Robust predictive sliding mode control for input rate-constrained discrete-time system

Robust predictive sliding mode control for input rate-constrained discrete-time system Abstract This paper investigates a class of uncertain linear discrete-time systems subject to input rate saturation. A predictive sliding mode control approach is proposed which guarantees the control inputs remain bounded in the input rate saturation. Furthermore, the disturbance observer is developed to compensate for the system uncertainty and disturbance. Finally, the simulations demonstrate the effectiveness of the proposed predictive sliding mode control scheme. 1. Introduction In recent years, the control problem of input constraint has received considerable attention and different design approaches have been proposed. In the study by Wang et al. (2011), an integral sliding surface and a special sliding mode controller have been designed to overcome the input saturation constraint. In the study by Grimm et al. (2003), the $$H_{2}$$ performance properties have been employed, and extensive linear matrix inequalities (LMIs)-based control method has been proposed for a class of linear input saturated systems. In the study by Hu et al. (2006), the systematic Lyapunov method was proposed for closed-loop regional stability control and the good control performance of saturated system was obtained. In the study by Zhu et al. (2011), an adaptive sliding mode controller has been proposed to guarantee system stability when the input saturations exist. To solve the tracking control problem with actuator saturations, one extended state observer-based sliding mode control approach using backstepping technique has been proposed in the study by Lu et al. (2013). It can be seen that the study of control input constraint problem usually focuses on the input amplitude constraint but ignores the problem of input rate constraint (Kefferputz & Adamy, 2011). However, the problem of control rate constraint seriously affects the system performance. When the input signal of the system changes too fast, the actuator’s physical characteristics cannot meet the control requirements and even causes instability in the system. Thus, we will design a controller for a class of linear discrete-time systems with input rate constraint. Sliding mode control has been receiving increasing attention in many control fields (Furuta, 1990; Gao et al., 1995; Choi, 2008). Discrete-time sliding mode dynamics systems cannot be simply achieved according to their continuous counterparts. Thus, the study of discrete-time sliding mode control (DSMC) is important in the control area. As one of robust control methods, DSMC has been widely employed to cope with the uncertain systems. But, in the field of traditional sliding mode control, there exist some problems of low convergence speed, long convergence time and severe chattering (Xiao et al., 2007; Gao et al., 2009). In the study by Qu et al. (2014), a discrete-time reaching law with a disturbance compensator has been developed and the DSMC for the uncertain systems have been studied. In the study by Acary et al. (2012), a sliding mode control algorithm based on an implicit Euler method has been designed, which can avoid the chattering effects. In the study by Mayne et al. (2000), a chattering-free variable reaching law has been designed for DSMC system, which can make the controller’s velocity of input signal limited. However, in the traditional sliding mode control method, the undesirable chattering phenomenon will reduce the steady-state behaviour. In this paper, we will develop a predictive sliding mode control method to deal with the undesirable chattering problem. Model predictive control is one of the multi-step optimization algorithms, and it is mainly by integrating the measured signals and predictive signals to obtain the ideal control input signals (Mao, 2003). Some new predictive control approaches to deal with the constrained problem have been studied in Cuzzola et al. (2002) and Garcia-Gabin et al. (2009). By introducing the model predictive control method, the robustness of the system will be improved. In other words, the predictive sliding mode algorithm can handle set point changes and disturbance rejection. In the study by Liu et al. (2013), the predictive sliding mode control strategy has been applied into a real solar air conditioning plant. Furthermore, disturbance attenuation ability and robustness against the variation of system parameter are two important properties required for control methods to be successfully used in the actual systems. Compared with traditional sliding mode control, the prediction sliding mode control can effectively eliminate the chattering phenomenon introduced to predict the sliding mode control method. Model predictive control method can handle the uncertainties and disturbance for a long time. However, employing disturbance observer can effectively and fast compensate for uncertainties and disturbance, and the performance of the system can also be greatly improved. Therefore, in this paper, a predictive sliding mode control will be proposed for the discrete-time system with disturbance observer. The disturbance observer is a valid approach to deal with external perturbation Chen & Chen (2010). Due to simple structure and analysis, the disturbance observer based control method has been applied to many applications (Chen et al., 2000b; Chen, 2004; Wei & Guo, 2010). In the study by Back & Shim (2008), the authors not only prove the steady-state performance of the proposed disturbance observer but also study the closed-loop system transient performance. In the study by Chen et al. (2000a), a non-linear disturbance observer for robotic manipulators has been proposed, and condition for convergence has been established. In the study by Chen & Jiang (2013), by using disturbance observer to estimate the compounded disturbance, the robust attitude control scheme has been employed for near-space vehicles. In the study by Chen & Ge (2013), a direct adaptive neural control approach using disturbance observer has been proposed for the uncertain non-affine non-linear systems. Although there is a large number of researches about observer, there are few studies of predictive sliding mode control with disturbance observer. In this paper, the disturbance observer will be introduced to cope with the external disturbance, and it will be combined with the predictive sliding mode controller design to achieve a good control performance. Motivated by above analysis, we will develop a predictive sliding mode control technique for an input rate constraint system with parametric uncertainties and mismatched disturbances. In this paper, the main contributions can be summarized as follows: 1. A robust controller combining model predictive control and sliding mode control are proposed for the discrete-time system with input rate constraints. 2. The disturbance observer is employed into the controller design to enhance the performance for the composite disturbance. Besides, this paper is organized as follows. Section 2 describes the linear control problem of input rate constraints and a reaching law for DSMC method. The main results are about a new predictive sliding mode control method; furthermore, the construction of the predictive sliding mode surface, the compound sliding mode control laws and the sliding mode reaching laws are given in Section 3. In Section 4, some numerical examples illustrate the advantages of the presented control algorithm. Finally, Section 5 draws the conclusions of the paper. 2. Problem formulation Considering a class of uncertain discrete-time dynamic systems described as   \begin{align} x\left({k + 1} \right) &= Ax (k) + Bu (k) + d (k)\nonumber\\ y\left( k \right) &= Cx\left( k \right) \end{align} (2.1)where k means the discrete time index, $$x\left ( k \right ) \in{R^{n}}$$ represents the system state, $$u\left ( k \right ) \in{R^{m}}$$ denotes the input vector, and $$y\left( k \right) \in{R^m}$$ stands for the output vector. $$d\left ( k \right ) \in{R^{n}}$$ denotes the external disturbance, which contains the modeling uncertainty and external disturbance. $$A \in{R^{n \times n}}$$ indicates the system matrix, $$B \in{R^{n \times m}}$$ indicates the input matrix with full rank m, $$C \in{R^{m \times m}}$$ is the output matrix. Moreover, we assume that the control input increment $$\Delta u\left ( k \right ) \in{R^{m}}$$ is restricted to a class of admissible controls. In this paper, the control objective is to guarantee the system stable and control input rate bounded. In order to meet the system control input rate constraints, we introduce discrete-time sliding mode reaching law with constructed function. Firstly, the following assumptions and lemmas are needed. Assumption 1 (A, B) is stabilizable, and (A, C) is detectable. Assumption 2: The state coefficient matrix A is invertible. Assumption 3: Control gain matrix B satisfies that $$BB^{T}$$ is invertible. Assumption 4: The external disturbance d(k) is slowly time-varying signal. Lemma 2.1 For all $$k\geqslant 0$$, given the sliding mode surface s(k), if the sliding mode surface satisfies the following inequalities (Gao et al., 1995)  \begin{align} \left[{s\left({k + 1} \right) - s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right) &< 0\nonumber\\ \left[{s\left({k + 1} \right) + s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right)&> 0 \end{align} (2.2)then we can conclude that the discrete-time system has satisfied the reaching condition of sliding mode motion. Lemma 2.2 For the ideal performance function r(⋅), there exists a reaching law (2.3) that allows the discrete time system to satisfy the reaching condition of sliding mode motion (Liu et al., 2013)   \begin{equation} s\left({k + 1} \right) = s\left( k \right) + Tf\left({s\left( k \right)\!,r\left( \cdot \right)\!,T} \right) \end{equation} (2.3)where s(k) is the current sliding mode value, $$s(k+1)$$ is the $$k+1 step$$ sliding mode value. T denotes the sample period, and f(x, r(⋅), T) is defined as (Liu et al., 2013)   \begin{align} f\left({x,r,T} \right) &= - \left({\frac{{\left|{s\left( k \right)} \right|}}{{iT}} + \frac{{\left({i - 1} \right)rT}}{2}} \right){\textrm{sgn}} \left({s\left( k \right)} \right)\nonumber\\ i &= fix({i^{\prime}}) + 1\nonumber\\ i^{\prime} &= 0.5\left({\sqrt{1 + \frac{{8\left|{s\left( k \right)} \right|}}{{r{T^{2}}}}} - 1} \right) \end{align} (2.4)where fix(⋅) denotes the integer rounding function, sgn(⋅) denotes the signum function. Proof. Substituting (2.4) into (2.3), the following results were derived:   \begin{align} \left[{s\left({k + 1} \right) - s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right) &= Tf\left({s\left( k \right)\!,r\left( \cdot \right),T} \right){\textrm{sgn}} \left({s\left( k \right)} \right)\nonumber\\ &= - \left({\frac{{\left|{s\left( k \right)} \right|}}{i} + \frac{{\left({i - 1} \right)r\left( \cdot \right){T^{2}}}}{2}} \right)\!.\quad\qquad\ \end{align} (2.5)  \begin{align} \left[{s\left({k + 1} \right) + s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right) &= 2\left|{s\left( k \right)} \right| + Tf\left({s\left( k \right),r\left( \cdot \right),T} \right){\textrm{sgn}} \left({s\left( k \right)} \right)\nonumber\\ &= 2\left|{s\left( k \right)} \right| - \left({\frac{{\left|{s\left( k \right)} \right|}}{i} + \frac{{\left({i - 1} \right)r\left( \cdot \right){T^{2}}}}{2}} \right)\!. \end{align} (2.6)On the basis of definition of i, we have $$i \geqslant 1$$. According to (2.5), it is clear that we have   \begin{equation} \left[{s\left({k + 1} \right) - s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right) < 0. \end{equation} (2.7)To illustrate the situation $$\left [{s\left ({k + 1} \right ) + s\left ( k \right )} \right ]{\textrm{sgn}} \left ({s\left ( k \right )} \right )> 0 $$, we will discuss the points in the following three aspects. When $$\left |{s\left ( k \right )} \right | < r\left ( \cdot \right ){T^{2}}$$, i = 1, then we obtain   \begin{equation} \left[{s\left({k + 1} \right) + s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right) = \left|{s\left( k \right)} \right|> 0. \end{equation} (2.8)When $$\left |{s\left ( k \right )} \right | = r\left ( \cdot \right ){T^{2}}$$, i = 2, we have   \begin{equation} \left[{s\left({k + 1} \right) + s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right) = 2\left|{s\left( k \right)} \right| - \left({\frac{{\left|{s\left( k \right)} \right|}}{i} + \frac{{\left({i - 1} \right)\left|{s\left( k \right)} \right|}}{2}} \right) = \left|{s\left( k \right)} \right|> 0. \end{equation} (2.9)When $$\left |{s\left ( k \right )} \right |> r\left ( \cdot \right ){T^{2}}$$, then we obtain   \begin{align} i - 1 < 0.5\left({\sqrt{1 + \frac{{8\left|{s\left( k \right)} \right|}}{{r\left( \cdot \right){T^{2}}}}} - 1} \right) &\Leftrightarrow\nonumber\\{\left({2i - 1} \right)^{2}} < 1 + \frac{{8\left|{s\left( k \right)} \right|}}{{r\left( \cdot \right){T^{2}}}} &\Leftrightarrow\nonumber\\ i\left({i - 1} \right) < \frac{{2\left|{s\left( k \right)} \right|}}{{r\left( \cdot \right){T^{2}}}} &\Leftrightarrow\nonumber\\ \frac{{\left({i - 1} \right)r\left( \cdot \right){T^{2}}}}{2} < \frac{{\left|{s\left( k \right)} \right|}}{i}.& \end{align} (2.10)Substituting (2.10) into (2.6), then we have   \begin{equation} \left[{s\left({k + 1} \right) + s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right)> 0. \end{equation} (2.11)Invoking (2.8), (2.9) and (2.11), the condition $$\left [{s\left ({k + 1} \right ) + s\left ( k \right )} \right ]{\textrm{sgn}} \left ({s\left ( k \right )} \right )> 0$$ has been achieved. In a word, according to Lemma 2.1, it can be concluded that the reaching law (2.2) has satisfied the reaching condition of sliding mode motion. Lemma 2.3 Assume that a linear time-invariant system can be described as   \begin{equation} \dot{\!x}\left( t \right) =\, \bar{\!A}x\left( t \right) +\, \bar{\!B}u\left( t \right) \end{equation} (2.12)where $$\,\bar{\!A} \in{R^{n \times n}}$$ and $$\,\bar{\!B} \in{R^{n \times m}}$$ are the known constant matrices with approximate dimensions. $$x\left ( t \right ) \in{R^{n}}$$ and $$u\left ( t \right ) \in{R^{n}}$$ indicate the state and control input of continuous system, respectively. Then, the corresponding Euler approximate system can be represented as (Hao et al., 2001)   \begin{equation} x\left({k + 1} \right) = \left[{I + \tau \,\bar{\!A}} \right]x( k) + \tau \,\bar{\!B}u(k). \end{equation} (2.13)If there exists $$\tau \to 0$$, then we can obtain $$\mathop{\lim }\limits _{\tau \to 0} A = I \Rightarrow (A - I) = O\left ( T \right )$$, where $$A=I+\tau \,\bar{\!A}$$. Lemma 2.4 Assume that $$r\left ( \cdot \right )$$ is selected to make sure r > 0, then the reaching sliding mode can be satisfied by the equation as follows (Liu et al., 2013):   \begin{equation} \frac{1}{T}\left({\frac{{s\left({k + 1} \right) - 2s\left( k \right) + s\left({k - 1} \right)}}{T}} \right) = - r{\textrm{sgn}} \left({s\left( k \right)} \right)\!. \end{equation} (2.14) In order to extend the application of input rate constraint control method in practise, we need to enhance the robust performance of the control method when the system contains parametric uncertainties and mismatched disturbances. 3. Main results In this section, we provide a synthesis procedure of predictive sliding mode control algorithm. Letting $$x\left ({k + i} \right )$$ be the predicted value of system state at time $$k + i$$; $$u\left ({k + i} \right )$$ be the future control at time $$k + i$$. In addition, $$d\left ({k + i} \right )$$ stands for the future disturbance at time $$k + i$$. Then a j step ahead predictor can be derived:   \begin{equation} y\left({k + j} \right) = C{A^{{\hskip.6pt}j}}x\left( k \right) + \sum\limits_{i = 1}^{j}{C{A^{{\hskip.6pt}j - i}}Bu\left({k + i - 1} \right)} + \sum\limits_{i = 1}^{j}{C{A^{{\hskip.6pt}j - i}}d\left({k + i - 1} \right)}. \end{equation} (3.1)Thus, we obtain   \begin{equation} \hat{\!y}\left({k + j} \right) = C{A^{{\hskip.6pt}j}}\,\hat{\!x}\left( k \right) + \sum\limits_{i = 1}^{j}{C{A^{{\hskip.6pt}j - i}}Bu\left({k + i - 1} \right)} + C{A^{{\hskip.6pt}j - 1}}d\left( k \right)\!. \end{equation} (3.2) Now introduce the global sliding mode approach, define a sliding mode function as follows:   \begin{equation} s\left( k \right) ={C_{s}}x\left( k \right) - \beta{C_{s}}{x_{0}} \end{equation} (3.3)where $$C_{s}\in R^{m\times n}$$ and $$\beta $$ is a constant $$0<\beta <1$$. x0 indicates the initial condition of state. Furthermore, the sliding mode predictive model is constructed:   \begin{equation} \tilde{\!s}\left( k \right) ={C_{s}}x\left( k \right) -{\beta^{k}}{C_{s}}{x_{0}} \end{equation} (3.4)where k > 0. Assume that the disturbance does not exist, the sliding function value at future time $$k+p$$ (p is predictive horizon) can be derived:   \begin{align} \tilde{\!s}\left({k + p} \right) ={C_{s}}{A^{p}}x\left( k \right) + \sum\limits_{i = 1}^{p}{{C_{s}}{A^{i - 1}}Bu\left({k + p - i} \right) +\sum\limits_{i = 1}^{p}{{C_{s}}{A^{i-1}}}d\left({k + p - i} \right) -{C_{s}}{\beta^{k + p}}{x_{0}}}. \end{align} (3.5)In order to simplify the form of the sliding function, we define the following matrices and vectors as follows:   $$ \tilde{\!S}\left( k \right) = \left[{\begin{array}{@{}c@{}} {\tilde{\!s}\left({k + 1} \right)}\\{\tilde{\!s}\left({k + 2} \right)}\\ \vdots \\{\tilde{\!s}\left({k + p} \right)} \end{array}} \right]\!,\quad{\tilde{\!S}_{0}}\left( k \right) = \left[{\begin{array}{@{}c@{}} {CAx\left( k \right) -{\beta^{k + 1}}C{x_{0}}}\\{C{A^{2}}x\left( k \right) -{\beta^{k + 2}}C{x_{0}}}\\ \vdots \\{C{A^{p}}x\left( k \right) -{\beta^{k + p}}C{x_{0}}} \end{array}} \right]$$  $$ G = \left[{\begin{array}{@{}ccccc@{}} {CB}&0& \cdots & \cdots &0\\{CAB}&{CB}&0& \cdots &0\\ \vdots & \vdots & \cdots & \cdots & \vdots \\{C{A^{m - 1}}B}&{C{A^{m - 2}}B}& \cdots & \cdots &{CB}\\{C{A^{m}}B}&{C{A^{m - 1}}B}& \cdots & \cdots &{CAB + CB}\\ \vdots & \vdots & \cdots & \cdots & \vdots \\{C{A^{p - 1}}B}&{C{A^{p - 2}}B}& \cdots & \cdots &{\sum\limits_{j = 0}^{p - m}{C{A^{j}}B} } \end{array}} \right]\!,\quad U\left( k \right) = \left[{\begin{array}{@{}c@{}} {u\left( k \right)}\\{u\left({k + 1} \right)}\\ \vdots \\{u\left({k + m - 1} \right)}\\ \end{array}} \right]$$  $$ M = \left[{\begin{array}{@{}ccccc@{}} {{C_{s}}}&0& \cdots & \cdots &0\\{{C_{s}}A}&{{C_{s}}}&0& \cdots &0\\ \vdots & \vdots & \cdots & \cdots & \vdots \\{{C_{s}}{A^{m - 1}}}&{{C_{s}}{A^{m - 2}}}& \cdots & \cdots &{{C_{s}}}\\{{C_{s}}{A^{m}}}&{{C_{s}}{A^{m - 1}}}& \cdots & \cdots &{{C_{s}}A + C}\\ \vdots & \vdots & \cdots & \cdots & \vdots \\{{C_{s}}{A^{p - 1}}}&{{C_{s}}{A^{p - 2}}}& \cdots & \cdots &{\sum\limits_{j = 0}^{p - m}{{C_{s}}{A^{j}}} } \end{array}} \right]\!,\quad D\left( k \right) = \left[{\begin{array}{@{}c@{}} {d\left( k \right)}\\{d\left({k + 1} \right)}\\ \vdots \\{d\left({k + m - 1} \right)} \end{array}} \right]$$where $$\,\tilde{\!S}\left ( k \right )$$ means the p-step sliding mode surface predictive vector, G denotes the predictive control matrix, M represents the predictive state matrix, m represents the control horizon and matrices $$U\left ( k \right )$$, $$D\left ( k \right )$$ denote m-step input control signal and disturbance, respectively. Therefore, the equation (3.5) can be rewritten as   \begin{equation} \tilde{\!S}\left( k \right) =\,{\tilde{\!S}_{0}}\left( k \right) + GU\left( k \right) + MD\left( k \right)\!. \end{equation} (3.6) Now the error between the real value of sliding mode and predictive value of sliding mode can be defined as   \begin{equation} e\left( k \right) = s\left( k \right) -\, \tilde{\!s}\left({k|k - p} \right) \end{equation} (3.7)where $$\,\tilde{\!s}\left ({k|k - p} \right )$$ denotes the sliding mode surface step k − p predictive value for step k. Consider the feedback correction, the sliding function value at time $$k + j$$ can be defined as   \begin{equation} \hat{\!s}\left({k + j} \right) =\, \tilde{\!s}\left({k + j} \right) +{\tau_{j}}\left[{s\left( k \right) - \,\tilde{\!s}\left({k|k - j} \right)} \right] \end{equation} (3.8)where $${\tau _{j}}$$ is a design parameter, $${\tau _{1}}>{\tau _{2}} > \cdots >{\tau _{P}} > 0$$, the value of parameter $${\tau _{j}}$$ will decide the degree of smoothing. Now we can define   $$ \Gamma = \left[{\begin{array}{@{}cccc@{}} {{\tau_{1}}}&0&0&0 \\ 0&{{\tau_{2}}}&0&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&0&{{\tau_{p}}} \end{array}} \right]\!,\quad \bar{\!S}\left( k \right) = \left[{\begin{array}{@{}c@{}} {s\left( k \right) - \,\tilde{\!s}\left({k|k - 1} \right)}\\{s\left( k \right) - \,\tilde{\!s}\left({k|k - 2} \right)}\\ \vdots \\{s\left( k \right) - \,\tilde{\!s}\left({k|k - p} \right)} \end{array}} \right]\!.$$ To simplify above equation (3.8), it can be derived as   \begin{equation} \hat{S}\left( k \right) ={\,\tilde{\!S}_{0}}\left( k \right) + GU\left( k \right) + MD\left( k \right) + \Gamma\,\bar{\!S}\left( k \right) \end{equation} (3.9)where   $$ \hat{S}\left( k \right) = \left[{\begin{array}{@{}c@{}} {\hat s\left({k + 1} \right)}\\{\hat s\left({k + 2} \right)}\\ \vdots \\{\hat s\left({k + p} \right)} \end{array}} \right]\!.$$ The objective function used in this formulation is the squared two-norm given by   \begin{equation} J\left( k \right) = \left\{{\sum\limits_{j = 1}^{p}{{{\left({\hat s\left({k + j} \right) -{s_{r}}\left({k + j} \right)} \right)}^{2}} + \sum\limits_{j = 1}^{m}{\lambda{{\left({u\left({k + j} \right)} \right)}^{2}}} } } \right\} \end{equation} (3.10)where constant $$\lambda> 0$$ is weight coefficient. p indicates the costing horizon and m denotes the control horizon. We aim at achieving the following performance objective:   \begin{equation} \frac{{dJ\left( k \right)}}{{dU\left( k \right)}} = 0. \end{equation} (3.11)In addition, in order to simplify the description of the objective function, equation (3.10) can be rewritten:   \begin{equation} J\left( k \right) = {\left({\hat{S} -{S_{r}}} \right)^{T}}\left({\hat{S} -{S_{r}}} \right) + \Lambda{U^{T}}U \end{equation} (3.12)where $$\Lambda $$ is a weight matrix,   $$ \Lambda = \left[{\begin{array}{@{}cccc@{}} {\lambda_{1} }&0&0&0\\ 0&{\lambda_{1} }&0&0\\ \vdots&\vdots&\ddots&\vdots \\ 0&0&0&{\lambda_{m} } \end{array}} \right] $$ Substituting equation (3.9) into equation (3.11) yields   \begin{align} J\left( k \right) &={\left({\hat{S} -{S_{r}}} \right)^{T}}\left({\hat{S} -{S_{r}}} \right) + \Lambda{U^{T}}U\nonumber\\ \;\;\;\;\;\;\;\;\; &={\left({{{\,\tilde{\!S}}_{0}}\left( k \right) + GU\left( k \right) + \Gamma\,\bar{\!S}\left( k \right) + MD\left( k \right) -{S_{r}}} \right)^{T}}\nonumber\\ \;\;\;\;\;\;\;\;\;\;\; &\quad\cdot \left({{{\,\tilde{\!S}}_{0}}\left( k \right) + GU\left( k \right) + \Gamma\,\bar{\!S}\left( k \right) + MD\left( k \right) -{S_{r}}} \right) + \Lambda{U^{T}}U \end{align} (3.13)where $${s_{r}}\left ({k + j} \right )$$ denotes the value of reference sliding mode, and $$\hat s\left ({k + j} \right )$$ indicates the value of predictive sliding mode. The corresponding weighting matrix $$\Lambda \in R^{n\times n} $$ is positive definite. The optimal discrete predictive sliding mode control law can be obtained as   \begin{equation} U\left( k \right) ={\left({{G^{T}}G + \Lambda } \right)^{ - 1}}{G^{T}}\left({{S_{r}}\left( k \right) -{{\,\tilde{\!S}}_{0}}\left( k \right) - \Gamma\,\bar{\!S}\left( k \right) - MD\left( k \right)} \right) \end{equation} (3.14)Moreover, the present control input signal is implemented:   \begin{equation} u = \left[{1,0, \cdots,0} \right]U \end{equation} (3.15) From the control laws (3.14) and (3.15), we can observe that the input rate can be restricted to a certain range. More specifically, the rate function can make the control rate restricted to a specified value. So we study the characteristic of the increment of $$ {S_r}\left( k \right) - {{\tilde S}_0}\left( k \right) - \Gamma\bar S\left( k \right) - MD\left( k \right) $$ .   \begin{align*} &\Delta \left({{S_{ri}}\left( k \right) -{{\,\tilde{\!S}}_{0i}}\left( k \right) -{T_{i}}{{\,\bar{\!S}}_{i}}\left( k \right) -{M_{i}}{D_{i}}\left( k \right)} \right)\\[-8pt] &\quad= s\left({k + i - 1} \right) + Tdf\left({s,r\left( \cdot \right),T} \right) - C{A^{i}}x\left( k \right) -{\beta^{k + i}}C{x_{0}} - \sum\limits_{k = 1}^{i}{{C_{s}}{A^{k - i}}d\left({k + i - 1} \right)} \\[-8pt] &\qquad- s\left({k + i - 2} \right) + Tdf\left({s,r\left( \cdot \right),T} \right) - C{A^{i}}x\left({k - 1} \right) -{\beta^{k + i}}C{x_{0}} - \sum\limits_{k = 1}^{i}{{C_{s}}{A^{k - i}}d\left({k + i - 2} \right)} \\[-8pt] &\quad\leqslant\ \left({s\left({k + i - 1} \right) - s\left({k + i - 2} \right)} \right) - C{A^{i}}\left({x\left( k \right) - x\left({k - 1} \right)} \right)\\[-8pt] \;\;\;\;\;\;\;\;\;\; &\qquad- \sum\limits_{k = 1}^{i}{\left({{C_{s}}{A^{k - i}} +{C_{s}}{A^{k - i - 1}}} \right)\delta d\left({k + i - 1} \right)} + (1 - \beta ){\beta^{k + i - 1}}C{x_{0}}\\ &\quad={T^{2}}\left({\frac{{s\left({k + i} \right) - 2s\left({k + i - 1} \right) + s\left({k + i - 2} \right)}}{{{T^{2}}}} - C{A^{i - 1}}\left({A - I} \right)\frac{{x\left( k \right) - x\left({k - 1} \right)}}{{{T^{2}}}}} \right)\\[-8pt] \;\;\;\;\;\;\;\;\; &\qquad- \sum\limits_{k = 1}^{i}{\left({{C_{s}}{A^{k - i}} +{C_{s}}{A^{k - i - 1}}} \right)\delta d\left({k + i - 1} \right)} + (1 - \beta ){\beta^{k + i - 1}}C{x_{0}}\; \end{align*}where $$\delta d$$ denotes the increment of external interference value. Based on Lemmas 2.2 and 2.3, the equation can be rewritten as   \begin{equation} \left|{\frac{{u\left({k + 1} \right) - u\left( k \right)}}{T}} \right| = {\left({{G^{T}}G + \Lambda } \right)^{ - 1}}{G^{T}}\left({\left| r \right| - 2{C_{s}}\delta d + (1 - \beta ){\beta^{k}}C{x_{0}}} \right) \end{equation} (3.17)where T is the sampling period. From (3.14), we know that there are only constant parameters in the part of $${\left ({{G^{T}}G + \Lambda } \right )^{ - 1}}{G^{T}}$$. So the relation between $${\left ({{G^{T}}G + \Lambda } \right )^{ - 1}}{G^{T}}$$ and $$\left ({{S_{r}}\left ( k \right ) \!-\!{{\,\tilde{\!S}}_{0}}\left ( k \right ) \!-\! \Gamma \,\bar{\!S}\left ( k \right ) \!-\! MD\left ( k \right )} \right )$$ is linear. It is easy to verify that, if the above design procedure guarantees, the increment of input can be restricted in a required range.   \begin{align} s\left({k + 1} \right) &= \left[{1,0, \cdots,0} \right]\left[{{{\,\tilde{\!S}}_{0}}\left( k \right) + GU\left( k \right) + MD\left( k \right) + \Gamma\,\bar{\!S}\left( k \right)} \right]\nonumber\\ &= \left[{1,0, \cdots,0} \right]\left[{{{\,\tilde{\!S}}_{0}}\left( k \right) +{S_{r}}\left( k \right) -{{\,\tilde{\!S}}_{0}}\left( k \right) + MD\left( k \right) + \Gamma\,\bar{\!S}\left( k \right) - \Gamma\,\bar{\!S}\left( k \right)} \right]\nonumber\\ &= \left[{1,0, \cdots,0} \right]\left({{S_{r}}\left( k \right) - MD\left( k \right)} \right)\nonumber\\ &= s\left( k \right) + Tf\left({s\left( k \right)\!,r\left( \cdot \right)\!,T} \right) -{C_{s}}\tilde d\left( k \right). \end{align} (3.18)Furthermore, we have the following actual sliding mode value in the form of   \begin{equation} S\left( k \right) = Ex\left( k \right) + G{\left({{G^{T}}G + \Lambda } \right)^{ - 1}}{G^{T}}\left({{S_{r}}\left( k \right) -{{\,\tilde{\!S}}_{0}}\left( k \right) - \Gamma\,\bar{\!S}\left( k \right)} \right)\!. \end{equation} (3.19) So far, we have finished the controller designed without disturbance observer. To estimate the unknown disturbance $$d\left ( k \right )$$, a sliding mode disturbance observer can be constructed in the form of   \begin{align} \hat{\!\!d}\left( k \right) &= x\left( k \right) - Ax\left({k - 1} \right) - Bu\left({k - 1} \right) - C_{s}^{ - 1}{s_{d}}\left( k \right)\nonumber\\{s_{d}}\left({k + 1} \right) &={s_{d}}\left( k \right) + Tf\left({{s_{d}}\left( k \right)\!,r\left( \cdot \right)\!,T} \right) \end{align} (3.20)where $$\,\,\hat{\!\!d}\left ( k \right )$$ is the disturbance observer value, $$x\left ( k \right )$$, $$x\left ({k - 1} \right )$$ denote the current moment state and a previous step state, respectively. $${s_{d}}\left ( k \right )$$ is the sliding mode value in sliding reaching law, and the form of f refers to (2.4). $$C_{s}^{ - 1}$$ is generalized inverse matrix of $${C_{s}}$$. On the basis of the dynamic system model (2.1), the new equation can be obtained.   \begin{equation} x\left( k \right) - Ax\left({k - 1} \right) - Bu\left({k - 1} \right) = d\left({k-1} \right)\!. \end{equation} (3.21) Also, the first equation in (3.19) can be rewritten as   \begin{equation} {s_{d}}\left( k \right) ={C_{s}}\left({\,\,\hat{\!\!d}\left( k \right) - d\left( k \right)} \right)\!. \end{equation} (3.22) In accordance with the previous statement about the sliding mode reaching law, it satisfies the reaching conditions:   \begin{align} \left[{{s_{d}}\left({k + 1} \right) -{s_{d}}\left( k \right)} \right]{\textrm{sgn}} \left({{s_{d}}\left( k \right)} \right) &< 0\nonumber\\ \left[{{s_{d}}\left({k + 1} \right) +{s_{d}}\left( k \right)} \right]{\textrm{sgn}} \left({{s_{d}}\left( k \right)} \right) &> 0. \end{align} (3.23) Consider the following Lyapunov candidate:   \begin{equation} V\left( k \right) = \frac{1}{2}{s_{d}}{\left( k \right)^{2}}. \end{equation} (3.24) In accordance with (3.21), furthermore, we have   \begin{align} \Delta V\left( k \right) = {{s_{d}^{2}}}\left({k + 1} \right) - {{s_{d}^{2}}}\left( k \right) < 0. \end{align} (3.25)$${s_{d}}\left ( k \right ) = 0$$ is a globally asymptotically stable surface. In other words, the tracking error of disturbance observer value will converge to zero. The matrix $$\hat{D}$$ denotes $${\left [{{{\,\,\hat{\!\!d}}_{1}}, \ldots ,{{\,\,\hat{\!\!d}}_{M}}} \right ]^{T}}$$ and defines the error of disturbance observer value as $$\tilde D\left ( k \right ) = \hat{D}\left ( k \right ) - D\left ( k \right )$$. So until now, we have finished the disturbance observer design. Thus, the control input with disturbance estimation can be rewritten as   \begin{align} U\left( k \right) &={\left({{G^{T}}G + \Lambda } \right)^{ - 1}}{G^{T}}\left({{S_{r}}\left( k \right) -{{\,\tilde{\!S}}_{0}}\left( k \right) - \Gamma \,\bar{\!S}\left( k \right) - M\hat{D}\left( k \right)} \right)\nonumber\\ \;\;\;\;\;\;u &= \left[{1,0, \cdots,0} \right]U \end{align} (3.26)where u denotes the current control input and U represents the control sequence. 4. Simulation study In this section, we present the consequences of numerical calculations for two examples to illustrate the effectiveness of the proposed predictive sliding mode control method. Firstly, consider a class of second order discrete-time systems (Liu et al., 2013):   \begin{equation} \left[\begin{array}{@{}c@{}}x_{1}(k+1)\\x_{2}(k+1)\end{array}\right] =\left[\begin{array}{@{}cc@{}}1&0.0088\\0&0.7788\end{array}\right] \left[\begin{array}{@{}c@{}}x_{1}(k)\\x_{2}(k)\end{array}\right] +\left[\begin{array}{@{}c@{}}0.0061\\1.1768\end{array}\right]u(k) \end{equation} (4.1)We assume that the initial conditions are $$x (0) =[1\quad 1 ]^{T}$$, and the sample time T = 0.01s. Then, we focus on the predictive sliding mode control algorithm for the standard dynamics without uncertain in (4.1). The corresponding state responses under the nominal system are shown in Fig. 1. Figure 2 shows the change of sliding mode value. Additionally, the control input and control input increment are shown in Figs. 3 and 4, respectively. By comparison, the trajectories of proposed method in this paper have some overtones in the transient process, which is inevitable. Fig. 1. View largeDownload slide Trajectory of states $$x_{1},x_{2}$$ for nominal system. Fig. 1. View largeDownload slide Trajectory of states $$x_{1},x_{2}$$ for nominal system. Fig. 2. View largeDownload slide Trajectory of sliding mode variable s for nominal system. Fig. 2. View largeDownload slide Trajectory of sliding mode variable s for nominal system. Fig. 3. View largeDownload slide Trajectory of control input u for nominal system. Fig. 3. View largeDownload slide Trajectory of control input u for nominal system. Fig. 4. View largeDownload slide Trajectory of control input increment $$\Delta u$$ for nominal system. Fig. 4. View largeDownload slide Trajectory of control input increment $$\Delta u$$ for nominal system. For discrete time system with uncertain 10%A, the responses of closed-loop signals are depicted in Figs. 5–8. Figure 5 shows the changes of state. Figure 6 shows the response of sliding mode value. In addition, the control input and control input increment are given in Figs. 7 and 8, respectively. Fig. 5. View largeDownload slide Trajectory of states $$x_{1},x_{2}$$ with uncertainty. Fig. 5. View largeDownload slide Trajectory of states $$x_{1},x_{2}$$ with uncertainty. Fig. 6. View largeDownload slide Trajectory of sliding mode variable s with uncertainty. Fig. 6. View largeDownload slide Trajectory of sliding mode variable s with uncertainty. Fig. 7. View largeDownload slide Trajectory of control input u with uncertainty. Fig. 7. View largeDownload slide Trajectory of control input u with uncertainty. Fig. 8. View largeDownload slide Trajectory of control input increment $$\Delta u$$ with uncertainty. Fig. 8. View largeDownload slide Trajectory of control input increment $$\Delta u$$ with uncertainty. The predictive sliding mode control algorithm is effective for the system parameter uncertainty, and the control input simulation curve is smoothing. It shows the tracking trajectories can be seen that a good tracking performance is achieved. The simulation results show that the developed predictive sliding mode control scheme can achieve the desired system performance: closed-loop system stability and asymptotic convergence, despite with the disturbance or system uncertainties. Moreover, the system curves are diverging, when the sliding mode control approach is applied into the uncertain system with 10%A uncertainty. Finally, in order to further verify the disturbance rejection capability, we introduce the parameter uncertain from 0 to $$10\% \cdot \Delta A$$. And the simulation can be shown as follows. Figure 9 shows the changes of state. Figure 10 shows the response of sliding mode value. In addition, the control input and control input increment are given in Figs. 11 and 12, respectively. Fig. 9. View largeDownload slide Trajectory of states $$x_{1},x_{2}$$ for perturbed system. Fig. 9. View largeDownload slide Trajectory of states $$x_{1},x_{2}$$ for perturbed system. Fig. 10. View largeDownload slide Trajectory of sliding mode variable s for perturbed system. Fig. 10. View largeDownload slide Trajectory of sliding mode variable s for perturbed system. Fig. 11. View largeDownload slide Trajectory of control input u for perturbed system. Fig. 11. View largeDownload slide Trajectory of control input u for perturbed system. Fig. 12. View largeDownload slide Trajectory of control input increment $$\Delta u$$ for perturbed system. Fig. 12. View largeDownload slide Trajectory of control input increment $$\Delta u$$ for perturbed system. According to experimental results with and without disturbance, a comparison of the experimental and simulation results reveals that, while both are very similar for the reference system, the control algorithms can satisfy the control effects. But for a small disturbance of structural parameters, the sliding mode control algorithm cannot make the practical system stable; however, the proposed control algorithm combined sliding mode control with model predictive control can satisfy the control demand. 5. Conclusion This paper has studied the problem of the control with input rate constraint. There is a new control algorithm designed combining the sliding mode control algorithm with the predictive control algorithm and disturbance observer. The proposed predictive sliding mode control algorithm guarantees the system stable. Also simulation shows that performance of the formulated algorithm is improved. References Acary, V., Brogliato, B. & Orlov, Y. V. 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Robust predictive sliding mode control for input rate-constrained discrete-time system

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© The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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0265-0754
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1471-6887
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Abstract

Abstract This paper investigates a class of uncertain linear discrete-time systems subject to input rate saturation. A predictive sliding mode control approach is proposed which guarantees the control inputs remain bounded in the input rate saturation. Furthermore, the disturbance observer is developed to compensate for the system uncertainty and disturbance. Finally, the simulations demonstrate the effectiveness of the proposed predictive sliding mode control scheme. 1. Introduction In recent years, the control problem of input constraint has received considerable attention and different design approaches have been proposed. In the study by Wang et al. (2011), an integral sliding surface and a special sliding mode controller have been designed to overcome the input saturation constraint. In the study by Grimm et al. (2003), the $$H_{2}$$ performance properties have been employed, and extensive linear matrix inequalities (LMIs)-based control method has been proposed for a class of linear input saturated systems. In the study by Hu et al. (2006), the systematic Lyapunov method was proposed for closed-loop regional stability control and the good control performance of saturated system was obtained. In the study by Zhu et al. (2011), an adaptive sliding mode controller has been proposed to guarantee system stability when the input saturations exist. To solve the tracking control problem with actuator saturations, one extended state observer-based sliding mode control approach using backstepping technique has been proposed in the study by Lu et al. (2013). It can be seen that the study of control input constraint problem usually focuses on the input amplitude constraint but ignores the problem of input rate constraint (Kefferputz & Adamy, 2011). However, the problem of control rate constraint seriously affects the system performance. When the input signal of the system changes too fast, the actuator’s physical characteristics cannot meet the control requirements and even causes instability in the system. Thus, we will design a controller for a class of linear discrete-time systems with input rate constraint. Sliding mode control has been receiving increasing attention in many control fields (Furuta, 1990; Gao et al., 1995; Choi, 2008). Discrete-time sliding mode dynamics systems cannot be simply achieved according to their continuous counterparts. Thus, the study of discrete-time sliding mode control (DSMC) is important in the control area. As one of robust control methods, DSMC has been widely employed to cope with the uncertain systems. But, in the field of traditional sliding mode control, there exist some problems of low convergence speed, long convergence time and severe chattering (Xiao et al., 2007; Gao et al., 2009). In the study by Qu et al. (2014), a discrete-time reaching law with a disturbance compensator has been developed and the DSMC for the uncertain systems have been studied. In the study by Acary et al. (2012), a sliding mode control algorithm based on an implicit Euler method has been designed, which can avoid the chattering effects. In the study by Mayne et al. (2000), a chattering-free variable reaching law has been designed for DSMC system, which can make the controller’s velocity of input signal limited. However, in the traditional sliding mode control method, the undesirable chattering phenomenon will reduce the steady-state behaviour. In this paper, we will develop a predictive sliding mode control method to deal with the undesirable chattering problem. Model predictive control is one of the multi-step optimization algorithms, and it is mainly by integrating the measured signals and predictive signals to obtain the ideal control input signals (Mao, 2003). Some new predictive control approaches to deal with the constrained problem have been studied in Cuzzola et al. (2002) and Garcia-Gabin et al. (2009). By introducing the model predictive control method, the robustness of the system will be improved. In other words, the predictive sliding mode algorithm can handle set point changes and disturbance rejection. In the study by Liu et al. (2013), the predictive sliding mode control strategy has been applied into a real solar air conditioning plant. Furthermore, disturbance attenuation ability and robustness against the variation of system parameter are two important properties required for control methods to be successfully used in the actual systems. Compared with traditional sliding mode control, the prediction sliding mode control can effectively eliminate the chattering phenomenon introduced to predict the sliding mode control method. Model predictive control method can handle the uncertainties and disturbance for a long time. However, employing disturbance observer can effectively and fast compensate for uncertainties and disturbance, and the performance of the system can also be greatly improved. Therefore, in this paper, a predictive sliding mode control will be proposed for the discrete-time system with disturbance observer. The disturbance observer is a valid approach to deal with external perturbation Chen & Chen (2010). Due to simple structure and analysis, the disturbance observer based control method has been applied to many applications (Chen et al., 2000b; Chen, 2004; Wei & Guo, 2010). In the study by Back & Shim (2008), the authors not only prove the steady-state performance of the proposed disturbance observer but also study the closed-loop system transient performance. In the study by Chen et al. (2000a), a non-linear disturbance observer for robotic manipulators has been proposed, and condition for convergence has been established. In the study by Chen & Jiang (2013), by using disturbance observer to estimate the compounded disturbance, the robust attitude control scheme has been employed for near-space vehicles. In the study by Chen & Ge (2013), a direct adaptive neural control approach using disturbance observer has been proposed for the uncertain non-affine non-linear systems. Although there is a large number of researches about observer, there are few studies of predictive sliding mode control with disturbance observer. In this paper, the disturbance observer will be introduced to cope with the external disturbance, and it will be combined with the predictive sliding mode controller design to achieve a good control performance. Motivated by above analysis, we will develop a predictive sliding mode control technique for an input rate constraint system with parametric uncertainties and mismatched disturbances. In this paper, the main contributions can be summarized as follows: 1. A robust controller combining model predictive control and sliding mode control are proposed for the discrete-time system with input rate constraints. 2. The disturbance observer is employed into the controller design to enhance the performance for the composite disturbance. Besides, this paper is organized as follows. Section 2 describes the linear control problem of input rate constraints and a reaching law for DSMC method. The main results are about a new predictive sliding mode control method; furthermore, the construction of the predictive sliding mode surface, the compound sliding mode control laws and the sliding mode reaching laws are given in Section 3. In Section 4, some numerical examples illustrate the advantages of the presented control algorithm. Finally, Section 5 draws the conclusions of the paper. 2. Problem formulation Considering a class of uncertain discrete-time dynamic systems described as   \begin{align} x\left({k + 1} \right) &= Ax (k) + Bu (k) + d (k)\nonumber\\ y\left( k \right) &= Cx\left( k \right) \end{align} (2.1)where k means the discrete time index, $$x\left ( k \right ) \in{R^{n}}$$ represents the system state, $$u\left ( k \right ) \in{R^{m}}$$ denotes the input vector, and $$y\left( k \right) \in{R^m}$$ stands for the output vector. $$d\left ( k \right ) \in{R^{n}}$$ denotes the external disturbance, which contains the modeling uncertainty and external disturbance. $$A \in{R^{n \times n}}$$ indicates the system matrix, $$B \in{R^{n \times m}}$$ indicates the input matrix with full rank m, $$C \in{R^{m \times m}}$$ is the output matrix. Moreover, we assume that the control input increment $$\Delta u\left ( k \right ) \in{R^{m}}$$ is restricted to a class of admissible controls. In this paper, the control objective is to guarantee the system stable and control input rate bounded. In order to meet the system control input rate constraints, we introduce discrete-time sliding mode reaching law with constructed function. Firstly, the following assumptions and lemmas are needed. Assumption 1 (A, B) is stabilizable, and (A, C) is detectable. Assumption 2: The state coefficient matrix A is invertible. Assumption 3: Control gain matrix B satisfies that $$BB^{T}$$ is invertible. Assumption 4: The external disturbance d(k) is slowly time-varying signal. Lemma 2.1 For all $$k\geqslant 0$$, given the sliding mode surface s(k), if the sliding mode surface satisfies the following inequalities (Gao et al., 1995)  \begin{align} \left[{s\left({k + 1} \right) - s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right) &< 0\nonumber\\ \left[{s\left({k + 1} \right) + s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right)&> 0 \end{align} (2.2)then we can conclude that the discrete-time system has satisfied the reaching condition of sliding mode motion. Lemma 2.2 For the ideal performance function r(⋅), there exists a reaching law (2.3) that allows the discrete time system to satisfy the reaching condition of sliding mode motion (Liu et al., 2013)   \begin{equation} s\left({k + 1} \right) = s\left( k \right) + Tf\left({s\left( k \right)\!,r\left( \cdot \right)\!,T} \right) \end{equation} (2.3)where s(k) is the current sliding mode value, $$s(k+1)$$ is the $$k+1 step$$ sliding mode value. T denotes the sample period, and f(x, r(⋅), T) is defined as (Liu et al., 2013)   \begin{align} f\left({x,r,T} \right) &= - \left({\frac{{\left|{s\left( k \right)} \right|}}{{iT}} + \frac{{\left({i - 1} \right)rT}}{2}} \right){\textrm{sgn}} \left({s\left( k \right)} \right)\nonumber\\ i &= fix({i^{\prime}}) + 1\nonumber\\ i^{\prime} &= 0.5\left({\sqrt{1 + \frac{{8\left|{s\left( k \right)} \right|}}{{r{T^{2}}}}} - 1} \right) \end{align} (2.4)where fix(⋅) denotes the integer rounding function, sgn(⋅) denotes the signum function. Proof. Substituting (2.4) into (2.3), the following results were derived:   \begin{align} \left[{s\left({k + 1} \right) - s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right) &= Tf\left({s\left( k \right)\!,r\left( \cdot \right),T} \right){\textrm{sgn}} \left({s\left( k \right)} \right)\nonumber\\ &= - \left({\frac{{\left|{s\left( k \right)} \right|}}{i} + \frac{{\left({i - 1} \right)r\left( \cdot \right){T^{2}}}}{2}} \right)\!.\quad\qquad\ \end{align} (2.5)  \begin{align} \left[{s\left({k + 1} \right) + s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right) &= 2\left|{s\left( k \right)} \right| + Tf\left({s\left( k \right),r\left( \cdot \right),T} \right){\textrm{sgn}} \left({s\left( k \right)} \right)\nonumber\\ &= 2\left|{s\left( k \right)} \right| - \left({\frac{{\left|{s\left( k \right)} \right|}}{i} + \frac{{\left({i - 1} \right)r\left( \cdot \right){T^{2}}}}{2}} \right)\!. \end{align} (2.6)On the basis of definition of i, we have $$i \geqslant 1$$. According to (2.5), it is clear that we have   \begin{equation} \left[{s\left({k + 1} \right) - s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right) < 0. \end{equation} (2.7)To illustrate the situation $$\left [{s\left ({k + 1} \right ) + s\left ( k \right )} \right ]{\textrm{sgn}} \left ({s\left ( k \right )} \right )> 0 $$, we will discuss the points in the following three aspects. When $$\left |{s\left ( k \right )} \right | < r\left ( \cdot \right ){T^{2}}$$, i = 1, then we obtain   \begin{equation} \left[{s\left({k + 1} \right) + s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right) = \left|{s\left( k \right)} \right|> 0. \end{equation} (2.8)When $$\left |{s\left ( k \right )} \right | = r\left ( \cdot \right ){T^{2}}$$, i = 2, we have   \begin{equation} \left[{s\left({k + 1} \right) + s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right) = 2\left|{s\left( k \right)} \right| - \left({\frac{{\left|{s\left( k \right)} \right|}}{i} + \frac{{\left({i - 1} \right)\left|{s\left( k \right)} \right|}}{2}} \right) = \left|{s\left( k \right)} \right|> 0. \end{equation} (2.9)When $$\left |{s\left ( k \right )} \right |> r\left ( \cdot \right ){T^{2}}$$, then we obtain   \begin{align} i - 1 < 0.5\left({\sqrt{1 + \frac{{8\left|{s\left( k \right)} \right|}}{{r\left( \cdot \right){T^{2}}}}} - 1} \right) &\Leftrightarrow\nonumber\\{\left({2i - 1} \right)^{2}} < 1 + \frac{{8\left|{s\left( k \right)} \right|}}{{r\left( \cdot \right){T^{2}}}} &\Leftrightarrow\nonumber\\ i\left({i - 1} \right) < \frac{{2\left|{s\left( k \right)} \right|}}{{r\left( \cdot \right){T^{2}}}} &\Leftrightarrow\nonumber\\ \frac{{\left({i - 1} \right)r\left( \cdot \right){T^{2}}}}{2} < \frac{{\left|{s\left( k \right)} \right|}}{i}.& \end{align} (2.10)Substituting (2.10) into (2.6), then we have   \begin{equation} \left[{s\left({k + 1} \right) + s\left( k \right)} \right]{\textrm{sgn}} \left({s\left( k \right)} \right)> 0. \end{equation} (2.11)Invoking (2.8), (2.9) and (2.11), the condition $$\left [{s\left ({k + 1} \right ) + s\left ( k \right )} \right ]{\textrm{sgn}} \left ({s\left ( k \right )} \right )> 0$$ has been achieved. In a word, according to Lemma 2.1, it can be concluded that the reaching law (2.2) has satisfied the reaching condition of sliding mode motion. Lemma 2.3 Assume that a linear time-invariant system can be described as   \begin{equation} \dot{\!x}\left( t \right) =\, \bar{\!A}x\left( t \right) +\, \bar{\!B}u\left( t \right) \end{equation} (2.12)where $$\,\bar{\!A} \in{R^{n \times n}}$$ and $$\,\bar{\!B} \in{R^{n \times m}}$$ are the known constant matrices with approximate dimensions. $$x\left ( t \right ) \in{R^{n}}$$ and $$u\left ( t \right ) \in{R^{n}}$$ indicate the state and control input of continuous system, respectively. Then, the corresponding Euler approximate system can be represented as (Hao et al., 2001)   \begin{equation} x\left({k + 1} \right) = \left[{I + \tau \,\bar{\!A}} \right]x( k) + \tau \,\bar{\!B}u(k). \end{equation} (2.13)If there exists $$\tau \to 0$$, then we can obtain $$\mathop{\lim }\limits _{\tau \to 0} A = I \Rightarrow (A - I) = O\left ( T \right )$$, where $$A=I+\tau \,\bar{\!A}$$. Lemma 2.4 Assume that $$r\left ( \cdot \right )$$ is selected to make sure r > 0, then the reaching sliding mode can be satisfied by the equation as follows (Liu et al., 2013):   \begin{equation} \frac{1}{T}\left({\frac{{s\left({k + 1} \right) - 2s\left( k \right) + s\left({k - 1} \right)}}{T}} \right) = - r{\textrm{sgn}} \left({s\left( k \right)} \right)\!. \end{equation} (2.14) In order to extend the application of input rate constraint control method in practise, we need to enhance the robust performance of the control method when the system contains parametric uncertainties and mismatched disturbances. 3. Main results In this section, we provide a synthesis procedure of predictive sliding mode control algorithm. Letting $$x\left ({k + i} \right )$$ be the predicted value of system state at time $$k + i$$; $$u\left ({k + i} \right )$$ be the future control at time $$k + i$$. In addition, $$d\left ({k + i} \right )$$ stands for the future disturbance at time $$k + i$$. Then a j step ahead predictor can be derived:   \begin{equation} y\left({k + j} \right) = C{A^{{\hskip.6pt}j}}x\left( k \right) + \sum\limits_{i = 1}^{j}{C{A^{{\hskip.6pt}j - i}}Bu\left({k + i - 1} \right)} + \sum\limits_{i = 1}^{j}{C{A^{{\hskip.6pt}j - i}}d\left({k + i - 1} \right)}. \end{equation} (3.1)Thus, we obtain   \begin{equation} \hat{\!y}\left({k + j} \right) = C{A^{{\hskip.6pt}j}}\,\hat{\!x}\left( k \right) + \sum\limits_{i = 1}^{j}{C{A^{{\hskip.6pt}j - i}}Bu\left({k + i - 1} \right)} + C{A^{{\hskip.6pt}j - 1}}d\left( k \right)\!. \end{equation} (3.2) Now introduce the global sliding mode approach, define a sliding mode function as follows:   \begin{equation} s\left( k \right) ={C_{s}}x\left( k \right) - \beta{C_{s}}{x_{0}} \end{equation} (3.3)where $$C_{s}\in R^{m\times n}$$ and $$\beta $$ is a constant $$0<\beta <1$$. x0 indicates the initial condition of state. Furthermore, the sliding mode predictive model is constructed:   \begin{equation} \tilde{\!s}\left( k \right) ={C_{s}}x\left( k \right) -{\beta^{k}}{C_{s}}{x_{0}} \end{equation} (3.4)where k > 0. Assume that the disturbance does not exist, the sliding function value at future time $$k+p$$ (p is predictive horizon) can be derived:   \begin{align} \tilde{\!s}\left({k + p} \right) ={C_{s}}{A^{p}}x\left( k \right) + \sum\limits_{i = 1}^{p}{{C_{s}}{A^{i - 1}}Bu\left({k + p - i} \right) +\sum\limits_{i = 1}^{p}{{C_{s}}{A^{i-1}}}d\left({k + p - i} \right) -{C_{s}}{\beta^{k + p}}{x_{0}}}. \end{align} (3.5)In order to simplify the form of the sliding function, we define the following matrices and vectors as follows:   $$ \tilde{\!S}\left( k \right) = \left[{\begin{array}{@{}c@{}} {\tilde{\!s}\left({k + 1} \right)}\\{\tilde{\!s}\left({k + 2} \right)}\\ \vdots \\{\tilde{\!s}\left({k + p} \right)} \end{array}} \right]\!,\quad{\tilde{\!S}_{0}}\left( k \right) = \left[{\begin{array}{@{}c@{}} {CAx\left( k \right) -{\beta^{k + 1}}C{x_{0}}}\\{C{A^{2}}x\left( k \right) -{\beta^{k + 2}}C{x_{0}}}\\ \vdots \\{C{A^{p}}x\left( k \right) -{\beta^{k + p}}C{x_{0}}} \end{array}} \right]$$  $$ G = \left[{\begin{array}{@{}ccccc@{}} {CB}&0& \cdots & \cdots &0\\{CAB}&{CB}&0& \cdots &0\\ \vdots & \vdots & \cdots & \cdots & \vdots \\{C{A^{m - 1}}B}&{C{A^{m - 2}}B}& \cdots & \cdots &{CB}\\{C{A^{m}}B}&{C{A^{m - 1}}B}& \cdots & \cdots &{CAB + CB}\\ \vdots & \vdots & \cdots & \cdots & \vdots \\{C{A^{p - 1}}B}&{C{A^{p - 2}}B}& \cdots & \cdots &{\sum\limits_{j = 0}^{p - m}{C{A^{j}}B} } \end{array}} \right]\!,\quad U\left( k \right) = \left[{\begin{array}{@{}c@{}} {u\left( k \right)}\\{u\left({k + 1} \right)}\\ \vdots \\{u\left({k + m - 1} \right)}\\ \end{array}} \right]$$  $$ M = \left[{\begin{array}{@{}ccccc@{}} {{C_{s}}}&0& \cdots & \cdots &0\\{{C_{s}}A}&{{C_{s}}}&0& \cdots &0\\ \vdots & \vdots & \cdots & \cdots & \vdots \\{{C_{s}}{A^{m - 1}}}&{{C_{s}}{A^{m - 2}}}& \cdots & \cdots &{{C_{s}}}\\{{C_{s}}{A^{m}}}&{{C_{s}}{A^{m - 1}}}& \cdots & \cdots &{{C_{s}}A + C}\\ \vdots & \vdots & \cdots & \cdots & \vdots \\{{C_{s}}{A^{p - 1}}}&{{C_{s}}{A^{p - 2}}}& \cdots & \cdots &{\sum\limits_{j = 0}^{p - m}{{C_{s}}{A^{j}}} } \end{array}} \right]\!,\quad D\left( k \right) = \left[{\begin{array}{@{}c@{}} {d\left( k \right)}\\{d\left({k + 1} \right)}\\ \vdots \\{d\left({k + m - 1} \right)} \end{array}} \right]$$where $$\,\tilde{\!S}\left ( k \right )$$ means the p-step sliding mode surface predictive vector, G denotes the predictive control matrix, M represents the predictive state matrix, m represents the control horizon and matrices $$U\left ( k \right )$$, $$D\left ( k \right )$$ denote m-step input control signal and disturbance, respectively. Therefore, the equation (3.5) can be rewritten as   \begin{equation} \tilde{\!S}\left( k \right) =\,{\tilde{\!S}_{0}}\left( k \right) + GU\left( k \right) + MD\left( k \right)\!. \end{equation} (3.6) Now the error between the real value of sliding mode and predictive value of sliding mode can be defined as   \begin{equation} e\left( k \right) = s\left( k \right) -\, \tilde{\!s}\left({k|k - p} \right) \end{equation} (3.7)where $$\,\tilde{\!s}\left ({k|k - p} \right )$$ denotes the sliding mode surface step k − p predictive value for step k. Consider the feedback correction, the sliding function value at time $$k + j$$ can be defined as   \begin{equation} \hat{\!s}\left({k + j} \right) =\, \tilde{\!s}\left({k + j} \right) +{\tau_{j}}\left[{s\left( k \right) - \,\tilde{\!s}\left({k|k - j} \right)} \right] \end{equation} (3.8)where $${\tau _{j}}$$ is a design parameter, $${\tau _{1}}>{\tau _{2}} > \cdots >{\tau _{P}} > 0$$, the value of parameter $${\tau _{j}}$$ will decide the degree of smoothing. Now we can define   $$ \Gamma = \left[{\begin{array}{@{}cccc@{}} {{\tau_{1}}}&0&0&0 \\ 0&{{\tau_{2}}}&0&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&0&{{\tau_{p}}} \end{array}} \right]\!,\quad \bar{\!S}\left( k \right) = \left[{\begin{array}{@{}c@{}} {s\left( k \right) - \,\tilde{\!s}\left({k|k - 1} \right)}\\{s\left( k \right) - \,\tilde{\!s}\left({k|k - 2} \right)}\\ \vdots \\{s\left( k \right) - \,\tilde{\!s}\left({k|k - p} \right)} \end{array}} \right]\!.$$ To simplify above equation (3.8), it can be derived as   \begin{equation} \hat{S}\left( k \right) ={\,\tilde{\!S}_{0}}\left( k \right) + GU\left( k \right) + MD\left( k \right) + \Gamma\,\bar{\!S}\left( k \right) \end{equation} (3.9)where   $$ \hat{S}\left( k \right) = \left[{\begin{array}{@{}c@{}} {\hat s\left({k + 1} \right)}\\{\hat s\left({k + 2} \right)}\\ \vdots \\{\hat s\left({k + p} \right)} \end{array}} \right]\!.$$ The objective function used in this formulation is the squared two-norm given by   \begin{equation} J\left( k \right) = \left\{{\sum\limits_{j = 1}^{p}{{{\left({\hat s\left({k + j} \right) -{s_{r}}\left({k + j} \right)} \right)}^{2}} + \sum\limits_{j = 1}^{m}{\lambda{{\left({u\left({k + j} \right)} \right)}^{2}}} } } \right\} \end{equation} (3.10)where constant $$\lambda> 0$$ is weight coefficient. p indicates the costing horizon and m denotes the control horizon. We aim at achieving the following performance objective:   \begin{equation} \frac{{dJ\left( k \right)}}{{dU\left( k \right)}} = 0. \end{equation} (3.11)In addition, in order to simplify the description of the objective function, equation (3.10) can be rewritten:   \begin{equation} J\left( k \right) = {\left({\hat{S} -{S_{r}}} \right)^{T}}\left({\hat{S} -{S_{r}}} \right) + \Lambda{U^{T}}U \end{equation} (3.12)where $$\Lambda $$ is a weight matrix,   $$ \Lambda = \left[{\begin{array}{@{}cccc@{}} {\lambda_{1} }&0&0&0\\ 0&{\lambda_{1} }&0&0\\ \vdots&\vdots&\ddots&\vdots \\ 0&0&0&{\lambda_{m} } \end{array}} \right] $$ Substituting equation (3.9) into equation (3.11) yields   \begin{align} J\left( k \right) &={\left({\hat{S} -{S_{r}}} \right)^{T}}\left({\hat{S} -{S_{r}}} \right) + \Lambda{U^{T}}U\nonumber\\ \;\;\;\;\;\;\;\;\; &={\left({{{\,\tilde{\!S}}_{0}}\left( k \right) + GU\left( k \right) + \Gamma\,\bar{\!S}\left( k \right) + MD\left( k \right) -{S_{r}}} \right)^{T}}\nonumber\\ \;\;\;\;\;\;\;\;\;\;\; &\quad\cdot \left({{{\,\tilde{\!S}}_{0}}\left( k \right) + GU\left( k \right) + \Gamma\,\bar{\!S}\left( k \right) + MD\left( k \right) -{S_{r}}} \right) + \Lambda{U^{T}}U \end{align} (3.13)where $${s_{r}}\left ({k + j} \right )$$ denotes the value of reference sliding mode, and $$\hat s\left ({k + j} \right )$$ indicates the value of predictive sliding mode. The corresponding weighting matrix $$\Lambda \in R^{n\times n} $$ is positive definite. The optimal discrete predictive sliding mode control law can be obtained as   \begin{equation} U\left( k \right) ={\left({{G^{T}}G + \Lambda } \right)^{ - 1}}{G^{T}}\left({{S_{r}}\left( k \right) -{{\,\tilde{\!S}}_{0}}\left( k \right) - \Gamma\,\bar{\!S}\left( k \right) - MD\left( k \right)} \right) \end{equation} (3.14)Moreover, the present control input signal is implemented:   \begin{equation} u = \left[{1,0, \cdots,0} \right]U \end{equation} (3.15) From the control laws (3.14) and (3.15), we can observe that the input rate can be restricted to a certain range. More specifically, the rate function can make the control rate restricted to a specified value. So we study the characteristic of the increment of $$ {S_r}\left( k \right) - {{\tilde S}_0}\left( k \right) - \Gamma\bar S\left( k \right) - MD\left( k \right) $$ .   \begin{align*} &\Delta \left({{S_{ri}}\left( k \right) -{{\,\tilde{\!S}}_{0i}}\left( k \right) -{T_{i}}{{\,\bar{\!S}}_{i}}\left( k \right) -{M_{i}}{D_{i}}\left( k \right)} \right)\\[-8pt] &\quad= s\left({k + i - 1} \right) + Tdf\left({s,r\left( \cdot \right),T} \right) - C{A^{i}}x\left( k \right) -{\beta^{k + i}}C{x_{0}} - \sum\limits_{k = 1}^{i}{{C_{s}}{A^{k - i}}d\left({k + i - 1} \right)} \\[-8pt] &\qquad- s\left({k + i - 2} \right) + Tdf\left({s,r\left( \cdot \right),T} \right) - C{A^{i}}x\left({k - 1} \right) -{\beta^{k + i}}C{x_{0}} - \sum\limits_{k = 1}^{i}{{C_{s}}{A^{k - i}}d\left({k + i - 2} \right)} \\[-8pt] &\quad\leqslant\ \left({s\left({k + i - 1} \right) - s\left({k + i - 2} \right)} \right) - C{A^{i}}\left({x\left( k \right) - x\left({k - 1} \right)} \right)\\[-8pt] \;\;\;\;\;\;\;\;\;\; &\qquad- \sum\limits_{k = 1}^{i}{\left({{C_{s}}{A^{k - i}} +{C_{s}}{A^{k - i - 1}}} \right)\delta d\left({k + i - 1} \right)} + (1 - \beta ){\beta^{k + i - 1}}C{x_{0}}\\ &\quad={T^{2}}\left({\frac{{s\left({k + i} \right) - 2s\left({k + i - 1} \right) + s\left({k + i - 2} \right)}}{{{T^{2}}}} - C{A^{i - 1}}\left({A - I} \right)\frac{{x\left( k \right) - x\left({k - 1} \right)}}{{{T^{2}}}}} \right)\\[-8pt] \;\;\;\;\;\;\;\;\; &\qquad- \sum\limits_{k = 1}^{i}{\left({{C_{s}}{A^{k - i}} +{C_{s}}{A^{k - i - 1}}} \right)\delta d\left({k + i - 1} \right)} + (1 - \beta ){\beta^{k + i - 1}}C{x_{0}}\; \end{align*}where $$\delta d$$ denotes the increment of external interference value. Based on Lemmas 2.2 and 2.3, the equation can be rewritten as   \begin{equation} \left|{\frac{{u\left({k + 1} \right) - u\left( k \right)}}{T}} \right| = {\left({{G^{T}}G + \Lambda } \right)^{ - 1}}{G^{T}}\left({\left| r \right| - 2{C_{s}}\delta d + (1 - \beta ){\beta^{k}}C{x_{0}}} \right) \end{equation} (3.17)where T is the sampling period. From (3.14), we know that there are only constant parameters in the part of $${\left ({{G^{T}}G + \Lambda } \right )^{ - 1}}{G^{T}}$$. So the relation between $${\left ({{G^{T}}G + \Lambda } \right )^{ - 1}}{G^{T}}$$ and $$\left ({{S_{r}}\left ( k \right ) \!-\!{{\,\tilde{\!S}}_{0}}\left ( k \right ) \!-\! \Gamma \,\bar{\!S}\left ( k \right ) \!-\! MD\left ( k \right )} \right )$$ is linear. It is easy to verify that, if the above design procedure guarantees, the increment of input can be restricted in a required range.   \begin{align} s\left({k + 1} \right) &= \left[{1,0, \cdots,0} \right]\left[{{{\,\tilde{\!S}}_{0}}\left( k \right) + GU\left( k \right) + MD\left( k \right) + \Gamma\,\bar{\!S}\left( k \right)} \right]\nonumber\\ &= \left[{1,0, \cdots,0} \right]\left[{{{\,\tilde{\!S}}_{0}}\left( k \right) +{S_{r}}\left( k \right) -{{\,\tilde{\!S}}_{0}}\left( k \right) + MD\left( k \right) + \Gamma\,\bar{\!S}\left( k \right) - \Gamma\,\bar{\!S}\left( k \right)} \right]\nonumber\\ &= \left[{1,0, \cdots,0} \right]\left({{S_{r}}\left( k \right) - MD\left( k \right)} \right)\nonumber\\ &= s\left( k \right) + Tf\left({s\left( k \right)\!,r\left( \cdot \right)\!,T} \right) -{C_{s}}\tilde d\left( k \right). \end{align} (3.18)Furthermore, we have the following actual sliding mode value in the form of   \begin{equation} S\left( k \right) = Ex\left( k \right) + G{\left({{G^{T}}G + \Lambda } \right)^{ - 1}}{G^{T}}\left({{S_{r}}\left( k \right) -{{\,\tilde{\!S}}_{0}}\left( k \right) - \Gamma\,\bar{\!S}\left( k \right)} \right)\!. \end{equation} (3.19) So far, we have finished the controller designed without disturbance observer. To estimate the unknown disturbance $$d\left ( k \right )$$, a sliding mode disturbance observer can be constructed in the form of   \begin{align} \hat{\!\!d}\left( k \right) &= x\left( k \right) - Ax\left({k - 1} \right) - Bu\left({k - 1} \right) - C_{s}^{ - 1}{s_{d}}\left( k \right)\nonumber\\{s_{d}}\left({k + 1} \right) &={s_{d}}\left( k \right) + Tf\left({{s_{d}}\left( k \right)\!,r\left( \cdot \right)\!,T} \right) \end{align} (3.20)where $$\,\,\hat{\!\!d}\left ( k \right )$$ is the disturbance observer value, $$x\left ( k \right )$$, $$x\left ({k - 1} \right )$$ denote the current moment state and a previous step state, respectively. $${s_{d}}\left ( k \right )$$ is the sliding mode value in sliding reaching law, and the form of f refers to (2.4). $$C_{s}^{ - 1}$$ is generalized inverse matrix of $${C_{s}}$$. On the basis of the dynamic system model (2.1), the new equation can be obtained.   \begin{equation} x\left( k \right) - Ax\left({k - 1} \right) - Bu\left({k - 1} \right) = d\left({k-1} \right)\!. \end{equation} (3.21) Also, the first equation in (3.19) can be rewritten as   \begin{equation} {s_{d}}\left( k \right) ={C_{s}}\left({\,\,\hat{\!\!d}\left( k \right) - d\left( k \right)} \right)\!. \end{equation} (3.22) In accordance with the previous statement about the sliding mode reaching law, it satisfies the reaching conditions:   \begin{align} \left[{{s_{d}}\left({k + 1} \right) -{s_{d}}\left( k \right)} \right]{\textrm{sgn}} \left({{s_{d}}\left( k \right)} \right) &< 0\nonumber\\ \left[{{s_{d}}\left({k + 1} \right) +{s_{d}}\left( k \right)} \right]{\textrm{sgn}} \left({{s_{d}}\left( k \right)} \right) &> 0. \end{align} (3.23) Consider the following Lyapunov candidate:   \begin{equation} V\left( k \right) = \frac{1}{2}{s_{d}}{\left( k \right)^{2}}. \end{equation} (3.24) In accordance with (3.21), furthermore, we have   \begin{align} \Delta V\left( k \right) = {{s_{d}^{2}}}\left({k + 1} \right) - {{s_{d}^{2}}}\left( k \right) < 0. \end{align} (3.25)$${s_{d}}\left ( k \right ) = 0$$ is a globally asymptotically stable surface. In other words, the tracking error of disturbance observer value will converge to zero. The matrix $$\hat{D}$$ denotes $${\left [{{{\,\,\hat{\!\!d}}_{1}}, \ldots ,{{\,\,\hat{\!\!d}}_{M}}} \right ]^{T}}$$ and defines the error of disturbance observer value as $$\tilde D\left ( k \right ) = \hat{D}\left ( k \right ) - D\left ( k \right )$$. So until now, we have finished the disturbance observer design. Thus, the control input with disturbance estimation can be rewritten as   \begin{align} U\left( k \right) &={\left({{G^{T}}G + \Lambda } \right)^{ - 1}}{G^{T}}\left({{S_{r}}\left( k \right) -{{\,\tilde{\!S}}_{0}}\left( k \right) - \Gamma \,\bar{\!S}\left( k \right) - M\hat{D}\left( k \right)} \right)\nonumber\\ \;\;\;\;\;\;u &= \left[{1,0, \cdots,0} \right]U \end{align} (3.26)where u denotes the current control input and U represents the control sequence. 4. Simulation study In this section, we present the consequences of numerical calculations for two examples to illustrate the effectiveness of the proposed predictive sliding mode control method. Firstly, consider a class of second order discrete-time systems (Liu et al., 2013):   \begin{equation} \left[\begin{array}{@{}c@{}}x_{1}(k+1)\\x_{2}(k+1)\end{array}\right] =\left[\begin{array}{@{}cc@{}}1&0.0088\\0&0.7788\end{array}\right] \left[\begin{array}{@{}c@{}}x_{1}(k)\\x_{2}(k)\end{array}\right] +\left[\begin{array}{@{}c@{}}0.0061\\1.1768\end{array}\right]u(k) \end{equation} (4.1)We assume that the initial conditions are $$x (0) =[1\quad 1 ]^{T}$$, and the sample time T = 0.01s. Then, we focus on the predictive sliding mode control algorithm for the standard dynamics without uncertain in (4.1). The corresponding state responses under the nominal system are shown in Fig. 1. Figure 2 shows the change of sliding mode value. Additionally, the control input and control input increment are shown in Figs. 3 and 4, respectively. By comparison, the trajectories of proposed method in this paper have some overtones in the transient process, which is inevitable. Fig. 1. View largeDownload slide Trajectory of states $$x_{1},x_{2}$$ for nominal system. Fig. 1. View largeDownload slide Trajectory of states $$x_{1},x_{2}$$ for nominal system. Fig. 2. View largeDownload slide Trajectory of sliding mode variable s for nominal system. Fig. 2. View largeDownload slide Trajectory of sliding mode variable s for nominal system. Fig. 3. View largeDownload slide Trajectory of control input u for nominal system. Fig. 3. View largeDownload slide Trajectory of control input u for nominal system. Fig. 4. View largeDownload slide Trajectory of control input increment $$\Delta u$$ for nominal system. Fig. 4. View largeDownload slide Trajectory of control input increment $$\Delta u$$ for nominal system. For discrete time system with uncertain 10%A, the responses of closed-loop signals are depicted in Figs. 5–8. Figure 5 shows the changes of state. Figure 6 shows the response of sliding mode value. In addition, the control input and control input increment are given in Figs. 7 and 8, respectively. Fig. 5. View largeDownload slide Trajectory of states $$x_{1},x_{2}$$ with uncertainty. Fig. 5. View largeDownload slide Trajectory of states $$x_{1},x_{2}$$ with uncertainty. Fig. 6. View largeDownload slide Trajectory of sliding mode variable s with uncertainty. Fig. 6. View largeDownload slide Trajectory of sliding mode variable s with uncertainty. Fig. 7. View largeDownload slide Trajectory of control input u with uncertainty. Fig. 7. View largeDownload slide Trajectory of control input u with uncertainty. Fig. 8. View largeDownload slide Trajectory of control input increment $$\Delta u$$ with uncertainty. Fig. 8. View largeDownload slide Trajectory of control input increment $$\Delta u$$ with uncertainty. The predictive sliding mode control algorithm is effective for the system parameter uncertainty, and the control input simulation curve is smoothing. It shows the tracking trajectories can be seen that a good tracking performance is achieved. The simulation results show that the developed predictive sliding mode control scheme can achieve the desired system performance: closed-loop system stability and asymptotic convergence, despite with the disturbance or system uncertainties. Moreover, the system curves are diverging, when the sliding mode control approach is applied into the uncertain system with 10%A uncertainty. Finally, in order to further verify the disturbance rejection capability, we introduce the parameter uncertain from 0 to $$10\% \cdot \Delta A$$. And the simulation can be shown as follows. Figure 9 shows the changes of state. Figure 10 shows the response of sliding mode value. In addition, the control input and control input increment are given in Figs. 11 and 12, respectively. Fig. 9. View largeDownload slide Trajectory of states $$x_{1},x_{2}$$ for perturbed system. Fig. 9. View largeDownload slide Trajectory of states $$x_{1},x_{2}$$ for perturbed system. Fig. 10. View largeDownload slide Trajectory of sliding mode variable s for perturbed system. Fig. 10. View largeDownload slide Trajectory of sliding mode variable s for perturbed system. Fig. 11. View largeDownload slide Trajectory of control input u for perturbed system. Fig. 11. View largeDownload slide Trajectory of control input u for perturbed system. Fig. 12. View largeDownload slide Trajectory of control input increment $$\Delta u$$ for perturbed system. Fig. 12. View largeDownload slide Trajectory of control input increment $$\Delta u$$ for perturbed system. According to experimental results with and without disturbance, a comparison of the experimental and simulation results reveals that, while both are very similar for the reference system, the control algorithms can satisfy the control effects. But for a small disturbance of structural parameters, the sliding mode control algorithm cannot make the practical system stable; however, the proposed control algorithm combined sliding mode control with model predictive control can satisfy the control demand. 5. Conclusion This paper has studied the problem of the control with input rate constraint. There is a new control algorithm designed combining the sliding mode control algorithm with the predictive control algorithm and disturbance observer. The proposed predictive sliding mode control algorithm guarantees the system stable. Also simulation shows that performance of the formulated algorithm is improved. References Acary, V., Brogliato, B. & Orlov, Y. V. 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IMA Journal of Mathematical Control and InformationOxford University Press

Published: Mar 27, 2018

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