Delayed Antiwindup Control Using a Decoupling Structure

This paper investigates the antiwindup (AW) control problem for plants with input saturation.TheAWcompensator is not activated as soon as input saturation occurs as usual. A delayed decoupling structure is first proposed. Then, appropriate linear matrix inequalities (LMIs) are developed to determine a plant-order AW compensator. Effectiveness of the presented AW technique is illustrated by a fighter aircraft model.


Introduction
Input saturation is one of the most frequently encountered nonlinearities in control design for linear plants.Most controllers are designed assuming that the inputs have no limits.If input saturation appears, output performance often deteriorates.Slight performance degradation could be large overshoot or sluggish response.Severe performance deterioration could be persistent oscillation, sometimes even instability.
Multiagent control has become a hot topic of academic research in recent years.This is because it can be used in various fields, such as physics, biology, industry, and military applications.The two main applications of multiagent control in industry and military are the multirobot cooperative control and the multivehicle flight control, respectively.Although these two applications have been intensively studied, however, few results consider the case when actuators are limited [1].The well-known AW control is one of the most successful methodologies applied in practice to deal with plants with input saturation.As summed in [2,3], the essential idea of AW control is adding control modification to a nominal linear controller.The nominal linear controller is first synthesized for the plant ignoring saturation, and then an AW compensator is designed to improve output performance when saturation occurs.Sufficient conditions to design a static AW compensator were provided in [4] in terms of LMIs.Necessary and sufficient conditions for the existence of a plant-order AW compensator were given in [5].A novel, less conservative sector condition was presented in [6], and a static AW compensator for both stable and unstable plants was obtained therein.A switching AW design using multiple Lyapunov functions was proposed in [7].In [4,5], attention was focused on minimizing the influence of a generic disturbance signal  in the output.In [6,7], concern was on maximization of the region of attraction.Innovatively in [8], Weston and Postlethwaite proposed a structure, which established a specific formulation of the error between the plant outputs with and without input saturation.Then, in [9], two different AW compensators were designed to make the error as small as possible based on the structure.
Usually, the AW compensator was activated as soon as input saturation occurs.However, a static AW compensator designed in [10] was not activated immediately but belatedly.It was found in [10] that delayed activation of AW compensator can lead to a better performance compared with the immediate one.In order to improve AW control performance and derive a simpler LMI condition for the AW compensator construction, motivated by the structure proposed in [8], a modified delayed decoupling structure is developed in this paper.Then, sufficient conditions in terms of LMIs are obtained to minimize the error between the plant outputs with and without input saturation.The remainder of this paper is organized as follows.In Section 2, the structure, which has a decoupling feature, is introduced.Main results of this paper are presented in Section 3. A new modified delayed AW decoupling structure is first proposed.And the AW compensator is constructed for single-input plants by making a set of LMIs feasible.It is also extended to multi-input plants.A fighter craft model is used to illustrate the proposed AW technique.Conclusions are given in Section 4.

A Decoupling Structure
A general AW control structure is shown in Figure 1, where () is the plant, () is the nominal linear controller, sat(⋅) is the input saturation function, and block AWC is the AW compensator.() is a stable linear plant subject to input saturation as where ũ :=  − sat() is the error between the unsaturated input and saturated input,  :=  −  is the tracking error, and  is the reference to be tracked.The AW compensator is activated only when input saturation occurs.As mentioned earlier, the essential idea of AW control is adding control modification to ().The state and output equations of () are modified by the AWC's output  = (  1 ,   2 )  as An appealing AW structure, which is a special case of the general structure, proposed by Weston and Postlethwaite in [8], is shown in Figure 2(a), where the design of AW compensator turns into the conditioning of a transfer function ().With all signals labeled the same, the structure in Figure 2(a) can be equivalently redrawn as in Figure 2(b).Note that in Figure 2(b), a dead-zone function (⋅) is introduced, which maps the unsaturated input  to the input error ũ; that is, What is fascinating about the structure in Figure 2(b) is that it can be seen as two decoupled subsystems: the nominal linear closed-loop system (NLCS) and the nonlinear path.If NLCS and the nonlinear path are stable, respectively, the stability of the overall system in Figure 2(b) follows.The NLCS is stable, as it is guaranteed by ().The NLCS's output  lin is what one tries to recover when input saturation occurs.Thus, the AW compensator should be designed to ensure stability of the nonlinear path, further on, if possible, to make the output difference   =  lin −  as small as possible.
One selection for () in Figure 2(a) is a right coprime factor of (); that is, () = () −1 ().The state space descriptions of () and () can be chosen as The conditioning with () turns into the design of matrix .()() = (), and the state space description of AW compensator is Remark 1.Note that all the initial states and the control inputs of () and () are equal.Therefore the states of () and () are identical.Under this circumstance, although () and () are different transfer functions, they can still share the same state  aw .
Define the distance between a vector  and a compact set S as dist(, S) := inf ∈S ‖ − ‖.Then, the objective of this paper, which is based on the decoupling AW structure proposed in [8], is to solve the AW control problem defined as follows.
Definition 2 (see [9]).The AW compensator ( 5) is said to solve the AW control problem for plant (1) if the closedloop system in Figure 2(a) is well posed and the following conditions hold.
The AW compensator ( 5) is said to solve the strong AW control problem if in addition, the following condition is satisfied.
(3) The operator  :  lin  →   is well defined and the L 2 gain from  lin to   is less than ; namely, ‖  ‖ 2 < ‖ lin ‖ 2 .Remark 3. Note that condition 3 in Definition 2 implies condition 2. In this paper, only the strong AW control problem is considered.

Main Results
Immediate AW compensation may increase safety of the overall system, being away from instability, for example, but the performance may be no better than leaving the controller uncompensated during some moderate input saturation, as the numerical example demonstrated in [10].If the nominal controller is robust, the system still demonstrates good performance during this modest saturation stage.Thus, the delayed activation of AW compensation is considered.

Delayed Decoupling AW Structure.
To realize certain delay in activating the AW compensator, an artificial saturation function is added after the controller, and thus the structure in Figure 2(a) becomes as in Figure 3.This structure has two saturation nonlinearities, and thus it is not equivalent to the decoupling structure in Figure 2(b).In order to sustain a decoupling feature, Figure 3 needs to be modified.A new decoupling structure with delayed saturation is therefore presented in Figure 4(a).Note that a time-varying gain   = diag[ 1 (),  2 (), . . .,   ()] is introduced in Figure 4(a), where   () is defined as Here   is the delayed saturation index, 0 <   < 1.The AW compensator is activated when the input magnitude is higher than the delayed saturation level   /  , rather than Taking the same steps as in [11], where the decoupling AW structure is extended to LPV system, the equivalence between    for   =   .Furthermore, the designing matrix  for AW compensator (8) to guarantee L 2 performance requirement ‖  ‖ 2 < ‖ lin ‖ 2 is given by  =  −1 .
Proof.The operation of the saturated plant with delayed activation of AW compensator can be separated into two stages.
Stage 1 (|| ≤ /  ).The input is below the artificial saturation level.The input to AW compensator is zero, namely, ũ = 0, and   ∈ [  , 1].Then, following the standard approach, if inequality Then, following the similar steps as [9], if inequality ( 13) holds, the strong AW control problem is solved.Substitute (8) into inequality ( 13); one has Inequality ( 14) can be rewritten as where According to the standard Schur complement and exerting congruency transformation, one has LMI (10) with  =  −1 ,  =  −1 ,  = ; thus,  =  −1 .Similar to Stage 1, with the two subsystems being stable, the stability of the overall closed-loop system at Stage 2 follows.Note that LMI (10) for   =   implies that LMI (9) for   =   , thus only   = 1 is checked for LMI (9) in the theorem.And this completes the proof.
The section of the delayed saturation index   can be determined by an iterative algorithm, starting at an initial value and stopping at a point when the L 2 gain  is optimized.By applying the AW compensator designed using Theorem 4, the nominal linear performance is recovered as much as possible.The static and dynamic AW compensators designed in [10,12] used the states  = (   ,    ) and  = (   ,    ,   aw ) respectively, in the Lyapunov function for LMI derivation.In this paper, only  aw is used, and thus the LMIs to be solved here are much simpler than the resulting LMIs in [10,12].For a multi-input plant,   is no longer a scalar, but a  ×  diagonal matrix.Because each diagonal element of   varies between   and 1, thus   can be expressed as a linear combination of the vertices as follows: where ]; similarly, one has the following theorem.
Theorem 5.For a multi-input saturated plant, the strong AW control problem in Definition 2 is solved if there exist a symmetric positive definite matrix  ∈ R × , a diagonal positive definite matrix  ∈ R × , a matrix  ∈ R × , and a positive real scalar  such that LMI (9) holds for   =  for all  = 2, 3, . . ., 2  .Furthermore, the designing matrix  for AW compensator (8) to guarantee L 2 performance requirement ‖  ‖ 2 < ‖ lin ‖ 2 is given by  =  −1 .
Proof.The proof follows the same steps as Theorem 4, and thus it is omitted here.

Application Example.
Consider a linearized fighter aircraft model (matrices   ,   , and   can be found from equations (A.2)-(A.4) in Yee et al. [13]).The aircraft has six states (, V, , , , ).The first three states are airspeed in the -, -, and -axes of the body frame, respectively.The last three states are roll, pitch, and yaw rates about -, -, and -axes.The outputs ( μ ) are, respectively, stability-axis roll rate, angle-of-attack, and sideslip angle.The inputs ( ℎ ,   ,   ) are elevator, aileron, and rudder deflection angles, with control surface constraints −25 ∘ ≤  ℎ ≤ 25 ∘ , −20 ∘ ≤   ≤ 20 ∘ , and −30 ∘ ≤   ≤ 30 ∘ .The reference to be tracked is shown in Figure 5. Without consideration of input saturation, the aircraft demonstrates precise and fast tracking response, as shown in Figure 6(a), using a continuous-time robust H ∞ controller designed in [13] (matrices   ,   ,   , and   are obtained ), and the obtained optimal L 2 gain is  = 5.7294.The corresponding tracking performance is shown in Figure 8(a).The AW compensator designed in [14] which is based on immediate activation is further presented here for comparison.Solving LMI (13) in [14], the designing matrix is and the optimal L 2 gain is  = 5.7298.The corresponding tracking performance is shown in Figure 7(b).Our method is also compared with the delayed AW compensator proposed in [10].Using the method of [10], the tracking performance is shown in Figure 8(b).
Comparing the tracking performance in Figure 8(a) with the uncompensated one as in Figure 7(a), fluctuation in  channel decreases significantly in Figure 8(a).The AW compensator plays its role.From Figures 8(a) and 7(b) which are based on the immediate AW compensation, the output dynamic property in Figure 8(a) is preferable.This is because modest saturation can be seen as a small amount of plant uncertainty, and since the nominal controller is robust, the system still demonstrates good performance during this modest saturation stage.Immediate AW compensation may destroy this equilibrium.From Figures 8(a) and 8(b) with the static delayed AW compensator, the output performance in Figure 8(a) is no worse.This is because the AW compensator    is a plant-order dynamic one, while the delayed AW compensator proposed in [10] is static.

Conclusion
In this paper, the AW control problem based on the decoupling structure proposed by Weston and Postlewaite is investigated.Based on the decoupling AW structure, matrix  is to be designed to determine (), which is a right coprime factor of the plant ().Motivated by the novel idea of delaying the activation of the AW compensator, a new decoupling structure with delayed activation of AW compensator is developed.A time-varying gain   is introduced, and Ĝ() = ()  is treated as a new quasi-LPV plant to deal with the delayed AW control problem.For both single-input and multi-input plants, a set of small scale LMIs are derived to calculate the designing matrix  for AW compensator.Only simple LMIs are needed to be solved, while the LMIs in [14] to be solved are large.Most of all, the AW technique proposed here has a better performance applied in the application example than in [14] and has no worse performance than [10].

Figure 1 :
Figure 1: General structure of AW control.
The nominal linear closed-loop system ũ y lin u lin (b) An equivalent decoupling structure

Figure 6 :Figure 7 :
Figure 6: Linear tracking response with pulse reference in  (no saturation).