Linear Differential Inclusions Research Articles

A Novel Robotic GWO LDI Modeling and Control for Nonlinear Systems

This works aims to develop a new and improved GWO (Grey Wolf Optimizer), the so-called Robotic GWO (RGWO). First, to improve GWO's update formula position with an optimal learning strategy, we adapt the algorithm to real mobile environments, including robots, so that tracking robots can move prey toward targets. Then, the nonlinear active suspension (AS) control system is linearized by a neural network (NN) based linear differential inclusion (LDI) using feedback and feedforward linearization. In theory, it is found that the general SM (Sliding Mode) optimal control cannot provide sudden optimal results for the active linearized suspension system, so a method is proposed to improve the shortcomings of the active linearized suspension system. By constructing an extended SM-optimal manifold function, an improved SM-optimal controller is designed, which incorporates information on the entire structure and the expected performance of the suspension. For comparison purposes, the performance of three kinds of controls: SM optimal refinement control, logic-fuzzy SM control, and PS (passive suspension), shows the proposed controller's advantages . Finally, our improved SM optimal control for nonlinear AS systems, in general, can achieve the actual nominal optimal suspension performance, as confirmed by the simulation results. The results also show that the improved SM optimal control method provides better robustness even when the operating conditions or parameters of the structure vary.

Comparing Explicit and Implicit Methods for Estimating the Region of Attraction

The region of attraction (ROA) is an important tool in investigating control and dynamic systems. Numerous computational methods have been proposed for estimating the ROA within the Mathematics, Power systems, and Control community. These methods employ different methods such as invariant manifolds, linear differential inclusion, level set methods, the sum of squares decomposition, and many others. This paper provides new insights into these methods and divides them into two classes: explicit and implicit. The explicit methods discretize the ROA (or the boundary of the ROA) into a finite union of geometric elements and store them directly. The essential idea of the implicit methods is to present the ROA implicitly, meaning that it is feasible to find the ROA as the level set of some function of the state space variables. The key point is the iteration and optimization of this implicit function. The explicit and implicit methods are compared and discussed by investigating some numerical examples. In particular, we consider the determination of the boundary of the ROA using the stable manifolds, and the estimation of the ROA as backward reachable sets using a Hamilton-Jacobi-Isaacs partial differential equation.

Cascade Control for Two-Axis Position Mechatronic Systems

The current paper proposes an extension for two controller design procedures for a two-axis positioning mechatronic system, followed by a comparison between them. As such, the first method consists in formulating an optimization problem in terms of linear matrix inequalities (LMIs) in order to impose the location of the closed-loop poles, considering an uncertain model of such a system. The uncertain model is treated using various forms of linear differential inclusions (LDIs), namely, polytopic LDI (PLDI) and diagonal norm-bound LDI (DNLDI). Additionally, the problem regarding the command signal constraints is characterized in terms of LMIs. The imposed structure of the controller is a cascade one, with a PI controller for the position loop and a P controller for the velocity loop, having an additional feedforward term. On the other hand, the second method consists in designing a cascade controller with an inner P controller, as in the previous method, the outer controller being a fractional-order IλIDλD (FO–ID) controller. In terms of degrees of freedom, both methods present four degrees of freedom for each axis. The presented controller design procedures will be applied for a numerical example of such a positioning system, and a comparison of the obtained performance metrics will be performed.

A Polynomial Chaos Approach to Robust Static Output-Feedback Control With Bounded Truncation Error

This article considers the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_\infty$</tex-math></inline-formula> static output-feedback control for linear time-invariant uncertain systems with polynomial dependence on probabilistic time-invariant parametric uncertainties. By applying polynomial chaos theory, the control synthesis problem is solved using a high-dimensional expanded system, which characterizes stochastic state uncertainty propagation. A closed-loop polynomial chaos transformation is proposed to derive the closed-loop expanded system. The approach explicitly accounts for the closed-loop dynamics and preserves the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {L}_2$</tex-math></inline-formula> -induced gain, which results in smaller transformation errors compared to existing polynomial chaos transformations. The effect of using finite-degree polynomial chaos expansions is first captured by a norm-bounded linear differential inclusion, and then addressed by formulating a robust polynomial chaos based control synthesis problem. This proposed approach avoids the use of high-degree polynomial chaos expansions to alleviate the destabilizing effect of truncation errors, which significantly reduces computational complexity. A numerical example illustrates the effectiveness of the proposed approach.

A Robust Lyapunov Criterion for Nonoscillatory Behaviors in Biological Interaction Networks

We introduce the notion of nonoscillation, propose a constructive method for its robust verification, and study its application to biological interaction networks (also known as, chemical reaction networks). We begin by revisiting Muldowney’s result on the nonexistence of periodic solutions based on the study of the variational system of the second additive compound of the Jacobian of a nonlinear system. We show that exponential stability of the latter rules out limit cycles, quasi-periodic solutions, and broad classes of oscillatory behavior. We focus then on nonlinear equations arising in biological interaction networks with general kinetics, and we show that the dynamics of the aforementioned variational system can be embedded in a linear differential inclusion. We then propose algorithms for constructing piecewise linear Lyapunov functions to certify global robust nonoscillatory behavior. Finally, we apply our techniques to study several regulated enzymatic cycles, where available methods are not able to provide any information about their qualitative global behavior.

Gain-scheduled output-feedback control of uncertain input-delay LPV systems with saturation via polytopic differential inclusion

This paper examines the control design for parameter-dependent input-delay linear parameter-varying (LPV) systems with saturation constraints. A linear differential inclusion approach is implemented to formulate the saturation nonlinearity. Subsequently, a gain-scheduled dynamic output-feedback controller is structured to conform to the saturation representation. By means of a Lyapunov–Krasovskii functional, sufficient delay-dependent conditions are derived for the analysis and feedback control synthesis of the closed-loop system in the presence of uncertainties and exogenous disturbances. Disturbance rejection objectives following an induced -norm and an norm formulation are examined. Estimation of the domain of attraction with the parametrisation of the admissible LPV controllers is presented. Three different optimisation objectives with respect to disturbance rejection, disturbance tolerance, and domain of attraction expansion are formulated. The proposed design methodology is examined in the context of the air-fuel ratio (AFR) regulation in a spark ignition (SI) engine with a speed-dependent delay and model parameters in the presence of canister purge disturbances and AFR dynamics uncertainty. The injector is restricted in the amount of fuel it can deliver to the cylinder. The three-way catalytic converter's performance in terms of oxygen storage and emission reduction is taken into consideration. Closed-loop simulation results are provided to validate the methodology.

PID Control of Planar Nonlinear Uncertain Systems in the Presence of Actuator Saturation

This paper investigates PID control design for a class of planar nonlinear uncertain systems in the presence of actuator saturation. Based on the bounds on the growth rates of the nonlinear uncertain function in the system model, the system is placed in a linear differential inclusion. Each vertex system of the linear differential inclusion is a linear system subject to actuator saturation. By placing the saturated PID control into a convex hull formed by the PID controller and an auxiliary linear feedback law, we establish conditions under which an ellipsoid is contractively invariant and hence is an estimate of the domain of attraction of the equilibrium point of the closed-loop system. The equilibrium point corresponds to the desired set point for the system output. Thus, the location of the equilibrium point and the size of the domain of attraction determine, respectively, the set point that the output can achieve and the range of initial conditions from which this set point can be reached. Based on these conditions, the feasible set points can be determined and the design of the PID control law that stabilizes the nonlinear uncertain system at a feasible set point with a large domain of attraction can then be formulated and solved as a constrained optimization problem with constraints in the form of linear matrix inequalities (LMIs). Application of the proposed design to a magnetic suspension system illustrates the design process and the performance of the resulting PID control law.

H∞ Optimal Control for Maintaining the R2R Peeling Front

Dry transfer using Roll-to-Roll (R2R) mechanical peeling could significantly increase the throughput and efficiency of the production of 2D materials such as graphene and flexible electronics. Currently, such a R2R process does not exist in industry. For this dry transfer R2R process to be practical for industrial applications, the peeling angle between the growth substrate and the functional material needs to be precisely controlled. In this paper, a nonlinear state space representation of the R2R dry peeling process is formulated with the peeling front velocity variation as a disturbance input. This state space model is used to construct a linear parameter varying (LPV) representation of the system, and a methodology on how to bound the LPV representation within a convex polytopic linear differential inclusion (PLDI) set is presented. This PLDI representation is then used in a linear matrix inequality (LMI) optimization framework to design a full state feedback controller that minimizes the H∞ gain of the connection between the adhesion energy variation and the peeling front geometry. Simulation results demonstrate that this controller improves the precision of the R2R peeling angle, and this increase in precision enables higher web speed. Thus, this technique can be an enabling tool for making R2R mechanical peeling dry transfer of 2D materials a reality in industrial settings.

Smart structural stability and NN based intelligent control for nonlinear systems

This paper has proposed an intelligent Evolutionary Bat Algorithm (afterward, EBA) Fuzzy NN (Neural Network) controller used to ensure the asymptotic simulation stability of a mathematics nonlinear system for a smart structure. The smart evolutionary fuzzy NN model adopts an NN numerical model and the linear differential inclusion (LDI) concept. Denotation of the nonlinear dynamics is constructed by transforming the nonlinear model into a multi-rule-based sector nonlinear form of mathematics linear numerical models, and implementing a new sufficient mathematics condition whereby the asymptotic simulation stability of the intelligent structure is guaranteed by the Lyapunov mathematics function, linear matrix inequality (LMI). The high frequency is also injected as an auxiliary to stabilize these nonlinear systems. According to the relaxed method injected with dithered auxiliary, the nonlinear system can be guaranteed stable by appropriately regulating the parameters. Finally, there is a numerical resultant example with simulation results which is designated in order to precisely demonstrate the advantages of the smart intelligent controller and the proposed control scheme compared to previous schemes.

Design of PID control for planar uncertain nonlinear systems with input delay

AbstractThis article presents a PID control design for a class of planar uncertain nonlinear systems with bounded time‐varying input delay. The delay is unknown with a known bound. The class of systems is characterized by the bounds on the growth rates of the nonlinear function in the system model. Based on these bounds on the growth rates, the system is placed in a linear differential inclusion, a convex hull with four linear systems as vertices. For a given set of PID control coefficients, a Lyapunov–Krasovskii functional is constructed that leads to sufficient conditions, in the form of linear matrix inequalities in the bound on the time delay, a positive scalar and two positive definite matrices, under which the equilibrium solution corresponding to the desired set point of the outputs of all vertex linear systems and, hence, that of the original uncertain nonlinear system, is globally uniformly asymptotically stable. With the bound on the time delay as a variable, an optimization problem with linear matrix inequalities as constraints can be formulated to determine the largest admissible bound on the time delay for the closed‐loop system under the given PID controller. By viewing the PID control coefficients as additional free variables, the optimization problem is then adapted for the design of the PID coefficients for the largest admissible bound on the time delay. The design is then applied to a magnetic suspension system. Simulation results illustrate the design algorithm and the performance of the resulting PID controller.

Evolved auxiliary controller with applications to aerospace

PurposeTo guarantee the asymptotic stability of discrete-time nonlinear systems, this paper aims to propose an evolved bat algorithm fuzzy neural network (NN) controller algorithm.Design/methodology/approachIn evolved fuzzy NN modeling, the NN model and linear differential inclusion representation are established for the arbitrary nonlinear dynamics. The control problems of the Fisher equation and a temperature cooling fin for high-speed aerospace vehicles will be described and demonstrated. The signal auxiliary controlled system is represented for the nonlinear parabolic partial differential equation (PDE) systems and the criterion of stability is derived via the Lyapunov function in terms of linear matrix inequalities.FindingsThis representation is constructed by sector nonlinearity, which converts the nonlinear model to a multiple rule base for the linear model and a new sufficient condition to guarantee the asymptotic stability.Originality/valueThis study also injects high frequency as an auxiliary and the control performance to stabilize the nonlinear high-speed aerospace vehicle system.

Reinforcement learning-based nonlinear tracking control system design via LDI approach with application to trolley system

In this paper, a novel scheme for the tracking problem of nonlinear systems is proposed. First, as a new technology of neural network in control field, linear differential inclusion is used to approximate the nonlinear term for the entire system. Based on the equivalent linear system, tracking reference signal is given and a new augmented system is built. According to the mentioned value function, two reinforcement learning algorithms are proposed to design the optimal control law. Notice that the online algorithm does not involve the system dynamics and tracking dynamics. In the simulation section, the model of trolley system is given to prove the effectiveness and accuracy of the scheme proposed in this paper.

Adaptive optimization algorithm for nonlinear Markov jump systems with partial unknown dynamics

AbstractAn online adaptive optimal control problem for a class of nonlinear Markov jump systems (MJSs) is studied. It is worth noting that the dynamic information of MJSs is partially unknown. Applying the neural network linear differential inclusion techniques, the nonlinear terms in MJSs are approximately converted to linear forms. By using subsystem transformation schemes, we can transfer the nonlinear MJSs to N new coupled linear subsystems. Then a new online policy iteration algorithm is put forward to obtain the adaptive optimal controller. Some theorems are given afterward to ensure the convergence of the new algorithm. At last, a simulation example is provided to verify the applicability of the algorithm.

Stability Analysis of an LTI System with Diagonal Norm Bounded Linear Differential Inclusions

In this article, we present a stability analysis of linear time-invariant systems in control theory. The linear time-invariant systems under consideration involve the diagonal norm bounded linear differential inclusions. We propose a methodology based on low-rank ordinary differential equations. We construct an equivalent time-invariant system (linear) and use it to acquire an optimization problem whose solutions are given in terms of a system of differential equations. An iterative method is then used to solve the system of differential equations. The stability of linear time-invariant systems with diagonal norm bounded differential inclusion is studied by analyzing the Spectrum of equivalent systems.

The nonsmoth optimal control problem for ensamble of trajectories of dynamic system under conditions of indeterminaci

In the paper we consider the one model of dynamic system under conditions of indeterminacy – linear controllable differential inclusions. For the informational model of the control system the minimax control problem for ensemble trajectories is researched. This control problem is study with a methods nonsmooth and multi-value analysis. The necessary and sufficient conditions of optimality are obtained.

Quadratic Stability of Non-Linear Systems Modeled with Norm Bounded Linear Differential Inclusions

In this article we present an ordinary differential equation based technique to study the quadratic stability of non-linear dynamical systems. The non-linear dynamical systems are modeled with norm bounded linear differential inclusions. The proposed methodology reformulate non-linear differential inclusion to an equivalent non-linear system. Lyapunov function demonstrate the existence of a symmetric positive definite matrix to analyze the stability of non-linear dynamical systems. The proposed method allows us to construct a system of ordinary differential equations to localize the spectrum of perturbed system which guarantees the stability of non-linear dynamical system.

Control design with guaranteed transient performance: An approach with polyhedral target tubes

In this paper a novel approach is presented for control design with guaranteed transient performance for multiple-input multiple-output discrete-time linear polytopic difference inclusions. We establish a theorem that gives necessary and sufficient conditions for the state to evolve from one polyhedral subset of the state-space to another. Then we present an algorithm which constructs a time-varying output feedback law which guarantees that the state evolves within a time-varying polyhedral target-tube specifying the system’s desired transient performance. We present generalisations involving constraints on the control, and a bounded additive disturbance term. Our formulation is very general and includes reference tracking with any desired transient behaviour in the face of disturbances, as specified, for example, by the most popular step response specifications. The approach is demonstrated by an example involving the control of water levels in two coupled tanks.

Regional consensus of linear differential inclusions subject to input saturation

SummaryIn this article, we consider regional consensus problem for a group of identical linear systems represented by a linear differential inclusion over an undirected communication topology. Each vertex system of the linear differential inclusion is represented by a general linear system subject to input saturation, and hence only regional consensus can be achieved. For given saturated distributed linear control protocols, we establish a set of conditions under which these control protocols achieve regional consensus and a level set of a Laplacian quadratic function can be used as an estimate of the domain of consensus. These conditions are given in the form of matrix inequalities and involve the properties of the communication topology. Based on these matrix inequalities, we formulate a linear matrix inequalities based optimization problem for obtaining as large an estimate of the domain of consensus as possible. By viewing the gain matrix in the consensus algorithms as an additional variable, this optimization problem can be adapted for the design of the consensus protocols. Simulation results illustrate the effectiveness of our proposed approach.

RMPC for Uncertain Nonlinear Systems With Non-Additive Dynamic Disturbances and Noisy Measurements

In this paper, we present a robust, model-predictive control scheme for the general class of uncertain and constrained discrete-time nonlinear systems subject to noisy measurements. The relationships between the system's dynamics, uncertainties, disturbances and the measurement noise are nonlinear and not necessarily additive. In particular, the disturbance is the output of an uncertain system with an unknown input. This study serves the threefold ultimate objective of ensuring robust satisfaction of the state constraints, recursive feasibility and stability. To satisfy state constraints, the proposed algorithms adopt a constraints tightening approach using the restricted constraint sets computed online. Several bounds on the prediction level and rate are derived and the size of the terminal region is maximized using polytopic linear differential inclusions (PLDI). An explicit bound on the maximum allowable disturbance for recursive feasibility is also derived based on optimization of the one-step ahead controllable set to the terminal region. The disturbance and uncertainties are non-vanishing and therefore only Input-to-state practical stability (ISpS) can be ensured. A simulation example demonstrates the efficacy of the mathematical framework and algorithms developed in this work.

Adaptive optimal controller design for a class of LDI-based neural network systems with input time-delays

In this paper, a new online adaptive optimal controller design scheme is studied for a class of nonlinear systems with input time-delays. First, we linearize the original nonlinear systems by means of linear differential inclusion technique. Then the adaptive optimal controller of the linearized systems with input time-delays is obtained by online policy iteration algorithm. It also proves the convergence of the designed adaptive optimal control algorithm. Finally, the effectiveness of the proposed method is verified by two simulation examples.