3.1. Proposal of sliding tracking control

First, a linear sliding surface is defined as

(7) \begin{align}{s_0} = {K_1}{e_1} + {e_2} \in {R^{n \times 1}}\end{align}

where ${K_1} = {k_1}{I_n} \in {R^{n \times n}}$ with k ₁ > 0 being the user-chosen constant.

Second, by employing the previously defined linear surface, a composite sliding surface is established as

(8) \begin{align}s = {\dot s_0} + {K_2}{s_0} \in {R^{n \times 1}}\end{align}

where ${K_2} = {k_2}{I_n} \in {R^{n \times n}}$ , with k ₂ > 0 being the user-chosen constant. The linear sliding surface s ₀ combined with the composite sliding surface s is employed to realize the sliding control. As s reaches zero in finite time, both s ₀ and $\dot{s}_{0}$ are bound to reach zero, and then the tracking errors e ₁ and e ₂ can asymptotically converge to zero.

Substituting (7) into (8), yields

(9) \begin{align}s & = {K_1}{{\dot e}_1} + {{\dot e}_2} + {K_1}{K_2}{e_1} + {K_2}{e_2} \nonumber\\ & = {{\dot e}_2} + ({K_1} + {K_2}){e_2} + {K_1}{K_2}{e_1}\end{align}

Thus, we can select suitable K ₁ and K ₂ to realize that the tracking errors have favorable dynamics on the sliding surface.

Substitute (6) into (9), and the ultimate sliding surface can be further described as

(10) \begin{align}s = \Theta {\tau _a}(t) + d + ({K_1} + {K_2}){e_2} + {K_1}{K_2}{e_1} - {\ddot x_r}\end{align}

In addition, differentiating (10) and using (6), the first time derivative of the sliding surface can be expressed as

(11) \begin{align}\dot{s} &= \Theta {{\dot \tau }_a}(t) + \dot d + ({K_1} + {K_2})\Theta {\tau _a} + ({K_1} + {K_2})d + {K_1}{K_2}{e_2} - ({K_1} + {K_2}){{\ddot x}_r} - {{\dddot x}_r} \nonumber\\ & = \Theta {{\dot \tau }_a}(t) + \Im v + ({K_1} + {K_2})\Theta {\tau _a} + {K_1}{K_2}{e_2} - ({K_1} + {K_2}){{\ddot x}_r} - {{\dddot x}_r} \end{align}

where $v = {[{d^T},{\dot d^T}]^T} \in {R^{2n \times 1}}$ is the vector to be estimated, and $\Im = [{K_1} + {K_2},{I_n}] \in {R^{n \times 2n}}$ .

Following the exponential reaching law approach, [Reference Devika and Thomas19] the expression related to the control strategy can be formulated as

(12) \begin{align}\Theta {\dot \tau _a} + ({K_1} + {K_2})\Theta {\tau _a} = - {K_1}{K_2}{e_2} - \Im \hat{v} + {\dddot{x}_r} + ({K_1} + {K_2}){\ddot x_r} - Ks - (\Pi + K){\mathop{\rm sgn}} (s)\end{align}

where $\hat{v} \in {R^{2n \times 1}}$ is the estimate of v; $K = k{I_n} \in {R^{n \times n}}$ and $\Pi = \eta {I_n} \in {R^{n \times n}}$ are the user selectable positive definite matrices.

Then, by substituting (12) into (11), one can obtain that

(13) \begin{align}\dot s = \Im \tilde v - Ks - (\Pi + K){\mathop{\rm sgn}} (s)\end{align}

where $\tilde v = v - \hat{v}$ is the estimate error vector.

Define the error vector $e = {[e_1^T{\rm{, }}e_2^T]^T} \in {R^{2n \times 1}}$ , and thus an augmented system can be established by taking the control input as an auxiliary state, which is described as

(14) $${\left\{ {\matrix{ {\mathop {\overline e }\limits^ = \bar A\bar e + {{\bar B}_1}\varpi + {{\bar B}_2}d} \cr {y = {{\bar C}_1}\bar e} \cr } } \right.}$$

where $\bar e = {[{e^T}{\rm{, }}\tau _a^T{\rm{ }}]^T} \in {R^{3n \times 1}}$ , $\varpi = {\dot \tau _a}$ , $\bar A = \left[ {\begin{array}{*{20}{c@{\quad}c@{\quad}c}}0 & {{I_n}} & 0\\0 & 0 & \Theta \\0 & 0 & 0\end{array}} \right]$ , ${\bar B_1} = \left[ {\begin{array}{*{20}{c}}0\\0\\{{I_n}}\end{array}} \right]$ , ${\bar B_2} = \left[ {\begin{array}{*{20}{c}}0\\{{I_n}}\\0\end{array}} \right]$ , ${\bar C_1} = \left[ {\begin{array}{*{20}{c@{\quad}c@{\quad}c}}{{I_n}} & 0 & 0\\0 & {{I_n}} & 0\end{array}} \right]$ .

3.2. Establishment of RDO

The purpose of this section is to design RDO based on H_∞ and pole assignment theories and utilize it to estimate the unknown function vector. Sufficient LMI–based existence conditions will be presented with the following DO:

(15) \begin{align}\left\{ \begin{array}{l}\hat{v} = p + \aleph {e_2}\\\dot p = - \aleph \Xi + Q\hat{v}\end{array} \right.\end{align}

where $p \in {R^{2n \times 1}}$ , $\aleph \in {R^{2n \times n}}$ and $Q \in {R^{2n \times 2n}}$ .

Assumption 2. The lumped vector d and its time derivatives up to the second order are continuous and norm bounded, that is,

(16) \begin{align}\left\| {{d^{(j)}}(x,t,{\tau _a})} \right\| \le {\mu _j} \lt + \infty ,\,\,\,\textrm{for}\;j = 0,{\rm{ }}1,{\rm{ }}2\end{align}

where μ _j is a positive unknown number.

According to (6) and (15), the derivative of $\hat{v}$ can be expressed as

(17) \begin{align}\dot{\hat{v}} & = \dot p + \aleph {{\dot e}_2}\nonumber\\ & = - \aleph \Xi + Q\hat{v} + \aleph (\Xi + d)\nonumber\\ & = Q\hat{v} + \aleph d\end{align}

The estimate error vector $\tilde v$ is described as

(18) \begin{align}\tilde v = v - \hat{v} = \left[ {\begin{array}{*{20}{c}}d\\{\dot{\!d}}\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}{\hat {d}}\\ \;{\hat{\dot{\!\!d}}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{\tilde d}\\\;{\tilde{\dot{\!\!d}}}\end{array}} \right]\end{align}

Differentiating (18) and using (17), we have

(19) \begin{align}\dot{\tilde{v}} & = \dot v - \dot{\hat{v}} \nonumber\\ & = Fv + \Psi \ddot d - Q\hat{v} + Tv \nonumber\\ & = Q\tilde v + \Psi \ddot d\end{align}

where $F = \left[ {\begin{array}{*{20}{c@{\quad}c}}{{0_{n \times n}}} & {{I_{n \times n}}}\\{{0_{n \times n}}} & {{0_{n \times n}}}\end{array}} \right]$ , $\Psi = \left[ {\begin{array}{*{20}{c}}{{0_n}}\\{{I_n}}\end{array}} \right]$ , $T = [ - \aleph ,{0_{2n \times n}}]$ , Q = T+F.

Consequently, by choosing the estimate error as the controlled output, system (19) can be rewritten as

(20) \begin{align}\left\{ \begin{array}{l}\dot{\tilde{{v}}} = Q\tilde v + \Psi \vartheta \\ \!z = \tilde v\end{array} \right.\end{align}

where $\vartheta = \ddot d$ .

The RDO design problem under consideration is to find appropriate matrix Q in (20) such that the error system satisfies the following requirements:

1. (i) The error system (19) is asymptotically stable, and the H_∞ norm of the closed–loop transfer function from $\vartheta $ to z, that is, $||{T_{\vartheta z}}|{|_\infty }$ , should be as small as possible.

2. (ii) All the system poles in (19) should be located into a specified circular disk (q _c ,r _a ), which is included in the open left–half complex plane. Here, the circular disk (q _c ,r _a ) denotes the disk with center at (−q _c ,0) and radius r _a , where q _c > r _a > 0.

Lemma 1. By virtue of the Bounded Real Lemma [Reference Chen, Guo and Syst20, Reference Prempain and Postlethwaite21] for the estimate error system (20), the H_∞ norm of ${T_{\vartheta z}}$ is less than ${\rho}$ if and only if there exists a symmetric positive definite ${\Sigma _1} \in {R^{2n \times 2n}}$ with

(21) \begin{align}\left[ {\begin{array}{*{20}{c@{\quad}c@{\quad}c}}{{Q^T}{\Sigma _1} + {\Sigma _1}Q} & {{\Sigma _1}\Psi } & I\\ \ast & { - \rho I} & 0\\\ast & \ast & { - \rho I}\end{array}} \right] \lt 0\end{align}

Lemma 2. According to literature, [Reference Hu, Jiang and Yang22] all poles of system (20) are located into the specified circular disk (q _c ,r _a ), if and only if there exists a symmetric positive definite ${\Sigma _2} \in {R^{2n \times 2n}}$ with

(22) \begin{align}\left[ {\begin{array}{*{20}{c@{\quad}c}}{ - {r_a}{\Sigma _2}} & {{q_c}{\Sigma _2} + Q{\Sigma _2}}\\ \ast & { - {r_a}{\Sigma _2}}\end{array}} \right] \lt 0\end{align}

As a consequence of Lemmas 1 and 2, if matrix inequalities (21) and (22) are satisfied simultaneously, an optimal matrix Q can be obtained via the following minimization problem

(23) \begin{align}\begin{array}{l}\min {\rm{ }}\rho \\{\rm{subject\; to\; (21)\; and\; (22)}}{\rm{.}}\end{array}\end{align}

Hence, the performances (i) and (ii) can be satisfied in system (20) simultaneously.

3.3. Solution for optimization problem of RDO

From the observation of (19), it is easy to find that matrix Q is an additive combination of matrices T and F, leading to certain inconveniences in solving the optimization problem (23) by LMI directly. Furthermore, the assumption of Σ₁ = Σ₂ in (23) can inevitably cause some conservation in traditional LMI method. Thus, it is necessary to introduce a nonconservative algorithm to solve the optimization problem.

DE is a stochastic, population-based algorithm with the advantages of the global solution finding property and powerful search capability, which is firstly proposed by Storn and Price [Reference Storn and Price23, Reference Storn24]. Hence, a DE-based LMI which combines the effective solution of LMI and the randomized search of DE will be utilized to obtain the optimal matrix Q by solving the optimization problem (23). Since the standard DE can be found in several related refs. [Reference Gong, Cai, Yang, Li and Jian25, Reference Liu, Wang, Jin, Huang and Tang26], the procedures of the proposed optimization algorithm are outlined as follows.

Step 1: Population initialization. An initial population of N _p individuals for $\aleph $ is randomly generated one or more times until every corresponding matrix Q can satisfy the constraint requirement in (23).

Step 2: Objective function evaluation. For any initialized matrix Q, problem (23) turns to LMI which can be efficiently solved by MATLAB LMI Toolbox, and take each ρ as the objective value corresponding to every individual.

Step 3: Evolution process. In this step, three operations of mutation, crossover and selection are implemented in turn to select the optimal individual.

Step 4: If a stopping criterion is satisfied, then output the optimal individual as well as its objective value; otherwise return to Step 2.

3.4. Performance analysis for RDO

Since the eigenvalues of Q in system (20) are determined in the left half phase, thus it is always possible to find a positive definite matrix P $_{\xi}$ such that ${Q^T}{P_\xi } + {P_\xi }Q = - \Phi $ holds for any given positive definite matrix Φ.

Choose the Lyapunov function as ${V_1} = {\tilde v^T}{P_\xi }\tilde v$ , and then

(24) \begin{align}{{\dot V}_1} & = {{\dot{\tilde{{v}}}}^{\,T}}{P_\xi }\tilde v + {{\tilde v}^T}{P_\xi \,}\dot{\tilde{{v}}} \nonumber\\ & = {{\tilde v}^T}({Q^T}{P_\xi } + {P_\xi }Q)\tilde v + 2{{\tilde v}^T}{P_\xi }\Psi \vartheta \end{align}

Noting that

\begin{align*}2{\tilde v^T}{P_\xi }\Psi \vartheta \le \frac{1}{\tau }{\tilde v^T}{P_\xi }{P_\xi }\tilde v + \tau {(\Psi \vartheta )^T}\Psi \vartheta, \end{align*}

where $\tau$ is a tunable positive value, the following inequality can be obtained

(25) \begin{align}{{\dot V}_1} & \le - {{\tilde v}^T}\Phi \tilde v + \frac{1}{\tau }{{\tilde v}^T}{P_\xi }{P_\xi }\tilde v + \tau {(\Psi \vartheta )^T}\Psi \vartheta \nonumber\\[5pt] & \le - [{\lambda _{\min }}(\Phi ) - \frac{1}{\tau }\lambda _{\max }^2({P_\xi })]||\tilde v|{|^2} + \tau ||\Psi \vartheta |{|^2}\end{align}

Based on properly chosen Φ, P $_{\xi}$ and $\tau$ , ${\lambda _{\min }}(\Phi ) - \dfrac{1}{\tau }\lambda _{\max }^2({P_\xi }) \gt 0$ can be guaranteed. It is thus derived that the following inequality

(26) \begin{align}{\dot V_1} \le - \lambda _{\max }^{ - 1}({P_\xi }) \cdot \left[{\lambda _{\min }}(\Phi ) - \frac{1}{\tau }\lambda _{\max }^2({P_\xi })\right]{V_1} + \tau ||\Psi \vartheta |{|^2} \le - {\gamma _1}{V_1} + {\gamma _2}\end{align}

where ${\gamma _1}\;{\rm{ = }}\;\lambda _{\max }^{ - 1}({P_\xi }) \cdot \left[{\lambda _{\min }}(\Phi ) - \dfrac{1}{\tau }\lambda _{\max }^2({P_\xi })\right] \gt 0$ and ${\gamma _2}\;{\rm{ = }}\;\tau ||\Psi \vartheta |{|^2} \gt 0$ hold obviously. Then it can be further obtained that ${V_1} \le ({V_1}(0) - \frac{{{\gamma _2}}}{{{\gamma _1}}}){e^{ - {\gamma _1}t}} + \frac{{{\gamma _2}}}{{{\gamma _1}}}$ , and the norm of the estimation error is upper bounded, that is, $\left\| {\tilde v} \right\| \le \bar L$ .

4.1. Control law design

In this section, ASTC will be proposed based on RDO designed in Section 3. Substituting (10) into (12) yields

(27) \begin{align}\eqalign{ \Theta {{\dot \tau }_a} + (({k_1} + {k_2}) + k)\Theta {\tau _a} = - K{K_1}{K_2}{e_1} - ({K_1}{K_2} + K{K_1} + K{K_2}){e_2} - Kd \cr {\rm{ - }}\underbrace {(\Im \hat{v} + (\Pi + K){\mathop{\rm sgn}} (s) - {{\dddot x}_r} - ({K_1} + {K_2} + K){{\ddot x}_r})}_{{f_s}} \cr} \end{align}

where $ \,f_{s} = (\Im \hat{v} + (\Pi + K){\mathop{\rm sgn}} (s) - {{\dddot x}_r} - ({K_1} + {K_2} + K){{\ddot x}_r}) $ .

Substituting (27) into (14), one can obtain the following closed–loop augmented system

(28) \begin{align}\left\{ \begin{array}{l}\dot{\bar{{e}}} = {A_c}\bar e + {B_c}{\omega _c}\\y = {C_c}\bar e\end{array} \right.\end{align}

where ${A_c} = \left[ {\begin{array}{*{20}{c@{\quad}c@{\quad}c}}{{0_n}} & {{I_n}} & {{0_n}}\\{{0_n}} & {{0_n}} & {\chi {I_n}}\\{{\Upsilon _1}} & {{\Upsilon _2}} & {{\Upsilon _3}}\end{array}} \right]$ , ${B_c} = \left[ {\begin{array}{*{20}{c@{\quad}c}}{{0_n}} & {{0_n}}\\{{I_n}} & {{0_n}}\\{{{ - k{I_n}} \mathord{\left/ {\vphantom {{ - k{I_n}} \chi }} \right.} \chi }} & {{{ - {I_n}} \mathord{\left/{\vphantom {{ - {I_n}} \chi }} \right.} \chi }}\end{array}} \right]$ , ${C_c} = \left[ {\begin{array}{*{20}{c@{\quad}c@{\quad}c}}{{I_n}} & {{0_n}} & {{0_n}}\\{{0_n}} & {{I_n}} & {{0_n}}\end{array}} \right]$ , ${\omega _c} = \left[ {\begin{array}{*{20}{c}}d\\{{f_s}}\end{array}} \right]$ ; ${\Upsilon _1} = {{ - k{k_1}{k_2}{I_n}} \mathord{\left/{\vphantom {{ - k{k_1}{k_2}{I_n}} \chi }} \right. } \chi }$ , ${\Upsilon _2} = {{( - {k_1}{k_2} - k{k_1} - k{k_2}){I_n}} \mathord{\left/{\vphantom {{( - {k_1}{k_2} - k{k_1} - k{k_2}){I_n}} \chi }} \right.} \chi }$ , ${\Upsilon _3} = ( - k - {k_1} - {k_2}){I_n}$ .

By calculation, one can obtain that ${\lambda _{3i - 2}}({A_c}) = - {k_1}$ , ${\lambda _{3i - 1}}({A_c}) = - {k_2}$ , ${\lambda _{3i}}({A_c}) = - k$ , i = 1,2,…,n. Since the eigenvalues are definitely located in the open left–half complex plane, matrix A _c is Hurwitz.

Theorem 1. Considering the nonlinear uncertain system (2), if the system control law is designed as

(29) \begin{align} \left\{ \begin{array}{l} {\tau _a}(t) = {e^{ - ({K_1} + {K_2})t}}{\tau _a}(0) + \int_0^t {{e^{ - ({K_1} + {K_2})(t - \upsilon )}}} {B_\upsilon }(t - \upsilon )d\upsilon \\[4pt] {B_\upsilon } = {\Theta ^{ - 1}}({\rm{ - }}{K_1}{K_2}{e_2} - \Im \hat{v} + {{\dddot x}_r} + ({K_1} + {K_2}){{\ddot x}_r} - Ks - (\prod + K){\mathop{\rm sgn}} (s)) \hfill \end{array} \right. \end{align}

the controlled system will be globally stable and the tracking errors will converge to zero with the following adaptive laws:

(30) \begin{align}\left\{ \begin{array}{l}\dot k\;{\rm{ = }}\;\alpha ||s||\\\dot \eta = \beta ||s||\end{array} \right.\end{align}

where $\alpha$ and $\beta$ are the adaptive rates.

Proof. From (12), it can be obtained that the system control law (29) holds.

Define $\tilde k$ and $\tilde \eta $ as

(31) \begin{align}\tilde k = {k_0} - k,\,\,\,\tilde \eta = {\eta _0} - \eta \end{align}

where ${k_0}\;{\rm{ = }}\;{\eta _0}\;{\rm{ = }}\;{{{\rm{||}}\Im ||\bar L} \mathord{\left/{\vphantom {{{\rm{||}}\Im ||\bar L} 2}} \right. } 2}$ .

Choose a Lyapunov function candidate as

\begin{align}V = \frac{1}{2}{s^T}s + \frac{1}{{2\alpha }}{\tilde k^2} + \frac{1}{{2\beta }}{\tilde \eta ^2}\end{align}

It is obvious that V is a positive definite function. Taking the time derivative of the Lyapunov function, candidate V gives

(32) \begin{align}\dot V = {s^T}\dot s + \frac{1}{\alpha }({k_0} - k)( - \dot k) + \frac{1}{\beta }({\eta _0} - \eta )( - \dot \eta )\end{align}

In view of Eq. (13), it follows that

(33) \begin{align}\begin{array}{c}\dot V = {s^T}\Im \tilde v - {s^T}Ks - {s^T}(\Pi + K){\mathop{\rm sgn}} (s) + \dfrac{1}{\alpha }({k_0} - k)( - \dot k) + \dfrac{1}{\beta }({\eta _0} - \eta )( - \dot \eta )\\ = {s^T}\Im \tilde v - k||s|{|^2} - (\eta + k)||s|{|_1} + \dfrac{1}{\alpha }({k_0} - k)( - \dot k) + \dfrac{1}{\beta }({\eta _0} - \eta )( - \dot \eta )\end{array}\end{align}

Taking (30) into (33), one can obtain

(34) \begin{align} \dot V & \le ||s|| \cdot ||\Im || \cdot ||\tilde v|| - k||s|{|^2} - (\eta + k)||s|{|_1} - ||s||({k_0} - k) - ||s||({\eta _0} - \eta )\nonumber\\ & \;{\rm{ = }}\; - k||s||_2^2 - (\eta + k)||s|{|_1} + (\eta + k)||s|| \nonumber\\ & \le - k||s||_2^2 \end{align}

According to Eq. (34), it is clear that $\dot V \le 0$ is held. Hence, the proof is completed.

4.2 Control parameter design for H₂ performance

As previously mentioned, ASTC is proposed in Theorem 1, in which an essential step is to choose appropriate control parameters before practical application. In this section, control parameters will be determined by introducing a constructive design procedure under the prescribed H₂ performance of the closed-loop system to further restrain the adverse effects caused by actuator nonlinearity.

Denote the transfer function from ω _c to y as T _s = C _c(sI-A _c )- [Reference Han and Lee1] B _c. If ||T _s ||₂ has an upper bound, then we call that the closed-loop system (28) possesses an H₂ performance. Let k ₁ = k ₂ = g > 0 for the convenience of parameter design, and ϒ₁, ϒ₂ and ϒ₃ in matrix A _c can be reformulated as ${\Upsilon _1} = {{ - k{g^2}{I_n}} \mathord{\left/{\vphantom {{ - k{g^2}{I_n}} \chi }} \right. } \chi }$ , ${\Upsilon _2} = {{( - {g^2} - 2gk){I_n}} \mathord{\left/{\vphantom {{( - {g^2} - 2gk){I_n}} \chi }} \right.} \chi }$ , ${\Upsilon _3} = ( - k - 2g){I_n}$ , respectively.

Remark 7. System (28) is asymptotically stable and has an H₂ performance $\lambda^{{-}} \gt 0$ if and only if there exists a positive definite symmetric matrix Ω such that

(35) \begin{align}\left\{ \begin{array}{l}{A_c}\Omega + \Omega A_c^T + {B_c}B_c^T \lt 0\\{\rm{ }}Trace({C_c}\Omega C_c^T) \lt {\lambda^{{-}}{}^2}\end{array} \right.\end{align}

As is well known, (35) is usually utilized as an effective approach for H₂ performance design [Reference Chen, Xu and Feng29–Reference Lu, Du, Wang and Xue31] Unfortunately, A _c, B _c and Ω, existing in an nonlinear form in (35), are all unknown matrices to designers. Therefore, it is difficult to solve (35) directly by using LMI toolbox. To overcome this difficulty, a novel straight-forward approach employing the method of definition is alternatively presented to determine the control parameters under the H₂ performance constraint.

Proof. According to the definition of H₂ norm of a transfer function, [Reference Scherer, Gahinet and Chilali32] ||T||₂ can be easily computed as

(36) \begin{align}||T|{|_2} = \sqrt {\frac{n}{4} \cdot \frac{{8{g^3} + 6{g^2}k + 2g + k + 2{k^2}{g^3} + 5k{g^4}}}{{k{g^3}({k^2} + 2kg + {g^2})}}} ,\,\,n = 1,2,3 \cdots \end{align}

Let α = k/g, and (36) can be rewritten as

(37) \begin{align}||T|{|_2} = \sqrt {\frac{n}{4} \cdot \frac{{2{g^5}{\alpha ^2} + (6{g^3} + 5{g^5} + g)\alpha + (2g + 8{g^3})}}{{\alpha {{(\alpha + 1)}^2}{g^6}}}} \end{align}

Assume that there exists δ > 0 satisfying $||T||_2^2 \le \delta $ , that is,

(38) \begin{align}\frac{n}{4} \cdot \frac{{2{g^5}{\alpha ^2} + (6{g^3} + 5{g^5} + g)\alpha + (2g + 8{g^3})}}{{\alpha {{(\alpha + 1)}^2}{g^6}}} \le \delta ,\forall \alpha \gt 1\end{align}

Denote $l(\alpha )\;{\rm{ = }}\;4\delta {g^6}{\alpha ^3} + (8\delta {g^6} - 2{g^5}n){\alpha ^2} + (4\delta {g^6} - 6{g^3}n - 5{g^5}n - gn)\alpha + ( - 8{g^3}n - 2gn)$ , and then (38) is equivalent to the following problem

(39) \begin{align}l(\alpha ) \ge 0,{\rm{ }}\exists \delta \gt 0,{\rm{ }}\forall \alpha \gt 1\end{align}

Since l(0) < 0 holds obviously, (39) can always hold as long as l(1) ≥ 0, that is,

(40) \begin{align}16\delta {g^6} - 7{g^5}n - 14{g^3}n - 3gn \ge 0\end{align}

By solving inequality (40), $\delta$ is obtained as

(41) \begin{align}\delta \ge \frac{{(7{g^4} + 14{g^2} + 3)n}}{{16{g^5}}}\end{align}

Hence, the proof is completed.

Based on the design process explained in Sections 3 and 4, the block diagram of ASTC is presented in Fig. 1 and the whole flow chart of the algorithm is shown in Fig. 2.

Figure 1.

Block diagram of the proposed control strategy.

Figure 2.

Flow chart of the proposed adaptive sliding tracking control.

5.1. Design procedure

To summarize the analyses in Sections 3 and 4, the procedures of ASTC are outlined as follows:

Step 1. Selection of observer parameters. Setting q _c = 200 and r _a = 100, and using LMI toolbox of MATLAB to solve minimization problem (23), one can obtain the following optimal DO matrix

(42) \begin{align}Q = \left[ {\begin{array}{*{20}{c@{\quad}c}}{ - 250.87} & 1\\{ - 19951} & 0\end{array}} \right].\end{align}

Step 2. Selection of control parameters. Assume that the prescribed H₂ performance is 0.5, and control parameters k ₁and k ₂ are selected as k ₁ = k ₂ = 3. The initial values of k and $\eta$ are set as k(0) = 4 and $\eta$ (0) = 0. In addition, adaptive rates $\alpha$ = 0.03 and $\beta$ = 0.01 are chosen.

5.2. Example 1: System without actuator nonlinearity

In this section, one-link robotic manipulator system without actuator nonlinearity is considered for tracking its equilibrium point x _e = [0, 0] ^T in two cases. Thus, in model (3), Θ = 1 and ζ = 0 are selected for simulation. The system initial state is selected as x(0) = [0, 1] ^T .

• Case 1: Nominal system.

System (2) is employed for simulation with $f(x,t) = - x_1^2 - 5{x_2}$ , $b(x,t) = {x_1}{x_2} + 1$ and $\omega (x,t) = ({x_1}{x_2} + 1)(0.2(t - 3) + 0.8\sin(3t - 3) + 0.5{e^{ - t}}) + \Gamma (t)$ , where Γ(t) is a random noise of 0 mean and 0.5 variance.

It is observed from Figs. 3 and 4 that the proposed controller can stabilize the considered system quickly. Moreover, no chatter phenomenon can be observed in the control input in Fig. 4. In order to further validate the superiority and effectiveness of the proposed ASTC, the existing results in literature [Reference Zhang and Wang33, Reference Liu34] are presented for the objective of comparison.

Figure 3.

States x ₁ and x ₂ for Case 1 in Example 1.

Figure 4.

Control inputs for Case 1 in Example 1.

The proposed controller is firstly compared with nonsingular terminal sliding mode (NTSM) controller designed in ref. [Reference Zhang and Wang33] where b(x,t) and f(x,t) are assumed to be precisely obtained. From Fig. 3, we can see that both of the controllers can stabilize the system to the equilibrium state at the same rate, although NTSM control has the outstanding advantage of fast convergence rate by using an enhanced nonlinear sliding surface. However, NTSM controller exhibits serious chatter phenomenon in the control input and substantial increase in the control effort than that in the proposed ASTC in Fig. 4.

For comparison purpose, another class of sliding mode controller in ref. [Reference Liu34] is also considered in the simulation study, in which a robust adaptive sliding (RAS) control based on radial basis function neural networks (RBFNN) is put forward. Different from NTSM controller in ref. [Reference Liu34], the estimates of b(x,t) and f(x,t) by RBFNN are employed in ref. [Reference Liu34], whereas the upper bound of ξ(x,t) is still a necessary requirement. Hence, RAS controller cannot avoid the dependence on prior information on system dynamics. As can be seen in Fig. 3, the system states under controller [Reference Liu34] converge to zero with the slowest rate. In addition, the controller proposed in ref. [Reference Liu34] is also seriously affected by the undesirable chatter phenomenon.

In conclusion, by comparison with the other two SMC-based strategies, similar or better output performances can be achieved although no prior knowledge about the robotic manipulator can be employed in ASTC. Additionally, the proposed controller produces smooth control input with small control efforts, which is free from undesirable chatter phenomenon. Thus, ASTC is appropriate for real applications to NURSs from a practical point of view.

• Case 2: Perturbed system.

In this section, complex changes in f(x,t) and b(x,t) are taken into account to further demonstrate the effectiveness and practicability of ASTC. To this end, a piece-wise nonlinear function of time ћ(t) is introduced as follows:

(43) \begin{align} \hbar(t) = \left\{ \begin{array}{l@{\quad}c} 0.2 + 2t, & 0 \le t \le 2 \\[4pt] 5{e^{2 - t}} + 2\sin(0.2\pi t), & 2 \le t \le 4 \\[4pt] 3\cos(0.2\pi (t - 6)) \cdot \sin(0.3\pi t), & t \gt 4 \end{array} \right. \end{align}

According to (43), f(x, t) and b(x, t) are transformed as $f(x,\,t) = \hbar(t)\cdot(\!-x^{2}_{1}-5x_{2})$ and $b(x,\,t) = (x_{1}x_{2}+1)+\hbar(t)$ , and the output responses are presented in Figs. 5 and 6. It is noteworthy that ASTC can still offer satisfying output performances by applying a smooth control input as compared to the controllers in refs. [Reference Zhang and Wang33, Reference Liu34].

Figure 5.

States x ₁ and x ₂ for Case 2 in Example 1.

Figure 6.

Control inputs for Case 2 in Example 1.

To evaluate the smoothness of the control input, the total variance (TV) [Reference Skogestad35] of the control input is utilized, which is calculated as follows:

(44) \begin{align}TV = \sum\limits_{i = 1}^{{n^\dagger } - 1} {\left| {{u_{i + 1}} - {u_i}} \right|} \end{align}

where ${n^\dagger }$ is the number of samples. In addition, energy of the control input is calculated by the 2–norm method. It is obvious that both TV and control input energy are desired to be as small as possible.

Table I shows TV and energy of the system inputs obtained by different control strategies in Cases 1 and 2, which are calculated for the period from 0 to 5 s with a sampling time of 0.01 s. It is clear that the proposed ASTC is able to produce a smoother control input with minimal control energy in both cases.

Table I.

Performance comparison of control inputs in Example 1.

5.3. Example 2: System with typical actuator nonlinearities

In this section, another one-link robotic manipulator of system (2) subject to typical actuator nonlinearities is employed, in which

(45) \begin{align}\left\{ \begin{array}{l}f(x,t) = - {x_2} + 3.5x_2^2\cos {x_1} - 35\sin {x_1}\\b(x,t) = 1\\w(x,t) = 0.8{e^{ - 0.2t}}\sin t\end{array} \right.{\rm{ }}\end{align}

The desired trajectories are described as ${x_r} = 0.5(\sin t + \sin(0.5t))$ rad and ${\dot x_r} = 0.5(\cos t + 0.5\cos(0.5t))$ rad/s, respectively. The system initial state is selected as x(0) = [0, 0] ^T . The ASTC parameters carried out in this example are same as those in Example 1, and three different types of actuator nonlinearities are considered. For the objective of comparison, the existing results in ref. [Reference Liu, Dong, Xue and Zhang12] are also presented, in which a Nussbaum gain-based adaptive neural network control is fit to the cases that were applied on it.

• Case 1: Actuator with dead-zone nonlinearity described as

(46) \begin{align}u = \left\{ \begin{array}{l}(1 + 0.3\sin {\tau _a}){\rm{ }}({\tau _a} - 0.5),{\rm{ }}{\tau _a} \ge 0.5\\0,{\rm{ - 0}}{\rm{.3}} \le {\tau _a} \le 0.5\\(0.8 + 0.2\cos {\tau _a}){\rm{ }}({\tau _a} + 0.3),{\rm{ }}{\tau _a} \le - 0.3\end{array} \right.\end{align}

• Case 2: Actuator with hysteresis nonlinearity described as

(47) \begin{align}\frac{{du}}{{dt}} = \left| {\frac{{d{\tau _a}}}{{dt}}} \right|(3.2{\tau _a} - u) + 0.4\frac{{d{\tau _a}}}{{dt}}\end{align}

• Case 3: Actuator with backlash nonlinearity described as

(48) \begin{align}u = \left\{ \begin{array}{l}1.2({\tau _a} - 0.5),\;{\rm{ }}\dot u \gt 0\\1.2({\tau _a} + 0.6),\;{\rm{ }}\dot u \lt 0\\u({t_ - }),\;{\rm{ }}otherwise\end{array} \right.\end{align}

Simulation results in Figs. 7–9 indicate that the proposed ASTC shows a faster convergence speed and smaller steady-state error when compared with the method in ref. [Reference Liu, Dong, Xue and Zhang12]. Thus, ASTC can achieve better tracking performance no matter what the actuator nonlinearities are.

Figure 7.

Tracking errors with dead-zone actuators.

Figure 8.

Tracking errors with hysteresis actuators.

Figure 9.

Tracking errors with backlash nonlinearity actuators.

Figure 10.

Tracking errors with complex nonlinearity actuators.

5.4. Example 3: System with complex nonlinearity

• Case 1: Two-link robot system

In this case, a two-link robotic arm in ref. [Reference Chen, Liu, Zhang, Chen and Xie13] is employed to confirm the effectiveness of the proposed scheme in this study. The dynamics is described as

\begin{align*}\left[ {\begin{array}{*{20}{c@{\quad}c}}{{M_{11}}} {{M_{12}}}\\{{M_{21}}} {{M_{22}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{{\ddot r}_1}}\\{{{\ddot r}_2}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c@{\quad}c}}{{G_{11}}} & {{G_{12}}}\\{{G_{21}}} & {{G_{22}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{{\dot r}_1}}\\{{{\dot r}_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{u_1}}\\{{u_2}}\end{array}} \right],\end{align*}

where ${M_{11}} = {\xi _{\rm{1}}} + {\rm{2}}{\xi _{\rm{3}}}\cos({r_2})$ , ${M_{12}} = {M_{21}} = {\xi _{\rm{2}}} + {\xi _{\rm{3}}}\cos({r_2}){\rm{ + }}{\xi _{\rm{4}}}\sin({r_2})$ , $\eqalign{ & {M_{22}} = {\xi _{\rm{2}}}{G_{11}} = ( - {\xi _3}\sin ({r_2}) + \cr & {\xi _4}\cos ({r_2})){{\dot r}_2},{\mkern 1mu} {\mkern 1mu} {G_{12}} = ( - {\xi _3}\sin ({r_2}) + {\xi _4}\cos ({r_2}))({{\dot r}_1} + {{\dot r}_2}),{G_{21}} = ({\xi _3}\sin ({r_2}){\mkern 1mu} - {\xi _4}\cos ({r_2})){{\dot r}_1}\;{\rm{and}}\;{G_{22}}\; = \;0 \cr} $ with actual physical parameters defined as $\xi = {\left[ {{\xi _1},{\xi _2},{\xi _3},{\xi _4}} \right]^T} = {\left[ {3.3,0.97,1.04,0.6} \right]^T}$ . The desired joint and velocity trajectories are chosen as ${x_r} = \left[ {\begin{array}{*{20}{c}}{0.2\sin(t) + 0.1\cos(2t)}\\{ - 0.3\sin(t) - 0.1\cos(2t)}\end{array}} \right]$ rad and ${\dot x_r} = \left[ {\begin{array}{*{20}{c}}{0.2\cos(t) - 0.2\sin(2t)}\\{ - 0.3\cos(t) + 0.2\sin(2t)}\end{array}} \right]$ rad/s, respectively. The initial joint and velocity of the robotic arm are set as $r(0) = {[0,{\rm{0}}]^T}$ rad and $\dot r(0) = {[0,{\rm{0}}]^T}$ rad/s. Gains in sliding surface are designed as ${K_1} = {K_2} = 4{I_2}$ , $\Theta \;{\rm{ = }}\;0.1\;{I_2}$ and $Q = \left[ {\begin{array}{*{20}{c@{\quad}c}}{ - 250.87{I_2}} & {{I_2}}\\{ - 19951{I_2}} & {{0_2}}\end{array}} \right]$ . The initial values of k and η are set as k(0) = 10 and η(0) = 0, respectively. Moreover, adaptive rates $\alpha$ = 0.03 and $\beta$ = 0.01 are chosen. The complex unknown actuator nonlinearity is simulated as

(49) \begin{align}u = \left\{ \begin{array}{l}(2 + 0.1\cos({\tau _a}))\;{\rm{ }}({\tau _a} - 0.35),{\rm{ }}{\tau _a} \ge 0.35\\{e^{ - 2t}} + 0.2,{\rm{ }} - 0.45 \lt {\tau _a} \lt 0.35\\(2 - 0.8\sin({\tau _a})){\rm{ }}({\tau _a} + 0.45),{\rm{ }}{\tau _a} \le - 0.45\end{array} \right.\end{align}

Joint and velocity tracking errors of the robotic arm are presented in Fig. 10. Moreover, the results of an adaptive method based on new Nussbaum analysis for Multi-Input Multi-Output (MIMO) mechanical systems with unknown actuator nonlinearities in ref. [Reference Chen, Liu, Zhang, Chen and Xie13] are also given for the sake of comparison. It can be obviously observed that smaller steady-state errors are obtained from Fig. 10, illustrating the effectiveness and superiority of the proposed control approach in dealing with complex actuator nonlinearity.

Integral of absolute error (IAE) values of the tracking errors in Examples 2 and 3 are concluded in Tables II and III, respectively. It is evident that IAE values of ASTC are less than those of the methods provided in refs. [Reference Liu, Dong, Xue and Zhang12, Reference Chen, Liu, Zhang, Chen and Xie13] correspondingly, showing smaller steady-state errors and better tracking performances.

• Case 2: Single-link robot system

In this case, a single-link robotic arm in ref. [Reference Fang, Tsai, Yan and Guo36] is utilized to verify the effectiveness and validity of ASTC. For ease of comparison studies, both the desired joint trajectory and the complex actuator nonlinearity existing in the robot in ref. [Reference Fang, Tsai, Yan and Guo36] are utilized for simulation, which are described as

(50) \begin{align}{x_r} = \left\{ \begin{array}{l}0.5\sin\left(\dfrac{2}{5}\pi t\right),{\rm{ }}t \le 15,\\0,{\rm{ }}t \gt 15.\end{array} \right.\end{align}

and $u = (1 + 0.3\sin({\tau _a})) \cdot {\tau _a}$ rad, respectively. Gains in sliding surface are designed as K ₁ = K ₂ = 6I ₂, $\Theta \;{\rm{ = }}\;0.1{I_2}$ and $Q = \left[ {\begin{array}{*{20}{c@{\quad}c}}{ - 250.87{I_2}} & {{I_2}}\\{ - 19951{I_2}} & {{0_2}}\end{array}} \right]$ , and the simulation results are presented in Figs. 11 and12. From Fig. 11, it can be observed that ASTC has better tracking effect when compared with the proposed tracking controller in ref. [Reference Fang, Tsai, Yan and Guo36]. Furthermore, ASTC is able to eliminate the undesired chattering effectively with less control inputs in Fig. 12, demonstrating great advantages in coping with NURSs with unknown actuator nonlinearities.

Table II.

Integral of absolute error (IAE) comparison of typical actuator nonlinearities in Example 2.

Table III.

Integral of absolute error (IAE) comparison of complex actuator nonlinearity in Case 1 in Example 3.

Figure 11.

The trajectories of x _r (t) and x ₁(t). Inset: The trajectories of the pre-specified reference r ₁(t) and displacement y ₁(t) by using the proposed tracking controller [Reference Fang, Tsai, Yan and Guo36].

Figure 12.

(a) The control input responses of the proposed continuous tracking controller [Reference Fang, Tsai, Yan and Guo36]. (b) The control input responses of adaptive sliding tracking control.