Patent ID: 7039567

Claim:
A system identification apparatus for performing a fast real-time identification of a time-invariant or time-variant system, comprising a filter robust against disturbances, said filter is formed by setting, as an H ∞ evaluation criterion, a maximum energy gain from disturbances weighted by ρ=1−γ f −2 and Σ wi =γ f −2 P^ i+1|i to a filter error to be smaller than a predetermined upper limit γ f 2 , wherein said filter, for a state-space model as in the following equations (7) to (9), satisfies the evaluation criterion by expression (10) and is given by the following equations (11) to (15), {circumflex over (x)} k+1|k+1 ={circumflex over (x)} k|k +K s,k+1 ( y k+1 −H k+1 {circumflex over (x)} k|k ) where, {circumflex over (x)} k|k : The estimate of state x k at time k, obtained by using observation signals y 0 to y k y k : Observation signal K s,k+1 : Filter gain H k : Observation matrix x k+1 =x k +w k , w k , x k εR N (7) y k =H k x k +v k , y k , v k εR (8) z k =H k x k , z k εR, H k εR 1×N (9) sup x 0 , { ω i } , { υ i } ⁢ ∑ i = 0 k ⁢  e f , i  2 / ρ  x 0 - x ⋁ 0 | - 1  ∑ 0 - 1 2 + ∑ i = 0 k ⁢  ω i  ∑ ω i - 1 2 + ∑ i = 0 k ⁢  υ i  2 / ρ < γ f 2 ( 10 ) {hacek over (z)} k|k =H k {circumflex over (x)} k|k (11) {circumflex over (x)} k+1|k+1 ={circumflex over (x)} k|k +K s,k+1 ( y k+1 −H k+1 {circumflex over (x)} k|k ) Filter equation (12) K s,k+1 ={circumflex over (P)} k+1|k H k+1 T ( H k+1 {circumflex over (P)} k+1|k H k+1 T +ρ) −1 Filter gain (13) P ^ k + 1 ⁢  k = P ^ k  ⁢ k - 1 - P ^ k ⁢  k - 1 ⁡ [ H k T ⁢ H k T ] ⁢ R c , k - 1 ⁡ [ H k H k ] ⁢ P ^ k ⁢  k - 1 + ∑ ω k ⁢ Riccati ⁢ ⁢ equation ⁢ ⁢ ( 14 ) where, e j , i = z ^ i ⁢  i - H i ⁢ x i ⁢ ⁢ R e , k = R k + [ H k H k ] ⁢ P ^ k ⁢  k - 1 ⁡ [ H k T ⁢ H k T ] ⁢ ⁢ R k = [ ρ 0 0 - ργ f 2 ] , Σ wk = γ f - 2 ⁢ P ^ k + 1 ⁢  k ⁢ ⁢ P ^ k ⁢  k - 1 - 1 + H k T ⁢ H k > 0 , P ^ 1 ⁢  0 = ɛ 0 ⁢ I , ɛ 0 > 0 ⁢ ⁢ 0 < ρ = 1 - γ f - 2 ≤ 1 , γ f > 1 ( 15 ) and the existence condition is given by the inequality P^ k|k−1 −1 +H k T H k >0 which appears in (15), where, the notation is used as follows, x k : State vector or just state, unknown and to be estimated, x 0 : Initial state, unknown, w k : System noise, unknown, v k : Observation noise, unknown, y k : Observation signal, known and input to a filter, z k : Output signal, unknown, H k : Observation matrix, known, x^ k|k : State value of the state x k at time k, estimated by using observation signals y 0 to y k , given by the filter equation, x^ 0|0 : Initial estimate of a state, essentially unknown but set to 0 for convenience, K s,k+1 : Filter gain, obtained by matrix P^ k+1|k, Σ wk : Corresponds to the covariance matrix of the system noise, known in theory but unknown in advance, P^ k|k−1 : Corresponds to the covariance matrix of the error of x^ k|k−1 , given by a Riccati equation, P^ 1|0 : Corresponds to the covariance matrix of an error in the initial state, essentially unknown but set to ε 0 I for convenience.