Polynomial Chaos-Based H 2 -optimal Static Output Feedback Control of Systems with Probabilistic Parametric Uncertainties

Abstract-Methods based on polynomial chaos expansion allow to approximate the behavior of systems with uncertain parameters by deterministic dynamics. These methods are used in a wide range of applications, spanning from simulation of uncertain systems to estimation and control. For practical purposes the exploited spectral series expansion is typically truncated to allow for efficient computation, which leads to approximation errors. Despite the Hilbert space nature of polynomial chaos, there are only a few results in the literature that explicitly discuss and quantify these approximation errors. This work derives error bounds for polynomial chaos approximations of polynomial and non-polynomial mappings. Sufficient conditions are established, which allow investigating the question whether zero truncation errors can be achieved and which series order is required to achieve this. Furthermore, convex quadratic programs, whose argmin operator is a special case of a piecewise polynomial mapping, are studied due to their relevance in predictive control. Several simulation examples illustrate our findings.

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“…Proof: For polynomial f (·), the exact PCE for y is given from (12). Consequently, for +1=(n ξ +d z d f )!/ (n ξ !…”

Section: Truncation Errors For Non-polynomial Mapsmentioning

confidence: 99%

“…Using Galerkin projection, the approximated system is deterministic but of larger dimension than the original system. This expanded system has been used, for example, to design linear controllers [10][11][12], and has been exploited in model predictive control [7,[13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Comments on Truncation Errors for Polynomial Chaos Expansions

Mühlpfordt

Findeisen

Hagenmeyer

et al. 2018

IEEE Control Syst. Lett.

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“…Equation (27c) is a sufficient condition for ensuring (26c), and Π and Γ i are bilinear terms with regard to P, S, and L. These bilinear terms cannot be converted into linear terms via conventional change-of-variables due to the block-diagonal structure of L = I Np+1 ⊗ L [5], [14].…”

Section: Pce-based Synthesismentioning

confidence: 99%

Probability-Guaranteed Set-Membership State Estimation for Polynomially Uncertain Linear Time-Invariant Systems

Wan

Puig

Ocampo‐Martínez

et al. 2018

2018 IEEE Conference on Decision and Control (CDC)

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Conventional deterministic set-membership (SM) estimation is limited to unknown-but-bounded uncertainties. In order to exploit distributional information of probabilistic uncertainties, a probability-guaranteed SM state estimation approach is proposed for uncertain linear time-invariant systems. This approach takes into account polynomial dependence on probabilistic uncertain parameters as well as additive stochastic noises. The purpose is to compute, at each time instant, a bounded set that contains the actual state with a guaranteed probability. The proposed approach relies on the extended form of an observer representation over a sliding window. For the offline observer synthesis, a polynomial-chaos-based method is proposed to minimize the averaged H2 estimation performance with respect to probabilistic uncertain parameters. It explicitly accounts for the polynomial uncertainty structure, whilst most literature relies on conservative affine or polytopic overbounding. Online state estimation restructures the extended observer form, and constructs a Gaussian mixture model to approximate the state distribution. This enables computationally efficient ellipsoidal calculus to derive SM estimates with a predefined confidence level. The proposed approach preserves time invariance of the uncertain parameters and fully exploits the polynomial uncertainty structure, to achieve tighter SM bounds. This improvement is illustrated by a numerical example with a comparison to a deterministic zonotopic method.

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“…In this paper, we address the design of robust and optimal linear quadratic regulators with probabilistic uncertainty in system parameters and solve it using deterministic algorithms. The deterministic algorithms are based on polynomial chaos theory and does not suffer from confidence issues like the randomized algorithms …”

Section: Introductionmentioning

confidence: 99%

“…14 In this paper, we address the design of robust and optimal linear quadratic regulators with probabilistic uncertainty in system parameters and solve it using deterministic algorithms. The deterministic algorithms are based on polynomial chaos theory [15][16][17][18][19][20] and does not suffer from confidence issues like the randomized algorithms. 14 This problem was first formulated in the polynomial-chaos framework, as a nonconvex optimization problem with bilinear inequalities.…”

Section: Introductionmentioning

confidence: 99%

Robust LQR design for systems with probabilistic uncertainty

Bhattacharya

2019

Intl J Robust & Nonlinear

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Summary In this paper, we consider the design of robust quadratic regulators for linear systems with probabilistic uncertainty in system parameters. The synthesis algorithms are presented in a convex optimization framework, which optimize with respect to an integral cost. The optimization problem is formulated as a lower‐bound maximization problem and developed in the polynomial chaos framework. Two approaches are considered here. In the first approach, an exact optimization problem is formulated in the infinite‐dimensional space, which is solved approximately using polynomial‐chaos expansions. In the second approach, an approximate problem is formulated using a reduced‐order model and solved exactly. The robustness of the controllers from these two approaches are compared using a realistic flight control problem based on an F16 aircraft model. Linear and nonlinear simulations reveal that the first approach results in a more robust controller.

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Polynomial Chaos-Based H 2 -optimal Static Output Feedback Control of Systems with Probabilistic Parametric Uncertainties

Cited by 8 publications

References 7 publications

Comments on Truncation Errors for Polynomial Chaos Expansions

Comments on Truncation Errors for Polynomial Chaos Expansions

Probability-Guaranteed Set-Membership State Estimation for Polynomially Uncertain Linear Time-Invariant Systems

Robust LQR design for systems with probabilistic uncertainty

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