Dong W. Distributed optimal control of multiple systems[J]. International Journal of Control, 2010, 83(10): 2067-2079.

3.2 General communication

由于所有智能體 $i$ 的動力學狀態都是相同的，針對系統 $i$ 我們可以構造一個估計器來估算其它所有系統的狀態。針對 $j$ 的估算器（$i$ 來估算 $j$）形式如下：
$${\dot{x}}_j^i=A{x}_j^i+B{u}_j^i$$

為了便于表示，我們定義
$$x_i^i=x_i$$

$$u_i^i=u_i$$

整理狀態寫成一個緊湊的形式 （compact form）
$${\dot{x}}^i=\bar{A}{x}^i+\bar{B}{u}^i$$

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定理 3.2

分布式控制法則為：
$$u_i=((e_i{e}_i^T)?I_r)u^i$$

$$u^i=?R^{?1}{\bar{B}}^{T}P{x}^i?{\bar{B}}^{?1}\sum _{j\in N_i}{w}_{ij}(x^i?x^j)$$

$${\dot{x}}_j^i=A{x}_j^i+B((e_j{e}_j^T)?I_r)u^i,1\le j=i\le m$$

這里，$P$ 是黎卡提方程的對稱正定解，$w_{ij}>0$ 常量，$e_i$ 是 $n$ 維向量第 $i$ 個元素為 1 其他為 0。

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證明

有控制法則
$$u^i=?R^{?1}{\bar{B}}^{T}P{x}^i?{\bar{B}}^{?1}\sum _{j\in N_i}{w}_{ij}(x^i?x^j)$$

我們可以得到
$$\begin{array}{rl}{\dot{x}}^i& =\bar{A}{x}^i+\bar{B}{u}^i\\ & =\bar{A}{x}^i+\bar{B}(?R^{?1}{\bar{B}}^{T}P{x}^i?{\bar{B}}^{?1}\sum _{j\in N_i}{w}_{ij}(x^i?x^j))\\ & =\bar{A}{x}^i?\bar{B}{R}^{?1}{\bar{B}}^{T}P{x}^i?\sum _{j\in N_i}{w}_{ij}(x^i?x^j)\\ & =(\bar{A}?\bar{B}{R}^{?1}{\bar{B}}^{T}P)x^i?\sum _{j\in N_i}{w}_{ij}(x^i?x^j)\end{array}$$

令
$$z^i=x^i?x^{?}$$

那么有
$${\dot{z}}^i=(\bar{A}?\bar{B}{R}^{?1}{\bar{B}}^{T}P)z^i?\sum _{j\in N_i}{w}_{ij}(z^i?z^j)$$

令
$$y^i=e^{?(\bar{A}?\bar{B}{R}^{?1}{\bar{B}}^{T}P)t}{z}^i$$

那么有
$${\dot{y}}^i=?\sum _{j\in N_i}{w}_{ij}(y^i?y^j)$$

拉普拉斯矩陣定義為
$$\begin{array}{r}{l}_{ij}=?\begin{array}{rlr}& ?w_{ij},& j\in N_i\\ & \sum _{l\in N_i}?w_{il},& i=j\\ & 0,& j\in N_i\end{array}\end{array}$$

那么結合拉普拉斯矩陣的定義，${\dot{y}}^i$ 可以寫成
$$\dot{y}=?\bar{L}y$$

因此，
$$y(t)=e^{\bar{L}t}y(0)$$

結合其他引理可得
$$\underset{t\to \infty }{\lim }(y^i?c1)=0$$

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Simulations

%% Distributed optimal control of multiple systems % Author: Zhao-Jichao % Date: 2021-07-05 clear clc%% Define Initial States A = [0 11 1]; B = [0 00 1]; x1 = [ 5 2]'; x2 = [-3 3]'; x3 = [-4 2]'; x4 = [ 1 -1]'; x5 = [-2 7]'; Q1 = [2 11 2]; R1 = [1 00 1]; m = 5; % Number of agents Abar = kron(eye(5), A); Bbar = kron(eye(5), B); Q = kron(eye(5), Q1); R = kron(eye(5), R1); % R = eye(5); [P, l, g] = care(Abar, Bbar, Q, R);in = [x1', x2', x3', x4', x5']';u = -inv(R) * Bbar' * P * in;global dx dx = (Abar - Bbar * inv(R) * Bbar' * P);%% [t, out] = ode45(@odeFun, [0, 10], in);%% Draw Results subplot(2,3,1) plot(t, out(:,1), t,out(:,2), 'linewidth',1.5); hold on; grid on legend('x_1(1)','x_1(2)'); subplot(2,3,2) plot(t, out(:,3), t,out(:,4), 'linewidth',1.5); hold on; grid on legend('x_2(1)','x_2(2)'); subplot(2,3,3) plot(t, out(:,5), t,out(:,6), 'linewidth',1.5); hold on; grid on legend('x_3(1)','x_3(2)'); subplot(2,3,4) plot(t, out(:,7), t,out(:,8), 'linewidth',1.5); hold on; grid on legend('x_4(1)','x_4(2)'); subplot(2,3,5) plot(t, out(:,9), t,out(:,10),'linewidth',1.5); hold on; grid on legend('x_5(1)','x_5(2)');%% DDE Function function out = odeFun(~, in)global dxout = dx * in; end

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先把程序留著，下一步準備探討作者設計的分布式協議

%% Distributed optimal control of multiple systems % Author: Zhao-Jichao % Date: 2021-07-05 clear clc%% Define Initial States A = [0 11 1]; B = [0 00 1]; x1 = [ 5 2]'; x2 = [-3 3]'; x3 = [-4 2]'; x4 = [ 1 -1]'; x5 = [-2 7]'; Q1 = [2 11 2]; R1 = [1 00 1]; m = 5; % Number of agents Abar = kron(eye(5), A); Bbar = kron(eye(5), B); Q = kron(eye(5), Q1); R = kron(eye(5), R1); [P, l, g] = care(Abar, Bbar, Q, R);in = [x1', x2', x3', x4', x5']';u = -inv(R) * Bbar' * P * in;global dx dx = (Abar - Bbar * inv(R) * Bbar' * P);L = [2 0 0 -1 -1-1 1 0 0 00 -1 1 0 00 0 -1 1 00 -1 0 0 1];% L1 = [0 0 0 0 0 % 0 0 0 0 0 % 0 0 0 0 0 % 0 0 0 0 0 % 0 0 0 0 0]; % L2 = [0 0 0 0 0 % 0 0 0 0 0 % 0 0 0 0 0 % 0 0 0 0 0 % 0 0 0 0 0];%% [t, out] = ode45(@odeFun, [0, 10], in);%% Draw Results subplot(2,3,1) plot(t, out(:,1), t,out(:,2), 'linewidth',1.5); hold on; grid on legend('x_1(1)','x_1(2)'); subplot(2,3,2) plot(t, out(:,3), t,out(:,4), 'linewidth',1.5); hold on; grid on legend('x_2(1)','x_2(2)'); subplot(2,3,3) plot(t, out(:,5), t,out(:,6), 'linewidth',1.5); hold on; grid on legend('x_3(1)','x_3(2)'); subplot(2,3,4) plot(t, out(:,7), t,out(:,8), 'linewidth',1.5); hold on; grid on legend('x_4(1)','x_4(2)'); subplot(2,3,5) plot(t, out(:,9), t,out(:,10),'linewidth',1.5); hold on; grid on legend('x_5(1)','x_5(2)'); %% DDE Function function out = odeFun(~, in)global dxout = dx * in; end

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