In recent years, the control methods for systems affected by uncertainties and disturbances have been focused by researchers [1][7]. Compared to other control methods, sliding mode control (SMC) has attracted a significant interest due to its conceptual simplicity, easy implementation, and robustness to external disturbances and model uncertainties [8][10]. SMC is a nonlinear control strategy that forces the closed loop trajectories to the switching manifold in finitetime using a discontinuous feedback control action. Therefore, SMC has been widely used in many applications, such as motion control, process control, etc. However, the conventional sliding mode control has a violent chattering phenomenon in the process, which can degrade the system performance. Moreover, it can guarantee invariance only if the uncertainties and disturbances satisfy the matching conditions, and cannot attenuate mismatched uncertainties and disturbances effectively.
Note that the matched and mismatched uncertainties and disturbances widely exist in practical engineering, such as power systems [11], electronic systems [12], [13] and motor systems [14]. The sliding motion of the traditional SMC is severely affected by the mismatched uncertainties and disturbances, and the wellknown robustness of SMC does not hold any more. Algorithms like LMIbased control [15], [16], adaptive control [17], [18], back stepping based control [19], and integral slidingmode control [20], [21] are proposed to handle mismatched uncertainties in a robust way, but the price is that the nominal control performance is compromised.
As a practical alternative approach, disturbance observerbased control has been proven to be promising and effective in compensating the effects of unknown external disturbances and model uncertainties in control systems as well as it will not deteriorate the existing controller [22], [23]. It could completely remove the nonvanishing disturbances from system as long as they can be estimated accurately [24]. Recently, several authors introduced a disturbance observer (DOB) for SMC to alleviate the chattering problem and retain its nominal control performance [25][29]. The idea is to construct the control law by combining the SMC feedback with the disturbance estimation basedfeed forward compensation straightforwardly. However, these methods given in [25][29] are only available for the matched uncertain systems. A nonlinear extension of DO has been proposed by W. H. Chen, which estimates matched as well as mismatched disturbances [30], [31]. Reference [1] extends a recent result on sliding mode control for general nth order systems with a larger class of mismatched uncertainties by proposing an extended disturbance observer. Reference [13] investigates an extended state observer (ESO)based sliding mode control (SMC) approach for pulse width modulationbased DCDC buck converter systems subject to mismatched disturbances, the proposed method obtains a better disturbance rejection ability even the disturbances do not satisfy the socalled matching condition. A novel slidingmode control based on the disturbance estimation by a nonlinear disturbance observer (NDOB) based SMC in [14] is only proposed to deal with mismatched uncertainties, it can ensure the system performance and reduce the chattering.
In this paper, aiming to improve the performance of the system affected by mismatched/matched uncertainties and disturbances, a novel nonlinear control scheme is proposed, where the SMC scheme is integrated with NDOB. By fully taking into account the estimation value of disturbances, a new slidingmode surface is firstly designed which is insensitive to not only matched disturbances but also mismatched ones. In this paper, the contributions are listed as follows:
1) The control is proposed for a general system of
2) The novel sliding surface is extended for a general system of n order and modified to enable improvement in the performance of the system without causing a large increase in the control.
3) The proposed method exhibits the properties of nominal performance recovery and chattering reduction as well as excellent dynamic and static performance as compared with the traditional SMC.
The paper is organized as follows: the problem of nominal sliding mode controller design with mismatched and matched uncertainties and disturbances for a class of nonlinear system is stated in Section 2. Generalization of NDOB, novel sliding surface design and the stability analysis are derived in Section 3. The simulation results are presented in Section 4, followed by some concluding remarks in Section 5.
2 Problem StatementConsider a class of singleinput singleoutput dynamic systems with matched and mismatched uncertainties, depicted by
$ \begin{equation} \left\{\begin{array}{llllll} & \!\!\!\!\!\!\!{{\dot x}_i} ={x_{i + 1}} + {d_i}(t)\\ & ~\vdots\\ & \!\!\!\!\!\!\!{{\dot x}_n} =a(x) + b(x)u + {d_n}(t)\\ & \!\!\!\!\!\!\!y ={x_1} \end{array} \right. \ i= 1, \ldots, n  1\ \end{equation} $  (1) 
where
Taking a secondorder system as an illustration, we can get the control model of the system:
$ \begin{equation} \left\{ \begin{array}{lllll} & \!\!\!\!\!\!\!{{\dot x}_1} ={x_2} + {d_1}(t)\\ & \!\!\!\!\!\!\!{{\dot x}_2} = a(x) + b(x)u + {d_2}(t)\\ & \!\!\!\!\!\!\!y ={x_1} \end{array} \right.. \end{equation} $  (2) 
Assumption 1: The lumped disturbances
The sliding mode surface and control law of the traditional SMC is usually designed as follows:
$ \begin{equation} {s_1} = {x_2} + {k_1}{x_1} \end{equation} $  (3) 
$ \begin{equation} u =  {b^{  1}}(x)\left[{{k_1}{x_2} + {\eta _1}{\rm sgn}(s_1) + a(x)} \right] \end{equation} $  (4) 
where
$ \begin{equation} {\dot s_1} =  {\eta _1}{\rm sgn}(s_1) + {k_1}{d_1}(t) + {d_2}(t). \end{equation} $  (5) 
The states of (2) will reach the sliding mode surface
$ \begin{equation} {\dot x_1} + {k_1}{x_1} = {d_1}(t). \end{equation} $  (6) 
Remark 1: Equation (6) implies that if the
In this paper, the matched and mismatched disturbance rejection problem is considered for (1). A novel sliding mode controller based on NDOB is proposed by the following two steps. First, an NDOB is employed to estimate the matched and mismatched disturbances. A novel sliding mode controller is then designed for the system based on the disturbance observation. The control structure of the secondorder system is designed as Fig. 1.
A nonlinear disturbance observer (NDOB), which is adopted to estimate the disturbance in [14]. Consider a class of nonlinear systems with uncertainties and external disturbance:
$ \begin{equation} \left\{ \begin{array}{ccccc} \dot x =f(x) + {g_1}(x)u + {g_2}(x)d(t)\\ y =h(x)~~~~~~~~~~~~~~~~~~~~~~~~~~ \end{array} \right.. \end{equation} $  (7) 
A NDOB is used to estimate the compound disturbance of the system and compensate accordingly, in order to improve the robustness of the controller. The NDOB is introduced and depicted by
$ \begin{equation} \left\{ \begin{array}{lllllll} & \!\!\!\!\!\!\!\dot z =  l{g_2}(x)z  l\left[{{g_2}(x)lx + f(x) + {g_1}(x)u} \right]\\ & \!\!\!\!\!\!\!\hat d(t) = z + lx \end{array} \right. \end{equation} $  (8) 
where
Assumption 2: The derivative of the disturbance in system (1) is bounded and satisfies
$ \begin{equation*}\mathop {\lim }\limits_{t \to \infty } d_i^{(j)}(t) = 0, \quad \quad i = 1, \ldots, n; j = 1, \ldots, n  1.\end{equation*} $ 
Remark 2: This assumption satisfies the requirements of the simulation sample. An extended disturbance observer is proposed in [1], which can observe a class of systems with mismatched uncertainties and get
The disturbance estimation error of the NDOB is defined:
$ \begin{equation} {e_d}(t) = d(t)  \hat d(t). \end{equation} $  (9) 
The error dynamics can be derived as
$ \begin{align} & {{{\dot{e}}}_{d}}(t)=\dot{d}(t)\dot{\hat{d}}(t) \\ & \ \ \ \ \ \ \ \ \ =\dot{d}(t)\dot{z}l\left[f(x)+{{g}_{1}}(x)u+{{g}_{2}}(x)d(t) \right]. \\ \end{align} $  (10) 
Substituting the value of
$ \begin{equation} {\dot e_d}(t) = \dot d(t)  {lg_2}(x){e_d}(t). \end{equation} $  (11) 
Lemma 1 [7]: Suppose that Assumptions 1 and 2 are satisfied for (7). The disturbance estimation
Consider the system
$ \begin{equation} \dot x = f(t, x, u) \end{equation} $  (12) 
where
Definition 1: The system (12) is said to be inputtostate stable (ISS) if there exist a class
$ \begin{equation} \left\ {x(t)} \right\ \le \beta (\left\ {x({t_0})} \right\, t {t_0}) + \gamma (\mathop {\sup }\limits_{{t_0} \le \tau \le t} \left\ {u(\tau )} \right\). \end{equation} $  (13) 
Such a function
Lemma 2 [14]: Consider a nonlinear system
A novel sliding mode manifold for (2) under matched and mismatched disturbances is designed as
$ \begin{equation} {s_2} = {x_2} + {k_2}{x_1} + {\hat d_1}(t). \end{equation} $  (14) 
Theorem 1: Considering the above (2) with matched and mismatched disturbances, we proposed slidingmode surface (14), if the control law is designed as follows:
$ \begin{align} u = &  {b^{  1}}(x)\Big\{ {k_2}\left[{{x_2} + {{\hat d}_1}(t)} \right] + {\eta _2}{\rm sgn}({s_2})\nonumber\\ & + a(x) + {{\hat d}_2}(t) + {{\dot {\hat d}_1}}(t) \Big\}. \end{align} $  (15) 
Suppose the secondorder system satisfies Assumptions 1 and 2, the observer gain
Proof: Consider a candidate Lyapunov function as
$ \begin{equation} {V_1} = \frac{1}{2}s_2^T{s_2}. \end{equation} $  (16) 
Taking derivative of
$ \begin{equation} {\dot V_1} = {s_2}{\dot s_2} = {s_2}({\dot x_2} + {k_2}{\dot x_1} + \dot{\hat{d}}_1 (t). \end{equation} $  (17) 
Substituting (15) into (2) yields
$ \begin{align} {{\dot x}_2} = & a(x) + b(x)u + {d_2}(t)\nonumber\\ = & a(x)  b(x) \times {b^{  1}}(x)\Big\{ {k_2}\left[{{x_2} + {{\hat d}_1}(t)} \right] + {\eta _2}{\rm sgn}({s_2})\nonumber\\ & + a(x) + {{\hat d}_2}(t) + {{\dot {\hat d}_1}(t)}\Big\} + {d_2}(t). \end{align} $  (18) 
Substituting (18) into (17) yields
$ \begin{align} {\dot V}_1= & {s_2} \Big\{  {k_2}\left[{{x_2} + {{\hat d}_1}(t)} \right]  {\eta _2}{\rm sgn}(s)  {{\hat d}_2}(t)\nonumber\\ &  {{\dot {\hat d}_1}(t)} + {d_2}(t) + {k_2}{x_2} + {k_2}{d_1}(t) + {{\dot {\hat d}_1}(t)} \Big\}\nonumber\\[1mm] = & {s_2} \Big\{  {k_2}{{\hat d}_1}(t)  {\eta _2}{\rm sgn}(s)  {{\hat d}_2}(t) + {d_2}(t) + {k_2}{d_1}(t) \Big\}\nonumber\\ = & {s_2}\left( {  {\eta _2}{\rm sgn}({s_2}) + {k_2}{e_{d1}} + {e_{d2}}} \right)\nonumber\\[1mm] \le & \left[{{\eta _2} + {k_2}{e_{d1}} + {e_{d2}}} \right]\left {{s_2}} \right\nonumber\\[1mm] = &  \sqrt 2 \left[{{\eta _2}({k_2}{e_{d1}} + {e_{d2}})} \right]V_1^{\frac{1}{2}} \end{align} $  (19) 
where
It can be derived from (19) that the system states will reach the defined sliding surface
$ \begin{equation} {\dot x_1} =  {k_2}{x_1} + {e_{d1}}. \end{equation} $  (20) 
With this result, it can be derived that (20) is ISS. According to Lemma 2, we know that the system states satisfy
Remark 3: Since the matched and mismatched disturbances have been precisely estimated by the NDOB, the switching gain of the proposed method can be designed much smaller than those of the traditional SMC, because the magnitude of the estimation error
Consider the following thirdorder system, depicted by
$ \begin{equation} \left\{ \begin{array}{llll} {{\dot x}_1} = {x_2} + {d_1}(t)\\ {{\dot x}_2} = {x_3} + {d_2}(t)\\ {{\dot x}_3} = a(x) + b(x)u + {d_3}(t)\\ y = {x_1}. \end{array} \right. \end{equation} $  (21) 
A sliding mode manifold for (21) is designed as
$ \begin{equation} {s_3} = {x_3} + {x_2} + {k_3}{x_1} + {\hat d_1}(t) + {\hat d_2}(t) + \dot{\hat{d}}_1(t). \end{equation} $  (22) 
Theorem 2: Considering the above (21) with matched and mismatched disturbances, we proposed slidingmode surface (22), if the control law is designed as follows:
$ \begin{align} u= &  {b^{  1}}(x)\Big\{ {k_3}\left[{{x_2}+ {{\hat d}_1}(t)} \right] + {\eta _3}{\rm sgn}({s_3}) + {x_3} + a(x)\nonumber\\ & + {{\hat d}_2}(t)+ {{\hat d}_3}(t) + {{\dot {\hat d}_1}(t)}+{{\dot {\hat d}_2}(t)}+{{\ddot {\hat d}_1}(t)} \Big\}. \end{align} $  (23) 
Suppose the thirdorder system satisfies Assumptions 1 and 2, the observer gain
A sliding mode manifold for (1) in the case
$\begin{equation} {s_n} = \sum\limits_{i = 2}^n {{x_i}} + {k_n}{x_1} + \sum\limits_{j = 1}^{n  1} {\sum\limits_{i = 1}^{n  j} {\hat d_i^{(j  1)}(t)} }. \end{equation} $  (24) 
Theorem 3: Considering the above system (1) with matched and mismatched disturbances, we proposed slidingmode surface (24), if the control law is designed as follows:
$ \begin{align} u = &  {b^{  1}}(x)\Big\{ {k_n}\left[{{x_2} + {{\hat d}_1}(t)} \right] + {\eta _n}{\rm sgn}({s_n}) + \sum\limits_{i = 3}^n {{x_i}} \nonumber\\ & + a(x) + \sum\limits_{i = 2}^n {{{\hat d}_i}(t)} + \sum\limits_{j = 1}^{n  1} {\sum\limits_{i = 1}^{n  j} {\hat d_i^{(j)}(t)} } \Big\}. \end{align} $  (25) 
Suppose the general highorder system satisfies Assumptions 1 and 2, the observer gain
Proof: Consider a candidate Lyapunov function as
$ \begin{equation} {V_n} = \frac{1}{2}s_n^T{s_n}. \end{equation} $  (26) 
Taking derivative of
$ \begin{align} {{\dot V}_n} = & {s_n}{{\dot s}_n} = {s_n}\big(\sum\limits_{i = 2}^n {{{\dot x}_i}} + {k_n}{{\dot x}_1} + \sum\limits_{j = 1}^{n  1} {\sum\limits_{i = 1}^{n  j} {\hat d_i^{(j)}(t)} } \big) \nonumber\\ = & {s_n}\Big[ {x_3} + {d_2} + {x_4} + {d_3} + \cdots + {x_n} + {d_{n1}} + a(x)\nonumber\\ & + b(x)u + {d_n}(t) + {k_n}{x_2} + {k_n}{d_1} + \sum\limits_{j = 1}^{n1} {\sum\limits_{i = 1}^{nj} {\hat d_i^{(j)}(t)} } \Big]. \end{align} $  (27) 
Substituting (25) into (27) yields
$\begin{align}{{\dot V}_n}= & {s_n}\Bigg\{ {x_3} + {d_2} + {x_4} + {d_3} + \cdots + {x_n} + {d_{n  1}} + a(x)\nonumber\\ &  b(x)\!\! \times\!\! {b^{  1}}(x)\bigg\{ {k_n}\left[{{x_2} + {{\hat d}_1}(t)} \right] + {\eta _n}{\rm sgn}({s_n})\nonumber\\ & + \sum\limits_{i = 3}^n {{x_i}} + a(x) + \sum\limits_{i = 2}^n {{{\hat d}_i}(t)} + \sum\limits_{j = 1}^{n  1} {\sum\limits_{i = 1}^{n  j} {\hat d_i^{(j)}(t)} } \bigg\}\nonumber\\ & + {d_n}(t) + {k_n}{x_2} + {k_n}{d_1} + \sum\limits_{j = 1}^{n  1} {\sum\limits_{i = 1}^{n  j} {\hat d_i^{(j)}(t)} } \Bigg\}\nonumber\\ = & {s_n}\Bigg[{{\eta _n}{\rm sgn}({s_n}) + {k_n}{e_{d1}} + \sum\limits_{i = 2}^n {{e_{di}}} } \Bigg]\nonumber\\ \le & \left {{s_n}} \right(  {\eta _n} + {k_n}{e_{d1}} + \sum\limits_{i = 2}^n {{e_{di}}}). \end{align} $  (28) 
It can be derived from
$ \begin{equation} x_1^{(n  1)} + x_1^{(n  2)} + \ldots + {\dot x_1} + {k_n}{x_1}  \sum\limits_{j = 1}^{n  1} {\sum\limits_{i = 1}^{n  j} {e_{di}^{(j  1)}}} = 0. \end{equation} $  (29) 
We can define
$ \begin{equation} \left\{ {\begin{array}{*{20}{lll}} {{Y_1} = {x_1}}\\ {{Y_2} = {{\dot x}_1}}\\ ~~~~\vdots \\ {{Y_{n  1}} = x_1^{(n  2)}} \end{array}} \right.. \end{equation} $  (30) 
Equation (29) is given by
$ \begin{equation} \dot Y\! =\! \left[\! {\begin{array}{*{20}{c}} 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & 0 & 0\\ \vdots & \vdots & \ddots & \vdots & \\ 0 & 0 & \cdots & 0 & 1\\ {{k_n}} & {1} & {1} & \cdots & {  1} \end{array}} \right]Y + \left[{\begin{array}{*{20}{c}} 0\\ 0\\ \vdots \\ 0\\ {\sum\limits_{j = 1}^{n1} {\sum\limits_{i = 1}^{nj} {e_{di}^{(j1)}} } } \end{array}} \right]. \end{equation} $  (31) 
Equation (31) can also be expressed as
$ \begin{equation} \dot Y = AY + B\sum\limits_{j = 1}^{n  1} {\sum\limits_{i = 1}^{n  j} {e_{di}^{(j  1)}} } \end{equation} $  (32) 
where
In view of Assumption 1 and (11), we can conclude:
$ \begin{equation} \begin{array}{llllllll} \sum\limits_{j = 1}^{n  1} {\sum\limits_{i = 1}^{n  j} {e_{di}^{(j  1)}} } = \sum\limits_{i = 1}^{n  1} {{e_{di}}} + \sum\limits_{i = 1}^{n  2} {({{\dot d}_i}  {{\lg}_2}{e_{di}})} \\+ \sum\limits_{i = 1}^{n  3} {({{\ddot d}_i}  {{\lg}_2}{{\dot d}_i} + {l^2}g_2^2{e_{di}})} + \cdots \\+ \sum\limits_{i = 1}^2 {(d_i^{n  3}  {{\rm lg}_2}d_i^{n  4} + {l^2}g_2^2d_i^{n  5}  \cdots  {l^{n  3}}g_2^{n  3}{e_{di}})} \\ + (d_1^{n  2}  {{\rm lg}_2}d_1^{n  3} + {l^2}g_2^2d_1^{n  4}  \cdots  {l^{n  2}}g_2^{n  2}{e_{d1}}). \end{array} \end{equation} $  (33) 
Combining Lemma 1, (11) and (33) can get:
$ \begin{equation} \mathop {\lim }\limits_{t \to \infty } \sum\limits_{j = 1}^{n  1} {\sum\limits_{i = 1}^{n  j} {e_{di}^{(j  1)}} } = 0. \end{equation} $  (34) 
According to the principle of Hurwitz matrix, it can be designed
Remark 4: The above proof implies that states of system can be driven to the desired equilibrium point and the control law can force the system states to reach the slidingmode surface in finite time. This is the main reason why the proposed NDOBbased SMC method is insensitive to matched uncertainties and disturbances as well as mismatched uncertainties and disturbances.
4 SimulationsTo evaluate the effectiveness of the proposed method, two examples are given below.
4.1 Numerical Example$ \begin{equation} \left\{ \begin{array}{l} {{\dot x}_1} = {x_2} + {d_1}(t)\\ {{\dot x}_2} = {x_3} + {d_2}(t)\\ {{\dot x}_3} =  2{x_2}  {x_3} + {e^{{x_1}}} + u + {d_3}(t)\\ y = {x_1}. \end{array} \right. \end{equation} $  (35) 
In order to show the advantages of the NDOBbased SMC method proposed in this paper compared with the nominal sliding control, we will use simulations to compare the performance between them for (35). The control parameters of the two control methods are listed in Table Ⅰ. Consider the initial states of (35) as
The reference/estimation uncertainties and disturbances of the system are shown in Fig. 4. It can be seen from Fig. 4 that the proposed control gives a better estimation of the disturbance and a better performance.
4.2 Application to PMSM Speed Control SystemIn
$ \begin{equation} \dot \omega =  a\omega + b{i_q}  d \end{equation} $  (36) 
where
The state variable of speed error is defined as
$ \begin{equation} {\dot x_1} = {x_2} = {\dot \omega _{\rm ref}}  \dot \omega = {\dot \omega _{\rm ref}} + a\omega  b{i_q} + d \end{equation} $  (37) 
where
The speed error derivative dynamic equation of the motor can be expressed as follows, with the parameters variations taken into account:
$ \begin{eqnarray} {{\dot x}_1} & = & {{\dot \omega }_{\rm ref}} + a\omega + \Delta a\omega  b{i_q}  \Delta b{i_q} + d + \Delta d\nonumber\\ & = & {x_2} + \Delta a\omega  \Delta b{i_q} + \Delta d\nonumber\\ & = & {x_2} + {d_1} \end{eqnarray} $  (38) 
where
The secondorder model of speed error derivative dynamic equation of PMSM system is described by
$ \begin{equation} {\dot x_2} = {\ddot \omega _{\rm ref}}  \ddot \omega = {\ddot \omega _{\rm ref}} + a\dot \omega  b{\dot i_q} + \dot d. \end{equation} $  (39) 
$ \begin{eqnarray} {{\dot x}_2} & = & {{\ddot \omega }_{\rm ref}} + a\dot \omega + \Delta a\dot \omega  b{{\dot i}_q}  \Delta b{{\dot i}_q} + \dot d + \Delta \dot d\nonumber\\ & = &  a{x_2}  b{{\dot i}_q} + {d_2}\nonumber\\ & = &  a{x_2}  bu + {d_2} \end{eqnarray} $  (40) 
where
The secondorder model of the PMSM speed regulation system can be represented in the following statespace form:
$ \begin{equation} \left\{ \begin{array}{l} {{\dot x}_1} = {x_2} + {d_1}\\ {{\dot x}_2} =  a{x_2}  bu + {d_2} \end{array} \right.. \end{equation} $  (41) 
To demonstrate the efficiency of the proposed method, simulation studies are carried out in this section. The parameters of the PMSM are given as follows:
Fig. 5 depicts the variable curve, reference speed changes from 20 rad/s to 40 rad/s at 5 s, and 40 rad/s to 20 rad/s at 10 s. It can be observed that the proposed method exhibits a faster speed and smooth transition than the nominal SMC method.
The unknown load torque
The response curves of the PMSM under the proposed method in the presence of mechanical parameter variations are shown in Fig. 8. The moment of inertia is supposed to have variations in its nominal operation values, 2 J at 10 s, and 1.5 J at 15 s, respectively. It can be observed from Fig. 8 that the proposed method is insensitive to mismatched uncertainty, and has fine robustness performance, while the nominal SMC method is sensitive to mismatched uncertainty.
5 ConclusionIn this paper, the mismatched/matched uncertainties and disturbances rejection control problem have been studied for the second, third, and higherorder systems. A novel NDOBbased SMC approach has been proposed. The controller not only make the states of closedloop system obtain better tracking performance, but also enhance the disturbance attenuation and system robustness. The proposed method has exhibited nominal performance recovery and chattering reduction as compared with the nominal SMC. Simulation results reveal the effectiveness of the proposed method.
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