A Gonzalez

P Garcia

P Albertos

P Castillo

R Lozano

Robustness of a discrete-time predictor-based controller for time-varying measurement delay

Keywords: Time-varying delay, discrete-predictor, digital implementation, linear matrix inequality (LMI), robust stability

A predictor-based controller for time-varying delay systems is presented in this paper and its robustness properties for different uncertainties are analyzed. First, a time-varying delay dependent stability condition is expressed in terms of LMIs. Then, uncertainties in the knowledge of all plant-model parameters are considered and the resulting closed-loop system is shown to be robust with respect to these uncertainties. A significant improvement with respect to the same control strategy without predictor is achieved. The scheme is applicable to open-loop unstable plants and it has been tested in a real-time application to control the roll angle of a quad-rotor helicopter prototype. The experimental results show good performance and robustness of the proposed scheme even in the presence of long delay uncertainties.

Introduction

In designing any control system the process behavior imposes some unavoidable performance limitations (see, for instance, [START_REF] Seron | Fundamental limitations in filtering and control[END_REF]). This is clearly the case when dealing with systems with time delays, see for example [START_REF] Normey-Rico | Control of Dead-time Processes[END_REF]; [START_REF] Normey-Rico | An unified approach for dtc design using interactive tools[END_REF]; [START_REF] Ingimundarson | Robust tuning procedures of deadtime compensating controllers[END_REF] and the references therein. The Smith Predfictor (SP), [START_REF] Smith | Closer control of loops with dead time[END_REF], and the Finite Spectrum Assignment (FSA), [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF], may be considered as the main control methods for linear processes with time delay in either the input or the output (see also [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF]). A careful analysis of these methods and their modifications shows that they all use, in an implicit or explicit manner, a prediction of the state in order to achieve the control of the system. The more appealing characteristic of a predictor-based control is that, for a delayed plant without modelling error, the achieved performance approaches that of a delay-free system, [START_REF] Yue | Delayed feedback control of uncertain systems with time-varying input delay[END_REF], [START_REF] Normey-Rico | Dead-time compensators: A survey[END_REF], [START_REF] Guzman | Interactive tool for analysis of time-delay systems with dead-time compensators[END_REF].

As explained in [START_REF] Palmor | Time-delay compensationsmith predictor and its modifications[END_REF], the use of an explicit unstable prediction model in the SP approach determines the internal instability of the closedloop system. Palmor also suggested how to implement the control law using the so-called integral form. In [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF], this approach was also introduced in the framework of spectrum assignment with distributed delays. However, as shown in [START_REF] Mondié | Approximation of control laws with distributed delays: a necessary condition for stability[END_REF], the implementation of this control approach on a digital computer may result in an unstable behavior.

Nowadays, almost any control system application is implemented by using a computer, [START_REF] Astrom | Computer-Controlled Systems -Theory and Design[END_REF]. Thus, it is worth to analyze the effect of the delays in sampled data systems. On the other hand, it is wellknown that Discrete-Time (DT) delayed systems can be transformed into a delay-free system by using state augmentation techniques. However, in many cases, such delays are too large and time-varying, [START_REF] Yue | Delayed feedback control of uncertain systems with time-varying input delay[END_REF], [START_REF] Pan | Stabilization of remote control systems with unknown time varying delays by LMI techniques[END_REF]. In such cases, conventional state augmentation technique is usually not applicable, [START_REF] Gao | New results on stability of discrete-time systems with time-varying state delay[END_REF]. These considerations motivate the study of time-varying DT systems, [START_REF] Zhong | On distributed delay in linear control laws-part i: Discretedelay implementations[END_REF], [START_REF] Boukas | Discrete-time systems with time-varying time delay:stability and stabilization[END_REF]. Different control strategies have been proposed, such as a static state feedback [START_REF] Du | Delay-dependent robust stabilisation of uncertain discrete-time switched systems with time-varying state delay[END_REF]; [START_REF] Guo | New stability criterion for discrete-time systems with interval time-varying state delay[END_REF]; [START_REF] Meng | A delay-partitioning approach to the stability analysis of discrete-time systems[END_REF] and [START_REF] Guangdeng | Further results concerning delay-dependent h ∞ control for uncertain discrete-time systems with timevarying delay[END_REF] or a dynamic controller [START_REF] Liu | Delay-dependent robust stabilisation of discrete-time systems with time-varying delay[END_REF]; [START_REF] He | Output feedback stabilization for a discrete-time system with a time-varying delay[END_REF].

In [START_REF] Lozano | Robust prediction-based control for unstable delay systems: Application to the yaw control of a mini-helicopter[END_REF], a discrete-time controller for Continuous-Time (CT) plants with time delay is proposed and the closed-loop stability is proved. The proposed robust stability proof allows positive or negative variations in the delay but they must be bounded by the size of the sampling period. In [START_REF] Garcia | Robustness with respect to delay uncertainties of a predictor-observer based discrete-time controller[END_REF], the previous results were enlarged when the delay uncertainty is larger than the sampling period but, in both papers, time-varying delay variations smaller than the sampling period are only considered. Recently in [START_REF] Garcia | Robustness of a discrete-time predictor-based controller for time-varying measurement delay[END_REF], this framework was improved in order to analyze time-varying delays greater than the sampling period, although the LMI constraints developed to check closed-loop stability have been shown to be very conservative.

The objective of this paper is to analyze robust stability of DT predictorbased state-feedback controllers applied to open-loop unstable systems with time-varying measurement delay, when only delay bounds are assumed to be known. To the best of the authors' knowledge, the robust stability analysis under large but bounded time varying-delays has not been previously reported in the literature.

The paper is organized as follows: the problem statement is introduced in the next section. The stability condition to analyze time-varying delay systems is proposed in Section 3. Section 4 is devoted to present some numerical examples to compare the performance of the predictor-based and the equivalent memoryless control schemes. In section 5, robustness issues are analyzed. In order to validate the proposed control scheme in real time operation, several experiment tests are carried-out in Section 6. Conclusions are drawn in Section 7.

Problem formulation

Assume the following CT state space representation of the plant

ẋ(t) = A c x(t) + B c u(t) (1a) x(t) = 0, t ≤ 0 (1b)
where the nominal plant parameter matrices are A c ∈ ℜ n×n , B c ∈ ℜ n×m . Without loss of generality, and with the aim to simplify the analysis, null initial conditions are assumed.

As depicted in Figure 1, the plant state x(t) is fully accessible but there is an unknown measurement/transmission delay τ (t). That is

x(t) = x(t -τ (t)) (2)
This delay is assumed to be randomly varying, τ (t), but bounded by known values. Since a computer implementation is intended, the sampling period is defined as T k = t k+1t k , where t k is the sampling time instant, k ∈ Z + , and

τ (t k ) = τ k .
In this section, the following assumptions are made Then, a DT approximation of (1) is given by

x k+1 = Ax k + Bu k (3) x k = 0, k ≤ 0
where A = e AcT , B = T 0 e Acσ dσB c and d k is the discrete-time delay, defined in Assumption A.2.

A state feedback law is assumed

u k = -Kx k (4)
where K ∈ ℜ m×n is a static state feedback control law such that the controlled system is stable and fulfils some design requirements. Thus, A -BK is a Hurwitz matrix.

But if there is an output measurement time-varying delay, that is, for

u k = -K xk = -Kx k-d k (5)
the stability of the closed-loop is not guaranteed.

The robust stability of the closed-loop system (3)-( 5) has been studied and reported in the cited literature: given the control law in (4), the time delay interval {d m , d M } ensuring the system stability can be computed.

Discrete time model based predictor

To counteract the delay, a predicted state x (see Figure 1) will be used to provide the control input and its influence in the closed-loop robustness will be analyzed.

A measurement delay h ∈ Z + is assumed to compute the predicted state (usually h ∈ {d m , d M }), being a predictor design parameter. The proposed predictor-based control scheme is

u k = -K xk+h ( 6
)
where xk+h is an h-step ahead prediction, that is

xk+h = A h x k-d k + h-1 i=0 A h-i-1 Bu k+i-h = A h x k-d k + Σ h,k (7)
where x k-d k is the current state measurement. The actual delay d k is randomly varying and unknown (Assumption A.2).

Closed-loop behavior

Concerning the dynamics of the closed-loop system, the following Lemma can be stated.

Lemma 1 The predictor-based closed-loop controlled system composed of (3), ( 6) and ( 7), leads to

x k+1 = (A -BK)x k -BKA h x k-d k + BKA h x k-h (8)
Proof: Combining ( 6) and (3), and taking into account (7), it yields

x k+1 = Ax k -BKA h x k-d k -BKΣ h,k (9)
where Σ h,k is as defined in (7).

On the other hand, by recursively applying (3) h steps, it results

x k = A h x k-h + Σ h,k (10)
But Σ h,k can be eliminated between ( 10) and ( 9), leading to (8). This completes the proof.

Delay-dependent stability condition for time-varying delay systems

In this section, a new stability condition for the predictor-based controlled system (8), where the control is computed by using the assumed measurement delay h different from the actual one d k which is time-varying but bounded, is proposed.

The stability condition is based on a set of LMIs [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF]).

The solution of the LMI problem, if a feasible one exists, can be easily found by any standard available software such as Matlab's LMI Control Toolbox, [START_REF] Gahinet | The lmi control toolbox[END_REF]. For a given delay lower bound d m , the goal is to find the largest delay interval, δ = d Md m , ensuring the closed-loop stability.

To simplify the notation, the new matrices A 0 = (A -BK) and A 1 = BKA h are defined. Hence, from (8) it yields

x k+1 = A 0 x k + A 1 x k-h -A 1 x k-d k (11) Theorem 1 System (11) is asymptotically stable if there exist matrices P, Z 1 , Z 2 > 0; Q, Q m , Q M , Q h , Z M ≥ 0 and matrices S 1 , S 2 , T 1 , T 2 satisfying the following matrix inequalities Γ δS δS T -δZ M < 0 Γ δT δT T -δZ M < 0 ( 12
)
where

S = 0 S T 1 S T 2 0 0 0 0 T T = 0 T T 1 0 T T 2 0 0 0 T .
The symmetric matrix Γ is defined as follows4 :

Γ =           Γ 1 0 Γ 2 0 Γ 3 Γ 4 Γ 5 ( * ) Γ 6 Γ 7 Γ 8 0 Γ 9 Γ 10 ( * ) ( * ) Γ 11 0 Z 1 0 0 ( * ) ( * ) ( * ) Γ 12 0 0 0 ( * ) ( * ) ( * ) ( * ) Γ 13 Γ 14 Γ 15 ( * ) ( * ) ( * ) ( * ) ( * ) -P 0 ( * ) ( * ) ( * ) ( * ) ( * ) ( * ) -Z       0 h > d m Γ 14 = A T 1 P Γ 15 = A T 1 Z Z = ρ 2 1 Z 2 + (d m -h) 2 Z 1 + δZ M ρ 1 = min(d m , h) Note that δ = d M -d m is an analysis parameter.
Proof: Following the ideas in [START_REF] Guo | New stability criterion for discrete-time systems with interval time-varying state delay[END_REF] and [START_REF] Garcia | Robustness of a discrete-time predictor-based controller for time-varying measurement delay[END_REF], a Lyapunov-Krasovskii function candidate is defined as

V (k) = V 1 (k) + V 2 (k) + V 3 (k) + V 4 (k) + V 5 (k) (13)
where

V 1 (k) = x T k P x k V 2 (k) = k-1 i=k-d k x T i Qx i V 3 (k) = k-1 i=k-dm x T i Q m x i + k-1 i=k-d M x T i Q M x i + k-1 i=k-h x T i Q h x i V 4 (k) = -dm j=-d M +1 k-1 i=k+j x T i Qx i V 5 (k) = ρ 1 -1 i=-ρ 1 k-1 m=k+i ν T m Z 2 ν m + |d m -h| -ρ 1 -1 i=-ρ 2 k-1 m=k+i ν T m Z 1 ν m + + -dm-1 i=-d M k-1 m=k+i ν T m Z M ν m with ν k = x k+1 -x k , ρ 1 . = min(d m , h) and ρ 2 .
= max(d m , h). Then, the closed-loop system (11) will be asymptotically stable if the forward difference ∆V

(k) = V (k + 1) -V (k) satisfies ∆V (k) = ∆V 1 (k) + ∆V 2 (k) + ∆V 3 (k) + ∆V 4 (k) + ∆V 5 (k) < 0 (14)
After some lengthy algebraic manipulations (see appendix), it yields

Γ + (d k -d m )SZ -1 M S T + (d M -d k )T Z -1 M T T < 0 (15)
Note that the first term in the above inequality is linear affine function in the variable d k . Taking into account the same considerations as [START_REF] Guo | New stability criterion for discrete-time systems with interval time-varying state delay[END_REF] the equivalence between this inequality and the following LMI constraints can be proved

Γ + δSZ -1 M S T < 0 (16) Γ + δT Z -1 M T T < 0 (17)
Finally, applying Schur complement to the above inequalities result on the LMI constraints ( 12). This completes the proof. This theorem allows to check the improvement of using the state predictor in the feedback control. The following numerical examples, already reported in the literature, illustrate this advantage.

Numerical examples

The stability of a state delayed plant defined by

x k+1 = Ax k + A d x k-d k (18)
where d k fulfils Assumption A.2, has been widely studied in the literature (see for example [START_REF] Gao | New results on stability of discrete-time systems with time-varying state delay[END_REF]; [START_REF] Guo | New stability criterion for discrete-time systems with interval time-varying state delay[END_REF]; [START_REF] Meng | A delay-partitioning approach to the stability analysis of discrete-time systems[END_REF]). This system is equivalent to (3) with delayed state feedback (5), leading to the equivalence

A d = -BK (19)
Thus, the results achieved by using the proposed predictor are compared with those obtained by the above referenced approaches where a delayed state feedback control is used. [START_REF] Gao | New results on stability of discrete-time systems with time-varying state delay[END_REF]). In that paper, a sampling period of 30 ms was adopted. Here, a shorter sampling period is used (T = 10 ms), in order to better show the influence of the delay (which is expressed as multiples of T ). The ZOH-discretized model of the inverted pendulum is

Example 1 Consider the open-loop unstable model of an inverted pendulum (see Example 3 in

x k+1 = 1.0009 0.0100 0.1730 1.0009 x k + 10 -3 -0.0088 -1.7652 u k
The poles of the open-loop unstable system are {1.0425, 0.9593}. Following [START_REF] Gao | New results on stability of discrete-time systems with time-varying state delay[END_REF] assigning the poles of the delay-free closed-loop system (A-BK) at {0.9994, 0.8588}. Now, the existence of a time varying d k measurement delay is assumed (see Figure 1). Two options are considered:

1. The feedback control input is the delayed state (5),

u k = -Kx k-d k .
2. The predicted state (7) is used when computing the control (6),

u k = -K xk+h .
Given d m , the goal is to obtain the upper value of d M such that the closed-loop system remains stable. In Table, 1 comparative results using Theorem 1 in [START_REF] Guo | New stability criterion for discrete-time systems with interval time-varying state delay[END_REF], Theorem 1 and 3 in [START_REF] Gao | New results on stability of discrete-time systems with time-varying state delay[END_REF], Theorem 1 in [START_REF] Meng | A delay-partitioning approach to the stability analysis of discrete-time systems[END_REF] 5 , without the predictor, as well as the results of the proposed stability condition in Theorem 1, when the predictor is used, are summarized. The proposed approach has been applied assuming h = d m . As shown, the use of the predictor enlarges the stability interval. For a minimum delay d m = 10 it is not possible to prove the time-varying delay-dependent stability of the delayed state feedback controlled plant whereas it is ensured until a maximum delay of d M = 15 if the predictor ( 7) is used in the feedback control.

Example 2 Now, consider the following double integrator plant (which is a simple model of the experimental platform used afterwards)

ẋ(t) = 0 1 0 0 x(t) + 0 0.1050 u(t)
with a sampling period of T = 10ms, the following control law u k = -[109.9907 50.0020]x k stabilizes the plant assigning the poles of the delay-free closed-loop system at {0.9735 ± 0.0212i}. Then, following the same procedure as in the previous example, the results in Table 2 are obtained. Also in this case, better time-varying measurement delay robustness is achieved by using the state predictor in the feedback control. Note that, when d m = 30 is not possible to prove the stability of the delayed state feedback controlled plant whereas it is ensured until a maximum delay of d M = 41 if the predictor (7) is used in the feedback control.

Remark 1 The main objective of this work is to prove that the proposed predictor-controller scheme improves robustness against uncertainties in the time-varying measurement delay. Theorem 1 provides a conservative bound (which could be improved by choosing another h) in the robust stability. This is a matter of current research.

Robust-stability of the closed-loop system

In this section, robustness of the designed control scheme to variations in all the parameters of the system model are analyzed.

Removing the assumptions A.1, A.2, and A.3 and following the ideas in [START_REF] Lozano | Robust prediction-based control for unstable delay systems: Application to the yaw control of a mini-helicopter[END_REF]; [START_REF] Cloosterman | Controller synthesis for networked control systems[END_REF], the system (1) can be modeled in discrete-time as:

x k+1 = Ax k + Bu k + ∆f k (21)
where

f k = [x T k , u T k , u T k-1 ]
T is the vector of system variables6 and the uncertainty matrix ∆ is defined as:

∆ = ∆ A ∆ B ∆ B1 = γG∆ k H A H B H B1
where G, H A , H B , H B1 are some constant matrices of appropriate dimensions, ∆ k is a real uncertain matrix function with Lebesgue measurable elements satisfying ∆ T k ∆ k ≤ I, and the scalar γ > 0 determines the size of the uncertainties.

Define an augmented state representation of the system (21) together with the control law (6) and the predictor (7). The autonomous system can be represented as

7 xk+1 = Ācl (d i )x k (23) where xk = x T k • • • x T k- d u T k-1 • • • u T k-h T (24) d = max(d M , h)
and the state matrix Ācl (d i ), assuming an actual delay

d k = d i , is Ācl (d i ) = Ā(d i ) + γ Ḡ∆ k H(d i ) Ā(d i ) = Ā11 (d i ) Ā12 Ā21 (d i ) Ā22 Ā11 (d i ) =       A + BK 0 (d i ) BK 1 (d i ) • • • • • • BK d(d i ) I 0 • • • • • • 0 0 I • • • • • • 0 • • • • • • • • • • • • • • • 0 0 • • • I 0       Ā12 =     -BKB • • • -BKA h-1 B 0 • • • 0 • • • • • • • • • 0 • • • 0     Ā21 (d i ) =     K 0 (d i ) K 1 (d i ) • • • K d(d i ) 0 0 • • • 0 • • • • • • • • • • • • 0 0 • • • 0     Ā22 =       -KB • • • -KA h-2 B -KA h-1 B I • • • • • • 0 0 • • • • • • 0 • • • • • • • • • • • • 0 • • • I 0       Ḡ = G 0 • • • 0 T H(d i ) = H1 (d i ) H2 H1 (d i ) = H A + H B K 0 (d i ) H B K 1 (d i ) • • • H B K d(d i ) H2 = -H B KB + H B1 -H B KAB • • • -H B KA h-1 B K j (d i ) = -KA h j = d i 0 j = d i
By applying a similar procedure as in [START_REF] Gonzalez | Robustness analysis of discrete predictor-based controllers for input-delay systems[END_REF], the following Theorem is obtained:

Theorem 2 The closed-loop system (21) with predictor-based scheme (6) is robustly stable if there exists L > 0 such that the following LMI's hold for every

d i = d m , d m + 1, • • • , d M   -L + H(d i ) T H(d i ) Ā(d i ) T L 0 ( * ) -L L Ḡ ( * ) ( * ) -γ -2 I   < 0 (25)
Moreover, the maximum tolerance to uncertainties keeping stability can be obtained by solving the following convex optimization problem: min γ -2 subject to (25).

Proof: Propose the complete quadratic Lyapunov functional candidate V (k) = xT k P xk , P > 0 where the augmented state is defined in Equation ( 23). The closed-loop stability can be proved if the following LMI constraints hold for every

d i = d m , d m + 1, • • • , d M . -P + Ācl (d i ) T P Ācl (d i ) < 0 (26)
Applying Schur Complement, the following is obtained

-P Ācl (d i ) T P ( * ) -P < 0 ( 27
)
That can be put in the following form, taking into account that Ācl (

d i ) = Ā(d i ) + γ Ḡ∆ k H(d i ) -P Ā(d i ) T P ( * ) -P + γ 0 P Ḡ ∆ k H(d i ) 0 + (28) γ H(d i ) 0 T ∆ T k 0 P Ḡ T < 0
The previous inequality is satisfied for some scalar ǫ > 0, [START_REF] Xie | Output feedback H ∞ of systems with parameter uncertainty[END_REF], such that

-P Ā(d i ) T P ( * ) -P + ǫγ 2 0 P Ḡ 0 ḠT P + (29) ǫ -1 H(d i ) T 0 H(d i ) 0 < 0 Applying again the Schur Complement   -P + ǫ -1 H(d i ) T H(d i ) Ā(d i ) T P 0 ( * ) -P P Ḡ ( * ) ( * ) -γ -2 ǫ -1 I   < 0 (30)
and multiplying by ǫ

  -ǫ P + H(d i ) T H(d i ) ǫ Ā(d i ) T P 0 ( * ) -ǫ P ǫ P Ḡ ( * ) ( * ) -γ -2 I   < 0 (31)
Finally, by defining L = ǫ P the expression ( 25) is obtained. This completes the proof.

Example 3 Consider the process in Example 2 with uncertainties as described in Equation ( 21) with G = 10 -4 1 1

T , H A = 1 1 and H B = 1.

Assuming that both bound are fixed, δ = d Md m , the objective is to find the upper value of γ such that the closed-loop system remains stable. By using Theorem 2, for h = d m , the maximum γ is shown for some delay intervals in Table 3. 4). This result has been also verified by simulation. As previously mentioned, the optimal value of this parameter is a matter of current research. 15 18 γ(δ = 2) 1.0905 1.0702 1.0493 1.0302 1.0046 0.9837 γ(δ = 3) 1.0826 1.0616 1.0385 1.0036 0.9632 0.9436 γ(δ = 4) 1.0743 1.0505 1.0065 0.9807 0.9406 0.9267 γ(δ = 5) 1.0615 1.0321 1.0009 0.9655 0.8774 0.8359 Now, some simulation results are reported. Assume the initial condition to be x k = [1 1] T for k ≤ 0. In addition, let the delay d k change randomly between d m = 18 and d M = 23 as depicted in Figure 2, and the variable process uncertainties being defined by γ = 0.8359 and ∆ k changing randomly between -1 and 1. The predictor-based control assumes h = d M = 23. Then, the state response of the double integrator is given in Figure 3. It can be seen that the system is asymptotically stable.

Experimental results

The goal of this section is to illustrate the effectiveness and robustness of the proposed predictor-based control scheme in a real-time closed-loop system. For that purpose, the real-time platform shown in Figure 4 has been used. The platform is composed of a ground control unit (GCU), developed using Linux RT, and a quad-rotor prototype equipped with a wireless Integrated Measurement Unit (IMU) to measure the angular position and velocity (yaw, pitch and roll angles).

Assuming the pitch and yaw angles are zero, the roll axis torque is controlled by the forces generated by two propellers, and the following simplified model can be obtained, [START_REF] Castillo | Modelling and control of mini flying machines[END_REF]:

φ = K φ (V r -V l )
where V r , V l are the right and left motor voltages respectively, K φ = 0.1050 is a constant representing the inertia moment of the body and φ denotes the roll acceleration of the system. Define x = [φ φ] T as the state vector. Then, the roll angle dynamics can be written as ẋ with

(t) = A c x(t) + B c u(t)
A c = 0 1 0 0 ; B c = 0 0 K φ -K φ ;
and

u(t) = V r V l .
Note that this model has been already used in Example 2. Again, the sampling period for all experiments is assumed to be T = 10 ms. As before, a stabilizing control law as the one used in Example 2 is designed. Then8 , the following control input is assumed

u k =
-109.9907 -50.0020 109.9907 50.0020

x k (32)

Figure 4: Quad-rotor system with a wireless Integrated Measurement Unit (IMU).

In order to test the predictor scheme, a square wave function, with 5 • of magnitude, is proposed as the desired reference position and the following experiments are run.

First experiment:

The first experiment is realized applying the controller (32) without any additional delay. Figure 5 illustrates the behavior of the system. Notice in this figure that the system remains stable with a constant error e φ = 1 • .

Second experiment:

The second experiment is carried out adding a random time-varying delay in the state. No prediction is implemented. That is, the state feedback measurement signal is delayed as x k-d k , where d k = d m + δ k , d m is the fixed minimum discrete delay, and δ k is the random number of the delayed samples here considered9 (1 ≤ δ k ≤ 2). The output response when this delay is activated is shown in Figure 6 response in Figure 5 is reproduced. The delay is increased to 3 ≤ d k ≤ 5 at t = 8 s, the system remaining stable. If the additional delay is further increased, 5 ≤ d k ≤ 7 at t ≥ 18s, the system becomes unstable (see Figure 6).

The improvement achieved by using the predictor scheme as well as its robustness in presence of bigger delays is illustrated in the next experiment.

Third experiment:

In this experiment, the proposed predictor control scheme is implemented with the previous control law (32), and h = 8. Similar to the previous experiment, the measurement state is delayed as depicted in Figure 8, where d k ∈ {5, 11} and | d k+1d k |≤ 2. Note that under the same conditions as the previous experiment (t < 25s), the closed-loop system remains stable. Moreover, the behavior obtained using the proposed scheme is very similar to the delay-free case (see Figures 5 and7).

In order to prove the robustness of the predictor-based control scheme, the additional delay is increased. At t = 25s, the lower bound delay is set at d m = 7, the system behavior is not much affected. But, for longer delays, d m = 9 at time t = 43 ( d k ∈ {9, 11}), the transient response degrades (see Figure 7). This result illustrates the use of the predictor scheme to restore the nominal closed-loop behavior, even when the time-varying delay-range is subject to some uncertainties.

Conclusions

In this paper a DT predictor-based control scheme for time-varying measurementdelay systems has been analyzed. The prediction horizon h (7), is a design parameter, its optimal value being determined by de plant model and the time delays bound. Currently this is a matter of research. This control has been shown to be robust with respect to uncertain and time-varying bounded delays, besides the possible parametric model uncertainties, coping with long delays. Furthermore, the controlled plant performance remains similar to that of the nominal plant.

Other than the theoretical result providing a set of LMI constraints to analyze the stability of a predictor-based control applied to a time varying delay unstable plant, uncertainties in the knowledge of plant-model and sampling pattern parameters are also considered. The resulting closed-loop system is shown to be robust with respect to these uncertainties. The delay interval δ = d Md m is an analysis parameter allowing to determine an interval providing stability in which the prediction horizon h ∈ {d M , d m } should be selected.

One of the main contributions of this work concerns to the digital implementation and the experimental validation of the proposed algorithm by stabilizing the roll angle of a quad-rotor helicopter. Real-time experiments have enlightened the performance of the prediction based controller and have satisfactorily demonstrated its robustness with respect to inherent plant model uncertainties and additional delay measurement errors. Note that, this prototype is unstable with very fast dynamics where losing the information of one sampling period could be critical, and an unknown delay introduced by the wireless IMU is always present.

As already mentioned, the formal analysis of the influence of the predictor design parameter (h) is a matter of current research. By using the candidate Lyapunov function defined in (13), the increments of V 1 to V 5 can be written respectively, as:

∆V 1 (k) = x T k+1 P x k+1 -x T k P x k = [A 0 x k + B 1 x k-h -B 1 x k-d k ] T P [A 0 x k + B 1 x k-h -B 1 x k-d k ] -x T k P x k ∆V 2 (k) = x T k Qx k -x T k-d k Qx k-d k + k-d k i=k-d k+1 +1
x T i Qx i

≤ x T k Qx k -x T k-d k Qx k-d k + k-dm i=k-d M +1
x T i Qx i

∆V 3 (k) = x T k (Q m + Q M + Q h )x k -x T k-dm Q m x k-dm -x T k-d M Q M x k-d M -x T k-h Q h x k-h ∆V 4 (k) = δ[x k -x k-d M ] T Q[x k -x k-d M ] - k-dm i=k-d M +1 x T i Qx i ∆V 5 (k) = ν T k Zν k -ρ 1 k-1 i=k-ρ 1 ν T i Z 2 ν i -|d m -h| k-ρ 1 -1 i=k-ρ 2 ν T i Z 1 ν i - k-d k -1 i=k-d M ν T i Z M ν i - k-dm-1 i=k-d k ν T i Z M ν i
where

Z = ρ 2 1 Z 2 + (d m -h) 2 Z 1 + δZ M and δ = d M -d m
Applying the Tchebyschev inequality (Lemma 1 in [START_REF] Guo | New stability criterion for discrete-time systems with interval time-varying state delay[END_REF]) it yields

-ρ 1 k-1 i=k-ρ 1 ν T i Z 2 ν i ≤ -[ k-1 i=k-ρ 1 ν i ] T Z 2 [ k-1 i=k-ρ 1 ν i ] -|d m -h| k-ρ 1 -1 i=k-ρ 2 ν T i Z 1 ν i ≤ -[ k-ρ 1 -1 i=k-ρ 2 ν i ] T Z 1 [ k-ρ 1 -1 i=k-ρ 2 ν i ] k-d k -1 i=k-d M ν T i Z M ν i
Finally, the forward difference ∆V (k) can be put as

∆V (k) ≤ λ T k Ωλ k - k-dm-1 i=k-d k ν T i Z M ν i - k-d k -1 i=k-d M ν T i Z M ν i < 0 (33)
where the augmented state is defined as

λ T k = x k x k-d k x k-dm x k-d M x k-h . and the matrix Ω is Ω =       Ω 1 Ω 2 Ω 3 0 Ω 4 ( * ) Ω 5 S 1 -S T 2 -T 1 + T T 2 Ω 6 ( * ) ( * ) Ω 7 0 Z 1 ( * ) ( * ) ( * ) Ω 8 0 ( * ) ( * ) ( * ) ( * ) Ω 9      
with Ω 1 = A T 0 P A 0 + (A 0 -I) T Z(A 0 -I) -P + (δ + 1)

Q + Q M + Q m + Q h -Z 2 Ω 2 = -A T 0 P B 1 -(A 0 -I) T ZB 1 Ω 3 = 0 h ≤ d m Z 1 h > d m Ω 4 = A T 0 P B 1 + (A 0 -I) T ZB 1 + Z 2 h ≤ d m 0 h > d m Ω 5 = B T 1 P B 1 + B T 1 ZB 1 -Q Ω 6 = -B T 1 P B 1 -B T 1 ZB 1 Ω 7 = -Q m -Z 1 + 0 h ≤ d m -Z 2 h > d m Ω 8 = -Q M Ω 9 = B T 1 P B 1 + B T 1 ZB 1 -Q h -Z 1 + -Z 2 h ≤ d m 0 h > d m
Then, after some lengthy algebraic manipulations and Schur complement, the inequality (15) is obtained.

Following the baseline of reasoning in [START_REF] Guo | New stability criterion for discrete-time systems with interval time-varying state delay[END_REF], the LMIs in ( 16)-( 17) are obtained and the final result ( 12) is achieved.

  2 : Assumption A1. The sampling period is constant T k = T . Assumption A2. The output delay satisfies τ k = d k T , where d k ∈ Z + and d m ≤ d k ≤ d M , being d m and d M known delay bounds 3 . That means d k is also randomly varying. Assumption A3. There are no uncertainties in the process model (A c , B c ).

Figure 2 :

 2 Figure 2: The time-varying delay in Example 3.

Figure 3 :

 3 Figure 3: State evolution with predictor-based control (h = d M ), timevarying delay 18 ≤ d k ≤ 23 and γ = 0.8359.

Figure 5 :

 5 Figure5: Closed-loop system response without additional delay (nominal case) using a square wave function as a desired angle.

Figure 6 :

 6 Figure 6: Closed-loop system response without predictor.

Figure 7 :

 7 Figure 7: Closed-loop system response with predictor and additional timevarying delay.

Figure 8 :

 8 Figure 8: Additional random time-varying delays applied in the system.

Table 1 :

 1 Calculation of d M given d m

	state feedback			d m
			1 4	7	10
	delayed	Th.1 Guo(2009) 8 8	8 infeasible
	delayed	Th.1 Gao(2007) 8 8	8 infeasible
	delayed	Th.3 Gao(2007) 8 8	9 infeasible
	delayed	Th.1 Meng	8 8	9	10
	predicted	Th.1 (h = d m ) 8 10 13	15

Table 2 :

 2 Calculation of d M given d m

	state feedback			d m
			5 10	20	30
	delayed	Th.1 Guo(2009) 19 19 infeasible infeasible
	delayed	Th.1 Gao(2007) 19 19 infeasible infeasible
	delayed	Th.3 Gao(2007) 19 19 infeasible infeasible
	delayed	Th.1 Meng	19 19	20	infeasible
	predicted	Th.1 (h = d m ) 21 25	32	41

Table 3 :

 3 Allowable upper bound γ for some delay intervals (h = d m ) Note that, if another measurement delay h is considered (for example, h = d M ), robustness against model uncertainties is improved (see Table

	d m	3	6	9	12	15	18
	γ(δ = 2) 1.0440 1.0148 0.9867 0.9591 0.9291 0.8882
	γ(δ = 3) 1.0080 0.9751 0.9409 0.9062 0.8473 0.8296
	γ(δ = 4) 0.9712 0.9308 0.8904 0.8446 0.7802 0.7336
	γ(δ = 5) 0.9290 0.8797 0.8297 0.7729 0.7031 0.6428
	Remark 2

Table 4 :

 4 Allowable upper bound γ for some delay intervals (h = d M )

	d m	3	6	9	12

These assumptions will be removed in Section 5.

In general, d m is known and fixed whereas d M is the maximum estimated delay to get stable behavior of the controlled plant, as a result of the stability analysis (Section 3).

   whereΓ 1 = -P + (δ + 1)Q + Q M + Q m + Q h -Z 2 Γ 2 = 0 h ≤ d m Z 2 h > d m Γ 3 = Z 2 h ≤ d m 0 h > d m Γ 4 = A T 0 P Γ 5 = (A 0 -I) T Z Γ 6 = -Q + T 1 + T T 1 -S 1 -S T 1 Γ 7 = S 1 -S T 2 Γ 8 = -T 1 + T T 2 Γ 9 = -A T 1 P Γ 10 = -A T 1 Z Γ 11 = -Q m + S 2 + S T 2 -Z 1 + 0 h ≤ d m -Z 2 h > d m Γ 12 = -Q M -T 2 -T T 2 Γ 13 = -Q h -Z 1 + -Z 2 h ≤ d m

I (0) denotes the identity (zero) matrix of appropriate dimension and ( * ) represents a term induced by symmetry.

Where the less conservative delay-partitioning approach has been used.

Note that the previous input, u k-1 , is included in f k , as some variations in the sampling period, (Assumption A.1), may force this dependence.

7 Note that the closed-loop realization also involves all the past state and system input delayed variables.

Note that the purpose of this section is not to validate the performance of the control law, which is assumed to be given, but to illustrate the effectiveness of the proposed predictor-scheme with respect to random delays in a real-time application.

In practical applications this is a usual range of time delay variations. Nevertheless some other intervals can be considered.

Acknowledgments

The authors thank the Associate Editor and anonymous reviewers for their valuable comments and suggestions.

This work has been partially granted by Conselleria de Educación under PROME-TEO project number 2008-088, and CICYT no. DPI2008-06737-C02-01 from Spanish government.