An Improved Antiwindup Design Using an Anticipatory Loop and an Immediate Loop

We present an improved antiwindup design for linear invariant continuous-time systems with actuator saturation nonlinearities. In the improved approach, two antiwindup compensators are simultaneously designed: one activated immediately at the occurrence of actuator saturation and the other activated in anticipatory of actuator saturation. Both the static and dynamic antiwindup compensators are considered. Sufficient conditions for global stability and minimizing the induced L 2 gain are established, in terms of linear matrix inequalities (LMIs). We also show that the feasibility of the improved antiwindup is similar to the traditional antiwindup. Benefits of the proposed approach over the traditional antiwindup and a recent innovative antiwindup are illustrated with well-known examples.


Introduction
Actuator saturation, which may cause loss in performance and even instability, is a ubiquitous and inevitable fact in any practical control systems.One general method to reduce adverse effects of saturation is the so-called antiwindup (AW).In AW design, a linear controller which does not take the saturation nonlinearity into account is first designed.Then, an AW compensator is added to ensure that stability is maintained (at least in some region near the origin) and that less performance degradation occurs than no AW is used [1].Such an approach has received much attention in recent several decades due to its intuitive motivation and its effectiveness in practice (see [1][2][3][4] and references therein).The traditional AW scheme is depicted in Figure 1, where P, C, and AW are the plant, the linear controller, and the AW compensator, respectively.Typically, the linear controller can be designed using the well-established linear control theory, and various methods for designing the AW compensators have been proposed in the literature (see [5][6][7][8] for some representative examples).
One of the main features of the traditional AW is that the AW compensator activated as soon as the saturation is encountered (nearly all the AW designs were based on this paradigm).In a pair of recent papers [9][10][11][12], Sajjadi-Kia and Jabbari investigated the effects of deferring the activation of the AW compensator and a so-called delayed AW was proposed.Based on the assumption that the linear controller possesses a reasonable amount of performance robustness, the motivation of the delayed AW is to apply the AW compensator until the closed-loop performance faces substantial decrease.With several examples, the authors showed that the delayed AW renders better performance than the traditional AW.Motivated by Sajjadi-Kia and Jabbari's work and considering the dynamical nature of the system, Wu and Lin proposed a new AW which is called as anticipatory AW [13][14][15][16].The anticipatory AW is opposite to the delayed AW, and its basic idea is to activate the AW compensator in anticipation of actuator saturation.It was shown in [14] that the anticipatory AW has the potential of leading to significant improvement in the closed-loop performance, in terms of both the transient quality in reference tracking and the region of stability.
A further modified AW is to simultaneously design two AW loops, one for immediate activation and the other for delayed activation [17,18].The main idea is to separate the saturated zone and obtain more aggressive and effective AW gains in lower levels of saturation.Using such a multiloop AW, one can obtain better response for moderate levels of saturation, but for high saturation levels, the improvement diminishes (see the numerical example in [18]).Motivated by Wu and Lin's work, we are hopeful to achieve a better performance by adding an anticipatory AW loop to the traditional AW scheme to take a "precautionary" action before actuator saturation occurs.It has been confirmed in [14] that a single anticipatory AW loop can work better than a single delayed AW loop.The main observation, here, is to combine an immediate (traditional) AW loop and an anticipatory AW loop to further improve the closed-loop performance.The proposed AW scheme is depicted in Figure 2, where AW  is the anticipatory AW compensator.We show that the synthesis results can be cast as an optimization over LMIs, and the two sets of AW gains can be straightforward obtained.Since we focus on the global results, we will restrict ourselves to stable plants.
The rest of this paper is organized as follows.In Section 2, we provide a general description of the traditional AW and the proposed AW.In Section 3, we demonstrate the synthesis results in detail, with LMIs.We will first focus on static gains and then extend to dynamic gains.The feasibility of the resulting optimization problem will also be examined.In Section 4, we illustrate the benefits of the proposed AW through two examples.Finally, we conclude with Section 5.
Notation.The notation in this paper is standard.R is the set of real numbers.  is the transpose of a real matrix .The matrix inequality  >  ( ≥ ) means that  and  are square Hermitian matrices and - is positive (semi-) definite.A block diagonal matrix with submatrices  1 ,  2 , . . .,   in its diagonal will be denoted by diag{ 1 ,  2 , . . .,   }.  denotes the identity matrix of appropriate dimensions.To reduce clutter, off-diagonal entries in symmetric matrices are occasionally replaced by " * ".The sector condition used in this paper is defined as follows.
Definition 1 (sector condition [19]).A function  : R  → R  is said to belong to the sector

Problem Formulation
Consider the following stable plant: where has been designed to guarantee that the closed-loop system is stable and achieve some performance specifications in the absence of actuator saturation.Here,   ∈ R   is the controller state,  ∈ R   is the controller output, and A c , B cy ,   , C c , D cy , and   are real constant matrices of appropriate dimensions.
In the absence of actuator saturation, the unconstrained interconnection between the plant  and linear controller  is given by û = . ( If saturation is present at the input of the plant, the unconstrained interconnection (3) is no longer guaranteed and it will be replaced by where sat(⋅) is the standard decentralized saturation function defined as with sat(  ) = sign(  ) min{|  |,   lim }; here   lim is the saturation bound for the th input.
In order to mitigate the undesirable effects caused by actuator saturation, a correction term proportional to  =  − û is added to the linear controller; that is, where is the traditional static AW compensator.Defining  = [      ]  , the traditional AW closed-loop system can be written as where In the proposed AW scheme, an artificial saturation element with a lower saturation bound  lim /  is added; here   > 1 is a design variable specified by designer.We note that, when the magnitude of the controller output  satisfies  lim /  ≤  <  lim , only the added (anticipatory) AW loop is activated, and if the controller output  goes beyond  lim , both the immediate AW loop and the anticipatory AW loop are activated.In Figure 2, signals to motivate the immediate AW loop and the anticipatory AW loop are  =  − û and   =  −   , respectively.We first assume that all the AW gains are static; that is, Then, the closed-loop system depicted in Figure 2 can be written into the following equivalent state-space form: Consider that the anticipatory AW loop has dynamic gains; that is, where   ∈ R   is the dynamic AW compensator state and   ,   ,   , and   are real constant matrices of appropriate dimensions.Define x = [         ]  .Then, the closed-loop system can be described as where In this paper, the objective of AW design is to compute the AW gains to meet some performance requirements.Similar to much of the AW design techniques, we choose the induced  2 gain as the performance index.The induced  2 gain from the exogenous input  to the output  is defined as [20] sup

Main Results
For simplicity, we will first consider the single actuator system.The results can be readily expanded to the multiactuator case, and it will be discussed later.

Static AW Gains.
We state the following lemma that obtains the stabilizing gains Λ and Λ a .
Lemma 2. The closed-loop system depicted in Figure 2 is stable and the  2 gain from  to  is less than  if there exist positive scalars  and   , symmetric matrix  > 0, and matrices ,   such that the following LMI holds: where An upper bound on the induced  2 gain can be obtained by minimizing  subject to LMI (16).If the optimization problem is feasible, then the static AW gains Λ and Λ  can be calculated from Λ =  −1 and Λ  =    −1  .
Proof.Define  = () and   =   ().Note that  and   are the dead-zone nonlinearity function.It is straightforward to show that Thus, we have We construct a quadratic Lyapunov function in the form  =  T  −1 , where  > 0. To guarantee stability of the closed-loop system and estimate the induced  2 gain from  to , we require [21] Invoking S-procedure for some positive scalars  and   (for multiactuator case,  and   are some positive definite matrices), we get the following sufficient condition to guarantee (20): If all the AW gains are static, then, in view of the closedloop system equation (11), inequality (21) can be expanded as where Figure 3: Sector condition for the artificial saturation element.
One can obtain a more aggressive and effective anticipatory AW compensator by the following approach.We are now assuming that the controller output satisfies || <  lim .Under such an assumption, the immediate AW loop will never be activated, and the closed-loop system can be relative written as Define   = φ () and   = 1 − 1/  .As depicted in Figure 3, when the magnitude of the controller output is bound with  lim , the artificial saturation element satisfies the following sector condition: Thus we now have We then invoke S-procedure for some positive scalars   (for multiactuator case,   is some positive definite matrices) to obtain which is a sufficient condition for inequality (20).
Expanding inequality (27) with the closed-loop system matrices in (24), followed by application of Schur complement and then congruent transformations diag{, , , , } and diag{, , ,  −1  }, and finally with the substitution of   =  −1  and   = Λ    , we arrive at the following equivalent LMI: where where Φ 11 and Ψ 41 are counterparts of Φ 11 and Ψ 41 when  is replaced by   .Based on the above analysis, we arrive at the following theorem that guarantees global stability and characterizes the induced L 2 gain of the closed-loop system.Theorem 3. The closed-loop system depicted in Figure 2 is stable and the L 2 gain from  to  is less than  if there exist positive scalars  and   , symmetric matrices  > 0 and   > 0, and matrices ,   such that LMI (16) and (29) hold.The optimization problem is ( 16)  (29) . (30) If the optimization problem is feasible, then the static AW gains Λ and Λ  can be calculated from Λ =  −1 and Λ  =    −1  .
As the lemma suggests, using LMI ( 16) alone guarantees stability and a L 2 performance .LMI (29) is used to obtain a more aggressive and effective anticipatory AW compensator.We note that  is the only L 2 performance of the closed-loop system.The gain   is best described as a measure of the aggressiveness and effectiveness of the anticipatory AW loop.

Combination of Static Immediate AW and Dynamic
Anticipatory AW.The synthesis after letting the anticipatory AW to be dynamic is parallel to that in previous section.To convexify and simplify the synthesis results, we use the same change of variable approach in [18], which results in a dynamic AW with   =   +   .Theorem 4. The closed-loop system depicted in Figure 2 is stable and the L 2 gain from  to  is less than  if there exist positive scalars  and   , symmetric matrices  > 0,   > 0, and  > 0, and matrices X,  1 ,  2 ,  3 , and  4 such that the following LMI holds: where The optimization problem is where Proof.We rely on the Lyapunov matrix  = x  −1 x with  > 0 and partition  as follows: where  =   ∈ R   ×  and  =   ∈ R (  +  )×(  +  ) .Define a set of variables  1 =   ,  2 =   ,  3 =     ,  4 =     , and  = Λ.Expanding ( 21) (where  is replaced by x) in terms of closed-loop system matrices in (13) and with some proper congruence transformations leads to (31).
As before, we hope to obtain a more aggressive and effective anticipatory AW compensator.Assuming that || <  lim , the closed-loop system can be written as Note that the sector condition ( 26) is also guaranteed if the magnitude of the controller output satisfies || <  lim .Thus, inequality (27) (where  is replaced by x) still hold true.Followed by the substitution of the closed-loop system matrices in (35), application of some proper congruence transformations yields where To reduce conservative, we use a different Y in (36) (denoted by   ).Thus, we complete the proof of Theorem 4.
As before, LMI (31) alone guarantees stability and a L 2 performance .LMI (33) is used to obtain a more aggressive and effective dynamic AW compensator.
Remark 5.The parameters   , , and   will indeed affect the obtained closed-loop performance (in general, we can fix c = 1 and adjust   ).It is straightforward to see that a larger   will lead to a smaller   .The values of   and   can be determined by a trial and error procedure, based on the computational results.Take   as an example; we set the initial value of   as 1 +  for a small scalar  > 0. Then adjust  to 1.1 or 0.1 iteratively until a desired closed-loop performance is achieved [14].
Remark 6.The results can be readily extended to multiinput plants.In the multi-input case, the design variable   is replaced by a diagonal matrix   = diag{   }; here,    is the design point chosen for the th input.In addition, the positive scalars ,   , , and   in single-input case are now diagonal positive definite matrices.

Feasibility of the Improved AW.
In [18], the authors pointed out that their modified AW, which contains an immediate activation compensator and a delayed activation compensator, is feasible if the traditional AW has a solution.In this subsection, we will show that similar feasibility condition can be obtained for the modified AW proposed in this paper.
In traditional AW synthesis, the condition for stability and a L 2 performance level of  is as follows [22]: We assume that there exists a pair of solutions ( * ,  * ,  * ) satisfying the above inequality; that is, We can always find a small enough  *  ( *  > 0) such that (20) holds: Now consider (27).As 0 <   < 1, a sufficient condition to ensure (27 We note that if (39) holds then ( * ,  * ,  *  ) is a solution of (40) (i.e., a solution of (27)).Thus, ( * ,  * ,  * ,  *  ) is a solution of inequalities (20) and ( 27).Finally, we can conclude that the proposed AW is guaranteed to have a solution if the traditional AW counterpart is feasible.

Numerical Examples
For ease of comparison, we choose the two examples that are also considered in [18].
which guarantees a performance level of  = 85.78 was designed through traditional approach for this example.In [18], two AW compensators Λ = [−1512.018.9 79.5]  , Λ = [−97.36411.2207 0.9435]  (43) Simulation results for a small and a large reference signals (which are the same as that in [18]) are depicted in Figure 4. We note that I-AW represents the traditional AW which has a single immediate AW loop, ID-AW represents the modified AW which consists of an immediate AW loop and a delayed AW loop, and IA-AW represents the proposed modified AW which consists of an immediate AW loop and an anticipatory AW loop.As Figure 4 suggests, system with IA-AW achieves the best performance, no matter the reference signal is small or large.In addition, the performance obtained by IA-AW is much better than that obtained by a single delayed AW loop or a single anticipatory AW loop (see the numerical example in [14]).The time histories of u,   , and û are illustrated in Figure 5.We can see that both the anticipatory AW loop and the immediate AW loop are activated.
Example 2. Consider the following example taken from [22] with actuator saturation  lim = 5: For this plant, static AW is not feasible.In [22], a dynamic AW with   = 4 has been designed.In [18], an immediate static AW compensator and a delayed dynamic AW compensator were simultaneously designed.Using Theorem 4 with (46) The two compensators guarantee performance levels  = 186.3811and   = 23.2318.Simulation results for a step input of duration 0.1 s and magnitude 0.5 are depicted in Figure 6.We can see that the proposed improved AW achieves the best system response by forcing the system to leave the saturation zone earlier than both the I-AW and the ID-AW.

Conclusions
We have proposed an improved AW design approach for stable linear systems subject to actuator saturation.In the proposed approach, two AW compensators were simultaneously computed, one for immediate activation at the occurrence of saturation and the other for anticipatory activation.Using the induced  2 gain as the performance index, the synthesis results were formulated and solved as optimization problems over LMIs.Numerical examples confirmed the effectiveness of the proposed AW design method.
the exogenous input (reference signals, disturbances, and noise), û ∈ R   is the control input,  ∈ R   is the controlled output,  ∈ R   is the measurement output, and A p , B 1 , B 2 , C 1 , D 11 , D 12 , C 2 , D 21 , and D 22 are real constant matrices of appropriate dimensions.Pairs (A p , B 2 ) and (C 2 , A p ) are assumed to be controllable and observable, respectively.Without loss of generality, we will assume that D 22 = 0 henceforth.Considering plant , we assume that an   th-order linear dynamic controller  : { ẋ  =     +    +     =     +    + 41 = −      +      and Ψ 44 = −      −         − 2  .To guarantee the uniqueness of Λ  , we use the same   and   in (22) and (28); though, to reduce conservatism, we can use different Lyapunov matrix (denoted by   ) in (28).