Linear Feedback Systems Research Articles

Almost feedback linearizable systems of the class C1 and solving the constructive controllability problem

Abstract We propose a method of solving the controllability problem using almost linearizable control systems. Developing an idea of triangular systems, proposed by the first author of this paper in 1973, we introduce a wider class of almost triangular systems for which it is possible to find explicitly a control steering the system from one given point to another given point. We give conditions of almost linearizability for nonlinear single-input systems of the class $C^{1}$ and discuss a method of solving the controllability problem for such systems.

Nonlinear Model Predictive Control for a Dynamic Positioning Ship Based on the Laguerre Function

In this paper, we present a novel nonlinear model predictive control (NMPC) algorithm based on the Laguerre function for dynamic positioning ships to solve the problems of input saturation, unknown time-varying disturbances, and heavy computation. The nonlinear model of a dynamic positioning ship is presented as a linear model, transformed from a standard affine nonlinear state-space model by precise feedback linearization. The environmental disturbance is overcome using an integrator. The time cost of the proposed nonlinear control algorithm is decreased by inducing the Laguerre function to describe the feedback-linearization system input increments. The Laguerre function reduces the matrix dimensions of the nonlinear optimization problem. The simulation results for a DP supply vessel showed that the novel algorithm maintained the effective control performance of the original nonlinear model predictive control algorithm and had a reduced computation load to satisfy the requirements of real-time operation.

Learning to Control Known Feedback Linearizable Systems From Demonstrations

In this paper, we revisit the problem of learning a stabilizing controller from a finite number of demonstrations by an expert. By first focusing on feedback linearizable systems, we show how to combine expert demonstrations into a stabilizing controller, provided that demonstration trajectories are sufficiently long and there are at least <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n+1$</tex-math></inline-formula> of them, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the number of states of the system being controlled. When we have more than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n+1$</tex-math></inline-formula> demonstration trajectories, we discuss how to optimally choose the best <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n+1$</tex-math></inline-formula> demonstrations to construct the stabilizing controller. We then extend these results to a class of systems that can be embedded into a higher-dimensional system containing a chain of integrators. The feasibility of the proposed algorithm is demonstrated by applying it on a CrazyFlie 2.0 quadrotor.

Data-Based Control of Feedback Linearizable Systems

We present an extension of Willems' Fundamental Lemma to the class of multi-input multi-output discrete-time feedback linearizable nonlinear systems, thus providing a data-based representation of their input-output trajectories. Two sources of uncertainty are considered. First, the unknown linearizing input is inexactly approximated by a set of basis functions. Second, the measured output data is contaminated by additive noise. Further, we propose an approach to approximate the solution of the data-based simulation and output matching problems, and show that the difference from the true solution is bounded. Finally, the results are illustrated on an example of a fully-actuated double inverted pendulum.

Adaptive Force-Control Augmentation for Small Celestial Body Sampling

An adaptive force-control augmentation for small celestial body sampling for a variety of surface properties is developed. The control algorithm consists of a nominal robust controller augmented with an adaptive controller combined with feedback linearization. When a spacecraft makes contact with the surface, it must maintain a desired contact force in order to capture a sample. The properties of the surface are unknown or uncertain before contact with the surface is made. Because the nominal robust controller may have poor performance in some surface property regimes, the goal is to improve performance using an adaptive controller. The adaptive controller performs system identification online to create an input–output model of the feedback-linearized system. From the input–output model, a block-observable canonical form is realized and the control input augmentation is determined by model predictive control. The resulting augmentation is added to the input of the robust controller to improve the closed-loop performance and maintain a desired contact force despite the unknown surface properties. The approach is applied to a variety of surface properties with linear and nonlinear contact models and multiple surface sampling maneuvers.

Comparison of Bootstrap Methods for Estimating Causality in Linear Dynamic Systems: A Review.

In this study, we present a thorough comparison of the performance of four different bootstrap methods for assessing the significance of causal analysis in time series data. For this purpose, multivariate simulated data are generated by a linear feedback system. The methods investigated are uncorrelated Phase Randomization Bootstrap (uPRB), which generates surrogate data with no cross-correlation between variables by randomizing the phase in the frequency domain; Time Shift Bootstrap (TSB), which generates surrogate data by randomizing the phase in the time domain; Stationary Bootstrap (SB), which calculates standard errors and constructs confidence regions for weakly dependent stationary observations; and AR-Sieve Bootstrap (ARSB), a resampling method based on AutoRegressive (AR) models that approximates the underlying data-generating process. The uPRB method accurately identifies variable interactions but fails to detect self-feedback in some variables. The TSB method, despite performing worse than uPRB, is unable to detect feedback between certain variables. The SB method gives consistent causality results, although its ability to detect self-feedback decreases, as the mean block width increases. The ARSB method shows superior performance, accurately detecting both self-feedback and causality across all variables. Regarding the analysis of the Impulse Response Function (IRF), only the ARSB method succeeds in detecting both self-feedback and causality in all variables, aligning well with the connectivity diagram. Other methods, however, show considerable variations in detection performance, with some detecting false positives and others only detecting self-feedback.

Cloud Control of Connected Vehicle Under Both Communication-Induced Sensing and Control Delay: A Prediction Method

In this paper, we focus on the problem of cloud control of the connected vehicle with time-varying communication-induced sensing delay and control delay. First, the connected vehicle cloud control system containing communication-induced delay is modeled as the discrete linear output feedback system with bounded input and output delay. Second, corresponding to the linear output feedback system, the “predictor-observer” structured controller is introduced and redesigned to compensate for the delay in both output and input channels actively, without measurement of actual input delay as its original version. The stability of the close-looped prediction-based control system under a certain delay boundary is analyzed, and the sufficient stability condition is proposed. Third, with the help of Cone Complementary Linearization (CCL), an iterative algorithm is proposed to design stabilized controller and observer gains. Finally, the proposed method is verified through vehicle lateral and longitudinal control simulations, and the effectiveness and the performance of the proposed delay compensation strategy are innovatively evaluated under real vehicle control scenarios.

Compactly Restrictable Metric Policy Optimization Problems

We study policy optimization problems for deterministic Markov decision processes (MDPs) with metric state and action spaces, which we refer to as Metric Policy Optimization Problems (MPOPs). Our goal is to establish theoretical results on the well-posedness of MPOPs that can characterize practically relevant continuous control systems. To do so, we define a special class of MPOPs called Compactly Restrictable MPOPs (CR-MPOPs), which are flexible enough to capture the complex behavior of robotic systems but specific enough to admit solutions using dynamic programming methods such as value iteration. We show how to arrive at CR-MPOPs using forward-invariance. We further show that our theoretical results on CR-MPOPs can be used to characterize feedback linearizable control affine systems.

Governor-parameterized barrier function for safe output tracking with locally sensed constraints

This paper considers output tracking with time-varying constraints, obtained from online sensor measurements, relevant to autonomous system navigation in unknown environments. Inspired by reference governor techniques, we introduce a virtual governor system, whose state specifies an output regulation point for the actual system and is controlled to adaptively track an output reference without violating the constraints. Our main contribution is a governor-parameterized barrier function (PBF) that quantifies the trade-off between safety (distance from constraint violation) and system energy (output-regulation Lyapunov function). The PBF defines a local safe set that varies as the system-governor state or the sensor measurements change. This safe set induces a governor control law, which guarantees safe and stable output tracking for the real system. We demonstrate our adaptive output-tracking controller on a feedback-linearizable system navigating in an unknown environment, where obstacle distances are measured online.

Constructing Feedback Linearizable Discretizations for Continuous-Time Systems Using Retraction Maps

Control laws for continuous-time dynamical systems are most often implemented via digital controllers using a sample-and-hold technique. Numerical discretization of the continuous system is an integral part of subsequent analysis. Feedback linearizability of such sampled systems is dependent upon the choice of discretization map or technique. In this article, for feedback linearizable continuous-time systems, we utilize the idea of retraction maps to construct discretizations that are feedback linearizable as well. We also propose a method to functionally compose discretizations to obtain higher-order integrators that are feedback linearizable.

Data-driven nonlinear predictive control for feedback linearizable systems

We present a data-driven nonlinear predictive control approach for the class of discrete-time multi-input multi-output feedback linearizable nonlinear systems. The scheme uses a non-parametric predictive model based only on input and noisy output data along with a set of basis functions that approximate the unknown nonlinearities. Despite the noisy output data as well as the mismatch caused by the use of basis functions, we show that the proposed multi-step robust data-driven nonlinear predictive control scheme is recursively feasible and renders the closed-loop system practically exponentially stable. We illustrate our results on a model of a fully-actuated double inverted pendulum.

Implications of Regret on Stability of Linear Dynamical Systems

The setting of an agent making decisions under uncertainty and under dynamic constraints is common for the fields of optimal control, reinforcement learning, and recently also for online learning. In the online learning setting, the quality of an agent's decision is often quantified by the concept of regret, comparing the performance of the chosen decisions to the best possible ones in hindsight. While regret is a useful performance measure, when dynamical systems are concerned, it is important to also assess the stability of the closed-loop system for a chosen policy. In this work, we show that for linear state feedback policies and linear systems subject to adversarial disturbances, linear regret implies asymptotic stability in both time-varying and time-invariant settings. Conversely, we also show that bounded input bounded state stability and summability of the state transition matrices imply linear regret.

Control Lyapunov-Barrier Function Based Stochastic Model Predictive Control for COVID-19 Pandemic

In this paper, a stochastic model predictive control (MPC) is proposed to design a non-pharmacutical policy to control and prevent the COVID-19 pandemic. The system dynamics of COVID-19 is described by a stochastic SEIHR model subject to practical constraints, and the model is proved to be feedback linearizable. A stochastic Control Lyapunov-Barrier Function (CLBF) is constructed for the feedback linearizable system. Constraints on hospitalized individuals are regarded as the unsafe region to construct the corresponding stochastic CLBF. In the proposed stochastic MPC, the stochastic CLBF constraints are applied to improve the overall performance on controlling and preventing the epidemic. Both theoretical proof and simulation results imply that, with the CLBF-based stochastic MPC, the proposed policy is effective in controlling and preventing COVID-19 pandemic.

Time space deformation approach to prescribed-time stabilization: Synergy of time-varying and non-Lipschitz feedback designs

A novel approach is proposed to the prescribed-time stabilizing design of feedback linearizable controllable systems regardless of their initial conditions. The basic design idea consists in scaling an appropriate finite-time stabilizing non-Lipschitz feedback by means of a certain time deformation which squeezes the infinite time interval [0,∞) to the prescribed one [0,T), while also changing the state variables. The resulting nonsmooth feedback design relies on time-varying gains, uniformly bounded over the infinite horizon, thereby yielding an attractive implementation opportunity which is opposed to the original design (Song et al., 2017) with time-varying gains, escaping to infinity as time goes to the prescribed time instant. To facilitate the exposition the approach is first developed for a perturbed double integrator, driven by a twisting-wise controller, which is fed by a supertwisting-wise observer. Both the controller and observer, chosen for their robustness properties, are properly scaled according to the time space deformation to constitute an innovative prescribed-time stabilizing output feedback design of the double integrator, operating in the presence of uniformly bounded matched disturbances. To test the versatility of the approach it is then extended to the prescribed-time state feedback design of multi-input multi-output systems representable in the normal form. Theoretical results are additionally supported by simulations.

A Separation Principle for Linear Event-Triggered Output Feedback Systems

The purpose of this article is to revisit the well-known separation principle of classical control for systems implemented in an event-triggered fashion. We consider a scenario consistent of two independent digital channels for communications between a sensor and a controller, and a controller and an actuator, and propose a methodology to separately design controller and observer gains as well as the event conditions. Under the proposed design, we address the problem of restoring the classical separation principle for the event-based design. We show that fast enough sampling at the sensor-to-observer channel suffices to ensure a separation principle. In contrast, under fast sampling at the controller-to-actuator channel, a separation principle can be obtained provided that an additional mild assumption is imposed on the stability matrices. A numerical example is provided to support the theoretical findings.

Addendum to “Converse Theorems for Integral Quadratic Constraints”

Consider a linear time-invariant feedback system consisting of two open-loop stable subsystems. In this note, it is established that if such feedback system is stable for one subsystem being any arbitrarily passive system, then the other subsystem must be output strictly passive.

A singular perturbation based adaptive strategy for bounded controller design in feedback linearizable systems

SummaryThe saturation of actuators is a critical issue in numerous industrial and robotic applications. The article presents a singular perturbation based adaptive control technique for solving the tracking problem in uncertain feedback linearizable systems. A high gain dynamic controller technique is utilized to enforce time scale separation and facilitating dynamic inversion. The resulting closed loop singularly perturbed system is proved to be partially contracting, and the existence of a unique isolated slow manifold is assured via contraction technique. The parameter adaptation ensures uncertainties compensation and subsequently bounded tracking performance. Further, the technique is extended for multi input multi output systems like mobile robots. The numerical simulations present the usability of the proposed methodology and its effectiveness.

High-order fully actuated system approaches: Part X. Basics of discrete-time systems

A basic introduction to high-order fully actuated (HOFA) approaches for discrete-time systems is given. Firstly, it is shown that, different from the continuous-time systems, general dynamical discrete-time systems can be represented by two types of discrete-time HOFA models, namely, the step forward HOFA models and the step backward HOFA models. Secondly, controllers for the step backward HOFA models are designed, which result in constant linear closed-loop systems with arbitrarily assignable eigenstructures. The related problem of feasibility is also discussed. The well-known discrete-time feedback linearisable systems and strict-feedback systems are shown to be equivalent to both the step forward and step backward HOFA models. Finally, a generalised step backward HOFA model containing control vectors of different time instants is proposed and investigated and demonstrated with a type of proposed pseudo feed-forward systems. The contribution in this paper has laid a fundamental basis for discrete-time HOFA approaches. Further analysis and design problems can be naturally established parallel to the continuous-time system case.

Equivalence of LADRC and INDI controllers for improvement of LADRC in practical applications

This paper explores the equivalence between the linear active disturbance rejection control (LADRC) and incremental nonlinear dynamic inversion (INDI) controllers. The equivalence is verified using an n-order, single-input–single-output, perturbed, pure integration plant which represents a class of feedback linearizable systems. And the linear extended state observer (LESO) inside the LADRC is interpreted from a filter perspective, which shows that the core of the LESO is a low-pass filter. A better parameter tuning method is proposed for the LESO. Then, based on the equivalence, an actuator model is integrated into the LADRC to improve its performance on the plant with non-negligible actuator dynamics that are ignored by the original LADRC. Through the modification, compared with the original LADRC, the bandwidth of the LESO and the whole controller is extended, so that improved performance can be achieved on the plants with low-bandwidth actuators. The equivalence between two controllers and the effectiveness of the improved LADRC are both demonstrated by experiments conducted on a quadrotor.

Command-filtered backstepping robust adaptive emotional control of strict-feedback nonlinear systems with mismatched uncertainties

Emotional models have recently gained increasing attention in the design of intelligent control systems because of their desirable properties, such as universal approximation and fast response. However, stable emotional controllers have generally been for full-state feedback linearizable systems with only matched uncertainties. They also require knowing the reference signal and its higher derivatives. In contrast, we propose here a command-filtered backstepping H∞ robust adaptive emotional controller (CFBRAEC) for strict-feedback nonlinear systems with mismatched uncertainties. Our approach applies command filters to address the “explosion of complexity” in backstepping control and introduces a compensating filter consistent with the H∞-based design to deal with the approximation errors of command filters. The command-filtered backstepping approach handles matched/mismatched uncertainties and disturbances while requiring only the reference signal and its first derivative to be known and continuous. Accordingly, CFBRAEC is more suitable for practical applications and applicable to a broader group of dynamic systems. The boundedness of the tracking error and other closed-loop signals are established based on Lyapunov theory. The effectiveness of CFBRAEC is shown through simulations in the presence of measurement noise and matched/mismatched disturbances with unknown bounds on two different uncertain nonlinear systems: an inverted pendulum and an electromechanical motor.