On l∞ performance optimization: linear switched systems and systems with cone constraints

Naghnaeian, Mohammad

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Description

Title

On l∞ performance optimization: linear switched systems and systems with cone constraints

Author(s)

Naghnaeian, Mohammad

Issue Date

2016-04-20

Director of Research (if dissertation) or Advisor (if thesis)

Voulgaris, Petros

Doctoral Committee Chair(s)

Hovakimyan, Naira

Committee Member(s)

• Dullerud, Geir
• Başar, Tamer

Department of Study

Mechanical Sci & Engineering

Discipline

Mechanical Engineering

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Control Theory
• Linear Switched Systems
• L1 performance

Abstract

The l∞ performance of Linear Time-Invariant (LTI) systems has been one of the corner stones of the robust control theory for over the past 30 years. The l∞ performance has been studied mostly for LTI systems and the scarcity of the results for other types of systems is prominent in this area. This dissertation aims to depart from LTI systems and investigate the l∞ performance for other classes of systems. In particular, the l∞ performance of Linear Switched Systems (LSS) and of linear systems with cone constraints is studied in the first and second part of this dissertation, respectively. Part I: In Part I, we first consider the worst-case l∞ induced norm computation of LSS. That is, sup_σ‖G_σ‖, where G_σ is a LSS, σ is the switching sequence, and the norm, ‖.‖, is the l∞ induced norm. This problem can be linked to robustness of systems when the switching is arbitrary. We provide lower and upper bounds of this quantity. These bounds are hard to compute and in general conservative. Hence, we narrow our attention to special classes of LSS by defining the classes of input, output, and input-output LSS and show that for these classes, exact expressions for the worst-case l∞ induced norm can be found. Moreover, we introduce the class of generalized input-output LSS and show how their l∞ gains can be computed exactly via Linear Programming (LP). The class of generalized input-output LSS proves to be a sufficiently rich class as it is dense in the set of all stable LSS. We further derive new stability and stabilizability conditions and control synthesis in terms of LP utilizing generalized input-output LSS. The other extreme from the worst-case norm is the minimal norm, i.e., inf_σ ‖G_σ‖. The interest in this type of problem is motivated by situations where there may be limited sensor and/or actuator resources for filtering and control. We show that for Finite Impulse Response (FIR)switching systems the minimizing switching sequence can be chosen to be periodic. For input-only or output-only switching systems an exact characterization of the minimal l∞ gain is provided, and it is shown that the minimizing switching sequence is constant, which, as also shown, is not true for input-output switching. Moreover, we study Markov Linear Switched Systems (MLSS). These are LSS whose switching sequence is a Markov process. We introduce the notion of the stochastic l∞ gain and provide exact expression to compute it. However, this computation is challenging, as we show, and hence we resort to a more relaxed but tractable notion of l∞ mean performance. We provide tractable computation and control synthesis method with respect to the l∞ mean performance. Part II: Part II of this dissertation deals with the l∞ gain of linear systems with positivity type of constraints. The study of such systems is well justified as there are many physical problems in which some variables are restricted be non-negative (or non-positive); examples can be found in biology, economics, and many other areas. We consider the case when the output is forced to be in the positive l∞ cone when the input is in this cone. This reflects as, so-called, an external positivity constraint on the system. As we point out, if such a constraint is imposed on the closed loop map, finding an optimal controller is LP and hence a tractable problem. If, on the other hand, the constraint known as internal positivity is sought, we show that a dynamic controller offers no advantage over a static one. These results can be used to obtain an optimal (static) state feedback controller. However, designing an optimal output feedback controller (which is static) is a harder problem and in general leads to a bilinear program. We show that this bilinear program can be reduced to LP, if the null space of the measurement matrix is invariant under multiplication by diagonal matrices. Besides the positive systems mentioned above, we consider the case where only the input is restricted to be in the positive cone of l∞, denoted by l∞+, and seek to characterize the induced norm from l∞+ to l∞. We stress here that no positivity constraint is imposed on the system itself. As an example, consider a positive nonlinear system with positive input that is linearized about a point other than origin. The linearized model is no longer a positive system as it is not linearized about the origin. Its inputs, however, remain positive and hence fit into this class of problems. We obtain an exact characterization of this norm (the induced norm from l∞+ to l∞ which can be used to synthesis a controller minimizing the induced norm from l∞+ to l∞ via LP.

Graduation Semester

2016-05

Type of Resource

text

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http://hdl.handle.net/2142/90553

Copyright and License Information

Copyright 2016 Mohammad Naghnaeian

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