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Robustness of Nonlinear Predictor Feedback Laws to Time- and State-Dependent Delay Perturbations

Much recent progress has been achieved for stabilization of linear and nonlinear systems with input delays that are long and dependent on either time or the plant state---provided the dependence is known. In this paper we consider the delay variations as unknown and study robustness of nominal constant-delay predictor feedbacks under delay variations that depend on time and the state. We show that when the delay perturbation and its rate have sufficiently small magnitude, the local asymptotic stability of the closed-loop system, under the nominal predictor-based design, is preserved. For the special case of linear systems, and under only time-varying delay perturbations, we prove robustness of global exponential stability of the predictor feedback when the delay perturbation and its rate are small in any one of four different metrics. We present two examples, one that is concerned with the control of a DC motor through a network and one of a bilateral teleoperation between two robotic systems.


Introduction
Networked control systems are present in various engineering applications such as tele-robotics, remote surgery and automotive systems among others. One of the major reasons is that they are advantageous over traditional control systems in terms of flexibility, reliability, maintenance cost etc. [14]. However, often their performance can be severely degraded in the presence of delays induced by the network [3], [13]. When the delay is constant and known, the predictor-based controller compensates the network-induced delay [25]. Yet, the networked-induced delay might be subject to time-varying and state-dependent uncertainties, which, when they are not considered in the control design, not only can degrade the performance of the control system, but can also destabilize the network [32]. It is therefore crucial to quantify the robustness properties of the constant-delay predictor feedback in the presence of time-and state-dependent delay uncertainties.
Although, some of the first predictor-based designs for linear, unstable plants, with constant input delays go back to the 1980s [2], a Lyapunov construction has been unavailable until recently [29]. In addition, a Lyapunov functional is provided in [31], which is employed to the stability analysis of the proposed control design. Yet, the robustness properties of the nominal, constantdelay, predictor-based feedback under time-and statedependent delay perturbations remain unexplored.
We consider forward-complete nonlinear systems that are locally, exponentially stabilizable in the absence of the 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 of the constant-delay predictor-based feedback, under simultaneous timevarying 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 comprised of the plant, the predictor feedback and the observer for the actuator state. With the constructed Lyapunov functional, we prove that the closed-loop system remains locally asymptotically stable when the perturbation and its rate are small (Section 2).
We also deal with linear systems under time-varying delay perturbations. We show robustness of global exponential stability of the predictor feedback for the cases where the delay perturbation and its rate either have small magnitude, or their L 1 norm is small, or they converge to zero as the time goes to infinity or, finally, have a small moving average for large times (Section 3). Finally, we illustrate the robustness properties of the predictor feedback with two examples. The first one is an example of a DC motor which is controlled through a network. The network induces a delay which is comprised 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. The second example is concerned with the 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 (Section 4).
Notation: We use the common definition of class K, K ∞ and KL functions from [26]. For an n-vector, the norm | · | denotes the usual Euclidean norm. We say that a function ζ : R + × (0, 1) → R + belongs to class KC if it is of class K with respect to its first argument for each value of its second argument and continuous with respect to its second argument. It belongs to class KC ∞ if it is in KC and also in K ∞ with respect to its first argument. With ζ we denote the inverse of the function ζ with respect to its first argument for each value of its second argument.
2 Robustness to time-and state-dependent delay perturbations for nonlinear systems We consider nonlinear plants of the forṁ where f : C 2 (R n × R; R n ), satisfies f (0, 0) = 0,D > 0, under the nominal, constant-delay predictor feedback given by where for all t −D ≤ θ ≤ t is the estimate of the actual predictor state P * (θ), defined for all t −D − δ(t, X(t)) ≤ θ ≤ t as where σ, the actual predicted time (which should be compared with the estimated predicted time t +D) is An example of such a model together with the predictorbased controller is shown in Fig. 1.
The predictor state (3) is the certainty equivalent predictor for system (1). This becomes clear by setting δ = 0 in (4). Note thatP (θ) (or P * (θ)) should be viewed as the output of an operator, parametrized by t, acting on P (s) and U (s), t −D ≤ s ≤ θ (or t −D − δ(t, X(t)) ≤ s ≤ θ) in the same way that the solution X(t) to an ODE can be viewed as the output of an operator, parametrized by t 0 , acting on X(s) and the input U (s), t 0 ≤ s ≤ t. However,P is given implicitly since the plant is nonlinear (for the same reason that the solution X(t) to a nonlinear ODE is given implicitly). In the case of a linear plantẊ(t) = AX(t) + BU t −D , equation (3) for the predictor state can be solved explicitly dθ. This is the standard predictor (used for example in [2]), which is obtained using the variations of constants formula for the linear ODE satisfied by the plant. An equivalent representation of the signalP (θ) iŝ for all x ∈ [0, 1], whereû, is the estimation of the actuator state U (θ), t−D−δ (t, X(t)) ≤ θ ≤ t, which satisfieŝ that is, With this definition,p(x, t) is the output of an operator, parametrized by t, that acts onp(y, t) andû(y, t), y ∈ [0, x]. With this representationp(1, t) =P (t).
Note also that from relation (4) we see that for P * to be well-defined the denominator in (4) must satisfy the following condition for all θ ≥ t 0 − D (X(t 0 )) c > δ t (σ(θ), P * (θ)) +∇δ (σ(θ), P * (θ)) f (P * (θ), U (θ)) , for c ∈ (0, 1], which is a condition on the perturbation δ, the initial conditions and the solutions of the system. As it turns out later on, this condition is satisfied by appropriately restricting the perturbation δ and the initial conditions of the plant. We proceed by imposing the following assumptions on the delay-free plant.

Assumption 1
The plantẊ = f (X, ω) is strongly forward complete, that is, there exist a smooth positive definite function R and class K ∞ functions α 1 , α 2 and α 3 such that for all X ∈ R n and for all ω ∈ R.
Assumption 1 guarantees that for every initial condition and every measurable locally essentially bounded input signal, the corresponding solution of the system exists for all times. Forward-completeness is a natural requirement for nonlinear plants with input delay. In the absence of this assumption, i.e., when the plant exhibits a finite-escape time, the control signal might reach the plant "too late". The difference with standard forwardcompleteness from [1] lies in the fact that R in Assumption 1 is positive definite, in accordance to the fact that f (0, 0) = 0.
Theorem 1 Consider the closed-loop system consisting of the plant (1), control law (2), (3) and observer (14)- (15). Under Assumptions 1 and 2 there exist positive constants c 1 , c * * , class K ∞ functionsμ, α * , a class KC ∞ function ζ, and a class KL function β such that if the perturbation δ satisfies for all (t, ξ) ∈ [t 0 , ∞) × R n , then for all initial conditions which satisfy where it holds that for all t ≥ t 0 and some 0 < c < 1.
We now introduce the backstepping transformation.
Lemma 1 Define the backstepping transformation together with its inverse, whereρ is given for all w(y, t) + κ t +Dy,ρ(y, t) dy.
System (1) together with the control law (2), (3) can be written in the following forṁ wherẽ and r 1 is defined in Appendix A. The observer error 1 An important observation at this point is thatp in (13) andρ in (28) are identical. To see this, observe that, through the backstepping transformation, bothp andρ satisfy the same ODEs in the spatial variable x, with the same initial condition X. However,p is expressed in terms of the original variables (X,û) in the direct backstepping transformation, whereasρ, is expressed in terms of the transformed variables (X,ŵ) and is used in the inverse transformation. and where the function σ is defined in (8), and the function r is defined in Appendix A. Furthermore, and wherẽ and r 2 , r 3 , r 4 and r 5 are defined in Appendix A.
For π to be a meaningful propagation speed it should be positive and uniformly bounded from below and above. Using (38), one can conclude that δ, the initial conditions and the solutions of the system should satisfy, in addition to (17) which guarantees that 0 <σ(t), also which guarantees 0 < σ(t) − t. The two conditions (17), (50) incorporate the functions σ and P * , that is, they are not expressed in terms of the perturbation δ and the functional Π. We derive next a sufficient condition for (17), (50) to be satisfied, in terms of Π.
Lemma 2 There exist positive constants c 1 , c * , such that if the perturbation δ satisfies (22), then for all solutions of the system satisfying, they also satisfy for some 0 < c < 1, and hence, conditions (17) for 0 < c < 1, and (50) are satisfied.
Proof See Appendix C.
Lemma 3 There exist a continuously differentiable, positive definite function S * , a class K ∞ function α * and positive constants λ, c 1 , c * such that if the perturbation δ satisfies (22) then for all solutions of the system satisfying (51), the Lyapunov function Proof See Appendix D.
The next two lemmas relate the Lyapunov function V with the norm of the system in the original variables, represented with PDEs, and the norm in PDE representation with the norm in standard delay form.
Lemma 4 There exist a positive constant c * , a class KC ∞ function α 24 and a class K ∞ function α 25 such that for all solutions of the system satisfying (51) the following holds where Proof See Appendix E.
Lemma 5 There exists positive constants c * , c 1 and class KC ∞ functions ζ 1 and ζ 2 such that if the perturbation δ satisfies (22), then for all solutions of the system satisfying (51) the following holds Proof Using Lemma 2 and (8), (10) we get that With relations (16), (48), (24), (57) and applying the appropriate change of variables in the integrals the proof is immediate using (59), (60).

Robustness to time-varying delay perturbation for linear systems
We consider the following special case of system (1) where for the rest of the section δ is a function only of the time t. For this linear case, the predictor-based controller is given explicitly as Theorem 2 Consider the closed-loop system consisting of the plant (63), observer (14)- (15), and control law (64). There exists a positive δ 1 , such that if the perturbation δ satisfies then, the closed-loop system is exponentially stable, in the sense that there exist positive constants R and λ such that the following holds: The proof of Theorem 2 is based on the application of Lemmas 1, 3, 4 and 5 for the special case of plant (63) and the special perturbation δ as in (63). However, we give each of these lemmas specialized to the present case for two reasons. Firstly, in the special case of linear systems with only time-varying perturbation, we study stability of the closed-loop system in the H 1 norm. This is a consequence of the fact that when δ does not depend on the state, the conditions (17), (50) are satisfied without restricting the supremum norm of the real actuator state U (θ), for all φ(t) ≤ θ ≤ t. Secondly, in the linear case the control law, as well as the direct and inverse backstepping transformations are given explicitly and are globally well-defined.
When (65) does not hold, one can still derive exponential stability of the closed-loop system, by imposing other conditions on the perturbation δ, such us the ones from [26] Chapter 9.3. However, in this case one has to guarantee in addition that the propagation speed π is still uniformly bounded from above and below and strictly positive. Therefore, we make the following assumptions which δ has to a priori satisfy.
and is such that and is such that Theorem 3 Assume that δ satisfy Assumptions 3, 4. There exist positive δ 2 and δ 3 such that if the perturbation δ satisfies either or for some positive ∆ and nonnegative T , then, the closedloop system consisting of the plant (63), observer (14)- (15), and control law (64) is exponentially stable, in the sense that there exist positive constants R and λ such that the following holds: From Theorem 3 we observe that in the case of linear systems and when the perturbation depends only on time and not on the state, we prove robustness of global exponential stability of the predictor feedback under three alternative conditions on the delay perturbation rather than just restricting the magnitude of the delay and its rate. This is not possible in the case where the perturbation depends on the state because for the conditions (17) and (50) to be satisfied one has to necessarily restrict the magnitude of the perturbation and its rate such that they are both sufficiently small.
We introduce now the backstepping transformation of the estimated actuator state.
Lemma 8 There exists a positive δ 1 such that if the perturbation δ satisfies (65), then there exists a positive λ such that V in (98) satisfieṡ Proof See Appendix G Lemma 9 There exist positive constants M 5,L and M 6,L such that Proof See Appendix H.
Lemma 10 There exist positive constants M 7,L and M 8,L such that Proof From (16) we getû x (x, t) =DU ′ t+D(x − 1) . Applying a change of variables in (103), with (95) we get Hence, the lemma is proved with M 7,L = replaced by π * 0 and π * 1 respectively). The only difference with the proof of Theorem 2 is in the proof of Lemma 8. Towards that end, we solve (99) to get Consider that γ(t) = 1 π(0,t) −D . Applying the change of variables τ = φ(θ) in the integral and using the facts Analogously, for the case Hence, Lemma 8 is proved with r 2 δ 2 < min {π * 1 r 1 , and the fact that t t0 γ(τ )dτ ≤ ∞ t0 γ(τ )dτ . Note that in the present case, Lemma 8 can be proved directly from relation (99) using Lemma B.6 in [28].

Control of a DC motor over a network
We consider the following model of a field-controlled DC motor ( [44]) with negligible shaft damping where i f , i a are field and armature currents respectively, ω is angular velocity and a, b, c, θ are positive constants. The equilibria of the unforced system are (ω, i a , i f ) = ω 0 , k b , 0 for some constant ω 0 . The system is feedback linearizable for (ω, i a , i f ) ∈ D, D = (ω, i a , i f ) ∈ R 3 |ω > 0 and i a > k 2b . A delay-free design, based on full-state linearization, is (Chapter 13.3 in [26]) Shifting the equilibrium ω 0 , k b , 0 of the system to the origin and setting The motor is controlled through a network that induces a constant delayD (e.g. [11]). The known, constant delay, is subject to a time-varying perturbation due to the effect of transmission of control signals to other motors through the network. We further assume that the perturbation δ increases when the armature current increases. Define the estimated predictors of X 1 , X 2 and X 3 aŝ respectively. Setting in (113)-(118) ω = X 1 + ω 0 , i a = X 2 + k b , i f = X 3 and replacing X 1 , X 2 , X 3 by the predictors (122)-(124) we get the nominal predictor feedback.
In Fig. 2 we show the field and armature currents, and in Fig. 3 the input voltage and the angular velocity of the motor. The nominal predictor feedback achieves local stabilization of the closed-loop system at the desired equilibrium, despite the presence of the perturbation.

Bilateral teleoperation
In bilateral teleoperation [15], the operator (e.g. a human) controls a robotic system, called the master, at the one end of the communication network. The actions of the master are transmitted (through the network) to another robotic system, called the slave, at the other end of the network. The goal of the control algorithm is the slave manipulator to behave (in a certain sense) as the master manipulator. A model of two robotic systems, each one having n degrees of freedom, representing the master and the slave manipulators is written as ( [15]) where x m , x s ∈ R n are the degrees of freedom of the robotic systems and the torques τ m , τ s ∈ R n are to be designed such as coordination between the master and the slave is achieved asymptotically, i.e., x m − x s → 0 as t → ∞. The constant delayD represents the known, network-induced delay which is subject to time-varying perturbations that are often present due to congestion, distance etc. [12]. For simplicity we assume scalar x m , x s , τ m , τ s and we re-write (125), (126) aṡ where where r is the set-point for the positions of the manipulators. The predictor-based version of this controller is We choose the desired set-point for x m and x s as r = 2, the parameters of the controller as K p = B m = B s = 2, the known delay asD = 1, the initial condition of the plant as x m (0) =ẋ m =ẋ s = 0, x s (0) = 1, the initial actuator state as U (θ) = 0, −D − δ(0) ≤ θ ≤ 0 and the initial estimation of the actuator state as U (θ) = 0, −D ≤ θ ≤ 0. We illustrate the robustness properties of the predictor feedback under a time-varying delay perturbation that is neither in L 1 nor converges to zero as the time goes to infinity nor is small in magnitude. However, after some long period of time its mean is small. This disturbance is show in Fig. 4.
In Fig. 5 we show the difference between the positions of the master and the slave for the cases where either there is or there is not a perturbation δ. In both cases, under the nominal predictor feedback the position of the slave tracks the position of the master. In Fig. 6 we show the torques applied to the two robotic systems under the perturbation δ. The control efforts are oscillatory as a result of the effect of the oscillatory perturbation.

Conclusions
Looking into the details of the proofs, we note that in the case of nonlinear systems with state-dependent perturbations, there is a trade off between the achievable region of attraction and the size of the perturbation and its  rate, at the origin. For linear systems under time-varying perturbations, global exponential stability holds, but the size of the perturbation and its rate should be appropriately restricted. With the available Lyapunov functional, our next step is to study the inverse optimal redesign problem of predictor feedback for nonlinear systems.
One might raise the question of robustness to stochastic delay perturbations, since stochastic perturbations have some resemblances with the time-varying case. Yet, in our analysis we restrict not only the magnitude of the perturbation δ but also the magnitude of its derivative (which also guarantees the invertibility of φ = t−D −δ), which can be unbounded in the case where δ is white noise, or even when δ is a low pass version of white noise.
Lemma 12 There exists a class K ∞ function α 6 such that for all x ∈ [0, 1] Proof Differentiating (13) with respect to x and comparing the resulting ODE with the ODE in t for X, the proof is complete with Assumption 1 and the comparison principle after appropriately majorizing eDDα 3 < α * . The detailed proof can be found in [31] (Lemma 7).
Proof Let c * be as in Lemma 14. The proof is based on (A.1)-(A.6) combined with (20), the fact that f is twice differentiable and with similar calculations as in the proof of Lemma 13. Yet, we provide the proofs of (B.25), (B.31) as some of the steps are useful later on. Under Assumption 2 (which allows us to chooseα continuously differentiable without loss of generality), Lemma 14, and the facts that, |ŵ(x, t)| ≤ 1 0 |ŵ x (x, t)|dx, for all x ∈ [0, 1] (which follows from (31)), and that f : C 2 (R n × R; R n ), f (0, 0) = 0, which allows us to conclude |f (X, ω)| ≤ α 5 (|X| + |ω|) , (B.32) for some function α 5 ∈ K ∞ ∩ C 1 , we get from (A.2) that for some class KC ∞ function α r , continuously differentiable in its first argument. Analogously, differentiating (A.2) with respect to x and using (28) together with the fact thatDr(x, t) =û x (x, t) it is shown that for some positive constant µ * and some functions α 1,rx , α 2,rx ∈ KC ∞ which are continuously differentiable with respect to their first argument. With the Cauchy-Schwarz inequality we get (B.25), (B.31).

C Proof of Lemma 2
The proof of this lemma is based on the following fact. Proof The proof can be found in [5] (Lemma 4).

H Proof of Lemma 9
Using relations (77), (78) together with Young's and Cauchy-Schwarz's inequalities we get that where · is defined in (F.6) and