CN109856970B - Finite time stabilization method with network-induced bounded time lag and data loss - Google Patents

Finite time stabilization method with network-induced bounded time lag and data loss Download PDF

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CN109856970B
CN109856970B CN201811555666.8A CN201811555666A CN109856970B CN 109856970 B CN109856970 B CN 109856970B CN 201811555666 A CN201811555666 A CN 201811555666A CN 109856970 B CN109856970 B CN 109856970B
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谭冲
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Harbin University of Science and Technology
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Abstract

A finite time stabilization method with network-induced bounded time lag and data loss relates to the technical field of networked control systems. The method solves the problem that the control performance of the limited time stabilization method is influenced because the existing limited time stabilization method cannot actively compensate network-induced bounded time lag and data loss. The method predicts the state information of the networked control system at the current moment by using a prediction control method, and actively compensates the influence of network-induced bounded time lag and data loss on the limited-time control performance of the networked control system. Compared with the prior art, the finite time stabilization method can actively compensate the fixed network induced time lag, the time-varying bounded network induced time lag and the data loss of the forward channel and the feedback channel of the networked control system, obtain the finite time stabilization method based on the linear matrix inequality solution, and achieve the purpose of actively compensating the network induced bounded time lag and the data loss. The method is suitable for the problem of limited time stabilization of the linear networked dynamic system.

Description

Finite time stabilization method with network-induced bounded time lag and data loss
Technical Field
The invention belongs to the technical field of networked control systems, and particularly relates to a finite time stabilization method with network-induced bounded time lag and data loss.
Background
The limited-time stabilization is an important research problem in a control system, and the control system has the advantages of high convergence speed, high steady-state precision, good robustness and the like. The method is widely applied to the fields of missile systems, communication network systems, robot control systems and the like. From the optimization point of view, the time-limited stabilization control method is a time-optimal control method, namely, the time-limited stabilization control method is used for controlling the system to a balance point within a limited time, and researches show that the time-limited stabilization system has better performance under the condition that the system has interference and uncertainty.
Although the existing research on the limited time stabilization method has made a certain progress, the existing limited time stabilization method still cannot actively compensate network-induced bounded time lag and data loss, and further influences the control performance of the limited time stabilization method.
Disclosure of Invention
The invention aims to solve the problem that the control performance of the limited time stabilization method is influenced because the existing limited time stabilization method cannot actively compensate network-induced bounded time lag and data loss.
The invention relates to a finite time stabilization method with network-induced bounded time lag and data loss, which comprises the following specific steps:
establishing a linear dynamic model of a networked control system with network-induced bounded time lag and data loss;
step two, establishing a state prediction model of the linear dynamic model of the networked control system in the step one;
thirdly, designing a state feedback controller according to the state prediction model of the linear dynamic model of the networked control system established in the second step;
step four, obtaining a closed loop system equation of the networked control system according to the state feedback controller obtained in the step three;
step five, acquiring a state estimation gain matrix L and a state feedback gain matrix K by utilizing a closed loop system of a networked control system through the Lyapunov stability theorem;
and step six, substituting the state estimation gain matrix L obtained in the step five into the state prediction model in the step two, and substituting the state feedback gain matrix K into the state feedback controller in the step three, so as to realize the limited time stabilization of the networked control system with network-induced bounded time lag and data loss.
The invention has the beneficial effects that: the invention provides a finite time stabilization method with network-induced bounded time lag and data loss, which predicts the state information of a networked control system at the current moment by using a prediction control method and actively compensates the influence of the network-induced bounded time lag and the data loss on the finite time control performance of the networked control system.
Minimum value of epsilon obtained by the method of the inventionIs composed of
Figure BDA0001911817650000021
Less than the minimum epsilon obtained without network-induced bounded skew and data lossmin2.8, the method of the invention can effectively avoid the influence of network induced time lag and data loss on the control performance of the finite time stabilization method, and has the advantage of easy solution and realization.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 shows an embodiment of the present invention at a given δxOptimization of the State trajectory x of a closed-Loop System in ε1(t) and State trajectory x2(t) view;
wherein: deltax 2Is the square of the initial state threshold value, epsilon2Is the square of the terminal state threshold, x (0) represents the initial state, xT(0) Is the transpose of x (0), x (t) represents the initial state at time t, xT(t) is the transpose of x (t), R is the domain matrix;
Detailed Description
First embodiment, the present embodiment is described with reference to fig. 1, and the first embodiment describes a finite time stabilization method with network-induced bounded skew and data loss, which includes the following specific steps:
establishing a linear dynamic model of a networked control system with network-induced bounded time lag and data loss;
step two, establishing a state prediction model of the linear dynamic model of the networked control system in the step one;
thirdly, designing a state feedback controller according to the state prediction model of the linear dynamic model of the networked control system established in the second step;
step four, obtaining a closed loop system equation of the networked control system according to the state feedback controller obtained in the step three;
step five, acquiring a state estimation gain matrix L and a state feedback gain matrix K by utilizing a closed loop system of a networked control system through the Lyapunov stability theorem;
and step six, substituting the state estimation gain matrix L obtained in the step five into the state prediction model in the step two, and substituting the state feedback gain matrix K into the state feedback controller in the step three, so as to realize the limited time stabilization of the networked control system with network-induced bounded time lag and data loss.
The second specific embodiment, which is different from the first specific embodiment, is that the specific process of the first step is as follows:
establishing a linear dynamic model of a networked control system with network-induced bounded time lag and data loss, the linear dynamic model having a state space form of:
Figure BDA0001911817650000031
wherein: x (t) is a state variable of a linear dynamic model of the networked control system at the time t, x (t +1) is a state variable of the linear dynamic model of the networked control system at the time t +1, u (t) is a control input function of a controller at the time t, y (t) is a measurement output function of a sensor at the time t, A is a system matrix, B is an input matrix, and C is an output matrix. The matrix pair (a, C) is detectable;
the sensors are connected with the controller through a network, the actuators are also connected with the controller through the network, and data transmitted through the network are provided with time stamps. The upper bounds of the network time lags for the feedback channel (i.e., the channel from the sensor to the controller) and the forward channel (i.e., the channel from the controller to the actuator) are n, respectivelybAnd nfThe upper bound of the number of the continuous packet losses of the data packets of the feedback channel and the forward channel is nd；
The third specific embodiment, which is different from the second specific embodiment, is that the specific process of the second step is as follows:
establishing a state prediction model of a linear dynamic model of the networked control system in the first step, wherein the specific form of the state prediction model is as follows:
Figure BDA0001911817650000032
in the formula (I), the compound is shown in the specification,
Figure BDA0001911817650000033
a predicted value of x (t-k- τ + i) at time t-k- τ + i, i ═ 2,3, …, k + τ, obtained based on a measured output function y (t-k- τ) at time t-k- τ;
y (t-k-tau) is a measurement output function at the time of t-k-tau;
Figure BDA0001911817650000034
the predicted value of x (t-k-tau) at the t-k-tau moment is obtained based on the measurement output function y (t-k-tau-1) at the t-k-tau moment;
Figure BDA0001911817650000035
the predicted value of x (t-k-tau +1) at the time of t-k-tau +1 is obtained based on the measurement output function y (t-k-tau) at the time of t-k-tau;
Figure BDA0001911817650000036
the predicted value of x (t-k-tau + i-1) at the time of t-k-tau + i-1 is obtained based on the measurement output function y (t-k-tau) at the time of t-k-tau;
l is a state estimation gain; u (t-k- τ) is the control input function of the controller at time t-k- τ; u (t-k-tau + i-1) is a control input function of the controller at the time of t-k-tau + i-1;
the upper bounds of the network time lags for the feedback channel (i.e., the channel from the sensor to the controller) and the forward channel (i.e., the channel from the controller to the actuator) are n, respectivelybAnd nfThe upper bound of the number of the continuous packet losses of the data packets of the feedback channel and the forward channel is nd(ii) a Intermediate variable k ═ nb+ndThe intermediate variable τ being nf+nd。
A fourth specific embodiment, which is different from the third specific embodiment, is that the specific process of the third step is as follows:
designing a state feedback controller according to the state prediction model of the linear dynamic model of the networked control system established in the step two;
Figure BDA0001911817650000041
in the formula (I), the compound is shown in the specification,
Figure BDA0001911817650000042
and K is a state feedback gain, and is a predicted value of x (t) at the time t, which is obtained based on a measured output function y (t-K-tau) at the time t-K-tau.
A fifth specific embodiment, which is different from the fourth specific embodiment, is that the specific process of the fourth step is as follows:
substituting the formula (3) into the formula (1) to obtain a closed-loop system equation of the networked control system:
ξ(t+1)＝Acξ(t) (4)
in the formula, xi (t) is a state variable of a closed-loop system of the networked control system at the time t, and xi (t +1) is a state variable of the closed-loop system of the networked control system at the time t + 1; xi (t) ═ xT(t) ET(t)]T，xT(t) is the transposition of the state variable x (t), ET(t) is the transpose of the error vector e (t), e (t) [ e ]T(t) eT(t-1)...eT(t-k-τ+1)]T，eT(t) is the transpose of e (t), e (t) is the state estimation error at time t, and
Figure BDA0001911817650000045
Figure BDA0001911817650000046
predicted value of x (t) at time t obtained based on measured output function y (t-1) at time t-1, AcA system matrix that is a closed-loop system of the networked control system;
matrix array
Figure BDA0001911817650000043
Wherein, InIs an n-dimensional identity matrix, In(k+τ)Is an n (k + tau) -dimensional identity matrix,
Figure BDA0001911817650000044
is the tensor product of the matrix.
A sixth specific implementation manner, which is different from the fifth specific implementation manner in that the fifth step obtains a state estimation gain matrix L and a state feedback gain matrix K, and the specific process is as follows:
Figure BDA0001911817650000051
Figure BDA0001911817650000052
Figure BDA0001911817650000053
the feedback coupling matrix X is obtained by equations (5), (6) and (7)1And estimating a coupling matrix X2And X1And X2Are all symmetric positive definite matrixes; q1Representing an input coupling matrix, Q2Representing an output coupling matrix; r is a domain matrix;
intermediate variable matrix
Figure BDA0001911817650000054
Comprises the following steps:
Figure BDA0001911817650000055
Figure BDA0001911817650000056
representative matrix
Figure BDA0001911817650000057
The condition number of (a) is,
Figure BDA0001911817650000058
Figure BDA0001911817650000059
is composed of
Figure BDA00019118176500000510
Is determined by the maximum characteristic value of the image,
Figure BDA00019118176500000511
is composed of
Figure BDA00019118176500000512
Gamma is not less than 1, and gamma is a normal number, deltaxIs an initial state threshold value, epsilon is a terminal state threshold value, and delta is more than 0x< ε, and δxAnd ε are both normal numbers, N is a known positive integer;
calculating a state estimation gain matrix L by formula (8);
Figure BDA00019118176500000513
calculating a state feedback gain matrix K through a formula (9);
Figure BDA00019118176500000514
seventh embodiment, which is different from the first embodiment, the lyapunov stability theorem in the fifth step is as follows:
V1(x(t+1))＜γV1(x(t)) (10)
wherein the content of the first and second substances,
V1(x(t))＝xT(t)P1x(t) (11)
in the formula, V1(x (t)) is the Lyapunov function at time t, V1(x (t +1)) is the Lyapunov function at time t +1, xT(t) is the transpose of x (t), P1Is a Lyapunov symmetric positive definite matrix.
Examples
The method of the invention is adopted for simulation:
system parameters:
Figure BDA0001911817650000061
C＝[2 1]
further, given that N is 4,
Figure BDA0001911817650000062
δ x1, the initial value of epsilon is 30;
the method of the invention is utilized to carry out time-limited stabilization on the networked control system and optimize the parameter epsilon:
case 1: there is no induced network skew for both the feedback path and the forward path, and no packet loss for both the feedback path and the forward path, i.e., nb＝0，nf＝0，nd＝0；
Solving a state estimation gain matrix L and a state feedback gain matrix K:
solving the formula (5), the formula (6), the formula (7) and the formula (9) to obtain a state feedback gain matrix K in the form of
K＝[1.02 1.54]
The limited time control effect of the networked control system without network-induced bounded time lag and data loss is as follows:
solving the formula (5), the formula (6) and the formula (7) to obtain the minimum value of epsilon as epsilonmin＝2.8。
Case 2: the feedback channel and the forward channel both have network-induced bounded time lags, and the feedback channel and the forward channel both have packet losses, i.e., the upper bound of the network time lags of the feedback channel is nbThe upper bound of the network skew for the forward path is n, 3f2, the upper bound of the number of the continuous packet losses of the data packets of the feedback channel and the forward channel is nd＝1；
Solving a state estimation gain matrix L and a state feedback gain matrix K:
solving the formula (5), the formula (6), the formula (7), the formula (8) and the formula (9) to obtain a state estimation gain matrix L and a state feedback gain matrix K which are in the following forms
Figure BDA0001911817650000063
K＝[1.00 1.60]
Finite time control effect on networked control systems with network-induced bounded skew and data loss:
solving the formula (5), the formula (6) and the formula (7) to obtain the minimum value of epsilon
Figure BDA0001911817650000064
As can be seen from the above analysis results, when there is network-induced bounded skew and data loss, the minimum value of ε obtained by the method of the present invention is
Figure BDA0001911817650000065
Which is less than the minimum epsilon obtained without network-induced bounded skew and data lossmin2.8. Therefore, aiming at the networked control system with network-induced bounded time lag and data loss, the method can effectively and actively compensate the network-induced bounded time lag and the data loss, and the limited-time control effect of the networked control system is better than the situation without the network-induced time lag and the data loss. Furthermore, as shown in FIG. 2, the state trajectory x of the closed-loop networked control system with network-induced bounded time lag and data loss under the action of the state feedback controller designed by the method of the present invention is visually and vividly shown1(t) and x2And (t) the control effect of finite time stability is achieved, and the finite time stabilization method can effectively and actively compensate network-induced bounded time lag and data loss.

Claims (2)

1. A method of finite time stabilization with network-induced bounded skew and data loss, the method comprising the steps of:
establishing a linear dynamic model of a networked control system with network-induced bounded time lag and data loss;
the specific process of the step one is as follows:
establishing a linear dynamic model of a networked control system with network-induced bounded time lag and data loss, the linear dynamic model having a state space form of:
Figure FDA0003391332750000011
wherein: x (t) is a state variable of a linear dynamic model of the networked control system at the time t, x (t +1) is a state variable of the linear dynamic model of the networked control system at the time t +1, u (t) is a control input function of a controller at the time t, y (t) is a measurement output function of a sensor at the time t, A is a system matrix, B is an input matrix, and C is an output matrix;
step two, establishing a state prediction model of the linear dynamic model of the networked control system in the step one;
the specific process of the second step is as follows:
establishing a state prediction model of a linear dynamic model of the networked control system in the first step, wherein the specific form of the state prediction model is as follows:
Figure FDA0003391332750000012
in the formula (I), the compound is shown in the specification,
Figure FDA0003391332750000013
a predicted value of x (t-k- τ + i) at time t-k- τ + i, i ═ 2,3, …, k + τ, obtained based on a measured output function y (t-k- τ) at time t-k- τ;
y (t-k-tau) is a measurement output function at the time of t-k-tau;
Figure FDA0003391332750000014
the predicted value of x (t-k-tau) at the t-k-tau moment is obtained based on the measurement output function y (t-k-tau-1) at the t-k-tau moment;
Figure FDA0003391332750000015
the predicted value of x (t-k-tau +1) at the time of t-k-tau +1 is obtained based on the measurement output function y (t-k-tau) at the time of t-k-tau;
Figure FDA0003391332750000016
the predicted value of x (t-k-tau + i-1) at the time of t-k-tau + i-1 is obtained based on the measurement output function y (t-k-tau) at the time of t-k-tau;
l is a state estimation gain; u (t-k- τ) is the control input function of the controller at time t-k- τ; u (t-k-tau + i-1) is a control input function of the controller at the time of t-k-tau + i-1;
the upper bound of the network skew of the feedback path and the forward path is nbAnd nfThe upper bound of the number of the continuous packet losses of the data packets of the feedback channel and the forward channel is nd(ii) a Intermediate variable k ═ nb+ndThe intermediate variable τ being nf+nd；
Thirdly, designing a state feedback controller according to the state prediction model of the linear dynamic model of the networked control system established in the second step;
the specific process of the third step is as follows:
designing a state feedback controller according to the state prediction model of the linear dynamic model of the networked control system established in the step two;
Figure FDA0003391332750000021
in the formula (I), the compound is shown in the specification,
Figure FDA0003391332750000022
based on time instants t-k-tauMeasuring a predicted value of x (t) obtained by the output function y (t-K-tau) at the time t, wherein K is a state feedback gain;
step four, obtaining a closed loop system equation of the networked control system according to the state feedback controller obtained in the step three;
the specific process of the step four is as follows:
substituting the formula (3) into the formula (1) to obtain a closed-loop system equation of the networked control system:
ξ(t+1)＝Acξ(t) (4)
in the formula, xi (t) is a state variable of a closed-loop system of the networked control system at the time t, and xi (t +1) is a state variable of the closed-loop system of the networked control system at the time t + 1; xi (t) ═ xT(t) ET(t)]T，xT(t) is the transposition of the state variable x (t), ET(t) is the transpose of the error vector e (t), e (t) [ e ]T(t) eT(t-1) … eT(t-k-τ+1)]T，eT(t) is the transpose of e (t), e (t) is the state estimation error at time t, and
Figure FDA0003391332750000023
Figure FDA0003391332750000024
predicted value of x (t) at time t obtained based on measured output function y (t-1) at time t-1, AcA system matrix that is a closed-loop system of the networked control system;
matrix array
Figure FDA0003391332750000025
Wherein, InIs an n-dimensional identity matrix, In(k+τ)Is an n (k + tau) -dimensional identity matrix,
Figure FDA0003391332750000026
is the tensor product of the matrix;
step five, acquiring a state estimation gain matrix L and a state feedback gain matrix K by utilizing a closed loop system of a networked control system through the Lyapunov stability theorem;
in the fifth step, a state estimation gain matrix L and a state feedback gain matrix K are obtained, and the specific process is as follows:
Figure FDA0003391332750000031
Figure FDA0003391332750000032
Figure FDA0003391332750000033
the feedback coupling matrix X is obtained by equations (5), (6) and (7)1And estimating a coupling matrix X2And X1And X2Are all symmetric positive definite matrixes; q1Representing an input coupling matrix, Q2Representing an output coupling matrix; r is a domain matrix;
intermediate variable matrix
Figure FDA0003391332750000034
Comprises the following steps:
Figure FDA0003391332750000035
Figure FDA0003391332750000036
representative matrix
Figure FDA0003391332750000037
The condition number of (a) is,
Figure FDA0003391332750000038
Figure FDA0003391332750000039
is composed of
Figure FDA00033913327500000310
Is determined by the maximum characteristic value of the image,
Figure FDA00033913327500000311
is composed of
Figure FDA00033913327500000312
Gamma is not less than 1, and gamma is a normal number, deltaxIs an initial state threshold value, epsilon is a terminal state threshold value, and delta is more than 0x< ε, and δxAnd ε are both normal numbers, N is a known positive integer;
calculating a state estimation gain matrix L by formula (8);
Figure FDA00033913327500000313
calculating a state feedback gain matrix K through a formula (9);
Figure FDA00033913327500000314
and step six, substituting the state estimation gain matrix L obtained in the step five into the state prediction model in the step two, and substituting the state feedback gain matrix K into the state feedback controller in the step three, so as to realize the limited time stabilization of the networked control system with network-induced bounded time lag and data loss.
2. The time-limited stabilization method with network-induced bounded skew and data loss of claim 1, wherein the Lyapunov stability theorem in step five is:
V1(x(t+1))＜γV1(x(t)) (10)
wherein the content of the first and second substances,
V1(x(t))＝xT(t)P1x(t) (11)
in the formula, V1(x (t)) is the Lyapunov function at time t, V1(x (t +1)) is the Lyapunov function at time t +1, xT(t) is the transpose of x (t), P1Is a Lyapunov symmetric positive definite matrix.
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