H∞ filter

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  • \(H_\infty\) filter
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\(H_\infty\) filter

• Motivation
  • Kalman filter requires knowledge of all system parameters
  • What do we do when we don't know anything about the noise corrupting the system?
  • Solution: filter that controls the worst case error
• Why do we care about the Kalman filter?
  • It's the optimal filter for linear state space model with gaussian noise
  • It's not optimal anymore the moment we change assumptions (see EKF, UKF, etc.)
• \(H_\infty\) filter
  • This is minimax filter (more on this soon)
  • Assumes nature is playing against you to choose the most damaging noise
  • Still requries knowledge of the system matrices
• For what to do without knowledge of the system, check out Sample Complexity of Kalman Filtering for Unknown Systems

Adversarial state-space model

• Consider the state space model \[ \vecx_{k+1} = \matF_k \vecx_k x_k + \vecu_k\qquad \vecy_k = \matH_k \vecx_k+\vecv_k \] where \(\vecu_k\) and \(\vecv_k\) can be anything

• We shall attempt to design a filter that optimizes the cost function (over horizon \(n-1\)) \[ J_1\eqdef \frac{\sum_{i=0}^{n-1}\norm[\matS_k]{\hat{\vecx}_k-\vecx_k}}{\norm[\matP_0^{-1}]{\hat{\vecx}_0-\vecx_0}^2+\sum_{i=0}^{n-1}\left(\norm[\matQ_k^{-1}]{\vecw_k}^2+\norm[\matR_k^{-1}]{\vecv_k}^2\right)} \]
  • We attempt to minize \(J_1\) (over the choice of the linear filter for \(\vecx_k\))
  • Nature attempts to maximize \(J_1\) (over the choice of the noise)
  • The matrices \(\matP_0\), \(\matQ_k\) and \(\matR_k\) are design parameters

Modified cost-function

• Direct minimization is intractable
  • We only require instead that \(J_1<\frac{1}{\theta}\) or alternatively \[ J \eqdef -\frac{1}{\theta}\norm[\matP_0^{-1}]{\hat{\vecx}_0-\vecx_0}^2 + \sum_{i=0}^{n-1}\left(\norm[\matS_k]{\hat{\vecx}_k-\vecx_k}-\frac{1}{\theta}\left((\norm[\matQ_k^{-1}]{\vecw_k}^2+\norm[\matR_k^{-1}]{\vecv_k}^2\right)\right) <1 \]
  • We will attempt to solve for \[ J^* = \min_{\hat{\vecx}^k}\max_{\vecw_k,\vecy_k,\vecx_0} J \]
• Dynamic constrained obtimization
  • The function to optimize is of the form \[ J = \psi(\vecx_0) + \sum_{k=0}^{n-1}\calL_k \] where the minimization is subject to the constraints of the dynamic state space model

Dynamic constrained optimization

• Use Lagrange multipliers \[ J = \psi(\vecx_0) + \sum_{k=0}^{n-1}\left[\calL_k+\frac{2\lambda_{k+1}^T}{\theta}(\matF_k \vecx_k+\vecw_k-\vecx_{k+1})\right] \]

• This is a bit technical, but instructive \[ \matK_k = \matP_k[\matI-\theta\matP_k+\matH_k^T\matR_k^{-1}\matH_k\matP_k]^{-1}\matH_k^{T}\matR_k^{-1} \] \[ \hat{\vecx}_{k+1} = \matF_k\hat{\vecx}_k+\matF_k\matK_k(\vecy_k-\matH_k\hat{\vecx}_k) \] \[ \matP_{k+1}=\matF_k\matP_k[\matI-\theta\matP_k+\matH_k^T\matR_k^{-1}\matH_k\matP_k]^{-1}\matF_k^{T}+\matQ_k \]

• The Kalman filter is obtained for \(\theta=0\) and using the known covariance matrices of the noise