Recent Results on Nonlinear Delay Control Systems

Stabilization of nonlinear systems under delays is a central and challenging problem in control theory. It is also of considerable interest in engineering, because delay systems are prevalent in aerospace, biological, marine robotic, network control, and many other applications. Input delays naturally arise due to transport phenomena, time consuming information processing, and sensor designs, and they can produce complicated systems that are beyond the scope of standard frequency-domain or Lyapunov function methods. This has led to large control theoretic and engineering literatures on stabilization problems, spanning more than 40 years, based on backstepping, Lyapunov-Krasovskii functionals, prediction, and sampling controllers. In addition to input delays, there may also be state delays in the vector fields that define the system. This tutorial summarizes some recent work on stabilization under input or state delays and suggests future research directions.

This paper studies the application of a recently proposed control scheme to globally Lipschitz nonlinear systems for which the input is delayed and applied with zero order hold, the measurements are sampled and delayed, and only an output is measured (i.e., the state vector is not available). The control scheme consists of an observer for the delayed state vector, an inter-sample predictor for the output signal, an approximate predictor for the future value of the state vector, and the nominal feedback law applied with zero order hold and computed for the predicted value of the future state vector. The resulting closed-loop system is robust with respect to modeling and measurement errors and robust to perturbations of the sampling schedule.

We present results on stabilization of nonlinear systems in the strict-feedback form with delays affecting the virtual inputs. We consider systems with constant and time-varying delays, as well as systems with delays that depend on the current or past states, which arise in numerous applications, such as, for example, in cooling systems. The design methodology is based on the concepts of infinite-dimensional backstepping and nonlinear predictor feedback. Several illustrative examples are provided.

This study addresses the problem of delay compensation via a predictor-based output feedback for a class of linear systems subject to input delay which itself depends on the input. The equation defining the delay is implicit and involves past values of the input through an integral relation, the kernel of which is a polynomial function of the input. This modeling represents systems where transport phenomena take place at the inlet of a system involving a nonlinearity, which frequently occurs in the processing industry. The conditions of asymptotic stabilization require the magnitude of the feedback gain to comply with the initial conditions. Arguments for the proof of this novel result include general Halanay inequalities for delay differential equations and take advantage of recent advances in backstepping techniques for uncertain or varying delay systems.

This chapter deals with the problem of output feedback control of nonlinear systems affected by time-varying measurement delay. A control law is presented, which is made of an observer-controller cascade where the controller is a classic state-linearizing scheme, and the observer is a high-gain observer of chain-type. It is shown that under suitable conditions on the system, the observer is globally exponentially convergent, and the replacement of the true state with the observer state in the control law results in an exponentially stabilizing feedback scheme. The main limitation with a single observer is the presence of a delay bound that depends on the Lipschitz constant of the nonlinear system. To overcome this limitation it is possible to resort to a chain of observers that, at the cost of a growing realization space and convergence time, can in principle allow to compensate any delay. This design is straightforward when the delay is known and constant but its extension to time-varying delays requires special attention, in particular when the delay is not continuous with respect to time, as it frequently happens in the applications. We therefore introduce a classification of delay functions with respect to the available output information and illustrate how to design the cascade of elementary observers to solve the state reconstruction problem. We also characterize the class of delay functions for which this approach fails to provide a viable implementation.

The normal form is discussed for nonlinear systems affected by constant commensurate delays. Two different forms are argued. In particular, necessary and sufficient conditions are given under which a nonlinear time-delay system can be decomposed into a (weakly) observable subsystem and a non observable subsystem. Whenever such a decomposition exists, additional conditions are required to ensure the feedback linearization of the weakly observable subsystem. Finally, a full characterization is derived for the nonlinear time delay system to have an unobservable subsystem not directly affected by the input and a weakly observable subsystem which is linearizable by feedback. The performed analysis is carried out within a new geometric framework recently introduced in the literature.

Neuromuscular electrical stimulation (NMES), often called functional electrical stimulation (FES),is a prescribed treatment for various neuromuscular disorders. When applied to articulate a person’s limb, the respective skeletal muscle groups are known to rapidly fatigue compared to muscles activated by the nervous system. Recent results have shown that muscles have a delayed response to electrical stimulation, and more recent results indicate that this delayed response increases as the muscle fatigues. A NMES control method is developed in this chapter as a means to compensate for the varying input delay for the uncertain nonlinear dynamics for the lower limb. Experimental results are provided to demonstrate the performance of the developed controller.

The notion of weighted homogeneity is extended to time-delay systems. It is shown that the stability (resp., instability) of homogeneous functional systems on a sphere implies the global stability (resp., instability) of the system. The notion of local homogeneity is introduced, and a relation between stability and instability of the locally approximating dynamics and the original time-delay system is established using a Lyapunov-Razumikhin approach. An implication between homogeneity and input-to-state stability is investigated. Examples of applications of the proposed theory are given.

A sliding mode observer in the presence of known output delay and its application to robust fault reconstruction is studied. The observer is designed using a singular perturbation method for which sufficient conditions are given in the form of linear matrix inequalities (LMIs) to guarantee ultimate boundedness of the error dynamics. Though an ideal sliding motion cannot be achieved in the observer when the outputs are delayed, ultimately bounded solutions can be obtained, provided the delay is sufficiently small. The bound on the solution is proportional to the delay and the magnitude of the switching gain. The proposed observer design is applied to the problem of fault reconstruction under delayed outputs and system uncertainties. It is shown that actuator or sensor faults can be reconstructed reliably from the output error dynamics. An example of observer design for an inverted pendulum system is used to demonstrate the merit of the proposed methodology, compared with existing sliding mode observer design approaches.

This chapter presents a framework for verifying stability and robustness of dynamical networks consisting of neutral subsystems subject to disturbances in the spirit of integral input-to-state stability. In addition to neutral-type delays in subsystems, time-delays are allowed to reside in both subsystems and interconnection channels. A small-gain condition is proposed for constructing a Lyapunov-Krasovskii functional to establish stability and robustness of the network. No assumption is made on the network topology.

This chapter deals with the input-to-state stability of switched nonlinear systems with time delays. The proposed results demonstrate a connection between small-gain arguments in the context of input-to-state stability and the traditional Lyapunov-Razumikhin method for switched systems. By using the notion of average dwell time, it is shown that a switching among ISS systems with compatible Lyapunov-Razumikhin functions will not destroy the stability property if the switching is not too fast on average. Particularly, it is shown that the existence of a common Razumikhin function is sufficient to guarantee input-to-state stability for time-delayed systems under arbitrary switching.

We provide various extensions of recent results on the existence ofLyapunov–Krasovskii functionals for uncertain systems described by neutral functional differential equations. We consider nonlinear neutral delay systems for which the difference operator and the right-hand side of the differential equations are Lipschitz on bounded sets.

Hybrid systemswith memory are dynamical systems that exhibit both hybrid and delay phenomena, as seen in many physical and engineered applications. A prominent example is the use of delayed hybrid feedback in control systems. This chapter outlines a framework that allows studying hybrid systems with delays through generalized solutions and summarizes some recent results on basic existence and well-posedness of solutions and stability analysis using Lyapunov-based methods.

Fluid networks are characterized by complex interconnected flows, involving high order nonlinear dynamics and transport phenomena. Classical lumped models typically capture the interconnections and nonlinear effects but ignore the transport phenomena, which may strongly affect the transient response. To control such flows with regulators of reduced complexity, we improve a classical lumped model (obtained by combining Kirchhoff’s laws and graph theory) by introducing the effect of advection as a time delay. The model is based on the isothermal Euler equations to describe the dynamics of the fluid through the pipe. The resulting hyperbolic system of partial differential equations (PDEs) is diagonalized using Riemann invariants to find a solution in terms of delayed equations, obtained analytically using the method of the characteristics. Conservation principles are applied at the nodes of the network to describe the dynamics as a set of (possibly non linear) delay differential equations. Both linearized and nonlinear Euler equations are considered.

We consider the stabilization problem for nonlinear time-delay systems.First, we review the finite spectrum assignment method for a class of linear retarded systems and its extension for nonlinear systems. Next, relaxing required conditions for the method, we propose a new stabilizing technique for nonlinear delay systems. The technique includes the use of a state predictor which is based on the idea of anticipating synchronization. Furthermore, we discuss the relationship among other control design methods for a wider class of retarded systems and this technique.

In this of either the controller or the network is addressed for sampled-data systems with input saturation. Using modified sector conditions, an adequate looped functional, and the Wirtinger-based integral inequality, quasi-LMI conditions, with a scalar parameter to tune, are proposed in the regional (or local) context for both design problems. The associated convex optimizations are briefly described. Finally, some examples show the efficiency of the methods with respect to existing results .

The exact determination of the state of a finite dimensional linear system where a variable of interest, say y(t), and its delayed versions appear in an implicit relation with the delays themselves is addressed. It is assumed that the observed signal is not this variable of interest, but rather the delay (or multiple delay vector) itself. This implies that this delay must be state dependent (through y). An implicit relation with known \(F:\mathbb {R}^{N+1}\rightarrow \mathbb {R}^N\) is assumed. If \(x(t)\in \mathbb {R}^n\) is the state of the linear system, the observability problem is to determine this state x(t) from the knowledge of the system input u(t) and the delays \(\overline{\tau }(t)\). This differs form the well known observability problem where x(t) is to be determined from u(t) and y(t). In the problem at hand, an inversion is involved, to obtain y(t) from \(\overline{\tau }(t)\), rendering the problem nonlinear. Such problems are relevant when dealing with sonar, pertinent in robotics, where mobile systems must avoid hitting walls, and in underwater vehicles, for instance the soft “landing” problem on an ocean floor. The requisite observability/invertibility conditions are derived. The relevance of the restrictions \(\dot{\tau }_i<1\) for the problem to be well posed is illuminated from the physical context in the problem. In addition, the inversion of a special ‘autoregressive’ relation (in iterated function sense) obeyed by a delay, is solved. This is of interest in singular perturbation approaches to systems with state dependent delay.

A class of nonlinear time varying delay systems in the presence of time delay uncertainties is considered in this chapter. The entries of the system input distribution matrix may be nonlinear functions of the outputs and time. Under mild limitations on the uncertainty, an observer is synthesised using sliding mode techniques such that the error dynamics are ultimately uniformly bounded in the presence of uncertainties and time delay. Then, a nonlinear control scheme is developed based on the estimated states, and a set of sufficient conditions is presented such that the corresponding closed-loop systems are uniformly ultimately bounded, using the well-known Lyapunov-Razumikhin approach. It is not required that the structure of the uncertainty be known. Finally, a numerical example is presented to demonstrate the approach and simulation results show the effectiveness of the developed paradigm.

This chapter studies the stabilization of exponentially unstable linear systems subject to saturation and time-varying delay in the input. The proposed stabilizing controller is developed in a two step process. First, the stabilizing controller for the system without saturation is obtained from the optimal solution to an iterative LMI problem, such that the upper bound of the delay function is maximized. Second, the input saturation is included in the analysis, and the previous iterative LMI problem is updated in order to maximize the domain of attraction of the closed-loop system. The effectiveness of the proposed methods is demonstrated through numerical examples, and by experimental validation on a test rig that captures the main operating characteristics of active magnetic bearings.

Using the theory of non-commutative rings, the delay identification problem of nonlinear time-delay systems with unknown inputs is studied. Necessary and sufficient conditions are proposed to judge the identifiability of the delay, where two different cases are discussed for the dependent and independent outputs, respectively. After that, necessary and sufficient conditions are given to analyze the causal and non-causal observability for nonlinear time-delay systems with unknown inputs.