An Output Feedback Stabilizer for MIMO Nonlinear Systems with Uncertain Input Gain: Nonlinear Nominal Model

TL;DR

This paper introduces an output feedback stabilizer for MIMO nonlinear systems with uncertain input gain, capable of handling nonlinear gain functions and ensuring semi-global practical stability.

Contribution

It extends existing controllers to systems with nonlinear, state-dependent input gain matrices by modifying disturbance observer-based output feedback control.

Findings

The proposed controller achieves semi-global practical stability.

It effectively handles nonlinear, state-dependent input gain matrices.

The method exploits system nonlinearity rather than canceling it.

Abstract

This paper deals with the output feedback stabilization problem of nonlinear multi-input multi-output systems having an uncertain input gain matrix. It is assumed that the system has a well-defined vector relative degree and that the zero dynamics is input-to-state stable. Based on the assumption that there exists a state feedback controller which globally asymptotically stabilizes the origin of the nominal closed-loop system, we present an output feedback stabilizer which recovers the stability of the nominal closed-loop system in the semi-global practical sense. Compared to previous results, we allow that the nominal system can have a nonlinear input gain matrix that is a function of state and this is done by modifying the structure of the disturbance observer-based robust output feedback controller. It is expected that the proposed controller can be well applied to the case when the system's nonlinearity is to be exploited rather than canceled.

Figures

Equations

$$\begin{array}{rl}\dot{z}& =F_0(z,x)\\ \dot{x}& =Ax+B(F(z,x)+G(z,x,t)u)\\ y& =Cx\end{array}$$

$$A$$

$$B$$

$$C$$

$$A_i$$

$$\begin{array}{rl}\dot{\bar{z}}& =F_0(\bar{z},\bar{x})\\ \dot{\bar{x}}& =A\bar{x}+B(F(\bar{z},\bar{x})+\bar{G}(\bar{z},\bar{x})u_r)\end{array}$$

$$\Vert z(t)\Vert \le {\beta }_z(\Vert z(0)\Vert ,t)+{\gamma }_z\left(\underset{0\le s\le t}{\sup }\Vert u_z(s)\Vert \right),\forall t\ge 0.\diamond$$

$$\begin{split}&\alpha_{1}\left(\left\|[\bar{z};\bar{x}]\right\|\right)\leq\bar{V}(\bar{z},\bar{x})\leq\alpha_{2}\left(\left\|[\bar{z};\bar{x}]\right\|\right)\\ &\begin{multlined}\frac{\partial\bar{V}}{\partial\bar{x}}[A\bar{x}+B(F(\bar{z},\bar{x})+\bar{G}(\bar{z},\bar{x})U_{r}(\bar{z},\bar{x})]\\ +\frac{\partial\bar{V}}{\partial\bar{z}}F_{0}(\bar{z},\bar{x})\!\leq\!-\alpha_{3}(\|\bar{z};\bar{x}\|)\end{multlined}\frac{\partial\bar{V}}{\partial\bar{x}}[A\bar{x}+B(F(\bar{z},\bar{x})+\bar{G}(\bar{z},\bar{x})U_{r}(\bar{z},\bar{x})]\\ +\frac{\partial\bar{V}}{\partial\bar{z}}F_{0}(\bar{z},\bar{x})\!\leq\!-\alpha_{3}(\|\bar{z};\bar{x}\|)\end{split}$$

$$\Vert I-G(z,x,t){\bar{G}}^{-1}(z,x)\Vert \le \mu <1,\forall (z,x)\in {\mathbb{R}}^n,\forall t.\diamond$$

$$\begin{array}{rl}\dot{\bar{z}}& =F_0(\bar{z},\varphi (q))\\ \dot{q}& =A_{a\tau }q+B_{a\tau }^{q}y\\ \dot{p}& =A_{a\tau }p+B_{a\tau }^p\bar{G}(\bar{z},\varphi (q))u\\ w& =Cp+B^{\top }{B}_{a\tau }^q(Cq-y)+F(\bar{z},\varphi (q))\\ u& ={\bar{G}}^{-1}(\bar{z},\varphi (q))\Phi (w)+U_r(\bar{z},\varphi (q))\end{array}$$

$$A_{a_i\tau }$$

$$B_{a_i\tau }^q=\left[\begin{array}{c}\frac{a_{i{\nu }_i}}{\tau };\cdots ;\frac{a_{i1}}{{\tau }^{{\nu }_i}}\end{array}\right],B_{a_i\tau }^p=\left[\begin{array}{c}0;\cdots ;0;\frac{a_{i1}}{{\tau }^{{\nu }_i}}\end{array}\right].$$

$$s^{{\nu }_i-1}+a_{i{\nu }_i}{s}^{{\nu }_i-2}+\cdots +a_{i2}=0$$

$$H_i(s)=\frac{1}{s}\frac{a_{i1}}{s^{{\nu }_i-1}+a_{i{\nu }_i}{s}^{{\nu }_i-2}+\cdots +a_{i2}}$$

$$\begin{array}{rl}(\bar{z},\varphi (x))& =(\bar{z},x),\forall (\bar{z},x)\in {\Omega }_{(\bar{z},x)}(\bar{l})\\ \Phi (w)& =w,\forall w\in {\Omega }_w\\ \Vert \frac{\partial \varphi }{\partial x}(x)\Vert & \le 1,\forall x\in {\mathbb{R}}^{\nu }\mathrm{and}\Vert \frac{\partial \Phi }{\partial w}(w)\Vert \le 1,\forall w\in {\mathbb{R}}^m\end{array}$$

$${\Omega }_w$$

$$\displaystyle\quad\quad\bar{G}(\bar{z},x)G^{-1}(z,x,t)\big{[}F(\bar{z},x)-F(z,x)$$

$$\displaystyle\hskip 56.9055pt+(\bar{G}(\bar{z},x)-G(z,x,t))U_{r}(\bar{z},x)\big{]}+\tilde{\omega},$$

$$\Vert z\Vert \in l_z,(\bar{z},x)\in {\Omega }_{(\bar{z},x)}(l),$$

$$\displaystyle\hskip 56.9055pt{\rm all\ possible\ }G,\|\tilde{\omega}\|\leq\delta_{w}+L_{F}\delta_{1}\Big{\}}.$$

$$\Vert x(t)\Vert \le ϵ,\forall t\ge T_{ϵ}.$$

$$\xi$$

$$\eta$$

$$\begin{array}{rl}{\xi }_{ij}& =\frac{1}{{\tau }^{{\nu }_i-j}}(q_{ij}-x_{ij}),{\eta }_{ij}={\tau }^{j-1}(p_{i1}^{(j-1)}-q_{i{\nu }_i}^{(j)}).\end{array}$$

$$\begin{array}{rl}\tau \dot{\xi }& =A_{\xi }\xi -\tau B{\Theta }_{\xi }\\ \tau \dot{\eta }& =A\eta +B{\Theta }_{\eta }\end{array}$$

$$\displaystyle A_{\xi}=A_{a\tau}\big{|}_{\tau=1}$$

$${\Theta }_{\xi }(z,x,\bar{z},\xi ,\eta ,t)=F(z,x)+G(z,x,t)U_r(\bar{z},\varphi (x+\Delta \xi ))$$

$$+G(z,x,t){\bar{G}}^{-1}(\bar{z},\varphi (x+\Delta \xi ))\Phi ({\eta }_{[1]}+F(\bar{z},\varphi (x+\Delta \xi )))$$

$${\Theta }_{\eta }(z,x,\bar{z},\xi ,\eta ,t)=-\mathsf{a}\eta$$

$$+{\mathsf{a}}_{[1]}\{-F(z,x)+(\bar{G}(\bar{z},\varphi (x+\Delta \xi ))-G(z,x,t))u\}.\diamond$$

$$\begin{array}{rl}u& ={\bar{G}}^{-1}(\bar{z},\varphi (x+\Delta \xi ))\Phi ({\eta }_{[1]}+F(\bar{z},\varphi (x+\Delta \xi )))\\ & +U_r(\bar{z},\varphi (x+\Delta \xi )).\end{array}$$

$$\dot{\xi }$$

$$={\Delta }^{-1}\left\{A_{a\tau }q+B_{a\tau }^{q}Cx-Ax-B(F(z,x)+G(z,x,t)u)\right\}$$

$$\displaystyle\begin{multlined}=\Delta^{-1}\left\{(A-B_{a\tau}^{q}C)q-(A-B_{a\tau}^{q}C)x\right\}\\ \quad-\Delta^{-1}B(F(z,x)+G(z,x,t)u))\end{multlined}=\Delta^{-1}\left\{(A-B_{a\tau}^{q}C)q-(A-B_{a\tau}^{q}C)x\right\}\\ \quad-\Delta^{-1}B(F(z,x)+G(z,x,t)u))$$

$$={\Delta }^{-1}(A-B_{a\tau }^{q}C)\Delta \xi -{\Delta }^{-1}B{\Theta }_{\xi }$$

$$=\frac{1}{\tau }{A}_{\xi }\xi -B{\Theta }_{\xi }.$$

$$\tau {\dot{\eta }}_{ij}={\eta }_{i,j+1}.$$

$$p_{i1}^{({\nu }_i)}$$

Full text

I Introduction

In most practical control systems, only parts of system states can be measured. To control such systems, one can simply construct a static output feedback controller such as proportional controller, or design a dynamic output feedback controller such as proportional-integral controller. When the system model is available, we can construct a state observer to estimate the system state and use the state feedback controller with the estimated state. The latter idea is one of standard solutions for linear systems and the separation principle guarantees that the idea actually works [1].

The idea of combining a state feedback controller and a state estimator does not work well for general nonlinear systems and actually it is quite challenging as explained in books and papers including [2, 3, 4, 5]. Fortunately, it turns out that the separation principle holds true if we consider a (possibly very large) compact region of attraction rather than the whole state space and the basic idea is to employ a high-gain observer whose convergence speed can be tuned arbitrarily fast and devise a structure that prevents possibly very large overshoot of the observer from propagating to the system states; see the works, e.g., [6, 4, 7, 8, 9] for details.

The aim of this paper is to develop an output feedback stabilizing controller for multi-input multi-output (MIMO) nonlinear systems which contain uncertainties in the input gain matrix. Although uncertainties in the input gain have been considered in many studies including [10, 11, 12], most of them assume that state variables of the system are available for feedback or considered single input systems. In MIMO systems, the uncertainty in the input gain makes the control problem very hard because the actual direction of an input applied to the system as well as the magnitude can be different from what is expected while in the single input case the direction is fixed. When we consider the output feedback case, the problem becomes obviously more difficult.

Some relevant results on the problem are available in the literature. In [13], an adaptive output feedback controller is presented for MIMO nonlinear systems with input uncertainties. It is assumed that the system has an identical relative degree between inputs and outputs, and the input uncertainty is composed of an uncertain diagonal term that depends on the states and an unknown constant matrix. More recently, a robust output feedback stabilizer has been proposed for a partially feedback linearizable system having a vector relative degree and an uncertain input gain matrix in [14]. The uncertainty is assumed to be close to a constant known matrix (called ‘guess of input gain’) in some sense and an extended high gain observer is employed to estimate the high derivatives of outputs so that the effect of uncertainties can be estimated. A disturbance observer-based solution has been proposed in [15], where the uncertain input gain matrix is assumed to belong to a known sector and two filters with sufficiently high bandwidth work together to estimate the effect of uncertainties and external disturbances; see [16] for the idea of a disturbance observer, [17] for a recent survey, and [18] for a tutorial. It is shown that the real closed-loop system behaves like a nominal linear system for whole time horizon. As in the work [14], a constant gain matrix is introduced in [15] which can be regarded as a nominal gain or a target gain for the nominal closed-loop system.

In this paper, we would like to consider more general case where the nominal system has a nonlinear nominal gain matrix that depends on the system state rather than a constant one. In many systems such as robot manipulators and aerial vehicles, the input gain matrix depends on the posture or attitude of the system and thus the input gain is naturally nonlinear. If some uncertainties in system parameters such as length and mass are present, which is often the case, the input gain matrix becomes uncertain. In this case, if we apply control strategies that rely on a linear nominal model, then the discrepancy between the actual system and the nominal one will be considerably large, resulting unnecessarily large control effort to compensate for the effect of uncertainties. Thus, our motivation is to consider more realistic assumption on the systems and allow the compensated system to have possibly good intrinsic nonlinearities.

The contribution of this paper can be summarized as follows. Firstly, we present a new structure of the disturbance observer-based controller for MIMO nonlinear systems that have uncertainties in the input gain matrix whose nominal counterpart can be a nonlinear function. Secondly, a new type of the uncertainty on the input gain matrix involving the nonlinear nominal model is presented. Finally, it is shown that the separation principle holds in our case in the sense that the state feedback controller for the given nominal system and the observer or the filter that estimates system outputs and the effect of uncertainties can be designed separately.

The rest of this paper is organized as follows. The output feedback stabilization problem for MIMO nonlinear systems is formulated in Section II. A robust output feedback controller is proposed and the stability of the closed-loop system is analyzed in Section III. An example to validate the proposed method is given in Section IV. Finally, conclusions and outlooks are presented in Section V.

Notation: For two vectors $x\in\mathbb{R}^{n_{1}}$ and $y\in\mathbb{R}^{n_{2}}$, the concatenated vector $[x;y]$ is defined by $[x;y]=\begin{bmatrix}x^{\top}&y^{\top}\end{bmatrix}^{\top}$. Concatenation of multiple vectors (or scalars) are defined similarly. Given $n$ vectors $x_{1},\dots,x_{n}$ in $\mathbb{R}^{\nu}$, $x_{[i]}\in\mathbb{R}^{n}$ is defined by $x_{[i]}=\begin{bmatrix}x_{1i};x_{2i};\cdots;x_{ni}\end{bmatrix}$. For two sets $\Omega_{x}$ and $\Omega_{y}$, the Cartesian product of sets $\Omega_{x}\times\Omega_{y}$ is denoted by $\Omega_{(x,y)}$. For a $k$-times differentiable function $x(t)$, $x^{(k)}(t)$ denotes $\frac{d^{k}x}{dt^{k}}(t)$. The vector $0_{k}$ stands for the zero vector in ${\mathbb{R}}^{k}$.

II Problem Formulation

We consider a nonlinear system which has $m$ inputs and $m$ outputs and admits a well-defined vector relative degree $\{\nu_{1},\nu_{2},\dots,\nu_{m}\}$. This system can be written in the Byrnes-Isidori normal form [3]

[TABLE]

where $u\in{\mathbb{R}}^{m}$ is the control input, $y\in{\mathbb{R}}^{m}$ is the system output, and $x\in{\mathbb{R}}^{\nu}$ and $z\in{\mathbb{R}}^{n-\nu}$ are system states with $\nu=\nu_{1}+\cdots+\nu_{m}$. The matrices $A$, $B$, and $C$ are given by

[TABLE]

where

[TABLE]

For convenience, we decompose the state vector $x$ as $x=\begin{bmatrix}x_{1};x_{2};\cdots;x_{m}\end{bmatrix}$ with $x_{i}\in{\mathbb{R}}^{\nu_{i}}$ being given by $x_{i}=\begin{bmatrix}x_{i1};x_{i2};\dots;x_{i\nu_{i}}\end{bmatrix}$.

It is assumed that the functions $F_{0}(z,x)$ and $F(z,x)$ are smooth. We assume that, for any $(z,x,t)\in{\mathbb{R}}^{n+1}$, the input gain matrix $G(z,x,t)$ is smooth and invertible. It is noted that the uncertainty in $G$ is the main concern of this paper and we do not consider uncertainties in $F_{0}$ or $F$ for the sake of simplicity.

We introduce a nominal gain matrix $\bar{G}(z,x)$ which is a nominal function for $G(z,x,t)$ (hence known) and assume that it is smooth for any $(z,x)\in{\mathbb{R}}^{n}$. With $\bar{G}$, a nominal closed-loop system is defined by

[TABLE]

where $u_{r}$ is the control input that has been designed for the nominal system.

Assumption 1

The system $\dot{z}=F_{0}(z,u_{z})$ is input-to-state stable with respect to the input $u_{z}$ [19]. Precisely, for any initial state $z(0)\in\mathbb{R}^{n-\nu}$ and any piecewise continuous bounded input $u_{z}$, the solution $z(t)$ exists for all $t\geq 0$ and there exists a class $\mathcal{KL}$ function $\beta_{z}$ and a class $\mathcal{K}$ function $\gamma_{z}$ such that

[TABLE]

Assumption 2

The origin of the nominal system (2) can be made globally asymptotically stable by a state feedback control of the form $u_{r}=U_{r}(\bar{z},\bar{x})$. $\diamond$

By the converse Lyapunov theorem [2, Theorem 4.17], Assumption 2 ensures that there exists a smooth, positive definite function $\bar{V}(\bar{z},\bar{x})$ such that

[TABLE]

*where $\alpha_{1}$ and $\alpha_{2}$ are class $\mathcal{K}_{\infty}$ functions, and $\alpha_{3}$ is a class $\mathcal{K}$ function. *

The main concern of this paper is to deal with the uncertainty in the input gain matrix $G(z,x,t)$. We allow that the nominal input gain matrix, denoted by $\bar{G}(z,x)$, can be a nonlinear function of state. This can be regarded as a generalization of the assumption introduced in [14], saying that there exists a constant matrix ${\bf G}$ such that $\max_{\Lambda\,{\rm diagonal},|\Lambda|\leq 1}|[G(z,x)-{\bf G}]\Lambda{\bf G}^{-1}|<1,\forall(z,x)\in{\mathbb{R}}^{n}$.

Assumption 3

The uncertain input gain matrix $G(z,x,t)$ and its nominal one $\bar{G}(z,x)$ are close in the sense that

[TABLE]

The objective of this paper is to design a dynamic output-feedback controller which makes the origin of the closed-loop system semi-globally practically stable despite the uncertainty in the input gain matrix. By the semi-global practical stability, we mean that for any given initial condition sets $S_{z}$ and $S_{x}$ of $z$ and $x$, respectively, we can design the controller so that the trajectory of the closed-loop system is uniformly ultimately bounded and the size of the ultimate bound can be tuned arbitrarily.

III Main Result

In this section, we firstly propose a robust output feedback stabilizer with a design procedure and then analyze the stability of the closed-loop system.

III-A Proposed Controller

We present a controller given by

[TABLE]

where $\bar{z}\in\mathbb{R}^{n-\nu}$, $q\in\mathbb{R}^{\nu}$, and $p\in\mathbb{R}^{\nu}$ are internal states of the controller. Noting that $q$ and $p$ have the same dimension as $x$, we decompose $q$ and $p$ like $x$, e.g., $q=\begin{bmatrix}q_{1};\cdots;q_{m}\end{bmatrix}$ with $q_{i}=\begin{bmatrix}q_{i1};\cdots;q_{i\nu_{i}}\end{bmatrix}$.

The controller involves several matrices ($A_{a\tau}$, $B_{a\tau}^{q}$, $B_{a\tau}^{p}$) and functions ($\Phi$, $\phi$) to be determined and in what follows we explain how to construct them.

Firstly, we explain the dynamics of $q$ and $p$. They share the same system matrix given by $A_{a\tau}={\rm diag}\{A_{a_{1}\tau},\dots,A_{a_{m}\tau}\}$ where

[TABLE]

The input matrices $B_{a\tau}^{q}$ and $B_{a\tau}^{p}$ are given by $B_{a\tau}^{q}={\rm diag}\{B_{a_{1}\tau}^{q},\dots,B_{a_{m}\tau}^{q}\}$ and $B_{a\tau}^{p}={\rm diag}\{B_{a_{1}\tau}^{p},\dots,B_{a_{m}\tau}^{p}\}$ where

[TABLE]

The constants $a_{i1},\dots,a_{i\nu_{i}}$, and $\tau$ in the matrices are design parameters to be chosen later.

Remark 1

Note that the dynamics of $q$ can be decomposed into $\dot{q}_{i}=A_{a_{i}\tau}q_{i}+B_{a_{i}\tau}^{q}y_{i}$ and this is the well known high-gain observer which estimates $y_{i}$ and its time derivatives. In addition, the dynamics of $p_{i}$ is obtained as $\dot{p}_{i}=A_{a_{i}\tau}p_{i}+B_{a_{i}\tau}^{p}\bar{G}_{i}(\bar{z},\phi(q))u$ where $\bar{G}_{i}(\bar{z},\phi(q))$ is the $i$th row vector of $\bar{G}_{i}(\bar{z},\phi(q))$, and this subsystem takes $\bar{G}_{i}(\bar{z},\phi(q))u$ as the input, and the signal $p_{i1}=C_{i}p_{i}$ (can be regarded as the output of $p_{i}$-dynamics) together with the signal $B_{i}^{\top}B_{a_{i}\tau}^{q}(C_{i}q_{i}-y_{i})$ which is nothing but, $\dot{q}_{i\nu_{i}}$ is used to generate a feedback signal $w_{i}$. This signal can be regarded as an estimate of the effect of the discrepancy between $G$ and $\bar{G}$; see for example [15]. Fig. 1 shows the structure of the proposed controller and the flow of signals. Here, $P$ denotes the system (1). $\diamond$

Now, we provide the details on the design of parameters $a_{i1},\dots,a_{i\nu_{i}}$. For simplicity, let $a_{i}=\begin{bmatrix}a_{i1};a_{i2};\cdots;a_{i\nu_{i}}\end{bmatrix}$, $i=1,\dots,m$. For each $i$, choose $a_{i2},\dots,a_{i\nu_{i}}$ such that

[TABLE]

has all roots in the left half complex plane. With $a_{i2},\dots,a_{i\nu_{i}}$ fixed, we choose $a_{i1}$ as follows. Define $D(1-\mu,1+\mu)$ by the closed disk in the complex plane whose center and diameter are the midpoint of $-1/(1-\mu)+j0$ and $-1/(1+\mu)+j0$ and the distance between them, respectively. Let

[TABLE]

and find a positive constant $a_{i1}$ such that the Nyquist plot of $H_{i}(s)$ lies strictly outside of the disk $D(1-\mu,1+\mu)$ and does not surround the disk. Note that such $a_{i1}$ always exists since the Nyquist plot of $H_{i}(s)$ is bounded to the left half plane and sufficiently small $a_{i1}$ will reduce the size of the Nyquist plot so that the disk $D(1-\mu,1+\mu)$ does not touch the Nyquist plot, where $\mu$ is given in Assumption 3. From the Nyquist stability theory, $a_{i1}$ guarantees that all the roots of $s^{\nu_{i}}+a_{i\nu_{i}}s^{\nu_{i}-1}+\dots+a_{i2}s+a_{i1}=0$ belong to the left half complex plane.

We are now ready to determine the functions $\phi$ and $\Phi$. They are chosen as continuously differentiable saturation functions and the saturation levels are found considering the dynamics of the nominal system, the sizes of initial condition sets, and the Lyapunov function associated to the nominal closed-loop system.

To proceed, let $\delta_{\bar{z}}>0$, $\delta_{z}>0$, and $\delta_{x}>0$ be given and define $S_{\bar{z}}=\{\bar{z}\in{\mathbb{R}}^{n-\nu}|\|\bar{z}\|\leq\delta_{\bar{z}}\}$, $S_{z}=\{z\in{\mathbb{R}}^{n-\nu}|\|z\|\leq\delta_{z}\}$, and $S_{x}=\{x\in{\mathbb{R}}^{\nu}|\|x\|\leq\delta_{x}\}$. For a given constant $\delta>0$, define the level set \Omega_{(\bar{z},x)}(\delta)=\{[\bar{z};x]\in{\mathbb{R}}^{n}\big{|}\bar{V}(\bar{z},x)\leq\delta\} where $\bar{V}$ is the Lyapunov function from (3).

Suppose the initial conditions $\bar{z}(0)$, $z(0)$ and $x(0)$ belong to $S_{\bar{z}}$, $S_{z}$, $S_{x}$, respectively. Let $l>0$ be any constant such that the level set \Omega_{(\bar{z},x)}(l)=\{[\bar{z};x]\in{\mathbb{R}}^{n}\big{|}\bar{V}(\bar{z},x)\leq l\} strictly contains $S_{\bar{z}}\times S_{x}$ in a way that the boundaries of $\Omega_{(\bar{z},x)}(l)$ and $S_{\bar{z}}\times S_{x}$ are disjoint.

From Assumption 1, we have $\|z(t)\|\leq\beta_{z}(\delta_{z},0)+\gamma_{z}(\alpha_{1}^{-1}(l))=:l_{z},\forall t\geq 0$ provided that $(\bar{z}(t);x(t))\in\Omega_{\bar{z},x}(l)$, where $\alpha_{1}$ comes from (3). Let $\delta_{1}$ and $\delta_{w}$ be positive constants. Define $\bar{l}=\alpha_{2}(\alpha_{1}^{-1}(l)+\delta_{1})$ and let $L_{F}$ be the Lipschitz constant of $F(\bar{z},x)$ over $\Omega_{(\bar{z},x)}(\bar{l})$. With $\Omega_{(\bar{z},x)}(\cdot)$ and the bound of $z(t)$, we define differentiable saturation functions $\phi$ and $\Phi$ by

[TABLE]

where

[TABLE]

One can find the saturation level for $\Phi$ by computing the maximum of $w(\bar{z},z,x)$ over all possible $(\bar{z},z,x)$ and $G$ under consideration. In fact, any bounded saturation level which can cover the set $\Omega_{w}$ defined above will do the job.

Based on the discussion so far, we state the main stability result of this paper.

Theorem 1

Consider the closed-loop system (1) and (4). Let $S_{p}$ and $S_{q}$ be the compact sets containing the origin and assume that $p(0)\in S_{p}$ and $q(0)\in S_{q}$. Let $l>0$ be such that $S_{\bar{z}}\times S_{x}\subset\Omega_{(\bar{z},x)}(l)$ and the boundaries of $S_{\bar{z}}\times S_{x}$ and $\Omega_{(\bar{z},x)}(l)$ are disjoint. Under Assumptions 1-3, the controller (4) whose design parameters are chosen from the procedure described in this section ensures the following. For a given $\epsilon>0$, there exists $\tau^{*}>0$ and $T_{\epsilon}>0$ such that, for each $0<\tau<\tau^{*}$, the solution of the closed-loop system denoted by $[\bar{z}(t);z(t);x(t);q(t);p(t)]$ is uniformly bounded and the solution $x(t)$ satisfies

[TABLE]

$\square$**

III-B Stability analysis of the closed-loop system

In this subsection, a proof of Theorem 1 is provided. We start by rewriting the system in new coordinates $(\xi,\eta)$ defined by

[TABLE]

where $i=1,\dots,m$, and the $j$th components of $\xi_{i}$ and $\eta_{i}$ are given by

[TABLE]

For a given $\tau>0$, let $\Delta_{i}={\rm diag}\{\tau^{\nu_{i}-1},\dots,\tau,1\}$. With $\Delta_{i}$, the variable $\xi_{i}$ is expressed compactly as $\xi_{i}=\Delta_{i}^{-1}(q_{i}-x_{i})$, and $\xi$ can be written as $\xi=\Delta^{-1}(q-x)$ where $\Delta={\rm diag}\{\Delta_{1},\dots,\Delta_{m}\}$.

For convenience, we define $\mathsf{a}={\rm diag}\{a_{1}^{\top},\dots,a_{m}^{\top}\}$ and $\mathsf{a}_{[1]}={\rm diag}\{a_{11},\dots,a_{m1}\}$.

Lemma 1

The dynamics of $(\xi,\eta)$ is given by

[TABLE]

where

[TABLE]

Proof:

From the dynamics of $q$, one has $\dot{q}_{i\nu_{i}}=-\frac{a_{i1}}{\tau^{\nu_{i}}}(q_{i1}-y_{i})=-B_{i}^{\top}B_{a_{i}\tau}^{q}(C_{i}q_{i}-y_{i})$, and using this fact we can express the signal $w$ as $w=\eta_{[1]}+F(\bar{z},\phi(q))$. Then, the control input $u$ becomes

[TABLE]

The dynamics of $\xi$ is derived as

[TABLE]

Meanwhile, the dynamics of $\eta$ can be obtained as follows. For $j=1,\dots,\nu_{i}-1$, the definition of $\eta_{ij}$ results in

[TABLE]

To derive the dynamics of $\eta_{i\nu_{i}}$, we compute

[TABLE]

Using these relations, one has

[TABLE]

Collecting the dynamics of $\eta_{ij}$ obtained above, we have the dynamics of $\eta$ in (10). Hence, the proof is complete. ∎

With (10), the closed-loop system is rewritten as

[TABLE]

This equation (III-B) is the standard singular perturbation form with the time separation parameter $\tau$ [2]. If $\tau$ is sufficiently small, the fast variables $\xi$ and $\eta$ approach to their quasi-steady states $\xi^{*}$ and $\eta^{*}$ which are functions of slow variables such as $z$ and $x$.

Lemma 2

Suppose that $\|z\|\leq l_{z}$, $(\bar{z},x)\in\Omega_{(\bar{z},x)}(l)$. Then, the quasi-steady states $(\xi^{*},\eta^{*})$ of the $(\xi,\eta)$-dynamics is given by

[TABLE]

$\diamond$* *

Proof:

The quasi-steady states $(\xi^{*},\eta^{*})$ are determined from the right hand side of the $(\xi,\eta)$-dynamics in (III-B) with $\tau=0$, namely

[TABLE]

Since $A_{\xi}$ is Hurwitz, we have $\xi^{*}=0$. From the structure of $A$ and $B$, it follows that $\eta_{i2}^{*}=\dots=\eta_{i\nu_{i}}^{*}=0$, $i=1,\dots,m$. To find $\eta_{11}^{*},\dots,\eta_{m1}^{*}$, we consider the dynamics of $\eta_{1\nu_{1}},\dots,\eta_{m\nu_{m}}$. Substituting the results on the components of $(\xi^{*},\eta^{*})$ obtained above yields

[TABLE]

Since the parameters $a_{i}$ are coefficients of stable polynomials, all of them are positive numbers. Hence, $\mathsf{a}_{[1]}$ is invertible. Moreover, by construction of $\Phi$ and $\phi$, the assumptions $\|z\|\leq l_{z}$ and $(\bar{z},x)\in\Omega_{\bar{z},x}(l)$ ensure that $\phi(x)=x$ and $\Phi(\eta_{[1]}^{*}+F(\bar{z},\phi(x)))=\eta_{[1]}^{*}+F(\bar{z},x)$. From these facts, we have

[TABLE]

and one can obtain $\eta_{[1]}^{*}$ as in (13). Thus, the proof is complete. ∎

With $\tilde{\xi}=\xi-\xi^{*}$ and $\tilde{\eta}=\eta-\eta^{*}$, the dynamics (III-B) becomes

[TABLE]

where $\Theta_{\tilde{\xi}}(z,x,\bar{z},\tilde{\xi},\tilde{\eta},t)=\Theta_{\xi}(z,x,\bar{z},\tilde{\xi},\tilde{\eta}+\eta^{*},t)$ and $\Theta_{\tilde{\eta}}(z,x,\bar{z},\tilde{\xi},\tilde{\eta},t)=\Theta_{\eta}(z,x,\bar{z},\tilde{\xi},\tilde{\eta}+\eta^{*},t)$.

Lemma 3

The function $\Theta_{\tilde{\eta}}$ can be decomposed as

[TABLE]

where $\bar{\mathsf{a}}=\mathsf{a}-\mathsf{a}_{[1]}C$,

[TABLE]

and the function $\tilde{\Theta}_{\tilde{\eta}}(\tilde{\xi},t)$ is continuously differentiable with respect to its arguments and $\tilde{\Theta}_{\tilde{\eta}}(0,t)=0$. In addition, suppose $(\bar{z},x)\leq\Omega_{(\bar{z},x)}(l)$. Then, under Assumption 3, the function $\Psi_{t}$ belongs to the sector [1-\mu,1+\mu\big{]}, i.e., $(\Psi_{t}(\zeta,t)\!-\!(1-\mu)\zeta)^{\top}(\Psi_{t}(\zeta,t)\!-\!(1+\mu)\zeta)\leq 0$, $\forall\zeta\in\mathbb{R}^{m}$. $\diamond$

Proof:

Noting that $\Theta_{\tilde{\eta}}(z,x,\bar{z},0,0,t)=0$, one has

[TABLE]

One can easily compute that the second line of this equation becomes $-\bar{\mathsf{a}}\tilde{\eta}-a_{[1]}\Psi_{t}(\tilde{\eta}_{[1]},t)$ while the first line becomes zero when $\tilde{\xi}=0$ and differentiable with respect to $\tilde{\xi}$ and $t$.

To prove the property of $\Psi_{t}$, we compute

[TABLE]

From Assumption 3 and the facts that $\phi(x)=x$ and $\|(\partial\Phi)/(\partial\tilde{\eta}_{[1]})\leq 1\|$, it follows that

[TABLE]

Finally, we can obtain

[TABLE]

which implies that $\Psi_{t}$ belongs to the sector $(1-\mu,1+\mu)$. ∎

To proceed, we define $\chi_{1}=[\bar{z};x]$ and $\chi_{2}=[\tilde{\xi};\tilde{\eta}]$ for the sake of simplicity. From the facts that the dynamics of $z,x,\bar{z}$ is uniformly bounded with respect to $\tau$, $\tilde{\xi}$, and $\tilde{\eta}$, and that the boundaries of $\Omega_{\chi_{1}}(l)$ and $S_{\bar{z}}\times S_{x}$ are disjoint, there exists $T_{0}>0$, independent of $\tau$, such that $\chi_{1}\in\Omega_{\chi_{1}}(l)$ and $\|z(t)\|\leq l_{z}$, $\forall 0\leq t\leq T_{0}$.

Consider the Lyapunov function of $\chi_{2}$ defined by

[TABLE]

where $P_{2}={\rm diag}\{P_{\tilde{\xi}},\gamma P_{\tilde{\eta}}\}$ with $\gamma$ being a positive constant to be chosen. The matrices $P_{\tilde{\xi}}$ and $P_{\tilde{\eta}}$ are symmetric positive definite matrices that are explained below. The matrix $P_{\tilde{\xi}}$ is determined from $P_{\tilde{\xi}}A_{\xi}+A_{\xi}^{\top}P_{\tilde{\xi}}=-I$, while $P_{\tilde{\eta}}$ is associated to the stability of $\tilde{\eta}$-dynamics expressed in a fast time scale $\sigma:=t/\tau$, namely

[TABLE]

We assume for now that $\chi_{1}\in\Omega_{\chi_{1}}(l)$ and investigate the stability of the system $\tilde{\eta}^{\prime}=(A-B\bar{\mathsf{a}})\tilde{\eta}+B\mathsf{a}_{[1]}\{-\Psi_{t}(\tilde{\eta}_{[1]},t)\}$ (the dynamics (17) without $\tau\dot{\eta^{*}}$) by interpreting it as a feedback system $\tilde{\eta}^{\prime}=(A-B\bar{\mathsf{a}})\tilde{\eta}+B\mathsf{a}_{[1]}u^{\dagger}$ with the output $y^{\dagger}=\tilde{\eta}_{[1]}$ and the feedback $u^{\dagger}=-\Psi_{t}(y^{\dagger},t)$. Note that the transfer function matrix from $u^{\dagger}$ to $y^{\dagger}$ is $H(s)={\rm diag}\{H_{1}(s),\dots,H_{m}(s)\}$ where $H_{i}(s)$ is the transfer function defined in (6). By the way it is constructed, $H(s)[I_{m}+(1-\mu)H(s)]^{-1}$ is diagonal and Hurwitz, and $[I_{m}+(1+\mu)H(s)][I_{m}+(1-\mu)H(s)]^{-1}$ is diagonal and strictly positive real. Since $\chi_{1}\in\Omega_{\chi_{1}}(l)$, $\Psi_{t}(y,t)$ belongs to the sector $[1-\mu,1+\mu]$ by Lemma 3 and the circle criterion ensures that the feedback system is exponentially stable and admits a quadratic Lyapunov function $V_{\tilde{\eta}}=\tilde{\eta}^{\top}P_{\tilde{\eta}}\tilde{\eta}$, with $P_{\tilde{\eta}}=P_{\tilde{\eta}}^{\top}>0$, such that $V_{\tilde{\eta}}^{\prime}$ along the trajectory of $\tilde{\eta}$-dynamics (without $\tau\dot{\eta}^{*}$) satisfies $V_{\tilde{\eta}}^{\prime}\leq-\kappa_{\tilde{\eta}}\|\tilde{\eta}\|^{2}$ with $\kappa_{\tilde{\eta}}>0$.

Lemma 4

Suppose that $\chi_{1}(t)\in\Omega_{\chi_{1}}(l)$ and $\|z(t)\|\leq l_{z}$, $\forall t_{0}\leq t\leq T$ for some $t_{0}$ and $T$. Then, there exist positive constants $\rho,\lambda$, and $\tau^{*}_{1}\leq 1$ such that for any $0<\tau<\tau_{1}^{*}$ and $t_{0}\leq t\leq T$, it holds that

[TABLE]

$\hfill\diamond$**

Proof:

The time derivative of $V_{\chi_{2}}$ is computed as

[TABLE]

Since $\Theta_{\tilde{\xi}}(z,x,\bar{z},\tilde{\xi},\tilde{\eta},t)$, $\tilde{\Theta}_{\tilde{\eta}}(\tilde{\xi},t)$, $\partial\tilde{\Theta}_{\tilde{\eta}}(\tilde{\xi},t)/\partial\tilde{\xi}$, and $\dot{\eta^{*}}$ are uniformly bounded when $\|z\|\leq l_{z}$, $(\bar{z},x)\in\Omega_{l}$, and $\tau\leq 1$, it holds that

[TABLE]

where $k_{1}$, $k_{2}$, and $k_{3}$ are positive constants such that $\|\tilde{\Theta}_{\tilde{\eta}}(\tilde{\xi},t)\|\leq k_{1}\|\tilde{\xi}\|$, $\|2P_{\tilde{\xi}}B\Theta_{\tilde{\xi}}\|\leq k_{2}$, and $\|2P_{\tilde{\eta}}(\dot{\eta^{*}})\|\leq k_{3}$, respectively.

Take $\gamma<\kappa_{\tilde{\eta}}/(2k_{1}^{2})$. Then, one has

[TABLE]

where $\bar{\kappa}_{1}=2(\gamma\kappa_{\tilde{\eta}}-2\gamma^{2}k_{1}^{2})>0$. The time derivative (18) is rewritten as

[TABLE]

where $\kappa_{1}=\min\{1/2,\bar{\kappa}_{1}/2\}$ and $\kappa_{2}=\sqrt{k_{2}^{2}+\gamma^{2}k_{3}^{2}}$. The proof is complete with $\lambda=\kappa_{1}$ and $\rho=4\lambda_{\max}(P_{2})\frac{\kappa_{2}^{2}}{\kappa_{1}^{2}}$. ∎

Let \Omega_{\chi_{2}}=\left\{\chi_{2}\in{\mathbb{R}}^{2\nu}\big{|}V_{\chi_{2}}\leq\rho\tau^{2}\right\} which depends on $\tau$. Recall the existence of time interval $0\leq t\leq T_{0}$ during which $\chi_{1}(t)\in\Omega_{\chi_{1}}(l)$ and $\|z(t)\|\leq l_{z}$. By Lemma 4, with $t_{0}=0$ and $T=T_{0}$, there exists $\rho>0$, $\tau_{1}^{*}>0$ with $\tau_{1}^{*}\leq\min\{\sqrt{\lambda_{\min}(P_{2})/\rho}\delta_{\tilde{w}},1\}$, and $P_{2}=P_{2}^{\top}>0$ such that for any $0<\tau<\tau_{1}^{*}$, it holds that $\dot{V}_{\chi_{2}}(t)\leq-\frac{\lambda_{1}}{\tau}V_{\chi_{2}}(t)$ with $\lambda_{1}=\frac{\lambda}{2\lambda_{\max}(P_{2})}$, provided that $V_{\chi_{2}}(t)\geq\rho\tau^{2}$ and $0\leq t\leq T_{0}$. Meanwhile, considering the compactness of the sets of initial conditions and the definition of $\tilde{\xi}$ and $\tilde{\eta}$, one can prove that $V_{\chi_{2}}(0)\leq\frac{k}{\tau^{2r}}$ for some positive constant $k>0$ and $r=\max\{\nu_{i}\}$, $i=1,\dots,m$. Applying this bound for $V_{\chi_{2}}(0)$, one has $V_{\chi_{2}}(t)\leq e^{-\lambda_{1}\frac{t}{\tau}}V_{\chi_{2}}(0)\leq e^{-\lambda_{1}\frac{t}{\tau}}\frac{k}{\tau^{2r}}$. From the fact that $\lim_{\tau\rightarrow 0}e^{-\lambda_{1}\frac{T_{0}}{\tau}}\frac{k}{\rho\tau^{2r+2}}=0$, one can find $\tau_{2}^{*}>0$ such that for any $\tau<\tau_{2}^{*}$, it holds that $V_{\chi_{2}}(T_{0})\leq\rho\tau^{2}$. Thus, for any initial condition $(p(0),q(0))\in S_{p}\times S_{q}$, it holds that $\chi_{1}(t)\in\Omega_{l}$, $\forall t\leq T_{0}$, and $\chi_{2}(t)$ enters the set $\Omega_{\chi_{2}}$ in finite time $T_{0}$.

We now prove that there exists $\tau_{3}^{*}>0$ such that for any $\tau<\tau_{3}^{*}$, the set $\Omega_{\chi_{1}}(l)\times\Omega_{\chi_{2}}$ is positively invariant. This will be carried out by showing that

[TABLE]

To show the first property of (20), we start by noting that when $\tau\leq\min\{\tau_{1}^{*},\tau_{2}^{*}\}$, $V_{\chi_{1}}(\bar{z},x)\leq l$, and $\chi_{2}\in\Omega_{\chi_{2}}$, the saturation functions $\Phi(w)$ and $\phi(x)$ lose the effect, i.e., $\Phi(w)=w$ and $\phi(x)=x$. Then, the time derivative of $\bar{V}(\chi_{1})$ becomes

[TABLE]

By Assumption 2, we have

[TABLE]

With (11) and $\eta^{*}_{[1]}$ of (13), one has

[TABLE]

where

[TABLE]

Note that $\Pi(\tilde{\xi},t)=0$ when $\tilde{\xi}=0$.

Since $\chi_{1}$ and $\chi_{2}$ belong to compact sets, we have

[TABLE]

where $\kappa_{3}=\max_{\chi_{1}\in\Omega_{\chi_{1}}(l)}\left\|\frac{\partial\bar{V}}{\partial x}B\right\|$, $k_{\tilde{\eta}}$ is the maximum of $\|G(z,x,t)\bar{G}^{-1}(\bar{z},x+\Delta\tilde{\xi})\|$ on the region under consideration, $k_{\tilde{\xi}}$ is a constant such that $\Pi(\tilde{\xi},t)\|\leq k_{\tilde{\xi}}\|\tilde{\xi}\|$, and $\kappa=\kappa_{3}{\rm max}\{k_{\tilde{\eta}},k_{\tilde{\xi}}\}$. Let $\bar{\tau}_{3}^{*}=\frac{1}{\kappa}\sqrt{\frac{\lambda_{\min}(P_{2})}{\rho}}\alpha_{3}(\alpha_{2}^{-1}(l))$ and take $\tau\leq\min\{\tau_{1}^{*},\tau_{2}^{*},\bar{\tau}_{3}^{*}\}$. Since $\chi_{2}\in\Omega_{\chi_{2}}$, it follows that $\|\chi_{2}\|\leq\sqrt{\frac{\rho}{\lambda_{\min}(P_{2})}}\bar{\tau}_{3}^{*}\leq\alpha_{3}(\alpha_{2}^{-1}(l))/\kappa$. Applying this result and the condition $\bar{V}(\chi_{1})=l$ to (21), we have $\dot{\bar{V}}(\chi_{1})\leq 0$, \forall(\chi_{1},\chi_{2})\in\left\{\chi_{1}\big{|}\bar{V}(\chi_{1})=l\right\}\times\Omega_{\chi_{2}}.

The second equation of (20) follows from Lemma 4 with possibly new bound on $\tau$, say $\tilde{\tau}_{3}>0$. Taking $\tau_{3}^{*}=\min\{\tau_{1}^{*},\tau_{2}^{*},\bar{\tau}_{3}^{*},\tilde{\tau}_{3}^{*}\}=:\tau^{*}$ proves the invariance.

The proceeding arguments prove that the solution of the closed-loop system is uniformly bounded.

Now, we prove the practical stability (8). Let $\epsilon>0$ be given. Since the time derivative of $\bar{V}(\chi_{1})$ satisfies (21), we can achieve the objective by taking sufficiently small $\tau$ such that $\|\chi_{2}(t)\|$ remains as small as desired. In fact, let $\tau_{4}^{*}=\frac{1}{2\kappa}\sqrt{\lambda_{\min}(P_{2})/{\rho}}\alpha_{3}(\alpha_{2}^{-1}(\alpha_{1}(\epsilon)))$ and take $\tau\leq\min\{\tau_{3}^{*},\tau_{4}^{*}\}$. Then by invariance, the assumption of Lemma 4 holds true with $t_{0}=T_{0}$ and $T=\infty$, and there exists $T_{1}>T_{0}$ such that $\chi_{2}(t)\in\Omega_{\chi_{2}}|_{\tau=\tau_{4}^{*}}$, $\forall t\geq T_{1}$ which means that $\|\chi_{2}(t)\|\leq\alpha_{3}(\alpha_{2}^{-1}(\alpha_{1}(\epsilon)))/(2\kappa)$, $\forall t\geq T_{1}$. Applying this bound to (21) yields

[TABLE]

and this implies that $\dot{\bar{V}}\leq-\frac{1}{2}\alpha_{3}(\alpha_{2}^{-1}(\bar{V}))$ whenever $\bar{V}(\chi_{1})\geq\alpha_{1}(\epsilon)$. Therefore, there exists $T_{\epsilon}>0$ such that $\chi_{1}(t)\leq\epsilon$, $\forall t\geq T_{\epsilon}$. The proof is complete.

IV Example

In this section, we validate the performance of the proposed strategy through an numerical example. Consider a point mass satellite that is allowed to have two dimensional (planar) motion. In polar coordinate, the dynamics (see Fig. 2) is given as

[TABLE]

where $m$ is the mass of the satellite, $r$ is the radial distance, $\psi$ is the polar angle from the horizontal line, $u_{r}$ is the thrust of radial direction, $u_{\theta}$ is the thrust of tangential direction, and $k$ is a constant related the force field. We assume that the satellite does not have a rotational motion by the external force, i.e., the behavior of the satellite is decided by only the control inputs $u_{1}$ and $u_{2}$, and the outputs $r$ and $\psi$ can be measured. The relation between $(u_{r},u_{\psi})$ and $(u_{1},u_{2})$ is described as

[TABLE]

We suppose that $\theta(t)=\theta_{o}(t)+\tilde{\theta}(t)$ where $\theta_{0}(t)$ is known but $\tilde{\theta}(t)$ is unknown and satisfies $|\tilde{\theta}(t)|\leq c_{\tilde{\theta}}$ where the known constant $c_{\tilde{\theta}}$ is given by $|\tilde{\theta}(t)|<\pi/5$. It is noted that, without a controller $(u_{1}=u_{2}=0)$, the system (22) admits solutions $r(t)=r_{*}$ and $\psi(t)=\omega_{*}t$ with $k=r_{*}^{3}\omega_{*}^{2}$, where $r_{*}>0$ and $\omega_{*}$ are constants.

Define $x_{11}=r-r_{*}$, $x_{12}=v$, $x_{21}=r_{*}(\psi-\omega_{*}t)$, and $x_{22}=r_{*}(\omega-\omega_{*})$. Then, the dynamics becomes

[TABLE]

The relative degree vector $\nu$ of the system is $[2;2]$. To design the controller, we choose the input gain matrix $\bar{G}$ such that $\bar{G}(x,t)={\rm diag}\{1/\bar{m},r_{*}/\bar{m}(x_{11}+r_{*})\}R(\theta_{o}(t))$. Suppose the state feedback controller for the nominal system is given by $U_{r}=\bar{G}^{-1}(x,t)(-F(x)-Kx)$.

The simulation parameters are given as follows. $\bar{m}=1$, $m=1.2$, $k=5$, $r_{*}=1.5$, $\omega_{*}=\sqrt{k/r_{*}^{3}}$, $\theta_{o}=(\pi/2)\sin(\pi t)$, $\tilde{\theta}=(\pi/5)\sin(4\pi t)$, $|sI-(A-BK)|=(s+1)(s+3)^{2}(s+5)$, $\mu=0.001$, $a_{11}=a_{21}=15$, $a_{12}=a_{22}=8$, $x(0)=[1;-2;0;-0.8]$. The levels of saturation functions $\Phi$ and $\phi$ are taken as $100$ and $25$, respectively.

Fig. 3 shows the state trajectories of the nominal closed-loop system under the state feedback controller $U_{r}$. It is seen that the states of nominal system converge to zero.

In Fig. 4, the state trajectories of the closed-loop system under the proposed controller with a constant nominal gain matrix $\bar{G}_{c}={\rm diag}\{1/m,1/m\}$ is shown and it is seen that the trajectories are distorted from the nominal ones, but the trajectory converges to the origin with very small error.

In Fig. 5, the proposed strategy having the nonlinear gain matrix $\bar{G}(x,t)$ mentioned above is applied to the actual system. One can see that, the proposed controller makes the system stable and recovers the performance of the nominal system despite the presence of system uncertainties.

Fig. 6 shows the comparison of control inputs used in the simulations corresponding to Figs. 4 and 5. As expected, the controller with a nonlinear nominal input gain matrix requires smaller control effort than the other.

V Conclusions

In this paper, we have presented a robust output feedback controller for MIMO nonlinear systems having an uncertain input gain matrix. We allow that the nominal input gain matrix can be a nonlinear function of state, which is often the case in practical systems. By proposing a new controller structure, it is shown that the performance of the state feedback controller, designed for the nonlinear nominal system, is recovered in the steady state. Compared to previous results, the proposed controller can reduce unnecessarily large control effort that comes from the difference between the actual input gain and somewhat artificial linear nominal input gain. For the future work, we will apply the proposed controller to practical systems and generalize the result to consider system uncertainties and external disturbances.