Observer-based feedback linearization control of multi-input multi-output nonlinear system and application to double rotor system

Abstract

This study first proposes an observer-based feedback linearization control of nonlinear multi-input multi-output systems with the input-to-state stability and almost disturbance decoupling characteristics, and summarizes an efficient algorithm for deriving the almost disturbance decoupling control. The state variables for the nonlinear dynamic equations generally are not available, and a state observer is applied to estimate the state variables. The existing approach solves the almost disturbance decoupling problem under the difficult sufficient conditions including the nonlinearities which are not globally Lipschitz and disturbances which appears linearly but multiplied nonlinearities. On the contrary, this problem can be easily solved via the proposed approach without those difficult conditions in this study. The generality of this approach and implications, due to the fact that unlike Jacobian linearization feedback linearization is not only locally valid, are also shown. The double rotor multi-input multi-output system mechanical structure is a laboratory equipment for some flying viewpoints and is investigated in this study to demonstrate the practical value of the proposed results.

Introduction

The state feedback approach can be performed if the state variables are measurable in the general control theorems. If this condition does not hold, the state feedback controller can be built with the aid of a state estimator (observer) and the Luenberger observer, employing the measurable output, generally is performed. Based on this perspective, the main design task is to build observer-based state controllers and one generally utilize the separation principle and the two-stage procedure to calculate the observer-based gain matrices with inaccessible states. On the contrary, the unknown disturbance is another significant factor which will be considered into the observer design. The observers can be built by estimating both the system states and the disturbances and then the overall stability of the observer producing the state and disturbance estimation should be guaranteed.

Stabilization problem and tracking problem are important subjects in the research of nonlinear control, and the tracking problem is generally more complex than the stabilization problem. There currently exists many designing methods, including feedback linearization, variable structure control (sliding mode control), backstepping, model predictive control with a disturbance observer, regulation control, nonlinear $H^{\infty }$ control, internal model principle, and $H^{\infty }$ adaptive fuzzy control, of nonlinear systems for the tracking problem to be investigated. Variable structure control approach has been applied to solve the nonlinear system.^1–3 However, the chattering phenomenon generates the unmodeled high frequency and inevitably drive systems to be unstable for variable structure control approach. Backstepping approach has been an efficient tool for some class of nonlinear systems. However, a disadvantage of the backstepping approach needs to calculate the complex repeated differentiations of some nonlinear functions.^4,5 Model predictive control with a disturbance observer has been known to be one of the most important advanced control methods in industrial systems due to its optimized tracking performance and is applicable to arbitrary disturbance relative degree. ⁶ The research results of Yang and Zheng ⁶ are compared with this study, and the significant comparisons are summarized as follows: (1) the existing nonlinear model predictive control is not applicable to the mismatched disturbance situation. On the contrary, both Yang and Zheng ⁶ and this study can solve the problem of this situation; (2) for the proposed method of Yang and Zheng, ⁶ the disturbance can be completely eliminated from the output channel by appropriately designed interference compensation gain. Conversely, the use of this study can only almost eliminate the disturbance; (3) the estimator variables of Yang and Zheng ⁶ are disturbances, and conversely, the estimator variables of this article are the state variables of the system; (4) when the state variables of the system are unmeasurable, the research method using Yang and Zheng ⁶ cannot be solvable. On the contrary, the research method using this study is solvable; (5) solving the observer gain of Yang and Zheng ⁶ requires to solve the differential equation. Conversely, solving the observer gain of this study only requires to solve a simple algebraic equation. Famous output tracking approach applies the structure of the output regulation control approach.^7,8 However, the problem of the regulation control approach needs to solve the complicated partial-differential equation. Moreover, another problem of the regulation control approach is that the exosystem states, switching to describe changes in the output, will generate the transient errors. ⁹ Above limitations of the output regulation control approach have been relaxed by the new output regulation approach shown in Yang and Ding. ¹⁰ There is no need to suppose that the disturbances are governed by certain neutral stable exosystems, which relaxes the restrictions on disturbances. The research results of Yang and Ding ¹⁰ are compared with this study, and the significant comparisons are summarized as follows: (1) when the state variables of the system are unmeasurable, the research method using Yang and Ding ¹⁰ cannot be solvable. On the contrary, the research method using this study is solvable via the state observer; (2) the remarkable property of existing output regulation approaches is that the nonlinearities are usually assumed to be sufficiently smooth with respect to their arguments. ¹¹ Yang and Ding ¹⁰ exploits the fact that the nonlinear items in the system is not required to be sufficiently smooth. This study solves the problem without the difficult sufficient conditions including the nonlinearities which are not globally Lipschitz and disturbances which appears linearly but multiplied nonlinearities; (3) when the disturbances are unknown, the homogeneous growth condition of Yang and Ding ¹⁰ become impractical because this condition contains disturbances. In contrast, the homogeneous growth condition is not necessary for this study. Generally, the nonlinear $H^{\infty }$ control approach needs to solve the difficult partial-differential Hamilton-Jacobi equation,^12,13 and we can obtain the closed-form solution only for some special cases. ¹⁴ The internal model principle approach generally transforms the tracking problem to the regulation problem. This approach needs to solve the complex partial-differential equation of the center manifold. ⁷ However, the asymptotic solutions of this partial-differential equation have been exploited only for some special cases.^15,16 The $H^{\infty }$ adaptive fuzzy control has been presented to solve nonlinear systems efficiently. ¹⁷ The drawback of $H^{\infty }$ adaptive fuzzy control approach is that the complicated parameter updating criterion makes this approach be impractical. ¹⁸ Feedback linearization approach is one of the most important nonlinear control design strategy developed during the last few decades.^19,20. This approach may result in linearization which is valid for larger practical operating points of the system, as opposed to a local Jacobian linearization about an operating point, and it has been applied successfully to address many practical systems including the tracking control of nonholonomic mobile robot, ²¹ the passivity-based surge control of compressors, ²² three-phase grid-connected photovoltaic systems, ²³ the control of electromagnetic suspension system, ²⁴ and bank-to-turn missile system. ²⁵

The almost disturbance decoupling subject, designing a controller to reduce the influence of the disturbance on the output end, is started up for linear and nonlinear systems by Willems ²⁶ and Marino et al., ²⁷ respectively, and then many important related studies have been created.^28–30 Marino et al. ²⁷ investigates the single-input single-output (SISO) almost disturbance decoupling problem based on the state feedback approach and the singular perturbation approach. The almost disturbance decoupling problem is solved under some complicated sufficient conditions including the nonlinearities which are not globally Lipschitz and disturbances which appears linearly but multiplied nonlinearities. The complicated sufficient conditions need the condition that the nonlinearities multiplying the disturbances satisfy the structural triangular conditions. Marino et al. ²⁷ exploits the fact that the following SISO almost disturbance decoupling problem of some examples cannot be solved: ${\stackrel{·}{x}}_1(t)=x_2+{\Theta }_1(t)$ , ${\stackrel{·}{x}}_2(t)=x_2^3{\Theta }_2(t)+u$ , $y=x_1$ , where $u,y$ are the input and output, respectively, and ${\Theta }_1,{\Theta }_2$ are the disturbance terms. In contrast, this example can be easily solved without those difficult conditions via the proposed approach in this study. The proposed controller has a feedback linearization controller and a state observer. The input-to-state stability of the feedback-controlled system is then proved by the composite Lyapunov theory such that the globally asymptotical tracking is guaranteed with the almost disturbance decoupling performance. Most notably, the generality of this approach and implications, due to the fact that unlike Jacobian linearization feedback linearization is not only locally valid, are also shown. In order to exploit the practical applicability, this study also has successfully achieved the tracking controller with almost disturbance decoupling performance for a double rotor multi-input multi-output (MIMO) system:

Observer-based feedback linearization control: in this study, we designed an observer-based feedback linearization controller for the following MIMO nonlinear control system with disturbances

$$\begin{array}{c}\left[\begin{array}{c}{\stackrel{·}{x}}_1\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{x}}_2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\stackrel{·}{x}}_n}\end{array}\right]=\left[\begin{array}{c}{f}_1(x_1,x_2,\dots ,x_n)\hfill \\ \multicolumn{1}{c}{f_2(x_1,x_2,\dots ,x_n)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{f_n(x_1,x_2,\dots ,x_n)}\end{array}\right]+\left[\begin{array}{cccc}{g}_1\hfill & g_2\hfill & \cdots \hfill & g_m\hfill \end{array}\right]\hfill \\ \left[\begin{array}{c}{u}_1(x_1,x_2,\dots ,x_n)\hfill \\ \multicolumn{1}{c}{u_2(x_1,x_2,\dots ,x_n)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{u_m(x_1,x_2,\dots ,x_n)}\end{array}\right]+\sum _{j=1}^p{q}_j^{*}{\theta }_{\mathrm{jd}}\hfill \end{array}$$

(1)

$$\left[\begin{array}{c}{y}_1(x_1,x_2,\dots ,x_n)\hfill \\ \multicolumn{1}{c}{y_2(x_1,x_2,\dots ,x_n)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{y_m(x_1,x_2,\dots ,x_n)}\end{array}\right]=\left[\begin{array}{c}{h}_1(x_1,x_2,\dots ,x_n)\hfill \\ \multicolumn{1}{c}{h_2(x_1,x_2,\dots ,x_n)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{h_m(x_1,x_2,\dots ,x_n)}\end{array}\right]$$

(2)

that is

$$\stackrel{·}{X}(t)=f(X(t))+\mathrm{gu}+Q\theta$$

(3)

$$y(t)=h(X(t))$$

(4)

$$Q\equiv {\left[\begin{array}{cccc}{q}_1^{*}\hfill & q_2^{*}\hfill & \cdots \hfill & q_p^{*}\hfill \end{array}\right]}_{n\times p}$$

(5)

where $X(t)\equiv [x_1(t)x_2(t)\cdots x_n(t){]}^T\in {\Re }^n$ , $u\equiv [u_1{u}_2\cdots u_m{]}^T\in {\Re }^m$ , and $y\equiv [y_1{y}_2\cdots y_m{]}^T\in {\Re }^m$ are the state vector, input vector, and the output vector, respectively; $\theta \equiv [{\theta }_{1d}(t){\theta }_{2d}(t)\cdots {\theta }_{\mathrm{pd}}(t){]}^T$ is a bounded disturbances term, $f\equiv [f_1{f}_2\cdots f_n{]}^T\in {\Re }^n$ and $h\equiv [h_1{h}_2\cdots h_m{]}^T\in {\Re }^m$ are smooth functions, and $g\equiv [g_1{g}_2\cdots g_m]\in {\Re }^{n\times m}$ is a constant vector. The nominal system of the system equation (3) is assumed to have the vector relative degree $\{r_1,r_2,\dots ,r_m\}$ such that the properties hold: ³¹

• 1. For all $1\le i\le m$ , $1\le j\le m$ , $k<r_i-1$

$$L_{g_j}{L}_f^k{h}_i\left(X\right)=0$$

(6)

where the function L is the Lie derivative operator ³¹ and $r_1+r_2+\cdots +r_m=r$ .

• 2. The following square matrix owns the nonsingular property

$$A\equiv \left[\begin{array}{ccc}{L}_{g_1}{L}_f^{r_1-1}{h}_1(X)\hfill & \cdots \hfill & L_{g_m}{L}_f^{r_1-1}{h}_1(X)\hfill \\ \multicolumn{1}{c}{L_{g_1}{L}_f^{r_2-1}{h}_2(X)}& \cdots \hfill & L_{g_m}{L}_f^{r_2-1}{h}_2(X)\hfill \\ \multicolumn{1}{c}{⋮}& \hfill & ⋮\hfill \\ \multicolumn{1}{c}{L_{g_1}{L}_f^{r_m-1}{h}_m(X)}& \hfill & L_{g_m}{L}_f^{r_m-1}{h}_m(X)\hfill \end{array}\right]$$

(7)

The desired output trajectory $y_d(t)$ and its first r derivatives are all uniformly bounded and $||[y_d(t),y_d^{(1)}(t),\dots ,y_d^{(r)}(t)]||\le B_d$ where $B_d$ is some positive constant. The block diagram of complete observer-based feedback linearization control is demonstrated in the Figure 1.

Figure 1.

The block diagram of complete observer-based feedback linearization control.

The original nonlinear control system can be separated with linear part and nonlinear part as follows

$$\stackrel{·}{X}(t)=f\left(X\right)+\mathrm{gu}+Q\theta =F_{\ell }X(t)+F_n\left(X\right)+\mathrm{gu}+Q\theta$$

(8)

$$y(t)=H^{T}X(t)\equiv h(X(t))$$

(9)

where $F_{\ell }X$ and $F_n(X)$ are the linear part and nonlinear part of function $f(X)$ , respectively. Similarly, the desired observer can be separated with linear part and nonlinear part

$$\begin{array}{cc}\multicolumn{1}{c}{\stackrel{·}{\hat{X}}(t)}& =f(\hat{X})+\mathrm{gu}+L(y(t)-\hat{y}(t))\hfill \\ \multicolumn{1}{c}{}& =F_{\ell }\hat{X}(t)+F_n(\hat{X})+\mathrm{gu}+L(y(t)-\hat{y}(t))\hfill \end{array}$$

(10)

$$\hat{y}(t)=H^T\hat{X}(t)\equiv h(\hat{X}(t))$$

(11)

where L is the observer gain. Subtracting equation (10) from equation (8) and performing the derivative operation yield

$$\stackrel{·}{\stackrel{~}{X}}(t)=(F_{\ell }-L{H}^T)\tilde{X}(t)+(F_n(X)-F_n(\hat{X}))+Q\theta$$

(12)

where $\tilde{X}\equiv X(t)-\hat{X}(t)$ is the estimated error. It is well known that if $(F_{\ell },H)$ possesses observable property, the matrix $F_{\ell }-L{H}^T$ will be Hurwitze such that the following equation hold

$${(F_{\ell }-L{H}^T)}^T{P}_2+P_2(F_{\ell }-L{H}^T)=-I$$

(13)

where $P_2$ is a positive definite matrix. Assume that there is a constant $M_2\ge 0$ such that the Jacobian inequalities hold

$$\Vert \frac{\partial F_n(\hat{X})}{\partial \hat{X}}\Vert \le M_2$$

(14)

and

$$\Vert F_n(X)-F_n(\hat{X})\Vert \le M_2\Vert X-\hat{X}\Vert$$

(15)

that is, $F_n(\hat{X})$ possesses the Lipschitz property. ³² Isidori ³¹ has exploited the fact that if the distribution

$$G\equiv \mathrm{span}\{g_1,g_2,\dots ,g_m\}$$

(16)

possesses the involutive property and the vector fields

$$Y_j^k,1\le j\le m,1\le k\le r_j$$

(17)

are complete functions, where

$$Y_j^k\equiv {(-1)}^{k-1}{\mathrm{ad}}_{\tilde{f}}^{k-1}{\tilde{g}}_j,1\le j\le m,1\le k\le r_j$$

(18)

$$\tilde{g}\equiv \left[\begin{array}{cccc}{\tilde{g}}_1\hfill & {\tilde{g}}_2\hfill & \cdots \hfill & {\tilde{g}}_m\hfill \end{array}\right]\equiv g{A}^{-1}(\hat{X})$$

(19)

$$\tilde{f}(\hat{X})\equiv f(\hat{X})-g{A}^{-1}(\hat{X})b(\hat{X})$$

(20)

$$b(\hat{X})\equiv {\left[\begin{array}{cccc}{L}_f^{r_1}{h}_1(\hat{X})\hfill & L_f^{r_2}{h}_2(\hat{X})\hfill & \cdots \hfill & L_f^{r_m}{h}_m(\hat{X})\hfill \end{array}\right]}^T$$

(21)

$${\mathrm{ad}}_f^{k}g\equiv \left[\begin{array}{cc}f\hfill & {\mathrm{ad}}_f^{k-1}g\hfill \end{array}\right]$$

(22)

$$\left[\begin{array}{cc}f\hfill & g\hfill \end{array}\right]\equiv \frac{\partial g}{\partial \hat{X}}f(\hat{X})-\frac{\partial f}{\partial \hat{X}}g$$

(23)

then the function

$$\phi :{\Re }^n\to {\Re }^n$$

(24)

$$\begin{array}{c}{\hat{\xi }}_i\equiv {\left[\begin{array}{cccc}{\hat{\xi }}_1^i\hfill & {\hat{\xi }}_2^i\hfill & \cdots \hfill & {\hat{\xi }}_{r_i}^i\hfill \end{array}\right]}^T\equiv {\left[\begin{array}{cccc}{\varphi }_1^i\hfill & {\varphi }_2^i\hfill & \cdots \hfill & {\varphi }_{r_i}^i\hfill \end{array}\right]}^T\hfill \\ \equiv {\left[\begin{array}{cccc}{L}_f^0{h}_i(\hat{X})\hfill & L_f^1{h}_i(\hat{X})\hfill & \cdots \hfill & L_f^{r_i-1}{h}_i(\hat{X})\hfill \end{array}\right]}^T,\hfill \\ i=1,2,\dots ,m\hfill \end{array}$$

(25)

$${\phi }_k(\hat{X}(t))\equiv {\hat{\eta }}_k(t),k=r+1,r+2,\dots ,n$$

(26)

$$L_{g_j}{\phi }_k\left(\hat{X}\left(t\right)\right)=0,k=r+1,r+2,\dots ,n,1\le j\le m$$

(27)

is a diffeomorphism. For the sake of convenience and simplicity, let us give some definitions as follows

$${\hat{e}}_j^i\equiv {\hat{\xi }}_j^i-y_d^{i(j-1)},i=1,2,\dots ,m,j=1,2,\dots ,r_i$$

(28)

$${\hat{e}}^i\equiv {\left[\begin{array}{cccc}{\hat{e}}_1^i\hfill & {\hat{e}}_2^i\hfill & \cdots \hfill & {\hat{e}}_{r_i}^i\hfill \end{array}\right]}^T\in {\Re }^{r_i}$$

(29)

$$\overline{{\hat{e}}_j^i}\equiv {\epsilon }^{j-1}{\hat{e}}_j^i,i=1,2,\dots ,m,j=1,2,\dots ,r_i$$

(30)

$$\overline{{\hat{e}}^i}\equiv {\left[\begin{array}{cccc}\overline{{\hat{e}}_1^i}\hfill & \overline{{\hat{e}}_2^i}\hfill & \cdots \hfill & \overline{{\hat{e}}_{r_i}^i}(t)\hfill \end{array}\right]}^T\in {\Re }^{r_i}$$

(31)

$$\overline{\hat{e}}\equiv {\left[\begin{array}{cccc}\overline{{\hat{e}}^1}\hfill & \overline{{\hat{e}}^2}\hfill & \cdots \hfill & \overline{{\hat{e}}^m}\hfill \end{array}\right]}^T\in {\Re }^r$$

(32)

$$\hat{\xi }\equiv {\left[\begin{array}{cccc}{\hat{\xi }}_1\hfill & {\hat{\xi }}_2\hfill & \cdots \hfill & {\hat{\xi }}_r\hfill \end{array}\right]}^T\in {\Re }^r$$

(33)

$$\hat{\eta }(t)\equiv {\left[\begin{array}{cccc}{\hat{\eta }}_{r+1}(t)\hfill & {\hat{\eta }}_{r+2}(t)\hfill & \cdots \hfill & {\hat{\eta }}_n(t)\hfill \end{array}\right]}^T\in {\Re }^{n-r}$$

(34)

$$\begin{array}{cc}\multicolumn{1}{c}{q(\hat{\xi }(t),\hat{\eta }(t))}& \equiv {\left[\begin{array}{cccc}{L}_f{\phi }_{r+1}(t)\hfill & L_f{\phi }_{r+2}(t)\hfill & \cdots \hfill & L_f{\phi }_n(t)\hfill \end{array}\right]}^T\hfill \\ \multicolumn{1}{c}{}& \equiv {\left[\begin{array}{cccc}{q}_{r+1}\hfill & q_{r+2}\hfill & \cdots \hfill & q_n\hfill \end{array}\right]}^T\hfill \end{array}$$

(35)

$$B^i\equiv {{\left[\begin{array}{ccccc}0\hfill & 0\hfill & \cdots \hfill & 0\hfill & 1\hfill \end{array}\right]}^T}_{r_i\times 1},1\le i\le m$$

(36)

$$d_{\mathrm{ij}}\equiv L_{g_j}{L}_f^{r_i-1}{h}_i(\hat{X}),1\le i\le m,1\le j\le m$$

(37)

$$c_i\equiv L_f^{r_i}{h}_i(\hat{X}),1\le i\le m$$

(38)

$$A_c^i\equiv {\left[\begin{array}{ccccc}0\hfill & 1\hfill & 0\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{}& ⋮\hfill & \hfill & \hfill & ⋮\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & \cdots \hfill & 1\hfill \\ \multicolumn{1}{c}{-{\alpha }_1^i}& -{\alpha }_2^i\hfill & -{\alpha }_3^i\hfill & \cdots \hfill & -{\alpha }_{r_i}^i\hfill \end{array}\right]}_{r_i\times r_i},1\le i\le m$$

(39)

where ${\alpha }_1^i,{\alpha }_2^i,\dots ,{\alpha }_{r_i}^i$ are some adjustable variables such that $A_c^i$ possesses Hurwitz property. Let $P_1^i$ be the positive definite matrix solution of the following Lyapunov equations

$${\left(A_c^i\right)}^T{P}_1^i+P_1^i{A}_c^i=-I,1\le i\le m$$

(40)

$${\lambda }_{max}\left(P_1^i\right)\equiv \mathrm{the}\mathrm{maximum}\mathrm{eigenvalue}\mathrm{of}{P}_1^i,1\le i\le m$$

(41)

$${\lambda }_{min}\left(P_1^i\right)\equiv \mathrm{the}\mathrm{minimum}\mathrm{eigenvalue}\mathrm{of}{P}_1^i,1\le i\le m$$

(42)

$${\lambda }_{max}^{*}\equiv min\{{\lambda }_{max}\left(P_1^1\right),{\lambda }_{max}\left(P_1^2\right),\dots ,{\lambda }_{max}\left(P_1^m\right)\}$$

(43)

$${\lambda }_{min}^{*}\equiv min\{{\lambda }_{min}\left(P_1^1\right),{\lambda }_{min}\left(P_1^2\right),\dots ,{\lambda }_{min}\left(P_1^m\right)\}$$

(44)

Definition 1

Given the system $\stackrel{·}{x}=f(t,x,\theta )$ , where $f:[0,\infty )\times {\Re }^n\times {\Re }^n\to {\Re }^n$ possesses piecewise continuous in t and locally Lipschitz properties in x and $\theta$ . This system is defined to be input-to-state stable if there exists a class-K function $\gamma$ , a class-KL function $\beta$ , and positive numbers $k_1$ and $k_2$ such that under any initial condition $x(t_0)$ with $||x(t_0)||<k_1$ and any bounded input $\theta (t)$ with $\underset{t\ge t_0}{sup}||\theta (t)||<k_2$ , the following inequality holds ³²

$$\Vert x(t)\Vert \le \beta (\Vert x(t_0)\Vert ,t-t_0)+\gamma (\underset{t_0\le \tau \le t}{sup}\Vert \theta (\tau )\Vert ),t\ge t_0\ge 0$$

(45)

Definition 2

The almost disturbance decoupling property is defined to be globally solvable by the desired controller for the transformed-equivalent system by a global diffeomorphism, if the controller makes the following properties hold:

1. The transformed-equivalent system is input-to-state stable with respect to given disturbances.

2. For any given initial condition ${\overline{\hat{x}}}_{e0}\equiv [\overline{\hat{e}}(t_0)\hat{\eta }(t_0){]}^T$ ,

$$\begin{array}{c}|y_i(t)-y_d^i(t)|\le {\beta }_{11}(\Vert \hat{x}(t_0)\Vert ,t-t_0)\hfill \\ +\frac{1}{\sqrt{{\beta }_{22}}}{\beta }_{33}(\underset{t_0\le \tau \le t}{sup}\Vert \theta (\tau )\Vert )\hfill \end{array}$$

(46)

and

$$\begin{array}{c}\underset{t_0}{\overset{t}{\int }}{[y_i(\tau )-y_d^i(\tau )]}^{2}d\tau \le \frac{1}{{\beta }_{44}}\hfill \\ [{\beta }_{55}(\Vert {\overline{\hat{x}}}_{e0}\Vert )+\underset{t_0}{\overset{t}{\int }}{\beta }_{33}\left({\Vert \theta \left(\tau \right)\Vert }^2\right)d\tau ],1\le i\le m\hfill \end{array}$$

(47)

where ${\beta }_{22}$ , ${\beta }_{44}$ are some positive numbers; ${\beta }_{33}$ , ${\beta }_{55}$ are class-K functions; and ${\beta }_{11}$ is a class-KL function. ³³

Theorem 1

• The almost disturbance decoupling problem is globally solvable by the controller defined as

$$u=A^{-1}\left\{-b+v\right\}$$

(48)

$$b\equiv {\left[\begin{array}{cccc}{L}_f^{r_1}{h}_1\hfill & L_f^{r_2}{h}_2\hfill & \cdots \hfill & L_f^{r_m}{h}_m\hfill \end{array}\right]}^T$$

(49)

$$v\equiv {\left[\begin{array}{cccc}{v}_1\hfill & v_2\hfill & \cdots \hfill & v_m\hfill \end{array}\right]}^T$$

(50)

$$\begin{array}{c}{v}_i\equiv y_d^{(r_i)}-{\epsilon }^{-r_i}{\alpha }_1^i[L_f^0{h}_i(\hat{X})-y_d^i]\hfill \\ -{\epsilon }^{1-r_i}{\alpha }_2^i[L_f^1{h}_i(\hat{X})-{y_d^i}^{(1)}]-\cdots -\hfill \\ {\epsilon }^{-1}{\alpha }_{r_i}^i[L_f^{r_i-1}{h}_i(\hat{X})-{y_d^i}^{(r_i-1)}],1\le i\le m\hfill \end{array}$$

(51)

if exists a positive number $M_1$ and a continuously differentiable function $V:{\Re }^{n-r}\to {\Re }^{+}$ such that the following inequalities hold for all $\eta \in {\Re }^{n-r}$

$$(\mathrm{a})\Vert q_{22}(t,\hat{\eta },\overline{\hat{e}})-q_{22}(t,\hat{\eta },0)\Vert \le M_1(\Vert \overline{\hat{e}}\Vert )$$

(52)

$$(\mathrm{a}){\omega }_1\Vert \hat{\eta }{\Vert }^2\le V\left(\hat{\eta }\right)\le {\omega }_2\Vert \hat{\eta }{\Vert }^2,{\omega }_1,{\omega }_2>0$$

(53)

$$(\mathrm{b}){\nabla }_{t}V+{\left({\nabla }_{\hat{\eta }}V\right)}^T{q}_{22}(t,\hat{\eta },0)\le -2{\alpha }_x\Vert \hat{\eta }\Vert {,}^2{\alpha }_{\hat{x}}>0$$

(54)

$$(\mathrm{c})\Vert {\nabla }_{\hat{\eta }}V\Vert \le {\varpi }_3\Vert \hat{\eta }\Vert ,{\varpi }_3>0,$$

(55)

where $q_{22}(t,\hat{\eta },\overline{\hat{e}})\equiv q(\hat{\xi },\hat{\eta })$ , $\hat{\eta }\in {\Re }^{n-r}$ , and $\hat{\xi }\in {\Re }^r$ . Moreover, the tracking error of the transformed-equivalent system can be attenuated by increasing the numbers $N_2$

$$k_{11}\equiv 2{\alpha }_{\hat{X}}-{\omega }_3^2{M}_1^2$$

(56)

$$k_{22}\equiv -\frac{1}{4}+\frac{k_1}{2\epsilon }$$

(57)

$$k_{33}=\frac{k_2}{2}-k_2{M}_2\Vert P_2\Vert -k_2^2\Vert P_2{\Vert }^2\Vert Q{\Vert }^2$$

(58)

$$N_2\equiv min\{k_{11},k_{22},k_{33}\}>1$$

(59)

$$N_1\equiv \frac{1}{4}{(\underset{t_0\le \tau \le t}{sup}\Vert \theta \left(\tau \right)\Vert )}^2,1\le i\le m$$

(60)

and be exponentially attracted into a sphere $B_{\underset{¯}{r}}$ , $\underset{¯}{r}=\sqrt{(N_1/N_2)}$ , with an exponential rate of convergence

$$\frac{1}{2}(\frac{N_2}{{\mathrm{\Delta }}_{max}}-\frac{N_1}{{\mathrm{\Delta }}_{max}{\underset{¯}{r}}^2})\equiv \frac{1}{2}{\alpha }^{*}$$

(61)

$${\mathrm{\Delta }}_{max}\equiv max\{{\omega }_2,\frac{k_1}{2}{\lambda }_{max}^{*},\frac{k_2}{2}{\lambda }_{max}(P_2)\}$$

(62)

Proof

Applying the diffeomorphism equation (24) yields

$$\begin{array}{c}{\stackrel{·}{\hat{\xi }}}_1^1=\frac{\partial {\phi }_1^1}{\partial \hat{X}}\frac{d\hat{X}}{\mathrm{dt}}=\frac{\partial h_1}{\partial \hat{X}}[f+g·u]\hfill \\ =\frac{\partial h_1}{\partial \hat{X}}f+\frac{\partial h_1}{\partial \hat{X}}{g}_1{u}_1+\cdots +\frac{\partial h_1}{\partial \hat{X}}{g}_m{u}_m=\frac{\partial h_1}{\partial X}f={\hat{\xi }}_2^1\hfill \end{array}$$

(63)

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{\hat{\xi }}}_{r_1-1}^1}& =\frac{\partial {\phi }_{r_1-1}^1}{\partial \hat{X}}\frac{d\hat{X}}{\mathrm{dt}}=\frac{\partial L_f^{r_1-2}{h}_1}{\partial \hat{X}}[f+g·u]\hfill \\ \multicolumn{1}{c}{}& =\frac{\partial L_f^{r_1-2}{h}_1}{\partial \hat{X}}f+\frac{\partial L_f^{r_1-2}{h}_1}{\partial \hat{X}}{g}_1{u}_1+\cdots +\hfill \\ \multicolumn{1}{c}{}& \frac{\partial L_f^{r_1-2}{h}_1}{\partial \hat{X}}{g}_m{u}_m=L_f^{r_1-1}{h}_1={\hat{\xi }}_{r_1}^1\hfill \end{array}$$

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{\hat{\xi }}}_{r_1}^1}& =\frac{\partial {\phi }_{r_1}^1}{\partial \hat{X}}\frac{d\hat{X}}{\mathrm{dt}}=\frac{\partial L_f^{r_1-1}{h}_1}{\partial \hat{X}}[f+g·u]\hfill \\ \multicolumn{1}{c}{}& =\frac{\partial L_f^{r_1-1}{h}_1}{\partial \hat{X}}f+\frac{\partial L_f^{r_1-1}{h}_1}{\partial \hat{X}}{g}_1{u}_1+\cdots +\frac{\partial L_f^{r_1-1}{h}_1}{\partial \hat{X}}{g}_m{u}_m\hfill \\ \multicolumn{1}{c}{}& =L_f^{r_1}{h}_1+L_{g_1}{L}_f^{r_1-1}{h}_1{u}_1+\cdots +L_{g_m}{L}_f^{r_1-1}{h}_1{u}_m\hfill \end{array}$$

$$=c_1+d_{11}{u}_1+\cdots +d_{1m}{u}_m$$

(64)

$$\begin{array}{c}{\stackrel{·}{\hat{\xi }}}_1^m=\frac{\partial {\phi }_1^m}{\partial \hat{X}}\frac{d\hat{X}}{\mathrm{dt}}=\frac{\partial h_m}{\partial \hat{X}}[f+g·u]\hfill \\ =\frac{\partial h_m}{\partial \hat{X}}f+\frac{\partial h_m}{\partial \hat{X}}{g}_1{u}_1+\cdots +\frac{\partial h_m}{\partial \hat{X}}{g}_m{u}_m=L_f^1{h}_m={\hat{\xi }}_2^m\hfill \end{array}$$

(65)

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{\hat{\xi }}}_{r_m-1}^m}& =\frac{\partial {\phi }_{r_m-1}^m}{\partial \hat{X}}\frac{d\hat{X}}{\mathrm{dt}}=\frac{\partial L_f^{r_m-2}{h}_m}{\partial \hat{X}}[f+g·u]\hfill \\ \multicolumn{1}{c}{}& =\frac{\partial L_f^{r_m-2}{h}_m}{\partial \hat{X}}f+\frac{\partial L_f^{r_m-2}{h}_m}{\partial \hat{X}}{g}_1{u}_1+\cdots +\hfill \\ \multicolumn{1}{c}{}& \frac{\partial L_f^{r_m-2}{h}_m}{\partial \hat{X}}{g}_m{u}_m=L_f^{r_m-1}{h}_m={\hat{\xi }}_{r_m}^m\hfill \end{array}$$

(66)

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{\hat{\xi }}}_{r_m}^m}& =\frac{\partial {\phi }_{r_m}^m}{\partial \hat{X}}\frac{d\hat{X}}{\mathrm{dt}}=\frac{\partial L_f^{r_m-1}{h}_m}{\partial \hat{X}}[f+g·u]\hfill \\ \multicolumn{1}{c}{}& =\frac{\partial L_f^{r_m-1}{h}_m}{\partial \hat{X}}f+\frac{\partial L_f^{r_m-1}{h}_m}{\partial \hat{X}}{g}_1{u}_1+\cdots +\frac{\partial L_f^{r_m-1}{h}_m}{\partial \hat{X}}{g}_m{u}_m\hfill \\ \multicolumn{1}{c}{}& =L_f^{r_m}{h}_m+L_{g_1}{L}_f^{r_m-1}{h}_m{u}_1+\cdots +L_{g_m}{L}_f^{r_m-1}{h}_m{u}_m\hfill \\ \multicolumn{1}{c}{}& =c_m+d_{m1}{u}_1+\cdots +d_{\mathrm{mm}}{u}_m\hfill \end{array}$$

(67)

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{\hat{\eta }}}_k(t)}& =\frac{\partial {\phi }_k}{\partial \hat{X}}\frac{d\hat{X}}{\mathrm{dt}}=\frac{\partial {\phi }_k}{\partial \hat{X}}[f+g·u]\hfill \\ \multicolumn{1}{c}{}& =\frac{\partial {\phi }_k}{\partial \hat{X}}f+\frac{\partial {\phi }_k}{\partial \hat{X}}{g}_1{u}_1+\cdots +\frac{\partial {\phi }_k}{\partial \hat{X}}{g}_m{u}_m\hfill \\ \multicolumn{1}{c}{}& =L_f{\phi }_k==q_k,k=r+1,r+2,\cdots ,n\hfill \end{array}$$

(68)

Since

$$c_i(\hat{\xi }(t),\hat{\eta }(t))\equiv L_f^{r_i}{h}_i(\hat{X}(t)),1\le i\le m$$

(69)

$$d_{\mathrm{ij}}\equiv L_{g_j}{L}_f^{r_i-1}{h}_i\left(\hat{X}\right),1\le i\le m,1\le j\le m$$

(70)

$$q_k(\hat{\xi }(t),\hat{\eta }(t))=L_f{\phi }_k\left(\hat{X}\right),k=r+1,r+2,\dots ,n$$

(71)

The dynamic equations of the observer will be rewritten as follows

$${\stackrel{·}{\hat{\xi }}}_i^1(t)={\hat{\xi }}_{i+1}^1(t),i=1,2,\dots ,r_1-1$$

(72)

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{\hat{\xi }}}_{r_1}^1(t)}& =c_1(\hat{\xi }(t),\hat{\eta }(t))+d_{11}(\hat{\xi }(t),\hat{\eta }(t))u_1+\cdots +\hfill \\ \multicolumn{1}{c}{}& d_{1m}(\hat{\xi }(t),\hat{\eta }(t))u_m\hfill \end{array}$$

(73)

$${\stackrel{·}{\hat{\xi }}}_i^m(t)={\hat{\xi }}_{i+1}^m(t),i=1,2,\dots ,r_m-1$$

(74)

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{\hat{\xi }}}_{r_m}^m(t)=}& c_m(\hat{\xi }(t),\hat{\eta }(t))+d_{m1}(\hat{\xi }(t),\hat{\eta }(t))u_1+\cdots +\hfill \\ \multicolumn{1}{c}{}& d_{\mathrm{mm}}(\hat{\xi }(t),\hat{\eta }(t))u_m\hfill \end{array}$$

(75)

$${\stackrel{·}{\hat{\eta }}}_k(t)=q_k(\hat{\xi }(t),\hat{\eta }(t)),k=r+1,\dots ,n$$

(76)

$$y_i(t)={\hat{\xi }}_1^i(t),1\le i\le m$$

(77)

According to equations (28), (51), (69), and (70), the designed controller can be described as equation (48). Substituting equations (48) into (72)–(75) gets the dynamic equations of the observer

$$\left[\begin{array}{c}{\stackrel{.}{\widehat{\xi }}}_1^i(t)\\ {\stackrel{.}{\widehat{\xi }}}_2^i(t)\\ ⋮\\ {\stackrel{.}{\widehat{\xi }}}_{r_i-1}^i(t)\\ {\stackrel{.}{\widehat{\xi }}}_{r_i}^i(t)\end{array}\right]=\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& 0\cdots & 0\\ & ⋮& & & ⋮\\ 0& 0& 0& \cdots & 1\\ 0& 0& 0& \cdots & 0\end{array}\right]\left[\begin{array}{c}{\widehat{\xi }}_1^i(t)\\ {\widehat{\xi }}_2^i(t)\\ ⋮\\ {\widehat{\xi }}_{r_i-1}^i(t)\\ {\widehat{\xi }}_{r_i}^i(t)\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\\ 1\end{array}\right]v_i$$

(78)

$$\left[\begin{array}{c}{\stackrel{·}{\hat{\eta }}}_{r+1}(t)\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{\hat{\eta }}}_{r+2}(t)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{\eta }}}_{n-1}(t)}\\ \multicolumn{1}{c}{{\stackrel{·}{\hat{\eta }}}_n(t)}\end{array}\right]=\left[\begin{array}{c}{q}_{r+1}(t)\hfill \\ \multicolumn{1}{c}{q_{r+2}(t)}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{q_{n-1}(t)}\\ \multicolumn{1}{c}{q_n(t)}\end{array}\right],1\le i\le m$$

(79)

$$\begin{array}{c}{y}_i={\left[\begin{array}{ccccc}1\hfill & 0\hfill & \cdots \hfill & 0\hfill & 0\hfill \end{array}\right]}_{r\times 1}\hfill \\ {{\left[\begin{array}{ccccc}{\hat{\xi }}_1^i(t)\hfill & {\hat{\xi }}_2^i(t)\hfill & \cdots \hfill & {\hat{\xi }}_{r_i-1}^i(t)\hfill & {\hat{\xi }}_{r_i}^i(t)\hfill \end{array}\right]}_{1\times r}}^T={\hat{\xi }}_1^i(t),1\le i\le m\hfill \end{array}$$

(80)

Combining equations (28), (30), (31), (39), and (51) shows that equations (78)–(80) can be derived into the following forms

$$\stackrel{·}{\hat{\eta }}(t)=q(\hat{\xi }(t),\hat{\eta }(t))\equiv q_{22}(t,\hat{\eta }(t),\overline{\hat{e}})$$

(81)

$$\epsilon \stackrel{.}{\overline{{\hat{e}}^i}}(t)=A_c^i\overline{{\hat{e}}^i},1\le i\le m$$

(82)

$$y_i(t)={\hat{\xi }}_1^i(t),1\le i\le m$$

(83)

Define the function $L(\overline{\hat{e}},\hat{\eta },\tilde{X})$

$$\begin{array}{cc}\multicolumn{1}{c}{L(\overline{\hat{e}},\hat{\eta },\tilde{X})}& \equiv V(\hat{\eta })+k_1{W}_1(\overline{\hat{e}})+k_2{W}_2(\tilde{X})\hfill \\ \multicolumn{1}{c}{}& \equiv V(\hat{\eta })+k_1(W_1^1+\cdots +W_1^m)\hfill \\ \multicolumn{1}{c}{}& +k_2{W}_2(\tilde{X}),k_1>0,k_2>0\hfill \end{array}$$

(84)

where

$$W_1\equiv W_1^1+\cdots +W_1^m$$

(85)

to be a composite Lyapunov function of the subsystems equations (81) and (82),^34,35 where $W_1^i$ and $W_2$ are

$$W_1^i\equiv \frac{1}{2}{\overline{{\hat{e}}^i}}^T{P}_1^i\overline{{\hat{e}}^i},1\le i\le m$$

(86)

and

$$W_2\equiv \frac{1}{2}{\tilde{X}}^T{P}_2\tilde{X}$$

(87)

In view of equations (28) and (52), the derivative function of the composite Lyapunov function along the trajectories of equations (12), (81), and (82) is given by

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{L}}_y}& =[{\nabla }_{t}V+{\left({\nabla }_{\hat{\eta }}V\right)}^T\stackrel{·}{\hat{\eta }}]\hfill \\ \multicolumn{1}{c}{}& +\frac{k_1}{2}[{\left(\stackrel{•}{\overline{{\hat{e}}^1}}\right)}^T{P}_1^1\overline{{\hat{e}}^1}+{\left(\overline{{\hat{e}}^1}\right)}^T{P}_1^1\left(\stackrel{•}{\overline{{\hat{e}}^1}}\right)+\cdots +{\left(\stackrel{•}{\overline{{\hat{e}}^m}}\right)}^T{P}_1^m\overline{{\hat{e}}^m}+{\left(\overline{{\hat{e}}^m}\right)}^T{P}_1^m\left(\stackrel{•}{\overline{{\hat{e}}^m}}\right)]\hfill \\ \multicolumn{1}{c}{}& +\frac{k_2}{2}[{\stackrel{·}{\stackrel{~}{X}}}^T{P}_2\tilde{X}+{\tilde{X}}^T{P}_2\stackrel{·}{\stackrel{~}{X}}]\hfill \end{array}$$

$$\begin{array}{c}=[{\nabla }_{t}V+{\left({\nabla }_{\hat{\eta }}V\right)}^T\stackrel{·}{\hat{\eta }}]\hfill \\ +\frac{k_1}{2}[{\left(\frac{1}{\epsilon }{A}_c^1\overline{{\hat{e}}^1}\right)}^T{P}_1^1\overline{{\hat{e}}^1}+{\left(\overline{{\hat{e}}^1}\right)}^T{P}_1^1\left(\frac{1}{\epsilon }{A}_c^1\overline{{\hat{e}}^1}\right)+\cdots +{\left(\frac{1}{\epsilon }{A}_c^m\overline{{\hat{e}}^m}\right)}^T{P}_1^m\overline{{\hat{e}}^m}+{\left(\overline{{\hat{e}}^m}\right)}^T{P}_1^m\left(\frac{1}{\epsilon }{A}_c^m\overline{{\hat{e}}^m}\right)]\hfill \\ +\frac{k_2}{2}{\{[(F_{\ell }-L{H}^T)\tilde{X}+(F_n(X)-F_n(\hat{X}))+Q\theta ]}^T{P}_2\tilde{X}+{\tilde{X}}^T{P}_2[(F_{\ell }-L{H}^T)\tilde{X}+(F_n(X)-F_n(\hat{X}))+Q\theta ]\}\hfill \end{array}$$

$$\begin{array}{c}=[{\nabla }_{t}V+{\left({\nabla }_{\hat{\eta }}V\right)}^T(q_{22}(t,\hat{\eta }(t),\overline{\hat{e}}))]\hfill \\ +\{\frac{k_1}{2\epsilon }{\left(\overline{{\hat{e}}^1}\right)}^T[{\left(A_c^1\right)}^T{P}_1^1+P_1^1\left(A_c^1\right)]\overline{{\hat{e}}^1}+\cdots +\frac{k_1}{2\epsilon }{\left(\overline{{\hat{e}}^m}\right)}^T[{\left(A_c^m\right)}^T{P}_1^m+P_1^m\left(A_c^m\right)]\overline{{\hat{e}}^m}\}\hfill \\ +\frac{k_2}{2}\left\{{\tilde{X}}^T[{(F_{\ell }-L{H}^T)}^T{P}_2+P_2(F_{\ell }-L{H}^T)]\tilde{X}\right\}+\frac{k_2}{2}\{{(F_n(X)-F_n(\hat{X}))}^T{P}_2\tilde{X}+{\tilde{X}}^T{P}_2(F_n(X)-F_n(\hat{X}))\}\hfill \\ +\frac{k_2}{2}\{{\theta }^T{Q}^T{P}_2\tilde{X}+{\tilde{X}}^T{P}_{2}Q\theta \}\hfill \end{array}$$

$$\begin{array}{c}=[{\nabla }_{t}V+{\left({\nabla }_{\hat{\eta }}V\right)}^T(q_{22}(t,\hat{\eta }(t),\overline{\hat{e}}))]-\frac{k_1}{2\epsilon }\left[(\overline{{\hat{e}}^1}\overline{{\hat{e}}^1}+\cdots +{\left(\overline{{\hat{e}}^m}\right)}^T\overline{{\hat{e}}^m})\right]\hfill \\ -\frac{k_2}{2}{\tilde{X}}^T\tilde{X}+\frac{k_2}{2}\{{(F_n(X)-F_n(\hat{X}))}^T{P}_2\tilde{X}+{\tilde{X}}^T{P}_2(F_n(X)-F_n(\hat{X}))\}+\frac{k_2}{2}\left\{2{\theta }^T{Q}^T{P}_2\tilde{X}\right\}\hfill \end{array}$$

$$\begin{array}{c}\le [{\nabla }_{t}V+{\left({\nabla }_{\hat{\eta }}V\right)}^T{q}_{22}(t,\hat{\eta }(t),0)]+{\left({\nabla }_{\hat{\eta }}V\right)}^T[q_{22}(t,\hat{\eta }(t),\overline{\hat{e}})-q_{22}(t,\hat{\eta }(t),0)]\hfill \\ -\frac{k_1}{2\epsilon }\left[({\Vert \overline{{\hat{e}}^1}\Vert }^2+\cdots +{\Vert \overline{{\hat{e}}^m}\Vert }^2)\right]-\frac{k_2}{2}\Vert \tilde{X}{\Vert }^2+\frac{k_2}{2}\{2\Vert F_n(X)-F_n(\hat{X})\Vert \Vert P_2\Vert \Vert \tilde{X}\Vert \}+\frac{k_2}{2}\{2\Vert \theta \Vert \Vert Q\Vert \Vert P_2\Vert \Vert \tilde{X}\Vert \}\hfill \end{array}$$

$$\begin{array}{c}\le -2{\alpha }_{\hat{x}}\Vert \hat{\eta }{\Vert }^2+{\omega }_3\Vert \hat{\eta }\Vert M_1\Vert \overline{\hat{e}}\Vert -\frac{k_1}{2\epsilon }\Vert \overline{\hat{e}}{\Vert }^2-\frac{k_2}{2}\Vert \tilde{X}{\Vert }^2+\frac{k_2}{2}\{2M_2{\Vert \tilde{X}\Vert }^2\Vert P_2\Vert \}\hfill \\ +\frac{k_2}{2}\{2\Vert \theta \Vert \Vert Q\Vert \Vert P_2\Vert \Vert \tilde{X}\Vert \}\hfill \end{array}$$

$$\begin{array}{c}\le -2{\alpha }_{\hat{x}}\Vert \hat{\eta }{\Vert }^2+{\omega }_3^2{M}_1^2\Vert \hat{\eta }{\Vert }^2+\frac{1}{4}\Vert \overline{\hat{e}}{\Vert }^2-\frac{k_1}{2\epsilon }\Vert \overline{\hat{e}}{\Vert }^2-\frac{k_2}{2}\Vert \tilde{X}{\Vert }^2+k_2{M}_2\Vert P_2\Vert \Vert \tilde{X}{\Vert }^2\hfill \\ +k_2^2\Vert P_2{\Vert }^2\Vert Q{\Vert }^2\Vert \tilde{X}{\Vert }^2+\frac{1}{4}\Vert \theta {\Vert }^2\hfill \end{array}$$

$$=-\Vert \hat{\eta }{\Vert }^2(2{\alpha }_{\hat{x}}-{\omega }_3^2{M}_1^2)-\Vert \overline{\hat{e}}{\Vert }^2(-\frac{1}{4}+\frac{k_1}{2\epsilon })-\Vert \tilde{X}{\Vert }^2(\frac{k_2}{2}-k_2{M}_2\Vert P_2\Vert -k_2^2{\Vert P_2\Vert }^2{\Vert Q\Vert }^2)+\frac{1}{4}\Vert \theta {\Vert }^2$$

(88)

that is

$$\stackrel{·}{L}\le -k_{11}\Vert \overline{\hat{e}}{\Vert }^2-k_{22}\Vert \hat{\eta }{\Vert }^2-k_{33}\Vert \tilde{X}{\Vert }^2+\frac{1}{4}\Vert \theta {\Vert }^2$$

(89)

As reflected in equation (59), we get

$$\stackrel{·}{L}\le -N_2({\Vert \overline{\hat{e}}\Vert }^2+{\Vert \hat{\eta }\Vert }^2+{\Vert \tilde{X}\Vert }^2)+\frac{1}{4}\Vert \theta {\Vert }^2$$

(90)

Let

$$\overline{\hat{e}}\equiv \left[\begin{array}{c}\overline{{\hat{e}}^1}\hfill \\ \multicolumn{1}{c}{\overline{{\hat{e}}^2}}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{\overline{{\hat{e}}^m}}\end{array}\right]\equiv \left[\begin{array}{c}\overline{{\hat{e}}_1^1}\hfill \\ \multicolumn{1}{c}{\overline{{\hat{e}}_{\mathrm{rem}}^1}}\end{array}\right],\overline{{\hat{e}}_{\mathrm{rem}}^1}\in {\Re }^{m-1}$$

(91)

Consequently, we get the following inequality

$$\stackrel{·}{L}\le -N_2({\Vert \hat{\eta }\Vert }^2+{\Vert \overline{{\hat{e}}_1^1}\Vert }^2+{\Vert \overline{{\hat{e}}_{\mathrm{rem}}^1}\Vert }^2+{\Vert \tilde{X}\Vert }^2)+\frac{1}{4}\Vert \theta {\Vert }^2$$

(92)

Applying equation (92) easily yields

$$\underset{t_0}{\overset{t}{\int }}{(y_1(\tau )-y_d^1(\tau ))}^{2}d\tau \le \frac{L\left(t_0\right)}{N_2}+\frac{1}{4N_2}\underset{t_0}{\overset{t}{\int }}{\Vert \theta (\tau )\Vert }^{2}d\tau$$

(93)

Similarly, it is also to prove that

$$\begin{array}{c}\underset{t_0}{\overset{t}{\int }}{(y_i(\tau )-y_d^i(\tau ))}^{2}d\tau \le \frac{L\left(t_0\right)}{N_2}+\frac{1}{4N_2}\hfill \\ \underset{t_0}{\overset{t}{\int }}{\Vert \theta (\tau )\Vert }^{2}d\tau ,2\le i\le m\hfill \end{array}$$

(94)

So that the almost disturbance decoupling property equation (47) holds. From equation (90), we obtain

$$\stackrel{·}{L}\le -N_2\left({\Vert {\hat{y}}_{\mathrm{total}}\Vert }^2\right)+\frac{1}{4}\Vert \theta {\Vert }^2$$

(95)

where

$$\Vert {\hat{y}}_{\mathrm{total}}{\Vert }^2\equiv \Vert \overline{\hat{e}}{\Vert }^2+\Vert \hat{\eta }{\Vert }^2+\Vert \tilde{X}{\Vert }^2$$

(96)

From Khalil ³² (Theorem 5.2), equation (95) then implies the almost disturbance decoupling input-to-state stable property for the closed-loop system. Furthermore, it is easy to derive that

$$\begin{array}{c}{\mathrm{\Delta }}_{min}({\Vert \overline{\hat{e}}\Vert }^2+{\Vert \hat{\eta }\Vert }^2+{\Vert \tilde{X}\Vert }^2)\hfill \\ \le L\le {\mathrm{\Delta }}_{max}({\Vert \overline{\hat{e}}\Vert }^2+{\Vert \hat{\eta }\Vert }^2+{\Vert \tilde{X}\Vert }^2)\hfill \end{array}$$

(97)

that is

$${\mathrm{\Delta }}_{min}\left({\Vert {\hat{y}}_{\mathrm{total}}\Vert }^2\right)\le L\le {\mathrm{\Delta }}_{max}\left({\Vert {\hat{y}}_{\mathrm{total}}\Vert }^2\right)$$

(98)

where ${\mathrm{\Delta }}_{min}\equiv min\{{\omega }_1,\frac{k_1}{2}{\lambda }_{min}^{*},\frac{k_2}{2}{\lambda }_{min}(P_2)\}$ and ${\mathrm{\Delta }}_{max}\equiv max\{{\omega }_2,\frac{k_1}{2}{\lambda }_{max}^{*},\frac{k_2}{2}{\lambda }_{max}(P_2)\}$ . From equations (90) and (98), we obtain

$$\stackrel{·}{L}\le -\frac{N_2}{{\mathrm{\Delta }}_{max}}L+\frac{1}{4}{(\underset{t_0\le \tau \le t}{sup}\Vert \theta (\tau )\Vert )}^2$$

(99)

then

$$L(t)\le L(t_0)e^{-\frac{N_2}{{\mathrm{\Delta }}_{max}}(t-t_0)}+\frac{{\mathrm{\Delta }}_{max}}{4N_2}{(\underset{t_0\le \tau \le t}{sup}\Vert \theta (\tau )\Vert )}^2,t\ge t_0$$

(100)

which gets

$$\begin{array}{cc}\multicolumn{1}{c}{|y_1(t)-y_d^1(t)|}& \le \sqrt{\frac{2L(t_0)}{k_1{\lambda }_{min}^{*}}}{e}^{-\frac{N_2}{2{\mathrm{\Delta }}_{max}}(t-t_0)}\hfill \\ \multicolumn{1}{c}{}& +\sqrt{\frac{{\mathrm{\Delta }}_{max}}{2k_1{\lambda }_{min}^{*}{N}_2}}(\underset{t_0\le \tau \le t}{sup}\Vert \theta (\tau )\Vert )\hfill \end{array}$$

(101)

Similarly, it is easy to derive that

$$\begin{array}{c}|y_i(t)-y_d^i(t)|\le \sqrt{\frac{2L(t_0)}{k_1{\lambda }_{min}^{*}}}{e}^{-\frac{N_2}{2{\mathrm{\Delta }}_{max}}(t-t_0)}\hfill \\ +\sqrt{\frac{{\mathrm{\Delta }}_{max}}{2k_1{\lambda }_{min}^{*}{N}_2}}(\underset{t_0\le \tau \le t}{sup}\Vert \theta (\tau )\Vert ),2\le i\le m\hfill \end{array}$$

(102)

So that the almost disturbance decoupling property equation (46) is proved and then the almost disturbance decoupling problem is globally solved. Finally, we will show that the sphere $B_{\underset{¯}{r}}$ is a global attractor for the output tracking error system. From equations (95) and (60), we obtain

$$\stackrel{·}{V}\le -N_2\left({\Vert {\hat{y}}_{\mathrm{total}}\Vert }^2\right)+N_1$$

(103)

For $\Vert {\hat{y}}_{\mathrm{total}}\Vert >\underset{¯}{r}$ , we get $\stackrel{·}{V}<0$ . Then any sphere described by

$$B_{\underset{¯}{r}}:=\{\left[\begin{array}{c}\overline{\hat{e}}\hfill \\ \multicolumn{1}{c}{\hat{\eta }}\end{array}\right]:{\Vert \overline{\hat{e}}\Vert }^2+{\Vert \hat{\eta }\Vert }^2+{\Vert \tilde{X}\Vert }^2\le \underset{¯}{r}\}$$

(104)

is a global final attractor for the tracking error system. Furthermore, it is easy to show that, for ${\hat{y}}_{\mathrm{total}}\notin B_{\overline{r}}$ , we have

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{\stackrel{·}{L}}{L}}& \le \frac{-N_2{\Vert {\hat{y}}_{\mathrm{total}}\Vert }^2+N_1}{L}\le \frac{-N_2{\Vert {\hat{y}}_{\mathrm{total}}\Vert }^2+N_1}{{\mathrm{\Delta }}_{max}{\Vert {\hat{y}}_{\mathrm{total}}\Vert }^2}\hfill \\ \multicolumn{1}{c}{}& \le \frac{-N_2}{{\mathrm{\Delta }}_{max}}+\frac{N_1}{{\mathrm{\Delta }}_{max}{\underset{¯}{r}}^2}\equiv -{\alpha }^{*}\hfill \end{array}$$

(105)

that is

$$\stackrel{·}{L}\le -{\alpha }^{*}L$$

Utilizing the comparison theorem Miller and Michel ³⁶ gets

$$L({\hat{y}}_{\mathrm{total}}(t))\le L({\hat{y}}_{\mathrm{total}}(t_0))\exp [-{\alpha }^{*}(t-t_0)]$$

Therefore,

$$\begin{array}{c}{\mathrm{\Delta }}_{min}\Vert {\hat{y}}_{\mathrm{total}}{\Vert }^2\le L({\hat{y}}_{\mathrm{total}}(t))\le L({\hat{y}}_{\mathrm{total}}(t_0))\hfill \\ \exp [-{\alpha }^{*}(t-t_0)]\le {\mathrm{\Delta }}_{max}\Vert {\hat{y}}_{\mathrm{total}}(t_0){\Vert }^2\exp [-{\alpha }^{*}(t-t_0)]\hfill \end{array}$$

(106)

Then we obtain

$$\Vert {\hat{y}}_{\mathrm{total}}\Vert \le \sqrt{\frac{{\mathrm{\Delta }}_{max}}{{\mathrm{\Delta }}_{min}}}\Vert {\hat{y}}_{\mathrm{total}}(t_0)\Vert \exp [-\frac{1}{2}{\alpha }^{*}(t-t_0)]$$

and we can conclude that the convergence rate toward the sphere $B_{\underset{¯}{r}}$ is equal to $({\alpha }^{*}/2)$ .

Illustrative example

A double rotor MIMO system is investigated in this section to demonstrate the practical value of main proposed theorem. The double rotor MIMO system mechanical structure, shown in Figure 2, is a laboratory equipment for some flying viewpoints. The double rotor MIMO system, consisting of two electrical rotors that are orthogonal to each other, is a complicated cross-coupling nonlinear system and is fixed by a beam pivoted on a basic element such that the unit can rotate in pitch direction and yaw direction. The master and tail rotors are separately installed at both ends of the beam where the master motor drives vertically the beam to turn up or down at a pitch angle and the tail motor drives horizontally the beam to turn left or right at a yaw angle. The incremental encoders, placed at the base and pivot, are used to measure the pitch angle and yaw angle and then the tachogenerators are connected to get the angular velocities of the two driven DC motors. Weighting counterbalance device is installed to the beam for setting the equilibrium point of the double rotor MIMO system. The desired control inputs are the supplied voltage of driving both DC motors and then the rotation motor speed and the beam position are changed by adjusting the inputs.

Figure 2.

The double rotor multi-input multi-output (MIMO) system.

According to Su and Wang, ³⁷ the dynamic equations of the double rotor MIMO system can be described as follows

$$\frac{d{\alpha }_h}{\mathrm{dt}}={\mathrm{\Omega }}_h,{\mathrm{\Omega }}_h=\frac{S_h+J_{\mathrm{mr}}{\omega }_m\cos \left({\alpha }_v\right)}{J_h}$$

(107)

$$\frac{d{\alpha }_v}{\mathrm{dt}}={\mathrm{\Omega }}_v,{\mathrm{\Omega }}_v=\frac{S_v+J_{\mathrm{tr}}{\omega }_t}{J_v}$$

(108)

$$\frac{d{S}_h}{\mathrm{dt}}=l_t{F}_h\left({\omega }_t\right)+\cos {\alpha }_v-{\mathrm{\Omega }}_h{k}_h$$

(109)

$$\begin{array}{c}\frac{d{S}_v}{\mathrm{dt}}=l_m{F}_v\left({\omega }_m\right)+g((A-B)\cos {\alpha }_v-C\sin {\alpha }_v)\hfill \\ -{\mathrm{\Omega }}_v{k}_v-0.5(A+B+C){\mathrm{\Omega }}_h^2\sin \left(2{\alpha }_v\right)\hfill \end{array}$$

(110)

$$\frac{d{u}_{\mathrm{hh}}}{\mathrm{dt}}=\frac{1}{T_{\mathrm{tr}}}(-u_{\mathrm{hh}}+u_h),{\omega }_t=P_h\left(u_{\mathrm{hh}}\right)$$

(111)

$$\frac{d{u}_{\mathrm{vv}}}{\mathrm{dt}}=\frac{1}{T_{\mathrm{mr}}}(-u_{\mathrm{vv}}+u_v),{\omega }_m=P_v\left(u_{\mathrm{vv}}\right)$$

(112)

where ${\alpha }_v$ and ${\alpha }_h$ are, respectively, the pitch and yaw angles, ${\omega }_t$ and ${\omega }_m$ are the angular frequencies of the master DC motor and the tail DC motor, respectively, $T_{\mathrm{mr}}$ and $T_{\mathrm{tr}}$ are the time constants of the master DC motor and the tail DC motor, respectively, $J_{\mathrm{mr}}$ and $J_{\mathrm{tr}}$ are the moments of inertia in the master DC motor and the tail DC motor, respectively, $S_h$ and $S_v$ are the angular momentums in the horizontal and vertical plane, respectively, for the beam, $l_m,l_t,k_h,k_v,A,B,C,D,E,F$ are related physical parameters of system, $F_h({\omega }_t)$ and $F_v({\omega }_m)$ are nonlinear functions of ${\omega }_t$ and ${\omega }_m$ ; and $u_v$ and $u_h$ are the vertical and horizontal supplied voltage of DC motors, respectively. Define the input and state variables to be $u_1\equiv u_v$ , $u_2\equiv u_h$ , $x_1\equiv S_h$ , $x_2\equiv {\alpha }_h$ , $x_3\equiv u_{\mathrm{hh}}$ , $x_4\equiv S_v$ , $x_5\equiv {\alpha }_v$ , and $x_6\equiv u_{\mathrm{vv}}$ . Substituting the physical values into dynamic equations yields the following equations

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{x}}_1=}& 0.07254{({\tan }^{-1}3.2x_3)}^3-0.075899{({\tan }^{-1}3.2x_3)}^2\hfill \\ \multicolumn{1}{c}{}& +0.037587({\tan }^{-1}3.2x_3)-0.855x_1\hfill \end{array}$$

(113)

$${\stackrel{·}{x}}_2=90x_1$$

(114)

$${\stackrel{·}{x}}_3=-2.6028x_3+2.6028u_2$$

(115)

$$\begin{array}{cc}\multicolumn{1}{c}{{\stackrel{·}{x}}_4}& =0.072183{\left({\tan }^{-1}3x_6\right)}^3-0.0027708{\left({\tan }^{-1}3x_6\right)}^2\hfill \\ \multicolumn{1}{c}{}& +0.019151\left({\tan }^{-1}3x_6\right)-0.049629x_4\hfill \\ \multicolumn{1}{c}{}& -0.15457\cos x_5-0.11466\sin x_5\hfill \end{array}$$

(116)

$${\stackrel{·}{x}}_5=9.1x_4$$

(117)

$${\stackrel{·}{x}}_6=-0.69832x_6+0.69832u_1+{\theta }_1$$

(118)

$$y_1=x_2$$

(119)

$$y_2=x_5$$

(120)

where ${\theta }_1(t)\equiv 0.1\sin t[u_s(t-5)-u_s(t-12)]$ is assumed to be the disturbance and $u_s(t)$ denotes the unit step function. Now we will demonstrate how to build a controller that tracks the desired signals $y_d^1=y_d^2=\sin t$ based on the main proposed theorem in this study. Choose ${\alpha }_1^1={\alpha }_1^2=0.06$ , $A_c^1=A_c^2=-0.06$ , $P^1=P^2=(25/3)$ , and ${\lambda }_{min}^{*}={\lambda }_{max}^{*}=(25/3)$ . From equation (48), we obtain the desired controllers as follows

$$\begin{array}{cc}\multicolumn{1}{c}{u_1=}& \{-[\frac{-23.8431x_6{\left({\tan }^{-1}3x_6\right)}^2+0.60192x_6\left({\tan }^{-1}3x_6\right)-2.06454x_6}{1+{\left(3x_6\right)}^2}+{\mathrm{\Delta }}_{22}]+{\mathrm{\Delta }}_2\}\hfill \\ \multicolumn{1}{c}{}& \left[\frac{1+{\left(3x_6\right)}^2}{23.53131{\left({\tan }^{-1}3x_6\right)}^2-0.60192\left({\tan }^{-1}3x_6\right)+2.06454}\right]\hfill \end{array}$$

(121)

$$\begin{array}{cc}\multicolumn{1}{c}{u_2=}& \{-[\frac{-163.129x_3{({\tan }^{-1}3.2x_3)}^2+113.7887x_3({\tan }^{-1}3.2x_3)-28.1754x_3}{1+{(3.2x_3)}^2}+{\mathrm{\Delta }}_{12}]+{\mathrm{\Delta }}_1\}\hfill \\ \multicolumn{1}{c}{}& \left[\frac{1+{(3.2x_3)}^2}{163.129{({\tan }^{-1}3.2x_3)}^2-113.7887({\tan }^{-1}3.2x_3)+28.1754}\right]\hfill \end{array}$$

(122)

Where

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathrm{\Delta }}_1=}& -\cos t-{\left(\epsilon \right)}^{-3}(x_2-\sin t)-2{\left(\epsilon \right)}^{-2}(90x_1-\cos t)-3{\left(\epsilon \right)}^{-1}\hfill \\ \multicolumn{1}{c}{}& [6.5286{({\tan }^{-1}3.2x_3)}^3-6.8309{({\tan }^{-1}3.2x_3)}^2+3.3828({\tan }^{-1}3.2x_3)-76.95x_1+\sin t]\hfill \end{array}$$

(123)

$$\begin{array}{c}{\mathrm{\Delta }}_2=-\cos t-{\left(\epsilon \right)}^{-3}(x_5-\sin t)-2{\left(\epsilon \right)}^{-2}(9.1x_4-\cos t)-3{\left(\epsilon \right)}^{-1}\hfill \\ \left[\begin{array}{c}0.6568{\left({\tan }^{-1}3x_6\right)}^3-0.02521{\left({\tan }^{-1}3x_6\right)}^2+0.17426\left({\tan }^{-1}3x_6\right)-0.45162x_4\hfill \\ \multicolumn{1}{c}{-1.40658\cos x_5-1.0434\sin x_5+\sin t}\end{array}\right]\hfill \end{array}$$

(124)

$${\mathrm{\Delta }}_{12}=-5.58195{({\tan }^{-1}3.2x_3)}^3+5.84042{({\tan }^{-1}3.2x_3)}^2-2.89232({\tan }^{-1}3.2x_3)+65.79225x_1$$

(125)

$$\begin{array}{c}{\mathrm{\Delta }}_{22}=-0.032599{\left({\tan }^{-1}3x_6\right)}^3+0.00125{\left({\tan }^{-1}3x_6\right)}^2-0.00865\left({\tan }^{-1}3x_6\right)+0.02241x_4\hfill \\ +0.06981\cos x_5+0.05178\sin x_5+12.79988x_4\sin x_5-9.4949946x_4\cos x_5\hfill \end{array}$$

(126)

It is easy to prove that the related conditions of Theorem 1 are satisfied if we choose $\epsilon =0.8$ , $r_1=3$ , $r_2=3$ , $B_d^1=B_d^2=0$ , $M_2=2.528$ , $N_1=0.025$ , $k_1=2$ , $k_{22}=1.25$ , $k_2=100$ , $\Vert P_2\Vert =0.1921$ , $\Vert Q\Vert ={10}^{-3}$ , ${\mathrm{\Delta }}_{max}=9.6$ , ${\mathrm{\Delta }}_{min}=(25/3)$ , $N_2=1.25$

$$H^T=\left[\begin{array}{cccccc}0\hfill & 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill \end{array}\right]$$

(127)

$$L^T=\left[\begin{array}{cccccc}100\hfill & 9\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 10\hfill & 10\hfill & 0\hfill \end{array}\right]$$

(128)

and

$$P_2=\left[\begin{array}{cccccc}0.0968\hfill & -0.0046\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-0.0046}& 0.1071\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0.1921\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0.1492\hfill & -0.0541\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & -0.0541\hfill & 0.1041\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0.125\hfill \end{array}\right]$$

(129)

As discussed in the previous section, the tracking controllers will steer the output tracking errors of the closed-loop system to be exponentially attenuated to the neighborhood of zero by virtue of Theorem 1. The tracking errors of the output variables $x_2$ and $x_5$ without the designed controller are depicted in Figures 3 and 4, respectively. It is obvious to see that the tracking performances are not achieved under the case of no controller. Then, adding the designed controller to the system gets the desired tracking performances of the output variables $x_2$ and $x_5$ as shown in Figures 5 and 6, respectively. The designed controllers u1 and u2 are shown in Figures 7 and 8, respectively. As seen in Figures 9 and 10, the estimated errors $x_2-{\hat{x}}_2$ and $x_5-{\hat{x}}_5$ of observer states converge to zero and then the observer performances are achieved.

Figure 3.

The tracking errors of the output variable $x_2$ without the designed controller.

Figure 4.

The tracking errors of the output variable $x_5$ without the designed controller.

Figure 5.

The tracking errors of the output variable $x_2$ with the designed controller.

Figure 6.

The tracking errors of the output variable $x_5$ with the designed controller.

Figure 7.

The designed controller u1.

Figure 8.

The designed controller u2.

Figure 9.

The estimated errors $x_2-{\hat{x}}_2$ .

Figure 10.

The estimated errors $x_5-{\hat{x}}_5$ .

Conclusion

We propose an observer-based feedback control to solve globally the tracking problem with almost disturbance decoupling property for MIMO nonlinear systems in this study. The investigation and practical application of feedback linearization approach for MIMO nonlinear control systems by parameterized diffeomorphism have been presented. Moreover, a practical example of the double rotor MIMO system demonstrates the applicability of the proposed observer-based feedback linearization approach and the composite Lyapunov approach. The generality of this approach and implications, due to the fact that unlike Jacobian linearization feedback linearization is not only locally valid, are also shown. Simulation results exploit the fact that the proposed methodology is successfully applied to the desired tracking and almost disturbance decoupling problem.