Robust Tracking and Model Following Controller Based on Higher Order Sliding Mode Control and Observation: With an Application to MagLev System

The main task in the design of a model following controller is to come up with a control algorithm which compels the dynamics of a specified plant to follow the model dynamics. The tracking error between the model and plant outputs should asymptotically converge to zero making sure that the plant outputs follow the model output perfectly. This approach has the complete freedom to specify all the design criteria through model to ensure the minimization of error between model and plant output [1]–[2].
In the model based control strategy the mismatch between the model of the plant and the actual plant dynamics exists due to the presence of external disturbances, parametric uncertainties and unmodeled dynamics resulting in declination of performance of the closed loop system. This necessitates the design of various robust controllers. Therefore, the problem of RTMF for a class of dynamical system has gathered good amount of attention amongst many researchers in the recent past [3]–[14].

In [3], a nonlinear control scheme is proposed to achieve robust tracking of dynamical signals. Linear state feedback controller is developed for robust tracking and model following of uncertain linear system in [4]. In [5], the problem of RTMF is considered for a class of dynamical systems which contain uncertain nonlinear terms and bounded unknown disturbances. However, these robust state feedback tracking controllers do not produce asymptotic tracking; instead, the so-called practical tracking is achieved.

Sliding mode control (SMC) technique has been recognized as one of the efficient tools to design robust controller for systems operating under various uncertainty conditions [1], [15]–[17]. It has many attractive features such as insensitive to matched uncertainties, order reduction, invariance and simplicity in design. The two step design procedure of sliding mode controller consists of design of a sliding manifold which reflects the desired performance during sliding and a discontinuous control law which makes sure that the state trajectories evolving in the sliding manifold [1] stays there for all future time. Based on SMC theory, a class of variable structure tracking controllers has been proposed in [6]–[14] for robust tracking and model following of dynamical signals for a class of uncertain dynamical systems. The main disadvantage of these SMC based strategies are the so called “chattering effect” which affects the actuator performance in practical implementation [15]–[17]. Moreover, most of these works in [6]–[14] have been tested only in simulation and no hardware implementation has been found so far in any literature.

In contrast to that, the controller based on STA is the most suitable choice to eliminate chattering due to continuous nature of control signal [17], [18]. It achieves finite-time convergence of not only the sliding variable but its first derivative as well. In the family of HOSM, the most effective SOSM algorithm is the STA because same algorithm can be used as controller [19], observer known as STO [20], [21] and robust exact differentiator [22]. Controller based on STA generates continuous control signal and consequently compensates chattering which ensures all the properties of first order SMC in addition to that it achieves second order sliding motion (SOSM) in the presence of bounded uncertainties.

With this motivation this paper presents a new class of RTMF controllers based on STA and then the proposed control algorithm is implemented in magnetic levitation (MagLev) apparatus (Feedback Instruments Ltd. UK, Model No. 33-210).

Industrial application of MagLev continues to grow world wide and successfully implemented for many applications like high-speed MagLev passenger trains, frictionless bearing, superconductor rotor suspension of gyroscopes, rocket-guiding projects and vibration isolation systems in semiconductor manufacturing. The position control of the levitated ball is a challenging control problem due to the fact that the system is inherently nonlinear and open loop unstable. In past many advanced controllers like robust sliding mode control [23]–[28], feedback linearization based control strategy [29]–[31], PID control [32]–[34], adaptive control [35]–[37], generalised proportional integral (GPI) control [38]–[39] have been proposed for compensation of magnetic levitation system.

The implementation of the proposed RTMF controller based on STA in MagLev system requires the information of position and velocity of the ball, but direct measurement of ball velocity is not available. For estimation of velocity, the measured position data is numerically differentiated in most of the literature dealing with control law implementation for MagLev system. This process is not suitable from the implementation point of view because differentiation will enhance the high frequency noise in the position measurement. In [29], a nonlinear observer with linear error dynamics is designed to estimate the speed. An model based integral re-constructor is proposed in [39] for online ball velocity estimation. However, these methodologies suffer from a drawback is that the estimators are not robust against the uncertainties. This fact was the motivation to design STO which is insensitive with respect to uncertainties to estimate the ball velocity of MagLev which is complemented with proposed RTMF controller based on STA. But it has been observed that with this RTMF controller based on STA with STO it is not possible to achieve SOSM using continuous control. As a remedy, continuous RTMF controller based on STA is implemented with a HOSMO. With this observation it can be said that the performance of RTMF based on STA with STO as well as with HOSMO has been validated experimentally first time with magnetic levitation apparatus.
The contribution can be summarized as follows:

• •

  The design of RTMF based on STA is proposed for a class of uncertain LTI systems which will generate continuous control signal, and consequently suppress the effect of chattering.

• •

  To validate the performance of this proposed controller, the implementation is done in MagLev system.

• •

  To avoid the implementation difficulties, the velocity of the levitated ball is estimated using STO and the RTMF based on STA is designed and implemented based on this estimated velocity. With this method it is not possible to achieve continuous control for chosen sliding surface. As a remedy, continuous RTMF controller based on STA is implemented with a HOSMO.

The rest of the paper is organized as follows. Section 2 gives some preliminaries and problem statement. Section 3 presents the design steps of RTMF controller based on STA for an uncertain LTI system. In section 4 dynamic model of the MagLev system is derived. In selection 5 the selection of model is discussed along with the design of RTMF controller based on STO and HOSMO for MagLev. In section 6 both simulated and experimental results are presented. Finally, Section 7 concludes the paper.

Consider a $n^{\text{th}}$ order linear time invariant (LTI) uncertain dynamical system as

where $x(t)\in\mathcal{X}\subseteq\mathbb{R}^{n}$ is the state vector, the control input is given by the mapping $\mathbb{R}\ni t\mapsto u(t)\in\mathbb{U}\subset\mathbb{R}^{m}\neq\varnothing$ which is Lebesgue measureable, $y(t)\in\mathcal{Y}\subseteq\mathbb{R}^{p}$ is the output vector which is to track the reference input $y_{r}(t)\in\mathbb{R}^{p}$ and $A\in\mathbb{R}^{n\times n},B\in\mathbb{R}^{n\times m},C\in\mathbb{R}^{p\times n}$ are constant real matrices. The matrix $D\in\mathbb{R}^{n\times l}$ is known and the function $f(t,x):\mathbb{R}_{+}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{l}$ is unknown exogenous input. Throughout the paper the following assumptions hold.

From the structural assumption all uncertainties can be lumped and the system (II) can be rewritten as

where, $\mathbb{R}\times\mathcal{X}\ni(t,x)\mapsto w(t,x):=R(t,x)x(t)+E(t,x)u(t)+\mathcal{G}(t,x)f(t,x)\in\mathbb{R}^{m}$ is a lumped uncertain vector.

In this paper the objective is to design the control law to make the output of the system (3) to follow the output of the reference model. Consider the reference model is given by

where $x_{r}(t)\in\mathbb{R}^{n_{r}}$ is the state vector of the reference model, $y_{r}(t)\in\mathbb{R}^{p}$ has the same dimension as $y(t)$ and $A_{r},C_{r}$ are constant matrices of appropriate dimensions. The reference input $y_{r}(t)$ is assumed to be the output of the model. Further, it is assumed that the model states are bounded, i.e., there exists a finite positive constant $\mathcal{L}$ such that $\forall t\geq t_{0}$, $\|x_{r}\|\leq\mathcal{L}$. The reference model represents the ideal response that the controlled system must follow.

It has been shown in [4] that if there exist matrices $G\in\mathbb{R}^{n\times{n_{r}}}$ and $H\in\mathbb{R}^{m\times{n_{r}}}$ such that the following matrix algebraic relation holds

then the RTMF control based on STA proposed latter in (40) will ensure that the output of the system (3) will follow the output of (5). If a solution to (12) is not found, then a different reference model must be chosen. A method of solution to (12) is discussed in [4] which is revisited in the following discussion briefly. To solve (12), it is assumed that

This condition is satisfied if the nominal system is controllable and the number of outputs is less than or equal to the number of inputs, i.e., $p\leq m$. If this satisfy, then solution to (12) can be written as

where $\Omega_{11}$ is an $n\times n$ matrix, $\Omega_{12}$ is an $n\times p$ matrix, $\Omega_{21}$ is an $m\times n$ matrix, and $\Omega_{22}$ is an $m\times p$ matrix. Specifically, if $p=m$ then :

and $\Omega$ is partitioned accordingly in the same way. An equivalent form of (19) is

The above relations can be solved simultaneously for $G$ and $H$. Though there are many methods exist to solve for the above relation here we discuss the method given in [4]. Let $M$ be any $n\times m$ matrix

where $m_{i}$ denotes the $i^{\text{th}}$ row of $M$. We define a stacking operator $\Phi(M)$ for the matrix $M$ as

The vector is obtained by stacking the transpose of the rows of $M$. To solve (33), we write it as

This equation can be solved using the Kronecker product $\otimes$. The Kronecker product of two matrices $U\in\mathbb{R}^{n\times m}$ and $V\in\mathbb{R}^{q\times r}$ is defined as

The resultant matrix $U\otimes V$ is of the order $nq\times mr$. Using the stacking operator and the Kronecker product, (36) reduces to

which is a system of linear equations for $nm$ components of $\Phi(G)$. Then $G$ is found by unstacking $\Phi(G)$. Once $G$ is solved, substituting for $G$ into (34) yields $H$.

Our design objective can be stated as: given a desired model and its output, find a RTMF controller to force the uncertain system under consideration to behave as the ideal model, i.e., have the same or similar rise time, settling time, damping, etc. To achieve this, we propose a RTMF controller based on STA which can guarantee the output $y(t)$ of uncertain system (3) will follow the output $y_{r}(t)$ of reference model (5) ensuring asymptotic convergence of tracking error to zero. Let the tracking error be defined as

Then the tracking control law is proposed as

where $v(t)\in\mathbb{U}\subseteq\mathbb{R}^{m}$ is a super twisting control law. A new auxiliary state vector $z(t)\in\mathcal{X}$ is defined as follows

Using (12) and (41), the relation between tracking error $e(t)$ and new state vector $z(t)$ can be written as

From (42), we can see that $\|e(t)\|\leq\|C\|\|z(t)\|$, and as $\|C\|<\infty$, it’s evident that $\|z(t)\|\to 0\implies$ $\|e(t)\|\to 0$. So, it’s enough to only think about stability of $z(t)$. From (5), (40) and (41), the auxiliary system can be expressed as

A convenient way to solve this problem is to first transform the system (43) into a suitable canonical form. In order that, partition (43) as

where $B_{1}\in\mathbb{R}^{(n-m)\times{m}}$ and $B_{2}\in\mathbb{R}^{m\times{m}}$, and $\det B_{2}\neq 0$. Define the following orthogonal transformation

where $\mathbb{I}_{n-m}$ is a $(n-m)\times(n-m)$ indentity matrix, now define the new coordinate of transformation as

Therefore, (43) can be written with the help of above transformation in the regular form as

where $\eta\in\mathbb{R}^{n-m}$ and $\xi\in\mathbb{R}^{m}$. In order to achieve robustness against matched uncertainty, we design STA for the (60). Since STA is a first order control algorithm, so a hypersurface is designed such that the control appears in its first derivative. This technique achieves asymptotic stability of the system once this hypersurface is reached in finite time.

We define the sliding variable as

where $K\in\mathbb{R}^{m\times(n-m)}$ is chosen by some appropriate design procedure and $\sigma^{\top}=\begin{bmatrix}\sigma_{1}&\dotsb&\sigma_{m}\end{bmatrix}$. Further, define a linear change of coordinates by

If this transformation $T_{2}$ represents the new coordinates as

then dynamics in the new coordinates is

where

Design the control input $v(t)$ as

Substituting the control input in (73) the system equation becomes

where the components of $v^{\prime}(t)$ are

where $k_{1},k_{2}$ are the scalar gains. Let $L_{i}:=\Omega_{i}+w_{i}$ and substituting (91) in (87) then the closed loop system

where the vector $\lfloor\sigma\rceil^{\frac{1}{2}}=\begin{bmatrix}\lfloor\sigma_{1}\rceil^{\frac{1}{2}}&\dotsb&\lfloor\sigma_{m}\rceil^{\frac{1}{2}}\end{bmatrix}^{\top}$ with each component $\lfloor\sigma_{i}\rceil^{\frac{1}{2}}=|\sigma_{i}|^{\frac{1}{2}}\text{sign}(\sigma_{i})$ for $i=1,\dotsc,m$ and $\text{Sign}(\sigma)=\begin{bmatrix}\text{sign}(\sigma_{1})&\dotsb&\text{sign}(\sigma_{m})\end{bmatrix}^{\top}$. The closed loop dynamics (92)–(94) consist of a set of differential inclusion and the solution to this are understood in sense of Filippov [40].

Proof : For any $i^{\text{th}}$ input, the dynamics (93)-(94) can be rewritten as

To show (95) and (96) finite time stable, the Lyapunov function can be chosen as $V_{i}(\Psi)=\Psi_{i}^{\top}P_{i}\Psi_{i}$ where $\Psi_{i}^{\top}=\left[|\sigma_{i}|^{\frac{1}{2}}\text{sign}(\sigma_{i})~{}~{}L_{i}\right]$, then the trajectories of the system (95)-(96) will converge to origin in a finite time smaller than $t_{i}$ [41]

where

for positive and symmetric definite matrices $P_{i}$ and $Q_{i}$. Here the gains $k_{ji}$ for $j=1,2$ can be selected to achieve $\sigma_{i}$ and $L_{i}$ zero in some finite time.

Select the gains $k_{1}$ and $k_{2}$ as $k_{1}=\min_{i}|k_{1i}|$ and $k_{2}=\max_{i}|k_{2i}|$. The gains $k_{1}$ and $k_{2}$ are enough to bring $\sigma\equiv 0$ in finite time. Hence from (92), the reduced order dynamics or zero dynamics of the system with respect to $\sigma$ becomes

The pair $(A_{11},A_{12})$ is controllable pair and $K$ can be designed in such way that $\text{spec}\left(A_{11}+A_{12}K\right)\in\mathbb{C}^{-}$ i.e (97) becomes globally asymptotically stable. This ends the proof. $\blacksquare$

Proof : From Theorem 1, it is shown that the sliding surface $\sigma^{\top}=[\sigma_{1}\dotsb\sigma_{m}]$ goes to zero in finite time. Using (97), it can be concluded that

Now using (64) and (98), it yields

Then, it follows immediately from (49), (98) and (99) that

Recalling relationship between $e(t)$ and $z(t)$, i.e. $e(t)=Cz(t)$, we obtain that the tracking error $e(t)$ also decreases asymptotically to zero. $\blacksquare$

The real time implementation of the proposed control law for magnetic levitation system is developed in the following sections.

A simplified schematic diagram of the MagLev system is shown in Fig.1. The system is mainly computer controlled and user’s specific control law can be implemented and applied on the system in realtime using MATLAB/SIMULINK environment through a PCI 1711 AD card.

Our main task here is vertical position control of the levitated ball by application of a suitable current i(t) by means of applying a voltage v(t) in the coil. R and L represents the resistance and inductance of the coil, respectively. Here $x_{\text{ml}}(t)$ denotes the distance by which the coil magnet and the ball’s center are separated. The nonlinear force applied by the magnet is $\mathcal{K}\frac{i^{2}}{x^{2}_{\text{ml}}}$, where $\mathcal{K}$ is a positive constant. The differential equation capturing dynamics of the system is of the form

where $m$ denotes mass of the levitated ferromagnetic ball, $g$ is the gravitational acceleration and ${\mathcal{K}}_{1}$ is a proportionality constant [42] which depends upon the input voltage and coil current. To design the proposed robust model following controller, the model is linearized around an equilibrium point $(x_{0},i_{0})$ and the transfer function of the system is found as

where ${\mathcal{K}}_{i}=\frac{2mg}{i_{0}}$, $\mathcal{K}_{x}=-\frac{2mg}{x_{0}}$. The ball position $x_{\text{ml}}$ is measured by the IR sensor in voltage $x_{v}$ and they are linearly related with a proportionality constant $\mathcal{K}_{2}$. Finally with $x_{v}$, the above transfer function can be rewritten as

The nominal system parameters are shown in Table I. From the open loop transfer function it is clear that the system is severely unstable.

In state space representation (104) can be written as

From (105), it is clear that the MagLev is a second order electro-mechanical system whose relative degree is two [43].

In this section the proposed robust tracking and model following control strategy is designed for the MagLev system which is affected by model uncertainties and external disturbances, then (105) can be rewritten as

where the lumped disturbance applied to the plant is chosen as $w(t):=5\sin(t)$.

The task of the robust model following controller is to ensure the asymptotic convergence of tracking error between model and the plant output to zero. The essential part for the controller design is to select a suitable model which the plant needs to follow. The selection of a suitable model for the MagLev system is discussed in the following subsection.