A hamiltonian formulation of risk-sensitive Linear/quadratic/gaussian control

Whittle, Peter; Kuhn, Jonathan R. D.

doi:10.1080/00207178608933445

Cited by 34 publications

(22 citation statements)

References 13 publications

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“…Then, we form the composite controller as in (24), which leads to (25). We now summarize this result below, as a corollary to Theorem 1.…”

mentioning

confidence: 83%

“…(24) /z* (t, x) =/z* 0 (t, x) %-/0 (t x) cO where/Zs*0 and/z were defined by (18) and (22), respectively, for 0 < Os. After some manipulations, this composite controller can be written as (25) # THEOREM 1. For the singularly perturbed system (1) with state-feedback information and the cost function (3), let the relevant parts of assumptions A1-A2 be satisfied, the pairs (A0, Bo) and (A2, Bz) be controllable, and the pair (A0, Qll QIQ2-1Q21) be observable.…”

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confidence: 99%

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Model Simplification and Optimal Control of Stochastic Singularly Perturbed Systems under Exponentiated Quadratic Cost

Pan¹,

Başar²

1996

SIAM J. Control Optim.

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We study the optimal control of a general class of stochastic singularly perturbed linear systems with perfect and noisy state measurements under positively and negatively exponentiated quadratic cost. The (expected) cost function to be minimized is actually taken as the long-term time average of the logarithm of the expected value of an exponentiated quadratic loss. We identify appropriate "slow" and "fast" subproblems, obtain their optimum solutions (compatible with the corresponding measurement structure), and subsequently study the performances they achieve on the full-order system as the singular perturbation parameter e becomes sufficiently small, with the expressions given in all cases being exact to within O(v/). It is shown that the composite controller (obtained by appropriately combining the optimum slow and fast controllers) achieves a performance level close to the optimal one whenever the full-order problem has a solution. The slow controller, on the other hand, achieves (asymptotically, as e --> 0) only a finite performance level (but not necessarily optimal), provided that the fast subsystem is open-loop stable. If the intensity of the noise in the system dynamics decreases to zero, however, the slow controller also achieves a performance level close to the optimal one. The paper also presents a more direct derivation (than heretofore available) of the solution to the linear exponential quadratic Gaussian (LEQG) problem under noisy state measurements, which allows for a general quadratic cost (with cross terms) in the exponent and correlation between system and measurement noises, and obtains both necessary and sufficient conditions for existence of an optimal solution. Such a general LEQG problem is encountered in the slow-fast decomposition of the full-order problem, even if the original problem does not feature correlated noises.In this general context, the paper also establishes a complete equivalence between the LEQG problem and the H aoptimal control problem with measurement feedback, though this equivalence does not extend to the slow and fast subproblems arrived at after time-scale separation. AMS subject classifications. 93E20, 90D25, 90A46, 93C80 1. Introduction. The problem of optimal control of stochastic linear systems under exponentiated quadratic loss (the so-called linear exponential quadratic Gaussian (LEQG) problem) has been studied extensively in the literature, with new interest aroused on the topic due to the recently established relationship with the H-optimal control of similar systems (but with deterministic disturbances) under quadratic loss. Perhaps the first formulation of the LEQG problem was given by Jacobson [8], in both discrete and continuous time, and using perfect state measurements, motivated by the fact that the exponentiated quadratic cost captures risk-seeking or risk-averse behavior, not obtainable using the linear quadratic Gaussian (LQG) formulation (which is risk neutral). Indeed it was discovered in [8] that the LEQG formulation with a positive exponent i...

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“…Then, we form the composite controller as in (24), which leads to (25). We now summarize this result below, as a corollary to Theorem 1.…”

mentioning

confidence: 83%

mentioning

confidence: 99%

Model Simplification and Optimal Control of Stochastic Singularly Perturbed Systems under Exponentiated Quadratic Cost

Pan¹,

Başar²

1996

SIAM J. Control Optim.

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“…One of the interesting method to include risk during decision-making process was proposed by Whittle and Kuhn (1986). In this method the expected quadratic-cost replaced by a risk-sensitive benchmark of exponentialquadratic form (Yang and Maciejowski, 2015).…”

Section: Risk-averse Predictive Controlmentioning

confidence: 99%

“…The conjugate variable or auxiliary variables of the Hamiltonian formulation has an interpretation in terms of the predicted course of process and observation noise. The RSCEP, in fact, provides a stochastic minimum principle for which all variables have a clear interpretation and the desired measurable properties (see Whittle and Kuhn (1986)). …”

Section: Risk-sensitive Minimum Principlementioning

confidence: 99%

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Risk-averse Stochastic Nonlinear Model Predictive Control for Real-time Safety-critical Systems

2017

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Stochastic nonlinear model predictive control has been developed to systematically find an optimal decision with the aim of performance improvement in dynamical systems that involve uncertainties. However, most of the current methods are risk-neutral for safety-critical systems and depend on computationally expensive algorithms. This paper investigates on the risk-averse optimal stochastic nonlinear control subject to real-time safety-critical systems. In order to achieve a computationally tractable design and integrate knowledge about the uncertainties, bounded trajectories generated to quantify the uncertainties. The proposed controller considers these scenarios in a risk-sensitive manner. A certainty equivalent nonlinear model predictive control based on minimum principle is reformulated to optimise nominal cost and expected value of future recourse actions. The capability of proposed method in terms of states regulations, constraints fulfilment, and real-time implementation is demonstrated for a semi-autonomous ecological advanced driver assistance system specified for battery electric vehicles. This system plans for a safe and energy-efficient cruising velocity profile autonomously.

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Introduction and Literature Survey of Statistical Control: Going Beyond the Mean

Won

Diersing

Yurkovich

2008

Advances in Statistical Control, Algebraic Systems Theory, and Dynamic Systems Characteristics

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A hamiltonian formulation of risk-sensitive Linear/quadratic/gaussian control

Cited by 34 publications

References 13 publications

Model Simplification and Optimal Control of Stochastic Singularly Perturbed Systems under Exponentiated Quadratic Cost

Model Simplification and Optimal Control of Stochastic Singularly Perturbed Systems under Exponentiated Quadratic Cost

Risk-averse Stochastic Nonlinear Model Predictive Control for Real-time Safety-critical Systems

Introduction and Literature Survey of Statistical Control: Going Beyond the Mean

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