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 $\widehat{x}$ and y only. The estimator is constructed on the assumption that the asymptotic value of $\widehat{x}$ 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:

${\Sigma }^{da}:\{\begin{array}{l}{x}_{k+1}=f\left(x_k\right)+g_1\left(x_k\right)w_k;x\left(k_0\right)=x^0\\ y_k=h_2\left(x\right)+k_{21}\left(x_k{w}_k\right)\end{array}$

(8.78)

where x ∈ $\mathcal{X}$ is the state vector, w ∈ W ⊂ ℓ₂([k₀, ∞), ℜ^r) is the disturbance or noise signal, which belongs to the set W of admissible disturbances and noise signals, the output y ∈ Y ⊂ ℜ^m is the measured output of the system, which belongs to the set Y of measured outputs, while z ∈ ℜ^s is the output to be estimated. The functions f : χ → χ, g₁ : χ → M^n×r, where M^i×j(χ) is the ring of i×j matrices over $\mathcal{X}$, h₂ : $\mathcal{X}$ → ℜ^m, and k₂₁ : $\mathcal{X}$ → M^m×r($\mathcal{X}$ ) 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

$\begin{array}{l}{k}_{21}\left(x\right)g_1^T\left(x\right)=0\\ k_{21}\left(x\right)k_{21}^T\left(x\right)=I.\end{array}$

Again, the discrete-time ${\mathcal{H}}_{\infty }$ nonlinear filtering problem (D H I N L F P ) is to synthesize a filter, F_k, for estimating the state x_k from available observations Y_k ≜ {y_i, i ≤ k} over a time horizon [k₀, ∞), such that

${\widehat{x}}_{k+1}={ℱ}_k\left(Y_k\right),k\in \left[k_0,\infty |\right),$

and the ℓ₂-gain from the disturbance/noise signal w to the estimation error output $\tilde{z}$ (to be defined later) is less than or equal to a desired number γ > 0, i.e.,

$k={\displaystyle \sum _{k=k_0}^{\infty }{\Vert {\tilde{z}}_k\Vert }^2\le {\text{γ}}^2{\displaystyle \sum _{k=k_0}^{\infty }{\Vert w_k\Vert }^2,}k\in z}$

(8.79)

for all w ∈ W, for all x₀ ∈ O ⊂ $\mathcal{X}$.

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 x_k1, x_k2 ∈ U ⊂ X and input w(.) ≡ 0,

$y\left(.,x_{k_1},w\right)\equiv y\left(.,x_{k_2},w\right)\Rightarrow x_{k_1}=x_{k_2}$

where y(., x_ki, w), i = 1, 2 is the output of the system with the initial condition x(k₀) = x_ki. Moreover, the system is said to be zero-input observable if it is locally zero-input observable at each x⁰ ∈ $\mathcal{X}$ or U = $\mathcal{X}$.

We now propose the following class of estimators:

${\Sigma }_1^{dacef}:\{\begin{array}{l}{\dot{\widehat{x}}}_{k+1}=f\left({\widehat{x}}_k\right)+g_1\left(\widehat{x}\right){\widehat{w}}_k^{\star }+L\left({\widehat{x}}_k,y_k\right)\left(y_k-h_2\left({\widehat{x}}_k\right)-k_{21}\left({\widehat{x}}_k\right){\widehat{w}}_k^{\star }\right)\\ {\widehat{z}}_k=h_2\left({\widehat{x}}_k\right)\\ {\tilde{z}}_k=y_k-h_2\left({\widehat{x}}_k\right)\end{array}$

(8.80)

where $\widehat{z}=\widehat{y}\in {\Re }^m$ is the estimated output, ${\tilde{z}}_k\in {\Re }^m$ is the new estimation error or penalty variable, ${\widehat{w}}^{\star }$ is the estimated worst-case system noise and $\widehat{L}:\mathcal{X}\times \mathcal{Y}\to {\mathcal{M}}^{n\times m}$ is the gain-matrix of the filter. We first determine w^⋆, and accordingly, define the Hamiltonian function H : X × Y × W × M^n×m × ℜ → ℜ corresponding to (8.80) and the cost functional

$J=\underset{L}{\min }\underset{w}{\max }{\displaystyle \sum _{k=k_0}^{\infty }\left[{\Vert {\tilde{z}}_k\Vert }^2-{\text{γ}}^2{\Vert w_k\Vert }^2\right]},k\in Z$

(8.81)

by

$H\left(\widehat{x},y,w,L,V\right)=V(f\left(\widehat{x}\right)+g_1\left(\widehat{x}\right)w+L\left(\widehat{x},y\right)\left(y-h_2\left(\widehat{x}\right)-k_{21}\left(\widehat{x}\right)w,y\right)-V\left(\widehat{x},y_{k-1}\right)+\frac{1}{2}\left(\Vert \tilde{z}\Vert -{\text{γ}}^2{\Vert w\Vert }^2\right)$

(8.82)

for some smooth function V : X × Y → ℜ, where x = x_k, y = y_k, w = w_k and the adjoint variable p is set as p = V. Applying now the necessary condition for the worst-case noise, we get

$\begin{array}{l}{\frac{{\partial }^{T}H}{\partial w}|}_{w={\widehat{w}}^{\star }}={\left(g_1^T\left(x\right)-k_{21}^T\left(\widehat{x}\right)L^T\left(\widehat{x},y\right)\right)\frac{{\partial }^{T}V\left(\text{λ,y}\right)}{\partial \text{λ}}|}_{\text{λ=}{f}^{\star }\left(\widehat{x},y,{\widehat{w}}^{\star }\right)}\\ -{\gamma }^2{\widehat{\omega }}^{\star }=0,\end{array}$

(8.83)

where

$f^{\star }(\widehat{x},y,{\widehat{w}}^{\star })=f(\widehat{x})+g_1(\widehat{x}){\widehat{w}}^{\star }+L(\widehat{x},y)(y_k-h_2(\widehat{x})-k_{21}(\widehat{x}){\widehat{w}}^{\star }),$

and

${\widehat{w}}^{\star }=\frac{1}{{\text{γ}}^2}\left(g_1^T\left(\widehat{x}\right)-k_{21}^T\left(\widehat{x}\right)L^T\left(\widehat{x},y\right)\right){\frac{{\partial }^{T}V\left(\text{λ,y}\right)}{\partial \text{λ}}|}_{\text{λ=}{f}^{\star }\left(\widehat{x},y,{\widehat{w}}^{\star }\right)}:={\alpha }_0\left(\widehat{x},{\widehat{w}}^{\star },y\right).$

(8.84)

Moreover, since

${\frac{{\partial }^{2}H}{\partial w^2}|}_{w={\widehat{w}}^{\star }}={\left(g_1^T\left(\widehat{x}\right)-k_{21}^T\left(\widehat{x}\right)L^T\left(\widehat{x},y\right)\right)\frac{{\partial }^{2}V\left(\text{λ,y}\right)}{\partial \text{λh2}}|}_{\text{λ=}{f}^{\star }\left(\widehat{x},{\widehat{w}}^{\star },y\right)}\left(g_1\left(\widehat{x}\right)-L\left(\widehat{x},y\right)k_{21}\left(\widehat{x}\right)\right)-{\text{γ}}^{2}I$

is nonsingular about ($\widehat{x}$, w, y) = (0, 0, 0), equation (8.84) has a unique solution ŵ^⋆ = α($\widehat{x}$, 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

$H\left(\widehat{x},y,{\widehat{w}}^{\star },L,V\right)=V(f\left(\widehat{x}\right)+g_1\left(\widehat{x}\right){\widehat{w}}^{\star }+L\left(\widehat{x},y\right)\left(y-h_2\left(\widehat{x}\right)-k_{21}\left(\widehat{x}\right){\widehat{w}}^{\star },y\right)-V\left(\widehat{x},y_{k-1}\right)+\frac{1}{2}\left(\Vert \tilde{z}\Vert -{\text{γ}}^2{\Vert {\widehat{w}}^{\star }\Vert }^2\right)$

and let

$L^{\star }=\arg \underset{L}{\min }\left\{H\left(\widehat{x},y,{\widehat{w}}^{\star },L,V\right)\right\}.$

(8.85)

Then by Taylor’s theorem [267], we can expand H(., ., ., .) about $(L^{\star },{\widehat{w}}^{\star })$ as

$\begin{array}{l}H\left(\widehat{x},y,w,L,V\right)=H\left(\widehat{x},y,{\widehat{w}}^{\star },L^{\star },V\right)+\frac{1}{2}{\left(w-{\widehat{w}}^{\star }\right)}^T\frac{{\partial }^{2}H}{\partial w^2}\left(w,L^{\star }\right)\left(w-{\widehat{w}}^{\star }\right)+\\ \frac{1}{2}{T}_r\left\{\left[I_n\otimes {\left(L-L^{\star }\right)}^T\right]\frac{{\partial }^{2}H}{\partial L^2}\left({\widehat{w}}^{\star },L\right)\left[I_m\otimes {\left(L-L^{\star }\right)}^T\right]\right\}+\\ O\left({\Vert w-{\widehat{w}}^{\star }\Vert }^3+{\Vert L-L^{\star }\Vert }^3\right).\end{array}$

(8.86)

Thus, taking L^⋆ as in (8.85) and ŵ^⋆ = α($\widehat{x}$, y) and if the conditions

${\frac{{\partial }^{2}H}{\partial w^2}\left(\widehat{x},y,w,L^{\star },V\right)|}_{\left(\widehat{x}=0,w=0\right)}<0,$

(8.87)

${\frac{{\partial }^{2}H}{\partial L^2}\left(\widehat{x},y,{\widehat{w}}^{\star },L,V\right)|}_{\left(\widehat{x}=0,w=0\right)}>0$

(8.88)

hold, we see that the saddle-point conditions

$H\left(w,L^{\star }\right)\le H\left({\widehat{w}}^{\star },L^{\star }\right)\le H\left(w^{\star },L\right),\forall L\in {ℳ}^{n\times m},\forall w\in \mathcal{W}$

are locally satisfied. Moreover, setting

$H\left(\widehat{x},y,{\widehat{w}}^{\star },L^{\star },V\right)=0$

gives the DHJIE:

$V(f\left(\widehat{x}\right)+g_1\left(\widehat{x}\right)\alpha \left(\widehat{x},y\right)+L\left(\widehat{x},y\right)\left(y-h_2\left(\widehat{x}\right)-k_{21}\left(\widehat{x}\right)\alpha \left(\widehat{x},y\right)),y\right)-V\left(\widehat{x},y_{k-1}\right)+\frac{1}{2}\Vert \tilde{z}\left(\widehat{x}\right)\Vert -\frac{1}{2}{\text{γ}}^2{\Vert \alpha \left(\widehat{x},y\right)\Vert }^2=0,V\left(0,0\right)=0\widehat{x},y\in N\times Y.$

(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 Σ^da 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 ⊂ $\mathcal{X}$ × Y of the origin ($\widehat{x}$, y) = (0, 0), and a matrix function L : N × Y → M^n×m, satisfying the DHJIE (8.89) together with the side-conditions (8.85), (8.87), (8.88). Then the filter ${\sum }_1^{dacef}$ 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 $({\widehat{w}}^{\star },L^{\star })$ 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 $\widehat{x}$ = 0, by the Inverse-function Theorem [234], there exists a neighborhood U ⊂ N × W of ($\widehat{x}$, w) = (0, 0) for which they are also satisfied. Consequently, we immediately have the important inequality

$\begin{array}{l}H\left(\widehat{x},y,w,L^{\star },V\right)\le H\left(\widehat{x},y,{\widehat{w}}^{\star },L^{\star },V\right)=0\forall \widehat{x}\in N,\forall y\in \mathcal{Y},\forall w\in \mathcal{W}\\ \iff V\left({\widehat{x}}_{k+1},y_k\right)-V\left({\widehat{x}}_k,y_{k-1}\right)\le \frac{1}{2}{\text{γ}}^2{\Vert w_k\Vert }^2-\frac{1}{2}{\Vert {\stackrel{⌣}{z}}_k\Vert }^2.\end{array}$

(8.90)

Summing now from k = k₀ to ∞, we get that the ℓ₂-gain condition (8.79) is satisfied:

$V\left({\widehat{x}}_{\infty },y_{\infty }\right)+\frac{1}{2}{\displaystyle \sum _{k_0}^{\infty }{\Vert {\stackrel{⌣}{z}}_k\Vert }^2}\le \frac{1}{2}{\displaystyle \sum _{k=k_0}^{\infty }{\text{γ}}^2}{\Vert w_k\Vert }^2+V\left(x_{k_0},y_{k_0-1}\right).$

(8.91)

Moreover, setting w_k ≡ 0 in (8.90), implies that $V\left({\widehat{x}}_{k+1},y_k\right)-V\left({\widehat{x}}_k,y_{k-1}\right)\le -\frac{1}{2}{\Vert {\tilde{z}}_k\Vert }^2$ and hence the estimator dynamics is stable. In addition,

$V\left({\widehat{x}}_{k+1},y_k\right)-V\left({\widehat{x}}_k,y_{k-1}\right)\equiv 0\Rightarrow \tilde{z}\equiv 0\Rightarrow y=h_2\left(\widehat{x}\right)=\widehat{y}.$

By the zero-input observability of the system Σ^da, this implies that x = $\widehat{x}$. □

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 y_k − y_k−1 is available (or equivalently y_k−1 is available), and consider the following class of filters:

${\Sigma }_2^{dacef}:\{\begin{array}{l}{\stackrel{\prime }{x}}_{k+1}=f\left({\stackrel{\prime }{x}}_k\right)+g_1\left({\stackrel{\prime }{x}}_k\right){\stackrel{\prime }{w}}^{\star }\left({\stackrel{\prime }{x}}_k\right)+{\stackrel{\prime }{L}}_1\left({\stackrel{\prime }{x}}_k,y_k,y_{k-1}\right)(y_k-h_2\left({\stackrel{\prime }{x}}_k\right)\\ k_{21}\left({\stackrel{\prime }{x}}_k\right){\stackrel{\prime }{w}}^{\star }\left({\stackrel{\prime }{x}}_k\right))+{\stackrel{\prime }{L}}_2\left({\stackrel{\prime }{x}}_k,y_k,y_{k-1}\right)(y_k-y_{k-1}-h_2\left({\stackrel{\prime }{x}}_k\right)-h_2\left({\stackrel{\prime }{x}}_{k-1}\right))\\ \stackrel{\prime }{z_k}=\left[\begin{array}{l}{h}_2\left({\stackrel{\prime }{x}}_k\right)\\ h_2\left({\stackrel{\prime }{x}}_k\right)-h_2\left({\stackrel{\prime }{x}}_{k-1}\right)\end{array}\right]\\ {\tilde{z}}_k=\left[\begin{array}{l}{y}_k-h_2\left({\stackrel{\prime }{x}}_k\right)\\ \left(y_k-y_{k-1}\right)-\left(h_2\left({\stackrel{\prime }{x}}_k\right)-h_2\left({\stackrel{\prime }{x}}_{k-1}\right)\right)\end{array}\right]\end{array}$

where ź ∈^m is the estimated output of the filter, $\tilde{z}$ ∈ ℜ^s is the error or penalty variable, while ${\stackrel{\prime }{L}}_1:\mathcal{X}\times \mathcal{X}\times \mathcal{Y}\times \mathcal{Y}\to {ℳ}^{n\times m},{\stackrel{\prime }{L}}_2:\mathcal{X}\times \mathcal{X}\times \mathcal{Y}\times \mathcal{Y}\to {ℳ}^{n\times m},$ 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 $\stackrel{\prime }{H}:\mathcal{X}\times \mathcal{X}\times \mathcal{Y}\times \mathcal{Y}\to {ℳ}^{n\times m}\times {ℳ}^{n\times m}\times \Re \to \Re$ for the filter as

$\stackrel{\prime }{H}\left(\stackrel{\prime }{x},\stackrel{\prime }{w},{\stackrel{\prime }{L}}_1,{\stackrel{\prime }{L}}_2,\stackrel{\prime }{V}\right)=\stackrel{\prime }{V}\left(\stackrel{\prime }{f}\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y,y_{k-1}\right),\stackrel{\prime }{x},y\right)-\stackrel{\prime }{V}\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y_{k-1}\right)+\frac{1}{2}\left({\Vert \tilde{z}\Vert }^2-{\text{γ}}^2{\Vert \stackrel{\prime }{w}\Vert }^2\right)$

(8.92)

for some smooth function $\tilde{V}\text{:}\mathcal{X}\times \mathcal{X}\times \mathcal{Y}\times \mathcal{Y}\to \Re$ and where

$\begin{array}{l}\stackrel{\prime }{f}\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y,y_{k-1}\right)=f\left(\stackrel{\prime }{x}\right)+g_1\left(\stackrel{\prime }{x}\right)\stackrel{\prime }{w}+{\stackrel{\prime }{L}}_1\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y_k,y_{k-1}\right)\left[y-h_2\left(\stackrel{\prime }{x}\right)-k_{21}\left(\stackrel{\prime }{x}\right)\stackrel{\prime }{w}\right]+\\ {\stackrel{\prime }{L}}_2\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y_k,y_{k-1}\right)\left[y_k-y_{k-1}-h_2\left({\stackrel{\prime }{x}}_k\right)-h_2\left({\stackrel{\prime }{x}}_{k-1}\right)\right].\end{array}$

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 $\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y,y_{k-1,}\stackrel{\prime }{w}$, etc.

Similarly, applying the necessary condition for the worst-case noise, we have

$\begin{array}{l}{\stackrel{\prime }{w}}^{\star }=\frac{1}{{\text{γ}}^2}\left(g_1^T\left(\stackrel{\prime }{x}\right)-k_{21}^T\left(\stackrel{\prime }{x}\right){\stackrel{\prime }{L}}_1^T\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y,y_{k-1}\right)\right){\frac{{\partial }^T\stackrel{\prime }{V}\left(\text{λ,}\stackrel{\prime }{x},y\right)}{\partial y}|}_{\text{λ=}\stackrel{\prime }{f}\left(\stackrel{\prime }{x},.,.,.\right)}\\ :={\alpha }_1\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},{{\omega }^{\prime }}^{\star },y,y_{k-1}\right)\end{array}$

(8.93)

Morever, since

$\frac{\partial \stackrel{\prime }{H}}{\partial w^2}=\left(g_1^T\left(\stackrel{\prime }{x}\right)-k_{21}^T\left(\stackrel{\prime }{x}\right){\stackrel{\prime }{L}}_1^T\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y,y_{k-1}\right)\right){\frac{{\partial }^T\stackrel{\prime }{V}\left(\text{λ,}\stackrel{\prime }{x},y\right)}{\partial y}|}_{\text{λ=}\stackrel{\prime }{f}\left(\stackrel{\prime }{x},.,.,.\right)}\left(g_1\left(\stackrel{\prime }{x}\right)-{\stackrel{\prime }{L}}_1\left(\stackrel{\prime }{x},.,.,.\right)k_{21}\left(\stackrel{\prime }{x}\right)\right)-{\text{γ}}^{2}I$

is nonsingular about $\stackrel{\prime }{(x},{\stackrel{\prime }{x}}_{k-1},\stackrel{\prime }{w},y,y_{k-1,})$ = (0, 0, 0, 0, 0), then again by the Impilicit-function Theorem, (8.93) has a unique solution ${\stackrel{\prime }{w}}^{\star }=\stackrel{\prime }{(x},{\stackrel{\prime }{x}}_{k-1},y,y_{k-1,}),\stackrel{\prime }{\alpha (0,0,0,0)=0}$ locally about $\stackrel{\prime }{(x},{\stackrel{\prime }{x}}_{k-1},\stackrel{\prime }{w},y,y_{k-1,})$ = (0, 0, 0, 0, 0).

Substitute now ${\stackrel{\prime }{w}}^{\star }$ into (8.92) to get

$\stackrel{\prime }{H}\left(\stackrel{\prime }{x},\stackrel{\prime }{w},{\stackrel{\prime }{L}}_1,{\stackrel{\prime }{L}}_2,\stackrel{\prime }{V}\right)=\stackrel{\prime }{V}\left(\stackrel{\prime }{f}\left(\stackrel{\prime }{x},.,.,.\right),\stackrel{\prime }{x},y\right)-\stackrel{\prime }{V}\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y_{k-1}\right)+\frac{1}{2}\left({\Vert \tilde{z}\Vert }^2-{\text{γ}}^2{\Vert {\stackrel{\prime }{w}}^{\star }\Vert }^2\right)$

(8.94)

and let

$\left[{\stackrel{\prime }{L}}_1^{\star }{\stackrel{\prime }{L}}_2^{\star }\right]=\arg \underset{{\stackrel{\prime }{L}}_1,{\stackrel{\prime }{L}}_2}{\min }\left\{\stackrel{\prime }{H}\left(\stackrel{\prime }{x},\stackrel{\prime }{w^{\star }},{\stackrel{\prime }{L}}_1,{\stackrel{\prime }{L}}_2,\stackrel{\prime }{V}\right)\right\}.$

(8.95)

Then, it can be shown using Taylor-series expansion similar to (8.86) and if the conditions

${\frac{\partial \stackrel{\prime }{H}}{\partial w^2}\left(\stackrel{\prime }{x},\stackrel{\prime }{w^{\star }},{\stackrel{\prime }{L}}_1^{\star },{\stackrel{\prime }{L}}_2^{\star },\stackrel{\prime }{V}\right)|}_{\left(\stackrel{\prime }{x}=0,\stackrel{\prime }{w=0}\right)}<0$

(8.96)

${\frac{\partial \stackrel{\text{'}}{H}}{\partial L_1^2}\left(\stackrel{\text{'}}{x},\stackrel{\text{'}}{w^{\star }},{\stackrel{\text{'}}{L}}_1,{\stackrel{\prime }{L}}_2^{\star },\stackrel{\prime }{V}\right)|}_{\left(\stackrel{\prime }{x}=0,\stackrel{\prime }{w=0}\right)}>0$

(8.97)

${\frac{\partial \stackrel{\prime }{H}}{\partial L_2^2}\left(\stackrel{\prime }{x},\stackrel{\prime }{w^{\star }},{\stackrel{\prime }{L}}_1^{\star },{\stackrel{\prime }{L}}_2,\stackrel{\prime }{V}\right)|}_{\left(\stackrel{\prime }{x}=0,\stackrel{\prime }{w=0}\right)}>0$

(8.98)

are locally satisfied, then the saddle-point conditions

$\stackrel{\prime }{H}\left(\stackrel{\prime }{w},{\stackrel{\prime }{L}}_1^{\star },{\stackrel{\prime }{L}}_2^{\star }\right)\le \stackrel{\prime }{H}\left(\stackrel{\prime }{w^{\star }},{\stackrel{\prime }{L}}_1^{\star },{\stackrel{\prime }{L}}_2^{\star }\right)\le \stackrel{\prime }{H}\left(\stackrel{\prime }{w^{\star }},{\stackrel{\prime }{L}}_1^{\star },{\stackrel{\prime }{L}}_2^{\star }\right)\forall {\stackrel{\prime }{L}}_1,{\stackrel{\prime }{L}}_2\in {ℳ}^{n\times m},\forall w\in \mathcal{W},$

are locally satisfied also.

Finally, setting

$\stackrel{\prime }{H\left(\stackrel{\prime }{x},\stackrel{}{w^{\star }},{\stackrel{}{L}}_1^{\star },{\stackrel{\prime }{L}}_2^{\star },{\stackrel{\prime }{V}}_{\stackrel{\prime }{x}}^T\right)}=0$

yields the DHJIE

$\stackrel{\prime }{V}\left(\stackrel{\prime }{f^{\star }}\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y,y_{k-1}\right),\stackrel{\prime }{x},y\right)-\stackrel{\prime }{V}\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y_{k-1}\right)+\frac{1}{2}\left({\Vert \tilde{z}\Vert }^2-{\text{γ}}^2{\Vert \stackrel{\prime }{w^{\star }}\Vert }^2\right)=0,\stackrel{\prime }{V}\left(0,0,0\right)=0,$

(8.99)

where

$\begin{array}{l}\stackrel{\prime }{f}\star (\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y,y_{k-1})=f\left(\stackrel{\prime }{x}\right)+g_1\left(\stackrel{\prime }{x}\right)\stackrel{\prime }{w^{\star }}+{\stackrel{\prime }{L}}_1^{\star }\left(\stackrel{\prime }{x},x_{k-1},y_k,y_{k-1}\right)\left[y-h_2\left(\stackrel{\prime }{x}\right)-k_{21}\left(\stackrel{\prime }{x}\right)\stackrel{\prime }{w^{\star }}\right]+\\ {\stackrel{\prime }{L}}_2^{\star }\left(\stackrel{\prime }{x},x_{k-1},y_k,y_{k-1}\right)\left[y_k-y_{k-1}-(h_2\left(\stackrel{\prime }{x}\right)-h_2\left({\stackrel{\prime }{x}}_{k-1}\right)\right].\end{array}$

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 Σ^da is locally asymptotically stable about the equilibrium point x = 0 and zero-input observable. Further, suppose there exists a C² positive-semidefinite function $\stackrel{\prime }{V}$ : Ń × Ń × Ý → ℜ₊ locally defined in a neighborhood Ń × Ń × Ý × Ý of the origin $\stackrel{\prime }{(x},{\stackrel{\prime }{x}}_{k-1},y)$ = (0, 0, 0), and matrix functions Ĺ₁, Ĺ₂ ∈ M^n×m, satisfying the DHJIE (8.99) together with the side-conditions (8.93), (8.95), (8.97)-(8.98). Then the filter ${\sum }_2^{\mathrm{dacef}}$ 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, Ĺ_i, 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 ${\sum }_1^{\text{dacef}}$. Accordingly, consider the Hamiltonian function H(., ., ., .) given by (8.82) and expand it in Taylor-series about f($\widehat{x}$) up to first-order. Denoting this approximation by $\widehat{H}$(., ., ., .) and the corresponding values of L, V and w by $\widehat{L},\widehat{V}$, and ŵ respectively, we get

$\begin{array}{l}\widehat{H}\left(\widehat{x},y,\widehat{w},\widehat{L},\widehat{V}\right)\\ =\left\{\widehat{V}\left(f\left(\widehat{x}\right),y\right)+{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)\left[g_1\left(\widehat{x}\right)\widehat{w}+\widehat{L}\left(\widehat{x},y\right)\left(y-h_2\left(\widehat{x}\right)-k_{21}\left(\widehat{x}\right)\widehat{w}\right)\right]+O\left({\Vert \widehat{\text{υ}}\Vert }^2\right)\right\}-\widehat{V}\left(\widehat{x},y_{k-1}\right)+\frac{1}{2}\left({\Vert \tilde{z}\Vert }^2-{\text{γ}}^2{\Vert \widehat{w}\Vert }^2\right)\end{array}$

(8.100)

where

$\begin{array}{l}\widehat{\text{υ}}\text{ =}{\text{g}}_1\left(\widehat{x}\right)\widehat{w}+L\left(\widehat{x},y\right)\left(y-h_2\left(\widehat{x}\right)-k_{21}\left(\widehat{x}\right)\widehat{w}\right),\\ \underset{\text{υ}\to 0}{\lim }\frac{O\left({\Vert \widehat{\text{υ}}\Vert }^2\right)}{{\Vert \widehat{\text{υ}}\Vert }^2}=0.\end{array}$

Now applying the necessary condition for the worst-case noise, we get

$\begin{array}{l}{\frac{\partial \widehat{H}}{\partial w}|}_{\widehat{w}={\widehat{w}}^{\star }}=g_1^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(\widehat{f}\left(\widehat{x}\right),y\right)-k_{21}^T\left(x\right){\widehat{L}}^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(f\left(\widehat{x}\right),y\right)-{\text{γ}}^2{\widehat{w}}^{\star }=0\\ \Rightarrow {\widehat{w}}^{\star }:=\frac{1}{{\text{γ}}^2}\left[g_1^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(\widehat{f}\left(\widehat{x}\right),y\right)-k_{21}^T\left(x\right){\widehat{L}}^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(f\left(\widehat{x}\right),y\right)\right].\end{array}$

(8.101)

Consequently, substituting ŵ^⋆ into (8.100) and assuming the conditions of Assumption 8.5.1 hold, we get

$\begin{array}{l}\widehat{H}\left(\widehat{x},y,{\widehat{w}}^{\star },\widehat{L},\widehat{V}\right)\approx \widehat{V}\left(f\left(\widehat{x}\right),y\right)+\frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)g_1\left(\widehat{x}\right)g_1^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(\widehat{f}\left(\widehat{x}\right),y\right)-\\ \widehat{V}\left(\widehat{x},y_{k-1}\right)+\widehat{V}\left(f\left(\widehat{x}\right),y\right)\widehat{L}\left(\widehat{x},y\right)\left(y-h_2\left(\widehat{x}\right)\right)+\\ \frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)\widehat{L}\left(\widehat{x},y\right){\widehat{L}}^T\left(\widehat{x},y\right){\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)+\frac{1}{2}{\Vert \tilde{z}\Vert }^2.\end{array}$

Next, we complete the squares with respect to $\widehat{L}$ in $\widehat{H}$(., ., ŵ^⋆ , .) to minimize it, i.e.,

$\begin{array}{l}\left(8.102\right)\\ \widehat{H}\left(\widehat{x},y,{\widehat{w}}^{\star },\widehat{L},\widehat{V}\right)\approx \widehat{V}\left(f\left(\widehat{x}\right),y\right)+\frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)g_1\left(\widehat{x}\right)g_1^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(f\left(\widehat{x}\right),y\right)-\widehat{V}\left(\widehat{x},y_{k-1}\right)+\\ \frac{1}{2{\text{γ}}^2}{\Vert {\widehat{L}}^T\left(\widehat{x},y\right){\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right)\right)+{\text{γ}}^2\left(y-h_2\left(\widehat{x}\right)\right)\Vert }^2-\frac{{\text{γ}}^2}{2}{\Vert \left(y-h_2\left(\widehat{x}\right)\right)\Vert }^2+\frac{1}{2}{\Vert \tilde{z}\Vert }^2\end{array}$

Thus, setting ${\widehat{L}}^{*}$ as

$\widehat{V}\left(f\left(\widehat{x}\right),y\right){\widehat{L}}^{\star }\left(\widehat{x}\right)=-{\text{γ}}^2{\left(y-h_2\left(\widehat{x}\right)\right)}^T,\widehat{x}\in \widehat{N}$

(8.102)

minimizes Ĥ (., ., ., .) and renders the saddle-point condition

$\widehat{H}\left({\widehat{w}}^{\star },{\widehat{L}}^{\star }\right)\le \widehat{H}\left({\widehat{w}}^{\star },\widehat{L}\right)\forall \widehat{L}\in {ℳ}^{n\times m}$

satisfied.

Substitute now ${\widehat{L}}^{*}$ as given by (8.102) in the expression for Ĥ (., ., ., .) and complete the squares in ŵ to obtain:

$\begin{array}{l}\widehat{H}\left(\widehat{x},{\widehat{w}}^{\star },{\widehat{L}}^{\star },\widehat{V}\right)=\widehat{V}\left(f\left(\widehat{x}\right),y\right)-\frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)\widehat{L}\left(\widehat{x},y\right){\widehat{L}}^T\left(\widehat{x},y\right){\widehat{V}}_{\widehat{x}}^T\left(f\left(\widehat{x}\right),y\right)-V\left(\widehat{x},y_{k-1}\right)+\\ \frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)g_1\left(\widehat{x}\right)g_1^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(\widehat{f}\left(\widehat{x}\right),y\right)+\frac{1}{2}{\Vert \tilde{z}\Vert }^2-\\ \frac{{\text{γ}}^2}{2}{\Vert \widehat{w}-\frac{1}{{\text{γ}}^2}{g}_1^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(\widehat{f}\left(\widehat{x}\right),y\right)-\frac{1}{{\text{γ}}^2}{k}_{21}^T\left(\widehat{x}\right){\widehat{L}}^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(\widehat{f}\left(\widehat{x}\right),y\right)\Vert }^2.\end{array}$

Similarly, substituting ŵ = ŵ^⋆ as given in (8.101), we see that the second saddle-point condition

$\widehat{H}\left({\widehat{w}}^{\star },{\widehat{L}}^{\star }\right)\ge \widehat{H}\left(\widehat{w},{\widehat{L}}^{\star }\right),\forall \widehat{w}\in \mathcal{W}$

is also satisfied. Therefore, the pair (${\widehat{w}}^{*},{\widehat{L}}^{*}$) constitute a unique saddle-point solution to the two-person zero-sum dynamic game corresponding to the Hamiltonian Ĥ (., ., ., .). Finally, setting

$\widehat{H}\left(\widehat{x},{\widehat{w}}^{\star },{\widehat{L}}^{\star },\widehat{V}\right)=0$

yields the following DHJIE:

$\begin{array}{l}\widehat{V}\left(f\left(\widehat{x}\right),y\right)-\widehat{V}\left(\widehat{x},y_{k-1}\right)+\frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)g_1\left(\widehat{x}\right)g_1^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(\widehat{f}\left(\widehat{x}\right),y\right)-\\ \frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)\widehat{L}\left(\widehat{x},y\right){\widehat{L}}^T\left(\widehat{x},y\right){\widehat{V}}_{\widehat{x}}^T\left(f\left(\widehat{x}\right),y\right)+\\ \frac{1}{2}{\left(y-h_2\left(\widehat{x}\right)\right)}^T\left(y-h_2\left(\widehat{x}\right)\right)=0,\tilde{V}\left(0,0\right)=0,\widehat{x}\in \widehat{N}\end{array}$

(8.103)

or equivalently the DHJIE:

$\begin{array}{l}\widehat{V}\left(f\left(\widehat{x}\right),y\right)-\widehat{V}\left(\widehat{x},y\right)+\frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)g_1\left(\widehat{x}\right)g_1^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(\widehat{f}\left(\widehat{x}\right),y\right)-\\ \frac{{\text{γ}}^2}{2}{\left(y-h_2\left(\widehat{x}\right)\right)}^T\left(y-h_2\left(\widehat{x}\right)\right)+\frac{1}{2}{\left(y-h_2\left(\widehat{x}\right)\right)}^T\left(y-h_2\left(\widehat{x}\right)\right)=0,\tilde{V}\left(0,0\right)=0.\end{array}$

(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 Σ^da is locally asymptotically-stable about the equilibrium point x = 0 and zero-input observable. Further, suppose there exists a C¹ positive-semidefinite $\widehat{V}:\widehat{N}\times \widehat{Y}\to {\Re }_{+}$ locally defined in a neighborhood $\widehat{N}\times \widehat{Y}\subset X\times Y$ of the origin ($\widehat{x},y$) = (0, 0), and a matrix function $\widehat{L}:\widehat{N}\times \widehat{Y}\to M^{n\times m}$, satisfying the following DHJIE (8.103) or (8.104) together with the side-conditions (8.102). Then, the filter ${\sum }_1^{\mathrm{dacef}}$ solves the D H I N L F P for the system locally in $\widehat{N}$.

Proof: The first part of the theorem on the existence of the saddle-point solutions (${\widehat{w}}^{*},{\widehat{L}}^{*}$) has already been shown above. It remains to show that the ℓ₂-gain condition (8.79) is satisfied and the filter provides asymptotic estimates.

Accordingly, assume there exists a smooth solution $\widehat{V}$ ≥ 0 to the DHJIE (8.103), and consider the time variation of $\widehat{V}$ along the trajectories of the filter ${\sum }_1^{\mathrm{dacef}}$, (8.80), with $\widehat{L}={\widehat{L}}^{\star }$, i.e.,

$\begin{array}{l}\widehat{V}\left({\widehat{x}}_{k+1},y\right)=\widehat{V}\left(f\left(\widehat{x}\right)+\widehat{\text{υ}}\right),\forall \widehat{x}\in \widehat{N},\forall y\in \widehat{Y},\forall \widehat{w}\in \mathcal{W}\\ \approx \widehat{V}\left(f\left(\widehat{x}\right),y\right)+{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)g_1\left(\widehat{x}\right)\widehat{w}+{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)\left[{\widehat{L}}^{\star }\left(\widehat{x},y\right)\left(y-h_2\left(\widehat{x}\right)-k_{21}\left(\widehat{x}\right)\widehat{w}\right)\right]\\ =\widehat{V}\left(f\left(\widehat{x}\right),y\right)+\frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)g_1\left(\widehat{x}\right)g_1^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(\widehat{f}\left(\widehat{x}\right),y\right)+\\ {\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right){\widehat{L}}^{\star }\left(\widehat{x},y\right)\left(y-h_2\left(\widehat{x}\right)\right)-\frac{{\text{γ}}^2}{2}{\Vert \widehat{w}-{\widehat{w}}^{\star }\Vert }^2+\\ \frac{{\text{γ}}^2}{2}{\Vert \widehat{w}\Vert }^2+\frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right)\widehat{L}\left(\widehat{x},y\right){\widehat{L}}^{{\star }^T}\left(\widehat{x},y\right){\widehat{V}}_{\widehat{x}}^T\left(f\left(\widehat{x}\right),y\right)\\ =\widehat{V}\left(f\left(\widehat{x}\right),y\right)+\frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right)\right)g_1\left(\widehat{x}\right)g_1^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(\widehat{f}\left(\widehat{x}\right),y\right)+\frac{{\text{γ}}^2}{2}{\Vert \widehat{w}\Vert }^2-\\ \frac{{\text{γ}}^2}{2}{\Vert \widehat{w}-{\widehat{w}}^{\star }\Vert }^2-\frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}\left(f\left(\widehat{x}\right),y\right){\widehat{L}}^{\star }\left(\widehat{x}\right){\widehat{L}}^{{\star }^T}\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(f\left(\widehat{x}\right),y\right)\\ \le \widehat{V}\left(\widehat{x},y_{k-1}\right)+\frac{{\text{γ}}^2}{2}{\Vert \widehat{w}\Vert }^2-\frac{1}{2}{\Vert \tilde{z}\Vert }^2\forall \widehat{x}\in \widehat{N},y\in \widehat{Y},\forall \stackrel{}{w}\in \mathcal{W},\end{array}$

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]:

$\widehat{V}\left({\widehat{x}}_{k+1},y_k\right)-\widehat{V}\left(\widehat{x},y_{k-1}\right)\le \frac{1}{2}{\text{γ}}^2{\Vert \widehat{w}\Vert }^2-\frac{1}{2}{\Vert \tilde{z}\Vert }^2\forall \widehat{x}\in \widehat{N},\mathrm{\forall y}\in \widehat{Y},\forall \widehat{w}\in \mathcal{W}.$

Therefore, the filter (8.80) provides locally ℓ₂-gain from $\widehat{w}$ to $\tilde{z}$ 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 ${\sum }_1^{\mathrm{dacef}}$ 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 Σ^da is locally asymptotically-stable about the equilibrium point x = 0 and zero-input observable. Further, suppose there exists a C¹ positive-semidefinite function $\widehat{V}:\widehat{N}\times \widehat{N}\times \widehat{Y}\to {\Re }_{+}$ locally defined in a neighborhood $\widehat{N}\times \widehat{N}\times \widehat{Y}\times \widehat{Y}$ of the origin $\stackrel{\prime }{(x},{\stackrel{\prime }{x}}_{k-1},y)=(0,0,0)$, and matrix functions ${\widehat{L}}_1\in M^{n\times m},{\widehat{L}}_2\in M^{n\times m}$, satisfying the DHJIE:

$\begin{array}{l}\widehat{V}\left(f\left(\stackrel{\prime }{x}\right),\stackrel{\prime }{x},y_k\right)+\frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\stackrel{\prime }{x}}\left(f\left(\stackrel{\prime }{x}\right),.,.,.\right)g_1\left(\stackrel{\prime }{x}\right)g_1^T\left(\stackrel{\prime }{x}\right){\widehat{V}}_{\stackrel{\prime }{x}}^T\left(f\left(\stackrel{\prime }{x}\right),.,.,.\right)-\\ \widehat{V}\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y_{k-1}\right)+\frac{\left(1-{\text{γ}}^2\right)}{2}{\left(y-h_2\left(\stackrel{\prime }{x}\right)\right)}^T\left(y-h_2\left(\stackrel{\prime }{x}\right)\right)-\\ \frac{1}{2}{\left(\mathrm{\Delta }y-\mathrm{\Delta }{h}_2\left(\stackrel{\prime }{x}\right)\right)}^T\left(\mathrm{\Delta }y-\mathrm{\Delta }{h}_2\left(\stackrel{\prime }{x}\right)\right)=0,\widehat{V}\left(0,.,0\right)=0\end{array}$

(8.105)

together with the coupling conditions

$\widehat{V_{\stackrel{\prime }{x}}}\left(\stackrel{\prime }{x},.,.\right){\widehat{L}}_1^{\star }\left(\stackrel{\prime }{x},.,.\right)=-{\text{γ}}^2{\left(y-h_2\left(\stackrel{\prime }{x}\right)\right)}^T$

(8.106)

$\widehat{V_{\stackrel{\prime }{x}}}\left(\stackrel{\prime }{x},.,.\right){\widehat{L}}_2^{\star }\left(\stackrel{\prime }{x},.,.\right)=-{\left(\mathrm{\Delta }y-\mathrm{\Delta }{h}_2\left(\stackrel{\prime }{x}\right)\right)}^T$

(8.107)

$where\mathrm{\Delta }y=y_k-y_{k-1},\mathrm{\Delta }{h}_2\left(x\right)=h_2\left(x_k\right)-h_2\left(x_{k-1}\right).$ Then the filter ${\sum }_1^{\mathrm{dacef}}$ solves the DHINLF P for the system locally in $\widehat{N}$.

Proof: (Sketch) We can similarly write the first-order Taylor-series approximation of $\stackrel{\prime }{H}$ as

$\begin{array}{l}\widehat{H}\left(\stackrel{\prime }{x},\widehat{w},{\widehat{L}}_1,{\widehat{L}}_2,\widehat{V}\right)=\widehat{V}\left(f\left(\stackrel{\prime }{x}\right),\stackrel{\prime }{x},y\right)+{\widehat{V}}_{\stackrel{\prime }{x}}\left(f\left(\stackrel{\prime }{x}\right),.,.,.\right)g_1\left(\stackrel{\prime }{x}\right)\stackrel{\prime }{w}+\\ \widehat{V}\left(f\left(\stackrel{\prime }{x}\right),\stackrel{\prime }{x},y\right){\widehat{L}}_1\left(\stackrel{\prime }{x},.,.,.\right)\left(y-h_2\left(\stackrel{\prime }{x}\right)-k_{21}\left(\stackrel{\prime }{x}\right)\stackrel{\prime }{w}\right)+\\ {\widehat{V}}_{\stackrel{\prime }{x}}\left(f\left(\stackrel{\prime }{x}\right),.,.,.\right){\widehat{L}}_2\left(\stackrel{\prime }{x},.,.,.\right)\left(\mathrm{\Delta }y-\mathrm{\Delta }{h}_2\left(\stackrel{\prime }{x}\right)\right)-\\ \widehat{V}\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y_{k-1}\right)+\frac{1}{2}\left({\Vert {\tilde{z}}_k\Vert }^2-{\text{γ}}^2{\Vert \stackrel{\prime }{w}\Vert }^2\right)\end{array}$

where $\widehat{w},{\widehat{L}}_1$ and ${\widehat{L}}_2$ are the corresponding approximate values of $\stackrel{\prime }{w},\stackrel{\prime }{L_1}$ and $\stackrel{\prime }{L_2}$ respectively. Then we can also calculate the approximate estimated worst-case system noise, ${\widehat{w}}^{\star }\approx \stackrel{\prime }{w^{\star }}$, as

${\widehat{w}}^{\star }=\frac{1}{{\text{γ}}^2}\left[g_1^T\left(\stackrel{\prime }{x}\right){\widehat{V}}_{\stackrel{\prime }{x}}^T\left(f\left(\stackrel{\prime }{x}\right),.,.,.\right)-k_{21}^T\left(\stackrel{\prime }{x}\right){\widehat{L}}_1^T\left(\stackrel{\prime }{x},.,.,.\right){\widehat{V}}_{\stackrel{\prime }{x}}^T\left(f\left(\stackrel{\prime }{x}\right),.,.,.\right)\right].$

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

$\widehat{H}\left(\stackrel{\prime }{x},{\widehat{w}}^{\star },{\widehat{L}}_1^{\star },{\widehat{L}}_2^{\star },{\widehat{V}}_{\stackrel{\prime }{x}}^T\right)=0$

yields the DHJIE (8.105). □

We consider a simple example.

Example 8.5.1 We consider the following scalar system

$\begin{array}{l}{x}_{k+1}=x_k^{\frac{1}{5}}+x_k^{\frac{1}{3}}\\ y_k=x_k+w_k\end{array}$

where w_k = w_0k + 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. $\underset{\_}{\text{1-DOF Filter}}$

Let γ = 1 and since g₁(x) = 0, we assume ${\tilde{V}}^0(\widehat{x},y)=\frac{1}{2}\left({\widehat{x}}^2+y^2\right)$ and compute

${\widehat{V}}^1\left(x,y\right)=\frac{1}{2}{\left({\widehat{x}}^{\frac{1}{5}}+{\widehat{x}}^{\frac{1}{3}}\right)}^2+\frac{1}{2}{y}^2$

${\widehat{V}}_x^1\left(x_k,y_k\right)=\left({\widehat{x}}_k^{\frac{1}{5}}+{\widehat{x}}_k^{\frac{1}{3}}\right)\left(\frac{1}{5}{\widehat{x}}_k^{\frac{-4}{5}}+\frac{1}{3}{\widehat{x}}_k^{\frac{-2}{3}}\right)$

Therefore,

$L\left({\widehat{x}}_k,y_k\right)=-\frac{y_k-h_2\left({\widehat{x}}_k\right)}{\left({\widehat{x}}_k^{\frac{1}{2}}+{\widehat{x}}_k^{\frac{1}{3}}\right)\left(\frac{1}{5}{\widehat{x}}_k^{\frac{-4}{5}}+\frac{1}{3}{\widehat{x}}_k^{\frac{-2}{3}}\right)}.$

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.

$\underset{}{\text{2-DOF Filter}}$

Similarly, we compute an approximate solution of the DHJIE (8.105) starting with the initial guess $\widehat{V}(\stackrel{\prime }{x},y)=\frac{1}{2}\left({\stackrel{\prime }{x}}^2+y^2\right)$ 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):

FIGURE 8.8
1-DOF Discrete-Time ${\mathcal{H}}_{\infty }$-Filter Performance with Unknown Initial Condition; Reprinted from Int. J. of Robust and Nonlinear Control, vol. 20, no. 7, pp. 818-833, © 2010, “2-DOF Discrete-time nonlinear ${\mathcal{H}}_{\infty }$ filtering,” by M. D. S. Aliyu and E. K. Boukas, with permission from Wiley Blackwell.

$\begin{array}{l}{\widehat{V}}^1\left(\stackrel{\prime }{x},{\stackrel{\prime }{x}}_{k-1},y\right)=\frac{1}{2}{\left({\stackrel{\prime }{x}}^{\frac{1}{2}}+{\stackrel{\prime }{x}}^{\frac{1}{3}}\right)}^2+\frac{1}{2}{\stackrel{\prime }{x}}_{k-1}^2+\frac{1}{2}{y}_{k-1}^2\\ \Rightarrow V_x^1\left({\stackrel{\prime }{x}}_k,{\stackrel{\prime }{x}}_{k-1},y_{k-1}\right)=\left({\widehat{x}}_k^{\frac{1}{2}}+{\widehat{x}}_k^{\frac{1}{3}}\right)\left(\frac{1}{2}{\stackrel{\prime }{x}}_k^{\frac{-1}{2}}+\frac{1}{2}{\stackrel{\prime }{x}}_k^{\frac{-2}{3}}\right)\end{array}$

using an iterative procedure [23], and compute the filter gains as

$L_1\left({\stackrel{\prime }{x}}_k,{\stackrel{\prime }{x}}_{k-1},y_k,y_{k-1}\right)=-\frac{y_k-h_2\left({\stackrel{\prime }{x}}_k\right)}{\left({\stackrel{\prime }{x}}_k^{\frac{1}{2}}+{\stackrel{\prime }{x}}_k^{\frac{1}{3}}\right)\left(\frac{1}{2}{\stackrel{\prime }{x}}_k^{\frac{-1}{2}}+\frac{1}{3}{\stackrel{\prime }{x}}_k^{\frac{-2}{3}}\right)}$

(8.108)

$L_2\left({\stackrel{\prime }{x}}_k,{\stackrel{\prime }{x}}_{k-1},y_k,y_{k-1}\right)=-\frac{\left(\mathrm{\Delta }{y}_k-\mathrm{\Delta }{h}_2\left({\stackrel{\prime }{x}}_k\right)\right)}{\left({\stackrel{\prime }{x}}_k^{\frac{1}{2}}+{\stackrel{\prime }{x}}_k^{\frac{1}{3}}\right)\left(\frac{1}{2}{\stackrel{\prime }{x}}_k^{\frac{-1}{2}}+\frac{1}{3}{\stackrel{\prime }{x}}_k^{\frac{-2}{3}}\right)}.$

(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 ${\mathcal{H}}_{\infty }$-Filtering

In this section, we discuss the robust ${\mathcal{H}}_{\infty }$-filtering problem for a class of uncertain nonlinear discrete-time systems described by the following model, and defined on $\mathcal{X}$ ⊂ ℜⁿ:

${\Sigma }_{\mathrm{\Delta }}^{ad}\text{}\text{}:\{\begin{array}{l}{x}_{k+1}=\left[A+\mathrm{\Delta }{A}_k\right]x_k+Gg\left(x_k\right)+B{w}_k;x_{k_0}=x^0,k\in z\\ z_k=C_1{x}_k\\ y_k=\left[C_2+\mathrm{\Delta }{C}_{2,k}\right]x_k+Hh\left(x_k\right)+D{w}_k\end{array}$

(8.110)

FIGURE 8.9
2-DOF Discrete-Time ${\mathcal{H}}_{\infty }$-Filter Performance with Unknown Initial Condition; Reprinted from Int. J. of Robust and Nonlinear Control, vol. 20, no. 7, pp. 818-833, © 2010,“2-DOF Discrete-time nonlinear ${\mathcal{H}}_{\infty }$ filtering,” by M. D. S. Aliyu and E. K. Boukas, with permission from Wiley Blackwell.

where all the variables have their previous meanings. In addition, A, ΔA_k ∈ ℜ^n×n, G ∈ ℜ^n×n1, g : $\mathcal{X}$ → ℜⁿ¹, B ∈ ℜ^n×r, C₁ ∈ ℜ^s×n, C₂, ΔC_2,k ∈ ℜ^m×n, and H ∈ ℜ^m×n2, h : $\mathcal{X}$ → ℜⁿ¹. ΔA_k is the uncertainty in the system matrix A while ΔC_2,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:

${\Xi }_{d,\mathrm{\Delta }}=\left\{\mathrm{\Delta }{A}_k,\mathrm{\Delta }{C}_{2,k}|\mathrm{\Delta }{A}_k=H_1{F}_{k}E,\mathrm{\Delta }{C}_{2,k}=H_2{F}_k,E,F_k^T{F}_k\le I\right\}$

where H₁, H₂, F_k, E are real constant matrices of appropriate dimensions, and F_k 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 ∈ $\mathcal{X}$, there exist constant matrices Γ₁, Γ₂ such that:

$g\left(0\right)=0$

(8.111)

$\Vert g\left(x_1\right)-g\left(x_2\right)\Vert \le \Vert {\Gamma }_1\left(x_1-x_2\right)\Vert ,$

(8.112)

$\Vert h\left(x_1\right)-h\left(x_2\right)\Vert \le \Vert {\Gamma }_2\left(x_1-x_2\right)\Vert .$

(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 F_k such that the ℓ₂-gain from (w, x⁰) to the filtering error (to be defined later), $\tilde{z}$, is attenuated by γ, i.e.,

${\Vert \tilde{z}\Vert }_{l_2}\le {\text{γ}}^2\left\{{\Vert w\Vert }_{l_2}+x^0\tilde{R}{x}^0\right\},$

and the error-dynamics (to be defined) is globally exponentially-stable for all (0, 0) = (w, x⁰) ∈ ₂[k₀, ∞) ⊕ X and all ΔA_k, ΔC_2,k ∈ Ξ^d_Δ, k ∈ Z where $\tilde{R}={\tilde{R}}^T>0$ > 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:

${\Sigma }^{dl}:\{\begin{array}{l}{\dot{\stackrel{⌣}{x}}}_{k+1}={\stackrel{⌣}{A}}_k{\stackrel{⌣}{x}}_k+{\stackrel{⌣}{B}}_k{\stackrel{⌣}{w}}_k,{\stackrel{⌣}{x}}_{k_0}={\stackrel{⌣}{x}}^0\\ {\stackrel{⌣}{z}}_k={\stackrel{⌣}{C}}_k{\stackrel{⌣}{x}}_k\end{array}$

(8.114)

where ${\stackrel{⌣}{x}}_k\in X$ ∈ $\mathcal{X}$ is the state vector, ${\stackrel{⌣}{w}}_k$ ∈ ℓ₂([k₀, ∞), ℜ^r) is the input vector, ž_k ∈ ℜ^s is the controlled output, and ${\stackrel{⌣}{A}}_k,{\stackrel{⌣}{B}}_k,{\stackrel{⌣}{C}}_k$ are bounded time-varying matrices. The induced ℓ_∞-norm (or ${\mathcal{H}}_{\infty }$(jℜ)-norm in the context of our discussion) from $(\stackrel{⌣}{w},{\stackrel{⌣}{x}}^0)$ to ž_k for the above system is defined by:

${\Vert {\Sigma }^{dl}\Vert }_{l_{\infty }}\triangleq \underset{\left(\stackrel{⌣}{w},{\stackrel{⌣}{x}}_0\right)\in l_2\otimes \mathcal{X}}{\sup }\frac{{\Vert \stackrel{⌣}{z}\Vert }^2}{{\Vert \stackrel{⌣}{w}\Vert }^2+{\stackrel{⌣}{x}}^{0T}\tilde{R}{\stackrel{⌣}{x}}^0}.$

(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 ‖Σ^dl‖_ℓ∞ < γ;

(b)  there exists a bounded time-varying matrix function Q_k = Q^T_k ≥ 0, ∀k ≥ k₀ satisfying:

$\begin{array}{l}{\stackrel{⌣}{A}}_k{Q}_k{\stackrel{⌣}{A}}_k^T-Q_{k+1}+{\text{γ}}^{-2}{\stackrel{⌣}{A}}_k{Q}_k{\stackrel{⌣}{C}}_k^T{\left(I-{\text{γ}}^{-2}{\stackrel{⌣}{C}}_k{Q}_k{\stackrel{⌣}{C}}_k^T\right)}^{-1}{\stackrel{⌣}{C}}_k{Q}_k{\stackrel{⌣}{A}}_k^T+{\stackrel{⌣}{B}}_k{B}_k^T=0;Q_{k_0}{\tilde{R}}^{-1},\\ I-{\text{γ}}^{-2}{\stackrel{⌣}{C}}_k{Q}_k{\stackrel{⌣}{C}}_k^T>0\forall k\ge k_0\end{array}$

and the closed-loop system

${\stackrel{⌣}{x}}_{k+1}=\left[{\stackrel{⌣}{A}}_k+{\text{γ}}^{-2}{\stackrel{⌣}{A}}_k{Q}_k{\stackrel{⌣}{C}}_k^T{\left(I-{\text{γ}}^{-2}{\stackrel{⌣}{C}}_k{Q}_k{\stackrel{⌣}{C}}_k^T\right)}^{-1}{\stackrel{⌣}{C}}_k\right]{\stackrel{⌣}{x}}_k$

is exponentially stable;

(c)  there exists a scalar δ₁ > 0 and a bounded time-varying matrix function ${\stackrel{⌣}{P}}_{k=}{\stackrel{⌣}{P}}_k{}^T$ ∀k ≥ k₀ satisfying:

$\begin{array}{l}{\stackrel{⌣}{A}}_k{\stackrel{⌣}{P}}_{k+1}{\stackrel{⌣}{A}}_k^T-{\stackrel{⌣}{P}}_k{\text{+γ}}^{-2}{\stackrel{⌣}{A}}_{k+1}^T{\stackrel{⌣}{B}}_k{\left(I-{\text{γ}}^{-2}{\stackrel{⌣}{B}}_k^T{\stackrel{⌣}{P}}_{k+1}{\stackrel{⌣}{B}}_k\right)}^{-1}{\stackrel{⌣}{B}}_k^T{\stackrel{⌣}{P}}_{k+1}{\stackrel{⌣}{A}}_k+{\stackrel{⌣}{C}}_k^T{\stackrel{⌣}{C}}_k+{\delta }_{1}I<0;\\ {\stackrel{⌣}{P}}_{k_0}<{\text{γ}}^2\tilde{R},\\ I-{\text{γ}}^{-2}{\stackrel{⌣}{B}}_k^T{\stackrel{⌣}{P}}_{k+1}{\stackrel{⌣}{B}}_k>0\forall k\ge k_0.\end{array}$

Proof: (a) ⇔ (b) ⇒ (c) has been shown in Reference [280], Theorem 3.1. To show that (a) ⇒ (c), we consider the extended output $z^b=\left[\frac{{\stackrel{⌣}{C}}_k}{\sqrt{{\delta }_1}I}\right]x_k$. By exponential stability of the system, and the fact that ${\Vert {\sum }_0^{dl}\Vert }_{l_{\infty }}<\text{γ}$, there exists a sufficiently small number δ₁ > 0 such that ${\Vert {\sum }_{}^{dl}\Vert }_{l_{\infty }}<\text{γ}$ for all (w, x₀) ∈ ℓ₂ ⊕ X to z^b. 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 ${\sum }_{\mathrm{\Delta }}^{ad}$ (8.110) satisfying Assumptions 8.6.1, 8.6.2. For a given γ > 0 and a weighting matrix $\tilde{R}={\tilde{R}}^T>0$, the system is globally exponentially-stable and

${\Vert z\Vert }_{l_2}^2<\text{γ}\left\{{\Vert w\Vert }_{l_2}^2+x^{0^T}\tilde{R}{x}^0\right\}$

for any non-zero (x₀, w) ∈ $\mathcal{X}$ ⊕ ℓ₂ and for all ΔA_k if there exists scalars ϵ > 0, δ₁ > 0 and a bounded time-varying matrix function Q_k = Q^T_k > 0, ∀k ≥ k₀ satisfying:

$\begin{array}{l}{A}^T{Q}_{k+1}A-Q_k+{\text{γ}}^{-2}{A}^T{Q}_{k+1}{B}_1{\left(I-{\text{γ}}^{-2}{B}_1^T{Q}_{k+1}{B}_1\right)}^{-1}{B}_1^T{Q}_{k+1}A+C_1^T{C}_1+\\ {ϵ}^2{E}^{T}E+{\Gamma }_1^T{\Gamma }_1+{\delta }_{1}I<0;\\ Q_{k_0}<{\text{γ}}^2\tilde{R},\\ I-{\text{γ}}^{-2}{B}_1^T{Q}_{k+1}{B}_1>0\forall k\ge k_0,\end{array}$

(8.116)

where

$B_1=\left[B\frac{\text{γ}}{ϵ}{H}_1\text{γG}\right].$

Proof: The inequality (8.116) implies that Q_k > δ₁I ∀k ≥ k₀. Moreover, since Q_k is bounded, there exists a scalar δ₂ > 0 such that Q_k ≤ δ₂I ∀k ≥ k₀. Now consider the Lyapunov-function candidate:

$V\left(x,k\right)=x_k^T{Q}_k{x}_k$

such that

${\delta }_1{\Vert x\Vert }^2\le V\left(x,k\right)\le {\delta }_2{\Vert x\Vert }^2.$

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 w_k = 0 ∀k ≥ k₀,

$\mathrm{\Delta }V\left(x,k\right)=V\left(k+1,x\right)-V\left(x,k\right)\le -{\delta }_1{\Vert x_k\Vert }^2.$

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 ${\sum }_{\mathrm{\Delta }}^{ad}$. 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 $\tilde{R}={\tilde{R}}^T>0$, 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:

$A^{T}PA-P+{\text{γ}}^{-2}{A}^{T}P\tilde{B}{\left(I-{\text{γ}}^{-2}{\tilde{B}}^{T}P\tilde{B}\right)}^{-1}{\tilde{B}}^{T}PA+E_1^T{E}_1+vI=0$

(8.117)

such that $P<{\text{γ}}^2\tilde{R}$ and $I-{\text{γ}}^2{\tilde{B}}^{T}P\tilde{B}>0$ where

$E_1={\left({\epsilon }^2{E}^{T}E+{\Gamma }_1^T{\Gamma }_1\right)}^{\frac{1}{2}},\tilde{B}=\left[B\frac{\text{γ}}{\epsilon }{H}_1\text{γ}G\right].$

(b) there exists a bounded time-varying matrix S_k = S^T_k ≥ 0 ∀k ≥ k₀, satisfying

$\begin{array}{l}{S}_{k+1}=\widehat{A}{S}_k{A}^T-\left(\widehat{A}{S}_k{\widehat{C}}_1^T+\widehat{B}{\widehat{D}}_1^T\right){\left({\widehat{C}}_1{S}_k{\widehat{C}}_1^T+\widehat{R}\right)}^{-1}\left({\widehat{C}}_1{S}_k{\widehat{A}}^T+{\widehat{D}}_1{\widehat{B}}^T\right)+\widehat{B}{\widehat{B}}^T;\\ S_0={\left(\tilde{R}-{\text{γ}}^{2}P\right)}^{-1},\\ I-{\text{γ}}^{-2}{\widehat{M}}^T{S}_k{\widehat{M}}^T>0\forall k\ge k_0,\end{array}$

(8.118)

and the system

${\rho }_{k+1}:A_{2k}{\rho }_k=\left[\widehat{A}-\left(\widehat{A}{S}_k{\widehat{C}}_1^T+\widehat{B}{\widehat{D}}_1^T\right){\left({\widehat{C}}_1{S}_k{\widehat{C}}_1^T+\widehat{R}\right)}^{-1}{\widehat{C}}_1\right]{\rho }_k$

(8.119)

is exponentially-stable, where

$\begin{array}{l}{\rho }_{k+1}:A_{2k}{\rho }_k=\left[\widehat{A}-\left(\widehat{A}{S}_k{\widehat{C}}_1^T+\widehat{B}{\widehat{D}}_1^T\right){\left({\widehat{C}}_1{S}_k{\widehat{C}}_1^T+\widehat{R}\right)}^{-1}{\widehat{C}}_1\right]{\rho }_k\\ \widehat{A}=A+\delta A_e:A+{\text{γ}}^{-2}\overline{B}{\overline{B}}^T{\left(P^{-1}-{\text{γ}}^{-2}\overline{B}{\overline{B}}^T\right)}^{-1}A\\ {\widehat{C}}_2=C_2+\delta C_{2e}:=C_2+\overline{D}{\overline{B}}^T{\left(P^{-1}-{\text{γ}}^{-2}\overline{B}{\overline{B}}^T\right)}^{-1}A\\ \widehat{B}=\left[\overline{B}Z\text{γG}\text{0}\right],\widehat{D}=\left[\overline{D}Z0\text{γH}\right]\\ {\widehat{C}}_1=\left[\begin{array}{l}{\text{γ}}^{-1}\widehat{M}\\ {\widehat{C}}_2\end{array}\right],{\widehat{D}}_1=\left[\begin{array}{l}0\\ \widehat{D}\end{array}\right],\widehat{R}=\left[\begin{array}{cc}-I& 0\\ 0& \widehat{D}{\widehat{D}}^T\end{array}\right]\\ \overline{B}=\left[\begin{array}{cc}B& \frac{\text{γ}}{ϵ}{H}_1\end{array}\right],\widehat{D}=\left[\begin{array}{cc}D& \frac{\text{γ}}{ϵ}{H}_2\end{array}\right]\\ \widehat{M}={\left[C_1^T{C}_1+{\Gamma }_1^T{\Gamma }_1\right]}^{\frac{1}{2}},\Gamma ={\left[\begin{array}{cc}{\Gamma }_1^T& {\Gamma }_2^T\end{array}\right]}^T\\ Z={\left[I+{\text{γ}}^{-2}{\overline{B}}^T{\left(P^{-1}-{\text{γ}}^{-2}\overline{B}{\overline{B}}^T\right)}^{-1}\overline{B}\right]}^{\frac{1}{2}}.\end{array}$

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

${\sum }^{daf}:\{\begin{array}{l}{\widehat{x}}_{k+1}=\widehat{A}\widehat{x}+Gg\left(\widehat{x}\right)+{\widehat{L}}_k\left[y_k-{\widehat{C}}_2\widehat{x}-Hh\left(\widehat{x}\right)\right],{\widehat{x}}_{k_0}=0\\ \widehat{z}=C_1\widehat{x},\end{array}$

(8.120)

where ˆL_k is the gain-matrix and is given by

$\begin{array}{l}{\widehat{L}}_k=\left(\widehat{A}{S}_k{\widehat{C}}_2^T+\widehat{B}{\widehat{D}}_1^T\right){\left({\widehat{C}}_2{\widehat{S}}_k{\widehat{C}}_2^T+\widehat{D}{\widehat{D}}^T\right)}^{-1}\\ {\widehat{S}}_k=S_k+{\text{γ}}^{-2}Sk{\widehat{M}}^T{\left(I-{\text{γ}}^{-2}\widehat{M}{S}_k{\widehat{M}}^T\right)}^{-1}\widehat{M}{S}_k.\end{array}$

(8.121)

Proof: We note that, $P^{-1}-{\text{γ}}^{-}{}^2\tilde{B}{\tilde{B}}^T$ is positive-definite since $I-{\text{γ}}^{-}{}^2{\tilde{B}}^{T}P\tilde{B}>0$.

Thus, Z is welldefined. Similarly, $I-{\text{γ}}^{-}{}^2\widehat{M}{S}_k{\widehat{M}}^T>0\forall k\ge k_0$ and together with Assumption 8.6.3:(a2), imply that Ĉ₁S_kC^T₁ + $\widehat{R}$ is nonsingular for all k ≥ k₀. Consequently, equation (8.118) is welldefined.

Next, consider the filter Σ^daf and rewrite its equation as:

$\begin{array}{l}{\widehat{x}}_{k+1}=\left(A+\delta A_e\right)\widehat{x}+Gg\left(\widehat{x}\right)+{\widehat{L}}_k\left[y_k-\left(C_2+\delta C_{2e}\right)\widehat{x}-Hh\left(\widehat{x}\right)\right],{\widehat{x}}_{k_0}=0\\ {\widehat{z}}_k=c_1\widehat{x}\end{array}$

where δA_e and δC_2e are defined above, and represent the uncertain and time-varying components of A and C₂ (i.e., ΔA_k and ΔC_2k) respectively, that are compensated in the estimator.

Then the dynamics of the state estimation error ${\tilde{x}}_k:=x_k-{\widehat{x}}_k$ is given by

$\{\begin{array}{l}{\tilde{x}}_{k+1}=\left[A+\delta A_e-{\widehat{L}}_k\left(C_2+\delta C_{2e}\right)\right]\tilde{x}+\left[\left(\mathrm{\Delta }A-\delta A_e\right)-{\widehat{L}}_k\left(C_2+\delta C_{2e}\right)\right]x_k\\ +\left(B-{\widehat{L}}_{k}D\right)w_k+G\left[g\left(x_k\right)-g\left(\widehat{x}\right)\right]-{\widehat{L}}_{k}H\left[h\left(x_k\right)-h\left(\widehat{x}\right)\right],{\tilde{x}}_{k_0}=x_0\\ \tilde{z}=C_1\tilde{x}\end{array}$

(8.122)

where $\tilde{z}:=z-\widehat{z}$ is the output estimation error. Now combine the system (8.110) and the error-dynamics (8.122) into the following augmented system:

$\begin{array}{l}{\eta }_{k+1}=\left(A_a+H_a{F}_k{E}_a\right){\eta }_k+G_a{g}_a\left(x_k,\widehat{x}\right)+B_a{w}_k;{\eta }_{k_0}={\left[\begin{array}{cc}{x}^{0^T}& x^{0^T}\end{array}\right]}^T\\ e_k=C_a{\eta }_k\end{array}$

where

$\begin{array}{l}\eta ={\left[\begin{array}{cc}{x}^T& {\tilde{x}}^T\end{array}\right]}^T\\ A_a=\left[\begin{array}{cc}A& 0\\ -\left(\delta A_e-{\widehat{L}}_k\delta C_{2e}\right)& A+\delta A_e-{\widehat{L}}_k\left(C_2+\delta C_{2e}\right)\end{array}\right]\\ B_a=\left[\begin{array}{l}B\\ B-{\widehat{L}}_{k}D\end{array}\right],H_a=\left[\begin{array}{l}{H}_1\\ H_1-{\widehat{L}}_k{H}_2\end{array}\right]\\ G_a=\left[\begin{array}{l}\begin{array}{ccc}G& 0& 0\end{array}\\ \begin{array}{ccc}0& G& -{\widehat{L}}_{k}H\end{array}\end{array}\right],g_a\left(x_k,{\widehat{x}}_k\right)=\left[\begin{array}{l}g\left(x_k\right)\\ g\left(x_k\right)-g\left({\widehat{x}}_k\right)\\ h\left(x_k\right)-h\left({\widehat{x}}_k\right)\end{array}\right]\\ C_a=\left[\begin{array}{cc}0& C_1\end{array}\right],E=\left[\begin{array}{cc}E& 0\end{array}\right].\end{array}$

Then, by Assumption 8.6.2

$\Vert g_a\left(x_k,{\widehat{x}}_k\right)\le \Vert \widehat{\Gamma }{\eta }_k\Vert ,\text{with}\widehat{\Gamma }=Blockdiag\left\{{\Gamma }_1,\Gamma \right\}.$

Further, define

$\mathrm{\Pi }=\left[\begin{array}{cc}{\mathrm{\Pi }}_{11}& {\mathrm{\Pi }}_{12}\\ {\mathrm{\Pi }}^T{}_{21}& {\mathrm{\Pi }}_{22}\end{array}\right]=A_a{x}_k{A}_a^T-X_{k+1}+A_a{x}_k{\widehat{C}}_k^T{\left(I-{\widehat{C}}_a{X}_k{\widehat{C}}_a^T\right)}^{-1}{\widehat{C}}_a{X}_k{A}_a^T+{\widehat{B}}_a{\widehat{B}}_a^T$

(8.123)

where

${\widehat{B}}_a=\left[{\text{γ}}^{-1}{B}_a{ϵ}^{-1}{H}_a{G}_a\right],{\widehat{C}}_a=\left[\begin{array}{cc}{\widehat{E}}_1& 0\\ 0& \widehat{M}\end{array}\right]$

and Ê₁ is such that

${\widehat{E}}_1^T{\widehat{E}}_1={ϵ}^2{E}^{T}E+V_1^T{V}_1+vI$

Also, let Q_k = γ⁻²S_k and

$X_k=\left[\begin{array}{cc}{P}^{-1}& 0\\ 0& Q_k\end{array}\right].$

Then by standard matrix manipulations, it can be shown that

${\mathrm{\Pi }}_{11}=A{P}^{-1}{A}^T-P^{-1}+A{P}^{-1}{\widehat{E}}_1{\left(I-{\widehat{E}}_1{P}^{-1}{\widehat{E}}_1^T\right)}^{-1}{\widehat{E}}_1{P}^{-1}{A}^T+{\text{γ}}^{-2}\tilde{B}{\tilde{B}}^T$

and

$\begin{array}{l}{\mathrm{\Pi }}_{12}=-A{\left(P-{\widehat{E}}_1^T{\widehat{E}}_1\right)}^{-1}{\left(\delta A_e-{\widehat{L}}_k\delta C_{2e}\right)}^T+{\text{γ}}^{-2}B{B}^T+\\ {ϵ}^{-2}{H}_1{H}_1^T-\left({\text{γ}}^{-2}B{D}^T+{ϵ}^{-2}{H}_1{H}_1^T\right){\widehat{L}}_k^T.\end{array}$

Moreover, since A is nonsingular, in view of (8.117) and the definition of δA_e, δC_2e, it implies that

${\mathrm{\Pi }}_{11}=0,{\mathrm{\Pi }}_{12}=0.$

It remains to show that Π₂₂ = 0. Using similar arguments as in Reference [89] (Theorem 3.1), it follows from (8.123) that Q_k satisfies the DRE:

$Q_{k+1}=\widehat{A}{\widehat{Q}}_k{\widehat{A}}^T-\left(\widehat{A}{\widehat{Q}}_k{\widehat{C}}^T+{\text{γ}}^{-2}\widehat{B}{\widehat{D}}^T\right){\left(\widehat{C}{\widehat{Q}}_k{\widehat{C}}^T+{\text{γ}}^{-2}\widehat{D}{\widehat{D}}^T\right)}^{-1}\left(\widehat{C}{\widehat{Q}}_k{\widehat{A}}^T+{\text{γ}}^{-2}\widehat{D}{\widehat{B}}^T\right)+{\text{γ}}^{-2}\widehat{B}{\widehat{B}}^T;Q_{{}_{k_0}}={\text{γ}}^{2}R-P$

(8.124)

where

${\widehat{Q}}_k=Q_k+Q_k{\widehat{M}}^T{\left(I-\widehat{M}{Q}_k{\widehat{M}}^T\right)}^{-1}\widehat{M}{Q}_k.$

Now, from (8.123) using some matrix manipulations we get

$\begin{array}{l}{\mathrm{\Pi }}_{22}=\widehat{A}{\widehat{Q}}_k{\widehat{A}}^T-\left(\widehat{A}{\widehat{Q}}_k{\widehat{C}}^T+{\text{γ}}^{-2}\widehat{B}{\widehat{D}}^T\right){\widehat{L}}_k-{\widehat{L}}_k\left(\widehat{C}{\widehat{Q}}_k{\widehat{A}}^T+{\text{γ}}^{-2}\widehat{D}{\widehat{B}}^T\right)+\\ {\widehat{L}}_k\left(\widehat{C}{\widehat{Q}}_k{\widehat{C}}^T+{\text{γ}}^{-2}\widehat{D}{\widehat{D}}^T\right){\widehat{L}}_k^T+{\text{γ}}^{-2}\widehat{B}{\widehat{B}}^T\end{array}$

and the gain matrix ${\widehat{L}}_k$ from (8.121) can be rewritten as

${\widehat{L}}_k=\left(\widehat{A}{\widehat{Q}}_k{\widehat{C}}^T+{\text{γ}}^{-2}\widehat{B}{\widehat{D}}^T\right){\left(\widehat{C}{\widehat{Q}}_k{\widehat{C}}^T+{\text{γ}}^{-2}\widehat{D}{\widehat{D}}^T\right)}^{-1}.$

(8.125)

Thus, from (8.124) and (8.125), it follows that T₂₂ = 0, and hence we conclude from (8.123) that

$A_a{x}_k{A}_a^T-X_{k+1}+A_a{x}_k{\widehat{C}}_a^T{\left(I-{\widehat{C}}_a{X}_k{\widehat{C}}_a^T\right)}^{-1}{\widehat{C}}_a{X}_k{A}_a^T+{\widehat{B}}_a{\widehat{B}}_a^T=0;x_0={\widehat{R}}^{-1}$

(8.126)

where

$\widehat{R}=\left[\begin{array}{cc}P& 0\\ 0& {\text{γ}}^2\tilde{R}-P\end{array}\right]>0.$

Next, we show that, $\mathcal{X}$_k is such that the time-varying system

${\rho }_{k+1}={\widehat{A}}_a{\rho }_k=\left[{\widehat{A}}_a+({\widehat{A}}_a{X}_k{\widehat{C}}_a^T{\left(I-{\widehat{C}}_a{X}_k{\widehat{C}}_a\right)}^{-1}{\widehat{C}}_a\right]{\rho }_k$

(8.127)

is exponentially-stable. Let

${\widehat{A}}_a:=\left[\begin{array}{cc}\overline{A}& 0\\ \star & A_{2k}\end{array}\right]$

where A_2k is as defined in (8.119), ‘⋆’ denotes a bounded but otherwise unimportant term, and

$\overline{A}=A+{\text{γ}}^{-2}\tilde{B}{\left(I-{\text{γ}}^{-2}\tilde{B}P\tilde{B}\right)}^{-1}{\tilde{B}}^{T}PA.$

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, $\mathcal{X}$_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 Y_k = Y ^T_k > 0 ∀k ≥ k₀ such that

$A_a^T{Y}_k{A}_a-Y_k+A_a^T{Y}_{k+1}{\widehat{B}}_a{\left(I-{\widehat{B}}_a^T{Y}_{k+1}{\widehat{B}}_a\right)}^{-1}{\widehat{B}}_a^T{Y}_{k+1}{A}_a+{\widehat{C}}_a^T{\widehat{C}}_a+{\delta }_{1}I<0;Y_{k_0}<\widehat{R}.$

Noting that

${\widehat{C}}_a^T{\widehat{C}}_a={\widehat{C}}_a^T{\widehat{C}}_a+{ϵ}^2{E}_a^T{E}_a+{\widehat{\Gamma }}^T\widehat{\Gamma }+\left[\begin{array}{cc}vI& 0\\ 0& 0\end{array}\right],$

we see that Y_k satisfies the following inequality:

$\begin{array}{l}{A}_a^T{Y}_k{A}_a-Y_k+A_a^T{Y}_{k+1}{\widehat{B}}_a{\left(I-{\widehat{B}}_a^T{Y}_{k+1}{\widehat{B}}_a\right)}^{-1}{\widehat{B}}_a^T{Y}_{k+1}{A}_a+{\widehat{C}}_a^T{\widehat{C}}_a+{ϵ}^2{E}^{T}E+\\ {\widehat{\Gamma }}^T\widehat{\Gamma }+{\delta }_{1}I<0,Y_{k_0}<\widehat{R}.\end{array}$

In addition,

${\eta }_0^T\widehat{R}{\eta }_0={\text{γ}}^2{x}_0^T\tilde{R}{x}_0.$

Finally, application of Lemma 8.6.2 and using the definition of ${\widehat{B}}_a$ imply that the error-dynamics (8.122) are exponentially-stable and

${\Vert \tilde{z}\Vert }_{l_2}^2\le \text{γ}\left\{{\Vert w\Vert }_{l_2}^2+x_0^T\tilde{R}{x}_0\right\}$

for all $\left(0,0\right)\ne \left(x^0,w\right)\in \mathcal{X}\otimes l_2$ and all $\mathrm{\Delta }{\text{A}}_k,\mathrm{\Delta }{C}_{2,k}\in {\Xi }_{d,\mathrm{\Delta }}.\square$

8.7    Notes and Bibliography

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 ${\mathcal{H}}_{\infty }$ 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.