Keywords

Introduction

Time-delays are important components of many systems from engineering, economics, and the life sciences, due to the fact that the transfer of material, energy, and information is mostly not instantaneous. They appear, for instance, as computation and communication lags, they model transport phenomena and heredity, and they arise as feedback delays in control loops. An overview of applications, ranging from traffic flow control and lasers with phase-conjugate feedback, over (bio)chemical reactors and cancer modeling, to control of communication networks and control via networks, is included in Sipahi et al. (2011).

The aim of this contribution is to describe some fundamental properties of linear control systems subjected to time-delays and to outline principles behind analysis and synthesis methods. Throughout the text, the results will be illustrated by means of the scalar system

$$\displaystyle{ \dot{x}(t) = u(t-\tau ), }$$
(1)

which, controlled with instantaneous state feedback, \(u(t) = -\mathit{kx}(t)\), leads to the closed-loop system

$$\displaystyle{ \dot{x}(t) = -kx(t-\tau ). }$$
(2)

Although this didactic example is extremely simple, we shall see that its dynamics are already very rich and shed a light on delay effects in control loops.

In some works, the analysis of (2) is called the hot shower problem, as it can be interpreted as a (over)simplified model for a human adjusting the temperature in a shower: x(t) then denotes the difference between the water temperature and the desired temperature as felt by the person, the term – kx(t) models the reaction of the person by further opening or closing taps, and the delay is due to the propagation with finite speed of the water in the ducts.

Basis Properties of Time-Delay Systems

Functional Differential Equation

We focus on a model for a time-delay system described by

$$\displaystyle{ \dot{x}(t) = A_{0}x(t) + A_{1}x(t-\tau ),\quad x(t) \in \mathbb{R}^{n}. }$$
(3)

This is an example of a functional differential equation (FDE) of retarded type. The term FDE stems from the property that the right-hand side can be interpreted as a functional evaluated at a piece of trajectory. The term retarded expresses that the right-hand side does not explicitly depend on \(\dot{x}\).

As a first difference with an ordinary differential equation, the initial condition of (3) at t = 0 is a function ϕ from \([-\tau,0]\ \mathrm{to}\ \mathbb{R}^{n}\). For all \(\phi \in \mathcal{C}\left (\left [-\tau,0\right ], \mathbb{R}^{n}\right )\), where \(\mathcal{C}\left (\left [-\tau,0\right ], \mathbb{R}^{n}\right )\) is the space of continuous functions mapping the interval [−τ, 0] into \(\mathbb{R}^{n}\), a forward solution x(ϕ) exists and is uniquely defined. In Fig. 1, a solution of the scalar system (2) is shown.

Control of Linear Systems with Delays, Fig. 1
figure 327figure 327

Solution of (2) for \(\tau = 1,k = 1\), and initial condition \(\phi \equiv 1\)

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The discontinuity in the derivative at t = 0 stems from \(A_{0}\phi (0) + A_{1}\phi (-\tau )\neq \lim _{\theta \rightarrow 0}\dot{\phi }\). Due to the smoothing property of an integrator, however, at \(t = n \in \mathbb{N}\), the discontinuity will only be present in the (n + 1)th derivative. This illustrates a second property of functional differential equations of retarded type: solutions become smoother as time evolves. As a third major difference with ODEs, backward continuation of solutions is not always possible (Michiels and Niculescu 2007).

Reformulation in a First-Order Form

The state of system (3) at time t is the minimal information needed to continue the solution, which, once again, boils down to a function segment \(x_{t}(\phi )\mathrm{where}\ x_{t}(\phi )(\theta ) = x(t+\theta ),\theta \in [-\tau,0]\) (in Fig. 1, the function x t is shown in red for t = 5). This suggests that (3) can be reformulated as a standard ordinary differential equation over the infinite-dimensional space \(\mathcal{C}([-\tau,0], \mathbb{R}^{n})\). This equation takes the form

$$\displaystyle{ \frac{d} {dt}z(t) = \mathcal{A}z(t),\,z(t) \in \mathcal{C}\left (\left [-\tau,0\right ], \mathbb{R}^{n}\right ) }$$
(4)

where operator \(\mathcal{A}\) is given by

$$\displaystyle\begin{array}{rcl} & & \mathcal{D}\left (\mathcal{A}\right ) = \left \{\begin{array}{l@{\quad }l} \phi \in \mathcal{C}\left (\left [-\tau _{m},0\right ],\, \mathbb{R}^{n}\right ) :\quad &\dot{\phi }\in \mathcal{C}\left (\left [-\tau _{m},0\right ],\, \mathbb{R}^{n}\right ) \\ \quad &\dot{\phi }\left (0\right ) = A_{0}\phi \left (0\right ) + A_{1}\phi \left (-\tau \right )\\ \quad \end{array} \right \}, \\ & & \quad \mathcal{A}\phi \qquad \qquad \qquad \qquad \qquad \qquad = \frac{d\phi } {d\theta }. {}\end{array}$$
(5)

The relation between solutions of (3) and (4) is given by \(z(t)(\theta ) = x(t+\theta ),\theta \in [-\tau,0]\). Note that all system information is concentrated in the nonlocal boundary condition describing the domain of \(\mathcal{A}\). The representation (4) is closely related to a description by an advection PDE with a nonlocal boundary condition (Krstic 2009).

Asymptotic Growth Rate of Solutions and Stability

The reformulation of (3) into the standard form (4) allows us to define stability notions and to generalize the stability theory for ordinary differential equations in a straightforward way, with the main change that the state space is \(\mathcal{C}([-\tau,0], \mathbb{R}^{n})\). For example, the null solution of (3) is exponentially stable if and only if there exist constants C > 0 and γ > 0 such that

$$\displaystyle{\forall \phi \in \mathcal{C}\left (\left [-\tau _{m},0\right ],\, \mathbb{R}^{n}\right )\left \|x_{ t}\left (\phi \right )\right \|_{s} \leq Ce^{-\gamma \text{t}}\left \|\phi \right \|_{ s},}$$

where \(\vert \vert \cdot \vert \vert _{s}\) is the supremum norm and \(\vert \vert \phi \vert \vert _{s} =\mathrm{ sup}_{\theta \in [-\tau,0]}\vert \vert \phi (\theta )\vert \vert _{2}\). As the system is linear, asymptotic stability and exponential stability are equivalent. A direct generalization of Lyapunov’s second method yields:

Theorem 1

The null solution of linear system (3) is asymptotically stable if there exist a continuous functional\(V : \mathcal{C}([-\tau,0], \mathbb{R}^{n}) \rightarrow \mathbb{R}\)(a so-called Lyapunov-Krasovskii functional) and continuous nondecreasing functions\(u,v,w : \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\)with

$$\displaystyle\begin{array}{rcl} u(0) = v(0) = w(0) = 0\ \mathrm{and}\ u(s)> 0,& & {}\\ v(s)> 0,w(s)>\ \mathrm{ for}\ s> 0,& & {}\\ \end{array}$$

such that for all\(\phi \in \mathcal{C}([-\tau,0], \mathbb{R}^{n})\)

$$\displaystyle\begin{array}{rcl} u\left (\left \|\phi \right \|_{s}\right ) \leq V (\phi ) \leq v\left (\left \|\phi ()\right \|_{2}\right ),& & {}\\ \dot{V }(\phi ) \leq -w\left (\left \|\phi ()\right \|_{2}\right ),& & {}\\ \end{array}$$

where

$$\displaystyle{\dot{V }(\phi ) =\mathop{ \lim \,\sup }\limits _{h\rightarrow 0+} \frac{1} {h}[V (x_{h}(\phi )) - V (\phi )].}$$

Converse Lyapunov theorems and the construction of the so-called complete-type Lyapunov-Krasovskii functionals are discussed in Kharitonov (2013). Imposing a particular structure on the functional, e.g., a form depending only on a finite number of free parameters, often leads to easy-to-check stability criteria (for instance, in the form of LMIs), yet as price to pay, the obtained results may be conservative in the sense that the sufficient stability conditions might not be close to necessary conditions. As an alternative to Lyapunov functionals, Lyapunov functions can be used as well, provided that the condition \(\dot{V } <0\) is relaxed (the so-called Lyapunov-Razumikhin approach); see, for example, Gu et al. (2003).

Delay Differential Equations as Perturbation of ODEs

Many results on stability, robust stability, and control of time-delay systems are explicitly or implicitly based on a perturbation point of view, where delay differential equations are seen as perturbations of ordinary differential equations. For instance, in the literature, a classification of stability criteria is often presented in terms of delay-independent criteria (conditions holding for all values of the delays) and delay-dependent criteria (usually holding for all delays smaller than a bound). This classification has its origin at two different ways of seeing (3) as a perturbation of an ODE, with as nominal system \(\dot{x}(t) = A_{0}x(t)\ \mathrm{and}\ \dot{x}(t) = (A_{0} + A_{1})x(t)\) (system for zero delay), respectively. This observation is illustrated in Fig. 2 for results based on input-output- and Lyapunov-based approaches.

Control of Linear Systems with Delays, Fig. 2
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The classification of stability criteria in delay-independent results and delay-dependent results stems from two different perturbation viewpoints. Here, perturbation terms are printed in red

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The Spectrum of Linear Time-Delay Systems

Two Eigenvalue Problems

The substitution of an exponential solution in (3) leads us to the nonlinear eigenvalue problem

$$\displaystyle{ (\lambda I - A_{0} - A_{1}e^{-\lambda \tau })v = 0,\lambda \in \mathbb{C},v \in \mathbb{C}^{n},v\neq 0. }$$
(6)

The solutions of the equation \(\det (\lambda I - A_{0} - A_{1}e^{-\lambda \tau }) = 0\) are called characteristic roots. Similarly, formulation (4) leads to the equivalent infinite-dimensional linear eigenvalue problem

$$\displaystyle\begin{array}{rcl} (\lambda I\,-\,\mathcal{A})u = 0,\lambda \in \mathbb{C},u \in \mathcal{C}([-\tau,0], \mathbb{C}^{n}),u\not\equiv 0.& &{}\end{array}$$
(7)

The combination of these two viewpoints lays at the basis of most methods for computing characteristic roots; see Michiels (2012). On the one hand, discretizing (7), i.e., approximating \(\mathcal{A}\) with a matrix, and solving the resulting standard eigenvalue problems allow to obtain global information, for example, estimates of all characteristic roots in a given compact set or in a given right half plane. On the other hand, the (finitely many) nonlinear equations (6) allow to make local corrections on characteristic root approximations up to the desired accuracy, e.g., using Newton’s method or inverse residual iteration. Linear time-delay systems satisfy spectrum-determined growth properties of solutions. For instance, the zero solution of (3) is asymptotically stable if and only if all characteristic roots are in the open left half plane.

In Fig. 3 (left), the rightmost characteristic roots of (2) are depicted for k τ = 1. Note that since the characteristic equation can be written as \(\lambda \tau +\mathit{k}\tau \mathit{e}^{-\lambda \tau } = 0,k\) and τ can be combined into one parameter. In Fig. 3 (right), we show the real parts of the characteristic roots as a function of k τ. The plots illustrate some important spectral properties of retarded-type FDEs. First, even though there are in general infinitely many characteristic roots, the number of them in any right half plane is always finite. Second, the individual characteristic roots, as well as the spectral abscissa, i.e., the supremum of the real parts of all characteristic roots, continuously depend on parameters. Related to this, a loss or gain of stability is always associated with characteristic roots crossing the imaginary axis. Figure 3 (right) also illustrates the transition to a delay-free system as k τ → 0+.

Control of Linear Systems with Delays, Fig. 3
figure 329figure 329

(Left) Rightmost characteristic roots of (2) for k τ = 1. (Right) Real parts of rightmost characteristic roots as a function of k τ

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Critical Delays: A Finite-Dimensional Characterization

Assume that for a given value of k, we are looking for values of the delay τ c for which (2) has a characteristic root j ω c on the imaginary axis. From \(\mathit{j}\omega = -\mathit{ke}^{-j\omega \tau }\), we get

$$\displaystyle\begin{array}{rcl} \omega _{c}& =& k,\;\tau _{c} = \frac{ \frac{\pi }{2} + l2\pi } {\omega _{c}},\;l \\ & =& 0,1,\ldots,\mathfrak{R}\left \{\frac{d\lambda } {d\tau }\vert _{(\tau _{c},j\omega _{c})}\right \}^{-1} = \frac{1} {\omega _{c}^{2}}.{}\end{array}$$
(8)

Critical delay values τ c are indicated with green circles on Fig. 3 (right). The above formulas first illustrate an invariance property of imaginary axis roots and their crossing direction with respect to delay shifts of 2π∕ω c . Second, the number of possible values of ω c is one and thus finite. More generally, substituting λ = j ω in (6) and treating τ as a free parameter lead to a two-parameter eigenvalue problem

$$\displaystyle{ (j\omega I - A_{0} - A_{1}z)v = 0, }$$
(9)

with ω on the real axis and \(z :=\mathrm{ exp}(-\mathit{j}\omega \tau )\) on the unit circle. Most methods to solve such a problem boil down to an elimination of one of the independent variables ω or z. As an example of an elimination technique, we directly get from (9)

$$\displaystyle\begin{array}{rcl} j\omega \in \sigma (A_{0} + A_{1}z),\;-j\omega \in \sigma (A_{0}^{{\ast}} + A_{ 1}^{{\ast}}z^{-1})& & {}\\ \Rightarrow \det \left ((A_{0} + A_{1}z) \oplus (A_{0}^{{\ast}} + A_{ 1}^{{\ast}}z^{-1})\right ) = 0,& & {}\\ \end{array}$$

where \(\sigma (\cdot )\) denotes the spectrum and \(\oplus\) the Kronecker sum. Clearly, the resulting eigenvalue problem in z is finite dimensional.

Control of Linear Time-Delay System

Limitations Induced by Delays

It is well known that delays in control loop may lead to a significant degradation of performance and robustness and even to instability (Niculescu 2001; Richard 2003). Let us return to example (2). As illustrated with Fig. 3 and expressions (8), the system loses stability if τ reaches the value π∕2k, while stability cannot be recovered for larger delays. The maximum achievable exponential decay rate of the solutions, which corresponds to the minimum of the spectral abscissa, is given by \(-1/\tau\); hence, large delays can only be tolerated at the price of a degradation of the rate of convergence. It should be noted that the limitations induced by delays are even more stringent if the uncontrolled systems are exponentially unstable, which is not the case for (2).

The analysis in the previous sections gives a hint why control is difficult in the presence of delays: the system is inherently infinite dimensional. As a consequence, most control design problems which involve determining a finite number of parameters can be interpreted as reduced-order control design problems or as control design problems for under-actuated systems, which both are known to be hard problems.

Fixed-Order Control

Most standard control design techniques lead to controllers whose dimension is larger or equal to the dimension of the system. For infinite-dimensional time-delay system, such controllers might have a disadvantage of being complicated and hard to implement. To see this, for a system with delay in the state, the generalization of static state feedback, u(t) = k(x), is given by \(u(t) =\int _{ -\tau }^{0}x(t+\theta )d\mu (\theta )\), where μ is a function of bounded variation. However, in the context of large-scale systems, it is known that reduced-order controllers often perform relatively well compared to full-order controllers, while they are much easier to implement.

Recently, new methods for the design of controllers with a prescribed order (dimension) or structure have been proposed (Michiels 2012). These methods rely on a direct optimization of appropriately defined cost functions (spectral abscissa, \(\mathcal{H}_{2}/\mathcal{H}_{\infty }\) criteria). While \(\mathcal{H}_{2}\) criteria can be addressed within a derivative-based optimization framework, \(\mathcal{H}_{\infty }\) criteria and the spectral abscissa require targeted methods for non-smooth optimization problems. To illustrate the need for such methods, consider again Fig. 3 (right): minimizing the spectral abscissa for a given value of τ as a function of the controller gain k leads to an optimum where the objective function is not differentiable, even not locally Lipschitz, as shown by the red circle. In case of multiple controller parameters, the path of steepest descent in the parameter space typically has phases along a manifold characterized by the non-differentiability of the objective function.

Using Delays as Controller Parameters

In contrast to the detrimental effects of delays, there are situations where delays have a beneficial effect and are even used as controller parameters; see Sipahi et al. (2011). For instance, delayed feedback can be used to stabilize oscillatory systems where the delay serves to adjust the phase in the control loop. Delayed terms in control laws can also be used to approximate derivatives in the control action. Control laws which depend on the difference \(x(t) - x(t-\tau )\), the so-called Pyragas-type feedback, have the property that the position of equilibria and the shape of periodic orbits with period τ are not affected, in contrary to their stability properties. Last but not least, delays can be used in control schemes to generate predictions or to stabilize predictors, which allow to compensate delays and improve performance (Krstic 2009; Zhong 2006). Let us illustrate the main idea once more with system (1).

System (1) has a special structure, in the sense that the delay is only in the input, and it is advantageous to exploit this structure in the context of control. Coming back to the didactic example, the person who is taking a shower is – possibly after some bad experiences – aware about the delay and will take into account his/her prediction of the system’s reaction when adjusting the cold and hot water supply. Let us, to conclude, formalize this. The uncontrolled system can be rewritten as \(\dot{x}(t) = v(t)\), where \(v(t) = u(t-\tau )\). We know u up to the current time t; thus, we know v up to time t +τ, and if x(t) is also known, we can predict the value of x at time t +τ,

$$\displaystyle\begin{array}{rcl} x_{p}(t+\tau )& =& x(t) +\int _{ t}^{t+\tau }v(s)ds {}\\ & =& x(t) +\int _{ t-\tau }^{t}u(s)ds, {}\\ \end{array}$$

and use the predicted state for feedback. With the control law \(u(t) = -\mathit{kx}_{p}(t+\tau )\), there is only one closed-loop characteristic root at \(\lambda = -k\), i.e., as long as the model used in the predictor is exact, the delay in the loop is compensated by the prediction. For further reading on prediction-based controllers, see, e.g., Krstic (2009) and the references therein.

Conclusions

Time-delay systems, which appear in a large number of applications, are a class of infinite-dimensional systems, resulting in rich dynamics and challenges from a control point of view. The different representations and interpretations and, in particular, the combination of viewpoints lead to a wide variety of analysis and synthesis tools.

Cross-References