Nonlinear Time-invariant Systems Research Articles

Adaptive Control With Global Exponential Stability for Parameter-Varying Nonlinear Systems Under Unknown Control Gains.

It is nontrivial to achieve exponential stability even for time-invariant nonlinear systems with matched uncertainties and persistent excitation (PE) condition. In this article, without the need for PE condition, we address the problem of global exponential stabilization of strict-feedback systems with mismatched uncertainties and unknown yet time-varying control gains. The resultant control, embedded with time-varying feedback gains, is capable of ensuring global exponential stability of parametric-strict-feedback systems in the absence of persistence of excitation. By using the enhanced Nussbaum function, the previous results are extended to more general nonlinear systems where the sign and magnitude of the time-varying control gain are unknown. In particular, the argument of the Nussbaum function is guaranteed to be always positive with the aid of nonlinear damping design, which is critical to perform a straightforward technical analysis of the boundedness of the Nussbaum function. Finally, the global exponential stability of parameter-varying strict-feedback systems, the boundedness of the control input and the update rate, and the asymptotic constancy of the parameter estimate are established. Numerical simulations are carried out to verify the effectiveness and benefits of the proposed methods.

Coordinate Transformation-Based Nonlinear Power Decoupling Control for Differential Buck Grid-Tied Inverter

Nonlinear time-varying systems complicate the analysis and controller design compared to nonlinear time-invariant systems. Single-phase differential grid-tied inverter is a classic nonlinear time-varying system. In this article, an input-output linearization (IOL) based coordinate transformation and a nonlinear power decoupling control scheme are proposed for the single-phase differential grid-tied inverter. Because of the transformation, the nonlinear periodic time-varying system is transformed into a nonlinear time-invariant one. Based on the nonlinear time-invariant model, a nonlinear power decoupling control scheme based on the idea similar to back-stepping control is designed, which realizes good grid current control and power decoupling. Furthermore, the small-signal stability of the inverter is proved by eigenvalue analysis method in the linear time-invariant (LTI) framework. Experiment results validate the effectiveness of the proposed control strategy and stability analysis.

On the Existence of Invariantly Differentiable Functionals for Globally Exponentially Stable Time-Delay Systems

This note argues over the existence of invariantly differentiable functionals for globally exponentially stable time-delay systems. We show that the existence of invariantly differentiable, Lipschitz on bounded sets Lyapunov-Krasovskii functional is necessary and sufficient condition for the global exponential stability of time-invariant nonlinear systems described by retarded functional differential equations with discrete type delay. The presented converse results are in terms of the invariant derivative definition. To prove converse results, both point-wise and history-wise dissipation conditions are provided. The validity of the results is illustrated by an example.

Goursat Normal Form and Linearization of Single-Input Nonlinear Control Systems via Time Scaling and Prolongation

We address the problem of linearizing a single-input time-invariant nonlinear control system by invertible state and input transformations and invertible time scaling. We prove a necessary and sufficient condition for linearizability of single-input control systems in the above-mentioned class of transformations. We show that the problem can be solved by constructing the derived flag of the Pfaffian system I, which is obtained by eliminating the time differential from the Pfaffian system naturally associated with the control system. We prove that, for regular points of the derived flag of I, a single-input nonlinear control system is linearizable in the above-mentioned class of transformations if and only if I is diffemorphic to the Goursat normal form and an additional condition holds for the (n - 2)th derived system of I where n is the dimension of the state manifold. We prove that if I is diffeomorphic to the Goursat normal form but the additional condition is not met, the control system can be linearized in the above-mentioned class of transformations after a prolongation.

Extension of the Observability Rank Condition to Time-Varying Nonlinear Systems

This article provides an extension of the observability rank condition to time-varying nonlinear systems. Previous conditions to check the state observability only hold for nonlinear systems that do not explicitly depend on time, or, for time-varying systems, they only account for the linear case. In this article, a general analytic condition is provided. The new condition reduces to the well-known observability rank condition in the two simpler cases of time-varying linear systems and time-invariant nonlinear systems. The proposed condition is applied to aerospace robotics to study the observability of a lunar-module-type system equipped with monocular vision and inertial sensors.

Exceeding the classical time-bandwidth product in nonlinear time-invariant systems

The classical “time-bandwidth” limit for linear time-invariant (LTI) devices in physics and engineering asserts that it is impossible to store broadband propagating waves (large $${\Delta }\omega$$ ’s) for long times (large Δt’s). For standing (non-propagating) waves, i.e., vibrations, in particular, this limit takes on a simple form, $${\Delta }t \,{\Delta }\omega = 1$$ , where $${\Delta }\omega$$ is the bandwidth over which localization (energy storage) occurs, and $${\Delta }t$$ is the storage time. This is related to a well-known result in dynamics, namely that one can achieve a high Q-factor (narrowband resonance) for low damping, or small Q-factor (broadband resonance) for high damping, but not simultaneously both. It thus remains a fundamental challenge in classical wave physics and vibration engineering to try to find ways to overcome this limit, not least because that would allow for storing broadband waves for long times, or achieving broadband resonance for low damping. Recent theoretical studies have suggested that such a feat might be possible in LTI terminated unidirectional waveguides or LTI topological “rainbow trapping” devices, although an experimental confirmation of either concept is still lacking. In this work, we consider a nonlinear but time-invariant mechanical system and demonstrate experimentally that its time-bandwidth product can exceed the classical time-bandwidth limit, thus achieving values both above and below unity, in an energy-tunable way. Our proposed structure consists of a single-degree-of-freedom nonlinear oscillator, rigidly coupled to a nondispersive waveguide. Upon developing the full theoretical framework for this class of nonlinear systems, we show how one may control the nonlinear flow of energy in the frequency domain, thereby managing to disproportionately decrease (increase) $${\Delta }t$$ , the storage time in the resonator, as compared with an increase (decrease) of the system’s bandwidth $${\Delta }\omega$$ . Our results pave the way toward conceiving and harnessing hitherto unattainable broadband and simultaneously low-loss wave-storage devices, both linear and nonlinear, for a host of key applications in wave physics and engineering.

Introducing a Novel Model-Free Multivariable Adaptive Neural Network Controller for Square MIMO Systems.

In this study, a novel Multivariable Adaptive Neural Network Controller (MANNC) is developed for coupled model-free n-input n-output systems. The learning algorithm of the proposed controller does not rely on the model of a system and uses only the history of the system inputs and outputs. The system is considered as a ‘black box’ with no pre-knowledge of its internal structure. By online monitoring and possessing the system inputs and outputs, the parameters of the controller are adjusted. Using the accumulated gradient of the system error along with the Lyapunov stability analysis, the weights’ adjustment convergence of the controller can be observed, and an optimal training number of the controller can be selected. The Lyapunov stability of the system is checked during the entire weight training process to enable the controller to handle any possible nonlinearities of the system. The effectiveness of the MANNC in controlling nonlinear square multiple-input multiple-output (MIMO) systems is demonstrated via three simulation studies covering the cases of a time-invariant nonlinear MIMO system, a time-variant nonlinear MIMO system, and a hybrid MIMO system, respectively. In each case, the performance of the MANNC is compared with that of a properly selected existing counterpart. Simulation results demonstrate that the proposed MANNC is capable of controlling various types of square MIMO systems with much improved performance over its existing counterpart. The unique properties of the MANNC will make it a suitable candidate for many industrial applications.

A Predictor Based Linearization Approach for Local Stabilization of Nonlinear Systems With Input Delay

In this paper, a predictor based linearization approach is presented to solve the local asymptotic stabilization (LAS) problem of time-invariant nonlinear systems with a large constant input delay. Under stabilizability and detectability of the linearized time-delay system, we prove that LAS is achievable by either predictor based state or output feedback controllers, which can be designed directly based on the linearization. The predictor based linearization technique developed in this research complements the work [1] and improves its result by removing the forward completeness condition and a number of technical requirements as well as by expanding the class of nonlinear systems with input delay to the general case. An example is given to demonstrate the application of the predictor based linearization technique.

Research on Photovoltaic Grid-Connected Control Strategy Based on Active Disturbance Rejection of Adaptive Extended State Observer

In the photovoltaic inverter grid-connected power generation system, the output power of photovoltaic panels is affected by illumination and temperature. The change of output power of photovoltaic panels will lead to the fluctuation of DC bus voltage. If the control is improper, it will directly affect the regular operation of the system. In order to improve the performance of the grid-connected inverter system, an active disturbance rejection control method based on adaptive extended state observer (ESO) is proposed. Firstly, a feedforward PI current inner loop controller is designed, which simplifies the structure of the control system and improves the tracking performance of the current. Then, the DC bus voltage outer loop ADRC is designed, and a conversion method that ignores the essential difference between nonlinear/time-varying and time-varying/linear is proposed. Through the conversion of time-invariant nonlinear system and time-varying linear system, the stability of the extended state observer is proved by the Routh criterion. Secondly, to solve the problem of mutual restriction between the stability and observation accuracy of the extended state observer, an adaptive function online automatic tuning ESO parameter method is proposed. Finally, the simulation results show that the proposed method has better dynamic and static performance, and the grid-connected voltage and current harmonics are small, which proves the correctness and effectiveness of the proposed method.

Lyapunov–Perron Transformation for Quasi-Periodic Systems and Its Applications

Abstract This paper depicts the application of symbolically computed Lyapunov–Perron (L–P) transformation to solve linear and nonlinear quasi-periodic systems. The L–P transformation converts a linear quasi-periodic system into a time-invariant one. State augmentation and the method of normal forms are used to compute the L–P transformation analytically. The state augmentation approach converts a linear quasi-periodic system into a nonlinear time-invariant system as the quasi-periodic parametric excitation terms are replaced by “fictitious” states. This nonlinear system can be reduced to a linear system via normal forms in the absence of resonances. In this process, one obtains near identity transformation that contains fictitious states. Once the quasi-periodic terms replace the fictitious states they represent, the near identity transformation is converted to the L–P transformation. The L–P transformation can be used to solve linear quasi-periodic systems with external excitation and nonlinear quasi-periodic systems. Two examples are included in this work, a commutative quasi-periodic system and a non-commutative Mathieu–Hill type quasi-periodic system. The results obtained via the L–P transformation approach match very well with the numerical integration and analytical results.

A Novel Real-Time Filtering Method to General Nonlinear Filtering Problem Without Memory

In this paper, the filtering problem for the general time-invariant nonlinear state-observation system is considered. Our work is based on the Yau-Yau filtering framework developed by S.-T. Yau and the third author in 2008. The key problem of Yau-Yau filtering framework is how to compute the solution to forward Kolmogorov equation (FKE) off-line effectively. Motivated by the supervised learning in machine learning, we develop an efficient method to numerically solve the FKE off-line from the point of view of optimization. Specifically, for the off-line computation part, the computation of the solution to a FKE is reduced to computing a linear system of equations by making the temporal inverse transformation and the loss function optimization, and we store the results for the preparation of on-line computation. For the on-line computation part, the unnormalized density function is approximated by a complete polynomial basis, and then the estimation of the state is computed using the stored off-line data. Our method has the merits of easily implementing, real-time and memoryless. More importantly, it can be applicable for moderate-high dimensional cases. Numerical experiments have been carried out to verify the feasibility of our method. Our algorithm outperforms extended Kalman filter, unscented Kalman filter and particle filter both in accuracy and costing time.

Computing robust control invariant sets of constrained nonlinear systems: A graph algorithm approach

This paper deals with the computation of the largest robust control invariant sets (RCISs) of constrained nonlinear systems. The proposed approach is based on casting the search for the invariant set as a graph theoretical problem. Specifically, a general class of discrete-time time-invariant nonlinear systems is considered. First, the dynamics of a nonlinear system is approximated with a directed graph. Subsequently, the condition for robust control invariance is derived and an algorithm for computing the robust control invariant set is presented. The algorithm combines the iterative subdivision technique with the robust control invariance condition to produce outer approximations of the largest robust control invariant set at each iteration. Following this, we prove convergence of the algorithm to the largest RCIS as the iterations proceed to infinity. Based on the developed algorithms, an algorithm to compute inner approximations of the RCIS is also presented. A special case of input affine and disturbance affine systems is also considered. Finally, two numerical examples are presented to demonstrate the efficacy of the proposed method.

Novel robust disturbance observer

In this paper, a predictive disturbance observer (DOB) design is presented. The proposed DOB does not require a direct model of the process without the transport delay for prediction purposes. The primary advantage of the proposed DOB design is its simplicity due to usage of the prediction block (PB), which approximates the ideal anticipation object. Furthermore, the entire DOB becomes robust to external disturbances because the DOB is based on the approximation of the anticipation object. The response accuracy of the proposed predictive DOB is directly dependent on the inverse model parameters and prediction coefficients in the PB. Notwithstanding the simplification, convenient implementation, identification, and optimization structure, the proposed solution allows for direct implementation of process models describing discontinuous nonlinear time-invariant systems.

Series Bridge Converter: Energy Rating Optimality for VSC-HVDC Applications

This letter investigates the optimality of the series bridge converter (SBC) voltage source converter-high voltage direct current (VSC-HVdc), which constitutes a nonlinear time-invariant (NTI) system. The contribution of this letter is the proof that SBC VSC-HVdc is the optimal NTI topology in the sense of energy rating. Namely, it is shown that SBC VSC-HVdc constitutes an optimal lower bound in the amount of joules installed in the converter hardware per dc output volt in the field of nonlinear converter topologies, and it may only be outperformed by time-varying VSC-HVdc topologies.

A Method for Solving a Local Boundary-Value Problem for a Nonlinear Controlled System

A wide class of nonlinear time-invariant systems of ordinary differential equations is considered. A rather convenient algorithm for constructing a differentiable control function that performs a guaranteed transition of such systems from an initial state to a given terminal state of the state space under control constraints is proposed. A constructive criterion that guarantees the above-mentioned translation is obtained. The efficiency of this algorithm is illustrated by numerical solution of a specific practical problem.

Topological entropy and minimal data rate for state observation of LTV systems

In this paper, we show that the topological entropy of linear time-varying systems coincide with their minimal data rate for state observation, thereby extending the well-known “observation data rate” theorem for LTI systems and time-invariant nonlinear systems with compact domain. This result is relevant for the problem of controlling and observing dynamical systems via limited-capacity communication networks, as it provides a tight bound on the data rate required for the state observation of these systems. This bound, which relies only on the topological properties of the system, can thus be used as a benchmark for the comparison of different implementations of coders–decoders observing the system.

Frequency domain response under arbitrary excitation for fading memory nonlinear systems

For dynamic systems, the steady-state system response to periodic excitation is well understood for both linear and certain nonlinear system classes. When the excitation is not periodic, however, the measured response will contain both transient and steady-state contributions. For linear systems, these transient contributions have been thoroughly explored, while no equivalent analysis exists for the nonlinear case. In this paper, we derive an expression in the frequency domain for the system response of all discrete time-invariant nonlinear systems which have fading memory, using the Volterra series representation. The expression contains both steady-state and transient contributions at each nonlinear order, revealing a highly structured view of nonlinear system response. For the nonlinear case, the transient expressions at higher nonlinear orders have a more complex structure than those generated by linear systems, which provides valuable insight for systems theory and identification purposes.

Model reference adaptive state-dependent Riccati equation control of nonlinear uncertain systems: Regulation and tracking of free-floating space manipulators

This work proposes a model reference adaptive state-dependent Riccati equation (SDRE) controller for nonlinear time-invariant systems considering uncertainty in the plant. The SDRE is vulnerable to uncertainty of nonlinear model, hence the adaptive structure is to compensate for the difference between a reference model and a real uncertain system. The application of the proposed method is dedicated to controlling a free-floating space manipulator (FFSM), a robotics system with a base in two modes: an inactive (no actuation or thrust) state, and a base reaction torque mode. A non-actuated heavy base FFSM probably performs a regulation or a tracking task precisely though that might not be a good solution. A new design is proposed for FFSM control: a heavy base is selected for base of a reference model and the uncertain system with proper weight of base will follow the reference system; and considering three motors for rotating back the base when it reorients from initial position. The new design has improved the precision of the FFSM and reduced the weight of the robot. A planar two degree-of-freedom and a 3D Stanford arm were modeled, simulated and analyzed to assess the performance of the proposed structure and adaptive SDRE controller; that successfully confirmed the claimed expectations.

Control of nonlinear systems with compensation of disturbances under measurement noises

ABSTRACTThe control algorithm for nonlinear time-invariant systems with compensation of unknown disturbances under measurement noises is proposed. The dimension of noises is equal to the state vector dimension of the plant. Disturbances can be presented in any equation of the plant model. The novel control law is based on the noise estimator and the disturbance compensator. The accuracy in the steady state depends on the first derivative of disturbance and the smallest component of noise vector. A sufficient condition in terms of linear matrix inequality provides feasibility of the proposed algorithm. The simulations show the efficiency of the proposed method compared with some existing ones.

Minimal realizations of nonlinear systems

The nonlinear realization theory is recasted for time-varying single-input single-output nonlinear systems. The concept of realization has been extended to cover also the realizations with order greater than the order of input–output equation. The minimal realization problem is studied. The state realization is said to be minimal if it is either accessible and observable or its state dimension is minimal. In the linear case the two definitions are equivalent, but not for nonlinear time-invariant systems. It is shown that the two definitions remain equivalent for nonlinear systems under certain technical assumptions. Two alternative methods are presented for finding the minimal realization.