Adaptive output-feedback control for uncertain nonlinear systems with function control coefficients

Abstract

This article addresses the global adaptive output-feedback state regulation problem for a class of uncertain nonlinear systems with generalized control coefficients. Remarkably, the system under investigation simultaneously possesses function control coefficients, input matching uncertainty, and unknown growth rate, which is essentially different from the closely related existing literature. Such nonlinearities from function control coefficients and uncertainties from unknown growth rate bring great technical difficulties to controller design. A novel dynamic gain is introduced to deal with the extra system nonlinearities from function control coefficients and unknowns from the growth rate. In addition, the extended state observer is developed to reconstruct the unmeasured states of the system. By integrating the dynamic gain and extended state observer, an adaptive output-feedback controller is designed, which ensures that the states of the system globally converge to zero, and the estimation of the input matching uncertainty converges to its actual value. Two simulation examples are given to demonstrate the effectiveness of the proposed method.

Keywords

Uncertain nonlinear systems function control coefficients input matching uncertainty adaptive output feedback

Introduction

As is well known, many scholars have been attaching great importance to the analysis and control design of different systems, and have achieved a series of research results.^1–21 A feedback domination design method that is not based on the separation principle was proposed when the growth rate of the system is a known constant.¹ Using this design method, a universal adaptive output-feedback controller was constructed, and the control coefficients of the system are only known constants (equal to 1).^2,3 A new robust static output-feedback $H_{\infty}$ controller design method was proposed for linear systems with constant control coefficients.⁸ Especially, based on Krishnamurthy and Khorrami,^4,5 the problem of global output-feedback control for nonlinear systems with generalized control coefficients was solved.^9,10 The event-triggering mechanism was introduced to study the nonlinear control problem.^13,14,19

It is well known that practical physical systems are inevitably affected by various uncertainties/noises. Therefore, in the past few decades, a large number of scholars have devoted effort to the state regulation problem for a class of nonlinear systems that can be used to describe more physical objects, and have more nonlinearities.^22–30 In practical industrial applications, input interference can sometimes be regarded as a constant input matching uncertainty. Furthermore, in some special cases, the control of systems with input matching uncertainty is the basis for other control problems.^22,26 The output-feedback control problem of the system was investigated, where the nonlinearities of the system are only related to the output of the system.²⁷ For systems with weaker conditions for growth rate, a compact design scheme was proposed, and the control coefficients of the system are constants.²⁸ Guo and Xie^29,30 considered the control problem of time-delay and feed-forward systems with input matching uncertainty, but the control coefficients of the systems mentioned above are constants.

As far as we know, when the control coefficients of the system are the functions of the output or when the system has input matching uncertainty, the output-feedback control problem of the system has been studied, respectively. However, when the system has both the function control coefficients and the input matching uncertainty, the adaptive output-feedback state regulation problem of the system has not been studied so far.

This article is devoted to proposing an adaptive output-feedback scheme to achieve the state regulation for the system with function control coefficients, input matching uncertainty, and unknown growth rate. We introduce a novel dynamic gain to counteract the negative effects of extra nonlinearities and unknowns in the system. By combining the dynamic gain, an extended state observer is built to estimate the input matching uncertainty and reconstruct the unmeasured states of the system. Based on this, an adaptive output-feedback controller is designed to ensure that all states of the system globally converge to zero, and the estimation of the input matching uncertainty ultimately converges to its actual value.

The highlights of the article can be summarized as follows:

The system allows more nonlinearities and unknowns than those in the related literature. Specifically, the control coefficients of the system are the functions of output, rather than known constants as in the previous literature.^6,7,26,28,29 The growth rate of the system nonlinearities is an unknown constant, unlike the case of known constant growth rate.¹ In addition, the system also has input matching uncertainty, which makes the system different from those in the relevant literature.^9,10 The existence of multiple uncertainties and nonlinearities in the system will inevitably bring some substantial obstacles to control design and analysis.

A novel dynamic gain is introduced to deal with the additional nonlinearities from the function control coefficients and the serious unknowns from the growth rate. This is essentially different from the related works.^9,28 Particularly, compared with constant control coefficients of Huang and Liu,²⁸ the gain in this article can be used to compensate the function control coefficients well.

The remainder of the article is organized as follows. The “Problem formulation and preliminaries” section formulates the system model. The “Output-feedback controller design” section designs an output-feedback controller to solve the global state regulation problem. The “Main results” section summarizes the main research results of this article. The “Proofs of Proposition 1 and Lemma 1” section gives detailed proofs of a proposition and a lemma. The “Simulation examples” section shows two simulation examples, and the “Conclusion” section addresses some concluding remarks.

Problem formulation and preliminaries

In this article, we consider the following global adaptive output-feedback state regulation problem for uncertain nonlinear systems

{\begin{matrix} {\overset{\cdot}{x}}_{i} = g_{i} (y) x_{i + 1} + f_{i} (t, x), i = 1, \dots, n - 1, \\ {\overset{\cdot}{x}}_{n} = g_{n} (y) (u + ω) + f_{n} (t, x), \\ y = x_{1} \end{matrix}

(1)

where $x = [x_{1}, \dots, x_{n}]^{T} \in R^{n}$ is the system state with the initial condition $x (0) = x_{0}$ ; $u \in R$ and $y \in R$ are the control input and system output, respectively; $ω \in R$ is an unknown constant, representing the input matching uncertainty; $g_{i} (y), i = 1, 2, \dots, n$ are known and locally Lipschitz functions. Functions $f_{i} (t, x), i = 1, 2, \dots, n$ are continuous in the first argument and locally Lipschitz in the rest.

In what follows, we suppose that only the system output $y$ is measurable, and the following assumptions are made on system equation (1).

Assumption 1

There exist known positive constants $\underline{g}, \bar{g}, \underline{a}, \bar{a}$ , and $p$ , such that for $\forall y \in R$

\begin{matrix} 0 < \underline{g} \leq | g_{i} (y) | \leq \bar{g} (1 + | y |^{p}), i = 1, \dots, n, \\ \underline{a} \leq | g_{i} (y) / g_{i - 1} (y) | \leq \bar{a}, i = 2, \dots, n \end{matrix}

Assumption 2

For system equation (1), there exists a positive unknown constant $θ$ such that

| f_{i} (t, x) | \leq θ (| x_{1} | + \dots + | x_{i} |)

holds for all $i = 1, \dots, n$ .

We would like to illustrate the difference between the state regulation problem in this article and those in the existing related works.^{1,6,7,26–29} Assumption 1 indicates that the control coefficients of system equation (1) are nonlinear functions of output, which is essentially different from the previous work, where the control coefficients of the system are known constants.^6,7,26–29 Assumption 2 shows that the growth of system equation (1) depends on the unmeasured states and the growth rate is an unknown constant, which is different from the related literature where the growth rate is only a known constant.¹ Hence, it is worth noting that in addition to the input matching uncertainty, namely the existence of unknown $ω$ , system equation (1) also has extra nonlinearities from function control coefficients and the unknown growth rate.

Output-feedback controller design

Denote $ω = x_{n + 1}$ for notational convenience. To rebuild the unmeasured states of system equation (1) and estimate the input matching uncertainty, we construct the following adaptive extended state observer

{\begin{matrix} \overset{\cdot}{\hat{x_{i}}} = g_{i} (y) {\hat{x}}_{i + 1} + h_{i} (y) L^{i} (x_{1} - {\hat{x}}_{1}), i = 1, \dots, n - 1, \\ \overset{\cdot}{\hat{x_{n}}} = g_{n} (y) {\hat{x}}_{n + 1} + g_{n} (y) u + h_{n} (y) L^{n} (x_{1} - {\hat{x}}_{1}), \\ \overset{\cdot}{\hat{x_{n + 1}}} = h_{n + 1} (y) L^{n + 1} (x_{1} - {\hat{x}}_{1}) \end{matrix}

(2)

with $L$ being a dynamic gain updated by

\overset{\cdot}{L} = \frac{ρ (ε_{1}, z_{1}) {(x_{1} - {\hat{x}}_{1})}^{2}}{L}, L (0) = 1

(3)

where $ρ (ε_{1}, z_{1}) = 1 + (1 + (ε_{1} + z_{1})^{2})^{\frac{p}{2}}$ , $ε_{1} = (x_{1} - {\hat{x}}_{1}) / L$ , $z_{1} = {\hat{x}}_{1} / L$ , and $p$ is the same as in Assumption 1. $h_{i} (y), i = 1, \dots, n + 1$ are functions that will be determined later. $\hat{x} = [{\hat{x}}_{1}, \dots, {\hat{x}}_{n + 1}]^{T}$ and ${\hat{x}}_{[i]} = [{\hat{x}}_{1}, \dots, {\hat{x}}_{i}]^{T},$ $i = 1, \dots, n$ .

Based on the above observer, we design the following output-feedback controller

u = - \frac{1}{g_{n} (y)} \sum_{i = 1}^{n} L^{n - i + 1} k_{i} (y) {\hat{x}}_{i} - {\hat{x}}_{n + 1}

(4)

where $k_{i} (y), i = 1, 2, \dots, n$ are functions determined later.

Remark 1

It is worth mentioning that system equation (1) has input matching uncertainty, which is different from Jin and Liu,⁹ and thus, the control design methods by Jin and Liu⁹ cannot be used to deal with the problems in this article. Moreover, the control coefficients of system equation (1) are functions of output, rather than constants as by Huang and Liu,²⁸ and this also makes the existing design methods inapplicable.

Let $e_{i} = x_{i} - {\hat{x}}_{i}, i = 1, 2, \dots, n + 1$ , and introduce the following scaling transformation

{\begin{matrix} ε_{i} = \frac{e_{i}}{L^{i}}, i = 1, \dots, n + 1, \\ z_{i} = \frac{{\hat{x}}_{i}}{L^{i}}, i = 1, \dots, n \end{matrix}

(5)

Using equations (1), (2), and (4), we have

{\begin{matrix} \overset{\cdot}{ε} = LA (y) ε - \frac{\overset{\cdot}{L}}{L} D_{n + 1} ε + F (t, x, L), \\ \overset{\cdot}{z} = LK (y) z - \frac{\overset{\cdot}{L}}{L} D_{n} z + L H_{n} (y) ε_{1} \end{matrix}

(6)

where $ε = [ε_{1}, \dots, ε_{n + 1}]^{T}, z = [z_{1}, \dots, z_{n}]^{T}$ , $D_{i} = diag {1, 2, \dots, i}$ , $H_{i} (y) = [h_{1} (y), \dots, h_{i} (y)]^{T}, i = n, n + 1$ , $F (t, x, L) = {[f_{1} (t, x) / L, f_{2} (t, x) / L^{2}, \dots, f_{n} (t, x) / L^{n}, 0]}^{T}$ , and

\begin{matrix} A (y) = (\begin{matrix} - h_{1} (y) & g_{1} (y) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ - h_{n} (y) & 0 & \dots & g_{n} (y) \\ - h_{n + 1} (y) & 0 & \dots & 0 \end{matrix}), \\ K (y) = (\begin{matrix} 0 & g_{1} (y) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & g_{n - 1} (y) \\ - k_{1} (y) & - k_{2} (y) & \dots & - k_{n} (y) \end{matrix}) . \end{matrix},

Choose $h_{i} (y), i = 1, \dots, n + 1$ and $k_{i} (y), i = 1, 2, \dots, n$ are locally Lipschitz functions, and $h_{i} (y), i = 1, \dots, n + 1$ are linear constant-coefficient combinations of $g_{i} (y), i = 1, \dots, n$ . There exist symmetric positive-definite matrices $P$ and $Q$ satisfying

{\begin{matrix} A^{T} (y) P + PA (y) \leq - d_{1} | g_{1} (y) | I_{n + 1}, \\ K^{T} (y) Q + QK (y) \leq - d_{2} | g_{1} (y) | I_{n}, \\ D_{n + 1} P + P D_{n + 1} \geq d_{3} I_{n + 1}, \\ D_{n} Q + Q D_{n} \geq d_{4} I_{n} \end{matrix}

(7)

where $d_{i}, i = 1, \dots, 4$ are positive constants. From Krishnamurthy and Khorrami⁵ and Praly and Jiang,²³ we know that such choice can be achieved.

By Assumption 1, there exists a known positive constant $\bar{h}$ such that

∥ H_{i} (y) ∥ = \sqrt{\sum_{j = 1}^{i} h_{j}^{2} (y)} \leq \bar{h} | g_{1} (y) |, i = n, n + 1

(8)

Before showing the main results of this article, we give the following proposition, which will play an important role in the subsequent proof.

Proposition 1. Let $V (ε, z) = V_{1} (ε) + m V_{2} (z) = ε^{T} P ε + m z^{T} Qz$ with $m = (d_{1} \cdot d_{2}) / (4 {\bar{h}}^{2} ∥ Q ∥^{2})$ . Then, we have

\overset{\cdot}{V} (t) \leq - β (L - Θ) (∥ ε ∥^{2} + m ∥ z ∥^{2})

(9)

where $V (t)$ simply denotes $V (ε, z)$ , $β = \underline{g} min {d_{1}, d_{2}} / 2$ , and $Θ$ is an unknown positive constant.

Proof

See the “Proof of Proposition 1” subsection later.

Main results

By the definition of $\overset{\cdot}{L}$ , we can see that $L (t) \geq 1$ is true on the interval of existence, and $\overset{\cdot}{L}$ is locally Lipschitz in $(x_{1}, {\hat{x}}_{1}, L)$ . We can see that the right-hand side of the closed-loop system is locally Lipschitz with respect to $(x, \hat{x}, L)$ in an open neighborhood of the initial condition. Therefore, by the theorem of existence and uniqueness of solution,³¹ the resulting closed-loop system has a unique solution on a small interval $[0, t_{f})$ . Meanwhile, by the theorem of continuation of solutions,³¹ the unique solution of the closed-loop system can be extended to the maximal interval $[0, T_{f})$ , where $T_{f} < \infty$ means $lim_{t \to T_{f}} (∥ x (t) ∥ + ∥ \hat{x} (t) ∥ + L (t)) = + \infty$ . We are now ready to give the main results of this article.

Theorem 1

Consider system equation (1) satisfying Assumptions 1 and 2 if the designed functions $h_{i} (y), i = 1, \dots, n + 1$ and $k_{i} (y), i = 1, \dots, n$ are chosen to satisfy equation (7), then, based on observer equation (2) and update law equation (3), the adaptive output-feedback controller equation (4) guarantees that for any initial condition, the solution of the closed-loop system is unique and bounded on $[0, + \infty)$ , and $lim_{t \to + \infty} (x (t), {\hat{x}}_{[n]} (t), {\hat{x}}_{n + 1} (t), u (t)) = (0, 0, ω, - ω)$ .

Before proving this theorem, we first propose the following lemma, whose proof is given in the “Proof of Lemma 1” subsection for compactness.

Lemma 1

For any initial condition of system equation (1), the controller equation (4) guarantees that the gain $L (t)$ and the states $ε (t) and z (t)$ are bounded on $[0, T_{f})$ , and

\int_{0}^{T_{f}} (∥ ε ∥^{2} + ∥ z ∥^{2}) dt < + \infty

(10)

Proof of Theorem 1. For any initial condition $(x_{0}, {\hat{x}}_{0})$ , let $(x (t), \hat{x} (t), L (t))$ be the unique solution of the resulting closed-loop system on the maximal interval of existence $[0, T_{f})$ . From Lemma 1, we know that $L (t), ε (t)$ , and $z (t)$ are bounded on $[0, T_{f})$ , and equation (10) is true. Noting equation (5), we have

{\begin{matrix} x_{i} = L^{i} (ε_{i} + z_{i}), i = 1, \dots, n, \\ {\hat{x}}_{i} = L^{i} z_{i}, i = 1, \dots, n, \\ {\hat{x}}_{n + 1} = ω - L^{n + 1} ε_{n + 1} \end{matrix}

(11)

We can deduce that $x (t)$ and $\hat{x} (t)$ are bounded on $[0, T_{f})$ . Hence $T_{f} = + \infty$ , which can be done by a contradiction argument. Assuming that $T_{f}$ is finite, which means that the maximal interval of existence is finite. Then, at least one of $x (t)$ and $\hat{x} (t)$ is unbounded on $[0, T_{f})$ , which leads to a contradiction. Using equations (3) and (6), Assumption 1 and the boundedness of $L (t)$ , $ε (t)$ , and $z (t)$ , we have that $\overset{\cdot}{ε} (t)$ and $\overset{\cdot}{z} (t)$ are bounded on $[0, + \infty)$ . Meanwhile, combined with the Barbălat Lemma,³² we obtain $lim_{t \to + \infty} (ε (t), z (t)) = (0, 0)$ . Furthermore, by equation (11), we can deduce that $lim_{t \to + \infty} (x (t), {\hat{x}}_{[n]} (t), {\hat{x}}_{n + 1} (t)) = (0, 0, ω)$ . In addition, using the definition of $u$ , we get that $lim_{t \to + \infty} u (t) = - ω$ .

Proofs of Proposition 1 and Lemma 1

In this section, we will give the detailed proofs of Proposition 1 and Lemma 1.

Proof of Proposition 1

Keeping in mind that $V (ε, z) = V_{1} (ε) + m V_{2} (z) = ε^{T} P ε + m z^{T} Qz$ and using equation (6), we have

\begin{matrix} \overset{\cdot}{V} (t) \leq - d_{1} L | g_{1} (y) | \cdot ∥ ε ∥^{2} - m d_{2} L | g_{1} (y) | \cdot ∥ z ∥^{2} \\ + 2 ε^{T} PF + 2 mL ε_{1} z^{T} Q H_{n} (y) \end{matrix}

(12)

By equation (8), the last term on the right-hand side of the above inequality satisfies

\begin{matrix} 2 mL ε_{1} z^{T} Q H_{n} (y) \leq 2 mL | ε_{1} | \cdot ∥ z ∥ \cdot ∥ Q ∥ \cdot \bar{h} | g_{1} (y) | \\ \leq \frac{m d_{2}}{2} L | g_{1} (y) | ∥ z ∥^{2} \\ + \frac{2 m {\bar{h}}^{2} ∥ Q ∥^{2}}{d_{2}} L | g_{1} (y) | ∥ ε ∥^{2} \end{matrix}

(13)

From Assumption 2, we obtain

\begin{matrix} \frac{| f_{i} |}{L^{i}} \leq \frac{θ (| x_{1} | + \dots + | x_{i} |)}{L^{i}} \\ \leq \frac{θ (| e_{1} | + | {\hat{x}}_{1} | + \dots + | e_{i} | + | {\hat{x}}_{i} |)}{L^{i}} \\ \leq θ \sqrt{n} (∥ ε ∥ + ∥ z ∥) \end{matrix}

Thus,

\begin{matrix} 2 ε^{T} PF \leq 2 n \sqrt{n} θ ∥ P ∥ \cdot ∥ ε ∥ \cdot (∥ ε ∥ + ∥ z ∥) \\ \leq 4 n \sqrt{n} θ ∥ P ∥ (∥ ε ∥^{2} + ∥ z ∥^{2}) \end{matrix}

(14)

Substituting equations (13) and (14) into equation (12), noting $m = (d_{1} \cdot d_{2}) / (4 {\bar{h}}^{2} ∥ Q ∥^{2})$ , and using Assumption 1, we arrive that

\begin{matrix} \overset{\cdot}{V} (t) \leq - d_{1} L | g_{1} (y) | \cdot ∥ ε ∥^{2} - m d_{2} L | g_{1} (y) | \cdot ∥ z ∥^{2} \\ + 4 n \sqrt{n} θ ∥ P ∥ (∥ ε ∥^{2} + ∥ z ∥^{2}) \\ + \frac{2 m {\bar{h}}^{2} ∥ Q ∥^{2}}{d_{2}} L | g_{1} (y) | \cdot ∥ ε ∥^{2} \\ + \frac{m}{2} d_{2} L | g_{1} (y) | \cdot ∥ z ∥^{2} \\ \leq - \frac{1}{2} d_{1} L | g_{1} (y) | \cdot ∥ ε ∥^{2} - \frac{1}{2} d_{2} L | g_{1} (y) | m ∥ z ∥^{2} \\ + 4 n \sqrt{n} θ ∥ P ∥ max {1, \frac{1}{m}} (∥ ε ∥^{2} + m ∥ z ∥^{2}) \\ \leq - \frac{\underline{g} min {d_{1}, d_{2}}}{2} L (∥ ε ∥^{2} + m ∥ z ∥^{2}) \end{matrix}

\begin{matrix} + 4 n \sqrt{n} θ ∥ P ∥ max {1, \frac{1}{m}} (∥ ε ∥^{2} + m ∥ z ∥^{2}) \\ \leq - β (L - Θ) (∥ ε ∥^{2} + m ∥ z ∥^{2}) \end{matrix}

(15)

where $Θ = 8 n \sqrt{n} θ ∥ P ∥ \max {1 / \underline{g} d_{1}, 1 / \underline{g} d_{2}, 1 / \underline{g} m d_{1}, 1 / \underline{g} m d_{2}}$ . The proof of Proposition 1 is completed.

Proof of Lemma 1

Boundedness of the gain $L (t)$ on the maximally extended interval $[0, T_{f})$ Assuming that $lim_{t \to T_{f}} L (t) = + \infty$ . Then, there exists a finite time $t_{1} \in [0, T_{f})$ such that

L (t) \geq 2 Θ, \forall t \in [t_{1}, T_{f})

(16)

From equation (9), we get

\begin{matrix} \overset{\cdot}{V} (t) \leq - β (L - Θ) (∥ ε ∥^{2} + m ∥ z ∥^{2}) \\ \leq - \frac{β}{2} L (∥ ε ∥^{2} + m ∥ z ∥^{2}) \\ \leq - \frac{β L}{2 max {λ_{max} (P), λ_{max} (Q)}} V \end{matrix}

(17)

which follows that $V (t)$ is bounded on $[t_{1}, T_{f})$ and $V (t) \leq V (t_{1}), \forall t \in [t_{1}, T_{f})$ . Recalling the definition of $V (t)$ , it implies that $ε_{1} (t)$ and $z_{1} (t)$ are bounded on $[t_{1}, T_{f})$ . From the definition of $\overset{\cdot}{L}$ , we have

\begin{matrix} \int_{t_{1}}^{T_{f}} ρ (ε_{1}, z_{1}) L (t) ε_{1}^{2} (t) dt = \int_{t_{1}}^{T_{f}} \overset{\cdot}{L} (t) dt \\ = L (T_{f}) - L (t_{1}) = + \infty \end{matrix}

By equation (17), we arrive that

\begin{matrix} \frac{β}{2} \int_{t_{1}}^{T_{f}} L (t) (∥ ε (t) ∥^{2} + m ∥ z (t) ∥^{2}) dt \\ \leq V (t_{1}) - V (T_{f}) < + \infty \end{matrix}

which is a contradiction.

Furthermore, $L (t)$ is continuous on the interval $[0, t_{1})$ . Therefore, $L (t)$ is well defined and bounded on $[0, T_{f})$ . Finally, we denote $lim_{t \to T_{f}} L (t) = L_{f}$ .

Boundedness of the states $ε (t)$ and $z (t)$ on the maximally extended interval $[0, T_{f})$ . First of all, we show that $z (t)$ is bounded on the interval $[0, T_{f})$ .

We define $V_{2} (z) = z^{T} Qz$ . From equations (6) and (7), we have

{\overset{\cdot}{V}}_{2} (z) \leq - d_{2} L | g_{1} (y) | \cdot ∥ z ∥^{2} + 2 L ε_{1} z^{T} Q H_{n} (y)

(18)

Moreover, noting that

\begin{matrix} 2 L ε_{1} z^{T} Q H_{n} (y) \leq 2 L | ε_{1} | \cdot ∥ z ∥ \cdot ∥ Q ∥ \cdot \bar{h} | g_{1} (y) | \\ \leq \frac{d_{2}}{2} L | g_{1} (y) | \cdot ∥ z ∥^{2} \\ + \frac{2 {\bar{h}}^{2} ∥ Q ∥^{2}}{d_{2}} L | g_{1} (y) | ε_{1}^{2} \end{matrix}

(19)

From Assumption 1 and $L (t) \geq 1$ , we can see that

\begin{matrix} g_{1} (y) \leq \bar{g} (1 + | y |^{p}) \\ \leq \bar{g} L^{p} (1 + | ε_{1} + z_{1} |^{p}) \\ \leq \bar{g} L^{p} (1 + {(1 + {(ε_{1} + z_{1})}^{2})}^{\frac{p}{2}}) \\ \leq \bar{g} L^{p} ρ (ε_{1}, z_{1}) \end{matrix}

(20)

Substituting equations (19) and (20) into equation (18), we obtain

\begin{matrix} {\overset{\cdot}{V}}_{2} (z) \leq - \frac{d_{2}}{2} L | g_{1} (y) | \cdot ∥ z ∥^{2} + \frac{2 {\bar{h}}^{2} ∥ Q ∥^{2}}{d_{2}} L | g_{1} (y) | ε_{1}^{2} \\ \leq - \frac{\underline{g} d_{2}}{2} ∥ z ∥^{2} + \frac{2 \bar{g} {\bar{h}}^{2} ∥ Q ∥^{2}}{d_{2}} L^{p + 1} ρ (ε_{1}, z_{1}) ε_{1}^{2} \\ = - \frac{\underline{g} d_{2}}{2} ∥ z ∥^{2} + \frac{2 \bar{g} {\bar{h}}^{2} ∥ Q ∥^{2}}{d_{2}} L^{p} \overset{\cdot}{L} \\ \leq - α_{1} V_{2} + α_{2} L^{p} \overset{\cdot}{L} \end{matrix}

(21)

with $α_{1} = \underline{g} d_{2} / 2 λ_{max} (Q)$ and $α_{2} = 2 \bar{g} {\bar{h}}^{2} ∥ Q ∥^{2} / d_{2}$ . Thus, we obtain

{\overset{\cdot}{V}}_{2} e^{α_{1} t} + α_{1} V_{2} e^{α_{1} t} \leq α_{2} L^{p} \overset{\cdot}{L} e^{α_{1} t}

Integrating both sides of the above inequality, we have

\int_{0}^{t} (V_{2} e^{α_{1} s})' ds \leq \int_{0}^{t} α_{2} L^{p} \overset{\cdot}{L} e^{α_{1} s} ds

that is, for all $t \in [0, T_{f})$ ,

\begin{matrix} V_{2} (z (t)) e^{α_{1} t} \leq V_{2} (z (0)) + α_{2} \int_{0}^{t} L^{p} (s) e^{α_{1} s} dL (s) \\ \leq V_{2} (z (0)) + \frac{α_{2}}{p + 1} L_{f}^{p + 1} e^{α_{1} t} \end{matrix}

Furthermore, we get

V_{2} (z (t)) \leq V_{2} (z (0)) e^{- α_{1} t} + \frac{α_{2}}{p + 1} L_{f}^{p + 1}

(22)

Therefore, $z (t)$ is bounded on the interval $[0, T_{f})$ . Besides, we can deduce that $\int_{0}^{T_{f}} ∥ z (t) ∥^{2} dt < + \infty$

Next, we will prove the boundedness of $ε (t)$ on the interval $[0, T_{f})$ . We introduce the coordinate transformation

ξ_{i} = \frac{ε_{i}}{{(L^{*})}^{i}}, i = 1, 2, \dots, n + 1

(23)

where $L^{*}$ is a positive constant satisfying $L^{*} = max {16 n \sqrt{n} θ ∥ p ∥ / d_{1} \underline{g}, 1}$ .

By equation (6), we have

{\begin{matrix} {\overset{\cdot}{ξ}}_{i} = L^{*} L g_{i} (y) ξ_{i + 1} - h_{i} (y) L^{*} L ξ_{1} + h_{i} (y) L^{*} L ξ_{1} \\ - h_{i} (y) \frac{L}{{(L^{*})}^{i - 1}} ξ_{1} - i \frac{\overset{\cdot}{L}}{L} ξ_{i} + \frac{f_{i} (t, x)}{{(L^{*} L)}^{i}}, \\ i = 1, 2, \dots, n, \\ {\overset{\cdot}{ξ}}_{n + 1} = - h_{n + 1} (y) L^{*} L ξ_{1} + h_{n + 1} (y) L^{*} L ξ_{1} \\ - h_{n + 1} (y) \frac{L}{{(L^{*})}^{n}} ξ_{1} - (n + 1) \frac{\overset{\cdot}{L}}{L} ξ_{n + 1} \end{matrix}

which can be written as

\begin{matrix} \overset{\cdot}{ξ} = L^{*} LA (y) ξ + L^{*} L H_{n + 1} (y) ξ_{1} - R^{*} L H_{n + 1} (y) ξ_{1} \\ - D_{n + 1} \frac{\overset{\cdot}{L}}{L} ξ + F^{*} (t, x, L, L^{*}) \end{matrix}

(24)

where

ξ = [ξ_{1}, \dots, ξ_{n + 1}]^{T}, R^{*} = diag {1, \frac{1}{L^{*}}, \dots, \frac{1}{{(L^{*})}^{n}}},

and

F^{*} (t, x, L, L^{*}) = {[\frac{f_{1} (t, x)}{L^{*} L}, \dots, \frac{f_{n} (t, x)}{{(L^{*} L)}^{n}}, 0]}^{T} .

Let $V_{3} (ξ) = ξ^{T} P ξ$ . By equations (6), (7), and (24), we obtain

\begin{matrix} {\overset{\cdot}{V}}_{3} (ξ) \leq - d_{1} | g_{1} (y) | L^{*} L ∥ ξ ∥^{2} + 2 L^{*} L ξ^{T} P H_{n + 1} (y) ξ_{1} \\ - 2 L ξ^{T} P R^{*} H_{n + 1} (y) ξ_{1} - d_{3} \frac{\overset{\cdot}{L}}{L} ∥ ξ ∥^{2} \\ + 2 ξ^{T} P F^{*} \\ \leq - d_{1} | g_{1} (y) | L^{*} L ∥ ξ ∥^{2} + 2 L^{*} L ξ^{T} P H_{n + 1} (y) ξ_{1} \\ - 2 L ξ^{T} P R^{*} H_{n + 1} (y) ξ_{1} + 2 ξ^{T} P F^{*} \end{matrix}

(25)

Noting that

\begin{matrix} 2 L^{*} L ξ^{T} P H_{n + 1} (y) ξ_{1} \leq 2 L^{*} L | ξ_{1} | \cdot ∥ P ∥ \cdot ∥ ξ ∥ \bar{h} | g_{1} (y) | \\ \leq \frac{d_{1}}{4} | g_{1} (y) | L^{*} L ∥ ξ ∥^{2} \\ + \frac{4 {\bar{h}}^{2} ∥ P ∥^{2}}{d_{1} L^{*}} | g_{1} (y) | L ε_{1}^{2}, \end{matrix}

\begin{matrix} - 2 L ξ^{T} P R^{*} H_{n + 1} (y) ξ_{1} \leq \frac{d_{1}}{4} | g_{1} (y) | L^{*} L ∥ ξ ∥^{2} \\ + \frac{4 {\bar{h}}^{2} ∥ P ∥^{2}}{d_{1} {(L^{*})}^{3}} | g_{1} (y) | L ε_{1}^{2}, \end{matrix}

and

\begin{matrix} 2 ξ^{T} P F^{*} \leq 2 n \sqrt{n} θ ∥ P ∥ \cdot ∥ ξ ∥ \cdot (∥ ξ ∥ + ∥ z ∥) \\ \leq n \sqrt{n} θ ∥ P ∥ (4 ∥ ξ ∥^{2} + ∥ z ∥^{2}) . \end{matrix}

Substituting the above inequalities into equation (25), and noting $L^{*} L \geq 16 n \sqrt{n} θ ∥ p ∥ / d_{1} \underline{g}$ , we obtain

\begin{matrix} {\overset{\cdot}{V}}_{3} (ξ) \leq - \frac{d_{1}}{2} | g_{1} (y) | L^{*} L ∥ ξ ∥^{2} + n \sqrt{n} θ ∥ P ∥ \\ (4 ∥ ξ ∥^{2} + ∥ z ∥^{2}) \\ + (\frac{4 {\bar{h}}^{2} ∥ P ∥^{2}}{d_{1} L^{*}} + \frac{4 {\bar{h}}^{2} ∥ P ∥^{2}}{d_{1} {(L^{*})}^{3}}) | g_{1} (y) | \times L ε_{1}^{2} \\ \leq - \frac{d_{1} \underline{g}}{4} L^{*} L ∥ ξ ∥^{2} + n \sqrt{n} θ ∥ P ∥ ∥ z ∥^{2} \\ + (\frac{4 {\bar{h}}^{2} ∥ P ∥^{2}}{d_{1} L^{*}} + \frac{4 {\bar{h}}^{2} ∥ P ∥^{2}}{d_{1} {(L^{*})}^{3}}) \bar{g} L^{p} ρ (ε_{1}, z_{1}) L ε_{1}^{2} \\ \leq - \frac{d_{1} \underline{g}}{4 λ_{max} (P)} V_{3} (ξ) + n \sqrt{n} θ ∥ P ∥ sup_{t \in [0, T_{f}]} ∥ z (t) ∥^{2} \\ + (\frac{4 {\bar{h}}^{2} ∥ P ∥^{2}}{d_{1} L^{*}} + \frac{4 {\bar{h}}^{2} ∥ P ∥^{2}}{d_{1} {(L^{*})}^{3}}) \bar{g} L^{p} \overset{\cdot}{L} \\ \leq - γ_{1} V_{3} (ξ) + γ_{2} L^{p} \overset{\cdot}{L} + γ_{3} \end{matrix}

(26)

where $γ_{1} = d_{1} \underline{g} / 4 λ_{max} (P)$ , $γ_{2} = ((4 {\bar{h}}^{2} ∥ P ∥^{2} / d_{1} L^{*}) + (4 {\bar{h}}^{2} ∥ P ∥^{2} / d_{1} (L^{*})^{3})) \bar{g}$ , and $γ_{3} = n \sqrt{n} θ ∥ P ∥ sup_{t \in [0, T_{f})} ∥ z (t) ∥^{2}$ , which implies that $ξ (t)$ is bounded on $[0, T_{f})$ . Thus, from the definition of $ξ (t)$ , we can see that $ε (t)$ is bounded on the maximally extended interval $[0, T_{f})$ , and $\int_{0}^{T_{f}} ∥ ε (t) ∥^{2} dt < + \infty$ .

Simulation examples

In this section, we give two examples to illustrate the validity of the theoretical results.

Example 1. Consider the following second-order nonlinear system

{\begin{matrix} {\overset{\cdot}{x}}_{1} = (1 + y^{2}) x_{2}, \\ {\overset{\cdot}{x}}_{2} = (1 + y^{2}) (u + ω) + θ (x_{1} + x_{2}), \\ y = x_{1} \end{matrix}

(27)

It can be verified that system equation (27) satisfies Assumption 1 with $p = 2$ and Assumption 2 with $θ = 0.8$ and $ω = 1.5$ .

We choose $h_{1} (y) = h_{2} (y) = 3 (1 + y^{2}), h_{3} (y) = (1 + y^{2}),$ $k_{1} (y) = 1.3 (1 + y^{2})$ , and $k_{2} (y) = 1.5 (1 + y^{2})$ . There exist $P$ and $Q$ defined as

P = [\begin{matrix} 7 & - 3 & 1 \\ - 3 & 2 & - 1.3 \\ 1 & - 1.3 & 3 \end{matrix}], Q = [\begin{matrix} 2.8 & 0.2 \\ 0.2 & 2.1 \end{matrix}] .

$d_{1} = 0.1, d_{2} = d_{3} = 0.2$ , and $d_{4} = 1$ that make equation (7) holds. According to the design scheme in the “Output-feedback controller design” section, an adaptive output-feedback controller of system equation (27) can be obtained as follows

{\begin{matrix} u = - 1.3 L^{2} {\hat{x}}_{1} - 1.5 L {\hat{x}}_{2} - {\hat{x}}_{3}, \\ {\begin{matrix} \overset{\cdot}{\hat{x_{1}}} = (1 + y^{2}) {\hat{x}}_{2} + 3 (1 + y^{2}) L (x_{1} - {\hat{x}}_{1}), \\ \overset{\cdot}{\hat{x_{2}}} = (1 + y^{2}) ({\hat{x}}_{3} + u) + 3 (1 + y^{2}) L^{2} (x_{1} - {\hat{x}}_{1}), \\ \overset{\cdot}{\hat{x_{3}}} = (1 + y^{2}) L^{3} (x_{1} - {\hat{x}}_{1}), \\ \overset{\cdot}{L} (t) = (\frac{2}{L} + \frac{x_{1}^{2}}{L^{3}}) {(x_{1} - {\hat{x}}_{1})}^{2}, L (0) = 1 \end{matrix} \end{matrix}

(28)

For simulation, we set the initial conditions of the closed-loop system by $x_{1} (0) = 3.3, x_{2} (0) = - 4, {\hat{x}}_{1} (0) = - 3.6, {\hat{x}}_{2} (0) = 3$ , and ${\hat{x}}_{3} (0) = 1.8$ . Then, we obtain the trajectories of all signals of the closed-loop system composed of equations (27) and (28), as shown in Figures 1 –5. Specifically, Figures 1 and 2 show that $x_{1} (t)$ and $x_{2} (t)$ converge to zero, thus achieving the desired goal.

Figure 1.

The trajectories of states $x_{1}$ and ${\hat{x}}_{1}$ .

Figure 2.

The trajectories of states $x_{2}$ and ${\hat{x}}_{2}$ .

Figure 3.

The trajectory of state ${\hat{x}}_{3}$ .

Figure 4.

The trajectory of gain $L$ .

Figure 5.

The trajectory of $u$ .

Example 2. Consider the nonzero set-point regulation of the following controlled pendulum system

ml \overset{\cdot\cdot}{ξ} + kl \overset{\cdot}{ξ} + mg \sin ξ = u

(29)

where $m and l$ are the mass and length of the rod, respectively; $k$ and $g$ are the friction coefficient and the acceleration of gravity; $ξ$ is the angle subtended by the rod and the vertical axis through the pivot point. Suppose that only $ξ$ is measurable.

The regulating objective is to make $ξ$ converge to a specified nonzero constant $ξ_{0}$ . Let $η_{1} = ξ - ξ_{0}$ and $η_{2} = \overset{\cdot}{ξ} + (k / m) (ξ - ξ_{0})$ . System equation (29) can be written as

{\begin{matrix} {\overset{\cdot}{η}}_{1} = η_{2} - \frac{k}{m} η_{1}, \\ {\overset{\cdot}{η}}_{2} = \frac{1}{ml} (u - mg \sin ξ_{0}) - \frac{g}{l} (\sin (η_{1} + ξ_{0}) - \sin ξ_{0}), \\ y = η_{1} \end{matrix}

(30)

where $- mg \sin ξ_{0}$ is the input matching uncertainty of system equation (30). Thus, the regulation of system equation (29) can be achieved by solving the global output-feedback state regulation of system equation (30).

By setting $θ = max {(k / m), (g / l)}$ , Assumptions 1 and 2 hold. Then, an adaptive output-feedback controller of system equation (30) can be designed as follows

{\begin{matrix} u = - 3 ml L^{2} {\hat{η}}_{1} - 3 mlL {\hat{η}}_{2} - {\hat{η}}_{3}, \\ {\begin{matrix} \overset{\cdot}{\hat{η_{1}}} = {\hat{η}}_{2} + 2 L (η_{1} - {\hat{η}}_{1}), \\ \overset{\cdot}{\hat{η_{2}}} = \frac{1}{ml} {\hat{η}}_{3} + \frac{1}{ml} u + 3 L^{2} (η_{1} - {\hat{η}}_{1}), \\ \overset{\cdot}{\hat{η_{3}}} = L^{3} (η_{1} - {\hat{η}}_{1}), \\ \overset{\cdot}{L} (t) = (\frac{2}{L} + \frac{η_{1}^{2}}{L^{3}}) {(η_{1} - {\hat{η}}_{1})}^{2}, L (0) = 1 \end{matrix} \end{matrix}

(31)

The matrices $P$ and $Q$ can be obtained by

P = [\begin{matrix} 2 & - 0.2 & - 0.05 \\ - 0.2 & 0.6 & - 0.7 \\ - 0.05 & - 0.7 & 4 \end{matrix}], Q = [\begin{matrix} 1 & 0.06 \\ 0.06 & 0.3 \end{matrix}],

and equation (7) holds when $d_{1} = d_{2} = 0.1, d_{3} = 0.2$ , and $d_{4} = 1$ .

Let $m = 0.25, l = 4, k = 0.25, g = 10$ , and $ξ_{0} = π / 4$ . Choose the initial conditions as $η_{1} (0) = (16 - 5 π) / 20,$ $η_{2} (0) = (36 + 10 π) / 40, {\hat{η}}_{1} (0) = 0, {\hat{η}}_{2} (0) = 1$ , and ${\hat{η}}_{3} (0) = 0.8$ . Using MATLAB, Figures 6–10 are obtained to show the trajectories of the closed-loop system. From Figure 6, we can see that $η_{1} (t)$ converges to zero, and in other words, $ξ$ converges to $ξ_{0}$ .

Figure 6.

The trajectories of states $η_{1}$ and ${\hat{η}}_{1}$ .

Figure 7.

The trajectories of states $η_{2}$ and ${\hat{η}}_{2}$ .

Figure 8.

The trajectory of state ${\hat{η}}_{3}$ .

Figure 9.

The trajectory of gain $L$ .

Figure 10.

The trajectory of $u$ .

Conclusion

In this article, the global state regulation of a class of uncertain nonlinear systems with generalized control coefficients is achieved by adaptive output feedback. The key is to deal with the nonlinearities and serious unknowns of the system caused by the function control coefficients, the unknown constant growth rate, and input matching uncertainty. It is worth noting that the growth rate of the system is only an unknown constant. Apparently, the current approach is inapplicable to the case where the growth rate of the system is the polynomial function of the output, for which the exploration can be attempted in the future. In addition, we will try to investigate the adaptive event-triggered output-feedback problem for nonlinear systems with input matching uncertainty and function control coefficients.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant no. 62073153) and Natural Science Foundation of Shandong Province (grant no. ZR2021MF004).

ORCID iD

Shaoli Jin

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