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Este documento presenta el diseño de un controlador para el péndulo con rueda de reacción usando una representación discreta a través de la técnica de control óptimo inverso desde el punto de vista de análisis basado en pasividad. La principal ventaja del controlador propuesto es que este permite garantizar estabilidad asintótica en el sentido de Lyapunov a través de una función cuadrática. Los resultados numéricos demuestran que el diseño de control óptimo inverso tiene in desempeño superior en comparación con enfoques continuos basados en Lyapunov y control basado en pasividad por inyección de interconexión y amortiguamiento. Una ventaja adicional del método de control óptimo inverso es su fácil implementación, ya que no requiere de la inclusión de estados adicionales (acciones integrales) y sólo requiere una discretización básica empleado un único paso hacia atrás. Todas las simulaciones presentadas en este trabajo han sido implementadas en el software MATLAB/OCTAVE empleando código en la ventana de desarrollo. In this paper it is presented the design of a controller for a reaction wheel pendulum using a discrete-time representation via optimal control from the point of view of passivity-based control analysis. The main advantage of the proposed approach is that it allows to guarantee asymptotic stability convergence using a quadratic candidate Lyapunov function. Numerical simulations show that the proposed inverse optimal control design permits to reach superior numerical performance reported by continuous approaches such as Lyapunov control functions and interconnection, and damping assignment passivity-based controllers. An additional advantage of the proposed inverse optimal control method is its easy implementation since it does not employ additional states. It is only required a basic discretization of the time-domain dynamical model based on the backward representation. All the simulations are carried out in MATLAB/OCTAVE software using a codification on the script environment. Fil: Montoya Giraldo, Oscar Danilo. Universidad Distrital Francisco José de Caldas; Colombia Fil: Gil González, Walter. Universidad Tecnologica de Bolivar; Colombia Fil: Serra, Federico Martin. Universidad Nacional de San Luis. Facultad de Ingeniería y Ciencias Agropecuarias. Laboratorio de Control Automático; Argentina

Optimal control is a powerful approach to achieve optimal behavior. However, it typically requires a manual specification of a cost function which often contains several objectives, such as reaching goal positions at different time steps or energy efficiency. Manually trading-off these objectives is often difficult and requires a high engineering effort. In this paper, we present a new approach to specify optimal behavior. We directly specify the desired behavior by a distribution over future states or features of the states. For example, the experimenter could choose to reach certain mean positions with given accuracy/variance at specified time steps. Our approach also unifies optimal control and inverse optimal control in one framework. Given a desired state distribution, we estimate a cost function such that the optimal controller matches the desired distribution. If the desired distribution is estimated from expert demonstrations, our approach performs inverse optimal control. We evaluate our approach on several optimal and inverse optimal control tasks on non-linear systems using incremental linearizations similar to differential dynamic programming approaches.

This paper proposes a systematic formulation of inverse optimal control (IOC) law based on a rather straightforward reduction of control Lyapunov function (CLF), applicable to a class of second-order nonlinear systems affine in the input. This method exploits the additional design degrees of freedom resulting from the non-uniqueness of the state dependent coefficient (SDC) formulation, which is widely used in pseudo-linear control techniques. The applicability of the proposed approach necessitates an apparently effortless SDC formulation satisfying an SDC matrix criterion in terms of the structure and characteristics of the state matrix, [Formula: see text]. Subsequently, a sufficient condition for the global asymptotic stability (g.a.s) of the closed-loop system is established. The SDC formulations conforming to the sufficient condition ensure the existence and determination of a smooth radially unbounded polynomial CLF of the form [Formula: see text], while offering a benevolent choice for the gain matrix [Formula: see text], in the CLF. The direct relationship between the gain matrix [Formula: see text] and state weighing matrix [Formula: see text] ensures optimization of an equivalent [Formula: see text]. This feature enables one to rightfully choose the gain matrix [Formula: see text] as per the performance requisites of the system. Finally, the application of the proposed methodology for the speed control of a permanent magnet synchronous motor validates the efficacy and design flexibility of the methodology.

We consider bilevel optimization problems which can be interpreted as inverse optimal control problems. The lower-level problem is an optimal control problem with a parametrized objective function....

This paper studies the inverse optimal control using Legendre-Fenchel (in short, LF) translation method for regime-switching jump diffusions. Our approach is to first design inverse pre-optimal stabilization controllers and then obtain inverse optimal stabilizers, which avoids solving a Hamilton-Jacobi-Bellman equation. Finally, an application to stochastic Hamiltonian systems with Markov regime-switching is studied in detail for illustration.

The object of this paper is to study the uniqueness of solutions of inverse control problems in the case where the dynamics is given by a control-affine system without drift and the costs are length and energy functionals.

In this paper an inverse optimal control problem in the form of a mathematical program with complementarity constraints MPCC is considered and numerical experiences are discussed. The inverse optimal control problem arises in the context of human navigation where the body is modelled as a dynamical system and it is assumed that the motions are optimally controlled with respect to an unknown cost function. The goal of the inversion is now to find a cost function within a given parametrized family of candidate cost functions such that the corresponding optimal motion minimizes the deviation from given data. MPCCs are known to be a challenging class of optimization problems typically violating all standard constraint qualifications CQs. We show that under certain assumptions the resulting MPCC fulfills CQs for MPCCs being the basis for theory on MPCC optimality conditions and consequently for numerical solution techniques. Finally, numerical results are presented for the discretized inverse optimal control problem of locomotion using different solution techniques based on relaxation and lifting.

In this paper, the problem of finite horizon inverse optimal control (IOC) is investigated, where the quadratic cost function of a dynamic process is required to be recovered based on the observati ...

In this note, we consider the problem of computing the parameters (or weights) of an optimal control objective function given optimal closed-loop state and control trajectories. We establish a method of inverse optimal control that exploits the discrete-time minimum principle. Under a testable matrix rank condition, our proposed method is guaranteed to recover the unknown objective-function parameters of finite-horizon discrete-time nonlinear optimal control problems.

In this paper, we propose an inverse optimal design method for the position control of a quadrotor. First, we derive the dynamics of a quadrotor using the Newton-Euler formulation. Second, we present the state transformation technique to derive the position dynamics from the kinematic and dynamic models of a quadrotor. Then, we present the nonlinear inverse optimal position control of a quadrotor. The stability analysis based on Lyapunov theorem shows that the proposed control method can realize a quadrotor system that is exponentially stabilized. In addition, we show the inverse optimality of the proposed inverse optimal controller for the position control of a quadrotor. The inverse optimality can simply and clearly be shown using the state transformation technique. Finally, we present some simulation results to verify the effectiveness of the proposed control method.

This paper presents an inverse optimal control approach in order to achieve stabilization of discrete-time nonlinear systems, avoiding the need to solve the associated Hamilton–Jacobi–Bellman equation, and minimizing a cost functional. Then, the proposed approach is extended to discrete-time disturbed nonlinear systems. The synthesized stabilizing optimal controller is based on a discrete-time control Lyapunov function. The applicability of the proposed approach is illustrated via simulations.

La optimización aplicada al control automático permite obtener acciones de control que satisfacen no solo el objetivo de control, sino también la minimización de un determinado funcional de costo. Dinámicas complejas dificultan hallar la solución explícita de un problema de control óptimo. El control óptimo inverso evita la solución explícita de la ecuación de Hamilton-Jacobi-Bellman para determinar la ley óptima de control. Ilustrar la potencialidad del control óptimo inverso como alternativa para resolver problemas complejos de optimización en control. Se describe el problema de control óptimo para motivar el control óptimo inverso. Se formulan resultados matemáticos generales y se ilustra su aplicación a través de casos de ejemplo. Las formulaciones matemáticas presentadas son probadas analíticamente en casos del tipo óptimo cuadrático lineal (LQR) y óptimo inverso basado en funciones de control de Lyapuvov (LCF). Es posible formular un problema de control óptimo para sistemas de tipo no lineal, sin abordar la solución explícita del problema de optimización, mediante control óptimo inverso.

Inverse optimal trajectory tracking via a control Lyapunov function (CLF) for discrete time non-linear systems is developed and applied to type 1 diabetes mellitus patients control. The control law calculates the insulin delivery rate in order to prevent hyperglycemia and hypoglycemia levels. To synthesize the inverse optimal control law a quadratic candidate CLF is used. The proposed algorithm is tuned to follow a desired trajectory; this trajectory reproduces the glucose absorption of a healthy person. Simulation results applied for two different patients illustrate the applicability of the control law and a comparison with inverse optimal neural control via passivity is included.

This paper analyzes the chaotic phenomenon of a permanent magnet synchronous motor (PMSM) when it is turned off and what effect research parameters of PMSM have on chaos in the PMSM. Linear feedback controller is designed based on the inverse optimal control method. This method is simple and avoids a great amount of computation. In theory, a chaotic system can be asymptotically stable to zero by the controller. Numerical simulations further test the effectiveness of the theoretical analysis.

Differentiating the ability from the motivation to act is of central importance to psychiatric disorders in general and depression in particular. However, it has been difficult to develop quantitative approaches to relate depression to poor motor performance in goal-directed tasks. Here, we use an inverse optimal control approach to provide a computational framework that can be used to infer and factorize performance deficits into three components: sensorimotor speed, goal setting and motivation. Using a novel computer-simulated driving experiment, we found that (1) severity of depression is associated with both altered sensorimotor speed and motivational function; (2) moderately to severely depressed individuals show an increased distance from the stop sign indicating aversive learning affecting goal setting functions. Taken together, the inverse optimal control framework can disambiguate on an individual basis the sensorimotor from the motivational dysfunctions in depression, which may help to develop more precisely targeted interventions.

In this paper, we consider the problem of estimating the parameters of an optimal control objective function based on measurements of the closed loop system. In contrast to previous work on inverse optimal control, we consider measurements that are noise-corrupted and contain only partial-state information. We propose an inverse optimal control method based on a new soft-optimality constrained methodology of state estimation. We establish a sufficient condition for recovery of the unknown objective function parameters given complete-state information, and develop results characterising the performance of our method for linear systems. We illustrate our proposed soft-optimality approach through simulations of a nonlinear and fully-actuated mechanical system.

In this paper, we stablish robustness of discrete-time inverse optimal control for nonlinear systems trajectory tracking; we analyze in detail disk margin which includes sector and gain margins. We illustrate the applicability of this result by means of a linear induction motor example.

International audience; We address the inverse problem of Lagrangian identification based on trajecto-ries in the context of nonlinear optimal control. We propose a general formulation of the inverse problem based on occupation measures and complementarity in linear programming. The use of occupation measures in this context offers several advan-tages from the theoretical, numerical and statistical points of view. We propose an approximation procedure for which strong theoretical guarantees are available. Finally, the relevance of the method is illustrated on academic examples.

This paper is focused on the problem of the finite-time H ∞ inverse optimal control for affine nonlinear systems. Based on the finite-time control Lyapunov function, we derive a sufficient condition for the existence of time-invariant, continuous, finite-time stabilizing and inverse optimal state feedback control law, and propose a universal formula for constructing the finite-time H ∞ inverse optimal control law. We investigate the relationship between the finite-time stabilization and the finite-time H ∞ inverse optimal control. Finally, some examples are given to illustrate the effectiveness of the presented results.

In this paper, we use the definition of control Lyapunov functions to study finite time inverse optimal control for affine nonlinear systems. Based on control Lyapunov functions, a finite time universal control formula is presented, which can ensure the closed-loop system is finite time stable. From this, less conservative conditions for the finite time inverse optimal control are derived. We design a finite time inverse optimal control law, which minimizes the cost functional. A numerical example verifies the validity of the proposed method.

This paper introduces in cross standard form (CSF) as a solution to the inverse optimal control problem. That is, the CSF is a canonical standard problem whose unique H ∞ or H 2 optimal controller is a given controller. From the control design point of view, the general idea is to apply the CSF to a given controller in order to set up a standard problem which can be completed to handle frequency domain H 2 or H ∞ specification. The analytical formulation of the CSF proposed in this paper can be applied to reduced-, full- or augmented-order compensators or two-degree of freedom compensations. Numerical and academic examples are given.

In this paper, the Cross Standard Form (CSF) is introduced as a solution to the inverse optimal control problem. That is, the CSF is a canonical augmented standard plant whose unique H ∞ or H 2 optimal controller is a given controller. From the control design point of view, the general idea is to apply the CSF to a given controller in order to set up a standard problem which can be completed to handle frequency-domain H 2 or H ∞ specifications. The analytical formulation of the CSF proposed in this paper can be applied to reduced-, full- or augmented-order compensators. An academic example is given.

In this paper, the problem of chaotifying the continuous-time fuzzy hyperbolic model (FHM) is studied. By tracking the dynamics of a chaotic system, a controller based on inverse optimal control and adaptive parameter tuning methods is designed to chaotify the FHM. Simulation results show that for any initial value the FHM can track a chaotic system asymptotically.

This work proposes a robust inverse optimal controller design for a class of nonlinear systems with bounded, time-varying uncertain variables. The basic idea is that of re-shaping the scalar nonlinear gain of an LgV controller, based on Sontag's formula, so as to guarantee certain uncertainty attenuation properties in the closed-loop system. The proposed gain re-shaping is shown to yield a control law that enforces global boundedness of the closed-loop trajectories, robust asymptotic output tracking with an arbitrary degree of attenuation of the effect of uncertainty on the output, and inverse optimality with respect to a meaningful cost that penalizes the tracking error and the control action. The performance of the control law is illustrated through a simulation example and compared with other controller designs. Copyright © 2003 John Wiley & Sons, Ltd.

A class of optimal control problems having an optimal solution of a specified property is investigated. Examples from irreversible thermodynamics and microeconomics are given. Such a formulation, as has been shown, can be regarded as an inverse optimal control problem.

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In a companion paper (Nonlinear Impulsive Dynamical Systems. Part I: Stability and Dissipativity) Lyapunov and invariant set stability theorems and dissipativity theory were developed for non-linear impulsive dynamical systems. In this paper we build on these results to develop general stability criteria for feedback interconnections of non-linear impulsive systems. In addition, a unified framework for hybrid feedback optimal and inverse optimal control involving a hybrid non-linear-non-quadratic performance functional is developed. It is shown that the hybrid cost functional can be evaluated in closed-form as long as the cost functional considered is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability of the non-linear closed-loop impulsive system. Furthermore, the Lyapunov function is shown to be a solution of a steady-state, hybrid Hamilton‐Jacobi‐Bellman equation.

25 pages; International audience; This paper is devoted to the general problem of reconstructing the cost from the observation of trajectories, in a problem of optimal control. It is motivated by the following applied problem, concerning HALE drones: one would like them to decide by themselves for their trajectories, and to behave at least as a good human pilot. This applied question is very similar to the problem of determining what is minimized in human locomotion. These starting points are the reasons for the particular classes of control systems and of costs under consideration. To summarize, our conclusion is that in general, inside these classes, three experiments visiting the same values of the control are needed to reconstruct the cost, and two experiments are in general not enough. The method is constructive. The proof of these results is mostly based upon the Thom's transversality theory.

In this chapter we study an inverse optimal control problem in discrete-time stochastic control. We give necessary and sufficient conditions for a solution to a system of stochastic difference equations to be the solution of a certain OCP. Our results extend to the stochastic case the work of Dechert [21]. In particular, we present a stochastic version of an important principle in welfare economics. The presentation of this chapter is based on Gonzalez–Sanchez and Hernandez–Lerma [36].

This paper is devoted to the general problem of reconstructing the cost from the observation of trajectories, in a problem of optimal control. It is motivated by the following applied problem, concerning HALE drones: one would like them to decide by themselves for their trajectories, and to behave at least as a good human pilot. This applied question is very similar to the problem of determining what is minimized in human locomotion. These starting points are the reasons for the particular classes of control systems and of costs under consideration. To summarize, our conclusion is that in general, inside these classes, three experiments visiting the same values of the control are needed to reconstruct the cost, and two experiments are in general not enough. The method is constructive. The proof of these results is mostly based upon the Thom's transversality theory.

International audience; This paper discusses forward and inverse optimal control problems for bipedal human-like running, with a focus on inverse optimal control. The (forward) optimal control problem looks for the optimal solution for a problem formulation, i.e. given objective function and given dynamic constraints. The inverse optimal control problem is more challenging and consists in determining the objective function and potentially unknown parts in the dynamic model that best reproduce a solution that is known from measurements. Periodic running motions are modeled as hybrid dynamic models with multiple phases and discontinuities, based on a three-dimensional multibody system model with 25 degrees of freedom.We investigate a recorded running motion on a treadmill at 10 km/h running speed and identify the best possible objective function based on some hypotheses for potential contributions to this objective function. For this, we apply a previously developed inverse optimal control technique which uses a combination of a direct multiple shooting method and a derivative-free optimization techniqueand we demonstrate here that it also works for problems of the given complexity.

This paper deals with finite time inverse optimal control for affine nonlinear systems. Based on control Lyapunov functions, a universal control formula is presented to ensure that the closed-loop system is finite time stable. From this, less conservative conditions for the finite time inverse optimal control are derived. A finite time inverse optimal control law that minimizes the cost functional is designed. A numerical example is used to illustrate the effectiveness of the proposed design technique.

This paper presents a speed-gradient-based inverse optimal control approach for the asymptotic stabilization of discrete-time nonlinear systems. With the solution presented, we avoid to solve the associated Hamilton-Jacobi-Bellman equation, and a meaningful cost function is minimized. The proposed stabilizing optimal controller uses the speed-gradient algorithm and is based on the proposal of what is called a discrete-time control Lyapunov function. This combined approach is referred to as the speed-gradient inverse optimal control. An example is used to illustrate the methodology. Several simulations are provided.

In this paper, we propose a generalized inverse optimal controller for nonlinear systems with input constraints. We clarify that directional vectors of inverse optimal controllers determine the lower limits of sector margins. Under input constraints, sector margins and domains of attraction have trade-off relations. Therefore, domains in which inverse optimal controllers exist may be small, or inverse optimal controllers may not guarantee enough sector margins. Finally, we provide a robust controller for adjusting sector margins and domains of attraction.

This paper introduces the Cross Standard Form (CSF) as a solution to the generalized inverse optimal control problem. The CSF is a canonical augmented standard plant whose H∞ or H2 optimal controller is a given controller. The definition of the CSF relies on the possibility to construct minimal observer-based realization of a given controller for a given model. From the multi-objective or robust control point of view, the general idea is to apply the CSF to a first controller satisfying nominal performances in order to initialize an H∞ design procedure to handle frequency-domain constraints or robustness specifications.

The problem of chaotifying the continuous-time fuzzy hyperbolic model (FHM) is studied in this paper. By tracking the dynamics of a chaotic system, a controller based on inverse optimal control and adaptive parameter tuning methods is designed to chaotify the FHM. Simulation results show that for any initial value the FHM can track a chaotic system asymptotically.

In this paper, a control strategy based on the inverse optimal control approach is applied for pinning weighted complex networks with chaotic systems at their nodes; additionally, a cost functional is minimized. This control strategy does not require to have the same coupling strength for all node connections.

The main objective of this paper is to formulate and resolve a number of inverse optimal control problems for linear quadratic (LQ) regulators subject to nonlinear time-varying perturbations and decentralized information structure constraints. As a by-product, we will establish standard robustness properties of LQ regulators for broad classes of nonlinear plants. In particular, it is shown when a decentralized LQ control of interconnected systems is optimal and can tolerate a wide range of nonlinear effects in the control and interconnection structure.

A linear discrete time-invariant system with a quadratic cost is considered and conditions are found for existence of non - optimal control laws which generate the optimal input and state trajectories. The problem of invariance of the quadratic cost on a set of non-optimal control laws is studied and its connection with the inverse optimal control problem is discussed.

Four distinct, though closely related, inverse optimal control problems are considered. Given a closed, convex setU in a real Hilbert spaceX and an elementu0 inU, it is desired to find all functionals of the form (u,Ru) such that (i)R is a self-adjoint positive operator and (u,Ru) is minimized over the setU at the pointu0, (ii)R is self-adjoint, positive definite and (u,Ru) is minimized overU atu0, (iv)R is self-adjoint, positive definite and (u,Ru) is uniquely minimized overU atu0. The interrelationships among the sets of solutions of these problems are pointed out. Necessary and sufficient conditions which explicitly characterize the solutions to each of these problems are derived. The question of existence of a solution (namely, Given a particular setU and a particular elementu0, under what conditions does there exist an operatorR having certain required properties?) is discussed. The results derived are illustrated by an example.