3.5    Dissipativity and Passive Properties of Discrete-Time Nonlinear Systems

In this section, we discuss the discrete-time counterparts of the previous sections. For this purpose, we again consider a discrete-time nonlinear state-space system defined on $\mathcal{X}$ ⊆ ℜⁿ containing the origin x = {0} in coordinates (x = x₁,…, x_n):

${\sum }^d:\{\begin{array}{l}{x}_{k+1}=f(x_k,u_k),x(k_0)=x^0\\ y_k=h(x_k,u_k)\end{array}$

(3.59)

where x_k ∈ $\mathcal{X}$ is the state vector, u_k ∈ $\mathcal{U}$ ⊆ ℜ^m is the input function belonging to an input space $\mathcal{U}$, y_k ∈ $\mathcal{Y}$ ⊆^m is the output function which belongs to the output space $\mathcal{Y}$ (i.e., Σ^d is square). The functions f : $\mathcal{X}$ × $\mathcal{U}$ → $\mathcal{X}$ and h : $\mathcal{X}$ × $\mathcal{U}$ → $\mathcal{Y}$ are real C^r functions of their arguments such that there exists a unique solution x(k, k₀, x₀, u_k) to the system for any x₀ ∈ $\mathcal{X}$ and u_k ∈ $\mathcal{U}$.

Definition 3.5.1 A function s(k) = s(u_k, y_k) is a supply-rate to the system Σ^d if it is locally absolutely summable, i.e.,${\sum }_{k=k_0}^{k_1}\left|s(k)\right|<\infty$

for all (k₀, k₁) ∈ Z × Z.

Then we have the following definition of dissipativity for the system Σ^d.

Definition 3.5.2 The nonlinear system Σ^d is locally dissipative with respect to the supply-rate s(u_k, y_k) if there exists a C⁰ positive-semidefinite storage-function Ψ : N ⊂ $\mathcal{X}$ → ℜ such that

$\mathrm{\Psi }(x_{k_1})-\mathrm{\Psi }(x_{k_0})\le {\displaystyle \sum _{k=k_0}^{k_1}s(u_k,y_k)}$

(3.60)

for all k₁ ≥ k₀, u_k ∈ $\mathcal{U}$, and x(k₀), x(k₁) ∈ N. The system is said to be dissipative if it is locally dissipative for all x_k0, x_k1 ∈ $\mathcal{X}$ .

Again, we can move from the integral (or summation) version of the dissipation-inequality (3.60) to its differential (or infinitesimal) version by differencing to get

$\mathrm{\Psi }(x_{k+1})-\mathrm{\Psi }(x_k)\le s(u_k,y_k),$

(3.61)

which is the discrete-time equivalent of (3.17). Furthermore, we can equally define the dissipation-inequality as in (3.1.3) by

${\displaystyle \sum _{k=k_0}^{k_1}s(u_k,y_k)}\ge 0,x^0=0\forall k_1\ge k_0.$

(3.62)

The available-storage and required-supply of the system ${\sum }_a^d$ can also be defined as follows:

${\mathrm{\Psi }}_a(x)=\underset{\begin{array}{l}{x}^0=x,u_k\in \mathcal{U},\\ K\ge 0\end{array}}{\sup }-{\displaystyle \sum _{k=0}^{k}s(u_k,y_k)}$

(3.63)

${\mathrm{\Psi }}_r(x)=\underset{\begin{array}{l}{x}^e\to x,\\ u_k\in \mathcal{U}\end{array}}{\text{inf}}{\displaystyle \sum _{k=k_{-1}}^{k=0}s(u_k,y_k)},\forall k_{-1}\le 0,$

(3.64)

where x^e = arg inf_{x∈$\mathcal{X}$} Ψ(x). Consequently, the equivalents of Theorems 3.1.1, 3.1.2 and 3.1.3 can be derived for the system Σ^d. However, of particular interest to us among discrete-time dissipative systems, are those that are passive because of their nice properties which are analogous to the continuous-time case. Hence, for the remainder of this section, we shall concentrate on this class and the affine nonlinear discrete-time system:

${\sum }^{da}:\{\begin{array}{l}{x}_{k+1}=f(x_k)+g(x_k)u_k,x(k_0)=x_0\\ y_k=h(x_k)+d(x_k)u_k\end{array}$

(3.65)

where g : $\mathcal{X}$ → M^n×m($\mathcal{X}$ ), d : $\mathcal{X}$ → ℜ^m, and all the other variables have their previous meanings. We shall also assume for simplicity that f(0) = 0 and h(0) = 0. We proceed with the following definition.

Definition 3.5.3 The system Σ^da is passive if it is dissipative with supply-rate s(u_k, y_k) = yT^ku_k and the storage-function Ψ : $\mathcal{X}$ → ℜ satisfies Ψ(0) = 0. Equivalently, Σ^da is passive if, and only if, there exists a positive-semidefinite storage-function Ψ satisfying

$\mathrm{\Psi }(x_{k_1})-\mathrm{\Psi }(x_{k_0})\le {\displaystyle \sum _{k=k_0}^{k_1}{y}_k^T{u}_k,}\mathrm{\Psi }(0)=0$

(3.66)

for all k₁ ≥ k₀ ∈ Z, u_k ∈ $\mathcal{U}$, or the infinitesimal version:

$\mathrm{\Psi }(x_{k+1})-\mathrm{\Psi }(x_k)\le y_k^T{u}_k,\mathrm{\Psi }(0)=0$

(3.67)

for all k₁ ≥ k₀ ∈ Z, u_k ∈ $\mathcal{U}$.

For convenience of the presentation, we shall concentrate on the infinitesimal version of the passivity-inequality (3.67).

Remark 3.5.1 Similarly, the discrete-time system Σ^da is strictly-passive if the strict inequality in (3.66) or (3.67) is satisfied, or there exists a positive-definite function Ξ : $\mathcal{X}$ → ℜ₊ such that

$\mathrm{\Psi }(x_{k+1})-\mathrm{\Psi }(x_k)\le y_k^T{u}_k-\text{Ξ}(x_k)\forall u_k\in \mathcal{U},\forall k\in \mathbf{Z}.$

(3.68)

We also have the following definition of losslessness.

Definition 3.5.4 A passive system Σ^da with a storage-function Ψ(.), is lossless if

$\mathrm{\Psi }(x_{k+1})-\mathrm{\Psi }(x_k)=y_k^T{u}_k\forall u_k\in \mathcal{U},\forall k\in Z.$

(3.69)

The following lemma is the discrete-time version of the nonlinear KYP lemma given in Theorem 3.3.1 and Definition 3.3.4 for lossless systems.

Lemma 3.5.1 The nonlinear system Σ^da with a C² storage-function Ψ(.) is lossless if, and only if,

(i)  

$\mathrm{\Psi }(f(x))=\mathrm{\Psi }(x)$

(3.70)

$\frac{\partial \mathrm{\Psi }(\alpha )}{\partial \alpha }{|}_{\alpha =f(x)g(x)}=h^T(x)$

(3.71)

$g^T(x)\frac{{\partial }^2\mathrm{\Psi }(\alpha )}{\partial {\alpha }^2}{|}_{\alpha =f(x)g(x)}=d^T\left(x\right)+d(x)$

(3.72)

(ii)  Ψ(f(x) + g(x)u) is quadratic in u.

Proof: (Necessity): If Σ^da is lossless, then there exists a positive-semidefinite storage-function Ψ(.) such that

$\mathrm{\Psi }(f(x)+g(x)u)-\mathrm{\Psi }(x)={\left(h\left(x\right)+d\left(x\right)u\right)}^{T}u.$

(3.73)

Setting u = 0, we immediately get (3.70). Differentiating both sides of the above equation (3.73) with respect to u once and twice, and setting u = 0 we also arrive at the equations (3.71) and (3.72), respectively. Moreover, (ii) is also obvious from (3.72). (Sufficiency:) If (ii) holds, then there exist functions p, q, r : $\mathcal{X}$ → ℜ such that

$\mathrm{\Psi }(f(x)+g(x)u)=p(x)+q(x)u+u^{T}r(x)u\forall u\in \mathcal{U}$

(3.74)

where the functions p(.), q(.), r(.) correspond to the Taylor-series expansion of Ψ(f(x) + g(x)u) about u:

$p(x)=\mathrm{\Psi }(f(x))$

(3.75)

$q(x)={\frac{\partial \mathrm{\Psi }(f(x)+g(x)u)}{\partial u}|}_{u=0}={\frac{\partial \mathrm{\Psi }(\alpha )}{\partial \alpha }|}_{\alpha =f(x)}g(x)$

(3.76)

$r(x)=\frac{1}{2}{\frac{{\partial }^2\mathrm{\Psi }(f(x)+g(x)u)}{\partial u^2}|}_{u=0}=g(x){\frac{\partial \mathrm{\Psi }(\alpha )}{\partial \alpha }|}_{\alpha =f(x)}g(x).$

(3.77)

Therefore equation (3.74) implies by (3.70)-(3.72),

$\begin{array}{l}\mathrm{\Psi }(f(x)+g(x)u)-\mathrm{\Psi }(x)=(q(x)+u^{T}r(x))u\forall u\in \mathcal{U}\\ =\left\{h^T(x)+\frac{1}{2}{u}^T(d^T(x)+d(x))\right\}u\\ =y^{T}u\forall u\in \mathcal{U}.\end{array}$

Hence Σ^da is lossless. □

Remark 3.5.2 In the case of the linear discrete-time system

${\sum }^{dl}:\{\begin{array}{l}{x}_{k+1}=A{x}_k+B{u}_k\\ y_k=C{x}_k+D{u}_k\end{array}$

where A, B, C and D are matrices of appropriate dimensions. The system is lossless with C² storage-function $\text{ψ}\left(x_k\right)=\frac{1}{2}{x}_k^{T}P{x}_k$, P = P ^T and the KYP equations (3.70)–(3.72) yield:

$\begin{array}{l}{A}^{T}PA=P\\ B^{T}PA=C\\ B^{T}PB=D^T+D.\end{array}\}$

(3.78)

Remark 3.5.3 Notice from the above equations (3.78) that if the system matrix D = 0, then since B ≠ 0, it implies that C = 0. Thus Σ^dl can only be lossless if y_k = 0 when D = 0. In addition, if rank(B) = m, then Σ^dl is lossless only if P ≥ 0.

Furthermore, D^T + D ≥ 0 and D + D^T > 0 if and only if rank(B) = m [74]. Thus, D > 0 and hence nonsingular. This assumption is also necessary for the nonlinear system Σ^da, i.e., d(x) > 0 and will consequently be adopted in the sequel.

The more general result of Lemma 3.5.1 for passive systems can also be stated.

Theorem 3.5.1 The nonlinear system Σ^da is passive with storage-function Ψ(.) which is positive-definite if, and only if, there exist real functions l : $\mathcal{X}$ → ℜ ⁱ, W : $\mathcal{X}$ → ℜ M^i×j($\mathcal{X}$ ) of appropriate dimensions such that

$\mathrm{\Psi }(f(x))-\mathrm{\Psi }(x)=-\frac{1}{2}{l}^T(x)l(x)$

(3.79)

${\frac{\partial \mathrm{\Psi }(\alpha )}{\partial \alpha }|}_{\alpha =f(x)}g(x)+l^T(x)W(x)=h^T(x)$

(3.80)

$d^T(x)+d(x)-g^T(x){\frac{{\partial }^2\mathrm{\Psi }(\alpha )}{\partial {\alpha }^2}|}_{\alpha =f(x)}g(x)=W^T(x)W(x).$

(3.81)

Proof: The proof follows from Lemma 3.5.1. □

Remark 3.5.4 In the case of the linear system Σ^dl the equations (3.79)-(3.81) reduce to the following wellknown conditions for Σ^dl to be positive-real:

$\begin{array}{l}{A}^{T}PA=-L^{T}L\\ A^{T}PB+L^{T}W=C^T\\ D^T+D-B^{T}PB=W^{T}W.\end{array}$

The KYP lemma above can also be specialized to discrete-time bilinear systems. Furthermore, unlike for the general nonlinear case above, the assumption that Ψ(f(x) + g(x)u) is quadratic in u can be removed. For this purpose, let us consider the following bilinear state-space system defined on $\mathcal{X}$ ⊆ ℜⁿ:

${\sum }^{dbl}:\{\begin{array}{l}{x}_{k+1}=A{x}_k+\left[B{u}_k+E\right]u_k\\ y_k=h(x_k)+d(u_k)u_k,h(0)=0\end{array},$

(3.82)

where all the previously used variables and functions have their meanings, with A ∈ ℜ^n×n, E ∈ ℜ^n×m, C ∈ ℜ^m×n constant matrices, while B : $\mathcal{X}$ → $\mathcal{M}$^n×m($\mathcal{X}$ ) is a smooth map defined by

$B\left(x\right)=\left[B_{1}x,\dots ,B_{m}x\right],$

where B_i ∈ ℜ^n×n, i = 1,…, m. We then have the following theorem.

Theorem 3.5.2 (KYP-lemma) The bilinear system Σ^dbl is passive with a positive-definite storage-function, $\mathrm{\psi }\left(x_k\right)=\frac{1}{2}{x}^{T}Px$ if, and only if, there exists a real constant matrix L ∈ ℜ^n×q and a real function W : $\mathcal{X}$ → $\mathcal{X}$^i×j($\mathcal{X}$ ) of appropriate dimensions, satisfying

$\begin{array}{l}{A}^{T}PA-P=-L^{T}L\\ x^T[A^{T}P(B(x)+E)+L^{T}W(x)=h^T(x)\\ d^T(x)+d(x)-\left(B\left(x\right)+E\right){}^{T}P\left(B\left(x\right)+E\right)=W^T(x)W(x)\end{array}$

for some symmetric positive-definite matrix P .

3.6    ℓ₂-Gain Analysis for Discrete-Time Dissipative Systems

In this section, we again summarize the relationship between the ℓ₂-gain of a discrete-time system and its dissipativeness, as well as the implications on its stability analogously to the continuous-time case discussed in Section 3.2. We consider the nonlinear discrete-time system Σ^d given by the state equations (3.59). We then have the following definition.

Definition 3.6.1 The nonlinear system Σ^d is said to have locally in N ⊂ $\mathcal{X}$, finite ℓ₂-gain less than or equal to γ > 0 if for all u_k ∈ℓ₂[k₀, K]

${\displaystyle \sum _{k=k_0}^K{\Vert y_k\Vert }^2}\le {\gamma }^2{\displaystyle \sum _{k=k_0}^K{\Vert u_k\Vert }^2}+{\beta }^{\prime }(x_0)$

(3.83)

for all K ≥ k₀, and all x₀ ∈ N, y_k = h(x_k, u_k), k ∈ [k₀, K], and for some function β^′ : $\mathcal{X}$ → ℜ₊. Moreover, the system is said to have ℓ₂-gain ≤ γ if N = $\mathcal{X}$ .

Without any loss of generality, we can take k₀ = 0 in the above definition. We now have the following important theorem which is the discrete-time equivalent of Theorems 3.2.1, 3.2.2 or the Bounded-real lemma for discrete-time nonlinear systems.

Theorem 3.6.1 Consider the discrete-time system Σ^d. Then Σ^d has finite ℓ₂-gain ≤ γ if Σ^d is finite-gain dissipative with supply-rate $s\left(u_{k,}{y}_k\right)=\left({\Vert u_k\Vert }^2-{\Vert y_k\Vert }^2\right)$, i.e., it is dissipative with respect to s(., .) and γ < ∞. Conversely, Σ^d is finite-gain dissipative with supply-rate s(u_k, y_k) if Σ^d has ℓ₂-gain ≤ γ and is reachable from x = 0.

Proof: By dissipativity (Definition 3.5.2)

$0\le V(x_{k+1})\le {\displaystyle \sum _{k=0}^K{\text{(γ}}^2{\Vert u_k\Vert }^2}-{\Vert y_k\Vert }^2)+V(x_0).$

(3.84)

$\Rightarrow {\displaystyle \sum _{k=0}^K{\Vert y_k\Vert }^2}\le {\gamma }^2{\displaystyle \sum _{k=0}^K{\Vert u_k\Vert }^2}+V(x_0).$

(3.85)

Thus, Σ^d has locally ℓ₂-gain ≤ γ by Definition 3.6.1. Conversely, if Σ^d has ℓ₂-gain ≤ γ, then

$\begin{array}{l}{\displaystyle \sum _{k=0}^K\left({\Vert y_k\Vert }^2\right)-}{\gamma }^2{\Vert u_k\Vert }^2\le {\beta }^{\prime }(x_0)\\ \Rightarrow V_a(x)=-\underset{\begin{array}{l}{x}^0=x,u_k\in \mathcal{U},\\ K\ge 0\end{array}}{\inf }{\displaystyle \sum _{k=0}^K({\Vert u_k\Vert }^2-}{\Vert y_k\Vert }^2)\le {\beta }^{\prime }(x_0)<\infty ,\end{array}$

the available-storage, is well defined for all x in the reachable set of Σ^d from x(0) with u_k ∈ ℓ₂, and for some function β^′. Consequently, Σ^d is finite-gain dissipative with some storage-function V ≥ V ^a and supply-rate s(u_k, y_k). □

The following theorem is the discrete-time counterpart of Theorems 3.2.1, 3.2.2 relating the dissipativity of the discrete-time system (3.65), its ℓ₂-gain and its stability, which is another version of the Bounded-real lemma.

Theorem 3.6.2 Consider the affine discrete-time system Σ^da. Suppose there exists a C² positive-definite function V : U →ℜ₊ defined locally in a neigborhood U of x = 0 satisfying

(H1)

$g^T(0)\frac{{\partial }^{2}V}{\partial x^2}(0)g(0)+d^T(0)d(0)-{\gamma }^{2}I<0;$

(H2)

$0=V(f(x)+g(x)\mu (x))-V(x)+\frac{1}{2}({\Vert h(x)+d(x)\mu (x)\Vert }^2-{\gamma }^2{\Vert u(x)\Vert }^2)$

(3.86)

where u = μ(x), μ(0)= -is the unique soution of

${\frac{\partial V}{\partial \lambda }|}_{\text{λ=}f(x)+g(x)u}g(x)+u^T(d^T(x)d(x)-{\gamma }^{2}I=-h^T(x)d(x);$

(H3) the affine system (3.65) is zero-state detectable, i.e.$y_k|u_k=0=h\left(x_k\right)=0\Rightarrow {\underset{}{\lim }}_{k\to \infty }{x}_k=0,$then Σ^da is locally asymptotically-stable and has ℓ₂-gain ≤ γ.

Proof: Consider the Hamiltonian function for (3.65) under the ℓ₂ cost criterion J =${\displaystyle {\sum }_{k=0}^{\infty }\left[{\Vert yk\Vert }^2-{\gamma }^2{\Vert wk\Vert }^2\right]:}$

$H(x,u)=V(f\left(x\right)+g\left(x\right)u,-V\left(x\right)+\frac{1}{2}({\Vert y\Vert }^2-{\gamma }^2{\Vert u\Vert }^2).$

Then

$\frac{\partial H}{\partial u}(x,u)={\frac{\partial V}{\partial \lambda }|}_{\lambda =f\left(x\right)+g\left(x\right)u}g\left(x\right)+u^T\left(d^T\left(x\right)-{\gamma }^{2}I\right)+h^T\left(x\right)d\left(x\right)$

(3.87)

and

$\frac{{\partial }^{2}H}{\partial u^2}(x,\mu (x))=g^T(x){\frac{{\partial }^{2}V}{\partial {\lambda }^2}|}_{\lambda =f(x)+g(x)u}g(x)+d^T(x)d(x)-{\gamma }^{2}I.$

It is easy to check that the point (x⁰, u⁰) = (0, 0) is a critical point of H(x, u) and the Hessian matrix of H is negative-definite at this point by hypothesis (H1). Hence, $\frac{{\partial }^{2}H}{\partial u^2}$ (0, 0) is nonsingular at (x⁰, u⁰) = (0, 0). Thus, by the Implicit-function Theorem, there exists an open neighborhood N ⊂ $\mathcal{X}$ of x⁰ = 0 and an open neighborhood U ⊂ $\mathcal{U}$ of u⁰ = 0 such that there exists a C¹ solution u = μ(x), μ : N → U, to (3.87).

Expanding H(., .) about (x⁰, u⁰) = (0, 0) using Taylor-series formula yields

$H(x,u)=H(x,\mu (x))+\frac{1}{2}{\left(u-\mu \left(x\right)\right)}^T\left(\frac{{\partial }^{2}H}{\partial u^2}\left(x,\mu \left(x\right)\right)+O\left(\Vert u-\mu \left(x\right)\Vert \right)\right)\left(u-\mu \left(x\right)\right).$

Again hypothesis H1 implies that there exists a neighborhood of x⁰ = 0 such that

$\frac{{\partial }^{2}H}{\partial u^2}\left(x,\mu \left(x\right)\right)={g^T\left(x\right)\frac{{\partial }^{2}V}{\partial {\lambda }^2}|}_{\lambda =f\left(x\right)+g\left(x\right)\mu \left(x\right)}g\left(x\right)+d^T\left(x\right)d\left(x\right)-{\gamma }^{{}^2}I<0.$

This observation together with the hypothesis (H2) implies that u = μ(x) is a local maximum of H(x, u), and that there exists open neighborhoods N₀ of x⁰ and U₀ of u⁰ such that

$H(x,u)\le H(x,\mu (x))=0\forall x\in N_0\text{and}\text{}\forall u\in U_0,$

or equivalently

$V(f(x)+g(x)u(x))-V(x)\le \frac{1}{2}({\gamma }^2|\left|u\right|{|}^2-\left|\right|h(x)+d(x)u|{|}^2).$

(3.88)

Thus, Σ^da is finite-gain dissipative with storage-function V with respect to the supply-rate s(u_k, y_k) and hence has ℓ₂-gain ≤ γ. To show local asymptotic stability, we substitute x = 0, u = 0 in (3.88) to see that the equilibrium point x = 0 is stable and V is a Lyapunov-function for the system. Moreover, by hypothesis (H3) and LaSalle's invariance-principle, we conclude that x = 0 is locally asymptotically-stable with u = 0. □

Remark 3.6.1 In the case of the linear discrete-time system Σ^dl, the conditions (H1), (H2) in the above theorem reduce to the algebraic-equation and DARE respectively:

$\begin{array}{l}{B}^{T}PB+D^{T}D-{\gamma }^{2}I\le 0,\\ A^{T}PA-P-{\left(B^{T}PA+D^{T}C\right)}^T{\left((B^{T}PB+D^{T}D-{\gamma }^2\right)}^{-1}\left(B^{T}PA+D^{T}C\right)+C^{T}C=0.\end{array}$

3.7    Feedback-Equivalence to a Discrete-Time Lossless Nonlinear System

In this section, we derive the discrete-time analogs of the results of Subsection 3.4; namely, when can a discrete-time system of the form Σ^da be rendered lossless (or passive) via smooth state-feedback? We present necessary and sufficient conditions for feedback-equivalence to a lossless discrete-time system, and the results could also be modified to achieve feedback-equivalence to a passive system. The results are remarkably analogous to the continuous-time case, and the necessary conditions involve some mild regularity conditions and the requirement of lossless zero-dynamics. The only apparent anomaly is the restriction that d(x) be nonsingular for x in $\mathcal{O}$ ∍{0}.

We begin by extending the concepts of relative-degree and zero-dynamics to the discerete-time case.

Definition 3.7.1 The nonlinear system Σ^da is said to have vector relative-degree {0,…, 0} at x = 0 if d(0) is nonsingular. It is said to have uniform vector relative-degree {0,…, 0} if d(x) is nonsingular for all x ∈ $\mathcal{X}$ .

Definition 3.7.2 If the system Σ^da has vector relative-degree {0,…, 0} at x = {0}, then there exists an open neighborhood $\mathcal{O}$ of x = {0} such that d(x) is nonsingular for all x ∈ $\mathcal{O}$ and hence the control

$u_k^{*}=-d^{-1}(x_k)h(x_k),\forall x\in \mathcal{O}$

renders y_k ≡ 0 and the resulting dynamics of the system

${\sum }_0^{da}*:x_{k+1}=f^{*}(x_k)=f(x_k)+g(x_k)u_k^{*},\forall x_k\in \mathcal{O}\subseteq \mathcal{X}$

(3.89)

is known as the zero-dynamics of the system. Furthermore, the n-dimensional submanifold.

$Z^{*}=\left\{x\in O:y_k=0,k\in Z\right\}\equiv O\subseteq \mathcal{X}$

is known as the zero-dynamics submanifold.

Remark 3.7.1 If Σ^da has uniform vector relative-degree {0,…, 0}, then

$Z^{*}=\left\{x\in \mathcal{O}:y_k=0,k\in Z\right\}\equiv \mathcal{O}\equiv \mathcal{X}\text{.}$

Notice that unlike in the continuous-time case with vector relative degree {1,…,1} in which the zero-dynamics evolves on an n−m-dimensional submanifold, for the discrete-time system, the zero-dynamics evolves on an n-dimensional submanifold.

The following notion of lossless zero-dynamics replaces that of minimum-phase for feedback-equivalence to a lossless system.

Definition 3.7.3 Suppose d(0) is nonsingular. Then the system Σ^da is said to have locally lossless zero-dynamics if there exists a C² positive-definite Lyapunov-function V locally defined in the neighborhood $\mathcal{O}$ of x = {0} such that

(i)  $V\left(f^{*}\left(x\right)\right)V\left(x\right),\forall x\in \mathcal{O}.$

(ii)  $V\left(f^{*}\left(x\right)+g\left(x\right)u\right)$ is quadratic in u.

The system is said to have globally lossless zero-dynamics if d(x) is nonsingular for all x ∈ ℜⁿ and there exists a C² positive-definite Lyapunov-function V that satisfies the conditions (i), (ii) above for all x ∈ ℜⁿ.

The following lemma will be required in the sequel.

Lemma 3.7.1 (Morse-lemma [1]): Let p be a nondegenerate critical point for a real-valued function υ. Then there exists a local coordinate system (y¹,…, yⁿ) in a neighborhood N about p such that y_i(p) = 0 for i = 1,…, n and υ is quadratic in N:

$\upsilon \text{(}y\text{)=}\upsilon \text{(}p)-y_1^2-\dots -y_1^2+y_{l+1}^2+\mathrm{\dots }+y_n^2$

for some integer 0 ≤ l ≤ n.

Lemma 3.7.2 Consider the zero-dynamics ${\sum }_0^{da}*$ of the system Σ^da. Suppose there is a Lyapunov-function V which is nondegenerate at x = 0 and satisfies V (f^∗(x)) = V (x) for all x ∈ $\mathcal{O}$ neighborhood of x = 0. Then there exists a change in coordinates $\tilde{x}=\phi \left(x\right)$ such that ${\sum }^{da}*$ is described by

${\tilde{x}}_{k+1}={\tilde{f}}^{*}({\tilde{x}}_k),$

where $\tilde{f}:=\phi \circ f^{*}\circ {\phi }^{-1}$ , and has a positive-definite Lyapunov-function $\tilde{V}$ which is quadratic in $\tilde{x}$, i.e., $\tilde{V}\left(\tilde{x}\right)={\tilde{x}}^{T}P\tilde{x}$ for some P > 0, and satisfies $\tilde{V}\left({\tilde{f}}^{*}\left(\tilde{x}\right)\right)=\tilde{V}\left(\tilde{x}\right)$.

Proof: Applying the Morse-lemma, there exists a local change of coordinates $\tilde{x}=\phi \left(x\right)$ such that

$\tilde{V}(\tilde{x})=V({\phi }^{-1}(\tilde{x}))={\tilde{x}}^{T}P\tilde{x}$

for some P > 0. Thus,

$\tilde{V}({\tilde{f}}^{*}(\tilde{x}))=V(f^{*}°{\phi }^{-1}(\tilde{x}))=V({\phi }^{-1}(\tilde{x}))=\tilde{V}(\tilde{x}).\text{□}$

Since the relative-degree of the system Σ^da is obviously so important for the task at hand, and as we have seen in the continuous-time case, the passivity (respectively losslessness) of the system has implications on its vector relative-degree, therefore, we shall proceed in the next few results to analyze the relative-degree of the system. We shall present sufficient conditions for the system to be lossless with a {0,…, 0} relative-degree at x = 0. We begin with the following preliminary result.

Proposition 3.7.1 Suppose Σ^da is lossless with a C² positive-definite storage-function Ψ which is nondegenerate at x = 0. Then

(i)  rank{g(0)} = m if and only if d(0) + d^T (0) > 0.

(ii)  Moreover, as a consequence of (i) above, d(0) is nonsingular, and Σ^da has vector relative-degree {0,…, 0} at x = 0.

Proof: (i)(only if): By Lemma 3.5.1, substituting x = 0 in equation (3.72), we have

$g^T(0){\frac{{\partial }^2\mathrm{\Psi }(\text{λ)}}{\partial {\lambda }^2}|}_{\text{λ=0}}g(0)=d^T(0)+d(0).$

Since the Hessian of V evaluated at λ = 0 is positive-definite, then

$x^T\left(d^T\left(0\right)+d\left(0\right)\right)x={{\left(g\left(0\right)x\right)}^T\frac{{\partial }^2\mathrm{\Psi }}{\partial {\lambda }^2}|}_{\lambda =0}\left(g\left(0\right)x\right).$

Hence, rank{g(0)} = m implies that d(0) + d^T (0) > 0.

(if): Conversely if d(0) + d^T (0) > 0, then

${\left(g\left(0\right)x\right)}^T{\frac{{\partial }^2\mathrm{\Psi }}{\partial {\lambda }^2}|}_{\lambda =0}\left(g\left(0\right)x\right)>0\Rightarrow g\left(0\right)x\ne 0\forall x\in {\mathcal{\Re }}^m\left\{0\right\}.$

Therefore g(0)x = 0 has a unique solution x = 0 when rank{g(0)} = m.

Finally, (ii) follows from (i) by positive-definiteness of d(0) + d^T (0). □

Remark 3.7.2 Notice that for the linear discrete-time system Σ^dl with a positive-definite storage-function$\mathrm{\Psi }\left(x\right)=\frac{1}{2}{x}^{T}Px,P>0,$ is always nondegenerate at x = 0 since the Hessian of V is P > 0.

The following theorem is the counterpart of Theorem 3.4.1 for the discrete-time system Σ^da.

Theorem 3.7.1 Suppose that rank{g(0)} = m and the system Σ^da is lossless with a C² positive-definite storage-function Ψ(.) which is nondegenerate at x = 0. Then Σ^da has vector relative-degree {0,…, 0} and a lossless zero-dynamics at x = 0.

Proof: By Proposition 3.7.1 Σ^da has relative-degree {0,…, 0} at x = 0 and its zero-dynamics exists locally in the neighborhood O of x = 0. Furthermore, since Σ^da is lossless

$\mathrm{\Psi }(f(x)+g(x)u)-\mathrm{\Psi }(x)=y^{T}u,\forall u\in \mathcal{U}.$

(3.90)

Setting u = u^∗ = −d⁻¹(x)h(x) gives

$\mathrm{\Psi }(f^{*}(x))=\mathrm{\Psi }(x)\forall x\in Z^{*}.$

Finally, if we substitute u = u^∗ + ũ in (3.74), it follows that Ψ(f^∗(x) + g(x)ũ) is quadratic in ũ. Thus, the zero-dynamics is lossless. □

Remark 3.7.3 A number of interesting corollaries which relate the losslessness of the system Σ^da with respect to a positive-definite storage-function which is nondegenerate at x = {0}, versus the rank{g(0)} and its relative-degree, could be drawn. It is sufficient here however to observe that, under mild regularity conditions, the discrete-time system Σ^da with d(x) ≡ 0 cannot be lossless; that Σ^da can only be lossless if it has vector relative-degree {0,…, 0} and d(x) is nonsingular. Conversely, under some suitable assumptions, if Σ^da is lossless with a positive-definite storage-function, then it necessarily has vector relative-degree {0,…, 0} at x = 0.

We now present the main result of the section; namely, a necessary and sufficient condition for feedback-equivalence to a lossless system using regular static state-feedback control of the form:

$u_k=\alpha (x_k)+\beta (x_k){\upsilon }_k,\alpha (0)=0$

(3.91)

for some smooth C^∞ functions α : N ⊆ $\mathcal{X}$ → ℜ^m, β : N → $\mathcal{M}$^m×m($\mathcal{X}$ ), 0 ∈ N, with β invertible over N.

Theorem 3.7.2 Suppose Ψ is nondegenerate at x = {0} and rank{g(0)} = m. Then Σ^da is locally feedback-equivalent to a lossless system with a C² storage-function which is positivedefinite, if and only if, Σ^da has vector relative-degree {0,…, 0} at x = {0} and lossless zero-dynamics.

Proof: (Necessity): Suppose there exists a feedback control of the form (3.91) such that Σ^da is feedback-equivalent to the lossless system

$\begin{array}{l}{\tilde{\sum }}^{da}:x_{x+1}=\tilde{f}(x_k)+\tilde{g}(x_k)w_k\hfill \\ y_k=\tilde{h}(x_k)+\tilde{d}(x_k)w_k\hfill \end{array}$

with a C² storage-function Ψ, where w_k is an auxiliary input and

$\tilde{f}(x)=f(x)+g(x)\alpha (x),\tilde{g}(x)=g(x)\beta (x)$

(3.92)

$\tilde{h}(x)=h(x)+d(x)\alpha (x),\tilde{d}(x)=d(x)\beta (x).$

(3.93)

Since β(0) is nonsingular, then $rank\left\{\tilde{g}\left(0\right)\right\}=m.$ Similarly, by Proposition 3.7.1 $\tilde{d}\left(0\right)$ is nonsingular, and therefore d(0). Hence Σ^da has relative-degree {0,…, 0} at x = 0. Next we show that Σ^da has locally lossless zero-dynamics.

The zero-dynamics of the equivalent system ${\tilde{\sum }}^{da}$ are governed by

${\tilde{\sum }}_0^{da}*:x_{k+1}={\tilde{f}}^{*}(x_k),x\in {Ƶ}^{*}$

where ${\tilde{f}}^{*}\left(x\right):=\tilde{f}-\tilde{g}\left(x\right){\tilde{d}}^{-1}\left(x\right)\tilde{h}\left(x\right)$. These are identical to the zero-dynamics of the original system Σ^da with f^∗(x) = f(x) − g(x)d⁻¹(x)h(x) since they are invariant under static state-feedback. But ˜Σ^da lossless implies by Theorem 3.7.1 that

(i)  $\mathrm{\Psi }\left({\tilde{f}}^{*}\left(x\right)\right)=\mathrm{\Psi }\left(x\right)$

(ii)  $\mathrm{\Psi }\left({\tilde{f}}^{*}\left(x\right)+\tilde{g}\left(x\right)w\right)$ is quadratic in w.

Therefore, by the invariance of the zero-dynamics, we have Ψ(f^∗(x)) = Ψ(x) and Ψ(f^∗(x)+ g(x)u) is also quadratic in u. Consequently, Σ^da has locally lossless zero-dynamics. (Sufficiency): If Σ^da has relative-degree {0,…, 0} at x = 0, then there exists a neighborhood $\mathcal{O}$ of x = {0} in which d⁻¹(x) is well defined. Applying the feedback

$u_k=u_k^{*}+d^{-1}(x_k)v_k=-d^{-1}(x_k)\text{h}\left(x_k\right)+d^{-1}\left(x_k\right){\upsilon }_k$

changes Σ^da into the system

$\begin{array}{l}{x}_{k+1}=f^{*}(x_k)+g^{*}(x_k){\upsilon }_k\\ y_k={\upsilon }_k\end{array}$

where g^∗(x) := g(x)d⁻¹(x). Now let the output of this resulting system be identical to that of the lossless system ${\tilde{\sum }}^{da}$ and have the equivalent system

$\begin{array}{l}{\tilde{\sum }}_1^{da}:x_{k+1}=f^{*}(x_k)+g^{*}(x_k)\tilde{h}(x_k)+g^{*}(x_k)\tilde{d}(x_k)w_k\\ y_k=\tilde{h}(x_k)+\tilde{d}(x_k)u_k.\end{array}$

By assumption, Ψ is nondegenerate at x = 0 and rank{g(0)} = m, therefore there exists a neighborhood N of x = 0 such that ${\left(g^{*}\left(x\right)\right)}^T{\frac{{\partial }^2\mathrm{\Psi }}{\partial {\lambda }^2}|}_{\lambda =f^{*}\left(x\right)}{g}^{*}\left(x\right)$ is positive-definite for all x ∈ N. Then

$\tilde{d}(x):={\left(\frac{1}{2}{(g^{*}(x))}^T{\frac{{\partial }^2\mathrm{\Psi }}{\partial {\lambda }^2}|}_{\lambda =f^{*}(x)}{g}^{*}(x)\right)}^{-1}$

(3.94)

$\tilde{h}(x):=\tilde{d}(x){\left({\frac{\partial \mathrm{\Psi }}{\partial \lambda }|}_{\lambda =f^{*}(x)}{g}^{*}(x)\right)}^T$

(3.95)

are well defined in N. It can now be shown that, with the above construction of $\tilde{d}\left(x\right)$ and $\tilde{h}\left(x\right)$ the system ${\tilde{\sum }}_1^{da}$

is lossless with a C² storage-function Ψ.

By assumption, Ψ(f^∗(x)+g^∗(x)u) is quadratic in u, therefore by Taylor-series expansion about f^∗(x), we have

$\begin{array}{l}\mathrm{\Psi }(f^{*}(x)+g^{*}(x)\tilde{h}(x))=\mathrm{\Psi }(f^{*}(x))+{\frac{\partial \mathrm{\Psi }}{\partial \lambda }|}_{\lambda =f^{*}(x)}{g}^{*}(x)\tilde{h}(x)+\\ \frac{1}{2}{\tilde{h}}^T(x){\left(g^{*}(x)\right)}^T{\frac{{\partial }^2\mathrm{\Psi }}{\partial {\lambda }^2}|}_{\lambda =f^{*}(x)}{g}^{*}(x)\tilde{h}(x).\end{array}$

(3.96)

It can then be deduced from the above equation (using again first-order Taylor’s expansion) that

${\frac{\partial \mathrm{\Psi }}{\partial \lambda }|}_{\lambda =f^{*}(x)+g^{*}(x)\tilde{h}(x)}{g}^{*}(x)={\frac{\partial \mathrm{\Psi }}{\partial \alpha }|}_{\lambda =f^{*}(x)}{g}^{*}(x)+{\tilde{h}}^T(x){\left(g^{*}\right)}^T(x){\frac{{\partial }^2\mathrm{\Psi }}{\partial {\lambda }^2}|}_{\lambda =f^{*}(x)}{g}^{*}(x)$

(3.97)

and

${\left(g^{*}(x)\right)}^T{\frac{{\partial }^2\mathrm{\Psi }}{\partial {\lambda }^2}|}_{\lambda =f^{*}(x)+g^{*}(x)\tilde{h}(x)}{g}^{*}(x)={\left(g^{*}(x)\right)}^T{\frac{{\partial }^2\mathrm{\Psi }}{\partial {\lambda }^2}|}_{\lambda =f^{*}(x)}{g}^{*}(x).$

(3.98)

Using (3.94) and (3.95), (3.98), we get

${\frac{\partial \mathrm{\Psi }}{\partial \lambda }|}_{\lambda =f^{*}(x)+g^{*}(x)\tilde{h}(x)}{g}^{*}(x)\tilde{d}(x)={\tilde{h}}^T(x)$

(3.99)

and

${\left(g^{*}(x)\tilde{d}(x)\right)}^T{\frac{{\partial }^2\mathrm{\Psi }}{\partial {\lambda }^2}|}_{\lambda =f^{*}(x)+g^{*}(x)\tilde{h}(x)}(g^{*}(x)\tilde{d}(x))={\tilde{d}}^T(x)+\tilde{d}(x).$

(3.100)

Similarly, substituting (3.94) and (3.95) in (3.96) gives

$\begin{array}{l}\mathrm{\Psi }(f^{*}(x)+g^{*}(x)\tilde{h}(x))=\mathrm{\Psi }(f^{*}(x))-{\tilde{h}}^T(x){\tilde{d}}^{-1}(x)\tilde{h}(x)+{\tilde{h}}^T(x){\tilde{d}}^{-1}(x)\tilde{h}(x)\\ =\mathrm{\Psi }(f^{*}(x)).\end{array}$

(3.101)

Recall now that $\left(f^{*}\left(x\right)+g^{*}\left(x\right)\tilde{h}\left(x\right)\right)$ is the zero-dynamics of the system Σ^d_a1. Together with the fact that $\psi \left(f^{*}\left(x\right)+g^{*}\left(x\right)\tilde{h}\left(x\right)+g^{*}\left(x\right)\tilde{d}\left(x\right)w\right)$ is quadratic in w and (3.99)-(3.101) hold, then by Lemma 3.5.1 and Definition 3.7.3, we conclude that the system ${\tilde{\sum }}_1^{da}$ is lossless with a C² storage-function Ψ. □

Remark 3.7.4 In the case of the linear discrete-time system Σ^dl, with rank(B) = m. The quadratic storage-function $\Psi \left(x\right)=\frac{1}{2}{x}^{T}Px$, P > 0 is always nondegenerate at x = 0, and Ψ(Ax + Bu) is always quadratic in u. It therefore follows from the above theorem that any linear discrete-time system of the form Σ^dl is feedback-equivalent to a lossless linear system with a positive-definite storage-function $\mathrm{\Psi }\left(x\right)=\frac{1}{2}{x}^{T}Px$ if, and only if, there exists a positive-definite matrix P such that

${\left(A-B{D}^{-1}C\right)}^{T}P\left(A-B{D}^{-1}C\right)=P.$

Remark 3.7.5 A global version of the above theorem also exists. It is easily seen that, if the local properties of the system are replaced by their global versions, i.e., if the system has globally lossless zero-dynamics and uniform vector relative-degree {0,…, 0}, then it would be globally feedback-equivalent to a lossless system with a C² positive-definite storage-function.

Remark 3.7.6 What is however missing, and what we have not presented in this section, is the analogous synthesis procedure for feedback-equivalence of the discrete-time system Σ^da to a passive one similar to the continuous-time case. This is because, for a passive system of the form Σ^da, the analysis becomes more difficult and complicated. Furthermore, the discrete-time equivalent of the KYP lemma is not available except for the restricted case when Ψ(f(x) + g(x)u) is quadratic in u (Theorem 3.5.1). But Ψ(f(x) + g(x)u) is in general not quadratic in u, and therefore the fundamental question: “When is the system Σ^da passive, or can be rendered passive using smooth state-feedback?” cannot be answered in general.

3.8    Notes and Bibliography

The basic definitions and fundamental results of Section 3.1 of the chapter on continuous-time systems are based on the papers by Willems [274] and Hill and Moylan [131]-[134]. The results on stability in particular are taken from [131, 133], while the continuous-time Bounded-real lemma is from [131, 202]. The stability results for feedback interconnection of dissipative systems are based on [133], and the connection between dissipativity and finite-gain are discussed in References [132, 264] but have not been presented in the chapter. This will be discussed however in Chapter 5.

The results of Sections 3.3, 3.4 on passivity and local feedback-equivalence to a passive continuous-time system are from the paper by Byrnes et al. [77]. Global results can also be found in the same reference, and a synthesis procedure for the linear case is given in [233, 254].

All the results on the discrete-time systems are based on the papers by Byrnes and Lin [74]-[76], while the KYP lemma for bilinear discrete-time systems is taken from Lin and Byrnes [185]. Finally, results on stochastic systems with Markovian-jump disturbances can be found in [8, 9], while in [100] the results for controlled Markov diffusion processes are discussed. More recent developments in the theory of dissipative systems in the behavioral context can be found in the papers by Willems and Trentelman [275, 228, 259], while applications to stabilization of electrical and mechanical systems can be found in the book by Lozano et al. [187] and the references [204, 228].