Carleman State Feedback Control Design of a Class of Nonlinear Control Systems

Amini, Arash; Sun, Qiyu; Motee, Nader

doi:10.1016/j.ifacol.2019.12.163

Cited by 11 publications

(8 citation statements)

References 8 publications

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“…Nevertheless, it is clear that for some initial conditions, the original system is unstable. This system's error bound is valid for both stable and unstable cases, as can be visualized in figure (2).…”

Section: Van Der Pole Oscillatormentioning

confidence: 76%

“…A variation of the theorem 4.1 have been proved in [19,2]. Because the truncation matrix (4.19) is always block upper triangular the theorem shows that if the Jacobian matrix of (2.1) is Hurwtiz and the system is time invariant then the truncated system for any truncation length is always stable.…”

Section: Finite Sectioning Of the Carleman Linearnizationmentioning

confidence: 99%

“…We performed a 1000 simulations for time interval of τ * = 0.2 with initial condition randomly picked from the set {x ∈ R 2 | x ∞ ≤ 2} with uniform distribution. Figure (2) shows the result for this example. The solid red line indicates the error bound obtained by theorem 6.1.…”

Section: Van Der Pole Oscillatormentioning

confidence: 99%

“…Unless there are some useful structures, handling systems with unbounded infinite-dimensional matrices is exceptionally challenging. A common remedy is to truncate the infinite-dimensional system and then utilize the truncated system for analysis and control purposes [2]. Several stories reported about the successful employment of Carleman linearization in the control systems community.…”

Section: Introductionmentioning

confidence: 99%

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Error Bounds for Carleman Linearization of General Nonlinear Systems

Amini¹,

Sun²,

Motee³

2021

2021 Proceedings of the Conference on Control and Its Applications

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In his 1932 paper, Carleman proposed a linearization method to transform a given finite-dimensional nonlinear system with analytic right-hand into an equivalent infinitedimensional linear system with (usually) unbounded operators. Finite truncation of the transformed system has been used to study dynamic properties, learning, and control of nonlinear systems. One of the remaining outstanding problems in this context is quantifying the effectiveness of such finitely truncated models. Intuitively, one expects that as the truncation length increases, the truncated system will approximate the original nonlinear more effectively. In this paper, we provide explicit error bounds and prove that the trajectory of the truncated system stays close to that of the original nonlinear system over a quantifiable time interval. This is particularly important in applications, such as Model Predictive Control, to choose proper truncation lengths for a given sampling period and employ the resulting truncated system as a good approximation of the nonlinear system. Several examples are discussed to support our theoretical estimates.

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Section: Van Der Pole Oscillatormentioning

confidence: 76%

Section: Finite Sectioning Of the Carleman Linearnizationmentioning

confidence: 99%

Section: Van Der Pole Oscillatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Error Bounds for Carleman Linearization of General Nonlinear Systems

Amini¹,

Sun²,

Motee³

2021

2021 Proceedings of the Conference on Control and Its Applications

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“…For deterministic systems, it has been used for approximation of nonlinear models by linear ones (Bellman and Richardson, 1963;Steeb and Wilhelm, 1980;Al-Tuwaim et al, 1998). Recently, Amini et al (2019) used a Carleman linearization approach to design state feedback controllers for continuous-time nonlinear polynomial systems. Moreover, Hashemian and Armaou (2019) used Carleman linearization in model predictive control of continuous-time deterministic nonlinear systems.…”

Section: Related Workmentioning

confidence: 99%

Moment Propagation Through Carleman Linearization with Application to Probabilistic Safety Analysis

Pruekprasert¹,

Dubut²,

Takisaka³

et al. 2022

Preprint

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We develop a method to approximate the moments of a discrete-time stochastic polynomial system. Our method is built upon Carleman linearization with truncation. Specifically, we take a stochastic polynomial system with finitely many states and transform it into an infinitedimensional system with linear deterministic dynamics, which describe the exact evolution of the moments of the original polynomial system. We then truncate this deterministic system to obtain a finite-dimensional linear system, and use it for moment approximation by iteratively propagating the moments along the finite-dimensional linear dynamics across time. We provide efficient online computation methods for this propagation scheme with several error bounds for the approximation. Our result also shows that precise values of certain moments can be obtained when the truncated system is sufficiently large. Furthermore, we investigate techniques to reduce the offline computation load using reduced Kronecker power. Based on the obtained approximate moments and their errors, we also provide probability bounds for the state to be outside of given hyperellipsoidal regions. Those bounds allow us to conduct probabilistic safety analysis online through convex optimization. We demonstrate our results on a logistic map with stochastic dynamics and a vehicle dynamics subject to stochastic disturbance.

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