Implicit discrete-time twisting controller without numerical chattering: Analysis and experimental results
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 of a time interval, with the explicit discretization, at the time instant tk, the argument of the signum function is the value of the sliding variable at tk, whereas with the implicit discretization it is the value at . 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 . The control value changes at time instants tk, defined as for all with . The scalar h is called the sampling period. Let and for all . The tilded variants and denote variables used in the controller. Let sgn be the classical single-valued signum function: for all and . Definition 1 Multivalued signum function Let . The multivalued signum function is defined as: If , then the vector-valued signum function is defined as .
Section snippets
Continuous-time twisting
The twisting algorithm was one of the first second-order sliding mode controllers presented in the literature Levant (1993). It requires the control input u to be of relative degree 2 with respect to the sliding variable σ, that iswith the following bounds: for all , The control law for the twisting controller isand with the conditionsthe state of the closed-loop system
System dynamics, actuators and sensors
We start with a description of the physical system, actuators and sensors as shown in Fig. 1. The electropneumatic system of the IRCCyN lab (Ecole Centrale de Nantes, France), depicted in Fig. 2, has two actuators. On the left-hand side, there is a double acting electropneumatic actuator (the “main” one) controlled by two servodistributors and composed of two chambers denoted P and N. The piston diameter is 80 mm and the rod diameter is 25 mm. With a source pressure equal to 7 bars, the maximum
Experimental results
This section is devoted to the analysis of the experimental results obtained on the electropneumatic setup. Recall that the control objective is to make the position of the piston track a sinusoidal trajectory. In the following, the desired trajectory is The controller was implemented as a Simulink model and then transferred onto a DS1005 dSpace board. We were able to get results with the sampling period h in the range [3, 100] ms and with the gain G in the range
Parameters selection
We mentioned at the beginning of Section 4 that the tuning of the sliding surface parameter α and of the two filtered differentiator constants (τv and τa) is important and may drastically affects the closed-loop behavior. Let us motivate the necessity of tuning α by looking at Fig. 14 to see how the value of α yielding the smallest average tracking error varies with the sampling period. With our experimental data, the selected values of α span from 25 for h=3 ms to 6500 for h=100 ms. With an
Comparison to the classical first-order sliding mode controller
Let us present some results with an implicit equivalent control based sliding mode controller (ECB-SMC) instead of a twisting controller. For a comparison between explicitly and implicitly discretized controller for the ECB-SMC, see (Wang et al., 2015), where it is shown that the implicit controller gives much better results than the explicit one. The implicit controller in the first-order sliding mode case has the following structure: The relative degree between the output y and
Conclusion
In this article we presented the results of a study of two discrete-time twisting controllers: the implicit and the explicit one. Extensive experiments were conducted in the context of a position tracking problem. The analysis of the data reveals that on this electropneumatic setup, the implicit twisting controller outperforms the explicit one on 3 criteria: the tracking error and both the input and output chattering. Despite the complexity of the control loop arising from the high relative
Acknowledgement
The authors acknowledge the support of the ANR Grant CHASLIM (ANR-11-BS03-0007).
References (19)
- et al.
Implicit Euler numerical scheme and chattering-free implementation of sliding mode systems
Systems and Control Letters
(2010) Robust exact differentiation via sliding mode technique
Automatica
(1998)- et al.
Lyapunov function design for finite-time convergence analysis“Twisting” controller for second-order sliding mode realization
Automatica
(2009) - et al.
A novel adaptive-gain supertwisting sliding mode controllerMethodology and application
Automatica
(2012) - et al.
Pneumatic actuator controlSolution based on adaptive twisting and experimentation
Control Engineering Practice
(2013) - et al.
Chattering-free digital sliding-mode control with state observer and disturbance rejection
IEEE Transactions on Automatic Control
(2012) - Acary, V., Brémond, B., Huber, O., & Pérignon, F. (2014). An introduction to Siconos. Rapport Technique RT-0340, INRIA....
- et al.
A pivotal method for affine variational inequalities
Mathematics of Operations Research
(1996) - Cottle, R. W., Pang, J. -S., & Stone, R. E. (2009). The linear Complementarity problem. In Number 60 in classics in...
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