Systems Of Relative Degree Research Articles

On derivative-free sample-and-hold control with prescribed performance

A feedback controller is proposed to perform output reference tracking with prescribed performance for nonlinear continuous-time systems of relative degree two. The controller is of sampled-data type, i.e., measurements are available only at sampling times – a typical situation in real systems when sensors are involved. Furthermore, only output information is available, i.e., neither the full state nor derivatives can be used for feedback. A sufficient uniform sampling rate is derived and the control consists of piecewise constant signals on the sampling intervals, i.e., zero-order hold. Feasibility of the controller and satisfaction of the control objective are rigorously proven. The controller is illustrated by a numerical example.

Modeling and Detection Scheme for Zero-Dynamics Attack on Wind Power System

In a wind farm, wind power system is key to maintaining continuous generation of point energy. Since Zero-Dynamics attack poses a major risk to future wind farms, investigation on the effect of Zero-Dynamics attack against wind power systems has both theoretical and practical importance. In this paper, we consider a scenario in which a wind power system is subject to Zero-Dynamics attack in power generation chain. Firstly, we transform a semi-direct D-PMSG into an equivalent circuit to establish mathematical model of wind power system under Zero-Dynamics attack. Then, we analyze and prove the advantages and limitations of Zero-Dynamics attack by divide the equation of state into a stable part and an unstable part with the help of system relative degree ρ. In addition, to better achieve desired attack effect, adjustments are made for Zero-Dynamics attack based on its limitations. Finally, we combine an Multiple Linear Regression (MRL) predictive control model with Byrnes-Isidori normal form to protect wind power system. The theoretical results were verified on designed small wind power system.

Discrete-Time Control Barrier Function: High-Order Case and Adaptive Case.

This article proposes the novel concepts of the high-order discrete-time control barrier function (CBF) and adaptive discrete-time CBF. The high-order discrete-time CBF is used to guarantee forward invariance of a safe set for discrete-time systems of high relative degree. An optimization problem is then established unifying high-order discrete-time CBFs with discrete-time control Lyapunov functions to yield a safe controller. To improve the feasibility of such optimization problems, the adaptive discrete-time CBF is designed, which can relax constraints on system control input through time-varying penalty functions. The effectiveness of the proposed methods in dealing with high relative degree constraints and improving feasibility is verified on the discrete-time system of a three-link manipulator.

Dynamic ILC for Linear Repetitive Processes Based on Different Relative Degrees

The current research on iterative learning control focuses on the condition where the system relative degree is equal to 1, while the condition where the system relative degree is equal to 0 or greater than 1 is not considered. Therefore, this paper studies the monotonic convergence of the corresponding dynamic iterative learning controller systematically for discrete linear repetitive processes with different relative degrees. First, a 2D discrete Roesser model of the iterative learning control system is presented by means of 2D systems theory. Then, the monotonic convergence condition of the controlled system is analyzed according to the stability theory of linear repetitive process. Furthermore, the sufficient conditions of the controller existence are given in linear matrix inequality format under different relative degrees, which guarantees the system dynamic performance. Finally, through comparison with static controllers under different relative degrees, the simulation results show that the designed schemes are effective and feasible.

Output feedback control with prescribed performance via funnel pre-compensator

We study output reference tracking of systems with high relative degree via output feedback only; this is, tracking where the output derivatives are unknown. To this end, we prove that the conjunction of the funnel pre-compensator with a minimum phase system of arbitrary relative degree yields a system of the same relative degree which is minimum phase as well. The error between the original system’s output and the pre-compensator’s output evolves within a prescribed performance funnel; moreover, the derivatives of the funnel pre-compensator’s output are known explicitly. Therefore, output reference tracking with prescribed transient behaviour of the tracking error is possible without knowledge of the derivatives of the original system’s output; via funnel control schemes for instance.

Control barrier functions for stochastic systems

Control Barrier Functions (CBFs) aim to ensure safety by constraining the control input at each time step so that the system state remains within a desired safe region. This paper presents a framework for CBFs in stochastic systems in the presence of Gaussian process and measurement noise. We first consider the case where the system state is known at each time step, and present reciprocal and zero CBF constructions that guarantee safety with probability 1. We extend our results to high relative degree systems and present both general constructions and the special case of linear dynamics and affine safety constraints. We then develop CBFs for incomplete state information environments, in which the state must be estimated using sensors that are corrupted by Gaussian noise. We prove that our proposed CBF ensures safety with probability 1 when the state estimate is within a given bound of the true state, which can be achieved using an Extended Kalman Filter when the system is linear or the process and measurement noise are sufficiently small. We propose control policies that combine these CBFs with Control Lyapunov Functions in order to jointly ensure safety and stochastic stability. Our results are validated via numerical study on a multi-agent collision avoidance scenario.

Bridging the gap between optimal trajectory planning and safety-critical control with applications to autonomous vehicles

We address the problem of optimizing the performance of a dynamic system while satisfying hard safety constraints at all times. Implementing an optimal control solution is limited by the computational cost required to derive it in real time, especially when constraints become active, as well as the need to rely on simple linear dynamics, simple objective functions, and ignoring noise. The recently proposed Control Barrier Function (CBF) method may be used for safety-critical control at the expense of sub-optimal performance. In this paper, we develop a real-time control framework that combines optimal trajectories generated through optimal control with the computationally efficient CBF method providing safety guarantees. We use Hamiltonian analysis to obtain a tractable optimal solution for a linear or linearized system, then employ High Order CBFs (HOCBFs) and Control Lyapunov Functions (CLFs) to account for constraints with arbitrary relative degrees and to track the optimal state, respectively. We further show how to deal with noise in arbitrary relative degree systems. The proposed framework is then applied to the optimal traffic merging problem for Connected and Automated Vehicles (CAVs) where the objective is to jointly minimize the travel time and energy consumption of each CAV subject to speed, acceleration, and speed-dependent safety constraints. In addition, when considering more complex objective functions, nonlinear dynamics and passenger comfort requirements for which analytical optimal control solutions are unavailable, we adapt the HOCBF method to such problems. Simulation examples are included to compare the performance of the proposed framework to optimal solutions (when available) and to a baseline provided by human-driven vehicles with results showing significant improvements in all metrics.

Identification of time-varying signals in quantum systems

The identification of time-varying parameters (e.g., directly inaccessible in situ signals in vacuum and low-temperature environments) is prevalent for characterizing the dynamics of quantum processes. Under certain circumstances, they can be identified from time-resolved measurements via Ramsey interferometry experiments, but only with specially designed probe systems can the parameters be explicitly read out, and a rigorous identifiability analysis is lacking, i.e., whether the measurement data are sufficient for unambiguous identification. In this paper we formulate this problem as the invertibility of the input-output mapping associated with the quantum system for which an algebraic identifiability criterion is derived based on the system's relative degree. The invertibility analysis also leads to an inversion-based algorithm for numerically identifying the parameters, which is computationally much more efficient than nonlinear regression methods. The effectiveness of the criterion are demonstrated by numerical examples.

Direct Adaptive Fuzzy Control Scheme With Guaranteed Tracking Performances for Uncertain Canonical Nonlinear Systems

In our recent work, we propose an indirect adaptive fuzzy control scheme for uncertain unparametrizable nonlinear systems, which ensures that the number of adaptive laws does not increase with the number of fuzzy rules, and the time derivative of the chosen Lyapunov function is negative semidefinite. However, the scheme involves a class of high-order smooth functions and their time derivatives, which can make the controller structure become sophisticated especially when the relative degree of system is quite large. To overcome this problem, in the article, we propose a new direct adaptive fuzzy control scheme based on a class of reduced-order smooth functions. With the scheme, no partial-derivative term is involved in controller and virtual controllers, only one adaptive law is used regardless of the increase of fuzzy rules, and also the time derivative of Lyapunov function can be ensured negative semidefinite. It is proved that all closed-loop signals are bounded, and the output tracking error converges to a prescribed interval asymptotically. The transient tracking performance and robustness of the proposed control scheme are also considered. The effectiveness of the obtained results is illustrated by two practical control systems.

Event-Triggered Sliding Mode Control of Fuzzy Systems via Artificial Time-Delay Estimation

In this article, we show that a fuzzy system of relative degree two can be stabilized by proportion-integration-differentiation (PID) sliding mode control depending on the outputs and their derivatives. The main focus of this article is to formulate a novel event-based fuzzy PID sliding surface under the imperfect premise matching. First, the measured output is adequately sampled for frequently checking a delay-dependent event-triggered condition, which reduces the number of data transmissions and controller updates. Then, an artificial time-delay estimator is used to approximate the output derivative. Based on linear matrix inequalities, a heuristic algorithm is investigated to design an event-triggered sliding mode controller, which reveals how to choose the maximum sampling interval that maintains the estimation accuracy and preserves the stability under fast event triggering. In simulation, the effectiveness of the proposed design method is verified in a cart and pendulum system subject to system uncertainties.

Contraction Mapping-Based Robust Convergence of Iterative Learning Control With Uncertain, Locally Lipschitz Nonlinearity

This paper studies the output tracking control problems for multiple-input, multiple-output (MIMO) locally Lipschitz nonlinear (LLNL) systems subject to iterative operation and uncertain, iteration-varying external disturbances and initial conditions. Under the assumption of a linear, P-type iterative learning control (ILC) update law, a double-dynamics analysis (DDA) approach is proposed to show the convergence of the ILC process in the presence of the locally Lipschitz nonlinearities and iteration-varying uncertainties. The DDA approach results in a contraction mapping-based convergence condition that guarantees both: 1) the boundedness of all system trajectories and 2) the robust convergence of the output tracking error. Further, a basic system relative degree condition is given that provides a necessary and sufficient (NAS) guarantee of the convergence of the ILC process. As a corollary, it is noted that in the absence of iteration-varying uncertainties, the results likewise provide an NAS convergence guarantee for MIMO LLNL systems. The simulations are presented to illustrate the ideas.

Adaptive stabilization of minimum-phase systems under quantized measurements

Abstract We construct an adaptive controller for a linear minimum-phase system of an arbitrary relative degree with an unknown bounded disturbance and dynamically quantized measurements. The key novelty is the extension of the shunting method (parallel feedforward compensator) to plants with bounded disturbances. This method leads to an augmented system of relative degree one that is stabilized by a passification-based adaptive controller. Moreover, we design a switching procedure for the controller parameters and the quantizer’s zoom that ensures the state convergence from an arbitrary set to an ellipsoid whose size depends on the disturbance bound. The results are demonstrated by an example of an aircraft flight control.

A New Class of Uniform Continuous Higher-Order Sliding Mode Controllers

Abstract This paper proposes a new class of uniform continuous higher-order sliding mode algorithm (UCHOSMA) for the arbitrary relative degree systems. The proposed methodology is a combination of two controllers where one of the components is a uniform super-twisting control which acts as the disturbance compensator and the second part gives the uniform finite time convergence for the disturbance free system. This algorithm provides uniform finite time convergence of the output and its higher derivatives using an absolutely continuous control signal and thus alleviating the chattering phenomenon. The attractive feature of the proposed controller is that irrespective of the different initial conditions, the control is able to bring the states of the system to the equilibrium point uniformly in finite time. The effectiveness of the proposed controller has been demonstrated with both simulation and experimental results.

Practical stability margins in continuous higher order sliding mode control systems

The finite time convergence of a sliding variable to zero in systems of arbitrary relative degree can be provided by Continuous Higher Order Sliding Mode (C-HOSM) control regardless of external bounded perturbations and without artificially increasing relative degree. In this work, metrics of robustness are introduced for C-HOSM control with respect to cascade unmodeled dynamics that are presented in any practical system. The frequency domain analysis of chattering in a dynamically perturbed system controlled by the C-HOSM controller is accomplished numerically using the Describing Function-Harmonic Balance technique via Newton–Raphson method. The robustness of C-HOSM controller to cascade unmodeled dynamics is quantified in terms of Practical Stability Phase and Gain Margins using the parameters of chattering (frequency and amplitude of the main harmonic) that may occur due to the unmodeled dynamics. The proposed methodologies are validated via simulations.

Funnel control for nonlinear systems with arbitrary relative degree using high-gain observers

We propose a new algorithm for controlling the transient behavior of the tracking error for nonlinear systems with arbitrary relative degree. In tracking problems, the transient behavior of the error is more important than the behavior of its derivatives. We introduce a concept of virtual output, which converts an arbitrary relative degree system into a relative degree one system. The virtual output is chosen in such a way that the difference between the virtual output and the tracking error can be made arbitrarily small all the time. Assuming that the state derivatives are not available for feedback we use a high-gain observer to estimate the derivatives and construct the virtual output.

Continuous Full-Order Nonsingular Terminal Sliding Mode Control for Systems With Matched and Mismatched Disturbances

A continuous full-order nonsingular terminal sliding mode control method is investigated for a general dynamic system subject to both matched and mismatched disturbances in this paper. Firstly, the essential dynamics of the original system is established based on the relative degree of system, where the original mismatched disturbances are turned into matched disturbances. Then, a composite control scheme based on the finite-time observer and full-order nonsingular terminal sliding mode control approach is developed to reject the lumped disturbances. Finally, some simulation results demonstrate the effectiveness of the proposed control strategy.

Optimized Iterative Learning Control for Linear Discrete-Time-Invariant Systems

In this paper, an optimized first-order iterative learning control (OILC) scheme is explored for a class of linear discrete-time-invariant systems with Markov parameters available and the system relative degree being unity. For the OILC scheme, the iteration-time-variable derivative learning-gain vector is argued by sequentially minimizing the sum of the tracking error energy and the learning effort intensity amplified by an iteration-wise tuning factor. In virtue of the optimization criterion, the existence and the uniqueness of the iteration-time-variable learning-gain vector is achieved. Then, by making use of the elementary transformations which exchange the rows and columns of a matrix and by taking advantage of the positivity relationship of the eigenvalues with the matrix-weighing quadratic function, the strictly monotone convergence of the OILC scheme is derived, which conveys that the strict monotonicity is guaranteed without any requirement to the system Markov parameters and the convergence rate is adjustable by scaling the tuning factor. Furthermore, an optimized higher-order iterative learning control mechanism is developed for the system relative degree is larger than unity, for which the existence and the uniqueness of the optimized higher-order iteration-time-variable learning-gain vector are discussed and the strictly non-conditional monotone convergence is analyzed. The numerical simulations demonstrate the validity and effectiveness.

Sampled-Data Implementation of Derivative-Dependent Control Using Artificial Delays

We study a sampled-data implementation of linear controllers that depend on the output and its derivatives. First, we consider an LTI system of relative degree $r\ge 2$ that can be stabilized using $r-1$ output derivatives. Then, we consider PID control of a second order system. In both cases, the Euler approximation is used for the derivatives giving rise to a delayed sampled-data controller. Given a derivative-dependent controller that stabilizes the system, we show how to choose the parameters of the delayed sampled-data controller that preserves the stability under fast enough sampling. The maximum sampling period is obtained from LMIs that are derived using the Taylor's expansion of the delayed terms with the remainders compensated by appropriate Lyapunov-Krasovskii functionals. Finally, we introduce the event-triggering mechanism that may reduce the amount of sampled control signals used for stabilization.

An improved time-delay implementation of derivative-dependent feedback

We consider an LTI system of relative degree r≥2 that can be stabilized using r−1 output derivatives. The derivatives are approximated by finite differences leading to a time-delayed feedback. We present a new method of designing and analyzing such feedback under continuous-time and sampled measurements. This method admits essentially larger time-delay/sampling period compared to the existing results and, for the first time, allows to use consecutively sampled measurements in the sampled-data case. The main idea is to present the difference between the derivative and its approximation in a convenient integral form. The kernel of this integral is hard to express explicitly but we show that it satisfies certain properties. These properties are employed to construct the Lyapunov–Krasovskii functional that leads to LMI-based stability conditions. If the derivative-dependent control exponentially stabilizes the system, then its time-delayed approximation stabilizes the system with the same decay rate provided the time-delay (for continuous-time measurements) or the sampling period (for sampled measurements) are small enough.

Finite‐time sliding mode stabilization using dirty differentiation and disturbance compensation

SummaryNovel robust finite‐time stabilizing algorithms are developed side by side for the state and output feedback designs. Being initially developed for a nonlinear cascade second‐order system, these algorithms are straightforwardly extendible to electromechanical systems of relative degree two. The proposed synthesis is based on the disturbance compensation, relying on the dirty differentiation and sliding mode approach, and it is applicable to a wider class of disturbances than that addressed in the literature. Simulation results illustrate the efficiency of the resulting synthesis procedure and support analytical results.