5.3    Robust Nonlinear ${\mathcal{H}}_{\infty }$ State-Feedback Control

In this section, we consider the state-feedback ${\mathcal{H}}_{\infty }$-control problem for the affine nonlinear system Σ^a in the presence of unmodelled dynamics and/or parameter variations. This problem has been considered in many references [6, 7, 209, 223, 245, 261, 265, 284]. The approach presented here is based on [6, 7] and is known as guaranteed-cost control. It is an extension of quadratic-stabilization, and was first developed by Chang [78] and later popularized by Petersen [226, 227, 235]. For this purpose, the system is represented by the model:

${\mathrm{\Sigma }}_{\text{Δ}}^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+\text{Δ}f\left(x,\theta ,t\right)+g_1\left(x\right)w+\left[g_2\left(x\right)+\text{Δ}{g}_2\left(x,\theta ,t\right)\right]u;\\ x\left(t_0\right)=x_0\\ y=x\\ z=\left[\begin{array}{l}{h}_1\left(x\right)\\ u\end{array}\right]\end{array}$

(5.41)

where all the variables and functions have their previous meanings and in addition $\text{Δ}f:\mathcal{X}\to V^{\infty }(\mathcal{X}),\text{Δ}{g}_2:\mathcal{X}\to {ℳ}^{n\times p}(\mathcal{X})$ are unknown functions which belong to the set Ξ of admissible uncertainties, and $\theta \in \mathrm{\Theta }\subset {\Re }^r$ are the system parameters which may vary over time within the set Θ. Usually, a knowledge of the sets Ξ and Θ is necessary in order to be able to synthesize robust control laws.

Definition 5.3.1 (Robust State-Feedback Nonlinear ${\mathcal{H}}_{\infty }$ -Control Problem (RSFBNLHICP)) Find (if possible!) a static control function of the form

$\tilde{u}=\beta \left(x,t\right),\beta :N\times \Re \to {\Re }^P$

(5.42)

for some smooth function β depending on x and t only, such that the closed-loop system:

${\mathrm{\Sigma }}_{\text{Δ},cl}^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+\text{Δ}f\left(x,\theta ,t\right)+g_1\left(x\right)w+\left[g_2\left(x\right)+\text{Δ}{g}_2\left(x,\theta ,t\right)\right]\beta \left(x,t\right);\\ x\left(t_0\right)=x_0\\ z=\left[\begin{array}{l}{h}_1\left(x\right)\\ \beta \left(x,t\right)\end{array}\right]\end{array}$

(5.43)

has locally ${ℒ}_2$ -gain from the disturbance signal w to the output z less than or equal to some prescribed number γ^⋆ > 0 with internal-stability, or equivalently, the closed-loop system achieves local disturbance-attenuation less than or equal to γ^⋆ with internal stability, for all perturbations $\text{Δ}f,\text{Δ}{g}_2\in \text{Ξ}$ and all parameter variations in Θ.

To solve the above problem, we first characterize the sets of admissible uncertainties of the system Ξ and Θ.

Assumption 5.3.1 The admissible uncertainties of the system are structured and matched, and they belong to the following sets:

$\begin{array}{l}{\text{Ξ}}_{\text{Δ}}=\{\text{Δ}f,\text{Δ}{g}_2|\text{Δ}f\left(x,\theta ,t\right)=H_2\left(x\right)F\left(x,\theta ,t\right)E_1\left(x\right),\text{Δ}{g}_2\left(x,\theta ,t\right)=g_2\left(x\right)F\left(x,\theta ,t\right)E_2\left(x\right),\\ {\Vert F\left(x,\theta ,t\right)\Vert }^2\le 1\forall x\in \mathcal{X},\theta \in \mathrm{\Theta },t\in \Re \}\\ \mathrm{\Theta }=\left\{\theta |0\le \theta \le {\theta }_u\right\}\end{array}$

where H₂(.), F(.,.), E₁(.), E₂(.) have appropriate dimensions.

Remark 5.3.1 The conditions of Assumption 5.3.1 are called “matching-conditions” and these types of uncertainties are known as “structured uncertainties.”

Now, define the following cost functional:

$J_{gc}\left(u,w\right)=\frac{1}{2}{\displaystyle {\int }_{t_0}^T\left({\Vert z\left(t\right)\Vert }^2-{\gamma }^2{\Vert w\left(t\right)\Vert }^2\right)dt,T>t_0.}$

(5.44)

Then we have the following definition:

Definition 5.3.2 The function β(., .) is a guaranteed-cost control for the system (5.43) if there exists a positive-(semi)definite C¹ function $V:N\subset \mathcal{X}\to {\Re }_{+}$ that satisfies the inequality

$\begin{array}{l}\frac{\partial V\left(x,t\right)}{\partial t}+\frac{\partial V\left(x,t\right)}{\partial x}\left\{f\left(x\right)+\text{Δ}f\left(x,\theta ,t\right)+g_1\left(x\right)w+\left[g_2\left(x\right)+\text{Δ}{g}_2\left(x,\theta ,t\right)\right]\beta \left(x,t\right)\right\}+\\ \frac{1}{2}\left({\Vert z\Vert }^2-{\gamma }^2{\Vert w\Vert }^2\right)\le 0\forall x\in N,\forall w\in {ℒ}_2\left[0,\infty \right),\forall \text{Δ}f,\text{Δ}g2\in {\text{Ξ}}_{\text{Δ}},\theta \in \mathrm{\Theta }.\end{array}$

(5.45)

It can now be observed that, since the cost function J_gc is exactly the ${\mathcal{H}}_{\infty }$-control cost function (equation (5.4)), then a guaranteed cost control β(., .) which stabilizes the system (5.43) clearly solves the robust ${\mathcal{H}}_{\infty }$-control problem. Moreover, integrating the inequality (5.45) from t = t₀ to t = t₁ > t₀ and starting at x(t₀), we get the dissipation inequality (5.23). Thus, a guaranteed-cost control with cost function (5.44) renders the system (5.43) dissipative with respect to the supply-rate s(w, z). Consequently, the guaranteed-cost framework solves the RSFBNLHICP for the nonlinear uncertain system (5.43) from both perspectives. In addition, we also have the following proposition for the optimal cost of this policy.

Proposition 5.3.1 If the control law β(.,.) satisfies the guaranteed-cost criteria, then the optimal cost $J_{gc}^{\star }\left(u^{\star },w^{\star }\right)$ of the policy is bounded by

$V^a\left(x,t\right)\le J_{gc}^{\star }\left(u^{\star },w^{\star }\right)\le V\left(t_0,x_0\right)$

where

$V^a\left(x,t\right)=\underset{x\left(0\right)=x,u\in \mathcal{U},t\ge 0}{\sup }-{\displaystyle {\int }_0^{t}s\left(w\left(\tau \right),z\left(\tau \right)\right)}d\tau$

is the available-storage of the system defined in Chapter 3.

Proof: Taking the supremum of −J_gc(., ) over $\mathcal{U}$ and starting at x₀, we get the lower bound. To get the upper bound, we integrate the inequality (5.45) from t = t₀ to t = t₁ to get the disssipation inequality (5.23) which can be expressed as

$V\left(t_1,x\left(t_1\right)\right)+{\displaystyle {\int }_{t_0}^{t_1}\frac{1}{2}\left({\Vert z\Vert }^2-{\gamma }^2{\Vert w\Vert }^2\right)dt\le V\left(t_0,x\left(t_0\right)\right),\forall x\left(t_0\right),x\left(t_1\right)\in N.}$

Since V(., .) ≥ 0, the result follows. □

The following lemma is a nonlinear generalization of [226, 227] and will be needed in the sequel.

Lemma 5.3.1 For any matrix functions H₂(.), F(., ., .) and E(.) of appropriate dimensions such that $\Vert F(x,\theta ,t)\Vert \le 1forallx\in \mathcal{X},\theta \in \mathrm{\Theta },andt\in \Re$, then

$\begin{array}{l}{V}_x\left(x\right)H_2\left(x\right)F\left(x,\theta ,t\right)E\left(x\right)\le \frac{1}{2}\left[V_X\left(x\right)H_2\left(x\right)H_1^T\left(x\right)V_x^T\left(x\right)+E^T\left(x\right)E\left(x\right)\right]\\ F\left(x,\theta ,t\right)E\left(x\right)\le I+\frac{1}{4}{E}^T\left(x\right)E\left(x\right)\end{array}$

for some C¹ function $V:\mathcal{X}\to \Re ,forallx\in \mathcal{X},\theta \in \mathrm{\Theta }andt\in \Re .$

Proof: For the first inequality, note that

$\begin{array}{l}0\le {\Vert H_2^T\left(x\right)V_x^T\left(x\right)-F\left(x,\theta ,t\right)E\left(x\right)\Vert }^2=V_X\left(x\right)H_2\left(x\right)H_2^T\left(x\right)V_x^T\left(x\right)-\\ 2V_X\left(x\right)H_2\left(x\right)F\left(x,\theta ,t\right)E\left(x\right)+E^T\left(x\right)E\left(x\right),\end{array}$

from which the result follows. Similarly, to get the second inequality, we have

$0\le {\Vert I-\frac{1}{2}F\left(x,\theta ,t\right)E\left(x\right)\Vert }^2=I-F\left(x,\theta ,t\right)E\left(x\right)+\frac{1}{4}{E}^T\left(x\right)E\left(x\right)$

and the result follows. □

Next we present one approach to the solution of the RSFBNLHICP which is the main result of this section.

Theorem 5.3.1 Consider the nonlinear uncertain system ${\sum }_{\text{Δ}}^a$ and the problem of synthesizing a guaranteed-cost control β(., .) that solves the RSFBNLHICP locally. Suppose the system is smoothly-stabilizable, zero-state detectable, and there exists a positive-(semi)definite C¹ function $\tilde{V}:N\subset \mathcal{X}\to \Re ,0\in N$ satisfying the following HJIE(inequality):

$\begin{array}{l}{\tilde{V}}_x\left(x\right)f\left(x\right)+\frac{1}{2}{\tilde{V}}_x\left(x\right)\left[\frac{1}{{\gamma }^2}{g}_1\left(x\right)g_1^T\left(x\right)+H_2\left(x\right)H_2^T\left(x\right)-g_2\left(x\right)g_2^T\left(x\right)\right]{\tilde{V}}_x^T\left(x\right)+\\ \frac{1}{2}{h}_1^T\left(x\right)h_1\left(x\right)+\frac{1}{2}{E}_1\left(x\right)E_1^T\left(x\right)\le 0,\tilde{V}\left(0\right)=0,x\in N.\end{array}$

(5.46)

Then the problem is solved by the control law

$\tilde{u}=\beta \left(x\right)=-g_2^T\left(x\right){\tilde{V}}_x^T\left(x\right).$

(5.47)

Proof: Consider the left-hand-side (LHS) of the inequality (5.45). Using the results of Lemma 5.3.1 and noting that it is sufficient to have the function $\tilde{V}$ dependent on x only, then

$\begin{array}{l}LHS={\tilde{V}}_x\left(x\right)f\left(x\right)+\frac{1}{2}{\tilde{V}}_x\left(x\right)H_2\left(x\right)H_2^T\left(x\right){\tilde{V}}_x^T\left(x\right)+\frac{1}{2}{E}_1^T\left(x\right)E_1\left(x\right)+\\ {\tilde{V}}_x\left(x\right)g_1\left(x\right)w+{\tilde{V}}_x\left(x\right)g_2\left(x\right)\left(2I+\frac{1}{4}{E}_2^T\left(x\right)E_2\left(x\right)\right)\tilde{u}+\frac{1}{2}{\Vert z\Vert }^2-{\gamma }^2{\Vert w\Vert }^2\\ ={\tilde{V}}_x\left(x\right)f\left(x\right)+\frac{1}{2}{\tilde{V}}_x\left(x\right)\left[g_1\left(x\right)g_1^T\left(x\right)+H_2\left(x\right)H_2^T\left(x\right)-g_2\left(x\right)g^T\left(x\right)\right]{\tilde{V}}_x^T\left(x\right)+\\ \frac{1}{2}{h}_1^T\left(x\right)h_1\left(x\right)+\frac{1}{2}{E}_1^T\left(x\right)E_1\left(x\right)-\frac{1}{2}{\Vert w-\frac{1}{{\gamma }^2}{g}_1^T\left(x\right){\tilde{V}}_x^T\left(x\right)\Vert }^2-\\ \frac{1}{2}{\tilde{V}}_x{g}_2\left(x\right)\left[3I+\frac{1}{2}{E}_2^T\left(x\right)E_2\left(x\right)\right]g_2^T\left(x\right){\tilde{V}}_x^T\left(x\right),\forall x\in N,\forall w\in {\mathcal{L}}_2\left[0,\infty \right).\end{array}$

Now using the HJIE (5.46), we obtain

$\begin{array}{l}LHS=-\frac{1}{2}{\Vert w-\frac{1}{{\gamma }^2}{g}_1^T\left(x\right){\tilde{V}}_x^T\left(x\right)\Vert }^2-\frac{1}{2}{\tilde{V}}_x{g}_2\left(x\right)\left[3I+\frac{1}{2}{E}_2^T\left(x\right)E_2\left(x\right)\right]g_2^T\left(x\right){\tilde{V}}_x^T\left(x\right)\le 0,\\ \forall x\in N,\forall w\in {\mathcal{L}}_2\left[0,\infty \right),\end{array}$

which implies that the control law (5.47) is a guaranteed-cost control for the system (5.41) and hence solves the RSFBNLHICP. □

Theorem 5.3.1 above gives sufficient conditions for the existence of a guaranteed-cost control law for the uncertain system ${\sum }_{\text{Δ}}^a$. With a little more effort, one can obtain a necessary and sufficient condition in the following theorem.

Theorem 5.3.2 Consider the nonlinear uncertain system ${\sum }_{\text{Δ}}^a$ and the problem of synthesizing a guaranteed-cost control β(., .) that solves the RSFBNLHICP locally. Assume the system is smoothly-stabilizable and zero-state detectable. Then, a necessary and sufficient condition for the existence of such a control law is that there exists a positive-(semi)definite C¹ function $\tilde{V}:N\subset \mathcal{X}\to \Re ,0\in N$ satisfying the following HJIE (inequality)

$\begin{array}{l}{\tilde{V}}_x\left(x\right)f\left(x\right)+\frac{1}{2}{\tilde{V}}_x\left(x\right)[\frac{1}{{\gamma }^2}{g}_1\left(x\right)g_1^T\left(x\right)+H_2\left(x\right)H_2^T\left(x\right)-g_2\left(x\right)(2I+\\ \frac{1}{4}{E}_2^T\left(x\right)E_2\left(x\right))g_2^T\left(x\right)]{\tilde{V}}_x^T\left(x\right)+\frac{1}{2}{h}_1^T\left(x\right)h_1\left(x\right)+\frac{1}{2}{E}_1\left(x\right)E_1^T\left(x\right)\le 0,\tilde{V}\left(0\right)=0.\end{array}$

(5.48)

Moreover, the problem is solved by the optimal feedback control law:

${\tilde{u}}^{\star }=\beta \left(x\right)=-g_2^T\left(x\right){\tilde{V}}_x^T\left(x\right).$

(5.49)

Proof: We shall only give the proof for the necessity part of the theorem only. The sufficiency part can be proven similarly to Theorem 5.3.1. Define the Hamiltonian of the system H:$T^{\star }\mathcal{X}\times \mathcal{U}\times \mathcal{W}\to \Re$ by

$\begin{array}{l}H\left(x,{\tilde{V}}_x^T,u,w\right)={\tilde{V}}_x\left(x\right)\left[f\left(x\right)+\text{Δ}f\left(x,\theta ,t\right)+g_1\left(x\right)w+\text{Δ}{g}_2(x)\left(x,\theta ,t\right)u\right]+\\ \frac{1}{2}{\Vert z\Vert }^2-{\gamma }^2{\Vert w\Vert }^2.\end{array}$

Then from Theorem 5.1.1, a necessary condition for an optimal control is that the inequality

$\underset{\mathcal{U}}{\min }\underset{\mathcal{W},{\text{Ξ}}_{\text{Δ}},\mathrm{\Theta }}{\sup }H\left(x,{\tilde{V}}_x,u,w\right)\le 0,$

(5.50)

or equivalently, the saddle-point condition:

$J_{gc}\left(u^{\star },w\right)\le J_{gc}\left(u^{\star },w^{\star }\right)\le J_{gc}\left(u,w^{\star }\right)\forall u\in \mathcal{U},w\in \mathcal{W},\text{Δ}f,\text{Δ}{g}_2\in {\text{Ξ}}_{\text{Δ}},\theta \in \mathrm{\Theta },$

(5.51)

be satisfied for all admissible uncertainties. Further, by Lemma 5.3.1,

$\begin{array}{l}\underset{{\text{Ξ}}_{\text{Δ}},\mathrm{\Theta }}{\sup }H(x,{\tilde{V}}_x,u,w)\le {\tilde{V}}_x(x)f(x)+\frac{1}{2}{\tilde{V}}_x(x)H_2(x)H_2^T(x){\tilde{V}}_x^T(x)+\frac{1}{2}{E}_1^T(x)E_1(x)+\\ {\tilde{V}}_x(x)g_1(x)w+{\tilde{V}}_x(x)g_2(x)\left(2I+\frac{1}{4}{E}_2^T(x)E_2(x)\right)u+\\ \frac{1}{2}(h_1^T(x)h_1(x)+u^{T}u)-\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2.\end{array}$

Setting the above inequality to an equality and differentiating with respect to u and w respectively, results in the optimal feedbacks:

$\begin{array}{l}{u}^{\star }=-{\left(2I+\frac{1}{4}{E}_2^T(x)E_2(x)\right)}^T{g}_2^T(x){\tilde{V}}_x^T(x)\\ w^{\star }=\frac{1}{{\gamma }^2}{g}_1^T(x){\tilde{V}}_x^T(x).\end{array}$

It can also be checked that the above feedbacks satisfy the saddle-point conditions (5.51) and that u^⋆ minimizes J_gc while w^⋆ maximizes it. Finally, substituting the above optimal feedbacks in (5.50) we obtain the HJIE (5.48). □

The above result, Theorem 5.3.1, can be specialized to linear uncertain systems of the form

${\mathrm{\Sigma }}_{\text{Δ}}^l:\{\begin{array}{l}\dot{x}=\left[F+\text{Δ}F\left(x,\theta ,t\right)\right]x+G_1\omega +\left[G_2+\text{Δ}{G}_2\left(x,\theta ,t\right)\right]u;x\left(0\right)=x_0\\ z=\left[\begin{array}{l}{H}_{1}x\\ u\end{array}\right],\end{array}$

(5.52)

where the matrices F, G₁, G₂ and H₂ are as defined in (5.21) and ΔF(.,.,.), Δg₂(.,.,.) ∈ Ξ_Δ have compatible dimensions. Moreover, Ξ_Δ, Θ in this case are defined as

$\begin{array}{l}{\text{Ξ}}_{\text{Δ},l}=\{\text{Δ}F,\text{Δ}{G}_2|\text{Δ}F\left(x,\theta ,t\right)=H_2\tilde{F}\left(x,\theta ,t\right)E_1,\text{Δ}{G}_2\left(x,\theta ,t\right)=G_2\tilde{F}\left(x,\theta ,t\right)E_2,\\ {\Vert \tilde{F}\left(x,\theta ,t\right)\Vert }^2\le 1\forall x\in \mathcal{X},\theta \in \mathrm{\Theta }t\in \Re \},\\ {\mathrm{\Theta }}_l=\left\{\theta |0\le \theta \le {\theta }_u\right\},\end{array}$

where $H_2,\tilde{F}(.,.),E_1,E_2(.)$ have appropriate dimensions. Then we have the following corollary to the theorem.

Corollary 5.3.1 Consider the linear uncertain system (5.52) and the RSFBNLHICP for it. Assume (F, G₂) is stabilizable and (H, F) is detectable. Suppose there exists a symmetric positive-semidefinite solution P ≥ 0 to the ARE:

$F^{T}P+PF+P\left[\frac{1}{{\gamma }^2}{G}_1{G}_1^T+H_2{H}_2^T-G_2{G}_2^T\right]P+H_1^T{H}_1+E_1^T{E}_1=0.$

(5.53)

Then the control law

$\tilde{u}=-G_2^{T}Px$

solves the RSFBNLHICP for the system ${\mathrm{\Sigma }}_{\text{Δ}}^l$.

The above results for the linear uncertain system can be further extended to a class of nonlinear uncertain systems with nonlinearities in the system matrices ΔF, ΔG₂ and some additional unknown C⁰ drift vector-field $f:\mathcal{X}\times \Re \to V^{\infty }({\Re }^q)$, which is Caratheodory,¹ as described by the following model:

${\mathrm{\Sigma }}_{\text{Δ}\sigma }^l:\{\begin{array}{l}\dot{x}=\left[F+\text{Δ}F\left(x,\theta ,t\right)\right]x+G_1\omega +\left[G_2+\text{Δ}{G}_2\left(x,\theta ,t\right)\right]u+G_{2}f\left(x,t\right);\\ x\left(t_0\right)=x_0\\ z=\left[\begin{array}{l}{H}_{1}x\\ u\end{array}\right],\end{array}$

(5.54)

where all the variables and matrices have their previous meanings, and in addition, the set of admissible uncertainties is characterized by

$\begin{array}{l}{\text{Ξ}}_{{\text{Δ}}_f}=\{\text{Δ}F,\text{Δ}{G}_2,f|\text{Δ}F\left(x,\theta ,t\right)=H_2\tilde{F}\left(x,\theta ,t\right)E_1,\text{Δ}{G}_2\left(x,\theta ,t\right)=G_{2}R(x,\\ \theta ,t),{\Vert \tilde{F}\left(x,\theta ,t\right)\Vert }^2\le \zeta \forall t>t_0,\Vert f\left(x,\theta ,t\right)\Vert \le \rho \left(x,t\right)\in \kappa \text{wrt}x,\\ \text{positive}\text{wrt}t\text{and}\underset{t\to \infty }{\lim }\rho \left(x,t\right)<\infty ,{\Vert R\left(x,\theta ,t\right)\Vert }_{\infty }\le \eta ,0\le \eta <1\\ \forall x\in \mathcal{X},\theta \in \mathrm{\Theta }\}.\end{array}$

We can now state the following theorem.

Theorem 5.3.3 Consider the nonlinear uncertain system ${\sum }_{\text{Δ}f}^a$ and the RSFBNLHICP for it. Assume the nominal system (F, G₂) is stabilizable and (H₁, F) is detectable. Suppose further there exists an $ϵ>0$ and a symmetric positive-definite matrix $Q\in {\Re }^{n\times n}$ such that the ARE:

$F^{T}P+PF+P\left[\in \zeta H_2{H}_2^T-2G_2{G}_2^T+{\gamma }^{-2}{G}_1{G}_1^T\right]P+\frac{1}{\in }{E}_1^T{E}_1+H_1^T{H}_1+Q=0$

(5.55)

has a symmetric positive-definite solution P. Then the control law

$\begin{array}{l}{u}_r=-K_x-\frac{1}{\left(1-\eta \right)}{\varphi }_r\left(x,t\right)\\ {\varphi }_r\left(x,t\right)=\frac{{\left(\rho \left(x,t\right)+\eta \Vert Kx\Vert \right)}^2}{Kx\left(\rho \left(x,t\right)+\eta \Vert Kx\Vert \right)+{\in }^{\star }{\Vert x\Vert }^2}Kx\\ K=G_2^{T}P\\ {\in }^{\star }<\frac{{\lambda }_{\min }\left(Q\right)}{2}\end{array}\}$

(5.56)

solves the RSFBNLHICP for the system.

Proof: Suppose there exist a solution P > 0 to the ARE (5.55) for some $ϵ>0,Q>0$. Let

$V\left(x\left(t\right)\right)=x^T\left(t\right)Px\left(t\right)$

be a Lyapunov function candidate for the closed-loop system (5.54), (5.56). We need to show that the closed-loop system is dissipative with this storage-function and the supply-rate $s\left(w,z\right)=\left({\gamma }^2{\Vert w\Vert }^2-{\Vert z\Vert }^2\right)$ for all admissible uncertainties and disturbances, i.e.,

$\dot{V}\left(x\left(t\right)\right)-\left({\gamma }^2{\Vert w\Vert }^2-{\Vert z\Vert }^2\right)\le 0,\forall w\in \mathcal{W},\forall \text{Δ}F,\text{Δ}{G}_2,f\in {\text{Ξ}}_{\text{Δ}f},\theta \in \mathrm{\Theta }.$

Differentiating V (.) along a trajectory of this closed-loop system and completing the squares, we get

$\begin{array}{l}\dot{V}\left(x\left(t\right)\right)=x^T\left[{\left(F+\text{Δ}F\right)}^{T}P+\left(F+\text{Δ}F\right)\right]x+2x^{T}P\left(G_2+\text{Δ}{G}_2\right)u+\\ \triangleq 2x^{T}P{G}_{1}w+2x^{T}P{G}_{2}f\left(x,t\right)A_1\left(x,t\right)+A_2\left(x,t\right)+A_3\left(x,t\right)-\\ {\Vert z\Vert }^2+{\gamma }^2{\Vert w\Vert }^2\end{array}$

where

$\begin{array}{l}{A}_1\left(x,t\right)=x^T[{\left(F+\text{Δ}F\right)}^{T}P+P\left(F+\text{Δ}F\right)+{\gamma }^{2}P{G}_1{G}_1^{T}P+H_1^T{H}_1-\\ 2P{G}_2{G}_2^{T}P]x\end{array}$

$A_2\left(x,t\right)=-2x^{T}P\left[\left(G_2+\text{Δ}{G}_2\right)\frac{1}{1-\eta }{\varphi }_r\left(x,t\right)+\text{Δ}{G}_2{G}_2^{T}Px-G_{2}f\left(x,t\right)\right]$

$A_3\left(x,w\right)=-{\gamma }^2{\left[w-{\gamma }^{-2}{G}_1^{T}Px\right]}^T\left[w-{\gamma }^{-2}{G}_1^{T}Px\right].$

Consider the terms A₁(x, t), A₂(x, t), A₃(x, t), and noting that

$x^{T}P\text{Δ}Fx=x^{T}P{H}_{2}F\left(x,t\right)E_{1}x\le \frac{1}{2}{x}^T\left\{\in \zeta P{H}_2{H}_2^{T}P+\frac{1}{\in }{E}_1^T{E}_1\right\}x\forall x\in {\Re }^n,$

$ϵ>0$. Then,

$\begin{array}{l}{A}_1\left(x,t\right)\le x^T[(F^{T}P+PF+\in \zeta P{H}_2{H}_2^{T}P+\frac{1}{\in }{E}_1^T{E}_1+{\gamma }^{2}P{G}_1{G}_1^{T}P+H_1^T{H}_1-\\ 2P{G}_2{G}_2^{T}P]x\\ \le -x^{T}Qx,\end{array}$

where the last inequality follows from the ARE (5.53). Next,

$\begin{array}{l}{A}_2\left(x,t\right)=2x^{T}P{G}_2\left(I+R\left(x,t\right)\right)\frac{1}{1-\eta }{\varphi }_r\left(x,t\right)-2x^{T}P{G}_2\left[R\left(x,t\right)Kx-f\left(x,t\right)\right]\\ \le -2x^{T}P{G}_2{\varphi }_r\left(x,t\right)-2x^{T}P{G}_2\left[R\left(x,t\right)Kx-f\left(x,t\right)\right]\\ \le 2x^{T}P{G}_2\left[f\left(x,t\right)-R\left(x,t\right)K\left(x,t\right)\right]-\frac{{\Vert Kx\Vert }^2{\left(\rho \left(x,t\right)+\eta \Vert Kx\Vert \right)}^2}{\Vert Kx\Vert \left(\rho \left(x,t\right)+\eta \Vert Kx\Vert \right)+{ϵ}^{\star }{\Vert x\Vert }^2}\\ \le \left\{\Vert Kx\Vert \left(\rho \left(x,t\right)+\eta \Vert Kx\Vert \right)-\frac{{\Vert Kx\Vert }^2{\left(\rho \left(x,t\right)+\eta \Vert Kx\Vert \right)}^2}{\Vert Kx\Vert \left(\rho \left(x,t\right)+\eta \Vert Kx\Vert \right)+{ϵ}^{\star }{\Vert x\Vert }^2}\right\}\\ \le 2\left\{\frac{\Vert Kx\Vert \left(\rho \left(x,t\right)+\eta \Vert Kx\Vert \right){ϵ}^{\star }{\Vert x\Vert }^2}{\Vert Kx\Vert \left(\rho \left(x,t\right)+\eta \Vert Kx\Vert \right)+{ϵ}^{\star }{\Vert x\Vert }^2}\right\}\\ \le 2{ϵ}^{\star }{\Vert x\Vert }^2.\end{array}$

Therefore,

$\begin{array}{l}\dot{V}\left(x\left(t\right)\right)\le -x^{T}Qx+2{ϵ}^{\star }{x}^{T}x-{\Vert z\Vert }^2+{\gamma }^2{\Vert w\Vert }^2\\ =-{\Vert z\Vert }^2+{\gamma }^2{\Vert w\Vert }^2-x^T\left(Q-2{ϵ}^{\star }I\right)x\end{array}$

(5.57)

and

$\dot{V}\left(x\left(t\right)\right)+{\Vert z\Vert }^2-{\gamma }^2{\Vert w\Vert }^2\le -x^T\left(Q-2{ϵ}^{\star }I\right)x\le 0.$

Thus, the closed-loop system is dissipative and hence has ${ℒ}_2-\text{gain}\le \gamma$ from w to z for all admissible uncertainties and all disturbances $w\in \mathcal{W}$. Moreover, from the above inequality, with w = 0, we have

$\dot{V}\left(x\left(t\right)\right)\le -x^T\left(Q-2{ϵ}^{\star }I\right)x<0$

since ${ϵ}^{\star }<\frac{{\lambda }_{\min }\left(Q\right)}{2}$. Consequently, by Lyapunov’s theorem, we have exponential-stability of the closed-loop system. □

Example 5.3.1 [209]. Consider the system (5.54) with

$\begin{array}{l}F=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],G_1=\left[\begin{array}{l}1\\ 0\end{array}\right],G_2=\left[\begin{array}{l}1\\ 0\end{array}\right],H_2=\left[\begin{array}{l}0.1\\ 0.2\end{array}\right]\\ H_1=\left[\begin{array}{cc}0.5& 0\end{array}\right],E_1=\left[\begin{array}{cc}0.1& 0.2\end{array}\right],\Vert f\left(x,t\right)\Vert \le |\frac{1}{3}{x}_2{|}^3.\end{array}$

(5.58)

Applying the result of Theorem 5.3.3 with $\gamma \mathrm{=}1\mathrm{,}ϵ=1$ and

$Q=\left[\begin{array}{cc}0.5& 0\\ 0& 0.5\end{array}\right],$

the ARE (5.55) has a symmetric positive-definite solution

$P=\left[\begin{array}{cc}3.7136& 2.7514\\ 2.7514& 2.6671\end{array}\right]\Rightarrow K=\left[\begin{array}{cc}2.7514& 2.6671\end{array}\right].$

5.4    State-Feedback ${\mathcal{H}}_{\infty }$-Control for Time-Varying Affine Nonlinear Systems

In this section, we consider the state-feedback ${\mathcal{H}}_{\infty }$-control problem for affine nonlinear time-varying systems (TV SFBNLHICP). Such systems are called nonautonomous, as the dynamics of the system is explicitly a function of time and can be represented by the model:

${\mathrm{\Sigma }}_t^a:\{\begin{array}{l}\dot{x}\left(t\right)=f\left(x,t\right)+g_1\left(x,t\right)w+g_2\left(x.t\right)u;x\left(0\right)=x_0\\ y\left(t\right)=x\left(t\right)\\ z\left(t\right)=\left[\begin{array}{c}h\left(x,t\right)\\ u\end{array}\right]\end{array}$

(5.59)

where all the variables have their usual meanings, while the functions $f:\mathcal{X}\times \Re \to V^{\infty },g_1:\mathcal{X}\times \Re \to {ℳ}^{n\times r}(\mathcal{X}\times \Re ),g_2:\mathcal{X}\times \Re \to {ℳ}^{n\times p}(\mathcal{X}\times \Re ),$ $h:\mathcal{X}\times \Re \to {\Re }^s,\text{and}{h}_2:\mathcal{X}\times \Re \to {\Re }^m\text{are}\text{real}{\text{C}}^{\infty ,0}\left(\mathcal{X}\times \Re \right)$ functions, i.e., are smooth with respect to x and continuous with respect to t. We also assume without any loss of generality that x = 0 is the only equilibrium point of the system with u = 0, w = 0, and is such that f(0, t) = 0, h(0, t) = 0.

Furthermore, since the system is time-varying, we are here interested in finding control laws that solve the TVSFNLHICP for any finite time-horizon [0, T]. Moreover, since most of the results on the state-feedback ${\mathcal{H}}_{\infty }$-control problem for the time-invariant case carry through to the time-varying case with only slight modifications to account for the time variation, we shall only summarize here the main result [189].

Theorem 5.4.1 Consider the nonlinear system (5.59) and the TV SFBNLHICP for it. Assume the system is uniformly (for all t) smoothly-stabilizable and uniformly zero-state detectable, and for some γ > 0, there exists a positive-definite function $V:N_1\subset \mathcal{X}\times \left[0,T\right]\to {\Re }_{+}suchthatV(x,T)\ge P_T(x)andV(x,0)\le P_0(x)\forall x\in N_1$, for some nonnegative functions $P_T,P_0:\mathcal{X}\to {\Re }_{+},with{P}_T(0)=P_0(0)=0,$ which satisfies the time-varying HJIE (inequality):

$\begin{array}{l}{V}_t\left(x,t\right)+V_x\left(x,t\right)f\left(x,t\right)+\frac{1}{2}{V}_x\left(x,t\right)\left[\frac{1}{{\gamma }^2}{g}_1\left(x,t\right)g_1^T\left(x,t\right)-g_2\left(x,t\right)g_2^T\left(x,t\right)\right]V_x^T\left(x,t\right)+\\ \frac{1}{2}{h}^T\left(x,t\right)h\left(x,t\right)\le 0,x\in N_1,t\in \left[0,T\right].\end{array}$

(5.60)

Then, the problem is solved by the feedback

$u^{\star }=-g_2^T\left(x,t\right)V_x^T\left(x,t\right),$

and

$w^{\star }=\frac{1}{{\gamma }^2}{g}_1^T\left(x,t\right)V_x^T\left(x,t\right)$

is the worst-case disturbance affecting the system.

Similarly, a parametrization of a set of full-information state-feedback controls for such systems can be given as

$\begin{array}{l}{K}_{F{I}_T}=\{u|u=-g_2^T\left(x,t\right)V^T\left(x,t\right)+Q\left(t\right)\left(w-g_1^T\left(x,t\right)V^T\left(x,t\right)\right),Q\in ℱ{\mathcal{G}}_T,\\ Q:inputs\mapsto outputs\}\end{array}$

where

$\begin{array}{l}ℱ{\mathcal{G}}_T\triangleq \{{\mathrm{\Sigma }}_t^a|{\mathrm{\Sigma }}_t^a\left(u=0,w=0\right)\text{is}\text{uniformly}\text{asymptotically}\text{stable}\text{and}\text{has}\\ {ℒ}_2\left(\left[0,T\right]\right)-\text{gain}\le \gamma \forall T>0\}.\end{array}$

In the next section, we consider the state-feedback problem for affine nonlinear systems with state-delay.

5.5    State-Feedback ${\mathcal{H}}_{\infty }$ -Control for State-Delayed Affine Nonlinear Systems

In this section, we present the state-feedback ${\mathcal{H}}_{\infty }$-control problem and its solution for affine nonlinear state-delayed systems (SFBNLHICPSDS). These are system that have memory and in which the dynamics of the system is affected by past values of the state variable. Therefore their qualitative behavior is significantly richer, and their analysis is more complicated. The dynamics of this class of systems is also intimately related to that of time-varying systems that we have studied in the previous section.

We consider at the outset the following autonomous affine nonlinear state-space system with state-delay defined over an open subset $\mathcal{X}\text{of}{\Re }^n\text{with}\mathcal{X}$ containing the origin x = 0:

${\mathrm{\Sigma }}_{d_0}^a:\{\begin{array}{l}\dot{x}\left(t\right)=f\left(x\left(t\right),x\left(t-d_0\right)\right)+g_1\left(x\left(t\right)\right)w\left(t\right)+g_2\left(x\left(t\right)\right)u\left(t\right);\\ x\left(t\right)=\varphi \left(t\right),t\in \left[t_0-d_0,t_0\right],x\left(t_0\right)=\varphi \left(t_0\right)=x_0,\\ y\left(t\right)=x\left(t\right),\\ z\left(t\right)=h_1\left(x\left(t\right)\right)+k_{12}\left(x\left(t\right)\right)u\left(t\right)+k_{13}\left(x\left(t\right)\right)u\left(t-d_0\right),\end{array}$

(5.61)

where x(.) $\in \mathcal{X}$ is the state vector, u(.)$\in \mathcal{U}\subseteq {\Re }^p$ is the p-dimensional control input, which belongs to the set of admissible controls $\mathcal{U},w(.)\in \mathcal{W}$ is the disturbance signal which has to be tracked or rejected and belongs to the set $\mathcal{W}\subset {\Re }^r$ of admissible disturbances, the output $y\in {\Re }^m$ is the measured output of the system, $z(.)\in {\Re }^s$ is the output to be controlled, d₀ > 0 is the state delay which is constant, and $\varphi (t)\in C\left[-d,t_0\right]$ is the initial function. Further, the functions $f(.,.):\mathcal{X}\times \mathcal{X}\to V^{\infty },g_1(.):\mathcal{X}\to {\Re }^{n\times r}(\mathcal{X}),g_2:\mathcal{X}\to {\Re }^{n\times p}(\mathcal{X}),$$h_1:\mathcal{X}\to {\Re }^s,h_2:\mathcal{X}\to {\Re }^m\text{and}{k}_{12},k_{13}:\mathcal{X}\to {\Re }^{s\times p}\text{are}\text{real}{\text{C}}^{\infty }$ functions of x(.) such that the system (5.61) is well defined. That is, for any initial states x(t₀ −d), x(t₀) $\in \mathcal{X}$ and any admissible input $u(t)\in \mathcal{U}$, there exists a unique solution $x(t,t_0,x_0,x_{t_0-d},u)$ to (5.61) on [t₀, ∞) which continuously depends on the initial data, or the system satisfies the local existence and uniqueness theorem for functional differential equations. Without any loss of generality we also assume that the system has an equilibrium at x = 0, and is such that f(0, 0) = 0, h₁(0) = 0.

We introduce the following definition of ${ℒ}_2$-gain for the affine delay system (5.61).

Definition 5.5.1 The system (5.61) is said to have ${ℒ}_2$-gain from u(t) to y(t) less than or equal to some positive number $\text{γ'}\text{>}\text{0}$, if for all (t₀, t ₁) ∈ [−d, ∞), initial state vector $x_0\in \mathcal{X}$ the response of the system z(t) due to any $u(t)\in {ℒ}_2\left[t_0,t_1\right]$ satisfies

${\displaystyle {\int }_{t_0}^{t_1}{\Vert z\left(t\right)\Vert }^{2}dt}\le \frac{{{\gamma }^{\prime }}^2}{2}{\displaystyle {\int }_{t_0}^{t_1}\left({\Vert w\left(t\right)\Vert }^2+{\Vert w\left(t-d_0\right)\Vert }^2\right)dt+\beta \left(x_0\right)}\forall t_1\ge t_0$

(5.62)

and some nonnegative function $\beta :\mathcal{X}\to {\Re }_{+},\beta \left(0\right)=0.$

The following definition will also be required in the sequel.

Definition 5.5.2 The system (5.61) with u(t) ≡ 0 is said to be locally zero-state detectable, if for any trajectory of the system with initial condition $x(t_0)\in U\subset \mathcal{X}$ and a neighborhood of the origin, such that $z(t)\equiv 0\forall t\ge t_0\Rightarrow {\lim }_{t\to \infty }x(t,t_0,x_0,x_{t_0-d_0},0)=0.$.

The problem can then be defined as follows.

Definition 5.5.3 (State-Feedback Suboptimal ${\mathcal{H}}_{\infty }$ -Control Problem for State-Delayed Systems (SFBHICPSDS)). The problem is to find a smooth state-feedback control law of the form

$u\left(t\right)=\alpha \left(t,x\left(t\right),x\left(t-d_0\right)\right),\alpha \in C^2\left(\Re \times N\times N\right),\alpha \left(t,0,0\right)=0\forall t,N\subset \mathcal{X},$

which is possibly time-varying, such that the closed-loop system has ${ℒ}_2$ -gain from w(.) to z(.) less than or equal to some prescribed positive number γ > 0, and is locally asymptotically-stable with w(t) = 0.

For this purpose, we make the following simplifying assumptions on the system.

Assumption 5.5.1 The matrices h₁(.), k₁₂(.), k₁₃(.) of the system (5.61) are such that $\forall t,i\ne j,i,j=2,3$

$h_1\left(x\left(t\right)\right)k_{1j}\left(x\left(t\right)\right)=0,k_{1j}^T\left(x\left(t\right)\right)k_{1j}\left(x\left(t\right)\right)=I,k_{1i}^T\left(x\left(t\right)\right)k_{1j}\left(x\left(t\right)\right)=0.$

Under the above assumption, we can without loss of generality represent z(.) as

$z\left(t\right)=\left[\begin{array}{c}{h}_1\left(x\left(t\right)\right)\\ u\left(t\right)\\ u\left(t-d_0\right)\end{array}\right].$

Assumption 5.5.2 The system ${\sum }_{d0}^a$ or the pair [f, g₂] is locally smoothly-stabilizable, if there exists a feedback control function $\alpha =\alpha (t,x(t),x(t-d_0))$such that $\dot{x}(t)=f(x(t),x(t-d_0))+g_2(x(t))\alpha (t,x(t),x(t-d_0))$ is locally asymptotically stable for all initial conditions in some neighborhood U of x = 0.

The following theorem then gives sufficient conditions for the solvability of this problem. Note also that in the sequel we shall use the notation x_t for x(t) and $x_{t-d_0}$ for $x\left(t-d_0\right)$for convenience.

Theorem 5.5.1 Consider the nonlinear system ${\sum }_{d0}^a$ and assume it is zero-state detectable. Suppose the Assumptions 5.5.1, 5.5.2 hold, and for some γ > 0 there exists a smooth positive-(semi)definite solution to the Hamilton-Jacobi-Isaacs inequality (HJII):

$\begin{array}{l}{V}_t\left(t,x_t,x_{t-d_0}\right)+V_{x_t}\left(t,x_t,x_{t-d_0}\right)f\left(x_t,x_{t-d_0}\right)+V_{x_{t-d_0}}\left(t,x_t,x_{t-d_0}\right)f\left(x_{t-d_0},x_{t-2d}\right)+\\ \frac{1}{2}\{\frac{1}{{\gamma }^2}\left[V_{x_t}{g}_1\left(x_t\right)g_1^T\left(x_t\right)V_{x_t}^T+V_{x_{t-d_0}}{g}_1\left(x_{t-d_0}\right)g_1^T\left(x_{t-d_0}\right)V_{x_{t-d_0}}^T\right]-\\ \left[V_{x_t}{g}_2\left(x_t\right)g_2^T\left(x_t\right)V_{x_t}^T+V_{x_{t-d_0}}{g}_2\left(x_{x-d_0}\right)g_2^T\left(x_{t-d_0}\right)V_{x_{t-d_0}}^T\right]\}+\frac{1}{2}{h}_1^T\left(x_t\right)h_1\left(x_t\right)\le 0,\\ V\left(t,0,0\right)=0\forall x_{t,}{x}_{t-d_0}\in \mathcal{X}.\end{array}$

(5.63)

Then, the control law

$u^{\star }\left(t\right)=-g_2^T\left(x\left(t\right)\right)V_{x_t}^T\left(t,x\left(t\right),x\left(t-d_0\right)\right)$

(5.64)

solves the SFBNLHICPSDS for the system ${\sum }_{d0}^a$.

Proof: Rewriting the HJII (5.63) as

$\begin{array}{l}{V}_t\left(t,x_t,x_{t-d_0}\right)+V_{x_t}\left(t,x_t,x_{t-d_0}\right)\left[f\left(t,x_t,x_{t-d_0}\right)+g_1\left(x_t\right)w\left(t\right)+g_2\left(x_t\right)u\left(t\right)\right]+\\ V_{x_{t-d_0}}\left[f\left(x_{t-d_0},x_{t-2d}\right)+g_1\left(x_{t-d_0}\right)w\left(t-d_0\right)+g_2\left(x_{t-d_0}\right)u\left(t-d_0\right)\right]\le \\ \frac{1}{2}\Vert u\left(t\right)-u^{\star }\Vert +\frac{1}{2}{\Vert u\left(t-d_0\right)-u^{\star }\left(t-d_0\right)\Vert }^2-\frac{{\gamma }^2}{2}{\Vert w\left(t\right)-w^{\star }\left(t\right)\Vert }^2-\\ \frac{{\gamma }^2}{2}{\Vert w\left(t-d_0\right)-w^{\star }\left(t-d_0\right)\Vert }^2-\frac{1}{2}{\Vert u\left(t\right)\Vert }^2-\frac{1}{2}{\Vert u\left(t-d_0\right)\Vert }^2+\frac{{\gamma }^2}{2}{\Vert w\left(t\right)\Vert }^2+\\ \frac{{\gamma }^2}{2}{\Vert w\left(t-d_0\right)\Vert }^2-\frac{1}{2}{\Vert h_1\left(x_t\right)\Vert }^2\end{array}$

where

$\begin{array}{l}{w}^{\star }\left(t\right)=\frac{1}{{\gamma }^2}{g}_1^T\left(x\left(t\right)\right)V_{x_t}^T\left(x\left(t\right),x\left(t-d_0\right)\right),\\ u^{\star }\left(t-d_0\right)=-g_2^T\left(x_t-d_0\right)V_{x_t-d_0}^T\left(t,x\left(t\right),x\left(t-d_0\right)\right),\end{array}$

and in the above, we have suppressed the dependence of $u^{\star }(.),w^{\star }(.)$ on x(t) for convenience. Then, the above inequality further implies

$\begin{array}{l}{V}_t\left(t,x_t,x_{t-d_0}\right)+V_{x_t}\left(t,x_t,x_{t-d_0}\right)\left[f\left(x_t,x_{t-d_0}\right)+g_1\left(x_t\right)w\left(t\right)+g_2\left(x_t\right)u\left(t\right)\right]+\\ V_{x_t-d_0}\left[f\left(x_{t-d_0},x_{t-2d}\right)+g_1\left(x_{t-d_0}\right)w\left(t-d_0\right)+g_2\left(x_{t-d_0}\right)u\left(t-d_0\right)\right]\le \\ \frac{1}{2}{\Vert u\left(t\right)-u^{\star }\left(t\right)\Vert }^2+\frac{1}{2}{\Vert u\left(t-d_0\right)-u^{\star }\left(t-d_0\right)\Vert }^2-\frac{1}{2}{\Vert u\left(t\right)\Vert }^2-\frac{1}{2}{\Vert u\left(t-d_0\right)\Vert }^2+\\ \frac{{\gamma }^2}{2}{\Vert w\left(t\right)\Vert }^2+\frac{{\gamma }^2}{2}{\Vert w\left(t-d_0\right)\Vert }^2-\frac{1}{2}{\Vert h_1\left(x_t\right)\Vert }^2.\end{array}$

Substituting now u(t) = u^$\star$ (t) and integrating from t = t₀ to t = t₁ ≥ t₀, starting from x₀, we get

$\begin{array}{l}V\left(t_1,x\left(t_1\right),x\left(t_1-d\right)\right)-V\left(t_0,x_0,x\left(t_0-d_0\right)\right)\le \frac{1}{2}\{{\gamma }^2({\Vert w\left(t\right)\Vert }^2+\\ {\Vert w\left(t-d_0\right)\Vert }^2)-{\Vert h_1\left(x_t\right)\Vert }^2-{\Vert u^{\star }\left(t\right)\Vert }^2-{\Vert u^{\star }\left(t-d_0\right)\Vert }^2\}dt,\end{array}$

which implies that the closed-loop system is dissipative with respect to the supply rate $s\left(w\left(t\right),z\left(t\right)\right)=\frac{1}{2}\left[{\gamma }^2\left({\Vert w\left(t\right)\Vert }^2+{\Vert w\left(t-d_0\right)\Vert }^2\right)-{\Vert z\left(t\right)\Vert }^2\right]$, and hence has ${ℒ}_2$-gain from w(t) to z(t) less than or equal to γ. Finally, for local asymptotic-stability, differentiating V from above along the trajectories of the closed-loop system with w(.) = 0, we get

$\dot{V}\le -\frac{1}{2}{\Vert z\left(t\right)\Vert }^2$

which implies that the system is stable. In addition the condition when $\dot{V}\equiv 0\forall t\ge t_s$, corresponds to $z\left(t\right)\equiv 0\forall t\ge t_s$. By zero-state detectability, this implies that lim_t→∞ x(t) = 0, and thus by LaSalle’s invariance-principle, we conclude asymptotic-stability. □

A parametrization of all stabilizing state-feedback ${\mathcal{H}}_{\infty }$-controllers for the system in the full-information (FI) case (when the disturbance can be measured) can also be given.

Proposition 5.5.1 Assume the nonlinear system ${\sum }_d^a$ satisfies Assumptions 5.5.1, 5.5.2 and is zero-state detectable. Suppose the SFBNLHICPSDS is solvable and the disturbance signal $w\in {ℒ}_2[0,\infty )$ is fully measurable. Let $ℱ\mathcal{G}$ denote the set of finite-gain (in the ${ℒ}_2$ sense) asymptotically-stable (with zero input and disturbances) input-affine nonlinear plants of the form:

${\sum }^a:\{\begin{array}{l}\dot{\xi }=a\left(\xi \right)+b\left(\xi \right)u\text{'}\\ \upsilon =c\left(\xi \right)\end{array}$

(5.65)

Then the set

$K_{FI}=\left\{u\left(t\right)|u\left(t\right)=u^{\star }\left(t\right)+Q\left(w\left(t\right)-w^{\star }\left(t\right)\right),Q\in ℱ\mathcal{G},Q:inputs\mapsto outputs\right\}$

(5.66)

is a paremetrization of all FI-state-feedback controllers that solves (locally) the SFBNLHICPSDS for the system ${\sum }_d^a$.

Proof: Apply u(t) ∈ K _{F I} to the system ${\sum }_{d0}^a$ resulting in the closed-loop system:

${\mathrm{\Sigma }}_d^a\left(u^{\star }\left(Q\right)\right):\{\begin{array}{l}\dot{x}\left(t\right)=f\left(x\left(t\right),x\left(t-d_0\right)\right)+g_1\left(x\left(t\right)\right)w\left(t\right)+g_2\left(x\left(t\right)\right)(u^{\star }\left(t\right)+\\ Q\left(w\left(t\right)-w^{\star }\left(t\right)\right);x\left(t_0\right)=x_0\\ z\left(t\right)=\left[\begin{array}{c}{h}_1\left(x\left(t\right)\right)\\ u\left(t\right)\\ u\left(t-d_0\right)\end{array}\right].\end{array}$

(5.67)

If Q = 0, then the result follows from Theorem 5.5.1. So assume $Q\ne 0$, and since $Q\in ℱ\mathcal{G},r\left(t\right)\triangleq Q\left(w\left(t\right)-w^{\star }\left(t\right)\right)\in {ℒ}_2[0,\infty )$. Then differentiating V (., ., .) along the trajectories of the closed-loop system (5.67),(5.66) and completing the squares, we have

$\begin{array}{l}\frac{d}{dt}V=V_t+V_{x_t}\left[f\left(x_t,x_{t-d_0}\right)+g_1\left(x_t\right)w\left(t\right)-g_2\left(x_t\right)g_2^T\left(x_t\right)V_{x_t}^T+g_2\left(x_t\right)r\left(t\right)\right]+\\ V_{x_{t-d_0}}[f\left(x_{t-d_0},x_{t-2d_0}\right)+g_1\left(x_{t-d_0}\right)w\left(t-d_0\right)-g_2\left(x_{t-d_0}\right)g_2^T\left(x_{t-d_0}\right)\times \\ V_{x-d_0}^T\left(t,x_t,x_{t-do}\right)+g_2\left(x_{t-d_0}\right)r\left(t-d_0\right)]\\ =V_t\left(t,x_t,x_{t-d_0}\right)+V_{x_t}\left(t,x_t,x_{t-d_0}\right)f\left(x_t,x_{t-d_0}\right)+\\ V_{x_{t-d_0}}\left(t,x_t,x_{t-d_0}\right)f\left(x_{t-d_0},x_{t-d_0}\right)-\frac{1}{2}{\gamma }^2{\Vert w\left(t\right)-w^{\star }\left(t\right)\Vert }^2+\frac{1}{2}{\gamma }^2{\Vert w\left(t\right)\Vert }^2+\\ \frac{1}{2}{\gamma }^2{\Vert w^{\star }\left(t\right)\Vert }^2-\frac{1}{2}{\Vert r\left(t\right)-g_2^T\left(x_t\right)V_{x_t}^T\left(t,x_t,x_{t-d_0}\right)\Vert }^2+\\ \frac{1}{2}{\Vert r\left(t\right)\Vert }^2-\frac{1}{2}{V}_{x_t}{g}_2\left(x_t\right)g_2\left(x_t\right)V_{x_t}^T-\frac{1}{2}{\gamma }^2{\Vert w\left(t-d_0\right)-w^{\star }\left(t-d_0\right)\Vert }^2+\\ \frac{1}{2}{\gamma }^2{\Vert w\left(t-d_0\right)\Vert }^2+\frac{1}{2}{\gamma }^2{\Vert w^{\star }\left(t-d_0\right)\Vert }^2-V_{x_{t-d_0}}{g}_2\left(x_{t-d_0}\right)g_2^T\left(x_{t-d_0}\right)V_{x_{t-d_0}}^T\\ -\frac{1}{2}{\Vert r\left(t-d_0\right)-g_2^T\left(x_{t-d_0}\right)V_{x_{t-d_0}}^T\Vert }^2+\frac{1}{2}{\Vert r\left(t-d_0\right)\Vert }^2+\frac{1}{2}{\Vert u^{\star }\left(t-d_0\right)\Vert }^2.\end{array}$

Using now the HJI-inequality (5.63), in the above equation we get

$\begin{array}{l}\frac{d}{dt}V\le -\frac{1}{2}\gamma {\Vert w\left(t\right)-w^{\star }\left(t\right)\Vert }^2+\frac{1}{2}{\gamma }^2{\Vert w\left(t\right)\Vert }^2-\frac{1}{2}{\Vert u\left(t\right)\Vert }^2+\frac{1}{2}{\Vert r\left(t\right)\Vert }^2-\\ \frac{1}{2}{\gamma }^2{\Vert w\left(t-d_0\right)-w^{\star }\left(t-d_0\right)\Vert }^2+\frac{1}{2}{\gamma }^2{\Vert w\left(t-d_0\right)\Vert }^2-\frac{1}{2}{\Vert u\left(t-d_0\right)\Vert }^2+\\ \frac{1}{2}{\Vert r\left(t-d_0\right)\Vert }^2-\frac{1}{2}{\Vert h_1\left(t\right)\Vert }^2.\end{array}$

Integrating now the above inequality from t = t₀ to t = t₁ > t₀, starting from x(t₀), and using the fact that

${\displaystyle {\int }_{t_0}^{t_1}{\Vert r\left(t\right)\Vert }^{2}dt\le {\gamma }^2{\displaystyle {\int }_{t_0}^{t_1}{\Vert w\left(t\right)-w^{\star }\left(t\right)\Vert }^2,}}$

we get

$\begin{array}{l}V\left(t_1,x\left(t_1\right),x\left(t_1-d\right)\right)-V\left(t_0,x_0,x\left(t_0-d_0\right)\right)\le \frac{1}{2}{\displaystyle {\int }_{t_0}^{t_1}\{{\gamma }^2{(\Vert w\left(t\right)\Vert }^2+}\\ {\Vert w\left(t-d_0\right)\Vert }^2-{\Vert y\left(t\right)\Vert }^2-{\Vert u\left(t\right)\Vert }^2-{\Vert u\left(t-d_0\right)\Vert }^2\}dt,\end{array}$

which implies that the closed-loop system is dissipative with respect to the supply-rate $s\left(w\left(t\right),z\left(t\right)\right)=\frac{1}{2}\left[{\gamma }^2\left({\Vert w\left(t\right)\Vert }^2+{\Vert w\left(t-d_0\right)\Vert }^2\right)-{\Vert z\left(t\right)\Vert }^2\right]$, and therefore, the system has ${ℒ}_2$-gain from w(t) to z(t) less than or equal to γ. Finally, asymptotic-stability of the system can similarly be concluded as in 5.5.1. □

5.6    State-Feedback ${\mathcal{H}}_{\infty }$-Control for a General Class of Nonlinear Systems

In this section, we look at the state-feedback problem for a more general class of nonlinear systems which is not necessarily affine. We consider the following class of nonlinear systems defined on a manifold $\mathcal{X}\subseteq {\Re }^n$ containing the origin in local coordinates x = (x₁,…, x_n) :

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

(5.68)

where all the variables have their previous meanings, while $F:\mathcal{X}\times \mathcal{W}\times \mathcal{U}\to V^{\infty }$ is the state dynamics function and $Z:\mathcal{X}\times \mathcal{U}\to {\Re }^s$ is the controlled output function. Moreover, the functions F (., ., .) and Z(., .) 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 and is such that F(0, 0, 0) = 0, Z(0, 0) = 0. The following assumption will also be required in the sequel.

Assumption 5.6.1 The linearization of the function Z(x, u) is such that

$rank\left(D_{21}\right)=rank\left(\frac{\partial Z}{\partial u}\left(0,0\right)\right)=p.$

Define now the Hamiltonian function for the above system $\tilde{H}:T^{\star }\mathcal{X}\times \mathcal{W}\times \mathcal{U}\to \Re$ as

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

(5.69)

Then it can be seen that the above function is locally convex with respect to u and concave with respect to w about (x, p, w, u) = (0, 0, 0, 0), and therefore has a unique local saddle-point (w, u) for each (x, p) in a neighborhood of this point. Thus, by Assumption 5.6.1 and the Implicit-function Theorem, there exist unique smooth functions $w^{\star }\left(x,p\right)\text{and}{u}^{\star }\left(x,p\right)$, defined in a neighborhood of (0, 0) such that $w^{\star }\left(0,0\right)=0,u^{\star }\left(0,0\right)=0$ and satisfying

$\frac{\partial \tilde{H}}{\partial w}\left(x,p,w^{\star }\left(x,p\right),u^{\star }\left(x,p\right)\right)=0,$

(5.70)

$\frac{\partial \tilde{H}}{\partial u}\left(x,p,w^{\star }\left(x,p\right),u^{\star }\left(x,p\right)\right)=0.$

(5.71)

Moreover, suppose there exists a nonnegative C¹-function $\tilde{V}:\mathcal{X}\to \Re ,\tilde{V}\left(0\right)=0$ which satisfies the inequality

${\tilde{H}}^{\star }\left(x,{\tilde{V}}_x^T\left(x\right)\right)=\tilde{H}\left(x,{\tilde{V}}_x^T\left(x\right),w^{\star }\left(x,{\tilde{V}}_x^T\left(x\right)\right),u^{\star }\left(x,{\tilde{V}}_x^T\left(x\right)\right)\right)\le 0,$

(5.72)

and define the feedbacks

$u^{\star }=\alpha \left(x\right)=u^{\star }\left(x,{\tilde{V}}_x^T\left(x\right)\right).$

(5.73)

$w^{\star }=w^{\star }\left(x.{\tilde{V}}_x^T\left(x\right)\right).$

(5.74)

Then, substituting u^$\star$ in (5.68) yields a closed-loop system satisfying

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

which is dissipative with respect to the supply-rate $\tilde{s}\left(w,z\right)=\frac{1}{2}\left({\gamma }^2{\Vert w\Vert }^2-{\Vert z\Vert }^2\right)$ with storage-function $\tilde{V}$ in the neighborhood of (x, w) = (0, 0).

The local asymptotic-stability of the system with w = 0 can also be proven as in the previous sections if the system is assumed to be zero-state detectable, or satisfies the following hypothesis.

Assumption 5.6.2 Any bounded trajectory x(t) of the system

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

satisfying

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

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

We summarize the above results in the following theorem.

Theorem 5.6.1 Consider the nonlinear system (5.68) and the SFBNLHICP for it. Assume the system is smoothly-stabilizable and zero-state detectable or satisfies Assumptions 5.6.1 and 5.6.2. Suppose further there exists a C¹ nonnegative function $\tilde{V}:N\subset \mathcal{X}\to {\Re }_{+}$ locally defined in a neighborhood N of x = 0 with $\tilde{V}\left(0\right)=0$ satisfying the following HJI-inequality:

${\tilde{V}}_x\left(x\right)F\left(x,w^{\star },u^{\star }\right)+\frac{1}{2}{\Vert Z\left(x,u^{\star }\right)\Vert }^2+\frac{1}{2}{\gamma }^2{\Vert w^{\star }\Vert }^2\le 0,\tilde{V}\left(0\right)=0,x\in N.$

(5.75)

Then the feedback control law (5.73) solves the SFBNLHICP for the system.

Proof: It has been shown in the preceding that if $\tilde{V}$ exists and locally solves the HJI-inequality (5.75), then the system is dissipative with $\tilde{V}$ as storage-function and supply-rate $\tilde{s}\left(w,z\right)$. Consequently, the system has the local disturbance-attenuation property. Finally, if the system is zero-state detectable or satisfies Assumption 5.6.2, then local asymptotic-stability can be proven along the same lines as in Sections 5.1–5.5. □

Remark 5.6.1 Similarly, the function $w^{\star }=w^{\star }\left(x,{\tilde{V}}_x^T\left(x\right)\right)$ is also interpreted as the worst-case disturbance affecting the system.

5.7    Nonlinear ${\mathcal{H}}_{\infty }$ Almost-Disturbance-Decoupling

In this section, we discuss the state-feedback nonlinear ${\mathcal{H}}_{\infty }$ almost-disturbance-decoupling problem (SFBNLHIADDP ) which is very closely related to the ${ℒ}_2$-gain disturbance-attenuation problem except that it is formulated from a geometric perspective. This problem is an off-shoot of the geometric (exact) disturbance-decoupling problem which has been discussed extensively in the literature [140, 146, 212, 277] and was first formulated by Willems [276] to characterize those systems for which disturbance-decoupling can be achieved approximately with an arbitrary degree of accuracy.

The SFBNLHIADDP is more recently formulated [198, 199] and is defined as follows. Consider the affine nonlinear system (5.1) with single-input single-output (SISO) represented in the form:

$\begin{array}{l}\dot{x}=f\left(x\right)+g_2\left(x\right)u+{\mathrm{\Sigma }}_{i=1}^r{g}_{1i}\left(x\right)w_i\left(t\right);x\left(t_0\right)=x_0\\ y=h\left(x\right)\end{array}\},$

(5.76)

where all the variables have their previous meanings with $u\in \mathcal{U}\subseteq \Re$ and output function $h:\mathcal{X}\to \Re$.

Definition 5.7.1 (State-Feedback ${ℒ}_2$-Gain Almost-Disturbance-Decoupling Problem (SFB ${ℒ}_2$-gainADDP)). Find (if possible!) a parametrized set of smooth state-feedback controls

$u=u\left(x,\lambda \right),\lambda \in {\Re }_{+}\lambda \mathrm{arbitrarily\; large}$

such that for every t ∈ [t₀, T],

${\displaystyle {\int }_{t_0}^t{y}^2\left(\tau \right)d\tau }\le \frac{1}{\lambda }{\displaystyle {\int }_{t_0}^t{\Vert w\left(\tau \right)\Vert }^2}d\tau$

(5.77)

for the closed-loop system with initial condition x₀ = 0 and for any disturbance function w(t) defined on an open interval [t₀, T) for which there exists a solution for the system (5.76).

Definition 5.7.2 (State-Feedback Nonlinear ${\mathcal{H}}_{\infty }$ Almost-Disturbance-Decoupling Problem (SFBNLHIADDP)). The SFBNLHIADDP is said to be solvable for the SISO system (5.76) if the SFB${ℒ}_2$-gainADDP for the system is solvable with $u=u\left(x,\lambda \right),u\left(0,\lambda \right)=0\forall \lambda \in {\Re }_{+}$ and the origin is globally asymptotically-stable for the closed-loop system with w(t) = 0.

In the following, we shall give sufficient conditions for the solvability of the above two problems for SISO nonlinear systems that are in the “strict-feedback” form. It would be shown that, if the system possesses a structure such that it is strictly feedback-equivalent to a linear system, is globally minimum-phase and has zero-dynamics that are independent of the disturbances, then the SFBNLHIADDP is solvable. We first recall the following definitions [140, 212].

Definition 5.7.3 The strong control characteristic index of the system (5.43) is defined as the integer ρ such that

$\begin{array}{l}{L}_{g_2}{L}_f^{i}h\left(x\right)=0,0\le i\le \rho -2,\forall x\in \mathcal{X}\\ L_{g_2}{L}_f^{\rho -1}h\left(x\right)\ne 0,\forall x\in \mathcal{X}.\end{array}$

Otherwise, $\rho =\infty if{L}_{g_2}{L}_f^{i}h\left(x\right)=0\forall i,\forall x\in \mathcal{X}.$

Definition 5.7.4 The disturbance characteristic index of the system (5.43) is defined as the integer ν such that

$\begin{array}{l}{L}_{g_1{}_j}{L}_f^{i}h\left(x\right)=0,1\le j\le r,0\le i\le v-2,\forall x\in \mathcal{X}\\ L_{g_1{}_j}{L}_f^{v-1}h\left(x\right)\ne 0,forsomex\in \mathcal{X},andsomej,1\le j\le r.\end{array}$

We assume in the sequel that ν ≤ ρ.

Theorem 5.7.1 Suppose for the system (5.76) the following hold:

(i)  ρ is well defined;

(ii)  ${\mathcal{G}}_{r-1}=span\left\{g_2,a{d}_f{g}_2,\mathrm{\dots },a{d}_f^{\rho -1}{g}_2\right\}$ is involutive and of constant rank $\rho in\mathcal{X}$;

(iii)  $a{d}_{g{1}_i}{\mathcal{G}}_j\subset {\mathcal{G}}_j,1\le i\le r,0\le j\le \rho -2,with{\mathcal{G}}_j=span\left\{g_2,a{d}_f{g}_2,\mathrm{\dots },a{d}_f^j{g}_2\right\};$

(iv)   the vector-fields

$\tilde{f}=f-\frac{1}{L_{g_2}{L}_F^{\rho -1}h}{L}_f^{\rho }h,{\tilde{g}}_2=\frac{1}{L_{g_2}{L}_f^{\rho -1}h}{g}_2$

are complete,

then the SFB ${ℒ}_2$-gain ADDP is solvable.

Proof: See Appendix A.

Remark 5.7.1 Conditions (ii),(iii) of the above theorem require the z_μ-dynamics of the system to be independent of z₂,…, z_ρ. Moreover, condition (iii) is referred to as the “strict-feedback” condition in the literature [153].

From the proof of the theorem, one arrives also at an alternative set of sufficient conditions for the solvability of the SFB${ℒ}_2$-gainADDP.

Theorem 5.7.2 Suppose for the system (5.76) the following hold:

(i)  ρ is well defined;

(ii)  $d\left(L_{g1i}{L}_f^i\right)\in span\left\{dh,d\left(L_{f}h\right),\mathrm{\dots },d\left(L_f^{i}h\right)\right\},v-1\le i\le \rho -1,1\le j\le r\forall x\in \mathcal{X};$

(iii)  the vector-fields

$\tilde{f}=f-\frac{1}{L_{g_2}{L}_f^{\rho -1}h}{L}_f^{\rho }h,{\tilde{g}}_2=\frac{1}{L_{g_2}{L}_f^{\rho -1}h}{g}_2$

are complete. Then the SFB${ℒ}_2$-gainADDP is solvable.

Proof: Conditions (i)-(iii) guarantee the existence of a global change of coordinates by augmenting the ρ linearly-independent set

$z_1=h\left(x\right),z_2=L_{f}h\left(x\right),\mathrm{\dots },z\rho =L_f^{\rho -1}h\left(x\right)$

with an arbitrary n − ρ linearly-independent set $z_{\rho +1}={\psi }_{\rho +1}\left(x\right),\mathrm{\dots },z_n={\psi }_n\text{with}{\psi }_i\left(0\right)=0,\langle d{\psi }_i,g_2\rangle =0,\rho +1\le i\le n.$ Then the state feedback

$u=\frac{1}{L_{g_2}{L}_f^{\rho -1}h\left(x\right)}\left(\upsilon -L_f^{\rho }h\left(x\right)\right)$

globally transforms the system into the form:

$\begin{array}{l}{\dot{z}}_i=z_{i+1}+{\mathrm{\Psi }}_i^T\left(z\right)w1\le i\le \rho -1,\\ {\dot{z}}_{\rho }=\upsilon +{\mathrm{\Psi }}_{\rho }^T\left(z\right)w\\ {\dot{z}}_{\mu }=\psi \left(z_1,z_{\mu }\right)+{\Pi }^T\left(z\right)w,\end{array}$

where $z_{\mu }=\left(z_{\rho +1},\mathrm{\dots },z_n\right)$. The rest of the proof follows along the same lines as Theorem 5.7.1. □

Remark 5.7.2 Condition (ii) in the above Theorem 5.7.2 requires that the functions ${\text{Ψ}}_1,\mathrm{\dots },{\text{Ψ}}_{\rho }$ do not depend on the z_μ dynamics of the system.

The following theorem now sums-up all the sufficient conditions for the solvability of the SFBNLHIADDP .

Theorem 5.7.3 Assume the conditions (i)-(iv) of Theorem 5.7.1 hold for the system (5.76), and the zero-dynamics

${\dot{z}}_{\mu }=\psi \left(0,z_{\mu }\right)$

are independent of w and globally asymptotically-stable about the origin z_μ = 0 (i.e. globally minimum-phase). Then the SFBNLHIADDP is solvable.

Proof: From the first part of the proof of Theorem 5.7.1 and the fact that the zero-dynamics are independent of w, (5.76) can be transformed into the form:

$\begin{array}{l}{\dot{z}}_i=z_{i+1}+{\text{Ψ}}_i^T\left(z_1,\mathrm{\dots },z_i\right)w1\le i\le \rho -1,\\ {\dot{z}}_{\rho }=\upsilon +{\text{Ψ}}_{\rho }^T\left(z_1,\mathrm{\dots },z_{\rho },z_{\mu }\right)w\\ {\dot{z}}_{\mu }=\psi \left(0,z_{\mu }\right)+z_1({\psi }_1\left(z_1,z_{\mu }\right)+{\Pi }_1{}^T\left(z_1,z_{\mu }\right)w),\end{array}\}$

(5.78)

for some suitable functions ${\psi }_1,{\Pi }_1$. Moreover, since the system is globally minimum-phase, by a converse theorem of Lyapunov [157], there exists a radially-unbounded Lyapunov-function $V_{\mu 0}\left(z_{\mu }\right)$ such that

$L_{\psi \left(0,z_{\mu }\right)}{V}_{\mu 0}=\langle d{V}_{\mu 0,}\psi \left(0,z_{\mu }\right)\rangle <0.$

In that case, we can consider the Lyapunov-function candidate

$V_{\mu 01}=V_{\mu 0}\left(z_{\mu }\right)+\frac{1}{2}{z}_1^2.$

Its time-derivative along the trajectories of (5.78) is given by

${\dot{V}}_{\mu 01}=\langle d{V}_{\mu 0,}\psi \left(0,z_{\mu }\right)\rangle +z_1\langle \left(d{V}_{\mu 0},{\psi }_1\right)\rangle +z_1{z}_2+z_1\left({\text{Ψ}}_1^T+\langle d{V}_{\mu 0,}{\Pi }_1^T\left(z_1,z_{\mu }\right)\rangle \right)w.$

(5.79)

Now let

$\overline{\text{Ψ}}\left(z_1,z_{\mu }\right)={\text{Ψ}}_1^T+\langle d{V}_{\mu 0},{\Pi }_1^T\left(z_1,z_{\mu }\right)\rangle ,$

and define

$z_{02}^{*}\left(z_1,z_{\mu }\right)=-\langle d{V}_{\mu 0},{\psi }_1)\rangle -z_1-\frac{1}{4}\lambda z_1\left(1+{\overline{\text{Ψ}}}_1^T{\overline{\text{Ψ}}}_1\right).$

Subsituting the above in (5.79) yields

${\dot{V}}_{\mu 01}\le -z_1^2+\frac{1}{\lambda }{\Vert w\Vert }^2+\langle d{V}_{\mu 0},\psi \left(0,z_{\mu }\right)\rangle .$

(5.80)

The last term in the above expression (5.80) is negative, and therefore by standard Lyapunov-theorem, it implies global asymptotic-stability of the origin z = 0 with w = 0. Moreover, upon integration from z(t₀) = 0 to some arbitrary value z(t), we get

$-{\displaystyle {\int }_{t_0}^t{y}^2\left(\tau \right)d\tau }+\frac{1}{\lambda }{\displaystyle {\int }_{t_0}^t{\Vert w\left(\tau \right)\Vert }^{2}d\tau }\ge V_{\mu 01}\left(z\left(t\right)\right)-V_{\mu 01}\left(0\right)\ge 0$

which implies the ${ℒ}_2$-gain condition holds, and hence the SFB${ℒ}_2$-gainADDP is solvable for the system with ρ = 1. Therefore the auxiliary control $\upsilon =z_{02}^{\star }\left(z_1,z_{\mu }\right)$ solves the SFBHIADDP for the system with ρ = 1. Using an inductive argument as in the proof of Theorem 5.7.1 the result can be shown to hold also for ρ > 1. □

Remark 5.7.3 It is instructive to observe the relationship between the SFBNLHIADDP and the SFBNLHICP discussed in Section 5.1. It is clear that if we take z = y = h(x) in the ADDP, and seek to find a parametrized control law $u={\alpha }_{\gamma }\left(x\right),\alpha \left(0\right)=0$ and the minimum $\gamma \text{=}\sqrt{\frac{1}{\lambda }}$ such that the inequality (5.77) is satisfied for all t ≥ 0 and all $w\in {ℒ}_2\left(0,t\right)$ with internal-stability for the closed-loop system, then this amounts to a SFBNLHICP. However, the problem is “singular” since the function z does not contain u. Moreover, making λ arbitrarily large λ → ∞ corresponds to making γ arbitrarily small, γ → 0.

5.8    Notes and Bibliography

The results presented in Sections 5.1 and 5.6 are mainly based on the valuable papers by Isidori et al. [138, 139, 145, 263, 264], while the controller parametrization is based on [188]. Results for the controller parametrization based on coprime factorizations can be found in [215, 214]. More extensive results on the SFBNLHICP for different system configurations along the lines of [92] are given in [223], while results on a special class of nonlinear systems is given in [191]. In addition, a J-dissipative approach is presented in [224]. Similarly, more results on the RSFBNLHICP which uses more or less similar techniques presented in Section 5.3 are given in the following references [192, 147, 148, 245, 261, 223, 284, 285]. Moreover, the results on the tracking problem presented in Section 5.2 are based on the reference [50].

The results for the state-delayed systems are based on [15], and for the general class of nonlinear systems presented in Section 5.6 are based on the paper by Isidori and Kang [145]. While results on stochastic systems, in particular systems with Markovian jump disturbances, can be found in the references [12, 13, 14].

Lastly, Section 5.7 on the SFBHIADDP is based on [198, 199].

¹A function $\sigma :{\Re }^n\times \Re \to {\Re }^q$ is a Caratheodory function if: (i) σ(x, .) is Lebesgue measurable for all $x\in {\Re }^n;$ (ii) σ(., t) is continuous for each $t\in \Re ;$ and (iii) for each compact set $O\subset {\Re }^n\times \Re ,$ there exists a Lebesgue integrable function $m:\Re \to \Re$ such that $\Vert \sigma \left(x,t\right)\Vert \le m\left(t\right)\forall \left(x,t\right)\in O.$