8.5 Discrete-Time Certainty-Equivalent Filters (CEFs)
In this section, we present the discrete-time counterpart of the the results of Section 8.3.1 which we referred to as “certainty-equivalent” worst-case estimators. It is also similarly apparent that the filter gain derived in equation (8.72) may depend on the estimated state, and this will present a serious stumbling block in implementation. Therefore, in this section we derive results in which the gain matrices are not functions of the state x, but of ˆx and y only. The estimator is constructed on the assumption that the asymptotic value of ˆx equals x. We first construct the estimator for the 1-DOF case, and then we discuss the 2-DOF case.
We reconsider the nonlinear discrete-time affine causal state-space system defined on a state-space X ⊆ ℜn with zero-input:
(8.78) |
where x ∈ X is the state vector, w ∈ W ⊂ ℓ2([k0, ∞), ℜr) is the disturbance or noise signal, which belongs to the set W of admissible disturbances and noise signals, the output y ∈ Y ⊂ ℜm is the measured output of the system, which belongs to the set Y of measured outputs, while z ∈ ℜs is the output to be estimated. The functions f : χ → χ, g1 : χ → Mn×r, where Mi×j(χ) is the ring of i×j matrices over X, h2 : X → ℜm, and k21 : X → Mm×r(X ) are real C∞ functions of x. Furthermore, we assume without any loss of generality that the system (8.78) has a unique equilibrium-point at x = 0 such that f(0) = 0, h2(0) = 0.
Assumption 8.5.1 The system matrices are such that
k21(x)gT1(x) = 0k21(x)kT21(x) = I.
Again, the discrete-time H∞ nonlinear filtering problem (D H I N L F P ) is to synthesize a filter, Fk, for estimating the state xk from available observations Yk ≜ {yi, i ≤ k} over a time horizon [k0, ∞), such that
ˆxk+1=ℱk(Yk), k∈[k0,∞|),
and the ℓ2-gain from the disturbance/noise signal w to the estimation error output ˜z (to be defined later) is less than or equal to a desired number γ > 0, i.e.,
(8.79) |
for all w ∈ W, for all x0 ∈ O ⊂ X.
There are various notions of observability, however for our purpose, we shall adopt the following which also generalizes the notion of “zero-state observability.”
Definition 8.5.1 For the nonlinear system (8.78), we say that it is locally zero-input observable if for all states xk1, xk2 ∈ U ⊂ X and input w(.) ≡ 0,
y(.,xk1,w) ≡ y(.,xk2,w)⇒xk1 = xk2
where y(., xki, w), i = 1, 2 is the output of the system with the initial condition x(k0) = xki. Moreover, the system is said to be zero-input observable if it is locally zero-input observable at each x0 ∈ X or U = X.
We now propose the following class of estimators:
Σdacef1: {˙ˆxk+1 = f(ˆxk)+g1(ˆx)ˆw⋆k+L(ˆxk,yk)(yk−h2(ˆxk)−k21(ˆxk)ˆw⋆k) ˆzk = h2(ˆxk)˜zk = yk−h2(ˆxk) |
(8.80) |
where ˆz=ˆy∈ℜm is the estimated output, ˜zk∈ℜm is the new estimation error or penalty variable, ˆw⋆ is the estimated worst-case system noise and ˆL:X×Y→Mn×m is the gain-matrix of the filter. We first determine w⋆, and accordingly, define the Hamiltonian function H : X × Y × W × Mn×m × ℜ → ℜ corresponding to (8.80) and the cost functional
(8.81) |
by
H(ˆx,y,w,L,V) = V(f(ˆx)+g1(ˆx)w+L(ˆx,y)(y−h2(ˆx)−k21(ˆx)w,y)−V(ˆx,yk−1)+12(‖˜z‖−γ2‖w‖2) |
(8.82) |
for some smooth function V : X × Y → ℜ, where x = xk, y = yk, w = wk and the adjoint variable p is set as p = V. Applying now the necessary condition for the worst-case noise, we get
∂TH∂w|w=ˆw⋆ = (gT1(x)−kT21(ˆx)LT(ˆx,y))∂TV(λ,y)∂λ|λ=f⋆(ˆx,y,ˆw⋆)−γ2ˆω⋆=0, |
(8.83) |
where
f⋆(ˆx,y,ˆw⋆)=f(ˆx)+g1(ˆx)ˆw⋆+L(ˆx,y)(yk−h2(ˆx)−k21(ˆx)ˆw⋆),
and
ˆw⋆ = 1γ2(gT1(ˆx)−kT21(ˆx)LT(ˆx,y))∂TV(λ,y)∂λ|λ=f⋆(ˆx,y,ˆw⋆) :=α0(ˆx,ˆw⋆,y). |
(8.84) |
Moreover, since
∂2H∂w2|w=ˆw⋆ = (gT1(ˆx)−kT21(ˆx)LT(ˆx,y))∂2V(λ,y)∂λh2|λ=f⋆(ˆx,ˆw⋆,y)(g1(ˆx)−L(ˆx,y)k21(ˆx))−γ2I
is nonsingular about (ˆx, w, y) = (0, 0, 0), equation (8.84) has a unique solution ŵ⋆ = α(ˆx, y), α(0, 0) = 0 in the neighborhood N × W × Y of (x, w, y) = (0, 0, 0) by the Implicit-function Theorem [234].
Now substitute ŵ⋆ in the expression for H(., ., ., .) (8.82), to get
H(ˆx,y,ˆw⋆,L,V) = V(f(ˆx)+g1(ˆx)ˆw⋆+L(ˆx,y)(y−h2(ˆx)−k21(ˆx)ˆw⋆,y)−V(ˆx,yk−1)+12(‖˜z‖−γ2‖ˆw⋆‖2)
and let
(8.85) |
Then by Taylor’s theorem [267], we can expand H(., ., ., .) about (L⋆,ˆw⋆) as
H(ˆx,y,w,L,V) = H(ˆx,y,ˆw⋆,L⋆,V)+12(w−ˆw⋆)T∂2H∂w2(w,L⋆)(w−ˆw⋆)+ 12Tr{[In⊗(L−L⋆)T]∂2H∂L2(ˆw⋆,L)[Im⊗(L−L⋆)T]}+ O(‖w−ˆw⋆‖3+‖L−L⋆‖3). |
(8.86) |
Thus, taking L⋆ as in (8.85) and ŵ⋆ = α(ˆx, y) and if the conditions
(8.87) |
(8.88) |
hold, we see that the saddle-point conditions
H(w,L⋆)≤H(ˆw⋆,L⋆)≤H(w⋆,L), ∀L∈ℳn×m,∀w∈W
are locally satisfied. Moreover, setting
H(ˆx,y,ˆw⋆,L⋆,V)=0
gives the DHJIE:
V(f(ˆx)+g1(ˆx)α(ˆx,y)+L(ˆx,y)(y−h2(ˆx)−k21(ˆx)α(ˆx,y)),y)−V(ˆx,yk−1)+12‖˜z(ˆx)‖−12γ2‖α(ˆx,y)‖2=0, V(0,0)=0 ˆx,y∈N×Y. |
(8.89) |
Consequently, we have the following result.
Proposition 8.5.1 Consider the discrete-time nonlinear system (8.78) and the D H I N L F P for it. Suppose Assumption 8.5.1 holds, the plant Σda is locally asymptotically-stable about the equilibrium point x = 0 and zero-input observable. Further, suppose there exists a C2 positive-semidefinite function V : N × Y → ℜ+ locally defined in a neighborhood N × Y ⊂ X × Y of the origin (ˆx, y) = (0, 0), and a matrix function L : N × Y → Mn×m, satisfying the DHJIE (8.89) together with the side-conditions (8.85), (8.87), (8.88). Then the filter ∑dacef1 solves the D H I N L F P for the system locally in N.
Proof: The first part of the theorem on the existence of the saddle-point solutions (ˆw⋆,L⋆) has already been shown above. It remains to show that the ℓ2-gain condition (8.79) is satisfied and the filter provides asymptotic estimates.
For this, let V ≥ 0 be a C2 solution of the DHJIE (8.89) and reconsider equation (8.86). Since the conditions (8.87) and (8.88) are satisfied about ˆx = 0, by the Inverse-function Theorem [234], there exists a neighborhood U ⊂ N × W of (ˆx, w) = (0, 0) for which they are also satisfied. Consequently, we immediately have the important inequality
H(ˆx,y,w,L⋆,V)≤H(ˆx,y,ˆw⋆,L⋆,V) =0 ∀ˆx∈N,∀y∈Y,∀w∈W ⇔V(ˆxk+1,yk)−V(ˆxk,yk−1)≤12γ2‖wk‖2−12‖⌣zk‖2. |
(8.90) |
Summing now from k = k0 to ∞, we get that the ℓ2-gain condition (8.79) is satisfied:
(8.91) |
Moreover, setting wk ≡ 0 in (8.90), implies that V(ˆxk+1,yk)−V(ˆxk,yk−1)≤−12‖˜zk‖2 and hence the estimator dynamics is stable. In addition,
V(ˆxk+1,yk)−V(ˆxk,yk−1)≡0⇒˜z≡0⇒y=h2(ˆx)=ˆy.
By the zero-input observability of the system Σda, this implies that x = ˆx. □
8.5.1 2-DOF Proportional-Derivative (PD) CEFs
Next, we extend the above certainty-equivalent design to the 2-DOF case. In this regard, we assume that the time derivative yk − yk−1 is available (or equivalently yk−1 is available), and consider the following class of filters:
Σdacef2 :{′xk+1 = f(′xk)+g1(′xk)′w⋆(′xk)+′L1(′xk,yk,yk−1)(yk−h2(′xk) k21(′xk)′w⋆(′xk))+′L2(′xk,yk,yk−1)(yk−yk−1−h2(′xk)−h2(′xk−1))′zk =[ h2(′xk)h2(′xk)−h2(′xk−1)]˜zk = [yk−h2(′xk)(yk−yk−1)−(h2(′xk)−h2(′xk−1))]
where ź ∈m is the estimated output of the filter, ˜z ∈ ℜs is the error or penalty variable, while ′L1:X×X×Y×Y→ℳn×m,′L2:X×X×Y×Y→ℳn×m, are the proportional and derivative gains of the filter respectively, and all the other variables and functions have their corresponding previous meanings and dimensions. As in the previous section, we can define the corresponding Hamiltonian function ′H:X×X×Y×Y→ℳn×m×ℳn×m×ℜ→ℜ for the filter as
′H(′x,′w,′L1,′L2,′V) = ′V(′f(′x,′xk−1,y,yk−1),′x,y)−′V(′x,′xk−1,yk−1)+12(‖˜z‖2−γ2‖′w‖2) |
(8.92) |
for some smooth function ˜V:X×X×Y×Y→ℜ and where
′f(′x,′xk−1,y,yk−1) = f(′x)+g1(′x)′w+′L1(′x,′xk−1,yk,yk−1)[y−h2(′x)−k21(′x)′w]+ ′L2(′x,′xk−1,yk,yk−1)[yk−yk−1−h2(′xk)−h2(′xk−1)].
Notice that, in the above and subsequently, we only use the subscripts k, k−1 to distinguish the variables, otherwise, the functions are smooth in the variables ′x,′xk−1,y,yk−1,′w, etc.
Similarly, applying the necessary condition for the worst-case noise, we have
′w⋆= 1γ2(gT1(′x)−kT21(′x)′LT1(′x,′xk−1,y,yk−1))∂T′V(λ,′x,y)∂y|λ=′f(′x,.,.,.):=α1(′x,′xk−1,ω′⋆,y,yk−1) |
(8.93) |
Morever, since
∂′H∂w2=(gT1(′x)−kT21(′x)′LT1(′x,′xk−1,y,yk−1))∂T′V(λ,′x,y)∂y|λ=′f(′x,.,.,.) (g1(′x)−′L1(′x,.,.,.)k21(′x))−γ2I
is nonsingular about ′(x,′xk−1,′w,y,yk−1,) = (0, 0, 0, 0, 0), then again by the Impilicit-function Theorem, (8.93) has a unique solution ′w⋆=′(x,′xk−1,y,yk−1,),′α(0,0,0,0)=0 locally about ′(x,′xk−1,′w,y,yk−1,) = (0, 0, 0, 0, 0).
Substitute now ′w⋆ into (8.92) to get
′H(′x,′w,′L1,′L2,′V) = ′V(′f(′x,.,.,.),′x,y)−′V(′x,′xk−1,yk−1)+12(‖˜z‖2−γ2‖′w⋆‖2) |
(8.94) |
and let
(8.95) |
Then, it can be shown using Taylor-series expansion similar to (8.86) and if the conditions
(8.96) |
(8.97) |
(8.98) |
are locally satisfied, then the saddle-point conditions
′H(′w,′L⋆1,′L⋆2)≤′H(′w⋆,′L⋆1,′L⋆2)≤′H(′w⋆,′L⋆1,′L⋆2) ∀′L1,′L2∈ℳn×m ,∀w∈W,
are locally satisfied also.
Finally, setting
′H(′x,w⋆,L⋆1,′L⋆2,′VT′x)=0
yields the DHJIE
′V(′f⋆(′x,′xk−1,y,yk−1),′x,y)−′V(′x,′xk−1,yk−1)+12(‖˜z‖2−γ2‖′w⋆‖2)=0, ′V(0,0,0)=0, |
(8.99) |
where
′f⋆(′x,′xk−1,y,yk−1) = f(′x)+g1(′x)′w⋆+′L⋆1(′x,xk−1,yk,yk−1)[y−h2(′x)−k21(′x)′w⋆]+ ′L⋆2(′x,xk−1,yk,yk−1)[yk−yk−1−(h2(′x)−h2(′xk−1)].
With the above analysis, we have the following result.
Theorem 8.5.1 Consider the discrete-time nonlinear system (8.78) and the DHINLF P for it. Suppose Assumption 8.5.1 holds, the plant Σda is locally asymptotically stable about the equilibrium point x = 0 and zero-input observable. Further, suppose there exists a C2 positive-semidefinite function ′V : Ń × Ń × Ý → ℜ+ locally defined in a neighborhood Ń × Ń × Ý × Ý of the origin ′(x,′xk−1,y) = (0, 0, 0), and matrix functions Ĺ1, Ĺ2 ∈ Mn×m, satisfying the DHJIE (8.99) together with the side-conditions (8.93), (8.95), (8.97)-(8.98). Then the filter ∑dacef2 solves the D H I N L F P for the system locally in Ń.
Proof: We simply repeat the steps of the proof of Proposition 8.5.1. □
8.5.2 Approximate and Explicit Solution
It is hard to appreciate the results of Sections 8.5, 8.5.1, since the filter gains L, Ĺi, i = 1, 2, are given implicitly. Therefore, in this subsection, we address this difficulty and derive approximate explicit solutions. More specifically, we shall rederive explicitly the results of Proposition 8.5.1 and Theorem 8.5.1. We begin with the 1-DOF filter ∑dacef1. Accordingly, consider the Hamiltonian function H(., ., ., .) given by (8.82) and expand it in Taylor-series about f(ˆx) up to first-order. Denoting this approximation by ˆH(., ., ., .) and the corresponding values of L, V and w by ˆL,ˆV, and ŵ respectively, we get
ˆH(ˆx,y,ˆw,ˆL,ˆV) = {ˆV(f(ˆx),y)+ˆVˆx(f(ˆx),y)[g1(ˆx)ˆw+ˆL(ˆx,y)(y−h2(ˆx)−k21(ˆx)ˆw)]+O(‖ˆυ‖2)}−ˆV(ˆx,yk−1)+12(‖˜z‖2−γ2‖ˆw‖2) |
(8.100) |
where
ˆυ = g1(ˆx)ˆw+L(ˆx,y)(y−h2(ˆx)−k21(ˆx)ˆw), limυ→0O(‖ˆυ‖2)‖ˆυ‖2=0.
Now applying the necessary condition for the worst-case noise, we get
∂ˆH∂w|ˆw=ˆw⋆=gT1(ˆx)ˆVTˆx(ˆf(ˆx),y)−kT21(x)ˆLT(ˆx)ˆVTˆx(f(ˆx),y)−γ2ˆw⋆=0 ⇒ˆw⋆ : = 1γ2[gT1(ˆx)ˆVTˆx(ˆf(ˆx),y)−kT21(x)ˆLT(ˆx)ˆVTˆx(f(ˆx),y)]. |
(8.101) |
Consequently, substituting ŵ⋆ into (8.100) and assuming the conditions of Assumption 8.5.1 hold, we get
ˆH(ˆx,y,ˆw⋆,ˆL,ˆV) ≈ ˆV(f(ˆx),y)+12γ2ˆVˆx(f(ˆx),y)g1(ˆx)gT1(ˆx)ˆVTˆx(ˆf(ˆx),y)− ˆV(ˆx,yk−1)+ˆV(f(ˆx),y)ˆL(ˆx,y)(y−h2(ˆx))+ 12γ2ˆVˆx(f(ˆx),y)ˆL(ˆx,y)ˆLT(ˆx,y)ˆVˆx(f(ˆx),y)+12‖˜z‖2.
Next, we complete the squares with respect to ˆL in ˆH(., ., ŵ⋆ , .) to minimize it, i.e.,
(8.102)ˆH(ˆx,y,ˆw⋆,ˆL,ˆV) ≈ ˆV(f(ˆx),y)+12γ2ˆVˆx(f(ˆx),y)g1(ˆx)gT1(ˆx)ˆVTˆx(f(ˆx),y)−ˆV(ˆx,yk−1)+ 12γ2‖ˆLT(ˆx,y)ˆVˆx(f(ˆx))+γ2(y−h2(ˆx))‖2−γ22‖(y−h2(ˆx))‖2+12‖˜z‖2
Thus, setting ˆL* as
(8.102) |
minimizes Ĥ (., ., ., .) and renders the saddle-point condition
ˆH(ˆw⋆,ˆL⋆)≤ˆH(ˆw⋆,ˆL) ∀ˆL∈ℳn×m
satisfied.
Substitute now ˆL* as given by (8.102) in the expression for Ĥ (., ., ., .) and complete the squares in ŵ to obtain:
ˆH(ˆx,ˆw⋆,ˆL⋆,ˆV) = ˆV(f(ˆx),y)−12γ2ˆVˆx(f(ˆx),y)ˆL(ˆx,y)ˆLT(ˆx,y)ˆVTˆx(f(ˆx),y)−V(ˆx,yk−1)+ 12γ2ˆVˆx(f(ˆx),y)g1(ˆx)gT1(ˆx)ˆVTˆx(ˆf(ˆx),y)+12‖˜z‖2− γ22‖ˆw−1γ2gT1(ˆx)ˆVTˆx(ˆf(ˆx),y)−1γ2kT21(ˆx)ˆLT(ˆx)ˆVTˆx(ˆf(ˆx),y)‖2.
Similarly, substituting ŵ = ŵ⋆ as given in (8.101), we see that the second saddle-point condition
ˆH(ˆw⋆,ˆL⋆)≥ˆH(ˆw,ˆL⋆), ∀ˆw∈W
is also satisfied. Therefore, the pair (ˆw*,ˆL*) constitute a unique saddle-point solution to the two-person zero-sum dynamic game corresponding to the Hamiltonian Ĥ (., ., ., .). Finally, setting
ˆH(ˆx,ˆw⋆,ˆL⋆,ˆV)=0
yields the following DHJIE:
ˆV(f(ˆx),y)−ˆV(ˆx,yk−1)+12γ2ˆVˆx(f(ˆx),y)g1(ˆx)gT1(ˆx)ˆVTˆx(ˆf(ˆx),y)− 12γ2ˆVˆx(f(ˆx),y)ˆL(ˆx,y)ˆLT(ˆx,y)ˆVTˆx(f(ˆx),y)+ 12(y−h2(ˆx))T(y−h2(ˆx)) = 0, ˜V(0,0) = 0, ˆx∈ˆN |
(8.103) |
or equivalently the DHJIE:
ˆV(f(ˆx),y)−ˆV(ˆx,y)+12γ2ˆVˆx(f(ˆx),y)g1(ˆx)gT1(ˆx)ˆVTˆx(ˆf(ˆx),y)−γ22(y−h2(ˆx))T(y−h2(ˆx)) +12(y−h2(ˆx))T(y−h2(ˆx)) = 0, ˜V(0,0) = 0. |
(8.104) |
Consequently, we have the following approximate counterpart of Proposition 8.5.1.
Proposition 8.5.2 Consider the discrete-time nonlinear system (8.78) and the DHINLF P for it. Suppose Assumption 8.5.1 holds, the plant Σda is locally asymptotically-stable about the equilibrium point x = 0 and zero-input observable. Further, suppose there exists a C1 positive-semidefinite ˆV:ˆN׈Y→ℜ+ locally defined in a neighborhood ˆN׈Y⊂X×Y of the origin (ˆx,y) = (0, 0), and a matrix function ˆL:ˆN׈Y→Mn×m, satisfying the following DHJIE (8.103) or (8.104) together with the side-conditions (8.102). Then, the filter ∑dacef1 solves the D H I N L F P for the system locally in ˆN.
Proof: The first part of the theorem on the existence of the saddle-point solutions (ˆw*,ˆL*) has already been shown above. It remains to show that the ℓ2-gain condition (8.79) is satisfied and the filter provides asymptotic estimates.
Accordingly, assume there exists a smooth solution ˆV ≥ 0 to the DHJIE (8.103), and consider the time variation of ˆV along the trajectories of the filter ∑dacef1, (8.80), with ˆL=ˆL⋆, i.e.,
ˆV(ˆxk+1,y) = ˆV(f(ˆx)+ˆυ), ∀ˆx∈ˆN, ∀y∈ˆY, ∀ˆw∈W ≈ˆV(f(ˆx),y)+ˆVˆx(f(ˆx),y)g1(ˆx)ˆw+ˆVˆx(f(ˆx),y)[ˆL⋆(ˆx,y)(y−h2(ˆx)−k21(ˆx)ˆw)] = ˆV(f(ˆx),y)+ 12γ2ˆVˆx(f(ˆx),y)g1(ˆx)gT1(ˆx)ˆVTˆx(ˆf(ˆx),y)+ ˆVˆx(f(ˆx),y)ˆL⋆(ˆx,y)(y−h2(ˆx))−γ22‖ˆw−ˆw⋆‖2+ γ22‖ˆw‖2+12γ2ˆVˆx(f(ˆx),y)ˆL(ˆx,y)ˆL⋆T(ˆx,y)ˆVTˆx(f(ˆx),y) =ˆV(f(ˆx),y)+ 12γ2ˆVˆx(f(ˆx))g1(ˆx)gT1(ˆx)ˆVTˆx(ˆf(ˆx),y)+γ22‖ˆw‖2− γ22‖ˆw−ˆw⋆‖2−12γ2ˆVˆx(f(ˆx),y)ˆL⋆(ˆx)ˆL⋆T(ˆx)ˆVTˆx(f(ˆx),y) ≤ ˆV(ˆx,yk−1)+γ22‖ˆw‖2−12‖˜z‖2 ∀ˆx∈ˆN, y∈ˆY, ∀w∈W,
where use has been made of the Taylor-series approximation, equation (8.102), and the DHJIE (8.103). Finally, the above inequality clearly implies the infinitesimal dissipation inequality [183]:
ˆV(ˆxk+1,yk)−ˆV(ˆx,yk−1)≤12γ2‖ˆw‖2−12‖˜z‖2 ∀ˆx∈ˆN, ∀y∈ˆY, ∀ˆw∈W.
Therefore, the filter (8.80) provides locally ℓ2-gain from ˆw to ˜z less or equal to γ. The remaining arguments are the same as in the proof of Proposition 8.5.1. □
Next, we extend the above approximation procedure to the 2-DOF filter ∑dacef1 to arrive at the following result which is the approximate counterpart of Theorem 8.5.1.
Theorem 8.5.2 Consider the discrete-time nonlinear system (8.78) and the DHINLFP for it. Suppose Assumption 8.5.1 holds, the plant Σda is locally asymptotically-stable about the equilibrium point x = 0 and zero-input observable. Further, suppose there exists a C1 positive-semidefinite function ˆV:ˆN׈N׈Y→ℜ+ locally defined in a neighborhood ˆN׈N׈Y׈Y of the origin ′(x,′xk−1,y)=(0,0,0), and matrix functions ˆL1∈Mn×m,ˆL2∈Mn×m, satisfying the DHJIE:
ˆV(f(′x),′x,yk)+12γ2ˆV′x(f(′x),.,.,.)g1(′x)gT1(′x)ˆVT′x(f(′x),.,.,.)− ˆV(′x,′xk−1,yk−1)+(1−γ2)2(y−h2(′x))T(y−h2(′x))− 12(Δy−Δh2(′x))T(Δy−Δh2(′x)) = 0, ˆV(0,.,0) = 0 |
(8.105) |
together with the coupling conditions
(8.106) |
(8.107) |
where Δy = yk−yk−1, Δh2(x) = h2(xk)−h2(xk−1). Then the filter ∑dacef1 solves the DHINLF P for the system locally in ˆN.
Proof: (Sketch) We can similarly write the first-order Taylor-series approximation of ′H as
ˆH(′x,ˆw,ˆL1,ˆL2,ˆV) = ˆV(f(′x),′x,y)+ˆV′x(f(′x),.,.,.)g1(′x)′w+ ˆV(f(′x),′x,y)ˆL1(′x,.,.,.)(y−h2(′x)−k21(′x)′w)+ ˆV′x(f(′x),.,.,.)ˆL2(′x,.,.,.)(Δy−Δh2(′x))− ˆV(′x,′xk−1,yk−1)+12(‖˜zk‖2−γ2‖′w‖2)
where ˆw,ˆL1 and ˆL2 are the corresponding approximate values of ′w,′L1 and ′L2 respectively. Then we can also calculate the approximate estimated worst-case system noise, ˆw⋆≈′w⋆, as
ˆw⋆= 1γ2[gT1(′x)ˆVT′x(f(′x),.,.,.)−kT21(′x)ˆLT1(′x,.,.,.)ˆVT′x(f(′x),.,.,.)].
Subsequently, going through the rest of the steps of the proof as in Proposition 8.5.2, we get (8.106), (8.107), and setting
ˆH(′x,ˆw⋆,ˆL⋆1,ˆL⋆2,ˆVT′x) = 0
yields the DHJIE (8.105). □
We consider a simple example.
Example 8.5.1 We consider the following scalar system
xk+1 = x15k+x13k yk = xk+wk
where wk = w0k + 0.1 sin(20πk) and w0 is a zero-mean Gaussian white-noise.
We compute the approximate solutions of the DHJIEs (8.104) and (8.105) using an iterative process and then calculate the filter gains respectively. We outline each case below. 1-DOF Filter_
Let γ = 1 and since g1(x) = 0, we assume ˜V0(ˆx,y)=12(ˆx2+y2) and compute
ˆV1(x,y) = 12(ˆx15+ˆx13)2+12y2
ˆV1x(xk,yk) = (ˆx15k+ˆx13k)(15ˆx−45k+13ˆx−23k)
Therefore,
L(ˆxk,yk) = −yk−h2(ˆxk)(ˆx12k+ˆx13k)(15ˆx−45k+13ˆx−23k).
The filter is then simulated with a different initial condition from the system, and the results of the simulation are shown in Figure 8.8.
2-DOF Filter
Similarly, we compute an approximate solution of the DHJIE (8.105) starting with the initial guess ˆV(′x,y)=12(′x2+y2) and γ = 1. Moreover, we can neglect the last term in (8.105) since it is negative; hence the approximate solution we obtain will correspond to the solution of the DHJI-inequality corresponding to (8.105):
ˆV1(′x,′xk−1,y) = 12(′x12+′x13)2+12′x2k−1+12y2k−1⇒V1x(′xk,′xk−1,yk−1) = (ˆx12k+ˆx13k) (12′x−12k+12′x−23k)
using an iterative procedure [23], and compute the filter gains as
L1(′xk,′xk−1,yk,yk−1)= −yk−h2(′xk)(′x12k+′x13k)(12′x−12k+13′x−23k) |
(8.108) |
L2(′xk,′xk−1,yk,yk−1)= −(Δyk−Δh2(′xk))(′x12k+′x13k)(12′x−12k+13′x−23k). |
(8.109) |
This filter is simulated with the same initial condition as the 1-DOF filter above and the results of the simulation are shown similarly on Figure 8.9. The results of the simulations show that the 2-DOF filter has slightly improved performance over the 1-DOF filter.
8.6 Robust Discrete-Time Nonlinear H∞-Filtering
In this section, we discuss the robust H∞-filtering problem for a class of uncertain nonlinear discrete-time systems described by the following model, and defined on X ⊂ ℜn:
ΣadΔ : {xk+1 = [A+ΔAk]xk+Gg(xk)+Bwk; xk0 = x0, k∈z zk = C1xk yk = [C2+ΔC2,k]xk+Hh(xk)+Dwk |
(8.110) |
where all the variables have their previous meanings. In addition, A, ΔAk ∈ ℜn×n, G ∈ ℜn×n1, g : X → ℜn1, B ∈ ℜn×r, C1 ∈ ℜs×n, C2, ΔC2,k ∈ ℜm×n, and H ∈ ℜm×n2, h : X → ℜn1. ΔAk is the uncertainty in the system matrix A while ΔC2,k is the uncertainty in the system output matrix C2 which are both time-varying. Moreover, the uncertainties are matched, and belong to the following set of admissible uncertainties.
Assumption 8.6.1 The admissible uncertainties of the system are structured and matched, and they belong to the following set:
Ξd,Δ = {ΔAk,ΔC2,k|ΔAk = H1FkE, ΔC2,k = H2Fk,E, FTkFk≤I}
where H1, H2, Fk, E are real constant matrices of appropriate dimensions, and Fk is an unknown time-varying matrix.
Whereas the nonlinearities g(.) and h(.) satisfy the following assumption.
Assumption 8.6.2 The nonlinearies g(.) and h(.) are Lipschitz continuous, i.e., for any x1, x2 ∈ X, there exist constant matrices Γ1, Γ2 such that:
(8.111) |
(8.112) |
(8.113) |
for some constant matrices Γ1, Γ2.
The problem is then the following.
Definition 8.6.1 (Robust Discrete-Time Nonlinear ℋ∞ (Suboptimal) Filtering Problem (RDNLHIFP)). Given the system (8.110) and a prescribed level of noise attenuation γ > 0, find a causal filter Fk such that the ℓ2-gain from (w, x0) to the filtering error (to be defined later), ˜z, is attenuated by γ, i.e.,
‖˜z‖l2≤ γ2{‖w‖l2+x0˜Rx0},
and the error-dynamics (to be defined) is globally exponentially-stable for all (0, 0) = (w, x0) ∈ 2[k0, ∞) ⊕ X and all ΔAk, ΔC2,k ∈ ΞdΔ, k ∈ Z where ˜R=˜RT>0 > 0 is some suitable weighting matrix.
Before we present a solution to the above filtering problem, we establish the following bounded-real-lemmas for discrete-time-varying systems (see also Chapter 3) that will be required in the proof of the main result of this section.
Consider the linear discrete-time-varying system:
(8.114) |
where ⌣xk∈X ∈ X is the state vector, ⌣wk ∈ ℓ2([k0, ∞), ℜr) is the input vector, žk ∈ ℜs is the controlled output, and ⌣Ak,⌣Bk,⌣Ck are bounded time-varying matrices. The induced ℓ∞-norm (or H∞(jℜ)-norm in the context of our discussion) from (⌣w,⌣x0) to žk for the above system is defined by:
(8.115) |
Then, we have the following lemma.
Lemma 8.6.1 For the linear time-varying discrete-time system (8.114) and a given γ > 0, the following statements are equivalent:
(a) the system is exponentially stable and ‖Σdl‖ℓ∞ < γ;
(b) there exists a bounded time-varying matrix function Qk = QTk ≥ 0, ∀k ≥ k0 satisfying:
⌣AkQk⌣ATk−Qk+1+γ−2⌣AkQk⌣CTk(I−γ−2⌣CkQk⌣CTk)−1⌣CkQk⌣ATk+⌣BkBTk = 0;Qk0˜R−1, I−γ−2⌣CkQk⌣CTk>0 ∀k≥k0
and the closed-loop system
⌣xk+1=[⌣Ak+γ−2⌣AkQk⌣CTk(I−γ−2⌣CkQk⌣CTk)−1⌣Ck]⌣xk
is exponentially stable;
(c) there exists a scalar δ1 > 0 and a bounded time-varying matrix function ⌣Pk=⌣PkT ∀k ≥ k0 satisfying:
⌣Ak⌣Pk+1⌣ATk−⌣Pk+γ−2⌣ATk+1⌣Bk(I−γ−2⌣BTk⌣Pk+1⌣Bk)−1⌣BTk⌣Pk+1⌣Ak+⌣CTk⌣Ck+δ1I<0; ⌣Pk0<γ2˜R, I−γ−2⌣BTk⌣Pk+1⌣Bk>0 ∀k≥k0.
Proof: (a) ⇔ (b) ⇒ (c) has been shown in Reference [280], Theorem 3.1. To show that (a) ⇒ (c), we consider the extended output zb=[⌣Ck√δ1I]xk. By exponential stability of the system, and the fact that ‖∑dl0‖l∞<γ, there exists a sufficiently small number δ1 > 0 such that ‖∑dl‖l∞<γ for all (w, x0) ∈ ℓ2 ⊕ X to zb. The result then follows again from Theorem 3.1, Reference [280]. □
The following lemma gives the bounded-real conditions for the system (8.110).
Lemma 8.6.2 Consider the nonlinear discrete-time system ∑adΔ (8.110) satisfying Assumptions 8.6.1, 8.6.2. For a given γ > 0 and a weighting matrix ˜R=˜RT>0, the system is globally exponentially-stable and
‖z‖2l2<γ{‖w‖2l2+x0T˜Rx0}
for any non-zero (x0, w) ∈ X ⊕ ℓ2 and for all ΔAk if there exists scalars ϵ > 0, δ1 > 0 and a bounded time-varying matrix function Qk = QTk > 0, ∀k ≥ k0 satisfying:
ATQk+1A−Qk+γ−2ATQk+1B1(I−γ−2BT1Qk+1B1)−1BT1Qk+1A+CT1C1+ ϵ2ETE+ΓT1Γ1+δ1I<0; Qk0<γ2˜R, I−γ−2BT1Qk+1B1>0 ∀k≥k0, |
(8.116) |
where
B1=[BγϵH1 γG].
Proof: The inequality (8.116) implies that Qk > δ1I ∀k ≥ k0. Moreover, since Qk is bounded, there exists a scalar δ2 > 0 such that Qk ≤ δ2I ∀k ≥ k0. Now consider the Lyapunov-function candidate:
V(x,k)=xTkQkxk
such that
δ1‖x‖2≤V(x,k)≤δ2‖x‖2.
Then, it can be shown (using similar arguments as in the proof of Theorem 4.1, Reference [279]) that along any trajectory of the free-system (8.110) with wk = 0 ∀k ≥ k0,
ΔV(x,k)=V(k+1,x)−V(x,k)≤−δ1‖xk‖2.
Therefore, by Lyapunov’s theorem [157], the free-system is globally exponentially-stable. □
We now present the solution to R D N L H I F P for the class of nonlinear discrete-time systems ∑adΔ. For this, we need some additional assumptions on the system matrices.
Assumption 8.6.3 The system (8.110) matrices are such that
(a1)(C2, A) is detectable.
(a2)[D H2 H] has full row-rank.
(a3)The matrix A is nonsingular.
Theorem 8.6.1 Consider the uncertain nonlinear discrete-time system (8.110) satisfying the Assumptions 8.6.1-8.6.3. Given γ > 0 and ˜R=˜RT>0, let ν > 0 be a small number and suppose the following conditions hold:
(a) for some constant number ϵ > 0, there exists a stabilizing solution P = P T > 0 to the stationary DARE:
(8.117) |
such that P<γ2˜R and I−γ2˜BTP˜B>0 where
E1=(ε2ETE+ΓT1Γ1)12, ˜B=[BγεH1 γG].
(b) there exists a bounded time-varying matrix Sk = STk ≥ 0 ∀k ≥ k0, satisfying
Sk+1 = ˆASkAT−(ˆASkˆCT1+ˆBˆDT1)(ˆC1SkˆCT1+ˆR)−1(ˆC1SkˆAT+ˆD1ˆBT)+ˆBˆBT;S0 = (˜R−γ2P)−1, I−γ−2ˆMTSkˆMT>0 ∀k≥k0, |
(8.118) |
and the system
(8.119) |
is exponentially-stable, where
ρk+1 : A2kρk=[ˆA−(ˆASkˆCT1+ˆBˆDT1)(ˆC1SkˆCT1+ˆR)−1ˆC1]ρkˆA = A+δAe : A+ γ−2ˉBˉBT(P−1−γ−2ˉBˉBT)−1AˆC2 = C2+δC2e : =C2+ˉDˉBT(P−1−γ−2ˉBˉBT)−1A ˆB = [ˉBZ γG 0], ˆD = [ˉDZ 0 γH] ˆC1 = [γ−1ˆM ˆC2] , ˆD1 = [0ˆD] , ˆR = [−I00ˆDˆDT]ˉB = [BγϵH1], ˆD = [DγϵH2]ˆM = [CT1C1+ΓT1Γ1]12, Γ=[ΓT1ΓT2]TZ=[I+γ−2ˉBT(P−1−γ−2ˉBˉBT)−1ˉB]12.
Then, the RNLHIF P for the system is solvable with a finite-dimensional filter. Moreover, if the above conditions are satisfied, a suitable filter is given by
∑daf : {ˆxk+1 = ˆAˆx+Gg(ˆx)+ˆLk[yk−ˆC2ˆx−Hh(ˆx)], ˆxk0=0 ˆz = C1ˆx, |
(8.120) |
where ˆLk is the gain-matrix and is given by
ˆLk = (ˆASkˆCT2+ˆBˆDT1)(ˆC2ˆSkˆCT2+ˆDˆDT)−1ˆSk = Sk+γ−2SkˆMT(I−γ−2ˆMSkˆMT)−1ˆMSk. |
(8.121) |
Proof: We note that, P−1−γ−2˜B˜BT is positive-definite since I−γ−2˜BTP˜B>0.
Thus, Z is welldefined. Similarly, I−γ−2ˆMSkˆMT>0∀k≥k0 and together with Assumption 8.6.3:(a2), imply that Ĉ1SkCT1 + ˆR is nonsingular for all k ≥ k0. Consequently, equation (8.118) is welldefined.
Next, consider the filter Σdaf and rewrite its equation as:
ˆxk+1 = (A+δAe)ˆx+Gg(ˆx)+ˆLk[yk−(C2+δC2e)ˆx−Hh(ˆx)], ˆxk0=0 ˆzk= c1ˆx
where δAe and δC2e are defined above, and represent the uncertain and time-varying components of A and C2 (i.e., ΔAk and ΔC2k) respectively, that are compensated in the estimator.
Then the dynamics of the state estimation error ˜xk:=xk−ˆxk is given by
{˜xk+1 = [A+δAe−ˆLk(C2+δC2e)]˜x+[(ΔA−δAe)−ˆLk(C2+δC2e)]xk +(B−ˆLkD)wk+G[g(xk)−g(ˆx)]−ˆLkH[h(xk)−h(ˆx)], ˜xk0=x0 ˜z = C1˜x |
(8.122) |
where ˜z:=z−ˆz is the output estimation error. Now combine the system (8.110) and the error-dynamics (8.122) into the following augmented system:
ηk+1 = (Aa+HaFkEa)ηk+Gaga(xk,ˆx)+Bawk; ηk0 = [x0Tx0T]T ek = Caηk
where
η = [xT˜xT]TAa=[A0−(δAe−ˆLkδC2e)A+δAe−ˆLk(C2+δC2e)]Ba = [ BB−ˆLkD], Ha=[ H1H1−ˆLkH2]Ga = [G000G−ˆLkH], ga(xk,ˆxk) = [ g(xk)g(xk)−g(ˆxk)h(xk)−h(ˆxk)]Ca = [0C1], E=[E0].
Then, by Assumption 8.6.2
‖ga(xk,ˆxk)≤‖ˆΓηk‖, with ˆΓ = Blockdiag{Γ1,Γ}.
Further, define
Π = [Π11Π12ΠT21Π22] = AaxkATa−Xk+1+AaxkˆCTk(I−ˆCaXkˆCTa)−1ˆCaXkATa+ˆBaˆBTa |
(8.123) |
where
ˆBa=[γ−1Ba ϵ−1Ha Ga], ˆCa=[ˆE100ˆM]
and Ê1 is such that
ˆET1ˆE1 = ϵ2ETE+VT1V1+vI
Also, let Qk = γ−2Sk and
Xk=[P−100Qk].
Then by standard matrix manipulations, it can be shown that
Π11 = AP−1AT−P−1+AP−1ˆE1(I−ˆE1P−1ˆET1)−1ˆE1P−1AT+γ−2˜B˜BT
and
Π12 = −A(P−ˆET1ˆE1)−1(δAe−ˆLkδC2e)T+γ−2BBT+ ϵ-2H1HT1−(γ−2BDT+ϵ−2H1HT1)ˆLTk.
Moreover, since A is nonsingular, in view of (8.117) and the definition of δAe, δC2e, it implies that
Π11 = 0, Π12 = 0.
It remains to show that Π22 = 0. Using similar arguments as in Reference [89] (Theorem 3.1), it follows from (8.123) that Qk satisfies the DRE:
Qk+1 = ˆAˆQkˆAT−(ˆAˆQkˆCT+γ−2ˆBˆDT)(ˆCˆQkˆCT+γ−2ˆDˆDT)−1(ˆCˆQkˆAT+γ−2ˆDˆBT)+γ−2ˆBˆBT; Qk0=γ2R−P |
(8.124) |
where
ˆQk = Qk+QkˆMT(I−ˆMQkˆMT)−1ˆMQk.
Now, from (8.123) using some matrix manipulations we get
Π22 = ˆAˆQkˆAT−(ˆAˆQkˆCT+γ−2ˆBˆDT)ˆLk−ˆLk (ˆCˆQkˆAT+γ−2ˆDˆBT)+ ˆLk(ˆCˆQkˆCT+γ−2ˆDˆDT)ˆLTk+γ−2ˆBˆBT
and the gain matrix ˆLk from (8.121) can be rewritten as
(8.125) |
Thus, from (8.124) and (8.125), it follows that T22 = 0, and hence we conclude from (8.123) that
(8.126) |
where
Next, we show that, k is such that the time-varying system
(8.127) |
is exponentially-stable. Let
where A2k is as defined in (8.119), ‘⋆’ denotes a bounded but otherwise unimportant term, and
Ā is Schur-stable2 since P is a stabilizing solution of (8.117). Moreover, by exponential-stability of the system (8.119), it follows that (8.127) is also exponentially-stable. Therefore, k is the stabilizing solution of (8.124). Consequently, by Lemma 8.6.1 there exists a scalar δ1 > 0 and a bounded time-varying matrix Yk = Y Tk > 0 ∀k ≥ k0 such that
Noting that
we see that Yk satisfies the following inequality:
In addition,
Finally, application of Lemma 8.6.2 and using the definition of imply that the error-dynamics (8.122) are exponentially-stable and
for all and all
The material of Section 8.1 is based on the Reference [66], while the material in Section 8.4 is based on the Reference [15]. An alternative to the solution of the discrete-time problem is also presented in Reference [244] under some simplifying assumptions. The materials of Sections 8.3 and 8.5 on 2-DOF and certainty equivalent filters are based on the References [22, 23, 24]. In particular continuous-time and discrete-time 2-DOF proportional-integral (PI) filters which are the counterpart of the PD filters presented in the chapter, are discussed in [22] and [24] respectively.
Furthermore, the results on R N L H I F P - Section 8.2 are based on the reference [211], while the discrete-time case in Section 8.6 is based on [279]. Lastly, comparison of simulation results between the filter and the extended-Kalman-filter can be found in the same references.
1A second-order Taylor-series approximation would be more accurate, but the solutions become more complicated. Moreover, the first-order method gives a solution that is close to the continuous-time case [66].
2Eigenvalues of Ā are inside the unit circle.