Stabilization of a class of upper-triangular systems without a priori knowledge of control directions under the homogeneous domination

Abstract

This paper studies the global asymptotic stabilization of a class of upper-triangular systems without a priori knowledge of control directions under the homogeneous domination. The homogeneous domination is used in this paper to design state feedback controllers. The Nussbaum-type gain method is also introduced to complete the design of the controller. The controller is not only used to ensure the global boundedness for all signals of the closed-loop system, but also to guarantee that the system state asymptotically converges to zero. Finally, the simulation results are given to illustrate the effectiveness of the proposed approach.

Keywords

Upper-triangular systems unknown control directions homogeneous domination Nussbaum-type gain

Introduction

In this paper, a class of upper-triangular systems with unknown control directions is considered and can be represented as

\begin{matrix} {\overset{\cdot}{x}}_{1} = g_{1} (t) x_{2}^{p_{1}} + f_{1} (x_{2}, x_{3}, \dots, x_{n}) \\ {\overset{\cdot}{x}}_{2} = g_{2} (t) x_{3}^{p_{2}} + f_{2} (x_{3}, \dots, x_{n}) \\ ⋮ \\ {\overset{\cdot}{x}}_{n - 1} = g_{n - 1} (t) x_{n}^{p_{n - 1}} + f_{n - 1} (x_{n}) \\ {\overset{\cdot}{x}}_{n} = g_{n} (t) u^{p_{n}} \end{matrix}

(1)

where $x = {(x_{1}, \dots, x_{n})}^{T} \in R^{n}$ is the system state, $u \in R$ is the system input, f_i(·) and $p_{i} \in R_{odd}^{+}, i = 1, 2, \dots, n$ , are nonlinear continuous functions and the ratios of positive odd integers, respectively, and p_i ≥ 1,i = 1,…,n, and g_i(t),i = 1,2,…,n, are bounded uncertain time-varying piecewise continuous parameters with g_i(t) ≠ 0.

In recent years the stability analysis of nonlinear cascade systems has attracted wide attention. Cascade systems are very important nonlinear control systems, and the control design problems in many systems can be converted to cascade control-system design problems. For high-order nonlinear systems the properties of cascade control systems are used to reduce the complexity of the design. Using cascade control design methods, the closed-loop system is transformed into two or more low-order cascaded form subsystems. It is a very effective method for obtaining the stability results of the original system through the stability analysis of the subsystems and the analysis of the interconnections for processing the analysis of the high-order control systems and the synthesis problems. The stability analysis problem needs to seek sufficient conditions to satisfy the stability of the system. Scholars have made much of the analysis stability results of cascade systems in the existing literature. The approach based on the cascade term growth conditions is a very effective way to deal with the stabilization problem of nonlinear cascade systems. (Imura et al., 1992; Panteley and LoriA, 2001).

This method has been widely used in cascade system stability analysis. Another effective method is based on the input-to-state stability (ISS) analysis (Sontag, 1989). ISS means that the bounded input leads to the bounded state. Assuming that the driving subsystem is globally asymptotically stable, if another status of the subsystem on the input to the driving state of the subsystem is stable, then the entire cascade system is globally asymptotically stable (Sontag, 1990).

In the existing literature on nonlinear control systems, many results are related to the lower-triangular systems. Compared with the stabilization control of lower-triangular systems, to the best of our knowledge, there are still few results available for the global stabilization problem of the upper-trangular systems (feedforwards systems).

Among the existing results on the stabilization of upper-triangular systems, the upper-triangular systems with strict feedback form are the main consideration. This system is a special form of the system considered in this paper, that is p_i = 1 and g_i = 1. For the upper-triangular systems with the strict feedback form, there are two main control methods, the nested saturation control method and the Lyapunov control method. The design ideas of the nested saturation method are given in Teel (1992, 1996). The Lyapunov method is another effective control method. By introducing a class of coordinate transformations and using the Lyapunov method, the stabilization problem of a class of upper-triangular systems is considered in Mazenc (1997) and Mazenc and Praly (1996). By designing the Lyapunov function with cross terms, the stability of a class of upper-triangular systems is analysed in Jankovic et al. (1996). In addition, there is some literature that considers the output feedback control problem of upper-triangular systems (Krishnamurthy and Khorrami, 2008; Qian and Li, 2006). If p_i > 1, the global stabilization problem of upper-triangular systems is more challenging. In this case, there exists some literature based on the limitations of p_i and f_i and some results have been obtained. In the high-order case, the problem of global stabilization for a class of upper-triangular systems which have unbounded or uncontrollable linearizations around the origin is considered in Ding et al. (2011). The global stabilization problem of system (1) satisfying g_i = 1 and under the conditions of p₁ ≥ p₂≥⋯ ≥ p_n ≥ 1 is considered in Nussbaum (1983) and Mazenc et al. (1999). By combining the nesting saturation method and the Lyapunov method, the conditions which f_i satisfies in Qian and Lin (1999) are weakened in Mazenc (1997) and the results are promoted (Mazenc, 1997); however, the results obtained in Mazenc (1997) and Qian and Lin (1999) require a stronger precondition, that is p₁ ≥ p₂ ≥ ⋯ ≥ p_n ≥ 1.

For the system

\begin{matrix} {\overset{\cdot}{x}}_{1} = g_{1} x_{2}^{3} + x_{3}^{5} \\ {\overset{\cdot}{x}}_{2} = g_{2} x_{3}^{7} \\ {\overset{\cdot}{x}}_{3} = u \end{matrix}

(2)

satisfying p₁ = 3, p₂ = 7 and p₁ < p₂, the methods that Mazenc (1997) and Qian and Lin (1999) proposed are no longer applicable. In this paper, the conditions that the high-order feedforward systems need to satisfy are weakened and we will consider the global stabilization of a class of more wider feedforward systems.

On the other hand, the stabilization problem of feedforward systems is more difficult. Many practical systems can be transformed into feedforward systems. The cart-pendulum system is a typical feedforward system. It can be described as (Mazenc and Praly, 1996)

\begin{matrix} {\overset{\cdot}{x}}_{0} = s_{0} \\ {\overset{\cdot}{s}}_{0} = u_{0} \\ {\overset{\cdot}{t}}_{0} = r_{0} \\ {\overset{\cdot}{r}}_{0} = \frac{2 t_{0} r_{0}^{2}}{1 + t_{0}^{2}} + t_{0} \sqrt{1 + t_{0}^{2}} - u_{0} \sqrt{1 + t_{0}} \end{matrix}

(3)

Therefore, it is necessary to investigate the stabilization problem of feedforward systems.

In the past few decades, the homogeneous system has been a widespread concern (Rosier, 1992). The homogeneous system is the vector fields of the system with homogeneity. One advantage of the homogeneous system is that for the autonomous system, if the system is locally asymptotically stable, the system is globally asymptotically stable (Rosier, 1992; Sepulchre et al., 1997). This means that local asymptotic stability and global asymptotic stability are equivalent. Therefore, the issue of discussing the global stability subsystems can be converted to discussing their local stability problems. The homogeneous domination approach is developed for the global output feedback stabilization of nonlinear systems in Qian (2005). Based on the concept of generalized homogeneity with monotone degree, a new design procedure to explicitly construct global stabilizers for a class of feedforward systems is developed in Zhang et al. (2013). The problem of global stabilization by smooth output feedback for a class of n-dimensional homogeneous systems whose Jacobian linearization is neither controllable nor observable is studied in Yang and Lin (2004); however, the control directions of the systems studied by the aforementioned literatures are known. In this paper, the concept of the homogeneous system is introduced and is combined with adding a power integrator approach to overcome the deficiencies of existing methods. At the same time, the Nussbaum-type gain function is introduced to compensate for the lack of control directions.

Since the early 1980s, scholars have begun to study adaptive control problems with unknown control directions and have achieved considerable results. The initial studies were carried out for linear systems and have proved that even though the high-frequency gain sign is unknown the adaptive control of linear systems is still viable. This groundbreaking work was first proposed in Nussbaum (1983) as the Nussbaum-type gain method and the control problem for a class of first-order linear systems is solved. Subsequently, the Nussbaum-type gain method is used to deal with the adaptive control problem of first-order nonlinear systems, nonlinear disturbances and high-order linear systems; however, because of the impact of interference, most of the existing controllers violently shake. To overcome this disadvantage, a number of results have been achieved in Ryan (1995), Ye (1999) and Ding (1998), and the requirements on the relative degree of the system are relaxed. The algorithm proposed in Ding (1998) and Ye (1999) holds for systems with arbitrary relative degree. In recent years, the control problem with unknown control directions of nonlinear systems has been studied widely. Applying the Nussbaum-type gain method, the strict feedback adaptive control of nonlinear systems with unknown constant parameters is considered in Ye et al. (1998). The decoupled backstepping algorithm is used in Ge and Wang (2002) to study the adaptive control of nonlinear systems with completely unknown control coefficients and strict feedback systems with uncertain parameters. When the uncertain system function satisfies certain conditions, the output problems of a class of time-varying uncertain nonlinear systems with unknown control coefficients is investigated in Ye (1999). The problem of adaptive tracking control is addressed in Zhang et al. (2014) for a class of nonlinear systems with unknown constant parameters and unknown actuator nonlinearity.

As far as we know, there are few pieces of literature which study the high-order feedforward system with unknown control directions. Whether applied to the high-order feedforward systems or to systems with unknown control directions, there is important significance to both theory and practical applications.

In this paper, we consider the global asymptotic stabilization of a class of feedforward systems with unknown control directions. We will combine the homogeneous domination and the Nussbaum-type gain method to design the state feedback controller. First, we consider a linear system with unknown control directions; in order to overcome the difficulties caused by the unknown control directions, the Nussbaum-type gain will be introduced in the design of the stabilization functions and the actual control laws. Then we consider a feedforward system and cascaded nonlinear system. The ISS Lypunov function and the idea of changing the energy function are used to solve the zero-dynamic subsystem issues. Finally, using Barbalat’s Lemma, we prove that the controller can not only ensure that all the closed-loop system signals are globally bounded, but it can also make the system states tend to zero asymptotically. The simulation examples are included to show the effectiveness of the design approach.

Problem formulation and preliminaries

Definition 1. (Qian, 2005)

The weighted homogeneity: for fixed coordinates $(x_{1}, \dots, x_{n}) \in R^{n}$ and real number r_i > 0,i = 1,…,n, the dilationΔ_ε(x) is defined by $Δ_{ε} (x) = (ε^{r_{1}} x_{1}, \dots, ε^{r_{n}} x_{n}), \forall ε > 0$ , where r_i is said to be the weight of the coordinates and we define the dilation weight asΔ_ε = (r₁,…,r_n).

A function $V \in C (R^{n}, R^{n})$ is said to be homogeneous of degree τ if there is a real number $τ \in R$ such that, $\forall x \in R^{n} ∖ 0, ε > 0$ , V(Δ_ε(x)) = ε^τV(x₁,…,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,…,n, $\forall x \in R^{n} ∖ 0, ε > 0$ , f_i(Δ_ε(x)) = ε^τ^+r_if_i(x).

A homogeneous p-norm is defined as $‖ x ‖_{Δ, p} = ({\sum_{i = 1}^{n} {| x_{i} |}^{\frac{p}{r_{i}}})}^{\frac{1}{p}}), \forall x \in R^{n}$ , for a constant p>1.

Lemma 1. (Qian, 2005) Given a dilation weightΔ = (r₁,…,r_n), suppose that V₁(x) and V₂(x) are homogeneous functions of degree τ₁and τ₂, respectively. Then V₁(x)V₂(x) is also homogeneous with respect to the same dilation weightΔ. Moreover, the homogeneous degree of V₁·V₂is τ₁ + τ₂.

Lemma 2. (Qian, 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:

(∂V)/(∂x_i) is still homogeneous of degree τ − r_i with r_i being the homogeneous weight of x_i; and

there is a constant c such that

V (x) \leq c ‖ x ‖_{Δ}^{τ}

Moreover, if V(x) is positive definite, $\underline{c} ‖ x ‖_{Δ}^{τ} \leq V (x)$ , for a positive constant c > 0.

Remark 1. A power integrator technique is used in this paper to design the homogeneous controller. The finite-time stability of homogeneous domination gives the direct link between the homogeneous and finite-time systems. The Lyapunov stability criterion of the finite-time system is given in Bhat and Bernstein (1998, 2000).

Lemma 3. (Qian and Lin, 2001) Let $a, b \in R$ , and let p ≥ 1 be a constant. Then the following inequality holds

{| a + b |}^{p} \leq 2^{p - 1} | a^{p} + b^{p} |

Lemma 4. (Qian and Lin, 2001) Let $x, y \in R$ , and let c, d be positive constants. Then there exists a real-valued smooth function γ(x, y) > 0, such that the following inequality holds

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

Lemma 5. (Lin and Qian, 2002) Let w ≥ 1, r ≥ 1 be integers and let $e, f \in R$ , e ≥ 0, f ≥ 0. Then the following inequality holds

e^{w - 1} f^{r} \leq e^{w} + f^{wr}

Definition 2. (Nussbaum, 1983) A Nussbaum-type function ν(·) is has the following properties

\begin{matrix} lim_{s \to \infty} inf \frac{1}{s} \int_{0}^{s} ν (η) d η = - \infty \\ lim_{s \to \infty} sup \frac{1}{s} \int_{0}^{s} ν (η) d η = \infty \end{matrix}

Remark 2. From Definition 2, it is easy to show that Nussbaum functions should have infinite gains and infinite switching frequencies. There are many functions that satisfy the earlier conditions, such as $\cos (\frac{π}{2} η) \exp (η^{2})$ and η ²cos(η), and we choose to use

ν (η) = \cos (\frac{π}{2} η) \exp (η^{2})

throughout the paper.

Assumption 1. There exists a positive constant ρ ≥ 0, τ ≥ such that

| f_{i} (x_{i + 1}, \dots, x_{n}) | \leq (| x_{i + 1} |^{\frac{a + τ}{r_{i + 1}}} + \dots + | x_{n} |^{\frac{a + τ}{r_{n}}})

(4)

where $r_{1} = 1, r_{i + 1} = \frac{r_{i} + τ}{p_{i}} > 0, i = 1, 2, \dots, n$ and a ≥ max_1≤i≤n(r_i,p_ir_i₊₁)

Assumption 2. The parameters g_i(t) which are time varying take values in the unknown closed intervals $I_{i} : = [l_{i}^{-}, l_{i}^{+}]$ with 0 ∉I_i, 1 ≤ i ≤ n, where $l_{i}^{-}, l_{i}^{+}$ are the unknown bounds of g_i(t). The signs of g_i(t),1 ≤ i ≤ n, called control directions, are unknown.

Design of the state feedback controller

High-order linear systems

In this section, we will present a recursive approach to design the virtual control laws and update laws for system (1). We first consider the following linear system

\begin{matrix} {\overset{\cdot}{x}}_{1} = g_{1} (t) x_{2}^{p_{1}} \\ {\overset{\cdot}{x}}_{2} = g_{2} (t) x_{3}^{p_{2}} \\ ⋮ \\ {\overset{\cdot}{x}}_{n - 1} = g_{n - 1} (t) x_{n}^{p_{n - 1}} \\ {\overset{\cdot}{x}}_{n} = g_{n} (t) u^{p_{n}} \end{matrix}

(5)

Step 1: Construct a Lyapunov function $V_{1} (x_{1}) = \int_{0}^{x_{1}} {(s^{\frac{a}{r_{1}}} - 0)}^{\frac{2 a - r_{1} - τ}{a}} ds$ .

The time derivative of V₁ is

\begin{matrix} {\overset{\cdot}{V}}_{1} & = x_{1}^{\frac{2 a - r_{1} - τ}{r_{1}}} {\overset{\cdot}{x}}_{1} \\ = g_{1} (t) x_{1}^{\frac{2 a - r_{1} - τ}{r_{1}}} x_{2}^{p_{1}} \\ \leq g_{1} (t) x_{1}^{\frac{2 a - r_{1} - τ}{r_{1}}} x_{2}^{* p_{1}} + g_{1} (t) x_{1}^{\frac{2 a - r_{1} - τ}{r_{1}}} (x_{2}^{p_{1}} - x_{2}^{* p_{1}}) \end{matrix}

(6)

The virtual control law $x_{2}^{*}$ and the update law η₁ that we construct are

\begin{matrix} x_{2}^{*} = ξ_{1}^{\frac{r_{2}}{a}} ν (η_{1}) β_{1} (x_{1}) \\ {\overset{\cdot}{η}}_{1} = ξ_{1}^{2} β_{1}^{p_{1}} (x_{1}) \end{matrix}

(7)

where $β_{1}^{p_{1}} (x_{1}) \geq 1$ is a smooth function, and $ξ_{1} = x_{1}^{\frac{a}{r_{1}}}$ .

Also, from Lemma 4, the following inequality holds

\begin{matrix} | g_{1} (t) x_{1}^{\frac{2 a - r_{1} - τ}{r_{1}}} (x_{2}^{p_{1}} - x_{2}^{* p_{1}}) | \\ \leq | g_{1} (t) ξ_{1}^{\frac{2 a - r_{1} - τ}{a}} | | {(x_{2}^{\frac{a}{r_{2}}})}^{\frac{p_{1} r_{2}}{a}} - {(x_{2}^{* \frac{a}{r_{2}}})}^{\frac{p_{1} r_{2}}{a}} | \\ \leq 2^{1 - \frac{p_{1} r_{2}}{a}} | ξ_{1} |^{\frac{2 a - r_{1} - τ}{a}} | g_{1} (t) | | ξ_{2} |^{\frac{r_{1} + τ}{a}} \\ = {(2^{\frac{a - p_{1} r_{2}}{p_{1} r_{2}}})}^{\frac{r_{1} + τ}{a}} | ξ_{1} |^{\frac{2 a - r_{1} - τ}{a}} | {\bar{g}}_{1}^{\frac{a}{r_{1} + τ}} |^{\frac{r_{1} + τ}{a}} | ξ_{2} |^{\frac{r_{1} + τ}{a}} \\ \leq \frac{\frac{2 a - r_{1} - τ}{a}}{\frac{2 a - r_{1} - τ}{a} + \frac{r_{1} + τ}{a}} γ | ξ_{1} |^{2} + \frac{\frac{r_{1} + τ}{a}}{\frac{2 a}{a}} γ^{- \frac{2 a - r_{1} - τ}{r_{1} + τ}} {(2^{\frac{a - p_{1} r_{1}}{p_{1} r_{1}}} {\bar{g}}_{1}^{\frac{a}{r_{1} + τ_{1}}})}^{\frac{2 a}{a}} | ξ_{2} |^{2} \end{matrix}

(8)

where we choose γ = 2a/(2a − r₁ − τ).

Then equation (10) is converted to

\begin{matrix} | g_{1} (t) x_{1}^{\frac{2 a - r_{1} - τ}{r_{1}}} (x_{2}^{p_{1}} - x_{2}^{* p_{1}}) | \\ \leq \frac{2 a - r_{1} - τ}{2 a} \frac{2 a}{2 a - r_{1} - τ} ξ_{1}^{2} \\ + \frac{r_{1} + τ}{2 a} {(\frac{2 a}{2 a - r_{1} - τ})}^{- \frac{2 a - r_{1} - τ}{r_{1} + τ}} {(2^{\frac{a - p_{1} r_{1}}{p_{1} r_{1}}} {\bar{g}}_{1}^{\frac{a}{r_{1} + τ_{1}}})}^{2} ξ_{2}^{2} \\ ≜ ξ_{1}^{2} + ξ_{2}^{2} {\hat{g}}_{1} \end{matrix}

(9)

where ${\hat{g}}_{1} = {\hat{g}}_{1} ({\bar{g}}_{1})$ is an unknown constant.

Substituting equations (7) and (9) into equation (6) yields

{\overset{\cdot}{V}}_{1} \leq g_{1} (t) {\overset{\cdot}{η}}_{1} ν^{p_{1}} (η_{1}) + ξ_{1}^{2} + ξ_{2}^{2} {\hat{g}}_{1}

(10)

Step 2: Consider the Lypunov function as $V_{2} (x_{1}, x_{2}) = \int_{x_{2}^{*}}^{x_{2}} {(s^{\frac{a}{r_{2}}} - x_{2}^{* \frac{a}{r_{2}}})}^{\frac{2 a - r_{2} - τ}{a}} ds$ .

The time derivative of V₂ is

{\overset{\cdot}{V}}_{2} = \frac{\partial V_{2}}{\partial x_{2}} {\overset{\cdot}{x}}_{2} + \frac{\partial V_{2}}{\partial x_{1}} {\overset{\cdot}{x}}_{1} + \frac{\partial V_{2}}{\partial η_{1}} {\overset{\cdot}{η}}_{1}

(11)

where

\frac{\partial V_{2}}{\partial x_{1}} = - \frac{2 a - r_{2} - τ}{a} \frac{\partial x_{2}^{* \frac{a}{r_{2}}}}{\partial x_{1}} \int_{x_{2}^{*}}^{x_{2}} {(s^{\frac{a}{r_{2}}} - x_{2}^{* \frac{a}{r_{1}}})}^{\frac{a - r_{2} - τ}{a}} ds

(12)

\frac{\partial V_{2}}{\partial η_{1}} = - \frac{2 a - r_{2} - τ}{a} \frac{\partial x_{2}^{* \frac{a}{r_{2}}}}{\partial η_{1}} \int_{x_{2}^{*}}^{x_{2}} {(s^{\frac{a}{r_{2}}} - x_{2}^{* \frac{a}{r_{1}}})}^{\frac{a - r_{2} - τ}{a}} ds

(13)

From Lemma 3, the following inequality holds

\begin{matrix} | \frac{\partial V_{2}}{\partial x_{1}} | & \leq | \frac{2 a - r_{2} - τ}{a} | | \frac{\partial x_{2}^{* \frac{a}{r_{2}}}}{\partial x_{1}} | | x_{2} - x_{2}^{*} | | ξ_{2} |^{\frac{a - r_{2} - τ}{a}} \\ \leq \frac{2 a - r_{2} - τ}{a} | \frac{\partial x_{2}^{* \frac{a}{r_{2}}}}{\partial x_{1}} | {(| ξ_{2} |^{\frac{a - r_{2} - τ}{a}} | {(x_{2}^{\frac{a}{r_{2}}})}^{\frac{r_{2}}{a}} - x_{2}^{* \frac{a}{r_{2}}})}^{\frac{r_{2}}{a}} | \\ \leq \frac{2 a - r_{2} - τ}{a} | \frac{\partial x_{2}^{* \frac{a}{r_{2}}}}{\partial x_{1}} | 2^{1 - \frac{r_{2}}{a}} | ξ_{2} |^{\frac{r_{2}}{a}} | ξ_{2} |^{\frac{a - r_{2} - τ}{a}} \\ \leq θ_{2} | \frac{\partial x_{2}^{* \frac{a}{r_{2}}}}{\partial x_{1}} | | ξ_{2} |^{\frac{a - τ}{a}} \end{matrix}

(14)

Similarly, we obtain

| \frac{\partial V_{2}}{\partial η_{1}} | \leq θ_{2} | \frac{\partial x_{2}^{* \frac{a}{r_{2}}}}{\partial η_{1}} | | ξ_{2} |^{\frac{a - τ}{a}}

(15)

where $θ_{2} = \frac{2 a - r_{2} - τ}{a} 2^{1 - \frac{r_{2}}{a}}$ .

Substituting equations (14) and (15) into equation (11) gives

{\overset{\cdot}{V}}_{2} \leq ξ_{2}^{\frac{2 a - r_{2} - τ}{a}} g_{2} (t) x_{3}^{p_{2}} + \frac{\partial V_{2}}{\partial x_{1}} g_{1} (t) x_{2}^{p_{1}} + \frac{\partial V_{2}}{\partial η_{1}} {\overset{\cdot}{η}}_{1}

(16)

where

\begin{matrix} \frac{\partial V_{2}}{\partial x_{1}} g_{1} (t) x_{2}^{p_{1}} & \leq θ_{2} | ξ_{2} |^{\frac{a - τ}{a}} | \frac{\partial x_{2}^{* \frac{a}{r_{2}}}}{\partial x_{1}} | | g_{1} (t) x_{2}^{p_{1}} | \\ \leq | ξ_{2}^{1 - \frac{τ}{a}} | (θ_{2} | \frac{\partial x_{2}^{* \frac{a}{r_{2}}}}{\partial x_{1}} | {\bar{g}}_{1} (| ξ_{2}^{\frac{r_{2} p_{1}}{a}} | + | ξ_{1}^{\frac{r_{2} p_{1}}{a}} ν^{p_{1}} (η_{1}) β_{1}^{p_{1}} (x_{1}) |) \\ \leq ξ_{2}^{2} {\tilde{ρ}}_{21} (x_{1}, x_{2}, η_{1}) + \frac{1}{2} ξ_{1}^{2} + ξ_{2}^{2} {\hat{ρ}}_{21} (x_{1}, x_{2}, η_{1}) \\ = \frac{1}{2} ξ_{1}^{2} + ρ_{21} (x_{1}, x_{2}, η_{1}) ξ_{2}^{2} \end{matrix}

(17)

and

| \frac{\partial V_{2}}{\partial η_{1}} | {\overset{\cdot}{η}}_{1} \leq \frac{1}{2} ξ_{1}^{2} + ρ_{22} (x_{1}, x_{2}, η_{1}) ξ_{2}^{2}

(18)

and where ρ₂₁(x₁,x₂,η₁) > 0, ρ₂₂(x₁,x₂,η₁) > 0 are smooth functions.

Then we have

{\overset{\cdot}{V}}_{2} \leq g_{2} (t) ξ_{2}^{\frac{2 a - r_{2} - τ}{a}} x_{3}^{p_{2}} + ξ_{1}^{2} + ξ_{2}^{2} β_{2}^{p_{2}} (x_{1}, x_{2}, η_{1})

(19)

where $β_{2}^{p_{2}} (x_{1}, x_{2}, η_{1}) = 1 + ρ_{21} (x_{1}, x_{2}, η_{1}) + ρ_{22} (x_{1}, x_{2}, η_{1}) \geq 1$ is a smooth function.

We construct the virtual control law $x_{3}^{*}$ and the update law η₂ as

\begin{matrix} x_{3}^{*} = ξ_{2}^{\frac{r_{3}}{a}} ν (η_{2}) β_{2} (x_{1}, x_{2}, η_{1}) \\ {\overset{\cdot}{η}}_{2} (x_{1}, x_{2}, η_{1}) = ξ_{2}^{2} β_{2}^{p_{2}} (x_{1}, x_{2}, η_{1}) \end{matrix}

(20)

and define the variable function $ξ_{3} = x_{3}^{\frac{a}{r_{3}}} - x_{3}^{* \frac{a}{r_{3}}}$ . Then equation (16) is transformed into

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq g_{2} (t) ξ_{2}^{\frac{2 a - r_{2} - τ}{a}} {(| x_{3}^{p_{2}} - x_{3}^{* p_{2}} |^{p_{2}})}^{\frac{1}{p_{2}}} + g_{2} (t) x_{3}^{* p_{2}} ξ_{2}^{\frac{2 a - r_{2} - τ}{a}} \\ + ξ_{1}^{2} + ξ_{2}^{2} β_{2}^{p_{2}} (x_{1}, x_{2}, η_{1}) \end{matrix}

(21)

From Lemma 3, the first term of equation (21) becomes

\begin{matrix} | g_{2} (t) ξ_{2}^{\frac{2 a - r_{2} - τ}{a}} | | {(x_{3}^{\frac{a}{r_{3}}})}^{\frac{p_{2} r_{3}}{a}} - {(x_{3}^{* \frac{a}{r_{3}}})}^{\frac{p_{2} r_{3}}{a}} | \\ \leq 2^{1 - \frac{p_{2} r_{3}}{a}} | ξ_{2} |^{\frac{2 a - r_{2} - τ}{a}} | g_{2} | | ξ_{3} |^{\frac{r_{2} + τ}{a}} \\ \leq ξ_{2}^{2} + ξ_{3}^{2} {\hat{g}}_{2} \end{matrix}

(22)

Then from equations (20) and (22), we have

{\overset{\cdot}{V}}_{2} \leq ξ_{1}^{2} + ξ_{2}^{2} + ξ_{3}^{2} {\hat{g}}_{2} + g_{2} (t) {\overset{\cdot}{η}}_{2} ν^{p_{2}} (η_{2}) + {\overset{\cdot}{η}}_{2}

(23)

where ${\hat{g}}_{2} = {\hat{g}}_{2} ({\bar{g}}_{2})$ is an unknown constant.

Step k: We suppose that at Step k − 1 there are a series of smooth functions β_i(x₁,…,x_i,η₁,…,η_i₋₁),i = 1,…,k − 1, that satisfy $β_{i}^{p_{i}} (\cdot) \geq 1$ such that under the virtual controllers defined by

\begin{matrix} x_{2}^{*} = ξ_{1}^{\frac{r_{2}}{a}} ν (η_{1}) β_{1} (x_{1}) \\ x_{3}^{*} = ξ_{2}^{\frac{r_{3}}{a}} ν (η_{2}) β_{2} (x_{1}, x_{2}, η_{1}) \\ ⋮ \\ x_{k}^{*} = ξ_{k - 1}^{\frac{r_{k}}{a}} ν (η_{k - 1}) β_{k - 1} (x_{1}, \dots, x_{k - 1}, η_{1}, \dots, η_{k - 2}) \end{matrix}

(24)

and the update laws defined by

\begin{matrix} {\overset{\cdot}{η}}_{1} (x_{1}, η_{1}) = ξ_{1}^{2} β_{1}^{p_{1}} (x_{1}) \\ {\overset{\cdot}{η}}_{2} (x_{1}, x_{2}, η_{1}) = ξ_{2}^{2} β_{2}^{p_{2}} (x_{1}, x_{2}, η_{1}) \\ ⋮ \\ {\overset{\cdot}{η}}_{k - 1} (x_{1}, x_{2}, \dots, x_{k - 1}, η_{1}, \dots, η_{k - 2}) \\ = ξ_{k - 1}^{2} β_{k - 1}^{p_{k - 1}} (x_{1}, \dots, x_{k - 1}, η_{1}, \dots, η_{k - 2}) \end{matrix}

(25)

the following inequality holds

{\overset{\cdot}{V}}_{k - 1} \leq \sum_{i = 1}^{k - 1} ξ_{k - 1}^{2} + {\hat{g}}_{k - 1} ξ_{k}^{2} + g_{k - 1} (t) {\overset{\cdot}{η}}_{k - 1} ν^{p_{k - 1}} (η_{k - 1}) + {\overset{\cdot}{η}}_{k - 1}

(26)

where the Lyapunov function is constructed as

\begin{matrix} V_{k - 1} (x_{1}, \dots, x_{k - 1}, η_{1}, \dots, η_{k - 1}) \\ = \int_{x_{k - 1}^{*}}^{x_{k - 1}} {(s^{\frac{a}{r_{k - 1}}} - x_{k - 1}^{* \frac{a}{r_{k - 1}}})}^{\frac{2 a - r_{k - 1} - τ}{a}} ds \end{matrix}

(27)

The variable functions are defined by

\begin{matrix} ξ_{1} = x_{1}^{\frac{a}{r_{1}}} \\ ξ_{2} = x_{2}^{\frac{a}{r_{2}}} - x_{2}^{* \frac{a}{r_{2}}} \\ ⋮ \\ ξ_{k - 1} = x_{k - 1}^{\frac{a}{r_{k - 1}}} - x_{k - 1}^{* \frac{a}{r_{k - 1}}} \end{matrix}

(28)

and $β_{i - 1}^{p_{i - 1}} (\cdot) \geq 1, i = \geq 1, \dots, k$ , are smooth functions. In the following, we will prove that equation (26) also holds if i = k.

Consider the following Lyapunov function

V_{k} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k}) = \int_{x_{k}^{*}}^{x_{k}} {(s^{\frac{a}{r_{k}}} - x_{k}^{* \frac{a}{r_{k}}})}^{\frac{2 a - r_{k} - τ}{a}} ds

(29)

The design approach is similar to Step 2, and we achieve

\begin{matrix} | \frac{\partial V_{k}}{\partial x_{i}} | & \leq | \frac{2 a - r_{k} - τ}{a} | | \frac{\partial x_{k}^{* \frac{a}{r_{k}}}}{\partial x_{i}} | | x_{k} - x_{k}^{*} | | ξ_{k} |^{\frac{a - r_{k} - τ}{a}} \\ \leq \frac{2 a - r_{k} - τ}{a} | \frac{\partial x_{k}^{* \frac{a}{r_{k}}}}{\partial x_{i}} | 2^{1 - \frac{r_{k}}{a}} | {(x_{k}^{\frac{a}{r_{k}}})}^{\frac{r_{k}}{a}} - {(x_{k}^{* \frac{a}{r_{k}}})}^{\frac{r_{k}}{a}} | | ξ_{k} |^{\frac{a - r_{k} - τ}{a}} \\ \leq \frac{2 a - r_{k} - τ}{a} | \frac{\partial x_{k}^{* \frac{a}{r_{k}}}}{\partial x_{i}} | 2^{1 - \frac{r_{k}}{a}} | ξ_{k} |^{\frac{r_{k}}{a}} | ξ_{k} |^{\frac{a - r_{k} - τ}{a}} \\ \leq θ_{k} | \frac{\partial x_{k}^{* \frac{a}{r_{k}}}}{\partial x_{i}} | | ξ_{k} |^{\frac{a - τ}{a}} \end{matrix}

(30)

Similarly, we obtain

| \frac{\partial V_{k}}{\partial η_{i}} | \leq θ_{k} | \frac{\partial x_{k}^{* \frac{a}{r_{k}}}}{\partial η_{i}} | | ξ_{k} |^{\frac{a - τ}{a}}

(31)

where

θ_{k} = \frac{2 a - r_{k} - τ}{a} 2^{1 - \frac{r_{k}}{a}}

Then, the time derivative of V_k is

\begin{matrix} {\overset{\cdot}{V}}_{k} & = \frac{\partial V_{k}}{\partial x_{k}} {\overset{\cdot}{x}}_{k} + \sum_{i = 1}^{k - 1} \frac{\partial V_{k}}{\partial x_{i}} {\overset{\cdot}{x}}_{i} + \sum_{i = 1}^{k - 1} \frac{\partial V_{k}}{\partial η_{i}} {\overset{\cdot}{η}}_{i} \\ \leq ξ_{2}^{\frac{2 a - r_{k} - τ}{a}} g_{k} (t) x_{k + 1}^{p_{k}} + \sum_{i = 1}^{k - 1} θ_{k} | \frac{\partial x_{k}^{* \frac{a}{r_{k}}}}{\partial x_{i}} | | ξ_{k} |^{\frac{a - τ}{a}} | {\overset{\cdot}{x}}_{i} | \\ + \sum_{i = 1}^{k - 1} θ_{k} | \frac{\partial x_{k}^{* \frac{a}{r_{k}}}}{\partial η_{i}} | | ξ_{k} |^{\frac{a - τ}{a}} | {\overset{\cdot}{η}}_{i} | \end{matrix}

(32)

Proposition 1. There exists a non-negative smooth function α_k,i(x₁,…,x_k,η₁,…η_k₋₁), for i = 1,2,…,k − 1, such that

| \frac{\partial x_{k}^{* \frac{a}{r_{k}}}}{\partial x_{i}} {\overset{\cdot}{x}}_{i} | \leq (| ξ_{1} |^{\frac{a + τ}{a}} + \dots + | ξ_{k} |^{\frac{a + τ}{a}}) α_{k, i} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1})

The proof of Proposition 1 is given in the Appendix.

From Proposition 1, we estimate the last two terms of equation (32) and it is easy to show that

\begin{matrix} \sum_{i = 1}^{k - 1} θ_{k} | \frac{\partial x_{k}^{*}}{\partial x_{i}} | | ξ_{k} |^{\frac{a - τ}{a}} | {\overset{\cdot}{x}}_{i} | \\ \leq θ_{k} | ξ_{k} |^{\frac{a - τ}{a}} (| ξ_{1} |^{\frac{a + τ}{a}} + \dots + | ξ_{k} |^{\frac{a + τ}{a}}) \\ \sum_{i = 1}^{k - 1} α_{k, i} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1}) \\ \leq \frac{1}{2} \sum_{i = 1}^{k - 1} ξ_{i}^{2} + ξ_{k}^{2} ρ_{k 1} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1}) \end{matrix}

(33)

and

\begin{matrix} \sum_{i = 1}^{k - 1} θ_{k} | \frac{\partial x_{k}^{*}}{\partial η_{i}} | | ξ_{k} |^{\frac{a - τ}{a}} | {\overset{\cdot}{η}}_{i} | \\ \leq θ_{k} | ξ_{k} |^{\frac{a - τ}{a}} (| ξ_{1} |^{\frac{a + τ}{a}} + \dots + | ξ_{k} |^{\frac{a + τ}{a}}) \\ \sum_{i = 1}^{k - 1} α_{k, i} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1}) \\ \leq \frac{1}{2} \sum_{i = 1}^{k - 1} ξ_{i}^{2} + ξ_{k}^{2} ρ_{k 2} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1}) \end{matrix}

(34)

where ρ_k₁(·) > 0, ρ_k₂(·) > 0 are smooth functions.

Substituting equations (33) and (34) into equation (32) yields

{\overset{\cdot}{V}}_{k} \leq g_{k} (t) ξ_{k}^{\frac{2 a - r_{k} - τ}{a}} x_{k + 1}^{p_{k}} + \sum_{i = 1}^{k - 1} ξ_{i}^{2} + ξ_{k}^{2} β_{k}^{p_{k}}

(35)

where

\begin{matrix} β_{k}^{p_{k}} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1}) = 1 \\ + ρ_{k 1} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1}) \\ + ρ_{k 2} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1}) \geq 1 \end{matrix}

is a smooth function.

Choose the virtual control law $x_{k + 1}^{*}$ and the update law η_k shown

\begin{matrix} x_{k + 1}^{*} = ξ_{k}^{\frac{r_{k + 1}}{a}} ν (η_{k}) β_{k} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1}) \\ {\overset{\cdot}{η}}_{k} = ξ_{k}^{2} β_{k}^{p_{k}} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1}) \end{matrix}

(36)

Then equation (35) becomes

{\overset{\cdot}{V}}_{k} \leq \sum_{i = 1}^{k} ξ_{i}^{2} + {\hat{g}}_{k} ξ_{k + 1}^{2} + g_{k} (t) {\overset{\cdot}{η}}_{k} ν^{p_{k}} (η_{k}) + {\overset{\cdot}{η}}_{k} .

(37)

where ${\hat{g}}_{k} = {\hat{g}}_{k} ({\bar{g}}_{k})$ is an unknown constant.

Step n: Consider the Lyapunov function

V_{n} (x_{1}, \dots, x_{n}, η_{1}, \dots, η_{n}) = \int_{x_{n}^{*}}^{x_{n}} {(s^{\frac{a}{r_{n}}} - x_{n}^{* \frac{a}{r_{n}}})}^{\frac{2 a - r_{n} - τ}{a}} ds

Similar to Step k, the time derivative of V_n is

{\overset{\cdot}{V}}_{n} \leq g_{n} (t) ξ_{n}^{\frac{2 a - r_{n} - τ}{a}} u^{p_{n}} + \sum_{i = 1}^{n - 1} ξ_{i}^{2} + ξ_{n}^{2} β_{n}

(38)

The control law u_n and the η_n are constructed as follows

\begin{matrix} u = ξ_{n}^{\frac{r_{n + 1}}{a}} ν (η_{n}) β_{n} (x_{1}, \dots, x_{n}, η_{1}, \dots, η_{n - 1}) \\ \overset{\cdot}{η} (x_{1}, \dots, x_{n}, η_{1}, \dots, η_{n - 1}) = ξ_{n}^{2} β_{n}^{p_{n}} (x_{1}, \dots, x_{n}, η_{1}, \dots, η_{n - 1}) \end{matrix}

(39)

Then equation (37) is converted to

{\overset{\cdot}{V}}_{n} \leq \sum_{i = 1}^{n} ξ_{n}^{2} + g_{n} (t) ν^{p_{n}} (η_{n}) {\overset{\cdot}{η}}_{n} + {\overset{\cdot}{η}}_{n}

(40)

High-order cascade nonlinear systems

In this section, a class of nonlinear systems with zero-dynamics are considered

\begin{matrix} \overset{\cdot}{z} = f (z, x_{1}) \\ {\overset{\cdot}{x}}_{1} = g_{1} (t) x_{2}^{p_{1}} + f_{1} (z, x_{2}, x_{3}, \dots, x_{n}) \\ {\overset{\cdot}{x}}_{2} = g_{2} (t) x_{3}^{p_{2}} + f_{2} (z, x_{3}, \dots, x_{n}) \\ ⋮ \\ {\overset{\cdot}{x}}_{n - 1} = g_{n - 1} (t) x_{n}^{p_{n - 1}} + f_{n - 1} (z, x_{n}) \\ {\overset{\cdot}{x}}_{n} = g_{n} (t) u^{p_{n}} \end{matrix}

(41)

where $z \in R^{r}$ is the unmeasurable state of the system.

Consider the following assumptions.

Assumption 3. There exists an ISS Lyapunov function W ₀(·) satisfying

\begin{matrix} \underline{k} (∥ z ∥) \leq W_{0} (z) \leq \bar{k} (∥ z ∥) \\ \frac{\partial W_{0} (z)}{\partial z} f (z, x_{1}) \leq - ∥ z ∥^{2} + k_{0} (x_{1}) \end{matrix}

(42)

where k(∥z∥) and $\bar{k} (‖ z ‖)$ are k_∞functions, k₀(x₁) ≥ 0 are $C^{2}$ function and k₀(0) = 0.

Assumption 4. There exists a positive constant ρ ≥ 0, τ ≥ such that

| f_{i} (z, x_{i + 1}, \dots, x_{n}) | \leq ρ (| z |^{\frac{a + τ}{a}} + | x_{i + 1} |^{\frac{a + τ}{r_{i + 1}}} + \dots + | x_{n} |^{\frac{a + τ}{r_{n}}})

(43)

where $r_{1} = 1, r_{i + 1} = \frac{r_{i} + τ}{p_{i}} > 0, i = 1, 2, \dots, n$ , and a ≥ max_1≤i≤n(r_i,p_ir_i₊₁).

Under Assumptions 3 and 4, we consider the cascaded nonlinear system (41). In order to deal with the unmeasured states, the energy change function design is applied in the following Lyapunov function

V_{0} (z) = \int_{0}^{W_{0} (z)} q_{s} ds, q (s) \geq 1, \forall s > 0

(44)

where q: [0,∞) → [1,∞) is a continuous monotonic non-decreasing function.

From Assumption 3, there exists a monotonic non-decreasing function q(·) ≥ 1 satisfying

\begin{matrix} \frac{\partial V_{0}}{\partial z} f (z, x_{1}) & = q (W_{0} (z)) \frac{\partial W_{0} (z)}{\partial z} f (z, x_{1}) \\ \leq - q (W_{0} (z)) ‖ z ‖^{2} + q (W_{0} (z)) α_{0} (x_{1}) \\ \leq - \frac{1}{2} q (W_{0} (z)) ‖ z ‖^{2} + \hat{α} (x_{1}) \end{matrix}

(45)

where $\hat{α} (x_{1})$ is $C^{2}$ non-negative function and $\hat{α} (0) = 0$ . From the properties of $\hat{α} (x_{1})$ , there exists a continuous function $\tilde{α} (x_{1}) \geq 0$ such that $\hat{α} (x_{1}) \equiv ξ_{1}^{2} \tilde{α} (x_{1})$ .

Consider the Lyapunov function

V'_{k} = V_{0} + V_{k} = \int_{0}^{W_{0} (z)} q_{s} ds + \int_{x_{k}^{*}}^{x_{k}} {(s^{\frac{a}{r_{k}}} - x_{k}^{* \frac{a}{r_{k}}})}^{\frac{2 a - r_{k} - τ}{a}} ds

(46)

and

\begin{matrix} {\overset{\cdot}{V}}_{k} \leq & \sum_{i = 1}^{k} ξ_{i}^{2} + g_{k} (t) {\overset{\cdot}{η}}_{k} ν^{p_{k}} (η_{k}) \\ + {\overset{\cdot}{η}}_{k} + {\hat{g}}_{k} ξ_{k + 1}^{2} + \frac{\partial V_{k}}{\partial x_{1}} f_{1} (\cdot) + \dots + \frac{\partial V_{k}}{\partial x_{k}} f_{k} (\cdot) \end{matrix}

(47)

It is easy to deduce from Assumption 4 that

\begin{matrix} \frac{\partial V_{k}}{\partial x_{1}} f_{1} (\cdot) + \dots + \frac{\partial V_{k}}{\partial x_{k}} f_{k} (\cdot) \\ \leq \frac{\partial V_{k}}{\partial x_{1}} ρ (| z |^{\frac{a + τ}{a}} + | x_{2} |^{\frac{a + τ}{r_{1}}} + \dots + | x_{n} |^{\frac{a + τ}{r_{n}}}) + \dots \\ + \frac{\partial V_{k}}{\partial x_{k}} ρ (| z |^{\frac{a + τ}{a}} + | x_{k + 1} |^{\frac{a + τ}{r_{k + 1}}} + \dots + | x_{n} |^{\frac{a + τ}{r_{n}}}) \end{matrix}

(48)

Then the following inequality holds

\begin{matrix} | \frac{\partial V_{k}}{\partial x_{i}} | | x_{j} |^{\frac{a + τ}{r_{j}}} \\ \leq θ_{k} | ξ_{k} |^{1 - \frac{τ}{a}} | \frac{\partial x_{k}^{* \frac{a}{r_{k}}}}{\partial x_{i}} | | x_{j} |^{\frac{a + τ}{r_{j}}} \\ = θ_{k} | ξ_{k} |^{1 - \frac{τ}{a}} | \frac{\partial (ξ_{k - 1} {(ν (η_{k - 1}) β_{k - 1} (x_{1}, \dots, x_{k - 1}, η_{1}, \dots, η_{k - 2}))}^{\frac{a}{r_{k}}})}{\partial x_{i}} | \\ {| ξ_{j} + ξ_{j - 1} ν^{\frac{a}{r_{j}}} (η_{j - 1}) β_{j - 1}^{\frac{a}{r_{j}}} (x_{1}, \dots, x_{j - 1}, η_{1}, \dots, η_{j - 2}) |}^{\frac{a + τ}{a}} \\ \leq θ_{k} | ξ_{k} |^{1 - \frac{τ}{a}} [| ξ_{k - 1} ν^{\frac{a}{r_{k}}} (η_{k - 1}) \frac{\partial β_{k - 1}^{\frac{a}{r_{k}}}}{\partial x_{i}} | + | {(ν (η_{k - 1}) β_{k - 1})}^{\frac{a}{r_{k}}} \frac{\partial x_{k - 1}^{* \frac{a}{r_{k - 1}}}}{\partial x_{i}} |] \\ {| ξ_{j} + ξ_{j - 1} ν^{\frac{a}{r_{j}}} (η_{j - 1}) β_{j - 1}^{\frac{a}{r_{j}}} (x_{1}, \dots, x_{j - 1}, η_{1}, \dots, η_{j - 2}) |}^{\frac{a + τ}{a}} \\ \leq | ξ_{k} |^{1 - \frac{τ}{a}} (| ξ_{1} |^{\frac{a + τ}{a}} + \dots + | ξ_{j} |^{\frac{a + τ}{a}}) c_{i, j} (x_{1}, \dots, x_{g}, η_{1}, \dots, η_{g - 1}) \end{matrix}

(49)

where g = max(k,j), i = 1,…,k, j = 2,…,n, and

\begin{array}{l} | \frac{\partial V_{k}}{\partial x_{i}} | | z |^{\frac{a + τ}{a}} \\ \leq θ_{k} | ξ_{k} |^{1 - \frac{τ}{a}} | \frac{\partial x_{k}^{* \frac{a}{r_{k}}}}{\partial x_{i}} | | z |^{\frac{a + τ}{a}} \\ \leq θ_{k} | ξ_{k} |^{1 - \frac{τ}{a}} [| ξ_{k - 1} ν^{\frac{a}{r_{k}}} (η_{k - 1}) \frac{\partial β_{k - 1}^{\frac{a}{r_{k}}} (x_{1}, \dots, x_{k - 1}, η_{1}, \dots, η_{k - 2}))}{\partial x_{i}} | \\ + | {(ν (η_{k - 1}) β_{k - 1} (\cdot))}^{\frac{a}{r_{k}}} \frac{\partial x_{k - 1}^{* \frac{a}{r_{k - 1}}}}{\partial x_{i}} |] | z |^{\frac{a + τ}{a}} \\ \leq \frac{1}{4} ‖ z ‖^{2} + c_{0} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1}) ξ_{k}^{2} \end{array}

(50)

Since

\begin{matrix} \overset{\cdot}{V}'_{k} & \leq \sum_{i = 1}^{k} ξ_{i}^{2} + g_{k} (t) {\overset{\cdot}{η}}_{k} ν^{p_{k}} (η_{k}) + {\overset{\cdot}{η}}_{k} + {\hat{g}}_{k} ξ_{k + 1}^{2} \\ + {\overset{\cdot}{V}}_{0} + \frac{\partial V_{k}}{\partial x_{1}} f_{1} (\cdot) + \dots + \frac{\partial V_{k}}{\partial x_{k}} f_{k} (\cdot) \end{matrix}

(51)

If we choose appropriately smooth functions C_n(·) ≥ 0, ${\bar{C}}_{n} (\cdot) \geq 0$ , c_i,j(·) ≥ 0, and λ_n(·) ≥ 1, then the following estimates hold

\begin{matrix} {\overset{\cdot}{V}}_{0} & + \frac{\partial V_{k}}{\partial x_{1}} f_{1} (\cdot) + \dots + \frac{\partial V_{k}}{\partial x_{k}} f_{k} (\cdot) \\ \leq & - \frac{1}{2} ‖ z ‖^{2} + \hat{α} (x_{1}) + \frac{1}{4} ‖ z ‖^{2} + C_{0} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1}) ξ_{k}^{2} \\ + | ξ_{k} |^{1 - \frac{τ}{a}} (| ξ_{1} |^{\frac{a + τ}{a}} \dots + | ξ_{j} |^{\frac{a + τ}{a}}) C_{n} (x_{1}, \dots, x_{n}, η_{1}, \dots, η_{n - 1}) \\ \leq & - \frac{1}{4} ‖ z ‖^{2} + ξ_{1}^{2} \tilde{α} (x_{1}) + C_{0} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1}) ξ_{k}^{2} \\ + (ξ_{1}^{2} + \dots + {\bar{C}}_{n} (x_{1}, \dots, x_{n}, η_{1}, \dots, η_{n - 1}) ξ_{k}^{2} + \dots + ξ_{n}^{2}) \\ \leq & - \frac{1}{4} ‖ z ‖^{2} + ξ_{1}^{2} \tilde{α} (x_{1}) + ξ_{1}^{2} + \dots \\ + {\bar{C}}_{n} (x_{1}, \dots, x_{n}, η_{1}, \dots, η_{n - 1}) ξ_{k}^{2} + \dots + ξ_{n}^{2}) \\ \leq & - \frac{1}{4} ‖ z ‖^{2} + ξ_{1}^{2} \tilde{α} (x_{1}) + ξ_{1}^{2} + \dots + ξ_{n - 1}^{2} \\ + λ_{n} (x_{1}, \dots, x_{n}, η_{1}, \dots, η_{n - 1}) ξ_{n}^{2} \end{matrix}

(52)

Through the procedures to acquire λ_n(·), $β_{n}^{p_{n}} (\cdot)$ and $β_{1}^{p_{1}} (\cdot)$ , if the conditions

\begin{matrix} \tilde{α} (x_{1}) \leq β_{1}^{p_{1}} (x_{1}) \\ λ_{n} (x_{1}, \dots, x_{n}, η_{1}, \dots, η_{n - 1}) \leq β_{n}^{p_{n}} (x_{1}, \dots, x_{n}, η_{1}, \dots, η_{n - 1}) \end{matrix}

are satisfied, then we have

\begin{matrix} {\overset{\cdot}{V}}_{0} + \frac{\partial V_{k}}{\partial x_{1}} f_{1} (\cdot) + \dots + \frac{\partial V_{k}}{\partial x_{k}} f_{k} (\cdot) \\ \leq - \frac{1}{4} ‖ z ‖^{2} + ξ_{1}^{2} \tilde{α} (x_{1}) + ξ_{1}^{2} + \dots + ξ_{n - 1}^{2} + λ_{n} (\cdot) ξ_{n}^{2} \\ \leq - \frac{1}{4} ‖ z ‖^{2} + β_{1}^{p_{1}} (x_{1}) ξ_{1}^{2} + ξ_{1}^{2} + \dots + ξ_{n - 1}^{2} + β_{n}^{p_{n}} (\cdot) ξ_{n}^{2} \\ = - \frac{1}{4} ‖ z ‖^{2} + {\overset{\cdot}{η}}_{1} + \sum_{i = 1}^{n - 1} ξ_{i}^{2} + {\overset{\cdot}{η}}_{n} \end{matrix}

(53)

Then equation (51) becomes

\begin{matrix} \overset{\cdot}{V}'_{k} \leq & - \frac{1}{4} ‖ z ‖^{2} + \sum_{i = 1}^{k} ξ_{i}^{2} + g_{k} (t) {\overset{\cdot}{η}}_{k} ν^{p_{k}} (η_{k}) \\ + {\overset{\cdot}{η}}_{k} + \sum_{i = 1}^{n - 1} ξ_{i}^{2} + {\overset{\cdot}{η}}_{n} + {\overset{\cdot}{η}}_{1} + {\hat{g}}_{k} ξ_{k + 1}^{2} \end{matrix}

(54)

where k = 1,…,n.

Remark 3. If the unmeasurable state z was not considered in this section, the cascaded nonlinear system (41) would come back to the feedforward system (1). The time derivative of V_k for system (1) would become

\begin{matrix} {\overset{\cdot}{V}}_{k} \leq \sum_{i = 1}^{k} ξ_{i}^{2} + g_{k} (t) {\overset{\cdot}{η}}_{k} ν^{p_{k}} (η_{k}) + {\overset{\cdot}{η}}_{k} \\ + \sum_{i = 1}^{n - 1} ξ_{i}^{2} + {\overset{\cdot}{η}}_{n} + {\overset{\cdot}{η}}_{1} + {\hat{g}}_{k} ξ_{k + 1}^{2} \end{matrix}

Under Assumption 1, the procedure to get ${\overset{\cdot}{V}}_{k}$ is similar to the process to get $\overset{\cdot}{V}'_{k}$ .

Main results

Theorem 1. Under Assumptions 2–4, suppose that the design procedure of the controller proposed in the section on high-order linear systems is applied to system (41), so that all signals of the closed-loop system are bounded on [0,∞) for all initial conditions. Moreover, the state of the system asymptotically converges to zero, i.e. $lim_{t \to \infty} x (t) = 0$ and $lim_{t \to \infty} z (t) = 0$ .

Proof. Suppose that the maximum interval of the closed-loop system is [0,t_f). From equations (36) and (54), and the definition of $β_{k}^{p_{k}} (\cdot)$ , we obtain

\begin{matrix} \overset{\cdot}{V}'_{k} & \leq - \frac{1}{4} ‖ z ‖^{2} + \sum_{j = 1}^{k} ξ_{j}^{2} + {\hat{g}}_{k} ξ_{k + 1}^{2} + g_{k} (t) ν^{p_{k}} (η_{k}) {\overset{\cdot}{η}}_{k} \\ + {\overset{\cdot}{η}}_{k} + \sum_{j = 1}^{n - 1} ξ_{j}^{2} + {\overset{\cdot}{η}}_{n} + {\overset{\cdot}{η}}_{1} \\ \leq g_{k} (t) ν^{p_{k}} (η_{k}) {\overset{\cdot}{η}}_{k} + a_{k} \sum_{j = 1}^{n} {\overset{\cdot}{η}}_{j} \end{matrix}

(55)

where $a_{k} = max (2, {\hat{g}}_{k}), k = 1, \dots, n$ . Integrating equation (55) gives

\begin{matrix} 0 \leq V'_{k} (t) \leq {\bar{c}}_{k} + \int_{0}^{t} g_{k} (τ) ν^{p_{k}} (η_{k}) {\overset{\cdot}{η}}_{k} dt \\ + a_{k} \sum_{j = 1}^{n} η_{j}, \forall t \in [0, t_{f}), k = 1, \dots, n, \end{matrix}

(56)

where ${\bar{c}}_{k}, k = 1, \dots, n$ are constants related to the initial data. To simplify the second term of equation (56), we introduce the following function

ν_{k} (η) = {\begin{matrix} | l_{k}^{+} | ν^{p_{k}} (η) & if η \in [4 m - 1, 4 m + 1), m is an integer \\ | l_{k}^{-} | ν^{p_{k}} (η) & if η \in [4 m + 1, 4 m + 3), m is an integer \end{matrix}

(57)

The function $ν (η) = e^{η^{2}} \cos (π / 2)$ is positive on intervals of the form (4m − 1, 4m + 1) and negative on intervals of the form (4m + 1,4m + 3), and ${\overset{\cdot}{η}}_{k} (t) \geq 0, \forall t \in [0, t_{f}), k = 1, \dots, n$ , hence

\begin{matrix} \int_{0}^{t} g_{k} (τ) ν^{p_{k}} (η_{k} (τ)) {\overset{\cdot}{η}}_{k} (τ) d τ & \leq sgn (g_{k}) \int_{0}^{t} ν_{k} (η_{k} (τ)) {\overset{\cdot}{η}}_{k} (τ) d τ \\ = sgn (g_{k}) \int_{0}^{η_{k} (t)} ν_{k} (s) ds + c'_{k}, \\ \forall t \in [0, t_{f}), k = 1, \dots, n, \end{matrix}

(58)

where $c'_{k}, k = 1, \dots, n$ , are constants related to the initial data. So equation (56) becomes

\begin{matrix} 0 \leq V'_{k} (t) \leq {\hat{c}}_{k} + sgn (g_{k}) \int_{0}^{η_{k} (t)} ν_{k} (s) ds \\ + a_{k} \sum_{j = 1}^{n} {\overset{\cdot}{η}}_{j}, \forall t \in [0, t_{f}), k = 1, \dots, n \end{matrix}

(59)

where ${\hat{c}}_{k} = {\bar{c}}_{k} + c'_{k}$ . Next we will prove by contradiction that η_k(t),k = 1,…,n are bounded on [0,t_f).

Suppose that there exist 1 ≤ j₁ < j₂ < ⋯ < j_q ≤ n, 1 ≤ q ≤ n such that $η_{j_{i}}$ ,1 ≤ i ≤ q are unbounded and the remaining (n − q)η_k are bounded. Then we choose the monotone increasing sequences $t_{m}^{i}, m = 1, 2, \dots, 1 \leq i \leq q$ , on the time interval [0,t_f) satisfying $lim_{m \to \infty} t'_{m} = t_{f}$ , and

η_{j_{i}} (t_{m}^{i}) = {\begin{matrix} 4 (m + m_{i}) + 1 & if sgn (g_{j_{i}}) = - 1 \\ 4 (m + m_{i}) + 3 & if sgn (g_{j_{i}}) = 1 \end{matrix}

(60)

where $m_{i} \geq η_{j_{i}}$ (0),1 ≤ i ≤ q, are constant positive integers to ensure the existence of $t_{1}^{i}$ for any initial $η_{j_{i}}$ . Then it follows that

r_{i} (m) = {\begin{matrix} \frac{e^{{(4 m + 4 m_{i} - 1)}^{2}}}{4 m + 4 m_{i} + 1} [2^{- \frac{1}{2}} | l_{j_{i}}^{+} | e^{4 m + 4 m_{i} - 0.75} - (4 m + 4 m_{i} - 1) (| l_{j_{i}}^{-} | + | l_{j_{i}}^{+} |)] & if sgn (g_{j_{i}}) = - 1 \\ \frac{e^{{(4 m + 4 m_{i} + 1)}^{2}}}{4 m + 4 m_{i} + 3} [2^{- \frac{1}{2}} | l_{j_{i}}^{+} | e^{4 m + 4 m_{i} + 1.25} - (4 m + 4 m_{i} + 1) (| l_{j_{i}}^{-} | + | l_{j_{i}}^{+} |)] & if sgn (g_{j_{i}}) = 1 \end{matrix}

(61)

Clearly, we have $lim_{m \to \infty} r_{i} (m) = \infty$ ,1 ≤ i ≤ q. According to the property of ν(η), the following inequality holds

sgn (g_{j_{i}}) \int_{0}^{η_{j_{i}} (t_{m}^{i})} ν_{j_{i}} (s) ds \leq - r_{i} (m) η_{j_{i}} (t_{m}^{i})

(62)

Then equation (59) becomes

\begin{matrix} 0 \leq V'_{j_{i}} (t_{m}^{i}) \leq - r_{i} (m) η_{j_{i}} (t_{m}^{i}) + {\hat{c}}_{j_{i}} \\ + a_{j_{i}} \sum_{h = 1}^{j_{i} + 1} η_{h} (t_{m}^{i}) \leq - r_{i} (m) η_{j_{i}} (t_{m}^{i}) + {\hat{c}}_{j_{i}} + a \sum_{h = 1}^{j_{i} + 1} η_{h} (t_{m}^{i}) \end{matrix}

(63)

where a = max(a₁,…,a_n) > 0,m = 1,2,…,1 ≤ i ≤ q. Since η_k(t),k = 1,…,n,k≠j_i,1 ≤ i ≤ q, is bounded and $lim_{t \to t_{f}} η_{j_{i}} (t) = \infty$ ,1 ≤ i ≤ q, there exists M > 0; if m > M, we have

0 \leq - {\bar{r}}_{i} (m) η_{j_{i}} (t_{m}^{i}) + \sum_{h = 1}^{q} η_{j_{h}} (t_{m}^{i}), 1 \leq i \leq q

(64)

where ${\bar{r}}_{i} (m) = r_{i} (m) / a$ . Obviously, $lim_{m \to \infty} {\bar{r}}_{i} (m) = \infty, 1 \leq i \leq q$ . Noting that $η_{j_{i}} (t_{m + 1}^{i}) = 4 + η_{j_{i}} (t_{m}^{i}), 1 \leq i \leq q$ , and $η_{j_{i}} (t)$ are monotone increasing, we obtain

\begin{matrix} 0 & \leq - {\bar{r}}_{i} (m) η_{j_{i}} (t_{m}^{i}) + \sum_{h = 1}^{q} η_{j_{h}} (t_{m}^{i}) = - {\bar{r}}_{i} (m) (η_{j_{i}} (t_{m + 1}^{i}) - 4) \\ + \sum_{h = 1}^{q} η_{j_{h}} (t_{m}^{i}) \\ \leq - {\bar{r}}_{i} (m) η_{j_{i}} (t - 4) + \sum_{h = 1}^{q} η_{j_{h}} (t_{m}^{i}) \\ \forall t \in [t_{m}^{i}, t_{m + 1}^{i}), m \geq M, 1 \leq i \leq q \end{matrix}

(65)

Since $lim_{m \to \infty} {\bar{r}}_{i} (m) = \infty, 1 \leq i \leq q$ , there exists $\bar{M} \geq M > 0$ ; if $m \geq \bar{M}$ , we get ${\bar{r}}_{i} (m) \geq n + 1, 1 \leq i \leq q$ . Then

\begin{matrix} 0 \leq - (n + 1) (η_{j_{i}} (t) - 4) + \sum_{h = 1}^{q} η_{j_{h}} (t) \\ \forall t \in [t^{*}, t_{f}), 1 \leq i \leq q \end{matrix}

(66)

where $t^{*} = max (t_{\bar{M}}^{1}, \dots, t_{\bar{M}}^{q})$ . Adding all of the inequalities in equation (66), we obtain

0 \leq - (n + 1 - q) \sum_{h = 1}^{q} η_{j_{h}} (t) + 4 (n + 1) q, \forall t \in [t^{*}, t_{f})

(67)

Take the limit of the right-hand side of equation (67) as t→t_f; hence $η_{j_{h}} (t) \to \infty, 1 \leq h \leq q$ , so the right-hand side of equation (67) tends to negative infinity and we have a contradiction. Consequently, η_k(t),1 ≤ k ≤ n, must also be bounded on [0,t_f), so that V_k(t),1 ≤ k ≤ n must be bounded on [0,t_f). Since $η_{k} (t), V'_{k} (t), 1 \leq k \leq n$ , is bounded on [0,t_f), ξ_k(t),1 ≤ k ≤ n, is bounded on [0,t_f). According to equation (36), $x_{k}^{*} (t)$ ,x_k(t) must be bounded on [0,t_f). Therefore, all the state variables of the closed-loop system are bounded on [0,t_f), and thus t_f = ∞.

Then we prove that z(t),V_k(t),2 ≤ k ≤ n, is bounded on [0,t_f). As

V_{k} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k}) = \int_{x_{k}^{*}}^{x_{k}} {(s^{\frac{a}{r_{k}}} - x_{k}^{* \frac{a}{r_{k}}})}^{\frac{2 a - r_{k} - τ}{a}} ds

is non-negative, $V_{0} (z), V'_{k} (t), 1 \leq k \leq n$ are positive definite, and $V'_{k} (t), 1 \leq k \leq n$ is bounded on [0,t_f), then z(t),V_k(t),2 ≤ k ≤ n are bounded on [0,t_f).

Since $u = x_{n + 1}^{*}$ is bounded, $‖ \overset{\cdot}{x} (t) ‖$ is bounded. In addition, since x_k(t),ξ_k(t), and η_k(t) are bounded, ${\overset{\cdot}{η}}_{k} (t), 1 \leq k \leq n$ , is also bounded, and hence $‖ \overset{\cdot}{ξ} (t) ‖$ is bounded, so therefore ∥ξ(t)∥² is uniformly continuous and

\begin{matrix} lim_{t \to \infty} \int_{0}^{t} ‖ ξ (τ) ‖^{2} d τ & = lim_{t \to \infty} \int_{0}^{t} \sum_{k = 1}^{n} ξ_{k}^{2} (τ) d τ \\ \leq lim_{t \to \infty} \int_{0}^{t} \sum_{k = 1}^{n} {\overset{\cdot}{η}}_{k} (τ) d τ \\ = lim_{t \to \infty} \sum_{k = 1}^{n} η_{k} (t) - \sum_{k = 1}^{n} η_{k} (0) < \infty \end{matrix}

(68)

implies by Barbalat’s Lemma that

lim_{t \to \infty} ‖ ξ (t) ‖ = 0, 1 \leq k \leq n

(69)

According to equation (36), the definition of ξ_k, and the boundedness of η_k, we have

lim_{t \to \infty} ‖ x (t) ‖ = 0, 1 \leq k \leq n

(70)

From equation (1), $‖ \overset{\cdot}{z} (t) ‖$ is bounded and hence ∥z(t∥² is uniformly continuous. From equation (45), we have

\begin{matrix} lim_{t \to \infty} \frac{1}{2} \int_{0}^{t} ‖ z (τ) ‖^{2} & \leq lim_{t \to \infty} \int_{0}^{t} \frac{1}{2} q (W_{0} (z)) ‖ z ‖^{2} d τ \\ = - lim_{t \to \infty} \int_{0}^{t} {\overset{\cdot}{V}}_{0} (z) d τ + lim_{t \to \infty} \int_{0}^{t} \hat{α} (x_{1}) d τ \\ \leq V_{0} (0) + lim_{t \to \infty} \int_{0}^{t} \hat{α} (x_{1}) d τ < \infty \end{matrix}

(71)

which implies by Barbalat’s Lemma that

lim_{t \to \infty} ‖ z (t) ‖ = 0, 1 \leq k \leq n

(72)

The proof of Theorem 1 is now complete. □

Remark 4. Theorem 1 was originally proved by Ye (1999). A global robust control for a class of time-varying uncertain nonlinear systems of arbitrary dynamic order with arbitrary unknown control is presented in Ye (1999). The global robust control is extended to a class of high-order cascaded systems in this paper. The system we consider is more general; if p_i = 1, i = 1,…,n, and the nonlinear terms are general nonlinear terms, the global robust control problem is similar to the problem studied in Ye (1999).

Remark 5. If the unmeasurable state z is not considered in Theorem 1, the cascaded nonlinear system (41) is converted to the feedforward system (1). We can also obtain a similar result to Theorem 1. Under Assumptions 1 and 2, the proof is similar to the proof of Theorem 1.

Example 1. Consider the system

\begin{matrix} \overset{\cdot\cdot}{x} + {yf}_{1} (y, \overset{\cdot}{y}) + a (t) y^{p} = 0 \\ \overset{\cdot\cdot}{y} + b (t) u^{q} = 0 \end{matrix}

(73)

where p > 1, q > 1 are known odd integers and f₁(0,0) = 0, a(t), b(t) are uncertain time-varying parameters.

We introduce the following change of coordinates

x_{1} = x, x_{2} = \overset{\cdot}{x}, x_{3} = - y, x_{4} = - \overset{\cdot}{y}

Then system (73) can be transformed into

\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = a (t) x_{3}^{p} + x_{3} f_{1} (- x_{3}, - x_{4}) \\ {\overset{\cdot}{x}}_{3} = x_{4} \\ {\overset{\cdot}{x}}_{4} = b (t) u^{q} \end{matrix}

(74)

According to the design approach in the section on the design of the state feedback controller, a continuous controller can be explicitly designed, making system (73) globally asymptotically stable if the nonlinear function f₁ satisfies Assumption 1.

Example 2. Consider the following nonlinear system

\begin{matrix} \overset{\cdot}{z} = - z + x_{1} \\ {\overset{\cdot}{x}}_{1} = g_{1} (t) x_{2}^{3} + {(\sin (x (2)))}^{8} + \sin (x (2)) z^{\frac{2}{3}} \\ {\overset{\cdot}{x}}_{2} = g_{2} (t) u \end{matrix}

(75)

where g₁(t) and g₂(t) are uncertain time-varying parameters and the g_i(t) values on unknown intervals $I_{i} = [l_{i}^{-}, l_{i}^{+}]$ and 0 ∉I_i, i = 1,2, i.e. the control directions, are unknown.

We choose $r_{1} = 1, τ = 2, r_{2} = \frac{r_{1} + τ}{p_{1}} = 1, r_{3} = \frac{r_{2} + τ}{p_{2}} = 3$ , and a = 3. Then we obtain

\begin{matrix} ξ_{1} = x_{1}^{3} \\ ξ_{2} = x_{2}^{3} - x_{2}^{* 3} \\ x_{2}^{*} = ξ_{1} ν (η_{1}) \\ ν (η) = e^{η^{2}} \cos (\frac{π}{2} η) \end{matrix}

(76)

According to the design process in the section on the design of the state feedback controller, we can construct the following controller

\begin{matrix} u = ξ_{2}^{\frac{r_{3}}{a}} ν (η_{2}) β_{2} (x_{1}, x_{2}, η_{1}) \\ {\overset{\cdot}{η}}_{1} = ξ_{1}^{2} β_{1}^{p_{1}} \\ {\overset{\cdot}{η}}_{2} = ξ_{2}^{2} β_{2}^{p_{2}} \end{matrix}

(77)

where

\begin{matrix} β_{1} = & 1, β_{2} (\cdot) = 1 + 2 θ_{2}^{2} ξ_{2}^{2 / 3} {| \frac{\partial x_{2}^{*}}{\partial x_{1}} |}^{2} ν^{6} (η_{1}) \\ + 2 θ_{2} ξ_{2}^{2 / 3} | \frac{\partial x_{2}^{*}}{\partial x_{1}} | + \frac{1}{3} x_{1}^{6} θ_{2} ξ_{2}^{2 / 3} | \frac{\partial x_{2}^{*}}{\partial η_{1}} | \end{matrix}

and $θ_{2} = \frac{2 a - r_{2} - τ}{a} 2^{1 - \frac{r_{2}}{a}}$ .

The simulation is implemented with the uncertain parameters g₁(t) = 1.5 + sin(t), g₂(t) = 1.5−sin(t) and the initial conditions x₁(0) = 1, x₂(0) = −1,η₁(0) = 1,η₂(0) = 1,z(0) = 2.

The simulation results shown in Figures 1 –3 show that the controller we have proposed in this paper is effective. It ensures that all signals of the closed-loop system are globally bounded on [0,∞), and that the system states x₁,x₂ and the system unmeasurable state z asymptotically converge to zero.

Remark 6. If the unmeasurable state z is ignored in Example 2, the design process is similar to the procedure to get the controller u in Example 2. The system states x₁, x₂also asymptotically converge to zero.

Conclusions

A state feedback controller has been proposed for a class of feedforward nonlinear systems with unknown control directions. The homogeneous domination is used to construct the controller and the problem of unknown control directions is solved by incorporating the Nussbaum-type gain into the virtual control laws and the actual control laws. We have proved that under the proposed controller the global boundedness for all signals of the closed-loop system can be achieved and the system state converges to zero, asymptotically. Finally, the simulation examples illustrate the effectiveness of our controller.

Footnotes

Appendix

Proof of Proposition 1. We apply mathematical deduction to complete the proof. If k = 2, we have

(78)

\begin{matrix} | \frac{\partial x_{2}^{* \frac{a}{r_{2}}}}{\partial x_{1}} {\overset{\cdot}{x}}_{1} | \\ = | \frac{\partial (x_{1}^{\frac{a}{r_{1}}} ν^{\frac{a}{r_{2}}} (η_{1}) β_{1}^{\frac{a}{r_{2}}} (x_{1}))}{\partial x_{1}} {\overset{\cdot}{x}}_{1} | \\ \leq | \frac{a}{r_{1}} x_{1}^{\frac{a - r_{1}}{r_{1}}} {(ν (η_{1}) β_{1} (x_{1}))}^{\frac{a}{r_{2}}} + ξ_{1} \frac{\partial {(ν (η_{1}) β (x_{1}))}^{\frac{a}{r_{2}}}}{\partial x_{1}} | \\ | g_{1} {(ξ_{2} + ξ_{1} {(ν β)}^{\frac{a}{r_{2}}})}^{\frac{r_{2} p_{1}}{a}} | \\ \leq | \frac{a}{r_{1}} ξ_{1}^{\frac{a - r_{1}}{a}} {(ν (η_{1}) β_{1} (x_{1}))}^{\frac{a}{r_{2}}} + ξ_{1} \frac{\partial {(ν (η_{1}) β (x_{1}))}^{\frac{a}{r_{2}}}}{\partial x_{1}} | \\ | g_{1} {(ξ_{2} + ξ_{1} {(ν β)}^{\frac{a}{r_{2}}})}^{\frac{r_{2} p_{1}}{a}} | \\ \leq (| ξ_{1} |^{\frac{a + τ}{a}} + | ξ_{2} |^{\frac{a + τ}{a}}) α_{2, 1} (x_{1}, x_{2}, η_{1}) \end{matrix}

where α_2,1(x₁, x₂, η₁) ≥ 0 is a smooth function.

Suppose that there exists a non-negative function α_k_−1,l(x₁,…,x_k₋₁,η₁,…,η_k₋₂) for l = 1,2,…,k−2 such that

(79)

\begin{matrix} | \frac{\partial x_{k - 1}^{* \frac{a}{r_{k - 1}}}}{\partial x_{l}} {\overset{\cdot}{x}}_{l} | \leq (| ξ_{1} |^{\frac{a + τ}{a}} + \dots + | ξ_{k - 1} |^{\frac{a + τ}{a}}) \\ α_{k - 1, l} (x_{1}, \dots, x_{k - 1}, η_{1}, \dots, η_{k - 2}) \end{matrix}

Then we proof that there exists a non-negative smooth function α_k,l(x₁,…,x_k,η₁,…,η_k₋₁) for l = 1,2,…,k, such that

(80)

\begin{matrix} | \frac{\partial x_{k}^{* \frac{a}{r_{k}}}}{\partial x_{l}} {\overset{\cdot}{x}}_{l} | \leq (| ξ_{1} |^{\frac{a + τ}{a}} a + \dots + | ξ_{k} |^{\frac{a + τ}{a}}) \\ α_{k, l} (x_{1}, \dots, x_{k}, η_{1}, \dots, η_{k - 1}) \end{matrix}

First consider the case when l = 1,2,…,k − 2, so that we obtain

(81)

\begin{matrix} | \frac{\partial x_{k}^{* \frac{a}{r_{k}}}}{\partial x_{l}} {\overset{\cdot}{x}}_{l} | \\ = | \frac{\partial (ξ_{k - 1} ν^{\frac{a}{r_{k}}} (η_{k - 1}) β_{k - 1}^{\frac{a}{r_{k}}} (\cdot))}{\partial x_{l}} {\overset{\cdot}{x}}_{l} | \\ \leq | ξ_{k - 1} ν^{\frac{a}{r_{k}}} (η_{k - 1}) \frac{\partial β_{k - 1}^{\frac{a}{r_{k}}} (\cdot)}{\partial x_{l}} {\overset{\cdot}{x}}_{l} | \\ + | {(ν (η_{k - 1}) β_{k - 1} (\cdot))}^{\frac{a}{r_{k}}} \frac{\partial (x_{k - 1}^{\frac{a}{r_{k - 1}}} - x_{k - 1}^{* \frac{a}{r_{k - 1}}})}{\partial x_{l}} {\overset{\cdot}{x}}_{l} | \\ \leq | ξ_{k - 1} ν^{\frac{a}{r_{k}}} (η_{k - 1}) (η_{k - 1}) \frac{\partial β_{k - 1}^{\frac{a}{r_{k}}} (\cdot)}{\partial x_{l}} {\overset{\cdot}{x}}_{l} | \\ + | {(ν (η_{k - 1}) β_{k - 1} (\cdot))}^{\frac{a}{r_{k}}} \frac{\partial x_{k - 1}^{* \frac{a}{r_{k}}}}{\partial x_{l}} {\overset{\cdot}{x}}_{l} | \\ \leq | {(ν (η_{k - 1}) β_{k - 1} (\cdot))}^{\frac{a}{r_{k}}} | (| ξ_{1} |^{\frac{a + τ}{a}} + \dots + | ξ_{k - 1} |^{\frac{a + τ}{a}}) α_{k - 1, l} (\cdot) \\ + | ξ_{k - 1} ν {(η_{k - 1})}^{\frac{a}{r_{k}}} (η_{k - 1}) \frac{\partial β_{k - 1}^{\frac{a}{r_{k}}} (\cdot)}{\partial x_{l}} {\overset{\cdot}{x}}_{l} | \\ \leq (| ξ_{1} |^{\frac{a + τ}{a}} + \dots + | ξ_{k - 1} |^{\frac{a + τ}{a}}) α_{k, l} (\cdot) \end{matrix}

Then consider the case that when l = k − 1, so that equation (80) also holds

(82)

\begin{matrix} | \frac{\partial x_{k}^{* \frac{a}{r_{k}}}}{\partial x_{k - 1}} {\overset{\cdot}{x}}_{k - 1} | \\ = | \frac{\partial {(ξ_{k - 1} (ν (η_{k - 1}) β_{k - 1} (\cdot)))}^{\frac{a}{r_{k}}}}{\partial x_{k - 1}} {\overset{\cdot}{x}}_{k - 1} | \\ \leq | ξ_{k - 1} \frac{\partial {(ν (η_{k - 1}) β_{k - 1} (\cdot))}^{\frac{a}{r_{k}}}}{\partial x_{k - 1}} {\overset{\cdot}{x}}_{k - 1} | \\ + | {(ν (η_{k - 1}) β_{k - 1} (\cdot))}^{\frac{a}{r_{k}}} \frac{\partial (x_{k - 1}^{\frac{a}{r_{k - 1}}} - x_{k - 2}^{* \frac{a}{r_{k - 1}}})}{\partial x_{k - 1}} {\overset{\cdot}{x}}_{k - 1} | \\ \leq | ξ_{k - 1} \frac{\partial {(ν (η_{k - 1}) β_{k - 1} (\cdot))}^{\frac{a}{r_{k}}}}{\partial x_{k - 1}} {\overset{\cdot}{x}}_{k - 1} | \\ + | \frac{a}{r_{k - 1}} x_{k - 1}^{\frac{a - r_{k - 1}}{r_{k - 1}}} {(ν (η_{k - 1}) β_{k - 1} (\cdot))}^{\frac{a}{r_{k}}} | | {\overset{\cdot}{x}}_{k - 1} | \\ \leq | ξ_{k - 1} \frac{\partial {(ν (η_{k - 1}) β_{k - 1} (\cdot))}^{\frac{a}{r_{k}}}}{\partial x_{k - 1}} {\overset{\cdot}{x}}_{k - 1} | \\ + | \frac{a}{r_{k - 1}} ξ_{k - 1}^{\frac{a - r_{k - 1}}{a}} {(ν (η_{k - 1}) β_{k - 1} (\cdot))}^{\frac{a}{r_{k}}} | | g_{k - 1} x_{k}^{p_{k - 1}} | \\ \leq | \frac{a}{r_{k - 1}} ξ_{k - 1}^{\frac{a - r_{k - 1}}{a}} {(ν (η_{k - 1}) β_{k - 1} (\cdot))}^{\frac{a}{r_{k}}} | | g_{k - 1} {(ξ_{k} + x_{k - 1}^{* \frac{a}{r_{k}}})}^{\frac{r_{k} p_{k - 1}}{a}} | \\ + | ξ_{k - 1} \frac{\partial {(ν (η_{k - 1}) β_{k - 1} (\cdot))}^{\frac{a}{r_{k}}}}{\partial x_{k - 1}} {\overset{\cdot}{x}}_{k - 1} | \\ \leq (| ξ_{1} |^{\frac{a + τ}{a}} + \dots + | ξ_{k} |^{\frac{a + τ}{a}}) {\hat{α}}_{k - 1} (\cdot) \end{matrix}

Combining equations (81) and (82), we know that, for all l = 1,2,…,k − 1, Proposition 1 always holds.□

Funding

This work was supported by the National Natural Science Foundation of PR China (grant numbers 61374153, 61174038 and 61374087) and the program for Changjiang scholars and innovative research team in Universities (grant number IRT13072).

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