Observers and state estimation Starting point Continuous-time - - PowerPoint PPT Presentation

Observers and state estimation

Starting point

◮ Continuous-time system:

˙ x(t) = Ax(t) + Bu(t) + Nv1(t), y(t) = Cx(t) + Du(t) + v2(t).

◮ Discrete-time system:

x(k + 1) = Fx(k) + Gu(k) + Nv1(k), y(k) = Hx(k) + Ju(k) + v2(k).

◮ v1 and v2 are zero mean white noise, with intensities

according to η = v1 v2

• ⇒

Φη(ω) = Rη = R1 R12 RT

12

R2

• .

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Continuous-time observers

The estimation error

◮ The observer:

˙ ˆ x(t) = Aˆ x(t) + Bu(t) + K(y(t) − Cˆ x(t) − Du(t))

◮ The estimation error ˜

x = x − ˆ x is governed by ˙ ˜ x(t) = (A − KC)˜ x(t) + Nv1(t) − Kv2(t) = (A − KC)˜ x(t) +

• N

−K

• =Bη

η(t)

◮ ˜

x(t) is a stationary stochastic process if A − KC is stable.

◮ The covariance of the estimation error, Π˜ x = E˜

x(t)˜ xT (t), is given by the continuous-time Lyapunov equation 0 = (A − KC)Π˜

x + Π˜ x(A − KC)T + BηRηBT η .

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Discrete-time observers

The estimation error

◮ The observer:

ˆ x(k + 1) = F ˆ x(k) + Gu(k) + K(y(k) − Hˆ x(k) − Ju(k))

◮ The estimation error ˜

x = x − ˆ x is governed by ˜ x(k + 1) = (F − KH)˜ x(k) + Nv1(k) − Kv2(k) = (F − KH)˜ x(k) +

• N

−K

• =Gη

η(k)

◮ ˜

x(k) is a stationary stochastic process if F − KH is stable.

◮ The covariance of the estimation error, Π˜ x = E˜

x(k)˜ xT (k), is given by the discrete-time Lyapunov equation Π˜

x = (F − KH)Π˜ x(F − KH)T + GηRηGT η .

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The Kalman filter

Continuous-time systems (Thm. 5.4)

◮ The optimal observer minimizes Π˜ x. ◮ Introduce the notation P = min K Π˜ x. ◮ The optimal observer gain is

K = (PCT + NR12)R−1

2 . ◮ The optimal observer is called the Kalman filter. ◮ P = P T ≥ 0 is the solution to the continuous-time algebraic

Riccati equation (CARE) 0 = AP+PAT +NR1NT −(PCT +NR12)R−1

2 (PCT +NR12)T ◮ Technical conditions:

◮ (A, C) detectable ⇔ ∃K such that A − KC is stable, ◮ R2 = RT

2 > 0,

◮ Rη = RT

η ≥ 0.

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The Kalman filter

Discrete-time systems (Thm. 5.6)

◮ Notation: ˆ

x(k|l) is the estimate of x(k) based on measurements up to the time l, i.e. . . . , y(l − 2), y(l − 1), y(l)

◮ The “standard” observer is

ˆ x(k+1|k) = F ˆ x(k|k−1)+Gu(k)+K(y(k)−Hˆ x(k|k−1)−Ju(k))

◮ Introduce the notation P = min K Π˜ x(k|k−1). ◮ The optimal observer gain is

K = (FPHT + NR12)(HPHT + R2)−1.

◮ P = P T ≥ 0 is the solution to the discrete-time algebraic

Riccati equation (DARE) P = FPF T + NR1NT − (FPHT + NR12)(HPHT + R2)−1(FPHT + NR12)T

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The Kalman filter

Discrete-time systems, cont’d

◮ ˆ

x(k + 1|k) is the optimal estimate of x(k + 1), given the measurements . . . , y(k − 1), y(k). Thus ˆ x(k + 1|k) is the

• ptimal one-step predictor.

◮ Refine the estimate of x(k) by use of the present

measurement y(k), i.e. get the optimal ˆ x(k|k), obtained as ˆ x(k|k) = ˆ x(k|k − 1) + ˜ K(y(k) − Hˆ x(k|k − 1) − Ju(k)) where ˜ K = PHT (HPHT + R2)−1.

◮ Predict m time-steps ahead — the optimal m-step predictor:

ˆ x(k + m|k) = F mˆ x(k|k) +

m−1

• l=0

F m−1−lGu(k + l)

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Innovations and innovations form

◮ Innovations: The update/feedback

ν(t) = y(t) − Cˆ x(t) − Du(t) = y(t) − ˆ y(t) in the Kalman filter is called the output innovations. (For discrete-time: ν(k) = y(k) − ˆ y(k|k − 1).)

◮ Innovations form: A state space representation of the form

˙ x(t) = Ax(t) + Bu(t) + Nv(t), y(t) = Cx(t) + Du(t) + v(t), is in innovations form if A − NC is stable. The Kalman filter can be written in innovations form, by the replacements

◮ ˆ

x ↔ x,

◮ K ↔ N, ◮ ν ↔ v. 7 / 8 hans.norlander@it.uu.se Kalman

Properties of the Kalman filter

The Kalman filter

◮ is a linear filter:

ˆ x(t) = F(p) y(t) u(t)

• ,

ˆ x(k|k − 1) = F(q) y(k) u(k)

• ◮ is always stable

◮ makes the innovations ν white noise with intensity

◮ R2 for continuous-time systems ◮ HPHT + R2 for discrete-time systems

◮ is the optimal linear filter for state estimation ◮ is the optimal state estimator if v1 and v2 are Gaussian ◮ is an observer, and R1, R2 and R12 may be regarded as

design parameters — trade-off between convergence speed and noise sensitivity

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