Nonlinear Control Under Nonconstant Delays

The front matter includes the title page, series page, copyright page, TOC, and preface.

1.1 Delay Systems

Delays are ubiquitous in applications. A nonexhaustive list includes network controlled systems, teleoperation, milling processes, rolling mills, cooling systems, chemical processes, traffic flow, supply networks, automotive engines, 3D printing/additive manufacturing, irrigation channels, and population dynamics. The presence of delays can severely degrade the performance of such systems and in some cases even lead to instability. The resulting poor performance in the former case and the catastrophic consequences for the physical system in the latter create the need for developing control techniques that explicitly take into account the delay.

Following the invention of the Smith Predictor in the late 1950s for dead compensation in stable linear systems, the first systematic feedback design for potentially unstable, linear plants with arbitrarily long input delays was introduced by Mayne [103] in 1968. This predictor-based approach reemerged, under the name finite spectrum assignment, in the widely known work by Manitius and Olbrot [102] in the late 1970s. Yet, the question of Lyapunov-based stability analysis of the plant under predictor feedback remained open for decades after these early contributions. In Section 2.1 we lay out some of the main ideas of predictor feedback. With the introduction of the infinite-dimensional backstepping transformation we construct a Lyapunov functional for the closed-loop system consisting of the plant and the predictor feedback law.

As it was illustrated in Chapter 2 for single-input systems with a lumped input delay, the introduction of the infinite-dimensional backstepping transformation of the actuator state allows one to construct a Lyapunov functional for the plant under predictor feedback. Yet the backstepping methodology is applicable neither in the case of single-input systems with distributed input delay nor in the case of multi-input systems with different delays in each individual input channel. This is because the system that is composed of the finite-dimensional state X and the infinite-dimensional actuator states U( s), s ∊ [ t − D, t], is not in strict-feedback form. In Section 3.1 we study multi-input multi-output linear systems, with distributed input or sensor delays that are different in each individual input or output channel. With the introduction of backstepping-forwarding transformations, we construct Lyapunov functionals which we use to prove closed-loop stability in the case of controller design, or convergence of the estimation error to zero in the case of observer design. We also design a control law for rejecting a matched, constant disturbance in the input by appropriately incorporating into the backstepping-forwarding transformations the estimation of the unknown disturbance in order to account for its effect.

In Section 3.2 we generalize the backstepping-forwarding transformations to the case where the parameters of the plant are unknown. One of the main challenges of this generalization is that one has to deal, in the case of the B matrix, with a vector of unknown functions, rather than just with a vector of unknown parameters. We resolve this challenge by constructing a Lyapunov functional with normalization and by employing an update law using projection on a projection set which can be either spherical or an infinite-dimensional hyper-rectangle. In addition, the gain kernels of these transformations are time varying, since they incorporate the estimations of the unknown parameters, and hence various technical difficulties arise when one proves that these kernels are bounded (which we need in our Lyapunov analysis).

This chapter is devoted to the application of our techniques for delay systems to automotive catalysts. Specifically, we consider the problem of controlling and analyzing the dynamics of the oxygen storage level in three-way catalytic converters. Our model consists of two coupled nonlinear first order hyperbolic PDEs, coupled with an infinite-dimensional (distributed) ODE. The relationship of this model with the models of systems with distributed delays can be explained as follows. By explicitly solving the two PDEs in terms of the state of the infinite-dimensional ODE and plugging these solutions into the equation for the ODE, we get an infinite-dimensional ODE which has, on the right-hand side, integral terms over its spatial domain. These integral terms have some resemblance to the distributed terms in the case of systems with distributed delays.

In this chapter we introduce the concepts of nonlinear predictor feedback and nonlinear infinite-dimensional backstepping transformation. One of the main challenges in developing predictor feedback designs for nonlinear plants is the determination of an implementable form for the future values of the state. Having determined the predictor state, the control law is then obtained by replacing the state in a nominal state feedback law (which stabilizes the delay-free plant) by the state's predictor. In addition, another major obstacle in designing globally stabilizing control laws for nonlinear systems with long input delays is the finite escape phenomenon. The input delay may be so large that the control signal cannot reach the plant before its state escapes to infinity. Therefore, in this chapter we assume that the plant is forward complete, that is, for every initial condition and every bounded input signal the corresponding solution is defined for all times. The last ingredient that one needs for achieving global stabilization is the existence of a (possibly time-varying) feedback law that globally stabilizes the delay-free plant.

We start in Section 5.1 with the design of the nonlinear predictor feedback. In Section 5.2 we introduce the nonlinear infinite-dimensional backstepping transformation, which allows the construction of a Lyapunov functional for the closed-loop system, which we use to prove stability of the closed-loop system in Section 5.3.

In this chapter we make the first step towards the design of control laws for systems with nonconstant delays. Two major challenges arise in the case of a time-varying delay. The first challenge is how to determine the predictor state. We resolve this challenge by computing the predictor over a nonconstant prediction horizon which we appropriately define. The predictor is then given explicitly by using the variation of constants formula starting with the current state as an initial condition. The second challenge lies in the stability analysis of the closed-loop system under predictor feedback. We resolve this challenge by introducing a backstepping transformation with time-varying kernels, and by appropriately defining a transport PDE representation of the actuator state using a time- and spatial-varying transport speed. This backstepping transformation enables one to construct a Lyapunov functional which proves the stability of the closed-loop system.

We start in the present chapter by considering linear systems with time-varying input delay, probably the simplest form of systems with nonconstant delays. For this class of systems, we design the predictor feedback law which compensates for the time-varying input delay in Section 6.1. We prove stability of the closed-loop system in Section 6.2. The stability analysis is based on the introduction of the backstepping transformation, which allows the construction of a Lyapunov functional for the closed-loop system. In this chapter we use the PDE representation of the actuator state, which reveals an explicit dependence of the transport speed coefficient from both the time and space.

In this chapter we highlight one of the benefits of the availability of a Lyapunov functional. We show robustness of nominal, constant-delay predictor feedbacks to delay perturbations that depend on time, when the delay perturbation and its rate are small in any one of four different metrics. Specifically, in Section 7.1 we show robustness, in the H₁ stability norm of the actuator state, of global exponential stability of the predictor feedback for the cases where either the delay perturbation and its rate have small magnitude, or their L₁ norm is small, or they converge to zero as the time goes to infinity, or, finally, they have a small moving average for large times. In Section 7.2 we present an example of bilateral teleoperation between two robotic systems through a network. The network induces a constant nominal delay which is subject to an unknown time-varying perturbation that has a small moving average after a long period of time.

We consider in this chapter nonlinear, forward-complete systems with time-varying input delay, and we design the nonlinear predictor feedback law for the compensation of time-varying input delays in Section 8.1. The major challenge that we resolve is the definition of the predictor state since the prediction horizon is not constant. We prove global asymptotic stability of general nonlinear plants under predictor feedback in the presence of a time-varying input delay in Section 8.2. The stability proof is based on a nonlinear infinite-dimensional backstepping transformation with time-varying kernels that we introduce, and which helps us construct a strict Lyapunov functional for the closed-loop system. We present in Section 8.3 an example of unicycle stabilization subject to a time-varying input delay.

In this chapter we “backstep” one time-varying integrator and design a globally stabilizing controller for nonlinear strict-feedback systems with time-varying delays on the virtual inputs. One of the challenges of designing the predictor feedback law for this class of systems is that the control signal reaches different states of the system after different time intervals. Hence, the predictor states for different states (in addition to being time varying) have to be determined for different prediction horizons. Moreover, the technical details of the stability analysis of the closed-loop system under the predictor feedback design are much more involved than the case of only input delay. The reason is that in the present case one has to assume forward completeness of an infinite-dimensional plant rather than of a finite-dimensional plant as in the case of plants with only input delay (even when there is no input delay, the plant is infinite-dimensional since there is a state delay). Finally, the predictor feedback design does not follow immediately from a nominal design for the delay-free plant, but one has to design the predictor feedback law directly on the delay system. This is achieved by constructing two infinite-dimensional backstepping transformations: one for the delayed internal state and one for the actuator state.

We design the nonlinear predictor feedback law in Section 9.1, and we prove stability of the closed-loop system with the introduction of an infinite-dimensional backstepping procedure in Section 9.2. We illustrate our design with a second order strict-feedback system with time-varying state delay in Section 9.3.

In this chapter we design the predictor feedback law for nonlinear systems with state-dependent delays. As was made evident throughout Part II of the book, which deals with time-varying delays, the prediction horizon over which the predictor is defined depends on the future values of the delay. Therefore, the state dependence of the delay makes the prediction horizon dependent on the future value of the state, which means that it is impossible to know a priori how far in the future the prediction is needed. In this chapter we resolve this key design challenge and provide the formula for the predictor state. We also highlight that due to a fundamental restriction on the allowable magnitude of the delay function's gradient (the control signal never reaches the plant if the delay rate is larger than one), global stabilization is not possible in general.

In this chapter we prove stability of the closed-loop system under the predictor feedback design presented in Chapter 10. Due to a fundamental restriction on the allowable magnitude of the delay function's gradient (the control signal never reaches the plant if the delay rate is larger than one), we obtain only a regional stability result, even for the case of forward-complete systems. We establish an estimate of the region of attraction for our control scheme in the state space of the infinite-dimensional closed-loop nonlinear system based on the construction of a strict, time-varying Lyapunov functional in Section 11.1. We present a global result for forward-complete systems under a restrictive Lyapunov-like condition, which has to be a priori verified, that the delay rate be bounded by unity irrespective of the values of the state and input in Section 11.3. We also deal with linear systems, treating them as a special case of the design for nonlinear systems, for which we prove exponential stability. We present several examples, including stabilization of the nonholonomic unicycle subject to distance-dependent input delay in Section 11.2.

In Section 11.1 we proved a local stability result under assumptions of global stabilizability (Assumption 11.3) and forward-completeness (Assumption 11.2) of the delay-free system. It is reasonable to ask whether a local stability result can be established under a less restrictive assumption of local stabilizability of the delay-free system. In this chapter we provide an affirmative answer to this question. Our proof of this result does not employ a Lyapunov construction and, as such, provides an illustrative alternative to the proof technique in Section 11.1.

In the case of forward-complete systems our results were local due to a fundamental limitation on the allowable magnitude of the delay function's gradient (the control signal never reaches the plant if the delay rate is larger than one) which, since it depends on the state, could be restricted by appropriately restricting the allowable initial conditions. In the case of locally stabilizable systems, the stability results that we obtain are local for an additional reason. One has to further restrict the initial conditions such that, when the control signal “kicks in,” the state is within the region of attraction of the nominal design. In achieving that, however, one has to estimate the time for the control signal to reach the plant. We provide an estimate of this time by constructing a bound on the norm of the open-loop solutions (i.e., before the control signal reaches the plant), which allows us to give an estimate of the region of attraction for our control scheme and prove asymptotic stability of the closed-loop system in Section 12.1. We also present an example of a system that can have a finite escape time and whose linearization is not stabilizable.

In this chapter we extend the technique for compensating state-dependent delays from systems with delayed inputs to systems with delayed states. We focus on predictor feedback design for nonlinear systems in the strict-feedback form having a state-dependent state delay on the virtual input. The two key challenges are the definition of the predictor state and the fact that the predictor design does not follow immediately from the delay-free design. We resolve these challenges and design a predictor-based control law in Section 13.1. As in the case of input delays, due to the fundamental limitation of the allowable magnitude of the delay function's gradient, we obtain only regional results. For forward-complete nonlinear systems we establish asymptotic stability of the resulting infinite-dimensional nonlinear system for general nonnegative-valued delay functions of the state and give an estimate the region of attraction of the proposed controller in Section 13.2. We also provide two examples in Section 13.3, including one dealing with the control of cooling systems.

We consider forward-complete nonlinear systems that are locally exponentially stabilizable in the absence of delay (by a possibly time-varying control law), for which we employ the predictor-based design. The predictor controller is designed assuming constant input delay and using only an estimation of the unmeasured (since the delay is not known) infinite-dimensional actuator state. We prove robustness in the H ₂ norm of the actuator state, of the constant-delay predictor-based feedback, under simultaneous time-varying and state-dependent perturbations on the delay. Specifically, using the nonlinear infinite-dimensional backstepping transformation, we construct a Lyapunov functional for the closed-loop system that is composed of the plant, the predictor feedback, and the observer for the actuator state. With the constructed Lyapunov functional, we prove in Section 14.1 that the closed-loop system remains locally asymptotically stable when the perturbation and its rate are small. We illustrate the robustness properties of the predictor feedback with an example of a DC motor which is controlled through a network in Section 14.2. The network induces a delay which is composed of a known constant part and an unknown time-varying perturbation on this nominal value. In addition, the delay is subject to a state-dependent perturbation that depends on the armature current.

In the previous chapters we developed a systematic methodology for the compensation of input delays that depend on the current state of the plant. In this chapter we consider a different problem in which the delay depends on past values of the state, at a time that depends on the delay itself. Since the delay is needed for computing the predictor state, the first design challenge that we resolve is computing the delay through this implicit relation. Since the prediction horizon, over which we design the predictor, depends on the state of the system, the second design challenge that we resolve is computing the predictor state. The principal difficulty of the problem is in the stability analysis of the closed-loop system. Due to an inherent limitation that the delay rate is larger than −1 (which ensures that the delay function is uniquely defined, and hence that the dynamical system which describes the dynamics of the plant is uniquely defined), and since the delay depends on the state, only local results are possible. We also consider nonlinear systems with state-dependent delays on the state, specifically systems in the strict-feedback form with a delay on the virtual input that depends on past values of the state of the system at which the control input enters.

We present the predictor feedback law for general nonlinear systems with state-dependent input delays that depend on delayed states in Section 15.1. In Section 15.2 we prove local asymptotic stability of the closed-loop system with the aid of a Lyapunov functional that we construct by introducing a backstepping transformation of the actuator state. We present a numerical example of a second order strict-feedforward system with a state-dependent input delay (Section 15.3). We also deal with nonlinear systems with state-dependent delays on the state in Section 15.4.

Basic Inequalities

Input-to-Output Stability

Lyapunov Stability, Forward-Completeness, and Input-to-State Stability

Our adaptive designs rely on the use of parameter projection in our identifiers. We provide a treatment of projection for a general convex parameter set. The treatment for some of our designs where projection is used for only a scalar estimate D̂ is easily deduced from the general case.

The back matter includes the bibliography and index.