A practical solution for the control of time-delayed and delay-free systems with saturating actuators

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A practical solution for the control of time-delayed and delay-free systems with saturating actuators Thiago A. Lima, Magno P. de Almeida Filho∗, Bismark C. Torrico, Fabrício G. Nogueira, Wilkley B. Correia Department of Electrical Engineering, Federal University of Ceará, 60455-760 Fortaleza, CE, Brazil

a r t i c l e

i n f o

Article history: Received 20 November 2018 Revised 8 April 2019 Accepted 27 June 2019 Available online xxx Recommended by S. Tarbouriech Keywords: Time-delay systems Delay-free systems Anti-windup Model-based control Discrete-time

a b s t r a c t This paper presents a predictor based structure that is suitable to deal with both time-delayed and delayfree systems with saturating actuators. Such condition, well known in the literature to cause the windup phenomenon, is usually coped within the control area by techniques that commonly lead to the controller augmentation. However, it seems to exist a lack of simple approaches to deal with the design of linear controllers for both systems with and without delay. Thus, this paper addresses the saturation problem by both proposing a linear controller design and an associated anti-windup compensator design. The controller tuning is realized in three distinct phases, initially considering set-point tracking adjustment, then robustness and disturbance rejection tuning, and finally the associated anti-windup compensator synthesis. Furthermore, linear matrix inequalities (LMIs) which include both performance and stability requirements are employed for the anti-windup synthesis. The proposed controller presents better results when compared with recently published anti-windup controllers as well as a constrained MPC algorithm. Experimental results on a neonatal intensive care unit are also presented in order to validate the usefulness of the proposed strategy. © 2019 European Control Association. Published by Elsevier Ltd. All rights reserved.

1. Introduction Dead-time compensators (DTCs) is a class of predictor based controllers that have been widely studied for about the past 25 years mainly due to their ability to improve the performance of classical proportional-integral (PI) and/or proportional-integralderivative (PID) controllers when the process presents time delay between the input and output. The first DTC was proposed by Smith [33], in 1957, also known in the literature as the Smith predictor (SP). Some drawbacks of the proposal have limited its applications as it is restricted to open-loop stable plants, while the slow poles of the plant dominate the disturbance rejection response [22]. Since then several extensions have been proposed to improve robustness, disturbance rejection, and measurement noise attenuation characteristics of the SP [1,2,8,9,12,13,17,18,23–27,29]. However, a problem with the aforementioned works is that they are not concerned with actuator saturation, which places a nonlinearity into the otherwise linear DTC structures, thus causing their

∗

Corresponding author. E-mail addresses: [email protected] (T.A. Lima), [email protected] (M.P. de Almeida Filho), [email protected] (B.C. Torrico), [email protected] (F.G. Nogueira), [email protected] (W.B. Correia).

effectiveness to drop. Such a condition is common in practical applications and can cause problems due to two reasons: Problem 1. Slow and/or integral poles in the primary controller have been historically related to windup issues. Problem 2. Differences between the controller output and the actual plant input can cause prediction errors; thus it is essential to include the model of the saturation to the control path. Some previous works have proposed different approaches to deal with saturating actuators in the control of time-delay systems. An appropriate solution of the modified Smith predictor [17,18] with anti-windup was proposed in [19], although an optimization procedure is necessary to define some desired robustness and noise sensitivity constraints. In [11] a predictive disturbance observer based filtered PI control for first order plus dead time (FOPDT) processes is presented. Recently, in [5] an anti-windup structure for the filtered SP (FSP) was proposed. However, the strategy was not applied to delay-free systems. Another alternative to deal with input constraints considers the use of model-based predictive controllers (MPCs) [3,22]. However, in the MPC case, a constrained quadratic problem needs to be solved at each sampling time. Non-model-based approaches have also successfully been applied in the control of time-delay systems with actuator saturation.

https://doi.org/10.1016/j.ejcon.2019.06.012 0947-3580/© 2019 European Control Association. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: T.A. Lima, M.P. de Almeida Filho and B.C. Torrico et al., A practical solution for the control of time-delayed and delay-free systems with saturating actuators, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.06.012

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As an example, Fridman et al. [6] deals with the regional stabilization problem by designing a constant state-feedback controller. On the other hand, the modern approach for dealing with actuator saturation considers the design of anti-windup structures that are only activated during saturation of the control signal. Thus, in this manner, very often the goal of the design is to improve performance by forcing the closed-loop out of the nonlinear region so that the system response is preserved as close as possible to the linear case. Some classical works with this goal can be visited in order to find well established anti-windup design techniques [31,42,45,47]. The inclusion of the saturation model leads to a nonlinear approach to the control synthesis, so that, many works related to nonlinear control have dealt with windup issues [7,21,37,38]. More recently, the design of nonlinear anti-windup techniques that improve the performance of previously designed low-order and full-order anti-windup controllers have been searched, as in [44]. Despite many works dealing with anti-windup compensator design, and even the design of anti-windup compensators for systems with time delays [32,35,36], relatively few papers treat the delay and the delay-free case in a simple and unified framework. Furthermore, many approaches do not consider, explicitly, practical issues such as tracking, robustness and disturbance rejection.

Furthermore, performance constraints related to the antiwindup compensator dynamics are included in the LMIs by using d-stability regions. This is a new approach for including prespecified dynamic requirements within the anti-windup design. It is emphasized that the approach not only deals with the joint problems of time-delays and saturation but also considers disturbances, tracking, and robustness. This makes the approach proposed here quite practical and useful for the engineer. Simulation results are used to analyze the tuning and establish a fair comparison with the anti-windup strategies presented in [5,15,44,48]. Also, a constrained model predictive controller (MPC) is considered where constraints are set to deal with the saturating issue. Furthermore, in order to test the applicability of the proposed controller to time-delay constrained systems, an experiment was performed for temperature control in a neonatal intensive care unit. 2. Preliminaries Consider a discrete-time LTI system P(z) described by the following equations



P (z ) ∼ 1.1. The simplified approach for the control of time-delay systems In [40], simple tuning rules were proposed for the FSP applied to the control of stable, integrative, and unstable first-order plus dead-time processes. The primary controller is free from integral action, differently from the traditional FSP, and ensures a good trade-off among disturbance rejection, robustness, and noise attenuation. In [41], the results obtained in [40] were extended to the case of multiple-delay SISO systems of any model order; this is referred to as the Simplified Dead-Time Compensator (SDTC), which was recently further explored in [39]. Works above, however, do not consider the design for delay-free or input saturated systems. 1.2. Contribution A gap has been noticed in the control field, whereby there does not appear to be a simple approach to design a linear controller for both the time-delay and delay-free case. This paper addresses this by developing the SDTC for time-delay and delay-free SISO systems and, also, providing LMIs for the design of anti-windup compensation if saturating actuators are present; this was not explored in [41] or [39]. To summarize, this work presents: • A new methodology for linear controller design which deals directly with actuator saturation. The key feature of this scheme is that it employs a predictor-based structure that implements integral action implicitly, thereby avoiding integrator wind-up traditionally associated with actuator saturation. • A unified framework which can be used to deal with the design of controllers for systems with actuator saturation, both with and without time-delays. • By means of an analysis structure which decouples the system into linear and nonlinear loops, it is shown that stability of the system under actuator saturation does not depend on the delay (which is compensated by the conditioning scheme) or the robustness filter. • Such nice aspects allow for the design of the controller in a three-degree of freedom fashion, such that set-point tracking, disturbance rejection, and anti-windup compensation are independently tuned. Therefore, although the whole proposed controller structure (including the anti-windup filter) can be seen as one entity, the tuning is realized in three distinct phases.

x p (k + 1 ) = Ax p (k ) + B[ua (k − d ) + w(k − d )], y(k ) = Cx p (k ) + D[ua (k − d ) + w(k − d )] + n(k ),

(1)

where x p (k ) ∈ Rn is the state vector, ua (k ) ∈ R is the actual plant input, y(k ) ∈ R is the measured output, w(k ) is a matched input disturbance, and n(k) is due to measurement noise, respectively. Matrices A ∈ Rn×n , B ∈ Rn , C ∈ Rn , D ∈ R, and delay d ≥ 0 are all constant and known. Following assumptions are taken: Assumption 1. The pair (A, B) is controllable. Assumption 2. The unknown disturbance signal is bounded by |w(k )| < Dw , and is locally integrable. Assumption 3. The process input, ua (k), is given by ua (k ) = sat (u(k )), where u(k) is the computed control action and the saturation function is defined as

sat (u(k )) = sign(u(k )) × min{|u(k )|, |u¯ |}, u¯ > 0,

(2)

where u¯ denotes the control amplitude bound. This work makes use of some critical conditions which are necessary for anti-windup synthesis with guaranteed regions of stability. Therefore, they are briefly reviewed as follows. 2.1. Deadzone operator and sector conditions Firstly, let us define the deadzone function as

Dz(u(k )) = u(k ) − sat (u(k )),

(3)

and the following inequality

Dz(u ) W[ϕ − Dz(u )] ≥ 0,

(4)

known as the generalized or “modified” sector condition, which was introduced in [31] in order to enlarge the region of attraction of closed-loop saturated systems. Such condition also allowed for the gathering of true LMI conditions for the design of controllers with guaranteed regions of stability under saturating actuators. Consider the classical case, with ϕ = Eu and the associated 1 set given by S(u, u¯ ) = {u ∈ reals; −ηu¯ ≤ u ≤ ηu¯ }, where η = 1−E , 0 ≤ E < 1 ∼∈ reals. Then, if u ∈ S, inequality (4) with ϕ = Eu holds. Furthermore, for SISO systems, the deadzone function is said to belong to Sector [0, E] if inequality (4) is satisfied for some some scalar W > 0.

Please cite this article as: T.A. Lima, M.P. de Almeida Filho and B.C. Torrico et al., A practical solution for the control of time-delayed and delay-free systems with saturating actuators, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.06.012

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3

2.2. Quadratic Lyapunov function Consider the discrete nonlinear system

x(k + 1 ) = f (x(k )). Then the origin of the above system is locally asymptotically stable if there exists a positive definite function (x(k)) such that

(x(k )) := (x(k + 1 )) − (x(k )) < 0, ∀x(k ) ∈  , where  is a region of the state-space. Furthermore, if  is the whole state-space, the system is globally asymptotically stable. In the work here, the Lyapunov function is chosen as

(x(k )) = x (k )Qx(k ),

Fig. 1. Proposed unified anti-windup scheme.

where Q = Q is a positive definite matrix. Furthermore, making use of the generalized sector condition, it is possible to define a modified functional for which regional stability of saturated systems holds, given by

 (x(k )) := (x(k )) + 2Dz(u ) W[Eu − Dz(u )] ≤ 0, such that the ellipsoid ε (Q, 1 ) = {x ∈ Rn ; x Qx ≤ 1} is a region of asymptotic stability (RAS) of the system f(x(k)). For further details, one is referred to the work in [31]. 2.3. D-stability Although the Lyapunov function previously defined is to guarantee stability conditions, it might be desired to some dynamical information to the anti-windup design. done in this work by using D-stability regions. So let us the following definition from [4].

enough include This is borrow

Definition 1. Let D be a domain on the complex plane, which is symmetric about the real axis. Then, a matrix A ∈ Rn×n is said to be D-stable if Fig. 2. Mismatch equivalent structure for analysis.

λi ∈ D, i = 1, 2 . . . n, where λi are the n eigenvalues of A. In this work, the following particular case of D-stability is employed

3. Unified anti-windup controller

D = Hζa ,ζb = {σ + jω

The proposed unified anti-windup SDTC (AW-SDTC) control structure to deal with both time-delayed and delay-free SISO systems with saturating actuators is depicted in Fig. 1, where Pn (z ) = Gn (z )z−dn is the nominal process, with Gn (z) and dn being the fast model and nominal dead time, respectively. Plant P (z ) = G(z )z−d is the model of the real process. Additive uncertainty enters the system such that P (e jω ) = Pn (e jω ) + P (e jω ). M(z) is the anti-windup conditioning filter, while V(z) and S(z) are calculated to improve set-point tracking, disturbance rejection and robustness properties of the controller. By using identity (3), where Dz(u(k )) = u˜ (k ) corresponds to a deadzone nonlinearity, the AW-SDTC scheme in Fig. 1 is redrawn in a equivalent form depicted by Fig. 2. The equivalency between the two structures is valid if and only if S(z ) = R(z ) + Gn (z )(F (z ) − V (z )z−d ), H (z ) = [1 + R(z ) + Gn (z )F (z )]−1 . From Fig. 2 note that the system is conveniently divided in three parts, namely the linear loop, the nonlinear loop and the disturbance filter, with output given by y(k ) = ylin (k ) − ynl (k ). This kind of decoupled structure is widely used for design and analysis of closed-loop systems under input saturation (see [20] and [43] for instance). Some of the aspects which make this structure attractive for the work herein are described as follows. First, note that the linear loop determines the system behavior in the absence of control saturation. This allow us to design the system to obtain a desired response in the nominal case, i.e. Pn (z ) = P (z ) and u(k ) = ua (k ). For this, the linear loop is tuned in a two-degree of freedom fashion, where R(z) and F(z) can be used to achieve a desired set-point tracking response, whilst V(z) can be

| ζa < σ < ζb },

(5)

such that the following proposition holds (see [4] for proof). Proposition 1. The matrix A ∈ Rn×n is Hζa ,ζb -stable if and only if there exists a positive definite matrix Q = Q such that



A Q + QA − 2ζa Q < 0, A Q + QA − 2ζb Q > 0.

(6)

2.4. l2 gain A nonlinear system with input ψ (k) and output (k) has an l2 gain of γ when

∀ψ ∈ l2 ,  < γ ψ + θ

(7)

where · denotes the standard Euclidean vector norm, and θ is a positive constant. The l2 gain, defined in equation (7), represents a bound on the root mean square energy gain of a nonlinear system. In order to ensure l2 gain of the nonlinear saturated system for all ψ while also holding stability guarantees, the following condition must be true

(x(k )) := (x(k )) +   − γ 2 ψ  ψ + 2Dz(u ) W[Eu − Dz(u )] ≤ 0,

(8)

which is a combination of the quadratic Lyapunov functional from Section 2.2, the sector condition (4) and the term   − γ 2 ψ  ψ which is used to ensure a level of l2 performance. The reader is referred to the work in [10] to see a detailed explanation of condition (8).

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adjusted to deal with disturbance rejection, robustness and noise attenuation characteristics of the proposed controller. On the other hand, note that both the disturbance filter and the nonlinear loop depend on the anti-windup conditioning filter M(z). From relation (3), it is clear that when saturation events end, the output of the dead-zone nonlinearity u˜ (k ) becomes null. Nevertheless, possible windup effects caused by the saturation are not instantaneously over, as the states of the disturbance filter are not instantaneously null. Therefore, the disturbance filter is responsible for determining both speed and manner of recovery of the closed-loop system after saturation ceases. On what concerns stability, by considering that the linear loop is stable (which obviously should be by proper design of R(z), F(z) and V(z)), the stability of the proposed structure depends solely on the stability of the nonlinear loop. Thereby, it is evident that the design of the conditioning filter M(z) should tackle both performance and stability issues of the system under saturation.

3.1. Tuning of the linear loop The input-output transfer functions for the nominal case are given as follows

Hyr (z ) =

Y (z ) Kr Pn (z ) = , Re f ( z ) 1 + R ( z ) + Gn ( z )F ( z )



(9)



Pn (z )V (z ) Y (z ) = Pn (z ) 1 − , Q (z ) 1 + R ( z ) + Gn ( z )F ( z )

(10)

Hun (z ) =

U (z ) −V (z ) = , N (z ) 1 + R ( z ) + Gn ( z )F ( z )

(11)

where U(z), Y(z), Ref (z), N(z) and Q(z) refer to the Z-transform of the control action ulin (k), process output ylin (k), reference ref (k), measurement noise n(k), and input load disturbance q(k), respectively.

| 1 + R(e jωTs ) + Gn (e jωTs )F (e jωTs ) | > δ P (e j ω ) | Gn (e jωTs )V (e jωTs ) |

solve an equation of the type c = ρ , with

⎡

1

(12)

is satisfied, where Ts is the sampling time, 0 < ω < π /Ts , the term δ P (e jω ) is an upper bound for the norm of the multiplicative uncertainty δ P(ejω ), and Ir (ejω ) is defined as the robustness index. The multiplicative uncertainty enters the system such that P (e jω ) = Pn (e jω ) + P (e jω ) = Pn (e jω )[1 + δ P (e jω )]. It is important to highlight from Eqs. (9) to (12) that Kr , R(z) and F(z) can be tuned in order to obtain a desired set-point tracking, while the filter V(z) is set to obtain a desired trade-off between disturbance rejection and robustness.

0

⎢ ⎢a 1 ⎢. ⎢. ⎢. ⎢ ⎢ a =⎢ ⎢ 0n ⎢ ⎢. ⎢ .. ⎢ ⎢ ⎣0  ⎡

r0

0 .. .

a1 .. . an

1 a1

0 .. . 0

a2 .. . an

⎤

⎢ .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢r ⎥ c = ⎢ n−1 ⎥, ρ ⎢ f0 ⎥ ⎢ . ⎥ ⎣ .. ⎦ fn−1

...

1

0

0



n ⎡

b0

0

b1 .. .

b0

bn 0 .. .

0 .. .

b1 .. . bn

b0 b1

0 .. . 0

b2 .. . bn

0 0



...



0



n

⎤



⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (15)

0 ⎢s1 − a1 ⎥ ⎢ . ⎥ ⎢ .. ⎥ ⎢ ⎥ ⎥, s − a =⎢ n n ⎢ ⎥ ⎢ sn+1 ⎥ ⎢ . ⎥ ⎣ . ⎦ . s2n−1

where  is a non-singular 2n square matrix, a1 . . . an and b0 . . . bn are the coefficients of

Gn ( z ) =

Hyq (z ) =

Ir (e jω ) =

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b0 + b1 z−1 + b2 z−2 . . . bn z−n , 1 + a1 z−1 + a2 z−2 . . . an z−n

(16)

and s1 . . . s2n−1 are the coefficients of the desired characteristic polynomial

1 + s1 z−1 + s2 z−2 + · · · + s2n−1 z−2n+1 = (1 − α1 z−1 )(1 − α2 z−1 ) . . . (1 − α2n−1 z−1 ).

(17)

Therefore, closed-loop poles α = [α1 . . . α2n−1 ], with 0 ≤ |α i | < 1, are the set-point tracking tuning parameter. For example, if faster set-point is desired, than smaller values of α i can be chosen, and vice-versa. The static gain Kr is then computed to yield zero steady-state error, given by

Kr =

1 + R ( 1 ) + Gn ( 1 )F ( 1 ) . Pn (1 )

(18)

3.1.2. Disturbance rejection and robustness Robustness filter V(z) is tuned aiming: (i) to reject step-like disturbances applied in the control signal; (ii) to eliminate the modes of the plant model Pn (z) from the disturbance rejection Hyq (z) response; (iii) to obtain a desired trade-off between robustness and disturbance rejection. Such goals can be met by applying the following filter

V (z ) =

n  1 vi z−i , (1 − β z−1 )n+1

(19)

i=0

3.1.1. Set-point tracking Tuning rules for the primary controller, defined by Kr , R(z), and F(z) are established in this subsection. For this, consider feedback FIR filters R(z) and F(z) as

R(z ) = r0 + r1 z−1 + r2 z−2 + · · · + rn−1 z−n+1 , F (z ) = f0 + f1 z−1 + f2 z−2 + · · · + fn−1 z−n+1 ,

(13)

where n is the number of poles of the process model and 0 < |β | < 1 is the filter tuning parameter. For the first objective (i) Hyq (z) must be zero at steady-state, i. e.,

V (1 ) =

1 + R ( 1 ) + Gn ( 1 )F ( 1 ) . Pn (1 )

(20)

For the second objective (ii), the poles pi of the process model must be canceled from Hyq (z), thus

(14)

where n is the order of the fast model Gn (z). Coefficients of R(z) and F(z) are obtained through standard pole placement design procedure in order to reach a desired set-point tracking response (9). In order to find the coefficients of R(z) and F(z), one must



Pn (z )V (z ) 1− 1 + R ( z ) + Gn ( z )F ( z )





= 0,

(21)

z= pi =1

Pn (z )V (z ) d 1− dz 1 + R ( z ) + Gn ( z )F ( z )



= 0.

(22)

z= pi =1

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Then coefficients vi are computed using (20)–(22). Such equations altogether set a linear system so that coefficients vi can be readily found. User adjustment of the β parameter can be made in order to establish a desired compromise between robustness and disturbance rejection, thus reach the objective (iii). From equation (12) note that V(z) appears in the denominator of Ir (e jω ), so that higher values of β provide enhanced robustness. On the other hand, according to equation (10) β also defines how fast disturbances can be rejected, which is faster for smaller values of β , thus illustrating the trade-off mentioned above. 3.2. Tuning of the nonlinear loop and the disturbance filter

M (z ) =

Dg ( z ) , (1 − ν1 z−1 )(1 − ν2 z−1 ) . . . (1 − νn z−1 )

(23)

with Gn (z ) = Ng (z )/Dg (z ). Note that the zeros of M(z) are equal to the poles of the process model while the ν i poles of M(z) can be defined in order to meet certain criteria. Such choice yield some interesting characteristics for the control strategy in this work. The equivalent disturbance filter is given by

Gn ( z )M ( z ) =

the goal of the proposed strategy is to design M(z) such that there are guaranteed regions of stability while also keeping desired output performance. In this work, the first goal is met by using classical Lyapunov theorem along with the generalized sector condition, while performance characteristics are introduced employing D-stability conditions (6), which allow the designer to allocate the poles of M(z) within a desired region. Herein, anti-windup design for both the regional and global stability cases is established using LMIs, which are suitable to assure stability conditions for stable, integrative and unstable open loop plants. Consider the state-space realization of the process fast model Gn (z), given by matrices A, B, C, D. The state-space equivalent representation of M(z) (23) is given as



Previous works have shown that most conditioning schemes for systems with actuator saturation can be interpreted as different choices of conditioning filter M(z) [46]. For the proposed control structure, such filter is defined as

Ng (z ) , (1 − ν1 z−1 )(1 − ν2 z−1 ) . . . (1 − νn z−1 )

so that the poles of Gn (z) are canceled by the numerator of M(z); therefore the disturbance filter is always stable with user-defined poles ν i . On the other hand, poles ν i define system recover manner after saturation while also playing a role on the stability of the nonlinear loop. Then, it is clear that there exists a trade-off between performance and stability of the saturated system. For |ν i | → 0, the system should present faster recover from saturation events. On the other hand, for poles |ν i | → 1 the nonlinear loop region of stability should increase. From Fig. 2, note that due to uncertainties in the system, the filter V(z) plays some influence in both the linear and the nonlinear loops. Even though this is true, notice that both the linear and nonlinear loops are decoupled, i.e., their responses are not feedbacked to each other. Furthermore, since robustness filter V(z) is previously designed to deal with uncertainty and yield robustness characteristics of the linear loop, the design of the anti-windup conditioning filter M(z) is done considering the nominal system behavior, i.e., P (z ) = 0. 3.2.1. Anti-windup synthesis Works using the mismatch equivalent representation (Fig. 2) often consider that the goal of the anti-windup design is to maintain the response of the system as close as possible to the response of the linear system, that is, the response of the system had saturation not been present. Moreover, previous studies have proven that global stability of the closed-loop in the presence of actuator saturation cannot be achieved by a bounded input in the case of unstable processes [14,34]. In the case of critically stable plants, closed-loop stability can only be achieved with nonlinear control laws [28]. Therefore, another common goal in anti-windup design is to seek a compensator which enlarges the region of attraction of the closed-loop system. Consider the analysis structure in Fig. 2. As previously explained, all the poles of the disturbance filter should be within the unit circle, therefore considering that the linear loop is stable through robust condition (12), stability under actuator saturation depends only upon the stability of the nonlinear loop. Therefore,

5

M (z ) ∼

 

A + BKB . K I

(24)

where the ν i poles of the conditioning filter M(z) are equal to the eigenvalues of A + BK. Therefore, computation of M(z) is accomplished by finding a suitable matrix K ∈ Rn which attends to pre-specified stability and performance design requirements. Since M (z ) − I is always strictly proper, no direct feedthrough exists from u˜ (k ) to ud (k), thus the system is well conditioned. In order to find the gain K, consider the τ pn : ulin → yd defined as

⎧ x(k + 1 ) = (A + BK )x(k ) + Bu˜ (k ) ⎪ ⎪ ⎨y (k ) = (C + DK )x(k ) + Du˜ (k ) τ pn  d , ⎪ud (k ) = Kx(k ) ⎪ ⎩ u˜ (k ) = f (ulin (k ))

(25)

where f (ulin (k )) = Dz(ulin (k ) − ud (k )). It is important to highlight that this mapping is defined for the nominal case, i.e. P (z ) = 0 in Fig. 2. Therefore, although the presence of delays indeed influence the closed-loop behavior in the uncertain case, they do not affect the anti-windup design. Then, the following theorems establish conditions to find the anti-windup conditioning filter M(z) with guaranteed regions of stability. Theorem 1. There exists an anti-windup conditioning filter M(z) if there exist matrices X > 0, J > 0, L, and scalar γ > 0 such that LMIs

⎡−X

−L E −2J   

⎢  ⎢  ⎣  



and

X 

0 E −γ I  

XC + L D JD 0 −γ I 

⎤

XA + L B JB ⎥ ⎥ < 0, 0 ⎦ 0 −X

(26)



L 2 ¯ 2 ≥ 0, η u

(27)

are feasible for η = 1/(1 − E ). Additionally, the poles of the conditioning filter M(z) will be within a Hζa ,ζb -region in the unit circle (see Fig. 3) if following LMIs hold along with (26) and (27)



XA + L B + AX + BL − 2ζa Q < 0, XA + L B + AX + BL − 2ζb Q > 0.

(28)

Then, a suitable K achieving finite l2 gain γ of the τ pn operator is given by K = LX. Moreover, the ellipsoid ε (Q, 1 ) = {x ∈ Rn ; x Qx ≤ 1}, with Q = X−1 , is a region of asymptotic stability (RAS) of the nonlinear loop in Fig. 2. Proof. Using the previously defined Lyapunov functional, sector condition, and l2 gain, a necessary condition for stability is written as [20]

(x(k )) + yd  yd − γ 2 ulin  ulin + 2u˜W[Eu − u˜] < 0.

(29)

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Corollary 1. The origin is globally asymptotically stable for the closed-loop system in Fig. 1 if there exist matrices X > 0 and J > 0 such that LMI

⎡−X ⎢  ⎢  ⎣  

−L −2J   

0 I −γ I  

XC + L D JD 0 −γ I 

⎤

XA + L B JB ⎥ ⎥<0 0 ⎦ 0 −X

(36)

is feasible. Furthermore, the poles of the conditioning filter M(z) will be within a Hζa ,ζb -region in the unit circle (see Fig. 3) if (36) holds along with (28). Then, a suitable K gain is given by K = LX. Proof. Proof of Corollary 1 is straightforward. In the global stability case E = I, then (26) becomes (36).  4. Simulation results

Fig. 3. Hζa ,ζb -region inside unit circle on the complex plane.





Then, by defining δ = x u˜ ulin  , we can write inequality (29) in an equivalent quadratic form as in (30), where ϒ11 = (A + BK ) Q(A + BK ) − Q + (C + DK ) (C + DK ) and ϒ12 = (A + BK ) QB + (C + DK ) D − K EW.

δ

 ϒ11  

ϒ12

0 WE −γ 2 I

B QB − 2W 



δ < 0.

(30)

By applying Schur complement twice to (30) it is obtained

⎡−Q ⎢  ⎢  ⎣  

−K EW −2W   

0 WE −γ I  

C + K D D 0 −γ I 

⎤

A + K B B ⎥ ⎥ < 0. 0 ⎦ 0 −Q−1

4.1. Delay-free examples

(31)

Using diag(Q−1 , W−1 , I, I, I ) to apply a congruence transformation with (31) it is obtained

⎡−Q−1 −Q−1 K E 0 Q−1 (C + K D ) Q−1 (A + K B )⎤ −2W−1 E W−1 D W−1 B ⎢  ⎥ ⎢  ⎥ < 0.  −γ I 0 0 ⎣ ⎦    −γ I 0 







−Q−1

(32) Q−1 ,

W−1 ,

Finally, by defining new variables X = J= and by making the change of variables L = KQ−1 , LMI (26) is obtained, with X > 0, J > 0 and γ > 0. Then, LMI (27) is used in order to help estimate the ellipsoidal region of stability  = {x ∈ Rn : xQx ≤ 1}. In order to ensure that the control signal is within the Sector[0, E], when ulin = 0, it is necessary that [30]

|ud | = |Kx| ≤ η2 u¯ 2 .

(33)

For x ∈  , this holds if

x K Kx ≤ x Qxη2 u¯ 2 ,

(34)

which is equivalent to



Q 



K . η2 u¯ 2

Some works dealing with the saturating issue point out that although it is possible to achieve global stability for open-loop stable systems, the local stability relaxation can often be preferred in order to obtain improved local performance (see [20], for instance). Furthermore, systems usually operate within a desired region of the space, so that the local assumption can often be enough. Therefore, the local approach was preferred for all the examples, even the open-loop stable ones.

(35)

LMI (27) is then obtained by applying a congruence transformation to (35) using diag(Q−1 , I ). Consider now equations (6). In order to obtain (28), replace A by A + BK, apply congruence transformation with Q−1 and make X = Q−1 , L = KQ−1 . This completes the proof.  Furthermore, for the case of stable processes, it is useful to establish the following corollary.

This subsection presents simulation results for systems without time-delay recently presented in [44] and [15]. 4.1.1. Example 1 - Comparison with nonlinear anti-windup Consider the example recently used in [44] to illustrate a nonlinear anti-windup technique where the plant is a third-order SISO system with input constraint given by u = 1. By using a zero-order hold and sampling time of T = 0.1s the following discrete-time transfer-function model is obtained:

P (z ) =

−0.1045z−1 + 0.1216z−2 − 0.03547z−3 . 1 − −2.309z−1 + 1.656z−2 − 0.3465z−3

The AW-SDTC was initially tuned to obtain set tracking response similar with the compared controller leading to α = [0.8 0.3 0 0 0] and β = 0.9. Then, for the tuning of the anti-windup conditioning filter M(z), Theorem 1 was used with E = 0.855, ζb = 0 and ζa = 1, yielding ν = [0.777 0.384 0.096] and γ = 87.3443. Fig. 4 shows output and control signals for a sequence of steps in the reference. From the output graph, one may notice that the AW-SDTC controller does not exhibit overshoots and undershoots while the nonlinear controller from [44] does. Hence, settling time is much smaller for the proposed controller herein. It is important to highlight that the control signal of the AW-SDTC does not present the high-frequency ripples of the controller from [44], thus being better behaved. Furthermore, it is evident that the AW-SDTC controller recovers faster from saturation events while presenting better output results. This behavior will be shown to exist in all further simulation examples. It should be highlighted that the better behavior of the approach herein comes with the cost of providing regional stability only, whereas approach from [44] guarantees global stability. This situation illustrates the tradeoff between local stability relaxation and performance that was mentioned in the beginning of Section 4. The region of stability obtained by using Theorem 1 is shown in Fig. 5.

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note that the AW-SDTC is the first controller to reject the input disturbance completely. As in [15], a second simulation (shown in Fig. 7) was realized in order to evaluate performance against output disturbance. Although both the AW-SDTC and the controller from [15] seem to present similar rejection of the disturbance, there is a slight advantage to the AW-SDTC, which completely rejects the disturbance faster and in a less oscillatory manner. It is worth to mention, once more, that the proposal herein preferred to use the local assumption for stability in order to prioritize performance. On the other hand, other approaches as the one from [45] are suitable to guarantee global stability for this plant. The ellipsoidal region of stability for this example obtained employing Theorem 1 is shown in Fig. 8. 4.2. Time-delay examples

Fig. 4. Simulation results for example 1.

This subsection presents two simulation case studies using the integrative plant presented in [48] and the unstable process recently presented in [5]. Comparison with a constrained MPC is also performed. Evaluated controllers are compared under input saturation, dead-time uncertainties, input and output disturbances, and step reference variations. 4.2.1. Example 3 - Integrating case This example presents comparative simulation results obtained with the following controllers: an input constrained MPC from [3], controller reported in [48], the recently published strategy in [5] and the proposed AW-SDTC. For this purpose, the following process model presented in [48] is considered:

P (s ) =

e−5s . s

The input limit is given by u¯ = 1. Parameters of the controller from [48] are defined as therein. The MPC uses the following model to compute predictions:

(1 − q−1 )y(t ) = 0.1u(t − 1 − d ) +

Fig. 5. Ellipsoidal region of stability for example 1.

4.1.2. Example 2 - Comparison with 4-degree-of-freedom anti-windup Consider the second-order SISO plant

G (s ) =

0.5s2 + 0.5s + 1 , s2 + 0.2s + 0.2

with input constraint such that u = 0.5. By using a zero-order hold and sampling time of T = 0.1s the following discrete-time transfer-function model is obtained:

P (z ) =

−0.5 + 0.941z−1 − 0.4549z−2 . 1 − 1.978z−1 + 0.9802z−2

In this example, comparison with the recent 4-degree-offreedom anti-windup from [15] and the classical low-order IMC anti-windup scheme from [45] are presented. Controllers from [15] and [45] are defined as therein. The AW-SDTC controller is  tuned with α = 0.947 + 0.128i 0.947 − −0.128i 0 , β = 0.7





and ν = 0.945 + 0.129i 0.945 − −0.129i . The poles of the antiwindup conditioning filter were found with E = 0.45, ζb = 0.946 and ζa = 0.948, yielding γ = 9.4416. Fig. 6 shows the results for a step change in the reference, followed by a negative input load disturbance of magnitude 0.5. For the set-point tracking the AW-SDTC and the controller from [15] exhibit very similar responses. However, it is possible to

(1 − σ q−1 )2 e (t ), (1 − q−1 ) n

(37)

where q−1 is the backward shift operator, σ = e−T /2.4 , d = Ln /T , T = 0.2 s is the sampling time, and en (t) is a white noise. The MPC tuning parameters are: the limits of the prediction window N1 = d + 1, N2 = d + 100, the control horizon Nu = 70, and the control weights defined as λ1 = 0 and λi = 250, i = 2, . . . , Nu . The MPC was tuned in order to present a step response quite similar to the controller from [48] in the nominal case without input saturation. For the AW-SDTC controller two different tunings are considered. The first one, named AW-SDTC1 , was tuned aiming to improve disturbance rejection under input saturation, leading to the following parameters α = 0.955, β = 0.865 and ν = 0.9548. The poles of the anti-windup conditioning filter were found with E = 0.7, ζb = 0.95, yielding γ = 113.1629. Figs. 9 and 10 show simulation results for a step reference. An input disturbance pulse with amplitude of −1.5 was applied from t = 40 s to t = 50 s. A step input disturbance with amplitude of −0.5 was applied at t = 100 s. Fig. 9 shows simulation results without uncertainties. As can be observed all the controllers follow the reference in a similar way. However, in the case of both input pulse disturbance and input saturation, the proposed AW-SDTC1 controller is the only one without overshoot at recovery. For the step input disturbance performance of all controllers are nearly the same as control action does not saturate. Fig. 10 shows simulation results using +20% dead-time uncertainty. It can be seen that the proposed AW-SDTC1 retains better output performance for pulse disturbance while may be taken

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2.5 2 1.5 1 0.5 0

0

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35

40

0

5

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40

0.2 0 -0.2 -0.4

Fig. 6. Simulation results for example 2 - Set-point tracking and input disturbance.

1

0.5

0

-0.5

0

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35

40

0

5

10

15

20

25

30

35

40

0.4 0.2 0 -0.2

Fig. 7. Simulation results for example 2 - Output disturbance rejection.

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10

3

5

Output

4

2

9

8.05

0

8 7.95

1

-5 0

50

100

150

0 1

Control Signal

-1 -2 -3

0.5

0 Flesch et al. [5] AW-SDTC 2

-4

0

-4

-3

-2

-1

0

1

2

3

50

4

100

150

Time(s) Fig. 11. Simulation results for example 3 (no uncertainties).

Fig. 8. Guaranteed ellipsoidal region of stability for Example 2. 10

Output

10

Output

5

5

0

Flesch et al. [5] AW-SDTC 2

Camacho and Bordons [3] Zhang and Jiang [48]

0

-5

AW-SDTC 1

0

50

0

50

100

150

100

150

-5 0

50

100

150 1

Control Signal

Control Signal

1

0.5

0.5

0

0

Time(s) 0

50

100

150

Time(s)

Fig. 12. Simulation results for example 3 (5% dead-time uncertainty).

Fig. 9. Simulation results for example 3 (no uncertainties).

Output

10

5 Camacho and Bordons [3 ] Zhang and Jiang [48] AW-SDTC 1

0

-5 0

50

0

50

100

150

100

150

Control Signal

1

0.5

0

Time(s) Fig. 10. Simulation results for example 3 (20% dead-time uncertainty).

as equivalent to the compared controllers for step disturbance. It is important to highlight that the proposed AW-SDTC achieves better performance than the MPC, which is a much more complex controller. In order to establish comparison with the recently published anti-windup strategy in [5], the AW-SDTC2 was tuned to yield a similar set-point tracking response by using the following parameters α = 0.5, β = 0.3 and ν = 0.493. The poles of the anti-windup conditioning filter were found with E = 0.96, and ζb = 0.45, yielding γ = 560.4146. Fig. 11 shows the results for the nominal case, while Fig. 12 presents the results using +5% dead-time uncertainty. Notice that, on the contrary to the controller from [5], the AW-SDTC2 does not present overshoot for either pulse or step disturbances in the nominal case. Also, note (from Fig. 12) that even in the uncertain case, the proposed controller had improved performance against the compared controller. 4.2.2. Example 4 - Unstable case This example also compares the proposed AW-SDTC with the controller proposed in [5]. The controlled process, which represents an isothermal chemical reactor, has been studied

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0.1

0.1

0.05

0.05

0

0 0

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200

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400

500

600

700

800

0

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800

0.05

0.05

Flesch et al. [5] AW-SDTC

Flesch et al. [5] AW-SDTC

0

0

-0.05

-0.05 0

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400

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700

0

100

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500

600

700

800

800

Fig. 14. Simulation results for unstable case with 10% dead-time uncertainty. Fig. 13. Example 4 - Simulation results for unstable case. Table 1 Output ISE and control variance for dead-time simulation examples.

in many works and is originally found in [16]. The following differential equation gives the nonlinear model

k1 C dC Q = (C f − C ) − , dt V ( k2 C + 1 )2 where Cf and C are the process control and output variables, respectively, given in mol/L. The operating parameters are given as k1 = 10 L/s, k2 = 10 L/mol, V = 1 L and Q = 0.03333 L/s. The bound on the process input is given by u¯ = 0.05. An unstable linearized model obtained around the steady-state condition with Cf = 3.288 mol/L and C = 1.316 mol/L is given by

3.433 P (s ) = e−20s , 103.1s − 1 where a measurement delay of L = 20 s is considered. By using a zero-order hold and sampling time of T = 5 s the discrete-time model is obtained as

P (z ) =

0.1706z−1 z−4 . 1 − 1.0497z−1

The controller from [5] is defined as therein. Once more, the AW-SDTC was tuned to obtain set tracking response similar to the compared controller. However, faster disturbance rejection was aimed with AW-SDTC controller tuned with α = 0.68, β = 0.65 and ν = 0.6804. The pole of the anti-windup conditioning filter was found with E = 0.5, and ζb = 0.65, yielding γ = 38.2798. The process nonlinear model is used in all simulations for a step reference of 0.1 mol/L at time t = 0 s, in addition to application of input and output disturbances of amplitude 0.015 mol/L and 0.02 mol/L in t = 200 s and t = 500 s, respectively. From Fig. 13, one may note the controller from [5] exhibits slower rejection for both input and output disturbances and undershoot in the rejection of the output disturbance due to the dominant zero imposed by its primary controller. On the other hand, the proposed AW-SDTC does not present any of these issues which may become inconvenient. It may be interesting to comment that the proposed AW-SDTC overall better performance might be related to the addition of the tuning parameters ζ a and ζ b , which allow fine adjustment of the poles of the anti-windup conditioning filter. On the other hand, the anti-windup technique from [5] does not use any tuning parameters. Consider now that the process measurement delay is actually 10% larger, L = 22 s. As can be seen in Fig. 14, the AW-SDTC maintains its performance characteristics related to disturbances rejection even under such circumstances, while issues mentioned

Example 3 Dead-time uncertainty (%)

0 2 5 10 20

Output Signal ISE

Control Signal Variance

AW-SDTC2

Ref. [5]

AW-SDTC2

Ref. [5]

1540.40 1543.88 1543.11 1502.72 2551.09

1550.51 1556.48 1563.34 1496.08 2619.83

0.1449 0.1475 0.1561 0.4308 0.7154

0.1549 0.1573 0.1652 0.3600 0.6131

Example 4 Dead-time uncertainty (%)

0 10 20 40 70

Output Signal ISE

Control Signal Variance

AW-SDTC

Ref. [5]

AW-SDTC

Ref. [5]

0.3953 0.4122 0.4609 0.5907 0.8141

0.3961 0.4140 0.4394 0.5117 0.8454

0.00040 0.00043 0.00059 0.00085 0.00083

0.00042 0.00044 0.00049 0.00065 0.00080

earlier for the nominal case remains unchanged to the comparing controller. It is important to highlight that for the proposed strategy such results are obtained by using simple gains in the primary controller, defined by f0 and Kr , and a first-order robustness filter. On the other hand, the strategy from [5] applies a PI as the primary controller, a first-order reference filter and a second-order robustness filter, which clearly shows a design advantage in favor of the AW-SDTC algorithm as it becomes much simpler than the compared one. Table 1 summarizes the results for other modeling uncertainty cases in Examples 3 and 4. The ISE of the output signal and the control signal variance were calculated. In most scenarios, the proposed controller presents betters results. 5. Experimental results This section intends to show the usefulness of the proposed AW-SDTC, applied for temperature control of a neonatal intensive care unit (NICU), depicted in Fig. 15. The plant has been identified using a step-test identification procedure [22], whose model is given by

Pn (s ) =

0.483e−10.2 s , 254.7s + 1

where the time is measured in minutes and the control is constrained within the range from 0 to 100 % as it is the current

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leading to an undershoot of 1.2 ◦ C. However, such disturbance was adequately rejected as set-point was achieved again within 30 min. Note that even though the control signal saturated again during the disturbance rejection phase, no windup issues emerged, which further demonstrates the controller ability to deal with the saturation condition. 6. Conclusion

Fig. 15. Picture of the neonatal intensive care unit.

30

25

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150

100

50

An anti-windup strategy applied to the control of both timedelayed and delay-free systems of any order with saturating actuators has been presented. LMIs which incorporate D-stability regions have been used to synthesize the anti-windup compensator in order to guarantee good performance of the design. Different from most previous saturation related works, this paper addresses the problem of saturation in a practical manner such that problems as tracking, robustness and disturbance rejection are explicitly considered. Furthermore, it has been shown that disturbance rejection response of the proposed AW-SDTC during saturation events presents better results when compared to anti-windup schemes previously proposed in the literature. The developed experiment effectively assures that the proposed controller does not present windup issues, while keeps the excellent performance of a conventional SDTC. Due to its inherent simplicity, the proposed scheme is supposed to present great potential to be implemented in practical applications, therefore being quite useful for control engineers. The discrete-time nature of the controller should provide a secure computer-based implementation. Acknowledgment

0 0

50

100

150

Fig. 16. Experimental results: temperature control of a NICU.

flowing through a heating resistor, which is limited by its maximum power. The following discrete-time model is obtained using a sampling time of T = 0.2 min

Pn (z ) =

0.0 0 03791 −51 z . z − 0.9992

The AW-SDTC control structure was tuned with α = 0.92 and β = 0.98 and ν = 0.9203. The pole of the anti-windup conditioning filter was found by running Corollary 1 with ζa = 0.925 and ζn = 0.915, thus global stability of the system has been guaranteed. It is important to notice that with this controller the nominal desired closed loop transfer function is chosen to obtain a much faster set-point tracking response than the open-loop plant. Furthermore, the robustness filter V(z) increases system robustness with a proper measurement noise attenuation. Experimental results using the proposed controller for a temperature set-point of 28 ◦ C are shown in Fig. 16. The initial temperature inside the NICU is 21.3 ◦ C, whilst the temperature of the room was kept around 18 ◦ C during the whole experiment. It is important to note that the acrylic dome which involves the NICU does not provide a good level of thermal isolation between the internal and external environments. Therefore, the external temperature is an unmodeled disturbance which is present during the whole experiment. Observe that, as expected, the output can track the reference without any oscillation. Furthermore, it is important to note that even though the plant input became saturated during the first 41 minutes, the controller did not present windup issues. In order to assess controller robustness, front portholes of the NICU were kept opened between t = 85 min and t = 95 min

The authors would like to thank Dr. Matthew C. Turner of the University of Leicester who provided insight and expertise that greatly assisted in the preparation of this paper. Also, financial support from the Brazilian funding agencies CAPES, CNPq, and FUNCAP is gratefully acknowledged. References [1] P. Albertos, P. García, Robust control design for long time-delay systems, J. Process Control 19 (10) (2009) 1640–1648. [2] K.J. Astrom, C.C. Hang, B.C. Lim, A new Smith predictor for controlling a process with a integrator and long dead-time, IEEE Trans. Autom. Control 39 (2) (1994) 343–345. [3] E.F. Camacho, C. Bordons, Model Predictive Control, Springer Verlag, 2004. [4] G. Duan, H. Yu, LMIs in Control Systems: Analysis, Design and Applications, Taylor & Francis, 2013. [5] R.C.C. Flesch, J.E. Normey-Rico, C.A. Flesch, A unified anti-windup strategy for SISO discrete dead-time compensators, Control Eng. Pract. 69 (2017) 50–60. [6] E. Fridman, A. Pila, U. Shaked, Regional stabilization and h control of time-delay systems with saturating actuators, Int. J. Robust Nonlinear Control 13 (9) (2003) 885–907. [7] S. Galeani, A.R. Teel, L. Zaccarian, Constructive nonlinear anti-windup design for exponentially unstable linear plants, Syst. Control Lett. 56 (5) (2007) 357–365. [8] P. García, P. Albertos, A new dead-time compensator to control stable and integrating processes with long dead-time, Automatica 44 (4) (2008) 1062–1071. [9] P. García, P. Albertos, T. Hagglund, Control of unstable non-minimum-phase delayed systems, J. Process Control 16 (10) (2006) 1099–1111. [10] G. Herrmann, M.C. Turner, I. Postlethwaite, Discrete-time and sampled-data anti-windup synthesis: stability and performance, Int. J. Syst. Sci. 37 (2) (2006) 91–113. [11] M. Huba, Comparing 2DOF PI and predictive disturbance observer based filtered pi control, J. Process Control 23 (2013) 1379–1400. [12] I. Kaya, A new smith predictor and controller for control of processes with long dead time, ISA Trans. 42 (2003) 101–110. [13] K. Kirtania, M.A.A.S. Choudhury, Set point weighted modified smith predictor for integrating and double integrating processes with time delay, J. Process Control 22 (3) (2012) 612–625. [14] J.B. Lasserre, Reachable, controllable sets and stabilizing control of constrained linear systems, Automatica 29 (2) (1993) 531–536. [15] Z.-S. Li, X.-Q. Mo, S.-J. Guo, W.-Y. Lan, C.-Q. Huang, 4-degree-of-freedom anti-windup scheme for plants with actuator saturation, J. Process Control 47 (2016) 111–120.

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[16] C.-T. Liou, C. Yu-Shu, The effect of nonideal mixing on input multiplicity in a CSTR, Chem. Eng. Sci. 46 (8) (1991) 2113–2116. [17] M.R. Mataušek, A.D. Micic´ , A modified smith predictor for controlling a process with a integrator and long dead-time, IEEE Trans. Autom. Control 41 (8) (1996) 1199–1203. [18] M.R. Mataušek, A.D. Micic´ , On the modified smith predictor for controlling a process with a integrator and long dead-time, IEEE Trans. Autom. Control 44 (8) (1999) 1603–1606. [19] M.R. Mataušek, A.I. Ribic´ , Control of stable, integrating and unstable processes by the modified smith predictor, J. Process Control 22 (1) (2012) 338– 343. [20] G.H. Matthew C. Turner, I. Postlethwaite, Anti-windup compensation using a decoupling architecture, in: G.G. S. Tarbouriech, A.H. Glattfelder (Eds.), Advanced Strategies in Control Systems with Input and Output Constraints, 2007, pp. 127–128,145,149. Berlin. [21] A. Megretski, L2 bibo output feedback stabilization with saturated control, IFAC Proc. Vol. 29 (1) (1996) 1872–1877. 13th World Congress of IFAC, 1996, San Francisco USA, 30 June - 5 July. [22] J.E. Normey-Rico, E.F. Camacho, Control of Dead-time Processes, Springer, Berlin, 2007. [23] J.E. Normey-Rico, E.F. Camacho, Simple robust dead-time compensator for first-order plus dead-time unstable processes, Ind. Eng. Chem. Res. 47 (14) (2008) 4784–4790. [24] F.S. de Oliveira, F.O. Souza, R.M. Palhares, Pid tuning for time-varying delay systems based on modified smith predictor, IFAC-PapersOnLine 50 (1) (2017) 1269–1274. 20th IFAC World Congress. [25] A.S. Rao, M. Chidambaram, Enhanced smith predictor for unstable processes with time delay, Ind. Eng. Chem. Res. 44 (22) (2005) 8291–8299. [26] A.S. Rao, M. Chidambaram, Analytical design of modified smith predictor in a two-degrees-of-freedom control scheme for second order unstable processes with time delay, ISA Trans. 47 (4) (2008) 407–419. [27] A.S. Rao, V.S.R. Rao, M. Chidambaram, Set point weighted modified smith predictor for integrating and double integrating processes with time delay, ISA Trans. 46 (1) (2007) 59–71. [28] S. Tarbouriech, G. Garcia, J.M.G. da Silva Jr., I. Queinnec, Stability and Stabilization of Linear Systems with Saturating Actuators, Springer, London, 2011. [29] R. Sanz, P. García, P. Albertos, A generalized smith predictor for unstable time-delay SISO systems, ISA Trans. 72 (2018) 197–204. [30] J.M.G. da Silva Jr., S. Tarbouriech, Anti-windup design with guaranteed regions of stability for discrete-time linear systems, Syst. Control Lett. 55 (3) (2006) 184–192. [31] J.M.G. da Silva Jr., S. Tarbouriech, Antiwindup design with guaranteed regions of stability: an lmi-based approach, IEEE Trans. Autom. Control 50 (1) (2005) 106–111.

[32] J.M.G. da Silva Jr., S. Tarbouriech, G. Garcia, Anti-windup design for time-delay systems subject to input saturation an lmi-based approach, Eur. J. Control 12 (6) (2006) 622–634. [33] O.J.M. Smith, Closed control of loops with dead-time, Chem. Eng. Progress 53 (1957) 217–219. [34] E.D. Sontag, An algebraic approach to bounded controllability of linear systems, Int. J. Control 39 (1) (1984) 181–188. [35] S. Tarbouriech, J.M.G. da Silva Jr., G. Garcia, Delay-dependent anti-windup strategy for linear systems with saturating inputs and delayed outputs, Int. J. Robust Nonlinear Control 14 (7) (2004) 665–682. [36] S. Tarbouriech, J.M.G. da Silva Jr., Synthesis of controllers for continuous-time delay systems with saturating controls via lmis, IEEE Trans. Autom. Control 45 (1) (20 0 0) 105–111. [37] A.R. Teel, A nonlinear small gain theorem for the analysis of control systems with saturation, IEEE Trans. Autom. Control 41 (9) (1996) 1256–1270. [38] A.R. Teel, N. Kapoor, The l lt;inf gt;2 lt;/inf gt; anti-winup problem: its definition and solution, in: Proceedings of the 1997 European Control Conference (ECC), 1997, pp. 1897–1902. [39] B.C. Torrico, M.P. de Almeida Filho, T.A. Lima, M.D. do N. Forte, R.C. Sá, F.G. Nogueira, Tuning of a dead-time compensator focusing on industrial processes, ISA Trans. 83 (2018) 189–198. [40] B.C. Torrico, M.U. Cavalcante, A.P.S. Braga, A.A.M. Albuquerque, J.E. Normey-Rico, Simple tuning rules for dead-time compensation of stable, integrative, and unstable first-order dead-time processes, Ind. Eng. Chem. Res. 52 (2013) 11646–11654. [41] B.C. Torrico, W.B. Correia, F.G. Nogueira, Simplified dead-time compensator for multiple delay SISO systems, ISA Trans. 60 (2016) 254–261. [42] M.C. Turner, G. Herrmann, I. Postlethwaite, Accounting for uncertainty in anti-windup synthesis, in: Proceedings of the 2004 American Control Conference, 6, 2004, pp. 5292–5297 vol.6. [43] M.C. Turner, G. Herrmann, I. Postlethwaite, Incorporating robustness requirements into antiwindup design, IEEE Trans. Autom. Control 52 (10) (2007) 1842–1855. [44] M.C. Turner, M. Kerr, A nonlinear modification for improving dynamic anti-windup compensation, Eur. J. Control 41 (2018) 44–52. [45] M.C. Turner, I. Postlethwaite, A new perspective on static and low order anti-windup synthesis, Int. J. Control 77 (1) (2004) 27–44. [46] P. Weston, I. Postlethwaite, Linear conditioning schemes for systems containing saturating actuators, IFAC Proc. Vol. 31 (17) (1998) 675–680. 4th IFAC Symposium on Nonlinear Control Systems Design 1998 (NOLCOS’98), Enschede, The Netherlands, 1–3 July. [47] P.F. Weston, I. Postlethwaite, Linear conditioning for systems containing saturating actuators, Automatica 36 (9) (20 0 0) 1347–1354. [48] M. Zhang, C. Jiang, Problem and its solution for actuator saturation of integrating process with dead time, ISA Trans. 47 (2008) 80–84.

Please cite this article as: T.A. Lima, M.P. de Almeida Filho and B.C. Torrico et al., A practical solution for the control of time-delayed and delay-free systems with saturating actuators, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.06.012