JP3611180B2 - Adaptive control method for periodic signals - Google Patents

Description

【００２６】
以上の数７の導き出し方については、後ほど実施例１の導出の項で段階を踏んで詳しく説明する。
前記数７では、直交化係数ｐ（ｎ），ｑ（ｎ）の両更新式は、ｓｉｎ，ｃｏｓ の違いがあるだけで、数式の次元も構成も同じで同格である。それゆえ、両ステップサイズパラメータμ_ｐ，μ_ｑが互いに同等であれば、両直交化係数ｐ（ｎ），ｑ（ｎ）の適応速度は互いに同等である。その結果、適応係数ベクトル更新アルゴリズムとして前記数７を採用すれば、両直交化係数ｐ（ｎ），ｑ（ｎ）がほぼ同時に適応し、一方の適応速度が他方よりも遅れてしまうことがないので、適応速度が向上するという効果がある。
【００２７】
また、前記数７では、直交化係数ｐ（ｎ），ｑ（ｎ）の更新式は最も簡素であり、適応係数ベクトル更新アルゴリズムの演算負荷が軽減される。したがって、前記数７を適応係数ベクトル更新アルゴリズムとして採用すれば、前述の理由で適応速度が向上するばかりではなく、演算手段にかかる負荷が軽減されるという効果もある。
【００２８】
ここで、前記数７の右辺第二項を正規化する第一法としては、伝達特性の推定ゲインをＡ＾（ｎ）として、Ａ＾^２（ｎ）＋γ≡Ｐ＾^２（ｎ）＋Ｑ＾^２（ｎ）＋γで割るという手段がある。ここで、γはゼロ以上の発散防止定数である。
すると、ほぼ適応した状態では伝達特性のゲインに比例する傾向がある誤差信号ε（ｎ）と同じ傾向がある推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）との積を、ゲイン推定値Ａ＾^２（ｎ）≡Ｐ＾^２（ｎ）＋Ｑ＾^２（ｎ）で正規化していることになる。それゆえ、伝達特性のゲインが低下した場合にも、正規化された適応係数ベクトル更新アルゴリズムの適応速度はあまり低下することがない。そればかりではなく、ゲイン推定値Ａ＾（ｎ）に正の発散防止定数γを足して正規化すれば、ゲイン推定値Ａ＾（ｎ）がゼロ付近にまで低下することがあっても、適応係数ベクトル更新アルゴリズムが発散することは防止される。
【００２９】
それゆえ、適応係数ベクトル更新アルゴリズムに前記第一法の正規化が施されていれば、伝達特性のゲインが低下しても、適応係数ベクトル更新アルゴリズムの適応速度が衰えないという効果がある。そればかりではなく、伝達特性の推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）が両方ともゼロ付近になっていても、適応係数ベクトル更新アルゴリズムの発散が防止されるという効果もある。
【００３０】
なお、前記数７の右辺第二項には、明瞭に推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）が含まれているばかりではなく、誤差信号ε（ｎ）にも推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）が含まれているものと考え得る。それゆえ、前記数７の右辺第二項をＰ＾^２（ｎ）＋Ｑ＾^２（ｎ）＋γで割るという手段は、推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）の無次元化の視点からしても妥当である。
【００３１】
また、前記数７の右辺第二項を正規化する第二法として、√｛ｐ^２（ｎ）＋ｑ^２（ｎ）｝＋γ_２ または√｛ｐ^２（ｎ）＋ｑ^２（ｎ）＋γ_２ ｝で右辺第二項を割るという手段もある。ここで、γ_２ はゼロ以上の発散防止定数である。右辺第二項中に含まれる誤差信号ε（ｎ）は、適応信号ｙ（ｎ）＋周期性信号ｆ（ｎ）であるから、周期性信号ｆ（ｎ）が小さい場合には、適応信号ｙ（ｎ）が適正に収束していなくても、周期性信号ｆ（ｎ）が小さいことに起因して誤差信号ε（ｎ）の絶対値が小さい場合があり得る。そこで、この第二法の手段をとれば、周期性信号ｆ（ｎ）が小さい場合にも適応速度が速やかになり誤差信号ε（ｎ）の収束が速くなるという効果がある。
【００３２】
あるいは、前記数７の右辺第二項を正規化する手段の第三法として、前記第一法および前記第二法を併用するという手段もある。この第三法によれば、前記第一法および前記第二法の効果を併せ持つという効果がある。
（第３手段）
本発明の第３手段は、請求項３記載の周期性信号の適応制御方法である。
【００３３】
すなわち本手段では、前記伝達特性推定アルゴリズムは、次の数８および数９の一対の更新式と該数８および該数９の右辺第二項を正規化した一対の更新式とのうち一方に従って、伝達特性の推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）を更新する。
【００３４】
【数８】

【００３５】
【数９】

【００３６】
【数１０】

以上の数８および数９の導き出し方は、後ほど実施例１の導出の項で順を追って具体的に説明する。
【００３７】
前記数８および前記数９では、伝達特性の推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）の両更新式は、ｓｉｎ，ｃｏｓ の違いがあるだけで、数式の次元も構成も同じで同格である。それゆえ、両ステップサイズパラメータμ_Ｐ，μ_Ｑが互いに同等であれば、両推定直交化係数の適応速度は同等である。その結果、適応係数ベクトル更新アルゴリズムとして前記数８および数９を採用すれば、両推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）がほぼ同時に適応し、一方の適応速度が他方よりも遅れてしまうことがないので、適応係数ベクトル更新アルゴリズムの適応速度が向上するという効果がある。
【００３８】
そればかりではなく、本手段の伝達特性の直交化表現した推定値である推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）の推定が前記数８および前記数９に則って行われる場合には、第一手段で述べた効果に加え、比較的演算負荷が小さくてすむという効果がある。
ここで、前記数８および前記数９の右辺第二項を正規化する第一法としては、両数式の右辺第二項を√｛Ｐ＾^２（ｎ）＋Ｑ＾^２（ｎ）｝＋γ’または√｛Ｐ＾^２（ｎ）＋Ｑ＾^２（ｎ）＋γ’｝で割るという手段がある。ここで、γ’はゼロ以上の発散防止定数である。すると、伝達特性のゲインが低下した場合にも適応速度はあまり低下することがない。そればかりではなく、ゲイン推定値Ａ＾（ｎ）≡√｛Ｐ＾^２（ｎ）＋Ｑ＾^２（ｎ）｝に正の発散防止定数γ’を足して正規化しているわけであるから、ゲイン推定値Ａ＾（ｎ）がゼロ付近にまで低下することがあっても、適応係数ベクトル更新アルゴリズムが発散することは防止されている。それゆえ、適応係数ベクトル更新アルゴリズムに前記第一法の正規化が施されていれば、伝達特性のゲインが低下しても適応速度が衰えないばかりではなく、ゲイン推定値Ａ＾（ｎ）≡｛Ｐ＾^２（ｎ）＋Ｑ＾^２（ｎ）｝がゼロ付近になっても発散が防止されるという効果もある。
【００３９】
また、前記数８および前記数９の右辺第二項を正規化する第二法として、ｐ^２（ｎ）＋ｑ^２（ｎ）＋γ”で右辺第二項を割るという手段もある。ここで、γ”はゼロ以上の発散防止定数である。すなわち、右辺第二項中には、ｐ^２（ｎ），ｑ^２（ｎ），ｐ（ｎ）ｑ（ｎ）の項が含まれているので、周期性信号ｆ（ｎ）の振幅が小さいなどの理由で制御信号ｙ（ｎ）の直交化係数ｐ（ｎ），ｑ（ｎ）の絶対値も小さい場合がありうる。また、制御信号ｙ（ｎ）の直交化係数ｐ（ｎ），ｑ（ｎ）の初期値がゼロに設定される場合もあり得る。しかしながら、この第二法によれば、このような場合にも、前記数８および前記数９が発散することは防止される。
【００４０】
あるいは、前記数８および前記数９の右辺第二項を正規化する手段の第三法として、前記第一法および前記第二法を併用するという手段もある。この第三法によれば、演算負荷こそ若干増えるが、前記第一法および前記第二法の効果を併せ持つという効果がある。
（第４手段）
本発明の第４手段は、請求項４記載の周期性信号の適応制御方法である。
【００４１】
すなわち本手段では、抑制すべき特定周波数成分の角振動数ω_ｋ （１≦ｋ≦Ｋ）の個数Ｋと、誤差信号ε_ｌ（ｎ）（１≦ｌ≦Ｌ）の個数Ｌと、前記適応信号ｙ_ｍ（ｎ）（１≦ｍ≦Ｍ）の個数Ｍとのうち、少なくとも一つは複数である。換言すれば本手段は、前述の第１手段をＫ成分・Ｌ入力・Ｍ出力に拡張した周期性信号の適応制御方法である。それゆえ本手段は、多成分多入力多出力の複雑な系に対しても適用することができる。
【００４２】
したがって、本手段を多成分多入力多出力の複雑な系に対して適用した場合、適応信号ｙ_ｍ（ｎ）から誤差信号ε_ｌ（ｎ）に至る伝達特性の変動にも柔軟に対応することができながら、適応速度がより向上するという効果がある。
なお、本手段の具体的な各アルゴリズムの数式については、後ほど実施例２の項で説明する。
【００４３】
【発明の実施の形態】
本発明の周期性信号の適応制御方法の実施の形態については、当業者に実施可能な理解が得られるよう、以下の実施例で明確かつ十分に説明する。
［実施例１］
（実施例１の構成）
本発明の実施例１としての周期性信号の適応制御方法は、図１に示すように、一入力一出力系の適応制御系において実施される。すなわち、適応制御系への入力としての誤差信号ε（ｎ）は一つであり、適応制御系からの出力としての適応信号ｙ（ｎ）も一つである。本実施例での適応制御系は、適応信号発生アルゴリズム１１と、適応係数ベクトル更新アルゴリズム１２と、伝達特性推定アルゴリズム１３と、伝達特性２３とをもつ。
【００４４】
ここで、適応制御系を含む全体的なシステム構成について、予備知識として説明しておく。
周期性信号ｆ（ｎ）は、単一の特定周波数成分を含む外乱信号であって、誤差信号ε（ｎ）を観測する観測点２４に入力される。本実施例の周期性信号の適応制御方法は、この周期性信号ｆ（ｎ）のうち前記特定周波数成分が観測点２４に与える影響を相殺し、観測点２４で検知される誤差信号ε（ｎ）を抑制することを制御目的とする。
【００４５】
ここで、周期性信号ｆ（ｎ）の前記特定周波数成分の角振動数ωは、工学的に十分精密に計測され、適応信号発生アルゴリズム１１および推定伝達特性１３に与えられるものとする。たとえば、周期性信号ｆ（ｎ）が自動車のエンジンに起因する振動加速度であるとすると、その角振動数ωは、点火パルス等の信号を観測することにより、リアルタイムで容易かつ正確に計測することができる。
【００４６】
適応信号発生アルゴリズム１１は、周期性信号ｆ（ｎ）の特定周波数成分の角振動数ωに基づいて、同特定周波数成分に同期した適応信号ｙ（ｎ）を発生させるアルゴリズムである。適応信号ｙ（ｎ）は、所定の伝達特性２３を介して相殺信号ｚ（ｎ）に変換され、観測点２４に加えられる。観測点２４では、周期性信号ｆ（ｎ）と相殺信号ｚ（ｎ）とが合成され、その結果としての誤差信号ε（ｎ）＝ｆ（ｎ）＋ｚ（ｎ）が観測される。前述の自動車のエンジンの例でいえば、観測点２４は乗員席の基台に固定された振動加速度センサに相当し、誤差信号ε（ｎ）は同センサの出力に相当する。
【００４７】
すなわち、本実施例の周期性信号の適応制御方法では、観測点２４に影響を及ぼす周期性信号ｆ（ｎ）に対し、周期性信号ｆ（ｎ）に同期している一つの特定周波数成分からなる適応信号ｙ（ｎ）が、伝達特性２３を介して逆位相で加えられる。こうすることによって、周期性信号ｆ（ｎ）のうち特定周波数成分が観測点２４へ及ぼす影響が能動的に除去され、その結果、観測点２４で検知される誤差信号ε（ｎ）が低減される。
【００４８】
以下、適応信号発生アルゴリズム１１、適応係数ベクトル更新アルゴリズム１２および伝達特性推定アルゴリズム１３を中心として、本実施例の周期性信号の適応制御方法についてより詳しく説明する。
適応信号発生アルゴリズム１１は、更新周期Ｔの離散時刻ｎにおいて、周期性信号ｆ（ｎ）のうち一つの特定周波数成分がもつ角振動数ωに基づき、次の数１１に従って直交化表現された正弦関数である適応信号ｙ（ｎ）を発生させる。
【００４９】
【数１１】
ｙ（ｎ） ＝ ｐ（ｎ）ｓｉｎ（ωＴｎ）＋ｑ（ｎ）ｃｏｓ（ωＴｎ）
一方、前記適応係数ベクトル更新アルゴリズム１２は、次の数１２に従って下記の適応係数ベクトルＷ（ｎ）を更新するアルゴリズムである。
【００５０】
【数１２】

【００５１】
ここで、適応係数ベクトルＷ（ｎ）は、次の数１３に定義するように、適応信号ｙ（ｎ）の特定周波数成分の直交化係数ｐ（ｎ），ｑ（ｎ）を要素とする二要素のベクトルである。
【００５２】
【数１３】

【００５３】
すなわち、適応係数ベクトル更新アルゴリズム１２は、適応係数ベクトルＷ（ｎ）を、伝達特性２３の一対の推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）と誤差信号ε（ｎ）とに基づいて、離散時刻ｎの経過毎に更新するアルゴリズムである。適応係数ベクトル更新アルゴリズム１２による適応係数ベクトルＷ（ｎ）の更新により、周期性信号ｆ（ｎ）の特定周波数成分の振幅、位相または角振動数ωの変動に対応して、適応係数ベクトルＷ（ｎ）の各要素は適応的に調整される。そして、適応係数ベクトル更新アルゴリズム１２により更新された適応係数ベクトルＷ（ｎ）の各要素ｐ（ｎ），ｑ（ｎ）をもって、適応信号発生アルゴリズム１１で発生する適応信号ｙ（ｎ）の特定周波数成分の直交化係数ｐ（ｎ），ｑ（ｎ）が更新される。
【００５４】
さらに、伝達特性推定アルゴリズム１３は、伝達特性２３の直交化推定値である推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）を適応的に調整して、周期性信号ｆ（ｎ）の特定周波数成分の角振動数ωに対応する伝達特性２３を推定する。すなわち、伝達特性２３の推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）を、角振動数ωの特定周波数成分の一周期分にあたる誤差信号ε（ｎ）の差分および直交化係数ｐ（ｎ），ｑ（ｎ）の差分に基づいて、離散時刻ｎの経過毎に更新する。ここで、タップ数（更新周期数）ｉは、Ｔｉが角振動数ωの特定周波数成分の一周期分に相当する時間になる値であるものとすると、前記誤差信号ε（ｎ）の差分は、｛ε（ｎ）−ε（ｎ−ｉ）｝である。同様に、前記直交化係数の差分は、｛ｐ（ｎ）−ｐ（ｎ−ｉ）｝および｛ｑ（ｎ）−ｑ（ｎ−ｉ）｝である。
【００５５】
このような伝達特性推定アルゴリズム１３は、次の数１４および数１５の組み合わせで表記される。
【００５６】
【数１４】

【００５７】
【数１５】

【００５８】
【数１６】

その結果、伝達特性推定アルゴリズム１３によって、抑制すべき特定周波数成分の角振動数ωに対応する伝達特性２３のゲインおよび位相に相当する推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）が推定される。そして、推定された伝達特性２３の推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）は、前述の適応係数ベクトル更新アルゴリズム１２の演算に提供されるに至る。
【００５９】
なお、前述のように、伝達特性推定アルゴリズム１３では、誤差信号の差分｛ε（ｎ）−ε（ｎ−ｉ）｝と両直交化係数の差分｛ｐ（ｎ）−ｐ（ｎ−ｉ）｝，｛ｑ（ｎ）−ｑ（ｎ−ｉ）｝とを使用している。このことを明瞭にするために、図１では、伝達特性推定アルゴリズム１３のブロックの外に一周期分の遅延ブロック１４，１５を描いたが、これらの遅延ブロック１４，１５は伝達特性推定アルゴリズム１３に含めて図示しても構わない。
【００６０】
（実施例１の導出）
以上の本実施例の周期性信号の適応制御方法は、以下のようにして導き出すことができる。
先ず、周期性信号ｆ（ｎ）のうち抑制すべき特定周波数成分の角振動数ωが、工学的に正確に計測されるものする。ここで、周期性信号ｆ（ｎ）の特定周波数成分の変動に従って、角振動数ωも時間変化しうるものとする。
【００６１】
そして、適応信号発生アルゴリズム１１は、直交化表現された次の数１７に従って、適応信号ｙ（ｎ）としての正弦波信号を発生させるものとする。
【００６２】
【数１７】
ｙ（ｎ） ＝ ｐ（ｎ）ｓｉｎ（ωＴｎ）＋ｑ（ｎ）ｃｏｓ（ωＴｎ）
すると、伝達特性２３を介して観測点２４に加えられる相殺信号ｚ（ｎ）は、適応信号ｙ（ｎ）が伝達特性Ｇ［Ａ（ω），Φ（ω）］を直交化表現したものによる直交化係数Ｐ，Ｑの影響を受けたものであるから、次の数１８で表される。ここで、直交化表現された伝達特性Ｇ［Ｐ（ω），Ｑ（ω）］は、ｊを虚数単位として複素表現すれば、Ｇ［Ｐ（ω），Ｑ（ω）］≡Ｐ（ω）＋ｊＱ（ω）と書き表される。
【００６３】
【数１８】

それゆえ、誤差信号ε（ｎ）は、観測点２４において周期性信号ｆ（ｎ）と相殺信号ｚ（ｎ）とが合成されたものであるから、次の数１９によって書き表される。
【００６４】
【数１９】

次に、適応係数ベクトルＷ（ｎ）を適応的に更新する適応係数ベクトル更新アルゴリズム１２は、勾配法に基づいて次の数２０で書き表される。
【００６５】
【数２０】
Ｗ（ｎ＋１） ＝ Ｗ（ｎ）−μ∇（ｎ）
ここで、適応係数ベクトルＷ（ｎ）は、次の数２１に示すように、適応信号ｙ（ｎ）の直交化係数ｐ（ｎ），ｑ（ｎ）を要素とする二要素ベクトルである。
【００６６】
【数２１】

【００６７】
なお、前記数２０において、μはステップサイズパラメータを表すが、単なるスカラーではなく、傾斜ベクトル∇（ｎ）≡∂Ｊ（ｎ）／∂Ｗ（ｎ）の各要素にかけられる調整係数μ_ｐ，μ_ｑを表すものとする。また、適応係数ベクトル更新アルゴリズム１２の直交化係数ｐ（ｎ），ｑ（ｎ）の更新式は、互いに同格であるから、通常、ステップサイズパラメータはμ_ｐ＝μ_ｑと設定して良い。
【００６８】
ここで、評価関数Ｊ（ｎ）を、次の数２２に示すように、誤差信号ε（ｎ）の二乗で定義する。
【００６９】
【数２２】
Ｊ（ｎ） ≡ ε^２（ｎ）
すると、最小二乗法による傾斜ベクトル∇（ｎ）＝∂Ｊ（ｎ）／∂Ｗ（ｎ）は、次の数２３のように表される。
【００７０】
【数２３】
∇（ｎ） ＝ ∂Ｊ（ｎ）／∂Ｗ（ｎ）
＝ ∂ε^２（ｎ）／∂Ｗ（ｎ）
＝ ２ε（ｎ）・∂ε（ｎ）／∂Ｗ（ｎ）
この数２３に前記数１９および前記数２１を代入すると、この数２３は次の数２４のように展開される。
【００７１】
【数２４】

【００７２】
ところで、現実においては、伝達特性２３のゲインＡおよび位相Φに相当する直交化係数Ｐ，Ｑは、真値であるから通常は直接知ることができない。そこで、伝達特性２３の推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）を推定し、この推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）をもって前記数２４中の伝達特性２３の直交化係数Ｐ，Ｑに代えることにより、前述の適応係数ベクトル更新アルゴリズム１２の基本形が得られる。すると、適応係数ベクトル更新アルゴリズム１２は、次の数２５のように書き表される。
【００７３】
【数２５】

【００７４】
ここでさらに、伝達特性２３の推定ゲインの二乗に相当する値に発散防止定数γを足したものでこの数２５の右辺第二項を割って正規化すれば、適応係数ベクトル更新アルゴリズム１２は、次の数２６で書き表される。
【００７５】
【数２６】

【００７６】
この際、伝達特性２３の推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）を求める目的で、伝達特性推定アルゴリズム１３を導入することが必要になる。
そこで最後に、伝達特性推定アルゴリズム１３は、次のようにして導き出すことができる。
先ず、推定偏差δ（ｎ）を次の数２７のように定義する。
【００７７】
【数２７】
δ（ｎ） ≡ ε（ｎ）−｛ｆ（ｎ）＋ｚ＾（ｎ）｝
＝｛ε（ｎ）−ｆ（ｎ）｝−ｚ＾（ｎ）
＝ ｚ（ｎ）−ｚ＾（ｎ）
ここで、推定相殺信号ｚ＾（ｎ）は、伝達特性２３の真値を知ることができないので、伝達特性２３の直交化係数Ｐ，Ｑの代わりに推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）を用いて、相殺信号ｚ（ｎ）を推定したものである。すなわち、推定相殺信号ｚ＾（ｎ）は次の数２８によって定義される。
【００７８】
【数２８】
ｚ＾（ｎ）≡
Ｐ＾（ｎ）｛ｐ（ｎ）ｓｉｎ（ωＴｎ）＋ｑ（ｎ）ｃｏｓ（ωＴｎ）｝
＋Ｑ＾（ｎ）｛ｐ（ｎ）ｃｏｓ（ωＴｎ）−ｑ（ｎ）ｓｉｎ（ωＴｎ）｝
次に、周期性信号ｆ（ｎ）のうち観測点２４への影響を消去すべき特定周波数成分の一周期に相当するタップ数ｉを定め、推定偏差δ（ｎ）の一周期分の差分である周期偏差Ｅ（ｎ）を次の数２９のように定義する。
【００７９】
【数２９】
Ｅ（ｎ）≡δ（ｎ）−δ（ｎ−ｉ）
＝ ε（ｎ） −｛ｆ（ｎ） ＋ｚ＾（ｎ） ｝
−［ε（ｎ−ｉ）−｛ｆ（ｎ−ｉ）＋ｚ＾（ｎ−ｉ）｝］
ここでさらに、周期性信号ｆ（ｎ）は一周期違いで同一の値を取るものと仮定すると、この数２９においてｆ（ｎ）−ｆ（ｎ−ｉ）＝０と置ける。この仮定は、周期性信号ｆ（ｎ）が定常状態にあるときには完全に正しく、周期性信号ｆ（ｎ）が遷移状態にあるときには近似的に正しい。ただし、周期性信号ｆ（ｎ）がステップ状に変動する瞬間には、この仮定は崩れるが、変動後に定常状態または遷移状態に入るならば、この仮定は再び完全にまたは近似的に正しくなる。それゆえ、周期性信号ｆ（ｎ）が急激に変動する瞬間には、伝達特性推定アルゴリズム１３の推定値は乱れるが、周期性信号ｆ（ｎ）の急変が収まり次第、伝達特性推定アルゴリズム１３の推定値は適正な値の付近で安定する。このことは、発明者が数値シミュレーションで確認し、本実施例の周期性信号の適応制御方法が正常に機能することを確認している。そこで前記仮定を受け入れることにすると、前記数２９は次の数３０のように展開される。
【００８０】
【数３０】
Ｅ（ｎ）
＝ ε（ｎ） −ｚ＾（ｎ）
−｛ε（ｎ−ｉ）−ｚ＾（ｎ−ｉ）｝
＝ ｛ε（ｎ）−ε（ｎ−ｉ）｝
−｛ｚ＾（ｎ）−ｚ＾（ｎ−ｉ）｝
＝ ｛ε（ｎ）−ε（ｎ−ｉ）｝
−［ Ｐ＾（ｎ）｛ｐ（ｎ）ｓｉｎ（ωＴｎ）＋ｑ（ｎ）ｃｏｓ（ωＴｎ）｝
＋Ｑ＾（ｎ）｛ｐ（ｎ）ｃｏｓ（ωＴｎ）−ｑ（ｎ）ｓｉｎ（ωＴｎ）｝］
＋［ Ｐ＾（ｎ−ｉ）｛ｐ（ｎ−ｉ）ｓｉｎ（ωＴ（ｎ−ｉ））＋ｑ（ｎ−ｉ）ｃｏｓ（ωＴ（ｎ−ｉ））｝
＋Ｑ＾（ｎ−ｉ）｛ｐ（ｎ−ｉ）ｃｏｓ（ωＴ（ｎ−ｉ））−ｑ（ｎ−ｉ）ｓｉｎ（ωＴ（ｎ−ｉ））｝］
ここでまた、伝達特性２３のゲインＡおよび位相Φが定常状態ないし準定常状態にある場合は、Ｐ＾（ｎ）＝Ｐ＾（ｎ−ｉ），Ｑ＾（ｎ）＝Ｑ＾（ｎ−ｉ）と置けるものと仮定する。実際のシステムに於いて、事故等の特別な場合を除きほとんどの場合には、伝達特性２３が急変することはまずないと考えて良いので、この仮定は概ね妥当である。また、発明者は数値シミュレーション等の手段により、実験的にこの仮定が成立し、この仮定に基づいて導かれた伝達特性推定アルゴリズム１３が適正に作用することを発見した。
【００８１】
さらに、ωＴ（ｎ−ｉ）＝ωＴｎ−ωＴｉ＝ωＴｎ−２πであるから、ｓｉｎ｛ωＴ（ｎ−ｉ）｝＝ｓｉｎ（ωＴｎ），ｃｏｓ｛ωＴ（ｎ−ｉ）｝＝ｃｏｓ（ωＴｎ）と置換することができる。
そこで、前記仮定とこの置換とを前記数３０に導入すると、周期偏差Ｅ（ｎ）は次の数３１のようにまとめられる。
【００８２】
【数３１】
Ｅ（ｎ）
＝ ｛ε（ｎ）−ε（ｎ−ｉ）｝
−Ｐ＾（ｎ）［ ｛ｐ（ｎ）−ｐ（ｎ−ｉ）｝ｓｉｎ（ωＴｎ）
＋｛ｑ（ｎ）−ｑ（ｎ−ｉ）｝ｃｏｓ（ωＴｎ）］
−Ｑ＾（ｎ）［ ｛ｐ（ｎ）−ｐ（ｎ−ｉ）｝ｃｏｓ（ωＴｎ）
−｛ｑ（ｎ）−ｑ（ｎ−ｉ）｝ｓｉｎ（ωＴｎ）］
伝達特性２３の推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）が適正に推定されていれば、周期偏差Ｅ（ｎ）はゼロに近づくはずである。そこで発明者は、周期偏差Ｅ（ｎ）の二乗を最小にするように最小二乗法に基づく勾配法を導入し、ゲイン推定値Ａ＾（ｎ）および位相推定値Φ＾（ｎ）の推定更新式を、それぞれ次の数３２および数３３のように導き出した。
【００８３】
【数３２】
Ｐ＾（ｎ＋１）
＝Ｐ＾（ｎ）−μ_Ｐ’・∂Ｅ^２（ｎ）／∂Ｐ＾（ｎ）
＝Ｐ＾（ｎ）−μ_ＰＥ（ｎ）・∂Ｅ（ｎ）／∂Ｐ＾（ｎ）
＝Ｐ＾（ｎ）
＋μ_ＰＥ（ｎ）［ ｛ｐ（ｎ）−ｐ（ｎ−ｉ）｝ｓｉｎ（ωＴｎ）
＋｛ｑ（ｎ）−ｑ（ｎ−ｉ）｝ｃｏｓ（ωＴｎ）］
【００８４】
【数３３】
Ｑ＾（ｎ＋１）
＝Ｑ＾（ｎ）−μ_Ｑ’・∂Ｅ^２（ｎ）／∂Φ＾（ｎ）
＝Ｑ＾（ｎ）−μ_ＱＥ（ｎ）・∂Ｅ（ｎ）／∂Φ＾（ｎ）
＝Ｑ＾（ｎ）
＋μ_ＱＥ（ｎ）［ ｛ｐ（ｎ）−ｐ（ｎ−ｉ）｝ｃｏｓ（ωＴｎ）
−｛ｑ（ｎ）−ｑ（ｎ−ｉ）｝ｓｉｎ（ωＴｎ）］
ただし、これら数３２および数３３において、μ_Ｐおよびμ_Ｑは、ステップサイズパラメータ（０＜μ_Ｐ，０＜μ_Ｑ）であり、推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）が互いに同格であるので、通常はμ_Ｐ＝μ_Ｑと置くことが望ましい。
【００８５】
以上のように、これら数３２および数３３の組み合わせをもって、前述の数１４および数１５の組み合わせからなる伝達特性推定アルゴリズム１３が導き出される。
（実施例１の作用効果）
本実施例の周期性信号の適応制御方法は、以上のように構成されているので、以下のような作用効果を発揮する。
【００８６】
前記数１１の適応信号発生アルゴリズム１１に基づいて、適応信号ｙ（ｎ）が生成され、伝達特性２３を介して観測点２４に加えられる。観測点２４では、角振動数ωの抑制すべき特定周波数成分を含む周期性信号ｆ（ｎ）に対し、角振動数ωの特定周波数成分からなる正弦波信号である適応信号ｙ（ｎ）が伝達特性２３を介して伝達された相殺信号ｚ（ｎ）が合成される。その結果、観測点２４では誤差信号ε（ｎ）＝ｆ（ｎ）＋ｚ（ｎ）が生成され、適応制御システムによって検知される。
【００８７】
すると先ず、伝達特性推定アルゴリズム１３によって、抑制すべき特定周波数成分の角振動数ωに対応する伝達特性２３の両直交化係数Ｐ，Ｑが推定される。この際、誤差信号の差分｛ε（ｎ）−ε（ｎ−ｉ）｝と、両直交化係数ｐ（ｎ），ｑ（ｎ）の差分値｛ｐ（ｎ）−ｐ（ｎ−ｉ）｝，｛ｑ（ｎ）−ｑ（ｎ−ｉ）｝とが、伝達特性推定アルゴリズム１３の演算に使用される。そして、抑制すべき特定周波数成分の角振動数ωに対応して推定された伝達特性２３の推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）は、適応係数ベクトル更新アルゴリズム１２に提供される。
【００８８】
次に、伝達特性２３の推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）は、適応係数ベクトル更新アルゴリズム１２の演算に使用され、同アルゴリズム１２によって次の離散時刻における適応信号ｙ（ｎ）の直交化係数ｐ（ｎ），ｑ（ｎ）が適応的に調整される。
最後に、適応係数ベクトル更新アルゴリズム１２から提供された直交化係数ｐ（ｎ），ｑ（ｎ）に基づいて、適応信号発生アルゴリズム１１が適応信号ｙ（ｎ）を生成する。その結果、適応信号ｙ（ｎ）は、伝達特性２３を介して伝達されて相殺信号ｚ（ｎ）となり、観測点２４に加えられる。そして、観測点２４に加えられた相殺信号ｚ（ｎ）は、同じく観測点２４に加わる周期性信号ｆ（ｎ）のうち抑制すべき特定周波数成分を相殺して、観測点２４で検知される誤差信号ε（ｎ）が抑制されるに至る。
【００８９】
この際、適応信号ｙ（ｎ）は、ｙ（ｎ）＝ｐ（ｎ）ｓｉｎ（ωＴｎ）＋ｑ（ｎ）ｃｏｓ（ωＴｎ）と直交化表現されているので、直交化係数ｐ（ｎ），ｑ（ｎ）が適応係数ベクトル更新アルゴリズム１２により同時に調整される。すると、変数としての性質が互いに同格な直交化係数ｐ（ｎ），ｑ（ｎ）が並行して調整されるので、両者の収束時間はほぼ同等になり、適応の過程において一方が他方の足を引っ張ることもなくなる。その結果、誤差信号ε（ｎ）の収束時間は短縮され、適応速度が向上するという効果がある。
【００９０】
また、伝達特性２３も直交化表現されているので、推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）が伝達位相特性同定アルゴリズム１３により同時に調整される。すると、変数としての性質が互いに同格な推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）が並行して調整されるので、両者の収束時間はほぼ同等になり、適応の過程において一方が他方の足を引っ張ることがなくなる。
【００９１】
それゆえ、適応係数ベクトル更新アルゴリズム１２は、伝達位相特性同定アルゴリズム１３によって速やかに収束した推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）を利用して、さらに速やかに直交化係数ｐ（ｎ），ｑ（ｎ）を収束させることができる。その結果、直交化された適応係数ベクトル更新アルゴリズム１２と、やはり直交化された伝達位相特性同定アルゴリズム１３との組み合わせにより、誤差信号ε（ｎ）の収束時間はさらに短縮され、適応速度がさらに向上するという効果がある。
【００９２】
したがって、本実施例の周期性信号の適応制御方法によれば、適応信号ｙ（ｎ）から観測点２３に至る伝達特性の変動にも柔軟に対応することができながら、よりいっそう適応速度が向上するという効果がある。
また、適応係数ベクトル更新アルゴリズム１２では、伝達特性２３のゲインに比例する傾向がある誤差信号ε（ｎ）を、ゲイン推定値Ａ＾（ｎ）の二乗に相当する値Ｐ＾^２（ｎ）＋Ｑ＾^２（ｎ）で正規化している。それゆえ、ある角振動数ωで伝達特性２３のゲインが低下した場合にも、適応係数ベクトル更新アルゴリズム１２の適応速度はあまり低下することがない。そればかりではなく、推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）の二乗和Ｐ＾^２（ｎ）＋Ｑ＾^２（ｎ）に正の発散防止定数γを足して正規化するものとすれば、Ｐ＾^２（ｎ）＋Ｑ＾^２（ｎ）がゼロ付近にまで低下することがあっても、適応係数ベクトル更新アルゴリズム１２が発散することは防止されている。したがって、本実施例の適応係数ベクトル更新アルゴリズム１２によれば、伝達特性２３のゲインが低下しても適応速度があまり衰えないうえに、Ｐ＾^２（ｎ）＋Ｑ＾^２（ｎ）がゼロ付近になっても発散が防止されるという効果もある。
【００９３】
（実施例１の数値シミュレーション）
本実施例の周期性信号の適応制御方法の効果を実証する目的で、発明者は数値シミュレーションによって本実施例の応答を検証してみた。この数値シミュレーションでの条件は以下の通りである。
先ず、周期性信号ｆ（ｎ）は、図２に示すように、抑制すべき特定周波数成分（正弦波）だけからなるものとし、その周波数は２０Ｈｚであり、その振幅は２．５√２≒０．３５４である。すなわち周期性信号ｆ（ｎ）は、次の数３４に従って観測点２４に加えられた。
【００９４】
【数３４】
ｆ（ｎ）＝０．２５｛ｓｉｎ（ωＴｎ）＋ｃｏｓ（ωＴｎ）｝
＝０．２５ｓｉｎ（ωＴｎ）＋０．２５ｃｏｓ（ωＴｎ）
ここで、影響を抑制すべき特性周波数成分の角振動数はω＝２０×２π［ｒａｄ／ｓ］であり、Ｔは後述の適応制御システムのサンプリング周期、ｎは離散化された時刻を示す整数である。
【００９５】
これに対し、適応係数ベクトル更新アルゴリズム１２（数１２参照）での直交化係数ｐ（ｎ），ｑ（ｎ）の初期値は、ｐ（ｎ）＝０，ｑ（ｎ）＝０と設定されていた。
次に、伝達特性２３は、周波数に関わりなく直交化係数Ｐ＝Ｑ＝０．３に設定されていた。これに対して伝達特性推定アルゴリズム１３（数１４および数１５参照）での両推定直交化係数の初期値は、Ｐ＾（０）＝Ｑ＾（０）＝−０．３に設定されていた。すなわち、伝達位相特性同定アルゴリズム１３において、推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）の初期値では、ゲインは真値と等価に設定されているが、位相は１８０°の誤差を持つように設定されていた。
【００９６】
なお、適応制御システムのサンプリング周期（更新周期）Ｔは１ミリ秒に設定されていた。このサンプリング周期は、サンプリング周波数に換算すると１，０００Ｈｚである。また、適応係数ベクトル更新アルゴリズム１２のステップサイズパラメータは、μ_ｐ＝μ_ｑ＝０．２に設定されており、発散防止定数はγ＝０．０２に設定されていた。一方、伝達特性推定アルゴリズム１３のステップサイズパラメータは、μ_Ｐ＝μ_Ｑ＝０．０３に設定されていた。
【００９７】
その結果、誤差信号ε（ｎ）の時間応答は、図３に示すように、約０．０８秒でおおむね収束し、約０．１秒で完全に収束している。この際、収束するまでの誤差信号ε（ｎ）の乱れは一往復半弱であり、外乱としての周期性信号ｆ（ｎ）の振幅を越えることが二度だけあるが、きわめて速やかにかつあまり乱れることなく誤差信号ε（ｎ）は収束している。
【００９８】
同様に、伝達特性推定アルゴリズム１３の推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）も、図４（ａ）〜（ｂ）に示すように、０．１秒弱で所定の値に収束して完全に安定している。そして、両推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）ともに、収束に当たってはほとんど乱れることがなく、素直に所定値に収束している。なお、両推定直交化係数Ｐ＾（ｎ），Ｑ＾（ｎ）の収束値は、真値の０．３とは若干異なる０．２４程度であるが、Ｐ＾（ｎ）≒Ｑ＾（ｎ）であるから伝達特性２３の位相の推定はおおむね正確である。一方、前述のように伝達特性２３のゲインの推定は正確ではないが、ゲイン推定値の不足分は適応係数ベクトル更新アルゴリズム１２が両直交化係数ｐ（ｎ），ｑ（ｎ）を大きくして、ゲイン推定値の不足分を補っている。
【００９９】
すなわち、伝達位相特性同定アルゴリズム１３に求められるのは位相推定の正確さであり、ゲイン推定にはそれほどの精度を要さない。ただし、伝達位相特性同定アルゴリズム１３を削除し、適応信号発生アルゴリズム１１および適応係数ベクトル更新アルゴリズム１２だけで適応制御システムを作動させると、誤差信号ε（ｎ）を収束させる作用は全く得られない。
【０１００】
一方、比較例として、前述の先行出願（特願平１０−３６５３９５号）の数値シミュレーション結果を引用し、図５ないし図７に示す。同先行出願は、適応信号発生アルゴリズム１１および適応係数ベクトル更新アルゴリズム１２は直交化されているが、伝達位相特性同定アルゴリズム１３は直交化されていない周期性信号の適応制御方法である。各パラメータおよび初期値の設定は前述の本実施例とは若干異なるが、発明者の労力の許す範囲でベストチューニングされており、伝達特性２３の推定値の初期位相差はむしろ本実施例よりも小さい。その結果、誤差信号ε（ｎ）の時間応答は、図５に示すように、幾度も大きく乱れており、完全に収束するまでに前述の本実施例よりもやや長く０．１秒強かかっている。また、伝達特性推定アルゴリズム１３のゲイン推定値Ａ＾（ｎ）および位相推定値Φ＾（ｎ）も、図６および図７にそれぞれ示すように、０．１秒強で所定の値に収束するまでにオーバーシュートなどの乱れが大きく、素直には収束していない。すなわち、この先行出願の技術では、本実施例の周期性信号の適応制御方法ほどに速やかで素直な誤差信号ε（ｎ）の収束特性は得られていない。
【０１０１】
したがって、本実施例の周期性信号の適応制御方法によれば、２０Ｈｚの特定周波数成分の影響を約０．１秒という短時間のうちに相殺して誤差信号ε（ｎ）を収束させることができる。すなわち、本実施例の周期性信号の適応制御方法は収束安定性がきわめて良好であり、よりいっそう高い適応速度を発揮することができるという効果がある。その結果、よりいっそう高い安定性が得られ、誤差信号ε（ｎ）はよりいっそう短時間のうちにあまり乱れることもなく収束するという効果が得られる。
【０１０２】
［実施例２］
（多入力多出力系の導出）
本項では、前述の実施例１の周期性信号の適応制御方法を、Ｋ成分・Ｌ入力・Ｍ出力に拡張する。すなわち、本項の対象とする場合は、抑制すべき特定周波数成分の角振動数ω_ｋ （１≦ｋ≦Ｋ）の個数Ｋと、誤差信号ε_ｌ（ｎ）（１≦ｌ≦Ｌ）の個数Ｌと、前記適応信号ｙ_ｍ（ｎ）（１≦ｍ≦Ｍ）の個数Ｍとのうち、少なくとも一つは複数である場合である。このような場合に対応でき、すなわち、多成分多入力多出力の複雑な系に対しても適用することができる周期性信号の適応制御方法を提供することを、本項の課題とする。
【０１０３】
したがって、本手段を多成分多入力多出力の複雑な系に対して適用した場合、適応信号ｙ_ｍ （ｎ）から誤差信号ε_ｌ （ｎ）に至る伝達特性２３’（図８参照）の変動にも柔軟に対応することができながら、適応速度がより向上するという効果がある。
Ｋ成分・Ｌ入力・Ｍ出力に対応した周期性信号の適応制御方法の各アルゴリズムの数式は、以下のようにして導出することができる。
【０１０４】
先ず、適応信号発生アルゴリズム１１’（図８参照）を次の数３５のように定義する。ここで、本適応制御システムはＭ出力であるから、ｍ＝１，・・，Ｍ（１≦Ｍ）である。
【０１０５】
【数３５】

【０１０６】
すると、各適応信号ｙ_ｍ （ｎ）の各直交化係数ｐ_ｋｍ（ｎ），ｑ_ｋｍ（ｎ）を要素としてもつそれぞれの適応係数ベクトルＷ_ｍ （ｎ）は、次の数３６に示すように定義される。ここで、本適応制御システムが抑制すべき特定周波数成分はＫ成分（ω_１，・・，ω_Ｋ）であるから、ｋ＝１，・・，Ｋ（１≦Ｋ）である。
【０１０７】
【数３６】

【０１０８】
また、観測点２４’（図８参照）で周期性信号ｆ（ｎ）に加えられて誤差信号ε_ｌ （ｎ）を生じる相殺信号ｚ_ｌ（ｎ）は、前記適応信号ｙ_ｍ（ｎ）が、直交化表現された伝達特性Ｇ_ｋｌｍ ［Ｐ_ｋｌｍ ，Ｑ_ｋｌｍ ］を介して伝達された信号である。それゆえ、相殺信号ｚ_ｌ （ｎ）は、次の数３７に示すように表記される。ここで、本適応制御システムはＬ入力であるから、ｌ＝１，・・，Ｌである。
【０１０９】
【数３７】

【０１１０】
次に、適応係数ベクトル更新アルゴリズムを導き出すために、評価関数Ｊ（ｎ）を次の数３８に示すように定める。
【０１１１】
【数３８】

【０１１２】
そして勾配法によって適応係数ベクトル更新アルゴリズムを導出すると、次の数３９に示すようになる。この際、伝達特性Ｇ_ｋｌｍ ［Ｐ_ｋｌｍ ，Ｑ_ｋｌｍ ］の真値は不可知であるので、伝達特性推定アルゴリズムによって推定された推定直交化係数Ｐ＾_ｋｌｍ（ｎ），Ｑ＾_ｋｌｍ（ｎ）をもって代替する。ここで、本適応制御システムが抑制すべき特定周波数成分はＫ成分（ω_１，・・，ω_Ｋ）であるから、ｋ＝１，・・，Ｋである。
【０１１３】
【数３９】

【０１１４】
ここでさらに、この数３９の右辺第二項をゲイン推定値Ａ＾_ｋｌｍ（ｎ）の二乗に相当するＰ＾_ｋｌｍ ^２（ｎ）＋Ｑ＾_ｋｌｍ ^２（ｎ）に発散防止定数γ_ｋｌｍ を加えたもので割って正規化することができる。こうすることにより、次の数４０に示す適応係数ベクトル更新アルゴリズム１２’（図８参照）を導き出すことができる。
【０１１５】
【数４０】

【０１１６】
この数４０の適応係数ベクトル更新アルゴリズムによれば、推定直交化係数Ｐ＾_ｋｌｍ（ｎ），Ｑ＾_ｋｌｍ（ｎ）がゼロに近い場合にも発散することなしに、収束性を向上させることができる。
最後に、伝達特性推定アルゴリズムは、前述の実施例１の場合と同様に、以下のようにして導き出される。
【０１１７】
始めに、推定偏差δ_ｌ（ｎ）および周期偏差Ｅ_ｌ（ｎ）について定義する。ここで、本適応制御システムはＬ入力系であるから、ｌ＝１，・・，Ｌ（１≦Ｌ）である。
推定偏差δ_ｌ（ｎ）は、次の数４１に従って定義される。
【０１１８】
【数４１】
δ_ｌ（ｎ） ≡ ε_ｌ（ｎ）−｛ｆ_ｌ（ｎ）＋ｚ＾_ｌ（ｎ）｝
また、周期偏差Ｅ_ｌ（ｎ）は次の数４２によって定義される。
【０１１９】
【数４２】
Ｅ_ｌ（ｎ） ≡ δ_ｌ（ｎ）−δ_ｌ（ｎ−ｉ）
ここで、この数４２に前述の実施例１と同様の仮定をたてて簡略化を施すことにより、周期偏差Ｅ_ｌ（ｎ）は次の数４３に示すように整理される。
【０１２０】
【数４３】

【０１２１】
周期偏差Ｅ（ｎ）の二乗を評価関数として最小二乗法に基づく勾配法を取り、前述の実施例１と同様にして整理すれば、伝達特性推定アルゴリズム１３’（図８参照）は、次の数４４および数４５の組み合わせとして定義される。
【０１２２】
【数４４】
Ｐ＾_ｋｌｍ（ｎ＋１）
＝ Ｐ＾_ｋｌｍ（ｎ）＋μ_ＰｋｌｍＥ_ｌ（ｎ）×
｛ （ｐ_ｋｍ（ｎ）−ｐ_ｋｍ（ｎ−ｉ））ｓｉｎ（ω_ｋＴｎ）
＋（ｑ_ｋｍ（ｎ）−ｑ_ｋｍ（ｎ−ｉ））ｃｏｓ（ω_ｋＴｎ）｝］
【０１２３】
【数４５】
Ｑ＾_ｋｌｍ（ｎ＋１）
＝ Ｑ＾_ｋｌｍ（ｎ）＋μ_ＱｋｌｍＥ_ｌ（ｎ）×
｛ （ｐ_ｋｍ（ｎ）−ｐ_ｋｍ（ｎ−ｉ））ｃｏｓ（ω_ｋＴｎ）
−（ｑ_ｋｍ（ｎ）−ｑ_ｋｍ（ｎ−ｉ））ｓｉｎ（ω_ｋＴｎ）｝］
以上のようにして、Ｋ成分・Ｌ入力・Ｍ出力の複雑な系に対しても、本実施例の周期性信号の適応制御方法を適用できるようになり、収束安定性および適応速度を向上させることができる。
【０１２４】
（実施例２の構成および効果）
本発明の実施例２としての周期性信号の適応制御方法は、図８に示すように、１成分２入力２出力の適応制御システムにおいて実施される。なお、この図８は、本実施例の適応制御システムを概念的に図示したものであって、図の複雑化を避けるためにある程度簡素化されている。たとえば、誤差信号ε_１（ｎ），ε_２（ｎ）の一周期分の差分と、両直交化係数ｐ_ｋｍ（ｎ），ｑ_ｋｍ（ｎ）の一周期分の差分とが混然と扱われているが、これは各信号の大まかな流れ方向を示すものとして御宥恕願いたい。
【０１２５】
本実施例の周期性信号の適応制御方法によっても、実施例１と同様に、収束安定性および適応速度が向上するという効果がある。
【図面の簡単な説明】
【図１】実施例１の適応制御システムの構成を示すブロック図
【図２】実施例１での周期性信号の波形を示すグラフ
【図３】実施例１での誤差信号の時間応答を示すグラフ
【図４】実施例１での伝達特性の推定直交化係数の収束を示す組図
（ａ）推定直交化係数Ｐ＾（ｎ）の時間応答を示すグラフ
（ｂ）推定直交化係数Ｑ＾（ｎ）の時間応答を示すグラフ
【図５】比較例での誤差信号の時間応答を示すグラフ
【図６】比較例でのゲイン推定値の時間応答を示すグラフ
【図７】比較例での位相推定値の時間応答を示すグラフ
【図８】実施例２の適応制御システムの構成を概念的に示すブロック図
【符号の説明】
１１，１１’：適応信号発生アルゴリズム
１２，１２’：適応係数ベクトル更新アルゴリズム
１３，１３’：伝達特性推定アルゴリズム
１４，１５，１４’，１５’：一周期分の遅延ブロック
２３，２３’：伝達特性 ２４，２４’：観測点
Ｐ，Ｑ：角振動数ωに対応する伝達特性の直交化係数
Ｐ＾（ｎ），Ｑ＾（ｎ）：角振動数ωに対応する伝達特性の推定直交化係数
ｐ（ｎ），ｑ（ｎ）：適応信号ｙ（ｎ）の直交化係数
ｙ（ｎ）：適応信号 ｚ（ｎ）：相殺信号
ε（ｎ）：誤差信号 ｆ（ｎ）：周期性信号
Ｔ：サンプリング周期（タップ周期または更新周期）
ω：角振動数 ｉ：一周期分の遅延タップ数
１≦ｋ≦Ｋ：角振動数ω_ｋの数（抑制すべき特定周波数成分の数）
１≦ｌ≦Ｌ：誤差信号ε_ｌ（ｎ）の数（センサの数）
１≦ｍ≦Ｍ：適応信号ｙ_ｍ（ｎ）の数（アクチュエータの数）[0001]
BACKGROUND OF THE INVENTION
The present invention belongs to the technical field of control technology, and more particularly to the technical field of active suppression technology for periodic signals. For example, if the periodic signal is vibration, it belongs to the technical field of active damping, and if the periodic signal is noise, it belongs to the technical field of active noise suppression. The range of applications is widening.
[0002]
[Prior art]
As a prior art for the present invention, an adaptive control method for a periodic signal filed by the same applicant as this application is disclosed in Japanese Patent Application Laid-Open No. 9-319403 (Japanese Patent Application No. 8-132090).
The prior art includes an adaptive signal generation algorithm, an adaptive coefficient vector update algorithm, and a transfer characteristic estimation algorithm, and each algorithm is digitally calculated each time the discrete time n of the update period T is updated. Here, the adaptive signal generation algorithm generates an adaptive signal y (n) = a (n) sin {ωTn + φ (n)} by a sine function, and adds the adaptive signal y (n) to the observation point via the transfer characteristic. Thus, the algorithm cancels the periodic signal f (n) at the observation point. The adaptive coefficient vector update algorithm also uses the amplitude a of the adaptive signal y (n) based on the error signal ε (n), the gain estimation value A ^ (n) and the phase estimation value Φ ^ (n) of the transfer characteristic. This is an algorithm for adaptively adjusting (n) and phase φ (n). On the other hand, the transfer characteristic estimation algorithm is an algorithm that adaptively adjusts the gain estimation value A ^ (n) and the phase estimation value Φ ^ (n) of the transfer characteristic that are used in the calculation of the adaptive coefficient vector update algorithm. .
[0003]
According to this conventional technique, it is possible to flexibly cope with a change in transfer characteristics from the adaptive signal y (n) to the observation point, and therefore, it is possible to provide an adaptive control method for a periodic signal that is more stable. I was able to.
[0004]
[Problems to be solved by the invention]
However, although the above-described conventional technology has excellent stability and relatively rapid convergence, it is desirable that the adaptive speed is still high in such a periodic signal adaptive control method. Not too long. Here, the improvement in the adaptation speed means that the error signal observed at the observation point converges more quickly.
[0005]
Accordingly, the present invention provides an adaptive control method for a periodic signal that can flexibly cope with fluctuations in transfer characteristics from the adaptive signal y (n) to the observation point, while further improving the adaptation speed. It is a problem to be solved.
[0006]
[Means for Solving the Problems]
In order to solve this problem, the inventor considered a factor that regulates the adaptation speed of the conventional technology.
As a result, the variables defining the adaptive signal y (n) = a (n) sin {ωTn + φ (n)} are the amplitude a (n) and the phase φ (n), and are different in character. I thought of it. That is, since the amplitude a (n) and the phase φ (n), which are two different variables having different personalities, are simultaneously adjusted, there is a difference in convergence time between one variable and the other variable, and the slower one It is considered that there is a problem that the error signal is converged with a convergence time of.
[0007]
Therefore, the inventor changes the adaptive signal y (n) to an orthogonal expression, and y (n) = p (n) sin (ωTn) + q (n) cos (ωTn), and the orthogonalization coefficients p (n), q It was decided to adjust both (n) simultaneously. Then, the orthogonalization coefficients p (n) and q (n) having the same characteristics as variables are adjusted at the same time, so that the convergence times of the two become substantially equal, and one of them pulls the other leg in the adaptation process. Also disappear. As a result, the inventor has obtained an insight that the adaptation speed of the adaptive control method of the present means should be improved and the convergence time of the error signal should be further shortened.
[0008]
Further, the inventor estimates the gain estimated value A ^ (n) and the phase estimated value Φ ^ (n) estimated by the transfer characteristic estimation algorithm when estimating the transfer characteristic from the adaptive signal y (n) to the observation point. I came to think that personality is a different variable. That is, in the transfer characteristic estimation algorithm, the gain estimation value A ^ (n) and the phase estimation value Φ ^ (n), which are two different variables having different characteristics, are adjusted simultaneously. Therefore, there is a difference in the convergence time between one variable and the other variable, and the error signal is converged with the later convergence time. Therefore, there is a problem that prevents the convergence time from being shortened.
[0009]
Therefore, the inventor changed the estimated value of the transfer characteristic to the orthogonalized expression and expressed the transfer characteristic with the estimated orthogonalization coefficients P ^ (n) and Q ^ (n). We decided to adjust P ^ (n) and Q ^ (n) simultaneously. Here, the relationship between the gain estimation value A ^ (n) and the phase estimation value Φ ^ (n) of the transfer characteristic and the estimated orthogonalization coefficients P ^ (n), Q ^ (n) is given by A ^ (n) = √ {P ^²(N) + Q ^²(N)}, tan Φ ^ (n) = Q ^ (n) / P ^ (n). Further, P ^ (n) = A ^ (n) cos {Φ ^ (n)}, Q ^ (n) = A ^ (n) sin {Φ ^ (n)}.
[0010]
Then, in the transfer characteristic estimation algorithm, the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) whose properties as variables are equivalent to each other are adjusted at the same time. Thus, one does not pull the other leg during the adaptation process. As a result, since the convergence speed of the transfer characteristic estimation algorithm is improved, the inventor believes that the adaptation speed of the adaptive control method of this means is further improved and the convergence time of the error signal should be further shortened. Obtained.
[0011]
Furthermore, the inventor derived a transfer characteristic estimation algorithm using the above-described orthogonalization coefficients p (n) and q (n) and estimated orthogonalization coefficients P ^ (n) and Q ^ (n). At that time, the number of taps i corresponding to Ti is introduced into one period of the specific frequency component of the periodic signal f (n), and sin (ωTn) and sin {ωT (n−i)} are approximately equivalent. I found that I can simplify the formula. Of course, the same applies to cos (ωTn) and cos {ωT (n−i)}, and it has been found that the mathematical expressions can be simplified by placing both of them approximately equivalent. The simplification of the above formulas will be specifically described later in the section on derivation of the first embodiment. As a result, since the transfer characteristic estimation algorithm can be simplified, the load on the calculation means can be reduced.
[0012]
The inventor has invented the following means based on the above insight and discovery.
(First means)
The first means of the present invention is the adaptive control method for periodic signals according to claim 1.
That is, the means includes an adaptive signal generation algorithm, an adaptive coefficient vector update algorithm, and a transfer characteristic estimation algorithm.
[0013]
The adaptive signal generation algorithm is a sinusoidal function expressed orthogonally based on the angular frequency ω of a specific frequency component to be suppressed among the periodic signal f (n) applied to the observation point at the discrete time n of the update period T. An adaptation signal y (n) is generated according to the following equation (6).
[0014]
[Formula 6]
y (n) = p (n) sin (ωTn) + q (n) cos (ωTn)
The adaptive signal y (n) is applied to the observation point via a predetermined transfer characteristic, cancels the specific frequency component to be suppressed in the periodic signal f (n), and is detected as an error signal ε at the observation point. (N) is suppressed. That is, the orthogonalization coefficients p (n) and q (n) of the adaptive signal y (n) are appropriately adjusted by the adaptive coefficient vector update algorithm, and the adaptive signal y (n) reduces the error signal ε (n). Acts like
[0015]
The adaptive coefficient vector update algorithm also orthogonalizes the specific frequency component of the adaptive signal y (n) so that the adaptive signal y (n) consequently cancels the specific frequency component of the periodic signal f (n). The coefficients p (n) and q (n) are adjusted adaptively. That is, an adaptive coefficient vector W (n) having orthogonalization coefficients p (n) and q (n) as elements is an estimated orthogonality that is an orthogonal expression of the error signal ε (n) and the estimated value of the transfer characteristic. Based on the conversion coefficients P ^ (n), Q ^ (n), the information is updated every time the discrete time n elapses. Then, each element of the adaptive coefficient vector W (n) is adjusted in accordance with the fluctuation of the amplitude, phase or angular frequency ω of the specific frequency component included in the periodic signal f (n) and the fluctuation of the transfer characteristic. .
[0016]
As a result, the adaptive signal y (n) transmitted to the observation point via the transfer characteristic cancels each other out of the periodic signal f (n) applied to the observation point and is detected at the observation point. Error signal ε (n) is suppressed.
Further, the transfer characteristic estimation algorithm adaptively adjusts the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristic to thereby change the angular frequency of the specific frequency component of the periodic signal f (n). The gain and phase of the transfer characteristic corresponding to ω are estimated. That is, the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristics are set to the difference of the error signal and the difference of the orthogonalization coefficient corresponding to one period or a plurality of periods of the specific frequency component of the angular frequency ω. Based on this, it is updated every time the discrete time n elapses.
[0017]
Here, assuming that the number of taps (update cycle number) i is a value where Ti is a time corresponding to one cycle or a plurality of cycles of the specific frequency component of the angular frequency ω, the error signal ε (n ) Is {ε (n) −ε (n−i)}. Similarly, the difference between the orthogonalization coefficients is {p (n) −p (n−i)} and {q (n) −q (n−i)}.
[0018]
As a result, the gain and phase of the transfer characteristic corresponding to the angular frequency ω of the specific frequency component to be suppressed can be estimated by the transfer characteristic estimation algorithm. Then, the estimated orthogonalization coefficients P ^ (n) and Q ^ (n), which are orthogonalized expressions of the estimated transfer characteristics, are provided to the above-described adaptive coefficient vector update algorithm.
[0019]
In this way, in this means, first, the estimated orthogonalization coefficient P ^ (n) corresponding to the gain and phase of the transfer characteristic corresponding to the angular frequency ω of the specific frequency component to be suppressed is determined by the transfer characteristic estimation algorithm. , Q ^ (n) is estimated. Next, the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristics are used in the calculation of the adaptive coefficient vector update algorithm, and the orthogonality of the adaptive signal y (n) at the next discrete time by the algorithm. The conversion factors p (n) and q (n) are adaptively adjusted. As a result, the adaptive signal y (n) transmitted to the observation point through the transfer characteristic cancels the specific frequency component to be suppressed in the periodic signal f (n) applied to the observation point, and the same observation is performed. The error signal ε (n) detected at the point is suppressed.
[0020]
At this time, since the adaptive signal y (n) is expressed orthogonally as y (n) = p (n) sin (ωTn) + q (n) cos (ωTn), the orthogonalization coefficients p (n), q (N) is adjusted simultaneously by the adaptive coefficient vector update algorithm. Then, since the orthogonalization coefficients p (n) and q (n) whose properties as variables are equivalent to each other are adjusted in parallel, the convergence times of the two become substantially equal, and one of them is in the process of adaptation. No longer pulls. As a result, the adaptation speed is improved, and the convergence time of the error signal ε (n) is shortened.
[0021]
The estimated values of the transfer characteristics are expressed orthogonally by estimated orthogonal coefficients P ^ (n), Q ^ (n), and the estimated orthogonal coefficients P ^ (n), Q ^ (n) are transmitted. Simultaneously adjusted by the characteristic estimation algorithm. Then, since both estimated orthogonalization coefficients P ^ (n) and Q ^ (n) having the same characteristics as variables are adjusted in parallel, the convergence times of the two become substantially equal, and one of them in the adaptation process. No longer pulls the other leg. As a result, the adaptation speed of the transfer characteristic estimation algorithm is improved, and the convergence time of the error signal ε (n) is further shortened.
[0022]
That is, since the convergence speed of the adaptive coefficient vector update algorithm and the convergence speed of the transfer characteristic estimation algorithm are improved, the adaptation speed of the entire adaptive control system is further improved by the interaction between the two.
Therefore, according to the adaptive control method of the periodic signal of this means, the adaptation speed can be further improved while being able to flexibly cope with the variation of the transfer characteristic from the adaptation signal y (n) to the observation point. effective.
[0023]
Note that the number of taps i may be a value corresponding to one cycle of the specific frequency component in a normal case. However, in some cases, it is possible to obtain a control result that is preferably a value corresponding to a delay corresponding to a plurality of cycles. This exceptional case is, for example, a case where the specific frequency component repeats increasing and decreasing the same pattern at a predetermined cycle. Therefore, except for special cases, the number of taps i may be a value where Ti corresponds to one period of the specific frequency component.
[0024]
(Second means)
The second means of the present invention is the adaptive control method for a periodic signal according to claim 2.
That is, in this means, the adaptive coefficient vector update algorithm calculates the adaptive coefficient vector W (n) according to one of the following update formula (7) and the update formula obtained by normalizing the second term on the right side of the formula (7). Update.
[0025]
[Expression 7]

[0026]
The method of deriving the above equation 7 will be described in detail later by taking steps in the derivation section of the first embodiment.
In Equation (7), the update formulas for the orthogonalization coefficients p (n) and q (n) are the same, with the same dimensionality and configuration of the formulas, only the difference between sin and cos. Therefore, both step size parameters μ_p, Μ_qAre equal to each other, the adaptive speeds of both orthogonalization coefficients p (n) and q (n) are equal to each other. As a result, if Equation 7 is adopted as the adaptive coefficient vector update algorithm, the two orthogonalization coefficients p (n) and q (n) are adapted almost simultaneously, and one adaptation speed is not delayed from the other. Therefore, there is an effect that the adaptation speed is improved.
[0027]
In Equation 7, the update formula for the orthogonal coefficients p (n) and q (n) is the simplest, and the calculation load of the adaptive coefficient vector update algorithm is reduced. Therefore, adopting Equation 7 as an adaptive coefficient vector update algorithm has the effect of not only improving the adaptation speed for the reason described above but also reducing the load on the computing means.
[0028]
Here, as a first method for normalizing the second term on the right-hand side of Equation 7, the estimated gain of the transfer characteristic is A ^ (n), and A ^²(N) + γ≡P ^²(N) + Q ^²There is a means of dividing by (n) + γ. Here, γ is a divergence prevention constant of zero or more.
Then, in an almost adapted state, the product of the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) having the same tendency as the error signal ε (n) that tends to be proportional to the gain of the transfer characteristic is obtained as the gain estimation. Value A ^²(N) ≡P ^²(N) + Q ^²It is normalized by (n). Therefore, even when the gain of the transfer characteristic decreases, the adaptation speed of the normalized adaptive coefficient vector update algorithm does not decrease much. In addition, if the gain estimation value A ^ (n) is normalized by adding a positive divergence prevention constant γ to the gain estimation value A ^ (n), even if the gain estimation value A ^ (n) may be reduced to near zero, it is adaptive. Divergence of the coefficient vector update algorithm is prevented.
[0029]
Therefore, if the normalization of the first method is applied to the adaptive coefficient vector update algorithm, there is an effect that the adaptation speed of the adaptive coefficient vector update algorithm does not decrease even if the gain of the transfer characteristic is lowered. In addition, there is an effect that the divergence of the adaptive coefficient vector update algorithm is prevented even if the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristics are both near zero.
[0030]
Note that the second term on the right side of Equation 7 clearly includes the estimated orthogonalization coefficients P ^ (n) and Q ^ (n), and the error signal ε (n) is also estimated orthogonalized. It can be considered that the coefficients P ^ (n) and Q ^ (n) are included. Therefore, P ^²(N) + Q ^²The means of dividing by (n) + γ is also appropriate from the viewpoint of non-dimensionalization of the estimated orthogonalization coefficients P ^ (n), Q ^ (n).
[0031]
As a second method for normalizing the second term on the right side of Equation 7, √ {p²(N) + q²(N)} + γ₂Or √ {p²(N) + q²(N) + γ₂}, There is also a means of dividing the second term on the right side. Where γ₂Is a divergence prevention constant of zero or more. Since the error signal ε (n) included in the second term on the right side is the adaptive signal y (n) + the periodic signal f (n), when the periodic signal f (n) is small, the adaptive signal y Even if (n) does not converge properly, the absolute value of the error signal ε (n) may be small due to the small periodic signal f (n). Therefore, if the second method is employed, the adaptation speed is increased even when the periodic signal f (n) is small, and the convergence of the error signal ε (n) is accelerated.
[0032]
Alternatively, as a third method of normalizing the second term on the right side of Equation 7, there is also a means of using the first method and the second method in combination. According to the third method, there is an effect of having both the effects of the first method and the second method.
(Third means)
According to a third aspect of the present invention, there is provided an adaptive control method for a periodic signal according to claim 3.
[0033]
That is, in this means, the transfer characteristic estimation algorithm is performed according to one of a pair of update expressions of the following equations 8 and 9, and a pair of update expressions obtained by normalizing the second term on the right side of the expressions 8 and 9. The estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristics are updated.
[0034]
[Equation 8]

[0035]
[Equation 9]

[0036]
[Expression 10]

The method of deriving the above equations 8 and 9 will be specifically described later in order in the derivation section of the first embodiment.
[0037]
In the equations (8) and (9), both update equations of the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristic are different in sin and cos, and have the same dimension and configuration. It is equivalent. Therefore, both step size parameters μ_P, Μ_QAre equal to each other, the adaptive speeds of both estimated orthogonalization coefficients are equal. As a result, if Equation 8 and Equation 9 are adopted as the adaptive coefficient vector update algorithm, both estimated orthogonalization coefficients P ^ (n) and Q ^ (n) are adapted almost simultaneously, and one adaptation speed is higher than that of the other. Since there is no delay, there is an effect that the adaptation speed of the adaptive coefficient vector update algorithm is improved.
[0038]
In addition, estimation of estimated orthogonalization coefficients P ^ (n) and Q ^ (n), which are estimated values obtained by orthogonalizing the transfer characteristics of the present means, is performed in accordance with Equation 8 and Equation 9. In addition to the effect described in the first means, there is an effect that the calculation load is relatively small.
Here, as a first method for normalizing the second term on the right side of the equations (8) and (9), the second term on the right side of both equations is represented by √ {P ^²(N) + Q ^²(N)} + γ ′ or √ {P ^²(N) + Q ^²There is a means of dividing by (n) + γ ′}. Here, γ 'is a divergence prevention constant of zero or more. Then, even when the gain of the transfer characteristic decreases, the adaptation speed does not decrease so much. Not only that, the gain estimate A ^ (n) ≡√ {P ^²(N) + Q ^²(N)} is normalized by adding a positive divergence prevention constant γ ′, so that the adaptive coefficient vector update algorithm can be used even if the gain estimate A ^ (n) decreases to near zero. Is prevented from emanating. Therefore, if the normalization of the first method is applied to the adaptive coefficient vector update algorithm, not only does the adaptation speed not decrease even if the gain of the transfer characteristic decreases, but the gain estimated value A ^ (n) ≡ {P ^²(N) + Q ^²Even if (n)} is near zero, there is an effect that divergence is prevented.
[0039]
In addition, as a second method for normalizing the second term on the right side of Equation 8 and Equation 9, p²(N) + q²There is also a means of dividing the second term on the right side by (n) + γ ″. Here, γ ″ is a divergence prevention constant of zero or more. That is, in the second term on the right side, p²(N), q²Since the terms (n), p (n) q (n) are included, the orthogonalization coefficient p (n) of the control signal y (n) is because the amplitude of the periodic signal f (n) is small. , Q (n) may also be small. In addition, the initial values of the orthogonalization coefficients p (n) and q (n) of the control signal y (n) may be set to zero. However, according to the second method, the divergence of the formula 8 and the formula 9 is prevented even in such a case.
[0040]
Alternatively, as a third method of normalizing the second term on the right side of the equation 8 and the equation 9, there is also a method of using the first method and the second method in combination. According to the third method, although the calculation load is slightly increased, there is an effect that the effects of the first method and the second method are combined.
(Fourth means)
According to a fourth aspect of the present invention, there is provided the adaptive control method for periodic signals according to the fourth aspect.
[0041]
That is, in this means, the angular frequency ω of the specific frequency component to be suppressed_kThe number K of (1 ≦ k ≦ K) and the error signal ε_l(N) The number L of (1 ≦ l ≦ L) and the adaptive signal y_m(N) At least one of the numbers M (1 ≦ m ≦ M) is plural. In other words, this means is an adaptive control method for periodic signals in which the first means is expanded to K component, L input, and M output. Therefore, this means can also be applied to a complex system having multiple components, multiple inputs and multiple outputs.
[0042]
Therefore, when this means is applied to a complex system with multiple components, multiple inputs and multiple outputs, the adaptive signal y_mFrom (n), the error signal ε_lThere is an effect that the adaptation speed can be improved while being able to flexibly cope with the fluctuation of the transfer characteristic up to (n).
Note that specific mathematical formulas for each algorithm of this means will be described later in the section of the second embodiment.
[0043]
DETAILED DESCRIPTION OF THE INVENTION
The embodiments of the adaptive control method for periodic signals of the present invention will be described clearly and sufficiently in the following examples so that a person skilled in the art can understand that the embodiments can be implemented.
[Example 1]
(Configuration of Example 1)
As shown in FIG. 1, the periodic signal adaptive control method according to the first embodiment of the present invention is implemented in a one-input one-output adaptive control system. That is, there is one error signal ε (n) as an input to the adaptive control system, and there is also one adaptive signal y (n) as an output from the adaptive control system. The adaptive control system in the present embodiment has an adaptive signal generation algorithm 11, an adaptive coefficient vector update algorithm 12, a transfer characteristic estimation algorithm 13, and a transfer characteristic 23.
[0044]
Here, the overall system configuration including the adaptive control system will be described as preliminary knowledge.
The periodic signal f (n) is a disturbance signal including a single specific frequency component, and is input to the observation point 24 where the error signal ε (n) is observed. In the periodic signal adaptive control method of this embodiment, the influence of the specific frequency component on the observation point 24 in the periodic signal f (n) is canceled, and an error signal ε (n (n) detected at the observation point 24 is detected. ) Is a control purpose.
[0045]
Here, it is assumed that the angular frequency ω of the specific frequency component of the periodic signal f (n) is measured with sufficient precision in engineering and is given to the adaptive signal generation algorithm 11 and the estimated transfer characteristic 13. For example, if the periodic signal f (n) is vibration acceleration caused by an automobile engine, the angular frequency ω can be easily and accurately measured in real time by observing a signal such as an ignition pulse. Can do.
[0046]
The adaptive signal generation algorithm 11 is an algorithm for generating an adaptive signal y (n) synchronized with the specific frequency component based on the angular frequency ω of the specific frequency component of the periodic signal f (n). The adaptive signal y (n) is converted into a cancellation signal z (n) via a predetermined transfer characteristic 23 and applied to the observation point 24. At the observation point 24, the periodic signal f (n) and the cancellation signal z (n) are combined, and the resulting error signal ε (n) = f (n) + z (n) is observed. In the example of the automobile engine described above, the observation point 24 corresponds to a vibration acceleration sensor fixed to the base of the passenger seat, and the error signal ε (n) corresponds to the output of the sensor.
[0047]
That is, in the periodic signal adaptive control method of this embodiment, the periodic signal f (n) affecting the observation point 24 is detected from one specific frequency component synchronized with the periodic signal f (n). The adaptation signal y (n) is added via the transfer characteristic 23 in antiphase. By doing so, the influence of the specific frequency component on the observation point 24 in the periodic signal f (n) is actively removed, and as a result, the error signal ε (n) detected at the observation point 24 is reduced. The
[0048]
In the following, the adaptive control method for periodic signals according to the present embodiment will be described in more detail with a focus on the adaptive signal generation algorithm 11, the adaptive coefficient vector update algorithm 12, and the transfer characteristic estimation algorithm 13.
The adaptive signal generation algorithm 11 is a sine expressed orthogonally according to the following equation 11 based on the angular frequency ω of one specific frequency component of the periodic signal f (n) at the discrete time n of the update period T. An adaptive signal y (n), which is a function, is generated.
[0049]
[Expression 11]
y (n) = p (n) sin (ωTn) + q (n) cos (ωTn)
On the other hand, the adaptive coefficient vector update algorithm 12 is an algorithm for updating the following adaptive coefficient vector W (n) according to the following equation 12.
[0050]
[Expression 12]

[0051]
Here, the adaptive coefficient vector W (n) is defined as two elements having orthogonal coefficients p (n) and q (n) of the specific frequency component of the adaptive signal y (n) as defined in the following equation (13). A vector of elements.
[0052]
[Formula 13]

[0053]
That is, the adaptive coefficient vector update algorithm 12 calculates the adaptive coefficient vector W (n) based on the pair of estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristic 23 and the error signal ε (n). Thus, the algorithm is updated every time the discrete time n elapses. By updating the adaptive coefficient vector W (n) by the adaptive coefficient vector updating algorithm 12, the adaptive coefficient vector W (() corresponds to the fluctuation of the amplitude, phase or angular frequency ω of the specific frequency component of the periodic signal f (n). Each element of n) is adjusted adaptively. The specific frequency of the adaptive signal y (n) generated by the adaptive signal generation algorithm 11 with the elements p (n) and q (n) of the adaptive coefficient vector W (n) updated by the adaptive coefficient vector update algorithm 12 The component orthogonalization coefficients p (n) and q (n) are updated.
[0054]
Furthermore, the transfer characteristic estimation algorithm 13 adaptively adjusts the estimated orthogonalization coefficients P ^ (n) and Q ^ (n), which are orthogonalized estimated values of the transfer characteristic 23, to generate the periodic signal f (n). A transfer characteristic 23 corresponding to the angular frequency ω of the specific frequency component is estimated. That is, the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristic 23 are set to the difference of the error signal ε (n) corresponding to one cycle of the specific frequency component of the angular frequency ω and the orthogonalization coefficient p ( n) and updated every time the discrete time n passes based on the difference between q (n). Here, assuming that the number of taps (update cycle number) i is a value where Ti is a time corresponding to one cycle of the specific frequency component of the angular frequency ω, the difference of the error signal ε (n) is , {Ε (n) −ε (n−i)}. Similarly, the difference between the orthogonalization coefficients is {p (n) −p (n−i)} and {q (n) −q (n−i)}.
[0055]
Such a transfer characteristic estimation algorithm 13 is expressed by a combination of the following equations 14 and 15.
[0056]
[Expression 14]

[0057]
[Expression 15]

[0058]
[Expression 16]

As a result, the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) corresponding to the gain and phase of the transfer characteristic 23 corresponding to the angular frequency ω of the specific frequency component to be suppressed are determined by the transfer characteristic estimation algorithm 13. Presumed. Then, the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the estimated transfer characteristic 23 are provided to the calculation of the adaptive coefficient vector update algorithm 12 described above.
[0059]
As described above, in the transfer characteristic estimation algorithm 13, the difference between the error signals {ε (n) −ε (n−i)} and the difference between the orthogonalization coefficients {p (n) −p (n−i) }, {Q (n) −q (n−i)}. In order to clarify this, in FIG. 1, the delay blocks 14 and 15 for one cycle are drawn in addition to the block of the transfer characteristic estimation algorithm 13. It may be included in the figure.
[0060]
(Derivation of Example 1)
The above-described adaptive control method for periodic signals according to the present embodiment can be derived as follows.
First, it is assumed that the angular frequency ω of a specific frequency component to be suppressed in the periodic signal f (n) is accurately measured in engineering. Here, it is assumed that the angular frequency ω can also change with time in accordance with the fluctuation of the specific frequency component of the periodic signal f (n).
[0061]
Then, the adaptive signal generation algorithm 11 generates a sine wave signal as the adaptive signal y (n) according to the following expression 17 expressed by orthogonalization.
[0062]
[Expression 17]
y (n) = p (n) sin (ωTn) + q (n) cos (ωTn)
Then, the cancellation signal z (n) applied to the observation point 24 via the transfer characteristic 23 is based on the adaptive signal y (n) representing the transfer characteristic G [A (ω), Φ (ω)] in an orthogonal manner. Since it is influenced by the orthogonalization coefficients P and Q, it is expressed by the following equation (18). Here, the transfer characteristic G [P (ω), Q (ω)] expressed orthogonally can be expressed as G [P (ω), Q (ω)] ≡P (ω ) + JQ (ω).
[0063]
[Expression 18]

Therefore, since the error signal ε (n) is a combination of the periodic signal f (n) and the cancellation signal z (n) at the observation point 24, the error signal ε (n) is expressed by the following equation (19).
[0064]
[Equation 19]

Next, the adaptive coefficient vector update algorithm 12 for adaptively updating the adaptive coefficient vector W (n) is expressed by the following equation 20 based on the gradient method.
[0065]
[Expression 20]
W (n + 1) = W (n) −μ∇ (n)
Here, the adaptive coefficient vector W (n) is a two-element vector having the orthogonalization coefficients p (n) and q (n) of the adaptive signal y (n) as elements, as shown in the following equation (21).
[0066]
[Expression 21]

[0067]
In Expression 20, μ represents a step size parameter, but is not a mere scalar, and is an adjustment coefficient μ applied to each element of the gradient vector ∇ (n) ≡∂J (n) / ∂W (n)._p, Μ_q. Also, since the update formulas of the orthogonalization coefficients p (n) and q (n) of the adaptive coefficient vector update algorithm 12 are equivalent to each other, the step size parameter is usually μ._p= Μ_qYou can set.
[0068]
Here, the evaluation function J (n) is defined by the square of the error signal ε (n) as shown in the following Expression 22.
[0069]
[Expression 22]
J (n) ≡ ε²(N)
Then, the gradient vector ∇ (n) = ∂J (n) / ∂W (n) by the least square method is expressed as the following Expression 23.
[0070]
[Expression 23]
∇ (n) = ∂J (n) / ∂W (n)
= ∂ε²(N) / ∂W (n)
= 2ε (n) · ∂ε (n) / ∂W (n)
By substituting the equation 19 and the equation 21 into the equation 23, the equation 23 is expanded as the following equation 24.
[0071]
[Expression 24]

[0072]
However, in reality, the orthogonalization coefficients P and Q corresponding to the gain A and the phase Φ of the transfer characteristic 23 are normally true values and thus cannot be directly known. Therefore, the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristic 23 are estimated, and the transfer characteristics 23 in the equation 24 are calculated using the estimated orthogonalization coefficients P ^ (n) and Q ^ (n). The basic form of the adaptive coefficient vector update algorithm 12 is obtained by substituting the orthogonal coefficients P and Q. Then, the adaptive coefficient vector update algorithm 12 is expressed as the following Expression 25.
[0073]
[Expression 25]

[0074]
Here, if the value corresponding to the square of the estimated gain of the transfer characteristic 23 is added to the divergence prevention constant γ and the second term on the right side of the number 25 is divided and normalized, the adaptive coefficient vector update algorithm 12 This is expressed by the following equation (26).
[0075]
[Equation 26]

[0076]
At this time, it is necessary to introduce the transfer characteristic estimation algorithm 13 for the purpose of obtaining the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristic 23.
Therefore, finally, the transfer characteristic estimation algorithm 13 can be derived as follows.
First, the estimated deviation δ (n) is defined as in the following Expression 27.
[0077]
[Expression 27]
δ (n) ≡ ε (n)-{f (n) + z ^ (n)}
= {Ε (n) −f (n)} − z ^ (n)
= Z (n) -z ^ (n)
Here, since the estimated cancellation signal z ^ (n) cannot know the true value of the transfer characteristic 23, the estimated orthogonalization coefficient P ^ (n), Q is used instead of the orthogonal coefficient P, Q of the transfer characteristic 23. ^ (N) is used to estimate the cancellation signal z (n). That is, the estimated cancellation signal z ^ (n) is defined by the following equation (28).
[0078]
[Expression 28]
z ^ (n) ≡
P ^ (n) {p (n) sin (ωTn) + q (n) cos (ωTn)}
+ Q ^ (n) {p (n) cos (ωTn) −q (n) sin (ωTn)}
Next, the number of taps i corresponding to one period of the specific frequency component that should eliminate the influence on the observation point 24 in the periodic signal f (n) is determined, and the difference of one period of the estimated deviation δ (n) A certain period deviation E (n) is defined as the following equation 29.
[0079]
[Expression 29]
E (n) ≡δ (n) −δ (ni)
= Ε (n)-{f (n) + z ^ (n)}
− [Ε (n−i) − {f (n−i) + z ^ (n−i)}]
Further, assuming that the periodic signal f (n) takes the same value with one period difference, f (n) −f (n−i) = 0 can be set in this equation 29. This assumption is perfectly correct when the periodic signal f (n) is in a steady state and is approximately correct when the periodic signal f (n) is in a transition state. However, at the moment when the periodic signal f (n) fluctuates stepwise, this assumption breaks down, but if the steady state or transition state is entered after the fluctuation, this assumption again becomes completely or approximately correct. Therefore, at the moment when the periodic signal f (n) suddenly fluctuates, the estimated value of the transfer characteristic estimation algorithm 13 is disturbed, but as soon as the sudden change of the periodic signal f (n) stops, the transfer characteristic estimation algorithm 13 The estimated value stabilizes near the correct value. This is confirmed by the inventor through a numerical simulation, and it is confirmed that the adaptive control method for the periodic signal of this embodiment functions normally. Therefore, if the assumption is accepted, the equation 29 is expanded as the following equation 30.
[0080]
[30]
E (n)
= Ε (n) -z ^ (n)
− {Ε (n−i) −z ^ (n−i)}
= {Ε (n) −ε (n−i)}
-{Z ^ (n) -z ^ (ni)}
= {Ε (n) −ε (n−i)}
-[P ^ (n) {p (n) sin (ωTn) + q (n) cos (ωTn)}
+ Q ^ (n) {p (n) cos (ωTn) −q (n) sin (ωTn)}]
+ [P ^ (n−i) {p (n−i) sin (ωT (n−i)) + q (n−i) cos (ωT (n−i))}
+ Q ^ (n−i) {p (n−i) cos (ωT (n−i)) − q (n−i) sin (ωT (n−i))}]
Here, when the gain A and the phase Φ of the transfer characteristic 23 are in a steady state or a quasi-steady state, P ^ (n) = P ^ (ni), Q ^ (n) = Q ^ (n- Assume that i). In an actual system, except for special cases such as accidents, in most cases, it can be considered that the transfer characteristic 23 is unlikely to change suddenly, so this assumption is generally valid. Further, the inventor has found that this assumption is experimentally established by means such as numerical simulation, and that the transfer characteristic estimation algorithm 13 derived based on this assumption works properly.
[0081]
Furthermore, since ωT (n−i) = ωTn−ωTi = ωTn−2π, sin {ωT (n−i)} = sin (ωTn), cos {ωT (n−i)} = cos (ωTn) Can be replaced.
Therefore, when the assumption and this replacement are introduced into the equation 30, the period deviation E (n) can be summarized as the following equation 31.
[0082]
[31]
E (n)
= {Ε (n) −ε (n−i)}
-P ^ (n) [{p (n) -p (n-i)} sin (ωTn)
+ {Q (n) -q (n-i)} cos (ωTn)]
-Q ^ (n) [{p (n) -p (n-i)} cos (ωTn)
− {Q (n) −q (n−i)} sin (ωTn)]
If the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristic 23 are properly estimated, the period deviation E (n) should approach zero. Therefore, the inventor introduces a gradient method based on the least square method so as to minimize the square of the period deviation E (n), and estimates and updates the gain estimated value A ^ (n) and the phase estimated value Φ ^ (n). Equations were derived as the following equations 32 and 33, respectively.
[0083]
[Expression 32]
P ^ (n + 1)
= P ^ (n) -μ_P’・ ∂E²(N) / ∂P ^ (n)
= P ^ (n) -μ_PE (n) · ∂E (n) / ∂P ^ (n)
= P ^ (n)
+ Μ_PE (n) [{p (n) −p (n−i)} sin (ωTn)
+ {Q (n) -q (n-i)} cos (ωTn)]
[0084]
[Expression 33]
Q ^ (n + 1)
= Q ^ (n) -μ_Q’・ ∂E²(N) / ∂Φ ^ (n)
= Q ^ (n) -μ_QE (n) · ∂E (n) / ∂Φ ^ (n)
= Q ^ (n)
+ Μ_QE (n) [{p (n) −p (n−i)} cos (ωTn)
− {Q (n) −q (n−i)} sin (ωTn)]
However, in these equations 32 and 33, μ_PAnd μ_QIs the step size parameter (0 <μ_P, 0 <μ_Q) And the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) are equivalent to each other._P= Μ_QIt is desirable to put
[0085]
As described above, the transfer characteristic estimation algorithm 13 composed of the combination of the above-described formulas 14 and 15 is derived from the combinations of the formulas 32 and 33.
(Operational effect of Example 1)
Since the periodic signal adaptive control method of the present embodiment is configured as described above, the following operational effects are exhibited.
[0086]
An adaptive signal y (n) is generated based on the adaptive signal generation algorithm 11 of Equation 11 and applied to the observation point 24 via the transfer characteristic 23. At the observation point 24, an adaptive signal y (n), which is a sine wave signal composed of a specific frequency component of the angular frequency ω, is generated with respect to the periodic signal f (n) including the specific frequency component to be suppressed of the angular frequency ω. The cancellation signal z (n) transmitted through the transfer characteristic 23 is synthesized. As a result, an error signal ε (n) = f (n) + z (n) is generated at the observation point 24 and detected by the adaptive control system.
[0087]
Then, first, the transmission characteristic estimation algorithm 13 estimates both orthogonalization coefficients P and Q of the transmission characteristic 23 corresponding to the angular frequency ω of the specific frequency component to be suppressed. At this time, the difference {ε (n) −ε (n−i)} of the error signal and the difference value {p (n) −p (n−i) between the two orthogonalization coefficients p (n) and q (n). }, {Q (n) −q (n−i)} are used for the calculation of the transfer characteristic estimation algorithm 13. The estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristic 23 estimated corresponding to the angular frequency ω of the specific frequency component to be suppressed are provided to the adaptive coefficient vector update algorithm 12. The
[0088]
Next, the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristic 23 are used in the calculation of the adaptive coefficient vector update algorithm 12, and the adaptive signal y (n at the next discrete time is calculated by the algorithm 12. ) Of the orthogonalization coefficients p (n) and q (n) are adaptively adjusted.
Finally, the adaptive signal generation algorithm 11 generates an adaptive signal y (n) based on the orthogonalization coefficients p (n) and q (n) provided from the adaptive coefficient vector update algorithm 12. As a result, the adaptive signal y (n) is transmitted via the transfer characteristic 23 to become a cancellation signal z (n) and is applied to the observation point 24. The cancellation signal z (n) applied to the observation point 24 cancels the specific frequency component to be suppressed among the periodic signal f (n) applied to the observation point 24, and is detected at the observation point 24. The error signal ε (n) is suppressed.
[0089]
At this time, since the adaptive signal y (n) is expressed orthogonally as y (n) = p (n) sin (ωTn) + q (n) cos (ωTn), the orthogonalization coefficients p (n), q (N) is simultaneously adjusted by the adaptive coefficient vector update algorithm 12. Then, since the orthogonalization coefficients p (n) and q (n) whose properties as variables are equivalent to each other are adjusted in parallel, the convergence times of the two become substantially equal, and one of them is in the process of adaptation. No longer pulls. As a result, the convergence time of the error signal ε (n) is shortened, and the adaptation speed is improved.
[0090]
Further, since the transfer characteristic 23 is also expressed by orthogonalization, the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) are simultaneously adjusted by the transfer phase characteristic identification algorithm 13. Then, the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) whose properties as variables are equivalent to each other are adjusted in parallel, so that the convergence times of the two become substantially equal, and one of them is in the adaptation process. The other leg will not be pulled.
[0091]
Therefore, the adaptive coefficient vector update algorithm 12 uses the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) that have been quickly converged by the transfer phase characteristic identification algorithm 13, so that the orthogonalization coefficient p ( n) and q (n) can be converged. As a result, the convergence time of the error signal ε (n) is further shortened and the adaptation speed is further improved by the combination of the orthogonalized adaptive coefficient vector update algorithm 12 and the orthogonalized transmission phase characteristic identification algorithm 13. There is an effect of doing.
[0092]
Therefore, according to the adaptive control method for periodic signals of this embodiment, the adaptation speed can be further improved while flexibly responding to fluctuations in the transfer characteristics from the adaptation signal y (n) to the observation point 23. There is an effect of doing.
Further, in the adaptive coefficient vector update algorithm 12, an error signal ε (n) that tends to be proportional to the gain of the transfer characteristic 23 is converted into a value P ^ corresponding to the square of the gain estimation value A ^ (n).²(N) + Q ^²Normalized by (n). Therefore, even when the gain of the transfer characteristic 23 decreases at a certain angular frequency ω, the adaptive speed of the adaptive coefficient vector update algorithm 12 does not decrease much. Not only that, but the sum of squares P ^ of estimated orthogonalization coefficients P ^ (n), Q ^ (n)²(N) + Q ^²If (n) is normalized by adding a positive divergence constant γ, P ^²(N) + Q ^²Even if (n) may drop to near zero, the adaptive coefficient vector update algorithm 12 is prevented from diverging. Therefore, according to the adaptive coefficient vector update algorithm 12 of the present embodiment, the adaptation speed does not decrease much even if the gain of the transfer characteristic 23 decreases, and P ^²(N) + Q ^²There is also an effect that divergence is prevented even when (n) is near zero.
[0093]
(Numerical simulation of Example 1)
In order to verify the effect of the adaptive control method for periodic signals of this embodiment, the inventor tried to verify the response of this embodiment by numerical simulation. The conditions in this numerical simulation are as follows.
First, as shown in FIG. 2, the periodic signal f (n) is composed of only a specific frequency component (sine wave) to be suppressed, its frequency is 20 Hz, and its amplitude is 2.5√2≈. 0.354. That is, the periodic signal f (n) was applied to the observation point 24 according to the following equation 34.
[0094]
[Expression 34]
f (n) = 0.25 {sin (ωTn) + cos (ωTn)}
= 0.25sin (ωTn) + 0.25cos (ωTn)
Here, the angular frequency of the characteristic frequency component whose influence should be suppressed is ω = 20 × 2π [rad / s], T is a sampling period of an adaptive control system described later, and n is an integer indicating a discretized time. It is.
[0095]
On the other hand, the initial values of the orthogonalization coefficients p (n) and q (n) in the adaptive coefficient vector update algorithm 12 (see Equation 12) are set as p (n) = 0 and q (n) = 0. It was.
Next, the transfer characteristic 23 was set to the orthogonalization coefficient P = Q = 0.3 regardless of the frequency. In contrast, the initial value of both estimated orthogonalization coefficients in the transfer characteristic estimation algorithm 13 (see Equations 14 and 15) was set to P ^ (0) = Q ^ (0) = − 0.3. . That is, in the transfer phase characteristic identification algorithm 13, the initial value of the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) is set to be equal to the true value, but the phase has an error of 180 °. Was set to have.
[0096]
Note that the sampling period (update period) T of the adaptive control system was set to 1 millisecond. This sampling period is 1,000 Hz in terms of sampling frequency. The step size parameter of the adaptive coefficient vector update algorithm 12 is μ_p= Μ_q= 0.2, and the divergence prevention constant was set to γ = 0.02. On the other hand, the step size parameter of the transfer characteristic estimation algorithm 13 is μ_P= Μ_Q= 0.03.
[0097]
As a result, as shown in FIG. 3, the time response of the error signal ε (n) generally converges in about 0.08 seconds and completely converges in about 0.1 seconds. At this time, the disturbance of the error signal ε (n) until convergence is slightly less than one reciprocal half and may exceed the amplitude of the periodic signal f (n) as a disturbance only twice. The error signal ε (n) converges without being disturbed.
[0098]
Similarly, the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristic estimation algorithm 13 are also set to predetermined values in a little less than 0.1 seconds, as shown in FIGS. It has converged and is completely stable. Both the estimated orthogonalization coefficients P ^ (n) and Q ^ (n) are almost undisturbed during convergence, and are smoothly converged to a predetermined value. The convergence value of both estimated orthogonalization coefficients P ^ (n) and Q ^ (n) is about 0.24 which is slightly different from the true value of 0.3, but P ^ (n) ≈Q ^ ( n), the phase estimation of the transfer characteristic 23 is generally accurate. On the other hand, although the estimation of the gain of the transfer characteristic 23 is not accurate as described above, the adaptive coefficient vector update algorithm 12 increases both the orthogonalization coefficients p (n) and q (n) when the gain estimation value is insufficient. This compensates for the shortage of gain estimates.
[0099]
That is, what is required of the transfer phase characteristic identification algorithm 13 is the accuracy of the phase estimation, and the gain estimation does not require so much accuracy. However, if the transfer phase characteristic identification algorithm 13 is deleted and the adaptive control system is operated only by the adaptive signal generation algorithm 11 and the adaptive coefficient vector update algorithm 12, the effect of converging the error signal ε (n) cannot be obtained.
[0100]
On the other hand, as a comparative example, the numerical simulation results of the above-mentioned prior application (Japanese Patent Application No. 10-365395) are cited and shown in FIGS. This prior application is an adaptive control method for a periodic signal in which the adaptive signal generation algorithm 11 and the adaptive coefficient vector update algorithm 12 are orthogonalized, but the transfer phase characteristic identification algorithm 13 is not orthogonalized. The setting of each parameter and initial value is slightly different from the above-described present embodiment, but is best tuned within the range allowed by the inventor's labor, and the initial phase difference of the estimated value of the transfer characteristic 23 is rather than the present embodiment. small. As a result, as shown in FIG. 5, the time response of the error signal ε (n) is greatly disturbed several times, and it takes a little longer than 0.1 second to be completely converged before the present embodiment. Yes. Further, the gain estimation value A ^ (n) and the phase estimation value Φ ^ (n) of the transfer characteristic estimation algorithm 13 converge to predetermined values in just over 0.1 seconds, as shown in FIGS. By the time, the disturbance such as overshoot is large, and it does not converge smoothly. That is, with the technique of this prior application, the convergence characteristic of the error signal ε (n) that is as quick and straightforward as the periodic signal adaptive control method of this embodiment is not obtained.
[0101]
Therefore, according to the periodic signal adaptive control method of this embodiment, the error signal ε (n) can be converged by canceling the influence of the specific frequency component of 20 Hz within a short time of about 0.1 seconds. it can. That is, the periodic signal adaptive control method of this embodiment has extremely good convergence stability, and has the effect of being able to exhibit an even higher adaptation speed. As a result, higher stability can be obtained, and the effect that the error signal ε (n) converges without much disturbance in a shorter time can be obtained.
[0102]
[Example 2]
(Derivation of multi-input multi-output system)
In this section, the adaptive control method for a periodic signal according to the first embodiment is extended to K component, L input, and M output. That is, when the target of this section, the angular frequency ω of the specific frequency component to be suppressed_kThe number K of (1 ≦ k ≦ K) and the error signal ε_l(N) The number L of (1 ≦ l ≦ L) and the adaptive signal y_m(N) At least one of the numbers M (1 ≦ m ≦ M) is a plurality. An object of this section is to provide an adaptive control method for a periodic signal that can cope with such a case, that is, can be applied to a complex system having multiple components, multiple inputs, and multiple outputs.
[0103]
Therefore, when this means is applied to a complex system with multiple components, multiple inputs and multiple outputs, the adaptive signal y_mFrom (n), the error signal ε_lWhile adapting to fluctuations in the transfer characteristic 23 ′ (refer to FIG. 8) leading to (n), it is possible to flexibly cope with the effect of improving the adaptation speed.
Equations for each algorithm of the adaptive control method for periodic signals corresponding to K component, L input, and M output can be derived as follows.
[0104]
First, the adaptive signal generation algorithm 11 '(see FIG. 8) is defined as in the following Expression 35. Here, since this adaptive control system has M outputs, m = 1,..., M (1 ≦ M).
[0105]
[Expression 35]

[0106]
Then, each adaptive signal y_mEach orthogonalization coefficient p of (n)_km(N), q_kmEach adaptive coefficient vector W having (n) as an element_m(N) is defined as shown in Equation 36 below. Here, the specific frequency component to be suppressed by the adaptive control system is the K component (ω₁, ..._K), K = 1,..., K (1 ≦ K).
[0107]
[Expression 36]

[0108]
Further, the error signal ε is added to the periodic signal f (n) at the observation point 24 ′ (see FIG. 8)._lCancellation signal z yielding (n)_l(N) is the adaptive signal y_m(N) is the transfer characteristic G expressed orthogonally._klm[P_klm, Q_klm] Is transmitted through the signal. Therefore, the cancellation signal z_l(N) is expressed as shown in Equation 37 below. Here, since this adaptive control system has L inputs, l = 1,.
[0109]
[Expression 37]

[0110]
Next, in order to derive an adaptive coefficient vector update algorithm, an evaluation function J (n) is determined as shown in the following equation (38).
[0111]
[Formula 38]

[0112]
Then, when the adaptive coefficient vector update algorithm is derived by the gradient method, the following equation 39 is obtained. At this time, transfer characteristic G_klm[P_klm, Q_klm] Is unknown, so the estimated orthogonalization coefficient P ^ estimated by the transfer characteristic estimation algorithm_klm(N), Q ^_klmSubstitute with (n). Here, the specific frequency component to be suppressed by the adaptive control system is the K component (ω₁, ..._K), K = 1,.
[0113]
[39]

[0114]
Here, further, the second term on the right side of Equation 39 is used as the gain estimated value A ^._klmP ^ corresponding to the square of (n)_klm ²(N) + Q ^_klm ²(N) divergence prevention constant γ_klmCan be normalized by dividing by By doing so, the adaptive coefficient vector update algorithm 12 ′ (see FIG. 8) expressed by the following equation 40 can be derived.
[0115]
[Formula 40]

[0116]
According to the adaptive coefficient vector update algorithm of the number 40, the estimated orthogonalization coefficient P ^_klm(N), Q ^_klmConvergence can be improved without divergence even when (n) is close to zero.
Finally, the transfer characteristic estimation algorithm is derived as follows in the same manner as in the first embodiment.
[0117]
First, the estimated deviation δ_l(N) and period deviation E_l(N) is defined. Here, since the adaptive control system is an L input system, l = 1,..., L (1 ≦ L).
Estimated deviation δ_l(N) is defined according to the following equation (41).
[0118]
[Expression 41]
δ_l(N) ≡ ε_l(N)-{f_l(N) + z ^_l(N)}
In addition, the period deviation E_l(N) is defined by the following equation (42).
[0119]
[Expression 42]
E_l(N) ≡ δ_l(N) -δ_l(Ni)
Here, the periodic deviation E is obtained by simplifying the equation 42 by making the same assumption as in the first embodiment._l(N) is arranged as shown in the following equation (43).
[0120]
[Equation 43]

[0121]
If the gradient method based on the least square method is taken using the square of the periodic deviation E (n) as an evaluation function and arranged in the same manner as in the first embodiment, the transfer characteristic estimation algorithm 13 '(see FIG. 8) It is defined as a combination of Equation 44 and Equation 45.
[0122]
(44)
P ^_klm(N + 1)
= P ^_klm(N) + μ_PklmE_l(N) x
{(P_km(N) -p_km(Ni)) sin (ω_kTn)
+ (Q_km(N) -q_km(Ni)) cos (ω_kTn)}]
[0123]
[Equation 45]
Q ^_klm(N + 1)
= Q ^_klm(N) + μ_QklmE_l(N) x
{(P_km(N) -p_km(Ni)) cos (ω_kTn)
-(Q_km(N) -q_km(Ni)) sin (ω_kTn)}]
As described above, the adaptive control method of the periodic signal of this embodiment can be applied to a complex system of K component, L input, and M output, and the convergence stability and the adaptation speed are improved. be able to.
[0124]
(Configuration and effect of Example 2)
The adaptive control method for periodic signals as the second embodiment of the present invention is implemented in a one-component two-input two-output adaptive control system as shown in FIG. FIG. 8 conceptually illustrates the adaptive control system of the present embodiment, and is simplified to some extent to avoid complication of the drawing. For example, the error signal ε₁(N), ε₂(N) The difference for one period and the orthogonalization coefficient p_km(N), q_km(N) Although the difference for one cycle is handled in a mixed manner, this should be regarded as indicating the rough flow direction of each signal.
[0125]
The adaptive control method for periodic signals according to the present embodiment also has the effect of improving the convergence stability and the adaptation speed as in the first embodiment.
[Brief description of the drawings]
FIG. 1 is a block diagram illustrating a configuration of an adaptive control system according to a first embodiment.
2 is a graph showing the waveform of a periodic signal in Example 1. FIG.
3 is a graph showing a time response of an error signal in Example 1. FIG.
FIG. 4 is a set diagram illustrating convergence of estimated orthogonalization coefficients of transfer characteristics in the first embodiment.
(A) A graph showing the time response of the estimated orthogonalization coefficient P ^ (n)
(B) A graph showing the time response of the estimated orthogonalization coefficient Q ^ (n)
FIG. 5 is a graph showing a time response of an error signal in a comparative example.
FIG. 6 is a graph showing a time response of an estimated gain value in a comparative example.
FIG. 7 is a graph showing a time response of a phase estimation value in a comparative example.
FIG. 8 is a block diagram conceptually showing the structure of the adaptive control system in the second embodiment.
[Explanation of symbols]
11, 11 ': Adaptive signal generation algorithm
12, 12 ': Adaptive coefficient vector update algorithm
13, 13 ': Transfer characteristic estimation algorithm
14, 15, 14 ', 15': Delay block for one cycle
23, 23 ': Transfer characteristics 24, 24': Observation point
P, Q: Orthogonalization coefficient of transfer characteristics corresponding to angular frequency ω
P ^ (n), Q ^ (n): Estimated orthogonalization coefficient of transfer characteristic corresponding to angular frequency ω
p (n), q (n): orthogonalization coefficient of the adaptive signal y (n)
y (n): adaptive signal z (n): cancellation signal
ε (n): error signal f (n): periodic signal
T: Sampling cycle (tap cycle or update cycle)
ω: angular frequency i: number of delay taps for one period
1 ≦ k ≦ K: angular frequency ω_kNumber (number of specific frequency components to be suppressed)
1 ≦ l ≦ L: error signal ε_lNumber of (n) (number of sensors)
1 ≦ m ≦ M: adaptive signal y_mNumber of (n) (number of actuators)

Claims (4)

With respect to the periodic signal f (n) affecting the observation point, an adaptive signal y (n) composed of a specific frequency component synchronized with the periodic signal f (n) is inverted in phase through a predetermined transfer characteristic. By adding the periodicity signal f (n), the periodicity of actively removing the influence of the specific frequency component on the observation point and reducing the error signal ε (n) detected at the observation point In an adaptive signal control method,
At the discrete time n of the update period T, the adaptive signal y (n), which is a sine function expressed orthogonally based on the angular frequency ω of the specific frequency component of the periodic signal f (n), is obtained. An adaptive signal generation algorithm to be generated according to the following equation 1,
An adaptive coefficient vector W (n) whose elements are orthogonalization coefficients p (n) and q (n) of the specific frequency component of the adaptive signal y (n) is expressed as the error signal ε (n) and the transfer characteristic. Based on the estimated orthogonalization coefficients P ^ (n), Q ^ (n), updated every time the discrete time n passes, the amplitude, phase or angle of the specific frequency component included in the periodic signal f (n) An adaptive coefficient vector update algorithm for adjusting each element of the adaptive coefficient vector W (n) to adapt to fluctuations in the frequency ω and fluctuations in the transfer characteristic;
The estimated orthogonalization coefficients P ^ (n) and Q ^ (n) of the transfer characteristic are set as the difference {ε (n) of the error signal corresponding to one period or a plurality of periods of the specific frequency component of the angular frequency ω. ) −ε (n−i)} and the difference {p (n) −p (n−i)}, {q (n) −q (n−i)} between the orthogonalization coefficients, the discrete time n And a transfer characteristic estimation algorithm for updating the transfer characteristic and estimating the transfer characteristic.
The identification of the adaptive signal y (n) generated by the adaptive signal generation algorithm with the elements p (n) and q (n) of the adaptive coefficient vector W (n) updated by the adaptive coefficient vector update algorithm The orthogonalization coefficients p (n) and q (n) of the frequency component are updated,
An adaptive control method for periodic signals.
[Expression 1]
y (n) = p (n) sin (ωTn) + q (n) cos (ωTn) The adaptive coefficient vector update algorithm is an algorithm for updating the adaptive coefficient vector W (n) according to one of the following update formula of Formula 2 and an update formula obtained by normalizing the second term on the right side of Formula 2. ,
The adaptive control method of the periodic signal according to claim 1.

The transfer characteristic estimation algorithm may perform the estimation orthogonalization according to one of a pair of update expressions of the following equations 3 and 4, and a pair of update expressions obtained by normalizing the second term on the right side of the equations 3 and 4. An algorithm for updating the coefficients P ^ (n) and Q ^ (n).
The adaptive control method of the periodic signal according to claim 1.

The angular frequency ω _k (1 ≦ k ≦ K) of the specific frequency component, the error signal ε _l (n) (1 ≦ l ≦ L), and the adaptive signal y _m (n) (1 ≦ m ≦ M). At least one of which is a plurality,
The adaptive control method of the periodic signal in any one of Claims 1-3.

Images