2.1. Dynamic description

A typical structure of two-link flexible robotic manipulator is shown in Fig. 1. On the premise of small deflection, the dynamic model of manipulator system obtained via assumed mode method and Lagrange’s equation can be written as ref. [Reference Gao, He, Zhou and Sun51]

(1) \begin{equation} M(q)\ddot{q}+B(q,\dot{q})\dot{q}+K(q)=u(t) \end{equation}

where $q=[\theta,\phi ]^{T}$ , $\theta =[\theta _{1},\theta _{2}]^{T}$ is the vector of joint angular position, $\phi =[\phi _{11},\ldots,\phi _{1n_{1}},\phi _{21},\ldots,\phi _{2n_{2}}]^{T}$ represents the flexible generalized coordinate vector, and $n=n_{1}+n_{2}$ is the total number of flexible variables. $M(q)\in R^{(n+2)\times (n+2)}$ is the positive inertia matrix, $B(q,\dot{q})\in R^{(n+2)\times (n+2)}$ denotes the centripetal and Coriolis forces, and $K(q)\in R^{(n+2)}$ is the vector of gravity and elasticity forces. $u=[u_{1},u_{2},0,\cdots,0]^{T}\in R^{(n+2)}$ is the input torque vector.

Fig. 1. Diagram of the two-link flexible manipulator.

The actuator fault is mathematically modeled as

(2) \begin{equation} u(t)=D\tau (t)+\bar{u} \end{equation}

where $\tau (t)\in R^{n+2}$ denotes the desired control torque, $D=\textrm{diag}\{l_{1},\cdots,l_{n+2}\}\in R^{(n+2)\times (n+2)}$ represents the actuator effectiveness matrix with $0\lt \eta _{i}\leq l_{i}\leq 1,i=1,2,\cdots,n+2$ , and $\bar{u}=[\bar{u}_{1},\cdots,\bar{u}_{n+2}]^{T}\in R^{n+2}$ is the additive actuator fault vector.

2.2. Prescribed tracking performance

Let $x_{1}=q,x_{2}=\dot{q}$ , the dynamic model (1) can be rewritten as

(3) \begin{equation} \dot{x}_{1} = x_{2} \end{equation}

(4) \begin{equation} \dot{x}_{2}= M^{-1}(x_{1})[u(t)-B(x_{1},x_{2})x_{2}-K(x_{1})] \end{equation}

Define the desired trajectory as $x_{d}=[x_{d1},\cdots,x_{d(n+2)}]^{T}$ , which is a continuously bounded and known variable. The tracking error vector can be expressed as $e_{1}=x_{1}-x_{d}=[e_{11},\cdots,e_{1(n+2)}]^{T}$ .

The given performance specifications of tracking error vector $e_{1}(t)$ are set as

(5a) \begin{equation} |e_{1i}(t)|\lt \varepsilon,\quad \forall t\geq T\gt t_{0} \end{equation}

(5b) \begin{equation} \lim \limits _{t\to \infty }|e_{1i}(t)|\lt \mu \lt \varepsilon \end{equation}

(5c) \begin{equation} -\delta _{1}\lt e_{1i}(t)\lt \delta _{2}, \quad e_{1i}(t_{0})\geq 0 \end{equation}

(5d) \begin{equation} -\delta _{2}\lt e_{1i}(t)\lt \delta _{1}, \quad e_{1i}(t_{0})\lt 0 \end{equation}

where $T\gt t_{0}$ , $\varepsilon \gt 0$ , $\mu \gt 0$ , $\delta _{2}\gt |e_{1i}(t_{0})|\gt \delta _{1}\gt 0$ are the specific parameters to be set, $t_{0}\geq 0$ is the initial time. The above descriptions constrain the transient and steady-state performance of the tracking error, respectively. More specifically, $e_{1i}(t)$ converges to a small region $(\!-\varepsilon,\varepsilon )$ within finite time $T$ and finally shrinks into a smaller range $(\!-\mu,\mu )$ . By (5c), we know that $-\delta _{1}$ and $\delta _{2}$ are the upper and lower bounds of the error $e_{1i}(t)$ when $e_{1i}(t_{0})\geq 0$ , for all $t\geq t_{0}$ . The same analysis applies when $e_{1i}(t_{0})\lt 0$ in (5d). Take $e_{1i}(t_{0})\geq 0$ as an example, the PP curve is shown as Fig. 2.

Fig. 2. Diagram of tracking error with preset performance.

2.3. Error transformation

It is difficult to ensure that the transient and steady-state performance meeting the above requirements if the error is directly controlled. Thus, we introduce two error transfer functions to control overshoot and steady-state error, respectively. For the overshoot, we define the function as

(6) \begin{equation} f(e_{1i})=\frac{\gamma e_{1i}^{m}}{(e_{1i}^{2m}+g)^{\frac{1}{2}}} \end{equation}

where $g$ is a small positive constant, $m$ is an odd integer with $m\geq \textrm{max}\{3,n+2\}$ , $\gamma =\gamma _{1}\gt 1$ for all $e_{1i}\geq 0$ and $\gamma =\gamma _{2}\gt 1$ for all $e_{1i}\lt 0$ . $\dot{f}(e_{1i})=gm\gamma e_{1i}^{m-1}/(e_{1i}^{2m}+g)^{\frac{3}{2}}\geq 0$ shows that $f(e_{1i})$ is monotonously increasing with respect to $e_{1i}$ . Since $f(e_{1i})$ is monotonous and

(7) \begin{equation} \lim \limits _{e_{1i}\to +\infty }f(e_{1i}) =\gamma _{1}, \lim \limits _{e_{1i}\to -\infty }f(e_{1i}) =-\gamma _{2} \end{equation}

we can obtain $f(e_{1i})\in (\!-\gamma _{2},\gamma _{1})$ .

In terms of steady-state error, a behavior-shaping function is imported as

(8) \begin{equation} Q(t)=\frac{k_{bi}}{(k_{bi}-\xi \varepsilon )\kappa ^{-1}(t)+\xi \varepsilon } \end{equation}

where $\kappa (t)\geq \kappa (t_{0})=1$ is a rate function with $\dot{\kappa }(t)\gt 0$ and $\ddot{\kappa }(t)\gt 0$ for all $t\geq t_{0}$ ; $k_{bi}\gt 1$ , $0\lt \xi \lt 1,0\lt \varepsilon \lt 1$ are parameters to be designed. It is different from the traditional PP function. $\xi$ and $\varepsilon$ represent the transient accuracy and steady-state accuracy of the error, respectively, and $\kappa (t)$ specifies the time of error convergence. More properties of function $Q(t)$ are described in Lemma 2.3.

2.4. Broad learning system

As shown in Fig. 3, different from the general NNs structure, the hidden layer of the BLS includes feature nodes and enhancement nodes. For the presented input vector $X$ , the feature mapping of $X$ can be extracted by following $r_{1}$ feature mappings

(9) \begin{equation} F_{xi}=\phi _{xi}(XW_{xi}+\epsilon _{xi}), \quad i=1,\cdots,r_{1} \end{equation}

where $\phi _{xi}$ is the linear or nonlinear activation function, $W_{xi}$ and $\epsilon _{xi}$ are the weight and bias generated randomly and then remained constant during training. Define the feature vector as $F^{r_{1}}=[F_{x1},\cdots,F_{xr_{1}}]$ , and the $r_{2}$ enhancement nodes are obtained by following feature mappings

(10) \begin{equation} H_{fj}=\phi _{fj}(F^{r_{1}}W_{fj}+\epsilon _{fj}), \quad j=1,\cdots,r_{2} \end{equation}

where $\phi _{fj}$ is the nonlinear activation function, $W_{fi}$ and $\epsilon _{fi}$ are the weight and bias generated randomly. Define the enhanced node vector as $H^{r_2}=[H_{f1},\cdots,H_{fr_{2}}]$ and define $A=[F^{r_{1}}|H^{r_{2}}]$ , then the output of BLS can be expressed as $Y=AW^{(r_{1}+r_{2})}$ with $W^{(r_{1}+r_{2})}$ is the weight from the hidden layer to the output layer.

Fig. 3. Neural networks structure with BLS.

2.5. RBFNNs with broad learning theory

In this paper, we use RBFNNs to approximate the uncertain dynamic system described as (1). Mathematically, RBFNNs function is expressed as

\begin{equation*} F(Z)=W^{T}R(Z)+\epsilon (Z) \end{equation*}

\begin{equation*} R_{i}(Z)=\textrm {exp}\bigg [\!-\frac {(Z-\varphi _{i})^{T}(Z-\varphi _{i})}{\eta _{i}^2}\bigg ],(i=1,\cdots,l) \end{equation*}

where $F(Z)\in R^{n+2}$ is the output of NNs, $Z\in R^{m_{z}}$ is the NN input vector, $W\in R^{l\times (n+2)}$ is the NNs weights matrix with $l$ being the number of NNs nodes, $\epsilon (Z)\in R^{n+2}$ is the approximation error vector, $R(Z)\in R^{l}$ is the basis function vector with $R_{i}(Z)$ is the Gaussian function, and $\varphi =[\varphi _{1},\cdots,\varphi _{l}]\in R^{m_{z}\times l}$ and $\eta _{i}$ denote the centers and widths.

Inspired by the ideas of BLS [Reference Chen and Liu25] and incremental learning theory [Reference Huang, Zhang, Yang and Chen27], an improved RBFNNs learning algorithm is presented in this section. In the traditional RBFNNs control, the designed NNs node space cannot guarantee to completely cover the input vector, so it is valuable to add nodes close to the input vector and discard nodes far away from the input vector.

Define the NNs node information as $[\varphi _{t_{0}},\eta _{t_{0}},W_{t_{0}}]$ when $t=t_{0}$ and the added NNs node information as $[\varphi _{\textrm{new}},\eta _{\textrm{new}},W_{\textrm{new}}]$ . Select $n_{p}$ center vectors close to the input vector $c_{\textrm{min}}=\{c_{1},\cdots,c_{n_{p}}\}\in \varphi$ and the mean is $a_v=\sum _{i=1}^{n_{p}}c_{i}/n_{p}$ , then the added node can be designed by following rule [Reference Huang, Zhang, Yang and Chen27]

(11) \begin{equation} \varphi _{\textrm{new}}=\bar{c}_{\textrm{min}}+k_{z}(Z-\bar{c}_{\textrm{min}}) \end{equation}

(12) \begin{equation} \eta _{\textrm{new}}=\eta _{t_{0}},W_{\textrm{new}}=[0,\cdots,0]\in R^{m_{z}} \end{equation}

where $k_{z}\in R^{m_{z}\times m_{z}}$ is a positive diagonal matrix, $\bar{c}_{\textrm{min}}\in R^{m_{z}}$ denotes the average distance vector to the closest centers $c_{\textrm{min}}$ , which is defined as $\bar{c}_{\textrm{min}}=[a_{v},\cdots,a_{v}]\in R^{m_{z}}$ . Thus, the NNs center is updated as

(13) \begin{equation} \varphi _{t_{0}+t_{s}}=\left \{ \begin{array}{c}\begin{aligned} &[\varphi _{t_{0}},\varphi _{\textrm{new}}],\quad \Vert \bar{c}_{\textrm{min}}\Vert \gt \varXi \ \textrm{and} \ l\lt \bar{l} \\ \\[-9pt] &\varphi _{t_{0}},\qquad \qquad \textrm{otherwise}.\\ \end{aligned}\end{array} \right. \end{equation}

where $t_{s}$ is the sampling interval, $\varXi$ is the preset threshold, and $\bar{l}$ is the preset maximum number of neural nodes. According to the idea of broad learning, we choose the hidden layer of NNs as $G(t_{0}+t_{s})=[Z|R(Z)]\in R^{l+m_{z}}$ . Meanwhile, initialize the weight from the hidden layer to the output layer with $W(t_{0}+t_{s})=[W_{t_{0}}|W_{\textrm{new}}]\in R^{(l+m_{z})\times (n+2)}$ . Consequently, the output of BNNs can be expressed as

(14) \begin{equation} Y(t_{0}+t_{s})=W(t_{0}+t_{s})^{T}G(t_{0}+t_{s}) \end{equation}

2.6. Useful technical lemmas

Lemma 2.1. [Reference Zhang, Dong, Ouyang, Yin and Peng53] For $\forall x\in R$ , there is a positive constant $\zeta \in R$ satisfying that $|x|\leq |\zeta |$ with $\zeta$ being the constraint value, the following inequality holds:

(15) \begin{equation} \textrm{ln}\frac{\zeta ^2}{\zeta ^{2}-x^2}\leq \frac{x^2}{\zeta ^{2}-x^2} \end{equation}

Lemma 2.2. [Reference Smaeilzadeh and Golestani31] For any constants $a_{1},a_{2},\cdots,a_{n}$ and $0\leq x\leq 1$ , the following inequality holds:

(16) \begin{equation} (|a_{1}|+\cdots +|a_{n}|)^{x}\leq |a_{1}|^{x}+\cdots +|a_{n}|^{x} \end{equation}

3.1. Control objectives

In this section, the backstepping method is performed to obtain an adaptive NNs FT controller.

1. (1) The desired reference angular vector $x_{d}$ can be tracked within the prescribed time. And the tracking errors of two flexible manipulator joints can satisfy the prescribed requirements in terms of overshoot, transient, and steady-state performance.

2. (2) The actuator faults described in (2) can be compensated by a PFTC scheme.

3. (3) All the close-loop error signals are uniformly ultimately bounded.

3.2. Adaptive NN PPC with actuator faults

To facilitate the controller design, we define the first error variable $z_{1}$ as

(17) \begin{equation} z_{1}=f(e_{1})Q(t) \end{equation}

where $f(e_{1})=[f(e_{1i}),\cdots,f(e_{(n+2)i})]^{T}$ . If $|z_{1i}|\lt k_{bi}(i=1,2,\cdots,n+2)$ with $k_{bi}\lt \textrm{min}\{\delta _{1},\delta _{2}\}$ , we get

(18) \begin{equation} |f(e_{1i})|=Q^{-1}(t)|z_{1i}|\lt (k_{b}-\xi \varepsilon )\kappa ^{-1}(t)+\xi \varepsilon \end{equation}

when $t\geq T$ , the $\kappa (T)$ is chosen as $(k_{b}-\xi \varepsilon )/(\varepsilon -\xi \varepsilon )$ , then

(19) \begin{equation} |f(e_{1i})||_{t\geq T}\lt (k_{bi}-\xi \varepsilon )\kappa ^{-1}(T)+\xi \varepsilon =\varepsilon \end{equation}

Considering that $\kappa (t)$ is monotonically increasing function with $\kappa (t_{0})=1$ , we get $\lim \limits _{t\to +\infty }\kappa ^{-1}(t)=0$ ; thus, we can further obtain

(20) \begin{equation} \lim \limits _{t\to +\infty }|f(e_{1i})|\lt \xi \varepsilon =\mu, \quad 0\lt \xi \lt 1 \end{equation}

Next, we discuss the relationship between $e_{1}$ and $f(e_{1})$ . From (6), we can get the inverse function of $f(e_{1})$

(21) \begin{equation} e_{1i}(f)=\frac{g^{\frac{1}{2m}}f^{\frac{1}{m}}}{(\gamma ^{2}-f^{2})^{\frac{1}{2m}}}, \quad f\in (\!-\gamma _{2},\gamma _{1}) \end{equation}

Differentiating $e_{1i}(f)$ with respect to $f$ yields $\dot{e}_{1i}=f^{(\frac{1}{m}-1)}g^{\frac{1}{m}}\gamma ^{2}(\gamma ^{2}-f^{2})^{(\frac{1}{2m}-1)}/m(\gamma ^{2}-f^{2})^{\frac{1}{m}}\geq 0$ with $g\gt 0$ and $m$ is an odd integer which is not less than $3$ . Thus, $e_{1i}(f)$ is strictly increasing for $f\in (\!-\gamma _1,\gamma _2)$ . Selecting $g=(\gamma ^{2}-\varepsilon ^{2})\varepsilon ^{(2m-2)}$ and combining (19) and (21), we conclude

(22) \begin{equation} |e_{1i}||_{t\geq T}\lt \frac{(\gamma ^{2}-\varepsilon ^{2})^{\frac{1}{2m}}\varepsilon ^{(1-\frac{1}{m})}\varepsilon ^{\frac{1}{m}}}{(\gamma ^{2}-\varepsilon ^{2})^{\frac{1}{2m}}}=\varepsilon \end{equation}

Substituting (20) into (21) yields

(23) \begin{equation} \lim \limits _{t\to +\infty }|e_{1i}|\lt \frac{(\gamma ^{2}-\varepsilon ^{2})^{\frac{1}{2m}}\varepsilon ^{(1-\frac{1}{m})}\xi ^{\frac{1}{m}}\varepsilon ^{\frac{1}{m}}}{(\gamma ^{2}-\varepsilon ^{2})^{\frac{1}{2m}}}=\xi ^{\frac{1}{m}}\varepsilon \end{equation}

Thus, the constraints of $e_{1i}$ in (5aa) and (5ab) are guaranteed.

According to the properties of $Q(t)$ , we know $0\lt Q^{-1}(t)\lt 1$ and further get

(24) \begin{equation} |f(e_{1i})|=Q^{-1}(t)|z_{1i}|\lt |z_{1i}|\lt k_{bi} \end{equation}

where $k_{bi}\lt \textrm{min}\{\gamma _{1},\gamma _{2}\}$ .

when $e_{1i}(t_{0})\geq 0$ , $\gamma =\gamma _{1}$ , we can get from (21) and (24)

(25) \begin{equation} e_{1i}(t)=\frac{g^{\frac{1}{2m}}f^{\frac{1}{m}}}{(\gamma _{1}^{2}-f^{2})^{\frac{1}{2m}}}\lt \frac{g^{\frac{1}{2m}}k_{bi}^{\frac{1}{m}}}{(\gamma _{1}^{2}-k_{bi}^{2})^{\frac{1}{2m}}}=P_{u} \end{equation}

when $e_{1i}(t_{0})\lt 0$ , $\gamma =\gamma _{2}$ , similarly, we get

(26) \begin{equation} e_{1i}(t)=\frac{g^{\frac{1}{2m}}f^{\frac{1}{m}}}{(\gamma _{2}^{2}-f^{2})^{\frac{1}{2m}}}\gt \frac{-g^{\frac{1}{2m}}k_{bi}^{\frac{1}{m}}}{(\gamma _{2}^{2}-k_{bi}^{2})^{\frac{1}{2m}}}=-P_{l} \end{equation}

where $P_{u}, -P_{l}$ are the upper and lower bounds of the tracking error, respectively. More specifically,

1. (1) When $0\leq e_{1i}(t_{0}) \leq P_{u}$ , the size of overshoot is determined by $P_{l}$ . Considering the definitions of $P_{l}$ and $g$ , we find that once $k_{bi}$ and $\varepsilon$ are fixed, the overshoot is only determined by $\gamma$ . According to the definition of $\gamma$ , it is easy to know that the overshoot is only determined by $\gamma _{1}$ .

2. (2) When $-P_{l}\leq e_{1i}(t_{0}) \lt 0$ , the size of overshoot is determined by $P_{u}$ . Considering the definitions of $P_{u}$ and $g$ , we find that once $k_{bi}$ and $\varepsilon$ are fixed, the overshoot is only determined by $\gamma$ . Similarly, it is easy to know that the overshoot is only determined by $\gamma _{2}$ .

The above analysis hold under the condition $|z_{1i}|\lt k_{bi}$ ; the preset performance can be achieved by transforming the error $e_{1i}$ twice with (6) and (8). In the following controller design process, a BLF $\sum _{i=1}^{n+2}\textrm{ln}\frac{k_{bi}^{2}}{k_{bi}^{2}-z_{1i}^2}$ is considered to guarantee that $z_{1i}$ remain into the region $|z_{1i}| \lt k_{bi}$ [Reference Ren, Ge, Tee and Lee54].

Define the second error variable $z_{2}$ as

(27) \begin{equation} z_{2}=x_{2}-\alpha \end{equation}

where $\alpha =[\alpha _{1},\cdots,\alpha _{n+2}]^{T}$ is the virtual error variable. Take the derivative of $z_{1}$ with respect to time, we have

(28) \begin{align} \dot{z}_{1i}& =\frac{gm\gamma e_{1i}^{m-1}(z_{2i}+\alpha _{i}-\dot{x}_{di})}{(e_{1i}^{2m}+g)^{\frac{3}{2}}}Q(t)+\dot{Q}(t)Q^{-1}(t)z_{1i} \nonumber \\& =L_{i}(z_{2i}+\alpha _{i}-\dot{x}_{di})Q(t)+\dot{Q}(t)Q^{-1}(t)z_{1i} \end{align}

where

\begin{equation*} L_{i}=\frac{gm\gamma e_{1i}^{m-1}}{(e_{1i}^{2m}+g)^{\frac{3}{2}}}, \quad i=1,\cdots,n+2 \end{equation*}

Since the term $z_{2}^{T}Mz_{2}$ is positive and continuously differentiable, we choose the Lyapunov function candidate as

(29) \begin{equation} V_{1}=\frac{1}{2}\sum _{i=1}^{n+2}\textrm{ln}\frac{k_{bi}^{2}}{k_{bi}^{2}-z_{1i}^2}+\frac{1}{2}z_{2}^{T}Mz_{2} \end{equation}

Taking the time derivative of $V_{1}$ along (4) and (28), we obtain

(30) \begin{align} \dot{V}_{1}&=\sum _{i=1}^{n+2}\frac{z_{1i}\dot{z}_{1i}}{k_{bi}^{2}-z_{1i}^2}+\frac{1}{2}z_{2}^{T}\dot{M}z_{2}+z_{2}^{T}M(\dot{x}_{2}-\dot{\alpha }_{1})\nonumber\\ \nonumber\\[-8pt]& =\sum _{i=1}^{n+2}\frac{z_{1i}}{k_{bi}^{2}-z_{1i}^2}\big (L_{i}(z_{2i}+\alpha _{i}-\dot{x}_{di})Q+\dot{Q}Q^{-1}z_{1i}\big ) \nonumber\\ \nonumber\\[-8pt]&\quad +z_{2}^{T}\big (D\tau +\bar{u}-B(q,\dot{q})\alpha -K(q)-M(q)\dot{\alpha }\big )+z_{2}^{T}(\frac{1}{2}\dot{M}(q)-B(q,\dot{q}))z_{2} \end{align}

According to Property 2, the term $z_{2}^{T}(\frac{1}{2}\dot{M}(q)-B(q,\dot{q}))z_{2}=0$ . To proceed, we define virtual control law $\alpha$ as

(31) \begin{equation} \alpha =\dot{x}_{d}-K_{1}M_{1}-\dot{Q}Q^{-1}M_{1} \end{equation}

with $K_{1}=\textrm{diag}\{K_{11},\cdots,K_{1(n+2)}\}$ is a positive-definite gain matrix and

\begin{equation*} M_{1}=\left[\frac{e_{11}(e_{11}^{2m}+g)}{gm},\cdots,\frac{e_{1i}(e_{1i}^{2m}+g)}{gm}\right]^{T} \end{equation*}

Combining $L_{i}, M_{1i}$ , and $Q$ , we get

(32) \begin{equation} \begin{aligned} M_{1i}L_{i}Q&=\frac{\gamma e_{1i}^{m}}{(e_{1i}^{2m}+g)^{\frac{1}{2}}}Q=z_{1i} \end{aligned} \end{equation}

Substituting (31) and (32) into (30), one has

(33) \begin{align} \dot{V}_{1}& =\sum _{i=1}^{n+2}\frac{z_{1i}}{k_{bi}^{2}-z_{1i}^2}(L_{i}z_{2i}Q-K_{1i}z_{1i})+z_{2}^{T}\big (D\tau +\bar{u}-B(q,\dot{q})\alpha -K(q)-M(q)\dot{\alpha }\big ) \nonumber\\[3pt]& =z_{1}^TwLQz_{2}-\sum _{i=1}^{n+2}w_{i}K_{1i}z_{1i}^2+z_{2}^{T}(D\tau +\bar{u}-B(q,\dot{q})\alpha -K(q)-M(q)\dot{\alpha }) \end{align}

where $w=\textrm{diag}\{w_{1},\cdots,w_{i}\}$ with $w_{i}=\frac{1}{k_{bi}^{2}-z_{1i}^2}$ , $L=\textrm{diag}\{L_{1},\cdots,L_{i}\}$ .

Since uncertainties in $M(q), B(q,\dot{q}), K(q)$ are difficult to obtain, we cannot realize the model-based control actually. Therefore, BNNs are utilized to approximate the uncertainties of system model. Design the adaptive controller as

(34) \begin{equation} \begin{aligned} \tau _{1}=\hat{P}\tau _{1}^{*}=\hat{P}\big [wLQz_{1}-K_{2}z_{2}+\hat{W}^{T}R(Z)-\textrm{sgn}(z_{2})u_{c}\big ] \end{aligned} \end{equation}

where $\hat{P}=\textrm{diag}\{\hat{P}_{1},\cdots,\hat{P}_{n+2}\}$ is the estimation matrix of $P$ with $P=D^{-1}$ , $\tau _{1}^{*}=[\tau _{11}^{*},\tau _{12}^{*},0,\cdots,0]^{T}\in R^{n+2}$ . $\hat{W}$ is the weight of NNs and $R(Z)$ is the basis function. The term $\hat{W}^{T}R(Z)$ denotes the estimation of ${W}^{*T}R(Z)$ , while ${W}^{*T}R(Z)$ represents the approximation of the uncertain item in the control (34) with

(35) \begin{equation} W^{*T} R(Z)=B(q,\dot{q})\alpha +K(q)+M(q)\dot{\alpha }-\epsilon (Z) \end{equation}

where $Z=[q^{T},\dot{q}^{T},\alpha ^{T},\dot{\alpha }^{T}]$ is the input of the basis function to the NNs and $\epsilon (Z)\in R^{n+2}$ is the approximation error. Define the NNs adaptive laws as

(36) \begin{equation} \dot{\hat W}_{i}=-\Gamma _{i}[R_{i}(Z)z_{2i}+\sigma _{i}\hat{W}_{i}] \end{equation}

where $\Gamma _{i}\in R^{(n+2)(n+2)}$ is the constant positive-definite gain matrix; $\sigma _{i}$ is a small positive constant.

Let $P_{i}=\frac{1}{l_{i}}, (i=1,\cdots,n+2)$ , and the adaptive laws of $\hat{P}_{i}$ are designed as

(37) \begin{equation} \begin{aligned} \dot{\hat{P}}_{i}=-z_{2i}\tau _{1i}^{*}-\hat{P}_{i} \end{aligned} \end{equation}

Then, a theorem that guarantees prescribed tracking performance under actuation faults is proposed as the following.

Theorem 3.1. For the system described by (1), the controller ( 34 ) with the adaptive law ( 36 ) and ( 37 ), if the initial conditions $ (q(0),\dot{q}(0),\hat{W}_{i}(0),\hat{P}_{i}(0) )$ are bounded and $z_{1i}(0)\lt k_{bi}$ , $i=1,2,\cdots,n+2$ , the semi-global stability of the system can be achieved. The error signals $z_{1}, z_{2}, \tilde{W}$ , and $\tilde{P}_{i}$ will remain within $\Omega _{z_{1}},\Omega _{z_{2}},\Omega _{W}$ , and $\Omega _{P}$ , respectively, which are defined as

\begin{align*} &\Omega _{z1}=\left \{z_{1}\in R^{n+2}||z_{1i}|\leq k_{bi}\sqrt{ \big (1-e^{-2J})\big )}\right \} \\[3pt]&\Omega _{z2}=\left \{z_{2}\in R^{n+2}|\Vert z_{2}\Vert \leq \sqrt{\frac{2J}{\lambda _{\textrm{min}}(M)}}\right \} \\[3pt]&\Omega _{W}=\left \{\tilde{W}\in R^{l\times (n+2)}|\Vert \tilde{W}\Vert \leq \sqrt{\frac{2J}{\lambda _{\textrm{min}}(\varGamma ^{-1}_{i})}}\right \} \\[3pt]& \Omega _{P_{i}}=\left \{\tilde{P_{i}}\in R^{n+2}||\tilde{P}_{i}| \leq \sqrt{\frac{2J}{\textrm{min}(l_{i})}} \right \} \end{align*}

with $i=1,\ldots,n+2$ and $J=V_{2}(0)+\frac{l_{0}}{p_{0}}$ with $l_{0}$ and $p_{0}$ are two positive constants.

$Proof:$ The Lyapunov function candidate is considered as

(38) \begin{equation} \begin{aligned} V_{2}&=\frac{1}{2}\sum _{i=1}^{n+2}\textrm{ln}\frac{k_{bi}^{2}}{k_{bi}^{2}-z_{1i}^2}+\frac{1}{2}z_{2}^{T}Mz_{2}+\frac{1}{2}\sum _{i=1}^{n+2}\tilde{W_{i}}^T\varGamma _{i}^{-1}\tilde{W_{i}}+\frac{1}{2}\sum _{i=1}^{n+2}l_{i}\tilde{P}_{i}^2 \end{aligned} \end{equation}

where $\tilde{P_{i}}=\hat{P}_{i}-P_{i}$ , $\tilde{W_{i}}=\hat{W}_{i}-W_{i}^{*}$ , and $\tilde{W_{i}}$ , $\hat{W}_{i}$ , $W_{i}^{*}$ are the NNs weight error, estimate value, and actual value, respectively.

Differentiating $V_{2}$ with respect to time leads to

(39) \begin{align} \dot{V}_{2}&=\sum _{i=1}^{n+2}\frac{z_{1i}}{k_{bi}^{2}-z_{1i}^2}\big (L_{i}(z_{2i}+\alpha _{i}-\dot{x}_{di})Q+\dot{Q}Q^{-1}z_{1i}\big ) \nonumber\\[3pt]&\quad +z_{2}^{T}[D\tau _{1}+\bar{u}-B(q,\dot{q})\alpha -K(q)-M(q)\dot{\alpha }]-\sum _{i=1}^{n+2}\tilde{W_{i}}^T[R_{i}(z)z_{2i}+\sigma _{i}\hat{W}_{i}]+\sum _{i=1}^{n+2}l_{i}\tilde{P}_{i}\dot{\hat{P}}_{i} \end{align}

According to the control law (34), NNs (35) and (37), we get

(40) \begin{align} &\quad z_{2}^{T}\big (D\tau _{1}-B(q,\dot{q})\alpha -K(q)-M(q)\dot{\alpha }\big )+\sum _{i=1}^{n+2}l_{i}\tilde{P_{i}}\dot{\hat{P}}_{i} \nonumber\\[3pt]& =z_{2}^{T}\big (D(\hat{P}-\tilde{P})\tau _{1}^{*}-W^{*T}R(Z)-\epsilon (Z)\big )-\sum _{i=1}^{n+2}l_{i}\tilde{P_{i}}\hat{P}_{i} \end{align}

Note that $l_{i}(\hat{P}_{i}-\tilde{P}_{i})=l_{i}P_{i}=1$ . Substituting (31), (34), and (40) into (39), we get

(41) \begin{align} \dot{V}_{2}&=-\sum _{i=1}^{n+2}w_{i}K_{1}z_{1i}^{2} +z_{2}^{T}\big (\!-K_{2}z_{2}-\epsilon (Z)+\bar{u}-\textrm{sgn}(z_{2})u_{c}\big )-\sum _{i=1}^{n+2}l_{i}\tilde{P}_{i}\hat{P}_{i}-\sum _{i=1}^{n+2}\sigma _{i}\tilde{W_{i}}^T\hat{W}_{i} \nonumber\\[3pt] & \leq -\sum _{i=1}^{n+2}w_{i}K_{1}z_{1i}^{2}-z_{2}^{T}K_{2}z_{2}-z_{2}^{T}\epsilon (Z) -\sum _{i=1}^{n+2}\sigma _{i}\tilde{W_{i}}^T\hat{W}_{i}-\sum _{i=1}^{n+2}l_{i}\tilde{P}_{i}\hat{P}_{i} \end{align}

By applying Young’s inequality, the following inequalities $-\sigma _{i}\tilde{W}_{i}^{T}\hat{W}_{i}\leq -\frac{\sigma _{i}}{2}(\Vert \tilde{W}_{i}\Vert ^{2}-\Vert{W_{i}^{*}}\Vert ^{2}),$ $-\tilde{P}_{i}\hat P_{i}\leq -\frac{1}{2}(\tilde P_{i}^2-P_{i}^2), -z_{2}^{T}\epsilon (Z)\leq \frac{1}{2}z_{2}^{T}z_{2}+\frac{1}{2}\Vert \epsilon (Z)\Vert ^{2}$ hold. And considering Lemmas 2.1 and 2.2, (41) can be written as

(42) \begin{align} \dot{V}_{2}& \leq -\sum _{i=1}^{n+2}K_{1i}\textrm{ln}\frac{k_{bi}^{2}}{k_{bi}^{2}-z_{1i}^2}-z_{2}^{T}\big (K_{2}-\frac{1}{2}I_{(n+2)(n+2)}\big )z_{2} -\frac{\sigma _{i}}{2}\sum _{i=1}^{n+2}\Vert \tilde{W}_{i}\Vert ^{2} \nonumber\\[3pt]&\quad -\frac{l_{i}}{2}\sum _{i=1}^{n+2}\tilde P_{i}^2+ \frac{\sigma _{i}}{2}\sum _{i=1}^{n+2}\Vert{W_{i}^{*}}\Vert ^{2}+\frac{l_{i}}{2}\sum _{i=1}^{n+2}P_{i}^2+\frac{1}{2}\Vert \epsilon (Z)\Vert ^{2} \nonumber\\[3pt]& \leq -l_{0}V_{2}+p_{0} \end{align}

where $I\in R^{(n+2)(n+2)}$ is an identity matrix, $l_{0}$ and $p_{0}$ are two constants defined as

(43) \begin{equation} \begin{aligned} l_{0}=\textrm{min}\bigg (\lambda _{\textrm{min}}(2K_{1}),\frac{\lambda _{\textrm{min}}(2K_{2}-I_{(n+2)(n+2)})}{\lambda _{\textrm{max}}(M)}, \textrm{min}\big (\frac{\sigma _{i}}{{\lambda _{\rm max}}(\varGamma _{i}^{-1})}\big ), 1\bigg ) \end{aligned} \end{equation}

(44) \begin{equation} p_{0}=\frac{\sigma _{i}}{2}\sum _{i=1}^{n+2}\Vert{W_{i}^{*}}\Vert ^{2}+\frac{l_{i}}{2}\sum _{i=1}^{n+2}P_{i}^2+\frac{1}{2}\Vert \epsilon (Z)\Vert ^{2} \end{equation}

with $\lambda _{\textrm{min}}(\!*\!)$ and $\lambda _{\textrm{max}}(\!*\!)$ refer to the minimum and maximum eigenvalues of the matrix $*$ , respectively. In order to ensure that $l_{0}\gt 0$ , the gains $K_{1}$ and $K_{2}$ should be satisfied

(45) \begin{equation} \lambda _{\textrm{min}}(2K_{1})\gt 0,\quad \lambda _{\textrm{min}}(2K_{2}-I_{(n+2)(n+2)})\gt 0 \end{equation}

Multiplying (42) by $e^{l_{0}t}$ yields

(46) \begin{equation} \frac{d}{dt}(V_{2}e^{l_{0}t})\leq p_{0}e^{l_{0}t} \end{equation}

Integrating $t$ on both sides of inequality (46), we have

(47) \begin{equation} V_{2}(t)\leq \bigg (V_{2}(0)-\frac{l_{0}}{p_{0}}\bigg )e^{l_{0}t}+\frac{l_{0}}{p_{0}}\leq V_{2}(0)+\frac{l_{0}}{p_{0}} \end{equation}

Combining (38) and (47), we get $z_{1i}^{2}\leq k_{bi}^2\big (1-e^{-2(V_{2}(0)+\frac{l_{0}}{p_{0}}})\big )$ , $\Vert z_{2}\Vert ^{2}\leq 2\big ( V_{2}(0)+\frac{l_{0}}{p_{0}}\big )\lambda _{\textrm{min}}^{-1}(M)$ , $\Vert \tilde{W}\Vert ^{2}\leq 2\big ( V_{2}(0)+\frac{l_{0}}{p_{0}}\big )\lambda _{\textrm{min}}^{-1}(\Gamma ^{-1})$ , $\tilde{P}_{i}^{2}\leq 2\big ( V_{2}(0)+\frac{l_{0}}{p_{0}}\big )(\textrm{min}(l_{i}))^{-1}$ . Therefore, we can conclude that $z_{1},z_{2},\tilde{W}$ , and $\tilde{P}$ all converge into the compact set shown in Theorem 1. The proof has been finished.

According to Theorem 1, the error variable $z_{1}$ is uniformly ultimately bounded with respect to the set $\Omega _{z1}$ . Thus, the preset performance described in (5a) can be achieved. Moreover, the error signals $z_{2}$ , $\tilde{W_{i}}$ , and $\tilde{P_{i}}$ are uniformly ultimately bounded. Because $W^{*}_{i}$ and $P_{i}$ are bounded with $\tilde{W_{i}}=\hat{W}_{i}-W_{i}^{*}$ , $\tilde{P_{i}}=P_{i}-\hat{P}_{i}$ , $\hat{W}_{i}$ and $\hat{P}_{i}$ must be bounded.

4.1. Simulation results without actuator faults

First, we prove the effectiveness of the proposed PP controller. Considering the initial state of errors of first joint and second joint is negative, so we choose $\gamma =\gamma _{2}$ . For the proposed controller, the specific parameters are set as: $K_{1}=\textrm{diag}\{12,12,1,1,1,1\}$ , $K_{2}=\textrm{diag}\{11,11,1,1,1,1\}$ , $m=3$ , $\gamma _{1}=\gamma _{2}=2$ , $\xi =0.5$ , $k_{bi}=1.2(i=1,\cdots,6)$ .

For the RBFNNs with BLS, the initial number of nodes in the NNs is $2^{6}=64$ , the initial center parameters are chosen as $-1$ or $1$ , and the widths are set as $\eta =2$ . The initial weights are $\hat{W}_{i}(0)=0(i=1,\cdots,64)$ , $\Gamma _{i}=0.1(i=1,\cdots,6)$ , $\sigma _{i}=0.02(i=1,\cdots,6)$ , the threshold is $\varXi =1.2$ , and the maximum number of neural nodes is $\bar{l}=100$ . According the definition and properties of rate function $\kappa (t)$ , we choose the $\kappa (t)$ as $e^{9.21t}$ . The simulation results are shown in Fig. 4.

Fig. 4. Tracking performance of the two joints.

From the tracking performance of the first and second joints shown in Fig. 4 (black dotted line), we find the tracking error can converge into the prescribed small region within $0.5\ \text{s}$ and then shrink to a smaller range at $t=1\ \text{s}$ , but the overshoot is too large to satisfy the prescribed requirement in (5d). Then we select a more larger value of $\gamma$ . So we set $\gamma =5$ in the next step, and the simulation results are shown as the red lines. We can clearly find that the tracking errors of two joints meet the PP described in (5a)–(5d) with small overshoot. Figure 5 shows that the uncertain function in (35) is better approximated under the improved RBFNNs and the number of neural nodes is only $100$ . Therefore, the proposed RBFNNs algorithm with BLS is effective. The first subfigure in Fig. 6 shows that the control inputs of both joints are bounded and smooth. The second subfigure in Fig. 6 is the trajectory change process of two joints in three-dimensional space, which shows the vibration of the flexible rod is effectively suppressed.

Fig. 5. Approximation effect of neural networks to unknown function.

Fig. 6. Control inputs and tip positions of two joints.

Second, to further demonstrate the superiority of the proposed NNs PP controller, the traditional PD control and the method proposed in ref. [Reference Karayiannidis and Doulgeri55] which has combined PD and PP are used as comparisons.

Design the PD control as

(49) \begin{equation} \tau =-K_{p}e_{1}-K_{d}\dot{e}_{1},\quad i=1,2 \end{equation}

where $e_{1}=x_{1}-x_{d}$ is the tracking error, and $K_{p}$ and $K_{d}$ are proportional and differential gains matrices. Considering the transient and steady-state performance of two joints, we get the optimal parameters with $K_{p}=10I_{6\times 6}$ and $K_{d}=10I_{6\times 6}$ .

For the model-free control with PP proposed in ref. [Reference Karayiannidis and Doulgeri55], the exponential performance function is set as

(50) \begin{equation} \rho (t)=(\rho _{0}-\rho _{\infty })\textrm{exp}(\!-lt)+\rho _{\infty } \end{equation}

with $\rho _{0}=1.1$ , $\rho _{\infty }=0.01$ , and $l=2$ . Taking into account the characteristics of the desired trajectory, we set the performance function as $\rho (t-10)$ at $t=10\,\text{s}$ .

Then, designing the model-free PPC $\tau$ as

(51) \begin{equation} \tau =-K_{p}e-K_{d}\dot{e}-K_{\varepsilon }J_{T}(x,t)\varepsilon (x) \end{equation}

where $K_{p},K_{d},K_{\varepsilon }$ are positive diagonal gain matrices, $x=\frac{e(t)}{\rho (t)}$ , $\varepsilon (x)$ is transformation function. For better comparison, we also choose $K_{p}=10I_{6\times 6}$ , $K_{d}=10I_{6\times 6}$ , and $K_{\varepsilon }=2I_{6\times 6}$ . See ref. [Reference Karayiannidis and Doulgeri55] for some other specific parameters. The simulation results of three cases are shown as Figs. 7 and 8.

Fig. 7. Tracking performance comparison of two joints.

Fig. 8. Tracking errors comparison of two joints.

From Fig. 7, we notice that although PD control can keep the system stable, it is much worse in terms of speed. Under the same proportional and derivative gains, the controller in ref. [Reference Karayiannidis and Doulgeri55] can significantly improve the rapidity of the system and keep $e(t)$ staying between $\rho (t)$ and $-\rho (t)$ , but still cannot meet our preset convergence time $T$ . Observing the proposed controller in Fig. 7, it can be found that the proposed NNs PP controller not only has higher stability accuracy but also faster convergence speed, which meet our preset requirements and effectively improve the transient and steady-state performance of flexible robotic manipulator. Therefore, the superiority of the proposed adaptive NNs PPC has been proved.

4.2. Simulation results with actuator faults

This part is mainly to verify the performance of the proposed FT controller in the case of actuator faults. The actuator faults in (2) include multiplicative and additive faults which are described as follows:

(52) \begin{equation} l_{1}(t)=\left \{ \begin{array}{l@{\quad}l} 1& t\lt 5\ \text{s}, \\ \\[-9pt]0.6& t\geq 5\ \text{s}.\\ \end{array} \right. \quad l_{2}(t)=\left \{ \begin{array}{l@{\quad}l} 1& t\lt 5\ \text{s}, \\ \\[-9pt]0.3& t\geq 5\ \text{s}.\\ \end{array} \right. \quad \bar u_{i}(t)=\left \{ \begin{array}{l@{\quad}l} 0& t\lt 5\ \text{s}, \\ \\[-9pt]0.3 sin(0.8t)+0.2& t\geq 5\ \text{s}.\\ \end{array} \right. \end{equation}

Through the above actuator faults settings, it is noted that the actuator for the first joint and the second joint lose $40\%$ and $70\%$ of its effectiveness, respectively, and both experience an additive bias fault after $5\ \text{s}$ . For time-varying additive fault, the upper bound of the fault is $u_{ci}=0.5, i=1,2$ . For the NNFT PPC (34), the parameters are same as described in section $B$ . Simulation results are shown as Figs. 9 and 10.

Fig. 9. Tracking performance of two joints with actuator faults.

Fig. 10. Control inputs of two joints with actuator faults.

From Fig. 9, the proposed adaptive FT control can still tracking the desired trajectories rapidly when actuators $\tau _{1}$ and $\tau _{2}$ occur faults. It is noted that the red line and blue line represent control input $u$ and control signal $\tau$ , respectively. Thus, we can conclude that the proposed NNFT control (34) possesses strong robustness against the actuator faults. In Fig. 10, when actuator faults occur at $5\ \text{s}$ , the proposed controller reacts rapidly to offset the adverse effects, so that the signal amplitude of the tracking error converges the prescribed range.