CN116227324A - A Fractional Order Memristive Neural Network Estimation Method Under Variance Constraint - Google Patents
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Abstract
本发明公开了一种方差受限下的分数阶忆阻神经网络估计方法，所述包括如下步骤：步骤一、建立分数阶忆阻神经网络动态模型；步骤二、放大转发协议下对分数阶忆阻神经网络动态模型进行状态估计；步骤三、计算分数阶忆阻神经网络的误差协方差矩阵的上界及H∞性能约束条件；步骤四、利用随机分析方法，并通过解线性矩阵不等式求解出估计器增益矩阵Kk的解，实现对放大转发协议下分数阶忆阻神经网络动态模型的状态估计，判断k+1是否达到总时长N，若k+1＜N，则执行步骤二，反之结束。本发明解决了现有状态估计方法不能同时处理放大转发协议下具有H∞性能约束及方差受限分数阶忆阻神经网络的状态估计导致的估计性能准确率低的问题，从而提高了估计性能的准确率。
The invention discloses a method for estimating a fractional-order memristive neural network under variance limitation, which comprises the following steps: Step 1, establishing a dynamic model of a fractional-order memristive neural network; The dynamic model of the memristive neural network is used to estimate the state; Step 3, calculate the upper bound of the error covariance matrix of the fractional order memristive neural network and the H ∞ performance constraints; Step 4, use the stochastic analysis method and solve the linear matrix inequality to obtain The solution of the estimator gain matrix K k realizes the state estimation of the dynamic model of the fractional-order memristive neural network under the amplification and forwarding protocol, and judges whether k+1 reaches the total duration N. If k+1<N, execute step 2, otherwise Finish. The invention solves the problem that the existing state estimation method cannot simultaneously deal with the low accuracy of the estimation performance caused by the state estimation of the fractional-order memristive neural network with H ∞ performance constraints and variance constraints under the amplification and forwarding protocol, thereby improving the estimation performance. Accuracy.
Description
技术领域Technical Field
本发明涉及一种神经网络的状态估计方法，具体涉及一种放大转发协议下具有H∞性能约束及方差受限的分数阶忆阻神经网络的状态估计方法。The invention relates to a state estimation method of a neural network, and in particular to a state estimation method of a fractional-order memristor neural network with H∞ performance constraints and limited variance under an amplify-and-forward protocol.
背景技术Background Art
神经网络是根据人脑中的神经细胞结构和功能模拟出来的信息处理系统，具有较强的联想能力、自适应性和容错能力等优势。在现实的许多网络中，这类网络能高效地解决模式识别、信号处理和图像识别等实际系统建模和分析方面。Neural networks are information processing systems that simulate the structure and function of nerve cells in the human brain. They have the advantages of strong associative ability, adaptability and fault tolerance. In many real networks, this type of network can efficiently solve practical system modeling and analysis aspects such as pattern recognition, signal processing and image recognition.
在过去的几十年里，递归神经网络的状态估计问题已成为一个引人关注的课题，它已成功地应用于联想记忆、模式识别和组合优化等广泛领域。然而，在实际应用过程中，神经元的信息往往是不完全可测的，因此需要使用有效的估计方法来估计它们。到目前为止，已经研究了许多不同类型的神经网络状态估计问题。但值得注意的是，目前的结果仅适用于定常的情况下，这可能会导致应用存在的局限性。In the past few decades, the state estimation problem of recurrent neural networks has become an interesting topic, which has been successfully applied to a wide range of fields such as associative memory, pattern recognition and combinatorial optimization. However, in practical applications, the information of neurons is often not completely measurable, so effective estimation methods are needed to estimate them. So far, many different types of neural network state estimation problems have been studied. But it is worth noting that the current results are only applicable to steady-state cases, which may lead to limitations in application.
目前已有的状态估计方法不能同时处理在方差受限下具有H∞性能约束及放大转发协议的分数阶忆阻神经网络的状态估计问题，导致估计性能准确率低。The existing state estimation methods cannot simultaneously handle the state estimation problem of fractional-order memristor neural networks with H∞ performance constraints and amplify-and-forward protocols under variance constraints, resulting in low estimation performance accuracy.
发明内容Summary of the invention
本发明针对时变系统进行研究，提供了一种方差受限下的分数阶忆阻神经网络估计方法。该方法解决了现有状态估计方法不能同时处理放大转发协议下具有H∞性能约束的分数阶忆阻神经网络的状态估计问题，从而导致估计精度准确率低，以及在放大转发协议下存在信息无法接收到其他时刻信息的情况下，导致估计性能准确率低的问题，可用于忆阻神经网络状态估计领域。The present invention studies time-varying systems and provides a fractional-order memristor neural network estimation method under variance constraints. The method solves the problem that the existing state estimation method cannot simultaneously process the state estimation problem of the fractional-order memristor neural network with H ∞ performance constraints under the amplification and forwarding protocol, resulting in low estimation accuracy, and the problem that when there is information under the amplification and forwarding protocol that cannot receive information at other times, the estimation performance accuracy is low, and can be used in the field of memristor neural network state estimation.
本发明的目的是通过以下技术方案实现的：The objective of the present invention is achieved through the following technical solutions:
一种方差受限下的分数阶忆阻神经网络估计方法，包括如下步骤：A variance-constrained fractional-order memristor neural network estimation method comprises the following steps:
步骤一、建立放大转发协议下的分数阶忆阻神经网络动态模型；Step 1: Establish a dynamic model of fractional-order memristor neural network under amplification and forwarding protocol;
步骤二、放大转发协议下对步骤一建立的分数阶忆阻神经网络动态模型进行状态估计；Step 2: performing state estimation on the dynamic model of the fractional-order memristor neural network established in step 1 under the amplification and forwarding protocol;
步骤三、给定H∞性能指标γ、半正定矩阵一号
步骤四、利用随机分析方法，并通过解线性矩阵不等式求解出估计器增益矩阵Kk的解，实现对放大转发协议下的分数阶忆阻神经网络动态模型的状态估计，判断k+1是否达到总时长N，若k+1＜N，则执行步骤二，反之结束。Step 4: Use the random analysis method and solve the linear matrix inequality to solve the solution of the estimator gain matrix Kk to realize the state estimation of the dynamic model of the fractional-order memristor neural network under the amplification and forwarding protocol, and judge whether k+1 reaches the total duration N. If k+1＜N, execute
本发明中，所述神经网络可以为车辆悬挂构成的网络、质点弹簧构成的网络、航天器构成的网络或雷达构成的网络，在生物学、数学、计算机以及联想记忆、模式识别、组合优化和图像处理等多学科领域具有重要应用。In the present invention, the neural network can be a network composed of vehicle suspension, a network composed of mass springs, a network composed of spacecraft or a network composed of radar, and has important applications in multiple disciplines such as biology, mathematics, computers, associative memory, pattern recognition, combinatorial optimization and image processing.
相比于现有技术，本发明具有如下优点：Compared with the prior art, the present invention has the following advantages:
1、本发明同时考虑了放大转发协议下具有H∞性能约束和方差受限对状态估计性能的影响，利用不等式处理技术以及随机分析方法，全面考虑了估计误差协方差矩阵的有效信息，与现有的神经网络状态估计方法相比，本发明的分数阶忆阻神经网络状态估计方法同时考虑在放大转发协议下具有H∞性能约束及方差受限的分数阶忆阻神经网络的状态估计问题，得到了误差系统同时满足估计误差协方差有上界和给定的H∞性能要求的分数阶忆阻神经网络状态估计方法，同时达到了抑制扰动，并且提高了估计精度的目的，而且目前的结果仅适用于定常的情况下，这可能会导致应用存在的局限性，本发明考虑了时变神经网络，更能接近实际。1. The present invention simultaneously considers the influence of H∞ performance constraint and variance constraint on state estimation performance under the amplification and forwarding protocol, and comprehensively considers the effective information of the estimation error covariance matrix by using inequality processing technology and random analysis method. Compared with the existing neural network state estimation method, the fractional-order memristor neural network state estimation method of the present invention simultaneously considers the state estimation problem of the fractional-order memristor neural network with H∞ performance constraint and variance constraint under the amplification and forwarding protocol, and obtains the fractional-order memristor neural network state estimation method in which the error system simultaneously satisfies the upper bound of the estimation error covariance and the given H∞ performance requirements, and achieves the purpose of suppressing disturbances and improving the estimation accuracy. Moreover, the current results are only applicable to steady-state situations, which may lead to limitations in application. The present invention considers time-varying neural networks and is closer to reality.
2、本发明解决了现有状态估计方法不能同时处理放大转发协议下具有H∞性能约束及方差受限分数阶忆阻神经网络的状态估计导致的估计性能准确率低的问题，从而提高了估计性能的准确率。从仿真图可以看出，功率越小，分数阶忆阻神经网络的状态估计性能逐渐降低，估计误差相对较大。此外，验证了本发明所提出的状态估计方法的可行性和有效性。2. The present invention solves the problem that the existing state estimation method cannot simultaneously handle the state estimation of the fractional-order memristor neural network with H∞ performance constraints and variance constrained under the amplification and forwarding protocol, resulting in low estimation performance accuracy, thereby improving the estimation performance accuracy. It can be seen from the simulation diagram that the smaller the power, the state estimation performance of the fractional-order memristor neural network gradually decreases, and the estimation error is relatively large. In addition, the feasibility and effectiveness of the state estimation method proposed in the present invention are verified.
附图说明BRIEF DESCRIPTION OF THE DRAWINGS
图1是本发明放大转发协议下的分数阶忆阻神经网络状态估计方法的流程图；FIG1 is a flow chart of a method for estimating a state of a fractional-order memristor neural network under an amplify-and-forward protocol of the present invention;
图2是分数阶忆阻神经网络实际状态轨迹zk在两个不同情形下状态估计轨迹
图3是神经网络控制输出估计误差轨迹图在两个不同情形下的误差对比图；其中
图4是神经网络实际状态误差协方差和误差协方差上界第一个分量的轨迹图；其中
图5是神经网络实际状态误差协方差和误差协方差上界第二个分量的轨迹图；其中
具体实施方式DETAILED DESCRIPTION
下面结合附图对本发明的技术方案作进一步的说明，但并不局限于此，凡是对本发明技术方案进行修改或者等同替换，而不脱离本发明技术方案的精神和范围，均应涵盖在本发明的保护范围中。The technical solution of the present invention is further described below in conjunction with the accompanying drawings, but is not limited thereto. Any modification or equivalent replacement of the technical solution of the present invention without departing from the spirit and scope of the technical solution of the present invention should be included in the protection scope of the present invention.
本发明提供了一种方差受限下的分数阶忆阻神经网络估计方法，所述方法利用随机分析方法和不等式处理技术，首先，分别考虑估计误差系统满足H∞性能约束条件及误差协方差有上界的充分条件；然后，再同时得到估计误差系统满足H∞性能约束条件及误差协方差有上界的判别条件；最后，通过求解一系列线性矩阵不等式得到估计器增益矩阵的值，实现了在放大转发协议下具有H∞性能约束以及方差受限同时发生的情况下性能估计不受影响，从而提高了估计准确率。如图1所示，具体包括如下步骤：The present invention provides a fractional-order memristor neural network estimation method under variance constraint. The method utilizes a random analysis method and an inequality processing technique. First, sufficient conditions for the estimated error system to satisfy the H∞ performance constraint and the error covariance to have an upper bound are considered respectively; then, the judgment conditions for the estimated error system to satisfy the H∞ performance constraint and the error covariance to have an upper bound are obtained simultaneously; finally, the value of the estimator gain matrix is obtained by solving a series of linear matrix inequalities, so that the performance estimation is not affected when the H∞ performance constraint and variance constraint occur simultaneously under the amplification and forwarding protocol, thereby improving the estimation accuracy. As shown in FIG1 , the method specifically comprises the following steps:
步骤一、建立放大转发协议下分数阶忆阻神经网络动态模型。具体步骤如下：Step 1: Establish a dynamic model of fractional-order memristor neural network under the amplify-and-forward protocol. The specific steps are as follows:
首先，介绍Grunwald-Letnikov分数阶导数定义，这是一种适合于数值实现和应用的形式。该定义的离散形式表示为：First, we introduce the Grunwald-Letnikov fractional derivative definition, which is a form suitable for numerical implementation and application. The discrete form of this definition is expressed as:
式中，Δα表示α阶的Grunwald-Letnikov分数阶导数定义，h为相应的采样间隔，假设采样间隔为1，k为采样时刻，
根据Grunwald-Letnikov分数阶导数定义，分数阶忆阻神经网络动态模型的状态空间形式为：According to the definition of Grunwald-Letnikov fractional-order derivative, the state space form of the dynamic model of the fractional-order memristor neural network is:
式中：Where:
这里，
状态依赖矩阵参数ai(xi,k)、aij,d(xi,k)和bij(xi,k)满足：The state dependence matrix parameters ai (xi ,k ), aij ,d (xi ,k ) and bij (xi ,k ) satisfy:
式中，ai(xi,k)、aij,d(xi,k)和bij(xi,k)分别为A(xk)，Ad(xk)和B(xk)的第i个分量，Ωi＞0为已知的切换阈值，
定义：definition:
式中，
容易得出A(xk)∈[A-,A+]、
式中，
式中，ΔAk为满足范数有界不确定性的第一号矩阵，ΔAdk为满足范数有界不确定性的第二号矩阵，ΔBk为满足范数有界不确定性的第三号矩阵，
步骤二、放大转发协议下对步骤一建立的分数阶忆阻神经网络动态模型进行状态估计。具体步骤如下：Step 2: Perform state estimation on the dynamic model of the fractional-order memristor neural network established in step 1 under the amplification and forwarding protocol. The specific steps are as follows:
步骤二一、为了顺利完成远程数据传输的任务，在无线网络信道中安置一种放大－转发中继器以此来补充数据传输所消耗的能量。令ps,k和ns,k分别表示传感器和放大－转发中继器具有的随机能量，放大－转发中继器的输出信号由
式中，
式中，Pr{·}表示的是数学概率，
放大－转发中继器的输出值可表示为：The output value of the amplify-and-forward repeater can be expressed as:
式中，χk＞0表示在第k时刻的放大系数，
式中，
非线性函数f(s)满足如下扇形有界条件：The nonlinear function f(s) satisfies the following fan-shaped bounded conditions:
式中，
步骤二二、基于可获得的测量信息，构造如下的时变状态估计器：Step 22: Based on the available measurement information, construct the following time-varying state estimator:
式中，
步骤二三、定义估计误差
式中，
本步骤的主要目的是设计一个基于放大转发协议下的时变状态估计器(2)，使得估计误差系统同时满足以下两个性能约束要求：The main purpose of this step is to design a time-varying state estimator (2) based on the amplify-and-forward protocol so that the estimation error system satisfies the following two performance constraints at the same time:
(1)令扰动衰减水平γ＞0，半正定矩阵一号和半正定矩阵二号分别为
式中，N为有限的节点个数，
(2)估计误差协方差满足如下的上界约束条件：(2) The estimated error covariance satisfies the following upper bound constraints:
式中，
步骤三、给定H∞性能指标γ、半正定矩阵一号
步骤三一、按照下式证明出H∞性能分析问题并给出相应的易于求解的判别准则：Step 3: 1. Prove the H∞ performance analysis problem according to the following formula and give the corresponding easy-to-solve judgment criteria:
式中：Where:
式中，γ为给定的正标量；
步骤三二、探讨协方差矩阵χk的上界约束问题，并给出如下充分条件：Step 3.2: Discuss the upper bound constraint of the covariance matrix χk and give the following sufficient conditions:
Sk+1≥Ω(Sk), (4)S k+1 ≥Ω(S k ), (4)
式中，In the formula,
式中，ek为在第k时刻的误差矩阵；
通过对上述两个结果的分析，得到了保证估计误差系统满足给定的H∞性能要求和误差协方差有界性的充分条件。By analyzing the above two results, sufficient conditions are obtained to ensure that the estimation error system meets the given H ∞ performance requirements and the error covariance is bounded.
步骤四、利用随机分析方法，并通过解一系列线性矩阵不等式求解出估计器增益矩阵Kk的解，实现对放大转发协议下的分数阶忆阻神经网络动态模型进行状态估计；判断k+1是否达到总时长N，若k+1＜N，则执行步骤二，反之结束。Step 4: Use the random analysis method and solve a series of linear matrix inequalities to solve the solution of the estimator gain matrix Kk to realize the state estimation of the dynamic model of the fractional-order memristor neural network under the amplification and forwarding protocol; determine whether k+1 reaches the total duration N. If k+1<N, execute
本步骤中，通过求解(5)～(7)一系列递推线性矩阵不等式，给出估计误差系统同时满足H∞性能要求和误差协方差有上界的充分条件，即可计算出估计器增益矩阵的值：In this step, by solving a series of recursive linear matrix inequalities (5) to (7), sufficient conditions are given for the estimation error system to simultaneously meet the H∞ performance requirements and the upper bound of the error covariance, and the value of the estimator gain matrix can be calculated:
Sk+1-Ωk+1≤0 (7)S k+1 -Ω k+1 ≤0 (7)
更新矩阵为：The updated matrix is:
式中：Where:
Ω22＝diag{-ε1,kI,-ε2,kI,-ε2,kI,-ε3,kI,-ε3,kI},Ω 22 =diag{-ε 1,k I,-ε 2,k I,-ε 2,k I,-ε 3,k I,-ε 3,k I},
Ω33＝diag{-ε4,kI,-ε4,kI,-ε5,kI,-ε5,kI},Ω 33 =diag{-ε 4,k I,-ε 4,k I,-ε 5,k I,-ε 5,k I},
式中，Ω11是第1行第1列分块矩阵，Ω12是第1行第2列分块矩阵，Ω13是第1行第3列分块矩阵，Ω22是第2行第2列分块矩阵，Ω33是第3行第3列分块矩阵，
分别为
本发明中，步骤三与步骤四中所述理论为：In the present invention, the theory described in step 3 and step 4 is:
首先，证明出H∞性能分析问题并给出相应的易于求解的判别准则；其次，探讨协方差矩阵Xk的上界约束问题，并给出如下充分条件；通过对上述两个结果的分析，得到了保证估计误差系统满足给定的H∞性能要求和误差协方差有界性的充分条件，通过解一系列线性矩阵不等式求解出估计器增益矩阵的解，并计算出估计器增益矩阵Kk的解。Firstly, the H∞ performance analysis problem is proved and the corresponding easy-to-solve judgment criterion is given; secondly, the upper bound constraint problem of the covariance matrix Xk is discussed, and the following sufficient conditions are given; through the analysis of the above two results, sufficient conditions are obtained to ensure that the estimation error system meets the given H∞ performance requirements and the boundedness of the error covariance. The solution of the estimator gain matrix is solved by solving a series of linear matrix inequalities, and the solution of the estimator gain matrix Kk is calculated.
实施例：Example:
本实施例以具有H∞性能约束及方差约束的分数阶忆阻神经网络为例，此外还可以应用在联想记忆、模式识别和组合优化中，采用本发明所述方法针对语音识别案例进行仿真：This embodiment takes a fractional-order memristor neural network with H∞ performance constraints and variance constraints as an example. In addition, it can also be applied to associative memory, pattern recognition and combinatorial optimization. The method of the present invention is used to simulate a speech recognition case:
放大转发协议下具有H∞性能约束及方差约束的分数阶忆阻神经网络状态模型、测量输出模型及被控输出模型的相关系统参数选取如下：The relevant system parameters of the fractional-order memristor neural network state model, measurement output model and controlled output model under the amplification and forwarding protocol with H∞ performance constraints and variance constraints are selected as follows:
根据人的声音的状态给定相应的调节矩阵为：According to the state of human voice, the corresponding adjustment matrix is given as:
C1k＝[-1.2-0.35sin(2k)]T,C 1k ＝[-1.2-0.35sin(2k)] T ,
测量调节矩阵为：The measurement adjustment matrix is:
C2k＝[-0.2-0.1sin(3k)]T,
被控输出调节矩阵为：The controlled output adjustment matrix is:
Hk＝[-0.01-0.01sin(2k)]H k =[-0.01-0.01sin(2k)]
状态权重矩阵为：The state weight matrix is:
非线性函数的权重矩阵和调节参数为：The weight matrix and adjustment parameters of the nonlinear function are:
Case I：给出以下传输功率的概率分布：Case I: Given the following probability distribution of transmission power:
Prob{pt,k＝1}＝0.1,Prob{pt,k＝1.5}＝0.3,Prob{p t, k = 1} = 0.1, Prob {p t, k = 1.5} = 0.3,
Prob{pt,k＝2}＝0.6,Prob{nt,k＝1}＝0.2,Prob{p t, k = 2} = 0.6, Prob {n t, k = 1} = 0.2,
Prob{nt,k＝1.5}＝0.4,Prob{nt,k＝2}＝0.4,Prob{n t,k ＝1.5}＝0.4,Prob{n t,k ＝2}＝0.4,
Case II：给出以下传输功率的概率分布：Case II: Given the following probability distribution of transmission power:
Prob{pt,k＝1}＝0.6,Prob{pt,k＝1.5}＝0.3,Prob{p t, k = 1} = 0.6, Prob {p t, k = 1.5} = 0.3,
Prob{pt,k＝2}＝0.1,Prob{nt,k＝1}＝0.4,Prob{p t, k = 2} = 0.1, Prob {n t, k = 1} = 0.4,
Prob{nt,k＝1.5}＝0.4,Prob{nt,k＝2}＝0.2.Prob{n t,k =1.5}=0.4, Prob{n t,k =2}=0.2.
激励函数取为：The activation function is taken as:
式中，xk＝[x1,kx2,k]T是神经元的状态向量，放大系数为χs＝1，x1,k为在第k时刻xk的第一个分量比重矩阵，x2,k为在第k时刻xk的第二个分量比重矩阵。Where xk = [ x1, k x2 , k ] T is the state vector of the neuron, the amplification factor is χs = 1, x1, k is the first component weight matrix of xk at the kth moment, and x2 , k is the second component weight matrix of xk at the kth moment.
其它仿真初始值选取如下：Other simulation initial values are selected as follows:
扰动衰减水平γ＝0.7，半正定矩阵一号
利用递推线性矩阵不等式，求解线性矩阵不等式(5)～(7)，部分数值如下：情形一(Case I)：Using the recursive linear matrix inequality, we solve linear matrix inequalities (5) to (7). Some numerical values are as follows: Case I:
情形二(Case II)：Case II:
状态估计器效果：State Estimator Effects:
由图2可见，针对放大转发协议下具有H∞性能约束及方差约束的分数阶忆阻神经网络，所发明的状态估计器设计方法可有效地估计出目标状态。As can be seen from FIG2 , for the fractional-order memristor neural network with H ∞ performance constraint and variance constraint under the amplify-and-forward protocol, the invented state estimator design method can effectively estimate the target state.
由图3、图4、图5可以看出，针对每个时刻，随着功率的减少，估计误差效果变差。It can be seen from FIG3 , FIG4 , and FIG5 that, at each moment, as the power decreases, the estimation error effect becomes worse.
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