Brief paperParameter space optimization towards integrated mechatronic design for uncertain systems with generalized feedback constraints☆
Introduction
Research and development for designing a mechatronic system, including both mechanical and control subsystems, is usually a sequential, iterative and time-consuming process (Bishop, 2005). The control specialists usually design and tune the controllers after the mechanical prototypes are fabricated. Such sequential design practice has been widely applied to design high-precision stages (Butler & de Hoon, 2013). However, the mechanical system and the control system are not optimally synthesized in this approach because interactions between them are not considered in the early design stage. As a result, it usually admits conservative design parameters, which downgrade the achievable performance of mechatronic systems (Butler, 2011, Geromel and Bernussou, 1982).
The integrated design approach for mechanical and control subsystems is developed by considering key design factors such as inertia, damping, resonant mode, sensor placement and coupling force (Affi et al., 2007, De Silva, 2004, Villarreal-Cervantes, 2017). For controller synthesis, optimal control theory has attracted considerable attention for many years due to its effectiveness. In Linear Quadratic Regulator (LQR) and problems, the optimal state feedback controller gain can be obtained by the use of algebraic Riccati equations (AREs)(Anderson & Moore, 1971). To employ such methods to integrated mechatronic design problems, the composite feedback gain matrix (CFGM) is derived through the augmentation of mechanical parameters and controller gains. However, the CFGM cannot be obtained by solving AREs. The first reason is that certain factors of design, such as disturbance, computational capacity, reconfigurability, ease of cabling and tuning, and so on, are application specific and they change between industrial environments(Bakule, 2008, Dolk et al., 2017). In some seminal works, the design of an optimal decentralized controller subjected to prescribed sparsity pattern has been carried out (Wang et al., 2018, Wang et al., 2019). Specially, for fully decentralized control systems in which measurement and control of subsystems are independent of each other, the off-diagonal elements in the CFGM are constrained to be zero. The second reason is that the mechanical components naturally introduce certain structural constraints to the CFGM (Ma et al., 2017, Ma, Chen, Kamaldi et al., 2018), such as stiffness and viscosity on the joints or contact surfaces (Tehrani et al., 2017, Zhu et al., 2016). Consequently, due to the specific control structure and the naturally imported constraints, two types of constraints are commonly encountered: (1) Some elements are zero in the CFGM; (2) Some elements in the CFGM are equal or opposite. Indeed, these constraints make the decentralized optimal control problem more challenging than its unconstrained centralized counterpart, and the constrained optimization problem becomes NP-hard (Fattahi and Lavaei, 2017, Tsitsiklis and Athans, 1985). Thus, it leaves an open problem on design of a linear full state feedback controller to optimize a pre-defined performance index under the above two types of constraints.
To deal with this issue, projection gradient methods are introduced, which convert the original optimization problem to an approximated equivalent counterpart, such that the Euclidean distance between the projection gradient in the constrained hyperplane and its unconstrained counterpart is minimized(Geromel & Bernussou, 1982). In Rosen (1960), the projection operator is computed analytically to speed up the optimization. In Zoutendijk (1966), the converted projection gradient problem is solved by means of linear programming. In Ma, Chen, Kamaldi et al. (2018) and Ma et al. (2017), linear–quadratic-based and -based constrained optimization algorithms are proposed accordingly, based on direct computation of the projection gradient matrix and line search of the optimal incremental step. It is worthwhile to mention that both the structural constraints and the closed-loop stability are preserved through the optimization process. However, these methods only find the local optimums (Geromel & Bernussou, 1982). Extensive achievements have been made on decentralized optimal controller synthesis based on convex reformulation (Bamieh and Voulgaris, 2005, De Castro and Paganini, 2002, Dvijotham et al., 2013, Fazelnia et al., 2017). On the basis of convex relaxations, the constraints on the controller can be added to the framework to obtain the suboptimal solution with performance guarantees, where an objective function is constructed to mathematically quantify the closeness of the given centralized and the decentralized control systems (Fattahi, Fazelnia, Lavaei, & Arcak, 2019). However, the model uncertainties are not explicitly handled.
Many researches reveal the robustness and the non-fragility issues (Keel & Bhattacharyya, 1997), but linear quadratic and robust non-fragile syntheses result in computationally expensive Linear Matrix Inequality (LMI) non-convex constraints (Famularo, Dorato, Abdallah, Haddad, & Jadbabaie, 2000). In recent researches, non-fragility notion is used to obtain sparse stabilizing state feedback controller in the vicinity of a given dense centralized state feedback controller (Lin, Fardad, & Jovanović, 2013). However, the drawbacks of this approach include the difficulty in catering for the equality constraints in the controller and the limited rate of sparsification (Bahavarnia, Somarakis, & Motee, 2017). Cooperative distributed model predictive control has been proven to be effective in constrained dynamical systems too (Giselsson and Rantzer, 2014, Stewart et al., 2010). Remarkably, the objective function is formulated using the optimizer of a robust optimization problem (Darivianakis, Fattahi, Lygeros, & Lavaei, 2018). Robust decentralized controllers have shown their applicability in multi-variable large-scale network systems with model uncertainties and design constraints (Hou et al., 2016, Pipelzadeh et al., 2013, Yao et al., 2014). Moreover, robust decentralized controller can be synthesized in the extended parameter space (Barmish, 1985), either in regulation (Bernussou et al., 1989, Geromel et al., 1994, Geromel et al., 1991, Geromel et al., 1992) or tracking applications (Ma, Chen, Liang et al., 2019). Such method has been extended to robust output feedback controller synthesis (Geromel et al., 1993, Peres et al., 1993). However, for the CFGM under generalized structural constraints, the controller synthesis technique is yet to be developed, which is highly desired for integrated mechatronic design problems under model uncertainties.
In this work, we aim to convert the integrated mechatronic design problem for an uncertain system to an optimization problem such that the -norm upper bound of the closed-loop system transmittance from the exogenous disturbance to the regulated variables is minimized over the convex-bounded uncertain domain. The original contributions of this paper are as follows: (1) Development of a series of algorithms to handle the structural constraints in the CFGM, where the non-decentralized constrained gain optimization problem is converted to its decentralized unconstrained counterpart as illustrated in Theorem 1. (2) Development of a comprehensive method to project all the CFGMs that satisfy the structural constraints onto a series of convex and non-convex hyperplanes as presented in Theorem 2, Theorem 3, Theorem 4. Subsequently, an equivalent optimization problem is formed to minimize the -norm’s upper bound of the closed-loop system transmittance from the exogenous disturbance to the regulated variables for both precisely known and uncertain dynamical systems as summarized in Theorem 5, Theorem 6. (3) The cutting-plane generation techniques are illustrated in Theorem 7, Theorem 8 against scenarios of constraint violation in the projected parameter matrix.
The paper is organized as follows: Section 2 presents the parameter space optimization for uncertain systems with structural constraints. In order to solve this kind of optimization problem under both convex and non-convex constraints, Section 3 proposes cutting-plane-based numerical procedures to obtain the global optimum. Section 4 presents an illustrative example based on a flexure-linked biaxial gantry stage. Lastly, Section 5 concludes the paper with salient points.
Section snippets
Problem formulation
Consider a linear time invariant system with , is the state vector, is the control input vector, is the exogenous disturbance input, is the controlled output vector, is the feedback gain matrix. In practice, can be the CFGM with structural constraints, which can be formed by augmenting all the control and mechanical parameters that affect generalized control efforts. Matrices , , , , are assumed to
Numerical procedures
The matrix that admits the optimal gain matrix lies in , which is the intersection of convex set and non-convex set , but is not explicitly known. To overcome the difficulty, linear programming is conducted within a pre-defined polytope . If is located outside , cutting plane technique is used and then a half space is constructed for separation purpose and update of the polytope. Utilizing “Optimization-Checking-Cutting” strategy, the following numerical procedures are
Problem formulation
A flexure-linked biaxial gantry stage is used as an example in this section, where the schematic of simplified reduced-order model is demonstrated in Ma et al. (2017). It is known that maintaining good tracking performance of two carriages via either controller or mechanical design is necessary, but the drawback is the induced resonant modes due to flexure-based mechanism, causing the end-effector being sensitive to chattering control signals. In this work, , , and represent the control
Conclusion
In this work, a systematic optimization method is proposed for a class of integrated mechatronic design problems, where the structural constraints with some elements either being zero or with equal or opposite relationships are imposed on the CFGM. As supported by relevant theoretical results, various mechanical and controller constraints, as well as model uncertainties on the state space are converted to the sets on a projected parameter space, so that the -norm upper bound of the
Acknowledgments
The authors would like to thank Prof José C. Geromel for his sharing and discussion. S. -L. Chen would like to acknowledge the support from National Natural Science Foundation of China (Funding No. 51875554) for co-authorship of this work.
Jun Ma received the B.Eng. (1st Class Hons.) degree in Electrical and Electronic Engineering from the Nanyang Technological University, Singapore, in 2014, and the Ph.D. degree in Electrical and Computer Engineering from the National University of Singapore, Singapore, in 2018. From 2018 to 2019, he was a Research Fellow with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. He is currently a Research Associate with the Department of Electronic
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Jun Ma received the B.Eng. (1st Class Hons.) degree in Electrical and Electronic Engineering from the Nanyang Technological University, Singapore, in 2014, and the Ph.D. degree in Electrical and Computer Engineering from the National University of Singapore, Singapore, in 2018. From 2018 to 2019, he was a Research Fellow with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. He is currently a Research Associate with the Department of Electronic and Electrical Engineering, University College London, London, U.K. His research interests include control and optimization, precision mechatronics, robotics, and medical technology. He was a recipient of the Singapore Commonwealth Fellowship in Innovation.
Si-Lu Chen received the B.Eng. and the Ph.D. degrees in Electrical Engineering from the National University of Singapore (NUS), in 2005 and 2010 respectively. From 2010 to 2011, he was with the Manufacturing Integration Technology Ltd, a Singapore-based semiconductor machine designer, as a senior engineer on motion control. From 2011 to 2017, he was a scientist in the Mechatronics group, Singapore Institute of Manufacturing Technology (SIMTech), Agency for Science, Technology and Research (A*STAR). During this period, he also acted as co-PI of the SIMTech-NUS Joint Lab on Precision Motion Systems, adjunct assistant professor of NUS, and PhD co-advisor for A*STAR Graduate School. Since 2017, he has been with the Ningbo Institute of Material Technology and Engineering, Chinese Academy of Sciences, as a professor. His current research interests include design and optimization of high-speed motion control systems, and beyond-rigid-body control for compliant light-weight systems. He is currently serving as technical reviewers for IEEE/ASME Transactions of Mechatronics, IFAC Journal of Mechatronics, and ISA Transactions.
Chek Sing Teo received the Ph.D. degree in Electrical Engineering from the National University of Singapore in 2008, under the Agency for Science Technology and Research (A*STAR) Scholarship Scheme, working on ”Accuracy Enhancement for High Precision Gantry Stage”. His research interests are in the application of advanced control techniques to precision mechatronic system and instrumentation, to enhance performance in motion control and measurement. His current work includes using mechatronics stiffness to reduce jerk reaction in high speed motion stage and sensor placement for adaptronics. He is currently working in the Singapore Institute of Manufacturing Technology (SIMTech) leading the Precision Mechatronics Team within the Mechatronics Group, as well as the co-Director of the SIMTech-NUS Joint Lab on Precision Motion Systems.
Arthur Tay received the B.Eng. (1st Class Hons.) and the Ph.D. degrees in Electrical Engineering from the National University of Singapore, Singapore, in 1995 and 1998, respectively. He was a visiting scholar with the Information System Laboratory at Stanford University, Stanford, CA, from 1998 to 2000. He is currently an Associate Professor with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. His research interests include applications of mathematical system science tools in healthcare, semiconductor manufacturing, and process control.
Abdullah Al Mamun received the B.Tech. (Hons.) degree in Electronics and Electrical Communication Engineering from the Indian Institute of Technology, Kharagpur, India, in 1985, and the Ph.D. degree in Electrical Engineering from the National University of Singapore, Singapore, in 1997. He is currently an Associate Professor with the National University of Singapore. His research interests include precision mechatronics, servomechanism in data storage devices, intelligent control, and mobile robots.
Kok Kiong Tan received the B.Eng. (1st Class Hons.) and the Ph.D. degrees in Electrical Engineering from the National University of Singapore, Singapore, in 1992 and 1995, respectively. Prior to joining the National University of Singapore, he was a Research Fellow at the Singapore Institute of Manufacturing Technology, a national R&D institute spearheading the promotion of R&D in local manufacturing industries, where he was involved in managing industrial projects. He is currently a Professor with the National University of Singapore, Singapore. He has authored or co-authored more than 200 journal papers to date and has written 14 books, all resulting from research in these areas. He has attracted research funding in excess of S$18 million to date and has received several research awards. His current research interests include precision motion control and instrumentation, advanced process control and auto-tuning, and general industrial automation.
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This work is supported by the SIMTech-NUS Joint Lab on Precision Motion Systems (Funding No. U12-R-024JL). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Angelo Alessandri under the direction of Editor Thomas Parisini.