Sampled-Data Output Feedback Control for Nonlinear Uncertain Systems Using Predictor-Based Continuous-Discrete Observer

In this article, we investigate the problem of sampled-data robust output feedback control for a class of nonlinear uncertain systems with time-varying disturbance and measurement delay based on continuous–discrete observer. An augmented system that includes the nonlinear uncertain system and disturbance model is first found, and by using the delayed sampled-data output, we then propose a novel predictor-based continuous–discrete observer to estimate the unknown state and disturbance information. After that, in order to attenuate the undesirable influences of nonlinear uncertainties and disturbance, a sampled-data robust output feedback controller is developed based on disturbance/uncertainty estimation and attenuation technique. It shows that under the proposed control method, the states of overall hybrid nonlinear system can converge to a bounded region centered at the origin. The main benefit of the proposed control method is that in the presence of measurement delay, the influences of time-varying disturbance and nonlinear uncertainties can be effectively attenuated with the help of feedback domination method and prediction technique. Finally, the effectiveness of the proposed control method is demonstrated via the simulation results of a numerical example and a practical example.


I. INTRODUCTION
I N RECENT decades, digital computer and communication network have been widely applied in modern industrial plants [1]- [4].Such a definite trend leads to considerable attention for sampled-data control where only sampled-data information is available in controller design.Due to complexity of nonlinearity, sampled-data control for continuoustime nonlinear system is more challenging than that for linear case [5]- [9].For continuous-time nonlinear systems, the existing results on sampled-data control can be roughly classified into two categories, including indirect method and direct method, which are briefly introduced as follows.
1) Indirect Method: A discrete-time controller is first designed for the discrete-time approximate description of continuous-time nonlinear system, and then, the stability analysis is conducted for the discrete-time closed-loop control model.This discretization method brings two issues.On the one hand, the existence of model approximation errors results in that only local or semiglobal result can be obtained.On the other hand, dynamical behavior over the intersample time interval is inevitably lost [10], [11].Different from the aforementioned method, the other indirect method is proposed in a manner of emulation, where a continuous-time controller is first developed to guarantee the stability for continuous-time nonlinear system, and after that, a discretization implementation is given by discretizing the continuous-time controller.It shows that by prudently determining the sampling period, the emulation controller can guarantee a certain of stability of some nonlinear systems [7], [12], [13].2) Direct Method: By using sampled-data information, a sampled-data controller is directly designed for continuous-time nonlinear system [5], [14], [15], and a continuous-discrete observer is often utilized to estimate unknown state when only sampled-data output is accessible [16]- [19].A rigorous stability analysis needs to be given to guarantee the performance for the resulting hybrid control system.In addition, various disturbances and uncertainties that widely exist in practical systems often have undesirable effects on the control performance [20]- [23], such as friction torque and parameter uncertainties in permanent magnet synchronous motor [20], and uncertain kinematics and complex nonlinearities in tracking control of inertially stabilized platform [24], [25].Existing results have shown that disturbance/ uncertainty estimation and attenuation (DUEA) technique is one promising robust method to improve the control performance [20], [26]- [29].The key idea of DUEA is that various disturbances and uncertainties are viewed as lumped disturbance, which can be effectively estimated if a disturbance observer is well designed.Then, the undesirable effect caused by the lumped disturbance can be effectively compensated or rejected by employing a compensate term in controller design.Based on the DUEA technique, many different robust control methods, such as active disturbance rejection control (ARDC) [26]- [28] and disturbance observerbased control [29]- [31], have been proposed in recent two decades and have shown notable contributions on improving disturbance rejection performance in many practical applications, such as permanent magnet synchronous motor [20], inertially stabilized platform [25], and dc-dc converter [32].
Note that existing works on sampled-data control based on DUEA are relatively limited in contrast to continuous-time counterpart.For example, Xue and Huang [33] investigated the sampled-data ADRC design problem for nonlinear uncertain systems.In [34], the problem of disturbance-observer-based sampled-data adaptive control was considered for a class of uncertain nonlinear systems.By using prediction technique, Tomizuka et al. [35], Cuenca et al. [36], and Zheng et al. [37] proposed prediction-correction methods to reconstruct the unknown states and disturbances, where the initial states of predictors need to be corrected by using the latest sampled outputs.Based on the intersample output predictor, a new sampled-data ADRC method for the high-precision control of voice coil motor servo systems was developed in [38].However, most of the aforementioned results did not consider the tough cases of multirate and measurement delay.The control systems with multirate exist in many practical applications since different controllers and sensors may have asynchronous updating/sampling periods [39]- [41].Output delay that often has serious influence on control performance may be caused by communication transmission or sensor hardware restriction [42].
In this article, we are concerned with the sampled-data output feedback control problem for a class of nonlinear uncertain systems subject to time-varying disturbance and measurement delay.The disturbance considered in this article is illustrated by an exogenous model, which covers a wide range of timevarying signals.When only delayed output is available in discrete time, we propose a new continuous-discrete observer with predictor to estimate the unknown state and disturbance information.Based on DUEA and feedback domination techniques, we then design the sampled-data robust output feedback control method by using the estimates.The overall hybrid control system is finally presented, and all the variables of the hybrid control system globally exponentially converge to a bounded region via a delicate stability analysis.The simulation results of a numerical example and a practical example are presented to illustrate the effectiveness of the proposed control method.
Different from the output feedback control design for linear systems, where the separation principle holds, the proposed control method does not obey the principle due to the existence of nonlinear uncertainties.Hence, the stability analysis tools for linear systems cannot be directly applied to the closedloop control systems considered in this article.To handle this, we employ a scaling gain in control design to dominate the nonlinear uncertainties.The main contributions of this article are concluded in the following.1) By using DUEA and prediction technique, we propose a predictor-based continuous-discrete observer, which can accurately estimate the information of unknown state and time-varying disturbance.Compared with existing results [5], [8], [9], [14], where the sampled-data output keeps the same over the intersample time intervals in the observer design, this article utilizes the intersample prediction technique to predict the output dynamics and compensates for the influences of output delay and sampling of continuous-time output.2) Most of the existing results only can solve the sampleddata control problems for nonlinear systems in the presence of single rate [5], [8], [43].In this article, the considered nonlinear systems are subject to multirate, which allows that the sensor and controller have asynchronous sampling/updating periods.The rest of this article is organized as follows.Section II gives the preliminaries and Section III presents the control method design.The stability analysis and synthesis are shown in Section IV.Section V describes the simulation results to verify the effectiveness of the proposed control method.Finally, some conclusions are given in Section VI.

A. Notations
R and R + denote the sets of real and nonnegative real numbers, respectively.Let N designate the nonnegative integer set.For a given real matrix or vector set {A i } i=1,...,n with compatible dimensions, col{A 1 , . . ., A n } = [A T 1 , . . ., A T n ] T and diag{A 1 , . . ., A n } denotes the diagonal matrix.The superscript T stands for the transpose.• denotes the Euclidean norm of a vector and the corresponding induced matrix norm.Given a symmetric matrix A, λ M (A) and λ m (A) represent the maximum and minimum eigenvalues of matrix A, respectively.C 0 (I ∈ R; R n ) denotes the space of continuous functions f : I → R n .

B. Problem Formulation
In this article, we are concerned with a class of networked nonlinear control systems subject to time-varying disturbance and nonlinear uncertainties as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where x(t) ∈ R n and u(t) ∈ R designate the system state and control input, respectively, y(t) ∈ R is the output from sensor, and o(t) ∈ R is the delayed output received by controller.The nonlinear functions ∈ R n are the unknown nonlinearities of system (1), and ζ(t) can be regarded as the unknown disturbances.A x ∈ R n×n , B x ∈ R n , and C x ∈ R n are the known system matrices.d(t) ∈ R represents the external disturbance generated by exogenous system.For system (1), the initial condition is given by x The external disturbance d(t) is supposed to be generated by the following exogenous system: where and C w ∈ R m are the known disturbance model matrices, and δ(t) ∈ R is the unknown term.
For dynamics ( 1) and ( 2), we suppose that both the tuples (A x , B x , C x ) and (A w , B w , C w ) have the chain form, which are described by .
The external disturbance d(t) modeled as system (2) is a kind of time-varying signal, which includes constant and ramp signals.Many practical examples include the kind of disturbances.Some mild assumptions on the nonlinear uncertainties in (1) and disturbance model (2) are presented in the following.
It should be underlined that the linear growth condition given in Assumption 1 is quite common in many existing results on nonlinear control systems [8], [14], and many practical processes, including dc-dc buck converter [43] and singlelink robot arm system [14], can be modeled as system (1).In this article, we assume that the sensor and controller are allowed to have asynchronous sampling period due to the sensor hardware restriction.Let T o and T u denote the sampling periods of sensor and controller, respectively.For system (1), only the delayed sampled-data output o(kT o ), i.e., y(kT o − τ ), is accessible for the controller, ∀k ∈ N. The actuator is eventdriven, i.e., the actuator updates whenever the data are received from the controller.
The aim of this article is to develop a new sampled-data robust output feedback control method for system (1), so as to enhance the control performance and the disturbance rejection ability in the presence of time-varying disturbance and delayed sampled-data measurement output.
Remark 1: It should be pointed out that multirate is quite common in vision-based tracking control systems [24], networked control system [40], and so on.For example, due to the hardware restrictions, the vision sensor has a much slower period than the controller in vision-based tracking system of inertially stabilized platform.Another practical example is networked control system, where multirate exists for saving limited communication bandwidth and output delay is caused by network transmission.In addition, this article does not need any restriction on the relationship of the sampling period T u and the updating period T o , which relaxes the condition that T o is a multiple of T u given in [38].Moreover, there does not exist any relationship of the sampling period T o and output delay τ in this article, which is more general than the condition [14].

III. PREDICTOR-BASED SAMPLED-DATA ROBUST
OUTPUT FEEDBACK CONTROL DESIGN By combining ( 1) and ( 2) together, we let η = col{x, w} and get an augmented nonlinear system as where ] T and the matrices

A. Predictor-Based Continuous-Discrete Observer Design
For system (3), the delay-free output y(t) and the delayed output o(t) can be further represented by In dynamics (4), the term CEδ disappears since CE = 0 by the definitions given in (3).
When only the delayed sampled-data output o(kT o ) is available, a predictor-based continuous-discrete observer for Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
augmented system (3) is developed as where η(t) = [η 1 (t), . . ., ηn+m (t)] T is the estimate of η(t), y p (t) denotes the predicted value of y(t), and z p (t) designates the predicted value of o(t) over the intersample time interval Remark 2: The main idea of prediction technique in ( 5) is that by using the delayed measurement output o(kT o ), we first obtain the predicted value y p (t) of the current output y(t), and then, the predicted value is used in the observer design such that the influences of time delay and sampling of output can be compensated.It should be highlighted that the prediction technique used in ( 5) is helpful to enhance the observer performance compared with the traditional sampleddata observer via zero-order holder, i.e., the sampled-data output holds the constant over the intersample time intervals (see [8], [9], [14]).

B. Sampled-Data Robust Output Feedback Controller Design
According to the definition of η, the estimates of x and w can be obtained by the proposed observer (5).Let x and ŵ stand for the estimate values of x and w, respectively.We design the sampled-data controller as follows: where Substituting the proposed controller (7) into system (1) and in the time interval t ∈ [kT u , (k + 1)T u ), we arrive at where Remark 3: Actually, we can transform the multirate control system (1) into single-rate one by defining the least common multiple of the sampling period and updating period, and thus, many traditional sampled-data control methods with single rate (see [8], [44]) can be used to solve the control problem of system (1).However, the intersample behavior is inevitably ignored for the single-rate controller, which seriously degrades the control system property.Moreover, the instability of the sample-data control system may be induced when the least common multiple of the sampling period and updating period is too large.

IV. MAIN RESULTS
Let x = x, ε = − ε, and X = col{ x, ε}.Recalling dynamics ( 6) and ( 8), the overall hybrid control system is written as According to Assumption 1, it can be easily deduced that with α = c(2n(n + 3)) 1/2 .Define two new variables φ X (t) : R + → R 2n+m and φ w (t) : Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
The overall hybrid control system (9) can be given in the time interval t ∈ R + as Ẋ(t) = L 1 X (t) + L 2 φ X (t) + 3 φ w (t) + L 4 e y (t)

ζ(t), X (t)). (12)
In what follows, we introduce one lemma about the estimates of e z and e y .(e c 0 s X (s)) where ρ = √ 2 max{L, c} and c 0 is a positive constant to be determined later.
Proof: We start the proof of Lemma 1 with estimating e z .Under Assumption 1, the following inequality holds: In what follows, the estimate of e y is deduced with the similar procedure as the above process.we first have that t t−τ e −c 0 s ds = −((e −c 0 t − e −c 0 (t−τ ) )/c 0 ).Similar to the deduction in (16), in the time interval t ∈ R + , one obtains from ( 6) and ( 15) that e c 0 t e y (t) By (18), there holds  (e c 0 s X (s)) This completes the proof.Remark 4: In inequality ( 13), it can be seen that sup 0≤s≤t (e c 0 s e z (s)) = 0 when T o = 0 since the error e z is induced by the sampling of the continuous-time output.However, in inequality (14), sup 0≤s≤t (e c 0 s e y (s)) is equal to zero when both T o and τ become zero because the error e y is caused by the output delay and the sampling of the continuoustime output.
The estimate of φ X is then obtained by the following lemma.Lemma 2: Consider dynamics ( 12) under Assumptions 1 and 2, and if the updating period T u satisfies (e c 0 s e y (s)) where β = max(L 1 + α, L 2 , L 4 ), c 0 is the same with that given in Lemma 1, θ 1 is a positive and bounded constant, and α has been defined in (10).
Proof: The analysis is first given in the time interval t ∈ [kT u , (k + 1)T u ).We obtain the following from (9): By Assumption 2, it can be concluded that δ(t) ≤ δ and w(t) − w(kT u ) ≤ T u δ, ∀t ∈ [kT u , (k + 1)T u ).With the definitions of 3 and 5 given in (9) where θ 1 is a bounded constant independent of L. Inserting ( 10) and ( 23) into (22) gives Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Integrating X from kT to t ∈ [kT u , (k + 1)T u ), we have Given a nonnegative function o : R + → R + and a positive constant c 0 , it can be easily obtained that e c 0 t t kT u e −c 0 s e c 0 s o(s)ds ≤ ((e c 0 T u − 1)/c 0 ) sup kT u ≤s≤t (e c 0 s o(s)).With the help of the above conclusion, multiplying both sides of ( 25) by e c 0 t yields (e c 0 s e y (s)) Recalling the definition of φ X given in (11), the conclusion in ( 26) can be extended from [kT u , (k + 1)T u ) to [0, t) and one has sup 0≤s≤t (e c 0 s φ X (s)) + β e c 0 T u − 1 c 0 sup 0≤s≤t (e c 0 s e y (s)) Selecting T u satisfying β((e c 0 T u − 1)/c 0 ) < 1, we obtain the following from (27) for t ∈ [0, ∞): (e c 0 s e y (s)) This completes the proof of Lemma 2.
For the overall hybrid control system (12), it is obvious that the matrix 1 can be tuned to be Hurwitz when the control gains K and H are appropriately chosen.Thus, there exists a positive definite matrix Next, the following theorem presents the main conclusions of this article.
Proof: To begin with, we construct a quadratic Lyapunov function V (X (t)) = X T (t)P X (t) for ( 12) and have Note that the estimate of (t, ζ(t), X (t)) has been given in (10) and λ m (P)X 2 ≤ V (X) ≤ λ M (P)X 2 .In addition, the scaling gain L can be selected to satisfy the condition L > 2Pα.
Furthermore, one gets from (33) that (e c 0 s e y (s)) By Lemmas 1 and 2, recalling the estimates of sup 0≤s≤t (e c 0 s φ X (s)) and sup 0≤s≤t (e c 0 s e y (s)), we arrive at where λ 1 has been defined in Theorem 1 and (e c 0 s X (s)) From (35), when λ 1 < 1, we have Note that e c 0 t X (t) ≤ sup 0≤s≤t (e c 0 s X (s)), and one has from (37) that Finally, we can conclude from (38) that when λ 1 < 1, X (t) globally exponentially converges a bounded region around the origin as t approaches infinity.In addition, when L is selected to be large enough and T u is small enough, λ 3 is arbitrarily small.The proof is completed.
Remark 5: It should be mentioned that the stability conditions given in Theorem 1 are conservative to some content since they are only sufficient rather than necessary.When the conditions do not hold, it is possible that the closed-loop control system is still exponentially bounded stable under the proposed control method.Hence, in order to get the better control performance, we suggest to tune the control parameters by the manner of trial and error in simulation and experiment.
Remark 6: For the proposed control method ( 5) and ( 7), the scaling gain L is one degree-of-freedom that can be regulated to guarantee the stability of the overall hybrid control system, and we can select L such that L > 2Pα.
In addition, the parameter c 0 can be chosen to satisfy c 0 < κ = ((L − 2Pα)/(2λ M (P))).From Theorem 1, it can be observed that when T u = 0, we cannot select L to be arbitrary large since the stability conditions given in (29) cannot be satisfied when L is too large.However, if T u = 0, we can set L to be arbitrary large.The scaling gain L should be tuned to get better the control performance.
Remark 7: According to (37), we can view λ 1 as a nonnegative function of the form It can be seen that the function λ 1 (0, 0, 0) = 0 and it is nonnegative, which means that the stability condition λ 1 < 1 can be achieved when T o , T u , and τ are not so large.Moreover, it can be seen that the maximum allowable sampling period and updating period of the control systems are bounded and become small when L increases.Therefore, if T o or T u is too large, the control systems cannot be stabilized by the proposed control method.
V. SIMULATION RESULTS To demonstrate the effectiveness of the proposed control method, we employ a numerical example and a practical example and present the comparative simulation results of two other control methods in this section.For the convenience of comparison, we assume that the sampling period of sensor is a multiple of the updating period of control in the two examples.In the first control method, the delayed sampled-data output is directly used in the observer design via zero-order holder over the intersample time intervals, and the controller updates at its own updating period, that is, the first control method is a kind of multirate control method without prediction technique.For the second control method [44], the designed observer is the same as that of the first control method, while the controller updates at the same period with that of sensor, that is, the second control method is a kind of single-rate control method without prediction technique.For simplicity, we call the first existing control method as Method 1 and the second existing control method as Method 2.Then, the comparative simulation results of the two examples are presented as follows.
The parameters of the proposed control method, Methods 1 and 2, are given as follows.The sampling period of sensor T o = 0.02 s, the output delay τ = 0.02 s, the sampling period of controller by T u = 0.002 s, the high gain and the observer parameters H = [60, 450, 1000] T .In Fig. 1, the trajectories of states x 1 and x 2 and the control inputs u are plotted under the proposed control method, Method 1, and Method 2. It can be observed from Fig. 1 that under all the three control methods, the states can asymptotically converge to zero and the disturbance rejection ability can be guaranteed.However, the proposed control method has the best control performance due to the prediction technique.Fig. 2 shows the trajectories of the estimation errors e 1 = x 1 − x1 , e 2 = x 2 − x2 , and e 3 = d − d.It shows that the estimation errors asymptotically converge to zero when t approaches infinity under all the three control methods.In addition, we can observe that the dynamic performances of the estimation errors under the proposed control method are the best.

B. Practical Example
Derived from [14], a practical example of a single-link robot arm system is employed in this section and depicted as The trajectories of states x, control input u, and estimation errors e are shown in Figs. 3 and 4. It can be seen that under all the three control methods, the states and estimation errors converge to zero asymptotically, while the proposed control method has the best dynamic performance than the other two control methods, since the influences of output delay and sampling of output are effectively compensated under the proposed control method.
As a conclusion, the simulation results show that the proposed control method can effectively improve the control Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.performance and the disturbance rejection ability compared with the traditional control methods without using the prediction technique.

VI. CONCLUSION
In this article, we have proposed a sampled-data robust output feedback control method for nonlinear uncertain systems subject to time-varying disturbance and measurement delay.When only the delayed sampled-data output is known, a predictor-based continuous-discrete observer has been proposed to estimate the unknown state and disturbance information.To handle the influences of measurement delay and sampling of output, output predictors have been designed to get the future output information in the proposed observer.
By using DUEA and feedback domination techniques, we have developed a sampled-data robust output feedback controller to globally exponentially stabilize the nonlinear uncertain system.Finally, the simulation results of two examples have verified the superiorities of the proposed robust control method.In the future, we consider the output feedback control problem for delayed control systems with strong nonlinearities.

Fig. 1 .
Fig. 1.Trajectories of states x 1 and x 2 and the control input u under the proposed control method, Method 1, and Method 2.