12

$Mixed{\mathscr{H}}_2/{\mathscr{H}}_{\infty }$ Nonlinear Filtering

The ${\mathcal{H}}_{\mathrm{\infty }}$ nonlinear filter has been discussed in Chapter 8, and its advantages over the Kalman-filter have been mentioned. In this chapter, we discuss the mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$-criterion approach for estimating the states of an affine nonlinear system in the spirit of Reference [179]. Many authors have considered mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$-filtering techniques for linear systems [162, 257], [269]-[282], which enjoy the advantages of both Kalman-filtering and ${\mathcal{H}}_{\mathrm{\infty }}$-filtering. In particular, the paper [257] considers a differential game approach to the problem, which is attractive and transparent. In this chapter, we present counterpart results for nonlinear systems using a combination of the differential game approach with a dissipative system’s perspective. We discuss both the continuous-time and discrete-time problems.

12.1  Continuous-Time Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ Nonlinear Filtering

The general set-up for the mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$-filtering problem is shown Figure 12.1, where the plant is represented by an affine nonlinear system Σ^a, while F is the filter. The filter processes the measurement output y from the plant which is also corrupted by the noise signal w, and generates an estimate $\widehat{z}$ of the desired variable z. The noise signal w entering the plant is comprised of two components,

FIGURE 12.1
Set-Up for Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ Nonlinear Filtering

$w=\left(\begin{array}{l}{w}_0\\ w_1\end{array}\right)$, with w₀ ∈ $\mathcal{S}$ a bounded-spectral signal (e.g., a white Gaussian noise signal); and w₁ ∈ P is a bounded-power signal or $\mathcal{L}$₂-signal. Thus, the induced-norm (or gain) from the input w₀ to $\stackrel{⌣}{z}$ (the output error) is the $\mathcal{L}$₂-norm of the interconnected system F ◦ Σ^a, i.e.,

$\Vert F\circ {\sum }^a{\Vert }_{{\mathcal{L}}_2}\triangleq \underset{0\ne w_0\in \mathcal{S}}{\sup }\frac{\Vert \stackrel{⌣}{z}\Vert p}{\Vert w_0\Vert \mathcal{S}},$

(12.1)

while the induced norm from w₁ to $\stackrel{⌣}{z}$ is the $\mathcal{L}$_∞-norm of P^a, i.e.,

$\Vert F\circ {\sum }^a{\Vert }_{{\mathcal{L}}_{\infty }}\triangleq \underset{0\ne w_0\in \mathcal{P}}{\sup }\frac{\Vert \stackrel{⌣}{z}{\Vert }_2}{\Vert w_1{\Vert }_2}.$

(12.2)

The objective is to synthesize a filter, $\mathcal{F}$, for estimating the state x(t) or a function of it, z = h₁(x), from observations of y(τ) up to time t over a time horizon [t₀, T ], i.e.,

${\text{Y}}_t\triangleq \left\{y\left(\tau \right):\tau \le t\right\},t\in \left[t_{0,}T\right],$

such that the above pair of norms (12.1), (12.2) are minimized, while at the same time achieving asymptotic zero estimation error with w ≡ 0. In this context, the above norms will be interpreted as the ${\mathcal{H}}_2$ and ${\mathcal{H}}_{\mathrm{\infty }}$-norms of the interconnected system.

However, in this chapter, we do not solve the above problem, instead we solve an associated ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$-filtering problem in which there is a single exogenous input w ∈ L₂[t₀, T ] and an associated ${\mathcal{H}}_2$-cost which represents the output energy of the system in $\stackrel{⌣}{z}$. More specifically, we seek to synthesize a filter, $\mathcal{F}$, for estimating the state x(t) or a function of it, z = h₁(x), from the observations Y_t such that the $\mathcal{L}$₂-gain from the input w to the penalty function $\stackrel{⌣}{z}$ (to be defined later) as well as the output energy defined by $\parallel \stackrel{⌣}{z}{\parallel }_{{\mathcal{H}}_2}$ are minimized, and at the same time achieving asymptotic zero estimation error with w ≡ 0.

Accordingly, the plant is represented by an affine nonlinear causal state-space system defined on a manifold $X\subset {\Re }^n$ with zero control input:

${\sum }^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+g_1\left(x\right)w;x\left(t_0\right)=x_0\\ z=h_1\left(x\right)\\ y=h_2\left(x\right)+k_{21}\left(x\right)w,\end{array}$

(12.3)

where $x\in \mathcal{X}$ is the state vector, $w\in \mathcal{W}\subseteq {\mathcal{L}}_2([t_0,\infty ),{\Re }^r)$ is an unknown disturbance (or noise) signal, which belongs to $\mathcal{W}$, the set of admissible noise/disturbance signals, $y\in \mathcal{Y}\subset {\Re }^m$ is the measured output (or observation) of the system, and belongs to $\mathcal{Y}$, the set of admissible outputs, $z\in {\Re }^s$ is the output of the system that is to be estimated.

The functions $f:\mathcal{X}\to V^{\infty }(\mathcal{X}),g_1:\mathcal{X}\to M^{n\times r}(\mathcal{X}),h_1:\mathcal{X}\to {\Re }^s,h_2:\mathcal{X}\to {\Re }^m$ and $k_{21}:X\to {\mathcal{M}}^{n\times r}(\mathcal{X})$ are real C^∞-functions of x. Furthermore, we assume without any loss of generality that the system (12.3) has a unique equilibrium-point at x = 0 such that f(0) = 0, h₁(0) = 0, h₂(0) = 0. We also assume that there exists a unique solution x(t, t₀, x₀, w), ∀t ∈ ℜ for the system for all initial conditions x₀ and all w ∈ $\mathcal{W}$.

Again, since it is difficult to minimize exactly the ${\mathcal{H}}_{\mathrm{\infty }}$-norm, in practice we settle for a suboptimal problem, which is to minimize $\parallel \stackrel{⌣}{z}{\parallel }_{\mathcal{H}}{}_{{}_2}$ while rendering $\parallel F\circ {\sum }^a{\parallel }_{\mathcal{H}\infty }\le {\gamma }^{\star }.$ For nonlinear systems, this ${\mathcal{H}}_{\mathrm{\infty }}$-constraint is interpreted as the $\mathcal{L}$₂-gain constraint and is defined as

${\displaystyle {\int }_{t_0}^T{\Vert \stackrel{⌣}{z}\left(\tau \right)\Vert }^{2}dt\le {\gamma }^{\text{2}}}{\displaystyle {\int }_{t_0}^T{\Vert w\left(\tau \right)\Vert }^{2}d\tau ,T}>0.$

(12.4)

More formally, we define the local nonlinear (suboptimal) mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$-filtering or state estimation problem as follows.

Definition 12.1.1 (Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ (Suboptimal) Nonlinear Filtering Problem (MH2HINLFP)). Given the plant Σ^a and a number γ^⋆ > 0, find a filter $\mathcal{F}:\mathcal{Y}\to \mathcal{X}$ such that

$\widehat{x}(t)=ℱ(Y_{\text{t}})$

and the output energy $\parallel \stackrel{⌣}{z}{\parallel }_{{\mathcal{L}}_2}$ is minimized while the constraint (12.4) is satisfied for all γ ≥ γ^⋆, for all w ∈ W and for all initial conditions x(t₀) ∈ O. In addition with w ≡ 0, lim_t→∞ $\stackrel{⌣}{z}$(t) = 0.

Moreover, if the above conditions are satisfied for all x(t₀) ∈ X, we say that F solves the MH2HINLFP globally.

Remark 12.1.1 The problem defined above is the finite-horizon filtering problem. We have the infinite-horizon problem if we let T →∞.

12.1.1  Solution to the Finite-Horizon Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ Nonlinear Filtering Problem

To solve the MH2HINLFP, we similarly consider the following Kalman-Luenberger filter structure

${\sum }^{af}:\{\begin{array}{l}\dot{\widehat{x}}=f\left(\widehat{x}\right)+L\left(\widehat{x},t\right)\left[y-h_2\left(\widehat{x}\right)\right],\widehat{x}\left(t_0\right)={\widehat{x}}_0\hfill \\ \widehat{z}=h_1\left(\widehat{x}\right)\hfill \end{array}$

(12.5)

where $\widehat{x}\in X$ is the estimated state, L(., .) ∈ $\mathcal{M}$^n×m($\mathcal{X}$ ×$\Re$) is the error-gain matrix which is smooth and has to be determined, and $\widehat{z}$ ∈ ℜ^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:

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

Then we combine the plant (12.3) and estimator (12.5) dynamics to obtain the following augmented system:

$\begin{array}{l}\dot{\stackrel{⌣}{x}}=\stackrel{⌣}{f}(\stackrel{⌣}{x})+\stackrel{⌣}{g}(\stackrel{⌣}{x})w,\stackrel{⌣}{x}(t_0)={\left(\begin{array}{cc}{x}_0^T& {\widehat{x}}_0^T\end{array}\right)}^T\\ \stackrel{⌣}{z}=\stackrel{⌣}{h}(\stackrel{⌣}{x})\end{array}\},$

(12.6)

where

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

The problem can then be formulated as a two-player nonzero-sum differential game (from Chapter 3, see also [59]) with two cost functionals:

$J_1\left(L,w\right)=\frac{1}{2}{\displaystyle {\int }_{t_0}^T\left[{\gamma }^{\text{2}}{\Vert w\left(\tau \right)\Vert }^2-{\Vert \stackrel{⌣}{z}\left(\tau \right)\Vert }^2\right]}d\tau ,$

(12.7)

$J_2\left(L,w\right)=\frac{1}{2}{\displaystyle {\int }_{t_0}^T{\Vert \stackrel{⌣}{z}\left(\tau \right)\Vert }^{2}d}\tau .$

(12.8)

Here, the first functional is associated with the ${\mathcal{H}}_{\mathrm{\infty }}$-constraint criterion, while the second functional is related to the output energy of the system or ${\mathcal{H}}_2$-criterion. It can easily be seen that, by making J₁ ≥ 0 then the ${\mathcal{H}}_{\mathrm{\infty }}$-constraint ‖F ◦ P^a‖_{${\mathcal{H}}_{\mathrm{\infty }}$} ≤ γ is satisfied. A Nash-equilibrium solution [59] to the above game is said to exist if we can find a pair of strategies (L^⋆, w^⋆) such that

$J_1\left(L^{\star },w^{\star }\right)\le J_1\left(L^{\star },w\right)\forall w\in \mathcal{W},$

(12.9)

$J_2\left(L^{\star },w^{\star }\right)\le J_2\left(L,w^{{}^{\star }}\right)\forall L\in {\mathcal{M}}^{n\times m}.$

(12.10)

To arrive at a solution to this problem, we form the Hamiltonian function H_i : T ^⋆ ($\mathcal{X}$ × $\mathcal{X}$)× $\mathcal{W}$ × $\mathcal{M}$^n×m → ℜ, i = 1, 2, associated with the two cost functionals:

$H_1\left(\stackrel{⌣}{x},w,L,Y_{\stackrel{⌣}{x}}^T\right)=Y_{\stackrel{⌣}{x}}\left(\stackrel{⌣}{x},t\right)\left(\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)+\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)w\right)+\frac{1}{2}\left({\gamma }^2\Vert w{\Vert }^2-\Vert \stackrel{⌣}{z}{\Vert }^2\right),$

(12.11)

$H_2\left(\stackrel{⌣}{x},w,L,V_{\stackrel{⌣}{x}}^T\right)=V_{\stackrel{⌣}{x}}\left(\stackrel{⌣}{x},t\right)\left(\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)+\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)w\right)+\frac{1}{2}{\Vert \stackrel{⌣}{z}\Vert }^2$

(12.12)

with the adjoint variables $p_1=Y_{\stackrel{⌣}{x}}^T,p_2=V_{\stackrel{⌣}{x}}^T$ respectively, for some smooth functions Y, V : $\mathcal{X}$ × $\mathcal{X}$ × ℜ → ℜ and where $Y_{\stackrel{⌣}{x}},V_{\stackrel{⌣}{x}}$ are the row-vectors of first partial-derivatives of the functions with respect to (wrt) $\stackrel{⌣}{x}$ respectively. The following theorem from Chapter 3 then gives sufficient conditions for the existence of a Nash-equilibrium solution to the above game.

Theorem 12.1.1 Consider the two-player nonzero-sum dynamic game (12.7)-(12.8),(12.6) of fixed duration [t₀, T ] under closed-loop memoryless perfect state information pattern. A pair of strategies (w^⋆ , L^⋆ ) provides a feedback Nash-equilibrium solution to the game, if there exists a pair of C¹-functions (in both arguments) Y, V: $\mathcal{X}$ × $\mathcal{X}$ × ℜ → ℜ, satisfying the pair of HJIEs:

$Y_t\left(\stackrel{⌣}{x},t\right)=-\underset{w\in \mathcal{W}}{\inf }{H}_1\left(\stackrel{⌣}{x},w,L^{\star },Y_{\stackrel{⌣}{x}}\right),Y\left(\stackrel{⌣}{x},T\right)=0,$

(12.13)

$V_t\left(\stackrel{⌣}{x},t\right)=-\underset{L\in {\mathcal{M}}^{n\times m}}{\min }{H}_2\left(\stackrel{⌣}{x},w^{\star },L,V_{\stackrel{⌣}{x}}\right),V\left(\stackrel{⌣}{x},T\right)=0.$

(12.14)

Based on the above theorem, it is now easy to find the above Nash-equilibrium pair for our game. The following theorem gives sufficient conditions for the solvability of the MH2HINLFP. For simplicity we make the following assumption on the plant.

Assumption 12.1.1 The system matrices are such that

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

Remark 12.1.2 The first of the above assumptions means that the measurement noise and the system noise are independent.

Theorem 12.1.2 Consider the nonlinear system (12.3) and the MH2HINLFP for it. Suppose the function h₁ is one-to-one (or injective) and the plant Σ^a is locally asymptotically-stable about the equilibrium point x = 0. Further, suppose there exists a pair of C¹ (with respect to both arguments) negative and positive-definite functions Y, V : N × N × ℜ → ℜ respectively, locally defined in a neighborhood N × N ⊂ X × X of the origin $\stackrel{⌣}{x}$ = 0, and a matrix function L : N × ℜ → $\mathcal{M}$^n×m satisfying the following pair of coupled HJIEs:

$\begin{array}{l}{Y}_t(\stackrel{⌣}{x},t)+Y_x(\stackrel{⌣}{x},t)f(x)+Y_{\widehat{x}}(\stackrel{⌣}{x},t)f(\widehat{x})-\frac{1}{2{\gamma }^2}{Y}_x(\stackrel{⌣}{x})g_1(x)g_1^T(x)Y_x^T(\stackrel{⌣}{x})-\\ \frac{1}{2{\gamma }^2}{Y}_{\widehat{x}}(\stackrel{⌣}{x})L(\widehat{x},t)L^T(\widehat{x},t)Y_{\widehat{x}}^T(\stackrel{⌣}{x})-\frac{1}{{\gamma }^2}{Y}_{\widehat{x}}(\stackrel{⌣}{x})L(\widehat{x},t)L^T(\widehat{x},t)V_{\widehat{x}}^T(\stackrel{⌣}{x})-\\ \frac{1}{2}{\left(h_1(x)-h_1(\widehat{x})\right)}^T\left(h_1(x)-h_1(\widehat{x})\right)=0,Y(\stackrel{⌣}{x})=0\end{array}$

(12.15)

$\begin{array}{l}{V}_t(\stackrel{⌣}{x},t)+V_x(\stackrel{⌣}{x},t)f(x)+V_{\widehat{x}}(\stackrel{⌣}{x},t)f(\widehat{x})-\frac{1}{{\gamma }^2}{V}_x(\stackrel{⌣}{x})g_1(x)g_1^T(x)Y_x^T(\stackrel{⌣}{x})-\\ \frac{1}{{\gamma }^2}{V}_{\widehat{x}}(\stackrel{⌣}{x})L(\widehat{x},t)L^T(\widehat{x},t)Y_{\widehat{x}}^T-{\gamma }^2{\left(h_2(x)-h_2(\widehat{x})\right)}^T-\left(h_2(x)-h_2(\widehat{x})\right)+\\ \frac{1}{2}{\left(h_1(x)-h_1(\widehat{x})\right)}^T\left(h_1(x)-h_1(\widehat{x})\right)=0,V(\stackrel{⌣}{x},T)=0,\end{array}$

(12.16)

together with the coupling condition

$V_{\widehat{x}}\left(\stackrel{⌣}{x},t\right)L\left(\widehat{x},t\right)=-{\gamma }^2{\left(h_2\left(x\right)-h_2\left(\widehat{x}\right)\right)}^T,x,\widehat{x}\in N.$

(12.17)

Then:

(i)  there exists locally a Nash-equilibrium solution (w^⋆, L^⋆) for the game (12.7), (12.8), (12.6);

(ii)  the augmented system (12.6) is dissipative with respect to the supply-rate $s(w,\stackrel{⌣}{z})=\frac{1}{2}({\gamma }^2\parallel w{\parallel }^2-\parallel \stackrel{⌣}{z}{\parallel }^2)$ and hence has finite $\mathcal{L}$₂-gain from w to $\stackrel{⌣}{z}$ less or equal to γ;

(iii)  the optimal costs or performance objectives of the game are $J_1^{\star }(L^{\star },w^{\star })=Y({\stackrel{⌣}{x}}_0,t_0)and{J}_2^{\star }(L^{\star },w^{\star })=V({\stackrel{⌣}{x}}_0,t_0);$

(iv)  the filter Σ^af with the gain matrix $L(\widehat{x},t)$ satisfying (12.17) solves the infinite-horizon MH2HINLF P for the system locally in N.

Proof: Assume there exist definite solutions Y, V to the HJIEs (12.15)-(12.16), and (i) consider the Hamiltonian function H₁(., ., ., .) first:

$\begin{array}{l}{H}_1\left(\stackrel{⌣}{x},w,L,Y_{\stackrel{⌣}{x}}^T\right)=Y_{x}f\left(x\right)+Y_{\widehat{x}}f\left(\widehat{x}\right)+Y_{\widehat{x}}L\left(\widehat{x},t\right)\left(h_2\left(x\right)-h_2(\widehat{x})\right)+Y_x{g}_1\left(x\right)w+\hfill \\ Y_{\widehat{x}}L\left(\widehat{x},t\right)k_{21}\left(x\right)w-\frac{1}{2}\Vert \stackrel{⌣}{z}{\Vert }^2+\frac{1}{2}{\gamma }^2\Vert w{\Vert }^2\hfill \end{array}$

where some of the arguments have been suppressed for brevity. Since it is quadratic and convex in w, we can apply the necessary condition for optimality, i.e.,

${\frac{\partial H_1}{\partial w}|}_{w=w^{\star }}=0,$

to get

$w^{\star }:=-\frac{1}{{\gamma }^2}\left(g_1^T\left(x\right)Y_x^T\left(\stackrel{⌣}{x},t\right)+k_{21}^T\left(x\right)L^T\left(\text{}\widehat{x},t\right)Y_{\widehat{x}}{}^T\left(\text{}\stackrel{⌣}{x},t\right)\right).$

(12.18)

Substituting now w^⋆ in the expression for H₂(., ., ., .) (12.12), we get

$\begin{array}{l}{H}_2\left(\stackrel{⌣}{x},w^{\star },L,V_{\stackrel{⌣}{x}}^T\right)=V_{x}f\left(x\right)+V_{\widehat{x}}f\left(\widehat{x}\right)+V_{\widehat{x}}L\left(\widehat{x},t\right)\left(h_2\left(x\right)-h_2\left(\widehat{x}\right)\right)-\frac{1}{{\gamma }^2}{V}_x{g}_1\left(x\right)g_1^T\left(x\right)Y_x^T-\hfill \\ \frac{1}{{\gamma }^2}{V}_{\widehat{x}}L\left(\widehat{x},t\right)L^T\left(\widehat{x},t\right)Y_{\widehat{x}}^T+\frac{1}{2}{\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right)}^T\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right).\hfill \end{array}$

Then completing the squares for L in the above expression for H₂(., ., ., .), we get

$\begin{array}{l}{H}_2\left(\stackrel{⌣}{x},w^{\star },L,V_{\stackrel{⌣}{x}}^T\right)=V_{x}f\left(x\right)+V_{\widehat{x}}f\left(\widehat{x}\right)-\frac{1}{{\gamma }^{\text{2}}}{V}_x{g}_1\left(x\right)g_1^T\left(x\right)Y_x^T+\\ \frac{1}{2}{\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right)}^T\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right)+\\ \frac{1}{2{\gamma }^{\text{2}}}{\Vert L^T\left(\widehat{x},t\right)V_{\widehat{x}}^T+{\gamma }^{\text{2}}\left(h_2\left(x\right)-h_2\left(\widehat{x}\right)\right)\Vert }^2-\frac{{\gamma }^{\text{2}}}{2}{\Vert h_2\left(x\right)-h_2\left(\widehat{x}\right)\Vert }^2-\\ \frac{1}{2{\gamma }^2}{\Vert L^T\left(\widehat{x},t\right)V_{\widehat{x}}^T+L^T\left(\widehat{x},t\right)Y_{\widehat{x}}^T\Vert }^2+\frac{1}{2{\gamma }^{\text{2}}}{\Vert L^T\left(\widehat{x},t\right)Y_{\widehat{x}}^T\Vert }^2.\end{array}$

Thus, choosing L^⋆ according to (12.17) minimizes H₂(., ., ., .) and renders the Nash-equilibrium condition

$H_2(w^{\star },L^{\star })\le H_2(w^{\star },L)\forall L\in {\mathcal{M}}^{n\times m}$

satisfied. Moreover, substituting (w^⋆, L^⋆ ) in (12.14) gives the HJIE (12.16).

Next, substitute L^⋆ as given by (12.17) in the expression for H₁(., ., ., .) and complete the squares to obtain:

$\begin{array}{l}{H}_1\left(\stackrel{⌣}{x},w,L^{\star },Y_{\stackrel{⌣}{x}}^T\right)=Y_{x}f\left(x\right)+Y_{\widehat{x}}f\left(\widehat{x}\right)-\frac{1}{{\gamma }^2}{Y}_{\widehat{x}}L\left(\widehat{x},t\right)L^T\left(\widehat{x},t\right)V_{\widehat{x}}^T-\hfill \\ \frac{1}{2{\gamma }^2}{Y}_x{g}_1\left(x\right)g_1^T(x)Y_x^T-\frac{1}{2{\gamma }^2}{Y}_{\widehat{x}}L\left(\widehat{x}\right)L^T\left(\widehat{x}\right)Y_{\widehat{x}}^T+\hfill \\ \frac{{\gamma }^{\text{2}}}{2}\Vert w+\frac{1}{{\gamma }^{\text{2}}}{g}_1^T\left(x\right)Y_x^T+\frac{1}{{\gamma }^2}{k}_{21}^T\left(x\right)L^T(\widehat{x},t)Y_{\widehat{x}}^T{\Vert }^2-\frac{1}{2}\Vert z{\Vert }^2.\hfill \end{array}$

Substituting now w = w^⋆ as given by (12.18), we see that the second Nash-equilibrium condition

$H_1\left(w^{\star },L^{\star }\right)\le H_1\left(w,L^{\star }\right),\forall w\in \mathcal{W}$

is also satisfied. Therefore, the pair (w^⋆, L^⋆ ) constitute a Nash-equilibrium solution to the two-player nonzero-sum dynamic game. Moreover, substituting (w^⋆, L^⋆ ) in (12.13) gives the HJIE (12.15).

(ii) Consider the HJIE (12.15) and rewrite as:

$\begin{array}{l}\{Y_t\left(\stackrel{⌣}{x},t\right)+Y_{x}f\left(x\right)+Y_{\widehat{x}}f\left(\widehat{x}\right)+Y_{\widehat{x}}L\left(\widehat{x},t\right)(h_2(x)-h_2\left(\widehat{x}\right)+Y_x{g}_1\left(x\right)w+\\ Y_{\widehat{x}}L\left(\widehat{x},t\right)k_{21}\left(x\right)w-\frac{1}{2}{\Vert \stackrel{⌣}{z}\Vert }^2+\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2\}-\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2-Y_x{g}_1\left(x\right)w-\\ Y_{\widehat{x}}L\left(\widehat{x},t\right)k_{21}\left(x\right)w-\frac{1}{2{\gamma }^2}{Y}_x{g}_1\left(x\right)g_1^T\left(x\right)Y_x^T-\frac{1}{2{\gamma }^2}{Y}_{\widehat{x}}L\left(\widehat{x},t\right)L^T\left(\widehat{x},t\right)Y_{\widehat{x}}^T=0\\ \iff \left\{Y_t\left(\stackrel{⌣}{x},t\right)+Y_{\widehat{x}}\left[\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)+\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)w\right]+\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2-\frac{1}{2}{\Vert \stackrel{⌣}{z}\Vert }^2\right\}-\frac{1}{2}{\gamma }^2{\Vert w-w^{\star }\Vert }^2=0\\ \Rightarrow \left\{{\stackrel{⌣}{Y}}_t\left(\stackrel{⌣}{x},t\right)+{\stackrel{⌣}{Y}}_{\stackrel{⌣}{x}}\left[\stackrel{⌣}{f}\left(\stackrel{⌣}{x}\right)+\stackrel{⌣}{g}\left(\stackrel{⌣}{x}\right)w\right]\right\}\le \frac{1}{2}{\gamma }^2{\Vert w\Vert }^2-\frac{1}{2}{\Vert \stackrel{⌣}{z}\Vert }^2\end{array}$

(12.19)

for some function $\stackrel{⌣}{Y}=-Y>0.$ Integrating now the last expression above from t = t₀ and $\stackrel{⌣}{x}$(t₀) to t = T and $\stackrel{⌣}{x}$(T), we get the dissipation-inequality:

$\stackrel{⌣}{Y}\left(\stackrel{⌣}{x}\left(T\right),T\right)-\stackrel{⌣}{Y}\left(\stackrel{⌣}{x}\left(t_0\right),t_0\right)\le \frac{1}{2}{\displaystyle {\int }_{t_0}^T\left\{{\gamma }^2{\Vert w\left(t\right)\Vert }^2-{\Vert \stackrel{⌣}{z}\left(t\right)\Vert }^2\right\}dt,}$

(12.20)

and hence the result.

(iii) Consider the cost functional J₁(L, w) first, and rewrite it as

$\begin{array}{l}{J}_1\left(L,w\right)+Y\left(\stackrel{⌣}{x}\left(T\right),T\right)-Y\left(\stackrel{⌣}{x}\left(t_0\right),t_0\right)\\ ={\displaystyle {\int }_{t_0}^T\left\{\frac{1}{2}{\gamma }^2{\Vert w\left(t\right)\Vert }^2-\frac{1}{2}{\Vert \stackrel{⌣}{z}(t)\Vert }^2+\dot{Y}\left(\stackrel{⌣}{x},t\right)\right\}dt}\\ ={\displaystyle {\int }_{t_0}^T{H}_1\left(\stackrel{⌣}{x},w,L,Y_{\stackrel{⌣}{x}}^T\right)dt}\\ ={\displaystyle {\int }_{t_0}^T\left\{\frac{{\gamma }^2}{2}{\Vert w-w^{\star }\Vert }^2+Y_{\widehat{x}}L\left(\widehat{x},t\right)\left(h_2\left(x\right)-h_2\left(\widehat{x}\right)\right)+\frac{1}{{\gamma }^2}{Y}_{\widehat{x}}L\left(\widehat{x},t\right)L^T\left(\widehat{x},t\right)V_{\widehat{x}}^T\right\}dt,}\end{array}$

where use has been made of the HJIE (12.15) in the above manipulations. Substitute now (L^⋆, w^⋆) as given by (12.17), (12.18) respectively to get the result.

Similarly, consider the cost functional J₂(L, w) and rewrite it as

$\begin{array}{l}{J}_2\left(L,w\right)-V\left(\stackrel{⌣}{x}\left(t_0\right),t_0\right)=\underset{t_0}{\overset{T}{{\displaystyle \int }}}\left\{\frac{1}{2}\Vert \stackrel{⌣}{z}(t){\Vert }^2+\dot{V}\left(\stackrel{⌣}{x},t\right)\right\}dt\hfill \\ =\underset{t_0}{\overset{T}{{\displaystyle \int }}}{H}_2\left(\stackrel{⌣}{x},w,L,Y_{\stackrel{⌣}{x}}^T\right)dt.\hfill \end{array}$

Then substituting (L^⋆, w^⋆) as given by (12.17), (12.18) respectively, and using the HJIE (12.16) the result also follows.

Finally, (iv) follows from (i)-(iii). □

Remark 12.1.3 Notice that by virtue of (12.17), the HJIE (12.16) can also be represented in the following form:

$\begin{array}{l}{V}_t\left(\stackrel{⌣}{x},t\right)+V_x\left(\stackrel{⌣}{x},t\right)f\left(x\right)+V_{\widehat{x}}\left(\stackrel{⌣}{x},t\right)f\left(\widehat{x}\right)-\frac{1}{{\gamma }^2}{V}_x\left(\stackrel{⌣}{x},t\right)g_1\left(x\right)g_1^T\left(x\right)Y_x^T\left(\stackrel{⌣}{x},t\right)-\\ \frac{1}{{\gamma }^2}{V}_{\widehat{x}}\left(\stackrel{⌣}{x},t\right)L\left(\widehat{x},t\right)L^T\left(\widehat{x},t\right)Y_{\widehat{x}}^T\left(\widehat{x},t\right)-\frac{1}{\gamma }{V}_{\widehat{x}}\left(\stackrel{⌣}{x},t\right)L\left(\widehat{x},t\right)L^T\left(\widehat{x},t\right)V_{\widehat{x}}^T\left(\stackrel{⌣}{x},t\right)+\\ \frac{1}{2}{\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right)}^T\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right)=0,V\left(\stackrel{⌣}{x},T\right)=0.\end{array}$

(12.21)

The above result (Theorem 12.1.2) can be specialized to the linear-time-invariant (LTI) system:

${\sum }^l:\{\begin{array}{l}\dot{x}=Ax+G_{1}w,x\left(t_0\right)=x_0\\ \stackrel{⌣}{z}=C_1\left(x-\widehat{x}\right)\\ y=C_{2}x+D_{21}w,\end{array}$

(12.22)

where all the variables have their previous meanings, and $A\in {\Re }^{n\times n},G_1\in {\Re }^{n\times r},C_1\in {\Re }^{s\times n},C_2\in {\Re }^{m\times n}$ and $D_{21}\in {\Re }^{m\times r}$ are real constant matrices. We have the following corollary to Theorem 12.1.2.

Corollary 12.1.1 Consider the LTI system Σ^l defined by (12.22) and the MH2HINLFP for it. Suppose C₁ is full-column rank and A is Hurwitz. Suppose further, there exist a negative and a positive-definite real-symmetric solutions P₁(t), P₂(t), t ∈ [t₀, T ] (respectively) to the coupled Riccati ordinary differential-equations (ODEs):

$\begin{array}{l}-{\dot{P}}_1\left(t\right)=A^T{P}_1\left(t\right)+P_1\left(t\right)A-\\ \frac{1}{{\gamma }^2}\left[\begin{array}{cc}{P}_1\left(t\right)& P_2\left(t\right)\end{array}\right]\left[\begin{array}{cc}L\left(t\right)L^T(t)+G_1{G}_1^T& L\left(t\right)L^T\left(t\right)\\ L\left(t\right)L^T\left(t\right)& 0\end{array}\right]\left[\begin{array}{l}{P}_1\left(t\right)\\ p_2\left(t\right)\end{array}\right]-\\ C_1^T{C}_1,P_1\left(T\right)=0,\end{array}$

(12.23)

$\begin{array}{l}-{\dot{P}}_2\left(t\right)=A^T{P}_2\left(t\right)+P_2\left(t\right)A-\\ \frac{1}{{\gamma }^2}\left[P_1\left(t\right)+P_2\left(t\right)\right]\left[\begin{array}{cc}0& L\left(t\right)L^T\left(t\right)+G_1{G}_1^T\\ L\left(t\right)L^T\left(t\right)+G_1{G}_1^T& 2L\left(t\right)L^T\left(t\right)\end{array}\right]\left[\begin{array}{l}{P}_1\left(t\right)\\ P_2\left(t\right)\end{array}\right]+\\ C_1^T{C}_1,P_2\left(T\right)=0,\end{array}$

(12.24)

together with the coupling condition

$P_2\left(t\right)L\left(t\right)=-{\gamma }^2{C}_2^T,$

(12.25)

for some n × m matrix function L(t) defined for all t ∈ [t₀, T ]. Then:

(i)  there exists a Nash-equilibrium solution (w^⋆, L^⋆) for the game given by

$\begin{array}{l}{w}^{\star }:=-\frac{1}{{\gamma }^2}\left(G_1^T+D_{21}^T\left(t\right)\right)P_1\left(t\right)\left(x-\widehat{x}\right),\hfill \\ {\left(x-\widehat{x}\right)}^T{P}_2\left(t\right)L^{\star }=-{\gamma }^2{\left(x-\widehat{x}\right)}^T{C}_2^T;\hfill \end{array}$

(ii)  the augmented system

${\sum }^{fl}:\{\begin{array}{l}\dot{\stackrel{⌣}{x}}=\left[\begin{array}{cc}A& 0\\ L^{\star }\left(t\right)C_2& A-L^{\star }{C}_2\end{array}\right]\stackrel{⌣}{x}+\left[\begin{array}{c}{G}_1\\ L^{\star }\left(t\right)D_{21}\end{array}\right]w,\\ \stackrel{⌣}{x}\left(t_0\right)=\left[\begin{array}{l}{x}_0\\ {\widehat{x}}_0\end{array}\right]\\ \stackrel{⌣}{z}=\left[C_1-C_1\right]\stackrel{⌣}{x}:=\stackrel{⌣}{C}\stackrel{⌣}{x}\end{array}$

has ${\mathcal{H}}_{\infty }$-norm from w to $\stackrel{⌣}{z}$ less than equal to γ

(iii)  the optimal costs or performance objectives of the game are $J_1^{\star }(L^{\star },w^{\star })=\frac{1}{2}{\left(x_0-{\widehat{x}}_0\right)}^T{P}_1(t_0)(x_0-{\widehat{x}}_0)and{J}_2^{\star }(L^{\star },w^{\star })=\frac{1}{2}{\left(x_0-{\widehat{x}}_0\right)}^T{P}_2(t_0)(x_0-{\widehat{x}}_0);$

(iv)  the filter Σ^fl with the gain matrix L(t) = L^⋆ (t) satisfying (12.25) solves the finite-horizon MH2HINLFP for the system.

Proof: Take

$\begin{array}{l}Y\left(\stackrel{⌣}{x},t\right)=\frac{1}{2}{\left(x-\widehat{x}\right)}^T{P}_1\left(t\right)\left(x-\widehat{x}\right),P_1=P_1^T<0,\hfill \\ V\left(\stackrel{⌣}{x},t\right)=\frac{1}{2}{\left(x-\widehat{x}\right)}^T{P}_2\left(t\right)\left(x-\widehat{x}\right),P_2=P_2^T>0,\hfill \end{array}$

and apply the result of the theorem. □

12.1.2  Solution to the Infinite-Horizon Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ Nonlinear Filtering

In this section, we discuss the infinite-horizon filtering problem, in which case we let T → ∞. In this case, we seek a time-independent gain matrix $\widehat{L}(\widehat{x})$ and functions $Y,V:\widehat{N}\times \widehat{N}\subset \mathcal{X}\times \mathcal{X}\to \Re$ such that the HJIEs:

$\begin{array}{l}{Y}_x\left(\stackrel{⌣}{x}\right)f\left(x\right)+Y_{\widehat{x}}f\left(\widehat{x}\right)-\frac{1}{2{\gamma }^2}{Y}_x\left(\stackrel{⌣}{x}\right)g_1\left(x\right)g_1^T\left(x\right)Y_x^T\left(\stackrel{⌣}{x}\right)-\frac{1}{2{\gamma }^2}{Y}_{\widehat{x}}\left(\stackrel{⌣}{x}\right)\widehat{L}\left(\widehat{x}\right){\widehat{L}}^T\left(\widehat{x}\right)Y_{\widehat{x}}^T\left(\stackrel{⌣}{x}\right)-\\ \frac{1}{{\gamma }^{\text{2}}}{Y}_{\widehat{x}}\left(\stackrel{⌣}{x}\right)\widehat{L}\left(\widehat{x}\right){\widehat{L}}^T\left(\widehat{x}\right)V_{\widehat{x}}^T\left(\stackrel{⌣}{x}\right)-\frac{1}{2}{\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right)}^T\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right)=0,\\ Y\left(0\right)=0\end{array}$

(12.26)

$\begin{array}{l}{V}_x\left(\stackrel{⌣}{x}\right)f\left(x\right)+V_{\widehat{x}}f\left(\stackrel{⌣}{x}\right)-\frac{1}{{\gamma }^2}{V}_x\left(\stackrel{⌣}{x}\right)g_1\left(x\right)g_1^T\left(x\right)Y_x^T\left(\stackrel{⌣}{x}\right)-\frac{1}{{\gamma }^2}{V}_{\widehat{x}}\left(\stackrel{⌣}{x}\right)\widehat{L}\left(\widehat{x}\right){\widehat{L}}^T\left(\widehat{x}\right)Y_{\widehat{x}}^T\left(\stackrel{⌣}{x}\right)-\\ {\gamma }^{\text{2}}{\left(h_2\left(x\right)-h_2\left(\widehat{x}\right)\right)}^T\left(h_2\left(x\right)-h_2\left(\widehat{x}\right)\right)+\frac{1}{2}{\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right)}^T\left(h_1\left(x\right)-h_1\left(\widehat{x}\right)\right)=0,\\ V\left(0\right)=0\end{array}$

(12.27)

are satisfied together with the coupling condition:

$V_{\widehat{x}}\left(\stackrel{⌣}{x}\right)\widehat{L}\left(\widehat{x}\right)=-{\gamma }^{\text{2}}{\left(h_2\left(x\right)-h_2\left(\widehat{x}\right)\right)}^T,x,\widehat{x}\in \widehat{N},$

(12.28)

where $\widehat{L}$ is the asymptotic value of L. It is also required in this case that the augmented system (12.6) is stable. Moreover, in this case, we can relax the requirement of asymptotic-stability for the original system (12.3) with a milder requirement of detectability which we define next.

Definition 12.1.2 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(t) is a trajectory of $\dot{x}(t)=f(x)$ satisfying $x(t_0)\in \mathcal{O}$, then h(x(t)) is defined for all t ≥ t₀ and h(x(t)) ≡ 0 for all t ≥ t_s, for some t_s ≥ t₀, implies lim_t→∞ x(t) = 0. Moreover, the system is zero-state detectable if $\mathcal{O}=\mathcal{X}$.

Remark 12.1.4 From the above definition, it can be inferred that, if h₁ is injective, then {f,h₁} zero-state detectable ⇒ {$\stackrel{⌣}{f}$,$\stackrel{⌣}{h}$ } zero-state detectable and conversely.

It is also desirable for a filter to be stable or admissible. The “admissibility” of a filter is defined as follows.

Definition 12.1.3 A filter F is admissible if it is asymptotically (or internally) stable for any given initial condition x(t₀) of the plant Σ^a, and with w ≡ 0

$\underset{t\to \infty }{\lim }\stackrel{⌣}{z}(t)=0.$

The following proposition can now be proven along the same lines as Theorem 12.1.2.

Proposition 12.1.1 Consider the nonlinear system (12.3) and the infinite-horizon MH2HINLFP for it. Suppose the function h₁ is one-to-one (or injective) and the plant Σ^a is zero-state detectable. Further, suppose there exists a pair of C¹ negative and positive-definite functions $Y,V:\widehat{N}\times \widehat{N}\to \Re$ respectively, locally defined in a neighborhood $\widehat{N}\times \widehat{N}\subset \mathcal{X}\times \mathcal{X}$ of the origin $\stackrel{⌣}{x}=0$, and a matrix function $\widehat{L}:\widehat{N}\to {\mathcal{M}}^{n\times m}$ satisfying the pair of coupled HJIEs (12.26), (12.27) together with (12.28). Then:

(i)  there exists locally a Nash-equilibrium solution $({\widehat{w}}^{\star },{\widehat{L}}^{\star })$ for the game;

(ii)  the augmented system (12.6) is dissipative with respect to the supply-rate $s(w,z)=\frac{1}{2}({\gamma }^2\parallel w{\parallel }^2-\parallel \stackrel{⌣}{z}{\parallel }^2)$ and hence has $\mathcal{L}$₂-gain from w to $\stackrel{⌣}{z}$ less or equal to γ;

(iii)  the optimal costs or performance objectives of the game are $J_1^{\star }({\widehat{L}}^{\star },{\widehat{w}}^{\star })=Y({\stackrel{⌣}{x}}_0)and{J}_2^{\star }({\widehat{L}}^{\star },{\widehat{w}}^{\star })=V({\stackrel{⌣}{x}}_0);$

(iv)  the filter Σ^af with the gain matrix $\widehat{L}(\widehat{x})={\widehat{L}}^{\star }(\widehat{x})$ satisfying (12.28) is admissible and solves the infinite-horizon MH2HINLFP locally in $\widehat{N}$.

Proof: Since the proof of items (i)-(iii) is the same as in Theorem 12.1.2, we only prove (iv) here. Using similar manipulations as in the proof of Theorem 12.1.2, we get an inequality similar to (12.19). This inequality implies that with w = 0,

$\dot{\stackrel{⌣}{Y}}\le -\frac{1}{2}{\Vert \stackrel{⌣}{z}\Vert }^2.$

(12.29)

Therefore, by Lyapunov’s theorem, the augmented system is locally stable. Furthermore, for any trajectory of the system $\stackrel{⌣}{x}(t)$ such that$\dot{\stackrel{⌣}{Y}}(\stackrel{⌣}{x})\equiv 0$for all t ≥ t_c for some t_c ≥ t₀, it implies that $\stackrel{⌣}{z}\equiv 0\forall \ge t_c$. This in turn implies $h_1(x)=h_1(\widehat{x})$, and $x(t)=\widehat{x}(t)\forall t\ge t_c$ by the injectivity of h₁. This further implies that $h_2(x)=h_2(\widehat{x})\forall t\ge t_c$ and it is a trajectory of the free system:

$\dot{\stackrel{⌣}{x}}=\left(\begin{array}{l}f\left(x\right)\\ f\left(\widehat{x}\right)\end{array}\right).$

By the zero-state detectability of {f, h₁}, we have ${\lim }_{t\to \infty }\stackrel{⌣}{x}(t)=0,$ and asymptotic-stability follows by LaSalle’s invariance-principle. On the other hand, if we have strict inequality, $\dot{\stackrel{⌣}{Y}}<-\frac{1}{2}\parallel \stackrel{⌣}{z}{\parallel }^2$, asymptotic-stability follows immediately from Lyapunov’s theorem, and lim_t→∞ z(t) = 0. Therefore, Σ^af is admissible. Combining now with items (i)-(iii), the conclusion follows. □

Similarly, for the linear system Σ^l (12.22), we have the following corollary.

Corollary 12.1.2 Consider the LTI system Σ^l defined by (12.22) and the MH2HINLFP for it. Suppose C₁ is full column rank and (A, C₁) is detectable. Suppose further, there exist a negative and a positive-definite real-symmetric solutions P₁, P₂, (respectively) to the coupled algebraic-Riccati equations (AREs):

$A^T{P}_1+P_{1}A-\frac{1}{{\gamma }^{\text{2}}}\left[\begin{array}{cc}{P}_1& P_2\end{array}\right]\left[\begin{array}{cc}\widehat{L}{\widehat{L}}^T+G_1{G}_1^T& \widehat{L}{\widehat{L}}^T\\ \widehat{L}{\widehat{L}}^T& 0\end{array}\right]\left[\begin{array}{l}{P}_1\\ P_2\end{array}\right]-C_1^T{C}_1=0$

(12.30)

$A^T{P}_2+P_{2}A-\frac{1}{{\gamma }^{\text{2}}}\left[\begin{array}{cc}{P}_1& P_2\end{array}\right]\left[\begin{array}{cc}0& \widehat{L}{\widehat{L}}^T+G_1{G}_1^T\\ \widehat{L}{\widehat{L}}^T+G_1{G}_1^T& 2\widehat{L}{\widehat{L}}^T\end{array}\right]\left[\begin{array}{l}{P}_1\\ P_2\end{array}\right]+C_1^T{C}_1=0,$

(12.31)

together with the coupling-condition

$P_2\widehat{L}=-{\gamma }^2{C}_2^T.$

(12.32)

Then:

(i)  there exists a Nash-equilibrium solution (w^⋆, L^⋆ ) for the game;

(ii)  the augmented system Σ^lf has ${\mathcal{H}}_{\mathrm{\infty }}$-norm from w to $\stackrel{⌣}{z}$ less than equal to γ;

(iii)  the optimal costs or performance objectives of the game are $J_1^{\star }({\widehat{L}}^{\star },{\widehat{w}}^{\star })=\frac{1}{2}{\left(x_0-{\widehat{x}}_0\right)}^T{P}_1(x_0-{\widehat{x}}_0)and{J}_2^{\star }({\widehat{L}}^{\star },{\widehat{w}}^{\star })=\frac{1}{2}{\left(x_0-{\widehat{x}}_0\right)}^T{P}_2(x_0-{\widehat{x}}_0);$

(iv)  the filter Σ_lf with gain-matrix $\widehat{L}={\widehat{L}}^{\star }\in {\Re }^{n\times m}$ satisfying (12.32) is admissible, and solves the infinite-horizon MH2HINLFP for the linear system.

We consider a simple example.

Example 12.1.1 We consider a simple example of the following scalar system:

$\begin{array}{l}\dot{x}=-x^3,x\left(0\right)=x_{0,}\hfill \\ z=x\hfill \\ y=x+w.\hfill \end{array}$

We consider the infinite-horizon problem and the associated HJIEs. It can be seen that the system satisfies all the assumptions of Theorem 12.1.2 and Proposition 12.1.1. Then, substituting in the HJIEs (12.21), (12.26), and coupling condition (12.28), we get

$\begin{array}{l}-x^3{y}_x-{\widehat{x}}^3{y}_{\widehat{x}}-\frac{1}{2{\gamma }^2}{l}^2{y}_{\widehat{x}}^2-\frac{1}{{\gamma }^2}{l}^2{\upsilon }_{\widehat{x}}{y}_{\widehat{x}}-\frac{1}{2}{\left(x-\widehat{x}\right)}^2=0,\\ -x^3{\upsilon }_x-{\widehat{x}}^3{\upsilon }_{\widehat{x}}-\frac{1}{{\gamma }^{\text{2}}}{l}^2{\upsilon }_{\widehat{x}}{y}_{\widehat{x}}-{\gamma }^2{\left(x-\widehat{x}\right)}^2+\frac{1}{2}{\left(x-\widehat{x}\right)}^2=0,\\ {\upsilon }_{\widehat{x}}l+{\gamma }^2\left(x-\widehat{x}\right)=0.\end{array}$

Looking at the above system of PDEs, we see that there are 5 variables: v_x, ${\upsilon }_{\widehat{x}}$, y_x, ${\upsilon }_{\widehat{x}}$, l and only 3 equations. Therefore, we make the following simplifying assumption. Let

${\upsilon }_x=-{\upsilon }_{\widehat{x}}.$

Then, the above equations reduce to

$x^3{y}_x+{\widehat{x}}^3{y}_{\widehat{x}}+\frac{1}{2{\gamma }^2}{l}^2{y}_{\widehat{x}}^2-\frac{1}{{\gamma }^2}{l}^2{\upsilon }_x{y}_{\widehat{x}}+\frac{1}{2}{\left(x-\widehat{x}\right)}^2=0,$

(12.33)

$x^3{\upsilon }_x+{\widehat{x}}^3{\upsilon }_x-\frac{1}{{\gamma }^2}{l}^2{\upsilon }_x{y}_{\widehat{x}}+{\gamma }^2{\left(x-\widehat{x}\right)}^2-\frac{1}{2}{\left(x-\widehat{x}\right)}^2=0,$

(12.34)

${\upsilon }_{x}l-{\gamma }^2\left(x-\widehat{x}\right)=0.$

(12.35)

FIGURE 12.2
Nonlinear ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$-Filter Performance with Unknown Initial Condition; Reprinted from Int. J. of Robust and Nonlinear Control, vol. 19, no. 4, pp. 394-417, © 2009, “Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ nonlinear filtering,” by M. D.S. Aliyu and E. K. Boukas, with permission from Wiley Blackwell.

Subtract now equations (12.33) and (12.34) to get

$\left(x^3{y}_x+{\widehat{x}}^3{y}_{\widehat{x}}\right)+\left({\widehat{x}}^3-x^3\right){\upsilon }_x+\frac{1}{2{\gamma }^2}{l}^2{y}_{\widehat{x}}^2+\left(1-{\gamma }^2\right){\left(x-\widehat{x}\right)}^2=0,$

(12.36)

${\upsilon }_{x}l-{\gamma }^2\left(x-\widehat{x}\right)=0.$

(12.37)

Now let

${\upsilon }_x=\left(x-\widehat{x}\right)\Rightarrow {\upsilon }_{\widehat{x}}=-\left(x-\widehat{x}\right),\upsilon \left(x,\widehat{x}\right)=\frac{1}{2}{\left(x-\widehat{x}\right)}^2,and\Rightarrow l_{2/\infty }^{\star }={\gamma }^2.$

Then, if we let γ = 1, the equation governing y_x (12.36) becomes

$y_{\widehat{x}}^2+2\left(x^3{y}_x+{\widehat{x}}^3{y}_{\widehat{x}}\right)+2\left({\widehat{x}}^3-x^3\right)\left(x-\widehat{x}\right)=0$

and

$y_x=-\left(x+\widehat{x}\right),y_{\widehat{x}}=-\left(x+\widehat{x}\right),$

approximately solves the above PDE (locally!). This corresponds to the solution

$y\left(x,\widehat{x}\right)=-\frac{1}{2}{\left(x+\widehat{x}\right)}^2.$

Figure 12.2 shows the filter performance with unknown initial condition and with the measurement noise w(t) = 0.5w₀ + 0.01 sin(t), where w₀ is zero-mean Gaussian-white with unit variance.

12.1.3  Certainty-Equivalent Filters (CEFs)

It should be observed from the previous two sections, 12.1.1, 12.1.2, that the filter gains (12.17), (12.28) may depend on the original state of the system which is to be estimated, except for the linear case where the gains are constants. As discussed in Chapter 8, this will make the filters practically impossible to construct. Therefore, the filter equation and gains must be modified to overcome this difficulty.

Furthermore, we observe that the number of variables in the HJIEs (12.15)-(12.27) is twice the dimension of the plant. This makes them more difficult to solve considering their notoriety, and makes the scheme less attractive. Therefore, based on these observations, we consider again in this section certainty-equivalent filters which are practically implementable, and in which the governing equations are of lower-order. We begin with the following definition.

Definition 12.1.4 For the nonlinear system (12.3), we say that it is locally zero-input observable if for all states x₁, x₂ ∈ U ⊂ X and input w(.) ≡ 0

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

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 observable at each x₀ ∈ X or U=X.

Consider now as in Chapter 8, the following class of filters:

${\tilde{\sum }}^{af}:\{\begin{array}{l}\dot{\widehat{x}}=f\left(\widehat{x}\right)+g_1\left(\widehat{x}\right)w^{\star }+\tilde{L}\left(\widehat{x},y\right)\left[y-h_2\left(\widehat{x}\right)-k_{21}\left(\widehat{x}\right){\tilde{w}}^{\star }\right],\widehat{x}(t_0)-{\widehat{x}}_0\\ \widehat{z}=h_2\left(\widehat{x}\right)\\ \widehat{z}=y-h_2\left(\widehat{x}\right),\end{array}$

(12.38)

where $\tilde{L}(.,.)\in M^{n\times m}$ is the filter gain matrix, ${\tilde{w}}^{\star }$ is the estimated worst-case system noise (hence the name certainty-equivalent filter) and $\tilde{z}$ is the new penalty variable. Then, consider the infinite-horizon mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ dynamic game problem with the cost functionals (12.7), (12.8), and with the above filter. We can define the corresponding Hamiltonians ${\tilde{H}}_i:T^{\star }\mathcal{X}\times \mathcal{W}\times T^{\star }\times \mathcal{Y}\times {\mathcal{M}}^{n\times m}\to \Re ,i=1,2$ as

$\begin{array}{l}{\tilde{H}}_1\left(\widehat{x},w,y,\tilde{L},{\tilde{Y}}_{\widehat{x}}^T,{\tilde{Y}}_y^T\right)={\tilde{Y}}_{\widehat{x}}\left(\widehat{x},y\right)\left[f\left(\widehat{x}\right)+g_1\left(\widehat{x}\right)w+\tilde{L}\left(\widehat{x},y\right)\left(y-h_2\left(\widehat{x}\right)-k_{21}\left(\widehat{x}\right)w\right)\right]+\\ {\tilde{Y}}_Y\left(\widehat{x},y\right)\dot{y}+\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2-\frac{1}{2}{\Vert \tilde{z}\Vert }^2,\\ {\tilde{H}}_2\left(\widehat{x},w,y,\tilde{L},{\tilde{V}}_{\widehat{x}}^T,{\tilde{V}}_y^T\right)={\tilde{V}}_{\widehat{x}}\left(\widehat{x},y\right)\left[f\left(\widehat{x}\right)+g_1\left(\widehat{x}\right)w+\tilde{L}\left(\widehat{x},y\right)\left(y-h_2\left(\widehat{x}\right)-k_{21}\left(\widehat{x}\right)w\right)\right]+\\ {\tilde{V}}_Y\left(\widehat{x},y\right)\dot{y}+\frac{1}{2}{\Vert \tilde{z}\Vert }^2,\end{array}$

for some smooth functions $\tilde{V},\tilde{Y}:\mathcal{X}\times \mathcal{Y}\to \Re$, and where the adjoint variables are set as ${\tilde{p}}_1={\tilde{Y}}_{\widehat{x}}^T,{\tilde{p}}_2={\tilde{V}}_{\widehat{x}}^T.$ Further, we have

${\frac{\partial {\tilde{H}}_1}{\partial w}|}_{w={\tilde{w}}^{\star }}=0\Rightarrow {\tilde{w}}^{\star }=-\frac{1}{{\gamma }^2}{\left[g_1\left(\widehat{x}\right)-\tilde{L}\left(\widehat{x},y\right)k_{21}\left(\widehat{x}\right)\right]}^T{\tilde{Y}}_{\widehat{x}}^T\left(\widehat{x},y\right),$

and repeating similar derivation as in the previous section, we can arrive at the following result.

Theorem 12.1.3 Consider the nonlinear system (12.3) and the MH2HINLFP for it. Suppose the plant Σ^a is locally asymptotically-stable about the equilibrium point x = 0 and zero-input observable. Further, suppose there exists a pair of C¹ (with respect to both arguments) negative and positive-definite functions $\tilde{Y},\tilde{V}:\tilde{N}\times \Upsilon \to \Re$ respectively, locally defined in a neighborhood $\tilde{N}\times \Upsilon \subset \mathcal{X}\times \mathcal{Y}$ of the origin $(\widehat{x},y)=(0,0)$, and a matrix function $\tilde{L}:\tilde{N}\times \Upsilon \to {\mathcal{M}}^{n\times m}$ satisfying the following pair of coupled HJIEs:

$\begin{array}{l}{\tilde{Y}}_{\widehat{x}}\left(\widehat{x},y\right)f\left(\widehat{x}\right)+{\tilde{Y}}_y\left(\widehat{x},y\right)\dot{y}-\frac{1}{2{\gamma }^2}{\tilde{Y}}_{\widehat{x}}\left(\widehat{x},y\right)g_1\left(\widehat{x}\right)g_1^T\left(\widehat{x}\right){\tilde{Y}}_{\widehat{x}}^T\left(\widehat{x},y\right)-\\ \frac{1}{2{\gamma }^2}{\tilde{Y}}_{\widehat{x}}\left(\widehat{x},y\right)\tilde{L}\left(\widehat{x},y\right){\tilde{L}}^T\left(\widehat{x},y\right){\tilde{Y}}_{\widehat{x}}^T\left(\widehat{x},y\right)-\frac{1}{{\gamma }^2}{\tilde{Y}}_{\widehat{x}}\left(\widehat{x},y\right)\tilde{L}\left(\widehat{x},y\right){\tilde{L}}^T\left(\widehat{x},y\right){\tilde{V}}_{\widehat{x}}^T\left(\widehat{x},y\right)-\\ \frac{1}{2}\left(y-h_2\left(\widehat{x}\right)\right)\left(y-h_2\left(\widehat{x}\right)\right)=0,\tilde{Y}\left(0,0\right)=0,\widehat{x}\in \tilde{N},y\in \Upsilon \end{array}$

(12.39)

$\begin{array}{l}{\tilde{V}}_{\widehat{x}}\left(\widehat{x},y\right)f\left(\widehat{x}\right)+{\tilde{V}}_y\left(\widehat{x},y\right)\dot{y}-\frac{1}{{\gamma }^2}{\tilde{V}}_{\widehat{x}}\left(\widehat{x},y\right)g_1\left(\widehat{x}\right)g_1^T\left(\widehat{x}\right){\tilde{Y}}_{\widehat{x}}^T\left(\widehat{x},y\right)-\\ \frac{1}{{\gamma }^2}{\tilde{V}}_{\widehat{x}}\left(\widehat{x},y\right)\tilde{L}\left(\widehat{x},y\right){\tilde{L}}^T\left(\widehat{x},y\right){\tilde{Y}}_{\widehat{x}}^T\left(\widehat{x},y\right)-\frac{1}{{\gamma }^2}{\tilde{V}}_{\widehat{x}}\left(\widehat{x},y\right)\tilde{L}\left(\widehat{x},y\right){\tilde{L}}^T\left(\widehat{x},y\right){\tilde{V}}_{\widehat{x}}^T\left(\widehat{x},y\right)+\\ \frac{1}{2}\left(y-h_2{\left(\widehat{x}\right)}^T(y-h_2\left(\widehat{x}\right)\right)=0,\tilde{V}\left(0,0\right)=0,\widehat{x}\in \tilde{N},y\in \Upsilon \end{array}$

(12.40)

together with the coupling condition

${\tilde{V}}_{\widehat{x}}\left(\widehat{x},y\right)\tilde{L}\left(\widehat{x},y\right)=-{\gamma }^2{\left(y-h_2\left(\widehat{x}\right)\right)}^T,\widehat{x}\in \tilde{N},y\in \Upsilon .$

(12.41)

Then, the filter ${\tilde{\mathrm{\Sigma }}}^{af}$ with the gain matrix $\tilde{L}(\widehat{x},y)$ satisfying (12.41) solves the infinite-horizon MH2HINLFP for the system locally in $\tilde{N}$.

Proof: (Sketch) Using similar manipulations as in Theorem 12.1.2 and Prop 12.1.1, it can be shown that the existence of a solution to the coupled HJIEs (12.39), (12.40) implies the existence of a solution to the dissipation-inequality

$\tilde{Y}\left(\widehat{x},y\right)-\tilde{Y}\left({\widehat{x}}_0,y_0\right)\le \frac{1}{2}{\displaystyle {\int }_{t_0}^{\infty }\left({\gamma }^2{\Vert w\Vert }^2-{\Vert \tilde{z}\Vert }^2\right)dt}$

for some smooth function $\tilde{Y}=-\tilde{Y}>0$. Again using similar arguments as in Theorem 12.1.2 and Prop 12.1.1, the result follows. In particular, with w ≡ 0 and $\dot{\tilde{Y}}(\widehat{x},y)\equiv 0\Rightarrow \tilde{z}\equiv 0,$ which in turn implies $x\equiv \widehat{x}$ by the zero-input observability of the system. □

Remark 12.1.5 Comparing the HJIEs (12.39)-(12.40) with (12.26)-(12.27) we see that the dimensionality of the former is half. This is indeed significant. Moreover, the filter gain corresponding to the new HJIEs (12.41) does not depend on x.

12.2  Discrete-Time Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ Nonlinear Filtering

The set-up for this case is the same as the continuous-time case shown in Figure 12.1 with the difference that the variables and measurements are discrete, and is shown on Figure 12.3. Therefore, the plant is similarly described by an affine causal discrete-time nonlinear state-space system with zero input defined on an n-dimensional space X ⊆ ℜⁿ in coordinates x = (x₁, …, x_n):

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

(12.42)

where x ∈ $\mathcal{X}$ is the state vector, w ∈ $\mathcal{W}$ is the disturbance or noise signal, which belongs to the set $\mathcal{W}\subset {\mathcal{l}}_2([k_0,\infty ){\Re }^r)$ of admissible disturbances or noise signals, the output $y\in {\Re }^m$ is the measured output of the system, while $z\in {\Re }^s$ is the output to be estimated. The functions $f:\mathcal{X}\to \mathcal{X},g_1:\mathcal{X}\to {\mathcal{M}}^{n\times r}(\mathcal{X}),h_1:\mathcal{X}\to {\Re }^s,h_2:\mathcal{X}\to {\Re }^m,$ and $k_{12}:\mathcal{X}\to {\mathcal{M}}^{s\times p}(\mathcal{X}),k_{21}:\mathcal{X}\to {\mathcal{M}}^{m\times r}(\mathcal{X})$ are real C^∞ functions of x. Furthermore, we assume without any loss of generality that the system (12.42) has a unique equilibrium-point at x = 0 such that f(0) = 0, h₁(0) = h₂(0) = 0.

FIGURE 12.3
Set-Up for Discrete-Time Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ Nonlinear Filtering

The objective is again to synthesize a filter, F_k, for estimating the state x_k (or more generally a function of it z_k = h₁(x_k)) from observations of y_i up to time k and over a time horizon [k₀, K], i.e., from

${\text{Y}}_k\triangleq \left\{y_i:i\le k\right\},k\in \left[k_0,K\right]$

such that the gains (or induced norms) from the inputs w₀ := {w_0,k}, w₁ := {w_1,k}, to the error or penalty variable $\stackrel{⌣}{z}$ defined respectively as the l₂-norm and l_∞-norm of the interconnected system F_k ◦ Σ^da respectively, i.e.,

${\Vert {\mathcal{F}}_k\circ {\sum }^{da}\Vert }_{{\mathcal{l}}_2}\triangleq \underset{0\ne w_0\in s\prime }{\sup }\frac{\Vert \stackrel{⌣}{z}\Vert p\prime }{\Vert w_0\Vert s\prime }$

Figure 12.43)

and

${\Vert {\mathcal{F}}_k\circ {\sum }^{da}\Vert }_{{\mathcal{l}}_{\infty }}\triangleq \underset{0\ne w_1\in \mathcal{P}\prime }{\sup }\frac{{\Vert \stackrel{⌣}{z}\Vert }_2}{{\Vert w_1\Vert }_2}$

Figure 12.44)

are minimized, where

$\begin{array}{l}\mathcal{P}\prime \triangleq \{w:w\in {\mathcal{l}}_{\infty },R_{ww}\left(k\right),S_{ww}\left(j\omega \right)\text{exist for all }k\text{and all }\omega \text{ resp}\text{.,}\\ {\Vert w\Vert }_{\mathcal{P}\prime }<\infty \}\\ \mathcal{S}\prime \triangleq \{w:w\in {\mathcal{l}}_{\infty },R_{ww}\left(k\right),S_{ww}\left(j\omega \right)\text{exist for all }k\text{and all }\omega \text{ resp}\text{.,}\\ {\Vert S_{ww}\left(j\omega \right)\Vert }_{\infty }<\infty \}\end{array}$

$\begin{array}{l}{\Vert z\Vert }_{\mathcal{P}\prime }^2\triangleq \underset{K\to \infty }{\lim }\frac{1}{2K}{\displaystyle \sum _{k=-K}^K{\Vert z\Vert }^2}\\ {\Vert w_o\Vert }_{\mathcal{S}\prime }^2=\Vert S_{w_0{w}_0}\left(j\omega \right)\Vert \infty ,{\Vert w_0\Vert }_{\mathcal{S}\prime }^2=\Vert S_{w_0{w}_0}\left(j\omega \right)\Vert \infty ,\end{array}$

and R_ww, S_ww(jω) are the autocorrelation and power spectral-density matrices of w [152]. In addition, if the plant is stable, we replace the induced ℓ-norms above by their equivalent ℋ-subspace norms.

The above problem is the standard discrete-time mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$-filtering problem. However, as pointed out in the previous section, due to the difficulty of solving the above problem, we do not solve it either in this section. Instead, we solve the associated problem involving a single noise/disturbance signal $w\in \mathcal{W}\subset {\ell }_2[k_0,\infty ),$ and minimize the output energy of the system, ${\Vert \stackrel{⌣}{z}\Vert }_{{\ell }_2}$, while rendering ${\Vert {\mathcal{F}}_k\circ {\sum }^{da}\Vert }_{{\mathcal{l}}_{\infty }}\le {\gamma }^{\star }$ for a given number ${\gamma }^{\star }>0$, for all $w\in \mathcal{W}$ and for all initial conditions $x^0\in O\subset \mathcal{X}$. In addition, we also require that with w_k ≡ 0, the estimation error converges to zero, i.e., ${\lim }_{k\to \infty }{\stackrel{⌣}{z}}_k=0$ = 0.

Again, for the discrete-time nonlinear system (12.42), the above ${\mathcal{H}}_{\infty }$ constraint is interpreted as the ℓ₂-gain constraint and is represented as

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

(12.45)

for all $w\in \mathcal{W}$, for all initial states $x^0\in O\subset \mathcal{X}$.

The discrete-time mixed-${\mathcal{H}}_2$/${\mathcal{H}}_{\mathrm{\infty }}$ filtering or estimation problem can then be defined formally as follows.

Definition 12.2.1 (Discrete-Time Mixed ${\mathcal{H}}_2$/${\mathcal{H}}_{\infty }$ (Suboptimal) Nonlinear Filtering Problem (DMH2HINLFP)). Given the plant Σ^da and a number ${\gamma }^{\star }>0$, find a filter ${ℱ}_k:\mathcal{Y}\to \mathcal{X}$ such that

${\widehat{x}}_{k+1}={ℱ}_k({\mathrm{Y}}_k)$

and the output energy ${\Vert \stackrel{⌣}{z}\Vert }_{{\ell }_2}$ is minimized, while the constraint (12.45) is satisfied for all $\gamma \ge {\gamma }^{\star }$, for all $w\in \mathcal{W}$ and all x⁰ ∈ O. In addition, with w_k ≡ 0, we have ${\lim }_{k\to \infty }{\stackrel{⌣}{z}}_k=0$.

Moreover, if the above conditions are satisfied for all x⁰ ∈ X, we say that F_k solves the D M H 2 H I N L F P globally.

Remark 12.2.1 The problem defined above is the finite-horizon filtering problem. We have the infinite-horizon problem if we let K → ∞.