a).
Detailed Description
The NCS structure with packet loss is shown in fig. 1. Using a random variable alpha of value 0 or 1 k And beta k Respectively describing S-C packet loss and C-A packet loss: alpha (alpha) ("alpha") k =0 denotes that S-C packet loss has occurred, α k =1 indicates that no S-C packet loss has occurred; beta is a k =0 denotes that C-se:Sub>A packet loss has occurred, β k =1 indicates that C-se:Sub>A packet loss has not occurred;
the controlled object is a linear system, and the state space is expressed as:
Figure BDA0002386868800000044
wherein x is k Is the system state vector, u k Is the system control input vector, ω k Is the system external input vector, y k Is the system output vector; a, B ω ,C,D ω Is a known real matrix.
Constructing an observer at a controller end:
Figure BDA0002386868800000045
wherein
Figure BDA0002386868800000046
Is the state of the observer and is,
Figure BDA0002386868800000047
is the output of the observer and,
Figure BDA0002386868800000048
the system output received by the observer is,
Figure BDA0002386868800000049
is the control input to the observer and L is the observer gain matrix to be determined.
Adopting a state feedback control law based on an observer:
Figure BDA00023868688000000410
where K is the controller gain matrix to be determined;
because of the S-C packet loss, the system output obtained by the controller at time k is:
Figure BDA00023868688000000411
due to the packet loss of C-se:Sub>A, the control quantity acting on the controlled object at time k is:
Figure BDA0002386868800000051
defining a state estimation error e k And an augmented vector ζ k ：
Figure BDA0002386868800000052
Obtaining a closed-loop system expression from the expressions (1) to (6):
Figure BDA0002386868800000053
wherein
Figure BDA0002386868800000054
And I is an identity matrix.
The closed loop system (7) consists of 4 subsystems:
1) When alpha is k ＝β k =0, i.e. S-C packet loss and C-se:Sub>A packet loss occur simultaneously, the closed-loop system (7) can be represented as:
Figure BDA0002386868800000055
wherein
Figure BDA0002386868800000056
N 1 ＝[-I I]。
2) When alpha is k ＝0,β k =1, i.e. S-C packet loss has occurred, the closed-loop system (7) can be represented as:
Figure BDA0002386868800000057
wherein
Figure BDA0002386868800000058
N 2 ＝[I -I]。
3) When alpha is k ＝1,β k =0, i.e. C-se:Sub>A packet loss has occurred, the closed-loop system (7) can be represented as:
Figure BDA0002386868800000059
wherein
Figure BDA00023868688000000510
N 3 ＝[-I I]，F 3 ＝[0 -C]，
Figure BDA00023868688000000511
4) When alpha is k ＝1,β k =1, i.e. no packet loss occurs, the closed-loop system (7) can be represented as:
Figure BDA00023868688000000512
wherein
Figure BDA00023868688000000513
N 4 ＝[-I I]，F 4 ＝[0 -C]，
Figure BDA00023868688000000514
With packet loss in different situations, the closed loop system (7) jumps among the 4 subsystems (8) - (11). Since the packet loss at the current time is closely related to the packet loss at the previous time, the closed-loop system (7) can be modeled as a Markov hopping system with 4 subsystems:
Figure BDA00023868688000000515
wherein { r k K ∈ Z } is a discrete Markov chain, takes values from the set M = {1,2,3,4}, and
Figure BDA00023868688000000516
Figure BDA00023868688000000517
r k the transition probability matrix is pi = [ pi ] pq ]，π pq ＝Pr{r k+1 ＝q|r k ＝p}，π pq ≥0，
Figure BDA0002386868800000061
p,q∈M。
Definition 1: when ω is k =0 if r is for any initial modality 0 Epsilon M and System initial State ζ 0 The presence of a positive definite matrix W > 0 such that
Figure BDA0002386868800000062
The closed loop system (12) is then randomly stable.
Note 1: different from the traditional point-to-point control system, the control input vector of the observer in the formula (2)
Figure BDA0002386868800000063
And the control input vector u of the controlled object in the formula (1) k 。
The objective of the invention is to design an observer (2) and an observer-based controller (3) such that se:Sub>A closed-loop system (12) with both S-C packet loss and C-A packet loss is randomly stable and meets the H ∞ performance index. That is, it should satisfy:
1) The closed loop system (12) is randomly stable;
2) At zero initial conditions for all ω k Not equal to 0, system output y k The following requirements should be satisfied:
Figure BDA0002386868800000064
where γ is the disturbance rejection performance indicator.
The invention will give sufficient requirements for the random stability of the closed loop system (12) and in the system mode r k The observer (2) and the controller (3) are designed under the condition that a part of unknown elements exist in the transition probability matrix pi. For convenience of writing, when r k When = p, r is k Denoted as p.
Theorem 1: the closed-loop system (12) is randomly stable if and only if a positive definite matrix P is present p ＞0,P q 0 and the matrix K, L is such that equation (14) is for all p , q ∈ M both hold:
Figure BDA0002386868800000065
and (3) proving that:
the sufficiency: defining a Lyapunov function
Figure BDA0002386868800000066
Wherein
Figure BDA0002386868800000067
When omega k =0, available from (12):
Figure BDA0002386868800000068
therefore, if (14) holds, we can:
Figure BDA0002386868800000069
wherein λ min The (- Θ) is the minimum eigenvalue of the matrix- Θ.
For any positive integer N ≧ 0, we can obtain:
Figure BDA00023868688000000610
Figure BDA0002386868800000071
according to definition 1, (12) is randomly stable.
The necessity: assuming that the closed loop system (12) is randomly stable, it is possible to obtain:
Figure BDA0002386868800000072
consider formula (16):
Figure BDA0002386868800000073
wherein
Figure BDA0002386868800000074
ζ k Not equal to 0. Is composed of(14) It can be known that
Figure BDA0002386868800000075
Is bounded, and:
Figure BDA0002386868800000076
since formula (17) is for any nonzero ζ k Is established, thus can obtain
Figure BDA0002386868800000077
Due to the fact that
Figure BDA0002386868800000078
From (17) to obtain
Figure BDA0002386868800000079
And:
Figure BDA00023868688000000710
let T → ∞, can obtain Θ <0, after verification.
Theorem 1 gives sufficient requirements for the observer (2) and controller (3) to be present, and theorem 2 will give sufficient conditions for the closed-loop system (12) to be randomly stable and have H ∞ performance.
Theorem 2: if a positive definite matrix P exists p ＞0,Y p > 0 and the matrix K, L is such that
Figure BDA00023868688000000711
Figure BDA00023868688000000713
Wherein
Figure BDA00023868688000000712
For all p , q ∈ M are all true, then the closed-loop system (12) is randomly stable and has H ∞ performance (13).
And (3) proving that: when ω is k Not equal to 0, obtainable from formula (12):
Figure BDA0002386868800000081
wherein
Figure BDA0002386868800000082
Figure BDA0002386868800000083
Figure BDA0002386868800000084
Figure BDA0002386868800000085
Order to
Figure BDA0002386868800000086
p is in the same order as M and can be obtained by Schur's theorem, omega p <0 is equivalent to (18).
Therefore, the following equations (18) and (20) can be obtained:
Figure BDA0002386868800000087
from (21), summing k from 0 to ∞ yields:
Figure BDA0002386868800000088
equation (13) is obtained from the stochastic stability of the closed-loop system and the zero initial condition, and is verified.
Theorem 2 assumes a system modality r k The transition probabilities of (c) are all known. However, it is often difficult to obtain the full transition probability. Considering in theorem 3 that the transition probability of a system modality is partially unknown, i.e. there are partially unknown elements in the matrix, for example Π may have the structure as shown in equation (22):
Figure BDA0002386868800000089
wherein "? "denotes an unknown element. Set M can be written as
Figure BDA00023868688000000810
Wherein
Figure BDA00023868688000000811
Figure BDA00023868688000000812
If it is used
Figure BDA00023868688000000813
Then
Figure BDA00023868688000000814
D is more than or equal to 1 and less than or equal to 4, wherein
Figure BDA00023868688000000815
Is the column index of the d known element of the pth row of the matrix Π.
Figure BDA00023868688000000816
Is marked as
Figure BDA00023868688000000817
Wherein
Figure BDA00023868688000000818
Of the 4 th-d unknown elements of the pth row of the matrix ΠColumn subscripts.
Theorem 3: the closed loop system (12) is randomly stable and has H ∞ performance (13) if a positive definite matrix P exists p ＞0，Y p > 0 and the matrix K, L is such that
Figure BDA00023868688000000819
Figure BDA0002386868800000091
Figure BDA00023868688000000910
Wherein
Figure BDA0002386868800000092
Figure BDA0002386868800000093
Figure BDA0002386868800000094
Figure BDA0002386868800000095
For p , q ∈ M is established.
And (3) proving that: due to the fact that
Figure BDA0002386868800000096
According to Schur theorem, Ω p <0 is equivalent to
Figure BDA0002386868800000097
Again using Schur, if equations (23) - (25) hold, then
Figure BDA0002386868800000098
After the test is finished.
Note 2: with respect to the unknown transition probability matrix, the method employed by the present invention is to separate the unknown probabilities from the known probabilities and then discard the unknown probabilities. Another approach is to separate the probabilities from the correlation matrix, e.g. by
Figure BDA0002386868800000099
This will greatly increase the conservatism of the conclusions.
Note 3: with respect to the method of separating the unknown probabilities from the known probabilities, as the number of unknown probabilities increases, the number of matrix inequalities that need to be solved will also increase. In theorem 3, if all transition probability matrices are true, the number of matrix inequalities to be solved is 4; if all transition probabilities are unknown, the number of matrix inequalities that need to be solved is 16.
Constraints (18) and (19) in theorem 3 are matrix inequalities having inverse matrix constraints, and can be solved by Cone Complementary Linearization (CCL). Controller gain matrix K, observer gain matrix L and minimum disturbance rejection performance index gamma min Can be converted into a nonlinear constraint problem with matrix inequality constraints:
Figure BDA0002386868800000101
constrained by (23), (24) and (26)
Figure BDA0002386868800000102
Controller gain matrix K, observer gain matrix L and minimum disturbance rejection performance index gamma min The solving algorithm of (1) is as follows:
the first step is as follows: given γ =γ 0 And maximum number of iterations R max ；
The second step is that: solving equations (23), (24) and (26) to obtain a set of feasible solutions
Figure BDA0002386868800000103
Let k =0;
the third step: solving the non-linear minimization problem:
Figure BDA0002386868800000104
is constrained by (23), (24) and (26) to
Figure BDA0002386868800000105
K k ＝K，L k ＝L；
The fourth step: checking whether the expressions (23) to (25) are satisfied, if so, reducing gamma appropriately, namely, making gamma = gamma-tau, tau be a positive real number, making k = k +1, and going to a third step; if the iteration number is more than R max The iteration is terminated;
the fifth step: check γ after iteration terminates: if γ = γ 0 If the optimization problem has no solution within the given iteration times; if gamma is<γ 0 Then γ min ＝γ+τ。
Numerical simulation
The parameters of the controlled object are:
Figure BDA0002386868800000106
the mode of the system is sigma along with the change of the packet loss k E {1,2,3,4}, in order to derive the relationship between the amount of transition probability information and the H ∞ performance of the system, consider the transition probability matrices for four different cases:
Figure BDA0002386868800000107
according to theorem 3, the minimum H infinity attenuation level gamma is obtained min As shown in table 1:
TABLE 1. Gamma. Under different transition probability matrices min Value of (2)
Transition probability matrix Λ 1 Λ 2 Λ 3 Λ 4
γ min 0.1434 0.0837 0.0593 0.0392
As can be seen from Table 1, the more elements of the transition probability matrix are known, the more the disturbance rejection performance index γ of the system min The smaller, i.e. the stronger the disturbance rejection capability of the system.
For having a transition probability matrix Λ 2 The NCS of (2) obtains a controller gain matrix K, an observer gain matrix L:
K＝[-0.1866 -0.5995],
Figure BDA0002386868800000108
initial state of the system
Figure BDA0002386868800000111
External disturbance is
Figure BDA0002386868800000112
FIG. 2 shows the transition values of the system mode, and FIG. 3 shows the state x of the closed-loop system 1 And its estimated value
Figure BDA0002386868800000113
Curve, fig. 4 closed loop system state x 1 And its estimated value
Figure BDA0002386868800000114
Curve line.
The invention constructs the observer for the NCS with S-C packet loss and C-A packet loss, models the closed-loop system as se:Sub>A Markov jump system, and the obtained observer and controller collaborative design method is not only suitable for the condition that the transition probability is totally known, but also suitable for the condition that the transition probability is partially unknown.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents and improvements made within the spirit and principle of the present invention are intended to be included within the scope of the present invention.

Claims (1)

1. A H-infinity control method of a network control system with packet loss comprises the following steps:
step 1: establishing a closed-loop system mathematical model:
using a random variable alpha having a value of 0 or 1 k And beta k Respectively describing S-C packet loss and C-A packet loss: alpha is alpha k =0 denotes that S-C packet loss has occurred, α k =1 means that no S-C packet loss has occurred; beta is a k =0 denotes that C-se:Sub>A packet loss has occurred, β k =1 indicates that C-se:Sub>A packet loss has not occurred;
the controlled object is a linear system, and the state space is expressed as:
Figure FDA0003833920910000011
whereinx k Is the system state vector, u k Is the system control input vector, ω k Is the system external input vector, y k Is the system output vector; a, B ω ,C,D ω Is a known real matrix;
constructing an observer at a controller end:
Figure FDA0003833920910000012
wherein
Figure FDA0003833920910000013
Is the state of the observer and is,
Figure FDA0003833920910000014
is the output of the observer and is,
Figure FDA0003833920910000015
is the system output received by the observer,
Figure FDA0003833920910000016
is the control input of the observer, L is the observer gain matrix to be determined;
adopting a state feedback control law based on an observer:
Figure FDA0003833920910000017
where K is the controller gain matrix to be determined;
because of the S-C packet loss, the system output obtained by the controller at time k is:
Figure FDA0003833920910000018
because of the packet loss of C-se:Sub>A, the control amount acting on the controlled object at time k is:
Figure FDA0003833920910000019
defining a state estimation error e k And an augmented vector xi k ：
Figure FDA00038339209100000110
Obtaining a closed-loop system expression from the expressions (1) to (6):
Figure FDA00038339209100000111
wherein
Figure FDA00038339209100000112
I is an identity matrix;
step 2: the closed-loop system is modeled into se:Sub>A Markov jump system with 4 modes through the influence of S-C packet loss and C-A packet loss on parameters of the closed-loop system:
1) When alpha is k ＝β k =0, i.e. S-C packet loss and C-se:Sub>A packet loss occur simultaneously, the closed-loop system (7) can be represented as:
Figure FDA00038339209100000113
wherein
Figure FDA00038339209100000114
N 1 ＝[-I I]；
2) When alpha is k ＝0,β k =1, i.e. S-C packet loss has occurred, the closed-loop system (7) can be represented as:
Figure FDA0003833920910000021
wherein
Figure FDA0003833920910000022
N 2 ＝[I -I]；
3) When alpha is k ＝1,β k =0, i.e. C-se:Sub>A packet loss has occurred, the closed-loop system (7) can be represented as:
Figure FDA0003833920910000023
wherein
Figure FDA0003833920910000024
N 3 ＝[-I I]，F 3 ＝[0 -C]，
Figure FDA0003833920910000025
4) When alpha is k ＝1,β k =1, i.e. no packet loss occurs, the closed-loop system (7) can be represented as:
Figure FDA0003833920910000026
wherein
Figure FDA0003833920910000027
N 4 ＝[I I]，F 4 ＝[0 -C]，
Figure FDA0003833920910000028
With the packet loss under different conditions, the closed-loop system (7) jumps among the 4 subsystems (8) - (11); the closed loop system (7) was modeled as a Markov hopping system with 4 subsystems:
Figure FDA0003833920910000029
wherein { r k K ∈ Z } is a discrete Markov chain, takes values from the set M = {1,2,3,4}, and
Figure FDA00038339209100000210
Figure FDA00038339209100000211
r k the transition probability matrix is pi = [ pi = pq ]，π pq ＝Pr{r k+1 ＝q|r k ＝p}，π pq ≥0，
Figure FDA00038339209100000212
Set M can be written as
Figure FDA00038339209100000213
Wherein
Figure FDA00038339209100000214
And step 3: giving the condition that the closed-loop system (12) is randomly stable and has H ∞ performance as shown in formula (13):
Figure FDA00038339209100000215
wherein γ is a disturbance rejection performance indicator;
if a positive definite matrix P exists p ＞0，Y p > 0 and the matrix K, L is such that
Figure FDA00038339209100000216
Figure FDA00038339209100000217
Y p P p ＝I (25)
Wherein
Figure FDA0003833920910000031
For p , q ∈ M all hold, then the closed-loop system (12) is randomly stable and has H ∞ performance (13);
and 4, step 4: giving out a controller gain matrix K, an observer gain matrix L and a minimum disturbance rejection performance index gamma min The solving algorithm of (2):
the first step is as follows: given γ = γ 0 And maximum number of iterations R max ；
The second step: solving equations (23), (24) and
Figure FDA0003833920910000032
obtain a set of feasible solutions
Figure FDA0003833920910000033
Let k =0;
the third step: solving the nonlinear minimization problem:
Figure FDA0003833920910000034
constrained by the formulae (23), (24) and
Figure FDA0003833920910000035
order to
Figure FDA0003833920910000036
K k ＝K，L k ＝L；
The fourth step: checking whether the formulas (23) to (25) are satisfied, if so, reducing gamma appropriately, namely, making gamma = gamma-tau and tau be a positive real number, and making k = k +1, and going to a third step; if not, directly turning to the third step;
the fifth step: if iterationNumber of times greater than R max The iteration is terminated; check γ after iteration ends: if γ = γ 0 Then the optimization problem has no solution within a given number of iterations; if gamma < gamma 0 Then γ is min ＝γ+τ。
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