6.4    Output Measurement-Feedback ${\mathcal{H}}_{\infty }$-Control for a General Class of Nonlinear Systems

In this section, we look at the output measurement-feedback problem for a more general class of nonlinear systems. For this purpose, we consider the following class of nonlinear systems defined on a manifold $\mathcal{X}\subseteq {\Re }^n$ containing the origin in coordinates x = (x₁, …, x_n):

${\mathrm{\Sigma }}^g:\{\begin{array}{l}\dot{x}=F\left(x,w,u\right);x\left(t_0\right)=x_0\\ z=Z\left(x,u\right)\\ y=Y\left(x,w\right)\end{array}$

(6.52)

where all the variables have their previous meanings, while $F:\mathcal{X}\times \mathcal{W}\times \mathcal{U}\to \mathcal{X}$ is the state dynamics function, $Z:\mathcal{X}\times \mathcal{U}\to {\Re }^s$ is the controlled output function and $Y:\mathcal{X}\times \mathcal{W}\to {\Re }^m$ is the measurement output function. Moreover, the functions F (., ., .), Z(., .) and $\mathcal{Y}$ (., .) are smooth C^r, r ≥ 1 functions of their arguments, and the point x = 0 is a unique equilibrium-point for the system Σ^g such that F (0, 0, 0) = 0, Z(0, 0) = 0, $\mathcal{Y}$ (0, 0) = 0. The following assumptions will also be adopted in the sequel.

Assumption 6.4.1 The linearization of the function Z(x, u) is such that

$rank\left(D_{12}\right)\triangleq rank\left(\frac{\partial Z}{\partial u}\left(0,0\right)\right)=p.$

The control action to Σ^g is to be provided by a controller of the form (6.3). Motivated by the results of Section 6.1, we can conjecture a dynamic-controller of the form:

${\mathrm{\Sigma }}_e^{gc}:\{\begin{array}{l}\dot{\zeta }=F\left(\zeta ,w,u\right)+\tilde{G}\left(\zeta \right)\left(y-Y\left(\zeta ,w\right)\right)\hfill \\ u=\widetilde{{\alpha }_2}\left(\zeta \right)\hfill \end{array}$

(6.53)

where ζ is an estimate of x and $\tilde{G}(.)$ is the output-injection gain matrix which is to be determined.

Again, since w is not directly available, we use its worst-case value in equation (6.53). Denote this value by $w^{\ast }={\tilde{\alpha }}_1(x)$, where ${\tilde{\alpha }}_1(.)$ is determined according to the procedure explained in Chapter 5. Then, substituting this in (6.53) we obtain

$\{\begin{array}{l}\dot{\zeta }=F\left(\zeta ,{\tilde{\alpha }}_1\left(\zeta \right),{\tilde{\alpha }}_2\left(\zeta \right)\right)+\tilde{G}\left(\zeta \right)(y-Y\left(\zeta ,\tilde{\alpha }\left(\zeta \right)\right)\\ u={\tilde{\alpha }}_2\left(\zeta \right).\end{array}$

(6.54)

Now, let ${\tilde{x}}^e=\left(\begin{array}{l}x\\ \zeta \end{array}\right)$ so that the closed-loop system (6.52) and (6.54) is represented by

$\{\begin{array}{l}{\dot{\tilde{x}}}^e=F^e\left({\tilde{x}}^e,w\right)\\ z=Z^e\left({\tilde{x}}^e\right)\end{array}$

(6.55)

where

$\begin{array}{l}{F}^e\left({\tilde{x}}^e,w\right)=\left(\begin{array}{l}F\left(x,w,{\tilde{\alpha }}_2\left(\zeta \right)\right)\\ F\left(\zeta ,{\tilde{\alpha }}_1\left(\zeta \right),{\tilde{\alpha }}_2\left(\zeta \right)\right)+\tilde{G}\left(\zeta \right)Y\left(x,w\right)-\tilde{G}\left(\zeta \right)Y\left(\zeta ,{\tilde{\alpha }}_1\left(\zeta \right)\right)\end{array}\right),\\ Z^e\left({\tilde{x}}^e\right)=Z\left(x,{\tilde{\alpha }}_2\left(\zeta \right)\right).\end{array}$

The objective then is to render the above closed-loop system dissipative with respect to the supply-rate $s\left(w,Z\right)=\frac{1}{2}\left({\gamma }^2{\Vert w\Vert }^2-{\Vert Z^e\left(x^e\right)\Vert }^2\right)$ with some suitable storage-function and an appropriate output-injection gain matrix $\tilde{G}(.)$. To this end, we make the following assumption.

Assumption 6.4.2 Any bounded trajectory x(t) of the system (6.52)

$\dot{x}\left(t\right)=F\left(x\left(t\right),0,u\left(t\right)\right)$

satisfying

$Z\left(x\left(t\right),u\left(t\right)\right)\equiv 0$

for all t ≥ t₀, is such that lim_t→∞ x(t) = 0.

Then we have the following proposition.

Proposition 6.4.1 Consider the system (6.55) and suppose Assumptions 6.4.1, 6.4.2 hold. Suppose also the system

$\dot{\zeta }=F\left(\zeta ,{\tilde{\alpha }}_1\left(\zeta \right),0\right)-\tilde{G}\left(\zeta \right)Y\left(\zeta ,{\tilde{\alpha }}_1\left(\zeta \right)\right)$

(6.56)

has a locally asymptotically-stable equilibrium-point at ζ = 0, and there exists a locally defined C¹ positive-definite function $\tilde{U}:\mathcal{O}\times \mathcal{O}\to {\mathcal{\Re }}_{+},$ $\tilde{U}\left(0\right)$ = 0, O ⊂ X, satisfying the dissipation inequality

${\tilde{U}}_{{\tilde{x}}^e}\left({\tilde{x}}^e\right)F^e\left({\tilde{x}}^e,w\right)+\frac{1}{2}{\Vert Z^e\left({\tilde{x}}^e\right)\Vert }^2-\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2\le 0$

(6.57)

for w = 0. Then, the system (6.55) has a locally asymptotically-stable equilibrium-point at ${\tilde{x}}^e=0.$

Proof: Set w = 0 in the dissipation-inequality (6.57) and rewrite it as

$\dot{\tilde{U}}\left({\tilde{x}}^e\left(t\right)\right)={\tilde{U}}_{x^e}\left({\tilde{x}}^e\right)F^e\left({\tilde{x}}^e,w\right)=-\frac{1}{2}\Vert Z\left(x,{\tilde{\alpha }}_2\left(\zeta \right)\right)\le 0,$

which implies that ${\tilde{x}}^e=0.$ is stable for (6.55). Further, any bounded trajectory ${\tilde{x}}^e=\mathrm{(t)}$ of (6.55) with w = 0 resulting in $\dot{\tilde{U}}\left({\tilde{x}}^e\left(t\right)\right)=0$ for all t ≥ t_s for some t_s ≥ t₀, implies that

$Z\left(x\left(t\right),{\tilde{\alpha }}_2\left(\zeta \left(t\right)\right)\right)\equiv 0\forall t\ge t_s.$

By Assumption 6.4.2 we have lim_t→∞ x(t) = 0. Moreover, Assumption 6.4.1 implies that there exists a smooth function u = u(x) locally defined in a neighborhood of x = 0 such that

$Z\left(x,u\left(t\right)\right)=0\text{and}u(0)=0.$

Therefore, lim_t→∞ x(t) = 0 and $Z\left(x\left(t\right),{\tilde{\alpha }}_2\left({\zeta }_2\left(t\right)\right)\right)=0\text{imply}{\text{lim}}_{t\to \infty }{\tilde{\alpha }}_2\left(\zeta \left(t\right)\right)=0.$ Finally, ${\text{lim}}_{t\to \infty }{\tilde{\alpha }}_2\left(\zeta \left(t\right)\right)=0\text{imply}{\text{lim}}_{t\to \infty }\zeta \left(t\right)=0$ since ζ is a trajectory of (6.56). Hence, by LaSalle’s invariance-principle, we conclude that x^e = 0 is asymptotically-stable. □

Remark 6.4.1 As a consequence of the above proposition, the design of the injectiongain matrix $\tilde{G}\left(\zeta \right)$ should be such that: (a) the dissipation-inequality (6.57) is satisfied for the closed-loop system (6.55) and for some storage-function $\tilde{U}\left(\zeta \right)$; (b) system (6.56) is asymptotically-stable. If this happens, then the controller (6.53) will solve the MFBNLHICP .

In the next several lines, we present the main result for the solution of the MFBNLHICP for the class of systems (6.52). We begin with the following assumption.

Assumption 6.4.3 The linearization of the output function Y (x, w) is such that

$rank\left(D_{21}\right)\triangleq rank\left(\frac{\partial Y}{\partial w}\left(0,0\right)\right)=m.$

Define now the Hamiltonian function K : T ^⋆ $\mathcal{X}$ × $\mathcal{W}$ × ℜ^m →ℜ by

$K\left(x,p,w,y\right)=p^{T}F\left(x,w,0\right)-y^{T}Y\left(x,w\right)+\frac{1}{2}{\Vert Z\left(x,0\right)\Vert }^2-\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2.$

(6.58)

Then, K is concave with respect to w and convex with respect to y by construction. This implies the existence of a smooth maximizing function ŵ(x, p, y) and a smooth minimizing function y_⋆(x, p) defined in a neighborhood of (0, 0, 0) and (0, 0) respectively, such that

${\left(\frac{\partial K\left(x,p,w,y\right)}{\partial w}\right)}_{w=\widehat{w}\left(x,p,y\right)}=0,\widehat{w}\left(0,0,0\right)=0,$

(6.59)

${\left(\frac{{\partial }^{2}K\left(x,p,w,y\right)}{\partial w^2}\right)}_{\left(x,p,w,y\right)=\left(0,0,0,0\right)}=-{\gamma }^{2}I,$

(6.60)

${\left(\frac{\partial K\left(x,p,\widehat{w}\left(x,p,y\right),y\right)}{\partial y^{\mathrm{}}}\right)}_{y=y_{*}\left(x,p\right)}=0,y_{*}\left(0,0\right)=0,$

(6.61)

${\left(\frac{{\partial }^{2}K\left(x,p,\widehat{w}\left(x,p,y\right),y\right)}{\partial y^2}\right)}_{\left(x,p,w,y\right)=\left(0,0,0,0\right)}=\frac{1}{{\gamma }^2}{D}_{21}{D}_{21}^T.$

(6.62)

Finally, setting

$w_{**}\left(x,p\right)=\widehat{w}\left(x,p,y_{*}\left(x,p\right)\right),$

we have the following main result.

Theorem 6.4.1 Consider the system (6.52) and let the Assumptions 6.4.1, 6.4.2, 6.4.3 hold. Suppose there exists a smooth positive-definite solution $\tilde{V}$ to the HJI-inequality (5.75) and the inequality

$K\left(x,W_x^T\left(x\right),w_{**}\left(x,W_x^T\left(x\right)\right),y_{*}\left(x,W_x^T\left(x\right)\right)\right)-{\tilde{H}}^{\star }\left(x,{\tilde{V}}_x^T\left(X\right)\right)<0,$

(6.63)

where ${\tilde{H}}^{\star }\left(.,.\right)$ is given by (5.72), has a smooth positive-definite solution W (x) defined in the neighborhood of x = 0 with W (0) = 0. Suppose also in addition that

(i)  $W\left(x\right)-\tilde{V}\left(x\right)>0\forall x\ne 0;$

(ii)  the Hessian matrix of

$K\left(x,W_x^T\left(x\right),w_{**}\left(x,W_x^T\left(x\right)\right),y_{*}\left(x,W_x^T\left(x\right)\right)\right)-{\tilde{H}}_{\star }\left(x,{\tilde{V}}_x^T\left(X\right)\right)$

is nonsingular at x = 0; and

(iii)  the equation

$\left(W_x\left(x\right)-\tilde{V}\left(x\right)\right)\tilde{G}\left(x\right)=y_{*}^T\left(x,W_x^T\left(x\right)\right)$

(6.64)

has a smooth solution $\tilde{G}\left(x\right).$

Then the controller (6.54) solves the MFBNLHICP for the system.

Proof: Let $Q\left(x\right)=W\left(x\right)-\tilde{V}\left(x\right)$ and define

$\tilde{S}\left(x,w\right)\triangleq Q_x\left[F\left(x,w,0\right)-G\left(x\right)Y\left(x,w\right)\right]+\tilde{H}\left(x,{\tilde{V}}_x^T,w,0\right)-{\tilde{H}}^{\star }\left(x,V_x^T\right)$

where $\tilde{H}\left(.,.,.,.\right)$ is as defined in (5.69). Then,

$\begin{array}{l}\tilde{S}\left(x,w\right)=W_{x}F\left(x,w,0\right)-y_{*}^T\left(x,W_x^T\right)Y\left(x,w\right)-{\tilde{V}}_{x}F\left(x,w,0\right)+\\ \tilde{H}\left(x,{\tilde{V}}_x^T,w,0\right)-{\tilde{H}}^{\star }\left(x,{\tilde{V}}_x^T\right)\\ =W_{x}F\left(x,w,0\right)-y_{*}^T\left(x,W_x^T\right)Y\left(x,w\right)+\frac{1}{2}\Vert Z\left(x,0\right)\Vert -\frac{1}{2}{\gamma }^2\Vert w\Vert -{\tilde{H}}^{\star }\left(x,{\tilde{V}}_x^T\right)\\ =K\left(x,W_x^T,w,y_{*}\left(x,W_x^T\right)\right)-{\tilde{H}}^{\star }\left(x,{\tilde{V}}_x^T\right)\\ \le K\left(x,W_x^T,w_{**}\left(x,W_x^T\right),y_{*}\left(x,W_x^T\right)\right)-{\tilde{H}}^{\star }\left(x,{\tilde{V}}_x^T\right)\\ =x^T\Upsilon \left(x\right)x\end{array}$

where ϒ(.) is some smooth matrix function which is negative-definite at x = 0.

Now, let

$\tilde{U}\left(x^e\right)=Q\left(x-\zeta \right)+\tilde{V}\left(x\right).$

Then, by construction, $\tilde{U}$ is such that the conditions (a), (b) of Remark 6.4.1 are satisfied. Moreover, if $\tilde{G}$ is chosen as in (6.64), then

$\begin{array}{l}{\tilde{U}}_{x^e}F\left(x^e,w\right)+\frac{1}{2}{\Vert Z^e\left(x^e\right)\Vert }^2-\frac{1}{2}{\Vert w\Vert }^2=Q_x\left(x-\zeta \right)[F\left(x,w,{\tilde{\alpha }}_2\left(\zeta \right)\right)-\\ F(\zeta ,{\tilde{\alpha }}_1\left(\zeta \right),{\tilde{\alpha }}_2\left(\zeta \right)-\tilde{G}\left(\zeta \right)Y\left(x,w\right)+\\ \tilde{G}\left(\zeta \right)Y\left(\zeta ,{\tilde{\alpha }}_1\left(\zeta \right)\right)]+{\tilde{V}}_{x}F\left(x,w,{\tilde{\alpha }}_2\left(\zeta \right)\right)+\\ \frac{1}{2}{\Vert Z\left(x,{\tilde{\alpha }}_2\left(\zeta \right)\right)\Vert }^2-\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2\\ \le Q_x\left(x-\zeta \right)[F\left(x,w,\tilde{\alpha }\left(\zeta \right)\right)-\\ F\left(\zeta ,{\tilde{\alpha }}_1\left(\zeta \right),{\tilde{\alpha }}_2\left(\zeta \right)\right)-\tilde{G}\left(\zeta \right)Y\left(x,w\right)+\\ \tilde{G}\left(\zeta \right)Y\left(\zeta ,\tilde{\alpha }\left(\zeta \right)\right)+\tilde{H}\left(x,{\tilde{V}}_x^T,w,\tilde{\alpha }\left(\zeta \right)\right)-\\ {\tilde{H}}^{\star }\left(x,V_x^T\right).\end{array}$

If we denote the right-hand-side (RHS) of the above inequality by L(x, ζ, w), and observe that this function is concave with respect to w, then there exists a unique maximizing function $\tilde{w}\left(x,\zeta \right)$ such that

${\left(\frac{\partial L\left(x,\zeta ,W\right)}{\partial w}\right)}_{w=\tilde{w}\left(x,\zeta \right)}=0,\tilde{w}\left(0,0\right)=0,$

for all (x, ζ, w) in the neighborhood of (0, 0, 0). Moreover, in this neighbohood, it can be shown (it involves some lengthy calculations) that $L\left(x,\zeta ,\tilde{w}\left(x,\zeta \right)\right)$ can be represented as

$L\left(x,\zeta ,\tilde{w}\left(x,\zeta \right)\right)={\left(x-\zeta \right)}^{T}R\left(x,\zeta \right)\left(x-\zeta \right)$

for some smooth matrix function R(., .) such that R(0, 0) = ϒ(0). Hence, R(., .) is locally negative-definite about (0, 0), and thus the dissipation-inequality

${\tilde{U}}_{x^e}F\left(x^e,w\right)+\frac{1}{2}{\Vert Z^e\left(x^e\right)\Vert }^2-\frac{1}{2}{\Vert w\Vert }^2\le L\left(x,\zeta ,\tilde{w}\left(x,\zeta \right)\right)\le 0$

is satisfied locally. This guarantees that condition (a) of the remark holds.

Finally, setting $w={\tilde{\alpha }}_1\left(x\right)$ in the expression for $\tilde{S}\left(x,w\right)$, we get

$0>S\left(x,{\tilde{\alpha }}_1\left(x\right)\right)\ge Q_x\left(F\left(x,{\alpha }_1\left(x\right),0\right)-G\left(x\right)Y\left(x,{\alpha }_1\left(x\right)\right)\right),$

which implies that Q(x) is a Lyapunov-function for (6.56), and consequently the condition (b) also holds. □

Remark 6.4.2 Notice that solving the MFBNLHICP involves the solution of two HJI-inequalities (5.75), (6.63) together with a coupling condition (6.64). This agrees well with the linear theory [92, 101, 292].

6.4.1    Controller Parametrization

In this subsection, we consider the controller parametrization problem for the general class of nonlinear systems represented by the model Σ^g. For this purpose, we introduce the following additional assumptions.

Assumption 6.4.4 The matrix $D_{11}^T{D}_{11}-{\gamma }^{2}I$ is negative-definite for some γ > 0, where $D_{11}=\frac{\partial Z}{\partial w}\left(0,0\right).$

Define similarly the following

$\tilde{H}\left(x,p,w,u\right)=p^{T}F\left(x,w,u\right)+\frac{1}{2}{\Vert Z\left(x,w,u\right)\Vert }^2-\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2$

(6.65)

$r_{11}\left(x\right)={\left(\frac{{\partial }^2\tilde{H}\left(x,{\tilde{V}}_x\left(x\right),w,u\right)}{\partial w^2}\right)}_{w={\tilde{\alpha }}_1\left(x\right),u={\tilde{\alpha }}_2\left(x\right)}$

(6.66)

$r_{12}\left(x\right)={\left(\frac{{\partial }^2\tilde{H}\left(x,{\tilde{V}}_x\left(x\right),w,u\right)}{\partial u\partial w}\right)}_{w={\tilde{\alpha }}_1\left(x\right),u={\tilde{\alpha }}_2\left(x\right)}$

(6.67)

$r_{21}\left(x\right)={\left(\frac{{\partial }^2\tilde{H}\left(x,{\tilde{V}}_x\left(x\right),w,u\right)}{\partial u\partial w}\right)}_{w={\tilde{\alpha }}_1\left(x\right),u={\tilde{\alpha }}_2\left(x\right)}$

(6.68)

$r_{22}\left(x\right)={\left(\frac{{\partial }^2\tilde{H}\left(x,{\tilde{V}}_x\left(x\right),w,u\right)}{\partial u^2}\right)}_{w={\tilde{\alpha }}_1\left(x\right),u={\tilde{\alpha }}_2\left(x\right)},$

(6.69)

where $\tilde{V}$ solves the HJI-inequality (5.75), while ${\tilde{\alpha }}_1$ and ${\tilde{\alpha }}_2$ are the corresponding worst-case disturbance and optimal feedback control as defined by equations (5.73), (5.74) in Section 5.6. Set also

$\tilde{R}\left(x\right)=\left[\begin{array}{l}\left(1-{\in }_1\right)r_{11}{r}_{12}\left(x\right)\\ r_{21}\left(x\right)\left(1+{\in }_2\right)r_{22}\left(x\right)\end{array}\right]$

where 0 < $\epsilon$₁ < 1 and $\epsilon$₂ > 0, and define also $\tilde{K}:T^{\star }\mathcal{X}\times {ℒ}_2\left({\Re }_{+}\right)\times {\Re }^m\to \Re$ by

$\begin{array}{l}\tilde{K}\left(x,p,w,y\right)=p^{T}F\left(x,w+{\alpha }_1\left(x\right),0\right)-y^{T}Y\left(x,w+{\alpha }_1\left(x\right)\right)+\\ \frac{1}{2}{\left[\begin{array}{l}w\\ -{\tilde{\alpha }}_2\left(x\right)\end{array}\right]}^T\tilde{R}\left(x\right)\left[\begin{array}{l}w\\ -{\tilde{\alpha }}_2\left(x\right)\end{array}\right].\end{array}$

Further, let the functions w₁(x, p, y) and y₁(x, p) defined in the neighborhood of (0, 0, 0) and (0, 0) respectively, be such that

${\left(\frac{\partial \tilde{K}\left(x,p,w,y\right)}{\partial w}\right)}_{w=w_1}=0,w_1\left(\mathrm{0.0.0}\right)=0,$

(6.70)

${\left(\frac{\partial \tilde{K}\left(x,p,w_1\left(x,p,y\right),y\right)}{\partial w}\right)}_{y=y_1}=0,y_1\left(0,0\right)=0.$

(6.71)

Then we make the following assumption.

Assumption 6.4.5 There exists a smooth positive-definite function $\tilde{Q}\left(x\right)$ locally defined in a neighborhood of x = 0 such that the inequality

$Y_2\left(x\right)=\tilde{K}\left(x,{\tilde{Q}}_x^T\left(x\right),w_1\left(x,{\tilde{Q}}_x^T\left(x\right),y_1\left(x,{\tilde{Q}}_x^T\left(x\right)\right),y_1(x,{\tilde{Q}}_x^T\left(x\right)\right)\right)<0$

(6.72)

is satisfied, and the Hessian matrix of the LHS of the inequality is nonsingular at x = 0.

Now, consider the following family of controllers defined by

${\mathrm{\Sigma }}_q^c:\{\begin{array}{l}\dot{\xi }=F\left(\xi ,{\tilde{\alpha }}_1\left(\xi \right),{\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)\right)+G\left(\xi \right)\left(y-Y\left(\xi ,{\tilde{\alpha }}_1\left(\xi \right)\right)\right)+{\widehat{g}}_1\left(\xi \right)c\left(q\right)+\\ {\widehat{g}}_1\left(\xi \right)d\left(q\right)\\ \dot{q}=a\left(q,y-Y\left(\xi ,{\tilde{\alpha }}_1\left(\xi \right)\right)\right)\\ u={\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right),\end{array}$

(6.73)

where ξ ∈ $\mathcal{X}$ and q ∈ℜ^v ⊂ $\mathcal{X}$ are defined in the neighborhoods of the origin in $\mathcal{X}$ and ℜ^v respectively, while G(.) satisfies the equation

${\tilde{Q}}_x\left(x\right)G\left(x\right)=y_1\left(x,{\tilde{Q}}_x^T\left(x\right)\right)$

(6.74)

with $\tilde{Q}(.)$ satisfying Assumption 6.4.5. The functions a(., .), c(.) are smooth, with a(0, 0) = 0, c(0) = 0, while ${\widehat{g}}_1(.),{\widehat{g}}_2(.)$ and d(.) are C^k, k ≥ 1 of compatible dimensions.

Define also the following Hamiltonian function J : ℜ^2n+v ×ℜ^2n+v ×ℜ^r →ℜ by

$\begin{array}{l}J\left(x_a,p_a,w\right)=p_a^T{F}_a\left(x_a,w\right)+\\ \frac{1}{2}{\left[\begin{array}{l}w-{\tilde{\alpha }}_1\left(x\right)\\ {\alpha }_2\left(\xi \right)+c\left(q\right)-{\tilde{\alpha }}_2\left(x\right)\end{array}\right]}^T\tilde{R}\left(x\right)\left[\begin{array}{l}w-{\tilde{\alpha }}_1\left(x\right)\\ {\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)-{\tilde{\alpha }}_2\left(x\right)\end{array}\right],\end{array}$

where

$x_a=\left[\begin{array}{l}x\\ \xi \\ q\end{array}\right],F_a\left(x_a,w\right)=\left[\begin{array}{l}F\left(x,w,{\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)\right)\\ \tilde{F}\left(\xi ,q\right)+G\left(\xi \right)Y\left(x,w\right)+{\widehat{g}}_1\left(\xi \right)c\left(q\right)+{\widehat{g}}_2\left(\xi \right)d\left(q\right)\\ a\left(q,Y\left(x,w\right)-Y\left(\xi ,{\tilde{\alpha }}_1\right)\right)\end{array}\right]$

and

$\tilde{F}\left(\xi ,q\right)=F\left(\xi ,{\tilde{\alpha }}_1,{\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)\right)-G\left(\xi \right)Y\left(\xi ,{\tilde{\alpha }}_1\left(\xi \right),{\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)\right).$

Note also that, since

${\left(\frac{{\partial }^{2}J\left(x_a,p_a,w\right)}{\partial w^2}\right)}_{\left(x_a,p_a,w\right)=\left(0,0,0\right)}=\left(1-{\in }_1\right)\left(G_{11}^T{D}_{11}-{\gamma }^{2}I\right),$

there exists a unique smooth solution w₂(x_a, p_a) defined on a neighborhood of (x_a, p_a) = (0, 0) satisfying

${\left(\frac{{\partial }^{2}J\left(x_a,p_a,w\right)}{\partial w^2}\right)}_{\left(x_a,p_a,w\right)=\left(0,0,0\right)}=\left(1-{\epsilon }_1\right)\left(G_{11}^T{D}_{11}-{\gamma }^{2}I\right),$

The following proposition then gives the parametrization of the set of output measurement-feedback controllers for the system Σ^g.

Proposition 6.4.2 Consider the system Σ^g and suppose the Assumptions 6.4.1-6.4.5 hold. Suppose the following also hold:

(H1) there exists a smooth solution $\tilde{V}$ to the HJI-inequality (5.75), i.e.,

$Y_1\left(x\right):=\tilde{H}\left(x,{\tilde{V}}_x^T\left(x\right),{\tilde{\alpha }}_1\left(x\right),{\tilde{\alpha }}_2\left(x\right)\right)\le 0$

for all x about x = 0;

(H2) there exists a smooth real-valued function M(x_a) locally defined in the neighborhood of the origin in ℜ^2n+v which vanishes at x_a = col(x, x, 0) and is positive everywhere, and is such that

$Y_3\left(x_a\right):J\left(x_a,M_{x_a}^T\left(x_a\right),w_2(x_a,M_{x_a}^T\left(x_a\right)\right)<0$

and vanishes at x_a = (x, x, 0).

Then the family of controllers ${\sum }_q^g$ solves the MFBNLHICP for the system Σ^g.

Proof: Consider the Lyapunov-function candidate

${\mathrm{\Psi }}_2\left(x_a\right)=\tilde{V}\left(x\right)+M\left(x_a\right)$

which is positive-definite by construction. Along the trajectories of the closed-loop system (6.1), (6.73):

${\mathrm{\Sigma }}_q^{gc}:\{\begin{array}{l}{\dot{x}}_a=F_a\left(x_a,w\right)\\ z=Z\left(x,{\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)\right),\end{array}$

(6.75)

and by employing Taylor-series approximation of $J\left(x_a,M_{x_a}^T,w_2\left(x_a,M_{x_a}^T\left(x_a\right)\right)\right)$about w = w₂(., .), we have

$\begin{array}{l}\frac{d{\mathrm{\Psi }}_2}{dt}+\frac{1}{2}{\Vert Z\left(x,{\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)\right)\Vert }^2-\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2=Y_1\left(x\right)+Y_3\left(x_a\right)+\\ \frac{1}{2}{\left[\begin{array}{l}w-{\tilde{\alpha }}_1\left(x\right)\\ {\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)-{\tilde{\alpha }}_2\left(x\right)\end{array}\right]}^T\left[\begin{array}{cc}{\epsilon }_1{r}_{11}\left(x\right)& 0\\ 0& -{\epsilon }_2{r}_{22}\left(x\right)\end{array}\right]\left[\begin{array}{l}w-{\tilde{\alpha }}_1\left(x\right)\\ {\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)-{\tilde{\alpha }}_2\left(x\right)\end{array}\right]+\\ \frac{1}{2}{\Vert w-w_2\left(x_a,M_{x_a}^T\left(x_a\right)\right)\Vert }_{\mathrm{\Gamma }\left(x_a\right)}^2+o\left(\Vert \begin{array}{l}w-{\tilde{\alpha }}_1\left(x\right)\\ {\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)-{\tilde{\alpha }}_2\left(x\right)\end{array}\Vert \right)+\\ o\left({\Vert w-w_2\left(x_a,M_{x_a}^T\left(x_a\right)\right)\Vert }^3\right)\end{array}$

(6.76)

where

$\mathrm{\Gamma }\left(x_a\right)={\left(\frac{{\partial }^{2}J\left(x_a,M_{x_a}^T\left(x_a\right),w\right)}{\partial w_2}\right)}_{w=w_2\left(x_a,M_{x_a}^T\left(x_a\right)\right)}$

and $\mathrm{\Gamma }\left(0\right)=\left(1-{\epsilon }_1\right)\left(D_{11}^T{D}_{11}-{\gamma }^{2}I\right).$ Further, setting w = 0 in the above equation, we get

$\begin{array}{l}\frac{d{\mathrm{\Psi }}_2}{dt}=\frac{1}{2}{\Vert Z\left(x,{\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)\right)\Vert }^2+Y_1\left(x\right)+Y_3\left(x_a\right)+\\ \frac{1}{2}{\left[\begin{array}{l}-{\tilde{\alpha }}_1\left(x\right)\\ {\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)-{\tilde{\alpha }}_2\left(x\right)\end{array}\right]}^T\left[\begin{array}{cc}{\epsilon }_1{r}_{11}\left(x\right)& 0\\ 0& -{\epsilon }_2{r}_{22}\left(x\right)\end{array}\right]\left[\begin{array}{l}-{\tilde{\alpha }}_1\left(x\right)\\ {\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)-{\tilde{\alpha }}_2\left(x\right)\end{array}\right]+\\ \frac{1}{2}{\Vert w_2\left(x_a,M_{x_a}^T\left(x_a\right)\right)\Vert }_{\mathrm{\Gamma }\left(x_a\right)}^2+\left(\Vert \begin{array}{l}-{\tilde{\alpha }}_1\left(x\right)\\ {\tilde{\alpha }}_2\left(\xi \right)+c\left(q\right)-{\tilde{\alpha }}_2\left(x\right)\end{array}\Vert \right)+\\ o\left({\Vert w_2\left(x_a,M_{x_a}^T\left(x_a\right)\right)\Vert }^3\right)\end{array}$

which is negative-semidefinite near x_a = 0 by hypothesis and Assumption 6.4.1 as well as the fact that r₁₁(x) < 0, r₂₂(x) > 0 about x_a = 0. This proves that the equilibrium-point x_a = 0 of (6.75) is stable.

To conclude asymptotic-stability, observe that any trajectory (x(t), ξ(t), q(t)) satisfying

$\frac{d{\mathrm{\Psi }}_2}{dt}\left(x\left(t\right),\xi \left(t\right),q\left(t\right)\right)=0\forall t\ge t_s$

(say!), is necessarily a trajectory of

$\dot{x}\left(t\right)=F\left(x,0,{\widetilde{\text{α}}}_2\left(\xi \right)+c\left(q\right)\right),$

such that x(t) is bounded and Z(x, 0, ${\tilde{\alpha }}_2$(ξ) + c(q)) = 0 for all t ≥ t_s. Furthermore, by hypotheses H1 and H2 and Assumption 6.4.1, ${\dot{\mathrm{\Psi }}}_2\left(x\left(t\right),\xi \left(t\right),q\left(t\right)\right)=0$ for all t ≥ t_s, implies x(t) = ξ(t) and q(t) = 0 for all t ≥ t_s. Consequently, by Assumption 6.4.2 we have lim_t→∞ x(t) = 0, lim_t→∞ ξ(t) = 0, and by LaSalle’s invariance-principle, we conclude asymptotic-stability. Finally, integrating the expression (6.76) from t = t₀ to t = t₁, starting at x(t₀), ξ(t₀), q(t₀), and noting that $\mathcal{Y}$₁(x) and $\mathcal{Y}$ (x_a) are nonpositive, it can be shown that the closed-loop system has locally ℒ₂-gain ≤ γ or the disturbance-attenuation property. □

The last step in the parametrization is to show how the functions ${\widehat{g}}_1(.),{\widehat{g}}_2(.),d(.)$ can be selected so that condition H2 in the above proposition can be satisfied. This is summarized in the following theorem.

Theorem 6.4.2 Consider the nonlinear system Σ^g and the family of controllers (6.73). Suppose the Assumptions 6.4.1-6.4.5 and hypothesis H1 of Proposition 6.4.2 holds. Suppose the following also holds:

(H3) there exists a smooth positive-definite function L(q) locally defined on a neighborhood of q = 0 such that the function

$Y_4\left(q,w\right)=L_q\left(q\right)a\left(q,Y\left(0,w\right)\right)+\frac{1}{2}{\left[\begin{array}{l}w\\ c\left(q\right)\end{array}\right]}^T\tilde{R}\left(0\right)\left[\begin{array}{l}w\\ c\left(q\right)\end{array}\right]$

is negative-definite at w = w₃(q) and its Hessian matrix is nonsingular at q = 0, where w₃(.) is defined on a neighborhood of q = 0 and is such that

${\left(\frac{\partial Y_4\left(q,w\right)}{\partial w}\right)}_{w=w_3\left(q\right)}=0,w_3\left(0\right)=0.$

Then, if ${\widehat{g}}_1(.),{\widehat{g}}_2(.)$ are selected such that

$\begin{array}{l}{\tilde{Q}}_x\left(x\right){\widehat{g}}_1\left(x\right)={\tilde{Q}}_x\left(x\right)\left(\frac{\partial F}{\partial u}\left(x,0,0\right)-G\left(x\right)\frac{\partial Y}{\partial u}\left(x,0,0\right)\right)+{\beta }^T\left(x,0,0\right)r_{12}\left(x\right)-\\ \left(1+{\in }_2\right){\tilde{\alpha }}_2^T\left(x\right)r_{22}\left(x\right)\end{array}$

and

${\tilde{Q}}_x\left(x\right){\widehat{g}}_2\left(x\right)=-Y^T\left(x,{\tilde{\alpha }}_1\left(x\right)+\beta \left(x,0,0\right)\right),$

respectively, where

$\beta \left(x,\xi ,q\right)=w_2\left(x_a,\left[{\tilde{Q}}_x\left(x-\xi \right)-{\tilde{Q}}_x\left(x-\xi \right)L_q\left(q\right)\right]\right)$

and let $d\left(q\right)=L_q^T\left(q\right)$. Then, each of the family of controllers (6.73) solves the MFBNLHICP for the system (6.52).

Proof: It can easily be shown by direct substitution that the function $M\left(x_a\right)=\tilde{Q}\left(x-\xi \right)+L\left(q\right)$ satisfies the hypothesis (H2) of Proposition 6.4.2. □

6.5    Static Output-Feedback Control for Affine Nonlinear Systems

In this section, we consider the static output-feedback (SOFB) stabilization problem (SOFBP) with disturbance attenuation for affine nonlinear systems. This problem has been extensively considered by many authors for the linear case (see [113, 121, 171] and the references contained therein). However, the nonlinear problem has received very little attention so far [45, 278]. In [45] some sufficient conditions are given in terms of the solution to a Hamilton-Jacobi equation (HJE) with some side conditions, which when specialized to linear systems, reduce to the necessary and sufficient conditions given in [171].

In this section, we present new sufficient conditions for the solvability of the problem in the general affine nonlinear case which we refer to as a “factorization approach” [27]. The sufficient conditions are relatively easy to satisfy, and depend on finding a factorization of the state-feedback solution to yield the output-feedback solution.

We begin with the following smooth model of an affine nonlinear state-space system defined on an open subset $\mathcal{X}\subset {\Re }^n$ without disturbances:

${\mathrm{\Sigma }}^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+g\left(x\right)u;x\left(t_0\right)=x_0\\ y=h\left(x\right)\end{array}$

(6.77)

where x ∈ $\mathcal{X}$ is the state vector, u ∈ ℜ^p is the control input, and y ∈ ℜ^m is the output of the system. The functions f : $\mathcal{X}$ → V^∞($\mathcal{X}$ ), and $g:\mathcal{X}\to {ℳ}^{n\times p},h:\mathcal{X}\to {\Re }^m$ are smooth C^∞ functions of x. We also assume that x = 0 is an equilibrium point of the system (6.77) such that f(0) = 0, h(0) = 0.

The problem is to find a SOFB controller of the form

$u=K\left(y\right)$

(6.78)

for the system, such that the closed-loop system (6.1)-(6.78) is locally asymptotically-stable at x = 0.

For the nonlinear system (6.77), it is well known that [112, 175], if there exists a C¹ positive-definite local solution V : ℜⁿ → ℜ₊ to the following Hamilton-Jacobi-Bellman equation (HJBE):

$V_x\left(x\right)f\left(x\right)-\frac{1}{2}{V}_x\left(x\right)g\left(x\right)g^T\left(x\right)V_x^T\left(x\right)+\frac{1}{2}{h}^T\left(x\right)h\left(x\right)=0,V\left(0\right)=0$

(6.79)

(or the inequality with “ ≤ ″), then the optimal control law

$u^{*}=-g^T\left(x\right)V_x^T\left(x\right)$

(6.80)

locally asymptotically stabilizes the equilibrium point x = 0 of the system and minimizes the cost functional:

$J_1\left(u\right)=\frac{1}{2}{\displaystyle \underset{t_0}{\overset{\infty }{\int }}\left({\Vert y\Vert }^2+{\Vert u\Vert }^2\right)dt.}$

(6.81)

Moreover, if V > 0 is proper (i.e., the set ${\mathrm{\Omega }}_c=\left\{x|0\le V\left(x\right)\le a\right\}$ is compact for all a > 0), then the above control law (6.80) globally asymptotically stabilizes the equilibrium point x = 0. The aim is to replace the state vector in the above control law by the output vector or some function of it, with some possible modifications to the control law. In this regard, consider the system (6.77) and suppose there exists a C¹ positive-definite local solution to the HJE (6.79) and a C⁰ function F : ℜ^m → ℜ^p such that:

$F\circ h\left(x\right)=-g^T\left(x\right)V_x^T\left(x\right),$

(6.82)

where “◦” denotes the composition of functions. Then the control law

$u=F\left(y\right)$

(6.83)

solves the SOFBP locally. It is clear that if the function F exists, then

$u=F\left(y\right)=F\circ h\left(x\right)=-g^T\left(x\right)V_x^T\left(x\right).$

This simple observation leads to the following result.

Proposition 6.5.1 Consider the nonlinear system (6.77) and the SOFBP. Assume the system is locally zero-state detectable and the state-feedback problem is locally solvable with a controller of the form (6.80). Suppose there exist C⁰ functions $F_1:\mathcal{Y}\subset {\Re }^m\to \mathcal{U},{\varphi }_1:X_0\subset \mathcal{X}\to {\Re }^p,{\eta }_1:\mathcal{X}\to {\Re }_{-}$(non-positive), ${\eta }_1\in ℒ\left[t_0,\infty \right)$ such that

$F_1\circ h\left(x\right)=-g^T\left(x\right)V_x^T\left(x\right)+{\varphi }_1\left(x\right)$

(6.84)

$V\left(x\right)g\left(x\right){\varphi }_1\left(x\right)={\eta }_1\left(x\right).$

(6.85)

Then:

(i)  the SOFBP is locally solvable with a feedback of the form

$u=F_1\left(y\right);$

(6.86)

(ii)  if the optimal cost of the state-feedback control law (6.80) is $J_{SFB}^{*}\left(u\right)$, then the cost J_SOFB (u) of the SOFB control law (6.86) is given by

$J_{1,SOBF}\left(u\right)=J_{1,SFB}^{*}\left(u^{\star }\right)+{\displaystyle \underset{t_0}{\overset{\infty }{\int }}{\eta }_1\left(x\right)dt.}$

Proof: Consider the closed-loop system (6.77), (6.86):

$\dot{x}=f\left(x\right)+g\left(x\right)F_1\left(h\left(x\right)\right).$

Differentiating the solution V > 0 of (6.79) along the trajectories of this system, we get upon using (6.84), (6.85) and (6.79):

$\begin{array}{l}\dot{V}=V_x\left(x\right)\left(f\left(x\right)+g\left(x\right)F_1(h\left(x\right)\right)=V_x\left(x\right)\left(f\left(x\right)-g\left(x\right)gT\left(x\right)V_x^T\left(x\right)+g\left(x\right){\varphi }_1\left(x\right)\right)\\ =V_x\left(x\right)f\left(x\right)-V_{x}g\left(x\right)g^T\left(x\right)V_x^T\left(x\right)+V_x\left(x\right)g\left(x\right){\varphi }_1\left(x\right)\\ =-\frac{1}{2}{V}_x\left(x\right)g\left(x\right)g^T\left(x\right)V^T\left(x\right)-\frac{1}{2}{h}^T\left(x\right)h\left(x\right)+{\eta }_1\left(x\right)\\ \le 0.\end{array}$

(6.87)

This shows that the equilibrium point x = 0 of the closed-loop system is Lyapunov-stable. Moreover, by local zero-state detectability of the system, we can conclude local asymptotic-stability. This establishes (i).

To prove (ii), integrate the last expression in (6.87) with respect to t from t = t₀ to ∞, and noting that by asymptotic-stability of the closed-loop system, V (x(∞)) = 0, this yields

$J_1\left(u^{*}\right):=\frac{1}{2}{\displaystyle \underset{t_0}{\overset{\infty }{\int }}\left({\Vert y\Vert }^2+{\Vert u^{*}\Vert }^2\right)dt}=V\left(x_0\right)+{\displaystyle \underset{t_0}{\overset{\infty }{\int }}{\eta }_1\left(x\right)dt}$

Since V (x₀) is the total cost of the policy, the result now follows from the fundamental theorem of calculus. □

Remark 6.5.1 Notice that if Σ^a is globally detectable, V > 0 is proper, and the functions ϕ₁(.) and η₁(.) are globally defined, then the control law (6.86) solves the SOFBP globally.

A number of interesting corollaries can be drawn from Proposition 6.5.1. In particular, the condition (6.85) in the proposition can be replaced by a less stringent one given in the following corollary.

Corollary 6.5.1 Take the same assumptions as in Proposition 6.5.1. Then, condition (6.85) can be replaced by the following condition

${\varphi }_1^T\left(x\right)F_1\circ h\left(x\right)\ge 0$

(6.88)

for all x ∈ $\mathcal{X}$₁ a neighborhood of x = 0.

Proof: Multiplying equation (6.84) by ${\varphi }_1^T\left(x\right)$ from the left and substituting equation (6.85) in it, the result follows by the non-positivity of η₁. □

Example 6.5.1 We consider the following scalar nonlinear system

$\begin{array}{l}\dot{x}=-x^3+\frac{1}{2}x+u\\ y=\sqrt{2}x.\end{array}$

Then the HJBE (6.79) corresponding to this system is

$\left(-x^3+\frac{1}{2}x\right)V_x\left(x\right)-\frac{1}{2}{V}_x^2\left(x\right)+x^2=0,$

and it can be checked that the function V (x) = x² solves the HJE. Moreover, the state-feedback problem is globally solved by the control law u = −2x. Thus, clearly,$u=-\sqrt{2}y$, and hence the function F₁(y) = −y solves the SOFBP globally.

In the next section we consider systems with disturbances.

6.5.1    Static Output-Feedback Control with Disturbance-Attenuation

In this section, we extend the simple procedure discussed above to systems with disturbances. For this purpose, consider the system (6.77) with disturbances [113]:

${\mathrm{\Sigma }}_2^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+g_1\left(x\right)w+g_2\left(x\right)u\\ y=h_2\left(x\right)\\ z=\left[\begin{array}{l}{h}_1\left(x\right)\\ u\end{array}\right]\end{array}$

(6.89)

where all the variables are as defined in the previous sections and $w\in \mathcal{W}\subset {ℒ}_2\left[t_0,\infty \right)$. We recall from Chapter 5 that the state-feedback control (6.80) where V : N → ℜ₊ is the smooth positive-definite solution of the Hamilton-Jacobi-Isaacs equation (HJIE):

$V_x\left(x\right)f\left(x\right)+\frac{1}{2}{V}_x\left(x\right)\left[\frac{1}{{\gamma }^2}{g}_1\left(x\right)g_1^T\left(x\right)-g_2\left(x\right)g_2^T\left(x\right)\right]V_x^T\left(x\right)+\frac{1}{2}{h}_1^T\left(x\right)h_1\left(x\right)=0,V\left(0\right)=0,$

(6.90)

minimizes the cost functional:

$J_2\left(u,w\right)=\frac{1}{2}{\displaystyle \underset{t_0}{\overset{\infty }{\int }}\left({\Vert z\left(t\right)\Vert }^2-{\gamma }^2{\Vert w\left(t\right)\Vert }^2\right)dt}$

and renders the closed-loop system (6.89), (6.80) dissipative with respect to the supply-rate $s\left(w,z\right)=\frac{1}{2}\left({\gamma }^2{\Vert w\Vert }^2-{\Vert z\Vert }^2\right)$ and hence achieves ℒ₂-gain ≤ γ. Thus, our present problem is to replace the states in the feedback (6.78) with the output y or some function of it. More formally, we define the SOFBP with disturbance attenuation as follows.

Definition 6.5.1 (Static Output-Feedback Problem with Disturbance Attenuation (SOFBPDA)). Find (if possible!) a SOFB control law of the form (6.78) such that the closed-loop system (6.89), (6.78) has locally ${\mathcal{L}}_2$-gain from w to z less or equal γ > 0, and is locally asymptotically-stable with w = 0.

The following theorem gives sufficient conditions for the solvability of the problem.

Theorem 6.5.1 Consider the nonlinear system (6.89) and the SOFBPDA. Assume the system is zero-state detectable and the state-feedback problem is locally solvable in N ⊂ $\mathcal{X}$ with a controller of the form (6.80) and V > 0 a smooth solution of the HJIE (6.90). Suppose there exist C⁰ functions $F_2:\mathcal{Y}\subset {\Re }^m\to \mathcal{U},{\varphi }_2:X_1\subset \mathcal{X}\to {\Re }^p,{\eta }_2:X_1\subset \mathcal{X}\to {\Re }_{-}$ (non-positive),${\eta }_2\in ℒ\left[t_0,\infty \right)$ such that the conditions

$F_2\circ h_2\left(x\right)=-g_2^T\left(x\right)V_x^T\left(x\right)+{\varphi }_2\left(x\right),$

(6.91)

$V_x\left(x\right)g_2\left(x\right){\varphi }_2\left(x\right)={\eta }_2\left(x\right),$

(6.92)

are satisfied. Then:

(i)  the SOFBP is locally solvable with the feedback

$u=F_2\left(y\right);$

(6.93)

(ii)  if the optimal cost of the state-feedback control law (6.80) is $J_{2,SFB}^{*}\left(u^{\star },w^{\star }\right)=V\left(x_0\right)$, then the cost J_2,SOFB(u) of the SOFBPDA control law (6.93) is given by

$J_{2,SOFB}\left(u\right)=J_{2,SFB}^{*}\left(u\right)+{\displaystyle \underset{t_0}{\overset{\infty }{\int }}{\eta }_2\left(x\right)dt.}$

Proof: (i) Differentiating the solution V > 0 of (6.90) along a trajectory of the closed-loop system (6.89) with the control law (6.93), we get upon using (6.91), (6.92) and the HJIE (6.90)

$\begin{array}{l}\dot{V}=V_x\left(x\right)\left[f\left(x\right)+g_1\left(x\right)w+g_2\left(x\right)F_2\left(h_2\left(x\right)\right)\right]\\ =V_x\left(x\right)\left[f\left(x\right)+g_1\left(x\right)w-g_2\left(x\right)g_2^T\left(x\right)V_x^T\left(x\right)+g_2\left(x\right)\varphi \left(x\right)\right]\\ =V_x\left(x\right)f\left(x\right)+V_x\left(x\right)g_1\left(x\right)w-V_x\left(x\right)g_2\left(x\right)g_2^T\left(x\right)V^T\left(x\right)+V_x\left(x\right)g_2\left(x\right){\varphi }_2\left(x\right)\\ \le -\frac{1}{2}{V}_x\left(x\right)g_2\left(x\right)g_2^T\left(x\right)V^T\left(x\right)-\frac{1}{2}{h}_1^T\left(x\right)h_1\left(x\right)+\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2-\frac{{\gamma }^2}{2}{\Vert w-\frac{g_1^T\left(x\right)V_x^T\left(x\right)}{{\gamma }^2}\Vert }^2\\ \le \frac{1}{2}\left({\gamma }^2{\Vert w\Vert }^2-{\Vert z\Vert }^2\right).\end{array}$

(6.94)

Integrating now from t = t₀ to t = T we have the dissipation inequality

$V\left(x\left(T\right)\right)-V\left(x_0\right)\le {\displaystyle \underset{t_0}{\overset{T}{\int }}\frac{1}{2}}\left({\gamma }^2{\Vert w\Vert }^2-{\Vert z\Vert }^2\right)dt,$

and therefore the system has ℒ₂-gain ≤ γ. In addition, with w ≡ 0, we get

$\dot{V}\le -\frac{1}{2}{\Vert z\Vert }^2,$

and therefore, the closed-loop system is locally stable. Moreover, the condition $\dot{V}\equiv 0$ for all t ≥ t_c, for some t_c, implies that z ≡ 0 for all t ≥ t_c, and consequently, lim_t→∞ x(t) = 0 by zero-state detectability. Finally, by the LaSalle’s invariance-principle, we conclude asymptotic-stability. This establishes (i).

(ii) Using similar manipulations as in (i) above, it can be shown that

$\dot{V}=\frac{1}{2}\left({\gamma }^2{\Vert w\Vert }^2-{\Vert z\Vert }^2\right)-\frac{{\gamma }^2}{2}{\Vert w-\frac{g_1^T\left(x\right)V_x^T\left(x\right)}{{\gamma }^2}\Vert }^2+{\eta }_2\left(x\right)$

integrating the above expression from t = t₀ to ∞, substituting w = w^⋆ and rearranging, we get

$J_2^{\star }\left(u^{\star },w^{\star }\right)=V\left(x_0\right)+{\displaystyle \underset{t_0}{\overset{\infty }{\int }}{\eta }_2\left(x\right)dt.\square }$

We consider another example.

Example 6.5.2 For the second order system

$\begin{array}{l}{\dot{x}}_1=x_1{x}_2\\ {\dot{x}}_2=-x_1^2+x_2+u+w\\ y={\left[x_1{x}_1+x_2\right]}^T\\ z={\left[x_{2}u\right]}^T,\end{array}$

the HJIE (6.90) corresponding to the above system is given by

$x_1{x}_2{V}_{x_1}+\left(-x_1^2+x_2\right)V_{x_2}+\frac{\left(1-{\gamma }^2\right)}{{\gamma }^2}{V}_{x_2}^2+\frac{1}{2}{x}_2^2=0.$

Then, it can be checked that for $\gamma \text{=}\sqrt{\frac{2}{5}}$, the function $V\left(x\right)=\frac{1}{2}\left(x_1^2+x_2^2\right)$, solves the HJIE. Consequently, the control law u = −x₂ stabilizes the system. Moreover, the function F₂(y) = y₁ − y₂ clearly solves the equation (6.91) with ϕ₂(x) = 0. Thus, the control law u = y₁ − y₂ locally stabilizes the system.

We can specialize the result of Theorem 6.5.1 above to the linear system:

${\mathrm{\Sigma }}^l:\{\begin{array}{l}\dot{x}=Ax+B_{1}w+B_{2}u,x\left(t_0\right)=0\\ y=c_{2}x\\ z=\left[\begin{array}{l}{c}_{1}x\\ u\end{array}\right]\end{array}$

(6.95)

where A ∈ ℜ^n×n, B₁ℜ^n×r, B₂ ∈ ℜ^n×p, C₂ ∈ ℜ^s×n, and C₁ ∈ ℜ^(s−p)×n are constant matrices. We then have the following corollary.

Corollary 6.5.2 Consider the linear system (6.95) and the SOFBPDA. Assume (A, B₂) is stabilizable, (C₁, A) is detectable and the state-feedback problem is solvable with a controller of the form $u=-B_1^{T}Px$, where P > 0 is the stabilizing solution of the algebraic-Riccati equation (ARE):

$A^{T}P+PA+P\left[\frac{1}{{\gamma }^2}{B}_1{B}_1^T-B_2{B}_2^T\right]P+C_1^T{C}_1=0.$

In addition, suppose there exist constant matrices ${\text{Γ}}_2\in {\Re }^{p\times m},{\text{Φ}}_2\in {\Re }^{p\times n},0\ge {\text{Λ}}_2\in {\Re }^{n\times n}$ such that the conditions

${\text{Γ}}_2{C}_2=-B_2^{T}P+{\text{Φ}}_2,$

(6.96)

$P{B}_2{\mathrm{\Phi }}_2={\text{Λ}}_2,$

(6.97)

are satisfied. Then:

(i)  the SOFBP is solvable with the feedback

$u={\mathrm{\Gamma }}_{2}y;$

(6.98)

(ii)  if the optimal cost of the state-feedback control law (6.80) is $J_{2,SFBl}^{*}\left(u^{\star },w^{\star }\right)=\frac{1}{2}{x}_0^{T}P{x}_0$, then the cost J_SOFB2(u,w) of the SOFBDA control law (6.93) is given by

$J_{2,SOFBl}^{*}\left(u^{\star },w^{\star }\right)=J_{2,SFBl}^{*}\left(u^{\star },w^{\star }\right)-x_0^T{\mathrm{\Lambda }}_2{x}_0.$

In closing, we give a suggestive adhoc approach for solving for the functions F₁, F₂, for of the SOFB synthesis procedures outlined above. Let us represent anyone of the equations (6.84), (6.91) by

$F\circ h_2\left(x\right)=\alpha \left(x\right)+\varphi \left(x\right)$

(6.99)

where α(.) represents the state-feedback control law in each case. Then if we assume for the time-being that ϕ is known, then F can be solved from the above equation as

$F=\alpha \circ h_2^{-1}+\varphi \circ h_2^{-1}$

(6.100)

provided $h_2^{-1}$ exists at least locally. This is the case if h₂ is injective, or h₂ satisfies the conditions of the inverse-function theorem [230]. If neither of these conditions is satisfied, then multiple or no solution may occur.

Example 6.5.3 Consider the nonlinear system

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

Then, it can be checked that the function $V\left(x\right)=\frac{1}{2}\left(x_1^2+x_2^2\right)$ solves the HJIE (6.90) corresponding to the state-feedback problem for the system. Moreover,

$\begin{array}{l}\alpha =-g_2^T\left(x\right)V_x^T\left(x\right)=-x_2,\\ h_2^{-1}\left(x\right)=h_2\left(x\right)=x_2.\end{array}$

Therefore, with $\varphi =0,F\left(x\right)=\alpha \circ h_2^{-1}\left(x\right)=-x_2\text{or}F\left(y\right)=-y$.

We also remark that the examples presented are really simple, because it is otherwise difficult to find a closed-form solution to the HJIE. It would be necessary to develop a computational procedure for more difficult problems.

Finally, again regarding the existence of the functions F₁, F₂, from equation (6.99) and applying the Composite-function Theorem [230], we have

$D_{x}F\left(h_2\left(x\right)\right)D_x{h}_2\left(x\right)=D_x\alpha \left(x\right)+D_x\varphi \left(x\right).$

(6.101)

This equation represents a system of nonlinear equations in the unknown Jacobian D_xF. Thus, if Dh₂ is nonsingular, then the above equation has a unique solution given by

$D_{x}F\left(h_2\left(x\right)\right)=\left(D_x\alpha \left(x\right)+D_x\varphi \left(x\right)\right){\left(D_x{h}_2\left(x\right)\right)}^{-1}.$

(6.102)

Moreover, if $h_2^{-1}$ exists, then F can be recovered from the Jacobian, DF (h₂), by composition. However, Dh₂ is seldom nonsingular, and in the absence of this, the generalized-inverse can be used. This may not however lead to a solution. Similarly, $h_2^{-1}$ may not usually exist, and again one can still not rule out the existence of a solution F. More investigation is still necessary at this point. Other methods for finding a solution using for example Groebner basis [85] and techniques from algebraic geometry are also possible.

6.6    Notes and Bibliography

The results of Sections 6.1, 6.3 of the chapter are based mainly on the valuable papers by Isidori et al. [139, 141, 145]. Other approaches to the continuous-time output-feedback problem for affine nonlinear systems can be found in the References [53, 54, 166, 190, 223, 224], while the results for the time-varying and the sampled-data measurement-feedback problems are discussed in [189] and [124, 213, 255, 262] respectively. The discussion on controller parametrization for the affine systems is based on Astolfi [44] and parallels that in [190]. It is also further discussed in [188, 190, 224], while the results for the general case considered in Subsection 6.4.1 are based on the Reference [289]. In addition, the results on the tracking problem are based on [50].

Similarly, the robust control problem is discussed in many references among which are [34, 147, 192, 208, 223, 265, 284]. In particular, the Reference [145] discusses the robust output-regulation (or tracking or servomechanism) problem, while the results on the reliable control problem are based on [285].

Finally, the results of Section 6.5 are based on [27], and a reliable approach using TSfuzzy models can be found in [278].