Robust stability of moving horizon estimation for continuous-time systems

Proof

The proof is straightforward and follows by noting that

Consequently, using the fact that a + b ≤ max{2a, 2b} for a, b ≥ 0, the RGAS property from (3) implies that

which is equivalent to (4) by setting ψ ( s , t ) ≔ β − 1 ( 2 β x ( s ) η t i ) and γ ( s ) ≔ β − 1 ( − 2 ln ⁡ η β w ( s ) ) .

When the state estimator is additionally RGES, we can exploit in (34) that β(s) ≥ C₁s ^r and β _x (s) ≤ C₂s ^r with C₁, C₂, r from Definition 2, which yields ψ ( s , t ) ≤ 2 C 2 C 1 1 r s η t i r and γ ( s ) ≤ − 2 C 1 ⁡ ln ⁡ η β w ( s ) 1 r (recall that s ↦ s 1 r is strictly increasing in s ≥ 0). Consequently, we can choose C ≔ 2 C 2 C 1 1 r and ρ ≔ η 1 r ∈ [ 0,1 ) . A suitable redefinition of γ establishes our claim and hence concludes this proof. □

Proof

Given any t ≥ 0 and its corresponding sampling time t _i = k(t), let l ≔ t − t i + T t i ∈ [ 0 , T t i ] and recall that x ̂ ( t ) = x ̄ t i * ( l ) = x l , χ ̄ t i * , u , w ̄ t i * by (12). Since x ̄ t i * satisfies (1) on [ 0 , T t i ] due to the constraints (8b)–(8e), we can invoke the i-iIOSS Lyapunov function from Assumption 1. To this end, we use that x ( t ) = x ( t , χ , u , w ) = x ( l , x ( t − T t i ) , u t i , w t i ) , where w t i : [ 0 , T t i ) → M W denotes the segment of w in the interval [ t i − T t i , t i ) defined by w t i ( τ ) ≔ w ( t i − T t i + τ ) , τ ∈ [ 0 , T t i ) . Then, we can evaluate the dissipation inequality (5b) with the trajectories x 1 ( l ) = x ( l , x ( t i − T t i ) , u t i , w t i ) and x 2 ( l ) = x ̄ t i * ( l ) = x l , χ ̄ t i * , u t i , w ̄ t i * and the corresponding outputs y 1 ( l ) = y ( l + t i − T t i ) = y t i ( l ) and y 2 ( l ) = y ̄ t i * ( l ) , which yields

where the last inequality follows by exploiting that l ≤ T t i , the definition of l, and the fact that x ̄ t i * ( 0 ) = χ ̄ t i * . Define

By Cauchy–Schwarz and Young’s inequality, we have

and

Then, U ̄ in (36) can be bounded using (37) and (38), yielding

By optimality, it follows that

Hence, (39) and (40) lead to

where the last equality follows by a change of coordinates. In combination, we obtain

Using | x ( t i − T t i ) − x ̂ ( t i − T t i ) | P 2 2 ≤ λ max ( P 2 , P 1 ) | x ( t i − T t i ) − x ̂ ( t i − T t i ) | P 1 2 and the first inequality in (5a) yields (15), which finishes this proof. □

Proof

For any pair of points x 1 , x 2 ∈ X , let Ξ(x₁, x₂) denote the set of piecewise smooth curves [ 0,1 ] → X connecting x₁ and x₂ such that c ∈ Ξ(x₁, x₂) satisfies c(0) = x₁ and c(1) = x₂.

Given any χ 1 , χ 2 ∈ X , u ∈ M U , and w 1 , w 2 ∈ M W satisfying Assumption 2, we consider the trajectories x _i (t) = x(t, χ _i , u, w _i ) and their output signals y _i (t) = y(t, χ _i , u, w _i ), t ≥ 0, i = 1, 2.

At any fixed time t = t^⋆ ≥ 0, let us consider the following smoothly parameterized paths for s ∈ [0, 1]: the path of states c(t, s) ∈ Ξ (x₁(t), x₂(t)) and the paths of disturbances ω(t, s) = w₁(t) + s(w₂(t) − w₁(t)). For t ∈ [t^⋆, t^⋆ + ϵ) (with ϵ > 0 arbitrarily small to guarantee local existence of solutions and continuity of u, w over t ∈ [t^⋆, t^⋆ + ϵ)) and each fixed s ∈ [0, 1], the path c(t, s) evolves according to (1) such that

where ζ(t, s) satisfies y₁(t) = ζ(t, 0) and y₂(t) = ζ(t, 1) for each t ∈ [t^⋆, t^⋆ + ϵ). Differentiating (43) with respect to s ∈ [0, 1] yields (after interchanging the order of differentiation of t and s)

for all t ∈ [t^⋆, t^⋆ + ϵ) using the substitutions δ _x ≔ dc/ds(t, s), δ _w ≔ dω/ds(t, s), and δ _y ≔ dζ/ds(t, s), and where the matrices A, B, C, D are the linearizations of f and h as in (30) evaluated at (c(t, s), u(t), ω(t, s)). Assuming that ( c ( t , s ) , u ( t ) , ω ( t , s ) ) ∈ X × U × W for all t ∈ [t^⋆, t^⋆ + ϵ), one can easily verify (by exploiting (44)) that satisfaction of the point-wise LMI condition (31) implies that

for all t ∈ [t^⋆, t^⋆ + ϵ). By integrating (45) over s ∈ [0, 1], interchanging integration and differentiation, and defining E ( c ( t , s ) ) ≔ ∫ 0 1 | d c / d s ( t , s ) | P 2 d s , we obtain

for t ∈ [t^⋆, t^⋆ + ϵ). The integration over t ∈ [t^⋆, t^⋆ + ϵ) yields

Note that E(c(t, s)) can be interpreted as the Riemannian energy of the path c(t, s). Since P is constant and X is convex, the shortest path γ(t, s) (with the minimum energy E(γ(t, s)) over all possible curves c(t, s) ∈ Ξ(x₁(t), x₂(t))) is, at each fixed t ≥ 0, always given by the straight line connecting x₁(t) and x₂(t), i.e., γ(t, s) = x₁(t) + s(x₂(t) − x₁(t)). Now let c(t^⋆, s) = γ(t^⋆, s) in (46) and note that E(γ(t, s)) ≤ E(c(t, s)) for all t ∈ [t^⋆, t^⋆ + ϵ). Therefore, by taking ϵ → 0, we have that

By construction of γ and ω (in particular, the fact that their derivatives with respect to s ∈ [0, 1] are constant in s ∈ [0, 1]), it follows that

where the last equality follows since h is affine in (x, w) by Assumption 2 (which implies that C and D are constant in s). Since t^⋆ ≥ 0 was arbitrary, using the definition U ( x 1 , x 2 ) ≔ | x 1 − x 2 | P 2 we can infer that

for each t ≥ 0. Note that U satisfies (5a) with P₁ = P₂ = P. In what follows, we will show that U also satisfies (5b). To this end, let v : R ≥ 0 → R ≥ 0 be the solution of the initial value problem

Solving for v yields

From the standard comparison theorem, we know that U(x₁(t), x₂(t)) ≤ v(t) for all t ≥ 0, which establishes (5b) with λ = e^−κ. Consequently, U is a quadratic (and hence smooth) i-iIOSS Lyapunov function that satisfies Assumption 1, which finishes this proof. □