6.4 Kalman Filter Equations
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1 6.4 Kalman Filter Equations Recap: Auxiliary variables Recall the definition of the auxiliary random variables x p k) and x m k): Init: x m 0) := x0) S1: x p k) := Ak 1)x m k 1) +uk 1) +vk 1) S2: z m k) := Hk)x p k) +wk) x m k) defined via its PDF f xmk)ξ) := f xpk) z mk)ξ zk)) ξ k = 1,2,... We proved last lecture that x p k) is identical to the random variable xk) conditioned on z1:k 1), and x m k) is identical to the random variable xk) conditioned on z1:k). That is, for all ξ and k = 1,2,..., f xpk)ξ) = f xk) z1:k 1)ξ z1:k 1)) f xmk)ξ) = f xk) z1:k)ξ z1:k)). Hence, x p k) and x m k) capture the prior update and the measurement update, respectively, of the Bayesian state estimator. We can work with these random variables in the following derivation of the Kalman filter. First, we show that x p k) and x m k) are GRVs. As a consequence, their PDFs are fully characterized by mean and variance, which we then compute as a second step. The resulting recursive equations for mean and variance are the Kalman filter equations. We use the following notation to denote mean and variance of the auxiliary variables: ˆx p k) := E[x p k)], ˆx m k) := E[x m k)], P p k) := Var[x p k)], P m k) := Var[x m k)] Auxiliary variables are GRVs Claim: x p k) and x m k) are GRVs. Proof: We prove by induction. The claim is true for x m 0) by definition. We assume x m k 1) is a GRV and show that this implies that x p k) and x m k) are GRVs. S1: By definition, x p k) is an affine combination of the random variables x m k 1) and vk 1). The process noise vk 1) is a GRV, and x m k 1) is a GRV by induction assumption. The random variables x m k 1) and vk 1) are independent since x m k 1) depends on x0), v0:k 2), and w1:k 1), which are independent of vk 1). Hence, x m k 1) and vk 1) are jointly Gaussian. Using Property 2 of last lecture Sec ), it follows that x p k) is a GRV. S2: To simplify the notation, we drop the time index k all random variables in question are indexed by k). We then have, by Bayes rule, f xm ξ) := f xp z m ξ z) = f z m x p z ξ)f xp ξ). f zm z) Date compiled: April 24,
2 Since x p is a GRV proved in S1 above), we have f xp ξ) exp 1 ) 2 ξ ˆx p) T Pp 1 ξ ˆx p ) and, by definition of z m, f zm x p z ξ) exp 1 ) 2 z Hξ)T R 1 z Hξ). Since z denotes an actual measurement f zm z) is a number and does not depend on ξ); we can therefore write: f xm ξ) exp 1 ξ ˆx p ) T Pp 1 ξ ˆx p )+ z Hξ) T R 1 z Hξ)) ). 6.7) 2 Note that the argument of the exponential is a quadratic in ξ. There exist therefore, µ and Σ such that f xm ξ) exp 1 ) 2 ξ µ)t Σ 1 ξ µ). 6.8) That is, x m k) is a GRV, which completes the induction Mean and variance of auxiliary variables We compute mean and variance of x p k) and x m k). S1: By direct calculation: ˆx p k) = E[x p k)] = Ak 1)E[x m k 1)]+uk 1) +E[vk 1)] = Ak 1)ˆx m k 1) +uk 1) P p k) = Var[x p k)] = E[x p k) ˆx p k))x p k) ˆx p k)) T ] = Ak 1)E[x m k 1) ˆx m k 1))x m k 1) ˆx m k 1)) T ]A T k 1) +E[vk 1)v T k 1)] = Ak 1)P m k 1)A T k 1) +Qk 1). Note that in the second to last equation, the cross-terms for example, E[x m k 1) ˆx m k 1))v T k 1)]) vanish because x m k 1) ˆx m k 1) and vk 1) are independent and zero-mean. S2: By 6.8), we have ˆx m k) = µ and P m k) = Σ. We can compute µ and Σ by comparing the quadratic and linear terms of 6.7) and 6.8). Quadratic term: ξ T P 1 p +H T R 1 H)ξ = ξ T Σ 1 ξ ξ Σ 1 = P 1 p +H T R 1 H ) for symmetric A, ξ T Aξ = 0 ξ if and only if A = 0) Linear term: ξ T P 1 p ˆx p +H T R 1 z) = ξ T Σ 1 µ ξ µ = Σ P 1 p ˆx p +H T R 1 z ) = Σ P 1 p ˆx p +H T R 1 z Hˆx p +Hˆx p ) ) = Σ Σ 1ˆx p +H T R 1 z Hˆx p ) ) = ˆx p +ΣH T R 1 z Hˆx p ). 2
3 6.4.4 Summary Kalman Filter equations) The Kalman filter is given by the recursive update equations for mean and variance above: Initialization: ˆx m 0) = x 0, P m 0) = P 0. Step 1 S1): Prior update/prediction step ˆx p k) = Ak 1)ˆx m k 1) +uk 1) P p k) = Ak 1)P m k 1)A T k 1) +Qk 1) Step 2 S2): A posteriori update/measurement update step Results from above, re-introducing time index k and shorthand notation in particular, zk) for zk)): P m k) = P 1 p k) +HT k)r 1 k)hk) ) 1 ˆx m k) = ˆx p k) +P m k)h T k)r 1 k)zk) Hk)ˆx p k)) Since the state xk) conditioned on past measurements z1:k) is a GRV, its PDF is fully characterized by its mean ˆx m k) and its variance P m k) and, analogously, the state xk) conditioned on z1:k 1)). This means that the Kalman Filter is the analytic solution to the Bayesian state estimation problem for a linear system with Gaussian distributions Alternative Equations An alternative form of the measurement update equations can be derived using the matrix inversion lemma): PSET 4: P2) Kk) = P p k)h T k) Hk)P p k)h T k) +Rk) ) 1 6.9) ˆx m k) = ˆx p k) +Kk)zk) Hk)ˆx p k)) 6.10) P m k) = I Kk)Hk) ) P p k) 6.11) = I Kk)Hk) ) P p k) I Kk)Hk) ) T +Kk)Rk)K T k) 6.12) The matrix Kk) is called the Kalman Filter gain. Note that implementing 6.12) is computationally more expensive than 6.11), but is less sensitive to numerical errors. This is known as the Joseph form of the covariance update. 6.5 Remarks Implementation If the problem data is known ahead of time i.e. Ak), Hk), Qk), and Rk) for all k as well as P 0 ), all KF matrices i.e. P p k), P m k), and Kk)) can be computed off-line. In particular, the KF gain does not depend on real-time measurement data! ˆx p k) = Ak 1)ˆx m k 1) +uk 1) ˆx m k) = ˆx p k) +Kk)zk) Hk)ˆx p k)) ˆx m k) = I Kk)Hk) ) Ak 1)ˆx m k 1) + I Kk)Hk) ) uk 1) +Kk)zk) = Âk)ˆx mk 1) + ˆBk)uk 1) +Kk)zk) Note that the filter is linear and time varying. uk 1) Kalman zk) Filter ˆx m k) 3
4 Estimation error Estimation error: ek) := xk) ˆx m k). ek) = xk) I Kk)Hk) ) Ak 1)ˆx m k 1) +uk 1) ) Kk)Hk)xk) Kk)wk) = I Kk)Hk) ) Ak 1)xk 1) +uk 1) +vk 1) ) ek) is a GRV. I Kk)Hk) ) Ak 1)ˆx m k 1) +uk 1) ) Kk)wk) = I Kk)Hk) ) Ak 1)ek 1) + I Kk)Hk) ) vk 1) Kk)wk) Error mean: E[ek)] = I Kk)Hk))Ak 1)E[ek 1)]. It follows that the filter is unbiased, i.e. E[ek)] = 0 assuming initialization with ˆx m 0) = x 0 ). Error variance: Var[ek)] = E[xk) ˆx m k))xk) ˆx m k)) T ] = P m k). Note that stability of the system E[ek)] = I Kk)Hk))Ak 1)E[ek 1)] is important; for example, for the mean to not diverge if we do not initialize properly E[e0)] 0). Asymptotic properties of the KF are discussed in more detail in the next lecture. Another interpretation Note that the equations 6.9) 6.12) are the recursive least squares RLS) equations! This is a second interpretation of the KF, which also applies to non-gaussian random variables: among the class of linear, unbiased estimators specifically, those that have the structure ˆx m k) = ˆx p k) +Kk)zk) Hk)ˆx p k)), compare RLS lecture #5), the KF is the one that minimizes the mean squared error. For a linear system and Gaussian noise, the KF is the best you can do it keeps track of the full conditional PDFs). The linear estimator structure is optimal. Otherwise, one must be careful. The KF is no longer optimal nonlinear may do better), but often reasonable it is the best linear estimator in the MMSE sense). Remark: Positive definiteness of variance matrices In the derivation of the KF, we assumed that x0), vk), and wk) are GRVs and therefore have positive definite variance matrices i.e. P 0 > 0, Qk) > 0, Rk) > 0). Note that the KF equations also make sense when some of the variance matrices are positive semidefinite i.e. 0), as long as the involved matrix inversions are well defined in the derivation of RLS, for example, we only require Hk)P p k)h T k) +Rk) > 0 in order to be invertible). Examples: P 0 = 0 means that the initial state is perfectly known. Q 0 with some zero eigenvalue means that there is no process noise in some direction of the state space. 4
5 6.6 Example Consider an object moving along a line and acted upon by an external force. external force 0 y The object can be modeled as ẏt) = pt) ṗt) = ft), where yt) and pt) are the object s location and velocity, and ft) is the external force. The external force has a known component uk) and is corrupted by noise v p k)with known variance), both of which we assume to only change every second: ft) = uk 1)+v p k 1), k 1 t < k, v p k 1) N0,0.1). Every second, the object s state is therefore pk) = pk 1)+uk 1)+v p k 1), yk) = yk 1)+pk 1)+0.5uk 1)+0.5v p k 1). A position measurement, corrupted by noise, is available every second: zk) = yk) + wk) with wk) N0,0.5). Initial distribution: y0), p0)) N0, 5I). System equations Let xk) := yk),pk)). Then [ ] [ ] [ xk) = xk 1) + uk 1) [ ] [ ] = xk 1) + uk 1) }{{}}{{} A B zk) = [ 1 0 ] }{{} H ] v p k 1), [ vy k 1) v p k 1) ], xk) +wk), wk) N0,R), R = 0.5. [ ] vy k 1) N0,Q), Q = v p k 1) A, B, H, Q, R, and initial mean and variance is all we need to implement the KF. Simulations Matlab files to simulate this example are provided on the class website.) Simulate true state xk) and estimate ˆx m k) KF provides estimates for position and velocity. 5 [ ] ,
6 P m k) captures uncertainty in estimates. For example, plot of mean plus/minus one standard deviation, ˆx i m k)± Pm iik): For any k, the probability of xi k) being in the interval [ˆx i m k) Pm iik), ˆxi m k)+ Pm ii k)] is 68.2%. P m k) converges to a constant matrix as k. To be discussed in the next lecture.) Run simulation multiple times and plot distribution of final error ek final ): a GRV with zero-mean and variance P m k final ), as expected. 6
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