Compensation of Input Delay for Linear, Nonlinear, Adaptive, and PDE Systems

We present a tutorial introduction to methods for stabilization of systems with long input delays. The methods are based on techniques originally developed for boundary control of partial differential equations. We start with a consideration of linear systems, first with a known delay and then subject to a small uncertainty in the delay. Then we study linear systems with constant delays that are completely unknown, which requires an adaptive control approach. For linear systems, we also present a method for compensating arbitrarily large but known time-varying delays. Finally, we consider nonlinear control problems in the presence of arbitrarily long input delays.

An enormous wealth of knowledge and research results exists for control of systems with state delays and input delays. Problems with long input delays, for unstable plants, represent a particular challenge. In fact, they were the first challenge to be tackled, in Otto J. M. Smith’s article [1], where the compensator known as the Smith predictor was introduced five decades ago. The Smith predictor’s value is in its ability to compensate for a long input or output delay in set point regulation or constant disturbance rejection problems. However, its major limitation is that, when the plant is unstable, it fails to recover the stabilizing property of a nominal controller when delay is introduced.

A substantial modification to the Smith predictor, which removes its limitation to stable plants was developed three decades ago in the form of finite spectrum assignment (FSA) controllers [2, 3, 4]. More recent treatment of this subject can also be found in the books [5, 6]. In the FSA approach, the system

1

$$ \dot {X}(t) = AX(t) + BU(t-D)\,, $$

where

X

is the state vector,

U

is the control input (scalar in our consideration here),

D

is an arbitrarily long delay, and (

A

,

B

) is a controllable pair, is stabilized with the infinite-dimensional predictor feedback

2

$$ U(t) = K\left[{\rm e}^{AD} X(t)+ \int_{t-D}^t {\rm e}^{A(t-\theta)} B U(\theta)d\theta\right]\,, $$

where the gain

K

is chosen so that the matrix

A

 + 

BK

is Hurwitz. The word ‘predictor’ comes from the fact that the bracketed quantity is actually the future state

X

(

t

 + 

D

), expressed using the current state

X

(

t

) as the initial condition and using the controls

U

(

θ

) from the past time window [

t

 − 

D

,

t

]. Concerns are raised in [7] regarding the robustness of the feedback law (2) to digital implementation of the distributed delay (integral) term but are resolved with appropriate discretization schemes [8,9].

One can view the feedback law (2) as being given implicitly, since

U

appears both on the left and on the right, however, one should observe that the input memory

U

(

θ

),

θ

 ∈ [

t

 − 

D

,

t

] is actually a part of the state of the overall infinite-dimensional system, so the control law is actually given by an explicit full-state feedback formula. The predictor feedback (2) actually represents a particular form of boundary control, commonly encountered in the context of control of partial differential equations.

Motivated by our recent efforts in solving boundary control problems for various classes of partial differential equations (PDEs) using the continuum version of the backstepping method [10,11], we review in this article various extensions to the predictor feedback design that we have recently developed, particularly for nonlinear and PDE systems. These extensions are the subject of our new book [12]. They include the extension of predictor feedback to nonlinear systems and PDEs with input delays, various robustness and inverse optimality results, a delay-adaptive design, an extension to time-varying delays, and observer design in the presence of sensor delays and PDE dynamics. This article is a tutorial introduction to these design tools and concludes with a brief review of some open problems and research opportunities.