Tracking Problem Of Nonlinear Systems Research Articles

Model reference output feedback tracking for nonlinear systems via LMIs

This work addresses the tracking problem for nonlinear systems with unmeasured variables. We offer a new design methodology that is composed of a two-degrees-of-freedom scheme. The reference model must track a time-varying signal, and the controller employed can be regarded as one-degree-of-freedom. Then, based on the error between the outputs of the reference model and the plant, another controller is designed to ensure that this mismatch converges to zero. The reference model and the system are described as Takagi–Sugeno (TS) fuzzy models with nonlinear consequents (N-TS), which allows for separating measured from unmeasured variables. An advantage of the proposed scheme is that the controller for the reference model can be carried out using several methodologies from the literature, whereas the controller that governs the output error can be static or dynamic, whose design is based on Linear Matrix Inequalities (LMIs). Two examples are used to illustrate the validity of the developed control methodology and to compare it to another recent approach in the literature.

Observer-based preview control for T-S fuzzy systems

PurposeConsidering the unmeasurable states of the systems and the previewed reference signal, a novel fuzzy observer-based preview controller, which is a mixed controller of the fuzzy observer-based controller, fuzzy integrator and preview controller, is considered to address the tracking control problem.Design/methodology/approachThe authors employ an augmentation technique to construct an augmented error system for uncertain T-S fuzzy discrete-time systems with time-varying uncertainties. Additionally, the authors obtain the corresponding linear matrix inequality (LMI) conditions for designing the preview controller.FindingsThis paper discusses the preview tracking problem for nonlinear systems. First, considering the unmeasurable states of the systems and the previewed reference signal, a novel fuzzy observer-based preview controller, which is a mixed controller of the fuzzy observer-based controller, fuzzy integrator, and preview controller, is considered to address the tracking control problem. Then, using the fuzzy Lyapunov functional with the linear matrix inequality (LMI) technique, new sufficient conditions for the asymptotic stability of the augmented system are derived by applying the LMI technique. The preview controller and fuzzy observer can be designed in one step. Finally, a numerical example is used to illustrate the effectiveness of the results.Originality/valueAn augmented error system is successfully constructed by the state augmentation approach. A novel preview controller is designed to address the tracking control problem. The preview controller and fuzzy observer can be designed in one step.

Prescribed-time adaptive output feedback tracking control for nonlinear systems with quantized input

This study concentrates on the prescribed-time output feedback tracking problem for nonlinear systems with unknown control gains and quantized input. A modified observer with the first-order scale function handles the unmeasurable problem of states while achieving accurate observation in the prescribed time. A simple controller is constructed by a new function transformation in which the computation burden is greatly reduced. The resulting controller not only drives that tracking error converges to zero but also ensures that all signals in the closed-loop system are bounded. Finally, the effectiveness of the control scheme is demonstrated by a numerical simulation and a practical simulation.

Adaptive Asymptotic Tracking Control for Input-Quantized Nonlinear Systems With Multiple Unknown Control Directions.

This study mainly concentrates on adaptive asymptotic tracking control for input-quantized strict-feedback nonlinear systems subjected to multiple unknown control directions. Novel improved lemmas, which relax the conditions for handling unknown control coefficients in the existing theoretical results, are certificated that can be applied to resolve the tracking problem for nonlinear systems under input quantification and unknown control directions simultaneously. Furthermore, by incorporating positive integral time-varying functions and the disintegration of the hysteresis quantizer into the controller design, the asymptotic tracking control is successfully achieved. Moreover, all signals in the closed-loop system are guaranteed to be bounded. Ultimately, a comparing numerical simulation and a practical simulation of a Nomoto ship model are presented to validate the feasibility of the proposed control algorithm.

Adaptive Output Consensus of Heterogeneous Nonlinear Multiagent Systems: A Distributed Dynamic Compensator Approach

Distributed dynamic compensators play a key role in the output consensus problem of heterogeneous nonlinear multi-agent systems. The existing distributed dynamic compensators require the compensators' information to be exchanged through communication networks, and to satisfy additional stability conditions in the controller design. This note proposes a distributed dynamic compensator to address the adaptive output consensus problem of heterogeneous nonlinear multi-agent systems with unknown parameters. The distributed dynamic compensator only requires the output information to be exchanged through communication networks. In addition, the distributed dynamic compensator converts the original consensus problem into asymptotic tracking problem of nonlinear systems, without requiring extra stability conditions in the literature. Then, well-established and standard control approaches are used to design tracking controllers and solve the standard tracking problems. The current work adopts the standard adaptive backstepping approach as an example to design an adaptive tracking controllers. It is further proved that all signals in the closed-loop system are globally uniformly bounded, and the proposed scheme enables the outputs of all agents to track the leader's signal asymptotically. A simulation is presented to illustrate the effectiveness of the design methodology.

Iterative Learning Control for Output Tracking of Nonlinear Systems With Unavailable State Information.

This work presents a novel design framework of adaptive iterative learning control (ILC) approach for a class of uncertain nonlinear systems. By using the closed-loop reference model that can be viewed as an observer, the proposed adaptive ILC approach can be adapted to deal with the output tracking problem of nonlinear systems with unavailable system states. In the systems considered, two classes of uncertainties are taken into account, including parametric input disturbances and input distribution uncertainties. To facilitate the controller design and convergence analysis, the composite energy function (CEF) methodology is employed. The design framework in this brief is novel and widely applicable, which extends the CEF-based ILC approach to output tracking control of nonlinear systems without requiring full knowledge of state information and complicated observer design process. To show the effectiveness of the proposed design framework and control algorithms, two numerical examples are illustrated.

Command filtered adaptive output feedback design with novel Lyapunov-based analysis for nonlinear systems with unmodeled dynamics

This paper presents a novel Lyapunov function-based backstepping controller design to tackle the tracking problems for nonlinear systems with unmodeled dynamics and unmeasurable states. The coexistence of unmodeled dynamics and unmeasurable states is the main challenge, which calls for novel techniques to take these two factors into account simultaneously. First, the classical Luenberger observer is extended with a novel transformation function to decouple the original system state and state estimation error. In this way, the effect of unmodeled dynamics on system stability can be separately considered. On this basis, a command-filtered controller is designed to simplify the backstepping design procedures. It is worthy to pointed out that, a novel Lyapunov function is developed to simplify the stability analysis with command filter, where the filter errors, the observer error, compensated tracking errors, and parameter estimation errors can be guaranteed to be semi-globally uniformly ultimate bounded. The simulation studies are investigated to validate the effectiveness of the presented design scheme.

Dynamic Event-Triggered Integral Sliding Mode Adaptive Optimal Tracking Control for Uncertain Nonlinear Systems

In this paper, we study the event-triggered integral sliding mode optimal tracking problem of nonlinear systems with matched and unmatched disturbances. The goal is to design an adaptive dynamic programming-based sliding-mode controller, which stabilizes the closed-loop system and guarantees the optimal performance of the sliding-mode dynamics. First, in order to remove the effects of the matched uncertainties, an event-triggered sliding mode controller is designed to force the state of the systems on the sliding mode surface without Zeno behavior. Second, another event-triggered controller is designed to suppress unmatched disturbances with a nearly optimal performance while also guaranteeing Zeno-free behavior. Finally, the benefits of the proposed algorithm are shown in comparison to several traditional triggering and learning-based mechanisms.

Constrained fast finite‐time exact tracking for disturbed nonlinear systems

AbstractThis article investigates the fast finite‐time exact tracking problem for nonlinear systems with time‐varying disturbances. A new composite controller with a finite‐time disturbance observer is proposed to achieve fast finite‐time exact tracking. Furthermore, a new time‐varying barrier power integrator technique is developed to design the controller. This can ensure not only asymmetric/symmetric time‐varying full‐state constraints but also singular‐free continuous control effort. To reduce the complexity of computation, fuzzy logic systems are adopted to approximate complex nonlinearities. Finally, an example of single‐link robot systems subject to time‐varying disturbances is presented to show the validity of the proposed method.

Control Refinement Using Particle Methods

The paper presents a control scheme for the real-time tracking problem of nonlinear systems subjected to hard nonlinearities. The proposed tracking controller introduces a refining component in the control input designed for the nominal plant model. The refining component compensates for tracking performance degradation caused by modelling uncertainties and external disturbances. The refining component is modelled as a random signal, the probability density function is expressed as a combination of finite weights typical of particle methods. The weights are updated based on sequential tracking error data. The proposed algorithm is simulated for an inverted pendulum affected by Coulomb friction. Comparison with existing techniques exhibits remarkably superior tracking performance.

Iterative Learning Fuzzy Control for Nonlinear Systems with Adaptive Gain and without Resetting Condition

In this paper, we present two ILC schemes. The first one is a PD-type iterative learning control with an initial state algorithm to solve the trajectory tracking problem for nonlinear systems with uncertainties and without satisfying the classical resetting condition. -norm method is used to prove the asymptotic stability of the closed loop system and the simulation results on perturbed nonlinear system have been given. The second approach is a simple P-type iterative learning fuzzy control scheme to solve the trajectory tracking problem for MIMO nonlinear systems. The control design is applicable to deal with a class of nonlinear systems without satisfying the global Lipschitz continuity condition, for which a fuzzy logic term is added to cope with unknown parameters. In addition, the swarm optimization algorithm is used to design the optimum iterative learning fuzzy control (ILFC). Using Lyapunov theory, the asymptotic stability of the closed loop system is guaranteed over the whole finite time. Finally, an illustrative example on two-link manipulator is provided to illustrate the effectiveness of the proposed controller.

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.

Iterative Learning Control for Output Tracking of Systems with Unmeasurable States

In this work, a new design framework of adaptive iterative learning control (ILC) approach for a class of uncertain nonlinear systems is presented. By making use of the closed-loop reference model which works as an observer, the developed adaptive ILC method is able to be adopted to deal with the output tracking problem of nonlinear systems without requiring the measurability of system states. In the system, the uncertainties are formed by the product of unknown parameters and state functions that are also unknown as the system states are not available. In order to facilitate the controller design and convergence analysis, the composite energy function (CEF) method is employed, and the accurate tracking task can be realized successfully. The proposed approach extends CEF-based ILC approach sucessfully to output tracking control of nonlinear systems without requiring the system states information and complicated observer design. The effectiveness of the proposed ILC scheme is verified through an illustrative numerical example.

Continuous Output Feedback TSM Control for Uncertain Systems With a DC–AC Inverter Example

This brief proposes a continuous output feedback terminal sliding mode (TSM) control method for the output tracking problem of nonlinear systems with mismatched uncertainties. This method is developed by two consecutive steps: 1) design a sliding mode observer to estimate the derivative and high-order derivatives of system output and 2) construct a continuous TSM controller based on system output and the estimations of its high-order derivatives. The proposed controller not only guarantees system output converges to its reference in finite time but also keeps the continuity of control action. Experiments on single-phase dc–ac inverter circuits are also carried out to show the effectiveness of the proposed controller.

Stabilization and robustness analysis for a chain of exponential integrators using strict Lyapunov functions

We provide a new method for building strict Lyapunov functions for two dimensional chains of exponential integrators, using nested exponential functions. One challenge is that the right sides of the systems saturate, so they are not completely controllable. The strictness of the Lyapunov functions is key to proving input-to-state stability (or ISS) properties with respect to additive uncertainty on the controls. We show how a large class of tracking problems for nonlinear systems with positivity constraints on the states can be solved using our theory.

Offset-free MPC explained: novelties, subtleties, and applications

This paper presents an updated and comprehensive description of offset-free MPC algorithms for nonlinear (and linear) discrete-time systems, with the intended objectives of clarifying the main concepts, showing new results, highlighting subtleties by means of challenging applications. First, the offset-free tracking problem for nonlinear systems is presented, putting a strong accent on the role of the disturbance model and observer, and then novel and stronger offset-free estimation results are presented. Next, recent advances in linear offset-free MPC are described, which show the equivalence of the velocity form algorithm (so far considered an alternative method) to a particular disturbance model and observer. Then, the concepts of offset-free estimation are exploited to design an offset-free economic MPC algorithm, which can asymptotically achieve the highest economic performance despite persistentmodel errors and disturbances. Extensive application results are presented to show the benefits of offsetfree MPC algorithms over standard ones, and to clarify misconceptions and design errors that can prevent constraint satisfaction, closed-loop stability, and offset-free performance.

A new sliding control strategy for nonlinear system solved by the Lie-group differential algebraic equation method

For a control problem of nonlinear system ẋ=f(x,u,t), the optimal control by minimizing a performance index is reformulated to be a set of differential algebraic equations (DAEs) with the Lagrangian being partially replaced by an exponentially time-decaying constraint: L1(x,t)=A0e-αt, and meanwhile the control force is bounded by |u|⩽umax. Then, we develop an implicit GL(n,R) Lie-group DAE (LGDAE) method to find u(t) by solving the DAEs: ẋ=f(x,u,t) and L1(x,t)-A0e-αt=0. Similarly, we propose a new sliding mode control (SMC) strategy by using the LGDAE to solve the control force, where in addition to the equivalent control force we add a compensated control force which is used to quickly steer and continuously enforce the state trajectory on the sliding surface. This novel SMC is robust and is chattering-free for regulator problem and finite-time tracking problem of nonlinear systems. Furthermore, we combine the above two methods as being a two-stage controller for the forced nonlinear Duffing oscillator by stabilizing it to an equilibrium point. The present SMC together with the LGDAE is also used to stabilize the state trajectory of some uncertain chaotic systems to a desired state point. Its robustness against uncertainty is obvious.

Direct feedback control design for nonlinear systems

We propose an approach for the direct design from data of controllers finalized at solving tracking problems for nonlinear systems. This approach, called Direct FeedbacK (DFK) design, overcomes relevant problems typical of the standard design methods, such as modeling errors, non-trivial parameter identification, non-convex optimization, and difficulty in nonlinear control design. Considering a Set Membership (SM) approach, we provide three main contributions. The first one is a theoretical framework for the stability analysis of nonlinear feedback control systems, in which the controller f̂ is an approximation identified from data of an ideal inverse model fo. In this framework, we derive sufficient conditions under which f̂ stabilizes the closed-loop system. The second contribution is a technique for the direct design of an approximate controller f∗ from data, having suitable optimality, stability, and sparsity properties. In particular, we show that f∗ is an almost-optimal controller (in a worst-case sense), and we derive a guaranteed accuracy bound, which can be used to quantify the performance level of the DFK control system. We also show that, when the number of data used for control design tends to infinity and these data are dense in the controller domain, the closed-loop stability is guaranteed for a set of trajectories of interest. The technique is based on convex optimization and sparse identification methods, and thus avoids the problem of local minima and allows an efficient online controller implementation in real-world applications. The third contribution is a simulation study, regarding the application of DFK to the challenging problem of control design for a class of airborne wind energy generators.

Hermite Polynomials for Iterative Output Replanning for Flat Systems Affected by Additive Noise

AbstractThe paper considers the output tracking problem for nonlinear systems whose performance output is also a flat output of the system itself. A desired output signal is sought on the actual performance output by using a feedforward inverse input that is periodically updated with discrete‐time feedback of the sampled state of the system. The proposed method is based on an iterative output replanning that uses the desired output trajectory and the sampled state to replan an output trajectory whose inverse input helps in reducing the tracking error. This iterative replanning exploits the Hermite interpolating polynomials to achieve an overall arbitrarily smooth input and a tracking error that can be made arbitrarily small if the state sampling period is sufficiently small and mild assumptions are considered. Some simulation results are presented for the cases of a unicycle and a one‐trailer system affected by additive noise.

Uniform global asymptotic stability of adaptive cascaded nonlinear systems with unknown high-frequency gains

We study adaptive tracking problems for nonlinear systems with unknown control gains. We construct controllers that yield uniform global asymptotic stability for the error dynamics, and hence tracking and parameter estimation for the original systems. Our result is based on a new explicit, global, strict Lyapunov function construction. We illustrate our work using a brushless DC motor turning a mechanical load. We quantify the effects of time-varying uncertainties on the motor electric parameters.