Decentralized global output feedback stabilization for a class of uncertain nonlinear systems

Abstract

This paper considers the problem of decentralized global stabilization via output feedback for a class of uncertain nonlinear systems with unknown output functions, in which the nonlinear terms are interconnected by unmeasurable states and the inputs of each subsystem. First, we design the controller for each nominal subsystem without the perturbing nonlinearities. Then, we apply the homogeneous domination approach to design scaled homogeneous observers and controllers with an appropriate choice of gain to render the nonlinear system globally asymptotically stable. Finally, two simulation examples are shown to illustrate the effectiveness of the proposed scheme.

Keywords

Decentralized control output feedback asymptotic nonlinear homogeneous

Introduction

In this paper, the problem of global stabilization via output feedback is considered for a class of uncertain nonlinear systems described by

\begin{matrix} {\overset{\cdot}{ζ}}_{i, 1} = ζ_{i, 2} + ψ_{i, 1} (ζ, ν, d (t)) \\ ⋮ \\ {\overset{\cdot}{ζ}}_{i, n - 1} = ζ_{i, n} + ψ_{i, n - 1} (ζ, ν, d (t)) \\ {\overset{\cdot}{ζ}}_{i, n} = υ_{i} + ψ_{i, n} (ζ, ν, d (t)) \\ y_{i} = h_{i} (ζ_{i, 1}) \end{matrix}

(1)

where $ζ = (ζ_{1, 1}, \dots, ζ_{1, n}, \dots, ζ_{m, 1}, \dots, ζ_{m, n}) \in ℝ^{m \times n}$ , $ν = (υ_{1}, \dots, υ_{m}) \in ℝ^{m}, y_{i} \in ℝ$ , $i = 1, \dots, m$ are system states, control input and output, respectively, and $ψ_{i, j} (\cdot)$ is a function of unmeasurable states, control inputs and uncertain bounded disturbance $d (t)$ . The output function $h_{i} (\cdot)$ is a $C^{1}$ function satisfying $h_{i} (0) = 0$ . The objective of this paper is to find the output feedback observers and controllers of the form

{\dot{ϖ}}_{i} = Γ_{i} (ϖ_{i}, y_{i}), υ_{i} = u_{i} (ϖ_{i}, y_{i}), ϖ_{i} \in ℝ^{n}, i = 1, \dots, m

(2)

with continuous functions $Γ_{i} : ℝ^{n + 1} \to ℝ^{n}$ and $u_{i} : ℝ^{n + 1} \to ℝ$ satisfying $Γ_{i} (0, 0) = 0, u_{i} (0, 0) = 0$ , such that the closed-loop system (1)–(2) is globally asymptotically stable.

Large-scale interconnected systems can be found in the real world. Examples of such systems include large-scale power systems, aerospace, decentralized large-scale networks, intelligent transportation and many other practical applications (Yan et al., 2010; Stipanović et al., 2004; Mazo and Tabuada, 2011; Pan, 2009; Feddema et al., 2002). Decentralized control uses local information at the level of each subsystem in the controller implementation for large-scale interconnected systems. Therefore, the decentralized controllers are of much simpler architecture and more practical than the centralized controllers which require sufficiently large communication bandwidth to exchange information between subsystems. The study of decentralized control dates back to the early 1990s (Siljak, 1991). With the development of modern nonlinear control theory, decentralized nonlinear control design has also attracted tremendous interest (see Ye et al., 2006, 2012; Lu et al., 2010; Liu, 2011; Jian and Khorrami, 1997; Jiang, 2000; Ye and Huang, 2003; Xi and Ding, 2007; Kalsi et al., 2010, and the references therein). Ye et al. (2006), Ye et al. (2012), and Lu et al. (2010) formulated the design of the decentralized stabilizing controllers by using the saturation approach. Liu (2011) proposed a new robust decentralized control scheme, based on sliding mode control theory, for a class of uncertain large-scale interconnected systems with matched and unmatched uncertainties. However, most of them require the availability of the states of each subsystems, which cannot be guaranteed in practice. This motivates researchers to develop decentralized dynamic output feedback controllers that incorporate local observers to estimate the states of the subsystems. An important fact about the global output feedback stabilization of nonlinear systems is that the nonlinear terms cannot grow too fast due to the finite escape time phenomenon as demonstrated by Mazenc et al. (1994). Counterexamples are given to indicate that global stabilization of nonlinear systems via output feedback is usually impossible without introducing extra growth conditions on unmeasurable states of the system. Therefore, the recent works of Jian and Khorrami (1997), Jiang (2000), Ye and Huang (2003), Xi and Ding (2007), Kalsi et al. (2010), Krishnamurthy and Khorrami (2010), Polendo and Qian (2007), and Xie et al. (2010) have focused on the decentralized output feedback stabilization for nonlinear systems with different growth conditions. The large-scale nonlinear systems of Jian and Khorrami (1997) and Jiang (2000) were interconnected via their outputs through linearly parameterized nonlinear functions. Ye and Huang (2003) and Xi and Ding (2007) paid more attention to the decentralized output regulation problem for large-scale nonlinear systems with nonlinear exosystem. While in recent work (Kalsi et al., 2010), the local sliding-mode observers were employed for the nonlinear interconnected systems, which consist of linear subsystems coupled by system inputs, and the feedback gain matrices of local controllers were obtained by solving a constrained optimization problem subject to two linear matrix inequalities. However, the above results for nonlinear large-scale systems considered only those interconnected by their outputs or inputs, with limited results for those that considered the interconnection of unmeasurable states. Polendo and Qian (2007) employed the concept of homogeneous domination to cover a larger class of large-scale nonlinear systems while relaxing the linear growth restriction to a polynomial one. Xie et al. (2010) proposed an $H_{\infty}$ output feedback controller for a class of large-scale interconnected systems with value bounded uncertainties in the states and inputs. Unfortunately, the output feedback control methods of Siljak (1991), Ye et al. (2006), Ye et al. (2012), Lu et al. (2010), Liu (2011), Jian and Khorrami (1997), Jiang (2000), Ye and Huang (2003), Xi and Ding (2007), Kalsi et al. (2010), Krishnamurthy and Khorrami (2010), Polendo and Qian (2007), and Xie et al. (2010) cannot work for a class of nonlinear systems with unknown output functions interconnected by unmeasurable states, even by their inputs. To solve this problem, we first provide a series of controllers and observers for the nominal systems without using the nonlinear parts. Then, using the homogeneous domination approach (Qian, 2005) with appropriate design parameters, a series of scaled observers and controllers can render the closed-loop systems globally asymptotically stable in two cases: lower-triangular case and upper-triangular case. Moreover, our design scheme can be extended to a general class of more complex large-scale interconnected systems, in which the gains in front of the states and inputs are uncertain constants.

The contributions of this paper are as follows: (i) this paper provides a decentralized controller design scheme via output feedback for system (1) by employing homogeneous domination approach; (ii) the use of homogeneity allows the nonlinear functions of each subsystem to be different from those of other subsystems, thus not requiring unnecessary symmetry in nonlinear interconnections.

Preliminaries

This section contains several useful definitions and lemmas which play important roles in this paper.

Definition 2.1 (Weighted homogeneity (Kawski, 1988)) For fixed coordinates $(x_{1}, \dots, x_{n}) \in ℝ^{n}$ and real numbers $r_{i} > 0, i = 1, \dots, n,$ :

the dilation $Δ_{ε} (x)$ is defined by $Δ_{ϵ} (x) = (ϵ^{r_{1}} x_{1}, \dots, ϵ^{r_{n}} x_{n})$ , $\forall ϵ > 0$ , with $r_{i}$ being called the weights of the coordinates, for simplicity, we define the dilation weight $Δ = (r_{1}, \dots, r_{n})$ ;

a function $V \in C (ℝ^{n}, ℝ)$ is said to be homogeneous of degree $τ$ if there is a real number $τ \in ℝ$ such that, $\forall x \in ℝ^{n} \ {0}$ , $ϵ > 0$ , $V (Δ_{ϵ} (x)) = ϵ^{τ} V (x_{1}, \dots, x_{n})$ ;

a vector field $f \in C (ℝ^{n}, ℝ^{n})$ is said to be homogeneous of degree $τ$ if there is a real number $τ \in ℝ$ such that, for $i = 1, \dots, n, \forall x \in ℝ^{n} \ {0}$ , $ϵ > 0$ , $f_{i} (Δ_{ε} (x)) = ϵ^{τ + r_{i}} f_{i} (x)$ ;

a homogeneous $p$ -norm is defined as $∥ x ∥_{Δ, p} = (\sum_{i = 1}^{n} | x_{i} |^{p / r_{i}})^{1 / p}$ , $\forall x \in ℝ^{n}$ , for a constant $p \geq 1$ ; for simplicity, in this paper, we choose $p = 2$ and write $∥ x ∥_{Δ}$ for $∥ x ∥_{Δ, 2}$ .

Lemma 2.1 (Hermes, 1991) If the trivial solution $x = 0$ of the $Δ_{ε}$ -homogeneous system

\begin{matrix} \overset{\cdot}{x} = f (x), f (0) = 0 \end{matrix}

(3)

is globally strongly stable, there exists a $Δ_{ε}$ -homogeneous Lyapunov function $V$ , which is positive definite and proper, such that

\overset{\cdot}{V} |_{(3)} = \frac{\partial V}{\partial x} f (x) < 0, \forall x \neq 0

(4)

Lemma 2.2 (Rosier, 1992) Let $f$ be a continuous vector field on $ℝ^{n}$ such that the origin is a locally asymptotically stable equilibrium point. Assume that $f$ is homogeneous of degree $k$ for some $r \in (0, \infty)^{n}$ . Then, for any $p \in ℕ^{*}$ and any $m > p \cdot max_{1 \leq i \leq n} {r_{i}}$ , there exists a strict Lyapunov function $V$ for (3), which is homogeneous of degree $m$ and of class $C^{p}$ . As a direct consequence, the time derivative $\overset{\cdot}{V}$ is homogeneous of degree $m + k$ .

Lemma 2.3 (Bacciotti and Rosier, 2005) Given a dilation weight $Δ = (r_{1}, \dots, r_{n})$ , suppose that $V_{1} (x)$ and $V_{2} (x)$ are homogeneous functions of degree $τ_{1}$ and $τ_{2}$ , respectively. Then $V_{1} (x) V_{2} (x)$ is also homogeneous with respect to the same dilation weight $Δ$ . Moreover, the homogeneous degree of $V_{1} \cdot V_{2}$ is $τ_{1} + τ_{2}$ .

Lemma 2.4 (Bacciotti and Rosier, 2005) Suppose that $V : ℝ^{n} \to ℝ$ is a homogeneous function of degree $τ$ with respect to the dilation weight $Δ$ . Then the following hold:

$(1)$ $\partial V / \partial x_{i}$ is homogeneous of degree $τ - r_{i}$ with $r_{i}$ being the homogeneous weight of $x_{i}; (2)$ there is a constant $\bar{c}$ such that

V (x) \leq \bar{c} ∥ x ∥_{Δ}^{τ}

Moreover, if $V (x)$ is positive definite, then

\underline{c} ∥ x ∥_{Δ}^{τ} \leq V (x)

where $\underline{c} > 0$ is a constant.

Lemma 2.5 (Qian and Lin, 2001a) Suppose that $c$ and $d$ are two positive real numbers. Given any real-valued function $γ (x, y) > 0$ , such that

\begin{matrix} | x |^{c} | y |^{d} \leq \frac{c}{c + d} γ (x, y) | x |^{c + d} + \frac{d}{c + d} γ^{- c / d} (x, y) | y |^{c + d} \end{matrix}

(5)

Lemma 2.6 (Qian and Lin, 2001b)For $x \in ℝ, y \in ℝ$ , and $p \geq 1$ , the following inequalities hold:

\begin{matrix} | x + y |^{p} \leq 2^{p - 1} | x^{p} + y^{p} | \\ (| x | + | y |)^{\frac{1}{p}} \leq | x |^{1 / p} + | y |^{1 / p} \leq 2^{(p - 1) / p} (| x | + | y |)^{1 / p} \end{matrix}

(6)

To solve the problem, the following condition is imposed on the unknown output function $h_{i} (ζ_{i, 1})$ .

Assumption 2.1 For $i = 1, \dots, m$ , there are two known positive constants ${\underline{θ}}_{i}$ and ${\bar{θ}}_{i}$ , such that

\begin{matrix} {\underline{θ}}_{i} \leq \frac{\partial h_{i} (s)}{\partial s} \leq {\bar{θ}}_{i}, s \in ℝ \end{matrix}

(7)

Remark 2.1 The simplest function satisfying (7) is a linear output $h_{i} (ζ_{i, 1}) = θ ζ_{i, 1}$ with an unknown constant $θ$ if the upper and lower bounds of $θ$ are known. In addition, some nonlinear output functions with bounded first derivatives, such as $h_{i} (ζ_{i, 1}) = 2 ζ_{i, 1} + \sin (ζ_{i, 1})$ , satisfy Assumption 2.1 as well.

Next, we will construct an output feedback stabilizer for the nominal system

\begin{matrix} {\overset{\cdot}{z}}_{i} & = z_{i + 1}, i = 1, \dots, n - 1 \\ {\overset{\cdot}{z}}_{n} & = u \\ y & = h (z_{1}) \end{matrix}

(8)

where $h (\cdot)$ is a $C^{1}$ function satisfying $h (0) = 0$ and $0 < \underline{θ} \leq \frac{\partial h (z_{1})}{\partial z_{1}} \leq \bar{θ}$ . Using the approach of Zhai and Qian (2012), we can design a homogeneous output feedback stabilizer for (8), which is described in the following lemma.

Lemma 2.7 For any constant $τ \geq 0$ , there exist constants $a_{i} > 0, β_{i} > 0, i = 1, \dots, n$ such that the homogeneous output feedback stabilizer

\begin{matrix} {\overset{\cdot}{\hat{z}}}_{i} = f_{i} ({\hat{z}}_{1}, {\hat{z}}_{i + 1}) = {\hat{z}}_{i + 1} - a_{i} {\hat{z}}_{1}^{r_{i + 1}}, i = 1, \dots, n - 1 \\ {\overset{\cdot}{\hat{z}}}_{n} = f_{n} ({\hat{z}}_{1}, u) = u - a_{n} {\hat{z}}_{1}^{r_{n + 1}} \\ u = - β_{n} {({\hat{z}}_{n}^{\frac{σ}{r_{n}}} + β_{n - 1}^{\frac{σ}{r_{n}}} ({\hat{z}}_{n - 1}^{\frac{σ}{r_{n - 1}}} + \dots + β_{2}^{\frac{σ}{r_{3}}} ({\hat{z}}_{2}^{\frac{σ}{r_{2}}} + β_{1}^{\frac{σ}{r_{2}}} y^{σ}) \dots))}^{\frac{r_{n} + τ}{σ}} \end{matrix}

(9)

where $σ = 1 + n τ$ and $r_{i}$ is defined as

\begin{matrix} r_{1} = 1, r_{i + 1} = r_{i} + τ \end{matrix}

(10)

renders system (8) globally asymptotically stable.

Proof. The proof is similar to Theorem 2.1 of Zhai and Qian (2012) with some modifications. For the sake of space, the detailed proof is omitted here. ■

Remark 2.2 Most of the existing results on global output feedback control of nonlinear systems are based on the form of $y = z_{1}$ , which is a special case of $y = h (z_{1})$ considered in this paper. Fortunately, we can find a method of how to tackle the problem of global output feedback stabilization for nonlinear systems in the presence of unknown function $h (\cdot)$ in Zhai and Qian (2012).

Denoting $Z : = (z_{1}, \dots, z_{n}, {\hat{z}}_{1}, \dots, {\hat{z}}_{n})^{T}$ , it is verified that the closed-loop system (8)–(9) can be rewritten

\begin{matrix} \overset{\cdot}{Z} = F (Z) = (z_{2}, \dots, z_{n}, u, f_{1}, \dots, f_{n})^{T} \end{matrix}

(11)

In fact, by choosing the dilation weight

\begin{matrix} Δ = (r_{1}, r_{2}, \dots, r_{n}, r_{1}, r_{2}, \dots, r_{n}) \\ = \underset{for z_{1}, \dots, z_{n}}{\underset{︸}{(1, 1 + τ, \dots, 1 + (n - 1) τ}}, \underset{for {\hat{z}}_{1}, \dots, {\hat{z}}_{n}}{\underset{︸}{1, 1 + τ, \dots, 1 + (n - 1) τ)}} \end{matrix}

it can be verified that $F (Z)$ is homogeneous of degree $τ$ with respect to $Δ$ .

According to Theorem 2 of Hermes (1991) and Lemmas 2.2 and 2.4, there exists a Lyapunov function $V$ with homogeneous degree $2 σ - τ$ such that

\begin{matrix} | \frac{\partial V (Z)}{\partial Z_{i}} | \leq {\bar{c}}_{i} ∥ Z ∥_{Δ}^{2 σ - τ - r_{i}}, {\bar{c}}_{i} > 0 \end{matrix}

(12)

where $∥ Z ∥_{Δ} = \sqrt{\sum_{i = 1}^{n} (| Z_{i} |^{\frac{2}{r_{i}}} + | Z_{n + i} |^{\frac{2}{r_{i}}})}$ . Similarly, by Lemmas 2.1 and 2.3, there is a constant ${\hat{c}}_{1} > 0$ such that

\begin{matrix} \overset{\cdot}{V} |_{(11)} = \frac{\partial V (Z)}{\partial Z} F (Z) \leq - {\hat{c}}_{1} ∥ Z ∥_{Δ}^{2 σ} \end{matrix}

(13)

Main results

In this section, the homogeneous domination approach will be employed to construct decentralized output feedback controllers for system (1). The results consist of two parts: (i) for the lower-triangular case, we first introduce a scaling gain $L \geq 1$ into a series of observers and controllers established in Lemma 2.7, which are only dependent on the information of its own subsystem. Then, by choosing a large enough gain $L$ , the proposed observers and controllers can render the lower-triangular systems globally asymptotically stable; (ii) for upper-triangular case, similar to the lower-triangular case, we introduce a scaling gain $0 < ε < 1$ into the observer and controller established in Lemma 2.7 for each subsystem. Then, a series of observers and controllers with small enough gain $ε$ can be obtained to render the upper-triangular systems globally asymptotically stable.

Controller design for lower-triangular systems

First, we introduce a generalized assumption under which it is possible to achieve global stabilization for a class of lower-triangular systems via output feedback.

Assumption 3.1 For $i = 1, \dots, m, j = 1, \dots, n$ , there exist constants $τ_{i} \geq 0$ and $μ > 0$ such that

\begin{matrix} | ψ_{i, j} (ζ, d (t)) | \leq μ ((| ζ_{1, 1} |^{\frac{1 + j τ_{i}}{1}} + \dots + | ζ_{1, j} |^{\frac{1 + j τ_{i}}{1 + (j - 1) τ_{1}}}) + \dots + (| ζ_{m, 1} |^{\frac{1 + j τ_{i}}{1}} + \dots + | ζ_{m, j} |^{\frac{1 + j τ_{i}}{1 + (j - 1) τ_{m}}})) \end{matrix}

(14)

with the constant $r_{i, j}$ defined as

\begin{matrix} r_{i, 1} = 1, r_{i, j + 1} = r_{i, j} + τ_{i} \end{matrix}

(15)

and $| τ_{l} - τ_{i} | < \frac{1}{n - 1} - \frac{1}{n}$ for $1 \leq i, l \leq m$ .

Remark 3.1 In this remark, an explanation is given to show that the relation between two homogeneous degrees is a necessary condition. In order to find a large enough scaling gain $L \geq 1$ to guarantee that the right-hand side of (34) is negative definite, the power of the scaling gain $L$ in the right-hand side of (27) is required to be less than 1, that is,

(k - 1) \frac{1 + j τ_{i}}{1 + (k - 1) τ_{l}} - (j - 1) < 1, for 1 \leq k \leq j \leq n, 1 \leq l, i \leq m

(16)

For the case of $k = 1$ , it is obvious that (16) holds under the condition $| τ_{l} - τ_{i} | < \frac{1}{n - 1} - \frac{1}{n}$ . For the case of $k \geq 2$ , a direct calculation yields

| τ_{i} - τ_{l} | < \frac{1}{k - 1} - \frac{1}{j}, for 2 \leq k \leq j, 2 \leq j \leq n, 1 \leq l, i \leq m

(17)

Since ${min}_{2 \leq k \leq j, 2 \leq j \leq n} {\frac{1}{k - 1} - \frac{1}{j}} = \frac{1}{n - 1} - \frac{1}{n}$ , then

\begin{matrix} | τ_{i} - τ_{l} | < \frac{1}{n - 1} - \frac{1}{n} \end{matrix}

(18)

holds for all $1 \leq l, i \leq m$ . Hence, $| τ_{i} - τ_{l} | < \frac{1}{n - 1} - \frac{1}{n}$ is a necessary condition to illustrate the system stability.

For simplicity, we assume that $τ_{i} = \frac{q_{i}}{p_{i}}$ with $q_{i}$ an even integer and $p_{i}$ an odd integer. Based on this, $r_{i, j} = 1 + (j - 1) τ_{i}, j = 1, \dots, n$ will be odd in both the denominator and numerator. Note that an equivalent result can be achieved for a real number $τ_{i}$ .

With the help of Lemma 2.7 and Assumption 3.1, we are now ready to use homogeneous domination approach to design global output feedback controllers for system (1).

Theorem 3.1 Under Assumption 3.1, the problem of decentralized global output feedback stabilization for the nonlinear system (1) can be solved by a series of homogeneous output feedback stabilizers.

Proof. The proof is divided into three steps: first, a state transformation is introduced, under which we can obtain a new system; then, we construct a series of observers and controllers with scaling gain $L$ ; finally, by choosing appropriate design parameters, the proposed observers and controllers can render the lower-triangular systems globally asymptotically stable.

We introduce the following change of coordinates

\begin{matrix} z_{i, j} = \frac{ζ_{i, j}}{L^{j - 1}}, i = 1, \dots, m, j = 1, \dots, n, u_{i} = \frac{υ_{i}}{L^{n}} \end{matrix}

(19)

with $L \geq 1$ being the scaling gain to be determined later, system (1) can be rewritten as

\begin{matrix} {\overset{\cdot}{z}}_{i, j} & = L z_{i, j + 1} + \frac{ψ_{i, j} (\cdot)}{L^{j - 1}}, j = 1, \dots, n - 1 \\ {\overset{\cdot}{z}}_{i, n} & = L u_{i} + \frac{ψ_{i, n} (\cdot)}{L^{n - 1}} \\ y_{i} & = h_{i} (z_{i, 1}) \end{matrix}

(20)

for $i = 1, \dots, m$ .

Next, we construct a series of observers with the scaling gain $L$

\begin{matrix} {\overset{\cdot}{\hat{z}}}_{i, j} & = L ({\hat{z}}_{i, j + 1} - a_{i, j} {\hat{z}}_{i, 1}^{r_{i, j + 1}}), j = 1, \dots, n - 1, i = 1, \dots, m \\ {\overset{\cdot}{\hat{z}}}_{i, n} & = L (u_{i} - a_{i, n} {\hat{z}}_{i, 1}^{r_{i, n + 1}}) \end{matrix}

(21)

where $a_{i, j}$ is the gain selected by Lemma 2.7. In addition, we design $υ_{i}$ using the same construction of (9), specifically,

\begin{matrix} υ_{i} & = L^{n} u_{i}, \\ u_{i} & = - β_{i, n} {({\hat{z}}_{i, n}^{\frac{\bar{σ}}{r_{i, n}}} + β_{i, n - 1}^{\frac{\bar{σ}}{r_{i, n}}} ({\hat{z}}_{i, n - 1}^{\frac{\bar{σ}}{r_{i, n - 1}}} + \dots + β_{i, 2}^{\frac{\bar{σ}}{r_{i, 3}}} ({\hat{z}}_{i, 2}^{\frac{\bar{σ}}{r_{i, 2}}} + β_{i, 1}^{\frac{\bar{σ}}{r_{i, 2}}} y_{i}^{\bar{σ}}) \dots))}^{\frac{r_{i, n} + τ_{i}}{\bar{σ}}} \end{matrix}

(22)

where $β_{i, j} > 0$ and $\bar{σ} = max_{1 \leq i \leq m} {1 + n τ_{i}}$ . Similar to (11), it can be seen that

\begin{matrix} Z_{i} & = (z_{i, 1}, \dots, z_{i, n}, {\hat{z}}_{i, 1}, \dots, {\hat{z}}_{i, n})^{T}, \\ {\overset{\cdot}{Z}}_{i} & = F_{i} (Z_{i}) = (z_{i, 2}, \dots, z_{i, n}, u_{i}, {\overset{\cdot}{\hat{z}}}_{i, 1}, \dots, {\overset{\cdot}{\hat{z}}}_{i, n})^{T} \end{matrix}

(23)

Therefore, the closed-loop system (20)–(22) can be written as

\begin{matrix} {\overset{\cdot}{Z}}_{i} = L F_{i} (Z_{i}) + (\frac{ψ_{i, 1} (\cdot)}{L^{0}}, \dots, \frac{ψ_{i, n} (\cdot)}{L^{n - 1}}, 0, \dots, 0)^{T} \end{matrix}

(24)

By Lemma 2.7, there exist the gains $a_{i, j}$ and $β_{i, j}$ such that ${\overset{\cdot}{Z}}_{i} = F_{i} (Z_{i})$ is globally asymptotically stable. According to Lemmas 2.1—2.4, there is a Lyapunov function $U_{i} (Z_{i})$ of degree $2 \bar{σ} - τ_{i}$ such that

\begin{matrix} \frac{\partial U_{i}}{\partial Z_{i}} F_{i} (Z_{i}) \leq - c_{i} ∥ Z_{i} ∥_{Δ_{i}}^{2 \bar{σ}} \end{matrix}

(25)

where $c_{i} > 0$ is a constant and $Δ_{i} = \underset{for z_{i, 1}, \dots, z_{i, n}}{\underset{︸}{(r_{i, 1}, \dots, r_{i, n}}}, \underset{for {\hat{z}}_{i, 1}, \dots, {\hat{z}}_{i, n}}{\underset{︸}{r_{i, 1}, \dots, r_{i, n})}}$ . With (25) in mind, it can be seen that

\begin{matrix} {\overset{\cdot}{U}}_{i} |_{(24)} = L \frac{\partial U_{i}}{\partial Z_{i}} F_{i} (Z_{i}) + \frac{\partial U_{i}}{\partial Z_{i}} (\frac{ψ_{i, 1} (\cdot)}{L^{0}}, \dots, \frac{ψ_{i, n} (\cdot)}{L^{n - 1}}, 0, \dots, 0)^{T} \\ \leq - L c_{i} ∥ Z_{i} ∥_{Δ_{i}}^{2 \bar{σ}} + \frac{\partial U_{i}}{\partial Z_{i}} (\frac{ψ_{i, 1} (\cdot)}{L^{0}}, \dots, \frac{ψ_{i, n} (\cdot)}{L^{n - 1}}, 0, \dots, 0)^{T} \end{matrix}

(26)

Next, under the change of coordinates (19), we deduce from Assumption 3.1 that

\begin{matrix} | \frac{ψ_{i, j} (\cdot)}{L^{j - 1}} | \leq μ (\frac{\sum_{l = 1}^{m} (\sum_{k = 1}^{j} | L^{k - 1} z_{l, k} |^{\frac{1 + j τ_{i}}{1 + (k - 1) τ_{l}}})}{L^{j - 1}}), j = 1, \dots, n \end{matrix}

(27)

The power of the scaling gain $L$ on the right-hand side of (27) is

\begin{matrix} (k - 1) \frac{1 + j τ_{i}}{1 + (k - 1) τ_{l}} - (j - 1) \\ = 1 - \frac{1 + (k - 1) τ_{l} - (k - 1) (1 + j τ_{i}) + (j - 1) (1 + (k - 1) τ_{l})}{1 + (k - 1) τ_{l}} \\ = 1 - \frac{j - (k - 1) + (k - 1) j (τ_{l} - τ_{i})}{1 + (k - 1) τ_{l}} \end{matrix}

(28)

Owing to $τ_{i} \geq 0, τ_{l} \geq 0, 1 \leq k \leq j \leq n$ and $| τ_{i} - τ_{l} | < \frac{1}{n - 1} - \frac{1}{n}$ , we can obtain

\begin{matrix} \frac{j - (k - 1) + (k - 1) j (τ_{l} - τ_{i})}{1 + (k - 1) τ_{l}} \\ > \frac{j - (k - 1) - (k - 1) j \frac{1}{n (n - 1)}}{1 + (k - 1) τ_{l}} \\ \geq \frac{1 - \frac{(k - 1) j}{n (n - 1)}}{1 + (k - 1) τ_{l}} \\ \geq 0 \end{matrix}

(29)

which means that we can find a constant ${\bar{α}}_{i, j} < 1$ such that

\begin{matrix} | \frac{ψ_{i, j} (\cdot)}{L^{j - 1}} | \leq μ L^{{\bar{α}}_{i, j}} \sum_{l = 1}^{m} (\sum_{k = 1}^{j} | z_{l, k} |^{\frac{1 + j τ_{i}}{1 + (k - 1) τ_{l}}}) \end{matrix}

(30)

Recall that for $j = 1, \dots, n$ , $\frac{\partial U_{i}}{\partial Z_{i, j}}$ is homogeneous of degree $2 \bar{σ} - τ_{i} - r_{i, j}$ . Then from Lemmas 2.3 and 2.4, it can be deduced that

\begin{matrix} \begin{matrix} | \frac{\partial U_{i}}{\partial Z_{i, j}} | \sum_{l = 1}^{m} (\sum_{k = 1}^{j} | z_{l, k} |^{\frac{1 + j τ_{i}}{1 + (k - 1) τ_{l}}}) \end{matrix} \end{matrix}

(31)

is homogeneous of degree $2 \bar{σ}$ . With (30) and (31) in mind, by Lemma 2.4, one can find a positive constant ${\bar{ρ}}_{i, j}$ such that

\begin{matrix} | \frac{\partial U_{i}}{\partial Z_{i, j}} \frac{ψ_{i, j} (\cdot)}{L^{j - 1}} | \leq {\bar{ρ}}_{i, j} L^{{\bar{α}}_{i, j}} ∥ Z ∥_{Δ}^{2 \bar{σ}} \end{matrix}

(32)

where $Z = (Z_{1}, Z_{2}, \dots, Z_{m})^{T}, Δ = (Δ_{1}, \dots, Δ_{m})$ .

Substituting (32) into (26) yields

\begin{matrix} {\overset{\cdot}{U}}_{i} |_{(24)} \leq - L c_{i} ∥ Z_{i} ∥_{Δ_{i}}^{2 \bar{σ}} + \sum_{j = 1}^{n} {\bar{ρ}}_{i, j} L^{{\bar{α}}_{i, j}} ∥ Z ∥_{Δ}^{2 \bar{σ}} \end{matrix}

(33)

Let the Lyapunov function be $U$ for the whole large-scale system, then it can be seen that the time derivative of $U$ is

\begin{matrix} \overset{\cdot}{U} = \sum_{i = 1}^{m} {\overset{\cdot}{U}}_{i} \\ \leq - L \sum_{i = 1}^{m} c_{i} ∥ Z_{i} ∥_{Δ_{i}}^{2 \bar{σ}} + \sum_{i = 1}^{m} \sum_{j = 1}^{n} {\bar{ρ}}_{i, j} L^{{\bar{α}}_{i, j}} ∥ Z ∥_{Δ}^{2 \bar{σ}} \\ \leq - L \bar{c} ∥ Z ∥_{Δ}^{2 \bar{σ}} + \sum_{i = 1}^{m} \sum_{j = 1}^{n} {\bar{ρ}}_{i, j} L^{{\bar{α}}_{i, j}} ∥ Z ∥_{Δ}^{2 \bar{σ}} \\ \leq - L (\bar{c} - \bar{ρ} L^{{\bar{α}}_{max} - 1}) ∥ Z ∥_{Δ}^{2 \bar{σ}} \end{matrix}

(34)

where ${\bar{α}}_{max} = max_{1 \leq i \leq m, 1 \leq j \leq n} {{\bar{α}}_{i, j}} < 1$ and $\bar{c} > 0, \bar{ρ} > 0$ do not depend on $L$ . From (34), we can choose a large enough gain $L$ such that the right-hand side of (34) is negative definite. Therefore, it can be verified that the scaled homogeneous output feedback stabilizers (21) and (22) render system (1) globally asymptotically stable. ■

Next, we show that Theorem 3.1 can be used to solve the global output feedback stabilization of large-scale lower-triangular systems.

Example 3.1 Consider the stabilization problem of the following uncertain nonlinear systems

\begin{matrix} Σ_{1} & : {\begin{matrix} {\overset{\cdot}{ζ}}_{1, 1} = ζ_{1, 2} + 40 ζ_{1, 1} ζ_{2, 1}^{4 / 5} \\ {\overset{\cdot}{ζ}}_{1, 2} = υ_{1} + d (t) ζ_{1, 1}^{6 / 5} ζ_{2, 2} + ζ_{1, 2} ζ_{2, 1}^{4 / 5} \\ y_{1} = θ_{1, 1} \sin (ζ_{1, 1}) + θ_{1, 2} ζ_{1, 1} \end{matrix}, \\ Σ_{2} & : {\begin{matrix} {\overset{\cdot}{ζ}}_{2, 1} = ζ_{2, 2} + d (t) ζ_{1, 1}^{3 / 5} ζ_{2, 1}^{4 / 5} \\ {\overset{\cdot}{ζ}}_{2, 2} = υ_{2} + d (t) ζ_{1, 1}^{4 / 5} ζ_{1, 2}^{5 / 9} + ζ_{2, 2} ζ_{2, 1}^{2 / 5} \\ y_{2} = θ_{2, 1} ζ_{2, 1} \end{matrix} \end{matrix}

(35)

where $0.1 \leq θ_{1, 1} \leq 1, 1 \leq θ_{1, 2} \leq 2, 1 \leq θ_{2, 1} \leq 3$ and $| d (t) | \leq 1$ . By choosing $τ_{1} = 4 / 5$ and $τ_{2} = 2 / 5$ , we can obtain that $r_{1, 2} = 9 / 5, r_{2, 2} = 7 / 5, \bar{σ} = 13 / 5$ and $τ_{1}, τ_{2}$ satisfy $| τ_{1} - τ_{2} | = 2 / 5 < 1 / 2$ . Moreover, by Young’s inequality, it can be verified that

\begin{matrix} | ζ_{1, 1} ζ_{2, 1}^{4 / 5} | \leq \frac{5}{9} | ζ_{1, 1} |^{9 / 5} + \frac{4}{9} | ζ_{2, 1} |^{9 / 5} \leq \frac{5}{9} (| ζ_{1, 1} |^{9 / 5} + | ζ_{2, 1} |^{9 / 5}), \\ | d (t) ζ_{1, 1}^{6 / 5} ζ_{2, 2} | \leq \frac{6}{13} | ζ_{1, 1} |^{13 / 5} + \frac{7}{13} | ζ_{2, 2} |^{13 / 7} \leq \frac{7}{13} (| ζ_{1, 1} |^{13 / 5} + | ζ_{2, 2} |^{13 / 7}), \\ | ζ_{1, 2} ζ_{2, 1}^{4 / 5} | \leq \frac{9}{13} | ζ_{1, 2} |^{13 / 9} + \frac{4}{13} | ζ_{2, 1} |^{13 / 5} \leq \frac{9}{13} (| ζ_{1, 2} |^{13 / 9} + | ζ_{2, 1} |^{13 / 5}), \\ | d (t) ζ_{1, 1}^{3 / 5} ζ_{2, 1}^{4 / 5} | \leq \frac{3}{7} | ζ_{1, 1} |^{7 / 5} + \frac{4}{7} | ζ_{2, 1} |^{7 / 5} \leq \frac{4}{7} (| ζ_{1, 1} |^{7 / 5} + | ζ_{2, 1} |^{7 / 5}), \\ | d (t) ζ_{1, 1}^{4 / 5} ζ_{1, 2}^{5 / 9} | \leq \frac{4}{9} | ζ_{1, 1} |^{9 / 5} + \frac{5}{9} | ζ_{1, 2} | \leq \frac{5}{9} (| ζ_{1, 1} |^{9 / 5} + | ζ_{1, 2} |), \\ | ζ_{2, 2} ζ_{2, 1}^{2 / 5} | \leq \frac{7}{9} | ζ_{2, 2} |^{9 / 7} + \frac{2}{9} | ζ_{2, 1} |^{9 / 5} \leq \frac{7}{9} (| ζ_{2, 2} |^{9 / 7} + | ζ_{2, 1} |^{9 / 5}) \end{matrix}

(36)

from which it is clear that $ψ_{1, 1}, ψ_{1, 2}, ψ_{2, 1}, ψ_{2, 2}$ satisfy Assumption 3.1.

Next, by the following change of coordinates

\begin{matrix} x_{1, 1} & = ζ_{1, 1}, x_{1, 2} = ζ_{1, 2} / L, u_{1} = υ_{1} / L^{2}, \\ x_{2, 1} & = ζ_{2, 1}, x_{2, 2} = ζ_{2, 2} / L, u_{2} = υ_{2} / L^{2} \end{matrix}

(37)

the scaled observers and controllers can be constructed as

\begin{matrix} {\overset{\cdot}{\hat{x}}}_{1, 1} & = L ({\hat{x}}_{1, 2} - a_{1, 1} {\hat{x}}_{1, 1}^{r_{1, 2}}), {\overset{\cdot}{\hat{x}}}_{1, 2} = L (u_{1} - a_{1, 2} {\hat{x}}_{1, 1}^{r_{1, 2} + τ_{1}}), \\ υ_{1} & = - L^{2} β_{1, 2} ({\hat{x}}_{1, 2}^{\bar{σ} / r_{1, 2}} + β_{1, 1}^{\bar{σ} / r_{1, 2}} y_{1}^{\bar{σ}})^{(r_{1, 2} + τ_{1}) / \bar{σ}}, \\ {\overset{\cdot}{\hat{x}}}_{2, 1} & = L ({\hat{x}}_{2, 2} - a_{2, 1} {\hat{x}}_{2, 1}^{r_{2, 2}}), {\overset{\cdot}{\hat{x}}}_{2, 2} = L (u_{2} - a_{2, 2} {\hat{x}}_{2, 1}^{r_{2, 2} + τ_{2}}), \\ υ_{2} & = - L^{2} β_{2, 2} ({\hat{x}}_{2, 2}^{\bar{σ} / r_{2, 2}} + β_{2, 1}^{\bar{σ} / r_{2, 2}} y_{2}^{\bar{σ}})^{(r_{2, 2} + τ_{2}) / \bar{σ}} \end{matrix}

(38)

with appropriate positive constants $a_{i, j}, β_{i, j}$ and $L$ . In the simulation, the unknown parameters are assumed to be $θ_{1, 1} = 0.1$ , $θ_{1, 2} = 1.2$ , $θ_{2, 1} = 1.1$ and $d (t) = 0.2 \sin (t)$ . The gains in (38) are chosen as

\begin{matrix} a_{1, 1} = 6, a_{1, 2} = 5, a_{2, 1} = 6, a_{2, 2} = 5, \\ β_{1, 1} = 2, β_{1, 2} = 3, β_{2, 1} = 2, β_{2, 2} = 3 \end{matrix}

and the scaling gain is chosen as $L = 6$ . If we choose the initial condition as

\begin{matrix} ζ_{1, 1} (0) = - 0.2, ζ_{1, 2} (0) = 0.4, {\hat{x}}_{1, 1} (0) = 0, {\hat{x}}_{1, 2} (0) = 0, \\ ζ_{2, 1} (0) = 0.3, ζ_{2, 2} (0) = 0.5, {\hat{x}}_{2, 1} (0) = 0, {\hat{x}}_{2, 2} (0) = 0 \end{matrix}

the simulation result is shown in Figure 1, which illustrates that the decentralized system (35) can be globally asymptotically stable by the decentralized output feedback stabilizers (38).

Controller design for upper-triangular systems

We present a homogeneous growth condition for a class of upper-triangular systems under which it is possible to achieve global stabilization of (1) via output feedback.

Assumption 3.2 For $i = 1, \dots, m, j = 1, \dots, n - 1$ , there exist constants $τ_{i} \geq 0$ and $c > 0$ such that

\begin{matrix} | ψ_{i, j} (ζ, ν, d (t)) | \leq c (\begin{matrix} (| ζ_{1, j + 2} |^{\frac{1 + j τ_{i}}{1 + (j + 1) τ_{1}}} + \dots + | ζ_{1, n} |^{\frac{1 + j τ_{i}}{1 + (n - 1) τ_{1}}} + | υ_{1} |^{\frac{1 + j τ_{i}}{1 + n τ_{1}}}) + \dots \\ + (| ζ_{m, j + 2} |^{\frac{1 + j τ_{i}}{1 + (j + 1) τ_{m}}} + \dots + | ζ_{m, n} |^{\frac{1 + j τ_{i}}{1 + (n - 1) τ_{m}}} + | υ_{m} |^{\frac{1 + j τ_{i}}{1 + n τ_{m}}}) \end{matrix}) \end{matrix}

(39)

with the constant $r_{i, j}$ defined as (15) and $| τ_{l} - τ_{i} | < \frac{1}{n - 1} - \frac{1}{n}$ for $1 \leq i, l \leq m$ .

Remark 3.2 In this remark, an explanation is given to show that the relation between two homogeneous degrees is a necessary condition. In order to find a small enough scaling gain $0 < ε < 1$ to guarantee that the right-hand side of (56) is negative definite, the power of the scaling gain $ε$ in the right-hand side of (48) is required to be larger than one, that is,

\begin{array}{l} (k - 1) \frac{1 + j τ_{i}}{1 + (k - 1) τ_{l}} - (j - 1) > 1, for \\ j + 2 \leq k \leq n, 1 \leq j \leq n - 2, 1 \leq l, i \leq m \end{array}

(40)

n \frac{1 + j τ_{i}}{1 + n τ_{l}} - (j - 1) > 1, for 1 \leq j \leq n - 1, 1 \leq l, i \leq m

From (40), a direct calculation yields

\begin{matrix} | τ_{i} - τ_{l} | < \frac{1}{j} - \frac{1}{k - 1}, \\ for j + 2 \leq k \leq n, 1 \leq j \leq n - 2, 1 \leq l, i \leq m \end{matrix}

(41)

| τ_{i} - τ_{l} | < \frac{1}{j} - \frac{1}{n}, for 1 \leq j \leq n - 1, 1 \leq l, i \leq m

Since $min_{j + 2 \leq k \leq n, 1 \leq j \leq n - 2} {\frac{1}{j} - \frac{1}{k - 1}} = \frac{1}{n - 2} - \frac{1}{n - 1}$ and ${min}_{1 \leq j \leq n - 1} {\frac{1}{j} - \frac{1}{n}} = \frac{1}{n - 1} - \frac{1}{n}$ , explicitly,

\begin{matrix} | τ_{i} - τ_{l} | < min {\frac{1}{n - 2} - \frac{1}{n - 1}, \frac{1}{n - 1} - \frac{1}{n}} = \frac{1}{n - 1} - \frac{1}{n} \end{matrix}

(42)

holds for all $1 \leq l, i \leq m$ . Hence, $| τ_{i} - τ_{l} | < \frac{1}{n - 1} - \frac{1}{n}$ is a necessary condition to illustrate the system stability.

With the help of the homogeneous controller and observer established in Lemma 2.7, we are ready to adopt the homogeneous domination approach to globally stabilize system (1) via output feedback.

Theorem 3.2 Under Assumption 3.2, the problem of the decentralized global output feedback stabilization for the nonlinear system (1) can be solved by a series of homogeneous output feedback stabilizers.

Proof. The proof is similar to that of Theorem 3.1 with some modifications. Following the same line, we first introduce the following change of coordinates

\begin{matrix} z_{i, j} = \frac{ζ_{i, j}}{ε^{j - 1}}, i = 1, \dots, m, j = 1, \dots, n, u_{i} = \frac{υ_{i}}{ε^{n}} \end{matrix}

(43)

with $0 < ε < 1$ being the scaling gain to be determined later, system (1) can be rewritten as

\begin{matrix} {\overset{\cdot}{z}}_{i, j} & = ε z_{i, j + 1} + \frac{ψ_{i, j} (\cdot)}{ε^{j - 1}}, i = 1, \dots, m, j = 1, \dots, n - 1 \\ {\overset{\cdot}{z}}_{i, n} & = ε u_{i} \\ y_{i} & = h_{i} (z_{i, 1}) \end{matrix}

(44)

Similar to the proof of Theorem 3.1, we construct a series of observers and controllers with the scaling gain $ε$

\begin{matrix} {\overset{\cdot}{\hat{z}}}_{i, j} & = ε ({\hat{z}}_{i, j + 1} - a_{i, j} {\hat{z}}_{i, 1}^{r_{i, j + 1}}), j = 1, \dots, n - 1, \\ {\overset{\cdot}{\hat{z}}}_{i, n} & = ε (u_{i} - a_{i, n} {\hat{z}}_{i, 1}^{r_{i, n + 1}}), \\ υ_{i} & = ε^{n} u_{i}, \\ u_{i} & = - β_{i, n} {({\hat{z}}_{i, n}^{\frac{\bar{σ}}{r_{i, n}}} + β_{i, n - 1}^{\frac{\bar{σ}}{r_{i, n}}} ({\hat{z}}_{i, n - 1}^{\frac{\bar{σ}}{r_{i, n - 1}}} + \dots + β_{i, 2}^{\frac{\bar{σ}}{r_{i, 3}}} ({\hat{z}}_{i, 2}^{\frac{\bar{σ}}{r_{i, 2}}} + β_{i, 1}^{\frac{\bar{σ}}{r_{i, 2}}} y_{i}^{\bar{σ}}) \dots))}^{\frac{r_{i, n} + τ_{i}}{\bar{σ}}} \end{matrix}

(45)

where $a_{i, j}, β_{i, j}$ , $i = 1, \dots, m$ , $j = 1, \dots, n$ , $\bar{σ}$ are the same as in Theorem 3.1. Using the same notation (23), the closed-loop system (44) and (45) can be written as

\begin{matrix} {\overset{\cdot}{Z}}_{i} = ε F_{i} (Z_{i}) + (\frac{ψ_{i, 1} (\cdot)}{ε^{0}}, \dots, \frac{ψ_{i, n - 1} (\cdot)}{ε^{n - 2}}, 0, \dots, 0)^{T} \end{matrix}

(46)

With (25) in mind, it can be seen that

\begin{matrix} {\overset{\cdot}{U}}_{i} |_{(46)} = ε \frac{\partial U_{i}}{\partial Z_{i}} F_{i} (Z_{i}) + \frac{\partial U_{i}}{\partial Z_{i}} (\frac{ψ_{i, 1} (\cdot)}{ε^{0}}, \dots, \frac{ψ_{i, n - 1} (\cdot)}{ε^{n - 2}}, 0, \dots, 0)^{T} \\ \leq - ε c_{i} ∥ Z_{i} ∥_{Δ_{i}}^{2 \bar{σ}} + \frac{\partial U_{i}}{\partial Z_{i}} (\frac{ψ_{i, 1} (\cdot)}{ε^{0}}, \dots, \frac{ψ_{i, n - 1} (\cdot)}{ε^{n - 2}}, 0, \dots, 0)^{T} \end{matrix}

(47)

Next, under the change of coordinates (43), we deduce from Assumption 3.2 that

\begin{array}{l} | \frac{ψ_{i, j} (\cdot)}{ε^{j - 1}} | \leq c (\frac{\sum_{l = 1}^{m} (\sum_{k = j + 2}^{n} | ε^{k - 1} z_{l, k} |^{\frac{1 + j τ_{i}}{1 + (k - 1) τ_{l}}} + | ε^{n} u_{l} |^{\frac{1 + j τ_{i}}{1 + n τ_{l}}})}{ε^{j - 1}}), \\ j = 1, \dots, n - 1 \end{array}

(48)

The power of the scaling gain $ε$ at the first part in the right-hand side of (48) is

\begin{matrix} (k - 1) \frac{1 + j τ_{i}}{1 + (k - 1) τ_{l}} - (j - 1) \\ = 1 + \frac{(k - 1) (1 + j τ_{i}) - j (1 + (k - 1) τ_{l})}{1 + (k - 1) τ_{l}} \\ = 1 + \frac{k - 1 - j + (k - 1) j (τ_{i} - τ_{l})}{1 + (k - 1) τ_{l}} \end{matrix}

(49)

Owing to $τ_{i} \geq 0$ , $τ_{l} \geq 0$ , $j + 2 \leq k \leq n$ and $| τ_{i} - τ_{l} | < \frac{1}{n - 1} - \frac{1}{n}$ , we can obtain

\begin{matrix} \frac{k - 1 - j + (k - 1) j (τ_{i} - τ_{l})}{1 + (k - 1) τ_{l}} \\ > \frac{k - 1 - j - (k - 1) j \frac{1}{n (n - 1)}}{1 + (k - 1) τ_{l}} \\ \geq \frac{1 - \frac{(k - 1) j}{n (n - 1)}}{1 + (k - 1) τ_{l}} \\ > 0 \end{matrix}

(50)

On the other hand, the power of the scaling gain $ε$ at the second part in the right-hand side of (48) is

\begin{matrix} n \frac{1 + j τ_{i}}{1 + n τ_{l}} - (j - 1) \\ = 1 + \frac{n (1 + j τ_{i}) - j (1 + n τ_{l})}{1 + n τ_{l}} \\ = 1 + \frac{n - j + nj (τ_{i} - τ_{l})}{1 + n τ_{l}} \end{matrix}

(51)

Similar to (50), it can be verified that $\frac{n - j + nj (τ_{i} - τ_{l})}{1 + n τ_{l}} > 0$ .

In summary, we can find a constant $α_{i, j} > 1$ such that

\begin{matrix} | \frac{ψ_{i, j} (\cdot)}{ε^{j - 1}} | \leq c ε^{α_{i, j}} \sum_{l = 1}^{m} (\sum_{k = j + 2}^{n} | z_{l, k} |^{\frac{1 + j τ_{i}}{1 + (k - 1) τ_{l}}} + | u_{l} |^{\frac{1 + j τ_{i}}{1 + n τ_{l}}}) \end{matrix}

(52)

Recall that for $j = 1, \dots, n - 1$ , $\frac{\partial U_{i}}{\partial Z_{i, j}}$ is homogeneous of degree $2 \bar{σ} - τ_{i} - r_{i, j}$ . Then from Lemmas 2.3, 2.4 and 2.6, it can be deduced that

\begin{matrix} \begin{matrix} | \frac{\partial U_{i}}{\partial Z_{i, j}} | \sum_{l = 1}^{m} (\sum_{k = j + 2}^{n} | z_{l, k} |^{\frac{1 + j τ_{i}}{1 + (k - 1) τ_{l}}} + | u_{l} |^{\frac{1 + j τ_{i}}{1 + n τ_{l}}}) \\ \leq c_{i, j} ∥ Z ∥_{Δ}^{2 \bar{σ} - τ_{i} - r_{i, j}} \sum_{l = 1}^{m} (\begin{matrix} \sum_{k = j + 2}^{n} | z_{l, k} |^{\frac{1 + j τ_{i}}{1 + (k - 1) τ_{l}}} + β_{l, n}^{\frac{1 + j τ_{i}}{\bar{σ}}} ({\hat{z}}_{l, n}^{\frac{\bar{σ}}{r_{l, n}}} + β_{l, n - 1}^{\frac{\bar{σ}}{r_{l, n}}} ({\hat{z}}_{l, n - 1}^{\frac{\bar{σ}}{r_{l, n - 1}}} + \dots \\ + β_{l, 2}^{\frac{\bar{σ}}{r_{l, 3}}} ({\hat{z}}_{l, 2}^{\frac{\bar{σ}}{r_{l, 2}}} + β_{l, 1}^{\frac{\bar{σ}}{r_{l, 2}}} y_{l}^{\bar{σ}}) \dots))^{\frac{1 + j τ_{i}}{\bar{σ}}} \end{matrix}) \\ \leq c_{i, j} ∥ Z ∥_{Δ}^{2 \bar{σ} - τ_{i} - r_{i, j}} \sum_{l = 1}^{m} (\sum_{k = j + 2}^{n} | z_{l, k} |^{\frac{1 + j τ_{i}}{1 + (k - 1) τ_{l}}} + β_{l, n}^{\frac{1 + j τ_{i}}{1 + n τ_{l}}} ({\hat{z}}_{l, n}^{\frac{1 + j τ_{i}}{r_{l, n}}} + A_{1} {\hat{z}}_{l, n - 1}^{\frac{1 + j τ_{i}}{r_{l, n - 1}}} + \dots \\ + A_{n - 2} z_{l, 2}^{\frac{1 + j τ_{i}}{r_{l, 2}}} + A_{n - 1} {\bar{θ}}_{l}^{1 + j τ_{i}} z_{l, 1}^{1 + j τ_{i}})) \\ \leq c_{i, j} ∥ Z ∥_{Δ}^{2 \bar{σ} - τ_{i} - r_{i, j}} d_{i, j} ∥ Z ∥_{Δ}^{1 + j τ_{i}} \\ \leq {\hat{ρ}}_{i, j} ∥ Z ∥_{Δ}^{2 \bar{σ}} \end{matrix} \end{matrix}

(53)

is homogeneous of degree $2 \bar{σ}$ where $c_{i, j}, d_{i, j}, {\hat{ρ}}_{i, j}$ , $i = 1, \dots, m$ , $j = 1, \dots, n - 1$ are positive constants and $A_{p} = β_{l, n - 1}^{\frac{1 + j τ_{i}}{r_{l, n}}} \dots β_{l, n - p}^{\frac{1 + j τ_{i}}{r_{l, n - p + 1}}}$ , $p = 1, \dots, n - 1$ . With (52) and (53) in mind, by Lemma 2.4, one can find a positive constant $ρ_{i, j}$ such that

| \frac{\partial U_{i}}{\partial Z_{i, j}} \frac{ψ_{i, j} (\cdot)}{ε^{j - 1}} | \leq ρ_{i, j} ε^{α_{i, j}} ∥ Z ∥_{Δ}^{2 \bar{σ}}

(54)

where $Z = (Z_{1}, Z_{2}, \dots, Z_{m})^{T}$ , $Δ = (Δ_{1}, \dots, Δ_{m})$ .

Substituting (54) into (47) yields

\begin{matrix} {\overset{\cdot}{U}}_{i} \leq - ε c_{i} ∥ Z_{i} ∥_{Δ_{i}}^{2 \bar{σ}} + \sum_{j = 1}^{n - 1} ρ_{i, j} ε^{α_{i, j}} ∥ Z ∥_{Δ}^{2 \bar{σ}} \end{matrix}

(55)

Let the Lyapunov function $U$ for the whole large-scale system, then it can be seen that the time derivative of $U$ is

\begin{matrix} \overset{\cdot}{U} = \sum_{i = 1}^{m} {\overset{\cdot}{U}}_{i} \\ \leq - ε \sum_{i = 1}^{m} c_{i} ∥ Z_{i} ∥_{Δ_{i}}^{2 \bar{σ}} + \sum_{i = 1}^{m} \sum_{j = 1}^{n - 1} ρ_{i, j} ε^{α_{i, j}} ∥ Z ∥_{Δ}^{2 \bar{σ}} \\ \leq - ε \tilde{c} ∥ Z ∥_{Δ}^{2 \bar{σ}} + \sum_{i = 1}^{m} \sum_{j = 1}^{n - 1} ρ_{i, j} ε^{α_{i, j}} ∥ Z ∥_{Δ}^{2 \bar{σ}} \\ \leq - ε (\tilde{c} - ρ ε^{α_{min} - 1}) ∥ Z ∥_{Δ}^{2 \bar{σ}} \end{matrix}

(56)

where $α_{min} = min_{1 \leq i \leq m, 1 \leq j \leq n - 1} {α_{i, j}} > 1$ and $\tilde{c} > 0$ , $ρ > 0$ do not depend on $ε$ . From (56), we can choose a small enough gain $ε$ such that the right-hand side of (56) is negative definite. Therefore, it can be verified that the scaled homogeneous output feedback stabilizers (45) render system (1) globally asymptotically stable. ■

In what follows, we show that Theorem 3.2 can still be extended to deal with a class of more general uncertain systems, in which gains in front of the states and control inputs are uncertain bounded constants.

Corollary 3.1 Under Assumption 3.2, we can find a series of observers and controllers of the form (45) to address the problem of global output feedback stabilization of large-scale nonlinear systems described as

\begin{matrix} {\overset{\cdot}{ζ}}_{i, 1} & = g_{i, 1} ζ_{i, 2} + ψ_{i, 1} (ζ, ν, d (t)) \\ {\overset{\cdot}{ζ}}_{i, 2} & = g_{i, 2} ζ_{i, 3} + ψ_{i, 2} (ζ, ν, d (t)) \\ ⋮ \\ {\overset{\cdot}{ζ}}_{i, n - 2} & = g_{i, n - 2} ζ_{i, n - 1} + ψ_{i, n - 2} (ζ, ν, d (t)) \\ {\overset{\cdot}{ζ}}_{i, n - 1} & = g_{i, n - 1} ζ_{i, n} + ψ_{i, n - 1} (ν, d (t)) \\ {\overset{\cdot}{ζ}}_{i, n} & = g_{i, n} υ_{i} \\ y_{i} & = h_{i} (ζ_{i, 1}) \end{matrix}

(57)

where $g_{i, j}, i = 1, \dots, m$ , $j = 1, \dots, n$ are uncertain bounded constants.

Proof. Consider the following transformation of the coordinates

\begin{matrix} x_{i, j} = \frac{ζ_{i, j}}{\sum_{k = j}^{n} g_{i, k} \dots g_{i, n}}, i = 1, \dots, m, j = 1, \dots, n \end{matrix}

(58)

Systems (57) becomes

\begin{matrix} {\overset{\cdot}{x}}_{i, 1} & = x_{i, 2} + ϕ_{i, 1} ({\bar{x}}_{1}, ν, d (t)) \\ {\overset{\cdot}{x}}_{i, 2} & = x_{i, 3} + ϕ_{i, 2} ({\bar{x}}_{2}, ν, d (t)) \\ ⋮ \\ {\overset{\cdot}{x}}_{i, n - 2} & = x_{i, n - 1} + ϕ_{i, n - 2} ({\bar{x}}_{n - 2}, ν, d (t)) \\ {\overset{\cdot}{x}}_{i, n - 1} & = x_{i, n} + ϕ_{i, n - 1} (ν, d (t)) \\ {\overset{\cdot}{x}}_{i, n} & = υ_{i} \\ y_{i} & = {\bar{h}}_{i} (x_{i, 1}) \end{matrix}

(59)

where the nonlinear terms $ϕ_{i, j} (\cdot)$ satisfy Assumption 3.2 with a new positive constant $\hat{c}$ . The rest of proof is the same as that of Theorem 3.2. For the sake of space, the detailed proof is omitted here. ■

Next, we present an example to illustrate the good performance of the output feedback stabilizers for a class of decentralized uncertain upper-triangular nonlinear systems.

Example 3.2 Consider the stabilization problem of the following uncertain nonlinear systems

\begin{array}{l} \begin{array}{l} Σ_{1} : {\begin{array}{l} {\dot{ζ}}_{1, 1} = ζ_{1, 2} + 40 ζ_{1, 1} ζ_{2, 1}^{4 / 5} \\ {\dot{ζ}}_{1, 2} = υ_{1} + d (t) ζ_{1, 1}^{6 / 5} ζ_{2, 2} + ζ_{1, 2} ζ_{2, 1}^{4 / 5} \\ y_{1} = θ_{1, 1} \sin (ζ_{1, 1}) + θ_{1, 2} ζ_{1, 1} \end{array}, \\ Σ_{2} : {\begin{array}{l} {\dot{ζ}}_{2, 1} = ζ_{2, 2} + d (t) ζ_{1, 1}^{3 / 5} ζ_{2, 1}^{4 / 5} \\ {\dot{ζ}}_{2, 2} = υ_{2} + d (t) ζ_{1, 1}^{4 / 5} ζ_{1, 2}^{5 / 9} + ζ_{2, 2} ζ_{2, 1}^{2 / 5} \\ y_{2} = θ_{2, 1} ζ_{2, 1} \end{array} \end{array} \end{array}

(60)

where $0.1 \leq θ_{1, 1} \leq 1, 1 \leq θ_{1, 2} \leq 2, 1 \leq θ_{2, 1} \leq 3$ and $| d (t) | \leq 1$ . By choosing $τ_{1} = 2 / 7$ and $τ_{2} = 2 / 9$ , we can obtain that $r_{1, 2} = 9 / 7$ , $r_{1, 3} = 11 / 7$ , $r_{2, 2} = 11 / 9$ , $r_{2, 3} = 13 / 9$ , $\bar{σ} = 13 / 7$ and $τ_{1}, τ_{2}$ satisfy $| τ_{1} - τ_{2} | = 4 / 63 < 1 / 6$ . By mean-value theory, we have

\begin{matrix} \ln (1 + ζ_{2, 3}^{2}) = \ln (1 + | ζ_{2, 3}^{2 / 7} |^{7}) \leq 7 \frac{s^{6}}{1 + s^{7}} | ζ_{2, 3} |^{2 / 7} \end{matrix}

(61)

for a $s$ between $[0, | ζ_{2, 3}^{2 / 7} |]$ . Moreover, by Young’s inequality, it can be verified that

with the help of above relations, it is clear that $ψ_{1, 1}, ψ_{1, 2}, ψ_{2, 1}, ψ_{2, 2}$ satisfy Assumption 3.2.

Next, by the following change of coordinates

\begin{array}{l} | d (t) ζ_{1, 3}^{5 / 9} ζ_{2, 3}^{2 / 7} + d (t) υ_{2}^{27 / 35} | \leq \frac{55}{81} | ζ_{1, 3} |^{9 / 11} + \frac{26}{81} | ζ_{2, 3} |^{81 / 91} + | υ_{2} |^{27 / 35}, \\ | d (t) ζ_{1, 3}^{17 / 33} \ln (1 + ζ_{2, 3}^{2}) + d (t) υ_{1}^{77 / 117} | \leq | ζ_{1, 3}^{17 / 33} \frac{1 + 6 s^{7}}{1 + s^{7}} ζ_{2, 3}^{2 / 7} | + | υ_{1} |^{77 / 117} \\ \leq 6 | ζ_{1, 3}^{17 / 33} ζ_{2, 3}^{2 / 7} | + | υ_{1} |^{77 / 117} \\ \leq \frac{306}{77} | ζ_{1, 3} |^{7 / 9} + \frac{156}{77} | ζ_{2, 3} |^{11 / 13} + | υ_{1} |^{77 / 117}, \\ | d (t) υ_{2}^{33 / 35} | \leq | υ_{2} |^{33 / 35}, \\ | d (t) υ_{1}^{7 / 9} | \leq | υ_{1} |^{7 / 9} \end{array}

(62)

\begin{matrix} x_{1, 1} = ζ_{1, 1}, x_{1, 2} = ζ_{1, 2} / ε, x_{1, 3} = ζ_{1, 3} / ε^{2}, u_{1} = υ_{1} / ε^{3}, \\ x_{2, 1} = ζ_{2, 1}, x_{2, 2} = ζ_{2, 2} / ε, x_{2, 3} = ζ_{2, 3} / ε^{2}, u_{2} = υ_{2} / ε^{3} \end{matrix}

(63)

the scaled observers and controllers can be constructed as

\begin{matrix} {\overset{\cdot}{\hat{x}}}_{1, 1} & = ε ({\hat{x}}_{1, 2} - a_{1, 1} {\hat{x}}_{1, 1}^{r_{1, 2}}), {\overset{\cdot}{\hat{x}}}_{1, 2} = ε ({\hat{x}}_{1, 3} - a_{1, 2} {\hat{x}}_{1, 1}^{r_{1, 3}}), \\ {\overset{\cdot}{\hat{x}}}_{1, 3} & = ε (u_{1} - a_{1, 3} {\hat{x}}_{1, 1}^{r_{1, 3} + τ_{1}}), \\ υ_{1} & = - ε^{3} β_{1, 3} ({\hat{x}}_{1, 3}^{\bar{σ} / r_{1, 3}} + β_{1, 2}^{\bar{σ} / r_{1, 3}} ({\hat{x}}_{1, 2}^{\bar{σ} / r_{1, 2}} + β_{1, 1}^{\bar{σ} / r_{1, 2}} y_{1}^{\bar{σ}}))^{(r_{1, 3} + τ_{1}) / \bar{σ}}, \\ {\overset{\cdot}{\hat{x}}}_{2, 1} & = ε ({\hat{x}}_{2, 2} - a_{2, 1} {\hat{x}}_{2, 1}^{r_{2, 2}}), {\overset{\cdot}{\hat{x}}}_{2, 2} = ε ({\hat{x}}_{2, 3} - a_{2, 2} {\hat{x}}_{2, 1}^{r_{2, 3}}), \\ {\overset{\cdot}{\hat{x}}}_{2, 3} & = ε (u_{2} - a_{2, 3} {\hat{x}}_{2, 1}^{r_{2, 3} + τ_{2}}), \\ υ_{2} & = - ε^{3} β_{2, 3} ({\hat{x}}_{2, 3}^{\bar{σ} / r_{2, 3}} + β_{2, 2}^{\bar{σ} / r_{2, 3}} ({\hat{x}}_{2, 2}^{\bar{σ} / r_{2, 2}} + β_{2, 1}^{\bar{σ} / r_{2, 2}} y_{2}^{\bar{σ}}))^{(r_{2, 3} + τ_{2}) / \bar{σ}} \end{matrix}

(64)

with appropriate positive constants $a_{i, j}, β_{i, j}$ and $ε$ . In the simulation, the unknown parameters are assumed to be $θ_{1, 1} = 0.1$ , $θ_{1, 2} = 1.2$ , $θ_{2, 1} = 1.1$ and $d (t) = 0.02 \sin (t)$ . The gains in (64) are chosen as

\begin{matrix} a_{1, 1} = 6, a_{1, 2} = 11, a_{1, 3} = 6, a_{2, 1} = 6, a_{2, 2} = 11, a_{2, 3} = 6, \\ β_{1, 1} = 3, β_{1, 2} = 4, β_{1, 3} = 5, β_{2, 1} = 3, β_{2, 2} = 4, β_{2, 3} = 5 \end{matrix}

and the scaling gain is chosen as $ε = 0.8$ . If we choose the initial condition as

\begin{matrix} ζ_{1, 1} (0) = 1, ζ_{1, 2} (0) = 1, ζ_{1, 3} (0) = - 1, {\hat{x}}_{1, 1} (0) = 0, {\hat{x}}_{1, 2} (0) = 0, {\hat{x}}_{1, 3} (0) = 0, \\ ζ_{2, 1} (0) = 1, ζ_{2, 2} (0) = - 1, ζ_{2, 3} (0) = 1, {\hat{x}}_{2, 1} (0) = 0, {\hat{x}}_{2, 2} (0) = 0, {\hat{x}}_{2, 3} (0) = 0 \end{matrix}

the simulation results are shown in Figure 2, which illustrate that the decentralized system (60) can be globally asymptotically stable by the output feedback stabilizer (64).

Conclusion

This paper considers the problem of decentralized global stabilization for a class of large-scale nonlinear systems with unknown output functions by output feedback. By using homogeneous domination approach, a series of output feedback stabilizers are designed to render the nonlinear systems globally asymptotically stable with appropriate design parameters. Moreover, under an appropriate transformation of the coordinates, the proposed design scheme can still be extended to a more general class of uncertain nonlinear systems with the uncertain bounded gains in front of the unmeasurable states and the inputs.

Footnotes

Funding

This work was supported in part by the National Natural Science Foundation of China (grant numbers 61104068 and 61273119), Natural Science Foundation of Jiangsu Province (grant number BK2010200), China Postdoctoral Science Foundation (grant number 2012M511176), Research Fund for the Doctoral Program of Higher Education of China (grant numbers 20090092120027 and 20110092110021), Excellent Young Teachers of Southeast University Grant Program.

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