Motion planning and tracking for tip displacement and deflection angle for flexible beams

Siranosian, Antranik Antonio; Krstić, Miroslav; Smyshlyaev, Andrey; Bement, Matt

doi:10.1109/acc.2008.4587122

Cited by 6 publications

(9 citation statements)

References 22 publications

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“…Setting ξ = 0 in (53), multiplying (53) by ε 2 (0), and using the facts that K vv (x, 0) = 0 for all x ∈ [0, 1] (see relation (31) in [34]) and that f is defined by 24, we get that f satisfies (51) for q = 0. The rest of the proof is similar to the case q = 0.…”

Section: Remarkmentioning

confidence: 99%

“…Our approach is different than the one in [34], in that we use backstepping for trajectory generation rather than stabilization, and the one in [26], in that we employ a different conceptual idea to a different class of systems. The idea of the backstepping-based trajectory generation for PDEs, which was conceived in [23], is applied to a beam PDE in [31] and the Navier-Stokes equations in [4], and is recently extended to general n × n linear hyperbolic systems in [22]. We apply this methodology to a wave PDE with indefinite in-domain and boundary damping by transforming (see, for example, [3]) the wave PDE to a 2 × 2 linear hyperbolic system coupled with a first-order ODE (Section 2.2).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Control of 2×2 linear hyperbolic systems: Backstepping-based trajectory generation and PI-based tracking

Lamare

Bekiaris-Liberis

2015

Systems & Control Letters

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We consider the problems of trajectory generation and tracking for general 2 × 2 systems of firstorder linear hyperbolic PDEs with anti-collocated boundary input and output. We solve the trajectory generation problem via backstepping. The reference input, which generates the desired output, incorporates integral operators acting on advanced and delayed versions of the reference output with kernels which were derived by Vazquez, Krstic, and Coron for the backstepping stabilization of 2 × 2 linear hyperbolic systems. We apply our approach to a wave PDE with indefinite in-domain and boundary damping. For tracking the desired trajectory we employ a PI control law on the tracking error of the output. We prove exponential stability of the closed-loop system, under the proposed PI control law, when the parameters of the plant and the controller satisfy certain conditions, by constructing a novel "non-diagonal" Lyapunov functional. We demonstrate that the proposed PI control law compensates in the output the effect of in-domain and boundary disturbances. We illustrate our results with numerical examples.

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Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Control of 2×2 linear hyperbolic systems: Backstepping-based trajectory generation and PI-based tracking

Lamare

Bekiaris-Liberis

2015

Systems & Control Letters

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“…(6), (7), and (4), respectively (to be exact, Àq :f(u(0,t), c 1 : c 0 and c 2 : c 1 ). When f is constant, the closedloop system (45), (46), and (49) is equivalent to the exponentially stable target system [27][28][29][30][31][32][33][34]…”

Section: Application To a String Pdementioning

confidence: 99%

“…We then turn our attention to some relevant basic mechanical PDE systems-the string and shear beam PDEs with Kelvin-nonlinearities. These designs are the extensions of results for the string [27][28][29][30] and shear beam [29][30][31][32]. The merits of these designs are highlighted by simulation.…”

Section: Introductionmentioning

confidence: 97%

Gain Scheduling-Inspired Boundary Control for Nonlinear Partial Differential Equations

Siranosian¹,

Krstić

Smyshlyaev

et al. 2011

Journal of Dynamic Systems, Measurement, and Control

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We present a control design method for nonlinear partial differential equations (PDEs) based on a combination of gain scheduling and backstepping theory for linear PDEs. A benchmark first-order hyperbolic system with an in-domain nonlinearity is considered first. For this system a nonlinear feedback law, based on gain scheduling, is derived explicitly, and a proof of local exponential stability, with an estimate of the region of attraction, is presented for the closed-loop system. Control designs (without proofs) are then presented for a string PDE and a shear beam PDE, both with Kelvin–Voigt (KV) damping and free-end nonlinearities of a potentially destabilizing kind. String and beam simulation results illustrate the merits of the gain scheduling approach over the linearization based design.

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“…More recently, a number of control techniques have emerged which use Lyapunov-based approaches to derive controllers for PDEs directly, without resorting to finite order approximations of the PDE [5], [6], [7], [10], [16], [17], [18]. These methods tend to do away with some limitations of the ODE-based approach, such as the occasional necessity for large order approximations, and the risk of spillover instabilities.…”

Section: Introductionmentioning

confidence: 99%

Sub-optimal boundary control of semilinear pdes using a dyadic perturbation observer

Paranjape

Chung

2016

2016 IEEE 55th Conference on Decision and Control (CDC)

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Abstract-In this paper, we present a sub-optimal controller for semilinear partial differential equations, with partially known nonlinearities, in the dyadic perturbation observer (DPO) framework. The dyadic perturbation observer uses a two-stage perturbation observer to isolate the control input from the nonlinearities, and to predict the unknown parameters of the nonlinearities. This allows us to apply well established tools from linear optimal control theory to the controlled stage of the DPO. The small gain theorem is used to derive a condition for the robustness of the closed loop system.

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Motion planning and tracking for tip displacement and deflection angle for flexible beams

Cited by 6 publications

References 22 publications

Control of 2×2 linear hyperbolic systems: Backstepping-based trajectory generation and PI-based tracking

Control of 2×2 linear hyperbolic systems: Backstepping-based trajectory generation and PI-based tracking

Gain Scheduling-Inspired Boundary Control for Nonlinear Partial Differential Equations

Sub-optimal boundary control of semilinear pdes using a dyadic perturbation observer

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