9

Singular Nonlinear ${\mathcal{H}}_{\infty }$-Control and ${\mathcal{H}}_{\infty }$-Control for Singularly-Perturbed Nonlinear Systems

In this chapter, we discuss the singular nonlinear ${\mathcal{H}}_{\infty }$-control problem. This problem arises when the full control signal is not available in the penalty variable due to some rankdeficiency of the gain matrix, and hence the problem is not well posed. The problem also arises when studying certain robustness issues in parametric or multiplicative uncertain systems.

Two approaches for solving the above problem are: (i) using the regular nonlinear ${\mathcal{H}}_{\infty }$-control techniques discussed in Chapters 5 and 6; and (ii) using high-gain feedback and/or converting the problem to the problem of “almost-disturbance-decoupling” discussed also in Chapter 5. We shall discuss both approaches in the chapter. We shall also discuss the measurement-feedback problem. However, only the continuous-time problem for affine nonlinear systems will be presented. Moreover, the approaches are extensions to similar techniques used for linear systems as discussed in References [237, 252, 253].

Another problem that is in the class of “singular” problems is that of ${\mathcal{H}}_{\infty }$-control of singularly-perturbed systems. These class of systems possess fast and slow modes which are weakly coupled. Such models are also used to represent systems with algebraic constraints, and the solution to the system with constraint is found as the asymptotic limit of the solution to an extended system without constraints. We shall study this problem in the later part of the chapter.

9.1    Singular Nonlinear ${\mathcal{H}}_{\infty }$-Control with State-Feedback

At the outset, we consider affine nonlinear systems defined on a state-space manifold $\chi \subset {\Re }^n$ defined in local coordinates (x₁, …, x_n):

${\tilde{\mathrm{\Sigma }}}^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+g_1\left(x\right)w+{\tilde{g}}_2\left(x\right)\tilde{u};x\left(t_0\right)=x_0\\ z=h_1\left(x\right)+{\tilde{k}}_{12}\left(x\right)\tilde{u}\\ y=x\end{array}$

(9.1)

where $u\in \mathcal{U}\subset {\Re }^p$ is the p-dimensional control input, which belongs to the set of admissible controls $\mathcal{U}\subset {ℒ}_2\left(\left[t_0,T\right],{\Re }^p\right),w\in \mathcal{W}$ is the disturbance signal, which belongs to the set $\mathcal{W}\subset {ℒ}_2([t_0,T],{\Re }^r)$ of admissible disturbances, the output $y\in {\Re }^m$ is the states-vector of the system which are measured directly, and $z\in {\Re }^s$ is the output to be controlled. The functions $f:\chi \to V^{\infty }\chi ,g_1:\chi \to {ℳ}^{n\times r}(\chi ),{\tilde{g}}_2:\chi \to {ℳ}^{n\times p}(\chi ),h_1:\chi \to {\Re }^s$, and ${\tilde{k}}_{12}:\chi \to {ℳ}^{p\times m}(\chi )$ are assumed to be real C^∞-functions of x. Furthermore, we assume that x = 0 is the only equilibrium of the system and is such that f(0) = 0, h₁(0) = 0. We also assume that the system is well defined, i.e., for any initial state $x(t_0)\in \chi$ and any admissible input $u\in \mathcal{U}$, there exists a unique solution x(t, t₀, x₀, u) to (9.1) on [t₀, ∞) which continuously depends on the initial conditions, or the system satisfies the local existence and uniqueness theorem for ordinary differential equations [157].

The objective is to find a static state-feedback control law of the form

$\tilde{u}=\tilde{\alpha }\left(x\right),\tilde{\alpha }\left(0\right)=0$

(9.2)

which achieves locally ${ℒ}_2$-gain from w to z less than or equal to γ^$\star$ > 0 for the closed-loop system (9.1), (9.2) and asymptotic-stability with w = 0.

The problem could have been solved by the techniques of Chapters 5 and 6, if not for the fact that the coefficient matrix in the penalty variable k₁₂ is not full-rank, and this creates the “singularity” to the problem. To proceed, let k = min rank{k₂₁(x)} < p assumed to be constant over the neighborhood $M\subset \chi$ of x = 0. Then, it is possible to find a local diffeomorphism¹ (a local coordinate-transformation)

$u=\phi \left(x\right)\tilde{u}$

which transforms the system (9.1) into

${\mathrm{\Sigma }}^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+g_1\left(x\right)w+g_2\left(x\right)u;x\left(t_0\right)=x_0\hfill \\ z=h_1\left(x\right)+k_{12}\left(x\right)u\hfill \\ y=x\hfill \end{array}$

(9.3)

with

$g_2\left(x\right)={\tilde{g}}_2\left(x\right){\phi }^{-1}\left(x\right),k_{12}\left(x\right)={\tilde{k}}_{12}\left(x\right){\phi }^{-1}\left(x\right)=\left[\begin{array}{cc}{D}_{12}& 0\end{array}\right]$

(9.4)

and $D_{12}\in {\Re }^{s\times \kappa },D_{12}^T{D}_{12}=I$. The control vector can now be partitioned conformably with the partition (9.4) so that $u=\left[\begin{array}{l}{u}_1\\ u_2\end{array}\right]$, where $u_1\in {\Re }^{\kappa },u_2\in {\Re }^{p-\kappa }$, and the system (9.3) is represented as

${\text{Σ}}^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+g_1\left(x\right)w+g_{21}\left(x\right)u_1+g_{22}\left(x\right)u_2;x\left(t_0\right)=x_0\hfill \\ z=h_1\left(x\right)+D_{12}{u}_1\hfill \\ y=x,\hfill \end{array}$

(9.5)

where g₂(x) = [g₂₁(x)   g₂₂(x)], and g₂₁, g₂₂ have compatible dimensions. The problem can now be more formally defined as follows.

Definition 9.1.1 (State-Feedback Singular (Suboptimal) Nonlinear ${\mathcal{H}}_{\infty }$-Control Problem (SFBSNLHICP)). For a given number γ^$\star$ > 0, find (if possible!) a state-feedbback control law of the form

$u=\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]=\left[\begin{array}{c}{\alpha }_{21}\left(x\right)\\ {\alpha }_{22}\left(x\right)\end{array}\right],{\alpha }_{2j}\left(0\right)=0,j=1,2,$

(9.6)

such that the closed-loop system (9.5), (9.6) is locally asymptotically-stable with w = 0, and has locally finite ${ℒ}_2$-gain from w to z less than or equal to γ^$\star$.

The following theorem gives sufficient conditions under which the SFBSNLHICP can be solved using the techniques discussed in Chapter 5.

Theorem 9.1.1 Suppose the state-feedback ${\mathcal{H}}_{\infty }$-control problem for the subsystem

$\begin{array}{l}\dot{x}=f\left(x\right)+g_1\left(x\right)w+g_{21}\left(x\right)u_1;x\left(t_0\right)=x_0\hfill \\ y=x\hfill \\ z=h_1\left(x\right)+k_{12}\left(x\right)u_1,k_{12}^T\left(x\right)k_{12}\left(x\right)=I,h_1^T\left(x\right)k_{12}\left(x\right)=0\hfill \end{array}\}$

(9.7)

is solvable for a given γ > 0, with the control law

$u_1^{\star }=-g_{21}^T\left(x\right)V_x^T\left(x\right),$

where $V:M\to \Re$ is a smooth positive-definite solution of the HJIE:

$V_x\left(x\right)f\left(x\right)+\frac{1}{2}{V}_x\left(x\right)\left[\frac{1}{{\gamma }^2}{g}_1\left(x\right)g_1^T\left(x\right)-g_{21}\left(x\right)-g_{21}^T\left(x\right)\right]V_x^T\left(x\right)+\frac{1}{2}{h}_1^T\left(x\right)h_1\left(x\right)=0,V\left(0\right)=0.$

(9.8)

In addition, suppose there exists a function ${\alpha }_{22}:M\to {\Re }^{p-k}$ such that

$V_x\left(x\right)g_{22}\left(x\right){\alpha }_{22}\left(x\right)\le 0\forall x\in M$

(9.9)

and the pair {f(x) + g₂₂(x)α₂₂(x), h₁(x)} is zero-state detectable. Then, the state-feedback control law

$u=\left[\begin{array}{c}{u}_1^{\star }\left(x\right)\\ u_2^{\star }\left(x\right)\end{array}\right]=\left[\begin{array}{l}-g_{21}^T\left(x\right)V_x^T\\ {\alpha }_{22}\left(x\right)\end{array}\right]$

(9.10)

solves the SFBSNLHICP for the system (9.1) in M.

Proof: Differentiating the solution V > 0 of (9.8) along a trajectory of the closed-loop system (9.5) with the control law (9.10), we get upon using the HJIE (9.8):

$\begin{array}{c}\dot{V}=V_x\left(x\right)\left[f\left(x\right)+g_1\left(x\right)w+g_{21}\left(x\right)u_1+g_{22}\left(x\right)u_2\right]\\ =V_x\left(x\right)\left[f\left(x\right)+g_1\left(x\right)w-g_{21}\left(x\right)g_{21}^T\left(x\right)V_x^T\left(x\right)+g_{22}\left(x\right){\alpha }_{22}\left(x\right)\right]\\ =V_x\left(x\right)f\left(x\right)+V_x\left(x\right)g_1\left(x\right)w-V_x\left(x\right)g_{21}\left(x\right)g_{21}^T\left(x\right)V^T\left(x\right)+V_x\left(x\right)g_{22}\left(x\right){\alpha }_{22}\left(x\right)\\ \le -\frac{1}{2}{V}_x\left(x\right)g_{21}\left(x\right)g_{21}^T\left(x\right)V_x^T\left(x\right)-\frac{1}{2}{h}^T\left(x\right)h\left(x\right)+\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2-\\ \frac{{\gamma }^2}{2}{\Vert w-\frac{g_1^T\left(x\right)V_x^T\left(x\right)}{{\gamma }^2}\Vert }^2\\ \le \frac{1}{2}\left({\gamma }^2{\Vert w\Vert }^2-{\Vert z\Vert }^2\right).\end{array}$

(9.11)

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

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

and therefore the system has ${ℒ}_2$-gain ≤ γ. Further, with w = 0, we get

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

and thus, the closed-loop system is locally stable. Moreover, the condition $\dot{V}\equiv 0$ for all t ≥ t_c, for some t_c ≥ t₀, implies that z ≡ 0 and h₁(x) ≡ 0, $u_1^{\star }\left(x\right)$ ≡ 0 for all t ≥ t_c.Consequently, it is a trajectory of $\dot{x}=f(x)+g_{22}(x){\alpha }_{22}(x)$. By the zero-state detectability of {f(x)+ g₂₂(x)α₂₂(x), h₁(x)}, this implies lim_{t →∞} x(t) = 0. Finally, by LaSalle’s invariance-principle, we conclude asymptotic-stability. □

Remark 9.1.1 The above theorem gives sufficient conditions for the solvability of the SFBSNLHICP for the case in which the regular SFBHICP for the subsystem (9.5) is solvable. Thus, it follows that the control input u₁, which is referred to as the “regular control,” has enough power to provide disturbance attenuation and stability for the system; while the remaining input u₂, which is referred to as the “singular control” can be utilized to achieve additional objectives such as transient performance. If however, this is not the case, then some other approach must be used for solving the singular problem. One approach converts the problem to the “almost-disturbance-decoupling problem” discussed in Chapter 5 [42], while another approach uses high-gain feedback. The latter approach will be discussed in the next subsection.

Remark 9.1.2 Note that the condition (9.9) is fulfilled with u₂ = 0. However, for any other function u₂ = α₂₂(x) ≠ 0 which satisfies (9.9), we have that

${\Vert {\sum }^a\left(u_2={\alpha }_{22}\left(x\right)\ne 0\right)\Vert }_{{ℒ}_2}<{\Vert {\sum }^a\left(u_2=0\right)\Vert }_{{ℒ}_2.}$

In fact, a better choice of u₂ is

$u_2=-R^{-1}{g}_{22}^T\left(x\right)V_x^T\left(x\right)$

for some weighting matrix R > 0.

9.1.1    State-Feedback Singular Nonlinear ${\mathcal{H}}_{\infty }$-Control Using High-Gain Feedback

In this subsection, we discuss an alternative approach to the SFBSNLHICP. For this purpose, rewrite the system (9.3) in the form

${\sum }^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+g_1\left(x\right)w+g_{21}\left(x\right)u_1+g_{22}\left(x\right)u_2;x\left(t_0\right)=x_0\\ z=\left[\begin{array}{l}{h}_1\left(x\right)\\ u_1\end{array}\right]\\ y=x,\end{array}$

(9.12)

where all the variables and functions have their previous meanings, and consider the following auxiliary ε-perturbed version of it:

${\sum }_{\epsilon }^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+g_1\left(x\right)w+g_{21}\left(x\right)u_1+g_{22}\left(x\right)u_2;x\left(t_0\right)=x_0\\ z_{\epsilon }=\left[\begin{array}{l}{h}_1\left(x\right)\\ u_1\\ \sqrt{\epsilon }{u}_2\end{array}\right]\\ y=x.\end{array}$

(9.13)

Then we have the following proposition.

Proposition 9.1.1 Suppose there exists a feedback of the form (9.6) which solves the SFBSNLHICP for the system (9.12) with finite ${ℒ}_2$-gain from w to α₂₂. Then, the closed-loop system (9.12), (9.6) has ${ℒ}_2$-gain ≤ γ if, and only if, the closed-loop system (9.13), (9.6) has ${ℒ}_2$-gain ≤ γ for some sufficiently small ε > 0.

Proof: (⇒) By assumption, the ${ℒ}_2$-gain from w to α₂₂ is finite (assume ≤ ρ < ∞). Thus,

${\displaystyle {\int }_{t_0}^T\epsilon \Vert {\alpha }_{22}\left(x\left(t\right)\right)\Vert }\le \epsilon p^2{\displaystyle {\int }_{t_0}^T{\Vert w\left(t\right)\Vert }^{2}dt\forall T>0.}$

Moreover, the closed-loop system (9.12), (9.6) has ${ℒ}_2$-gain ≤ γ, therefore

${{\displaystyle {\int }_{t_0}^T\Vert z\left(t\right)\Vert }}^2\le {\gamma }^2{{\displaystyle {\int }_{t_0}^T\Vert w\left(t\right)\Vert }}^{2}dt\forall T>0.$

Adding the above two inequalities, we get

${\displaystyle {\int }_{t_0}^T{\Vert z\left(t\right)\Vert }^{2}dt+\epsilon {\Vert {\alpha }_{22}\left(x\left(t\right)\right)\Vert }^2\le {\gamma }^2}{\displaystyle {\int }_{t_0}^T{\Vert w\left(t\right)\Vert }^{2}dt+\epsilon p^2}{{\displaystyle {\int }_{t_0}^T\Vert w\left(t\right)\Vert }}^{2}dt\forall T>0.$

Since $w\in {ℒ}_2[0,\infty )$ and ε is small, there exists a constant β > ∞ such that for all $w\in {ℒ}_2[0,\infty ),\epsilon {\rho }^2{\displaystyle {\int }_{t_0}^T{\Vert w(t)\Vert }^2}dt<\beta$. Therefore,

${{\displaystyle {\int }_{t_0}^T\Vert z_{\epsilon }\left(t\right)\Vert }}^{2}dt\le {\gamma }^2{{\displaystyle {\int }_{t_0}^T\Vert w\left(t\right)\Vert }}^{2}dt+\beta ,$

and hence the result.

(⇐) This is straight-forward. Clearly

$\begin{array}{c}{{\displaystyle {\int }_{t_0}^T\Vert z_{\epsilon }\left(t\right)\Vert }}^{2}dt\le {\gamma }^2{{\displaystyle {\int }_{t_0}^T\Vert w\left(t\right)\Vert }}^{2}dt\\ \Rightarrow {{\displaystyle {\int }_{t_0}^T\Vert z\left(t\right)\Vert }}^{2}dt\le {\gamma }^2{{\displaystyle {\int }_{t_0}^T\Vert w\left(t\right)\Vert }}^{2}dt\square \end{array}$

Based on the above proposition, we can proceed to design a regular stabilizing feedback for the system (9.12) using the system (9.13). For this, we have the following theorem.

Theorem 9.1.2 Consider the nonlinear system Σ^a defined by (9.12) and the SFBSNLHICP for it. Suppose there exists a C¹ solution V ≥ 0 to the HJIE:

$\begin{array}{l}\begin{array}{c}\\ V_x\left(x\right)f\left(x\right)+\frac{1}{2}{V}_x\left(x\right)\left[\frac{1}{{\gamma }^2}{g}_1\left(x\right)g_1^T\left(x\right)-g_{21}\left(x\right)-\frac{1}{\epsilon }{g}_{22}\left(x\right)g_{22}^T\left(x\right)\right]V_x^T\left(x\right)+\end{array}\\ \frac{1}{2}{h}_1^T\left(x\right)h_1\left(x\right)=0,V\left(0\right)=0,\end{array}$

(9.14)

for some γ > 0 and some sufficiently small ε > 0. Then the state-feedback

$u=\left[\begin{array}{l}{u}_1^{\star }\left(x\right)\\ u_2^{\star }\left(x\right)\end{array}\right]=-\left[\begin{array}{c}{g}_{21}^T\left(x\right)\\ \frac{1}{\epsilon }{g}_{22}^T\left(x\right)\end{array}\right]V_x^T\left(x\right)$

(9.15)

solves the SFBSNLHICP for the system.

Proof: It can be shown as in the proof of Theorem 9.1.1 that the state-feedback (9.15) when applied to the system ${\mathrm{\Sigma }}_{\epsilon }^a$ leads to a closed-loop system which is locally asmptotically-stable and has ${ℒ}_2$-gain ≤ γ from w to z_ε. The result then follows by application of Proposition 9.1.1. □

We can specialize the result of Theorem 9.1.2 to the linear system

${\sum }^l:\{\begin{array}{l}\dot{x}=Ax+B_{1}w+B_{21}{u}_1{B}_{22}{u}_2,x\left(t_0\right)=x_0\\ z=\left[\begin{array}{l}{c}_{1}x\\ u_1\end{array}\right]\\ y=x,\end{array}$

(9.16)

where all the variables have their previous meanings, and $A\in {\Re }^{n\times n},B_1\in {\Re }^{n\times r},B_{21}\in {\Re }^{n\times \kappa },B_{22}\in {\Re }^{n\times p-\kappa },C_1\in {\Re }^{s\times n}$ are real constant matrices. We have the following corollary to Theorem 9.1.2.

Corollary 9.1.1 Consider the linear system Σ^l defined by (9.16) and the SFBSNLHICP for it. Suppose there exists a symmetric solution P > 0 to the ARE:

$A^{T}P+PA+P\left[\frac{1}{{\gamma }^2}{B}_1{B}_1^T-B_{21}{B}_{21}^T-\frac{1}{\epsilon }{B}_{22}{B}_{22}^T\right]P+C_1^T{C}_1+ϵQ=0,$

(9.17)

for some matrix Q > 0, γ > 0 and some sufficiently small $ϵ>0$. Then, the state-feedback

$u_1=\left[\begin{array}{l}{u}_{1,l}^{\star }\\ u_{2,l}^{\star }\end{array}\right]=-\left[\begin{array}{l}{B}_{21}^T\\ \frac{1}{\epsilon }{B}_{22}^T\end{array}\right]Px$

(9.18)

solves the SFBSNLHICP for the system.

Remark 9.1.3 Note that there is no assumption of detectability of (C₁, A) in the above corollary. This is because, since $\epsilon Q+C_1^T{C}_1>0$, there exists a C such that $ϵQ+C_1^T{C}_1=C^{T}C$ and (C, A) is always detectable.

9.2    Output Measurement-Feedback Singular Nonlinear ${\mathcal{H}}_{\infty }$-Control

In this section, we discuss a solution to the singular nonlinear ${\mathcal{H}}_{\infty }$-control problem for the system Σ^a using a dynamic measurement-feedback controller of the form:

${\sum }_{dyn}^c:\{\begin{array}{l}\dot{\xi }=\eta \left(\xi \right)+\theta \left(\xi \right)y\\ u=\left[\begin{array}{l}{u}_1\\ u_2\end{array}\right]=\left[\begin{array}{l}{\alpha }_{21}\left(\xi \right)\\ {\alpha }_{22}\left(\xi \right)\end{array}\right],\end{array}$

(9.19)

where $\xi \in \Xi$ is the state of the compensator with $\Xi \subset \chi$ a neighborhood of the origin, $\eta :\Xi \to V^{\infty }(\Xi ),\eta (0)=0$, and $\theta :\Xi \to {ℳ}^{n\times m}$ are some smooth functions. The controller processes the measured variable y of the plant (9.1) and generates the appropriate control action u, such that the closed-loop system ${\mathrm{\Sigma }}^a\circ {\mathrm{\Sigma }}_{dyn}^c$ has locally ${ℒ}_2$-gain from the disturbance signal w to the output z less than or equal to some prescribed number γ^$\star$ > 0 with internal stability. This problem will be abbreviated as MFBSNLHICP.

To solve the above problem, we consider the following representation of the system (9.3) with disturbances and measurement noise:

${\tilde{\mathrm{\Sigma }}}^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+g_1\left(x\right)w_1+g_{21}\left(x\right)u_1+g_{22}\left(x\right)u_2;x\left(t_0\right)=x_0\\ z=\left[\begin{array}{l}{h}_1\left(x\right)\\ u_1\end{array}\right]\\ y=h_2\left(x\right)+w_2,\end{array}$

(9.20)

where $h_2:\mathcal{X}\to {\Re }^m$is a smooth matrix while all the other functions and variables have their previous meanings. As in the previous section, we can proceed to design the controller for the following auxiliary ε-perturbed version of the plant ${\tilde{\mathrm{\Sigma }}}^a$:

${\tilde{\mathrm{\Sigma }}}_{\epsilon }^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+g_1\left(x\right)w_1+g_{21}\left(x\right)u_1+g_{22}\left(x\right)u_2;x\left(t_0\right)=x_0\hfill \\ z=\left[\begin{array}{l}{h}_1\left(x\right)\hfill \\ u_1\hfill \\ \sqrt{\epsilon }{u}_2\hfill \end{array}\right]\hfill \\ y=h_2\left(x\right)+w_2.\hfill \end{array}$

(9.21)

Moreover, it can be shown similarly to Proposition 9.1.1, that under the assumption that the ${ℒ}_2$-gain from w to u₂ is finite, the closed-loop system (9.20), (9.19) has ${ℒ}_2$-gain ≤ γ if and only if the closed-loop system (9.21), (9.19) has ${ℒ}_2$-gain ≤ γ. We can then employ the techniques of Chapter 6 to design a “certainty-equivalent worst-case” dynamic controller ${\mathrm{\Sigma }}_{dyn}^c$ for the system. The following theorem gives sufficient conditions for the solvability of the problem.

Theorem 9.2.1 Consider the nonlinear system ${\tilde{\mathrm{\Sigma }}}^a$ and the MFBSNLHICP. Suppose for some sufficiently small ε > 0 there exist smooth C² solutions V ≥ 0 to the HJIE (9.14) and W ≥ 0 to the HJIE:

$W_x\left(x\right)f\left(x\right)+\frac{1}{2{\gamma }^2}{W}_x\left(x\right)g_1\left(x\right)g_1^T\left(x\right)W_x^T\left(x\right)+\frac{1}{2}{h}_1\left(x\right)h_1^T\left(x\right)-\frac{1}{2}{\gamma }^2{h}_2^T\left(x\right)h_2\left(x\right)=0,W\left(0\right)=0,$

(9.22)

on $M\subset \chi$ such that

$f-g_{21}{g}_{21}^T{V}_x^T-\frac{1}{\epsilon }{g}_{22}\left(x\right)g_{22}^T\left(x\right)V_x^T+\frac{1}{{\gamma }^2}{g}_1{g}_1^T{V}_x^T\mathrm{is\; exponentially\; stable,}$

$-\left(f+\frac{1}{{\gamma }^2}{g}_1{g}_1^T{W}_x^T\right)\mathrm{is\; exponentially\; stable,}$

$W_{xx}\left(x\right)>V_{xx}\left(x\right)\forall \in M.$

Then, the dynamic controller defined by

${\sum }_{dyn}^c:\{\begin{array}{l}\xi =f\left(\xi \right)-g_{21}\left(\xi \right)g_{21}^T\left(\xi \right)V_{\xi }^T\left(\xi \right)-\frac{1}{\epsilon }{g}_{22}\left(\xi \right)g_{22}^T\left(\xi \right)V_{\xi }^T\left(\xi \right)+\\ \frac{1}{{\gamma }^2}{g}_1\left(\xi \right)g_1^T\left(\xi \right)V_{\xi }^T\left(\xi \right)+{\gamma }^2{\left[W_{\xi \xi }\left(\xi \right)-V_{\xi \xi }\left(\xi \right)\right]}^{-1}\frac{\partial h_2}{\partial \xi }\left(\xi \right)\left(y-h_2\left(\xi \right)\right)\\ u=\left[\begin{array}{l}{u}_1\\ u_2\end{array}\right]=-\left[\begin{array}{l}{g}_{21}^T\left(\xi \right)\\ \frac{1}{\epsilon }{g}_{22}^T\left(\xi \right)\end{array}\right]V_{\xi }^T\left(\xi \right)\end{array}$

(9.23)

solves the MFBSNLHICP for the system locally on M.

Proof: (Sketch). For the purpose of the proof, we consider the finite-horizon problem on the interval [t₀, T ], and derive the solution of the infinite-horizon problem by letting T → ∞. Accordingly, we consider the cost-functional:

${}_{u\epsilon \mathcal{U}}^{\min }{}_{w\epsilon \mathcal{W}}^{\max }{J}_m\left(w,u\right)=\frac{1}{2}{\displaystyle {\int }_{t_0}^T\left\{{\Vert u_1\Vert }^2+\epsilon {\Vert u_2\Vert }^2+{\Vert h_1\left(x\right)\Vert }^2-{\gamma }^2{\Vert w_1\Vert }^2-{\gamma }^2{\Vert w_2\Vert }^2\right\}dt}$

where u(τ) depends on y(τ), τ ≤ t. As in Chapter 6, the problem can then be split into two subproblems: (i) the state-feedback subproblem; and (ii) the state-estimation subproblem.

Subproblem (i) has already been dealt with in the previous section leading to the feedbacks (9.15). For (ii), we consider the certainty-equivalent worst-case estimator:

$\dot{\xi }=f\left(\xi \right)+g_1\left(\xi \right){\alpha }_1\left(\xi \right)+g_{21}\left(\xi \right){\alpha }_{21}\left(\xi \right)+g_{22}\left(\xi \right){\alpha }_{22}\left(\xi \right)+G_s\left(\xi \right)\left(y-h_2\left(\xi \right)\right),$

where ${\alpha }_1\left(x\right)=w^{\star }=\frac{1}{{\gamma }^2}{g}_1\left(x\right)g_1^T\left(x\right)V_x^T\left(x\right)$ is the worst-case disturbance, ${\alpha }_{21}\left(x\right)=u_1^{\star }\left(x\right),{\alpha }_{22}\left(x\right)=u_2^{\star }\left(x\right)\text{and}{G}_s(.)$ is the output-injection gain matrix. Then, we can design the gain matrix G_s(.) to minimize the estimation error e = y − h₂(ξ) and render the closed-loop system asymptotically-stable.

Acordingly, let ${\tilde{u}}_1(t),{\tilde{u}}_2(t)$ and ${\tilde{y}}_1(t),t\in [t_0,\tau ]$ be a given pair of inputs and corresponding measured output. Then, we consider the problem of maximizing the cost functional

${\tilde{J}}_m\left(w,u\right)=V\left(x\left(\tau \right),\tau \right)+\frac{1}{2}{\displaystyle {\int }_{t_0}^{\tau }\left\{{\Vert {\tilde{u}}_1\Vert }^2+\epsilon {\Vert {\tilde{u}}_2\Vert }^2+{\Vert h_1\left(x\right)\Vert }^2-{\gamma }^2{\Vert w_1\Vert }^2-{\gamma }^2{\Vert w_2\Vert }^2\right\}dt,}$

with respect to x(t), w₁(t), w₂(t), where V (x, τ) is the value-function for the first optimization subproblem to determine the state-feedbacks, and subject to the constaint that the output of the system (9.21) equals $\tilde{y}(t)$. Moreover, since w₂ directly affects the observation y, we can substitute $w_2=h_2(x)-\tilde{y}$ into the cost functional ${\tilde{J}}_m(.,.)$ such that the above constraint is automatically satisfied. The resulting value-function for this maximization subproblem is then given by S(x, τ) = V (x, τ) − W (x, τ), where W ≥ 0 satisfies the HJIE

$\begin{array}{l}{W}_t\left(x,t\right)+W_x\left(x,t\right)\left[f\left(x\right)+g_{21}\left(x\right){\tilde{u}}_1\left(t\right)+g_{22}\left(x\right){\tilde{u}}_2\left(t\right)\right]+\frac{1}{2}{h}_1^T\left(x\right)h_1\left(x\right)\\ \frac{1}{2{\gamma }^2}{W}_x\left(x,t\right)g_1\left(x\right)g_1^T\left(x\right)W_x^T\left(x,t\right)+-\frac{1}{2}{\gamma }^2{h}_2^T\left(x\right)h_2\left(x\right)+{\gamma }^2{h}_2^T\left(x\right)\tilde{y}\left(t\right)-\\ \frac{1}{2}{\gamma }^2{\Vert \tilde{y}\left(t\right)\Vert }^2+\frac{1}{2}{\Vert {\tilde{u}}_1\left(t\right)\Vert }^2+\frac{1}{2}\epsilon {\Vert {\tilde{u}}_2\left(t\right)\Vert }^2=0,W\left(x,t_0\right)=0.\end{array}$

(9.24)

Assuming that the maximum of S(.,.) is determined by the condition S_x(ξ(t), t) = 0 and that the Hessian is nondegenerate,² then the corresponding state-equation for ξ can be found by differentiation of S_x(ξ(t), t) = 0.

Finally, we obtain the controller which solves the infinite-horizon problem by letting T → ∞ while imposing that x(t) → 0, and t₀ → −∞ while x(t₀) → 0. A finite-dimensional approximation to this controller is given by (9.23). □

9.3    Singular Nonlinear ${\mathcal{H}}_{\infty }$-Control with Static Output-Feedback

In this section, we briefly extend the static output-feedback approach developed in Section 6.5 to the case of singular control for the affine nonlinear system. In this regard, consider the following model of the system with the penalty variable defined in the following form

${\sum }_1^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+g_1\left(x\right)w+{\tilde{g}}_2\left(x\right)\tilde{u}\\ y=h_2\left(x\right)\\ z=\left[\begin{array}{l}{h}_1\left(x\right)\\ {\tilde{k}}_{12}\left(x\right)\tilde{u}\end{array}\right],\end{array}$

(9.25)

where all the variables and functions have their previous meanings and dimensions, and min rank{k₁₂(x)} = κ < p (assuming it is constant for all x). Then as discussed in Section 9.1, in this case, there exists a coordinate transformation $\phi$ under which the system can be represented in the following form [42]:

${\sum }_2^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+g_1\left(x\right)w+g_{21}\left(x\right)u_1+g_{22}\left(x\right)u_2\\ y=h_2\left(x\right)\\ z=\left[\begin{array}{l}{h}_1\left(x\right)\\ D_{12}{u}_1\end{array}\right],D_{12}^T{D}_{12}=I,\end{array}$

(9.26)

where $g_2=[g_{21}(x)g_{22}],D_{12}\in {\Re }^{s-k\times k},u_1\in {\Re }^k$ is the regular control, and $u_2\in {\Re }^{p-k}$, is the singular control. Furthermore, it has been shown that the state-feedback control law

$u=\left[\begin{array}{l}{u}_1\\ u_2\end{array}\right]=\left[\begin{array}{l}-g_{21}^T\left(x\right){\tilde{V}}_x^T\left(x\right)\\ {\alpha }_{22}\left(x\right)\end{array}\right],$

(9.27)

where α₂₂(x) is such that

$V_x\left(x\right)g_{22}\left(x\right){\alpha }_{22}\left(x\right)\le 0$

and $\tilde{V}>0$ is a smooth local solution of the HJIE (9.8), solves the state-feedback singular ${\mathcal{H}}_{\infty }$-control problem (SFBSNLHICP ) for the system (9.25) if the pair {f + g₂₂α, h₁ } is locally zero-state detectable. The following theorem then shows that, if the state-feedback singular ${\mathcal{H}}_{\infty }$-control problem is locally solvable with a regular control, then a set of conditions similar to (6.91), (6.92) are sufficient to guarantee the solvability of the SOFBP.

Theorem 9.3.1 Consider the nonlinear system (9.25) and the singular ${\mathcal{H}}_{\infty }$-control problem for this system. Suppose the SFBSNLHICP is locally solvable with a regular control and there exist C⁰ functions $F_3:\mathcal{Y}\subset {\Re }^m\to {\Re }^{\kappa },{\varphi }_3:X_2\subset \mathcal{X}\to {\Re }^p,{\psi }_3:\mathcal{Y}\to {\Re }^{p-\kappa }$ such that the conditions

$F_3\circ h_2\left(x\right)=-g_{21}^T\left(x\right){\tilde{V}}_x^T\left(x\right)+{\varphi }_3\left(x\right),$

(9.28)

${\tilde{V}}_x\left(x\right)\left[g_{21}\left(x\right){\varphi }_3\left(x\right)+g_{22}\left(x\right){\psi }_3\left(h_2\left(x\right)\right)\right]\le 0,$

(9.29)

are satisfied. In addition, suppose also the pair {f(x)+g₂₂(x)ψ₃ ◦h₂(x), h₁(x)} is zero-state detectable. Then, the static output-feedback control law

$u=\left[\begin{array}{l}{u}_1\\ u_2\end{array}\right]=\left[\begin{array}{l}{F}_3\left(y\right)\\ {\psi }_3\left(y\right)\end{array}\right]$

(9.30)

solves the singular ${\mathcal{H}}_{\infty }$-control problem for the system locally.

Proof: Differentiating the solution $\tilde{V}>0$ 0 of (9.8) along a trajectory of the closed-loop system (9.25) with the control law (9.30), we get upon using (9.28), (9.29) and the HJIE (9.8):

$\begin{array}{l}\dot{\tilde{V}}={\tilde{V}}_x\left(x\right)\left[f\left(x\right)+g_1\left(x\right)w+g_{21}\left(x\right)(F_3\left(h_2\left(x\right)\right)+g_{22}\left(x\right){\psi }_3\left(y\right)\right]\\ ={\tilde{V}}_x\left(x\right)\left[f\left(x\right)+g_1\left(x\right)w-g_{21}\left(x\right)g_{21}^T\left(x\right){\tilde{V}}_x^T\left(x\right)+g_{21}\left(x\right){\varphi }_3\left(x\right)+g_{22}\left(x\right){\psi }_3\left(y\right)\right]\\ ={\tilde{V}}_x\left(x\right)f\left(x\right)+{\tilde{V}}_x\left(x\right)w-{\tilde{V}}_x\left(x\right)g_{21}\left(x\right)g_{21}^T\left(x\right){\tilde{V}}^T\left(x\right)+\\ {\tilde{V}}_x\left(x\right)\left(g_{21}\left(x\right){\varphi }_3\left(x\right)+{\psi }_3\left(h_2\left(x\right)\right)\right)\\ \le -\frac{1}{2}{\tilde{V}}_x\left(x\right)g_{21}\left(x\right)g_{21}^T\left(x\right){\tilde{V}}^T\left(x\right)-\frac{1}{2}{h}_1^T\left(x\right)h_1\left(x\right)+\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2-\\ \frac{{\gamma }^2}{2}{\Vert w-\frac{g_1^T\left(x\right){\tilde{V}}_x^T\left(x\right)}{{\gamma }^2}\Vert }^2\\ \le \frac{1}{2}\left({\gamma }^2{\Vert w\Vert }^2-{\Vert z\Vert }^2\right).\end{array}$

(9.31)

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

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

and therefore the system has ${ℒ}_2$-gain ≤ γ. Moreover, with w = 0, we get

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

and thus, the closed-loop system is locally stable. Moreover, the condition $\dot{\tilde{V}}\equiv 0$ for all t ≥ t_c, for some t_c ≥ t₀, implies that z ≡ 0 and h₁(x) = 0, u (x) = 0 for all t ≥ t_c. Consequently, it is a trajectory of $\dot{x}=f\left(x\right)+g_{22}\left(x\right){\psi }_3\left(h_2\left(x\right)\right)$. By the zero-state detectability of $\{f(x)+g_{22}(x){\psi }_3(h_2(x)),h_1(x)\}$, this implies lim_t→∞ x(t) = 0, and using LaSalle’s invariance-principle we conclude asymptotic-stability. □

On the other hand, if the singular problem is not solvable using the regular control above, then it has been shown in Section 9.1.1 that a high-gain feedback can be used to solve it. Thus, similarly, the results of Theorem 9.1.2 can be extended straightforwardly to the static-output feedback design. It can easily be guessed that if there exist C⁰ functions $F_4:\mathcal{Y}\subset {\Re }^m\to \mathcal{U},{\varphi }_4:X_4\subset \chi \to {\Re }^p,{\psi }_4:\mathcal{Y}\subset {\Re }^{p-k}$, such that the conditions

$F_4\circ h_2\left(x\right)=-\left[\begin{array}{l}{g}_{21}^T\left(x\right)\\ \frac{1}{\epsilon }{g}_{22}^T\left(x\right)\end{array}\right]{\tilde{V}}_x^T\left(x\right)+{\varphi }_4\left(x\right),$

(9.32)

${\tilde{V}}_x\left(x\right)\left[g_{21}\left(x\right){\varphi }_4\left(x\right)+g_{22}\left(x\right){\psi }_4\left(h_2\left(x\right)\right)\right]\le 0,$

(9.33)

where $\tilde{V}$ is a local solution of the HJIE (9.8), are satisfied, then the output-feedback control law

$u=\left[\begin{array}{l}{F}_4\left(y\right)\\ {\psi }_4\left(y\right)\end{array}\right]$

(9.34)

solves the singular ${\mathcal{H}}_{\infty }$-control problem for the system locally.

9.4    Singular Nonlinear ${\mathcal{H}}_{\infty }$-Control for Cascaded Nonlinear Systems

In this section, we discuss the SFBSNLHICP for a fairly large class of cascaded nonlinear systems that are totally singular, and we extend some of the results presented in Section 9.1 to this case. This class of systems can be represented by an aggregate state-space model defined on a manifold $\chi \subset {\Re }^n:$

${\sum }_{agg}^a:\{\begin{array}{l}\dot{x}=f\left(x\right)+g_1\left(x\right)w+g_2\left(x\right)u;x\left(t_0\right)=x_0\\ z=h\left(x\right)\\ y=x,\end{array}$

(9.35)

where all the variables have their previous meanings, while $g_1:\chi \to {\mathcal{M}}^{n\times r},g_2:\chi \to {\Re }^{n\times p},h:\chi \to {\Re }^s$ are C^∞ function of x. We assume that the system has a unique equilibrium-point x = 0, and f(0) = 0, h(0) = 0. In addition, we also assume that there exist local coordinates $\left[\begin{array}{cc}{x}_1^T& x_2^T\end{array}\right]=\left[x^1,\mathrm{\dots },x^q,x^{q+1},\mathrm{\dots },x^n\right],1\le q<n$ such that the system ${\mathrm{\Sigma }}_{agg}^a$ can be decomposed into

${\sum }_{dec}^a:\{\begin{array}{l}{x}_1=f_1\left(x_1\right)+g_{11}\left(x_1\right)w+g_{21}\left(x_1\right)h_2\left(x_1,x_2\right)x_1\left(t_0\right)x=x_{10}\\ x_2=f_2\left(x_1,x_2\right)+g_{12}\left(x_1,x_2\right)w+g_{22}\left(x_1,x_2\right)u;x_2\left(t_0\right)=x_{20}\\ z=\left[\begin{array}{l}{z}_1\\ z_2\end{array}\right]=\left[\begin{array}{l}{h}_1\left(x_1\right)\\ h_2\left(x_1,x_2\right)\end{array}\right]\\ y=x\end{array}$

(9.36)

where z₁ ∈ ℜ^s1, z₂ ∈ℜ^s2, s₁ ≥ 1, s = s₁ + s₂. Moreover, in accordance with the representation (9.35), we have

$\begin{array}{l}f\left(x\right)=\left[\begin{array}{l}{f}_1\left(x_1\right)+g_{21}\left(x_1\right)z_2\\ f_2\left(x_1,x_2\right)\end{array}\right],g_1\left(x\right)=\left[\begin{array}{l}{g}_{11}\left(x_1\right)\\ g_{12}\left(x_1,x_2\right)\end{array}\right],\\ g_2\left(x\right)=\left[\begin{array}{l}0\\ g_{22}\left(x_1,x_2\right)\end{array}\right],h\left(x\right)=\left[\begin{array}{l}{h}_1\left(x_1\right)\\ h_2\left(x_1,x_2\right)\end{array}\right].\end{array}$

The decomposition (9.36) can also be viewed as a cascade of two subsystems:

$\begin{array}{l}{\sum }_1^a:\{\begin{array}{l}\dot{x}=f_1\left(x_1\right)+g_{11}\left(x_1\right)w+g_{21}\left(x_1\right)z_2;\\ z_1=h_1\left(x_1\right)\end{array}\\ {\sum }_2^a:\{\begin{array}{l}{\dot{x}}_2=f_2\left(x_1,x_2\right)+g_{12}\left(x_1,x_2\right)w+g_{22}\left(x_1,x_2\right)u\\ z_2=h_2\left(x_1,x_2\right)\end{array}\end{array}$

Such a model represents many physical dynamical systems we encouter everyday; for example, in mechnical systems, the subsystem ${\mathrm{\Sigma }}_1^a$ can represent the kinematic subsystem, while ${\mathrm{\Sigma }}_2^a$ represents the dynamic subsystem.

To solve the SFBSNLHICP for the above system, we make the following assumption.

Assumption 9.4.1 The SFBHICP for the subsystem ${\mathrm{\Sigma }}_1^a$ is solvable, i.e., there exists a smooth solution P : M₁ → ℜ, P ≥ 0, M₁ ⊂ $\mathcal{X}$ a neighborhood of x₁ = 0 to the HJI-inequality

$\begin{array}{l}{P}_{x_1}\left(x_1\right)f_1\left(x_1\right)+\frac{1}{2}{P}_{x_1}\left(x_1\right)\left[\frac{1}{{\gamma }^2}{g}_{11}\left(x_1\right)g_{11}^T\left(x_1\right)-g_{21}\left(x_1\right)g_{21}^T\left(x_1\right)\right]P_{x_1}^T\left(x_1\right)+\\ \frac{1}{2}{h}_1^T\left(x_1\right)h_1\left(x_1\right)\le 0,P\left(0\right)=0,\end{array}$

(9.37)

such that z₂(x) viewed as the control is given by

$z_2^{\star }=-g_{21}^T\left(x_1\right)P_{x_1}^T\left(x_1\right),$

and the worst-case disturbance affecting the subsystem is also given by

$w_1^{\star }=\frac{1}{{\gamma }^2}{g}_{11}^T\left(x_1\right)P_{x_1}^T\left(x_1\right).$

We can now define the following auxiliary system

${\overline{\sum }}_{dec}^a:\{\begin{array}{l}{\dot{x}}_1=f_1\left(x_1\right)+g_{11}\left(x_1\right)\overline{w}+g_{21}\left(x_1\right)z_{21}\left(x_1\right)z_2+\frac{1}{{\gamma }^2}{g}_{11}\left(x_1\right)P_{x_1}^T\left(x_1\right)\\ {\dot{x}}_2=f_2\left(x\right)+g_{12}\left(x\right)\overline{w}+g_{22}\left(x\right)u+\frac{1}{{\gamma }^2}{g}_{12}\left(x\right)g_{11}^T\left(x\right)P_{x_1}^T\left(x_1\right)\\ \overline{z}=z_2-z_2^{\star }=h_2\left(x\right)+g_{11}^T\left(x_1\right)P_{x_1}^T\left(x_1\right).\end{array}$

(9.38)

where $\overline{z}\in {\Re }^{s-s_1}$ and

$\tilde{w}=w-\frac{1}{{\gamma }^2}{g}_{11}^T\left(x_1\right)P_{x_1}^T\left(x_1\right).$

Now define

$\begin{array}{l}\overline{h}\left(x\right)=z=z_2-z_2^{\star }=h_2\left(x\right)+g_{21}^T\left(x_1\right)P_{x_1}^T\left(x_1\right)\\ \overline{f}\left(x\right)=\left[\begin{array}{l}{f}_1\left(x_1\right)+g_1\left(x_1\right)z_2+\frac{1}{{\gamma }^2}{g}_{11}\left(x_1\right)g_{11}^T{P}_{x_1}^T\left(x_1\right)\\ f_2\left(x\right)+\frac{1}{{\gamma }^2}{g}_{12}^T\left(x\right)g_{11}^T\left(x_1\right)P_{x_1}^T\left(x_1\right)\end{array}\right].\end{array}$

Then, (9.38) can be represented in the aggregate form

${\overline{\sum }}_{agg}^a:\{\begin{array}{l}\dot{x}=\overline{f}\left(x\right)+g_1\left(x\right)\overline{w}+g_2\left(x\right)u;\overline{f}\left(0\right)=0\\ \overline{z}=\overline{h}\left(x\right),\tilde{h}\left(0\right)=0,\end{array}$

(9.39)

and we have the following lemmas.

Lemma 9.4.1 For the system representations ${\mathrm{\Sigma }}_{agg}^a$ (9.35) and ${\overline{\mathrm{\Sigma }}}_{agg}^a$ (9.39), the following inequality holds:

$\frac{1}{2}{\Vert \overline{z}\Vert }^2\ge p_{x_1}\left(x_1\right)\left(f\left(x_1\right)+g_{21}\left(x_1\right)z_2\right)+\frac{1}{2{\gamma }^2}{\Vert g_{11}^T\left(x_1\right)P_{x_1}^T\left(x_1\right)\Vert }^2+\frac{1}{2}{\Vert z\Vert }^2.$

(9.40)

Proof: Note,

$\frac{1}{2}{\Vert \overline{z}\Vert }^2=\frac{1}{2}{\Vert z_2-z_2^{\star }\Vert }^2=P_x{}_1\left(x_1\right)g_{21}\left(x_1\right)z_2+\frac{1}{2}{\Vert z_2\Vert }^2+\frac{1}{2}{\Vert g_{21}^T\left(x_1\right)P_{x_1}^T\left(x_1\right)\Vert }^2,$

and from the HJI-inequality (9.37), we get

$\frac{1}{2}{\Vert g_{21}^T\left(x_1\right)P_{x_1}^T\left(x_1\right)\Vert }^2\ge P_{x_1}\left(x_1\right)f_1\left(x_1\right)+\frac{1}{2{\gamma }^2}{\Vert g_{11}^T\left(x_1\right)P_{x_1}^T\left(x_1\right)\Vert }^2+\frac{1}{2}{\Vert z_1\Vert }^2.$

Upon inserting this in the previous equation, the result follows. □

Lemma 9.4.2 Let $K_{\gamma }:T^{\star }\mathcal{X}\times \mathcal{U}\times \mathcal{W}\to \Re$ be the pre-Hamiltonian for the system ${\mathrm{\Sigma }}_{agg}^a$ defined by

$K_{\gamma }\left(x,p,u,w\right)=p^T\left(f\left(x\right)+g_1\left(x\right)w+g_2\left(x\right)u\right)+\frac{1}{2}{\Vert z\Vert }^2-\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2,$

where (x¹,…,xⁿ)^T, (p¹,…, pⁿ)^T are local coordinates for $T^{\star }\mathcal{X}$. Similarly, let ${\overline{K}}_{\gamma }:T^{\star }\mathcal{X}\times \mathcal{U}\times \mathcal{W}\to \Re$ be the pre-Hamiltonian for ${\overline{\mathrm{\Sigma }}}_{agg}^a$. Then

$K_{\gamma }\left(x,p+P_x^T,u,w\right)\le {\overline{K}}_{\gamma }\left(x,p,u,\overline{w}\right).$

(9.41)

Proof: We note that $\overline{f}(x)+g_1(x)\overline{w}=f(x)+g_1(x)w$. Then

$\begin{array}{l}{\overline{K}}_{\gamma }\left(x,p,u,\overline{w}\right)=p^T\left(\overline{f}\left(x\right)+g_1\left(x\right)\overline{w}+g_2\left(x\right)u\right)+\frac{1}{2}{\Vert \overline{z}\Vert }^2-\frac{1}{2}{\gamma }^2{\Vert \overline{w}\Vert }^2\\ =p^T\left(f\left(x\right)+g_1\left(x\right)w+g_2\left(x\right)u\right)+\frac{1}{2}{\Vert \overline{z}\Vert }^2-\frac{1}{2}{\gamma }^2{\Vert \overline{w}\Vert }^2\\ \ge p^T\left(f\left(x\right)+g_1\left(x\right)w+g_2\left(x\right)u\right)+P_{x_1}\left(x_1\right)\left(f_1\left(x_1\right)+g_1\left(x_1\right)z_2\right)+\\ \frac{1}{2{\gamma }^2}{\Vert g_{11}^T\left(x_1\right)P_{x_1}\left(x_1\right)\Vert }^2+\frac{1}{2}{\Vert z\Vert }^2-\frac{1}{2}{\gamma }^2{\Vert \overline{w}\Vert }^2,\end{array}$

(9.42)

where the last inequality follows from Lemma 9.4.1. Noting that

$\begin{array}{l}\frac{1}{2}{\gamma }^2{\Vert \overline{w}\Vert }^2=\frac{1}{2}{\gamma }^2{\left(w-\frac{1}{2{\gamma }^2}{g}_{11}^T\left(x_1\right)P_{x_1}^T\left(x_1\right)\right)}^T\left(w-\frac{1}{2{\gamma }^2}{g}_{11}^T\left(x_1\right)P_{x_1}^T\left(x_1\right)\right)\\ =\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2+\frac{1}{2{\gamma }^2}{\Vert g_{11}^T\left(x_1\right)P_{x_1}^T\left(x\right)\Vert }^2-P_{x_1}^T\left(x_1\right)g_{11}\left(x_1\right)w,\end{array}$

then substituting the above expression in the inequality (9.42), gives

$\begin{array}{l}{\overline{K}}_{\gamma }\left(x,p,u,\overline{w}\right)\ge p^T\left(f\left(x\right)+g_1\left(x\right)w+g_2\left(x\right)u\right)+P_{x1}\left(f_1\left(x_1\right)+g_1\left(x_1\right)z_2+g_{11}\left(x_1\right)w\right)+\\ \frac{1}{2}{\Vert z\Vert }^2-\frac{1}{2}{\gamma }^2{\Vert w\Vert }^2\\ =K_{\gamma }\left(x,p+P_{x_1}\left(x_1\right),u,w\right).\square \end{array}$

As a consequence of the above lemma, we have the following theorem.

Theorem 9.4.1 Let γ > 0 be given, and suppose there exists a static state-feedback control

$u=\alpha \left(x\right),\alpha \left(0\right)=0$

that solves the SFBSNLHICP for the system ${\overline{\mathrm{\Sigma }}}_{agg}^a$ such that there exists a C¹ solution $W:M\to \Re ,W\ge 0,M\subset \chi$ to the dissipation-inequality

${\overline{K}}_{\gamma }\left(x,W_x^T\left(x\right),\alpha \left(x\right),\overline{w}\right)\le 0,W\left(0\right),\forall \overline{w}\in \mathcal{W}.$

Then, the same control law also solves the SFBSNLHICP for the system ${\mathrm{\Sigma }}_{agg}^a$ and the dissipation-inequality

$K_{\gamma }\left(x,V_x^T\left(x\right),\alpha \left(x\right),w\right)\le 0,V\left(0\right)=0,\forall w\in \mathcal{W}$

is satisfied as well for a nonnegative C¹ function $V:M\to \Re$, such that V=W+P.

Proof: Substituting $W_x^T$ and u = α(x) in (9.41), we get

$K_{\gamma }\left(x,W_x^T\left(x\right)+P_x^T,\alpha \left(x\right),w\right)\le {\overline{K}}_{\gamma }\left(x,W_x^T\left(x\right),\alpha \left(x\right),\overline{w}\right)\le 0\forall \overline{w}\in \mathcal{W}$

(9.43)

$\Rightarrow {\overline{K}}_{\gamma }\left(x,V_x^T\left(x\right),\alpha \left(x\right),w\right)\le 0\forall w\in \mathcal{W}.$

(9.44)

Moreover, V (0) = 0 also, and a solution to the HJIE for the SFBHICP in Chapter 5. □

Remark 9.4.1 Again as a consequence of Theorem (9.4.1), we can consider the SFBSNLHICP for the auxiliary system ${\overline{\mathrm{\Sigma }}}_{agg}^a$ instead of that of ${\mathrm{\Sigma }}_{agg}^a$. The benefit of solving the former is that the penalty variable $\overline{z}$ has lower dimension than z. This implies that, while the s₂ × p matrix L_g2$\overline{h}$(x) may have full-rank, the s × p matrix L_g2h(x) is always rank deficient. Consequently, the former can be strongly input-output decoupled by the feedback

$u=L_{g_2}^{-1}\overline{h}\left(x\right)\left(\upsilon -L_f\overline{h}\left(x\right)\right).$

This feature will be used in solving the SFBSNLHICP.

The following theorem then gives a solution to the SFBSNLHICP for the system ${\mathrm{\Sigma }}_{agg}^a$.

Theorem 9.4.2 Consider the system ${\overline{\mathrm{\Sigma }}}_{agg}^a$ and the SFBSNLHICP. Suppose s₂ = p and the s₂ × s₂ matrix $L_{g2}\overline{h}(x)$ is invertible for all x ∈ M. Let

$C\left(x\right)=I+\frac{s_2}{4{\gamma }^2}D\left(x\right),$

where

$D\left(x\right)=diag\left\{{\Vert L_{g_1}^T{\overline{h}}_1\left(x\right)\Vert }^2,\mathrm{\dots },{\Vert L_{g_1}^T{\overline{h}}_{s_2}\left(x\right)\Vert }^2\right\}.$

Then, the state-feedback

$u=\alpha \left(x\right)=-L_{g_2}\overline{h}\left(x\right)\left[L_{\overline{f}}\overline{h}\left(x\right)+C\left(x\right)\overline{h}\left(x\right)\right]$

(9.45)

renders the differential dissipation-inequality

${\overline{K}}_{\gamma }\left(x,W_x^T\left(x\right),\alpha \left(x\right),\overline{w}\right)\le 0,W\left(0\right)=0,\forall \overline{w}\in \mathcal{W}$

satisfied for

$W\left(x\right)=\frac{1}{4}{\overline{h}}^T\left(x\right)\overline{h}\left(x\right).$

Consequently, by Theorem 9.4.1, the same feedback control also solves the SFBSNLHICP for the system ${\mathrm{\Sigma }}_{agg}^a$ with

$V\left(x\right)=\frac{1}{4}{\overline{h}}^T\left(x\right)\overline{h}\left(x\right)+P\left(x\right)$

satisfying

$K_{\gamma }\left(x,V_x^T\left(x\right),\alpha \left(x\right),w\right)\le 0,V\left(0\right)=0\forall w\in \mathcal{W}.$

Proof: Consider the closed-loop system (9.39), (9.45). Then using completion of squares (see also [199]), we have

$\begin{array}{l}{\overline{K}}_{\gamma }\left(x,W_x^T\left(x\right),\alpha \left(x\right),\overline{w}\right)=W_x\left(x\right)\left[\overline{f}\left(x\right)+g_1\left(x\right)\overline{w}+g_2\left(x\right)\alpha \left(x\right)\right]-\frac{1}{2}{\gamma }^2{\Vert \overline{w}\Vert }^2+\frac{1}{2}{\Vert \overline{z}\Vert }^2\\ =\frac{1}{2}{\overline{h}}^T\left(x\right)\left[L_{\overline{f}}\overline{h}\left(x\right)+L_{g_1}\overline{h}\left(x\right)\overline{w}+L_{g_2}\overline{h}\left(x\right)\alpha \left(x\right)\right]-\frac{1}{2}{\gamma }^2{\Vert \overline{w}\Vert }^2+\frac{1}{2}{\Vert \overline{z}\Vert }^2\\ =\frac{1}{2}{\overline{h}}^T\left(x\right)\left[-C\left(x\right)\overline{h}\left(x\right)+L_{g_1}\overline{h}\left(x\right)\overline{w}\right]-\frac{1}{2}{\gamma }^2{\Vert \overline{w}\Vert }^2+\frac{1}{2}{\Vert \overline{z}\Vert }^2\\ =-\frac{1}{2}\frac{s_2}{4{\gamma }^2}{\overline{h}}^T\left(x\right)D\left(x\right)\overline{h}\left(x\right)+\frac{1}{2}{\overline{h}}^T\left(x\right)L_{g_1}\overline{h}\left(x\right)\overline{w}-\frac{1}{2}{\gamma }^2{\Vert \overline{w}\Vert }^2\\ =-\frac{1}{2}\frac{s_2}{{\gamma }^2}{\displaystyle \sum _{i=1}^{s_2}\left[\frac{1}{4}{\overline{h}}_i^2\left(x\right){\Vert L_{g_1}^T\overline{h}\left(x\right)\Vert }^2-\frac{{\gamma }^2}{s_2}{\overline{h}}_i\left(x\right)L_{g_1}\overline{h}\left(x\right)\overline{w}+\frac{{\gamma }^4}{s_2^2}{\Vert \overline{w}\Vert }^2\right]}\\ =-\frac{1}{2}\frac{s_2}{{\gamma }^2}{\displaystyle \sum _{i=1}^{s_2}\left[{\Vert \frac{1}{2}{\overline{h}}_i\left(x\right)L_{g_1}^T\overline{h}\left(x\right)-\frac{{\gamma }^2}{s_2}\overline{w}\Vert }^2\right]}\le 0.\end{array}$

Moreover, W (0) = 0, since $\overline{h}(0)=0$, and the result follows. □

We apply the results developed above to a couple of examples.

Example 9.4.1 [88]. Consider the system ${\mathrm{\Sigma }}_{agg}^a$, and suppose the subsystem ${\mathrm{\Sigma }}_1^a$ is passive (Chapter 3), i.e., it satisfies the KYP property:

$V_{1,x_1}\left(x_1\right)f_1\left(x_1\right)\le 0,V_{1,x_1}\left(x_1\right)g_{21}\left(x_1\right)=h_1^T\left(x_1\right),V_1\left(0\right)=0$

for some storage-function V₁ ≥ 0. Let also

$g_{11}\left(x_1\right)=c{g}_{21}\left(x_1\right),c\in \Re ,and\gamma >\sqrt{\left|c\right|}.$

Then the function

$p_1\left(x_1\right)=\frac{\gamma }{\sqrt{{\gamma }^2-c^2}}{V}_1\left(x_1\right)$

solves the HJI-inequality (9.37) and

$z_2^{\star }=-\frac{\gamma }{\sqrt{{\gamma }^2-c^2}}{h}_1\left(x_1\right).$

Therefore,

$\overline{f}\left(x\right)=\left[\begin{array}{l}{f}_1\left(x_1\right)+g_{21}\left(x_1\right)\left(z_2+\frac{c^2}{\gamma \sqrt{{\gamma }^2-c^2}}{h}_1\left(x_1\right)\right)\\ f_2\left(x\right)+\frac{c^2}{\gamma \sqrt{{\gamma }^2-c^2}}{g}_{12}\left(x\right)h_1\left(x_1\right)\end{array}\right]$

(9.46)

$\overline{h}\left(x\right)=z_2+\frac{\gamma }{\sqrt{{\gamma }^2-c^2}}.$

(9.47)

Example 9.4.2 [88]. Consider a rigid robot dynamics given by the following state-space model:

${\sum }_r^a:\{\begin{array}{l}{\dot{x}}_1=x_2\\ {\dot{x}}_2=-{\overline{M}}^{-1}\left[C\left(x_1,x_2\right)x_2+e\left(x_1\right)\right]+{\overline{M}}^{-1}\left(x_1\right)w+{\overline{M}}^{-1}\left(x_1\right)u\\ z=\left[\begin{array}{l}{x}_1\\ x_2\end{array}\right],\end{array}$

(9.48)

where x₁x₂ ∈ ℜⁿ represent joint positions and velocities,

$\overline{M}$(x₁) is the positive-definite inertia matrix, C(x₁, x₂)x₂ is the coriolis and centrifugal forces, while e(x₁) is the gravity load.

The above model ${\mathrm{\Sigma }}_r^a$ of the robot can be viewed as a cascade system with a kinematic subsystem ${\mathrm{\Sigma }}_{\mathrm{r1}}^a$ represented by the state x₁, and a dynamic subsystem represented by the state x₂. Moreover, g₁₁(x₁) = 0, c = 0, and applying the results of Theorem 9.4.2, we get

$\begin{array}{l}\overline{f}\left(x\right)=\left[\begin{array}{l}{x}_2\\ -M^{-1}\left(x\right)(C\left(x_1,x_2\right)x_2+e\left(x_1\right)\end{array}\right]\\ g_2\left(x\right)=g_1\left(x\right)=\left[\begin{array}{l}0\\ M^{-1}\left(x\right)\end{array}\right],\overline{h}\left(x\right)=x_1+x_2.\end{array}$

Furthermore, s₂ = m = n and $L_{g_2}\overline{h}\left(x\right)={\overline{M}}^{-1}\left(x_1\right)>0$ is nonsingular. Thus,

$\begin{array}{l}{L}_f\overline{h}\left(x\right)=x_2-{\overline{M}}^{-1}\left(x_1\right)(C\left(x_1,x_2\right)x_2+e\left(x_1\right)),\\ L_{g_1}\overline{h}\left(x\right)={\overline{M}}^{-1}\left(x_1\right),\end{array}$

and

$D\left(x_1\right)=diag\left\{{\Vert {\overline{M}}_{.1}{}^{-1}\left(x_1\right)\Vert }^2,\mathrm{\dots },{\Vert {\overline{M}}_{.m}^{-1}\left(x_1\right)\Vert }^2\right\}.$

Therefore, by Theorem 9.4.2, the state-feedback

$\begin{array}{l}u=-\overline{M}\left(x_1\right)x_2+C\left(x_1,x_2\right)x_2+e\left(x_1\right)-\overline{M}\left(x_1\right)\left(I+\frac{n}{4{\gamma }^2}D\left(x_1\right)\right)\left(x_1+x_2\right)\\ =-k_1\left(x_1\right)x_1-k_2\left(x_1-x_2\right)x_2+e\left(x_1\right)\end{array}$

where the feedback gain matrices k₁, k₂ are given by

$\begin{array}{l}{k}_1\left(x_1\right)=\overline{M}\left(x_1\right)\left(I+\frac{n}{4{\gamma }^2}D\left(x_1\right)\right),\\ k_2\left(x_1,x_2\right)=-C\left(x_1,x_2\right)+\overline{M}\left(x_1\right)\left(2I+\frac{n}{4{\gamma }^2}D\left(x_1\right)\right),\end{array}$

solves the SFBSNLHICP for the robot system. Moreover, the function

$V\left(x\right)=\frac{1}{2}{x}_1^T{x}_1+\frac{1}{4}{\left(x_1+x_2\right)}^T\left(x_1+x_2\right)$

is positive-definite and proper. Consequently, the closed-loop system is globally asymptotically-stable at x = 0 with w = 0.

9.5    ${\mathcal{H}}_{\infty }$-Control for Singularly-Perturbed Nonlinear Systems

In this section, we discuss the state-feedback ${\mathcal{H}}_{\infty }$-control problem for nonlinear singularly-perturbed systems. Singularly-perturbed systems are those class of systems that are characterized by a discontinuous dependence of the system properties on a small perturbation parameter ε. They arise in many physical systems such as electrical power systems and electrical machines (e.g., an asynchronous generator, a dc motor, electrical converters), electronic systems (e.g., oscillators) mechanical systems (e.g., fighter aircrafts), biological systems (e.g., bacterial-yeast cultures, heart) and also economic systems with various competing sectors. This class of systems has multiple time-scales, namely, a “fast” and a “slow” dynamics. This makes their analysis and control more complicated than regular systems. Nevertheless, they have been studied extensively [157, 165]. The control problem for this class of systems is also closely related to the control problems that were discussed in the previous sections.

We consider the following affine class of singularly-perturbed systems defined on $\chi \subset {\Re }^n\text{:}$

${\sum }_{sp}^a:\{\begin{array}{l}{\dot{x}}_1=f_1\left(x_1,x_2\right)+g_{11}\left(x_1,x_2\right)w+g_{21}\left(x_1,x_2\right)u;x\left(t_0\right)=x_0\\ \epsilon {\dot{x}}_2=f_2\left(x_1,x_2\right)+g_{12}\left(x_1,x_2\right)w+g_{22}\left(x_1,x_2\right)u;\\ z=\left[\begin{array}{l}{h}_1\left(x_1,x_2\right)\\ u\end{array}\right]\\ y=x,\end{array}$

(9.49)

where $x_1\in {\Re }^{n_1}\subset \chi$ is the slow state, $x_2\in {\Re }^{n_2}\subset \chi$ is the fast state, $u\in \mathcal{U}\subset {\mathcal{L}}_2([t_0,T],{\Re }^p)$) is the control input, $w\in \mathcal{W}\subset {\mathcal{L}}_2([t_0,T],{\Re }^r)\text{)}$) is the disturbance input, $z\in {\Re }^s$ is the controlled output, while $y\in {\Re }^m$ is the measured output and ε is a small parameter. The functions $f_i:\mathcal{X}\times \mathcal{X}\to V^{\infty }\mathcal{X},g_{ij}:\mathcal{X}\times \mathcal{X}\to {ℳ}^{n\times \ast },i,j=1,2,\ast =r\text{if}j=1\text{or}\ast \text{=}p\text{if}j\text{=2,}\text{and}{h}_1:\mathcal{X}\to {\Re }^s$ are smooth C^∞ functions of $x=\left(x_1^T,x_2^T\right)$. We also assume that f₁(0, 0) = 0 and h₁(0, 0) = 0.

The problem is to find a static state-feedback control of the form

$u=\beta \left(x\right),\beta \left(0\right)=0,$

(9.50)

such that the closed-loop system (9.49), (9.50) has locally ${\mathcal{L}}_2$-gain from w to z less than or equal to a given number γ^$\star$ > 0 with closed-loop local asymptotic-stability. The following result gives a solution to this problem, and is similar to the result in Chapter 6 for the regular problem.

Proposition 9.5.1 Consider the system (9.49) and the state-feedback ${\mathcal{H}}_{\infty }$-control problem (SFBNLHICP) for it. Assume the system is zero-state detectable, and suppose for some γ > 0 and each ε > 0, there exists a C² solution$V:\tilde{M}\to \Re ,V\ge 0,\tilde{M}\subset \chi$ to the HJIE

$\begin{array}{l}{V}_{x_1}\left(x\right)f_1\left(x_1,x_2\right)+V_{x_2}\left(x\right)f_2\left(x\right)+\frac{1}{2}\left(V_{x_1}^T\left(x\right){\epsilon }^{-1}{V}_{x_2}^T\left(x\right)\right)\left(\begin{array}{cc}{s}_{11}\left(x\right)& s_{12}\left(x\right)\\ s_{21}\left(x\right)& s_{22}\left(x\right)\end{array}\right)\times \\ \left(\begin{array}{l}{V}_{x_1}\left(x\right)\\ {\epsilon }^{-1}{V}_{x_2}\left(x\right)\end{array}\right)+\frac{1}{2}{h}_1^T\left(x\right)h_1\left(x\right)=0,V\left(0\right)=0\end{array}$

(9.51)

where

$S_{ij}\left(x\right)={\gamma }^{-2}{g}_1{}_i\left(x\right)g_{1_j}^T\left(x\right)-g_{2_i}\left(x\right)g_{2_j}^T\left(x\right),i,j=1,2.$

Then the state-feedback control

$u^{\star }=-\left[g_{21}^T\left(x\right){\epsilon }^{-1}{g}_{22}^T\left(x\right)\right]V_x^T\left(x\right)$

(9.52)

solves the SFBNLHICP for the system on $\tilde{M}$, i.e., the closed-loop system (9.49), (9.52) has ${\mathcal{L}}_2$-gain from w to z less than or equal to γ and is asymptotically-stable with w= 0.

Proof: Proof follows along similar lines as in Chapter 5 with the slight modification for the presence of ε. □

The controller constructed above depends on ε which may present some computational difficulties. A composite controller that does not depend on ε can be constructed as an asymptotic approximation to the above controller as ε → 0. To proceed, define the Hamiltonian system corresponding to the SFBNLHICP for the system as defined in Section 5.1:

$\begin{array}{l}{\dot{x}}_1=\frac{\partial H_{\gamma }}{\partial p_1}=F_1\left(x_1,p_1,x_2{p}_2\right)\\ {\dot{p}}_1=-\frac{\partial H_{\gamma }}{\partial x_1}=F_2\left(x_1,p_1,x_2{p}_2\right)\\ \epsilon {\dot{x}}_2=\frac{\partial H_{\gamma }}{\partial p_2}=F_3\left(x_1,p_1,x_2{p}_2\right)\\ \epsilon {\dot{p}}_2=-\frac{\partial H_{\gamma }}{\partial x_2}{F}_4\left(x_1,p_1,x_2{p}_2\right)\end{array}\}$

(9.53)

with Hamiltonian function $H_{\gamma }:T^{\star }\mathcal{X}\to \Re$ defined by

$\begin{array}{l}{H}_{\gamma }\left(x_1,x_2,p_1,p_2\right)=p_1^T{f}_1\left(x_1,x_2\right)+p_2^T{f}_2\left(x\right)+\\ \frac{1}{2}\left(\begin{array}{cc}{p}_1^T& p_2^T\end{array}\right)\left(\begin{array}{cc}{s}_{11}\left(x\right)& s_{12}\left(x\right)\\ s_{21}\left(x\right)& s_{22}\left(x\right)\end{array}\right)\left(\begin{array}{l}{p}_1\\ p_2\end{array}\right)+\frac{1}{2}{h}_1^T\left(x\right)h_1\left(x\right).\end{array}$

Let now ε → 0 in the above equations (9.53), and consider the resulting algebraic equations:

$F_3\left(x_1,p_1,x_2,p_2\right)=0,F_4\left(x_1,p_2,x,p_2\right)=0.$

If we assume that H_γ is nondegenerate at (x, p) = (0, 0), then by the Implicit-function Theorem [157], there exist nontrivial solutions to the above equations

$x_2=\varphi \left(x_1,p_1\right),p_2=\psi \left(x_1,x_2\right).$

Substituting these solutions in (9.53) results in the following reduced Hamiltonian system:

$\{\begin{array}{l}{\dot{x}}_1=F_1\left(x_1,p_1,\varphi (x_1,p_1\right),\psi \left(x_1,p_1\right))\\ {\dot{p}}_1=F_2\left(x_1,p_1,\varphi \right)\left(x_1,p_1\right),\psi \left(x_1,p_1\right)).\end{array}$

(9.54)

To be able to analyze the asymptotic behavior of the above system and its invariant-manifold, we consider its linearization of (9.49) about x = 0. Let $A_{ij}=\frac{\partial f_i}{\partial x_j}(0,0),B_{ij}=g_{ij}(0,0),C_i=\frac{\partial h_1}{\partial x_j}(0),i,j=1,2$, so that we have the following linearization

${\sum }_{sp}^l:\{\begin{array}{l}{\dot{x}}_1=A_{11}{x}_1+A_{12}{x}_2+B_{11}w+B_{21}u;x\left(t_0\right)=x_0\\ \epsilon {\dot{x}}_2=A_{21}{x}_1+A_{22}{x}_2+B_{12}w+B_{22}u;\\ z=\left[\begin{array}{l}{C}_1{x}_1+C_2{x}_2\\ u\end{array}\right]\\ y=x.\end{array}$

(9.55)

Similarly, the linear Hamiltonian system corresponding to this linearization is given by

$\left[\begin{array}{l}\dot{x}\\ \dot{p}\end{array}\right]={\overline{H}}_{\gamma }\left[\begin{array}{l}x\\ p\end{array}\right],$

(9.56)

where ${\overline{H}}_{\gamma }$ is the linear Hamiltonian matrix corresponding to (9.55) which is defined as

${\overline{H}}_{\gamma }=\left[\begin{array}{cc}{H}_{11}& H_{12}\\ {\epsilon }^{-1}{H}_{21}& {\epsilon }^{-1}{H}_{22}\end{array}\right],$

and H_ij are the sub-Hamiltonian matrices:

$H_{ij}=\left[\begin{array}{cc}{A}_{ij}& -S_{ij}\left(0\right)\\ -C_i^T{C}_i& -A_{ji}^T\end{array}\right],$

with S_ij = γ⁻²B_1iB_1j −B_2iB_2j.

The Riccati equations corresponding to the fast and slow dynamics are respectively given by

$A_{22}^T{X}_f+X_f{A}_{22}-X_f{S}_{22}\left(0\right)X_f+C_2^T{C}_2=0,$

(9.57)

$A_0^T{X}_s+X_s{A}_0-X_0{S}_0{X}_0+Q_0=0,$

(9.58)

where

$\left[\begin{array}{cc}{A}_0& -S_0\\ -Q_0& -A_0^T\end{array}\right]=H_{11}-H_{12}{H}_{22}^{-1}{H}_{21.}$

Then it is wellknown [216] that, if the system (9.55) is stabilizable, detectable and does not have invariant-zeros on the imaginary axis (this holds if the Hamiltonian matrix ${\overline{H}}_{\gamma }$ is hyperbolic [194]), then the ${\mathcal{H}}_{\infty }$-control problem for the system is solvable for all small ε. In fact, the control

$\begin{array}{l}{u}_{cl}=-B_{21}^T{X}_s{x}_1-B_{22}^T\left(X_f{x}_2+X_c{x}_1\right),\\ X_c=\left[X_f-I\right]H_{22}^{-1}{H}_{21}\left[\begin{array}{c}I\\ X_s\end{array}\right]\end{array}$

(9.59)

which is ε-independent, is one such state-feedback. It is also well known that this feedback controller also stabilizes the nonlinear system (9.49) locally about x = 0. The question is: how big is the domain of validity of the controller, and how far can it be extended for the nonlinear system (9.49)?

To answer the above question, we make the following assumptions.

Assumption 9.5.1 For a given γ > 0, the ARE (9.57) has a symmetric solution X_f ≥ 0 such that A_cf = A₂₂ − S₂₂X_f is Hurwitz.

Assumption 9.5.2 For a given γ >0, the ARE (9.58) has a symmetric solution X_s ≥ 0 such that A_cs = A₀ − S₀X_s is Hurwitz.

Then, under Assumption 9.5.2 and the theory of ordinary differential-equations, the system (9.54) has a stable invariant-manifold

$M_s^{{\ast }^{-}}=\left\{x_1,p_1={\sigma }_s\left(x_1\right)=X_s{x}_1+O\left({\Vert x_1\Vert }^2\right)\right\},$

(9.60)

and the dynamics of the system (which is asymptotically-stable) restricted to this manifold is given by

${\dot{x}}_1=F_1\left(x_1,{\sigma }_s(x_1\right),\varphi \left(x_1,{\sigma }_s\left(x_1\right)\right),\psi \left(x_1,{\sigma }_s\left(x_1\right))\right),$

(9.61)

x₁ in a small neighborhood of x = 0. Moreover, from (9.54), the function σ satisfies the partial-differential equation (PDE)

$\begin{array}{l}\frac{\partial {\sigma }_s}{\partial x_1}{F}_1\left(x_1,{\sigma }_s(x_1\right),\varphi \left(x_1,{\sigma }_s\left(x_1\right)\right),\psi \left(x_1,{\sigma }_s\left(x_1\right))\right)=F_2\left(x_1,{\sigma }_s(x_1\right),\varphi \left(x_1,{\sigma }_s\left(x_1\right)\right),\\ \psi \left(x_1,{\sigma }_s\left(x_1\right))\right)\end{array}$

(9.62)

which could be solved approximately using power-series expansion [193]. Such a solution can be represented as

${\sigma }_s\left(x_1\right)=X_s{x}_1+{\sigma }_{s2}\left(x_1\right)+{\sigma }_{s3}\left(x_1\right)+\mathrm{\dots }$

(9.63)

where σ_si(.), i = 2, …, are the higher-order terms, and σ_s1 = X_s. In fact, substituting (9.63) in (9.62) and equating terms of the same powers in x₁, the higher-order terms could be determined recursively.

Now, substituting the solution (9.60) in the fast dynamic subsystem, i.e., in (9.53), we get for all x₁ such that $\tilde{M}$ exists,

${\dot{x}}_2={\overline{F}}_3\left(x_1,x_2,p_2\right),{\dot{p}}_2={\overline{F}}_4\left(x_1,x_2,p_2\right),$

where

$\begin{array}{l}{\overline{F}}_3\left(x_1,x_2,p_2\right)=F_3\left(x_1,{\sigma }_s\left(x_1\right),x_2+\varphi \left(x_1,{\sigma }_2\left(x_1\right),p_2+{\sigma }_2(x_1\right)\right),\\ {\overline{F}}_4\left(x_1,x_2,p_2\right)=F_4\left(x_1,{\sigma }_s\left(x_1\right),x_2+\varphi \left(x_1,{\sigma }_2\left(x_1\right),p_2+{\sigma }_2(x_1\right)\right).\end{array}$

The above fast Hamiltonian subsystem has a stable invariant-manifold

$M_f^{{\ast }^{-}}=\left\{x_2,p_2={\vartheta }_f\left(x_1,x_2\right)\right\}$

with asymptotically-stable dynamics

${\dot{x}}_2={\overline{F}}_3\left(x_1,{\sigma }_s(x_1\right),x_2,{\vartheta }_f\left(x_1,x_2\right)$

for all x₁, x₂ ∈ Ω neighborhood of x = 0. The function ϑ_f similarly has a representation about x = 0 as

${\vartheta }_f\left(x_1,x_2\right)=X_f{x}_2+O\left(\left(\Vert x_1\Vert +\Vert x_2\Vert \right)\Vert x_2\Vert \right)$

and satisfies a similar PDE:

$\frac{\partial \vartheta }{\partial x_2}{\overline{F}}_3\left(x_1,x_2{\vartheta }_f\left(x_1,x_2\right)\right)={\overline{F}}_4\left(x_1,x_2,{\vartheta }_f\left(x_1,x_2\right)\right).$

Consequently, ϑ_f (.,.) has the following power-series expansion in powers of x₂

${\vartheta }_f\left(x_1,x_2\right))=X_f{x}_2+{\vartheta }_{f2}\left(x_1,x_2\right)+{\vartheta }_{f3}\left(x_1,x_2\right)+\mathrm{\dots }$

where ϑ_{f i}, i = 2,3,… denote the higher-order terms.

Finally, we can define the following composite controller

$u_c=-g_{21}^T\left(x\right){\sigma }_s\left(x_1\right)-g_{22}^T\left(x\right)\left[\psi \left(x_1,{\sigma }_s\left(x_1\right)\right)+\vartheta \left(x_1,x_2-\varphi (x_1,{\sigma }_s\left(x_1\right)\right)\right]$

(9.64)

which is related to the linear composite controller (9.59) in the following way:

$u_c=u_{cl}+O\left(\Vert x_1\Vert +\Vert x_2\Vert \right).$

Thus, (9.64) solves the nonlinear problem locally in $\tilde{M}$. Summing up the above analysis, we have the following theorem.

Theorem 9.5.1 Consider the system (9.49) and the SFBNLHICP for it. Suppose Assumptions 9.5.1, 9.5.2 hold. Then there exist indices m₁, m₂ ∈Z₊ and an ε₀ > 0 such that, for all ε ∈ [0, ε₀) the following hold:

(i)  there exists a C² solution $V:{\Omega }_{m1}\times {\Omega }_{m2}\to {\Re }_{+},{\Omega }_{m1}\times {\Omega }_{m2}\subset \chi$ to the HJIE (9.51) whose approximation is given by

$V\left(x_1,x_2\right)=V_0\left(x_1\right)+O\left(\epsilon \right)$

for some function $V_0={\displaystyle \int {\sigma }_s\left(x_1\right)d{x}_1}$. Consequently, the control law (9.52) also has the approximation:

$u=u_c+O\left(\epsilon \right);$

(ii)   the SFBNLHICP for the system is locally solvable on Ω_m1 × Ω_m2 by the composite control (9.64);

(iii)  the SFBNLHICP for the system is locally solvable on Ω_m1×Ω_m2 by the linear composite control (9.59).

Proof: (i)  The existence of a C² solution to the HJIE (9.51) follows from linearization. Indeed, V₀ ≈ x^T X_sx ≥ 0 is C².

(ii)  Consider the closed-loop system (9.49), (9.64). We have to prove that Assumptions 9.5.1, 9.5.2 hold for the linearized system:

${\dot{x}}_1={\tilde{A}}_{11}{x}_1+{\tilde{A}}_{12}{x}_2+B_{11}w,{\dot{x}}_2={\tilde{A}}_{21}{x}_1+{\tilde{A}}_{22}{x}_2+B_{12}w,z=\left[{\begin{array}{cc}{\tilde{C}}_1& \tilde{C}\end{array}}_2\right]x,$

where ${\tilde{C}}_1=\left[{\stackrel{}{C}}_1{B}_{21}^T{X}_s-B_{22}^T{X}_c\right],{\tilde{C}}_2=\left[{\stackrel{}{C}}_2,-B_{22}^T{X}_f\right].$. This system corresponds to the closed-loop system (9.55), (9.59) whose Hamiltonian system is also given similarly by (9.56) with Hamiltonian matrix ${\tilde{H}}_{\gamma }$, and where all matrices are replaced by “tilde” superscript, with ${\tilde{S}}_{ij}=-{\gamma }^2{B}_{1i}{B}_{1j},i,j=1,2$

It should be noted that Assumptions 9.5.1, 9.5.2 imply that the fast and reduced subsystems have stable invariant-manifolds. Setting ε = 0 and substituting p₁ = X_s x₁, p₂ = X_c x₁ +X_f x₂ leads to the Hamiltonian system (9.56) with ε = 0. Therefore, p₂ = X_f x₂ and p₁ = X_s x₁ are the desired stable manifolds.

iii  This proof is similar to (ii). □

9.6    Notes and Bibliography

The results of Sections 9.1-9.2 are based on the References [42, 194]. A detailed proof of Theorem 9.2.1 can be found in [194]. Moreover, an alternative approach to the MFBSNLHICP can be found in [176]. In addition, the full-information problem for a class of nonlinear systems is discussed in [43].

The material of Section 9.4 is based on the Reference [88]. Application of singular ${\mathcal{H}}_{\infty }$-control to the control of a rigid spacecraft can also be found in the same reference.

The results presented in Section 9.5 are based on References [104]-[106]. Results on descriptor systems can also be found in the same references. In addition, robust control of nonlinear singularly-perturbed systems is discussed in Reference [218].

Finally, the results of Section 9.3 are based on [27].

¹A coordinate transformation that is bijective (therefore invertible) and smooth (or a smooth homeomorphism). A global diffeomorphism can be found if there exists a minor of ${\tilde{k}}_{12}$ with constant rank $\kappa$ for all $x\in \mathcal{X}$.

²The Hessian is nondegenerate if it is not identically zero.