Implicit discrete-time twisting controller without numerical chattering: Analysis and experimental results

Highlights

• Sliding mode twisting controllers for an electropneumatic plant are implemented.

• Implicit and explicit discrete-time twisting controllers are considered.

• Better tracking performance is achieved with the implicit method.

• Drastic chattering reducing is witnessed with the implicit controller.

• Selection process for some control parameters is discussed.

Abstract

In this paper, we present an implementation of the sliding mode twisting controller on an electropneumatic plant for a tracking control problem. To this end, implicitly and explicitly discretized twisting controllers are considered. We discuss their structure, properties and implementations, as well as the experimental results. The analysis of the performance sustains the theoretical superiority of the implicitly discretized version, as shown in previous works. The main advantages of the implicit method are better tracking performance and drastic reduction in the input and output chattering. This is achieved without modifying the structure of the controller compared to its continuous-time version. The tracking error cannot be used as the sliding variable: it has a relative degree 3 w.r.t. the control input. The tuning of the sliding surface has well as some other parameters in the control loop was instrumental in achieving good performance. We detail the selection procedure of those parameters and their influence on the closed-loop behavior. Finally we also present some results with an implicitly discretized EBC-SMC controller.

Introduction

Implementation of control laws is almost exclusively done using microcontrollers. This implies that the controller is in discrete-time rather than in continuous-time. In sliding mode control, this can induce a degradation of the performance by contributing to the chattering phenomenon. We call this the numerical chattering. An intense activity over the last 30 years was devoted to the reduction of this numerical chattering, mainly for equivalent control based sliding mode control (ECB-SMC). In the early 90s, second order sliding mode control concept was introduced in Levant (1993) and sparked the development of a large wealth of literature. One of the first controllers of this kind was the twisting controller which features a discontinuous control action w.r.t. the sliding variables. However, to the best of our knowledge, few discrete-time versions of the twisting controller have been proposed. The substitution of the signum function by a saturation, common trick to reduce the chattering for first order SMC, has no straightforward extension to the twisting algorithm. It is then fair to assume that the explicit discretization was used to get a discrete-time twisting controller, like in Taleb, Levant, and Plestan (2013).

The other discretization method we consider is the implicit method. It has been used for a long time in the nonsmooth mechanics community, but it was not applied in control theory until very recently (Acary and Brogliato, 2010, Acary et al., 2012, Huber et al., 2013a, Huber et al., 2013b). The implicit discretization of the twisting controller was first studied in Acary et al. (2012). Roughly speaking, the difference between the explicit and implicit methods in our context is the following: given a partition $\left\{t_k\right\}$ of a time interval, with the explicit discretization, at the time instant t_k, the argument of the signum function is the value of the sliding variable at t_k, whereas with the implicit discretization it is the value at $t_{k+1}$. Despite its name and formulation, the implicitly discretized twisting controller is non-anticipative and induces a well-defined behavior, as we shall see in Section 2. Its main features are the drastic reduction of the output chattering and the reduction of the control input chattering, that is the control input is no more of the high frequency “bang-bang” type. In the discrete-time sliding regime, the control input is also insensitive to an increase of the gain. To simplify the nomenclature, we refer to the discrete-time twisting controller with an implicit (resp. explicit) discretization as the implicit (resp. explicit) twisting controller.

In the following, we present results from an implementation of both explicit and implicit twisting controllers on an electropneumatic plant. The control problem at hand is the tracking of a sinusoidal trajectory for the position of the end of the piston. The analysis of the gathered data supports the theoretically predicted reduction of the chattering claimed in Acary et al. (2012) and also the claim that the numerical chattering can be the main source of chattering, see Huber et al. (2013a). This highlights the importance of the discretization process which is unfortunately often overlooked both in the analysis and in the implementation.

The second part of the paper is dedicated to the choice of three parameters: the first one defines the sliding variable and the two others are constants for two filtered differentiators.

The influence of those parameters is only visible with the implicit controller. With an explicit one, the performance is not good enough to always see a change when their values change. It appears that with an implicit controller the differentiators become the weakest component in the control loop. Empirical data suggest that the three parameters have to be tuned simultaneously. To help with the tuning, we present the selection procedure that we used. We also analyze how the experimental tracking performance varies with the choice of the sliding surface. We hope that this presentation raises awareness for the importance of tuning to get the best possible performance for systems with similar setup.

In the remainder of this section, we introduce the notations. In Section 2 we briefly recall the twisting controller in continuous-time as well as in discrete-time. The experimental setup is presented in Section 3 as well as the control scheme. Then the experimental results are analyzed in Section 4. In Section 5, we deal with the tuning of some control parameters and the impact it has on the performance. In Section 6 an experimental comparison between the twisting and a classical first order SMC is proposed. Conclusions end the paper in Section 7.

Notations: The sliding variable is denoted by σ, it is supposed to be at least twice differentiable and Σ denotes ${\left(\sigma \dot{\sigma }\right)}^T$. The control value changes at time instants t_k, defined as $t_k≔t_0+\mathit{kh}$ for all $k\in \mathbb{N}$ with $t_0,h\in {\mathbb{R}}_{+}$. The scalar h is called the sampling period. Let ${\sigma }_k≔\sigma \left(t_k\right)$ and ${\dot{\sigma }}_k≔\dot{\sigma }\left(t_k\right)$ for all $k\in \mathbb{N}$. The tilded variants $\tilde{\sigma },\tilde{\dot{\sigma }}$ and $\tilde{\Sigma }$ denote variables used in the controller. Let sgn be the classical single-valued signum function: for all $x>0,\mathrm{sgn}\left(\mathrm{x}\right)=1,\mathrm{sgn}(-\mathrm{x})=-1$ and $\mathrm{sgn}\left(0\right)=0$.

Definition 1 Multivalued signum function

Let $x\in \mathbb{R}$. The multivalued signum function $\mathrm{Sgn}:\mathbb{R}\rightrightarrows [-1,1]$ is defined as:

$$\mathrm{Sgn}\left(\mathrm{x}\right)=\{\begin{array}{cc}\left\{1\right\}\hfill & x>0\hfill \\ \{-1\}\hfill & x<0\hfill \\ [-1,1]\hfill & x=0.\hfill \end{array}$$

If $x\in {\mathbb{R}}^n$, then the vector-valued signum function $\mathrm{Sgn}:{\mathbb{R}}^n\rightrightarrows {[-1,1]}^n$ is defined as $\mathrm{Sgn}\left(\mathrm{x}\right)≔{\left(\mathrm{Sgn}\right({\mathrm{x}}_1),\dots ,\mathrm{Sgn}({\mathrm{x}}_n\left)\right)}^T$.