Prediction-Based Control for Nonlinear Systems with Input

This work has two primary objectives. First, it presents a state prediction strategy for a class of nonlinear Lipschitz systems subject to constant time delay in the input signal. As a result of a suitable change of variable, the state predictor asymptotically provides the value of the state τ units of time ahead. Second, it proposes a solution to the stabilization and trajectory tracking problems for the considered class of systems using predicted states. The predictor-controller convergence is proved by considering a complete Lyapunov functional. The proposed predictor-based controller strategy is evaluated using numerical simulations.


Introduction
The importance of time delay systems has increased in the control community.Time delay systems are found in many practical problems such as chemical processes, mechanical systems, biological systems, and communication systems, for example, in teleoperation control problems.Time delays are classified as dead time, communication, or measurement delays [1].The existence of delays in the control signal or states of the system is a major factor for closed loop stability.See, for instance, [2][3][4].
The control problem in time delay systems has been analyzed by several approaches, starting by ignoring its effects when the magnitude of the delay is small enough or by considering approximate predicted feedback obtained using Pade approximations for linear systems.A standard control design approach for time delayed systems is to develop prediction technics to compute the future value of the state.The estimation of future states in a dynamical system has deserved an increasing attention since the publication of the well-known Smith Predictor Compensator (SPC) [5].The SPC solves the control problem for open-loop stable linear systems subject to time delay at the input signal.Reference [6] reports several improvements to the SPC approach.Thus, by developing state predictors, it is possible to synthesize causal state feedback for linear systems [5,6].
It is mandatory first to analyze the state prediction problem, to solve, using causal state feedback, control problems for nonlinear systems with delay at the input [7,8].In most of the cases, the state prediction problem is related to the pioneering work of state observation [9] or its generalization presented in [10] for delay-free nonlinear systems.The necessity of causal state feedback to solve control problems for nonlinear systems with input delay has led to developing approximate state predictors, for example, [11].In [12] a robust controller is designed to compensate time varying perturbations through a disturbance estimator.The work [13] considers the stabilization problem for a class of linear systems with multidelays at the input.The proposed solution uses an infinite dimensional feedback based on predicted states.A sliding mode controller is presented in [14] for stabilization of systems with delayed input.A state predictor and a sliding surface are designed to minimize the effect of the delay.A robust strategy to stabilize linear input delayed systems under parametric uncertainties is presented in [15]; the solution is based on linear matrix inequalities; the same problem is addressed in [16] through a Lyapunov-Krasovskii functional.The case of linearizable feedback systems with bounded input delay is analyzed in [17] where the stabilization problem for a sufficiently small time delay is solved; the same problem for linear systems was considered in [18].
Another option, to tackle the control problem in time delay systems, is employing an infinite integral prediction strategy known as finite spectrum assignment proposed in [19] and its generalizations presented in [20].Infinite integral prediction strategies are analytical solutions that suffer from practical implementation problems; see, for instance, [21] for a comparison between the infinite integral solution and a simple static state feedback.The works in [22][23][24] elaborate on the limitations of the finite spectrum assignment.Authors of [25] and [26] propose truncated output feedback controllers for linear systems and a class of Lipschitz nonlinear systems, respectively, to avoid implementation problems.Truncated output feedback controllers safely ignore the infinite dimensional term.
This work addresses the prediction and control problem, avoiding integral infinite dimensional solutions, for a class of nonlinear input delayed systems.Considering the linear predictor proposed in [27], this article suggests a nonlinear predictor, for a class of nonlinear systems, to compute future values of the state,  units of time in advance.Under a Lipschitz condition assumption, it is possible to show that the proposed predictor asymptotically converges to future state values.The Lipschitz assumption is common in the literature for the delay-free nonlinear systems [9,10] and for a class of nonlinear time delayed systems [28,29].Additionally, the predicted future states are used to solve the trajectory tracking problem under similar assumptions to the ones for the prediction problem.The proposed predictor is equivalent to a delayed observer for an advanced system obtained using a particular change of coordinates.The convergence of prediction and trajectory tracking errors is shown by considering a complete Lyapunov-Krasovskii functional [30].The solution proposed in this paper is close to the work in [17].Following a different approach, the solution in [17] depends on the size of the time variant delay upper bound, while the solution developed in this paper provides an upper bound for the time delay size.
The rest of the work is organized as follows.Section 2 presents the class of dynamic systems discussed in this article.Section 3 states the prediction problem and defines the predictor dynamics.In Section 4, the solution to the trajectory tracking problem with predicted states is solved.The extension of the proposed results to full linearizable nonlinear systems is described in Section 5. Section 6 shows numerical simulations for the predictor-controller. Finally, Section 7 closes this work with concluding remarks.

Class of Nonlinear Input Delay Systems
This paper considers a class of nonlinear systems.The following differential equation describes this class of nonlinear systems: where  ∈  ⊂ R  is the state, ( − ) ∈ R  is the input signal,  ∈  ⊂ R  is the output signal, and  ∈ R + is a constant.It is assumed that the output signal () is available for measurement.Moreover, nonlinear system (1) satisfies the following assumptions.
Assumption 1. Φ(, ) is a locally Lipschitz function with respect to ; that is, for all , x in a certain region containing the origin with   ∈ R + .In the case that (2) is satisfied for all , x, function Φ(, ) is said to be globally Lipschitz.
Notice that any nonlinear system of the form ẋ = (, ( − )) can be expressed in the form (1) as long as (, ( − )) is differentiable with respect to .A detailed justification on the generality of this class of nonlinear system (1) is found, for instance, in [9,31,32].
Input delayed signals arise from the existence of deadtimes found in several industrial processes [2,4] or from teleoperated systems where the control platform is placed in a remote location as could be the case of mobile robots applications [33][34][35].

Prediction-Observation Problem
The prediction problem is stated as follows.Based on the knowledge of the output signal () and the input signal (), design a dynamic system whose states x( + ) converge asymptotically to future values of the states ( + ); that is, lim →∞ ( ( + ) − x ( + )) = 0. ( To motivate the importance of the prediction problem, assume that there exists a state feedback controller that solves the trajectory tracking problem for system (1) with  = 0. Due to the involved time delay, the closed loop dynamics (1)-(4) took the following form: thus, in general, the state feedback controller (4) no longer solves the trajectory tracking problem for (1).Assume that it is possible to design an ideal feedback based on future values of the state, this feedback, in terms of future states, will take the form Notice that the use of the ideal feedback (6) will produce ideal delay-free closed loop dynamics (1)-( 6).This fact motivates the design of a predictor capable of generating the estimated value x( + ).However, as expected, the best possible solution only achieves asymptotic convergence to future states.Moreover, due to the time delay effect, the closed loop dynamics will behave as in open-loop on [0, ] producing an undesirable and not compensable transient.
Instrument in addressing the prediction problem for the class of systems described in (1) is the following change of coordinates: In terms of new coordinates (7), dynamic system (1) is described by the following equations: This dynamic system represents the evolution of original system (1) and  units of time ahead.

Predictor Design.
The predictor dynamics can now be designed based on time advanced system (8).The dynamics for the predictor is proposed as a copy of the time advanced dynamics (8) plus a term injecting the delayed observer error: where  is the predictor matrix gain, and ŵ() represents the predicted evolution of the state () and consequently the prediction of x( + ) that represent the original state ()  time units ahead.Notice that, in (9), it is not possible to inject the signal () since according to (7) this is not available.

Predictor Convergence.
The convergence to the origin of the prediction error ( 10) is proven in two steps.First, notice that the stability of the perturbation-free system (11) corresponds to the stability of a linear system of the form with an initial condition function Notice that the prediction error initial condition depends on those of plant (1) and predictor (9).Since the parameters of matrix  are free, from the observability assumption of the pair (, ), it is always possible to locate the eigenvalues of the matrix  −  in the left half complex plane.The Hurwitz property of matrix  −  ensures the stability of differential difference equation (12) for a sufficiently small time delay  [21].The maximum time delay  * for which system (12) will be asymptotically stable depends on the choice of the observer gain  and its computation is not an easy task since its complexity rises as the dimension of system (12) increases.Thus, given a matrix gain , such that  −  is a Hurwitz matrix, it is always possible to asymptotically stabilize prediction error (12) for a constant time delay  <  * .This fact can be verified, for instance, by considering Proposition 5.2.2, Chapter 5 in [36], or reviewing the pole assignment procedure given in [21].
Based on the above arguments, it will be assumed, without loss of generality, that there exist  and  * such that system (12) is asymptotically stable for 0 ≤  <  * .It should also be noticed that the stability of ( 12) is not only asymptotic but also exponential [37].

Complete Type Functional.
Following the results presented in [30], to show the asymptotic stability of (11), it is possible to consider a complete type functional of the form where and  1 = −.The Lyapunov matrix (⋅) in ( 14) is given by where () is the fundamental matrix of system (12) without perturbation, and with positive definite matrices  0 and  1 .
It is possible now to state the prediction convergence result.
Proof.Considering the properties of the Lyapunov matrix (), after some manipulations, the time derivative of functional (13), along the solution of the prediction error (11), produces Then, it is possible to upper-bound the undefined sign terms in (19) as follows: Therefore, ( 19) can be upper-bounded in the following form: As a result, one has Notice that condition () can be rewritten as Then, one obtains since from condition ().Conditions ( 24) and ( 25) conclude the proof.Finally, notice that since the perturbation-free system (11) is globally stabilizable, if Φ(, ) is globally Lipschitz with respect to , the result will be also global.
Following [9], if the matrix gain  is chosen such that where  and  are symmetric, positive definite matrices associated with then, observer (9) (with  = 0) converges asymptotically to the real states, () = ().The optimal value of   is obtained for  = , [38].Thus, the observation problem is reduced to choose  such that Further analysis of this problem has been carried out in [31,32].
Remark 5. Notice also that, in the case of  = 0, property (P3) takes the form

Control Problem Based on Predicted State Feedback
The regulation or trajectory tracking problem for time delayed systems, based on predicted states, is more challenging than the prediction problem.First, the prediction errors link up with the regulation or trajectory tracking errors.Second, if the matching condition between the disturbances and the control input is not satisfied control design has to be addressed case by case.To illustrate the use of the proposed state predictor for control design a narrow class of nonlinear systems is considered.It is assumed that Φ(, ( − )) has the structure which satisfies Assumption 1.It is also considered that the following assumption holds.
From the choice of gains  and , it is clear that matrix   +  1 is also Hurwitz.From Assumption 1, function Φ (, x) is a Lipschitz function with respect to (); that is,
Therefore, it is possible to state the solution to the state prediction-based stabilization problem in the next theorem.
Theorem 7. Suppose that  in (9) and  in (32) are chosen such that the perturbation-free system ( 34) is exponentially stable.Then, the prediction-based stabilization problem for system ( 1)-( 31) has a solution if for ), and positive definite matrices  0 and  1 .
Proof.The proof of this theorem follows the same lines of Theorem 3. The complete Lyapunov functional is constructed with   ,  1 , and to build the fundamental matrix   () associated with the perturbation-free system, that is, system (34) with Φ (, x) = 0.The new Lyapunov matrix   () also satisfies the properties (P1), (P2), and (P3) taking care of the corresponding subscripts associated with system (34).

Full Linearizable Systems Case
To strengthen the importance of the class of nonlinear systems (1), notice a broad class of nonlinear systems of the form where  ∈ X ⊂ R  ,  ∈ R  ,  ∈ R  , smooth vector fields ,   : R  → R  ,  = 1, . . ., , and scalar valued functions ℎ  ,  = 1, . . .,  from R  to R can be transformed to a system of form (1) as long as system (39) has a well-defined relative degree  =  1 + ⋅ ⋅ ⋅ +   (see, [39]).The transformation of system (39) to a set of systems of form (1) can be done through the diffeomorphism, () = (()), (0) = ((0)) with As a result, system (39) can be rewritten in the form with for  = 1, . . .,  where   () are entries of the invertible decoupling matrix   () [39].It is clear that system (41) can be assumed to have the structure given by ( 1).Notice that, in the case of system (39), defining the time ahead representation of (41) is obtained as for  = 1, . . ., .The predictor for system (44) takes the form Defining the prediction errors as for  = 1, . . .,  and  = 1, . . .,   allows writing the prediction error dynamics in the form 8. Notice that for the input-state linearizable system (39), due to the change of coordinates () = ( + ) and the diffeomorphism, ( + ) = (( + )), the estimated future value is obtained as (48)

Control Problem for Full Linearizable Systems.
For control design in the case of full linearizable systems, the dynamics described by the set of differential equations in ( 39) is considered, or equivalently, a system of form (41) which has the structure of (1).
A feedback proposed for the solution of the delay-free case [39] can be modified based on future states as where the entries of matrix   ( ω) and vector ( ω) = [ 1 ⋅ ⋅ ⋅   ] ⊤ are given in (42) and the new input signal with  = 1, . . .,  where    () =    ( + ) is the desired reference for   ().
Notice that in the case of a stabilization problem for system (39) and (41), feedback (49) is rewritten considering    = 0 for  = 1, . . .,  and  = 1, . . .,   .The closed loop system (41), ( 42), ( 44)-(47), and (49) produces where Defining a set of tracking errors as the following is obtained: and, as a result, one has where Therefore, replacing the feedback (49) with (47), the complete closed loop dynamics is given by where   corresponds to the -row of vector .
System (57) can be rewritten in a vector form as follows: with and   = [0, . . ., 0,   ] ⊤ .Therefore, the following can be written: where Finally, one has where Based on the above developments, the trajectory tracking problem for system (39) and (41) based on predicted future values is stated under the same lines of Theorems 3 and 7.
From the parameters of the system the following bounds are computed:   = 0.1 and ‖ 1 ‖ = ‖−LC‖ = 7.3622.The bound V = max ∈[0,] ‖()‖ is obtained from the time evolution of matrix (), which can be numerically computed from the procedure proposed in [40].The evolution of () is depicted in Figure 1; thus one has V = 1.2597.
The prediction error evolution   () = () − ŵ( − ) is shown in Figure 2 where its convergence to the origin is clearly stated.
To stabilize system (63) at the origin, the predictor described before was considered and  = [−3.5 −3.75] that set the poles of  +  in {−1.75, −2}.Notice that in this case  0 = diag{0.79}, 1 = diag{2.1}from where  min ( 0 ) = 0.79,  min ( 1 ) = 2.1,   = 0.1414, and ‖ 1 ‖ = 7.3622.The computation of   () is depicted in Figure 3 from where V = 2.8232.Notice that the conditions of Theorem 7 are satisfied.The evolution of the state () is depicted in Figure 4 where appropriate convergence to the origin is obtained.The prediction errors are shown in Figure 5 while the tracking errors are depicted in Figure 6; both errors converge to the origin as expected.The corresponding predictorbased feedback is shown in Figure 7. Finally, the estimated perturbation signal is shown in Figure 8.

Conclusions
This work addresses and provides solutions to the state prediction and control problems of a class of nonlinear systems with known constant time delay at the input signal.An appropriate time shift of the state coordinates lets the predictor be designed as an observer.The predictor convergence depends on the time delay magnitude and the Lipschitz constant associated with the class of considered systems.The future estimated states solve the regulation and trajectory tracking problems.Numerical simulations evaluate the prediction strategy as well as the regulation and trajectory tracking controller based on predicted states.