Time-varying formation tracking control method based on dynamic average consistency

Abstract

The invention discloses a time-varying formation tracking control method based on dynamic average consistency. The method comprises the following steps: establishing a dynamic model of a swarm intelligence system, a target dynamic model and a corresponding system communication topology; according to the established dynamic model of the swarm intelligence system and the target dynamic model, obtaining an expected time-varying formation, and constructing an event triggering condition; determining a time-varying formation tracking control strategy based on dynamic average consistency according to event triggering conditions; and the intelligent agent of the swarm intelligent system is controlled by using the time-varying formation tracking control strategy, so that an expected time-varying formation can be formed, and a convex hull formed by multiple dynamic targets can be tracked. According to the method, all the agents determine a proper formation reference trajectory through comment neutralization, so that a more general exploration relationship between the agents and the target is realized, the requirement on the exploration capability of a single agent is further reduced, and the enthusiasm of all the agents on target exploration is aroused.

Description

Technical Field

[0001] The present invention relates to the fields of unmanned aerial vehicle cooperative control, multi-robot operation, aerospace, etc., and in particular to a time-varying formation tracking control method based on dynamic average consistency. Background Art

[0002] With the rapid development of unmanned systems, multi-agent systems play an increasingly important role in various fields. Multi-agent formation control refers to a group of agents forming a specific formation structure and achieving predetermined mission objectives through cooperation and coordination. Multi-agent adaptive time-varying formation tracking control technology has been widely studied and applied in recent years.

[0003] In the existing multi-target time-varying formation tracking control method, at least one quasi-central node is required as a well-informed agent to obtain information about all targets, while other uninformed agents cannot explore any targets. In practical applications, well-informed agents usually carry expensive communication and detection equipment, and are easily treated as high-value targets and attacked first, with poor substitutability and robustness.

[0004] In addition, for the multi-target time-varying formation tracking control problem under general linear systems, the agents have the same dynamic model and state dimension. The Kronecker product can be used to directly expand the dimension of the cluster system, and then obtain the feasibility conditions for system stability. However, in practical applications, agents with heterogeneous dynamics are more common. Each individual not only has differences in the parameters of the dynamic model, but also may have different dimensions of the state variables, and the physical meaning of each state may also be different. For example, unmanned vehicles and unmanned boats move in two dimensions, while drones move in three dimensions. For complex individuals such as manned / unmanned fighters, missiles, unmanned vehicles and unmanned boats, typical first-order and second-order models are difficult to accurately characterize their dynamics. Models of more than three orders may be required to describe their dynamic characteristics. Therefore, research on high-order models is more meaningful. Traditional linear homogeneous analysis methods are no longer applicable and cannot provide effective solutions for complex tasks across regions and multiple dimensions. In addition, the full state information of each individual is difficult to be fully observed. The difficulty and cost of observing the output information are much smaller than the full state. It is difficult for heterogeneous systems to achieve formation control in the full state space. Summary of the invention

[0005] Purpose of the invention: The purpose of the present invention is to provide a time-varying formation tracking control method based on dynamic average consistency, which solves the time-varying formation tracking control problem under general exploration relationship in the case of event-triggered discrete communication for heterogeneous linear systems, so that the intelligent agent can explore a partial number of targets according to its own capabilities.

[0006] In order to achieve the above object, the present invention provides a time-varying formation tracking control method based on dynamic average consistency, comprising the following steps:

[0007] S1. Establish the dynamic model, target dynamic model and corresponding system communication topology of the swarm intelligence system;

[0008] S2. According to the established dynamics model of the swarm intelligence system and the target dynamics model, the desired time-varying formation is obtained, and event triggering conditions are constructed;

[0009] S3. According to the event triggering conditions, the time-varying formation tracking control strategy is determined based on the dynamic average consistency;

[0010] S4. Use the time-varying formation tracking control strategy to control the intelligent agents of the swarm intelligence system to form the desired time-varying formation and track the convex hull formed by multiple dynamic targets.

[0011] A further preferred technical solution of the present invention is that the swarm intelligence system described in step S1 includes N agents for tracking M targets.

[0012] Preferably, the dynamic model of the swarm intelligence system established in step S1 is:

[0013]

[0014] y i (t) = C i x i (t), i=1, 2, ..., N;

[0015] in, represents the state of agent i, represents the control input of agent i, represents the output of agent i, and are constant matrices, representing the state transfer matrix of agent i, the control input matrix of agent i, and the output matrix of agent i respectively; (A i , B i ) is controllable, and

[0016] Where n i Represents the dimension of the state vector, m i represents the dimension of the control input; All represent real vectors, where n and m represent dimensions. Respectively represent n i ×n i Real matrix of order and n i dimensional real vector;

[0017] The target dynamics model established in step S1 is:

[0018]

[0019] in represents the state of target k, f k is the reference input of the target, expressed as represents the set of positive real numbers; is a constant matrix, and rank(B)=m.

[0020] Preferably, the specific method for obtaining the desired time-varying formation in step S2 is:

[0021] For a given matrix B i , choose a suitable non-singular matrix Make And meet Where T means transposing the matrix. I n Respectively represent m i The identity matrix of order n and order n;

[0022] At the same time, the formation feasibility conditions must also be met The desired formation tracking condition constructed is:

[0023]

[0024] in is the time-varying formation vector, is the piecewise continuously differentiable formation offset; ω k is a positive number, satisfying is a constant matrix;

[0025] When achieving the time-varying formation tracking control goal, the states of the N agents agree on the formation reference, i.e., the convex hull of the M target states, and maintain their time-varying offset

[0026] when Sometimes, there is That is, the linear combination of multiple target positions is located at the center of the multi-agent formation, then Defined as the desired time-varying formation to encircle or pursue M targets.

[0027] Preferably, the specific method of constructing the event triggering condition in step S2 is:

[0028] The estimation error of the convex hull formed by the estimator is set to in is the sampling information, is the kth triggering moment;

[0029] Then the estimator p i Design trigger function:

[0030]

[0031] Where c1>0, c2>0 is the coupling strength, Δ i is the degree of agent i, K is the feedback gain matrix; J i , ε i is any positive constant;

[0032] If the trigger condition is met Then p i (t) updates its current state and communicates the current state to its neighbors, is reset to 0; when the estimator obtains the latest status information of its neighbors, the estimator updates its own status;

[0033] Design the following trigger mechanism for agent i:

[0034]

[0035] in

[0036] In the formula, is the measurement error, ξ i (t) = y i (t)-h i (t), For the trigger moment, is a positive constant; when the trigger mechanism is satisfied When , the agent updates its state and transmits the output information to its neighbors. is reset to 0.

[0037] Preferably, in step S3, according to the event triggering condition, the time-varying formation tracking control strategy determined based on the dynamic average consistency is specifically:

[0038]

[0039] in is the Lumberg observer, used to observe the state x of the agent i ; Matrix L oi Satisfy A i +L oi C i It was Hurwitz's;

[0040] u i A controller, comprising a relative output and a first integral and a second integral of the relative output; Compensation input for formation tracking, satisfying is the local feedback gain and satisfies

[0041] It was Hurwitz's;

[0042] d i ,q i is the auxiliary variable of agent i, The purpose is to use proportional integral feedback to eliminate the communication of the auxiliary state; δ1>0, δ2>0 are coupling gains;

[0043] p i is an estimator used to estimate the convex hull formed by multiple dynamic targets; s i is the internal state of the estimator and satisfies w ik represents the exploration relationship of agent i to target k; when the agent is a partially informed agent Where N1 represents the number of partially informed agents, which can at least obtain information about one target. When an agent is not well-informed, m i =0, where the number of uninformed agents is N-N1, and they cannot obtain any information about their targets.

[0044] is the sampling information, represents the kth triggering moment;

[0045] is the coupling strength, in, f k (r k (t), t) is bounded; λ2 is the smallest non-zero eigenvalue of the Laplace matrix L;

[0046] K is the feedback gain matrix, which is obtained by solving the algebraic Riccati equation A T P+PA-PBB T P+Q=0, where Q>0, P>0, select K=-B T P;

[0047] γ 1i , γ 2i and is the solution to the following linear matrix equation:

[0048]

[0049] Beneficial effects: Based on dynamic average consistency, the present invention introduces partially informed agents, and all agents determine the appropriate formation reference trajectory through opinion neutralization, thereby realizing a more general exploration relationship between agents and targets, namely one-to-one, one-to-many, and many-to-one, further reducing the requirements for the exploration ability of a single agent and mobilizing the enthusiasm of all agents in target exploration. In addition, the present invention adopts an output regulation control method for heterogeneous linear multi-agent systems, and designs a multi-target time-varying formation tracking control method under the condition of event-triggered discrete communication.

[0050] In the present invention, all agents determine the appropriate formation reference trajectory through opinion neutralization, thereby realizing a more general exploration relationship between the agent and the target, namely one-to-one, one-to-many and many-to-one, further reducing the burden of a single agent in target exploration;

[0051] In practical applications, continuous communication between agents often consumes a lot of resources, which is not conducive to the execution of long-term tasks. This paper introduces an event trigger mechanism to deal with the time-varying formation tracking control problem. Compared with the algorithm of the continuous communication method, the proposed event-triggered time-varying formation tracking algorithm effectively reduces the communication frequency and optimizes system performance and resource utilization;

[0052] Since the full state information of each individual is difficult to be fully observed, the difficulty and cost of observing the output information is much lower than the full state, and it is difficult for heterogeneous systems to achieve formation control in the full state space. In many practical applications, it is not necessary for the system to form a full-state formation, so the research on output formation is more practical. The present invention focuses on output formation control, which reduces the difficulty and cost of observing full-state information. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 It is a flow chart of the time-varying formation tracking control method based on dynamic average consistency of the present invention;

[0054] Figure 2 A schematic diagram of the communication topology between the swarm intelligence system and the target in an embodiment of the present invention;

[0055] Figure 3 A schematic diagram of the state changes of the agent and the target during the time-varying formation tracking implementation in an embodiment of the present invention;

[0056] Figure 4 Schematic diagram of the estimation error of the target convex hull by the intelligent agent in the process of realizing time-varying formation tracking of the UAV in an embodiment of the present invention;

[0057] Figure 5 Schematic diagram of formation error in the process of time-varying formation tracking of UAVs in an embodiment of the present invention;

[0058] Figure 6 Schematic diagram of output formation error in the process of time-varying formation tracking of UAVs in an embodiment of the present invention;

[0059] Figure 7 Schematic diagram of the triggering time of the estimator in the control process of the drone in an embodiment of the present invention;

[0060] Figure 8 Schematic diagram of the triggering moment of the intelligent agent in the drone control process in an embodiment of the present invention. DETAILED DESCRIPTION

[0061] In order to make the purpose, technical solution and advantages of the present invention clearer, the technical solution of the present invention will be clearly and completely described below in conjunction with the drawings in the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments, and they should not be understood as limitations on the present invention. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention. In the description of the present invention, it should be understood that the terms used are only for descriptive purposes and cannot be understood as indicating or implying relative importance.

[0062] Combine the following Figure 1-Figure 8 The present invention describes a time-varying formation tracking control method based on dynamic average consistency.

[0063] Embodiment: This embodiment provides a time-varying formation tracking control method based on dynamic average consistency, comprising the following steps:

[0064] S1. Establish the dynamic model, target dynamic model and corresponding system communication topology of the swarm intelligence system;

[0065] S2. According to the established dynamics model of the swarm intelligence system and the target dynamics model, the desired time-varying formation is obtained, and event triggering conditions are constructed;

[0066] S3. According to the event triggering conditions, the time-varying formation tracking control strategy is determined based on the dynamic average consistency;

[0067] S4. Use the time-varying formation tracking control strategy to control the intelligent agents of the swarm intelligence system to form the desired time-varying formation and track the convex hull formed by multiple dynamic targets.

[0068] Specifically, this embodiment establishes a swarm intelligence system with N agents for tracking M targets.

[0069] The dynamic model of the swarm intelligence system established is:

[0070]

[0071] y i (t) = C i x i (t), i=1, 2, ..., N;

[0072] in, represents the state of agent i, represents the control input of agent i, represents the output of agent i, and are constant matrices, representing the state transfer matrix of agent i, the control input matrix of agent i, and the output matrix of agent i respectively; (A i , B i ) is controllable, and

[0073] where n i Represents the dimension of the state vector, m i represents the dimension of the control input; All represent real vectors, where n and m represent dimensions. Respectively represent n i ×n i Real matrix of order and n i dimensional real vector.

[0074] The target dynamics model established is:

[0075]

[0076] in represents the state of target k, f k is the reference input of the target, expressed as represents the set of positive real numbers; is a constant matrix, and rank(B)=m.

[0077] The specific method for obtaining the desired time-varying formation in step S2 is:

[0078] For a given matrix B i , choose a suitable non-singular matrix Make And meet Where T means transposing the matrix. I n Respectively represent m i The identity matrix of order n and order n;

[0079] At the same time, the formation feasibility conditions must also be met The desired formation tracking condition constructed is:

[0080]

[0081] in is the time-varying formation vector, is the piecewise continuously differentiable formation offset; ω k is a positive number, satisfying is a constant matrix;

[0082] When achieving the time-varying formation tracking control goal, the states of the N agents agree on the formation reference, i.e., the convex hull of the M target states, and maintain their time-varying offset

[0083] when Sometimes, there is That is, the linear combination of multiple target positions is located at the center of the multi-agent formation, then Defined as the desired time-varying formation to encircle or pursue M targets.

[0084] The specific method of constructing event trigger conditions is:

[0085] The estimation error of the convex hull formed by the estimator is set to in is the sampling information, is the kth triggering moment;

[0086] Then the estimator p i Design trigger function:

[0087]

[0088] Where c1>0, c2>0 is the coupling strength, Δ i is the degree of agent i, K is the feedback gain matrix; J i , ε i is any positive constant;

[0089] If the trigger condition is met Then pi(t) updates its current state and passes the current state to its neighbors, is reset to 0; when the estimator obtains the latest status information of its neighbors, the estimator updates its own status;

[0090] Design the following trigger mechanism for agent i:

[0091]

[0092] in

[0093] In the formula, is the measurement error, ξ i (t) = y i (t)-h i (t), For the trigger moment, is a positive constant; when the trigger mechanism is satisfied When , the agent updates its state and transmits the output information to its neighbors. is reset to 0.

[0094] In step S3, according to the event triggering condition, the time-varying formation tracking control strategy determined based on the dynamic average consistency is specifically:

[0095]

[0096] in is the Lumberg observer, used to observe the state x of the agent i ; Matrix L oi Satisfy A i +L oi C i It was Hurwitz's;

[0097] u i A controller, comprising a relative output and a first integral and a second integral of the relative output; Compensation input for formation tracking, satisfying is the local feedback gain and satisfies It was Hurwitz's;

[0098] d i ,q i is the auxiliary variable of agent i, The purpose is to use proportional integral feedback to eliminate the communication of the auxiliary state; δ1>0, δ2>0 are coupling gains;

[0099] p i is an estimator used to estimate the convex hull formed by multiple dynamic targets; s i is the internal state of the estimator and satisfies w ik represents the exploration relationship of agent i to target k; when the agent is a partially informed agent Where N1 represents the number of partially informed agents, which can at least obtain information about one target. When an agent is not well-informed, m i=0, where the number of uninformed agents is N-N1, and they cannot obtain any information about their targets.

[0100] is the sampling information, represents the kth triggering moment;

[0101] is the coupling strength, in, f k (r k (t), t) is bounded; λ2 is the smallest non-zero eigenvalue of the Laplace matrix L;

[0102] K is the feedback gain matrix, which is obtained by solving the algebraic Riccati equation A T P+PA-PBB T P+Q=0, where Q>0, P>0, select K=-B T P;

[0103] γ 1i , γ 2i and is the solution to the following linear matrix equation:

[0104]

[0105] Step S4, using the time-varying formation tracking control strategy to control the system's intelligent agent to form a desired time-varying formation and track the convex hull formed by multiple dynamic targets.

[0106] In order to verify the time-varying formation tracking control method based on dynamic average consistency proposed in the present invention, a simulation experiment is carried out:

[0107] Consider a swarm intelligence system consisting of 8 drones and 4 dynamic targets. The 8 drones surround and track the 4 dynamic targets in a predefined formation. The dynamic model of each drone is as follows:

[0108]

[0109] y i (t) = C i x i (t), i=1, 2, ..., 8;

[0110] The dynamic model of the dynamic target is:

[0111]

[0112] Among them, the values ​​of matrices A and B are:

[0113]

[0114] The entire drone swarm intelligence system formed a regular octagonal formation within 35 seconds and tracked the convex hull formed by the four targets.

[0115] Topology Figure 2 As shown, the blue circle {S1, S2, S3, S4} represents the target, the green circle {X1, X3, X5, X7} represents the uninformed agent, and the purple circle {X2, X4, X6, X8} represents the partially informed agent. It can be seen that none of the agents {X2, X4, X6, X8} has obtained the information of all targets, the agent X2 can obtain the information of the target {S3, S4}, the agent {X4, X6} can obtain the information of the target S2 at the same time, and the agent X8 only obtains the information of the target S1. This includes the one-to-one, one-to-many and one-to-zero exploration relationships between the agent and the target.

[0116] The desired time-varying formation is:

[0117]

[0118] Where r = 2, ω = 1. From h i From the structure of (t), if the desired formation tracking is achieved, the states of the eight agents will form a regular octagon centered on the convex hull of the four target states and maintain time-varying rotation.

[0119] choose The initial position of the agent is:

[0120] x1(0) = [13; 5; 7], x2(0) = [8; 16; 8], x3(0) = [15; 7; 13], x4(0) = [21; 8; 10],

[0121] x5(0)=[7;10;11], x6(0)=[5;10;14], x7(0)=[9;11;13], x8(0)=[12;12;12].

[0122] The initial position of the target is:

[0123] r1(0)=[13;5;10], r2(0)=[15;7;14], r3(0)=[7;10;10], r4(0)=[10;11;12].

[0124] The target input f k (r k (t), t) is:

[0125] f1(r1,t)=0.5sint;

[0126] f2(r2,t)=-1.5cos(|r 43 |)t+e -2t ;

[0127] f3(r3,t)=3sin(r 21 +πt / 3);

[0128] f4(r4,t)=2-sin(||r3||t);

[0129] Select R=I, Q=I, and solve the algebraic Riccati equation to obtain:

[0130]

[0131] The feedback gain matrix is ​​K = [-0.1231 -0.2910 -0.1931].

[0132] Parameter A i , B i , γ 1i , γ 2i , L oi The values ​​are:

[0133]

[0134]

[0135] The other parameters are selected as δ1=30, δ2=20, c1=20, c2=50, and:

[0136]

[0137] For convenience, we use represents the estimation error of the estimator on the target convex hull, represents the formation tracking error, Represents the output formation tracking error. Select parameters ε i =0.2. Figure 3 The relative positions of the agent and the target at t = 0s, 5s, 25s, and 35s are given. Figure 3 (a) and Figure 3 In (b), we can see that the agents are forming the desired time-varying formation configuration. Figure 3 (c) and Figure 3(d) shows that the agents form a regular octagonal formation configuration and track the convex hull formed by multiple dynamic targets while maintaining time-varying rotation. Figure 4 and Figure 5 They are respectively the error convergence curve of the convex hull formed by the estimator estimating multiple targets and the convergence curve of the formation tracking error. It can be seen that both have achieved convergence. Figure 6 It is the corresponding output formation tracking error convergence curve. Convergence is achieved at about t=35s. Due to the introduction of event triggering, the error curve has burrs and is not a smooth curve. Figure 7 and Figure 8 The triggering moments of the estimator and the agent in the first 3 seconds are given respectively, and it can be seen that there is no continuous communication. From the above results, it can be seen that the event-triggered time-varying formation tracking control method of the swarm intelligence system proposed in the present invention can flexibly realize the formation tracking control of the swarm intelligence system, which is conducive to better completing diversified formations.

[0138] As described above, although the present invention has been shown and described with reference to specific preferred embodiments, it should not be construed as limiting the present invention itself. Various changes in form and details may be made without departing from the spirit and scope of the present invention as defined in the appended claims.

Claims

1. A time-varying formation tracking control method based on dynamic average consistency, characterized in that: The steps include: S1. Establish the dynamic model, target dynamic model and corresponding system communication topology of the swarm intelligence system; S2. According to the established dynamics model of the swarm intelligence system and the target dynamics model, the desired time-varying formation is obtained, and event triggering conditions are constructed; S3. According to the event triggering conditions, the time-varying formation tracking control strategy is determined based on the dynamic average consistency; S4. Use the time-varying formation tracking control strategy to control the intelligent agents of the swarm intelligence system to form the desired time-varying formation and track the convex hull formed by multiple dynamic targets.

2. The time-varying formation tracking control method based on dynamic average consistency according to claim 1 is characterized in that: The swarm intelligence system described in step S1 includes N agents for tracking M targets.

3. The time-varying formation tracking control method based on dynamic average consistency according to claim 2 is characterized in that: The dynamic model of the swarm intelligence system established in step S1 is: y i (t)=C i x i (t),i=1,2,...,N; in, represents the state of agent i, represents the control input of agent i, y i (t)∈ represents the output of agent i, and are constant matrices, representing the state transfer matrix of agent i, the control input matrix of agent i, and the output matrix of agent i respectively; (A i ,B i ) is controllable, and n i Represents the dimension of the state vector, m i represents the dimension of the control input; All represent real vectors, where n and m represent dimensions. Respectively represent n i ×n i Real matrix of order and n i dimensional real vector; The target dynamics model established in step S1 is: in represents the state of target k, f k is the reference input of the target, expressed as represents the set of positive real numbers; is a constant matrix, and rank(B)=m.

4. The time-varying formation tracking control method based on dynamic average consistency according to claim 3 is characterized in that: The specific method for obtaining the desired time-varying formation in step S2 is: For a given matrix B i , choose a suitable non-singular matrix Make And meet Where T means transposing the matrix. I n Respectively represent m i The identity matrix of order n and n; At the same time, the formation feasibility conditions must also be met The desired formation tracking condition constructed is: in is the time-varying formation vector, is the piecewise continuously differentiable formation offset; ω k is a positive number, satisfying is a constant matrix; When achieving the time-varying formation tracking control goal, the states of the N agents agree on the formation reference, i.e., the convex hull of the M target states, and maintain their time-varying offset when Sometimes, there is That is, the linear combination of multiple target positions is located at the center of the multi-agent formation, then Defined as the desired time-varying formation to encircle or pursue M targets.

5. The time-varying formation tracking control method based on dynamic average consistency according to claim 4 is characterized in that: The specific method of constructing the event triggering condition in step S2 is: The estimation error of the convex hull formed by the estimator is set to in is the sampling information, is the kth triggering moment; Then the estimator p i Design trigger function: Where c1>0, c2>0 is the coupling strength, Δ i is the degree of agent i, K is the feedback gain matrix; J i , ε i is any positive constant; If the trigger condition is met Then p i (t) updates its current state and communicates the current state to its neighbors, is reset to 0; when the estimator obtains the latest status information of its neighbors, the estimator updates its own status; Design the following trigger mechanism for agent i: in In the formula, is the measurement error, ξ i (t) = y i (t)-h i (t), For the trigger moment, is a positive constant; when the trigger mechanism is satisfied When , the agent updates its state and transmits the output information to its neighbors. is reset to 0.

6. The time-varying formation tracking control method based on dynamic average consistency according to claim 5 is characterized in that: In step S3, according to the event triggering condition, the time-varying formation tracking control strategy determined based on the dynamic average consistency is specifically: in i=1,2,…,N is the Lumberg observer, which is used to observe the state x of the agent. i ; Matrix L oi Satisfy A i +L oi C i It was Hurwitz's; u i A controller, comprising a relative output and a first integral and a second integral of the relative output; Compensation input for formation tracking, satisfying is the local feedback gain and satisfies It was Hurwitz's; d i ,q i is the auxiliary variable of agent i, The purpose is to use proportional integral feedback to eliminate the communication of the auxiliary state; δ1>0, δ2>0 is the coupling gain; p i is an estimator used to estimate the convex hull formed by multiple dynamic targets; s i is the internal state of the estimator and satisfies w ik represents the exploration relationship of agent i to target k; when the agent is a partially informed agent Where N1 represents the number of partially informed agents, which can at least obtain information about one target. When an agent is not well-informed, m i = 0, where the number of uninformed agents is N-N1, and they cannot obtain any information about their targets; is the sampling information, represents the kth triggering moment; is the coupling strength, in, k=1,2,…,M,f k (r k (t),t) is bounded; λ2 is the smallest non-zero eigenvalue of the Laplace matrix L; K is the feedback gain matrix, which is obtained by solving the algebraic Riccati equation A T P+PA-PBB T P+Q=0, where Q>0, P>0, select K=-B T P; γ 1i , γ 2i and i is the solution to the following linear matrix equation:

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