CN103179596B

CN103179596B - A kind of LQG that optimizes controls the method with power of communications wastage in bulk or weight

Info

Publication number: CN103179596B
Application number: CN201310078930.4A
Authority: CN
Inventors: 章辉; 田垠
Original assignee: Zhejiang University ZJU
Current assignee: Zhejiang University ZJU
Priority date: 2013-03-12
Filing date: 2013-03-12
Publication date: 2016-01-27
Anticipated expiration: 2033-03-12
Also published as: CN103179596A

Abstract

The invention discloses a method for optimizing the total consumption of LQG control and communication power. The method aims at the control problem of guided vehicles in an automatic highway system, adopts a simple amplification transmission strategy, and makes LQG control performance by designing control signals and off-line design amplification coefficients. The sum of the power consumption of the communication and the communication is extremely small, thereby realizing the optimization of the total consumption of the LQG control and the communication power.

Description

A Method for Optimizing the Total Consumption of LQG Control and Communication Power

技术领域technical field

本发明属于网络化控制系统领域，涉及一种优化LQG控制与通信功率总消耗的方法。The invention belongs to the field of networked control systems and relates to a method for optimizing the total consumption of LQG control and communication power.

背景技术Background technique

在网络化控制系统中，传感器、控制器、执行器往往分布各处，并通过共享网络连结成一个整体。这使得系统结构灵活，网络建造和维护成本降低，因此在许多方面都有应用，如自动公路系统。在自动公路系统中，传感器对车辆运行及周边环境状态进行探测，并传输给车载计算机，计算机根据这些信息控制车辆的油门、转向和刹车，从而实现车辆的自动驾驶。由于信息传输采用无线通信技术，因此将面临多径衰落。In a networked control system, sensors, controllers, and actuators are often distributed everywhere and connected into a whole through a shared network. This makes the system structure flexible and reduces the cost of network construction and maintenance, so it has applications in many aspects, such as automatic highway systems. In the automatic road system, the sensor detects the vehicle's operation and the surrounding environment, and transmits it to the on-board computer. The computer controls the accelerator, steering and brake of the vehicle according to the information, so as to realize the automatic driving of the vehicle. Since information transmission adopts wireless communication technology, it will face multipath fading.

多径衰落主要是由无线信道的时变多径特征引起。根据信道幅度增益的概率分布多径衰落可分为瑞利衰落、莱斯衰落等；根据信道特征多径衰落又可分为平坦衰落和频率选择性衰落，慢衰落和快衰落。因为多径衰落的影响，信道的信噪比时刻在发生变化，当信噪比很小时，通信将变得十分不可靠。Multipath fading is mainly caused by the time-varying multipath characteristics of wireless channels. According to the probability distribution of channel amplitude gain, multipath fading can be divided into Rayleigh fading, Rice fading, etc.; according to channel characteristics, multipath fading can be further divided into flat fading, frequency selective fading, slow fading and fast fading. Due to the influence of multipath fading, the signal-to-noise ratio of the channel is changing all the time. When the signal-to-noise ratio is small, communication will become very unreliable.

通常在设计控制器时，由于噪声的影响，我们无法得到完美的状态信息，因此需要进行状态估计。而由于多径衰落，数字通信往往存在丢包现象，这容易导致状态估计器的不稳定。因此，一种可替代的方案是基于模拟通信的放大传输策略，此时信息经放大后采用模拟通信的方式进行传输。显然，放大系数越大，信道的信噪比就越大，控制性能就越好，但增大放大系数就意味着增大通信功率，因此放大系数的设计又受到有限功率的约束。基于此，通常设计放大系数的方法是给定最大通信功率，谋求最优控制性能，但这样做容易导致能源的浪费，即虽然没有达到功率限制，但若继续增大通信功率，控制性能的提升将十分有限。Usually when designing a controller, due to the influence of noise, we cannot get perfect state information, so state estimation is required. However, due to multipath fading, digital communication often has packet loss, which easily leads to instability of the state estimator. Therefore, an alternative solution is an amplified transmission strategy based on analog communication. At this time, the information is transmitted by means of analog communication after being amplified. Obviously, the larger the amplification factor, the greater the signal-to-noise ratio of the channel and the better the control performance, but increasing the amplification factor means increasing the communication power, so the design of the amplification factor is subject to the constraints of limited power. Based on this, the usual way to design the amplification factor is to give the maximum communication power to seek the optimal control performance, but this will easily lead to waste of energy, that is, although the power limit has not been reached, if the communication power continues to increase, the control performance will be improved. will be very limited.

发明内容Contents of the invention

本发明的目的是通过设计控制信号、离线设计放大系数使得LQG控制性能与通信功率消耗的总和达到极小，从而实现LQG控制与通信功率总消耗的优化。The purpose of the present invention is to minimize the sum of LQG control performance and communication power consumption by designing control signals and off-line amplification factors, thereby realizing the optimization of LQG control and total communication power consumption.

本发明的目的是通过以下技术方案来实现的：一种优化LQG控制与通信功率总消耗的方法，主要包括如下步骤：The object of the present invention is achieved through the following technical solutions: a method for optimizing LQG control and total communication power consumption, mainly comprising the following steps:

步骤1：对引导车辆的运动过程进行建模，得到状态空间模型；Step 1: Model the motion process of the guided vehicle to obtain a state space model;

步骤2：传感器发送导频信号到车载计算机，计算机根据接收信号估计信道脉冲响应，并由此得到信道幅度增益；Step 2: The sensor sends a pilot signal to the on-board computer, and the computer estimates the channel impulse response based on the received signal, and thus obtains the channel amplitude gain;

步骤3：传感器测量得到车辆速度，经放大后通过模拟通信装置传输给车载计算机；Step 3: The vehicle speed is measured by the sensor, which is amplified and transmitted to the on-board computer through the analog communication device;

步骤4：针对步骤1的状态空间模型，考虑LQG控制问题，提出LQG控制性能指标；Step 4: Aiming at the state space model in step 1, consider the LQG control problem, and propose the LQG control performance index;

步骤5：在步骤4的性能指标中进一步考虑通信功率消耗，得到新的性能指标；Step 5: further consider communication power consumption in the performance index of step 4 to obtain a new performance index;

步骤6：基于离线设计放大系数这个前提，根据步骤5的新性能指标设计控制信号；Step 6: Based on the premise of off-line design amplification factor, design the control signal according to the new performance index in step 5;

步骤7：基于步骤6的控制信号设计放大系数；Step 7: design an amplification factor based on the control signal in step 6;

步骤8：将步骤2、3的测量数据及步骤7的放大系数带入步骤6得到控制信号，此时步骤5的新性能指标将达到极小，LQG控制与通信功率总消耗将得到优化。Step 8: Bring the measurement data of steps 2 and 3 and the amplification factor of step 7 into step 6 to obtain the control signal. At this time, the new performance index of step 5 will be minimized, and the total power consumption of LQG control and communication will be optimized.

本发明的有益效果是，本发明采用模拟通信传输信息，从而避免了状态估计的不稳定。另外在设计放大系数时将控制与通信过程联合在一起考虑，通过设计控制信号、离线设计放大系数使得LQG控制性能与通信能量消耗的总和达到极小，从而实现了LQG控制与通信功率总消耗的优化。The beneficial effect of the present invention is that the present invention uses analog communication to transmit information, thereby avoiding instability of state estimation. In addition, when designing the amplification factor, the control and communication processes are considered together. By designing the control signal and off-line design of the amplification factor, the sum of the LQG control performance and communication energy consumption is minimized, thus realizing the balance between the LQG control and the total communication power consumption. optimization.

附图说明Description of drawings

图1是系统结构图；Fig. 1 is a system structure diagram;

图2是仿真结果图。Figure 2 is a graph of the simulation results.

具体实施方式detailed description

本发明针对自动公路系统中引导车辆的控制对象，采用简单的放大传输策略传输信号，整个系统结构如图1所示，其中，一种优化LQG控制与通信功率总消耗的方法如下：The present invention is aimed at the control object of the guided vehicle in the automatic highway system, and adopts a simple amplified transmission strategy to transmit signals. The entire system structure is shown in Figure 1, wherein a method for optimizing the total consumption of LQG control and communication power is as follows:

步骤1：对自动公路系统中引导车辆的运动过程进行建模，得到状态空间模型；Step 1: Model the movement process of the guided vehicle in the automatic highway system to obtain a state space model;

参考S.S.Stankovic,M.J.Stanojevic,D.D.Siljak:Stochasticinclusionprincipleappliedtodecentralizedoverlappingsuboptimallqgcontrolofaplatoonofvehicles,TheInternationalConferenceonComputerasaTool,EUROCON,318–321(2005)中的方法进行建模：Refer to the method in S.S.Stankovic, M.J.Stanojevic, D.D.Siljak: Stochasticinclusionprincipleappliedtodecentralizedoverlappingsuboptimallqgcontrolofaplatoonofvehicles, The International Conference on Computer as a Tool, EUROCON, 318–321 (2005) for modeling:

$dx dx ((t t)) = = [\begin{matrix} 00 & 11 \\ 00 & - - ζ ζ \end{matrix}] x x ((t t)) dt dt + + [\begin{matrix} 00 \\ ζ ζ \end{matrix}] u u ((t t)) dt dt + + dw dw ((t t)),,$

其中，t表示时间，x(t)=[v_L(t)a_L(t)]^T，v_L(t)是车辆速度,a_L(t)是车辆加速度，u(t)是车辆控制输入，w(t)是白噪声输入，ζ为车辆动力学系数，d表示微分。Among them, t represents the time, x(t)=[v _L (t)a _L (t)] ^T , v _L (t) is the vehicle speed, a _L (t) is the vehicle acceleration, u(t) is the vehicle control Input, w(t) is the white noise input, ζ is the vehicle dynamics coefficient, and d is the differential.

将上述模型离散化，可得：Discretizing the above model, we can get:

x(k+1)=Ax(k)+Bu(k)+w(k)，x(k+1)=Ax(k)+Bu(k)+w(k),

其中，k表示时刻，A、B为定常矩阵，可由连续时间模型求得:Among them, k represents the moment, A and B are constant matrices, which can be obtained by the continuous time model:

$A A = = exp exp (([\begin{matrix} 00 & 11 \\ 00 & - - ζ ζ \end{matrix}] \cdot &Center Dot; T T)),,$ $B B = = {{{&Integral; &Integral;}_{00}^{T T} exp exp (([\begin{matrix} 00 & 11 \\ 00 & - - ζ ζ \end{matrix}] \cdot &Center Dot; t t)) dt dt}} \cdot &Center Dot; [\begin{matrix} 00 \\ ζ ζ \end{matrix}];;$

其中，T为采样间隔，exp为指数函数，表示[0,T]区间上求积分。系统初始状态x(0)服从高斯分布，均值为m₀，协方差阵为Σ₀；w(k)为零均值高斯白噪声向量，协方差阵为 Among them, T is the sampling interval, exp is the exponential function, Represents the integral on the [0,T] interval. The initial state x(0) of the system obeys the Gaussian distribution, the mean is m ₀ , and the covariance matrix is Σ ₀ ; w(k) is a zero-mean Gaussian white noise vector, and the covariance matrix is

无线信道考虑瑞利平坦衰落信道，导频信号为s(k)，则接收信号为：The wireless channel considers the Rayleigh flat fading channel, and the pilot signal is s(k), then the received signal is:

$\overset{~ ~}{r r} ((k k)) = = \overset{~ ~}{h h} ((k k)) s the s ((k k)) + + \overset{~ ~}{n no} ((k k)),,$

其中，为等效低通接收信号；为等效低通信道脉冲响应，是零均值复高斯随机过程，协方差为为信道加性噪声，是协方差为的零均值复高斯白噪声。in, is the equivalent low-pass receiving signal; is the equivalent low-pass channel impulse response, which is a zero-mean complex Gaussian random process, and the covariance is is the channel additive noise, and the covariance is zero-mean complex white Gaussian noise.

车载计算机根据接收信号估计信道脉冲响应：The on-board computer estimates the channel impulse response from the received signal:

$\overset{\overset{^^}{~ ~}}{h h} ((k k)) = = \frac{{σ σ}_{h h}^{22} s the s ((k k))}{{σ σ}_{n no}^{22} + + {σ σ}_{h h}^{22} {s the s}^{22} ((k k))} \overset{~ ~}{r r} ((k k)),,$

其中，为信道脉冲响应估计值，它的模即为信道幅度增益h(k)。in, is the estimated value of the channel impulse response, and its modulus is the channel amplitude gain h(k).

传感器测量得到车辆速度：The sensor measures the vehicle speed:

y(k)=Cx(k)+v(k)，y(k)=Cx(k)+v(k),

其中，y(k)是传感器测量值；C=[10]，保证了系统的能观性；v(k)为观测噪声，是协方差为的零均值高斯白噪声。Among them, y(k) is the measured value of the sensor; C=[10], which ensures the observability of the system; v(k) is the observation noise, and the covariance is zero-mean Gaussian white noise.

测量值经放大后通过模拟通信装置传输给车载计算机：The measured values are amplified and transmitted to the on-board computer via an analog communication device:

z(k)=αh(k)y(k)+n(k)，z(k)=αh(k)y(k)+n(k),

其中，z(k)为接收信号；α为放大系数，n(k)为信道加性噪声，是协方差为的零均值高斯白噪声。Among them, z(k) is the received signal; α is the amplification factor, n(k) is the channel additive noise, and the covariance is zero-mean Gaussian white noise.

LQG性能指标为J^c，如下：The LQG performance index is J ^c , as follows:

${J J}^{c c} = = \underset{M m &RightArrow; &Right Arrow; \infty \infty}{lim lim} E E. {{{Σ Σ}_{k k = = 00}^{M m - - 11} [[{x x}^{T T} ((k k)) Qx Qx ((k k)) + + {Ru Ru}^{22} ((k k))]]}},,$

其中，M为终端时刻；Q、R为权重系数，Q≥0，R>0；E(.)表示期望运算；T表示矩阵的转置；Σ(.)表示求和运算；表示M趋于无穷时求极限。式中第一项是对状态跟踪情况的度量，第二项是对控制能量消耗的度量。Among them, M is the terminal moment; Q and R are weight coefficients, Q≥0, R>0; E(.) represents the expected operation; T represents the transposition of the matrix; Σ(.) represents the sum operation; Find the limit when M tends to infinity. The first term in the formula is a measure of state tracking, and the second term is a measure of control energy consumption.

步骤5：在步骤4的性能指标中进一步考虑通信功率消耗，得到新的性能指标。Step 5: further consider the communication power consumption in the performance index of step 4 to obtain a new performance index.

新性能指标为J，如下：The new performance index is J, as follows:

$J J = = \underset{M m &RightArrow; &Right Arrow; \infty \infty}{lim lim} E E. {{{Σ Σ}_{k k = = 00}^{M m - - 11} [[{x x}^{T T} ((k k)) Qx Qx ((k k)) + + {Ru Ru}^{22} ((k k)) + + {Sα Sα}^{22} {y the y}^{22} ((k k))]]}},,$

其中，第三项是对通信功率消耗的度量，S为权重系数，S>0。Among them, the third item is a measure of communication power consumption, S is a weight coefficient, and S>0.

步骤6：基于离线设计放大系数这个前提，根据步骤5的新性能指标设计控制信号。Step 6: Based on the premise of off-line design of the amplification factor, design the control signal according to the new performance index in step 5.

基于放大系数的离线设计，控制信号与放大系数可分离设计。此时采用动态规划方法可将控制信号设计为u^*(k)：Based on the off-line design of the amplification factor, the control signal and the amplification factor can be designed separately. At this time, the dynamic programming method can be used to design the control signal as u ^* (k):

${u u}^{* *} ((k k)) = = - - F f \overset{^^}{x x} ((k k | | k k)),,$

其中，F为反馈系数，为时刻k对当前时刻的最小二乘估计。Among them, F is the feedback coefficient, is the least squares estimate of time k to the current time.

F满足：F satisfies:

F=[B^TPB+R]^-1B^TPA，F=[B ^T P B +R] ^-1 B ^T P A,

P=Q+A^T{P-PB[B^TPB+R]^-1B^TP}A+Sα²C^TC，P=Q+A ^T {P-PB[B ^T PB+R] ^-1 B ^T P}A+Sα ² C ^T C,

其中，P为中间变量，可采用数值迭代法进行求解。Among them, P is an intermediate variable, which can be solved by numerical iteration method.

通过Kalman滤波器得到： Get through Kalman filter:

$\overset{^^}{x x} ((k k | | k k)) = = \overset{^^}{x x} ((k k | | k k - - 11)) + + K K ((k k)) [[z z ((k k)) - - {C C}^{' '} ((k k)) \overset{^^}{x x} ((k k | | k k - - 11))]],,$

Σ(k|k)=Σ(k|k-1)-K(k)C′(k)Σ(k|k-1)，Σ(k|k)=Σ(k|k-1)-K(k)C'(k)Σ(k|k-1),

$K K ((k k)) = = Σ Σ ((k k | | k k - - 11)) {C C}^{' ' T T} ((k k)) {[[{C C}^{' '} ((k k)) Σ Σ ((k k | | k k - - 11)) {C C}^{' ' T T} ((k k)) + + {σ σ}_{{v v}^{' '}}^{22} ((k k))]]}^{- - 11},,$

$\overset{^^}{x x} ((k k | | k k - - 11)) = = A A \overset{^^}{x x} ((k k - - 11 | | k k - - 11)) + + Bu Bu ((k k - - 11)),,$

$Σ Σ ((k k | | k k - - 11)) = = AΣ AΣ ((k k - - 11 | | k k - - 11)) {A A}^{T T} + + {σ σ}_{w w}^{22},,$

C′(k)=h(k)αC， $σ_{v^{'}}^{2} (k) = h^{2} (k) α^{2} σ_{v}^{2} + σ_{n}^{2};$ C'(k)=h(k)αC, $σ_{v^{'}}^{2} (k) = h^{2} (k) α^{2} σ_{v}^{2} + σ_{no}^{2};$

其中，是时刻k-1对时刻k的预测值，Σ(k|k-1)是预测误差方差，Σ(k|k)是时刻k对当前时刻的估计误差方差，K(k)为Kalman增益系数，C′(k)、为中间变量。算法初始值为Σ(0|-1)=Σ₀。in, is the predicted value of time k-1 to time k, Σ(k|k-1) is the forecast error variance, Σ(k|k) is the estimated error variance of time k to the current time, K(k) is the Kalman gain coefficient , C′(k), as an intermediate variable. The initial value of the algorithm is Σ(0|-1)=Σ ₀ .

步骤7：基于步骤6的控制信号设计放大系数。Step 7: Design an amplification factor based on the control signal in step 6.

当控制信号为u^*(k)时，步骤（5）中的新性能指标J将为J(u^*)：When the control signal is u ^* (k), the new performance index J in step (5) will be J(u ^* ):

$J J (({u u}^{* *})) = = S S {α α}^{22} {σ σ}_{v v}^{22} + + tr tr {{Q Q {σ σ}_{w w}^{22} + + S S {α α}^{22} {C C}^{T T} C C {σ σ}_{w w}^{22} + + Ψ Ψ \overset{&OverBar; &OverBar;}{Σ Σ}}},,$

其中，Ψ=F^T[B^TPB+R]F；tr(.)表示矩阵的迹；为在时刻趋于无穷时的极限值，表示Σ(k|k)对信道状态H_k={h(0),…,h(k)}求期望。Among them, Ψ=F ^T [B ^T PB+R]F; tr(.) represents the trace of the matrix; for The limit value when the time tends to infinity, Indicates that Σ(k|k) expects the channel state H _k ={h(0),…,h(k)}.

放大系数的设计应使J(u^*)达到极小。由于计算十分复杂，我们求取放大系数的次优解使J(u^*)的上界J′(u^*)达到极小。The design of the magnification factor should make J(u ^* ) extremely small. because Computationally complex, we find a suboptimal solution for the magnification factor Make the upper bound J′(u ^* ) of J(u ^* ) extremely small.

J′(u^*)可根据DeyS,LeongA:EvansJ,Kalmanfilteringwithfadedmeasurements,Automatica,2009,45(10):2223–2233得到：J′(u ^* ) can be obtained according to DeyS, LeongA: EvansJ, Kalmanfilteringwithfadedmeasurements, Automatica, 2009, 45(10):2223–2233:

${J J}^{' '} (({u u}^{* *})) = = S S {α α}^{22} {σ σ}_{v v}^{22} + + tr tr {{Q Q {σ σ}_{w w}^{22} + + S S {α α}^{22} {C C}^{T T} C C {σ σ}_{w w}^{22} + + ΨΓ ΨΓ}}$

其中，Γ是的上界，满足：where Γ is The upper bound of , satisfying:

$Γ Γ = = {A A}^{- - 11} ((Ω Ω - - {σ σ}_{w w}^{22})) {(({A A}^{- - 11}))}^{T T},,$

$Ω Ω = = {σ σ}_{w w}^{22} + + {AΩA AΩA}^{T T} - - \frac{{AΩC AΩC}^{T T} {CΩA CΩA}^{T T}}{{CΩC CΩC}^{T T} + + {σ σ}_{v v}^{22}} [[11 - - γexp γ exp ((γ γ)) {E E.}_{11} ((γ γ))]],,$

$γ γ = = \frac{{σ σ}_{n no}^{22}}{{σ σ}_{h h}^{22} {α α}^{22} [[CΩ CΩ {C C}^{T T} + + {σ σ}_{v v}^{22}]]};;$

其中，Ω、γ为中间变量；E₁(γ)为指数积分。Γ在计算时可采用数值迭代法进行求解。Among them, Ω and γ are intermediate variables; E ₁ (γ) is an exponential integral. Γ can be solved by numerical iteration method during calculation.

采用模式搜索法进行求解，可调用Matlab命令： Using the pattern search method to solve, you can call the Matlab command:

X=patternsearch(fun,x₀,a,b,aeq,beq,lb,ub)；X=patternsearch(fun,x ₀ ,a,b,aeq,beq,lb,ub);

其中，patternsearch是Matlab中采用模式搜索法寻找目标函数fun极小值点的命令，以x₀为初始值点，a、b、aeq、beq分别为不等式约束和等式约束条件的参数，lb、ub为变量取值范围的下界和上界，优化而得的极小值点为X。Among them, patternsearch is a command in Matlab to use the pattern search method to find the minimum value point of the objective function fun, with x ₀ as the initial value point, a, b, aeq, and beq are the parameters of the inequality constraints and equality constraints, respectively, lb, ub is the lower bound and upper bound of the value range of the variable, and the optimized minimum value point is X.

LQG控制与通信功率总消耗即为步骤（5）中的新性能指标J：The total power consumption of LQG control and communication is the new performance index J in step (5):

当放大系数α和控制信号u(k)分别取为和u^*(k)时，J将达到极小，此时LQG控制与通信功率总消耗将得到优化。When the amplification factor α and the control signal u(k) are respectively taken as When and u ^* (k), J will reach a minimum, and the total power consumption of LQG control and communication will be optimized at this time.

本发明在考虑自动公路系统中的信息传输时采用简单的放大传输策略，设备结构简单，并且避免了状态估计的不稳定问题。在讨论放大系数的设计时，本发明通过以LQG控制性能与通信功率消耗的总和为性能函数，设计控制信号与放大系数使之达到极小，将控制与通信过程联合在一起考虑，从而避免了一般设计方法所面临的能量浪费。虽然在设计放大系数时求取的是次优解，但仍然在一定程度上实现了LQG控制与通信功率总消耗的优化。The invention adopts a simple amplifying transmission strategy when considering the information transmission in the automatic road system, the device structure is simple, and the unstable problem of state estimation is avoided. When discussing the design of the amplification factor, the present invention uses the sum of the LQG control performance and communication power consumption as the performance function, designs the control signal and the amplification factor to make it extremely small, and considers the control and communication process together, thereby avoiding the Energy waste faced by general design methods. Although the suboptimal solution is obtained when designing the amplification factor, the optimization of LQG control and total communication power consumption is still achieved to a certain extent.

另外，对本发明做如下几点说明：In addition, the present invention is described as follows:

1）本发明讨论的是无限时域情形下的LQG控制问题，这主要考虑到在无限时域情形下，放大系数将取为常值，控制信号是定常反馈控制，较之于有限时域情形具有更大的应用价值。另外由于放大系数取为常值，因此能够显著地缩短计算时间。1) The present invention discusses the LQG control problem in the infinite time domain situation, which mainly considers that in the infinite time domain situation, the amplification factor will be taken as a constant value, and the control signal is a constant feedback control, compared with the limited time domain situation It has greater application value. In addition, since the amplification factor is taken as a constant value, the calculation time can be significantly shortened.

2）本发明主要讨论了状态回复到零平衡态的LQG控制问题，对于状态跟踪问题，设计方法类似。2) The present invention mainly discusses the LQG control problem of returning the state to the zero-balance state. For the state tracking problem, the design method is similar.

模型仿真model simulation

①仿真对象，取ζ=1，T=0.01，则系统离散模型为：①Simulation object, take ζ=1, T=0.01, then the discrete model of the system is:

$A A = = [\begin{matrix} 11 & 1.0101 1.0101 \\ 11 & 0.99 0.99 \end{matrix}],,$ $B B = = [\begin{matrix} 0.0101 0.0101 \\ 0.01 0.01 \end{matrix}];;$

初始状态及噪声协方差取为：The initial state and noise covariance are taken as:

${m m}_{00} = = [\begin{matrix} 11 \\ 11 \end{matrix}],,$ ${Σ Σ}_{00} = = [\begin{matrix} 11 & 00 \\ 00 & 44 \end{matrix}],,$ ${σ σ}_{w w}^{22} = = [\begin{matrix} 11 & 00 \\ 00 & 11 \end{matrix}],,$ ${σ σ}_{v v}^{22} = = 11,, {σ σ}_{n no}^{22} = = 44,, {σ σ}_{h h}^{22} = = 22;;$

②性能指标J中权重系数分别取为：② The weight coefficients in the performance index J are respectively taken as:

$Q = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}],$ R=1，S=1； $Q = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}],$ R=1, S=1;

③通过本发明方法可得到放大系数此时J(u^*)=1.074*10⁵。计算过程中涉及到的求取，这里我们通过计算50000个样本序列的平均值得到。③The magnification factor can be obtained by the method of the present invention At this time J(u ^* )=1.074*10 ⁵ . involved in the calculation The calculation of , here we get by calculating the average value of 50,000 sample sequences.

为了说明本文方法的有效性，我们求取一系列α所对应的性能J(u^*)。图2是上述系统在α取不同值时J(u^*)及其上界J′(u^*)的取值情况，其中□对应J(u^*)，*对应J′(u^*)，J(u^*)的起伏是由于的计算误差造成。对比两条曲线可以发现，J(u^*)取极小值时由此说明了本发明的有效性。In order to illustrate the effectiveness of the method in this paper, we obtain the performance J(u ^* ) corresponding to a series of α. Figure 2 shows the values of J(u ^* ) and its upper bound J′(u ^* ) when the above system takes different values of α, where □ corresponds to J(u ^* ), * corresponds to J′(u ^* ), J The ups and downs of (u ^* ) are due to caused by calculation errors. Comparing the two curves, it can be found that when J(u ^* ) takes the minimum value The effectiveness of the present invention has thus been illustrated.

Claims

1. A method for optimizing LQG control and total power consumption for communications, the method comprising the steps of:

(1) modeling the motion process of a guide vehicle in an automatic road system to obtain a state space model:

wherein t represents time, and x (t) ═ v_L(t)a_L(t)]^T，v_L(t) is the vehicle speed, a_L(t) is vehicle acceleration, u (t) is vehicle control input, w (t) is white noise input,d represents a differential for the vehicle dynamics coefficient;

discretizing the model to obtain:

x(k+1)＝Ax(k)+Bu(k)+w(k)，

where k denotes the time, A, B is a constant matrix, which can be obtained from a continuous time model:

where T is the sampling interval, exp is an exponential function,represents [0, T]Integrating over the interval; the initial state x (0) of the system follows Gaussian distribution, and the mean value is m₀The covariance matrix is ∑₀(ii) a w (k) is a zero-mean Gaussian white noise vector and the covariance matrix is

(2) The sensor sends a pilot signal to the vehicle-mounted computer, and the computer estimates channel impulse response according to the received signal and obtains channel amplitude gain;

the wireless channel considers a rayleigh flat fading channel, and the pilot signal is s (k), then the received signal is:

\tilde{r} (k) = \tilde{h} (k) s (k) + \tilde{n} (k),

wherein,is an equivalent low-pass received signal;is an equivalent low-pass channel impulse response, is a zero-mean complex Gaussian random process with a covariance of For additive noise of the channel, the covariance isZero-mean complex white gaussian noise;

the vehicle-mounted computer estimates the channel impulse response according to the received signal:

\overset{\hat{~}}{h} (k) = \frac{σ_{h}^{2} s (k)}{σ_{n}^{2} + σ_{h}^{2} s^{2} (k)} \tilde{r} (k),

wherein,the channel impulse response estimation value is obtained, and the modulus of the channel impulse response estimation value is the channel amplitude gain h (k);

(3) the sensor measures the vehicle speed, and the vehicle speed is amplified and transmitted to the vehicle-mounted computer through the analog communication device; the sensor measures the vehicle speed:

y(k)＝Cx(k)+v(k)，

wherein y (k) is a sensor measurement; c ═ 10]The visibility of the system is ensured; v (k) is observed noise, is covarianceZero mean gaussian white noise;

the measured value is amplified and then transmitted to the vehicle-mounted computer through the analog communication device:

z(k)＝αh(k)y(k)+n(k)，

where z (k) is the received signal, α is the amplification factor, n (k) is the channel additive noise, and the covariance isZero mean gaussian white noise;

(4) aiming at the state space model in the step (1), considering the problem of LQG control, providing an LQG control performance index, wherein the LQG is Linear quadratic Gaussian control;

LQG performance index is J^cThe following are:

J^{c} = \lim_{M &RightArrow; \infty} E {Σ_{k = 0}^{M - 1} [x^{T} (k) Qx (k) + R u^{2} (k)]},

wherein M is a terminal time; q, R is weight coefficient, Q is not less than 0, R is more than 0; e (.) represents a desired operation;

^Trepresents a transpose of a matrix; Σ (·) denotes a summation operation;the limit is calculated when M tends to be infinite; wherein the first term is a measure of the state tracking condition and the second term is a measure of the control energy consumption;

(5) further considering communication power consumption in the performance indexes of the step (4) to obtain new performance indexes; the new performance index is J, as follows:

J = \lim_{M &RightArrow; \infty} E {Σ_{k = 0}^{M - 1} [x^{T} (k) Qx (k) + R u^{2} (k) + S α^{2} y^{2} (k)]},

wherein the third term is a measure of communication power consumption, S is a weight coefficient, S > 0;

(6) designing a control signal according to the new performance index of the step (5) on the premise of designing an amplification factor off line;

based on the off-line design of the amplification factor, the control signal and the amplification factor can be designed separately; at this time, the control signal can be designed to be u by adopting a dynamic programming method^*(k)：

u^{*} (k) = - F \hat{x} (k | k),

Wherein F is a feedback coefficient,least squares estimation of the current time for time k;

f satisfies the following conditions:

F＝[B^TPB+R]^-1B^TPA，

P＝Q+A^T{P-PB[B^TPB+R]^-1B^TP}A+Sα²C^TC，

wherein, P is an intermediate variable and can be solved by adopting a numerical iteration method;

obtaining by a Kalman filter:

\hat{x} (k | k) = \hat{x} (k | k - 1) + K (k) [z (k) - C^{'} (k) \hat{x} (k | k - 1)],

Σ(k|k)＝Σ(k|k-1)-K(k)C′(k)Σ(k|k-1)，

K (k) = Σ (k | k - 1) C^{' T} (k) {[C^{'} (k) Σ (k | k - 1) C^{' T} (k) + σ_{v^{'}}^{2} (k)]}^{- 1},

\hat{x} (k | k - 1) = A \hat{x} (k - 1 | k - 1) + Bu (k - 1),

Σ (k | k - 1) = AΣ (k - 1 | k - 1) A^{T} + σ_{w}^{2},

C′(k)＝h(k)αC，

σ_{v^{'}}^{2} (k) = h^{2} (k) α^{2} σ_{v}^{2} + σ_{n}^{2};

wherein,is the predicted value of time k-1 to time k, Σ (k | k-1) is the prediction error variance, Σ (k | k) is the estimated error variance of time k to the current time, k (k) is the Kalman gain coefficient, C' (k)Is an intermediate variable; the initial value of the algorithm isΣ(0|-1)＝Σ₀；

(7) Designing an amplification factor based on the control signal of the step (6);

when the control signal is u^*(k) Then the new performance index J in step (5) will be J (u)^*)：

J (u^{*}) = S α^{2} σ_{v}^{2} + tr {Q σ_{w}^{2} + S α^{2} C^{T} C σ_{w}^{2} + Ψ \overset{&OverBar;}{Σ}},

Wherein Ψ ═ F^T[B^TPB+R]F; tr (.) denotes the trace of the matrix;is composed ofThe limit value at which the time tends to infinity,represents Σ (k | k) versus channel state H_kH (0), …, h (k) as desired;

the amplification factor is designed such that J (u)^*) The size is extremely small; due to the fact thatThe calculation is very complex, and a suboptimal solution of the amplification factor is obtainedLet J (u)^*) Upper bound J' (u)^*) The size is extremely small;

J′(u^*) Is obtained by the following formula:

J^{'} (u^{*}) = S α^{2} σ_{v}^{2} + tr {Q σ_{w}^{2} + S α^{2} C^{T} C σ_{w}^{2} + ΨΓ}

wherein is prepared fromThe upper bound of (c), satisfies:

Γ = A^{- 1} (Ω - σ_{w}^{2}) {(A^{- 1})}^{T},

Ω = σ_{w}^{2} + AΩ A^{T} - \frac{A {ΩC}^{T} CΩ A^{T}}{CΩ C^{T} + σ_{v}^{2}} [1 - γexp (γ) E_{1} (γ)],

γ = \frac{σ_{n}^{2}}{σ_{h}^{2} α^{2} [CΩ C^{T} + σ_{v}^{2}]};

wherein omega and gamma are intermediate variables; e₁(γ) is an exponential integration; during calculation, a numerical iteration method can be adopted for solving;

solving by adopting a mode search method, and calling a Matlab command:

X＝patternsearch(fun,x₀,a,b,aeq,beq,lb,ub)；

wherein, pattern search is a command for finding the minimum value point of the target function fun by adopting a pattern search method in Matlab, and x is used₀The method comprises the following steps that (1) an initial value point is obtained, a, b, aeq and beq are parameters of inequality constraint and equality constraint conditions respectively, lb and ub are a lower bound and an upper bound of a variable value range, and a minimum value point obtained through optimization is X;

(8) substituting the measurement data of the steps (2) and (3) and the amplification factor of the step (7) into the step (6) to obtain a control signal, wherein the new performance index of the step (5) is extremely small, and the total consumption of the LQG control and the communication power is optimized;

the total consumption of the LQG control and the communication power is the new performance index J in the step (5):

J = \lim_{M &RightArrow; \infty} E {Σ_{k = 0}^{M - 1} [x^{T} (k) Qx (k) + R u^{2} (k) + S α^{2} y^{2} (k)]},

when the amplification factor α and the control signal u (k) are taken asAnd u^*(k) J will be minimal and the overall LQG control and communication power consumption will be optimized.