8.3    Certainty-Equivalent Filters (CEFs)

We discuss in this section a class of estimators for the nonlinear system (8.1) which we refer to as “certainty-equivalent” worst-case estimators (see also [139]). The estimator is constructed on the assumption that the asymptotic value of $\widehat{x}$ equals x, and the gain matrices are designed so that they are not functions of the state vector x, but of $\widehat{x}$ and y only. We begin with the one degree-of-freedom (1-DOF) case, then we discuss the 2-DOF case. Accordingly, we propose the following class of estimators:

${\sum }_1^{acef}:\{\begin{array}{l}\dot{\widehat{x}}=f\left(\widehat{x}\right)+g_1\left(\widehat{x}\right){\widehat{w}}^{\star }+\widehat{L}\left(\widehat{x},y\right)\left(y-h_2\left(\widehat{x}\right)-k_{21}\left(\widehat{x}\right)w^{\star }\right)\\ \widehat{z}=h_2\left(\widehat{x}\right)\\ \tilde{z}=y-h_2\left(\widehat{x}\right)\end{array}$

(8.32)

where $\widehat{z}=\widehat{y}\in {\Re }^m$ is the estimated output, $\tilde{z}\in {\Re }^m$ is the new 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 for the filter.

Again, the problem is to find a suitable gain matrix $\widehat{L}$ (.,.) for the above filter, for estimating the state x(t) of the system (8.1) from available observations $Y_t\triangleq \left\{y(\tau ),\tau \le t\right\}$ t ∈ [t₀, ∞) such that the L₂-gain from the disturbance/noise signal w to the error output $\tilde{z}$ (to be defined later) is less than or equal to a desired number γ > 0, i.e.,

${\displaystyle {\int }_{t_0}^T{\Vert \tilde{z}\left(\tau \right)\Vert }^{2}d\tau \le {\text{γ}}^2}{\displaystyle {\int }_{t_0}^T{\Vert \stackrel{}{\omega }\left(\tau \right)\Vert }^{2}d\tau ,T>t_0},$

(8.33)

for all w ∈ W, for all x₀ ∈ O ⊂ $\mathcal{X}$. Moreover, with w = 0, we also have $\underset{t\to \infty }{\lim }\tilde{z}(t)=0$

The above problem can similarly be formulated by considering the cost functional

$J\left(\widehat{L},w\right)=\frac{1}{2}{\displaystyle {\int }_{t_0}^{\infty }\left({\Vert \tilde{z}\left(t\right)\Vert }^2-{\text{γ}}^2{\Vert w\left(\tau \right)\Vert }^2\right)d\tau .}$

(8.34)

as a differential game with the two contending players controlling L and w respectively. Again by making J ≤ 0, we see that the ${\mathcal{L}}_2$-gain constraint (8.33) is satisfied, and moreover, we seek for a saddle-point equilibrium solution, $({\widehat{L}}^{\star },w^{\star })$, to the above game such that

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

(8.35)

Before we proceed with the solution to the problem, we introduce the following notion of local zero-input observability.

Definition 8.3.1 For the nonlinear system Σ^a, we say that it is locally zero-input observable if for all states x₁, x₂ ∈ U ⊂ X and input w(.) ≡ 0

$y\left(.,x_1,w\right)=y\left(.,x_2,w\right)\Rightarrow x_1=x_2,$

where y(., x_i, w), i = 1, 2 is the output of the system with the initial condition x(t₀) = x_i. Moreover, the system is said to be zero-input observable, if it is locally zero-input observable at each x₀ ∈ $\mathcal{X}$ or U = $\mathcal{X}$.

Next, we first estimate ŵ^⋆, and for this purpose, define the Hamiltonian function H : $T^{\ast }\mathcal{X}\times \mathcal{W}\times {\mathcal{M}}^{n\times m}\to \Re$ for the estimator:

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

(8.36)

for some smooth function $\widehat{V}:\mathcal{X}\times \mathcal{Y}\to \Re$ and with the adjoint vector $p={\widehat{V}}_{\widehat{x}}^T\left(\widehat{x},y\right)$.

Applying now the necessary condition for the worst-case noise

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

Then substituting ŵ^⋆ into (8.36), we get

$\begin{array}{l}H\left(\widehat{x},{\widehat{w}}^{\star },\widehat{L},{\widehat{V}}_{\widehat{x}}^T\right)=V_{\widehat{x}}f\left(\widehat{x}\right)+{\widehat{V}}_y\dot{y}+\frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}{g}_1\left(\widehat{x}\right)g_1^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T\left(\widehat{x}\right)+{\widehat{V}}_{\widehat{x}}\widehat{L}\left(\widehat{x},y\right)\left(y-h_2\left(\widehat{x}\right)\right)+\\ \frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}\widehat{L}\left(\widehat{x},y\right){\widehat{L}}^T\left(\widehat{x},y\right){\widehat{V}}_{\widehat{x}}+\frac{1}{2}{\left(y-h_2\left(\widehat{x}\right)\right)}^T\left(y-h_2\left(\widehat{x}\right)\right).\end{array}$

(8.37)

Completing the squares now for $\widehat{L}$ in the above expression for H(., ŵ^⋆ , L, .), we get

$\begin{array}{l}H\left(\widehat{x},{\widehat{w}}^{\star },\widehat{L},{\widehat{V}}_{\widehat{x}}^T\right)=V_{\widehat{x}}f\left(\widehat{x}\right)+{\widehat{V}}_y\dot{y}+\frac{1}{2{\text{γ}}^2}{\widehat{V}}_{\widehat{x}}{g}_1\left(\widehat{x}\right)g_1^T\left(\widehat{x}\right){\widehat{V}}_{\widehat{x}}^T-\frac{{\text{γ}}^2}{2}{\left(y-h_2\left(\widehat{x}\right)\right)}^T\left(y-h_2\left(\widehat{x}\right)\right)+\\ \frac{1}{2{\text{γ}}^2}{\Vert {\widehat{L}}^T\left(\widehat{x},y\right){\widehat{V}}_{\widehat{x}}+{\text{γ}}^2\left(y-h_2\left(\widehat{x}\right)\right)\Vert }^2+\frac{1}{2}{\left(y-h_2\left(\widehat{x}\right)\right)}^T\left(y-h_2\left(\widehat{x}\right)\right).\end{array}$

Therefore, choosing ${\widehat{L}}^{\star }$ as

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

(8.38)

minimizes H(., ., ., ., .) and gurantees that the saddle-point conditions (8.35) are satisfied by (ŵ^⋆, ${\widehat{L}}^{\star }$). Finally, substituting this value in (8.37) and setting

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

yields the HJIE

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

(8.39)

or equivalently the HJIE

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

(8.40)

This is summarized in the following result.

Proposition 8.3.1 Consider the nonlinear system (8.1) and the N L H I F P for it. Suppose Assumption 8.1.1 holds, the plant Σ^a is locally asymptotically-stable about the equilibrium-point x = 0 and zero-input observable. Suppose further, there exists a C¹ positive semidefinite function $\widehat{V}:\widehat{N}\times \widehat{Y}\to {\Re }_{+}$ locally defined in a neighborhood $\widehat{N}\times \widehat{Y}\subset \mathcal{X}\times \mathcal{Y}$ 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 HJIE (8.39) or (8.40) together with the coupling condition (8.38). Then the filter ${\sum }_2^{acef}$ solves the N L H I F P for the system.

Proof: Let $\widehat{V}$ ≥ 0 be a C¹ solution of the H J I E (8.39) or (8.40). Differentiating $\widehat{V}$ along a trajectory of (8.32) with w in place of ŵ^⋆, it can be shown that

$\dot{\widehat{V}}\le \frac{1}{2}\left({\text{γ}}^2{\Vert \widehat{w}\Vert }^2-{\Vert \tilde{z}\Vert }^2\right).$

Integrating this inequality from t = t₀ to t = ∞, and since V ≥ 0 implies that the L₂-gain condition ${\displaystyle \underset{t_0}{\overset{\infty }{\int }}\Vert \widehat{z}}{(t)\Vert }^{2}dt\le {\gamma }^2{\displaystyle \underset{t_0}{\overset{\infty }{\int }}\Vert w}{(t)\Vert }^{2}dt$ is satisfied. Moreover, setting w = 0 in the above inequality implies that $\dot{\widehat{V}}\le -\frac{1}{2}{\Vert \tilde{z}\Vert }^2$ and hence the estimator dynamics is stable. In addition,

$\dot{\widehat{V}}=0\forall t\ge t_s\Rightarrow \tilde{z}\equiv 0\Rightarrow y=h_2\left(\widehat{x}\right)=\widehat{y}.$

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

Remark 8.3.1 Notice also that for γ = 1, a control Lyapunov-function $\widehat{V}$ [102, 170] for the system (8.1) solves the above HJIE (8.39).

8.3.1    2-DOF Certainty-Equivalent Filters

Next, we extend the design procedure in the previous section to the 2-DOF case. In this regard, we assume that the time derivative $\dot{y}$ is available as an additional measurement information. This can be obtained or estimated using a differentiator or numerically. Notice also that, with y = h₂(x) + k₂₁(x)w and with w ∈ ${\mathcal{L}}_2$(0, T ) for some T sufficiently large, then since the space of continuous functions with compact support C_c is dense in ${\mathcal{L}}_2$, we may assume that the time derivatives $\dot{y}$, ÿ exist a.e. That is, we may approximate w by piece-wise C¹ functions. However, this assumption would have been difficult to justify if we had assumed w to be some random process, e.g., a white noise process.

Accordingly, consider the following class of filters:

${\Sigma }_2^{acef}:\{\begin{array}{l}\stackrel{‵}{\dot{x}}=f\left(\stackrel{‵}{x}\right)+g_1\left(\stackrel{‵}{x}\right){\stackrel{‵}{w}}^{\star }+{\stackrel{‵}{L}}_1\left(\stackrel{‵}{x},y,\dot{y}\right)(y-h_2\left(\stackrel{‵}{x}\right)-k_{21}\left(\stackrel{‵}{x}\right){\stackrel{‵}{w}}^{\star })+\\ {\stackrel{‵}{L}}_2\left(\stackrel{‵}{x},y,\dot{y}\right)\left(\dot{y}-{\mathcal{L}}_{f\left(\stackrel{‵}{x}\right)}{h}_2\left(\stackrel{‵}{x}\right)\right)\\ \stackrel{‵}{z}=\left[\begin{array}{l}{h}_2\left(\stackrel{‵}{x}\right)\\ {\mathcal{L}}_{f\left(\stackrel{‵}{x}\right)}{h}_2\left(\stackrel{‵}{x}\right)\end{array}\right]\\ \tilde{z}=\left[\begin{array}{l}y-h_2\left(\stackrel{‵}{x}\right)\\ \dot{y}-{\mathcal{L}}_{f\left(\stackrel{‵}{x}\right)}{h}_2\left(\stackrel{‵}{x}\right)\end{array}\right]\end{array}$

where ‵z ∈ℜ^m is the estimated output of the filter, $\tilde{z}\in {\Re }^s$ is the error or penalty variable, L_f h₂ denotes the Lie-derivative of h₂ along f [268], while $\stackrel{‵}{L_1}:X\times TY\to M^{n\times m}$, $\stackrel{‵}{L_2}:X\times TY\to M^{n\times m}$ are the filter gains, and all the other variables and functions have their corresponding previous meanings and dimensions. As in the previous subsection, we can define the Hamiltonian $\stackrel{‵}{H}:T^{\star }\mathcal{X}\times \mathcal{W}\times {\mathcal{M}}^{n\times m}\times {\mathcal{M}}^{n\times m}\to \Re$: of the system as

$\begin{array}{l}\stackrel{‵}{H}\left(\stackrel{‵}{x},\stackrel{\text{'}}{w},\stackrel{\text{'}}{L_1},\stackrel{‵}{L_2},{\stackrel{‵}{V_{\stackrel{‵}{x}}}}^T\right)=\stackrel{‵}{V_{\stackrel{‵}{x}}}\left(\stackrel{‵}{x},y,\dot{y}\right)\{f\left(\stackrel{‵}{x}\right)+g_1\left(\stackrel{‵}{x}\right){\stackrel{‵}{w}}^{}+{\stackrel{‵}{L}}_1\left(\stackrel{‵}{x},y,\dot{y}\right)(y-h_2\left(\stackrel{‵}{x}\right)-k_{21}\left(\stackrel{‵}{x}\right){\stackrel{‵}{w}}^{})+\\ {\stackrel{‵}{L}}_2\left(\stackrel{‵}{x},y,\dot{y}\right)\left(\dot{y}-{ℒ}_{f\left(\stackrel{‵}{x}\right)}{h}_2\left(\stackrel{‵}{x}\right)\right)\}+\stackrel{‵}{V_y\dot{y}}+\stackrel{‵}{V_{\dot{y}}}\ddot{y}+\frac{1}{2}\left({\Vert \tilde{z}\Vert }^2+{\text{γ}}^2{\Vert \stackrel{‵}{w}\Vert }^2\right)\end{array}$

(8.41)

for some smooth function $\stackrel{‵}{V}$ : $\mathcal{X}$ × T $\mathcal{Y}$ → ℜ and by setting the adjoint vector $\stackrel{‵}{p=}\stackrel{‵}{V_{\stackrel{‵}{x^{}}}{}^T}$. Then proceeding as before, we clearly have

${\stackrel{‵}{w}}^{\star }=\frac{1}{{\text{γ}}^2}\left[g_1^T\left(\stackrel{‵}{x}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}\left(\stackrel{‵}{x},y,\dot{y}\right)-k_{21}^T{\stackrel{‵}{L}}_1\left(\stackrel{‵}{x},y,\dot{y}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}\left(\stackrel{‵}{x},y,\dot{y}\right)\right].$

Substitute now $\stackrel{‵}{w^{\star }}$ into (8.41) to get

$\begin{array}{l}\stackrel{‵}{H}\left(\stackrel{‵}{x},\stackrel{‵}{w},\stackrel{‵}{L_1},\stackrel{‵}{L_2},{\stackrel{‵}{V_{\stackrel{‵}{x}}}}^T\right)=\stackrel{‵}{V_{\stackrel{‵}{x}}}f\left(\stackrel{‵}{x}\right)+\stackrel{‵}{V_y\dot{y}}+\stackrel{‵}{V_{\dot{y}}}\ddot{y}+\frac{1}{2{\text{γ}}^2}\stackrel{‵}{V_{\stackrel{‵}{x}}}{g}_1\left(\stackrel{‵}{x}\right)g_1^T\left(\stackrel{‵}{x}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}+\\ \stackrel{\text{'}}{V_{\stackrel{\text{'}}{x}}}{\stackrel{‵}{L}}_1\left(\stackrel{‵}{x},y,\dot{y}\right)\left(y-h_2\left(\stackrel{‵}{x}\right)\right)+\frac{1}{2{\text{γ}}^2}\stackrel{‵}{V_{\stackrel{‵}{x}}}{\stackrel{‵}{L}}_1\left(\stackrel{‵}{x},y,\dot{y}\right){\stackrel{‵}{L}}_1^T\left(\stackrel{‵}{x},y,\dot{y}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}+\\ \stackrel{‵}{V_{\stackrel{‵}{x}}}{\stackrel{‵}{L}}_2\left(\stackrel{‵}{x},y,\dot{y}\right)\left(\dot{y}-{ℒ}_f{h}_2\left(\stackrel{‵}{x}\right)\right)+\frac{1}{2}{\left(\dot{y}-{ℒ}_f{h}_2\left(\stackrel{‵}{x}\right)\right)}^T\left(\dot{y}-{ℒ}_f{h}_2\left(\stackrel{‵}{x}\right)\right)+\\ \frac{1}{2}{\left(y-h_2\left(\stackrel{‵}{x}\right)\right)}^T\left(y-h_2\left(\stackrel{‵}{x}\right)\right)\end{array}$

and completing the squares for $\stackrel{‵}{L_1}$ and $\stackrel{‵}{L_2}$, we get

$\begin{array}{l}\stackrel{‵}{H}\left(\stackrel{‵}{x},{\stackrel{‵}{w}}^{\star },\stackrel{‵}{L_1},\stackrel{‵}{L_2},{\stackrel{‵}{V_{\stackrel{‵}{x}}}}^T\right)=\stackrel{‵}{V_{\stackrel{‵}{x}}}f\left(\stackrel{‵}{x}\right)+\stackrel{‵}{V_y\dot{y}}+\stackrel{‵}{V_{\dot{y}}}\ddot{y}+\frac{1}{2{\text{γ}}^2}\stackrel{‵}{V_{\stackrel{‵}{x}}}{g}_1\left(\stackrel{‵}{x}\right)g_1^T\left(\stackrel{‵}{x}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}-\\ \frac{{\text{γ}}^2}{2}{\left(y-h_2\left(\stackrel{‵}{x}\right)\right)}^T\left(y-h_2\left(\stackrel{‵}{x}\right)\right)+\frac{1}{2{\text{γ}}^2}{\Vert {\stackrel{‵}{L}}_1^T\left(\stackrel{‵}{x},y,\dot{y}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}+{\text{γ}}^2\left(y-h_2\left(\stackrel{‵}{x}\right)\right)\Vert }^2+\\ \frac{1}{2}{\Vert L_2^T\left(\stackrel{‵}{x},y,\dot{y}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}+\left(\dot{y}-{ℒ}_f{h}_2\left(\stackrel{‵}{x}\right)\right)\Vert }^2-\frac{1}{2}\stackrel{‵}{V_{\stackrel{‵}{x}}}{L}_2\left(\stackrel{‵}{x},y,\dot{y}\right)L_2^T\left(\stackrel{‵}{x},y,\dot{y}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}+\\ \frac{1}{2}{\left(y-h_2\left(\stackrel{‵}{x}\right)\right)}^T\left(y-h_2\left(\stackrel{‵}{x}\right)\right)\end{array}$

Hence, setting

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

(8.42)

$\stackrel{‵}{V_{\stackrel{‵}{x}}}\left(\stackrel{‵}{x},y,\dot{y}\right){\stackrel{‵}{L}}_1^{\star }\left(\stackrel{‵}{x},y,\dot{y}\right)=-{\left(\dot{y}-{ℒ}_f{h}_2\left(\stackrel{‵}{x}\right)\right)}^T$

(8.43)

minimizes $\stackrel{‵}{H}$ and gurantees that the saddle-point conditions (8.35) are satisfied. Finally, setting

$\stackrel{‵}{H}\left(\stackrel{‵}{x},\stackrel{‵}{w^{\star }},\stackrel{‵}{L_1^{\star }},\stackrel{‵}{L_2^T},\stackrel{‵}{V_{\stackrel{‵}{x}}^T}\right)=0$

yields the HJIE

$\begin{array}{l}\stackrel{‵}{V_{\stackrel{‵}{x}}}\left(\stackrel{‵}{x},y,\dot{y}\right)f\left(\stackrel{‵}{x}\right)+\stackrel{‵}{V_y}\left(\stackrel{‵}{x},y,\dot{y}\right)\dot{y}+\stackrel{‵}{V_{\dot{y}}}\left(\stackrel{‵}{x},y,\dot{y}\right)\ddot{y}+\\ \frac{1}{2{\text{γ}}^2}\stackrel{‵}{V_{\stackrel{‵}{x}}}\left(\stackrel{‵}{x},y,\dot{y}\right)g_1\left(\stackrel{‵}{x}\right)g_1^T\left(\stackrel{‵}{x}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}\left(\stackrel{‵}{x},y,\dot{y}\right)-\\ \frac{1}{2}{\left(\dot{y}-{ℒ}_f{h}_2\left(\stackrel{‵}{x}\right)\right)}^T\left(\dot{y}-{ℒ}_f{h}_2\left(\stackrel{‵}{x}\right)\right)+\\ \frac{\left(1-{\text{γ}}^2\right)}{2}{\left(y-h_2\left(\stackrel{‵}{x}\right)\right)}^T\left(y-h_2\left(\stackrel{‵}{x}\right)\right)=0,\stackrel{‵}{V}\left(0,0\right)=0.\end{array}$

(8.44)

Consequently, we have the following result.

Theorem 8.3.1 Consider the nonlinear system (8.1) and the N L H I F P for it. Suppose Assumption 8.1.1 holds, the plant Σ^a is locally asymptotically-stable about the equilibrium-point x = 0 and zero-input observable. Suppose further, there exists a C¹ positive-semidefinite function $\stackrel{‵}{V_{}}:\stackrel{‵}{N\times }\stackrel{‵}{Y}\to {\Re }_{+}$ locally defined in a neighborhood $\stackrel{‵}{N\times }\stackrel{‵}{Y}\subset X\times TY$ of the origin $\left(\stackrel{‵}{x},y,\dot{y}\right)$ = (0, 0, 0), and matrix functions $\stackrel{\text{'}}{L_1:N\times }\stackrel{‵}{Y}\to M^{n\times m},\stackrel{‵}{L_2:N\times }\stackrel{⌣}{Y}\to M^{n\times m}$, satisfying the HJIE (8.44) or equivalently the HJIE:

$\begin{array}{l}\stackrel{‵}{V_{\stackrel{‵}{x}}}\left(\stackrel{‵}{x},y,\dot{y}\right)f\left(\stackrel{‵}{x}\right)+\stackrel{‵}{V_y}\left(\stackrel{‵}{x},y,\dot{y}\right)\dot{y}+\stackrel{‵}{V_{\dot{y}}}\left(\stackrel{‵}{x},y,\dot{y}\right)\ddot{y}+\frac{1}{2{\text{γ}}^2}\stackrel{‵}{V_{\stackrel{‵}{x}}}\left(\stackrel{‵}{x},y,\dot{y}\right)g_1\left(\stackrel{‵}{x}\right)g_1^T\left(\stackrel{‵}{x}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}\left(\stackrel{‵}{x},y,\dot{y}\right)-\\ \frac{1}{2{\text{γ}}^2}\stackrel{‵}{V_{\stackrel{‵}{x}}}\left(\stackrel{‵}{x},y,\dot{y}\right){\stackrel{‵}{L}}_1\left(\stackrel{‵}{x},y,\dot{y}\right){\stackrel{‵}{L}}_1^T\left(\stackrel{‵}{x},y,\dot{y}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}\left(\stackrel{‵}{x},y,\dot{y}\right)-\\ \frac{1}{2}\stackrel{‵}{V_{\stackrel{‵}{x}}}\left(\stackrel{‵}{x},y,\dot{y}\right){\stackrel{‵}{L}}_2\left(\stackrel{‵}{x},y,\dot{y}\right){\stackrel{‵}{L}}_2^T\left(\stackrel{‵}{x},y,\dot{y}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}\left(\stackrel{‵}{x},y,\dot{y}\right)+\frac{1}{2}{\left(y-h_2\left(\stackrel{‵}{x}\right)\right)}^T\left(y-h_2\left(\stackrel{‵}{x}\right)\right)=0,\\ \stackrel{‵}{V}\left(0,0,0\right)=0\end{array}$

(8.45)

together with the coupling condition (8.42), (8.43). Then the filter ${\sum }_1^{acef}$ solves the H I N L F P for the system.

Proof: The first part of the proof has already been established above, that $\stackrel{‵}{(w^{\star }},\stackrel{‵}{[L_1^{\star }},\stackrel{‵}{L_2^{\star }}])$ constitute a saddle-point solution for the game (8.34), (8.41). Therefore, we only need to show that the ${\mathcal{L}}_2$-gain requirement (8.33) is satisfied, and the filter provides asymptotic estimates when w = 0.

Let $\stackrel{‵}{V_{}}\ge 0$ be a C¹ solution of the HJIE (8.44), and differentiating it along a trajectory of the filter ${\sum }_1^{acef}$ with $\stackrel{‵}{w^{}}$ in place of $\stackrel{‵}{w^{\star }}$, and $\stackrel{‵}{L_1^{\star }},\stackrel{‵}{L_2^{\star }}$, we have

$\begin{array}{l}\stackrel{‵}{V=\stackrel{‵}{V_{\stackrel{‵}{x}}}\left(\stackrel{‵}{x},y,\dot{y}\right)[f\left(\stackrel{‵}{x}\right)+g_1\left(\stackrel{‵}{x}\right)\stackrel{‵}{w}+{\stackrel{‵}{L}}_1\left(\stackrel{‵}{x},y\right)(y-h_2\left(\stackrel{‵}{x}\right)-k_{21}\left(\stackrel{‵}{x}\right)\stackrel{‵}{w})+}\\ L_2\left(\stackrel{‵}{x},y,\dot{y}\right)\left(\dot{y}-{ℒ}_f{h}_2\left(\stackrel{‵}{x}\right)\right)]+\stackrel{‵}{V_y\dot{y}}+\stackrel{‵}{V_{\dot{y}}}\ddot{y}\\ =\{\stackrel{‵}{V_{\stackrel{‵}{x}}}f\left(\stackrel{‵}{x}\right)+\stackrel{‵}{V_y\dot{y}}+\stackrel{‵}{V_{\dot{y}}}\ddot{y}+\frac{1}{2{\text{γ}}^2}\stackrel{‵}{V_{\stackrel{‵}{x}}}{g}_1\left(\stackrel{‵}{x}\right)g_1^T\left(\stackrel{‵}{x}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}+\\ \frac{\left(1-{\text{γ}}^2\right)}{2}{(y-h_2\left(\stackrel{‵}{x}\right)}^T(y-h_2\left(\stackrel{‵}{x}\right)-\frac{1}{2}{\left(\dot{y}-{ℒ}_f{h}_2\left(\stackrel{‵}{x}\right)\right)}^T\left(\dot{y}-{ℒ}_f{h}_2\left(\stackrel{‵}{x}\right)\right)\}+\\ \stackrel{‵}{V_{\stackrel{‵}{x}}}{g}_1\left(\stackrel{‵}{x}\right)\stackrel{‵}{w}-\stackrel{‵}{V_{\stackrel{‵}{x}}}\stackrel{‵}{L_1^{\star }}\left(\stackrel{‵}{x},y\right)k_{21}\left(\stackrel{‵}{x}\right)\stackrel{‵}{w}-\frac{1}{2{\text{γ}}^2}\stackrel{‵}{V_{\stackrel{‵}{x}}}{g}_1\left(\stackrel{‵}{x}\right)g_1^T\left(\stackrel{‵}{x}\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}-\\ \frac{{\text{γ}}^2}{2}{(y-h_2\left(\stackrel{‵}{x}\right)}^T(y-h_2\left(\stackrel{‵}{x}\right)+\stackrel{‵}{V_{\stackrel{‵}{x}}}{L}_2\left(\stackrel{‵}{x},y,\dot{y}\right)\left(\dot{y}-{ℒ}_f{h}_2\left(\stackrel{‵}{x}\right)\right)-\\ \frac{1}{2}{\left(y-h_2\left(\stackrel{‵}{x}\right)\right)}^T\left(y-h_2\left(\stackrel{‵}{x}\right)\right)+\frac{1}{2}{\left(\dot{y}-{ℒ}_f{h}_2\left(\stackrel{‵}{x}\right)\right)}^T\left(\dot{y}-{ℒ}_f{h}_2\left(\stackrel{‵}{x}\right)\right)\end{array}$

$\begin{array}{l}=-\frac{{\text{γ}}^2}{2}{\Vert w-\frac{1}{{\text{γ}}^2}{g}_1^T\left(\stackrel{‵}{x}\right){\stackrel{‵}{V_{\stackrel{‵}{x}}}}^T+\frac{1}{{\text{γ}}^2}{k}_{21}^T\left(\stackrel{‵}{x}\right)\stackrel{‵}{L_1^{{\star }^T}}\left(\stackrel{‵}{x},y\right)\stackrel{‵}{V_{\stackrel{‵}{x}}^T}\Vert }^2+\frac{{\text{γ}}^2}{2}{\Vert \stackrel{‵}{w}\Vert }^2-\frac{1}{2}{\Vert \tilde{z}\Vert }^2\\ \le \frac{1}{2}\left({\text{γ}}^2{\Vert \stackrel{‵}{w}\Vert }^2-\frac{1}{2}{\Vert \tilde{z}\Vert }^2\right).\end{array}$

Integrating the above inequality from t = t₀ to t = ∞, we get that the L₂-gain condition (8.33) is satisfied.

Similarly, setting $\stackrel{‵}{w^{\star }}$ = 0, we have $\stackrel{.}{\stackrel{`}{V}}=-\frac{1}{2}\Vert \tilde{z}{\Vert }^2$. Therefore, the filter dynamics is stable.

Moreover, the condition

$\dot{\stackrel{`}{V}}\equiv 0\forall t\ge t_s\Rightarrow \tilde{z}\equiv 0\Rightarrow y=h_2\left(\widehat{x}\right),\dot{y}\equiv {\mathcal{L}}_f{h}_2\left(\stackrel{\text{'}}{x}\right).$

By the zero-input observability of the system Σ^a, this implies that `x ≡ x ∀t ≥ t_s. □

We consider an example to illustrate and compare the performances of the 1-DOF and the 2-DOF.

Example 8.3.1 Consider the nonlinear system

$\begin{array}{l}{\dot{x}}_1=-x_1^3-x_2\\ {\dot{x}}_2=-x_1-x_2\\ y=x_1+w\end{array}$

where w = 5w₀ + 5 sin(t) and w₀ is a zero-mean Gaussian white-noise with unit variance. It can be checked that the system is locally zero-input observable and the functions $\widehat{V}(x)=\frac{1}{2}({\widehat{x}}_1^2+{\widehat{x}}_2^2),\stackrel{\text{'}}{V}(\stackrel{‵}{x})=\frac{1}{2}({\stackrel{‵}{x}}_1^2+{\stackrel{‵}{x}}_2^2)$ solve the HJI-inequalities corresponding to (8.39), (8.44) for the 1-DOF and 2-DOF certainty-equivalent filters respectively with γ = 1. Subsequently, we calculate the gains of the filters as

$\widehat{L}\left(\widehat{x},y\right)=\left[\begin{array}{l}1\\ -y/{\widehat{x}}_2\end{array}\right],{\stackrel{‵}{L}}_1\left(\stackrel{‵}{x},y,\dot{y}\right)=\left[\begin{array}{l}1\\ -y/{\widehat{x}}_2\end{array}\right],{\stackrel{‵}{L}}_2\left(\stackrel{‵}{x},y,\dot{y}\right)=\left[\begin{array}{l}-{\stackrel{‵}{x}}_2^2\\ -\left(\dot{y}-\stackrel{‵}{x_2}\right)/{\stackrel{‵}{x}}_2\end{array}\right]$

and construct the filters ${\sum }_1^{acef},{\sum }_2^{acef}$ respectively. The results of the individual filter performance with the same initial conditions but unknown system initial conditions are shown in Figure 8.5. It is seen from the simulation that the 2-DOF filter has faster convergence than the 1-DOF filter, though the latter may have better steady-state performance. The estimation errors are also insensitive or robust against a significantly high persistent measurement noise.

8.4    Discrete-Time Nonlinear ${\mathcal{H}}_{\infty }$-Filtering

In this section, we consider ${\mathcal{H}}_{\infty }$-filtering problem for discrete-time nonlinear systems. The configuration for this problem is similar to the continuous-time problem shown in Figure 8.1 but with discrete-time inputs and output measurements instead, and is shown in Figure 8.7. We consider an affine causal discrete-time state-space system with zero input defined on $\mathcal{X}\subseteq {\Re }^n$ in coordinates x = (x₁,&, x_n):

${\Sigma }^{da}:\{\begin{array}{l}{x}_{k+1}=f\left(x_k\right)+g_1\left(x_k\right)w_k;x\left(k_0\right)=x_0\\ z_k=h_1\left(x_k\right)\\ y_k=h_2\left(x_k\right)+k_{21}\left(x_k\right)w_k\end{array}$

(8.46)

FIGURE 8.5
1-DOF and 2-DOF Nonlinear ${\mathcal{H}}_{\infty }$-Filter Performance with Unknown Initial Condition

FIGURE 8.6
Configuration for Discrete-Time Nonlinear ${\mathcal{H}}_{\infty }$-Filtering

where x ∈ $\mathcal{X}$ is the state vector, w ∈ W ⊂ ℓ ₂[k₀, ∞) is the disturbance or noise signal, which belongs to the set W ⊂ ℜ^r of admissible disturbances or noise signals, the output y ∈ ℜ^m is the measured output of the system, while z ∈ ℜ^s is the output to be estimated. The functions f : χ → χ, g₁ : χ → M^n×r(χ ), h₁ : χ → ℜ ^s, h₂ : χ → ℜ ^m, and k₁₂ : χ → M^s×p(χ ), k₂₁ : χ → M^m×r(χ) are real C^∞ functions of x. Furthermore, we assume without any loss of generality that the system (8.46) has a unique equilibrium-point at x = 0 such that f(0) = 0, h₁(0) = h₂(0) = 0.

The objective is to synthesize a causal filter, F_k, for estimating the state x_k or a function of it, z_k = h₁(x_k), from observations of y_k up to time k ∈ Z₊, i.e., from

${\text{Y}}_k\triangleq \left\{y_i:i\le k\right\},$

such that the ℓ₂-gain from w to $\stackrel{⌣}{z}$, the error (or penalty variable), of the interconnected system defined as

${\Vert {\Sigma }^{daf}\circ {\Sigma }^{da}\Vert }_{l\infty }\triangleq \underset{0\ne w\in l_2}{\sup }\frac{{\Vert \stackrel{⌣}{z}\Vert }_2}{{\Vert w\Vert }_2},$

(8.47)

where w := {w_k}, $\stackrel{⌣}{z}:=\left\{{\stackrel{⌣}{z}}_k\right\}$, is rendered less or equal to some given positive number γ > 0. Moreover, with w_k ≡ 0, we have lim_k→∞ ${\stackrel{⌣}{z}}_k$ = 0.

For nonlinear systems, the above condition is interpreted as the ℓ₂-gain constraint and is represented as

${\displaystyle \sum _{k=k_0}^K{\Vert {\stackrel{⌣}{z}}_k\Vert }^2\le {\text{γ}}^2{\displaystyle \sum _{k=k_0}^k{\Vert w_k\Vert }^2,K>k_0\in z}}$

(8.48)

for all w_k ∈ W, and for all x⁰ ∈ O ⊂ χ .

The discrete-time nonlinear ${\mathcal{H}}_{\infty }$ suboptimal filtering or estimation problem can be defined formally as follows.

Definition 8.4.1 (Discrete-Time Nonlinear ℋ_∞ (Suboptimal) Filtering/Prediction Problem (DNLHIFP)). Given the plant Σ^da and a number γ^⋆ > 0. Find a filter F_k : Y → χ such that

${\widehat{x}}_{k+1}={\mathcal{F}}_k\left({\text{Y}}_k\right),k=k_0,\dots ,K$

and the constraint (8.48) is satisfied for all γ ≥ γ^⋆ , for all w ∈ W, and for all x⁰ ∈ O. Moreover, with w_k ≡ 0, we have lim_k→∞ ${\stackrel{⌣}{z}}_k$ = 0.

Remark 8.4.1 The problem defined above is the finite-horizon filtering problem. We have the infinite-horizon problem if we let K → ∞.

To solve the above problem, we consider the following class of estimators:

${\Sigma }^{daf}:\{\begin{array}{l}{\widehat{x}}_{k+1}=f\left({\widehat{x}}_k\right)+L\left({\widehat{x}}_k\text{,}k\right)\left[y-h_2\left({\widehat{x}}_k\right)\right],\widehat{x}\left(k_0\right)={\widehat{x}}_0\\ {\widehat{z}}_k=h_1\left({\widehat{x}}_k\right)\end{array}$

(8.49)

where ${\widehat{x}}_k\in X$ is the estimated state, L(., .) ∈ M^n×m($\mathcal{X}$ × Z) is the error-gain matrix which has to be determined, and $\stackrel{⌣}{z}\in {\Re }^s$ is the estimated output of the filter. We can now define the estimation error or penalty variable, $\stackrel{⌣}{z}$, which has to be controlled as:

${\stackrel{⌣}{z}}_k:z_k-{\widehat{z}}_k=h_1\left(x_k\right)-h_1\left({\widehat{x}}_k\right).$

Then we combine the plant (8.46) and estimator (8.49) dynamics to obtain the following augmented system:

$\begin{array}{l}{\stackrel{⌣}{x}}_{k+1}=\stackrel{⌣}{f}\left({\stackrel{⌣}{x}}_k\right)+\stackrel{⌣}{g}\left({\stackrel{⌣}{x}}_k\right)w_k,\stackrel{⌣}{x}\left(k_0\right)={\left(x^{0^T}{\widehat{x}}^{0^T}\right)}^T\\ {\stackrel{⌣}{z}}_k=\stackrel{⌣}{h}\left({\stackrel{⌣}{x}}_k\right)\end{array}\},$

(8.50)

where${\stackrel{⌣}{x}}_k={\left(x_k^T{\widehat{x}}_k^T\right)}^T,$

$\begin{array}{l}\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)=\left(\begin{array}{l}f\left(x_k\right)\\ f\left({\widehat{x}}_k\right)+L\left({\widehat{x}}_k,k\right)\left(h_2\left(x_k\right)-h_2\left({\widehat{x}}_k\right)\right)\end{array}\right),\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)=\left(\begin{array}{l}{g}_1\left(x_k\right)\\ L\left({\widehat{x}}_k,k\right)k_{21}\left(x_k\right)\end{array}\right),\\ \stackrel{⌣}{h}\left({\stackrel{⌣}{x}}_k\right)=h_1\left(x_k\right)-h_2\left({\widehat{x}}_k\right).\end{array}$

The problem is then formulated as a two-player zero-sum differential game with the cost functional:

$\underset{L}{\min }\underset{w}{\max }\tilde{J}\left(L,w\right)=\frac{1}{2}{\displaystyle \sum _{k=k_0}^K\left\{{\Vert {\stackrel{⌣}{z}}_k\Vert }^2-{\text{γ}}^2{\Vert w_k\Vert }^2\right\},}$

(8.51)

where w := {w_k}, L := L(x_k, k). By making J ≤ 0, we see that the ${\mathcal{H}}_{\infty }$ constraint ‖Σ^daf ◦ Σ^a‖_H∞ ≤ γ is satisfied. A saddle-point equilibrium solution to the above game is said to exist if we can find a pair (L^⋆ , w^⋆ ) such that

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

(8.52)

Sufficient conditions for the solvability of the above game are well known [59]. These are also given as Theorem 2.2.2 which we recall here.

Theorem 8.4.1 For the two-person discrete-time nonzero-sum game (8.51)-(8.50), under memoryless perfect information structure, a pair of strategies (L^⋆, w^⋆) provides a feedback saddle-point solution if, and ony if, there exist a set of K − k₀ functions V : $\mathcal{X}$ × Z → ℜ, such that the following recursive equations (or discrete-Hamilton-Jacobi-Isaac’s equations (DHJIE)) are satisfied for each k ∈ [k₀, K]:

$\begin{array}{l}V\left(\stackrel{⌣}{x},k\right)=\underset{L\in {ℳ}^{n\times m}}{\min }\underset{w\in \mathcal{W}}{\sup }\frac{1}{2}\left\{{\Vert {\stackrel{⌣}{z}}_k\Vert }^2-{\text{γ}}^2{\Vert w_k\Vert }^{2}V\left({\stackrel{⌣}{x}}_{k+1},k+1\right)\right\},\\ =\underset{w\in \mathcal{W}}{\sup }\underset{L\in {ℳ}^{n\times m}}{\min }\frac{1}{2}\left\{{\Vert {\stackrel{⌣}{z}}_k\Vert }^2-{\text{γ}}^2{\Vert w_k\Vert }^{2}V\left({\stackrel{⌣}{x}}_{k+1},k+1\right)\right\},\\ =\frac{1}{2}\left({\Vert {\stackrel{⌣}{z}}_k(\stackrel{⌣}{x})\Vert }^2-{\text{γ}}^2{\Vert w_k{}^{\star }(\stackrel{⌣}{x})\Vert }^2\right)+V\left({\stackrel{⌣}{x}}_{k+1}^{\star },k+1\right),V\left(\stackrel{⌣}{x},k+1\right)=0,\\ k=k_0,\dots ,K,where\stackrel{⌣}{x}={\stackrel{⌣}{x}}_k,and\\ {x_{k+1}^{\star }={\stackrel{⌣}{f}}^{\star }\left({\stackrel{⌣}{x}}_k\right)+{\stackrel{⌣}{g}}^{\star }\left({\stackrel{⌣}{x}}_k\right)w_k^{\star },{\stackrel{⌣}{f}}^{\star }{}_{}\left({\stackrel{⌣}{x}}_{{}_k}\right)=\stackrel{⌣}{f}\left({\stackrel{⌣}{x}}_k\right)|}_{L=L^{\star }},{\stackrel{⌣}{g}}^{\star }\left({\stackrel{⌣}{x}}_k\right)=\stackrel{⌣}{g}\left({\stackrel{⌣}{x}}_k\right)|L=L^{\star }.\end{array}$

(8.53)

Equation (8.53) is also known as Isaac’s equation.

Thus, we can apply the above theorem to derive sufficient conditions for the solvability of the D N L H I F P. To do that, we define the Hamiltonian function H : ($\mathcal{X}\times \mathcal{X}$ ) × W × ℳ^n×m × ℜ → ℜ associated with the cost functional (8.51) and the estimator dynamics as

$H\left(\stackrel{⌣}{x},w_k,L,V\right)=V\left(\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)+\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)w_k\left(\stackrel{⌣}{x}\right),k+1\right)-V\left(\stackrel{⌣}{x},k\right)+\frac{1}{2}{\Vert {\stackrel{⌣}{z}}_k\left(\stackrel{⌣}{x}\right)\Vert }^2-\frac{1}{2}{\text{γ}}^2{\Vert w_k\left(\stackrel{⌣}{x}\right)\Vert }^2,$

(8.54)

where the adjoint variable has been set to p = V. Then we have the following result.

Theorem 8.4.2 Consider the nonlinear system (8.46) and the DNLHIF P for it. Suppose the function h₁ is one-to-one (or injective) and the plant Σ^da is locally asymptotically-stable about the equilibrium-point x = 0. Further, suppose there exists a C¹ (with respect to both arguments) positive-definite function V : N × N × Z → ℜ locally defined in a neighborhood N ⊂ χ of the origin $\stackrel{⌣}{x}=0$, and a matrix function L : N × Z → M^n×m satisfying the following DHJIE:

$V\left(\stackrel{⌣}{x},k\right)=V\left({\stackrel{⌣}{f}}^{\star }\left({\stackrel{⌣}{x}}_k\right)+{\stackrel{⌣}{g}}^{\star }\left({\stackrel{⌣}{x}}_k\right)w_k^{\star }\left(\stackrel{⌣}{x}\right),k+1\right)+\frac{1}{2}{\Vert {\stackrel{⌣}{z}}_k\left(\stackrel{⌣}{x}\right)\Vert }^2-\frac{1}{2}{\text{γ}}^2{\Vert w_k^{\star }\left(\stackrel{⌣}{x}\right)\Vert }^2,V\left(\stackrel{⌣}{x},K+1\right)=0$

(8.55)

k = k₀,…, K, together with the side-conditions:

${w_k^{\star }\left(\stackrel{⌣}{x}\right)=\frac{1}{{\text{γ}}^2}{\stackrel{⌣}{g}}^T\left(\stackrel{⌣}{x}\right)\frac{{\partial }^{T}V\left(\text{λ,}k\text{+1}\right)}{\partial \text{λ}}|}_{\text{λ}=\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)+\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)w_k^{\star }},$

(8.56)

$L^{\star }\left(\widehat{x}\right)=\arg \underset{L}{\min }\left\{H\left(\stackrel{⌣}{x},w_k^{\star },L,V\right)\right\},$

(8.57)

${\begin{array}{l}\\ \frac{\partial H}{\partial w^2}H\left(\stackrel{⌣}{x},w_k,L^{\star },V\right)\end{array}|}_{\stackrel{⌣}{x}=0}<0$

(8.58)

${\begin{array}{l}\\ \frac{\partial H}{\partial w^2}H\left(\stackrel{⌣}{x},w_k^{\star },L,V\right)\end{array}|}_{\stackrel{⌣}{x}=0}>0.$

(8.59)

Then: (i) there exists a unique saddle-point equilibrium solution (w^⋆, L^⋆) for the game (8.51), (8.50) locally in N; and (ii) the filter Σ^daf with the gain-matrix $L({\widehat{x}}_k,k)$ satisfying (8.57) solves the finite-horizon D N L H I F P for the system locally in N.

Proof: Assume there exists positive-definite solution V (., k) to the DHJIEs (8.55) in N ⊂ $\mathcal{X}$ for each k, and (i) consider the Hamiltonian function H(., ., ., .). Apply the necessary condition for optimality, i.e.,

${\frac{{\partial }^{T}H}{\partial w}|}_{w=w^{\star }}{={\stackrel{⌣}{g}}^T\left(\stackrel{⌣}{x}\right)\frac{{\partial }^{T}V\left(\text{λ,}k\text{+1}\right)}{\partial \text{λ}}|}_{\text{λ}=\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)+\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)w_k^{\star }}-{\text{γ}}^2{w}_k^{\star }=0,$

to get

${w_k^{\star }=\frac{1}{{\text{γ}}^2}{\stackrel{⌣}{g}}^T\left(\stackrel{⌣}{x}\right)\frac{{\partial }^{T}V\left(\text{λ,}k\text{+1}\right)}{\partial \text{λ}}|}_{\text{λ}=\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)+\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)w_k^{\star }}:={\alpha }_0\left(\stackrel{⌣}{x},w_k^{\star }\right).$

(8.60)

Thus, w^⋆ is expressed implicitly. Moreover, since

$\frac{{\partial }^{T}H}{\partial w^2}{={\stackrel{⌣}{g}}^T\left(\stackrel{⌣}{x}\right)\frac{{\partial }^{T}V\left(\text{λ,}k\text{+1}\right)}{\partial {\text{λ}}^2}|}_{\text{λ}=\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)+\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)w_k^{\star }}\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)-{\text{γ}}^{2}I$

is nonsingular about $(\stackrel{⌣}{x},w)=(0,0)$, the equation (8.60) has a unique solution α₀($\stackrel{⌣}{x}$), α₀(0) = 0 in the neighborhood N × W of (x, w) = (0, 0) by the Implicit-function Theorem [234].

Now substitute w^⋆ in the expression for H(., ., ., .) (8.54), to get

$H\left(\stackrel{⌣}{x},w_k^{\star },L,V\right)=V\left(\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)+\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)w_k^{\star }\left(\stackrel{⌣}{x}\right),k+1\right)-V\left(\stackrel{⌣}{x},k\right)+\frac{1}{2}{\Vert {\stackrel{⌣}{z}}_k\left(\stackrel{⌣}{x}\right)\Vert }^2-\frac{1}{2}{\text{γ}}^2{\Vert w_k^{\star }\left(\stackrel{⌣}{x}\right)\Vert }^2$

and let

$L^{\star }=\arg \underset{L}{\min }\left\{H\left(\stackrel{⌣}{x},w_k^{\star },L,V\right)\right\}.$

Then, by Taylor’s theorem, we can expand H(., ., ., .) about (L^⋆ , w^⋆ ) [267] as

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

(8.61)

Thus, taking L^⋆ as in (8.57) and w^⋆ = α₀($\stackrel{⌣}{x}$, w^⋆) and if the conditions (8.58), (8.59) hold, we see that the saddle-point conditions

$H\left(w,L^{\star }\right)\le H\left(w^{\star },L^{\star }\right)\le H\left(w^{\star },L\right)\forall L\in {\mathcal{M}}^{n\times m},\forall w\in {\mathcal{l}}_2\left[k_0,K\right],k\in \left[k_0,K\right]$

are locally satisfied. Moreover, substituting (w^⋆ , L^⋆) in (8.53) gives the DHJIE (8.55).

(ii) Reconsider equation (8.61). Since the conditions (8.58) and (8.59) are satisfied about $\stackrel{⌣}{x}$ = 0, by the Inverse-function Theorem [234], there exists a neighborhood U ⊂ N × N of $\stackrel{⌣}{x}$ = 0 for which they are also satisfied. Consequently, we immediately have the important inequality

$H\left(\stackrel{⌣}{x},w,L^{\star },V\right)\le H\left(\stackrel{⌣}{x},w^{\star },L^{\star },V\right)=0\forall \stackrel{⌣}{x}\in U$

$\iff V\left({\stackrel{⌣}{x}}_{k+1},k+1\right)-V\left(\stackrel{⌣}{x},k\right)\le \frac{1}{2}{\text{γ}}^2{\Vert w_k\Vert }^2-\frac{1}{2}{\Vert {\stackrel{⌣}{z}}_k\Vert }^2,\forall \stackrel{⌣}{x}\in U,\forall w_k\in \mathcal{W}.$

(8.62)

Summing now from k = k₀ to k = K, we get the dissipation inequality [183]:

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

(8.63)

Thus, the system has ℓ₂-gain from w to $\stackrel{⌣}{z}$ less or equal to γ. □

8.4.1    Infinite-Horizon Discrete-Time Nonlinear ${\mathcal{H}}_{\infty }$-Filtering

In this subsection, we discuss the infinite-horizon discrete-time filtering problem, in which case we let K → ∞. Since we are interested in finding a time-invariant gain for the filter, we seek a time-independent function $V:\tilde{N}\times \tilde{N}\to \Re$ locally defined in a neighborhood $\tilde{N}\times \tilde{N}\subset \chi \times \chi$ of $(x,\widehat{x})=(0,0)$ such that the following stationary DHJIE:

$\tilde{V}\left({\stackrel{⌣}{f}}^{\star }\left(\stackrel{⌣}{x}\right)+{\stackrel{⌣}{g}}^{\star }\left(\stackrel{⌣}{x}\right)w^{\star }\left(\stackrel{⌣}{x}\right)\right)-\tilde{V}\left(\stackrel{⌣}{x}\right)+\frac{1}{2}{\Vert \stackrel{⌣}{z}\left(\stackrel{⌣}{x}\right)\Vert }^2-\frac{1}{2}{\text{γ}}^2{\Vert w^{\star }\left(\stackrel{⌣}{x}\right)\Vert }^2=0,\tilde{V}\left(0\right)=0,x,\widehat{x}\in \widehat{N}$

(8.64)

is satisfied together with the side-conditions:

${\tilde{w}}^{\star }\left(\stackrel{⌣}{x}\right)={\frac{1}{{\text{γ}}^2}{\stackrel{⌣}{g}}^T\left(\stackrel{⌣}{x}\right)\frac{{\partial }^T\tilde{V}\left(\text{λ}\right)}{\partial \text{λ}}|}_{\text{λ}=\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)+\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)w^{\star }}:={\text{α}}_1\left(\stackrel{⌣}{x},{\tilde{w}}^{\star }\right)$

(8.65)

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

(8.66)

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

(8.67)

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

(8.68)

where

${{\stackrel{⌣}{f}}^{\star }=\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)|}_{\tilde{L}={\tilde{L}}^{\star }},{{\stackrel{⌣}{g}}^{\star }\left(\stackrel{⌣}{x}\right)=\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)|}_{\tilde{L}={\tilde{L}}^{\star }},$

$\tilde{H}\left(\stackrel{⌣}{x},w_k,\tilde{L},\tilde{V}\right)=\tilde{V}\left(\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)+\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)w\right)-\tilde{V}\left(\stackrel{⌣}{x}\right)+\frac{1}{2}{\Vert {\stackrel{⌣}{z}}_k\Vert }^2-\frac{1}{2}{\text{γ}}^2{\Vert w_k\Vert }^2,$

(8.69)

and ${\tilde{w}}^{\star },{\tilde{L}}^{\star }$ are the asymptotic values of ${\stackrel{}{w}}^{\star }{}_k,{\stackrel{}{L}}_k{}^{\star }$ respectively. Again here, since the estimation is carried over an infinite-horizon, it is necessary to ensure that the augmented system (8.50) is stable with w = 0. However, in this case, we can relax the requirement of asymptotic-stability for the original system (8.46) with a milder requirement of detectability which we define next.

Definition 8.4.2 The pair {f, h} is said to be locally zero-state detectable if there exists a neighborhood $\mathcal{O}$ of x = 0 such that, if x_k is a trajectory of x_k+1 = f(x_k) satisfying x(k₀) ∈ $\mathcal{O}$, then h(x_k) is defined for all k ≥ k₀ and h(x_k) ≡ 0 for all k ≥ k_s, implies lim_k→∞ x_k = 0.

A filter is also required to be stable, so that trajectories do not blow-up for example in an open-loop system. Thus, we define the “admissibility” of a filter as follows.

Definition 8.4.3 A filter $\mathcal{F}$ is admissible if it is (internally) asymptotically (or exponentially) stable for any given initial condition x(k₀) of the plant Σ^da, and with w = 0

$\underset{x\to \infty }{\lim }{\stackrel{⌣}{z}}_k=0.$

The following proposition can now be proven along the same lines as Theorem 8.4.2.

Proposition 8.4.1 Consider the nonlinear system (8.46) and the infinite-horizon D N L H I F P for it. Suppose the function h₁ is one-to-one (or injective) and the plant Σ^da is zero-state detectable. Further, suppose there exist a C¹ positive-definite function $\tilde{V}:\tilde{N}\times \tilde{N}\to \Re$ locally defined in a neighborhood $\begin{array}{l}\text{where}{\stackrel{⌣}{x}}_k={\left(x_k^T{\widehat{x}}_k^T\right)}^T,\\ \tilde{N}\subset \chi \end{array}$ of the origin $\stackrel{⌣}{x}=0,\tilde{V}(0)=0$ and a matrix function $\tilde{L}:\tilde{N}\to M^{n\times m}$ satisfying the DHJIEs (8.64) together with (8.65)-(8.69). Then: (i) there exists locally a unique saddle-point solution $(\stackrel{}{{\tilde{w}}^{\star }},\stackrel{}{{\tilde{L}}^{\star })}$ for the game; and (ii) the filter Σ^daf with the gain matrix $\tilde{L}(\widehat{x})={\tilde{L}}^{\star }(\widehat{x})$ satisfying (8.66) solves the infinite-horizon D N L H I F P locally in $\tilde{N}$.

Proof: (Sketch). We only prove that the filter Σ^daf is admissible, as the rest of the proof is similar to the finite-horizon case. Using similar manipulations as in the proof of Theorem 8.4.1, it can be shown that with w = 0,

$\tilde{V}\left({\stackrel{⌣}{x}}_{k+1}\right)-\tilde{V}\left({\stackrel{⌣}{x}}_k\right)\le -\frac{1}{2}{\Vert {\stackrel{⌣}{z}}_k\Vert }^2.$

Therefore, the augmented system is locally stable. Further, the condition that $\tilde{V}({\stackrel{⌣}{x}}_{k+1})\equiv \tilde{V}({\stackrel{⌣}{x}}_k)\forall k\ge k_c$, for some k_c ≥ k₀, implies that ${\stackrel{⌣}{z}}_k\equiv 0\forall k\ge k_c$. Moreover, the zero-state detectability of the system (8.46) implies the zero-state detectability of the system (8.50) since h₁ is injective. Thus, by virtue of this, we have lim_k→∞ x_k = 0, and by LaSalle’s invariance-principle, this implies asymptotic-stability of the system (8.50). Hence, the filter Σ^daf is admissible. □

8.4.2    Approximate and Explicit Solution

In this subsection, we discuss how the D N L H I F P can be solved approximately to obtain explicit solutions [126]. We consider the infinite-horizon problem, but the approach can also be used for the finite-horizon problem. For simplicity, we make the following assumption on the system matrices.

Assumption 8.4.1 The system matrices are such that

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

Now, consider the infinite-horizon Hamiltonian function $\tilde{H}$(., ., ., .) defined by (8.69). Expanding it in Taylor-series about $\widehat{f}(\stackrel{⌣}{x})=\left(\begin{array}{l}f(x)\\ f(\widehat{x})\end{array}\right)$ up to first-order¹ and denoting this expansion by $\widehat{H}$(., ., ., .) and L by $\widehat{L}$, we get:

$\begin{array}{l}\widehat{H}\left(\stackrel{⌣}{x},w,\tilde{L},\tilde{V}\right)=\left\{\tilde{V}\left(\widehat{f}\left(\stackrel{⌣}{x}\right)\right)+{\tilde{V}}_x\left(\widehat{f}\left(\stackrel{⌣}{x}\right)g_1\left(x\right)w+{\tilde{V}}_{\widehat{x}}\left(\widehat{f}\left(\stackrel{⌣}{x}\right)\right)\right)\left[\widehat{L}\left(\widehat{x}\right)\left(h_2\left(x\right)-h_2\left(\widehat{x}\right)\right)+k_{21}\left(x\right)w\right]+O\left({\Vert \tilde{v}\Vert }^2\right)\right\}\\ -\tilde{V}\left(\stackrel{⌣}{x}\right)+\frac{1}{2}{\Vert {\stackrel{⌣}{z}}_k\Vert }^2-\frac{1}{2}{\text{γ}}^2{\Vert w\Vert }^2,\stackrel{⌣}{x}\in \widehat{N}\times \widehat{N},w\in \mathcal{W}\end{array}$

(8.70)

where $\stackrel{⌣}{x}={\stackrel{⌣}{x}}_k,\stackrel{⌣}{z}={\stackrel{⌣}{z}}_k,w=w_{k,}{\tilde{V}}_x,{\tilde{V}}_{\widehat{x}}$ are the row-vectors of the partial-derivatives of $\tilde{V}$ with respect to x, $\widehat{x}$ respectively,

$\tilde{v}=\left(\begin{array}{l}{g}_1\left(x\right)w\\ \widehat{L}\left(\widehat{x}\right)\left[h_2\left(x\right)-h_2\left(\widehat{x}\right)+k_{21}\left(x\right)w\right]\end{array}\right)$

and

$\underset{\tilde{v}\to 0}{\lim }\frac{O({\Vert \tilde{v}\Vert }^2)}{{\Vert \tilde{v}\Vert }^2}=0$

Then, applying the necessary condition for optimality, we get

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

(8.71)

Now substitute ${\widehat{w}}^{\star }$ in (8.70) to obtain

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

Completing the squares for $\widehat{L}$ in the above expression for $\widehat{H}$(., ., ., .), we get

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

Thus, taking ${\widehat{L}}^{\star }$ as

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

(8.72)

minimizes $\widehat{H}$(., ., ., .) 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 (8.72) in the expression for $\widehat{H}$(., ., ., .) and complete the squares in w to obtain:

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

Thus, substituting w = ŵ^⋆ as given in (8.71), we see that the second saddle-point condition

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

is also satisfied. Hence, the pair $({\widehat{w}}^{\star },{\widehat{L}}^{\star })$ constitutes a unique saddle-point solution to the game corresponding to the Hamiltonian $\widehat{H}$(., ., ., .). Finally, substituting $({\widehat{w}}^{\star },{\widehat{L}}^{\star })$ in the DHJIE (8.53), we get the following DHJIE:

$\begin{array}{l}\tilde{V}\left(\widehat{f}\left(\stackrel{⌣}{x}\right)\right)-\tilde{V}\left(\stackrel{⌣}{x}\right)+\frac{1}{2{\text{γ}}^2}{\tilde{V}}_x\left(\widehat{f}\left(\stackrel{⌣}{x}\right)\right)g_1\left(x\right)g_1^T\left(x\right){\tilde{V}}_x^T\left(\widehat{f}\left(\stackrel{⌣}{x}\right)\right)-\\ \frac{1}{2{\text{γ}}^2}{\tilde{V}}_{\widehat{x}}\left(\widehat{f}\left(\stackrel{⌣}{x}\right)\right)\widehat{L}\left(\widehat{x}\right){\widehat{L}}^T\left(\widehat{x}\right){\tilde{V}}_{\widehat{x}}^T\left(\widehat{f}\left(\stackrel{⌣}{x}\right)\right)+\\ \frac{1}{2}{\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right)}^T\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right)=0,\tilde{V}\left(0\right)=0,x,\widehat{x}\in \widehat{N.}\end{array}$

(8.73)

With the above analysis, we have the following theorem.

Theorem 8.4.3 Consider the nonlinear system (8.46) and the infinite-horizon D N L H I F P for this system. Suppose the function h₁ is one-to-one (or injective) and the plant Σ^da is zero-state detectable. Suppose further there exists a C¹ positive-definite function $\tilde{V}:\widehat{N}\times \widehat{N}\to \Re$ locally defined in a neighborhood $\widehat{N}\times \widehat{N}\subset X\times \mathcal{X}$ of the origin $\stackrel{⌣}{x}=0$, anda matrix function $\widehat{L}:\widehat{N}\times \widehat{N}\to M^{n\times m}$ the DHJIE (8.73) together with the side-condition (8.72). Then:

(i) there exists locally in $\widehat{N}$ a unique saddle-point solution $({\widehat{w}}^{\star },{\widehat{L}}^{\star })$ for the dynamic game corresponding to (8.51), (8.50);

(ii) the filter Σ^daf with the gain matrix $\widehat{L}(\widehat{x})={\widehat{L}}^{\star }(\widehat{x})$ satisfying (8.72) solves the infinite-horizon D N L H I F P for the system locally in $\widehat{N}$

Proof: Part (i) has already been shown above. For part (ii), consider the time variation of $\tilde{V}>0$ (a solution to the DHJIE (8.73)) along a trajectory of the augmented system (8.50) with $\widehat{L}={\widehat{L}}^{\star }$, i.e.,

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

where use has been made of the Taylor-series approximation, equation (8.72) and the DHJIE (8.73). Finally, the above inequality clearly implies the infinitesimal dissipation-inequality:

$\tilde{V}\left({\stackrel{⌣}{x}}_{k+1}\right)-\tilde{V}\left({\stackrel{⌣}{x}}_k\right)\le \frac{1}{2}{\text{γ}}^2{\Vert w_k\Vert }^2-\frac{1}{2}{\Vert {\stackrel{⌣}{z}}_k\Vert }^2,\forall \stackrel{⌣}{x}\in \widehat{N}\times \widehat{N},\forall w\in \mathcal{W}$

Therefore, the system (8.50) has locally ℓ₂-gain from w to $\stackrel{⌣}{z}$ less or equal to γ. The remaining arguments are the same as in the proof of Proposition 8.4.1. □

We now specialize the result of the above theorem to the linear-time-invariant (LTI) system:

${\Sigma }^{dl}:\{\begin{array}{l}{\dot{x}}_{k+1}=A_{x_k}+B_1{w}_k,x\left(k_0\right)=x^0\\ {\stackrel{⌣}{z}}_k=C_1\left(x_k-{\widehat{x}}_k\right)\\ y_k=C_2{x}_k+D_{21}{w}_k,D_{21}{B}_1^T=0,D_{21}^T{D}_{21}=I,\end{array}$

(8.74)

where all the variables have their previous meanings, and A ∈ ℜ ^n×n, B₁ ∈ ℜ^n×r, C₁ ∈ ℜ^s×n, C₂ ∈ ℜ^m×n and D₂₁ ∈ ℜ^m×r are constant real matrices. We have the following corollary to Theorem 8.4.3.

Corollary 8.4.1 Consider the discrete-LTI system Σ^dl defined by (8.74) and the D N L H I F P for it. Suppose C₁ is full column rank and A is Hurwitz. Suppose further, there exists a positive-definite real symmetric solution $\widehat{P}$ to the discrete algebraic-Riccati

equation (DARE):

$A^T\widehat{P}A-\widehat{P}-\frac{1}{2{\text{γ}}^2}{A}^T\widehat{P}{B}_1{B}_1^T\widehat{P}A-\frac{1}{2{\text{γ}}^2}{A}^T\widehat{P}\widehat{L}{\widehat{L}}^T\widehat{P}A+C_1^T{C}_1=0$

(8.75)

together with the coupling condition:

$A^T\widehat{P}{\widehat{L}}_i^{\star }=-{\text{γ}}^2{C}_2^T.$

(8.76)

Then:

(i)  there exists a unique saddle-point solution $({\widehat{w}}^{\star }{}_l,{\widehat{L}}^{\star }{}_l)$ for the game given by

$\begin{array}{l}{\widehat{w}}_l^{\star }=\frac{1}{{\text{γ}}^2}\left(B_1^T+D_{21}^T\widehat{L}\right)\widehat{P}A\left(x-\widehat{x}\right),\\ {\left(x-\widehat{x}\right)}^T{A}^T\widehat{P}{\widehat{L}}_l^{\star }=-{\text{γ}}^2\left(x-\widehat{x}\right)C_2^T;\end{array}$

(ii)  the filter F defined by

${\Sigma }_{ldf}:{\widehat{x}}_{k+1}=A{\widehat{x}}_k+\widehat{L}\left(y-C_2{x}_k\right),\widehat{x}\left(k_0\right)={\widehat{x}}^0$

with the gain matrix $\widehat{L}={\widehat{L}}^{\star }{}_l$ satisfying (8.76) solves the infinite-horizon D N L H I F P for the system.

Proof: Take:

$\tilde{V}\left(\stackrel{⌣}{x}\right)=\frac{1}{2}{\left(x-\widehat{x}\right)}^T\widehat{P}\left(x-\widehat{x}\right),\widehat{P}={\widehat{P}}^T>0,$

and apply the result of the theorem. □

Next we consider an example.

Example 8.4.1 We consider the following scalar example:

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

where w_k = w₀ + 0.1sin(2πk), and w₀ is zero-mean Gaussian white noise. We compute the solution of the DHJIE (8.73) using the iterative scheme:

$\begin{array}{l}{\tilde{V}}^{j+1}\left(\stackrel{⌣}{x}\right)={\tilde{V}}^k\left(\widehat{f}\left(\stackrel{⌣}{x}\right)\right)+\frac{1}{2{\text{γ}}^2}{\tilde{V}}_x^j\left(\widehat{f}\left(\stackrel{⌣}{x}\right)\right)g_1\left(x\right)g_1^T\left(x\right){\tilde{V}}_x^{T,j}\left(\widehat{f}\left(\stackrel{⌣}{x}\right)\right)-\\ \frac{1}{2{\text{γ}}^2}{\tilde{V}}_{\widehat{x}}^j\left(\widehat{f}\left(\stackrel{⌣}{x}\right)\right){\widehat{L}}^j\left(\widehat{x}\right){\widehat{L}}^{j,T}\left(\widehat{x}\right){\tilde{V}}_{\widehat{x}}^{T,j}\left(\widehat{f}\left(\stackrel{⌣}{x}\right)\right)+\frac{1}{2}{\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right)}^T\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right),\\ {\tilde{V}}^j\left(0\right)=0,x,\widehat{x}\in \widehat{N},j=0,1,\dots \end{array}$

(8.77)

starting with the initial guess ${\tilde{V}}^0(\stackrel{⌣}{x})=\frac{1}{2}{\left(x-\widehat{x}\right)}^2,\text{γ}=1$ and initial filter-gain l₀ = 1, after one iteration, we get ${\tilde{V}}^1(\stackrel{⌣}{x})=(x^{\frac{1}{3}}-{\widehat{x}}^{\frac{1}{3}})+\frac{1}{2}{\left(x-\widehat{x}\right)}^2$ and ${\tilde{V}}_{\widehat{x}}{}^1\left(\widehat{f}\left(\widehat{x}\right)\right)=-\frac{2}{3}\left(\frac{x^{\frac{1}{9}}-{\widehat{x}}^{\frac{1}{9}}}{{\widehat{x}}^{\frac{2}{9}}}\right)-(x^{\frac{1}{3}}-{\widehat{x}}^{\frac{1}{3}}).$. Then we proceed to compute the filter-gain using (8.72). The result of the simulation is shown in Figure 8.7.

FIGURE 8.7
Discrete-Time Nonlinear ${\mathcal{H}}_{\infty }$-Filter Performance with Unknown Initial Condition