Velocity controller design of a conveyor system in a fish sorting system using modified model reference adaptive control

Abstract

A modified model reference adaptive controller for velocity control of a conveyor system in a fish sorting system with uncertainty parameters, input saturation and bounded disturbances is proposed in this article. In general, an asymptotic tracking can be achieved in model reference adaptive controller systems. However, a tracking performance of model reference adaptive controller controllers can be quite poor as an adaptation rate is increased. To improve the tracking performance and robustness of the proposed controller in the presence of bounded disturbances, the followings are done. First, the reference model in the conventional model reference adaptive controller is replaced by a modified reference model in a modified model reference adaptive controller to reduce unexpected high-frequency oscillation in control input signal when the adaptation rate is increased. Second, estimated parameters in an adaptive law can be varied smoothly under bounded external disturbances and a projection operator is utilized in an adaptive law for the proposed modified model reference adaptive controller to be robust. Third, an auxiliary error vector is introduced for compensating the error dynamics of the system when the saturation input occurs. Finally, the experimental results are shown to verify the better effectiveness and performance of the proposed controller under the bounded disturbance and saturated input than that of a conventional model reference adaptive controller.

Introduction

In a fish sorting system, captured fishes are transported from a fish pump that pumps the captured fishes in a ship into a fish sorting line through a conveyor system. The captured fishes are sorted by an injured rate which is estimated using an image processing system. Thus, a conveyor system speed plays a key role in estimating injured rate of fishes with high accuracy. Accordingly, the velocity of the conveyor system should be controlled in a suitable speed sufficiently to obtain the reliable recognition of the image processing system. The conveyor system in the fish sorting system consists of three or more conveyors that the desired velocities are defined in trapezoidal velocity profiles as Thirachai et al. ¹ proposed. To control the closed-loop dynamics of the conveyor system close to the desired velocities, a system modelling is obtained to develop a model-based controller. However, the conveyor system has some uncertain parameters such as a friction factor, a belt elastic factor and pulling force to be unmeasured in the conveyor system. Many approaches have been suggested subject to controlling systems with model uncertainties such as an extended state observer by Yao et al., ² µ-synthesis-based adaptive robust control by Chen and Wang, ³ backstepping control by Ahn et al., ⁴ sliding mode control by Guan and Pan, ⁵ passivity-based control by Alleyne and Liu ⁶ and so on.

An adaptive controller with its important ability to system uncertainties without requiring explicit unknown plant parameter identification was considered by Astromn and Wittenmark. ⁷ A conventional model reference adaptive controller (CMRAC) tuning directly control parameter is one of the main schemes utilized in an adaptive control field such as Duong et al., ⁸ Nestorovic et al., ⁹ Karason and Annaswamy, ¹⁰ Wagg ¹¹ and Chen et al. ¹² Although asymptotic tracking could be achieved in CMRAC systems, the tracking performance in transient state could be poor in Zang ¹³ because it is impossible to achieve a small deviation of the tracking error in transient with an insufficient adaptation rate. To improve the tracking performance in the transient state, the adaptation rate should be increased, but this results in many unexpected high-frequency elements in a control input signal and leads to instability. There were several efforts modifying controller architecture by Sun ¹⁴ and Stepanyan et al. ¹⁵ and update laws by Datta and Ioannou, ¹⁶ Artega and Tang ¹⁷ for improving it. A novel approach was proposed by Stepanyan and Kalmanje ¹⁸ to modify a reference model in the CMRAC by feeding back the tracking error to the reference model. A tracking error feedback gain in concert with the adaptation rate could provide ability to increase not only the tracking performance, but also the control signal performance in transient phase.

In addition, estimated parameters in update laws can be varied smoothly in the presence of bounded disturbances. Therefore, some modifications in Stepanyan and Kalmanje ¹⁸ such as the dead-zone technique, e-modification and σ-modification were utilized to modify update laws. Another approach utilizing a projection operator was also proposed by Gibson et al. ¹⁹ and Yao et al. ²⁷

For engineering systems, it should be noted that the control input signal is frequently saturated and has proved to be a source of performance degradation. Abramovitch et al. ²⁰ and Sun et al. ²¹ showed that an input control signal saturation could lead to a poor control performance and even a closed-loop system instability if its effects were not considered in the design of the controller. An anti-windup block was added in the controller of Sun et al. ²² to adjust the control strategy in the presence of saturation. Another method utilized an auxiliary error vector added in a modelling error vector to compensate the effect of input control signal saturation in Duong et al. ²³

It can be seen that there are many issues to be considered in a practical system such as uncertain parameters, external disturbance and saturation input. For these issues, good tracking performance should be achieved in both transient and asymptotic states. However, all the above-mentioned approaches only focused on a few issues, whereas an approach developed for a practical system with all above issues has not been considered yet.

This article proposes a model reference adaptive control approach, which the reference model is modified by a feedback of a modelling error signal as in Stepanyan and Kalmanje ¹⁸ for velocity control of the conveyor system in a fish sorting system with uncertain parameters, saturated input and bounded disturbances. In the presence of bounded disturbance, a projection operator inspired by Gibson et al. ¹⁹ is utilized in update laws of the proposed controller to eliminate the drift phenomenon of control parameters. As well, an auxiliary error vector to compensate for its error dynamics is employed when saturation input occurs as in Duong et al. ²³ The contribution of this article is to apply the projection operator to the update laws in the presence of the auxiliary error vector. The closed-loop dynamics of the auxiliary error vector are proved to be bounded in this article, whereas Duong et al. ²³ did not prove it. Additionally, the boundedness of all signals of the modified model reference adaptive controller (M-MRAC) encountered with saturated input is proved, and the constraint of norm of modelling error vector, the feedback gain and the adaptation rate are demonstrated. Experimental results are shown to verify the effectiveness and the performance of the proposed controller.

System modelling

A typical conveyor system in the fish pump system shown in Figure 1 consists of an on-loading conveyor (first conveyor), a camera conveyor (second conveyor) for moving fishes pass the image processing system and a transition conveyor (third conveyor). Each conveyor consists of a mechanical subsystem and an electrical subsystem. The simplified model of the mechanical subsystem of the ith conveyor of the induction conveyor line system is shown in Figure 2 $(i=1,2,3)$ .

Figure 1.

Typical conveyor system in the fish pump system.

Figure 2.

Simplified model of the ith conveyor of the induction conveyor line system.

In Figure 2, $J_{i1}\mathrm{and}{J}_{i2}$ are the moments of inertia of the driving roller and the driven roller, ${\omega }_{i1}\mathrm{and}{\omega }_{i2}$ are the angular velocities of the driving roller and the driven roller, $f_{i1}\mathrm{and}{f}_{i2}$ are the friction coefficients of bearing inside the driving roller and the driven roller and $D_{i1}\mathrm{and}{D}_{i2}$ are diameters of the driving roller and the driven roller, respectively.

To simplify the mechanical subsystem modelling, assumption 1 is proposed as follows:

Assumption 1

• Connection between motor shaft and driving roller is rigid and short.

• Belt slippage on the rollers is negligible.

• Parcel slippage on the belt is negligible.

• The mass of parcel is negligible.

To drive the mechanical subsystem, the electrical subsystem is used. The inverter with direct current (DC) voltage input controls the induction motor to create sufficient torque to drive the mechanical subsystem as shown in Figure 2.

Under the above assumptions, the ith mechanical driven system can be expressed by the following

$${\tau }_i=J_i{\stackrel{·}{\omega }}_{i1}+f_i{\omega }_{i1}+{\tau }_{\mathrm{di}}(t)$$

(1)

where $J_i=J_{i1}+J_{i2}$ , $f_i=f_{i1}+f_{i2}$ , ${\tau }_{\mathrm{di}}(t)$ is a bounded external disturbance torque caused by an abruptly payload put on the ith conveyor and ${\tau }_i$ is a sufficient torque to drive the mechanical subsystem of the ith conveyor given as follows

$${\tau }_i=k_i{u}_i^{*}$$

(2)

where $k_i$ is an amplifier gain, $u_i^{*}$ is DC voltage input of the ith inverter to create the desired torque ${\tau }_i$ . $u_i^{*}$ is defined as a saturated control input of the ith conveyor as follows

$$u_i^{*}=\{\begin{array}{cc}{u}_{\mathrm{imin}}\hfill & \mathrm{for}{u}_i<u_{\mathrm{imin}}\hfill \\ \multicolumn{1}{c}{u_i}& \mathrm{for}{u}_{\mathrm{imin}}\le u_i\le u_{\mathrm{imax}}\hfill \\ \multicolumn{1}{c}{u_{\mathrm{imax}}}& \mathrm{for}{u}_i>u_{\mathrm{imax}}\hfill \end{array}$$

(3)

where $u_i^{*}$ is a designed control input for the ith conveyor by the proposed controller and $u_{\mathrm{imin}}$ and $u_{\mathrm{imax}}$ are limited thresholds of the ith designed control inputs.

A dynamic induction conveyor line system based on equations (1)–(3) can be expressed in the state space as follows

$$\stackrel{·}{\mathbf{x}}=\mathbf{Ax}+\mathbf{B}({\mathbf{u}}^{*}-\mathbf{d}(t))$$

(4)

where $\mathbf{x}=[\begin{array}{ccc}{\omega }_{11}\hfill & {\omega }_{21}\hfill & {\omega }_{31}\hfill \end{array}{]}^T$ is an angular velocity output vector of the induction conveyor line system measured by encoders attached to the driving rollers, ${\omega }_{i1}$ is an angular velocity of the driving roller of the ith conveyor, ${\mathbf{u}}^{*}=\mathrm{sat}(\mathbf{u})=[\begin{array}{ccc}{u}_1^{*}\hfill & u_2^{*}\hfill & u_3^{*}\hfill \end{array}{]}^T$ is a saturated control input vector, $\mathbf{d}(t)=[\begin{array}{ccc}{d}_1(t)\hfill & d_2(t)\hfill & d_3(t)\hfill \end{array}{]}^T$ is a bounded external disturbance vector with $d_i(t)=({\tau }_{\mathrm{di}}(t)/k_i)$ and unknown constant matrices $\mathbf{A},\mathbf{B}\in {\Re }^{3\times 3}$ are given as follows: $\mathbf{A}=\left[\begin{array}{ccc}{a}_{11}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& a_{22}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & a_{33}\hfill \end{array}\right]$ , $\mathbf{B}=\left[\begin{array}{ccc}{b}_{11}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& b_{22}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & b_{33}\hfill \end{array}\right]$ with $a_{\mathrm{ii}}=-(f_i/J_i)$ and $b_{\mathrm{ii}}=(k_i/J_i)$ .

Controller design

The control objective is to determine a designed control input vector $\mathbf{u}=[\begin{array}{ccc}{u}_1\hfill & u_2\hfill & u_3\hfill \end{array}{]}^T$ for a model reference adaptive system with the saturated inputs and bounded disturbance such that the angular velocity output vector tracks a output vector of a reference model.

Conventional model reference adaptive controller

The conventional reference model used such that its output vector tracks asymptotically a reference input vector $\mathbf{r}$ in trapezoidal type is chosen as follows

$${\stackrel{·}{\mathbf{x}}}_r={\mathbf{A}}_m{\mathbf{x}}_r+{\mathbf{B}}_m\mathbf{r}+\stackrel{·}{\mathbf{r}}$$

(5)

where ${\mathbf{x}}_r=[\begin{array}{ccc}{\omega }_{r1}\hfill & {\omega }_{r2}\hfill & {\omega }_{r3}\hfill \end{array}{]}^T$ is an angular velocity output vector of the reference model, ${\omega }_{\mathrm{ri}}$ is the ith reference angular velocity, $\mathbf{r}=[\begin{array}{ccc}{r}_1\hfill & r_2\hfill & r_3\hfill \end{array}{]}^T$ is the reference angular velocity input vector and ${\mathbf{A}}_{\mathbf{m}},{\mathbf{B}}_{\mathbf{m}}\in {\Re }^{3\times 3}$ are given as follows: ${\mathbf{A}}_m=\left[\begin{array}{ccc}{a}_{m1}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& a_{m2}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & a_{m3}\hfill \end{array}\right]$ , ${\mathbf{B}}_m=\left[\begin{array}{ccc}{b}_{m1}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& b_{m2}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & b_{m3}\hfill \end{array}\right]$ with $a_{\mathrm{mi}}$ and $b_{\mathrm{mi}}$ are reference model parameters and chosen to satisfy assumptions 2 and 3 as follows.

Assumption 2

Given a known Hurwitz matrix ${\mathbf{A}}_{\mathbf{m}}\in {\Re }^{3\times 3}$ and a known matrix ${\mathbf{B}}_{\mathbf{m}}\in {\Re }^{3\times 3}$ of full rank, there exists an unknown control gain matrix $\mathbf{K}\in {\Re }^{3\times 3}$ and an unknown positive definite diagonal constant matrix $\mathbf{\Lambda }\in {\Re }^{3\times 3}$ such that the following equations are held

$$\{\begin{array}{c}\mathbf{A}={\mathbf{A}}_{\mathbf{m}}-\mathbf{BK}\hfill \\ \multicolumn{1}{c}{\mathbf{B}={\mathbf{B}}_{\mathbf{m}}\mathbf{\Lambda }}\end{array}$$

(6)

Remark 1

The true knowledge of the matrix $\mathbf{K}$ is not required, and only its existence is assumed. The sign definiteness of $\mathbf{\Lambda }$ corresponds to the conventional sign condition on the high-frequency gain matrix of multiple input and multiple output (MIMO) system (see, for example, Narendra and Annaswamy ²⁴ ).

Assumption 3

A positive symmetric definite matrix $\mathbf{P}={\mathbf{P}}^T>0$ is the solution of the Lyapunov equation

$${\mathbf{A}}_m^T\mathbf{P}+\mathbf{P}{\mathbf{A}}_m=-{\mathbf{Q}}_m$$

(7)

where ${\mathbf{Q}}_m$ is the positive definite matrix.

Substituting equation (6) into equation (4), adding and subtracting ${\mathbf{B}}_m\mathbf{r},\stackrel{·}{\mathbf{r}}$ yields

$$\stackrel{·}{\mathbf{x}}={\mathbf{A}}_m\mathbf{x}+{\mathbf{B}}_m\mathbf{r}+\stackrel{·}{\mathbf{r}}+{\mathbf{B}}_m\mathbf{\Lambda }({\mathbf{u}}^{*}-{\mathbf{u}}_r)$$

(8)

where $\mathbf{\Phi }={\mathbf{\Lambda }}^{-1}$ , $\mathbf{\Omega }=({\mathbf{B}}_m\mathbf{\Lambda }{)}^{-1}$ and ${\mathbf{u}}_r$ is a conventional control input vector as follows

$${\mathbf{u}}_r=\mathbf{Kx}+\mathbf{\Phi }\mathbf{r}+\mathbf{\Omega }\stackrel{·}{\mathbf{r}}+\mathbf{d}(t)$$

(9)

By defining an error vector ${\mathbf{e}}_r=\mathbf{x}-{\mathbf{x}}_r$ , the first time derivative of ${\mathbf{e}}_r$ is given by

$${\stackrel{·}{\mathbf{e}}}_r={\mathbf{A}}_m{\mathbf{e}}_r+{\mathbf{B}}_m\mathbf{\Lambda }({\mathbf{u}}^{*}-{\mathbf{u}}_r)$$

(10)

If ${\mathbf{u}}^{*}={\mathbf{u}}_r$ , ${\stackrel{·}{\mathbf{e}}}_r={\mathbf{A}}_m{\mathbf{e}}_r$ . Because of the Hurwitz matrix ${\mathbf{A}}_m$ , it can be concluded that ${\mathbf{e}}_r\to 0$ as $t\to \infty$ . It implies that the plant in equation (4) can track asymptotically the reference model in equation (5). However, the ideal control input vector cannot be implemented since the matrices $\mathbf{K},\mathbf{\Phi },\mathbf{\Omega }$ and the disturbance vector $\mathbf{d}(t)$ are unknown. Therefore, a designed control input vector $\mathbf{u}$ as estimation of ${\mathbf{u}}_r$ is chosen in the form

$$\mathbf{u}=\hat{\mathbf{K}}\mathbf{x}+\hat{\mathbf{\Phi }}\mathbf{r}+\hat{\mathbf{\Omega }}\stackrel{·}{\mathbf{r}}+\hat{\mathbf{d}}(t)$$

(11)

where $\hat{\mathbf{K}},\hat{\mathbf{\Phi }},\hat{\mathbf{\Omega }}$ are estimations of unknown control gain matrices $\mathbf{K},\mathbf{\Phi },\mathbf{\Omega }$ , respectively, and $\hat{\mathbf{d}}\in {\Re }^3$ is an estimated vector of an unknown constant vector $\overline{\mathbf{d}}$ which is the average value vector of $\mathbf{d}(t)$ in equation (4). These estimated parameters are updated by conventional update laws of Stepanyan and Kalmanje ²⁵ as follows

$$\begin{array}{c}\stackrel{·}{\hat{\mathbf{K}}}=-\gamma {\mathbf{e}}_r^T\mathbf{P}{\mathbf{B}}_m\mathbf{x},\stackrel{·}{\hat{\mathbf{\Phi }}}=-\gamma {\mathbf{e}}_r^T\mathbf{P}{\mathbf{B}}_m\mathbf{r},\hfill \\ \multicolumn{1}{c}{\stackrel{·}{\hat{\mathbf{\Omega }}}=-\gamma {\mathbf{e}}_r^T\mathbf{P}{\mathbf{B}}_m\stackrel{·}{\mathbf{r}},{\stackrel{·}{\hat{\mathbf{d}}}}^T=-\gamma {\mathbf{e}}_r^T\mathbf{P}{\mathbf{B}}_m}\end{array}$$

(12)

where $\gamma >0$ is an adaptation rate.

Modified model reference adaptive controller

Values of the designed control input vector $\mathbf{u}$ are smaller than that of the ideal control input vector ${\mathbf{u}}_r$ in transient state due to initial large errors between the estimated control gains $\hat{\mathbf{K}},\hat{\mathbf{\Phi }},\hat{\mathbf{\Omega }},\hat{\mathbf{d}}$ and the unknown control gains $\mathbf{K},\mathbf{\Phi },\mathbf{\Omega },\mathbf{d}$ . So tracking performance along with the reference input vector $\mathbf{r}$ in transient state is poor. To improve the tracking performance in transient state, the adaptation rate is increased. However, this results in an unexpected high-frequency element in the control input signals of the control input vector $\mathbf{u}$ . Therefore, the reference model in equation (5) is changed to a modified reference model in the following equation (13)

$${\stackrel{·}{\mathbf{x}}}_m={\mathbf{A}}_m{\mathbf{x}}_m+{\mathbf{B}}_m\mathbf{r}+\stackrel{·}{\mathbf{r}}+\lambda \mathbf{e}$$

(13)

$$\mathbf{e}=\mathbf{x}-{\mathbf{x}}_m$$

(14)

where $\lambda >0$ is an error feedback gain, $\mathbf{e}$ is a modelling error vector, ${\mathbf{x}}_m=[\begin{array}{ccc}{\omega }_{m1}\hfill & {\omega }_{m2}\hfill & {\omega }_{m3}\hfill \end{array}{]}^T$ is an output vector of the modified reference model and ${\omega }_{\mathrm{mi}}$ is the ith reference angular velocity.

When the modelling error vector $\mathbf{e}$ goes to zero, the modified reference model in equation (13) becomes the reference model in equation (5). Thus, the plant asymptotically tracks not only the modified reference model in equation (13) but also the reference model in equation (5).

A saturated input error vector is defined as follows

$$\mathrm{\Delta }\mathbf{u}=\mathbf{u}-{\mathbf{u}}^{*}$$

(15)

From equations (8), (9), (11) and (13)–(15), the first time derivative of $\mathbf{e}$ is given as follows

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\mathbf{e}}=\stackrel{·}{\mathbf{x}}-{\stackrel{·}{\mathbf{x}}}_m=({\mathbf{A}}_m-\lambda \mathbf{I})\mathbf{e}}\\ +{\mathbf{B}}_m\mathbf{\Lambda }(\mathbf{u}-\mathbf{Kx}-\mathbf{\Phi }\mathbf{r}-\mathbf{\Omega }\stackrel{·}{\mathbf{r}}-\overline{\mathbf{d}})+{\mathbf{B}}_m\mathbf{\Lambda }(\overline{\mathbf{d}}-\mathbf{d}-\mathrm{\Delta }\mathbf{u})\hfill \\ =({\mathbf{A}}_m-\lambda \mathbf{I})\mathbf{e}+{\mathbf{B}}_m\mathbf{\Lambda }(\tilde{\mathbf{K}}\mathbf{x}+\tilde{\mathbf{\Phi }}\mathbf{r}+\tilde{\mathbf{\Omega }}\stackrel{·}{\mathbf{r}}+\tilde{\mathbf{d}})\hfill \\ +{\mathbf{B}}_m\mathbf{\Lambda }(\overline{\mathbf{d}}-\mathbf{d}-\mathrm{\Delta }\mathbf{u})\hfill \end{array}$$

(16)

where $\tilde{\mathbf{K}}=\hat{\mathbf{K}}-\mathbf{K}$ , $\tilde{\mathbf{\Phi }}=\hat{\mathbf{\Phi }}-\mathbf{\Phi }$ , $\tilde{\mathbf{\Omega }}=\hat{\mathbf{\Omega }}-\mathbf{\Omega }$ and $\tilde{\mathbf{d}}=\hat{\mathbf{d}}-\overline{\mathbf{d}}$ .

To remove the effect of the saturated input, an auxiliary error vector ${\mathbf{e}}_{\mathrm{\Delta }}$ is defined as follows

$${\stackrel{·}{\mathbf{e}}}_{\mathrm{\Delta }}=({\mathbf{A}}_m-\lambda \mathbf{I}){\mathbf{e}}_{\mathrm{\Delta }}-{\hat{\mathbf{K}}}_{\mathrm{\Delta }}\mathrm{\Delta }\mathbf{u}$$

(17)

where $({\mathbf{A}}_m-\lambda \mathbf{I})$ is a stable Hurwitz matrix and ${\hat{\mathbf{K}}}_{\mathbf{\Delta }}\in {\Re }^{3\times 3}$ is the adaptable parameter matrix.

Therefore, a new error vector is defined as follows

$${\mathbf{e}}_u=\mathbf{e}-{\mathbf{e}}_{\mathrm{\Delta }}$$

(18)

From equations (16) to (18), the first time derivative of ${\mathbf{e}}_u$ is given as follows

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{\mathbf{e}}}_{\mathbf{u}}=\stackrel{·}{\mathbf{e}}-{\stackrel{·}{\mathbf{e}}}_{\mathbf{\Delta }}}\\ =({\mathbf{A}}_m-\lambda \mathbf{I}){\mathbf{e}}_u+{\mathbf{B}}_m\mathbf{\Lambda }(\tilde{\mathbf{K}}\mathbf{x}+\tilde{\mathbf{\Phi }}\mathbf{r}+\tilde{\mathbf{\Omega }}\stackrel{·}{\mathbf{r}}+\tilde{\mathbf{d}})\hfill \\ +{\tilde{\mathbf{K}}}_{\mathrm{\Delta }}\mathrm{\Delta }\mathbf{u}+{\mathbf{B}}_m\mathbf{\Lambda }(\overline{\mathbf{d}}-\mathbf{d})\hfill \end{array}$$

(19)

where ${\tilde{\mathbf{K}}}_{\mathrm{\Delta }}={\hat{\mathbf{K}}}_{\mathrm{\Delta }}-{\mathbf{B}}_m\mathbf{\Lambda }$ .

Estimated control gains $\hat{\mathbf{K}},\hat{\mathbf{\Phi }},\tilde{\mathbf{\Omega }},\hat{\mathbf{d}}$ in equation (12) could be varied smoothly due to effect of bounded disturbances. So, robustness of update laws based on projection operators will be designed in the following Theorem 1.

Theorem 1

An M-MRAC system of equations (4) is stable as long as a designed control input vector of equation (11) is given and update laws using a projection operator are given as follows

$$\stackrel{·}{\hat{\mathbf{\Theta }}}={\gamma }_1\mathrm{Proj}(\hat{\mathbf{\Theta }},\mathbf{Y},\mathbf{f})$$

(20)

$${\stackrel{·}{\hat{\mathbf{K}}}}_{\mathrm{\Delta }}={\gamma }_2\mathrm{Proj}({\hat{\mathbf{K}}}_{\mathrm{\Delta }},\mathbf{R},\mathbf{g})$$

(21)

$$\stackrel{·}{\hat{\mathbf{d}}}={\gamma }_3\mathrm{Proj}(\hat{\mathbf{d}}(t),\mathbf{S},h)$$

(22)

where $\stackrel{·}{\hat{\mathbf{\Theta }}}=[\begin{array}{ccc}{\stackrel{·}{\hat{\mathbf{K}}}}^T\hfill & {\stackrel{·}{\hat{\mathbf{\Phi }}}}^T\hfill & {\stackrel{·}{\hat{\mathbf{\Omega }}}}^T\hfill \end{array}{]}^T$ , $\hat{\mathbf{\Theta }}=[\begin{array}{ccc}{\hat{\mathbf{K}}}^T\hfill & {\hat{\mathbf{\Phi }}}^T\hfill & {\hat{\mathbf{\Omega }}}^T\hfill \end{array}{]}^T$ , $\mathbf{Y}=[\begin{array}{ccc}{\mathbf{Y}}_1^T\hfill & {\mathbf{Y}}_2^T\hfill & {\mathbf{Y}}_3^T\hfill \end{array}{]}^T$ with ${\mathbf{Y}}_1=-{\mathbf{B}}_m^T\mathbf{P}{\mathbf{e}}_u{\mathbf{x}}^T$ , ${\mathbf{Y}}_2=-{\mathbf{B}}_m^T\mathbf{P}{\mathbf{e}}_u{\mathbf{r}}^T$ and ${\mathbf{Y}}_3=-{\mathbf{B}}_m^T\mathbf{P}{\mathbf{e}}_u{\stackrel{·}{\mathbf{r}}}^T$ , $\mathbf{R}=-\mathbf{P}{\mathbf{e}}_u\mathrm{\Delta }{\mathbf{u}}^T$ , $\mathbf{S}=-{\mathbf{B}}_m^T\mathbf{P}{\mathbf{e}}_u$ , $\mathbf{f},\mathbf{g}\in {\Re }^3$ are convex function vectors, h is a convex scalar function and ${\gamma }_1,{\gamma }_2,{\gamma }_3>0$ are adaptation rates.

Definition 1

A general form of the projection operator defined by (20) is a $n\times m$ matrix extending a vector projection operator definition as follows

$$\mathrm{Proj}(\hat{\mathbf{\Theta }},\mathbf{Y},\mathbf{f})=[\begin{array}{ccc}\mathrm{Proj}({\hat{\mathbf{\theta }}}_1,{\mathbf{y}}_1,f_1)\hfill & \cdots \hfill & \mathrm{Proj}({\hat{\mathbf{\theta }}}_m,{\mathbf{y}}_m,f_m)\hfill \end{array}]$$

(23)

where $\hat{\mathbf{\Theta }},\mathbf{Y}\in {\Re }^{n\mathrm{x}m}$ and $\mathbf{f}\in {\Re }^m$ is a convex function vector and $\mathrm{Proj}({\hat{\mathbf{\theta }}}_j,{\mathbf{y}}_j,f_j)$ is a projection operator for two vectors ${\hat{\mathbf{\theta }}}_j,{\mathbf{y}}_j\in {\Re }^n(j=1~m)$ introduced in Definition 2:

Definition 2

A projection operator for two vectors ${\mathbf{\theta }}_j=[\begin{array}{ccc}{\theta }_{j1}\hfill & \cdots \hfill & {\theta }_{\mathrm{jn}}\hfill \end{array}{]}^T,{\mathbf{y}}_j=[\begin{array}{ccc}{y}_{j1}\hfill & \cdots \hfill & y_{\mathrm{jn}}\hfill \end{array}{]}^T$ is introduced as

$$\begin{array}{c}\mathrm{Proj}({\mathbf{\theta }}_j,{\mathbf{y}}_j,f_j)=\hfill \\ \{\begin{array}{c}{\mathbf{y}}_j-\frac{\nabla f({\mathbf{\theta }}_j)(\nabla f({\mathbf{\theta }}_j{))}^T}{\parallel \nabla f({\mathbf{\theta }}_j){\parallel }^2}{\mathbf{y}}_{j}f({\mathbf{\theta }}_j)\mathrm{if}f({\mathbf{\theta }}_j)>0\wedge (\nabla f({\mathbf{\theta }}_j){)}^T{\mathbf{y}}_j>0\hfill \\ \multicolumn{1}{c}{{\mathbf{y}}_j\mathrm{otherwise}}\end{array}\hfill \end{array}$$

(24)

where $\nabla f_j({\mathbf{\theta }}_j)={[(\partial f_j({\mathbf{\theta }}_j)/\partial {\theta }_{j1})\cdots (\partial f_j({\mathbf{\theta }}_j)/\partial {\theta }_{\mathrm{jn}})]}^T$ , $\wedge \mathrm{is}$ an and operator and $f_j:{\Re }^n\to \Re$ is a smooth convex function defined as

$$f_j({\mathbf{\theta }}_j)=\frac{\parallel {\mathbf{\theta }}_j{\parallel }^2-{\theta }_{\mathrm{jmax}}^2}{{\epsilon }_j{\theta }_{\mathrm{jmax}}^2}$$

(25)

where ${\theta }_{\mathrm{jmax}}$ is a norm bound imposed on the parameter vector ${\mathbf{\theta }}_j$ and ${\epsilon }_j$ denotes a convergence tolerance.

One important property of the projection operator proven by Lavretsky et al. ²⁶ is as follows

$$({\mathbf{\theta }}_j-{\mathbf{\theta }}_j^{*}{)}^T(\mathrm{Proj}({\mathbf{\theta }}_j,{\mathbf{y}}_j,f_j)-{\mathbf{y}}_j)\le 0$$

(26)

where ${\mathbf{\theta }}_j^{*}\in {\Omega }_0$ and ${\Omega }_0\stackrel{\mathrm{\Delta }}{=}\{{\mathbf{\theta }}_j\in {\Re }^n|f_j({\mathbf{\theta }}_j)\le 0\}$

Lemma 1

Putting $\mathbf{F}=[\begin{array}{ccc}{f}_1\hfill & \cdots \hfill & f_m\hfill \end{array}{]}^T\in {\Re }^m$ to be a convex function vector and $\hat{\mathbf{\Theta }}=[\begin{array}{ccc}{\hat{\mathbf{\theta }}}_1\hfill & \cdots \hfill & {\hat{\mathbf{\theta }}}_m\hfill \end{array}]$ , $\mathbf{\Theta }=[\begin{array}{ccc}{\mathbf{\theta }}_1\hfill & \cdots \hfill & {\mathbf{\theta }}_m\hfill \end{array}]$ and $\mathbf{Y}=[\begin{array}{ccc}{\mathbf{y}}_1\hfill & \cdots \hfill & {\mathbf{y}}_m\hfill \end{array}]$ where $\hat{\mathbf{\Theta }},\mathbf{\Theta },\mathbf{Y}\in {\Re }^{n\times m}$ , the following is satisfied

$$\mathrm{trace}\{(\hat{\mathbf{\Theta }}-\mathbf{\Theta }{)}^T(\mathrm{Proj}(\hat{\mathbf{\Theta }},\mathbf{Y},\mathbf{f})-\mathbf{Y})\}\le 0$$

(27)

Proof

Using equation (25), the following is obtained as

$$\begin{array}{l}trace\{{(\widehat{\mathbf{\Theta }}-\mathbf{\Theta })}^T(\mathrm{Proj}(\widehat{\mathbf{\Theta }},Y,f)-Y)\}\\ ={\displaystyle \sum _{j=1}^m{({\widehat{\mathbf{\theta }}}_j-{\mathbf{\theta }}_j)}^T}(\mathrm{Proj}({\widehat{\mathbf{\theta }}}_j,y_j,f_j)-y_j)\le 0\end{array}$$

(28)

□

Proof of Theorem 1

A candidate Lyapunov function is chosen to analyse the stability of the system as follows

$$\begin{array}{c}\multicolumn{1}{c}{V(t)={\mathbf{e}}_u^T\mathbf{P}{\mathbf{e}}_u+\frac{1}{{\gamma }_1}\mathrm{trace}({\tilde{\mathbf{K}}}^{\mathbf{T}}\mathbf{\Lambda }\tilde{\mathbf{K}}+{\tilde{\mathbf{\Phi }}}^{\mathbf{T}}\mathbf{\Lambda }\tilde{\mathbf{\Phi }}+{\tilde{\mathbf{\Omega }}}^{\mathbf{T}}\mathbf{\Lambda }\tilde{\mathbf{\Omega }})}\\ +\frac{1}{{\gamma }_2}\mathrm{trace}({\tilde{\mathbf{K}}}_{\mathbf{\Delta }}^{\mathbf{T}}{\tilde{\mathbf{K}}}_{\mathbf{\Delta }})+\frac{1}{{\gamma }_3}{\tilde{\mathbf{d}}}^T\mathbf{\Lambda }\tilde{\mathbf{d}}\hfill \end{array}$$

(29)

The first time derivative of $V(t)$ is given as follows

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}(t)=-{\mathbf{e}}_u^T({\mathbf{Q}}_m+2\lambda \mathbf{P}){\mathbf{e}}_u+2{\mathbf{e}}_u^T\mathbf{P}{\mathbf{B}}_m\mathbf{\Lambda }\tilde{\mathbf{K}}\mathbf{x}}\\ +2{\mathbf{e}}_u^T\mathbf{P}{\mathbf{B}}_m\mathbf{\Lambda }\tilde{\mathbf{\Phi }}\mathbf{r}+2{\mathbf{e}}_u^T\mathbf{P}{\mathbf{B}}_m\mathbf{\Lambda }\tilde{\mathbf{\Omega }}\stackrel{·}{\mathbf{r}}+2{\mathbf{e}}_u^T\mathbf{P}{\mathbf{B}}_m\mathbf{\Lambda }\tilde{\mathbf{d}}\hfill \\ +2{\mathbf{e}}_u^T\mathbf{P}{\tilde{\mathbf{K}}}_{\mathrm{\Delta }}\mathrm{\Delta }\mathbf{u}+2{\mathbf{e}}_u^T\mathbf{P}{\mathbf{B}}_m\mathbf{\Lambda }(\mathbf{d}-\overline{\mathbf{d}})\hfill \\ +\frac{2}{{\gamma }_1}\mathrm{trace}({\tilde{\mathbf{K}}}^T\mathbf{\Lambda }\stackrel{·}{\hat{\mathbf{K}}}+{\tilde{\mathbf{\Phi }}}^T\mathbf{\Lambda }\stackrel{·}{\hat{\mathbf{\Phi }}}+{\tilde{\mathbf{\Omega }}}^T\mathbf{\Lambda }\stackrel{·}{\hat{\mathbf{\Omega }}})\hfill \\ +\frac{2}{{\gamma }_2}\mathrm{trace}({\tilde{\mathbf{K}}}_{\mathrm{\Delta }}^T{\stackrel{·}{\hat{\mathbf{K}}}}_{\mathrm{\Delta }})+\frac{2}{{\gamma }_3}{\tilde{\mathbf{d}}}^T\mathbf{\Lambda }\stackrel{·}{\hat{\mathbf{d}}}\hfill \\ \multicolumn{1}{c}{=-{\mathbf{e}}_u^T({\mathbf{Q}}_m+2\lambda \mathbf{P}){\mathbf{e}}_u}\\ +2\mathrm{trace}({\tilde{\mathbf{K}}}^T{\mathbf{\Lambda }\mathbf{B}}_m^T\mathbf{P}{\mathbf{e}}_u{\mathbf{x}}^T+{\tilde{\mathbf{\Phi }}}^T{\mathbf{\Lambda }\mathbf{B}}_m^T\mathbf{P}{\mathbf{e}}_u{\mathbf{r}}^T\hfill \\ +{\tilde{\mathbf{\Omega }}}^T{\mathbf{\Lambda }\mathbf{B}}_m^T\mathbf{P}{\mathbf{e}}_u{\stackrel{·}{\mathbf{r}}}^T)+2\mathrm{trace}({\tilde{\mathbf{K}}}_{\mathrm{\Delta }}^T\mathbf{P}{\mathbf{e}}_u\mathrm{\Delta }{\mathbf{u}}^T)\hfill \\ +2{\mathbf{e}}_u^T\mathbf{P}{\mathbf{B}}_m\mathbf{\Lambda }\tilde{\mathbf{d}}+2{\mathbf{e}}_u^T\mathbf{P}{\mathbf{B}}_m\mathbf{\Lambda }(\mathbf{d}-\overline{\mathbf{d}})\hfill \\ +\frac{2}{{\gamma }_1}\mathrm{trace}({\tilde{\mathbf{K}}}^T\mathbf{\Lambda }\stackrel{·}{\hat{\mathbf{K}}}+{\tilde{\mathbf{\Phi }}}^T\mathbf{\Lambda }\stackrel{·}{\hat{\mathbf{\Phi }}}+{\tilde{\mathbf{\Omega }}}^T\mathbf{\Lambda }\stackrel{·}{\hat{\mathbf{\Phi }}})\hfill \\ +\frac{2}{{\gamma }_2}\mathrm{trace}({\tilde{\mathbf{K}}}_{\mathrm{\Delta }}^T{\stackrel{·}{\hat{\mathbf{K}}}}_{\mathrm{\Delta }})+\frac{2}{{\gamma }_3}{\tilde{\mathbf{d}}}^T\mathbf{\Lambda }\stackrel{·}{\hat{\mathbf{d}}}\hfill \\ \multicolumn{1}{c}{=-{\mathbf{e}}_u^T({\mathbf{Q}}_m+2\lambda \mathbf{P}){\mathbf{e}}_u}\\ +\frac{2}{{\gamma }_1}\mathrm{trace}[{\tilde{\mathbf{K}}}^T\mathbf{\Lambda }(\stackrel{·}{\hat{\mathbf{K}}}+{\gamma }_1{\mathbf{B}}_m^T\mathbf{P}{\mathbf{e}}_u{\mathbf{x}}^T)\hfill \\ +{\tilde{\mathbf{\Phi }}}^T\mathbf{\Lambda }(\stackrel{·}{\hat{\mathbf{\Phi }}}+{\gamma }_1{\mathbf{B}}_m^T\mathbf{P}{\mathbf{e}}_u{\mathbf{r}}^T)\hfill \\ +{\tilde{\mathbf{\Phi }}}^T\mathbf{\Lambda }(\stackrel{·}{\hat{\mathbf{\Phi }}}+{\gamma }_1{\mathbf{B}}_m^T\mathbf{P}{\mathbf{e}}_u{\stackrel{·}{\mathbf{r}}}^T)]\hfill \\ +\frac{2}{{\gamma }_2}\mathrm{trace}[{\tilde{\mathbf{K}}}_{\mathrm{\Delta }}^T({\stackrel{·}{\hat{\mathbf{K}}}}_{\mathrm{\Delta }}+{\gamma }_2\mathbf{P}{\mathbf{e}}_u\mathrm{\Delta }{\mathbf{u}}^T)]\hfill \\ +\frac{2}{{\gamma }_3}[{\tilde{\mathbf{d}}}^T\mathbf{\Lambda }(\stackrel{·}{\hat{\mathbf{d}}}+{\gamma }_3{\mathbf{B}}_m^T\mathbf{P}{\mathbf{e}}_u)]\hfill \\ +2{\mathbf{e}}_u^T\mathbf{P}{\mathbf{B}}_m\mathbf{\Lambda }(\mathbf{d}-\overline{\mathbf{d}})\hfill \end{array}$$

(30)

Equation (30) can be reduced as follows

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}(t)=-{\mathbf{e}}_u^T({\mathbf{Q}}_m+2\lambda \mathbf{P}){\mathbf{e}}_u}\\ +\frac{2}{{\gamma }_1}\mathrm{trace}[(\hat{\mathbf{\Theta }}-\mathbf{\Theta }{)}^T\mathbf{\Lambda }(\stackrel{·}{\hat{\mathbf{\Theta }}}-{\gamma }_1\mathbf{Y})]\hfill \\ +\frac{2}{{\gamma }_2}\mathrm{trace}[({\hat{\mathbf{K}}}_{\mathrm{\Delta }}-{\mathbf{K}}_{\mathrm{\Delta }}{)}^T({\stackrel{·}{\hat{\mathbf{K}}}}_{\mathrm{\Delta }}-{\gamma }_2\mathbf{R})]\hfill \\ +\frac{2}{{\gamma }_3}{(\hat{\mathbf{d}}-\overline{\mathbf{d}})}^T\mathbf{\Lambda }(\stackrel{·}{\hat{\mathbf{d}}}-{\gamma }_{3}S)+2{\mathbf{e}}_u^T\mathbf{P}{\mathbf{B}}_m\mathbf{\Lambda }(\mathbf{d}-\overline{\mathbf{d}})\hfill \end{array}$$

(31)

where $\mathbf{\Theta }=[\begin{array}{ccc}{\mathbf{K}}^T\hfill & {\mathbf{\Phi }}^T\hfill & {\mathbf{\Omega }}^T\hfill \end{array}{]}^T$ .

Using update laws (20)–(22), definition 2 and Lemma 1, the followings equations are satisfied

$$\begin{array}{c}\frac{2}{{\gamma }_1}\mathrm{trace}[(\hat{\mathbf{\Theta }}-\mathbf{\Theta }{)}^T\mathbf{\Lambda }(\stackrel{·}{\hat{\mathbf{\Theta }}}-{\gamma }_1\mathbf{Y})]\hfill \\ =\frac{2}{{\gamma }_1}\mathrm{trace}[{\gamma }_1(\hat{\mathbf{\Theta }}-\mathbf{\Theta }{)}^T\mathbf{\Lambda }(\mathrm{Proj}(\mathbf{\Theta },\mathbf{Y},\mathbf{F})-\mathbf{Y})]\le 0\hfill \end{array}$$

(32)

$$\begin{array}{c}\frac{2}{{\gamma }_2}\mathrm{trace}[({\hat{\mathbf{K}}}_{\mathrm{\Delta }}-{\mathbf{K}}_{\mathrm{\Delta }}{)}^T({\stackrel{·}{\hat{\mathbf{K}}}}_{\mathrm{\Delta }}-{\gamma }_2\mathbf{R})]\hfill \\ =\frac{2}{{\gamma }_2}\mathrm{trace}[{\gamma }_2({\hat{\mathbf{K}}}_{\mathrm{\Delta }}-{\mathbf{K}}_{\mathrm{\Delta }}{)}^T(\mathrm{Proj}({\hat{\mathbf{K}}}_{\mathrm{\Delta }},\mathbf{R},\mathbf{G})-\mathbf{R})]\le 0\hfill \end{array}$$

(33)

$$\begin{array}{c}\frac{2}{{\gamma }_3}[(\hat{\mathbf{d}}-\overline{\mathbf{d}}{)}^T\mathbf{\Lambda }(\stackrel{·}{\hat{\mathbf{d}}}-{\gamma }_3\mathbf{S})]\hfill \\ =\frac{2}{{\gamma }_3}[{\gamma }_3(\hat{\mathbf{d}}-\overline{\mathbf{d}}{)}^T\mathbf{\Lambda }(\mathrm{Proj}(\hat{\mathbf{d}},\mathbf{S},\mathbf{g})-\mathbf{S})]\le 0\hfill \end{array}$$

(34)

From equations (32) to (34) and the Rayleigh principle, equation (31) can be rewritten as follows

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}(t)\le -{\mathbf{e}}_u^T({\mathbf{Q}}_m+2\lambda \mathbf{P}){\mathbf{e}}_u+2{\mathbf{e}}_u^T\mathbf{P}{\mathbf{B}}_m\mathbf{\Lambda }(\mathbf{d}-\overline{\mathbf{d}})}\\ \le -a_1\parallel {\mathbf{e}}_u{\parallel }^2+2\parallel {\mathbf{e}}_u\parallel d^{*}\hfill \\ \le -\parallel {\mathbf{e}}_u\parallel (a_1\parallel {\mathbf{e}}_u\parallel -2d^{*})\le -\eta \parallel {\mathbf{e}}_u{\parallel }^2\le 0\hfill \end{array}$$

(35)

where $a_1={\lambda }_{\min }({\mathbf{Q}}_m)+2{\lambda }_{\min }(\mathbf{P})\lambda >0$ , $a_1>\eta >0$ and $d^{*}=\parallel \mathbf{P}{\mathbf{B}}_{\mathbf{m}}\mathbf{\Lambda }\mathbf{(}\mathbf{d}-\overline{\mathbf{d}})\parallel \in l_{\infty }$

$\stackrel{·}{V}(t)$ of equation (35) is the negative semi-definite whenever

$$a_1\parallel {\mathbf{e}}_u\parallel -2d^{*}\ge \eta \parallel {\mathbf{e}}_u\parallel \Rightarrow \parallel {\mathbf{e}}_u\parallel \ge \frac{2d^{*}}{a_1-\eta }$$

(36)

This implies that ${\mathbf{e}}_u,\hat{\mathbf{K}},\hat{\mathbf{\Phi }},\hat{\mathbf{\Omega }},{\hat{\mathbf{K}}}_{\mathrm{\Delta }}$ and $\hat{\mathbf{d}}$ are bounded from equations (29), (35) and (36) and ${\mathbf{e}}_u\to 0$ as $t\to \infty$ by Barbalat’s lemma. Hence, $\mathbf{e}\to {\mathbf{e}}_{\mathrm{\Delta }}$ and $\mathbf{e}$ is bounded if and only if ${\mathbf{e}}_{\mathrm{\Delta }}$ is also bounded. The boundedness of ${\mathbf{e}}_{\mathrm{\Delta }}$ is proven as follows:

A candidate Lyapunov is chosen as

$$W={\mathbf{e}}_{\mathrm{\Delta }}^T\mathbf{P}{\mathbf{e}}_{\mathrm{\Delta }}\ge 0$$

(37)

Using Rayleigh principle, equation (37) yields

$$W\le \parallel {\mathbf{e}}_{\mathrm{\Delta }}{\parallel }^2{\lambda }_{\max }(\mathbf{P})$$

(38)

and the first time derivative of W is given as follows

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{W}=-{\mathbf{e}}_{\mathrm{\Delta }}^T({\mathbf{Q}}_m+2\lambda \mathbf{P}){\mathbf{e}}_{\mathrm{\Delta }}-2{\mathbf{e}}_{\mathrm{\Delta }}^T\mathbf{P}{\hat{\mathbf{K}}}_{\mathrm{\Delta }}\mathrm{\Delta }\mathbf{u}}\\ \le -\parallel {\mathbf{e}}_{\mathrm{\Delta }}{\parallel }^2{a}_1+{\lambda }_3\le {\lambda }_{2}W+{\lambda }_3\hfill \end{array}$$

(39)

where ${\lambda }_3=2\parallel -{\mathbf{e}}_{\mathbf{\Delta }}^{\mathbf{T}}{\mathbf{P}\hat{\mathbf{K}}}_{\mathbf{\Delta }}^{\mathbf{T}}\mathbf{\Delta }\mathbf{u}\parallel \ge 0$ and ${\lambda }_2=(a_1/{\lambda }_{\max }(\mathbf{P}))>0$ .

Using the Gronwall Bellman Inequality, equation (39) implies that

$$W\le [W({\mathbf{e}}_{\mathrm{\Delta }}(0))-\frac{{\lambda }_3}{{\lambda }_2}]\exp (-{\lambda }_{2}t)+\frac{{\lambda }_3}{{\lambda }_2}$$

(40)

Using equations (37) and (40), following equations are obtained

$$\underset{t\to \infty }{lim}{\mathbf{e}}_{\mathrm{\Delta }}^T\mathbf{P}{\mathbf{e}}_{\mathrm{\Delta }}\le \frac{{\lambda }_3}{{\lambda }_2}$$

(41)

$$\frac{{\lambda }_3}{{\lambda }_2}\ge \underset{t\to \infty }{lim}{\mathbf{e}}_{\mathrm{\Delta }}^T\mathbf{P}{\mathbf{e}}_{\mathrm{\Delta }}\ge {\lambda }_{\min }(\mathbf{P})\underset{t\to \infty }{lim}\parallel {\mathbf{e}}_{\mathrm{\Delta }}{\parallel }^2$$

(42)

$$\underset{t\to \infty }{lim}\parallel {\mathbf{e}}_{\mathrm{\Delta }}\parallel \le \sqrt{\frac{{\lambda }_3}{{\lambda }_2{\lambda }_{\min }(\mathbf{P})}}$$

(43)

It can be proven that ${\mathbf{e}}_{\mathrm{\Delta }}$ is also bounded.

E.O.D.

Constraint of the modelling error vector, the feedback gain and the adaptation rate

Since ${\hat{\mathbf{K}}}_{\mathbf{x}},{\hat{\mathbf{K}}}_{\mathbf{r}},{\hat{\mathbf{K}}}_{\mathbf{\Delta }},\hat{\mathbf{d}}$ are bounded, there exist a constant $\beta$ such that the following inequality is satisfied

$$\begin{array}{c}\frac{1}{{\gamma }_1}\mathrm{trace}({\tilde{\mathbf{K}}}^{\mathbf{T}}\mathbf{\Lambda }\tilde{\mathbf{K}}+{\tilde{\mathbf{\Phi }}}^{\mathbf{T}}\mathbf{\Lambda }\tilde{\mathbf{\Phi }}+{\tilde{\mathbf{\Omega }}}^{\mathbf{T}}\mathbf{\Lambda }\tilde{\mathbf{\Omega }})\hfill \\ +\frac{1}{{\gamma }_2}\mathrm{trace}({\tilde{\mathbf{K}}}_{\mathbf{\Delta }}^{\mathbf{T}}{\tilde{\mathbf{K}}}_{\mathbf{\Delta }})+\frac{1}{{\gamma }_3}{\tilde{\mathbf{d}}}^T\mathbf{\Lambda }\tilde{\mathbf{d}}\le \frac{\beta }{\gamma }\hfill \end{array}$$

(44)

where $\gamma$ is the maximum value of the set $\{{\gamma }_1,{\gamma }_2,{\gamma }_3\}$ .

From equations (29), (36) and Rayleigh principle, the following equation is satisfied

$$V(t)\le {\mathbf{e}}_u^T\mathbf{P}{\mathbf{e}}_u+\frac{\beta }{\gamma }\le \parallel {\mathbf{e}}_u{\parallel }^2{\lambda }_{\max }(\mathbf{P})+\frac{\beta }{\gamma }$$

(45)

To obtain a bound on $\parallel {\mathbf{e}}_u\parallel$ , a Lyapunov level set is defined as

$$L=\{({\mathbf{e}}_u,\tilde{\mathbf{K}},\tilde{\mathbf{\Phi }},\tilde{\mathbf{\Omega }},{\tilde{\mathbf{K}}}_{\mathrm{\Delta }},\tilde{\mathbf{d}}):V\le V^{*}\}$$

(46)

with $V^{*}\equiv {\lambda }_{\max }(\mathbf{P})(4d^{*2}/(a_1-\eta {)}^2)+(\beta /\gamma )$ .

Using the Rayleigh principle in equations (29), (45) and (46), the following equation is satisfied

$$\begin{array}{cc}{\lambda }_{\min }\hfill & (\mathbf{P})\parallel {\mathbf{e}}_u^2\parallel \le {\mathbf{e}}_u^T\mathbf{P}{\mathbf{e}}_u\le V\le V^{*}\hfill \\ \multicolumn{1}{c}{\Rightarrow {\lambda }_{\min }}& (\mathbf{P})\parallel {\mathbf{e}}_u^2\parallel \le {\lambda }_{\max }(\mathbf{P})\frac{4d^{*2}}{{(a_1-\eta )}^2}+\frac{\beta }{\gamma }\hfill \\ \multicolumn{1}{c}{\Rightarrow \parallel {\mathbf{e}}_u\parallel }& \le \sqrt{\frac{{\lambda }_{\max }(\mathbf{P})}{{\lambda }_{\min }(\mathbf{P})}\frac{4d^{*2}}{{(a_1-\eta )}^2}+\frac{\beta }{\gamma {\lambda }_{\min }(\mathbf{P})}}\hfill \end{array}$$

(47)

From equations (36) and (47), the following equation is obtained

$$0<\frac{2d^{*}}{a_1-\eta }\le \parallel {\mathbf{e}}_u\parallel \le \sqrt{\frac{{\lambda }_{\max }(\mathbf{P})}{{\lambda }_{\min }(\mathbf{P})}\frac{4d^{*2}}{{(a_1-\eta )}^2}+\frac{\beta }{\gamma {\lambda }_{\min }(\mathbf{P})}}$$

(48)

Hence, from equations (18), (44) and (48), the above limit of $\mathbf{e}$ can be obtained

$$\begin{array}{c}\parallel \mathbf{e}\parallel \le \parallel {\mathbf{e}}_{\mathbf{u}}\parallel +\parallel {\mathbf{e}}_{\mathbf{\Delta }}\parallel \hfill \\ \Rightarrow \parallel \mathbf{e}\parallel \le \sqrt{\frac{{\lambda }_{\max }(\mathbf{P})}{{\lambda }_{\min }(\mathbf{p})}\frac{4d^{*2}}{{(a_1-\eta )}^2}+\frac{\beta }{\gamma {\lambda }_{\min }(\mathbf{p})}}\hfill \\ +\sqrt{\frac{{\lambda }_3}{{\lambda }_2{\lambda }_{\min }(\mathbf{P})}}\hfill \end{array}$$

(49)

In inequality of equation (49), the bounded value of $\parallel \mathbf{e}\parallel$ can be decreased if the feedback gain $\lambda$ and the adaption rate $\gamma$ at first and second terms in the first square root are increased.

The block diagram of the proposed controller is shown in Figure 3.

Figure 3.

Block diagram of the proposed controller.

Experimental results

Experimental setup

To evaluate the effectiveness of the proposed controller (M-MRAC) and compare it with the CMRAC, a conveyor plant is established as shown in Figure 4. This plant consists of three conveyors and a TMS320F28069 digital signal processor (DSP)-based microcontroller. Each conveyor in the conveyor plant consists of an alternating current (AC) motor, an inverter, driven and driving rollers, a belt and an encoder to measure an angular velocity. The M-MRAC and the CMRAC are built into DSP chip.

Figure 4.

Experimental conveyor plant.

The initial values of the state variables and the controller inputs are set to zero. Input voltages of the inverters considered as control inputs of the proposed controller can vary in range from $u_{1\min }=u_{2\min }=u_{3\min }=0\mathrm{V}$ to $u_{1\max }=u_{2\max }=u_{3\max }=5\mathrm{V}$ .

The parameters of the modified model reference system are given by $a_{m1}=a_{m2}=a_{m3}=-30,b_{m1}=b_{m2}=b_{m3}=30$ . The error feedback gain is chosen as $\lambda =10$ , the controller gains are chosen as ${\gamma }_1=100,{\gamma }_2=3.3$ and the positive symmetric definite matrix $\mathbf{P}=\mathrm{diag}([{10}^{-8}{10}^{-8}{10}^{-8}])$ . The reference inputs for the conveyor plant and angular velocity inputs are given in Figure 5.

Figure 5.

Velocity profiles of all conveyors.

Comparative experimental results

In order to demonstrate the effectiveness of the proposed controller, the following three cases are considered.

Case 1

The adaptation rates of both CMRAC and M-MRAC are set to ${\gamma }_1=1.67$ . It can be seen that both the output of the proposed M-MRAC $x_1(t)$ and the output of the CMRAC $x_{1M}(t)$ for the first conveyor track the reference input $r_1(t)$ as shown in Figure 6. However, the output amplitude of the CMRAC varies more largely than that of the proposed M-MRAC. The high-frequency oscillation is generated in the control input signal $u_{1M}(t)$ of the CMRAC, while the control input signal $u_1(t)$ of the proposed M-MRAC does not almost change as shown in Figure 7 when the angular velocity output reaches 42.1 rad/s.

Figure 6.

Experimental output of the M-MRAC and the conventional MRAC controller for the first conveyor.

Figure 7.

Experimental control input of the M-MRAC and the conventional MRAC controller for the first conveyor.

Case 2

The adaptation rates of both CMRAC and M-MRAC are set to ${\gamma }_1=10$ . The angular velocity output $x_2(t)$ of the proposed M-MRAC for the second conveyor tracks the reference input $r_2(t)$ with tiny error (from −0.9 to +1.3 rad/s), while the angular velocity output $x_{2M}(t)$ of the CMRAC for the second conveyor tracks the reference input $r_2(t)$ with error from −2.3 to +2.5 rad/s in Figure 8. The control input $u_{2M}(t)$ of the CMRAC is also oscillated with higher frequency and amplitude than the control input $u_2(t)$ of the proposed M-MRAC as shown in Figure 9. The maximum and minimum values of the control input of the CMRAC are 4.1 and 3.72 V, respectively, while the average value of the control input is 3.88 V. The control input for the proposed M-MRAC varies slower than that for the CMRAC. Therefore, the performance of the proposed M-MRAC is better than that of the CMRAC in this case.

Figure 8.

Experimental output of the M-MRAC and the conventional MRAC controller for the second conveyor.

Figure 9.

Experimental control input of the M-MRAC and the conventional MRAC controller for the second conveyor.

Case 3

Similarly, the adaptation rates of both CMRAC and M-MRAC are set to ${\gamma }_1=6.67$ . The angular velocity output $x_3(t)$ of the proposed M-MRAC for the third conveyor also tracks the reference input $r_3(t)$ better than the angular velocity output $x_{3M}(t)$ of the CMRAC as shown in Figure 10. Because the reference input $r_3(t)$ is a step type, both the amplitudes of control input signals of the proposed M-MRAC and the CMRAC are large (15.5 and 12.5 V, respectively) in Figure 11. Therefore, the saturated control input $u_3^{*}(t)$ is set to $u_{3\max }=5\mathrm{V}$ , and the angular velocity outputs of both the proposed M-MRAC and the CMRAC reach 90.3 rad/s in Figure 10.

Figure 10.

Experimental output of the M-MRAC and the conventional MRAC controller for the third conveyor.

Figure 11.

Experimental control input of the M-MRAC and the conventional MRAC controller for the third conveyor.

The tracking performance of the proposed M-MRAC controller versus the CMRAC is given in Table 1.

Table 1.

Modelling error of M-MRAC versus CMRAC.

		M-MRAC (%)	CMRAC (%)
First conveyor, $\lambda =10$	Modelling error	1.4	5.2
Second conveyor, $\lambda =10$	Modelling error	1.8	3.5
Third conveyor, $\lambda =10$	Modelling error	0.7	3.1

M-MRAC: modified model reference adaptive controller; CMRAC: conventional model reference adaptive control.

Conclusion

An M-MRAC for belt conveyors in a fish sorting system with uncertainty parameters, input saturation and bounded disturbances was proposed. The feedback of modelling error signal in the proposed M-MRAC controller obtained smaller modelling error than that in the CMRAC (0.7% vs 3.1%). The tracking performance of the proposed M-MRAC has better improvement in both transient state and asymptotic state than that of the CMRAC and the high-frequency elements in control input signals were reduced in the proposed M-MRAC when the adaptation rate was increased. The error dynamics under the input saturation was compensated by the auxiliary output error. The experimental results showed that the proposed M-MRAC became more effective than the CMRAC when the adaptation rate was large and the error feedback gain was selected suitably.