CN111736465A - Wireless cloud control system scheduling method and system - Google Patents
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Abstract
The invention discloses a wireless cloud control system scheduling method and system, which comprises the steps of respectively establishing a system model according to each actual controlled object; configuring a cloud computing platform, and determining a prediction time domain of a model prediction control method, and requirements of successful transmission probability of a controller and a sensor; adjusting the sending probability of all sensors; determining a sending threshold according to the sending probability of each sensor; calculating an optimal control sequence according to whether respective sensor data is successfully received or not, and preparing to send the optimal control sequence to respective actuators through control channels; the central scheduler selects a controller to access the control channel; and each executor updates the content of the buffer according to whether the control sequence is successfully received or not, and selects a control quantity from the buffer for execution. The invention can make the sending probability of all sensors reach the high-efficiency Nash equilibrium point, save the sending power consumption of the sensors, avoid the collision of control signals among control channels and ensure the stability of all control systems.
Description
Technical Field
The invention relates to the technical field of control, in particular to a wireless cloud control system scheduling method and system.
Background
The wireless cloud control system is a control system for transmitting sensing data and control data through a wireless communication network, and a local controller is transferred to the cloud. With the rapid development of cloud computing that "centralizes" computing resources and wireless communication technologies such as 5G capable of reliable data transmission with ultra-low latency, wireless cloud control systems (which may include a large number of independent control systems) have received increasing attention in recent years from both academic and industrial industries. Existing examples of wireless cloud control systems can be found in drone surveillance, underwater navigation, etc., and are expected to find further applications in the near future.
The wireless cloud control system has unique functions. First, a wireless cloud control system usually has a plurality of independent control systems; second, these control systems must competitively use the wireless communication channel to transmit data; finally, all controllers use shared cloud computing resources and compute control signals at the cloud.
These unique functions present unique challenges to the design and analysis of wireless cloud control systems, one of which is efficient channel access scheduling for all control systems. In practice, individual control systems may greedily access the communication channels, because more data transfers for each control system generally means better control system performance. However, the capacity of the wireless channel is always limited, which means that the above-mentioned demand cannot be satisfied without compensation. Therefore, an effective channel scheduling strategy of the control system is crucial to design the wireless cloud control system.
The traditional centralized or decentralized scheduling strategy is not ideal for the channel access scheduling strategy in the wireless cloud control system, because the global information required by the scheduler in the wireless cloud control system is neither perfect nor nonexistent. In fact, it may not be feasible to install a centralized channel access mechanism in a wireless cloud control system, while installing a decentralized mechanism may be too conservative. Therefore, a new scheduling strategy is needed to take these characteristics of the wireless cloud control system into account.
Disclosure of Invention
The scheduling method of the wireless cloud control system can solve the technical problem that the traditional centralized or distributed scheduling strategy cannot meet the requirement of the wireless cloud control system.
In order to achieve the purpose, the invention adopts the following technical scheme:
a wireless cloud control system scheduling method is based on a cloud computing platform and comprises the following steps:
s100, respectively establishing a system model according to each actual controlled object;
s200, configuring a cloud computing platform according to the established system model to obtain corresponding model prediction controllers and a central scheduler;
s300, determining a prediction time domain of the model prediction control method, and successful transmission probability requirements of a controller and a sensor according to a system model of a controlled object;
s400, adjusting the sending probability of all the sensors according to the successful transmission probability requirement set by each sensor and the successful sending probability of the sensor at the current moment;
s500, determining a sending threshold according to the sending probability of each sensor; when receiving that the current channel gain is higher than the respective threshold, each sensor sends data to the controller;
s600, each controller calculates an optimal control sequence according to whether the sensor data is successfully received or not, and prepares to send the optimal control sequence to each actuator through a control channel;
s700, the central scheduler selects a controller to access the control channel according to the successful transmission probability requirement of the controller determined in S200;
and S800, each executor updates the content of the buffer according to whether the control sequence is successfully received or not, and selects a control quantity from the buffer for execution.
Further, the S100 respectively establishes a system model according to each actual controlled object;
the method specifically comprises the following steps:
setting n controlled objects to respectively establish the following nonlinear systems:
xi(k+1)=fi(xi(k),ui(k),wi(k)),i=1,…,n (1)
wherein: k is the system operation time, and k is more than or equal to 0;state vector at time k;is an input vector;is an external interference vector;is a bounded set and contains an origin; f. ofiIs a continuous function; w is ai(k) Are independent and identically distributed random variables and satisfy
Further, the S200 performs configuration of a cloud computing platform according to the established system model to obtain corresponding model predictive controllers and central schedulers;
wherein the model predictive controller configuration process specifically includes:
first, a state estimator is provided for each system i (i ═ 1, …, n), respectively
WhereinFor the estimated state of system i at time k, ui(k-1) represents the true input of system i at time k-1;
secondly, a model predictive controller is set, specifically, the following constraint optimal control is carried out:
wherein u isi(k)={ui,0(k),…,ui,N-1(k) The decision variables for the optimization problem are,for the estimated state of system i at time k, Li(x, u) and Fi(x) Respectively positive stage cost and terminal cost;
obtaining an optimal solution through optimization problem solving softwareNiIs a prediction time domain;
and finally, designing a buffer to store the control sequence with successful transmission and provide real control input, wherein the specific process is as follows:
wherein b isi(k) For the buffer content of system i at time k, dc,i(k) Being a binary variable, 1 indicates success of transmission of the control sequence of system i and 0 indicates failure of transmission;
Further, the S200 performs configuration of a cloud computing platform according to the established system model to obtain corresponding model predictive controllers and central schedulers;
the central scheduler randomly selects a controller to access the control channel according to the following probability:
wherein p isciFor the selected probability of the central scheduler i,is the successful transmission probability requirement of the sensor i to be designed.
Further, the S300 determines a prediction time domain of the model prediction control method, and successful transmission probability requirements of the controller and the sensor according to the system model of the controlled object;
the method specifically comprises the following steps:
if the successful transmission probability requirement of each sensor is determinedSuccessful transmission probability requirement of each controllerAnd the prediction time domain N of model predictive controli;
wherein, γiSatisfies Fi(fi(x,0,0))≤γiFi(x),μi=infx,uLi(x,u)/Fi(x)。
Further, in the step S400, the sending probabilities of all the sensors are adjusted according to the requirement of the successful transmission probability set by each sensor and the probability of successful sending of the sensor at the current time;
the method specifically comprises the following steps:
determining that each sensor i (i ═ 1, …, n) sends a status x to the central controller at time ki(k) The probability of (d); the transmission probability is adjusted according to the following formula:
wherein the initial value is set topsi(k) β∈ [0,1) is a weighting factor,the requirement of successful transmission probability of the sensor i needs to be set in advance, and q issi(psi(k),ps,-i(k) Is the probability of successful transmission, p, of sensor i at time ks,-i(k) The transmission probability of other sensors except the sensor i at the moment k is obtained; q. q.ssi(psi(k),ps,-i(k) Is estimated approximately by counting the ratio of the number of successful transmissions to the total number of transmissions.
Further, in step S500, a sending threshold is determined according to the sending probability of each sensor; when receiving that the current channel gain is higher than the respective threshold, each sensor sends data to the controller;
the method specifically comprises the following steps:
set threshold &siThen sensor i sends with probability 1 the channel gain that is and is only currently obtainedsi(k) Greater than &si;
And the threshold channel gain is obtained by the following equation:
wherein p issi(k) Obtained from the formula (3) osi(h) Is a channel gain probability density function of the measurement channel.
Further, the S700 hub scheduler selects a controller to access the control channel according to the successful transmission probability requirement of the controller determined in S200;
wherein,
the selection rule is as follows: defining a random variable theta, wherein theta is i and represents that the ith controller is selected; then a random number e (k) between 0,1 is generated uniformly at time k and the controller is selected accordingly:
on the other hand, the invention also discloses a wireless cloud control system scheduling system, which comprises n system models, wherein n is a natural number greater than 1; wherein,
each system model comprises a sensor, a model prediction controller, a central scheduler, a buffer and an actuator;
the controlled signal passes through the sensor, the model predictive controller, the central scheduler and the buffer to the actuator in sequence;
further comprising:
the sensor successful transmission probability determining unit is used for determining a prediction time domain of the model prediction control method, a controller and the successful transmission probability requirement of the sensor according to a system model of a controlled object;
a sensor transmission threshold determining unit for determining a transmission threshold according to the transmission probability of each sensor; when receiving that the current channel gain is higher than the respective threshold, each sensor sends data to the sensor;
wherein,
the model predictive controller calculates an optimal control sequence according to whether respective sensor data is successfully received or not, and prepares to send the optimal control sequence to respective actuators through control channels;
the central scheduler selecting one controller to access the control channel according to the determined successful transmission probability requirement of the model predictive controller;
and the executor updates the content of the buffer according to whether the control sequence is successfully received or not, and selects a control quantity from the content of the buffer for execution.
Further, the n system models are represented as follows:
setting n controlled objects to respectively establish the following nonlinear systems:
xi(k+1)=fi(xi(k),ui(k),wi(k)),i=1,…,n (1)
wherein: k is the system operation time, and k is more than or equal to 0;state vector at time k;is an input vector;is an external interference vector;is a bounded set and contains an origin; f. ofiIs a continuous function; w is ai(k) Are independent and identically distributed random variables and satisfy
According to the technical scheme, the wireless cloud control system scheduling method and system provided by the invention provide a dual scheduling strategy under a prediction control framework based on a data packet; the scheduling strategies include a decentralized scheduling strategy of the sensor, a centralized scheduling strategy of the controller, and a model predictive controller based on data packets. The solution allows on the one hand to explicitly take into account status and control constraints and on the other hand to actively compensate for unsuccessfully transmitted information by packet-based transmission, thereby achieving stability of all systems.
The invention provides a dual scheduling strategy based on a data packet framework under a wireless cloud control system with a plurality of controlled objects, reasonably selects a proper successful transmission probability demand and a prediction time domain, can enable the transmission probability of all sensors to reach a high-efficiency Nash equilibrium point, saves the transmission power consumption of the sensors, avoids the collision of control signals among control channels, and ensures the stability of all control systems.
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FIG. 1 is a flow chart of a method of the present invention;
fig. 2 is a schematic diagram of the system of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention.
As shown in fig. 1, the method for scheduling a wireless cloud control system according to this embodiment, based on a dual scheduling policy under a packet prediction control framework, includes the following steps:
s100, respectively establishing corresponding system models according to actual controlled objects;
s200, configuring a cloud computing platform according to the established system model to obtain frames of corresponding model prediction controllers and a central scheduler;
s300, determining a prediction time domain of a model prediction control method and successful transmission probability requirements of a controller and a sensor according to a system model of a controlled object;
s400, adjusting the sending probability of all the sensors according to the successful transmission probability requirement set by each sensor and the successful sending probability of the sensor at the current moment;
s500, determining a sending threshold according to the sending probability of each sensor; when receiving that the current channel gain is higher than the respective threshold, each sensor sends data to the controller;
s600, the controllers of all the systems calculate optimal control sequences according to whether the respective sensor data are successfully received or not, and prepare to send the optimal control sequences to respective actuators through control channels;
s700, the central scheduler randomly selects a controller to access the control channel according to the successful transmission probability requirement of the controller determined in S200;
and S800, updating the content of the buffer by the executors of all the systems according to whether the control sequence is successfully received, and selecting a control quantity from the buffer for execution.
The above steps are explained in detail as follows:
step 1: respectively establishing the following nonlinear systems for the n controlled objects;
xi(k+1)=fi(xi(k),ui(k),wi(k)),i=1,…,n (1)
wherein: k is the system operation time, and k is more than or equal to 0;state vector at time k;is an input vector;is an external interference vector.Is a closed set and contains the origin. f. ofiIs a continuous function.
Step 2: a remote cloud computing platform is configured, including a package-based model predictive controller and a central scheduler.
(1) A packet-based model predictive controller;
first, a state estimator is provided for each system i (i ═ 1, …, n), respectively
WhereinFor the estimated state of system i at time k, ui(k-1) represents the true input of system i at time k-1.
Secondly, a model predictive controller is set, specifically, the following constraint optimal control problem is set:
wherein u isi(k)={ui,0(k),…,ui,N-1(k) The decision variables for the optimization problem are,for the estimated state of system i at time k, Li(x, u) and Fi(x) Respectively positive stage cost and finalEnd cost. The optimal solution is obtained by commonly used optimization problem solving software (such as Matlab)NiIs the prediction time domain.
Finally, a buffer is required to be designed to store the control sequence of successful transmission and provide the actual control input, and the specific process is as follows:
wherein b isi(k) For the buffer content of system i at time k, dc,i(k) Being a binary variable, 1 indicates that the control sequence of system i was successfully transmitted and 0 indicates that the transmission failed.
And step 3: determining a successful transmission probability requirement for each sensorSuccessful transmission probability requirement of each controllerAnd the prediction time domain N of model predictive controli。
wherein, γiSatisfies Fi(fi(x,0,0))≤γiFi(x),μi=infx,uLi(x,u)/Fi(x)。
And 4, step 4: determining that each sensor i (i ═ 1, …, n) sends a status x to the central controller at time ki(k) The probability of (c). The transmission probability is adjusted according to the following formula
Wherein the initial value is set topsi(k) β∈ [0,1) is a weighting factor,for the successful transmission probability requirement of sensor i, the setting, q, needs to be implementedsi(psi(k),ps,-i(k) Is the probability of successful transmission, p, of sensor i at time ks,-i(k) Is the transmission probability at time k for other sensors than sensor i. q. q.ssi(psi(k),ps,-i(k) May be approximated by counting the ratio of the number of successful transmissions to the total number of transmissions.
And 5: and determining whether the sensor i sends or not according to a threshold strategy.
Given thresholdThen sensor i sends a channel gain of if and only if it is currently obtained with probability 1si(k) Greater than &si. And the threshold channel gain may be obtained by the following equation:
wherein p issi(k) Obtained from the formula (3) osi(h) Is a channel gain probability density function of the measurement channel. The above equation can be solved approximately by bisection
The current channel gain needs to be measured before the sensor sends datasi(k) The measurement process can be easily implemented by a short frame pilot signal.
Step 6: all systems i (i ═ 1, …, n) respectively solve model predictive controllers, and the obtainedReady to be sent to actuator i.
And 7: the central scheduler selects an appropriate control sequence to send to the actuators. The selection rule is as follows: defining a random variable θ, θ ═ i denotes selecting the ith controller. Then a random number e (k) between 0,1 is generated uniformly at time k and the controller is selected accordingly:
and 8: the actuators of the respective systems select appropriate control amounts and apply to the systems. The concrete way is by designing the buffer, and the updating of the content of the buffer is exactly the same as (2).
In the following, specific embodiments are discussed in connection with control examples of n mobile robots (n ═ 3):
if xi=[xi,yi,θi]TTo represent the state vector of the i-th robot, ui=[vi,wi]TFor corresponding control inputs, wdiFor the corresponding disturbance, the model of each robot is then a discrete-time nonlinear system as follows:
wherein (x)i,yi) Is the position of the ith robot, thetaiIs a direction, vi,wiLinear velocity and angular velocity, respectively, T ═ 0.6s is the sampling period, wdi(k)=sin(7k)×[0.12 0.14 0.1]TIs an external disturbance.
The model predictive controller and the corresponding central scheduler in the cloud control platform are designed according to the step 2, and related parameters can be respectively determined asAnd Ni=12(i=1,2,3)。
Updating the sending probability of the sensor, whether the sensor sends or not, calculating an optimal control sequence, randomly selecting a central scheduler and selecting a real control quantity can be respectively determined according to the steps 4 to 8.
On the other hand, the embodiment of the invention also discloses a wireless cloud control system scheduling system, which comprises n system models, wherein n is a natural number greater than 1; wherein,
each system model comprises a sensor, a model prediction controller, a central scheduler, a buffer and an actuator;
the controlled signal passes through the sensor, the model predictive controller, the central scheduler and the buffer to the actuator in sequence;
further comprising:
the sensor successful transmission probability determining unit is used for determining a prediction time domain of the model prediction control method, a controller and the successful transmission probability requirement of the sensor according to a system model of a controlled object;
a sensor transmission threshold determining unit for determining a transmission threshold according to the transmission probability of each sensor; when receiving that the current channel gain is higher than the respective threshold, each sensor sends data to the controller;
wherein,
the model predictive controller calculates an optimal control sequence according to whether respective sensor data is successfully received or not, and prepares to send the optimal control sequence to respective actuators through control channels;
the central scheduler selecting one controller to access the control channel according to the determined successful transmission probability requirement of the model predictive controller;
and the executor updates the content of the buffer according to whether the control sequence is successfully received or not, and selects a control quantity from the content of the buffer for execution.
Further, the n system models are represented as follows:
setting n controlled objects to respectively establish the following nonlinear systems:
xi(k+1)=fi(xi(k),ui(k),wi(k)),i=1,…,n (1)
wherein: k is the system operation time, and k is more than or equal to 0;state vector at time k;is an input vector;is an external interference vector;is a bounded set and contains an origin; f. ofiIs a continuous function; w is ai(k) Are independent and identically distributed random variables and satisfy
It is understood that the system provided by the embodiment of the present invention corresponds to the method provided by the embodiment of the present invention, and the explanation, the example and the beneficial effects of the related contents can refer to the corresponding parts in the method.
In summary, the invention provides a dual scheduling strategy based on a data packet framework under a wireless cloud control system with a plurality of controlled objects, reasonably selects a proper successful transmission probability requirement and a prediction time domain, can enable the transmission probability of all sensors to reach an efficient Nash equilibrium point, saves the transmission power consumption of the sensors, avoids the collision of control signals between control channels, and ensures the stability of all control systems.
The above examples are only intended to illustrate the technical solution of the present invention, but not to limit it; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.
Claims (10)
1. A wireless cloud control system scheduling method is based on a cloud computing platform and is characterized in that: the method comprises the following steps:
s100, respectively establishing a system model according to each actual controlled object;
s200, configuring a cloud computing platform according to the established system model to obtain corresponding model prediction controllers and a central scheduler;
s300, determining a prediction time domain of the model prediction control method, and successful transmission probability requirements of a controller and a sensor according to a system model of a controlled object;
s400, adjusting the sending probability of all the sensors according to the successful transmission probability requirement set by each sensor and the successful sending probability of the sensor at the current moment;
s500, determining a sending threshold according to the sending probability of each sensor; when receiving that the current channel gain is higher than the respective threshold, each sensor sends data to the controller;
s600, each controller calculates an optimal control sequence according to whether the sensor data is successfully received or not, and prepares to send the optimal control sequence to each actuator through a control channel;
s700, the central scheduler selects a controller to access the control channel according to the successful transmission probability requirement of the controller determined in S200;
and S800, each executor updates the content of the buffer according to whether the control sequence is successfully received or not, and selects a control quantity from the buffer for execution.
2. The wireless cloud control system scheduling method of claim 1, wherein: the S100 respectively establishes a system model according to each actual controlled object;
the method specifically comprises the following steps:
setting n controlled objects to respectively establish the following nonlinear systems:
xi(k+1)=fi(xi(k),ui(k),wi(k)),i=1,...,n (1)
wherein: k is the system operation time, and k is more than or equal to 0;state vector at time k;is an input vector;is an external interference vector;is a bounded set and contains an origin; f. ofiIs a continuous function; w is ai(k) Are independent and identically distributed random variables and satisfy
3. The wireless cloud control system scheduling method of claim 2, wherein: the S200 carries out configuration of the cloud computing platform according to the established system model to obtain corresponding model prediction controllers and a central scheduler;
wherein the model predictive controller configuration process specifically includes:
first, a state estimator is provided for each system i (i ═ 1.., n), respectively
WhereinFor the estimated state of system i at time k, ui(k-1) represents the true input of system i at time k-1;
secondly, a model predictive controller is set, specifically, the following constraint optimal control is carried out:
wherein u isi(k)={ui,0(k),...,ui,N-1(k) The decision variables for the optimization problem are,for the estimated state of system i at time k, Li(x, u) and Fi(x) Respectively positive stage cost and terminal cost;
obtaining an optimal solution through optimization problem solving softwareNiIs a prediction time domain;
and finally, designing a buffer to store the control sequence with successful transmission and provide real control input, wherein the specific process is as follows:
wherein b isi(k) For the buffer content of system i at time k, dc,i(k) Being a binary variable, 1 represents that the control sequence of system i is successfully transmitted and 0 represents that the transmission is failed;
4. the wireless cloud control system scheduling method of claim 3, wherein: the S200 carries out configuration of the cloud computing platform according to the established system model to obtain corresponding model prediction controllers and a central scheduler;
the central scheduler randomly selects a controller to access the control channel according to the following probability:
5. The wireless cloud control system scheduling method of claim 4, wherein: the S300 determines a prediction time domain of the model prediction control method, and successful transmission probability requirements of the controller and the sensor according to a system model of the controlled object;
the method specifically comprises the following steps:
if the successful transmission probability requirement of each sensor is determinedSuccessful transmission probability requirement of each controllerAnd the prediction time domain N of model predictive controli;
wherein, γiSatisfies Fi(fi(x,0,0))≤γiFi(x),μi=infx,uLi(x,u)/Fi(x)。
6. The wireless cloud control system scheduling method of claim 5, wherein: s400, adjusting the sending probability of all the sensors according to the successful transmission probability requirement set by each sensor and the successful sending probability of the sensor at the current moment;
the method specifically comprises the following steps:
determining that each sensor i (i 1.., n) sends a status x to the central controller at time ki(k) The probability of (d); the transmission probability is adjusted as followsA step of:
wherein the initial value is set topsi(k) β∈ [0,1) is a weighting factor,the requirement of successful transmission probability of the sensor i needs to be set in advance, and q issi(psi(k),ps,-i(k) Is the probability of successful transmission, p, of sensor i at time ks,-i(k) The transmission probability of other sensors except the sensor i at the moment k is obtained; q. q.ssi(psi(k),ps,-i(k) Is estimated approximately by counting the ratio of the number of successful transmissions to the total number of transmissions.
7. The wireless cloud control system scheduling method of claim 6, wherein: the S500 determines a sending threshold according to the sending probability of each sensor; when receiving that the current channel gain is higher than the respective threshold, each sensor sends data to the controller;
the method specifically comprises the following steps:
set threshold &siThen sensor i sends with probability 1 the channel gain that is and is only currently obtainedsi(k) Greater than &si;
And the threshold channel gain is obtained by the following equation:
wherein p issi(k) Obtained from the formula (3) osi(h) Is a channel gain probability density function of the measurement channel.
8. The wireless cloud control system scheduling method of claim 7, wherein: the S700 hub scheduler selecting a controller to access the control channel based on the successful transmission probability requirement of the controller determined in S200;
wherein,
the selection rule is as follows: defining a random variable theta, wherein theta is i and represents that the ith controller is selected; then a random number e (k) between 0,1 is generated uniformly at time k and the controller is selected accordingly:
9. a wireless cloud control system scheduling system which characterized in that:
the method comprises n system models, wherein n is a natural number greater than 1; wherein,
each system model comprises a sensor, a model prediction controller, a central scheduler, a buffer and an actuator;
the controlled signal passes through the sensor, the model predictive controller, the central scheduler and the buffer to the actuator in sequence;
further comprising:
the sensor successful transmission probability determining unit is used for determining a prediction time domain of the model prediction control method, a controller and the successful transmission probability requirement of the sensor according to a system model of a controlled object;
a sensor transmission threshold determining unit for determining a transmission threshold according to the transmission probability of each sensor; when receiving that the current channel gain is higher than the respective threshold, each sensor sends data to the controller;
wherein,
the model predictive controller calculates an optimal control sequence according to whether respective sensor data is successfully received or not, and prepares to send the optimal control sequence to respective actuators through control channels;
the central scheduler selecting one controller to access the control channel according to the determined successful transmission probability requirement of the model predictive controller;
and the executor updates the content of the buffer according to whether the control sequence is successfully received or not, and selects a control quantity from the content of the buffer for execution.
10. The wireless cloud control system scheduling system of claim 9, wherein:
the n system models are represented as follows:
setting n controlled objects to respectively establish the following nonlinear systems:
xi(k+1)=fi(xi(k),ui(k),wi(k)),i=1,,...,n (1)
wherein: k is the system operation time, and k is more than or equal to 0;state vector at time k;is an input vector;is an external interference vector;is a bounded set and contains an origin; f. ofiIs a continuous function; w is ai(k) Are independent and identically distributed random variables and satisfy
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