CN110198236B - Networked system robust control method based on dynamic event trigger mechanism - Google Patents

Networked system robust control method based on dynamic event trigger mechanism Download PDF

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CN110198236B
CN110198236B CN201910437083.3A CN201910437083A CN110198236B CN 110198236 B CN110198236 B CN 110198236B CN 201910437083 A CN201910437083 A CN 201910437083A CN 110198236 B CN110198236 B CN 110198236B
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dynamic event
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俞立
田栩帆
张丹
周京
陈鹏程
赵磊
廖义辉
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Zhejiang University of Technology ZJUT
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
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    • H04—ELECTRIC COMMUNICATION TECHNIQUE
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Abstract

A networked system robust control method based on a dynamic event trigger mechanism comprises the following steps: the method comprises the steps of firstly, modeling an inverted pendulum networked control problem into a state space model with uncertain parameters by considering network delay; secondly, obtaining sufficient conditions that the system is asymptotically stable and meets robust performance indexes based on a Lyapunov stability theory and a linear matrix inequality technology; and thirdly, providing a design method of the state feedback controller under the dynamic event trigger mechanism. The invention provides a networked system robust control method based on a dynamic event trigger mechanism, which can reduce the network data transmission times and enhance the system robust performance.

Description

Networked system robust control method based on dynamic event trigger mechanism
Technical Field
The invention belongs to the field of networked control, and designs a method which is applied to networked control, reduces network load and energy consumption, reduces the times of data transmission, stabilizes a system and can enhance the robustness of the system.
Background
Nowadays, with the rapid development and wide application of network technology, the trend of the control system towards networking, distribution, intellectualization and synthesis is increasingly shown. In a networked control system, the various elements of the system are connected by a common network, and signals are transmitted and exchanged over a communications network. The communication mode of information interaction through the network also provides new opportunities and challenges for analysis and design of the networked control system, such as problems of network delay, bandwidth limitation, signal quantization, disturbance, packet loss rate and the like, wherein the most prominent problems are network delay and limited bandwidth of a communication channel. Depending on the type of network, this delay may be constant, time-varying or random, but the presence of any delay may degrade the performance of the system or even cause the system to be unstable, making it difficult to apply the conventional control theory and method directly to the research of the network control system.
In the last 90 s of the century, event-based ideas were first applied to engine control. Many articles state the advantages of event-based control. It is noted that early event-triggered control is so-called continuous event triggering, requiring special hardware for continuous monitoring of the current state. To overcome this problem, Heemels proposes periodic event triggering. An important issue to be noted in event triggering is that a minimum time interval between any two event execution time points needs to be ensured, i.e. the minimum event interval time is strictly greater than zero. To address this problem, Yue proposes event triggering based on sampled data. Both periodic event triggering and event triggering based on sampled data belong to discrete event triggering, and there are some documents on stability analysis and controller design methods for discrete event triggering. With the increasing degree of system informatization, the control system scale is continuously enlarged, and in order to reduce the communication pressure between systems, more and more scholars pay attention to distributed event-triggered control and distributed event-triggered control of large-scale systems. Girard proposes a dynamic event trigger mechanism that can increase the minimum event interval time, even close to the allowable maximum transmission interval, compared to static event triggers.
However, the existing dynamic time mechanism cannot be applied to a system with uncertain parameters, and cannot solve the problem of network delay at the same time. In view of the limitation of the existing results, a dynamic event trigger mechanism suitable for a networked control system with uncertain parameters and network delay is provided, so that the system can be stabilized and the network data transmission times can be reduced.
Disclosure of Invention
In order to overcome the defects that the conventional networked control method cannot be applied to a system with uncertain parameters and has poor robustness, the invention introduces a dynamic event trigger mechanism, provides a networked system robust control method of the dynamic event trigger mechanism with uncertain parameters, and provides a state feedback controller design method under the dynamic event trigger mechanism based on a Lyapunov stability theory and a linear matrix inequality technique.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a networked system robust control method based on a dynamic event trigger mechanism comprises the following steps:
firstly, establishing a dynamic mathematical model of an inverted pendulum system:
Figure BDA0002070836520000021
the inverted pendulum system comprises a trolley, a pendulum rod connected to the top of the trolley and resistance for the trolley; in the formula, M is the mass of the trolley, M is the mass of the oscillating bar, l is the length from the rotating axis of the oscillating bar to the mass center, x represents the displacement of the trolley, theta is the included angle between the oscillating bar and the vertical downward direction, F is the external force received by the trolley, and b is the friction coefficient of the trolley;
establishing a state space model: selecting the position x and the speed of the trolley
Figure BDA0002070836520000027
Pendulum rod angle theta and pendulum rod angular speed
Figure BDA0002070836520000028
The four quantities are state quantities, and the state space equation is obtained as follows:
Figure BDA0002070836520000022
Figure BDA0002070836520000023
and secondly, constructing an inverted pendulum robust controller with uncertain parameters and network delay, wherein the process comprises the following steps:
2.1) dynamic event triggering mechanism as follows:
Figure BDA0002070836520000024
wherein: eta (t) is an event-triggered dynamic variable, and
Figure BDA0002070836520000025
Figure BDA0002070836520000026
is a positive definite matrix, j is 1,2,3, x (kh) is the trigger time vector, and x (k + j) h is the state vector.
2.2) considering that the parameters of the inverted pendulum system are uncertain, considering that the inverted pendulum system under network control has time delay, establishing the following system model by using the inverted pendulum model:
Figure BDA0002070836520000031
wherein: x (t) is belonged to RnFor the system state vector, u (t) is e.g. RmFor the system input vector, w (t) is e.g. RpFor the disturbance input of the system, y (t) ∈ RrOutputs vectors, A, B, C, B, for the systemwIs a parameter matrix of the corresponding dimension, Δ A, Δ B are norm-bounded parameter matrices, and
[ΔA ΔB]＝HF(t)[E1 E2] (5)
wherein: h, E1，E2Is a matrix of appropriate dimensions, and F (t) is an unknown matrix;
2.3) define a delay function:
Figure BDA0002070836520000032
defining an error signal e based on dynamic event triggering conditionsk(t)。
Figure BDA0002070836520000033
Wherein i is 1,2,3 … d-1; tau iskDelaying the system by tauM＝max{τkIs the maximum value of the time delay;
2.4) converting the initial dynamic event trigger condition into a dynamic event trigger form with delay:
ek(t)＝x(tkh)-x(t-τ(t)) (8)
Figure BDA0002070836520000034
the system model is converted into a system model with delay:
Figure BDA0002070836520000035
the third step: designing a constraint condition matrix of the controller with robust stability performance by adopting a Lyapunov stability analysis method according to the delay model:
the following theorem is given:
and (4) Lemma: for matrices R > 0 and XTX is-XR-1X≤ε2R-2 ε X, wherein ε is any constant;
studying the dynamic event triggering problem with robustness H ∞, given a disturbance attenuation coefficient γ, a state feedback controller is designed such that a system (10) that satisfies the dynamic event triggering mechanism (9) satisfies the following two requirements:
4.1) the closed-loop system (10) under w (t) ≡ 0 is robust index stable;
4.2) at zero initial conditions, for any non-zero w (t) e L2[0, ∞), the controller output z (t) all satisfy | | z (t) | luminance2≤γ||w(t)||2；
The lyapunov generalized method is established, and the following conclusion is obtained.
For a given parameter γ, υ,
Figure BDA0002070836520000041
And a parameter mu > 0, the system (7) under the trigger mechanism (9) and the feedback gain K ═ YX-1Robust exponential stability under H infinity norm boundary gamma, if any
Figure BDA0002070836520000042
Such that the following inequality holds:
Figure BDA0002070836520000043
Figure BDA0002070836520000044
Figure BDA0002070836520000045
Figure BDA0002070836520000046
Figure BDA0002070836520000047
Figure BDA0002070836520000048
Figure BDA0002070836520000049
Θ51＝[CX DY 0 DY]
Figure BDA0002070836520000051
Figure BDA0002070836520000052
selecting upsilon, gamma, epsilon,
Figure BDA0002070836520000053
Mu and tauMThen, the LMI toolbox is used for solving to obtain a feedback matrix K and trigger condition parameters
Figure BDA0002070836520000054
Further, in 2.2), the uncertain reasons of the parameters comprise neglecting nonlinear dynamics, mass and rod length measurement inaccuracy and pendulum rod flexibility.
The technical conception of the invention is as follows: firstly, considering the influence of time delay and network bandwidth, a traditional time period triggering mechanism is changed into a dynamic event triggering mechanism method; then, modeling the closed-loop system into a time-lag model of a parameter uncertain system, wherein the model is based on the Lyapunov stability theory and the linear matrix inequality technology and describes the mutual restriction relation between a dynamic event trigger mechanism, the communication network performance and the system stability; finally, a design method of the state feedback controller under the dynamic event trigger mechanism is provided.
The invention has the following beneficial effects: the dynamic event-triggered approach will reduce the number of "unnecessary" sampled signals sent over the network, which will result in high bandwidth usage of the communication. (1) The dynamic event-triggered control scheme may reduce computing resource, battery device energy and communication resource usage, reduce sensor release time and network communication burden. (2) A dynamic event trigger mechanism is introduced, so that trigger conditions are more diversified and operable, and the number of events is reduced better than that of event triggers. (3) Considering the uncertainty of the parameters of the system increases the robustness of the system. (4) The network delay problem is considered, and the method is more suitable for practical application. .
Drawings
Fig. 1 is a first-order inverted pendulum diagram.
Fig. 2 is a networked system robust control system model of a dynamic event triggering mechanism.
Fig. 3 is a network system triggering situation under the event triggering mechanism.
Fig. 4 is a network system triggering situation under the dynamic event triggering mechanism.
Fig. 5 is an inverted pendulum motion state under an event trigger mechanism.
Fig. 6 shows the motion state of the inverted pendulum under the dynamic event trigger mechanism.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1 and 2, a flow chart of networked system robust control of an inverted pendulum model and representative values of various parameters and a dynamic event triggering mechanism is respectively described.
A networked system robust control method based on a dynamic event trigger mechanism comprises the following steps:
step 1: measuring the mass M of the trolley to be 1.096kg, the mass M of the small ball to be 0.196kg and the length L of the swing rod to be 0.25M to obtain a state space model of the inverted pendulum and obtain a corresponding parameter matrix
Figure BDA0002070836520000061
C，BW(ii) a Order to
Figure BDA0002070836520000062
And combined with [ Delta A Delta B]＝HF(t)[E1 E2]；
Figure BDA0002070836520000063
Figure BDA0002070836520000064
Figure BDA0002070836520000065
Figure BDA0002070836520000066
Figure BDA0002070836520000067
Step 2: considering network time delay, the general model of the inverted pendulum is converted into a time delay model with uncertain parameters, the dynamic event trigger condition is converted into a dynamic event trigger condition with delay, and the tau is enabled to beM＝0.0014；
And step 3: designing a constraint condition matrix of networked system robust control with a dynamic event triggering mechanism by adopting a Lyapunov stability analysis method according to a delay model;
and 4, step 4: solving through an LMI tool kit to obtain a feedback matrix K and trigger condition parameters
Figure BDA0002070836520000068
Let e be 0.53, γ be 200, upsilon be 20,
Figure BDA0002070836520000069
mu is 0.1 to get:
K＝[9.7786 14.0975 -76.3929 -13.6949]
Figure BDA0002070836520000071
and 5: the feedback matrix K and the triggering condition parameters are obtained through the steps
Figure BDA0002070836520000072
The networked system robust control method using the dynamic event trigger mechanism is used for simulating the inverted pendulum system, and simultaneously comparing the simulation result with the event triggerAnd (5) controlling the networked inverted pendulum. As can be seen from the comparison of fig. 3 to fig. 6, the dynamic event trigger has a great advantage in the number of triggers without affecting the stability of the system.

Claims (2)

1. A networked system robust control method based on a dynamic event trigger mechanism is characterized by comprising the following steps:
firstly, establishing a dynamic mathematical model of an inverted pendulum system:
Figure FDA0003499582960000011
the inverted pendulum system comprises a trolley, a pendulum rod connected to the top of the trolley and resistance for the trolley; in the formula, M is the mass of the trolley, M is the mass of the oscillating bar, l is the length from the rotating axis of the oscillating bar to the mass center, x represents the displacement of the trolley, theta is the included angle between the oscillating bar and the vertical downward direction, F is the external force received by the trolley, and b is the friction coefficient of the trolley;
establishing a state space model: selecting the position x and the speed of the trolley
Figure FDA0003499582960000017
Pendulum rod angle theta and pendulum rod angular velocity
Figure FDA0003499582960000018
The four quantities are state quantities, and the state space equation is obtained as follows:
Figure FDA0003499582960000012
Figure FDA0003499582960000013
and secondly, constructing an inverted pendulum robust controller with uncertain parameters and network delay, wherein the process is as follows:
2.1) dynamic event triggering mechanism as follows:
Figure FDA0003499582960000014
wherein: eta (t) is a dynamic variable triggered by an event, 0 < eta (t) is less than or equal to 1,
Figure FDA0003499582960000015
Figure FDA0003499582960000016
is a positive definite matrix, j is 1,2,3, x (kh) is the trigger time vector, x (k + j) h is the state vector;
2.2) considering that the parameters of the inverted pendulum system are uncertain, considering that the inverted pendulum system under network control has time delay, establishing the following system model by using the inverted pendulum model:
Figure FDA0003499582960000021
wherein: x (t) is belonged to RnFor the system state vector, u (t) is e.g. RmFor the system input vector, w (t) is e.g. RpFor the disturbance input of the system, y (t) ∈ RrOutputs vectors, A, B, C, B, for the systemwIs a parameter matrix of the corresponding dimension, Δ A, Δ B are norm-bounded parameter matrices, and
[ΔA ΔB]＝HF(t)[E1 E2] (5)
wherein: h, E1，E2Is a matrix of appropriate dimensions, and F (t) is an unknown matrix;
2.3) define a delay function:
Figure FDA0003499582960000022
defining an error signal e based on dynamic event triggering conditionsk(t)：
Figure FDA0003499582960000023
Wherein i is 1,2,3 … d-1; tau iskDelaying the system by tauM＝max{τkIs the maximum value of the time delay;
2.4) converting the initial dynamic event trigger condition into a dynamic event trigger form with delay:
ek(t)＝x(tkh)-x(t-τ(t)) (8)
Figure FDA0003499582960000024
the system model is converted into a system model with delay:
Figure FDA0003499582960000025
the third step: designing a constraint condition matrix of the controller with robust stability performance by adopting a Lyapunov stability analysis method according to the delay model:
the following theorem is given:
and (4) Lemma: for matrices R > 0 and XTX is-XR-1X≤ε2R-2 ε X, wherein ε is an arbitrary constant;
studying the event triggering problem with a robust H ∞, given a disturbance attenuation coefficient γ, a state feedback controller is designed such that a system (10) that satisfies a dynamic event triggering mechanism (9) satisfies the following two requirements:
4.1) the closed-loop system (10) under w (t) ≡ 0 is robust index stable;
4.2) at zero initial conditions, for any non-zero w (t) e L2[0, ∞), the controller outputs z (t) all satisfy | | z (t) | luminance2≤γ||w(t)||2；
The method for establishing the Lyapunov general function obtains the following conclusion:
for a given parameter γ, υ,
Figure FDA0003499582960000031
And a parameter mu > 0, the system (7) under the trigger mechanism (9) and the feedback gain K ═ YX-1Robust exponential stability under H infinity norm boundary gamma, if any
Figure FDA0003499582960000032
The appropriate dimension of Y is such that the following inequality holds:
Figure FDA0003499582960000033
Figure FDA0003499582960000034
Figure FDA0003499582960000035
Figure FDA0003499582960000036
Figure FDA0003499582960000037
Figure FDA0003499582960000038
Figure FDA0003499582960000039
Θ51＝[CX DY 0 DY]
Figure FDA00034995829600000310
Figure FDA0003499582960000041
selecting upsilon, gamma, epsilon,
Figure FDA0003499582960000042
Mu and tauMThen, the LMI toolbox is used for solving to obtain a feedback matrix K and trigger condition parameters
Figure FDA0003499582960000043
2. The networked system robust control method based on the dynamic event trigger mechanism as claimed in claim 1, wherein in 2.2), the uncertain causes of parameters include neglecting of non-linear dynamics, mass and rod length measurement inaccuracy and pendulum rod flexibility.
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