Global Controllability by Nice Controls

Grasse, Kevin A.; Sussmann, Héctor J.

doi:10.1201/9780203745625-3

Cited by 15 publications

(7 citation statements)

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“…Control theory for sampled-data control systems has already been considered in several texts (see, e.g., [1,3,31,35,44]), often in the context of digital control. Optimal sampled-data control problems have also been investigated in the literature (see, e.g., [2,7,13,15,16,17]) with general terminal constraints on x(0) and x(T ), free final time, free sampling times, etc.…”

Section: A General Optimal Sampled-data Control Problem With Running mentioning

confidence: 99%

“…Sampled-data control systems are used in Automation, notably in model predictive control algorithms in which the control value at each sampling time is chosen as the first value of a finite sequence of control values optimizing the given cost on a fixed finite horizon (see, e.g., [36]). Numerous texts and articles have developed control theory for sampleddata control systems (see, e.g., [1,2,3,31,35,44] and references therein). Recently Bourdin and Trélat have obtained in [16] a version of the PMP for general nonlinear optimal sampled-data control problems (which is valid in the even more general framework of time scale calculus).…”

Section: Introductionmentioning

confidence: 99%

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Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon

Bourdin

Dhar

2020

Math. Program.

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Sampled-data control systems have steadily been gaining interest for their applications in Engineering and Automation. In the present paper we derive a Pontryagin maximum principle for general nonlinear optimal sampled-data control problems in the presence of running inequality state constraints. In particular we obtain a nonpositive averaged Hamiltonian gradient condition associated to an adjoint vector being a function of bounded variations. As a well-known challenge, theoretical and numerical difficulties may arise in optimal permanent control problems with state constraints due to the possible pathological behavior of the adjoint vector (jumps and singular part lying on parts of the trajectory in contact with the boundary of the restricted state space). However, in our case of sampled-data controls, we find that, under certain general hypotheses, the optimal trajectory only contacts the running inequality state constraints at most at the sampling times. In that context the adjoint vector only experiences jumps at most at the sampling times (and thus in a finite number and at precise instants) and its singular part vanishes. Hence, taking advantage of this bouncing trajectory phenomenon, we are able to implement a numerical indirect method which we use to solve three simple examples.

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Section: A General Optimal Sampled-data Control Problem With Running mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon

Bourdin

Dhar

2020

Math. Program.

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“…If we let U meas denote the set of measurable functions u : I ⊂ R → Ω, where I is an interval, then the class of admissible controls for a control system f is defined to be the subset U f ⊂ U meas containing the functions u with the property that the differential equationẋ(t) = f (x(t), u(t)) satisfies Carathéodory's theorem on the existence and uniqueness of solutions. If T f ω : TM × Ω → TTM is continuous 3 -a property we always assume-then U f contains the set U step of piecewise constant maps with a finite number of discontinuities [27,28] Because the objects of interest in the paper are real analytic control systems, in the remainder of the paper, and without loss of generality [27], the admissible controls will be elements of U step . This choice is implicit in the definitions of Sect.…”

Section: Basic Definitionsmentioning

confidence: 99%

“…To avoid cumbersome expressions, we sometimes write Γ ω E for the space of real analytic sections of a vector bundle (E, π, M). We begin by making precise what is meant by a "control system"; the definition is adopted from [27,28,47].…”

Section: Basic Definitionsmentioning

confidence: 99%

On the existence of stabilising feedback controls for real analytic small-time locally controllable systems

Isaiah

2015

Math. Control Signals Syst.

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It is shown that, for real analytic control systems of the form f : M × Ω (q, u) → f (q, u) ∈ T q M, where M is a real analytic manifold and Ω is a separable metric space, small-time local controllability from an equilibrium p ∈ M implies the existence of a piecewise analytic feedback control that locally stabilises f at p. The proof is similar in spirit to an earlier analogous result for globally controllable systems; however, it resolves several technical obstructions that emerge when the assumption of small-time local controllability is substituted for that of global controllability. In the light of a recent characterisation of small-time local controllability for homogeneous control systems, the main result of the paper implies that, for a large class of control systems that appear in applications and the literature, there is a computable sufficient condition for stabilisability by means of a piecewise analytic feedback control.

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“…Numerous texts and articles have developed control theory for sampled-data control systems (see, e.g., [1,2,3,18,25] and references therein). For instance, global controllability for sampled-data control systems has been investigated in [20]. Sampled-data control systems are used in Automation, notably in model predictive control algorithms in which the control value at each sampling time is chosen as the first value of a finite sequence of control values optimizing the given cost on a fixed finite horizon (see, e.g., [21]).…”

Section: Introductionmentioning

confidence: 99%

Continuity/constancy of the Hamiltonian function in a Pontryagin maximum principle for optimal sampled-data control problems with free sampling times

Bourdin

Dhar

2019

Math. Control Signals Syst.

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In a recent paper by Bourdin and Trélat, a version of the Pontryagin maximum principle (in short, PMP) has been stated for general nonlinear finite-dimensional optimal sampled-data control problems. Unfortunately their result is only concerned with fixed sampling times, and thus it does not take into account the possibility of free sampling times. The present paper aims to fill this gap in the literature. Precisely we establish a new version of the PMP that can handle free sampling times. As in the aforementioned work by Bourdin and Trélat, we obtain a first-order necessary optimality condition written as a nonpositive averaged Hamiltonian gradient condition. Furthermore, from the freedom of choosing sampling times, we get a new and additional necessary optimality condition which happens to coincide with the continuity of the Hamiltonian function. In an autonomous context, even the constancy of the Hamiltonian function can be derived. Our proof is based on the Ekeland variational principle. Finally a linear-quadratic example is numerically solved using shooting methods, illustrating the possible discontinuity of the Hamiltonian function in the case of fixed sampling times and highlighting its continuity in the instance of optimal sampling times.

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Global Controllability by Nice Controls

Cited by 15 publications

References 0 publications

Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon

Optimal sampled-data controls with running inequality state constraints: Pontryagin maximum principle and bouncing trajectory phenomenon

On the existence of stabilising feedback controls for real analytic small-time locally controllable systems

Continuity/constancy of the Hamiltonian function in a Pontryagin maximum principle for optimal sampled-data control problems with free sampling times

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