Input-affine Nonlinear Systems Research Articles

Approximate Optimal Control via Measurement Feedback for a Class of Nonlinear Systems

The approximate optimal control problem via measurement feedback for input-affine nonlinear systems is considered in this paper. In particular, a systematic method is provided for constructing stabilising output feedbacks that approximate - with the optimality loss explicitly quantifiable - the solution of the optimal control problem by requiring only the solution of algebraic equations. In fact, the combination of a classical state estimate with an additional dynamic extension permits the construction of a dynamic control law, without involving the solution of any partial differential equation or inequality. Moreover, provided a given sufficient condition is satisfied, the dynamic control law is guaranteed to be (locally) stabilising. A numerical example illustrating the method is provided.

LMI Relaxations for Quadratic Stabilization of Guaranteed Cost Control of T–S Fuzzy Systems

Less conservative condition is provided in this paper for quadratic stabilization of guaranteed cost control (GCC) of Takagi–Sugeno fuzzy systems with parallel distributed compensation (PDC). To derive the condition, firstly a parameter-dependent linear matrix inequality (PD-LMI) is established to find quadratically stable PDC controller gains of GCC. Secondly, by applying Polya’s theorem, evaluation of the PD-LMI is transformed into an equivalent problem of evaluation of a sequence of LMI relaxations. Different from other existing conditions, the LMI relaxations are sufficient and asymptotically reach necessity for evaluating the PD-LMI as a related scalar parameter, d, increases. The resulting guaranteed costs of PDC controllers are non-increasing with respect to the increase in the parameter d and converge to the global optimal value under quadratic stability at the limiting case. In addition, for input-affine nonlinear systems, the proposed condition is extended with the consideration of modeling errors, which helps to reduce the computational complexity of the LMI relaxations. Finally, simulations of two examples demonstrate the efficiency and feasibility of the proposed condition.

Strong iISS for a class of systems under saturated feedback

This paper proposes sufficient conditions under which nonlinear input-affine systems can be made Strongly iISS in the presence of actuator saturation. Strong iISS was recently proposed as a compromise between the strength of input-to-state-stability (ISS) and the generality of integral input-to-state stability (iISS). It ensures in particular that solutions are bounded provided that the disturbance magnitude is below a certain threshold, and that they tend to the origin in response to any vanishing disturbance. We propose a growth rate condition under which the bounded feedback law proposed by Lin and Sontag for disturbance-free nonlinear systems ensures Strong iISS in the presence of perturbations. We illustrate our findings on the angular velocity control of a spacecraft with limited-power thrusters. In the specific case of linear time-invariant systems with neutrally stable internal dynamics, we provide a simple static state-feedback that ensures Strong iISS in presence of actuator saturations. This second result is illustrated by the robust stabilization of the harmonic oscillator. In both cases, we provide an estimate of the maximum disturbance amplitude that can be tolerated without compromising solutions’ boundedness.

Stability analysis of some classes of input-affine nonlinear systems with aperiodic sampled-data control

In this paper we investigate the stability analysis of nonlinear sampled-data systems, which are affine in the input. We assume that a stabilizing controller is designed using the emulation technique. We intend to provide sufficient stability conditions for the resulting sampled-data system. This allows to find an estimate of the upper bound on the asynchronous sampling intervals, for which stability is ensured. The main idea of the paper is to address the stability problem in a new framework inspired by the dissipativity theory. Furthermore, the result is shown to be constructive. Numerically tractable criteria are derived using linear matrix inequality for polytopic systems and using sum of squares technique for the class of polynomial systems.

A Nash Game Approach to Mixed H2/H∞ Control for Input-Affine Nonlinear Systems

Abstract: With the aim of designing controllers to simultaneously ensure robustness and optimality properties, the mixed H 2 / H ∞ control problem is considered. A class of input-affine nonlinear systems is considered and the problem is formulated as a nonzero-sum differential game, similar to what has been done in the 1990s by Limebeer et al. for linear systems. A heuristic algorithm for obtaining solutions for the coupled algebraic Riccati equations which are characteristic of the linear quadratic problem is provided together with a systematic method for constructing approximate solutions for the general, nonlinear problem. A few numerical examples are provided.

A two-stage approach for managing actuators redundancy and its application to fault tolerant flight control

In safety-critical systems such as transportation aircraft, redundancy of actuators is introduced to improve fault tolerance. How to make the best use of remaining actuators to allow the system to continue achieving a desired operation in the presence of some actuators failures is the main subject of this paper. Considering that many dynamical systems, including flight dynamics of a transportation aircraft, can be expressed as an input affine nonlinear system, a new state representation is adopted here where the output dynamics are related with virtual inputs associated with the intended operation. This representation, as well as the distribution matrix associated with the effectiveness of the remaining operational actuators, allows us to define different levels of fault tolerant governability with respect to actuators’ failures. Then, a two-stage control approach is developed, leading first to the inversion of the output dynamics to get nominal values for the virtual inputs and then to the solution of a linear quadratic (LQ) problem to compute the solicitation of each operational actuator. The proposed approach is applied to the control of a transportation aircraft which performs a stabilized roll maneuver while a partial failure appears. Two fault scenarios are considered and the resulting performance of the proposed approach is displayed and discussed.

局所半凹制御Lyapunov関数による凸入力制約を考慮した非線形制御系設計

Semiconcave control Lyapunov functions (semiconcave CLFs) play important roles in nonlinear control theory. Many semiconcave CLF based stabilizing controllers were proposed. In this paper, we consider a local asymptotic stabilization problem of input affine nonlinear systems under convex input constraints. To design a stabilizing state feedback under the input constraints, we employ a locally semiconcave CLF and the convex optimization theory. Due to nonsmoothness of the semiconcave CLF, the proposed state feedback is discontinuous on the state space. We consider sample and hold solutions and guarantee the asymptotic stability of the closed loop system in the sense of sample stability.

Reconfigurable Control of Input Affine Nonlinear Systems under Actuator Fault

This paper proposes a fault tolerant control method for input-affine nonlinear systems using a nonlinear reconfiguration block (RB). The basic idea of the method is to insert the RB between the plant and the nominal controller such that fault tolerance is achieved without re-designing the nominal controller. The role of the RB is twofold: on one hand it transforms the output of the faulty system such that its behaviour is similar to that of the nominal one from the controller's viewpoint; on the other hand it modifies the control input to the faulty system such that the stability of the reconfigured loop is preserved. The RB is realized by a virtual actuator and a REFERENCE model. Using notions of incremental and input-to-state stability (ISS), it is shown that ISS of the closed-loop reconfigured system can be achieved by the separate design of the virtual actuator. The proposed method does not need any knowledge of the nominal controller and only assumes that the nominal closed-loop system is ISS. The method is demonstrated on a dynamic positioning system for an offshore supply vessel, where the virtual actuator is designed using backstepping.

Functional Series Expansions for Continuous-Time Switched Systems

The main objective of this paper is to describe a class of functional series expansions, known as Fliess operators, which admit inputs from a ball in an L p space as well as Poisson random processes. Conditions are given under which these functional series expansions converge absolutely in the mean. Then, it is shown that a continuous-time switched input-affine nonlinear system with a Poisson switching signal can be represented as a Fliess operator and that for certain cases a closed form solution can be obtained in terms of Poisson integrals.

Integral reinforcement learning for continuous-time input-affine nonlinear systems with simultaneous invariant explorations.

This paper focuses on a class of reinforcement learning (RL) algorithms, named integral RL (I-RL), that solve continuous-time (CT) nonlinear optimal control problems with input-affine system dynamics. First, we extend the concepts of exploration, integral temporal difference, and invariant admissibility to the target CT nonlinear system that is governed by a control policy plus a probing signal called an exploration. Then, we show input-to-state stability (ISS) and invariant admissibility of the closed-loop systems with the policies generated by integral policy iteration (I-PI) or invariantly admissible PI (IA-PI) method. Based on these, three online I-RL algorithms named explorized I-PI and integral Q -learning I, II are proposed, all of which generate the same convergent sequences as I-PI and IA-PI under the required excitation condition on the exploration. All the proposed methods are partially or completely model free, and can simultaneously explore the state space in a stable manner during the online learning processes. ISS, invariant admissibility, and convergence properties of the proposed methods are also investigated, and related with these, we show the design principles of the exploration for safe learning. Neural-network-based implementation methods for the proposed schemes are also presented in this paper. Finally, several numerical simulations are carried out to verify the effectiveness of the proposed methods.

Closed-form solution to finite-horizon suboptimal control of nonlinear systems

Finite-horizon optimal control of input-affine nonlinear systems with fixed final time is considered in this study. It is first shown that the associated Hamilton–Jacobi–Bellman partial differential equation to the problem is reducible to a state-dependent differential Riccati equation after some approximations. With a truncation in the control equation, a near optimal solution to the problem is obtained, and the global onvergence properties of the closed-loop system are analyzed. Afterwards, an approximate method, called Finite-horizon State-Dependent Riccati Equation (Finite-SDRE), is suggested for solving the differential Riccati equation, which renders the origin a locally exponentially stable point. The proposed method provides online feedback solution for controlling different initial conditions. Finally, through some examples, the performance of the resulting controller in finite-horizon control is analyzed. Copyright © 2014 John Wiley & Sons, Ltd.

Approximate finite-horizon optimal control for input-affine nonlinear systems with input constraints

The finite-horizon optimal control problem with input constraints consists in controlling the state of a dynamical system over a finite time interval (possibly unknown) minimising a cost functional, while satisfying hard constraints on the input. In this framework, the minimum-time optimal control problem and some related problems are of interest for both theory and applications. For linear systems, the solution of the problem often relies upon the use of bang-bang control signals. For nonlinear systems, the “shape” of the optimal input is in general not known. The control input can be found solving a Hamilton–Jacobi–Bellman (HJB) partial differential equation (PDE): it typically consists of a combination of bang-bang controls and singular arcs. In this paper, a methodology to approximate the solution of the HJB PDE is proposed. This approximation yields a dynamic state feedback law. The theory is illustrated by means of two examples: the minimum-time optimal control problem for an industrial wastewater treatment plant and the Goddard problem, i.e. a maximum-range optimal control problem.

Discontinuous Control of Nonlinear Systems with convex input constraint via Locally Semiconcave Control Lyapunov Functions

Recently, locally semiconcave control Lyapunov functions (LS-CLFs) play important roles in nonlinear control theory. Many LS-CLF based stabilizing controllers are proposed. In this paper, we consider the locally asymptotic stabilization problem of the input affine nonlinear systems with convex input constraints. To design a stabilizing state feedback under the input constraints, we employ the LS-CLF and convex optimization theory. Due to nonsmoothness of LS-CLF, the proposed state feedback is discontinuous on the state space. Therefore, we consider sample and hold solutions and guarantee the asymptotic stability of the closed loop system in the sense of sample stability. The effectiveness of the proposed method is confirmed by the numerical example.

An Iterative Predictive Learning Control Approach With Application to Energy Efficient Train Trajectory Tracking

An approach of iterative predictive learning control (IPLC) is studied with consideration to both precise train trajectory tracking and energy efficient operation. Through designing the predictive cost function, the IPLC approach for input-affine nonlinear systems is formulated and solved in this paper. Its application to train operation is detailed to compromise between punctuality and energy consumption. Rigorous theoretical analysis confirms that the proposed approach can guarantee the asymptotic convergence of train speed and position to desired profiles along iteration axis. Simulation result shows its effectiveness and enegry efficiency.

An analytical fuzzy-based approach to -gain optimal control of input-affine nonlinear systems using Newton-type algorithm

This paper is concerned with -gain optimisation of input-affine nonlinear systems controlled by analytic fuzzy logic system. Unlike the conventional fuzzy-based strategies, the non-conventional analytic fuzzy control method does not require an explicit fuzzy rule base. As the first contribution of this paper, we prove, by using the Stone–Weierstrass theorem, that the proposed fuzzy system without rule base is universal approximator. The second contribution of this paper is an algorithm for solving a finite-horizon minimax problem for -gain optimisation. The proposed algorithm consists of recursive chain rule for first- and second-order derivatives, Newton’s method, multi-step Adams method and automatic differentiation. Finally, the results of this paper are evaluated on a second-order nonlinear system.

Fixed-final-time optimal tracking control of input-affine nonlinear systems

In this study, approximate dynamics programming framework is utilized for solving the Bellman equation related to the fixed-final-time optimal tracking problem of input-affine nonlinear systems. Convergence of the weights of the neurocontroller in the proposed successive approximation based algorithms is provided and the network is trained to provide the optimal solution to the problems with (a) unspecified initial conditions (b) different time horizons, and (c) different reference trajectories under certain general conditions. Numerical simulations illustrate the versatility of the proposed neurocontroller.

A numerical framework for optimal control of switched input affine nonlinear systems subject to path constraint

In this paper, we address the problem of numerical implementation of optimal control for switched input affine nonlinear systems subject to path constraint. In order to properly solve the problem, a relaxed system is introduced and the connection between the solution of this system and the solution of the initial one is established. One of the main difficulties is then related to the fact that the optimal solution can be singular. We show that, using slack variables, a set of complementarity constraints can be used to take into account the singular nature of the solution. The optimal control problem is then reformulated as a constraint optimization problem over the Hamiltonian systems and solved via a direct method. This formulation does not require a priori knowledge on the structure (regular/singular) of the solution. In addition, state path constraints are included. Numerical simulations for boost, buck–boost and flying capacitor converters, both in continuous and discontinuous conduction mode, illustrate the effectiveness of the proposed methodology.

Dynamic Approximate Solutions of the HJ Inequality and of the HJB Equation for Input-Affine Nonlinear Systems

The solution of most nonlinear control problems hinges upon the solvability of partial differential equations or inequalities. In particular, disturbance attenuation and optimal control problems for nonlinear systems are generally solved exploiting the solution of the so-called Hamilton-Jacobi (HJ) inequality and the Hamilton-Jacobi-Bellman (HJB) equation, respectively. An explicit closed-form solution of this inequality, or equation, may however be hard or impossible to find in practical situations. Herein we introduce a methodology to circumvent this issue for input-affine nonlinear systems proposing a dynamic, i.e., time-varying, approximate solution of the HJ inequality and of the HJB equation the construction of which does not require solving any partial differential equation or inequality. This is achieved considering the immersion of the underlying nonlinear system into an augmented system defined on an extended state-space in which a (locally) positive definite storage function, or value function, can be explicitly constructed. The result is a methodology to design a dynamic controller to achieve <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -disturbance attenuation or approximate optimality, with asymptotic stability.

Nonlinear Adaptive Model Predictive Control via Immersion and Invariance Stabilizability

Adaptive control systems are designed to achieve the desired control performance when plant parameters are unknown or possibly slow-changing. In this paper, we propose an adaptive model predictive control (MPC) algorithm for a class of nonlinear input affine systems. The key idea is to combine the MPC algorithm with the adaptive Immersion and Invariance (I&I) control method. That is, MPC is used to calculate the input satisfying the assumption in the adaptive I&I control method and then the parameter update law in I&I depends on the state, estimated parameter, and input determined by the MPC algorithm. This strategy allows us to estimate the unknown parameters online and produce the control input at the same time. To modify the I&I method, we show a stability theorem for a linearly parameterized plant and then, numerical examples are given to demonstrate its effectiveness.

Direct modeling method of generalized Hamiltonian system and simulation simplified

Generalized Hamiltonian model can give dynamics mechanism of inner parameters connected, however, it will meet many difficult while an actual physical system is converted to Generalized Hamiltonian model. And due to complexity of the structure matrix and damping matrix of high order Hamiltonian system, analysis and application to inner parameters connected mechanism is not ease and convenience. In this paper one kind of Hamiltonian model, four order and single input affine nonlinear system, is given basic form, it is a formulation formulize form, that is generalize Hamiltonian model can be directly derived by formulate calculation. The hydro turbine Hamiltonian model is taken as case to introduce this modeling method. At last, a simulation simplified method based-physical background is proposed, and used to simplify the hydro turbine Hamiltonian model.