Generic stabilizability for time-delayed feedback control

Abstract

Time-delayed feedback control is one of the most successful methods to discover dynamically unstable features of a dynamical system in an experiment. This approach feeds back only terms that depend on the difference between the current output and the output from a fixed time T ago. Thus, any periodic orbit of period T in the feedback-controlled system is also a periodic orbit of the uncontrolled system, independent of any modelling assumptions. It has been an open problem whether this approach can be successful in general, that is, under genericity conditions similar to those in linear control theory (controllability), or if there are fundamental restrictions to time-delayed feedback control. We show that, in principle, there are no restrictions. This paper proves the following: for every periodic orbit satisfying a genericity condition slightly stronger than classical linear controllability, one can find control gains that stabilize this orbit with extended time-delayed feedback control. While the paper’s techniques are based on linear stability analysis, they exploit the specific properties of linearizations near autonomous periodic orbits in nonlinear systems, and are, thus, mostly relevant for the analysis of nonlinear experiments.

1. Introduction

Time-delayed feedback control was originally proposed by Pyragas in 1992 as a tool for discovery of unstable periodic orbits (one frequent building block in nonlinear systems with chaotic dynamics or multiple attractors) in experimental nonlinear dynamical systems [1]. Pyragas proposed that one take the output $x(t)\in {\mathbb{R}}^n$ of a dynamical system and feed back in real time the difference between this output and the output time T ago into an input $u(t)\in {\mathbb{R}}^{n_u}$ of the system (multiplied by some control gains $K^T\in {\mathbb{R}}^{n_u\times n}$):

$$u(t)=K^T[x(t-T)-x(t)].$$ 1.1

In a first experimental demonstration, Pyragas and Tamaševičius successfully identified and stabilized an unstable periodic orbit in a chaotic electrical circuit [2]. Socolar et al. in 1994 [3] introduced a generalization of time-delayed feedback (which is often used in place of (1.1) and is implemented as shown in figure 1 as a block diagram):

$$\begin{array}{rl}u(t)& =K^T[\tilde{x}(t)-x(t)],\text{where}\\ \tilde{x}(t)& =(1-\epsilon )\tilde{x}(t-T)+\epsilon x(t-T),\end{array}$$ 1.2

and ε∈(0,1], called extended time-delayed feedback. If ε=1, feedback law (1.2) reduces to time-delayed feedback (1.1), if ε=0 feedback law (1.2) degenerates to classical linear feedback with a fixed T-periodic reference signal $\tilde{x}(t)$ (see (1.3) for a discussion). Note that, for example in [3], the variable $\tilde{x}(t)$ was eliminated in the mathematical discussion by writing

$$u(t)=K^T(\epsilon [\sum _{j=1}^{\mathrm{\infty }}{(1-\epsilon )}^{j-1}x(t-jT)]-x(t)).$$

While this would suggest that knowledge of all history of x is required to initialize the system, in the experiment, the feedback control was implemented as shown in the block diagram in figure 1, which is equivalent to (1.2). By construction of the feedback laws (1.1) and (1.2), for ε>0 every periodic, orbit of period T of the dynamical system with feedback control is also a periodic orbit of the uncontrolled system (u=0).¹ However, the stability of the periodic orbit may change from unstable without control to asymptotically stable with control for appropriately chosen gains K.

Figure 1.

Block diagram for extended time-delayed feedback (1.2), as applied to an experiment, for example, in [3]. We prove generic stabilizability for the case when the output x is the whole internal state of the dynamical system and the input u is scalar. Triangle block symbols are multiplications of the signal by the factor in the block.

The delayed terms x(t−T) and $\tilde{x}(t-T)$ make extended time-delayed feedback control different from the classical linear feedback control, which has the form

$$u(t)=K^T[x_{\ast }(t)-x(t)],$$ 1.3

where x_*(t) is, for example, a known unstable periodic orbit of the dynamical system governing x. While the goal of (1.3) is to stabilize a known reference output (in this case, a periodic orbit), time-delayed feedback is able to stabilize and, thus, find a priori unknown periodic orbits. For this reason time-delayed feedback originated, and has found most attention, in the physics and science community, rather than in the control engineering community. It can be used to discover features of nonlinear dynamical systems inaccessible in conventional experiments, such as unstable equilibria, periodic orbits and their bifurcations, non-invasively. A few examples where time-delayed feedback (or its extended version) have been successfully used are control of chemical turbulence [4], all-optical control of unstable steady states and self-pulsations in semiconductor lasers [5–7], control of neural synchrony [8–10], control of the Taylor–Couette flow [11], atomic force microscopy [12] and (with further modifications) systematic bifurcation analysis in mechanical experiments in mechanical engineering [13–15].

One difficulty for time-delayed feedback is that there are until now no general statements guaranteeing the existence of stabilizing control gains K under some genericity condition on the dynamical system governing x and its input u, such as controllability. This is in contrast to the situation for classical linear feedback control (1.3), where the following is known [16]: if the periodic orbit x_* is linearly controllable by input u in p periods (this is a genericity condition), then one can assign its period-pT monodromy matrix to any matrix with positive determinant by pT-periodic feedback gains $K{(t)}^T\in {\mathbb{R}}^{n_u\times n}$.

The greater level of difficulty for (extended) time-delayed feedback is unsurprising, because the feedback-controlled system acquires memory. Let us assume that the measured quantity x is governed by an ordinary differential equation (ODE) $\dot{x}(t)=f(x(t),u(t))$ (which is autonomous without control (u=0) and non-autonomous with classical feedback control (1.3)). Then, x and $\tilde{x}$ will be governed by a delay differential equation (DDE) if u is given by time-delayed feedback (1.1), or by an ODE coupled to a difference equation if u is given by extended time-delayed feedback (1.2) with ε∈(0,1) (we will refer to both cases simply as DDEs). This means that the initial value for both, x and $\tilde{x}$, is a history segment, a function on [−T,0] with values in ${\mathbb{R}}^n$. In DDEs, periodic orbits have infinitely many Floquet multipliers.² Section 2 will review the development of analysis for the time-delayed feedback laws (1.1) and (1.2). This paper proves a first simple generic stabilizability result for extended time-delayed feedback control (1.2) with time-periodic gains K(t) (similar to results for classical linear feedback control).

Main result: the following theorem states that the classical approach to periodic feedback gain design by Brunovsky [17] can be applied to make (1.2) stable in the limit of small ε>0 in the simplest and most common case of a scalar input u (thus, n_u=1) and linear controllability of the periodic orbit by an input at a single time instant.

Theorem 1.1 (generic stabilizability with extended time-delayed feedback). —

Assume that the dynamical system

$$\dot{x}(t)=f(x(t),u(t))(\hspace{0.17em}f:{\mathbb{R}}^n\times \mathbb{R}\mapsto {\mathbb{R}}^n\mathrm{smooth})$$ 1.4

with u=0 has a periodic orbit x_*(t) of period T>0, and assume that the monodromy matrix³ P₀ of x_* from time 0 to T is controllable with b₀=∂_uf(x_*(0),0) that is, $det[b_0,P_0{b}_0,\dots ,P_0^{n-1}{b}_0]\ne 0$.

Then, there exist gains $K_0\in {\mathbb{R}}^n$ such that x_* as a periodic orbit of the feedback-controlled system (1.4) with see below for the definition of the function Δ_δ

$$u(t)={\mathrm{\Delta }}_{\delta }(t)K_0^T[\tilde{x}(t)-x(t)],\tilde{x}(t)=(1-\epsilon )\tilde{x}(t-T)+\epsilon x(t-T)$$ 1.5

has one simple Floquet multiplier at 1 and all other Floquet multipliers inside the unit circle for all sufficiently small ε and δ.

The function Δ_δ is zero except for a short interval of length δ every period T such that the feedback u has the form of a short but large near-impulse:⁴

$${\mathrm{\Delta }}_{\delta }(t)=\{\begin{array}{ll}1/\delta ,& \text{if }t{|}_{\mathrm{mod}[0,T)}\in [0,\delta ]\\ 0,& \text{if }t{|}_{\mathrm{mod}[0,T)}\notin [0,\delta ].\end{array}$$ 1.6

Remarks–constant gains: the gains as constructed are periodic. This is to be expected, because there are no general results for constant gains $K\in {\mathbb{R}}^n$ for the classical linear feedback case (1.3), either. Furthermore, simple examples show that the above statement can definitely not be made when we restrict ourselves to constant gains in (1.5): $u(t)=K^T[\tilde{x}(t)-x(t)]$ with $K\in {\mathbb{R}}^n$. See §5 for an example.

Properties of the spectrum of the linearization (see also figure 2 for an illustration): the claim of theorem 1.1 is about linear stability of the periodic orbit $(x(t),\tilde{x}(t))=(x_{\ast }(t),x_{\ast }(t))$ of (1.4)–(1.5). Thus, we have to consider the problem (1.4)–(1.5), linearized in $(x(t),\tilde{x}(t))=(x_{\ast }(t),x_{\ast }(t))$:

$$\begin{array}{rl}\dot{x}(t)& =A(t)x(t)+b(t){\mathrm{\Delta }}_{\delta }(t)K_0^T[\tilde{x}(t)-x(t)]\\ \mathrm{and}\tilde{x}(t)& =(1-\epsilon )\tilde{x}(t-T)+\epsilon x(t-T),\end{array}\}$$ 1.7

where A(t)=∂_xf(x_*(t),0) and b(t)=∂_uf(x_*(t),0).

Figure 2.

Illustration of Floquet multiplier spectrum for extended time-delayed feedback with single input. Using the appropriate control gains K₀, n Floquet multipliers can be freely assigned up to determinant restrictions (n=2 in the illustrated case). The other Floquet multipliers lie on a circle of radius ε/2 around 1−ε/2, accumulating at 1−ε. This spectrum is achieved asymptotically for sufficiently short and strong impulses (δ≪1) and small ε. A simple trivial multiplier at 1 is always present. (Online version in colour.)

The gains K₀ are identical to those chosen by Brunovsky [17] for the classical feedback spectrum assignment problem (note that Brunovsky made weaker assumptions on A(t) and b(t) than theorem 1.1). One can choose the gains K₀ to place the n Floquet multipliers λ_k (k=1,…,n) of

$$\dot{x}=A(t)x-b(t){\mathrm{\Delta }}_{\delta }(t)K_0^{T}x,$$

anywhere inside the unit circle subject to the restriction that they have to be eigenvalues of a real matrix with positive determinant.

However, DDEs such as (1.7) may have infinitely many Floquet multipliers. Theorem 1.1 rests on a perturbation argument for small ε>0 for the other, delay-induced, Floquet multipliers.⁵ At ε=0, the difference equation for $\tilde{x}$ in (1.7) simplifies to $\tilde{x}(t)=\tilde{x}(t-T)$. Thus, an arbitrary initial history $\tilde{x}$ with period T will not change under the time-T map of (1.7). This results for ε=0 in a spectrum of (1.7) consisting of

• — the finitely many assigned Floquet multipliers λ_k (k=1,…,n) as determined by the gains K₀, and (assuming all λ_k≠1)

• — the spectral point ${\mathrm{\lambda }}_{\mathrm{\infty }}=1$ with an infinite-dimensional eigenspace. Specifically, if we choose the space of continuous functions $C([-T,0];{\mathbb{R}}^n\times {\mathbb{R}}^n)$ as phase space for (1.7) then, for ε=0, the eigenspace for ${\mathrm{\lambda }}_{\mathrm{\infty }}=1$ is

  $$\begin{array}{rl}& \{(x,\tilde{x})\in C([-T,0];{\mathbb{R}}^n\times {\mathbb{R}}^n):\hspace{0.17em}x(0)=x(-T),\tilde{x}(0)=\tilde{x}(-T),\phantom{K_0^T}\\ & \dot{x}=A(t)x+b(t){\mathrm{\Delta }}_{\delta }(t)K_0^T[\tilde{x}(t)-x(t)]\}.\end{array}$$

  Note that, because λ_k≠1 for k=1,…,n, the ODE $\dot{x}=A(t)x+b(t){\mathrm{\Delta }}_{\delta }(t)K_0^T[\tilde{x}(t)-x(t)]$ has a unique periodic solution x for all periodic functions $\tilde{x}$. This means that for every T-periodic $\tilde{x}$ there is an eigenvector for ${\mathrm{\lambda }}_{\mathrm{\infty }}=1$ with this $\tilde{x}$-component.

The general theory for DDEs [18] ensures that for positive (small) ε the Floquet multipliers λ_k (k=1,…,n) are only slightly perturbed, and that the infinitely many Floquet multipliers emerging from ${\mathrm{\lambda }}_{\mathrm{\infty }}$ accumulate to the spectrum of the essential part, the difference equation in (1.7) with the $\tilde{x}$ terms only: $\tilde{x}(t)=(1-\epsilon )\tilde{x}(t-T)$. Specifically, the only accumulation point of the spectrum of (1.7) for ε∈(0,1) is at 1−ε and the stability of (1.7) is determined by the location of the Floquet mulitpliers emerging from the perturbation of ${\mathrm{\lambda }}_{\mathrm{\infty }}$ (of which at most finitely many can lie outside the unit circle). The detailed analysis in §3 will show that for small ε>0 the Floquet multipliers emerging from ${\mathrm{\lambda }}_{\mathrm{\infty }}$ lie close to a circle of radius ε/2 around 1−ε/2, inside the unit circle (except for the unit Floquet multiplier), as shown in figure 2 for n=2.

Trivial multiplier: the eigenvector to the trivial multiplier 1 is ${\dot{x}}_{\ast }(0)$, corresponding to the linearized phase shift (for every $s\in \mathbb{R}$, t↦x_*(t+s) is also a solution of the system with extended time-delayed feedback (1.4) and (1.5)). Section 4 gives a modification of theorem 1.1 with a function Δ_δ depending on x(t) instead of t switching the gains on and off. Then, the feedback-controlled system becomes autonomous. In this modified system with autonomous (but nonlinear) extended time-delayed feedback, the periodic orbit x_* is asymptotically stable in the classical sense.

Timing of impulse: in (1.6), we chose the timing of the impulse (the part of the period [0,T), where Δ_δ is non-zero) as [0,δ] without loss of generality. The genericity condition in its most general form requires that there must be a time t∈[0,T] such that the monodromy matrix from t to t+T and ∂_uf(x_*(t),0) are controllable. As the uncontrolled system is autonomous, we can shift the phase of the periodic orbit x_*, considering x_*(t+⋅) instead of x_*.

Practical considerations: the result gives precise control over the Floquet multipliers in the limit of small δ and ε. For small δ the feedback control corresponds to a sharp kick once per period, which is not practical for strongly unstable periodic orbits. However, the gains found with the help of theorem 1.1 provide a feasible starting point for optimization-based spectrum assignment methods (continuous pole placement) as constructed by Michiels et al. [19,20] and adapted to time-delayed feedback (1.1) [21–23]. In the context of continuation one can combine the gains provided by theorem 1.1 as starting points, continuous pole placement, and the automatic adjustment of the time delay T demonstrated in [24,25] to create a feedback control that non-invasively tracks a family of periodic orbits in a system parameter.

2. Review: analysis of (extended) time-delayed feedback

The initial proposals of time-delayed feedback (1.1) and its extended version (1.2) were accompanied with demonstrations in simulations and experiments, showing that this type of feedback control can be successful [1–3], but not with general necessary or sufficient conditions for applicability or with constructive ways to design the feedback gains.

However, it was quickly recognized that time-delayed feedback can be applied to periodic orbits that are weakly unstable owing to a period doubling bifurcation or torus bifurcation [26,27]. Hence, time-delayed feedback is often associated with control of chaos, because it can be used to suppress period doubling cascades. However, general sufficient criteria were rather restrictive [28], requiring full access to the state (x governed by $\dot{x}(t)=f(x(t))+u(t)$ with $u\in {\mathbb{R}}^n$). A first general result was negative, the so-called odd number limitation for periodically forced systems [29], showing that extended time-delayed feedback cannot stabilize periodic orbits in periodically forced systems with an odd number of Floquet multipliers λ with Reλ>1 (and no Floquet multiplier at 1). This theoretical limitation is not a severe restriction in practice, because one can extend the uncontrolled system with an artificial unstable degree of freedom before applying time-delayed feedback [30]. Fiedler et al. [31] showed that this limitation does not apply to autonomous periodic orbits [32]. Since then, general results have been proven for weakly unstable periodic orbits with a Floquet multiplier close to 1 (but larger than 1 [33]), or near subcritical Hopf bifurcations [34,35]. A review of developments up to 2010 is given in [36].

An extension of the odd number limitation to autonomous periodic orbits (with trivial Floquet multiplier) was given by Hooton & Amann [37,38] for both, time-delayed feedback (1.1) and its extension (1.2). However, these limitations merely impose restrictions on the gains K. They do not rule out feedback stabilizability a priori (which is in contrast to the statements about periodic orbits in forced systems).

3. Spectrum of linearization for extended time-delayed feedback-controlled system

Let us consider a feedback-controlled dynamical system with extended time-delayed feedback control and arbitrary time-dependent gains $K(t)\in {\mathbb{R}}^n$:

$$\begin{array}{rl}\dot{x}(t)& =f(x(t),u(t)),\end{array}$$ 3.1

$$\begin{array}{rl}u(t)& =K{(t)}^T[\tilde{x}(t)-x(t)]\end{array}$$ 3.2

$$\begin{array}{rl}\mathrm{and}\tilde{x}(t)& =(1-\epsilon )\tilde{x}(t-T)+\epsilon x(t-T).\end{array}$$ 3.3

This system is governed by an ODE without control (u=0) and a DDE with control. We assume that the uncontrolled system $\dot{x}(t)=f(x(t),0)$ has a periodic orbit x_* of period T. This periodic orbit x_* is also a periodic orbit of (3.1)–(3.3) if ε>0: $x(t)=\tilde{x}(t)=x_{\ast }(t)$. System (3.1)–(3.3) is a DDE with the phase space

$$\{(x,\tilde{x})\in C([-T,0];{\mathbb{R}}^n\times {\mathbb{R}}^n):\tilde{x}(0)=(1-\epsilon )\tilde{x}(-T)+\epsilon x(-T)\}.$$

Hale & Verduyn-Lunel [18] treated DDEs of the type of system (3.1)–(3.3) (which contains difference equations) as part of their discussion of neutral DDEs. The essential part of the semiflow generated by (3.1)–(3.3) is governed by the part of (3.3) containing $\tilde{x}$: $\tilde{x}(t)=(1-\epsilon )\tilde{x}(t-T)$, which is linear and has spectral radius 1−ε. Thus, it fits into the scope of the theory as described in the textbook by Hale & Verduyn-Lunel [18]. Specifically, the asymptotic stability of the periodic orbit given by $x(t)=\tilde{x}(t)=x_{\ast }(t)$ is determined by the point spectrum of the linearization of (3.1)–(3.3). Hence, the periodic orbit x_* is stable if all Floquet multipliers of the linearization along x_* except the trivial multiplier 1 are inside the unit circle (and the trivial Floquet multiplier 1 is simple). We denote the monodromy matrix⁶ of

$$\dot{x}=[A(t)-\mu b(t)K{(t)}^T]x(t),\text{where}\hspace{0.17em}A(t)={\mathrm{\partial }}_{x}f(x_{\ast }(t),0),b(t)={\mathrm{\partial }}_{u}f(x_{\ast }(t),0)$$ 3.4

for $\mu \in \mathbb{C}$ by P(μ). Thus, the monodromy matrix of the uncontrolled system $\dot{x}(t)=A(t)x(t)$ equals P(0), which we denote by

$$P_0=P(0).$$ 3.5

With this definition of P(μ), Floquet multipliers of the linearization of (3.1)–(3.3) in x_* different from 1−ε are given as roots of

$$h(\mathrm{\lambda };\epsilon ):=det[\mathrm{\lambda }I-P(1-\frac{\epsilon }{\mathrm{\lambda }-(1-\epsilon )})]$$

(I is the identity matrix; see appendix A(a) for detailed proof). Lemma 3.1 states that the gains K(t) can only stabilize a periodic orbit x_* with extended time-delayed feedback and small ε, if they are stabilizing with classical linear feedback (that is, when replacing the recursively determined signal $\tilde{x}$ by the target orbit x_*: u(t)=K(t)[x_*(t)−x(t)]). (Recall that A(t)=∂_xf(x_*(t),0), b(t)=∂_uf(x_*(t),0).)

Lemma 3.1 (extended time-delayed feedback stabilization implies classical stabilization). —

If the linear system

$$\dot{x}(t)=[A(t)-b(t)K{(t)}^T]x(t)$$ 3.6

has at least one Floquet multiplier outside the unit circle, then there exists a ${\epsilon }_{max}\in (0,1)$ such that the periodic orbit x_* is unstable for the extended time-delayed feedback (3.1)–(3.3) for all $\epsilon \in (0,{\epsilon }_{max})$.

Proof. —

The Floquet multipliers of (3.6) are given as roots of $h(\mathrm{\lambda };0)=det(\mathrm{\lambda }I-P(1))$. We denote the root with modulus greater than 1 by λ₀ such that h(λ₀;0)=0. Consequently, for all λ in the ball B_r(λ₀), where r=(|λ₀|−1)/2, the difference h(λ;ε)−h(λ;0) is uniformly bounded and analytic for all ε∈(0,1) and all λ in B_r(λ₀). Because λ₀ must have finite multiplicity as a root of h(⋅;0), h(λ;ε) must have a root in B_r(λ₀) for sufficiently small ε>0 (say, $\epsilon \in (0,{\epsilon }_{max})$), too. By choice of r, this root lies outside of the unit circle. ▪

Lemma 3.1 shows that gains K(t) that stabilize with extended time-delayed feedback with small ε also have to feedback-stabilize in the classical sense. Because K is an arbitrary periodic function, there are many ways to construct gains for the classical linear feedback control u(t)=K(t)[x_*(t)−x(t)] for periodic orbits x_* [16]. We choose the approach comprehensively treated by Brunovsky [17], which is particularly amenable to analysis in the extended time-delayed feedback case and for which one can then prove the converse of lemma 3.1:

extended time-delayed feedback is stabilizing for the same gains for which Brunovksy’s approach is stabilizing the classical feedback control.

Near-impulse feedback and its parametrized monodromy matrix: we pick state feedback control in the form of a single large but short impulse. That is, we consider a short time δ∈(0,T) and define the linear feedback control

$$u_{\delta }(t;y)={\mathrm{\Delta }}_{\delta }(t)K_0^{T}y,\text{where}\hspace{0.17em}{\mathrm{\Delta }}_{\delta }(t)=\{\begin{array}{ll}1/\delta ,& \text{if }t{|}_{\mathrm{mod}[0,T)}\in [0,\delta ]\\ 0,& \text{if }t{|}_{\mathrm{mod}[0,T)}\notin [0,\delta ].\end{array}$$ 3.7

where t|_mod[0,T) is the number τ∈[0,T) such that (t−τ)/T is an integer, and $K_0\in {\mathbb{R}}^n$ is a vector of constant control gains. Let us first look at classical feedback $u(t)={\mathrm{\Delta }}_{\delta }(t)K_0^T[x_{\ast }(t)-x(t)]$ (where we assume that we know the periodic orbit x_*). Using feedback law (3.7), the feedback-controlled system reads

$$\dot{x}(t)=f(x(t),{\mathrm{\Delta }}_{\delta }(t)K_0^T[x_{\ast }(t)-x(t)]).$$ 3.8

We define the nonlinear time-T map X(x;δ,K₀) as the solution at time T (the period of the periodic orbit x_*) of (3.8) when starting from x at time 0 (including the dependence on parameters δ and K₀ as additional arguments of X). Then, for small deviations y₀ from x_*(0), the map X(⋅;δ,K₀) has the form X(x_*(0)+y₀;δ,K₀)=y(T)+O(∥y₀∥²), where y satisfies the linear differential equation (recall that A(t)=∂_xf(x_*(t),0), b(t)=∂_uf(x_*(t),0))

$$\dot{y}(t)=[A(t)-b(t){\mathrm{\Delta }}_{\delta }(t)K_0^T]y(t),y(0)=y_0,$$ 3.9

and the term O(∥y₀∥²) is uniformly small (including its derivatives) for all δ . Let us introduce a complex parameter μ into (3.9), which will become useful later in our consideration of extended time-delayed feedback: define for a general complex μ with |μ|≤C (with an arbitrary fixed C>0) the linear ODE

$$\dot{y}(t)=[A(t)-\mu b(t){\mathrm{\Delta }}_{\delta }(t)K_0^T]y(t),y(0)=y_0.$$ 3.10

Denote the monodromy matrix of (3.10) from t=0 to t=T by P(μ;δ,K₀) to keep track of its dependence on the parameters δ∈(0,T) and $K_0\in {\mathbb{R}}^n$. Thus, P(μ;δ,K₀) refers to the same monodromy matrix as P(μ), defined by (3.4), for the special case K(t)=Δ_δ(t)K₀. Then, P(μ;δ,K₀) satisfies

$$P(\mu ;\delta ,K_0)=P_0\exp (-b(0)K_0^T\mu )+O(\delta ),$$ 3.11

where the error term O(δ) is uniform for |μ|≤C and bounded ∥K₀∥, including its derivatives with respect to all arguments. Hence, we can extend the definition of P(μ;δ,K₀) to δ=0:

$$\begin{array}{rl}P(\mu ;0,K_0)& =\underset{\delta \to 0}{lim}P(\mu ;\delta ,K_0)=P_0\exp (-b(0)K_0^T\mu ),\end{array}$$ 3.12

$$\begin{array}{rl}& =P_0[I-\sigma (\mu )b(0)K_0^T],\text{where}\\ \sigma (\mu )& =\{\begin{array}{ll}\frac{\exp (\mu K_0^{T}b(0))-1}{K_0^{T}b(0)},& \text{if }{K}_0^{T}b(0)\ne 0\\ \mu ,& \text{if }{K}_0^{T}b(0)=0.\end{array}\end{array}$$ 3.13

The limit is uniform for all μ with modulus less than C. For μ=0, P is the monodromy matrix P₀ of the uncontrolled system, and, thus, independent of δ and K₀.

Approximate spectrum assignment for finitely many Floquet multipliers: the control (3.7) is a simplification of the general case of finitely many (at most n) short impulses treated in [17]. Feedback of type (3.7) permits us to assign arbitrary spectrum approximately under the assumption that the pair (P₀,b(0)) is controllable (recall that, according to the definition of P₀ in (3.5), P₀ is the monodromy matrix of the uncontrolled system $\dot{x}(t)=f(x(t),0)$ along the periodic orbit x_*). This is a stronger assumption than the assumption made in [17], but it is still a genericity assumption.

Lemma 3.2 (approximate spectrum assignment for classical state feedback control, simplified from [17]). —

Let r>0 be arbitrary. If the pair (P₀,b(0)) is controllable i.e. the n×n controllability matrix $[b(0),P_{0}b(0),\dots ,P_0^{n-1}b(0)]$ is regular, then there exist a ${\delta }_{max}>0$ and a vector of control gains $K_0\in {\mathbb{R}}^n$ in (3.7) such that all Floquet multipliers of x_* for the differential equation (3.8) have modulus less than r for all $\delta \in (0,{\delta }_{max}),$ where Δ_δ is as defined in (3.7).

Note that the vector K₀ can be chosen independent of the $\delta \in (0,{\delta }_{max}),$ but it may depend on the radius r into which one wants to assign the spectrum. This result follows from classical linear feedback control theory ([17] proves a more general result). In short, linear feedback control theory [17] makes the following argument (thus, proving lemma 3.2): the linearization of X with respect to its initial condition can be expanded in δ as

$${\mathrm{\partial }}_{x}X(x_{\ast }(0);\delta ,K_0)=P(1;\delta ,K_0)=P_0\exp (-b(0)K_0^T)+O(\delta ),$$

(where P(⋅;δ,K₀) was the generalized monodromy matrix defined for (3.10)). Because $det{P}_0$ is positive, we can for every matrix R with positive determinant find a vector K₀ such that $\mathrm{spec}R=\mathrm{spec}(P_0\exp (-b(0)K_0^T))$ (using the assumption of controllability; see auxiliary lemma A.1, which is a special case from the more general treatment in [17] and [39] for a Matlab implementation). Hence, if we choose the spectrum of R inside a circle B_r/2(0) of radius r/2 around 0, then the spectrum of ∂_xX(x_*(0);δ,K₀) is also inside B_r(0) for sufficiently small δ>0.

Approximate spectrum for extended time-delayed feedback: we fix the control gains K₀ such that $\underset{\delta \to 0}{lim}P(1;\delta ,K_0)=P_0\exp (-b(0)K_0^T)$ has all eigenvalues inside B_r(0) for some r∈(0,1). Consider now again the extended time-delayed feedback control (3.1)–(3.3) with the particular choice of short impulse linear feedback law (3.7):

$$\dot{x}(t)=f(x(t),{\mathrm{\Delta }}_{\delta }(t)K_0^T[\tilde{x}(t)-x(t)])$$ 3.14

and

$$\tilde{x}(t)=(1-\epsilon )\tilde{x}(t-T)+\epsilon x(t-T),$$ 3.15

where ε∈(0,1).

Lemma 3.3 (Floquet multipliers of extended time-delayed feedback). —

Assume that the matrix $P_0\exp (-b(0)K_0^T)$ has all eigenvalues inside the ball B_r(0) with r<1. Then, for all sufficiently small ε and δ, the periodic orbit $x(t)=\tilde{x}(t)=x_{\ast }(t)$ of system (3.14), (3.15) has a simple Floquet multiplier λ=1 and all its other Floquet multipliers are inside the unit circle.

Outline of proof (details are given in appendix A(b)): eigenvalues λ of the linearization of (3.14)–(3.15) are roots of the function

$$h(\mathrm{\lambda };\epsilon ,\delta )=det[\mathrm{\lambda }I-P(1-\frac{\epsilon }{\mathrm{\lambda }-(1-\epsilon )};\delta ,K_0)].$$ 3.16

Roots of h with a non-small distance from 1−ε are close to the roots of $det(\mathrm{\lambda }I-P(1;\delta ,K_0))$, which are inside the unit circle by assumption. Roots λ of h close to 1−ε with modulus greater than 1−ε/2 have the form λ=1−ε+ε/κ, where |κ| is bounded away from 0 and infinity. The roots κ of h(1−ε+ε/κ;δ,ε) are small perturbations of the roots κ_ℓ,0 of $det(I-P_0-P_{0}b(0)K_0^T\sigma (\kappa -1))$, where σ is as defined in (3.12). These roots κ_ℓ,0 have the form

$${\kappa }_{\ell ,0}=1+\frac{2\pi \mathrm{i}\ell }{K_0^{T}b(0)}$$ 3.17

(if $K_0^{T}b(0)\ne 0$, otherwise, only a single root κ_0,0=1 exists). The roots κ_ℓ,0 have all modulus greater than unity (except for ℓ=0, which corresponds to the trivial eigenvalue λ=1) such that the corresponding roots λ_ℓ of h have modulus smaller than unity.

Remark—two types of Floquet multipliers: the proof of lemma 3.3 shows that there are two distinct types of roots: those approximating the spectrum assigned by the choice of control gains K₀, and those close to 1−ε (called λ_ℓ above). The roots λ_ℓ lie close to the circle of radius ε/2 around the centre 1−ε/2 in the complex plane and have the form

$${\mathrm{\lambda }}_{\ell }\approx 1-\frac{\epsilon }{2}+\frac{\epsilon }{2}\left[\frac{K_0^{T}b(0)-2\pi \mathrm{i}\ell }{K_0^{T}b(0)+2\pi \mathrm{i}\ell }\right](\ell \in \mathbb{Z}).$$

For ℓ=0, the expression is exact (giving the simple root at unity), for the others, the approximation is sufficiently accurate for small δ and ε to ensure that they stay inside the unit circle. The illustration in figure 2 shows the two distinct groups for the Hopf normal form example discussed in §5.

Importance of scalar input and trivial Floquet multiplier: the proof of lemma 3.3 hinges on one argument that depends on the presence of a trivial Floquet multiplier: we need to find the roots s_j of $s\mapsto det(I-P_0-P_{0}b(0)K_0^{T}s)$ and then find solutions κ of σ(κ−1)=s_j for all these roots s_j. Because $b(0)K_0^T$ has rank one we know that $s\mapsto det(I-P_0-P_{0}b(0)K_0^{T}s)$ is a first-order polynomial (see appendix A(b) for details). The presence of a trivial Floquet multiplier then ensures that this first-order polynomial has the root 0. Hence, 0 is its only root, restricting the possible location for the κ_ℓ,0 to the list in (3.17). This simple argument would not apply for cases where the uncontrolled periodic orbit x_* has no trivial Floquet multiplier, or for control with non-scalar inputs u, or for control with more than one kick per period.

4. Autonomous feedback control

The feedback control constructed in lemma 3.3 introduces an explicit time dependence into the system. The controlled system has the form

$$\begin{array}{rl}\dot{x}(t)& =f(x(t),{\mathrm{\Delta }}_{\delta }(t)K_0^T[\tilde{x}(t)-x(t)])\\ \mathrm{and}\tilde{x}(t)& =(1-\epsilon )\tilde{x}(t-T)+\epsilon x(t-T),\end{array}\}$$ 4.1

where Δ_δ is time-periodic with period T, but the system still has a Floquet multiplier λ=1. The neutrally stable direction corresponding to this Floquet multiplier is a phase shift: if $(x(t),\tilde{x}(t))=(x_{\ast }(t),x_{\ast }(t))$ is a periodic orbit of (4.1), then so is $(x(t),\tilde{x}(t))=(x_{\ast }(t+s),x_{\ast }(t+s))$ for any $s\in \mathbb{R}$. Hence, the controlled system with the gains K(t)=Δ_δ(t)K₀ is susceptible to arbitrarily small time-dependent perturbations (say, experimental disturbances): the phase s of the stabilized solution may drift until Δ_δ is non-zero at a time s where the gains K₀ are no longer stabilizing. This problem does not occur if, instead of applying the feedback $K_0^T[\tilde{x}(t)-x(t)]$ at a fixed time per period, we apply it in a strip in ${\mathbb{R}}^n$ close to a Poincaré section at x_*(0) (as illustrated in figure 3), putting a factor depending on x(t) in front of $K_0^T[\tilde{x}(t)-x(t)]$. Specifically, we let the function Δ_δ not depend explicitly on time t but on a function $\tilde{t}:{\mathbb{R}}^n\mapsto \mathbb{R}$, where the argument of $\tilde{t}$ is x(t). Then, the common notion of asymptotic stability of periodic orbits in autonomous dynamical systems applies. One would then always apply control near x_*(0) despite phase drift.

Figure 3.

Illustration of choice for strip and Poincaré section where gains should be non-zero.

A possible explicit expression for u is

$$\begin{array}{rl}u(t)& ={\mathrm{\Delta }}_{\rho ,\delta }(x(t))K_0^T[\tilde{x}(t)-x(t)],\text{where }{\mathrm{\Delta }}_{\rho ,\delta }(x)=J_{\rho }(x){\mathrm{\Delta }}_{\delta }(\tilde{t}(x)),\end{array}$$ 4.2

$$\begin{array}{rl}{J}_{\rho }(x)& =\{\begin{array}{ll}1& \text{if }|x-x_{\ast }(0)|<\rho ,\\ 0& \text{if }|x-x_{\ast }(0)|>2\rho ,\end{array}\tilde{t}(x)=\frac{{\dot{x}}_{\ast }{(0)}^T}{{\dot{x}}_{\ast }{(0)}^T{\dot{x}}_{\ast }(0)}[x-x_{\ast }(0)],\end{array}$$ 4.3

ρ>0 is a small radius and J_ρ is smooth. In (4.3), $\tilde{t}(x_{\ast }(t))=t+O(t^2)$ for |t|≪1, and J_ρ restricts control to the neighbourhood of radius ρ around x_*(0). With u as defined in (4.2), the right-hand side of the now autonomous system

$$\dot{x}(t)=f(x(t),{\mathrm{\Delta }}_{\delta ,\rho }(x(t))K_0^T[\tilde{x}(t)-x(t)]),\tilde{x}(t)=(1-\epsilon )\tilde{x}(t-T)+\epsilon x(t-T)$$ 4.4

has a right-hand side that depends discontinuously on x(t) (because Δ_δ is discontinuous in its argument. Because the general mathematical theory for DDEs coupled to difference equations is not well developed, one may replace the discontinuous Δ_δ in (4.2) with a smooth approximation of Δ_δ. This does not affect the final result, which we can state as a lemma (see appendices (c) and (d) for the details of the choice for ρ and the smoothing of Δ_δ,ρ).

Lemma 4.1 (autonomous stabilizability of periodic orbits with extended time-delayed feedback). —

Assume that the matrix $P_0\exp (-b(0)K_0^T),$ as used in lemma 3.3, has all eigenvalues inside the ball B_r(0) with r<1. Then, for all sufficiently small ρ, there exist ${\epsilon }_{max}>0$ and ${\delta }_{max}>0$ such that the periodic orbit $x(t)=\tilde{x}(t)=x_{\ast }(t)$ of system (4.4) is asymptotically exponentially stable for all $\epsilon \in (0,{\epsilon }_{max})$ and $\delta \in (0,{\delta }_{max})$.

Remark: other arguments for Δ_δ,ρ In (4.2), we can replace the argument x(t) of Δ_δ,ρ with $\tilde{x}(t)$, x(t−T) or $\tilde{x}(t-T)$ without changing the linearization in $x(t)=\tilde{x}(t)=x_{\ast }(t)$. Thus, (4.4) successfully stabilizes the periodic orbit x_* also with these modifications.

Robustness: we assumed perfect knowledge of the periodic orbit x_* and the right-hand side f in the construction of K₀ and Δ_δ,ρ. However, we know that stable periodic orbits persist under small perturbations. Thus, for gains near K₀ and functions close to Δ_δ,ρ, the periodic orbit of the controlled system persists. Owing to the non-invasive nature of extended time-delayed feedback, the periodic orbit of the system with perturbed K₀ and Δ_δ,ρ is still identical to x_*.

5. Illustrative example: Hopf normal form

The construction of gains as described in §4 has been implemented as a Matlab function (publically available at [39], depending on DDE-Biftool [40–42]). The electronic supplementary material demonstrates how one can find stabilizing gains for two examples:

• (i) a family of period-two unstable oscillations around the hanging-down position of the parametrically excited pendulum, and

• (ii) the unstable periodic orbits in the subcritical Hopf normal form.

We discuss example (ii) in more detail in this section, because for this example we can prove that stabilization with ETDF is not possible with constant gains and small ε. The subcritical Hopf bifurcation has also been used commonly in the literature as a benchmark example. Here we choose the Hopf normal form with constant speed of rotation (such that in polar coordinates the angle θ satisfies $\dot{\theta }=1$ and all periodic orbits have period 2π). Note that the control constructed by Fiedler et al. [31] depended on changing rotation and was stabilizing only in a small neighbourhood of the bifurcation. Flunkert & Schöll [32] analysed time-delayed feedback control (with ε=1) of the subcritical Hopf bifurcation completely, but also excluded the case of constant rotation and restricted themselves to a small neighbourhood of the bifurcation. Thus, even though example (ii) is seemingly simple, it shows that the method proposed in the paper is able to stabilize periodic orbits that are beyond the approaches previously suggested in the literature. Without loss of generality, we choose a linear control input b=[1,1]^T such that the system with control has the form

$$\begin{array}{rl}{\dot{x}}_1& =p{x}_1-x_2+x_1[x_1^2+x_2^2]+u\\ \mathrm{and}{\dot{x}}_2& =x_1+p{x}_2+x_2[x_1^2+x_2^2]+u,\end{array}\}$$ 5.1

where p<0. This system has for u=0 an unstable periodic orbit of the form $x_{\ast }(t)={[r\sin t,-r\cos t]}^T$ with radius $r=\sqrt{-p}$ and period T=2π. The monodromy matrix P₀ for the uncontrolled system along the periodic orbit x_* equals

$$P_0=\left[\begin{array}{cc}1& 0\\ 0& \exp (-4\pi p)\end{array}\right].$$

Because the derivative of the right-hand side with respect to the control input equals b(t)=b=[1,1]^T, the periodic orbit is controllable in time T. In fact, the pair (P₀,b) is controllable as required for the applicability of lemma 3.3. Extended time-delayed feedback control, applied to a two-dimensional system has the form

$$\begin{array}{rl}u(t)& =K_1(t)[{\tilde{x}}_1(t)-x_1(t)]+K_2(t)[{\tilde{x}}_2(t)-x_2(t)]\\ \mathrm{and}{\tilde{x}}_j(t)& =(1-\epsilon ){\tilde{x}}_j(t-T)+\epsilon x_j(t-T)(\hspace{0.17em}j=1,2).\end{array}\}$$ 5.2

We can state two simple corollaries from our general considerations. First, it is impossible to stabilize the periodic orbit x_* with extended time-delayed feedback using time-independent gains K₁ and K₂ for small ε.

Lemma 5.1 (lack of stabilizability for constant control gains). —

Let p<0 and let K₁(t) and K₂(t) be arbitrary constants (also calling them K₁ and K₂). Then, there exists an ${\epsilon }_{max}>0$ such that the periodic orbit x_* is unstable with the extended time-delayed feedback control (5.2) for all $\epsilon \in (0,{\epsilon }_{max})$.

Proof. —

Amann & Hooton [38] proved a general topological restriction on the gains K(t) for extended time-delayed feedback control: let $K(t)\in {\mathbb{R}}^n$ be arbitrary (continuous), θ∈[0,1] be arbitrary, and let u be of the form

$$u=\theta K{(t)}^T[\tilde{x}(t)-x(t)].$$

The scalar θ provides a homotopy from the uncontrolled system (θ=0) to the controlled system (θ=1). Assume that the trivial Floquet multiplier λ₁=1 of x_* is isolated for u=0 (which is the case for example (5.1) with p<0). Then, the Floquet multiplier λ₁ depends smoothly on θ at least for small θ and will be real: ${\mathrm{\lambda }}_1(\theta )\in \mathbb{R}$ for 0<θ≪1. A necessary condition for extended time-delayed feedback with gains K(t) to be stabilizing for x_* and arbitrary ε∈(0,1) is that λ₁′(0)≥0 if the number of Floquet multipliers in $\{z\in \mathbb{C}:Re\hspace{0.17em}z>1\}$ is odd for θ=0. If we denote an adjoint eigenvector for the trivial Floquet multiplier by ${\overline{x}}_{\ast }(t)$ (the right eigenvector is ${\dot{x}}_{\ast }(t)$), then this criterion can be simplified to

$$\frac{{\int }_0^T{\overline{x}}_{\ast }{(t)}^{T}b(t)K{(t)}^T{\dot{x}}_{\ast }(t)\hspace{0.17em}\mathrm{d}t}{{\int }_0^T{\overline{x}}_{\ast }{(t)}^T{\dot{x}}_{\ast }(t)\hspace{0.17em}\mathrm{d}t}\le 0,$$

where b(t)=∂_uf(x_*(t),0) (this simplifying criterion was formulated in general in [33]). For our particular example, we have

$${\dot{x}}_{\ast }(t)={\overline{x}}_{\ast }(t)=\left[\begin{array}{c}r\cos t\\ r\sin t\end{array}\right],b(t)=\left[\begin{array}{c}1\\ 1\end{array}\right],T=2\pi$$

and constant gains K₁ and K₂ such that the necessary condition of [33,38] is

$$K_1+K_2\le 0.$$ 5.3

On the other hand, if K₁+K₂≤0, the Jacobian of (5.1) with classical linear feedback control

$$u=K_1(x_{1,\ast }(t)-x_1)+K_2(x_{2,\ast }(t)-x_2)=K_1(r\sin t-x_1)+K_2(-r\cos t-x_2)$$ 5.4

along x=x_*(t) has the trace

$$\text{tr }{\mathrm{\partial }}_{x}f(x(t),K(x_{\ast }(t)-x(t))){|}_{x(t)=x_{\ast }(t)}=2p+4r^2-K_1-K_2=-2p-K_1-K_2.$$

Because p<0, this trace is positive if K₁+K₂≤0 for all t∈[0,2π] such that the classical linear feedback control (5.4) cannot be stabilizing for the periodic orbit $x_{\ast }={[r\sin t,-r\cos t]}^T$. Thus, lemma 3.1 implies that extended time-delayed feedback cannot be stabilizing either, for sufficiently small ε>0. ▪

Construction of gains: for the periodic control gains ${\mathrm{\Delta }}_{\delta }(t)K_0^T$ (or the autonomous nonlinear gains ${\mathrm{\Delta }}_{\delta ,\rho }(x(t))K_0^T$), the gains K₀ are constructed such the matrix $P_0\exp (-b(0)K_0^T)$ has all eigenvalues inside the unit circle (for our illustration, we choose the target location at ±i/2). Figure 4 shows the amplitude and unstable Floquet exponent of the periodic orbits and the gains obtained in this manner (called K₁ and K₂ in figure 4). Because the pair (P₀,b(0)) is not controllable at the Hopf point, the gains diverge to infinity for p→0. In particular, P₀=I for p=0 such that it cannot be linearly controllable with a single input.

Figure 4.

(a) Amplitude $r=\sqrt{-p}$ and unstable Floquet exponent (equals −2p) along family of periodic orbits. (b) Gains K₁ (note that K₁<0 always) and K₂ along family of periodic orbits in Hopf normal form (5.1) with feedback law (5.5). (Online version in colour.)

Illustration of asymptotics: figure 5 shows how the true Floquet multipliers approximate their asymptotic values when using the autonomous time-delayed feedback control (A 10) with gains depending on x:

$$\begin{array}{rl}u(t)& ={\mathrm{\Delta }}_{\delta ,\rho }(x(t))[K_1[{\tilde{x}}_1(t)-x_1(t)]+K_2[{\tilde{x}}_2(t)-x_2(t)]]\\ \mathrm{and}{\tilde{x}}_j(t)& =(1-\epsilon ){\tilde{x}}_j(t-T)+\epsilon x_j(t-T)(\hspace{0.17em}j=1,2).\end{array}\}$$ 5.5

For the construction of Δ_δ,ρ, we used the construction of $\tilde{t}$ proposed in (4.2)–(4.3):

$$\tilde{t}(x)=\frac{y_0^T}{y_0^T{y}_0}[x-x_0]\text{for }x\in B_{\rho }(x_0),$$ 5.6

where y₀=[r,0]^T, x₀=[0,−r]^T and $r=\sqrt{-p}$. At the particular parameter value p=−0.25 shown in figure 5, the uncontrolled periodic orbit is already strongly unstable: the unstable Floquet multiplier equals $\exp (-4\pi p)\approx 23.141$. The gains K₁ and K₂, designed to assign the Floquet mulitpliers ±i/2, are also large after division by δ. Hence, the range of δ and ε, for which stabilization is successful is small. The values for δ and ε used in the illustration are chosen such that deviations from the asymptotic limit are visible but small.

Figure 5.

Numerically computed versus asymptotic spectrum for p=−0.25 (ε= 0.04, δ=T/500, ρ=0.3, K₁=−0.258, K₂=4.786). (a) Unit circle in the complex plane. (b) Zoom into the circle around 1−ε or radius ε/2. Computed with DDE-Biftool [40–42], see electronic supplementary material and [39] for the code. (Online version in colour.)

6. Conclusion and outlook

The paper proves conclusively that there are no restrictions inherent in extended time-delayed (state) feedback control if one accepts time-periodic gains, whereas for constant gains, general positive results are unlikely (as they are absent for classical feedback control of linear time-periodic systems). The particulars of the gain construction presented here, following the approach of Brunovksy, are merely for the purpose of proving their existence analytically. While the result raises the possibility that more general assignment is feasible (because there is a lot of freedom in the choice of general time-periodic K(t)) the techniques for proving this may have to be different from those in the paper. The central argument of the paper rests on the rank-one nature of the control input, making it easy to locate all roots κ of the transcendental function $\kappa \mapsto det[I-P_0\exp (b(0)K_0^T(\kappa -1))]$ (the matrix $b(0)K_0^T$ has rank one in our case). The argument also exploits the presence of the trivial Floquet multiplier, thus, making the result (even if it is based entirely on linear theory) mostly relevant to the analysis of nonlinear systems.

Appendix A. Auxiliary lemmas and detailed arguments of proofs

Lemma A.1. —

Let (A,b) with $A\in {\mathbb{R}}^{n\times n}$ and $b\in {\mathbb{R}}^n$ be controllable, let $detA>0,$ and let $({\mathrm{\lambda }}_1,\dots {\mathrm{\lambda }}_n)\in {\mathbb{C}}^n$ be the spectrum of a real matrix with positive determinant. Then, one can find a vector $K\in {\mathbb{R}}^n$ such that $\mathrm{spec}(A\exp (b{K}^T))=({\mathrm{\lambda }}_1,\dots ,{\mathrm{\lambda }}_n)$.

The proof is given in [17]. A Matlab implementation of the explicit construction is SpecExpAssign.m in [39].

(a) Floquet multipliers for extended time-delayed feedback

Lemma A.2 (characteristic equation for Floquet multipliers). —

Let $A(t)\in {\mathbb{R}}^{n\times n},$ b(t), $K(t)\in {\mathbb{R}}^n$ be T-periodic and let ε be positive. Then, the Floquet multipliers different from 1−ε of the linear system

$$\dot{x}(t)=A(t)x(t)+b(t)K{(t)}^T[\tilde{x}(t)-x(t)]$$ A 1

and

$$\tilde{x}(t)=(1-\epsilon )\tilde{x}(t-T)+\epsilon x(t-T)$$ A 2

are roots of the function

$$h:\mathrm{\lambda }\mapsto det[\mathrm{\lambda }I-P(1-\frac{\epsilon }{\mathrm{\lambda }-(1-\epsilon )})].$$

For $\mu \in \mathbb{C},$ the matrix P(μ) was defined as the solution x at time T of $\dot{x}=[A(t)-\mu b(t)K{(t)}^T]x$ with x(0)=I (i.e. P(μ) is the monodromy matrix of $\dot{x}=[A(t)-\mu b(t)K{(t)}^T]x$).

Proof. —

Let $\mathrm{\lambda }\in \mathbb{C}$ be an eigenvalue of the time-T map M of (A 1)–(A 2). Let x₀, ${\tilde{x}}_0$ (both $[-T,0]\mapsto {\mathbb{R}}^n$) be the components of an eigenvector corresponding to λ, and let x₁, ${\tilde{x}}_1$ (also both $[-T,0]\mapsto {\mathbb{R}}^n$) be the corresponding components of $M[x,\tilde{x}]$. Then, ${\tilde{x}}_1(t)=\mathrm{\lambda }{\tilde{x}}_0(t)$ and, by definition of the time-T map M, ${\tilde{x}}_1(t)=(1-\epsilon ){\tilde{x}}_0(t)+\epsilon x_0(t)$. Hence, $\mathrm{\lambda }{\tilde{x}}_0(t)=(1-\epsilon ){\tilde{x}}_0(t)+\epsilon x_0(t)$, which implies (because λ≠1−ε)

$${\tilde{x}}_0(t)=\frac{\epsilon }{\mathrm{\lambda }-(1-\epsilon )}{x}_0(t)$$

for t∈[−T,0]. Using this relation, we can solve (A 1) on the interval [−T,0] as

$${\dot{x}}_0(t)=A(t)x_0(t)+b(t)K{(t)}^T[\frac{\epsilon }{\mathrm{\lambda }-(1-\epsilon )}-1]x_0(t)$$

with boundary condition x₀(0)=λx₀(−T). By definition of the monodromy matrix P, this is equivalent to

$$P(1-\frac{\epsilon }{\mathrm{\lambda }-(1-\epsilon )})x_0(-T)=\mathrm{\lambda }{x}_0(-T),$$

which has a non-trivial solution x₀(−T) if and only if the function h in lemma A.2 is non-zero. ▪

(b) Proof of lemma 3.3

The characteristic function, $h(\mathrm{\lambda };\epsilon ,\delta )=det[\mathrm{\lambda }I-P(1-\epsilon /(\mathrm{\lambda }-(1-\epsilon ));\delta ,K_0)]$, defined in (3.16), has a root λ=1 for all small δ and all ε∈(0,1): a nullvector of I−P(0;δ,K₀) is ${\dot{x}}_{\ast }(0)$ (corresponding to a linearized phase shift).

For λ with modulus larger than 1−ε/2, the term ε/(λ−(1−ε)) has modulus less or equal than 2. Let us pick δ₁>0 and a C₁∈(0,1) (both small) such that the polynomial

$$\mathrm{\lambda }\mapsto det(\mathrm{\lambda }I-P(1-\mu ;\delta ,K_0))$$

has all roots inside the ball B_(r+1)/2(0)⊂B₁(0) for all δ∈[0,δ₁] and μ with |μ|≤C₁. This is possible, because $\mathrm{\lambda }\mapsto det(\mathrm{\lambda }I-P(1;0,K_0))$ has all roots inside the ball B_r(0) by assumption of the lemma and the limit of P(1−μ;δ,K₀) for δ→0 was uniform for bounded μ (recall $\underset{\delta \to 0}{lim}P(1-\mu ;\delta ,K_0)=P_0\exp ((\mu -1)b(0)K_0^T)$). Thus, h(λ;ε,δ) cannot have roots λ on or outside the unit circle for which

$$\left|\frac{\epsilon }{\mathrm{\lambda }-(1-\epsilon )}\right|\le C_1$$

holds. Hence, for all δ∈[0,δ₁], all roots of h(⋅;ε,δ) on or outside of the unit circle must satisfy

$$C_1\le \left|\frac{\epsilon }{\mathrm{\lambda }-(1-\epsilon )}\right|\le 2.$$ A 3

We introduce the new variable $\kappa \in \mathbb{C}$ defined via

$$\mathrm{\lambda }=1-\epsilon +\epsilon /\kappa .$$ A 4

Restriction (A 3) for λ is equivalent to the restriction C₁≤|κ|≤2 for κ. Hence, for all δ∈[0,δ₁] and ε∈(0,1), every root λ of h(⋅;δ,ε) on or outside of the unit circle corresponds to a root κ of

$$\begin{array}{rl}g(\kappa ;\delta ,\epsilon )& =h(1-\epsilon +\epsilon /\kappa ;\delta ,\epsilon )\\ & =det[(1-\epsilon +\frac{\epsilon }{\kappa })I-P(1-\kappa ;\delta ,K_0)]\end{array}$$

with C₁≤|κ|≤2. This one-to-one correspondence of roots of h and g is given via relation (A 4) and includes multiplicity of the roots. Relation (A 4) also implies that |κ|≤1, because, otherwise, |λ|<1. The function g has a limit

$$g(\kappa ;\delta ,\epsilon )\to g(\kappa ;0,0)\text{for}\hspace{0.17em}(\delta ,\epsilon )\to 0$$

uniformly for κ with C₁≤|κ|≤2. Hence, the set of roots κ of g(⋅;δ,ε) with C₁≤|κ|≤2 is a small perturbation of the set of roots of g(⋅;0,0):

$$\begin{array}{rl}g(\kappa ;0,0)& =det[I-P_0\exp (b(0)K_0^T[\kappa -1])]\\ & =det[I-P_0-P_{0}b(0)K_0^T\sigma (\kappa -1)],\text{where }\\ \sigma (\kappa -1)& =\{\begin{array}{ll}\frac{\exp (K_0^{T}b(0)(\kappa -1))-1}{K_0^{T}b(0)},& \text{if }{K}_0^{T}b(0)\ne 0\\ \kappa -1,& \text{if }{K}_0^{T}b(0)=0.\end{array}\end{array}$$

The generalized eigenvalue problem for the matrix pair $(I-P_0,P_{0}b(0)K_0^T)$ with characteristic polynomial $\sigma \mapsto det(I-P_0-P_{0}b(0)K_0^T\sigma )$ is regular, because $det(I-P_0)=0$ but $det(I-P_0-P_{0}b(0)K_0^T\sigma (-1))$ is regular (because $P_0\exp (-b(0)K_0^T)$ has all eigenvalues inside the unit circle, $I-P_0\exp (-b(0)K_0^T)$ is regular). As $P_{0}b(0)K_0^T$ has rank 1, the characteristic polynomial corresponding to $\sigma \mapsto det(I-P_0-P_{0}b(0)K_0^T\sigma )$ has degree 1. Moreover, its only root equals 0 (which must be simple owing to the regularity of $det(I-P_0-P_{0}b(0)K_0^T\sigma )$). Hence, we know that g(κ;0,0)=0 if and only if σ(κ−1)=0.

Case $K_0^{T}b(0)=0$ If $K_0^{T}b(0)=0$: this implies that the only root κ of g(⋅;0,0) with C₁≤|κ|≤2 equals unity. Hence, also for sufficiently small ε and δ, the only root κ with C₁≤|κ|≤2 of g(⋅;δ;ε) equals unity (because g(1;δ;ε)=0).

Case $K_0^{T}b(0)\ne 0$ If $K_0^{T}b(0)\ne 0$: we have that g(κ;0,0)=0 if and only if $\exp (K_0^{T}b(0)(\kappa -1))=1$ such that the roots are

$${\kappa }_{\ell ,0}=1+\frac{2\pi \mathrm{i}\ell }{K_0^{T}b(0)}(\ell \in \mathbb{Z})$$

The first part of the subscript, ℓ, numbers the roots, the second part of the subscript, 0, indicates that δ=ε=0. Thus, the roots κ_ℓ,0 of g(⋅;0,0) have a modulus

$$|{\kappa }_{\ell ,0}|=\sqrt{1+\frac{4{\pi }^2{\ell }^2}{{(K_0^{T}b(0))}^2}}$$

such that only the roots κ_ℓ,0 with index

$$-{\ell }_{max}\le \ell \le {\ell }_{max},\text{where}\hspace{0.17em}{\ell }_{max}=\frac{\sqrt{3}}{2\pi }|K_0^{T}b(0)|,$$

are in the admissible range with C₁≤|κ_ℓ,0|≤2. (Hence, both cases, $K_0^{T}b(0)=0$ and $K_0^{T}b(0)\ne 0$ can be treated equally.) The admissible roots κ_ℓ,0 of g(⋅;0,0) are all simple. Hence, for sufficiently small δ and ε, g(⋅;δ,ε) will have roots κ_ℓ for $|\ell |\le {\ell }_{max}$ that are small perturbations of κ_ℓ,0, and these roots κ_ℓ are the only roots of g(⋅;δ,ε) with modulus in [C₁,2]. Because we know that g(1;δ,ε)=0, we know that κ₀=1 (hence, for ℓ=0 the perturbation is zero). Furthermore, for non-zero ℓ with $|\ell |\le {\ell }_{max}$, the modulus of κ_ℓ,0 is greater than 1. Hence, the perturbed roots κ_ℓ also have modulus greater then 1 for sufficiently small ε and δ, and non-zero $|\ell |\le {\ell }_{max}$.

Consequently, by relation (A 4), the only roots of h(⋅;δ,ε) that could be on or outside the unit circle are

$${\mathrm{\lambda }}_{\ell }=1-\epsilon +\frac{\epsilon }{{\kappa }_{\ell }},\text{where}\hspace{0.17em}|\ell |\le {\ell }_{max}.$$

However, these roots λ_ℓ are simple and satisfy λ₀=1 and |λ_ℓ|<1 for non-zero ℓ, because |κ_ℓ|>1 for non-zero ℓ.

(c) Details of construction for autonomous feedback gains: regularization of the short impulse Δ_δ(t)

To avoid discontinuous dependence of the right-hand side on the solution, we first regularize the discontinuity of the time-dependent gain K(t). Define for $\delta \in (0,\sqrt{T+1/16}-\frac{1}{4})$ (such that 2δ²+δ<T) the regularized version of Δ_δ:

$$\begin{array}{rl}{\mathrm{\Delta }}_{\delta }(t)& =\{\begin{array}{ll}1/\delta ,& \text{if }t{|}_{\mathrm{mod}[0,T)}\in [0,\delta ],\\ 0,& \text{if }t{|}_{\mathrm{mod}[0,T)}\in [\delta +{\delta }^2,T-{\delta }^2],\\ \frac{1}{\delta }\hspace{0.17em}m\left(\frac{\delta +{\delta }^2-t{|}_{\mathrm{mod}[0,T)}}{{\delta }^2}\right),& \text{if }t{|}_{\mathrm{mod}[0,T)}\in (\delta ,\delta +{\delta }^2)\\ \frac{1}{\delta }\hspace{0.17em}m\left(\frac{t{|}_{\mathrm{mod}[0,T)}-T+{\delta }^2}{{\delta }^2}\right),& \text{if }t{|}_{\mathrm{mod}[0,T)}\in (T-{\delta }^2,T),\end{array}\end{array}$$ A 5

where $m:\mathbb{R}\mapsto [0,1]$ is an arbitrary smooth monotone increasing function with m(s)=0 for s≤0 and m(s)=1 for s≥1. When using Δ_δ as defined in (A 5) instead of (1.6) to define the linear (now approximately) short-impulse feedback law

$$u_{\delta }(t;y)={\mathrm{\Delta }}_{\delta }(t)K_0^{T}y$$ A 6

the nonlinear time-T map is still linearizable. Denoting the monodromy matrix of the linear system (recall that A(t)=∂_xf(x_*(t),0), b(t)=∂_uf(x_*(t),0) and using definition (A 5) for Δ_δ)

$$\dot{y}(t)=[A(t)-\mu b(t){\mathrm{\Delta }}_{\delta }(t)K_0^T]\hspace{0.17em}y(t),y(0)=y_0$$

again by P(μ;δ,K₀) then the monodromy matrix of the smoothed system still satisfies (identical to (3.11))

$$P(\mu ;\delta ,K_0)=P_0\exp (-b(0)K_0^T\mu )+O(\delta ),$$ A 7

where the error term O(δ) is uniform for bounded $\mu \in \mathbb{C}$ and $K_0\in {\mathbb{R}}^n$. Hence, we can replace the discontinuous definition (1.6) for Δ_δ(t) by (A 5) in (3.14), and Lemma 3.3 still applies to the modified (regularized) system.

(d) State-dependent gains K(x(t))

Consider a sufficiently small radius ρ>0 such that the equation ${\dot{x}}_{\ast }{(0)}^T[x-x_{\ast }(t)]=0$ has a unique solution $t\in \mathbb{R}$ close to 0 for all $x\in B_{2\rho }(x_{\ast }(0))\subset {\mathbb{R}}^n$, thus defining implicitly a smooth function t_ρ:

$$t_{\rho }:\hspace{0.17em}{\mathbb{R}}^n\mapsto \mathbb{R},t_{\rho }(x)=\{\begin{array}{ll}\text{root }t\text{ of }{\dot{x}}_{\ast }{(0)}^T[x-x_{\ast }(t)]=0,& \text{if }x\in B_{2\rho }(x_{\ast }(0)),\\ \text{arbitrary such that }{t}_{\rho }\text{ is smooth},& \text{otherwise}.\end{array}$$ A 8

The function $\tilde{t}$, defined in (4.3), is approximately equal to t_ρ along the periodic orbit x_* and near x_*(0): $t_{\rho }(x_{\ast }(t))-\tilde{t}(x_{\ast }(t))=t-\tilde{t}(x_{\ast }(t))=O(t^2)$. We consider also a regularized indicator function for the neighbourhood of x_*(0)

$$J_{\rho }:\hspace{0.17em}{\mathbb{R}}^n\mapsto \mathbb{R},J_{\rho }=\{\begin{array}{ll}1,& \text{if }x\in B_{\rho }(x_{\ast }(0)),\\ 0,& \text{if }x\notin B_{2\rho }(x_{\ast }(0)),\\ \text{arbitrary such that }{J}_{\rho }\text{ is smooth},& \text{if }x\in B_{2\rho }(x_{\ast }(0))\setminus B_{\rho }(x_{\ast }(0)).\end{array}$$

and combine t_ρ and J_ρ with ${\tilde{\mathrm{\Delta }}}_{\delta }$ as defined in (A 5) to the smooth globally defined function

$${\tilde{\mathrm{\Delta }}}_{\delta ,\rho }:\hspace{0.17em}{\mathbb{R}}^n\mapsto \mathbb{R},{\tilde{\mathrm{\Delta }}}_{\delta ,\rho }(x)=J_{\rho }(x){\tilde{\mathrm{\Delta }}}_{\delta }(t_{\rho }(x))$$

When applying ${\tilde{\mathrm{\Delta }}}_{\delta ,\rho }$ to x_*(t), the result is identical to the timed impulse ${\tilde{\mathrm{\Delta }}}_{\delta }$ for small δ: if ∥x_*(t)−x_*(0)∥<ρ for all t∈[−δ²,δ+δ²], then

$${\tilde{\mathrm{\Delta }}}_{\delta ,\rho }(x_{\ast }(t))={\tilde{\mathrm{\Delta }}}_{\delta }(t)$$

for all $t\in \mathbb{R}$. Consequently, the system with extended time-delayed feedback and state-dependent gains

$$\dot{x}(t)=f(x(t),{\tilde{\mathrm{\Delta }}}_{\delta ,\rho }(x(t))K_0^T[\tilde{x}(t)-x(t)])$$ A 9

and

$$\tilde{x}(t)=(1-\epsilon )\tilde{x}(t-T)+\epsilon x(t-T),$$ A 10

which is now autonomous with a smooth right-hand side, has for sufficiently small ρ and δ+δ²<ρ exactly the same linearization along the periodic orbit $x(t)=\tilde{x}(t)=x_{\ast }(t)$ as system (3.14) and (3.15). The derivative of ${\tilde{\mathrm{\Delta }}}_{\delta ,\rho }$ with respect to its argument is multiplied by 0 if $\tilde{x}(t)=x(t)=x_{\ast }(t)$ for all times t in the term ${\tilde{\mathrm{\Delta }}}_{\delta ,\rho }(x(t))K_0^T[\tilde{x}(t)-x(t)]$ in (A 9).

The time reconstruction function t_ρ as defined in (A 8) satisfies t_ρ(x_*(t))=t as long as x_*(t)∈B_2ρ(x_*(0)). The definition of t_ρ as proposed in (4.3) in the main text is a O(t²) perturbation of (A 8) for t of order δ. Thus, continuity of the Flqouet multipliers implies that the perturbed version of t_ρ preserves the stability of the periodic orbit x_*.

Data accessibility

https://dx.doi.org/10.6084/m9.figshare.2993812.

Competing interests

I have no competing interests.

Funding

J.S.’s research has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement number 643073.