Input-affine Nonlinear Systems Research Articles

Output information-based intermittent optimal control for continuous-time nonlinear systems with unmatched uncertainties via adaptive dynamic programming

Intermittent control stands as a valuable strategy for resource conservation and cost reduction across diverse systems. Nonetheless, prevailing research is intractable to address the challenges posed by robust optimal intermittent control of nonlinear input-affine systems with unmatched uncertainties. This paper aims to fill this gap. Initially, we introduce an enhanced finite-time intermittent control approach to ensure stability within nonlinear dynamic systems harboring bounded errors. A neural networks (NNs) state observer is constructed to estimate system information. Subsequently, an optimal intermittent controller that operates within a finite time span, guaranteeing system stability by employing the Hamilton–Jacobi–Bellman (HJB) methodology. Furthermore, we devise an output information-based event-triggered intermittent (ETI) approach rooted in the robust adaptive dynamic programming (ADP) algorithm, furnishing an optimal intermittent control law. In this process, a critic NNs is introduced to estimate the cost function and optimal intermittent controller. Simulation results show that our proposed method is superior to existing intermittent control strategies.

Disturbance-observer-based event-triggered model predictive control of nonlinear input-affine systems

This paper proposes a novel disturbance-observer-based event-triggered model predictive control (DEMPC) framework for a class of nonlinear input-affine systems with state and control input constraints as well as unmatched disturbances to simultaneously enhance the robustness of standard MPC and reduce computational resource utilization. First, a nonlinear disturbance observer is designed to compensate for disturbances actively. Next, a constant-threshold-type event-triggering mechanism (CTETM) is designed in terms of the state prediction deviation caused by the remaining disturbances. Subsequently, a DEMPC algorithm is constructed using the disturbance estimation information, the CTETM, and the so-called dual-mode scheme. Furthermore, rigorous theoretical analysis is provided, involving robust constraint fulfillment, Zeno phenomenon prevention, recursive feasibility, and stability. Particularly for systems with only matched disturbances, it is proven that continuous disturbance compensation would result in a possibly lower triggering frequency and enhanced control performance for the DEMPC compared to conventional EMPC without introducing additional conservatism into the state constraints. In the end, simulation studies are performed to illustrate the effectiveness of the proposed framework.

Output feedback and adaptive output feedback control design for input-affine nonlinear systems based on Riemannian metrics

This paper deals with output feedback and adaptive output feedback control design for input-affine nonlinear systems on the basis of Riemannian metrics. Firstly, an observer for the considered class of nonlinear systems is proposed. Then, it is shown that by combining the introduced observer with state-feedback controllers in a suitable way, output tracking can be achieved. Moreover, the obtained results are extended to be used for adaptive output feedback stabilization of uncertain nonlinear systems. The introduced adaptive controller is more general than the existing adaptive output feedback ones because it does not assume any constraint on the structure of the system and the parametric uncertainty. Simulation results are presented to verify the effectiveness of the proposed control methods.

Making nonlinear systems negative imaginary via state feedback

This paper provides a state feedback stabilization approach for nonlinear systems of relative degree less than or equal to two by rendering them nonlinear negative imaginary (NI) systems. Conditions are provided under which a nonlinear system can be made a nonlinear NI system or a nonlinear output strictly negative imaginary (OSNI) system. Roughly speaking, an input-affine nonlinear system can be rendered nonlinear NI and nonlinear OSNI if it has a normal form with relative degree less than or equal to two after possible output transformation. In addition, if the internal dynamics of the normal form are input-to-state stable, then there exists a state feedback input that stabilizes the system. This stabilization result is then extended to achieve stability for systems with a nonlinear NI uncertainty. A numerical example is provided to illustrate the process of stabilizing an uncertain system using the proposed results.

Online Reinforcement Learning Control of Nonlinear Dynamic Systems: A State-action Value Function Based Solution

In this paper, we present an online reinforcement learning-based solution to the optimal control problem of continuous-time nonlinear input-affine systems. The proposed approach contains a concurrent identifier that estimates time derivatives of states of the system in some arbitrary points. The identifier is utilized to simulate a so-called Bellman error in some unvisited points. The simulated errors together with errors obtained along the trajectory of the system are used to estimate the state-action value function, which is then employed to derive the estimated optimal controller. The designed approach does not explicitly require the input dynamics, which is hard to segregate it from the drift dynamics in optimal regulation problems. In addition, the simulated Bellman errors relax the confining persistence of excitation condition, which is needed for convergence in deterministic systems. A Lyapunov-based analysis was conducted to derive convergence conditions. Simulation studies demonstrated the effectiveness of the developed control scheme.

Optimal Charging Control of Energy Storage Systems for Pulse Power Load Using Deep Reinforcement Learning in Shipboard Integrated Power Systems

In this paper, the charging control of the energy storage system for the pulse power load accommodation in a shipboard integrated power system (SIPS) is formulated as an optimal control problem. The SIPS is an input-affine nonlinear system with randomness and fast dynamics. The improved twin-delayed deep deterministic policy gradient (TD3) algorithm -one of the deep reinforcement learning (DRL) algorithms, is proposed to solve this optimal control problem. The proposed DRL-based control solution considers the issues regarding the reward function design and input and ramp rate constraints handling for control variables. The proposed approach linked the optimal control and DRL framework. Test cases demonstrated that we could utilize DRL algorithms to control the nonlinear system with fast dynamics by following the specific reward function design, data sampling, and constraints handling techniques.

Disturbance Prediction-Based Adaptive Event-Triggered Model Predictive Control for Perturbed Nonlinear Systems

A disturbance prediction-based adaptive event-triggered model predictive control scheme is proposed for nonlinear systems in the presence of slowly varying disturbance. The optimal control problem in the model predictive control scheme is formulated by taking advantage of a proposed central path-based disturbance prediction approach, and the event-triggered mechanism is designed to be adaptive to the triggering interval. As a result, the proposed scheme improves the state prediction precision and, hence, reduces greatly the triggering frequency. Furthermore, for input-affine nonlinear systems, the disturbance separation and compensation techniques are developed to further enlarge the triggering interval. The theoretical analysis of the algorithm feasibility and closed-loop stability, as well as numerical evaluations of the effectiveness of the proposed schemes, is also given.

Structure algorithms, normal forms and their role in the problem of feedback design for input-affine nonlinear MIMO systems

Structure algorithms, normal forms and their role in the problem of feedback design for input-affine nonlinear MIMO systems

Lifted Bilinear Model-based Linear Model Predictive Control with Scalability

We propose a novel linear model predictive control (MPC) using a lifted bilinear model based on Koopman theory, which is computationally scalable against the dimension of the target system and the prediction horizon length. In MPC, the accuracy of the prediction model determines control performance, but it is a challenge to reduce the computational cost especially when considering nonlinearity of the model. To address this, a method has been proposed which represents the nonlinear input affine system as a lifted bilinear model and utilizes linear approximation and prediction error correction regarding the lifted state to achieve a low-computational-cost linear MPC with equivalent performance to nonlinear MPC. However, although the previous studies have shown its effectiveness for relatively low-order systems, it has not been applied to practical systems with higher dimensions. In this study, we extend the conventional method and propose a scalable linear MPC using a lifted bilinear model and apply it to higher-order nonlinear systems. In this paper, a quadrotor system operating in three-dimensional space is considered and its analytical lifted bilinear model is derived. In the formulation of linear MPC using the lifted bilinear model, an error correction method is newly introduced to feedback error for adjustment of the numerical relationships among the elements in the lifted state. The effectiveness of the proposed method is demonstrated through numerical simulations.

Efficient Computation of Inner Approximations of Reachable Sets for a Verified Motion Planning Concept

Verification of control systems often involves the computation of reachable sets. In this context, reachable sets are used to verify that a system is capable of reaching a desired set of states or that it will not reach a set of undesired states. This contribution presents a novel method for computing convex inner approximations of reachable sets for non-linear input-affine systems with constrained inputs. The inner approximation is obtained using the reachable set of the linearized system that is shrunk by a factor depending on the linearization errors. The proposed method is integrated into a motion planning concept for mobile robots. In this way, the motion planning concept guarantees, that motions that are planned based on a simplified model of a robot can be reached by an accurate model of the robot. The application example demonstrates the computational efficiency of the proposed method as it is utilized to compute inner approximated reachable sets of a system that consists of 18 states and features eight inputs.

Disturbance Rejection MPC Framework for Input-Affine Nonlinear Systems

This article proposes a novel disturbance rejection model predictive control (DRMPC) framework to improve the robustness of model predictive control (MPC) for a broad class of input-affine nonlinear systems with constraints and state-dependent disturbances. The proposed controller includes two parts—a disturbance compensation input and an optimal MPC control input. The former one is designed to compensate for the matched disturbance actively. This is made possible via a disturbance observer that estimates the disturbance and by adopting a space decomposition method. The residual disturbance is then handled in the MPC optimization problem by appropriate tightening of the constraints and designing the terminal constraint. Under reasonable assumptions, recursive feasibility and regional input-to-state practical stability (regional ISpS) of the closed-loop system are shown. Furthermore, we extend the DRMPC framework toward the tracking problem and apply it to a nonholonomic mobile robot. The performance of the proposed approach is demonstrated by a numerical example of the nonholonomic mobile robot.

Model‐free adaptive optimal control of continuous‐time nonlinear non‐zero‐sum games based on reinforcement learning

In this paper, two novel algorithms to find the Nash equilibrium solution of the non-zero-sum games for continuous-time input-affine nonlinear systems are presented. Based on integral reinforcement learning method, the integral-exploration-coupled Hamilton-Jacobi (HJ) equations are derived, which does not contain any information of the system dynamics. Then, based on neural networks approximation, two different adaptive tuning law of weights are given to estimate the approximate solution of the coupled HJ equations. Both two algorithms can estimate the value function and the policy without knowing or identifying the system dynamics. The closed-loop system stability and the convergence of weights are guaranteed based on Lyapunov analysis. Finally, the simulation results of a two-player non-zero-sum game demonstrate the effectiveness of our algorithms.

Time-energy suboptimal control of nonlinear input-affine systems

In this paper, we derive the necessary conditions for time-energy optimal control of input-affine nonlinear systems using Hamilton–Jacobi–Bellman equation. The objective function is chosen as a weighted combination of control effort and time-horizon minimisation. The necessary conditions of optimality are transformed such that the resultant conditions are structurally similar to a differential matrix Riccati equation. A closed-form approximate solution to these necessary conditions is derived such that the resulting closed-loop dynamics is locally exponentially stable around the origin. Furthermore, it is also shown that the states converge within a ball of an arbitrarily small radius around the origin within a finite time. Finally, the performance of the proposed controller is tested using numerical simulations for Vander Pol's oscillator as well as spacecraft detumbling problem. Furthermore, the performance of the proposed controller is also tested in comparison to the infinite-horizon state-dependent Riccati equation controller.

Predictor-based safety control for systems with multiple time-varying delays

Control barrier functions (CBFs) have been recently considered for ensuring safety of nonlinear input-affine systems by means of appropriately designed controllers rendering a desired superlevel set of the CBF function forward invariant. In this work, we consider the safety control problem for nonlinear input-affine systems with multiple time-varying input delays. In order to ensure safety, we first design a set of predictors that estimate the state of the system at different future times by utilizing the control laws designed to ensure safety of the delay-free system. Under the assumption of perfect estimation of the future states, we show that under the designed controller, the closed-loop performance of the systems with and without the input delays is the same by the time the input with the largest delay acts on the system with delays for the first time.

A novel domain of attraction based synthesis of inverse optimal control

AbstractThis article proposes a novel and systematic methodology for the inverse optimal control (IOC) of a class of input affine nonlinear systems. First, the system is represented in the pseudo‐linear state‐dependent coefficient (SDC) form so as to facilitate the use of matrix algebraic concepts in the design, while preserving the nonlinear system dynamics. Second, the IOC problem which seeks the solution to the Hamilton–Jacobi–Bellman equation is translated to a diagonal stability (D‐stability) problem. Therefore, the general notion of D‐stability, which deals with the characterization of a matrix for the existence of a diagonal matrix such that , is redefined with respect to SDC factored matrices in this work. Consequently, the criteria for the existence (necessary condition) and determination of a diagonal solution (sufficient condition) are formulated in terms of the SDC matrices. It is shown in this work that these criteria can be readily met by the closed‐loop system matrix , for any arbitrary domain in state‐space, by manipulating only the diagonal elements using the diagonal matrix constituted by the IOC feedback. In short, the IOC feedback is designed such that the criteria for D‐stability by the closed‐loop system matrix is fulfilled. Hence this novel design methodology is meaningfully named as “Inverse optimal control via diagonal stabilization” (IOC‐D). The main advantages of IOC‐D include: (i) guaranteed estimate of the domain of attraction (DOA), (ii) Immense flexibility of design, and (iii) guaranteed closed form solution which makes the design procedure analytic and computationally efficient. Numerical simulations validate the theoretical developments and the overall efficiency of the proposed approach.

Krasovskii Passivity and μ-Synthesis Controller Design for Quasi-Linear Affine Systems

This paper presents an end-to-end method to design passivity-based controllers (PBC) for a class of input-affine nonlinear systems, named quasi-linear affine. The approach is developed using Krasovskii’s method to design a Lyapunov function for studying the asymptotic stability, and a sufficient condition to construct a storage function is given, along with a supply-rate function. The linear fractional transformation interconnection between the nonlinear system and the Krasovskii PBC (K-PBC) results in a system which manages to follow the provided input trajectory. However, given that the input and output of the closed-loop system do not have the same physical significance, a path planning is mandatory. For the path planning component, we propose a robust controller designed using the μ-synthesis mixed-sensitivity loop-shaping for the linearized system around a desired equilibrium point. As a case study, we present the proposed methodology for DC-DC converters in a unified manner, giving sufficient conditions for such systems to be Krasovskii passive in terms of Linear Matrix Inequalities (LMIs), along with the possibility to compute both the K-PBC and robust controller alike.

Higher order sliding mode inspired nonlinear discrete-time observer

This work proposes a design scheme for arbitrary order discrete-time sliding mode observers for input-affine nonlinear systems. The dynamics of the estimation errors are represented in a pseudo-linear form, where the coefficients of the characteristic polynomial comprise the nonlinearities of the algorithm. The design process is reduced to a state-dependent eigenvalue placement procedure. Moreover, two different discrete-time eigenvalue mappings are proposed. As basis for the eigenvalue mappings serves a modified version of the continuous-time uniform robust exact differentiator. Due on the chosen eigenvalue mapping the proposed algorithm does not suffer from discretization chattering. Global asymptotic stability of the estimation errors for observers of order 2 and 3 is proven and the method to prove stability for higher order observers is demonstrated. The performance of a 3-rd order observer is illustrated in simulation. Simulation studies indicate that proposed discrete-time observer might possess an upper bound of its convergence time independent of the initial conditions.

Formulation of nonlinear control problems with actuator saturation as linear programs

Control of input-affine nonlinear dynamical systems is investigated in this paper. In this regard, a system of linear inequalities (SLI) in the control inputs is developed, and it is proved that if this SLI has a solution at all times, applying it to the system leads to moving its state trajectories towards the desired point in space for all initial conditions. But, since it may happen that this SLI has infinitely many solutions or even no solution, the best (possibly, approximate) solution is obtained by solving a linear programming (LP) problem. Different LP problems with different properties and applications are proposed for this purpose. Formulation of the control problem as an LP makes it possible to take into account the effect of actuator saturation simply by adding bound constraints to the problem. We can also make the resulting control system robust to model uncertainties. Some numerical examples including the output voltage regulation of double-source DC-DC converter and robot path-planning, both by considering the effect of actuators saturation and taking into account the uncertainties in model, are also presented.

Computation of Controlled Invariants for Nonlinear Systems: Application to Safe Neural Networks Approximation and Control

Abstract In this paper, we consider the problem of computing multidimensional interval controlled invariants for nonlinear input-affine systems. We first present sufficient conditions for an interval to be controlled invariant. Then, we introduce the concept of local framers, based on which we present a sound algorithm to compute interval controlled invariants. Finally, we show how the proposed framework makes it possible to provide safety guarantees when using deep neural networks, either as a model or a controller of nonlinear systems. Illustrative examples are provided showing the merits of the proposed approach and its scalability properties.

Nonlinear Tracking on the Infinite Horizon: Optimal Robust State Feedback

Optimal robust state-feedback controllers that minimize tracking errors over all times t ≥ 0 are considered. It is shown that such controllers exist for nonlinear input-affine systems that satisfy a certain controllability condition. Easy-to-implement controllers that approximate optimal performance are presented.