Brief paperOutput feedback Continuous Twisting Algorithm☆
Introduction
Sliding Mode Control is a useful technique to design controllersand observers for uncertain systems, providing robustness, and even insensibility, against some sort of disturbances Levant (2003), Utkin et al. (2009). Consider, for example, the disturbed double integrator where is the state, is the available output, is the control input, is a Lipschitz disturbance, and is a bounded noise signal. For , first-order sliding mode controllers (Utkin et al., 2009) are able to stabilize exponentially the origin of (1) by confining in finite-time the system dynamics in a desired sliding surface. However, this is done by using a discontinuous control signal causing the (generally undesirable) phenomenon of chattering, and the disturbance must be bounded. To substitute the discontinuous control signal with a continuous one, Super-Twisting controller Levant (1993), Levant (1998) was suggested to stabilize exponentially the origin of (1) by reducing in finite-time the system dynamics in a desired sliding surface for the case when the disturbance is Lipschitz. Continuous controllers, such as those in Bernuau, Perruquetti, Efimov, and Moulay (2015) and Bhat and Bernstein (1998), achieve finite-time stability but are not able to reject Lipschitz disturbances .
Higher-order sliding mode (HOSM) controllers can be used to stabilize in finite-time the origin of (1). For example, Twisting, Terminal, Sub-Optimal and Quasi-Continuous controllers Bartolini et al. (1997), Levant (1993), Levant (2005b), Man et al. (1994) ensure finite-time stability of the system’s origin when is bounded, however, they also produce the discontinuous control action. The continuous HOSM controllers Edwards and Shtessel (2016), Kamal et al. (2016), Laghrouche et al. (2017), Torres-González et al. (2017), achieve finite-time stability of despite Lipschitz disturbances . Remarkably, this is attained by means of a continuous control signal. These features make continuous HOSM very appealing, however, the performance of such controllers should be analysed considering additional issues present in real applications. For example, under discretization, finite-time convergence to the origin cannot be obtained. Nonetheless, these controllers exhibit, under discretization, an accuracy in steady state of order three Kamal et al. (2016), Laghrouche et al. (2017), Torres-González et al. (2017). This is a guarantee of large steady state error reduction when the discretization step is reduced Levant (1993), Levant (2005a). Other important issues is that the measurement of can be unavailable and the measurement of can be noisy. In Chalanga, Kamal, Fridman, Bandyopadhyay, and Moreno (2016) the problem of output feedback for (1) is studied using the Super-Twisting controller by designing a sliding variable. It is shown that the Super-Twisting based observer cannot be applied for such a control strategy and a second order Robust Exact Differentiator (RED) (Levant, 2003) must be used to realize the controller’s properties. Therefore, the problem of output feedback control of the continuous HOSM deserves special attention. To solve such a problem, the following options can be considered:
(a) The uniform differentiator proposed in Angulo, Moreno, and Fridman (2013) and Cruz-Zavala, Moreno, and Fridman (2011) is able to compute the derivative of in a fixed time (this time does not depend on the initial error). Then, it is possible to maintain the control off and turn it on after the uniform differentiator has converged. The disadvantage with this strategy is that the convergence time for the differentiator is usually overestimated, producing large time transients.
(b) The proposal in Angulo, Fridman, and Levant (2012) consists in using the RED to estimate the derivatives of the output. In this strategy, the controller must be maintained off until an on-line algorithm detects that the differentiator has converged.
An additional problem is that, for the noise analysis, the results of ISS robustness for homogeneous systems are not useful (in general) for the case of controllers involving discontinuities, see e.g. Bernuau, Efimov, Perruquetti, and Polyakov (2014) and Perruquetti (2018).
This paper devoted to the output feedback control for Continuous Twisting Algorithm (CTA) (Torres-González et al., 2017) that is able, in absence of noise, to stabilize in finite-time the origin of (1), compensating exactly Lipschitz disturbances ensuring accuracy of order three with respect to the output during discretization (Torres-González et al., 2017). The contributions can be summarized as follows.
(1) The robustness properties of the CTA considering noise in the states are studied through a Lyapunov function (LF).
(2) For (1), two output feedback schemes based on the first and second order REDs are considered. The effect caused by a noisy output is investigated, and a separation principle based on input-to-state stability (ISS) properties of CTA and RED is provided.
(3) For the case of the second order RED a LF for the whole closed loop (CL) is designed. It is verified that in this case the conservation of homogeneity allows the output feedback scheme to preserve the robustness and accuracy properties of the state feedback CTA.
(4) It is shown how the scheme CTA–secondorder RED can be applied for output feedback control for the class of second order nonlinear systems that can be written as where are as above, and is a Lipschitz function, for which we know a model and a Lipschitz constant .
(5) Numerical simulations are performed illustrating that the usage of second order RED allows to realize the third order accuracy predicted for CTA for output based CTA.
Paper organization: In Section 2 we recall the state feedback CTA and two RED observers. In this section we also provide the result on the robustness of the CTA in presence of noise. The output feedback controllers are stated in Section 3. The application to nonlinear systems with drift term is presented in Section 4. A numerical example is shown in Section 5. Some concluding remarks are given in Section 6.
Notation: is the set of real numbers, and (analogously for ). For , denotes the Euclidean norm. denotes the matrix for some . For and , .
Section snippets
The state feedback CTA and two observers
The state feedback CTA, given by is able to drive the states of (1) to zero in finite-time rejecting disturbances with bounded derivative (Torres-González et al., 2017). Although the second equation in (3) is discontinuous, this is integrated through , allowing the control signal to be continuous. By defining the virtual state the CL (1), (3) is given by The third
Output feedback CTA schemes
If we replace in (3) the states , by the measured output and the estimation of , respectively, we obtain Hereafter we refer as CTA–1RED to the connection of (14) with (8), and as CTA–2RED to the connection of (14) with (10).
Theorem 6 Suppose that for , , and , vector satisfies Theorem 1 , and vector satisfies Theorem 3 , then:
for , the trajectories of (1) in CL with CTA–1RED
Output feedback CTA for a class of nonlinear systems
In this section we apply the output feedback scheme CTA–2REDto (2). Suppose that, for the function in (2), we know a Lipschitz constant and a model such that , for some . Now we propose the controller where is the CTA given by (14) and is the observed state provided by (10). To enunciate the stability result for the CL we require the following. Define the sets , , for some constants .
Numerical example
In this simulation example we consider (1) with a disturbance . We choose a bounded disturbance to be able to compare both output feedback schemes. Table 1 shows some admissible gains for CTA (Torres-González et al., 2017) and for (10). Although not all the gains in such table were reported in Sanchez et al. (2017) they have been designed with the same procedure proposed there. The gains for (8) can be designed with the methods reported in Cruz-Zavala and Moreno (2016)
Conclusions
Two output feedback strategies were proposed for the Continuous Twisting Algorithm. The algorithms provide, by means of a continuous signal, finite time convergence to the origin despite Lipschitz (Lipschitz and bounded for the first order RED) matched disturbances. Both output feedback schemes are robust in presence of noise in the measured output. It was shown that the scheme CTA–Second order RED can be used for output feedback control of a class of nonlinear systems, preserving the exact
Acknowledgement
The authors thank Víctor Torres-González who participated at the beginning of this research.
Tonametl Sanchez received his M. Eng. and Ph.D. in Electrical Engineering from National and Autonomous University of Mexico (UNAM), Mexico City, Mexico in 2012 and 2016, respectively. He was a postdoctoral researcher at Institute of Engineering of UNAM, and currently he is a postdoctoral researcher at INRIA-Lille, France. His research interests include Lyapunov function design methods, homogeneous control systems, time-delayed systems and High Order Sliding Mode control.
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Tonametl Sanchez received his M. Eng. and Ph.D. in Electrical Engineering from National and Autonomous University of Mexico (UNAM), Mexico City, Mexico in 2012 and 2016, respectively. He was a postdoctoral researcher at Institute of Engineering of UNAM, and currently he is a postdoctoral researcher at INRIA-Lille, France. His research interests include Lyapunov function design methods, homogeneous control systems, time-delayed systems and High Order Sliding Mode control.
Jaime A. Moreno was born in Colombia and he received his PhD degree (Summa cum Laude) in Electrical Engineering (Automatic Control) from the Helmut-Schmidt University in Hamburg, Germany in 1995. The Diploma-Degree in Electrical Engineering (Automatic Control) from the Universität zu Karlsruhe, Karlsruhe, Germany in 1990, and the Licentiate-Degree (with honors) in Electronic Engineering from the Universidad Pontificia Bolivariana, Medellin, Colombia in 1987. He is full Professor of Automatic Control and the Head of the Electrical and Computing Department at the Institute of Engineering from the National University of Mexico (UNAM), in Mexico City. He is a member of the Technical Board of IFAC and the author and editor of 8 books, 4 book chapters, 1 patent, and author and co-author of more than 300 papers in refereed journals and conference proceedings. His current research interests include robust and non-linear control with application to biochemical processes (wastewater treatment processes), the design of nonlinear observers and higher order sliding mode control.
Leonid M. Fridman received an M.S. degree in mathematics from Kuibyshev (Samara) State University, Samara, Russia, in 1976, a Ph.D. degree in applied mathematics from the Institute of Control Science, Moscow, Russia, in 1988, and a Dr. Sc. degree in control science from Moscow State University of Mathematics and Electronics, Moscow, Russia, in 1998. From 1976 to 1999, he was with the Department of Mathematics, Samara State Architecture and Civil Engineering University. From 2000 to 2002, he was with the Department of Postgraduate Study and Investigations at the Chihuahua Institute of Technology, Chihuahua, Mexico. In 2002, he joined the Department of Control Engineering and Robotics, Division of Electrical Engineering of Engineering Faculty at National Autonomous University of Mexico (UNAM), Mexico.
His research interests are Variable Structure Systems. He is currently a Chair of TC on Variable Structure and Sliding Mode Control of IEEE Control Systems Society. He is an author and editor of ten books and seventeen special issues devoted to the sliding mode control. He won a Scopus Prize for the best cited Mexican Scientists in Mathematics and Engineering 2010. He was working as an invited professor in more than 20 universities and research laboratories of Argentina, Australia, Austria, China, France, Germany, Italy, Israel, and Spain.
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The authors thank the financial support of CONACyT (Consejo Nacional de Ciencia y Tecnología): Projects 241171, 282013, and CVU 371652; PAPIIT-UNAM (Programa de Apoyo a Proyectos de Investigación e Innovación Tecnológica) IN113216, IN113614 and IN113617; Fondo de Colaboración del II-FI UNAM IISGBAS-100-2015. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Lorenzo Marconi under the direction of Editor Daniel Liberzon.