Small-gain Arguments Research Articles

Event-triggered boundary control of an unstable reaction diffusion PDE with input delay

In chemical, biological, or population (epidemiological) processes the feedback action may be considerably delayed by time-consuming chemical measurements or biological tests. With such large delays on the control action in mind, and motivated by the fact that in some of these systems only piecewise-constant inputs can be applied between time instants at which measurements trigger changes in control, we consider the problem of event-triggered stabilization of 1-D reaction–diffusion PDE systems with input delay. The approach relies on reformulating the delay problem as an actuated transport PDE which cascades into the reaction–diffusion PDE, and on the emulation of backstepping control. The paper proposes a static (state-dependent) triggering condition which establishes the time instants at which the control value needs to be updated. It is shown that under the proposed event-triggered boundary control, there exists a minimal dwell-time (independent of the initial conditions) between two triggering times which allows to guarantee the well-posedness of the closed-loop system, and the exponential stability. The stability analysis is based on Input-to-State stability theory for PDEs and small-gain arguments. A simulation example is presented to validate the theoretical results.

Periodic Event-Triggered Control for Networked Control Systems With External Disturbance and Input and Output Delays.

This article investigates the problem of periodic event-triggered output-feedback control for networked control systems in the presence of external disturbance and input and output delays. With the aid of the prediction technique, we first develop the predictor-based-extended state observer to reconstruct the system information, including the unknown state and disturbance. The periodic event-triggered output-feedback control law is then designed via the disturbance/uncertainty estimation and attenuation (DUEA) method, such that the communication times can be remarkably reduced and, at the same time, the disturbance rejection ability can be effectively enhanced. Under the predictor-based event-triggered control method, the influence of the time delays is effectively attenuated, and the effect of external disturbance is considerably attenuated due to the prediction technique and the DUEA method. By using the small-gain arguments, this article gives some sufficient stability conditions for the overall control system, and the explicit computations of sampling/updating period and time delays are presented as well. Finally, we employ a practical example and show some comparative simulation results to demonstrate the advantages of the predictor-based event-triggered control method proposed in this article.

Event-triggered control for linear time-varying systems using a positive systems approach

This paper presents a new design for the event-triggered output feedback control of uncertain linear time-varying systems. Departing from the traditional treatments of event-triggered linear time-invariant systems, it is shown that factoring of fundamental solutions as products of solutions of matrix differential equations, together with small-gain arguments and the notions of interval observers and positive systems, lead to a new class of robust event-triggered output-feedback controllers. A key innovation is our use of new event triggers, which use vectors of absolute values instead of the usual Euclidean 2-norms. An illustrative example of a curve tracking system from marine robotics validates the efficacy of the proposed design scheme.

Predictor-based periodic event-triggered control for nonlinear uncertain systems with input delay

In this paper we investigate the problem of predictor-based periodic event-triggered control for a class of nonlinear uncertain systems subject to input delay. When only sampled-data output is available, a novel predictor-based continuous-discrete observer is first designed to estimate system state, and an event-triggered controller is second proposed to globally exponentially stabilize the nonlinear uncertain system. Under the proposed control method, the effects of input delay and sampling of output can be considerably compensated thanks to prediction technique. As a byproduct, the Zeno behavior is avoided in nature since the proposed event-triggering mechanism is detected in the form of discrete-time. Some sufficient stability conditions on maximum allowable sampling period and input delay are presented by using feedback domination technique and small-gain argument.

Thrust vector control of constrained multibody systems

This paper presents a general control framework for the constrained control of multibody systems actuated by vectorized thrusters. A cascade control scheme augmented with a constraint-enforcement unit is proposed to stabilize the system while ensuring constraint satisfaction at all times. The cascade controller consists of an inner loop and an outer loop that are interconnected by a suitable mapping. The inner loop is tasked to control the attitude of the vectorized thrusters. The outer loop is designed to steer the task configuration of the system to a desired pose, while providing to the inner loop the desired attitude through a mapping. To prove the stability of the interconnected system, input-to-state stability (ISS) and small gain arguments are used. All stability properties are derived in the absence of constraints and are shown to be local. Hence, the control scheme is augmented with a Reference Governor (RG) to enforce constraints at all times. Simulations on an unmanned tiltrotor docked to a stationary platform for a refueling operation are carried out to demonstrate the effectiveness of the proposed control framework.

Event-triggered boundary control of constant-parameter reaction–diffusion PDEs: A small-gain approach

This paper deals with an event-triggered boundary control of constant-parameters reaction–diffusion PDE systems. The approach relies on the emulation of backstepping control along with a suitable triggering condition which establishes the time instants at which the control value needs to be updated. In this paper, it is shown that under the proposed event-triggered boundary control, there exists a minimal dwell-time (independent of the initial condition) between two triggering times and furthermore the well-posedness and global exponential stability are guaranteed. The analysis follows small-gain arguments and builds on recent papers on sampled-data control for this kind of PDE. A simulation example is presented to validate the theoretical results.

Asynchronous links that re-order samples: An L2 gain embedding

An embedding of the uncertain time-varying delay associated with a class of asynchronous sampled-data links is devised in terms of the L 2 gain of a related input-output operator. For links in the class considered, each output zero-order-hold update event occurs after an uncertain time-varying delay from a corresponding aperiodic input sampling event. The encapsulation of the effective link delay is parametrized by upper and lower bounds on the uncertain interval between sample events and the variable delay for the corresponding update events. The possibility of sample re-ordering at the output can arise when the update delay is allowed to vary freely within the specified bounds, which is accommodated in the analysis. The gain bound based embedding is relevant within the context of input-output based robustness analysis of asynchronous sampled-data networks via small-gain arguments and integral quadratic constraints more generally.

Robustness of constant-delay predictor feedback for in-domain stabilization of reaction–diffusion PDEs with time- and spatially-varying input delays

This paper discusses the in-domain feedback stabilization of reaction–diffusion PDEs with Robin boundary conditions in the presence of an uncertain time- and spatially-varying delay in the distributed actuation. The proposed control design strategy consists of a constant-delay predictor feedback designed based on the known nominal value of the control input delay and is synthesized on a finite-dimensional truncated model capturing the unstable modes of the original infinite-dimensional system. By using a small-gain argument, we show that the resulting closed-loop system is exponentially stable provided that the variations of the delay around its nominal value are small enough. The proposed proof actually applies to any distributed-parameter system associated with an unbounded operator that 1) generates a C0-semigroup on a weighted space of square integrable functions over a compact interval; and 2) is self-adjoint with compact resolvent.

Boundary-to-Displacement asymptotic gains for wave systems with Kelvin–Voigt damping

We provide estimates for the asymptotic gains of the displacement of a vibrating string with endpoint forcing, modelled by the wave equation with Kelvin–Voigt and viscous damping and a boundary disturbance. Two asymptotic gains are studied: the gain in the spatial norm and the gain in the spatial sup norm. It is shown that the asymptotic gain property holds in the norm of the displacement without any assumption for the damping coefficients. The derivation of the upper bounds for the asymptotic gains is performed by either employing an eigenfunction expansion methodology or by means of a small-gain argument, whereas a novel frequency analysis methodology is employed for the derivation of the lower bounds for the asymptotic gains. The graphical illustration of the upper and lower bounds for the gains shows that the asymptotic gain in the norm is estimated much more accurately than the asymptotic gain in the sup norm.

Observer Design for Nonlinear Networked Control Systems With Persistently Exciting Protocols

We study the design of state observers for nonlinear networked control systems (NCSs) affected by disturbances and measurement noise, via an emulation-like approach. That is, given an observer designed with a specific stability property in the absence of communication constraints, we implement it over a network, and we provide sufficient conditions on the latter to preserve the stability property of the observer. In particular, we provide a bound on the maximum allowable transmission interval (MATI) that guarantees an input-to-state stability (ISS) property for the corresponding estimation error system. The stability analysis is trajectory-based, utilizes small-gain arguments, and exploits a persistently exciting property of the scheduling protocols. This property is key in our analysis and allows us to obtain significantly larger MATI bounds in comparison to the ones found in the literature. Our results hold for a general class of NCSs; however, we show that these results are also applicable to NCSs implemented over a specific physical network called WirelessHART (WH). The latter is mainly characterized by its multihop structure, slotted communication cycles, and the possibility to simultaneously transmit over different frequencies. We show that our results can be further improved by taking into account the intrinsic structure of the WH–NCS model. That is, we explicitly exploit the model structure in our analysis to obtain an even tighter MATI bound that guarantees the same ISS property for the estimation error system. Finally, to illustrate our results, we present analysis and numerical simulations for a class of Lipschitz nonlinear systems and high-gain observers.

Stability and L 1 × ℓ1-to-L 1 × ℓ1 performance analysis of uncertain impulsive linear positive systems with applications to the interval observation of impulsive and switched systems with constant delays

ABSTRACT Solutions to the interval observation problem for delayed impulsive and switched systems with -performance are provided. The approach is based on first obtaining stability and -to- performance analysis conditions for uncertain linear positive impulsive systems in linear fractional form with norm-bounded uncertainties using a scaled small-gain argument involving time-varying D-scalings. Both range and minimum dwell-time conditions are formulated – the case of constant and maximum dwell-times can be directly obtained as corollaries. The conditions are stated as timer/clock-dependent conditions taking the form of infinite-dimensional linear programmes that can be relaxed into finite-dimensional ones using polynomial optimisation techniques. It is notably shown that under certain conditions, the scalings can be eliminated from the stability conditions to yield equivalent stability conditions on the so-called worst-case system, which is obtained by replacing the uncertainties by the identity matrix. These conditions are then applied to the special case of linear positive systems with delays, where the delays are considered as uncertainties, similarly to as in (Briat (2018a). Stability and performance analysis of linear positive systems with delays using input-output methods. International Journal of Control, 91(7), 1669–1692). As before, under certain conditions, the scalings can be eliminated from the conditions to obtain conditions on the worst-case system, coinciding here with the zero-delay system – a result that is consistent with all the existing ones in the literature on linear positive systems with delays. Finally, the case of switched systems with delays is considered. The approach also encompasses standard continuous-time and discrete-time systems, possibly with delays and the results are flexible enough to be extended to cope with multiple delays, time-varying delays, distributed/neutral delays and any other types of uncertain systems that can be represented as a feedback interconnection of a known system with an uncertainty.

Control of a quadrotor and a ground vehicle manipulating an object

This paper focuses on the control of a cooperative system composed of an Unmanned Aerial Vehicle (UAV) and an Unmanned Ground Vehicle (UGV) manipulating an object. The two units are subject to input saturations and collaborate to move the object to a desired pose characterized by its position and inclination. The dynamics are derived using the Euler–Lagrange method. A stabilizing control law is proposed where the UGV is tasked to deploy the object to a certain position whereas the UAV adjusts its inclination. In particular, a proportional–derivative control law is proposed for the UGV, and a cascade control is used for the UAV, where the inner loop controls the attitude of the UAV and the outer loop stabilizes the inclination of the object. The stability of the points of equilibrium is proven using small gain arguments. To ensure constraints satisfaction at all times, a reference governor unit is added to this stabilizing control law. Simulations and experimental results are provided to validate the effectiveness of the proposed control scheme.

Sampled-data observer design for delayed output-injection state-affine systems

ABSTRACT We are considering the problem of observer design for a general class of state-affine systems in presence of output delay and sampling. Two novelties characterize the considered class of systems: (i) the state equation is subject to output injection involving future output values (not accessible to measurement); (ii) the injected output values come in the state equation not only through a driving term but also through the state matrix. These novel characteristics entail the loss of the system state-affine nature and lead to a new observer design problem never investigated so far. The solution we develop in this paper is a sampled-data based observer of Kalman-like type, augmented with inter-sample predictors and signal saturations. Using both Lyapunov and small-gain arguments, we show that the observer is exponentially convergent for sufficiently small sampling interval and delay, provided an observability condition is satisfied.

Small-Gain-Based Boundary Feedback Design for Global Exponential Stabilization of One-Dimensional Semilinear Parabolic PDEs

This paper presents a novel methodology for the design of boundary feedback stabilizers for one-dimensional, semilinear, parabolic PDEs. The methodology is based on the use of small-gain arguments ...

Stabilization of Nonlinear Systems With Time-Varying Input and State Delays by Approximate Predictor

In this paper, the stabilization problem of nonlinear systems with time-varying input and state delays is considered by an approximate predictor. Due to the introduction of state delays, the nonlinear dynamic without state delays is used to predict the plant state, and a new error term is introduced into the approximation error between the prediction value and the future state of the plant. By Lyapunov–Krasovskii functional and small-gain argument, the stability of the whole closed-loop system is guaranteed provided that the sampling period, the predictor accuracy, and the upper bound of state delays satisfy an inequality constraint. The effectiveness of the proposed results is illustrated by a numerical example.

Robust Control Contraction Metrics: A Convex Approach to Nonlinear State-Feedback <inline-formula> <tex-math notation="LaTeX">${H}^\infty$ </tex-math> </inline-formula> Control

This letter proposes a new method for robust state-feedback control design for nonlinear systems. We introduce robust control contraction metrics (RCCM), extending the method of control contraction metrics from stabilization to disturbance attenuation and robust control. An RCCM is a Riemannian metric that verifies differential $L^{2}$ -gain bounds in closed-loop, and guarantees robust stability of arbitrary trajectories via small gain arguments. Numerical search for such a metric can be transformed to a convex optimization problem. We also show that the associated Riemannian energy can be used as a robust control Lyapaunov function. A simple computational example based on jet-engine surge illustrates the approach.

A Small-Gain Approach to Robust Event-Triggered Control of Nonlinear Systems

This paper presents a new approach to event-triggered control for nonlinear uncertain systems by using the notion of input-to-state stability (ISS) and the nonlinear small-gain theorem. The contribution of this paper is threefold. First, it is proved that infinitely fast sampling can be avoided if the system is input-to-state stabilizable with the sampling error as the external input and the corresponding ISS gain is locally Lipschitz. No assumption on the existence of known ISS-Lyapunov functions is made in the discussions. Moreover, the forward completeness problem with event-triggered control is studied systematically by using ISS small-gain arguments. Second, the proposed approach gives rise to a new self-triggered sampling strategy for a class of nonlinear systems subject to external disturbances. If an upper bound of the external disturbance is known, then the closed-loop system can be designed to be robust to the external disturbance, and moreover, the system state globally asymptotically converges to the origin if the external disturbance decays to zero. Third, a new design method is developed for event-triggered control of nonlinear uncertain systems in the strict-feedback form. It is particularly shown that the ISS gain with the sampling error as the input can be designed to satisfy the proposed condition for event-triggered control and self-triggered control.

Adaptive restricted trajectory tracking for a non-minimum phase hypersonic vehicle model

The design of a nonlinear robust controller for a non-minimum phase model of an air-breathing hypersonic vehicle is presented in this work. When flight-path angle is selected as a regulated output and the elevator is the only control surface available for the pitch dynamics, longitudinal models of the rigid-body dynamics of air-breathing hypersonic vehicles exhibit unstable zero-dynamics that prevent the applicability of standard inversion methods for control design. The approach proposed in this paper uses a combination of small-gain arguments and adaptive control techniques for the design of a state-feedback controller that achieves asymptotic tracking of a family of velocity and flight-path angle reference trajectories belonging to a given class of vehicle maneuvers, in spite of model uncertainties. The method reposes upon a suitable redefinition of the internal dynamics of a control-oriented model of the vehicle dynamics, and uses a time-scale separation between the controlled variables to manage the peaking phenomenon occurring in the system. Simulation results on a full nonlinear vehicle model that includes structural flexibility illustrate the effectiveness of the methodology.

A Framework for the Observer Design for Networked Control Systems

This technical note proposes a framework for the observer design for networked control systems (NCS) affected by disturbances, via an emulation-like approach. The proposed model formulation allows us to consider various static and dynamic time-scheduling protocols, in-network processing implementations and encompasses sampled-data systems as a particular case. Provided that the continuous-time observer is robust to the measurement errors (in an appropriate sense) we derive bounds on the maximum allowable transmission interval that ensure the convergence of the observation errors under network-induced communication constraints. The stability analysis is trajectory-based and utilizes small-gain arguments. A number of observers can be combined and used within our approach to obtain estimators for NCS.

Input-to-state stability analysis for interconnected difference equations with delay

Input-to-state stability (ISS) of interconnected systems with each subsystem described by a difference equation subject to an external disturbance is considered. Furthermore, special attention is given to time delay, which gives rise to two relevant problems: (i) ISS of interconnected systems with interconnection delays, which arise in the paths connecting the subsystems, and (ii) ISS of interconnected systems with local delays, which arise in the dynamics of the subsystems. The fact that a difference equation with delay is equivalent to an interconnected system without delay is the crux of the proposed framework. Based on this fact and small-gain arguments, it is demonstrated that interconnection delays do not affect the stability of an interconnected system if a delay-independent small-gain condition holds. Furthermore, also using small-gain arguments, ISS for interconnected systems with local delays is established via the Razumikhin method as well as the Krasovskii approach. A combination of the results for interconnected systems with interconnection delays and local delays, respectively, provides a framework for ISS analysis of general interconnected systems with delay. Thus, a scalable ISS analysis method is obtained for large-scale interconnections of difference equations with delay.