Control design for a class of multivariable nonlinear system with uncertain control direction: A laser cladding case study
Introduction
Control of nonlinear systems under uncertainties is an active area of research with several open problems. Adaptive control has been widely used to overcome parameter uncertainty. Recently, strategies such as neural networks [22], [14], [29] to approximate and compensate perturbations, adaptive fuzzy approaches [31], [30], [33], boundary control approaches, fault-tolerant controls [15], [20], as well as uncertainty observer-based control [34], [35], have been successfully applied to deal with uncertain systems. For uncertain linear systems some interesting results have been reported, for instance, see [9], [19] where linear matrix inequalities are used to tackle the problem.
The case of multiplicative uncertainties is especially challenging. Particularly, for multiple-input systems, it implies the presence of uncertainty in the control direction [3]. Several works devoted to the consensus problem have proposed solutions for single-input systems. For example, in [36] a solution for high order uncertain nonlinear strict-feedback systems is proposed. Nussbaum-type functions are used to deal with the uncertain control direction and both transient and steady performances are considered. In [37], the authors proposed new distributed control algorithms by using classical Nussbaum-type functions for nonlinear multiagent systems. Moreover, in [38], a solution based on adaptive control for agents of first-order, linearly parameterized with the unknown sign of the high-frequency gain is proposed.
The aforementioned solutions for the consensus problem consider multiple systems, all of them with a single input, i.e., . Their results are based on the Nussbaum-type functions, which ensure that the single output can be steered to zero [25]. An alternative PI-based approach with better robustness in presence of unmodelled dynamics is proposed in [26], [39]. In [39] a new class of distributed control algorithms for single and double integrator agents is proposed. They consider the sign of the unknown high-frequency gain. However, such control proposals consider single-input systems. This is not the case in our problem, where we have a system with multiple inputs, i.e., with .
For multiple input-multiple output (MIMO) systems, robust model predictive control (MPC) design has been applied for uncertain linear systems in presence of both multiplicative and additive disturbances [18], [5]. For nonlinear MIMO systems, the solutions proposed in [1], [2], [11] address the problem using sliding modes and simplex methods to ensure stability.
An alternative control proposal for nonlinear MIMO systems with uncertain control direction and additive uncertainty is presented in the current work. The presented controller does not require to estimate the uncertainties. It is worth remarking that the control matrix is considered uncertain rather than unknown, in contrast to single-input systems where the unknown high-frequency gain problem has been studied as mentioned before. We present a pair of conditions, ensuring the stability under the proposed control scheme, that relax some conditions presented in previous works, such as [1], [2], [11]. An additive manufacturing (AM) process called Laser Cladding (LC) is used as a case study.
AM technologies have become a new paradigm for the design and manufacturing of small-volume, high-added value, and complex geometric components. The LC process is multivariable, highly non-linear, and has several scientific and technological challenges concerning the modeling and monitoring of the laser-matter interactions, as well as the control applied to the process in real-time [28].
During the LC process, a feedstock material either in powder or wire form is melted, while it is being deposited [13] through the assistance of a focused heat source, creating a region called the melt pool that becomes a track deposited onto a defined substrate (Fig. 1 (a)). Control based multi-physics models for LC have been proposed before, e.g., [41], [10], [40]. Nevertheless, the most accurate multi-physics models are not suitable to be used for control design goals due to their highly complex nonlinear dynamics of the process. Controlling the melt pool features is very important as it is well-known that the final characteristics of the manufactured part are directly related to them, as well as to the thermal history of the solidified layers during the build-up.
Control strategies for the LC process have considered the laser power, scan velocity, and powder flow rate as the input variables, and as outputs the desired track geometry [10], as well as the melt pool temperature [23] and its cooling rate [12]. Most of the reported control schemes for LC are single-input single-output (SISO) schemes. For instance, the ones using laser power for controlling the track geometry. [41] reported the design of a sliding mode control, which was implemented to control the track height under uncertainties. In [24] a proportional-integral (PI) scheme was designed to control the melt pool width through the laser power. Another related work is reported by Liu et al. [21], in which a discrete-time MPC was proposed to control melt pool size (area), changing the voltage, which changes the laser power.
Other works using the SISO approach address the melt pool temperature control. Salehi and Brandt [27] reported a proportional integral derivative (PID) control scheme with static gains. The authors use laser power as the control signal and a pyrometer to measure the melt pool temperature. Moreover, [6] showed the effect of controlling the cooling and heating rates on reducing the crack generation through a thermal shock. They use a thermocouple attached to the substrate to monitor the melt pool temperature during the processing.
The development of MIMO schemes that simultaneously control the geometry and temperature of the track is necessary for guaranteeing the quality of the LC process. Nevertheless, only a few works using this approach have been reported so far, see for instance, [8], [43]. In [8] a predictive control with two inputs (laser power and scan velocity) and two outputs (height and temperature of the track) is proposed. The authors showed practical stability considering only additive uncertainties. To the best of our knowledge, there is a lack of research focused on the development of robust MIMO controllers for the LC based on formal stability analysis and considering both additive and multiplicative uncertainties.
In this work, a robust control scheme for a MIMO nonlinear system with uncertainties is proposed and applied to the LC process. The LC MIMO model reported by Wang et al. [40] was adopted. This model is based on physical principles, i.e., mass and energy balance. However, the model preserves simplicity as a result of being developed specifically for control purposes and was experimentally validated for metallic alloys, Inconel 718 (IN718) and Ti6Al4V. Additionally, this model was validated in our laboratory and the results are presented in Section 5.
The proposed control scheme is composed of a feedback term [17] and a sliding mode term, the latter is used with a boundary layer to avoid the chattering effect [32]. The theoretical results for our controller were also tested via simulation considering the deposition of Ti6Al4V and IN718 under different uncertainties. It is important to remark that these alloys are highly needed in many industrial applications, e.g., the aerospace and medical industries [42], [16].
In summary, the main contributions of this work are: the proposal of a simple solution for the control of a class of nonlinear MIMO systems with additive uncertainty and uncertain control direction. Conditions that ensure the controllability of the system, based on the eigenvalues of the uncertain distribution matrix, are given. The theoretical results were verified via simulations considering the LC process. The proposed controller is easy to implement. Finally, the same control scheme has been successfully applied to the two models corresponding to the two different materials. A preliminary work has been presented by our group in [4]. In that work, it was shown the results of a robust control scheme, where the control gains were obtained empirically by trial and error.
This paper is presented as follows: Section 2 describes the control objective. Section 3 shows the control design and stability analysis. The general model for the case is introduced in Section 4. In Section 5, an experimental model validation for IN718 is discussed. The simulation results are presented in Section 6. Finally, Section 7 gives the concluding remarks.
NOTE: Throughout the manuscript, the notation stands for the Euclidean norm if a is a vector, and the spectral norm if a is a matrix. refers to the minimum real part of the eigenvalues of matrix , i.e., . A superscript is used to indicate transpose.
Section snippets
Problem statement
Considering the class of uncertain nonlinear MIMO systems described by where , are the states and the control input, respectively. and are unknown functions. represents the general perturbations. In this work, perturbations are considered to encompass both parametric uncertainty and external disturbances.
The objective is to design a controller for the regulation of the system states, , described by Eq. (1). Such regulation task will consider a
Robust controller design
The proposed controller relies on a feedback linearization approach and can be seen as composed of two parts: The first one is precisely for feedback linearization, , and includes a linear state feedback term, the second part introduces a sliding mode control, , using the boundary layer technique. The feedback linearization controller is aimed at compensating the known system dynamics, which are also referred to as nominal system dynamics, whereas the sliding mode control term is aimed at
Case study: LC model description and assumptions
The LC process (see Fig. 1 (a)) involves complex physical phenomena such as laser-matter interactions, heat transfer, fluid flow, rapid solidification, among others [7].
For the case study, the mathematical MIMO model proposed by [40] is considered. The model includes the mass and energy balances assuming a homogeneous domain with two inputs (the laser power and scan velocity) and two outputs (the temperature and height of the tracks). The melt pool shape is assumed as half of an ellipsoid (
Model validation
Some experiments were carried out to validate the model described in Section 4 and verify that fits the Inputs-and-Outputs of our LC system.
Results: controller design for Ti6Al4V and IN718
Considering the case study of the LC nonlinear MIMO model presented in Section 4 expressed as the general equation by Eq. (1). For Ti6Al4V, is:
For IN718, due to the different approximation for , see Eq. (25), the following is obtained:
The function is the same for both Ti6A14V and IN718:
Conclusions
A controller for a class of MIMO nonlinear systems with uncertainties is proposed. This control problem is particularly challenging because the considered uncertainties imply an uncertain control direction. The stability analysis was made and conditions for the selection of gains are given in order to ensure the rejection of perturbations. The robust controller was applied to additive manufacturing, specifically the LC process, considering Ti6Al4V and IN718 superalloys as the feedstock
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
A. Bejarano-Rincón’s work was supported by CONACYT (the National Council for Science and Technology, México). The authors are additionally grateful for the financial support from the National Problems Project, grant no. 2015-01-863, and CONACYT-AEM, grant no. 275781. Also, the author thanks the financial support provided by CONACYT through the Programs Fordecyt (projects 297265 and 296384). Additionally, thanks are due to the CONACYT Consortium in Additive Manufacturing (CONMAD) for the use of
References (43)
- et al.
Control of heating and cooling for direct laser deposition repair of cast iron components
Proceedings of the International Symposium on Flexible Automation (ISFA)
(2016) - et al.
Robust static output feedback control design for linear systems with polytopic uncertainties
Syst. Control Lett.
(2015) - et al.
Hardware-in-the-loop control of additive manufacturing processes using temperature feedback
J. Laser Appl.
(2016) - et al.
Modeling and trajectory tracking control for flapping-wing micro aerial vehicles
IEEE/CAA J. Autom. Sin.
(2021) Nonlinear systems
(2002)- et al.
Robust control for networked control systems with randomly occurring uncertainties: observer-based case
ISA Trans.
(2018) - et al.
Modeling and adaptive control for a spatial flexible spacecraft with unknown actuator failures
Sci. China Inf. Sci.
(2020) - et al.
A feedforward controller for tuning laser cladding melt pool geometry in real time
Int. J. Adv. Manuf. Technol.
(2017) - et al.
Nonlinear PI control of uncertain systems: an alternative to parameter adaptation
Syst. Control Lett.
(2002) - et al.
Melt pool temperature control using LabVIEW in Nd:YAG laser blown powder cladding process
Int. J. Adv. Manuf. Technol.
(2006)
Adaptive cooperative control with guaranteed convergence in time-varying networks of nonlinear dynamical systems
IEEE Trans. Cybern.
Physics-based multivariable modeling and feedback linearization control of melt-pool geometry and temperature in directed energy deposition
J. Manuf. Sci. Eng
Simplex methods for nonlinear uncertain sliding-mode control
IEEE Trans. Autom. Control
Multi-input sliding mode control of nonlinear uncertain non-affine systems with mono-directional actuation
IEEE Trans. Autom. Control
Vector method of design of sliding motion and simplex algorithms
Avtomatika I Telemekhanika
Design of a controller of melt pool temperature and track height for Laser Cladding
Proceedings of the 3rd International Conference on Aeronautics
Robust controller for uncertain parameters systems
ISA Trans.
Laser Additive Manufacturing: Materials, Design, Technologies, and Applications
Robust multivariable predictive control for laser-aided powder deposition processes
J. Frankl. Inst.
Combined backstepping and HOSM control design for a class of nonlinear MIMO systems
Int. J. Robust. Nonlinear Control
Real-time control of microstructure in laser additive manufacturing
Int. J. Adv. Manuf. Technol.
Cited by (5)
Evolutionary design of marginally robust multivariable PID controller
2023, Engineering Applications of Artificial IntelligenceAn H<inf>∞</inf> Robust Decentralized PID Controller Design for Multi-Variable Chemical Processes Using Loop Shaping Technique
2023, Arabian Journal for Science and Engineering