On the Achievable Delay Margin Using LTI Control for Unstable Plants

Abstract-We show that stabilizing tracking proportional-integral (PI) controllers can be constructed for minimum-phase nonaffine-in-control systems. The constructed PI controller is an equivalent realization of an approximate dynamic inversion controller. This equivalence holds only for the time response when applied to the unperturbed system. Even when restricted to unperturbed minimum-phase linear time invariant systems, their closed loop robustness properties differ. This shows that in general, properties that do not define the equivalence relation for systems/controllers are not preserved under such equivalence transformations.

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“…For a stable LTI system with loop transfer function L(s), we use the definition of the time delay margin [20] …”

Section: B Time Delay Marginmentioning

confidence: 99%

Proportional-Integral Controllers for Minimum-Phase Nonaffine-in-Control Systems

Teo

How

Lavretsky

2010

IEEE Trans. Automat. Contr.

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“…Let us synthesize a robust controller for this example. Moreover, we will synthesize, as an example [58,59], a robust controller for a third-order system with uncertain delay rather than some parameters.…”

Section: Systems With Delaymentioning

confidence: 99%

Section: Uncertainty Of Delaymentioning

confidence: 99%

“…Let the delay t rather than the parameters of the system be uncertain, i.e., as in [59], we assume that t does not depend on time and [ ] t t Î 0 m . The maximum value of t m for an unstable plant with a dynamic controller was estimated in [59]. Let us use the example from [58,59] to show that the procedures described above can be successfully used to synthesize a static controller that stabilizes the plant for sufficiently great values of t m .…”

Section: Uncertainty Of Delaymentioning

confidence: 99%

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Stabilization problems for a system with output feedback (review)

Aliev

Larin

2011

Int Appl Mech

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Algorithms for solving the problem of design of static output feedback controllers for stationary linear systems with continuous and discrete time are reviewed. The inverse problem is considered. The algorithms of synthesis of output feedback controllers are generalized to the case of a periodic discrete-time system. To solve such problems, it might be more natural to use an approach based on multi-criterion optimization. It is also shown that these algorithms can be used for the optimal stabilization of unstable systems with delay. In this connection, the parameters of a controller with given structure for a controlled unstable scalar system with delay are optimized. To this end, the system is first approximated by a system without delay, with the exponent approximated by a fractionally rational function. Since the structure of the controller is given, the quality of approximation is estimated as the difference (in the space of controller coefficients) between the stability domains of the original and approximating systems. At the next stage, the gain coefficients of the controller for the reduced system are optimized. The efficiency of the thus synthesized controller is assessed through mathematical modeling of a system with delay whose feedback loop is defined by the gain coefficients found. The approach is illustrated by stabilizing an inverted simple pendulum with a proportional-derivative controller with delay. The problem of synthesis of a robust controller for this example is considered. Some examples of designing a robust controller, including for a third-order system in which the delay rather than some parameter is uncertain are presented Keywords: LQ-problem, inverse LQ-problem, static output feedback, periodic system, system with delay, proportional-derivative controller 1. Introduction. In many applied problems, including some control problems for mechanical systems [7, 51, 66, 68, 69, etc.], it is necessary to synthesize a controller when only some of the elements of the phase vector are observable. When only some elements of the phase vector are observable in the linear-quadratic problem (LQ-problem), a feedback loop is known [6,8] to be synthesized by solving two matrix algebraic Riccati equations. One of these equations describes a filter generating an estimate of the entire phase vector. The solution of the second equation yields the controller gain matrix that relates the controls and estimates of the phase vector. This is the so-called case of synthesis of dynamic feedback. It is more difficult to synthesize a static feedback loop [32, 57, etc.] where it is necessary to find a constant matrix that generates controls directly from the observable part of the phase vector (static output feedback controller problem [15]). A similar problem is the use of one static controller to stabilize several plants simultaneously [39]. This problem occupies an important place in the synthesis of robust controllers (other approaches to this problem are given in [23, 44, etc.]). In [15], the synthesis of a sta...

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“…In [2] (p. 154), the delay margin was examined for the first-order system with a constant delay stabilized by static feedback, while in [15], the stabilization was achieved by using PID controllers for first-order systems. Furthermore, for single-input single-output (SISO) systems with constant delays, the upper bound was determined in [16,17] for general LTI systems with an arbitrary number of unstable poles. These bounds consequently provide a limit beyond which no single LTI output feedback controller may exist to robustly stabilize a delay plant family within the delay margin.…”

Section: Introductionmentioning

confidence: 99%

On the Achievable Stabilization Delay Margin for Linear Plants with Time-Varying Delays

Zhu

2017

Mathematics

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Abstract:The paper contributes to stabilization problems of linear systems subject to time-varying delays. Drawing upon small gain criteria and robust analysis techniques, upper and lower bounds on the largest allowable time-varying delay are developed by using bilinear transformation and rational approximates. The results achieved are not only computationally efficient but also conceptually appealing. Furthermore, analytical expressions of the upper and lower bounds are derived for specific situations that demonstrate the dependence of those bounds on the unstable poles and nonminumum phase zeros of systems.

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On the Achievable Delay Margin Using LTI Control for Unstable Plants

Cited by 109 publications

References 20 publications

Proportional-Integral Controllers for Minimum-Phase Nonaffine-in-Control Systems

Proportional-Integral Controllers for Minimum-Phase Nonaffine-in-Control Systems

Stabilization problems for a system with output feedback (review)

On the Achievable Stabilization Delay Margin for Linear Plants with Time-Varying Delays

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