# Delayed Antiwindup Control Using a Decoupling Structure

Authors: Huawei Zhu; Guozheng Qin; Yingxin Yan; Zhichao Jiang; Zhisheng Duan
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper investigates the antiwindup (AW) control problem for plants with input saturation. The AW compensator 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.


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## Body

## 1. 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 w 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 Section2, 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.


## 2. A Decoupling Structure

A general AW control structure is shown in Figure1, where G(s) is the plant, K(s) is the nominal linear controller, sat(·) is the input saturation function, and block AWC is the AW compensator. G(s) is a stable linear plant subject to input saturation as
(1)G(s)~{x˙p=Apxp+Bpsat(u),y=Cpxp+Dpsat(u),
where (Ap, Bp) is stabilizable, xp∈ℝn,y∈ℝq,u∈ℝp, and sat(u) is the saturated input. The saturation function sat(·) satisfies sat(u)=[sat(u1),sat(u2),…,sat(up)]T, sat(ui)=sign(ui)min(|ui|,u-i). u-i is the amplitude limit, u-i>0,i=1,2,…,p. The state-space descriptions of K(s) and AW compensator are, respectively
(2)K(s)~{x˙c=Acxc+Bce,u=Ccxc+Dce,AWC~{x˙aw=Aawxaw+Bawu~,η=(η1T,η2T)T=Cawxaw+Dawu~,
where u~:=u-sat(u) is the error between the unsaturated input and saturated input, e:=r-y is the tracking error, and r 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 K(s). The state and output equations of K(s) are modified by the AWC’s output η=(η1T,η2T)T as
(3)x˙c=Acxc+Bce+η1,u=Ccxc+Dce+η2.Figure 1
General structure of AW control.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 M(s). 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 dz(·) is introduced, which maps the unsaturated input u to the input error u~; that is, dz(u)=u~. If u∈𝒰, where 𝒰∈ℝp is a compact set defined as 𝒰=[-u-1,u-1]×[-u-2,u-2]×⋯×[-u-p,u-p], then dz(u)=0. 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 K(s). The NLCS’s output ylin 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 yd=ylin-y as small as possible.A decoupling structure.
(a)
Conditioning withM(s)
(b)
An equivalent decoupling structureOne selection forM(s) in Figure 2(a) is a right coprime factor of G(s); that is, G(s)=N(s)M-1(s). The state space descriptions of M(s) and N(s) can be chosen as
(4)(M(s)N(s))~(Ap+BpFBpFICp+DpFDp).
The conditioning with M(s) turns into the design of matrix F. G(s)M(s)=N(s), and the state space description of AW compensator is
(5)x˙aw=(Ap+BpF)xaw+Bpu~,ud=Fxaw,yd=(Cp+DpF)xaw+Dpu~.Remark 1.
Note that all the initial states and the control inputs ofM(s) and N(s) are equal. Therefore the states of M(s) and N(s) are identical. Under this circumstance, although M(s) and N(s) are different transfer functions, they can still share the same state xaw.Define the distance between a vectorx and a compact set 𝒮 as dist(x, 𝒮):=infy∈𝒮∥x-y∥. 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 closed-loop system in Figure 2(a) is well posed and the following conditions hold.(1)
Ifdist(ulin,𝒰)=0 for all t≥0, then yd=0 for all t≥0 (assuming zero initial condition for AW compensator).(2)
Ifdist(ulin, 𝒰)∈ℒ2, then yd∈ℒ2.The AW compensator (5) is said to solve the strong AW control problem if in addition, the following condition is satisfied.(3)
The operatorτ:ulin↦yd is well defined and the ℒ2 gain from ulin to yd is less than γ; namely, ∥yd∥2<γ∥ulin∥2.Remark 3.
Note that condition 3 in Definition2 implies condition 2. In this paper, only the strong AW control problem is considered.


## 3. 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.


### 3.1. 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 Figure2(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 Gd=diag[g1(t),g2(t),…,gp(t)] is introduced in Figure 4(a), where gi(t) is defined as
(6)gi(t)={1,|ui|≤u-i,sgn(ui)u-iui,u-i<|u-i|<u-igd,gd,|ui|≥u-igd.
Here gd is the delayed saturation index, 0<gd<1. The AW compensator is activated when the input magnitude is higher than the delayed saturation level u-i/gd, rather than u-i. Now treat G^(s):=G(s)Gd as a new plant, which is linear parameter-varying (LPV). The time varying matrix gain Gd is continuous in the known bounds, Gd∈[gd,1]. Since the expressions for u, y, u~ are the same in Figures 4(a) and 4(b), the two figures are mathematically equivalent if the following equation holds:
(7)G^ulin=N^u~+G^(ulin-M^u~).
Taking the same steps as in [11], where the decoupling AW structure is extended to LPV system, the equivalence between Figures 4(a) and 4(b) follows. Note that in the nonlinear path, G^(s) replaces the original G(s). Therefore, the state space description of the AW compensator becomes
(8)x˙aw=(Ap+BpGdF)xaw+BpGdu~,ud=Fxaw,yd=(Cp+DpGdF)xaw+DpGdu~.Figure 3
Conditioning withM(s) (delayed).The delayed decoupling structure.
(a)
Conditioning withM(s) (modified)
(b)
The equivalent decoupling structure (modified)


### 3.2. Delayed AW Synthesis Using LMI

In order to analyze the behavior of the new system, for intuitiveness, single-input (p=1) plants are considered first. And one has the following theorem.Theorem 4.
For a single-input saturated plant, the strong AW control problem in Definition2 is solved if there exist a symmetric positive definite matrix Q∈ℝn×n, a diagonal positive definite matrix S∈ℝ1×1, a matrix R∈ℝ1×n, and a positive real scalar γ such that
(9)(ApQ+QApT+BpGdR+RTGdBpT**0-γI*CpQ+DpGdR0-γI)<0
for Gd=1 and(10)(ApQ+QApT+BpGdR+RTGdBpT***0-γI**CpQ+DpGdR0-γI*SGdBpT-RI(DpGdS)T-2S)<0for Gd=gd. Furthermore, the designing matrix F for AW compensator (8) to guarantee ℒ2 performance requirement ∥yd∥2<γ∥ulin∥2 is given by F=RQ-1.Proof.
The operation of the saturated plant with delayed activation of AW compensator can be separated into two stages.
Stage 1 (|u|≤u-/gd). The input is below the artificial saturation level. The input to AW compensator is zero, namely, u~=0, and Gd∈[gd,1]. Then, following the standard approach, if inequality
(11)V˙+γ-1ydTyd-γulinTulin<0
holds, it gives ∥yd∥2<γ∥ulin∥2, and the strong AW control problem is solved. Choose the Lyapunov function in (11) as V=xawTPxaw where P=PT>0, and substitute (8) into inequality (11); one has
(12)[(Ap+BpGdF)xaw]TPxaw+xawTP[(Ap+BpGdF)xaw]+γ-1[(Cp+DpGdF)xaw]T[(Cp+DpGdF)xaw]-γulinTulin<0.
Then, by Schur complement and congruency transformation, inequality (12) becomes LMI (9) in Theorem 4 with Q=P-1,R=FQ; thus, F=RQ-1. Because Gd∈[gd,1] at Stage 1, it suffices to check LMI (9) for Gd at the vertices, namely for Gd=gd and Gd=1. The stability of the nonlinear path is thus ensured. Note that during this stage, the “nominal” closed-loop system is linear time-varying as shown in Figure 4(b). Leaving the nominal controller K(s) undisturbed when |u|∈(u-,u-/gd], the stability of new NCLS is preserved by the robustness of controller K(s). With the two subsystems being stable, the stability of the overall closed-loop system at Stage 1 follows.
Stage 2 (|u|>u-/gd). During this stage, the controller’s output exceeds the delayed saturation level u-/gd. As a result, u~ is nonzero, the AW compensator is activated, and Gd=gd. Due to u~Tu≥u~Tu~, the sector condition 2u~TW(u-u~)≥0 always holds, where W is a diagonal positive definite matrix. Then, a sufficient condition for inequality (11) is
(13)V˙+γ-1ydTyd-γulinTulin+2u~TW(u-u~)<0.
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
(14)[(Ap+BpGdF)xaw+BpGdu~]TPxaw+xawTP[(Ap+BpGdF)xaw+BpGdu~]+γ-1[(Cp+DpGdF)xaw+DpGdu~]T×[(Cp+DpGdF)xaw+DpGdu~]-γulinTulin+2u~TW(ulin-Fxaw-u~)<0.
Inequality (14) can be rewritten as
(15)(xawTu~TulinT)[X-YT(-γ)-1Y](xawu~ulin)<0,
where (16)X=((Ap+BpGdF)TP+P(Ap+BpGdF)**BpTGdP-WF-2W*0W-γI),Y=(Cp+DpGdFDpGd0).According to the standard Schur complement and exerting congruency transformation, one has LMI (10) with Q=P-1,S=W-1,R=FQ; thus, F=RQ-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 Gd=gd implies that LMI (9) for Gd=gd, thus only Gd=1 is checked for LMI (9) in the theorem. And this completes the proof.The section of the delayed saturation indexgd can be determined by an iterative algorithm, starting at an initial value and stopping at a point when the ℒ2 gain γ 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 x=(xpT,xcT)T and x=(xpT,xcT,xawT)T respectively, in the Lyapunov function for LMI derivation. In this paper, only xaw 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,Gd is no longer a scalar, but a p×p diagonal matrix. Because each diagonal element of Gd varies between gd and 1, thus Gd can be expressed as a linear combination of the vertices as follows:
(17)Gd=∑j=12p‍αjGdj,∑j=12p‍αj=1,αj∈[0,1],Gdj∈𝒢,j=1,2,…,2p,
where 𝒢={diag[g1,g2,…,gp]:gi=gdor1,i=1,2,…,p}. Let Gd1=diag[1,1,…,1]; similarly, one has the following theorem.Theorem 5.
For a multi-input saturated plant, the strong AW control problem in Definition2 is solved if there exist a symmetric positive definite matrix Q∈ℝn×n, a diagonal positive definite matrix S∈ℝp×p, a matrix R∈ℝp×n, and a positive real scalar γ such that LMI (9) holds for Gd=Gd1 and(18)(ApQ+QApT+BpGdjR+RTGdjBpT***0-γI**CpQ+DpGdjR0-γI*SGdjBpT-RI(DpGdjS)T-2S)<0for all j=2,3,…,2p. Furthermore, the designing matrix F for AW compensator (8) to guarantee ℒ2 performance requirement ∥yd∥2<γ∥ulin∥2 is given by F=RQ-1.Proof.
The proof follows the same steps as Theorem4, and thus it is omitted here.


### 3.3. Application Example

Consider a linearized fighter aircraft model (matricesAp,Bp, and Cp can be found from equations (A.2)–(A.4) in Yee et al. [13]). The aircraft has six states (u,v,w,p,q,r). The first three states are airspeed in the X-, Y-, and Z-axes of the body frame, respectively. The last three states are roll, pitch, and yaw rates about X-, Y-, and Z-axes. The outputs (μ˙αβ) are, respectively, stability-axis roll rate, angle-of-attack, and sideslip angle. The inputs (δh, δa, δr) are elevator, aileron, and rudder deflection angles, with control surface constraints -25°≤δh≤25°,-20°≤δa≤20°, and -30°≤δr≤30°. The reference to be tracked is shown in Figure 5.Figure 5
Reference.Without consideration of input saturation, the aircraft demonstrates precise and fast tracking response, as shown in Figure6(a), using a continuous-time robust ℋ∞ controller designed in [13] (matrices Ac,Bc,Cc, and Dc are obtained by using the robust control toolbox). But the control inputs δa and δr actually exceed their amplitude limits as Figure 6(b) displays. Imposing amplitude limits at the aircraft’s inputs in the simulation, the tracking performance deteriorates, as shown in Figure 7(a). Thus, AW compensation should be considered. Using Theorem 5 and taking the delayed saturation index gd=0.65, the designing matrix is
(19)F=(-1.63990.03270.6927-0.24442413.88410.0219-0.22810.001345.8410.000338.7870.0155-0.20990.0010-10.3030.1976174.09),
and the obtained optimal ℒ2 gain is γ=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
(20)F=(-731.1316.222425.51-106.50981751634.731.960-329.782.433754547-76.7695924611.554-123.601.0543-5903.479.15310750),
and the optimal ℒ2 gain is γ=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).Linear tracking response with pulse reference inβ (no saturation).
(a)
Output response
(b)
Control inputsSaturated tracking response with pulse reference inβ (with saturation).
(a)
Without AW compensator
(b)
With AW compensatorSaturated tracking response with delayed AW compensator.
(a)
Delayed AW compensator
(b)
Delayed AW compensator proposed in [10]Comparing the tracking performance in Figure8(a) with the uncompensated one as in Figure 7(a), fluctuation in μ 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.


## 3.1. 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 Figure2(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 Gd=diag[g1(t),g2(t),…,gp(t)] is introduced in Figure 4(a), where gi(t) is defined as
(6)gi(t)={1,|ui|≤u-i,sgn(ui)u-iui,u-i<|u-i|<u-igd,gd,|ui|≥u-igd.
Here gd is the delayed saturation index, 0<gd<1. The AW compensator is activated when the input magnitude is higher than the delayed saturation level u-i/gd, rather than u-i. Now treat G^(s):=G(s)Gd as a new plant, which is linear parameter-varying (LPV). The time varying matrix gain Gd is continuous in the known bounds, Gd∈[gd,1]. Since the expressions for u, y, u~ are the same in Figures 4(a) and 4(b), the two figures are mathematically equivalent if the following equation holds:
(7)G^ulin=N^u~+G^(ulin-M^u~).
Taking the same steps as in [11], where the decoupling AW structure is extended to LPV system, the equivalence between Figures 4(a) and 4(b) follows. Note that in the nonlinear path, G^(s) replaces the original G(s). Therefore, the state space description of the AW compensator becomes
(8)x˙aw=(Ap+BpGdF)xaw+BpGdu~,ud=Fxaw,yd=(Cp+DpGdF)xaw+DpGdu~.Figure 3
Conditioning withM(s) (delayed).The delayed decoupling structure.
(a)
Conditioning withM(s) (modified)
(b)
The equivalent decoupling structure (modified)


## 3.2. Delayed AW Synthesis Using LMI

In order to analyze the behavior of the new system, for intuitiveness, single-input (p=1) plants are considered first. And one has the following theorem.Theorem 4.
For a single-input saturated plant, the strong AW control problem in Definition2 is solved if there exist a symmetric positive definite matrix Q∈ℝn×n, a diagonal positive definite matrix S∈ℝ1×1, a matrix R∈ℝ1×n, and a positive real scalar γ such that
(9)(ApQ+QApT+BpGdR+RTGdBpT**0-γI*CpQ+DpGdR0-γI)<0
for Gd=1 and(10)(ApQ+QApT+BpGdR+RTGdBpT***0-γI**CpQ+DpGdR0-γI*SGdBpT-RI(DpGdS)T-2S)<0for Gd=gd. Furthermore, the designing matrix F for AW compensator (8) to guarantee ℒ2 performance requirement ∥yd∥2<γ∥ulin∥2 is given by F=RQ-1.Proof.
The operation of the saturated plant with delayed activation of AW compensator can be separated into two stages.
Stage 1 (|u|≤u-/gd). The input is below the artificial saturation level. The input to AW compensator is zero, namely, u~=0, and Gd∈[gd,1]. Then, following the standard approach, if inequality
(11)V˙+γ-1ydTyd-γulinTulin<0
holds, it gives ∥yd∥2<γ∥ulin∥2, and the strong AW control problem is solved. Choose the Lyapunov function in (11) as V=xawTPxaw where P=PT>0, and substitute (8) into inequality (11); one has
(12)[(Ap+BpGdF)xaw]TPxaw+xawTP[(Ap+BpGdF)xaw]+γ-1[(Cp+DpGdF)xaw]T[(Cp+DpGdF)xaw]-γulinTulin<0.
Then, by Schur complement and congruency transformation, inequality (12) becomes LMI (9) in Theorem 4 with Q=P-1,R=FQ; thus, F=RQ-1. Because Gd∈[gd,1] at Stage 1, it suffices to check LMI (9) for Gd at the vertices, namely for Gd=gd and Gd=1. The stability of the nonlinear path is thus ensured. Note that during this stage, the “nominal” closed-loop system is linear time-varying as shown in Figure 4(b). Leaving the nominal controller K(s) undisturbed when |u|∈(u-,u-/gd], the stability of new NCLS is preserved by the robustness of controller K(s). With the two subsystems being stable, the stability of the overall closed-loop system at Stage 1 follows.
Stage 2 (|u|>u-/gd). During this stage, the controller’s output exceeds the delayed saturation level u-/gd. As a result, u~ is nonzero, the AW compensator is activated, and Gd=gd. Due to u~Tu≥u~Tu~, the sector condition 2u~TW(u-u~)≥0 always holds, where W is a diagonal positive definite matrix. Then, a sufficient condition for inequality (11) is
(13)V˙+γ-1ydTyd-γulinTulin+2u~TW(u-u~)<0.
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
(14)[(Ap+BpGdF)xaw+BpGdu~]TPxaw+xawTP[(Ap+BpGdF)xaw+BpGdu~]+γ-1[(Cp+DpGdF)xaw+DpGdu~]T×[(Cp+DpGdF)xaw+DpGdu~]-γulinTulin+2u~TW(ulin-Fxaw-u~)<0.
Inequality (14) can be rewritten as
(15)(xawTu~TulinT)[X-YT(-γ)-1Y](xawu~ulin)<0,
where (16)X=((Ap+BpGdF)TP+P(Ap+BpGdF)**BpTGdP-WF-2W*0W-γI),Y=(Cp+DpGdFDpGd0).According to the standard Schur complement and exerting congruency transformation, one has LMI (10) with Q=P-1,S=W-1,R=FQ; thus, F=RQ-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 Gd=gd implies that LMI (9) for Gd=gd, thus only Gd=1 is checked for LMI (9) in the theorem. And this completes the proof.The section of the delayed saturation indexgd can be determined by an iterative algorithm, starting at an initial value and stopping at a point when the ℒ2 gain γ 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 x=(xpT,xcT)T and x=(xpT,xcT,xawT)T respectively, in the Lyapunov function for LMI derivation. In this paper, only xaw 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,Gd is no longer a scalar, but a p×p diagonal matrix. Because each diagonal element of Gd varies between gd and 1, thus Gd can be expressed as a linear combination of the vertices as follows:
(17)Gd=∑j=12p‍αjGdj,∑j=12p‍αj=1,αj∈[0,1],Gdj∈𝒢,j=1,2,…,2p,
where 𝒢={diag[g1,g2,…,gp]:gi=gdor1,i=1,2,…,p}. Let Gd1=diag[1,1,…,1]; similarly, one has the following theorem.Theorem 5.
For a multi-input saturated plant, the strong AW control problem in Definition2 is solved if there exist a symmetric positive definite matrix Q∈ℝn×n, a diagonal positive definite matrix S∈ℝp×p, a matrix R∈ℝp×n, and a positive real scalar γ such that LMI (9) holds for Gd=Gd1 and(18)(ApQ+QApT+BpGdjR+RTGdjBpT***0-γI**CpQ+DpGdjR0-γI*SGdjBpT-RI(DpGdjS)T-2S)<0for all j=2,3,…,2p. Furthermore, the designing matrix F for AW compensator (8) to guarantee ℒ2 performance requirement ∥yd∥2<γ∥ulin∥2 is given by F=RQ-1.Proof.
The proof follows the same steps as Theorem4, and thus it is omitted here.


## 3.3. Application Example

Consider a linearized fighter aircraft model (matricesAp,Bp, and Cp can be found from equations (A.2)–(A.4) in Yee et al. [13]). The aircraft has six states (u,v,w,p,q,r). The first three states are airspeed in the X-, Y-, and Z-axes of the body frame, respectively. The last three states are roll, pitch, and yaw rates about X-, Y-, and Z-axes. The outputs (μ˙αβ) are, respectively, stability-axis roll rate, angle-of-attack, and sideslip angle. The inputs (δh, δa, δr) are elevator, aileron, and rudder deflection angles, with control surface constraints -25°≤δh≤25°,-20°≤δa≤20°, and -30°≤δr≤30°. The reference to be tracked is shown in Figure 5.Figure 5
Reference.Without consideration of input saturation, the aircraft demonstrates precise and fast tracking response, as shown in Figure6(a), using a continuous-time robust ℋ∞ controller designed in [13] (matrices Ac,Bc,Cc, and Dc are obtained by using the robust control toolbox). But the control inputs δa and δr actually exceed their amplitude limits as Figure 6(b) displays. Imposing amplitude limits at the aircraft’s inputs in the simulation, the tracking performance deteriorates, as shown in Figure 7(a). Thus, AW compensation should be considered. Using Theorem 5 and taking the delayed saturation index gd=0.65, the designing matrix is
(19)F=(-1.63990.03270.6927-0.24442413.88410.0219-0.22810.001345.8410.000338.7870.0155-0.20990.0010-10.3030.1976174.09),
and the obtained optimal ℒ2 gain is γ=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
(20)F=(-731.1316.222425.51-106.50981751634.731.960-329.782.433754547-76.7695924611.554-123.601.0543-5903.479.15310750),
and the optimal ℒ2 gain is γ=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).Linear tracking response with pulse reference inβ (no saturation).
(a)
Output response
(b)
Control inputsSaturated tracking response with pulse reference inβ (with saturation).
(a)
Without AW compensator
(b)
With AW compensatorSaturated tracking response with delayed AW compensator.
(a)
Delayed AW compensator
(b)
Delayed AW compensator proposed in [10]Comparing the tracking performance in Figure8(a) with the uncompensated one as in Figure 7(a), fluctuation in μ 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.


## 4. 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, matrixF is to be designed to determine M(s), which is a right coprime factor of the plant G(s). 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 Gd is introduced, and G^(s)=G(s)Gd 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 F 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].



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*Source: 248153-2013-05-22.xml*