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制御工学Advent Calendar 2019

Day 20
@HppyCtrlEngnrng

閉ループ同定

Last updated at Posted at 2019-12-23

どうも、コッラー川渡河中の幸福の制御工学です。

はじめに

閉ループ同定についてまとめます。

開ループ同定と閉ループ同定

システム同定を行う際には、外部から同定用の励振信号を入力し、励起された同定対象の出力を観測します。通常は図1のように開ループで実験を行えばよいのですが、同定対象が不安定であったり、制御系の運転効率を大きく落とせない場合には図2のようにフィードバックループを構成して制御を行いながら同定も実行する閉ループ同定を実施します。

image.png
図1 開ループ同定実験のブロック線図

path5892-2-1.png
図2 閉ループ同定実験のブロック線図

ここで問題になるのが、未知雑音$e(t)$がフィードバックループにより同定対象の入力に影響することです。これにより、雑音と入力の無相関性を仮定する同定手法は有効に機能せず、同定結果にバイアスを生じます。

これは直感的には以下のように説明できます。例として、励振信号$r(t)$をステップ信号とし、未知雑音$e(t)$が正弦波である場合について考えます。もし開ループで同定しているのであれば、図3のように雑音の影響下(灰線)でもステップ励振信号と相関のありそうな出力信号(橙線)をとらえることができます。

image.png
図3 開ループ同定における雑音の影響

一方、閉ループ同定においては図4のように、同定対象への入力$u(t)$に雑音がフィードバックされるため、$y(t)$からステップ応答を読み取ることが困難になっています。

image.png
図4 閉ループ同定における雑音の影響

閉ループ同定アルゴリズムと実機実験

閉ループ同定問題に対するアルゴリズムとしては、開ループ同定と同様に同定対象の入出力を用いる直接法のほか、一度閉ループ特性を開ループ同定してから同定対象を求める間接法などが提案されています。

ここでは、間接法の一種であるCL-MOESPを用いた実機実験を行います。CL-MOESPは間接法の一種で、閉ループ感度を開ループ同定したのちに同定対象を求める(のと同等の演算を行う)アルゴリズムです。

次元推定をさぼったMATLABコードはこちら
function P = clmoesp(y, u, r, s, n, N, dt)
    if ( size(y,1) > size(y,2) )
        y = y';
    end
    
    if ( size(u,1) > size(u,2) )
        u = u';
    end
    
    if ( size(r,1) > size(r,2) )
        r = r';
    end
    
    m = size(u, 1);
    l = size(y, 1);
    
    R_0_2s_N = blkhankel(r, 0, 2*s, N);
    U_0_2s_N = blkhankel(u, 0, 2*s, N);
    Y_s_s_N = blkhankel(y, s, s, N);
    
    L = lq([R_0_2s_N; U_0_2s_N; Y_s_s_N]);

    L31 = L(2*m*s+1:3*m*s, 1:m*s); L32 = L(2*m*s+1:3*m*s, m*s+1:2*m*s);
    L41 = L(3*m*s+1:4*m*s, 1:m*s); L42 = L(3*m*s+1:4*m*s, m*s+1:2*m*s);
    L51 = L(4*m*s+1:end, 1:m*s);   L52 = L(4*m*s+1:end, m*s+1:2*m*s);
    
    P1 = L31-(L31*L41'+L32*L42')/(L41*L41'+L42*L42')*L41;
    P2 = L32-(L31*L41'+L32*L42')/(L41*L41'+L42*L42')*L42;
    
    Gam1_2 = (L51*P1'+L52*P2')/sqrtm(P1*P1'+P2*P2');

    [U,~,~] = svd(Gam1_2);
    Un = U(:,1:n);
    Ur = U(:,n+1:end);
    Un_ = Un(1:l*(s-1),:);

    C = Un(1:l,:);
    A = Un_\Un(l+1:l*s,:);

    alpha = Ur';
    beta = Ur'*(L51*L41'+L52*L42')/(L41*L41'+L42*L42');

    Alpha = zeros(size(alpha,1)*s, size(alpha,2));
    for i = 1:s
        Alpha(size(alpha,1)*(i-1)+1:size(alpha,1)*i,:) = [alpha(:,l*(i-1)+1:end) zeros(l*s-n, l*(i-1))];
    end

    DB = (Alpha*blkdiag(eye(l), Un_)) \ reshape(beta, numel(beta),1);

    D = 0*DB(1:l,:);
    B = DB(l+1:end,:);

    P = ss(A,B,C,D,dt);
end

function X_i_j_N = blkhankel(x, i, j, N)
    l = size(x,1);
    X_i_j_N = zeros(l*j, N);
    
    for n = 1:N
        X_i_j_N(:,n) = reshape(x(:,i+n:i+j+n-1), l*j, 1);
    end
end

function [L, Q] = lq(A)
    [Q, R] = qr(A');
    L = R';
    Q = Q';
end

また、MATLABに閉ループ同定用のビルトイン関数がないか調査していたところ、こちらにPEMによる直接法が紹介されていましたのでこれも試してみます。

実機は手前味噌ですが去年の制御工学Advent Calendarの記事のArduinoを用いた実験キットを再投入しました。
image.png
図4 実験環境外観図
image.png
図5 回路図

開ループ同定実験

まずは開ループ同定を実施してみます。以下のコードの通り、サンプリング周期は10 msとし、励振信号$r$を更新周期100サンプル、振幅2 VのM系列、雑音$e$を振幅0.1 Vのビルトイン乱数として実験を実施しました。

開ループ同定実験用のArduinoコード
#include <TimerOne.h>

// ピン設定
#define INPUT_PIN 3
#define OUTPUT_PIN_1 6
#define OUTPUT_PIN_2 5

// 同定用M系列更新周期
#define MSEQ_WIDTH 100

// 状態変数
float V1, Vo;
// 入力
float Vi;

// シリアル出力
void serOut(){
  Serial.print(Vi, 4);
  Serial.print(',');
  Serial.print(V1, 4);
  Serial.print(',');
  Serial.println(Vo, 4);
}

char mseq(){
  // シフトレジスタ
  static unsigned char Xn = 0x55;
  static unsigned int cnt = MSEQ_WIDTH;
  unsigned ret;

  cnt--;
  ret = ((Xn & 64) >> 6) ^ (Xn & 1);
  if (cnt == 0){
    Xn = (Xn << 1) + ret;
    cnt = MSEQ_WIDTH;
  }

  return ret;
}

// AD値(0~1023)を-2.5~2.5Vに変換
float convAdc(float y){
  return (y - 511) / 1023.0 * 5.0;
}

// -2.5~2.5VをPWM値(0~255)に変換
int convPwm(float u){
  int ret = 255/5*u + 128;

  if (ret > 255)
    ret = 255;
  else if (ret < 0)
    ret = 0;

  return ret;
}

// 制御則の実装
void control() {
  V1 = convAdc(analogRead(OUTPUT_PIN_1));
  Vo = convAdc(analogRead(OUTPUT_PIN_2)) + 0.1*random(-1,1);

  // M系列を入力
  Vi = 4.0 * (mseq() - 0.5);

  analogWrite(INPUT_PIN, convPwm(Vi));
  serOut();
}

void setup() {
  Serial.begin(230400);

  Timer1.initialize(10000);
  Timer1.attachInterrupt(control);

  Timer1.start();
}

void loop() {
}

実験にて取得したデータを図6に示します。
image.png
図6 開ループ同定試験データ

取得したデータを開ループ同定アルゴリズムN4SIDにて同定したところ、図7の結果を得ました。同図より、開ループであれば雑音の影響下でも良好に同定できていることがわかります。
image.png
図7 N4SIDによる開ループ同定結果

MATLABによる開ループ同定スクリプト
clear;
close all;

dt = 10e-3;

dat = csvread('openloop.csv');
u = dat(:,1);
y = dat(:,3:-1:2);
idd = iddata(y, u, dt);

P_n4sid = n4sid(idd, 2);

P_nominal = struct2array(load('nominal.mat'));

bode(P_nominal, P_n4sid);
legend('whitebox', 'n4sid');

閉ループ同定実験

次に本題の閉ループ同定実験を行うため、比例ゲイン2.0、積分ゲイン0.6のPI制御器により直結フィードバック系を構成し、励振信号$r$を200サンプル周期、振幅1 VのM系列として再度実験します。雑音$e$に関しては前項と同様です。

閉ループ同定実験用のArduinoコード
#include <TimerOne.h>

// ピン設定
#define INPUT_PIN 3
#define OUTPUT_PIN_1 6
#define OUTPUT_PIN_2 5

// 同定用M系列更新周期
#define MSEQ_WIDTH 200

// 状態変数
float V1, Vo;
// 入力
float Vi;
// 励振信号
float Vr;

// シリアル出力
void serOut(){
  Serial.print(Vr, 4);
  Serial.print(',');
  Serial.print(Vi, 4);
  Serial.print(',');
  Serial.print(V1, 4);
  Serial.print(',');
  Serial.println(Vo, 4);
}

char mseq(){
  // シフトレジスタ
  static unsigned char Xn = 0x55;
  static unsigned int cnt = MSEQ_WIDTH;
  unsigned ret;

  cnt--;
  ret = ((Xn & 64) >> 6) ^ (Xn & 1);
  if (cnt == 0){
    Xn = (Xn << 1) + ret;
    cnt = MSEQ_WIDTH;
  }

  return ret;
}

// AD値(0~1023)を-2.5~2.5Vに変換
float convAdc(float y){
  return (y - 511) / 1023.0 * 5.0;
}

// -2.5~2.5VをPWM値(0~255)に変換
int convPwm(float u){
  return 255/5*u + 128;
}

// PI制御演算
float pi_ctrl(float err) {
  static float esum = 0.0;
  float u;
  
  esum += err * 10e-3;
  u =  2.0*err + 0.6*esum;

  
  if (u > 2.5)
    u = 2.5;
  else if (u < -2.5)
    u = -2.5;

  return u;
}

// 制御則の実装
void control() {
  V1 = convAdc(analogRead(OUTPUT_PIN_1));
  Vo = convAdc(analogRead(OUTPUT_PIN_2)) + 0.1*random(-1,1);

  // 制御演算
  Vr = 2.0 * (mseq() - 0.5);
  Vi = pi_ctrl(-Vo) + Vr;

  analogWrite(INPUT_PIN, convPwm(Vi));
  serOut();
}

void setup() {
  Serial.begin(230400);

  Timer1.initialize(10000);
  Timer1.attachInterrupt(control);

  Timer1.start();
}

void loop() {
}

実験にて取得したデータを図8に示します。フィードバックにより、入力$u$に雑音の影響が確認できます。

image.png
図8 閉ループ同定実験データ

このデータに対し、前項と同様にN4SIDにより同定を行った結果を図9に示します。雑音は前項と同じものにも関わらず、$V_1$の同定精度が大きく悪化しています。
image.png
図9 N4SIDによる閉ループ直接同定

次にCL-MOESPおよびPEMによる閉ループ同定を試みます。CL-MOESPは先述の通り、PEMに関しては以下の通りN4SIDにより1次同定し、その後PEMによる修正を行うようにしています。

P_pem = pem(idd, P_n4sid, 'dist', 'estim');
MATLABによる閉ループ同定スクリプト
clear;
close all;

dt = 10e-3;

dat = csvread('closedloop.csv');
r = dat(:,1);
u = dat(:,2);
y = dat(:,4:-1:3);
idd = iddata(y, u, dt);

P_n4sid = n4sid(idd, 2);
P_clmoesp = clmoesp(y, u, r, 100, 2, 4000, dt);
P_pem = pem(idd, P_n4sid, 'dist', 'estim');

P_nominal = struct2array(load('nominal.mat'));

figure();
bode(P_nominal, P_n4sid);
legend('whitebox', 'n4sid');

figure();
bode(P_nominal, P_clmoesp, P_pem);
legend('whitebox', 'CL-MOESP', 'PEM');

結果は図11のようになり、閉ループ制御下においても同定できていることがわかります。
image.png

まとめ

閉ループ環境下では雑音の影響がプラント入力に現れるため、適切な手法を取らなければいけない、というお話でした。以下に実験データを置いておくのでよろしければどうぞ。

開ループ同定実験データ
励振信号r, 出力V1, 出力Vo 2.0000,1.8671,0.5670 2.0000,1.8866,0.5718 2.0000,1.8719,0.4718 2.0000,1.8768,0.4767 2.0000,1.8719,0.5865 2.0000,1.8719,0.4914 2.0000,1.8817,0.5963 2.0000,1.8671,0.4963 2.0000,1.8719,0.5012 2.0000,1.8671,0.6012 2.0000,1.8671,0.6109 2.0000,1.8817,0.6207 2.0000,1.8768,0.6158 2.0000,1.8817,0.6256 2.0000,1.8768,0.6256 2.0000,1.8768,0.6354 2.0000,1.8817,0.6403 2.0000,1.8768,0.5403 2.0000,1.8817,0.5500 2.0000,1.8719,0.6500 2.0000,1.8817,0.5598 2.0000,1.8915,0.5696 2.0000,1.8817,0.6647 2.0000,1.8817,0.6745 2.0000,1.8817,0.6696 2.0000,1.8915,0.5843 2.0000,1.8964,0.6940 2.0000,1.8768,0.5843 2.0000,1.8915,0.6989 2.0000,1.8719,0.5940 2.0000,1.8866,0.7038 2.0000,1.8915,0.7087 2.0000,1.8915,0.6087 2.0000,1.8915,0.7185 2.0000,1.8866,0.6185 2.0000,1.8964,0.7283 2.0000,1.8964,0.7380 2.0000,1.8817,0.7331 2.0000,1.8866,0.6380 2.0000,1.8817,0.7380 2.0000,1.8915,0.7478 2.0000,1.8915,0.7527 2.0000,1.8866,0.6478 2.0000,1.8964,0.7576 2.0000,1.8915,0.6576 2.0000,1.8964,0.6674 2.0000,1.9013,0.7771 2.0000,1.8866,0.7722 2.0000,1.8964,0.6820 2.0000,1.8866,0.6820 2.0000,1.8915,0.6918 2.0000,1.8866,0.7918 2.0000,1.8915,0.6869 2.0000,1.9013,0.8016 2.0000,1.8915,0.7016 2.0000,1.8964,0.7065 2.0000,1.9013,0.7162 2.0000,1.8915,0.7113 2.0000,1.8964,0.8211 2.0000,1.8964,0.7260 2.0000,1.8915,0.8309 2.0000,1.8915,0.8358 2.0000,1.8964,0.8309 2.0000,1.9062,0.7456 2.0000,1.9013,0.7407 2.0000,1.9013,0.8456 2.0000,1.9062,0.7553 2.0000,1.8964,0.7553 2.0000,1.9013,0.8602 2.0000,1.8964,0.8651 2.0000,1.8964,0.8651 2.0000,1.9013,0.8749 2.0000,1.8964,0.8700 2.0000,1.9110,0.7895 2.0000,1.8964,0.7749 2.0000,1.9013,0.7847 2.0000,1.9159,0.8944 2.0000,1.8964,0.8895 2.0000,1.9110,0.9042 2.0000,1.9013,0.8993 2.0000,1.9013,0.9042 2.0000,1.9013,0.9140 2.0000,1.9013,0.8091 2.0000,1.9110,0.8140 2.0000,1.9110,0.8189 2.0000,1.9062,0.9238 2.0000,1.9208,0.8335 2.0000,1.9013,0.8286 2.0000,1.9110,0.9335 2.0000,1.9062,0.8384 2.0000,1.9110,0.9433 2.0000,1.9062,0.9433 2.0000,1.9013,0.8482 2.0000,1.9257,0.8580 2.0000,1.9110,0.8531 2.0000,1.9159,0.9580 2.0000,1.9208,0.9677 2.0000,1.9013,0.9580 2.0000,1.9110,0.9726 2.0000,1.9062,0.9677 2.0000,1.9110,0.9775 2.0000,1.9110,0.9775 2.0000,1.9013,0.8775 2.0000,1.9208,0.8873 2.0000,1.9159,0.9873 2.0000,1.9159,0.9922 2.0000,1.9257,0.9971 2.0000,1.9062,0.9922 2.0000,1.9159,0.9020 2.0000,1.9159,1.0068 2.0000,1.9110,1.0068 2.0000,1.9159,0.9166 2.0000,1.9013,1.0117 2.0000,1.9306,1.0215 2.0000,1.9208,0.9215 -2.0000,1.9208,1.0264 -2.0000,1.5200,0.9362 -2.0000,1.1437,1.0215 -2.0000,0.8113,1.0313 -2.0000,0.5181,0.9215 -2.0000,0.2639,1.0215 -2.0000,0.0489,1.0215 -2.0000,-0.1564,0.9117 -2.0000,-0.3226,0.9117 -2.0000,-0.4839,0.9971 -2.0000,-0.6207,0.9971 -2.0000,-0.7331,0.9020 -2.0000,-0.8456,0.8775 -2.0000,-0.9433,0.8775 -2.0000,-1.0362,0.8726 -2.0000,-1.1095,0.8629 -2.0000,-1.1730,0.9580 -2.0000,-1.2366,0.8433 -2.0000,-1.2805,0.8384 -2.0000,-1.3294,0.8238 -2.0000,-1.3685,0.9189 -2.0000,-1.4076,0.9091 -2.0000,-1.4418,0.7944 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