Inversion-based optimal output tracking–transition switching with preview for nonminimum-phase linear systems

doi:10.1016/j.automatica.2011.11.011

Automatica

Volume 48, Issue 7, July 2012, Pages 1364-1371

https://doi.org/10.1016/j.automatica.2011.11.011 Get rights and content

Abstract

In this article, the problem of nonperiodic tracking–transition switching with preview is considered. Such a control problem exists in applications including nanoscale material property mapping, robot manipulation, and probe-based nanofabrication, where the output needs to track the desired trajectory during the tracking sections, and rapidly transit to another point during the transition sections with no post-transition oscillations. Due to the coupling between the control of the tracking sections and that of the transition ones, and the potential mismatch of the boundary system state at the tracking–transition switching instants, these control objectives become challenging for nonminimum-phase systems. In the proposed approach, the optimal desired output trajectory for the transition sections is designed through a direct minimization of the output energy, and the needed control input that maintains the smoothness of both the output and the system state across all tracking–transition switching is obtained through a preview-based stable-inversion approach. The needed preview time is quantified by the characteristics of the system dynamics, and can be minimized via the recently developed optimal preview-based inversion technique. The proposed approach is illustrated through a nanomanipulation example in simulation.

Introduction

We present an inversion-based optimal control approach to solving the problem of nonperiodic tracking–transition switching with preview for nonminimum-phase linear systems. Such a problem arises in many applications, including nanoscale material property mapping (Butt, Cappella, & Kappl, 2005), hard-disk drive read/writing operation (Iamratanakul, Jordan, Leang, & Devasia, 2008), and nanomanipulation (Cavalcanti, 2003). In these applications involving multiple tracking–transition switching, the control objectives are to: (1) track, during each tracking section, the desired trajectory (to meet the specific needs of that application); and (2) transit, during the transition section immediately next, the output to the desired position rapidly with no induced post-oscillations. The tracking–transition switching is, in general, nonperiodic, and in many applications can be previewed for a finite length of time. We propose to combine the stable-inversion approach with the optimal control technique to maintain the smoothness of both the output and the system state across the tracking–transition switching, and achieve precision output tracking throughout the entire tracking–transition course.

The development of the stable-inversion theory (Devasia, Chen, & Paden, 1996) has solved the challenging problem of exact output tracking for nonminimum-phase systems, and has been extended recently to track online-generated desired output trajectory with preview (Zou, 2009, Zou and Devasia, 2007, Zou and Devasia, 1999), and to achieve other control objectives such as minimal-time regulation (Piazzi & Visioli, 2005), efficient numerical computation of the stable-inversion (Marro, Prattichizzo, & Zattoni, 2002), and causal feedforward control for static state transition (Graichen, Hagenmeyer, & Zeitz, 2005). Neither output transition nor tracking–transition switching, however, is allowed. On the contrary, the existing approaches to the output transition problem, including the conventional method of solving the output transition through the optimal state transition (OST) method (e.g., Lewis and Syrmos (1995)), the input-shaping technique, and the recently developed optimal output transition technique (OOT) (Perez and Devasia, 2003, Perez et al., 2004), only addressed the output tracking outside the transition period for special cases with either constant output outside the transition period (Graichen et al., 2005, Lau and Pao, 2003, Perez and Devasia, 2003, Piazzi and Visioli, 2005), or periodic tracking–transition switching (Perez et al., 2004). Clearly, there is a need to study more general tracking–transition switching problems.

Challenges exist in the preview-based output tracking of nonminimum-phase systems with nonperiodic tracking–transition switching. Although minimal post-transition oscillation is needed in many applications, the desired trajectory (for output transition) obtained by using the OST (Lewis & Syrmos, 1995) or the OOT (Perez and Devasia, 2003, Perez et al., 2004) techniques that are based on input energy minimization can be highly oscillatory when the system is lightly damped—to minimize the input energy, the frequency components of the optimal input are concentrated around the resonant peak(s) (Kim, Zou, & Su, 2008). Although the large oscillations can be mitigated by using pre-filter (Iamratanakul et al., 2008, Kim et al., 2008), direct minimization of the output energy is desirable. More importantly, smooth tracking across the switching instants requires the system state (not just the output) to be smooth across the switching, whereas in existing approaches (Lau and Pao, 2003, Perez and Devasia, 2003, Perez et al., 2004) the desired system state after switching is unknown, resulting in a mismatch of the boundary state across switching, and thereby, output oscillations after switching. Furthermore, the pre-actuation needed for output tracking of nonminimum-phase systems implies the control of tracking, and transition sections are coupled. These challenges motivate this work.

The main contribution of the article is an approach that extends the stable-inversion theory to the nonperiodic output tracking–transition switching with preview for nonminimum-phase systems. First, the desired output trajectory for the transition sections is designed via a direct output energy minimization that guarantees the smoothness of the desired output across all switching instants. Then secondly, the required control input is obtained through the preview-based stable-inversion approach. It is shown that the smoothness of the system state across the switching instants is maintained as the control input is updated, and furthermore, the tracking error caused by the finite preview exponentially decays with the increase of the preview time determined by the nonminimum-phase zeros of the system. Finally, the amount of preview time needed is further reduced by using the recently developed optimal preview-based stable-inversion method (Zou, 2009) when the amount of preview time is stringent (Waite, Zou, & Kelkar, 2008). A simulation example of nanomanipulation using a piezoelectric actuator is presented to illustrate the proposed approach. The comparison with the input-shaping approach (Lau & Pao, 2003) along with a proportional-integral (PI) feedback control demonstrates the efficacy of the proposed approach.

Section snippets

Problem formulation: preview-based optimal output tracking transition

Consider the following square linear time invariant (LTI) system: $\dot{x} = A x + B u, y = C x,$ with the same number of inputs and outputs, $u (\cdot)$ , $y (\cdot) \in ℜ^{p}$ , and $x (\cdot) \in ℜ^{n}$ . We assume:

Assumption 1

System (1) is controllable, observable, and hyperbolic (i.e., no zeros on the imaginary axis) with a well-defined relative degree $r ≜ {[r_{1}, r_{2}, \dots, r_{p}]}^{T}$ (Isidori, 1995).

With no loss of generality, we assign all transition sections, $T_{k}$ for $k \in N$ ( $N$ : the set of natural numbers), to be closed (see Fig. 1), and all tracking sections, $I_{k}$ to be

The output-tracking form of system (1)

Under Assumption 1, there exist (i) a state transformation, $T : ℜ^{n} \to ℜ^{n}$ and (ii) an input law for transforming system (1) into the output-tracking form. The needed state transformation $T$ is given by $[\begin{matrix} ξ (t) \\ η_{s} (t) \\ η_{u} (t) \end{matrix}] = T x (t) = [\begin{matrix} T_{ξ} \\ T_{η, s} \\ T_{η, u} \end{matrix}] x (t),$ where $ξ (t) = T_{ξ} x (t)$ are the output and its derivatives as in $ξ (t) = {[ξ_{1}^{T} (t), ξ_{2}^{T} (t), \dots, ξ_{p}^{T} (t)]}^{T}$ with $ξ_{k} (t)$ given by (10), and $η_{s}$ and $η_{u}$ are the stable and the unstable subspaces of the internal dynamics, respectively (specified immediately below in Eq. (18)). The input

A simulation example: nanomanipulation

We illustrate the proposed approach by means of a simulation example of nanomanipulation using piezoelectric actuator.

Conclusion

In this article, an inversion-based approach to achieving precision tracking in nonperiodic tracking–transition switching with preview has been proposed for nonminimum-phase linear systems. The desired output trajectory for the transition sections was designed through direct minimization of the output energy, and the required control input was obtained by using a preview-based stable-inversion approach. The proposed approach maintained the smoothness of the system state across the

Haiming Wang received his B.S. degree in Precision Instruments, Measurement and Control from Hefei University of Technology, Hefei, China, in 2005, and the M.S. degree in Precision Machinery and Precision Instrumentation from the University of Science and Technology of China, Hefei, China, in 2008. He is currently working toward the Ph.D. degree in the Department of Mechanical and Aerospace Engineering, Rutgers, the State University of New Jersey. His research interests include iterative

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Qingze Zou received his Ph.D. in Mechanical Engineering from the University of Washington, Seattle, WA, in Fall 2003. He obtained a M.S. degree in Mechanical Engineering from Tsinghua University, Beijing, China, in 1997 and a Bachelors degree in Automatic Control from the University of Electronic Science and Technology of China in 1994. Currently he is an Associate Professor in the Mechanical & Aerospace Engineering Department, Rutgers, the State University of New Jersey. Previously he had taught in the Mechanical Engineering Department of Iowa State University. His research interests include inversion-based output tracking and path-following, control tools for high-speed scanning probe microscope imaging, probe-based nanomanufacturing, and micromachining, and rapid broadband nanomechanical measurement and mapping of soft materials. He received the NSF CAREER award in 2009, and the O. Hugo Schuck Best Paper Award from the American Automatic Control Council (AACC) in 2010. Currently he is an Associate Editor for ASME Journal of Dynamic Systems, Measurement and Control.

Hongbing Xu received his Ph.D. in Circuits and Systems from the University of Electronic Science and Technology (UESTC), Chengdu, China, in 2000. He obtained a M.S. degree in Automatic Control from the Southeastern University, Nanjing, China, in 1991 and a Bachelors degree in Automatic Control from UESTC in 1988. Currently he is a professor in the School of Automation Engineering, University of Electronic Science and Technology of China. His research interests include advanced control techniques for renewable energy power systems, control and fault diagnosis for complex dynamic systems.

^☆: This work was supported by NSF CAREER award CMMI-1066055 and NSFC Grant 60972107. The material in this paper was partially presented at the 2010 American Control Conference (ACC), June 30–July 2, 2010, Baltimore, Maryland, USA. This paper was recommended for publication in revised form by Associate Editor Yoshikazu Hayakawa under the direction of Editor Toshiharu Sugie.

¹: Tel.: +1 110 5152949354; fax: +1 110 5152943261.

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Brief paperInversion-based optimal output tracking–transition switching with preview for nonminimum-phase linear systems☆

Abstract

Introduction

Section snippets

Problem formulation: preview-based optimal output tracking transition

The output-tracking form of system (1)

A simulation example: nanomanipulation

Conclusion

Surface Science Reports

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Automatica

Assembly automation with evolutionary nanorobots and sensor-based control applied to nanomedicine

IEEE Transactions on Nanotechnology

Nonlinear inversion-based output tracking

IEEE Transactions on Automatic Control

Brief paper
Inversion-based optimal output tracking–transition switching with preview for nonminimum-phase linear systems☆