Simultaneous anti-windup synthesis for linear systems subject to actuator saturation

Maopeng Ran,Qing Wang,Chaoyang Dong,and Maolin Ni

1.School of Automation Science and Electrical Engineering,Beihang University,Beijing 100191,China;

2.School of Aeronautic Science and Engineering,Beihang University,Beijing 100191,China;

3.Chinese Society of Astronautics,Beijing 100191,China

Simultaneous anti-windup synthesis for linear systems subject to actuator saturation

Maopeng Ran1,*,Qing Wang1,Chaoyang Dong2,and Maolin Ni3

1.School of Automation Science and Electrical Engineering,Beihang University,Beijing 100191,China;

2.School of Aeronautic Science and Engineering,Beihang University,Beijing 100191,China;

3.Chinese Society of Astronautics,Beijing 100191,China

A synthesis method for global stability and performance of input constrained linear systems,which uses a linear outputfeedback controller and a static anti-windup compensator is investigated.Different from the traditional two-step anti-windup design procedure,the proposed method synthesizes all controller parameters simultaneously.Suffcient conditions for global stability and minimizing the induced L2gain are formulated and solved as a linear matrix inequalities(LMIs)optimization problem,which also provides an opportunity to search for a better performance tradeoff between the linear controller and the anti-windup compensator. The well-posedness of the close-loop system is also guaranteed. Simulation results show the effectiveness of the proposed method.

actuator saturation,anti-windup,simultaneous synthesis,L2synthesis,linear matrix inequalities(LMIs).

1.Introduction

Actuator saturation is a ubiquitous fact in any real control design problem.It can lead to deterioration in the performance of the system,and may also induce instability.The phenomenon caused by saturation is called anti-windup, and compensation for preventing this is referred to as antiwindup[1].Generally speaking,the anti-windup design follows a two-step procedure,i.e.,frst design a linear controller which does not take the saturation nonlinearity into account,and then add an anti-windup compensator to address any adverse effects that saturation creates[2,3]. The main superiority of this method is that there are no restrictions placed on the linear controller design,and while the actuator is not reaching saturation,the system closedloop response coincides with the linear unconstrained response.Under the two-step anti-windup procedure,various methods have been developed to design the anti-windup compensator[4–8].

In nearly all two-step anti-windup design practices,the linear controller and the anti-windup compensator are designed separately.One disadvantage of the traditional twostep approach is that it is always suboptimal,as a result of the two-step design of the linear controller and the antiwindup compensator[9].A recent modifcation of the traditional two-step approach is to design the linear controller and the anti-windup compensator simultaneously.There are several instances which considered the simultaneous design.In[9],the simultaneous synthesis in the multiplier setting was addressed.A suffcient condition was established to make sure the closed-loop system guarantees a domain of attraction with a given ellipsoid.In[10],a systematic approach using deadzone loops was presented for the simultaneous construction of an output feedback controller and an anti-windup compensator,and an algorithm that minimizes the regional L2gain was developed.In[11], a linear output feedback controller and a static anti-windup compensator were simultaneously designed using multiobjective convex optimization.The synthesis method proposed in[11]was also extended to the delayed anti-windup scheme[12].In[13],the authors proposed a simultaneous synthesis method for a nonlinear controller,which contains a linear dynamic compensator,two static anti-windup loops,and an input saturating integrator system.All the simultaneous synthesis methods mentioned above relied on linear matrix inequalities(LMIs).

Although the simultaneous synthesis has been studied by several groups of researchers,some problems remain open.Here,we are interested in searching for a better performance tradeoff between the linear controller and the anti-windup compensator to achieve a further improved performance of the closed-loop system.Our work is basedon extending the simultaneous anti-windup design method in[11]and the output-feedback control synthesis in[14]. We consider the linear systems subject to actuator saturation nonlinearities,and develop a simultaneous synthesis method to obtain a controller which achieves a desirable performance.In particular,the controller consists of a linear output feedback controller and a static anti-windup compensator.The performance index adopted in this paper is the induced L2gain.

The rest of this paper is organized as follows.In Section 2,we provide a general description of the anti-windup design problem.In Section 3,we demonstrate the simultaneous design in details,and an LMI based optimization is established to search for the performance tradeoff between the linear controller and the anti-windup compensator.In Section 4,we illustrate the proposed method with a numerical example.Finally,we conclude the paper in Section 5.

We use standard notation throughout the paper.For a non-Hermitian real matrix A,He(A)=A+AT.To reduce clutter,off-diagonal entries in symmetric matrices will be replaced by“?”.

2.Problem statement and formulation

Consider the following linear plant P:

where xp∈ Rnp,?u ∈ Rnu,ω ∈ Rnω,y∈ Rnyand z∈Rnzare respectively the state vector,the control input vector,the exogenous input vector(possibly containing external disturbance,measurement noise and reference signals),the measurement output vector,and the controlled output vector.Ap,B1,B2,C1,D11,D12,C2and D21are matrices of suitable dimensions.Assume that an output-feedback controller K:

has been designed.Here xk∈Rnkand u∈Rnuare respectively the controller state vector and the controller output vector,and Ak,Bk,Ck,and Dkare matrices of suitable dimensions.The unconstrained interconnection between the plant P and the output feedback controller K is

When the saturation is encountered,the above equality will be replaced by

where sat(·):Rnu→Rnuis the standard decentralized saturation nonlinearity function defned as

where ulimis the amplitude limitation for each ui(i= 1,2,...,nu).

In the anti-windup design,a compensation term using the difference between the plant input signal and the controller output signal q=u??u is added to the linear controller to minimize performance degradation caused by windup,that is

The resulting overall nonlinear closed-loop system(1), (4),(7),(8)is illustrated in Fig.1,where P,C,and AW are the plant,the linear controller and the anti-windup compensator,respectively.

Fig.1 Typical anti-windup scheme

where

The aim of anti-windup synthesis is to design the linear controller and the anti-windup compensator to make sure that the closed-loop system is stable(at least in some regions near the origin)and meets some performance requirements.In the traditional anti-windup,the linear controller and the anti-windup compensator are designed separately.The goal of this paper is to compute the entire controller parameters(i.e.,Λ1,Λ2,Ak,Bk,Ckand Dk) in the anti-windup closed-loop simultaneously.Besides, the synthesis results also provide an opportunity to search for the performance relationships between the linear controller and the anti-windup compensator.

3.LMI-based anti-windup synthesis

In this section,we focus on the simultaneous anti-windup design.For simplicity,we frst consider the single input systems.The synthesis results can be readily extended to the multiple input case which will be discussed later.

3.1L2anti-windup design

The induced L2gain from ω to z is defned as

We construct a quadratic Lyapunov function V = xTPx with P ＞0.To guarantee asymptotic stability of the closed-loop system and the L2gain from ω to z less than γ[15],we require

For subsequent use,we give partition P and its inverse Q=P?1as

where P11and Q11are np×npand symmetric,and defne

In Fig.1,the relationship between q and u satisfes q(q?u)≤0,thus with a positive scalar W,we have

Invoking S-procedure,we get the following suffcient condition to ensure(11):

Substitution for various parameters in(15)using(9), (15)can be written as the following equivalent LMI:

where

where

We use the substitution(18),originally defned in[14], and the substitution(19),originally defned in[11],to calculate the linear controller and the anti-windup compensator,respectively.

Thus,(17)can be expanded into

where

An upper bound on the L2gain can be obtained by minimizing γ subject to the LMI(20)and

Here,one can compute all the controller parameters simultaneously.In this paper,we also expect the linear controller guarantees high-performance in the absence of saturation.Assume that|u(t)|＜ulim,thus the anti-windup compensator will not be activated and the closed-loop system degrades to a linear system.The following LMI and the LMI(21)establish stability and an L2gain to be less than?γ in this case[14].

We note that(19)and(20)guarantee the closed-loop system is stable with a performance level γ,no matter whether the saturation happens or not,while(21)and(22) ensure an L2gain of?γ if the controller output satisfes |u(t)|＜ulim.We now combine(20)and(22)and arrive at the following theorem.

Theorem 1The anti-windup closed-loop system depicted in Fig.1 is asymptotical stable and the L2gain from ω to z is less than γ,if there are matrices?Λ1,?Λ2,?A,?B, ?C,?D,P11and Q11such that the inequalities(20),(21) and(22)hold.The optimization problem is

If these LMIs are feasible,then the controller matrices Ak, Bk,Ckand Dkcan be calculated from(18)while Λ1and Λ2satisfy(19).

The optimization problem(23)is a standard LMI eigenvalues optimization[15].In the optimization(23),the positive scalars c and?c are weight factors which are specifed by the designer to balance the level of performance of the linear controller and the anti-windup compensator. It is straightforward to see that a larger?c will result in a smaller?γ,therefore,one can expect to obtain a more aggressive linear controller.We note that only γ is the L2performance guarantee of the closed-loop system.The gain?γ can be described as the measure of the aggressiveness and effectiveness of the linear controller.

We now consider the feasibility conditions of optimization(23).LMI(22)is the same as(1:4,1:4)block of(20) with γ=?γ,thus if(20)and(21)are feasible,then the optimization(23)will at least have one set of decision variables.We note that the optimization

is similar to the synthesis result obtained in[11].It was shown in[11]that a necessary and suffcient condition for the existence of a solution to the simultaneous synthesis is that all the eigenvalues of Aphave negative real parts. Thus,we can conclude that if Apis Hurwitz,then the optimization(23)is guaranteed to have a solution.Otherwise, the two-step anti-windup synthesis is not always feasible even the plant is open-loop stable[16].

For the extensions of the synthesis results proposed in this paper,we have the following remark.

Remark 1The synthesis results are readily extended to multiple input systems.For the multiple input systems, the only difference is that W and M in all the equations are diagonal positive defnite matrices.In this paper,we focus on the L2anti-windup synthesis.Various other performance indices,such as the peak-to-peak gain and the energy-to-peak gain,have been adopted in the anti-windup design.In[14],the authors presented an extensive catalog of linear performance indices,all of which can be extended to the simultaneous anti-windup design using the similar derivation steps from which we get Theorem 1.

3.2Well-posedness of the closed-loop system

In this subsection,we consider the well-posedness of the closed-loop system.The closed-loop system(9)is said to be well-posed if,for each pair of(x,ω),there is a unique u satisfying the algebraic equation[9].We show that the well-posedness of the closed-loop system is guaranteed by Theorem 1.

Theorem 2The closed-loop system(9)is well-posed under the LMI-based optimization problem(23).

ProofFor the proof of Theorem 2,we need the following lemma.

Lemma 1[17,18]Given a square matrix D,if?2I+ D+DT＜0,then I?DΔ is nonsingular for all Δ such that z?→Δz belongs to sector[0,I].

For the closed-loop system(8),we have

Defne

whereΔ∈[0,I]is the deadzone nonlinearity.Substituting (26)in(25)and moving u term to the left side result in

Thus the well-posedness of the closed-loop system is equivalently to the nonsingularity of matrix I+DuηΛΔ. Using Lemma 1,we get the following suffcient condition:

which can be rewritten as

Note that the left-hand side of the above inequality is the same as the(4,4)block of(17).Thus,we can conclude that the LMI-based optimization problem(23)guarantees the well-posedness of the closed-loop system. ?

4.Simulation results

We consider the spring-gantry system taken from[19].As depicted in Fig.2,the input of the system is the voltage applied to a DC motor that drives the cart,the output is the pendulum angle and cart position,and the external disturbance signal is the disturbance forces applied to the pendulum.The state xp,control input?u and external disturbance ω are defned as

where p is the position of the cart,m;˙p is the speed of the cart,m/s;θ is the angular position of the pendulum, rad;˙θ is the angular rate of the pendulum,rad/s.And?u is the volts of the DC motor,V(limited to[–5,5]);ω is the impulse on the pendulum,N·s.

Fig.2 The damped spring-gantry system

The system matrices can be obtained from[19].

It was shown in[19]that static anti-windup is not feasible for a specifed linear controller for this example,and a dynamic anti-windup compensator which guarantees an induced L2gain performance of 181.82 was designed by the two-step design procedure.Using Theorem 1,we frst evaluate the performance tradeoff between the linear controller and the anti-windup compensator.Fig.3 shows the values of γ and?γ versus?c with c=1.As we expect,by using a larger?c,one can obtain a smaller?γ and a larger over-all L2gain γ.

Fig.3 γ and?γ versus?c with c=1.

By choosing c=1 and?c=5,we get γ=365.452 2 and?γ=92.735 4.The resulting linear controller and antiwindup compensator are given in(30).Simulation results for a small tap(a constant force of 4 N with duration0.01s) and a larger tap(a constant force of 8 N with duration 0.01 s)are depicted in Fig.4 and Fig.5,respectively. For comparison,we evaluate the resulting linear controller and anti-windup compensator on the ideal system(systemwithout bounded actuator),and the system under bounded actuator and linear controller but without the anti-windup compensator.Also plotted in the fgures are the system responses of the dynamic anti-windup compensator designed in[19].As Fig.4 and Fig.5 suggest,for both small and large disturbances,the simultaneous anti-windup design method proposed in this paper leads to the best performance,and forces the system to leave the saturation zone earliest.We also note that the simultaneous anti-windup design avoids the control chattering that exists in the dynamic anti-windup design.

Fig.4 System response for a small disturbance

Fig.5 System response for a large disturbance

5.Conclusions

We present a simultaneous anti-windup synthesis for linear systems subject to actuator saturation,and the parameters of an output-feedback linear controller and a static anti-windup compensator could be simultaneously computed.The synthesis results are cast as an optimization over LMIs.In addition,the proposed framework,when compared with the existing results,is novel in that it provides an opportunity to search for a better performance tradeoff between the linear controller and the anti-windup compensator.Simulation results confrm the effectiveness of the proposed method.

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Biographies

Maopeng Ran was born in 1990.He received his B.E.degree in automatic control from Beihang University,in 2012.He is now a Ph.D.candidate in the School of Automation Science and Electrical Engineering,Beihang University.His research interests include missile guidance and control,and anti-windup techniques.

E-mail:rmppinbo@asee.buaa.edu.cn

Qing Wang is a professor in the School of Automation Science and Electrical Engineering at Beihang University.She received her Ph.D.degree in fight control,guidance and simulation from Northwestern Polytechnical University,in 1996.She has authored or co-authored more than 150 papers.Her current research interests are missile guidance and control, switch control,and fault detection.

E-mail:wangqing@buaa.edu.cn

Chaoyang Dong is a professor in the School of Aeronautic Science and Engineering at Beihang University.He received his Ph.D.degree in guidance,navigation and control in 1996 from Beihang University.He has authored or co-authored more than 100 papers.His research interests include dynamics and control of fight vehicles,modeling and simulation of aerospace vehicles,and synthesis of aerospace electrical system.

E-mail:dongchaoyang@buaa.edu.cn

Maolin Ni was born in 1963.He received his Ph.D.degree in automatic control from the China Academy ofSpace Technology,Beijing,in 1992.He was a research assistant at University of California, Davis,from 1997 to 1998,and a research fellow at Nanyang Technological University,Singapore,from 1998 to 2004.He was an engineer,senior engineer and professor at Beijing Institute of Control Engineering during 1992–1997 and 2004–2011.Currently,he is the deputy editor-in-chief of Chinese Journal of Astronautics at Chinese Society of Astronautics.His research interests include aerospace control and robust tolerant control.

E-mail:niml@bice.org.cn

10.1109/JSEE.2015.00016

Manuscript received November 06,2013.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61074027;61273083).