A Minimax Linear Quadratic Gaussian Method for Antiwindup Control Synthesis
aa r X i v : . [ c s . S Y ] A ug A Minimax Linear Quadratic Gaussian Method for AntiwindupControl Synthesis
Obaid ur Rehman, Ian R. Petersen and Barıs¸ Fidan
Abstract — In this paper, a dynamic antiwindup compensatordesign is proposed which augments the main controller andguarantees robust performance in the event of input saturation.This is a two stage process in which first a robust optimalcontroller is designed for an uncertain linear system whichguarantees the internal stability of the closed loop system andprovides robust performance in the absence of input saturation.Then a minimax linear quadratic Gaussian (LQG) compensatoris designed to guarantee the performance in certain domain ofattraction, in the presence of input saturation. This antiwindupaugmentation only comes into action when plant is subject toinput saturation. In order to illustrate the effectiveness of thisapproach, the proposed method is applied to a tracking controlproblem for an air-breathing hypersonic flight vehicle (AHFV).
I. INTRODUCTIONThe design of a controller for a linear system with inputsaturations is a challenging task. Input saturation can have adisastrous effects [1]. In general, it is common to considercontrol problem involving integral action and in the presenceof input saturations, the state of the integrator can wind upand affect the response of the underlying system. Engineersusing PI and PID control have solved the windup problemsusing ad hoc measures, such as by resetting the integralstates in the case of windup, or by using a modified digitalimplementation of the controller, which does not depend onthe integral error terms. However, these ad hoc measureslack mathematical rigor and are mostly heuristic [2], [3].Attempts have also been made by control system designersto penalize the control output so that actuator limits wouldnever be violated. These method are useful to some extentbut may result in a design which is too conservative andunable to utilize the full capability of the available controlauthority.In order to use full controller authority of the controllerand to recover the nominal performance of the system in theevent of actuator saturation, considerable attention has beengiven to antiwindup augmentation over the last decades. Themain advantage of using an antiwindup augmentation schemeis that the system will operate with its full capability interm of robustness and performance in the absence of inputsaturation. The antiwindup correction only come into effect
Obaid Ur Rehman is with school of Engineering and Information Tech-nology, University of New South Wales at the Australian Defence ForceAcademy, Canberra, Australia. ( [email protected]
Prof. Ian R. Petersen is with school of Engineering and InformationTechnology, University of New South Wales at the Australian Defence ForceAcademy, Canberra, Australia. ( [email protected] )Barıs¸ Fidan is with department of Mechanical and Mechatronics Engi-neering, University of Waterloo, Canada ( [email protected] ) in an event of input saturation. Antiwindup augmentation, asthe name suggests, is a two-stage design procedure. In thisdesign procedure, the requirement of small signal behavior ofthe nominal system is guaranteed by ignoring the saturation.Then antiwindup compensation is added to recover thenominal performance in the presence of saturation.Noting the importance of the antiwindup augmentation tohandle saturation, different methods have been appeared inthe literature over the decades. Most of the methods relyingon the H ∞ optimal control approach have considered theantiwindup problem as an L gain minimization problem[4], [5], [6], [7], [8]. In these papers, saturation nonlinearitiesare considered as a sector or dead-zone type nonlinearitiesand the antiwindup problem is solved by formulating it in aconvex optimal framework. In [6], [9], [10], linear fractionaltransformation (LFT) and linear parameter varying (LPV)approaches have been proposed for parameter-varying satu-rated systems. All of these methods employs linear matrixinequalities (LMIs) as a tool to compute global optima in asimplified way. The main drawback of the LMI frameworkis that in several situations LMI constraints are unfeasibleand hence no solution exists. Also, LMI methods can sufferfrom numerical and computational problems in the case ofhigh order system. These problems are solved to some extentin [8] by characterizing the antiwindup problem in terms ofa nonconvex feasibility problem, which reduces to a convexfeasibility problem when a certain rank constraint becomesinactive. Another solution of this problem is attempted in[11] where, the non-feasible problem was solved based on theapproximate solution of an LMI. The results of [8], [11] areonly applicable to linear stable plants. Antiwindup controllerdesign for an unstable system was considered in [5] whichis also based on the LMI method. However, the results areonly valid when no uncertainty exists in the original systemexcept for the uncertainty due to the actuator saturation.In this paper we have considered stabilizable unstablelinear systems subject to parameter uncertainties and inputsaturations. The formulation of the antiwindup problem isbased on an H ∞ framework. In this framework a dynamicantiwindup compensator is proposed, which solves a risk-sensitive control problem using the minimax LQG (a robustversion of LQG control) design method [12]. This is similarto the L gain optimization problem. The only differenceis that in minimax approach, we minimize the upper boundon a time averaged cost function and guarantee the robustperformance. This method not only allows for the uncer-tainty which arises from saturation nonlinearities but alsoaccounts for uncertainties present in the original system. Inhe propose method, a suitable robust controller is obtainedby ignoring actuator saturation and then the closed loopsystem is augmented by a robust antiwindup compensator.The proposed antiwindup controller is then applied tosolve the actuator saturation problem in an air-breathing hy-personic flight vehicle. The control problem in this exampleposes significant challenges as it offers a challenging trade-off among conflicting requirements. In high speed aircraft,the model is subject to parameter uncertainties due to alarge flight envelop and a good robust controller is requiredto be designed. However, in the absence of antiwindupaugmentation, limited control authority seldom allows therobust controller to work in its full capacity and thus degra-dation in the performance is unavoidable. The antiwindupcompensator proposed in this paper solves these problemsquite effectively and can be easily implemented on-boardthe aircraft.The paper is organized as follows. Section II describes theclass of uncertain linear systems and uncertainties consideredin the paper. Section III describes the general antiwindupproblem and the synthesis of an antiwindup compensatorusing the minimax LQG method. The application of theproposed method to an AHFV control problem along withsimulation results are presented in Section IV. The paperis concluded in Section V, with some final remarks on theproposed procedure.II. S YSTEM D EFINITION
Consider an unconstrained uncertain linear unstable plantgiven by ˙ x ( t ) = Ax ( t ) + Bu ( t ) + l X j =1 C j ζ j ; z i ( t ) = K i x ( t ) + G i u ( t ); i = 1 , , · · · , my ( t ) = ¯ C x ( t ) + D u ( t ) (1)where x ( t ) ∈ R n is the plant state, u ( t ) ∈ R n u is the controlinput, ζ j ∈ R n ζ is the uncertainty input, z i ( t ) ∈ R n q is theuncertainty output, y ∈ R n y and A , B , C j , K i , G i , ¯ C , and D are the matrices of suitable dimensions. Assume alsothat the uncertainty in the system satisfy following integralquadratic constraint condition (IQC) [12] Z ∞ ( k z j ( t ) k − k ζ j ( t ) k ) dt ≥ − x T (0) d j x (0) , (2)where d j > for each j = 1 , · · · , l is a given positivedefinite matrix. Also, assume a minimax optimal linearquadratic regulator (LQR) control [13] of the following formexists for the system (1) which is well posed and guaranteesinternal stability of the closed loop system: u ( t ) = − G τ x ( t ) , (3)where G τ = E − [ B T X τ + G T K ] . Here, X τ is obtained by solving a game type Riccatiequation ( A − BE − G T K ) T X τ + X τ ( A − BE − G T K )+ X τ ( CC T − BE − B T ) X τ + K T ( I − GE − G T ) K = 0 , (4)where K = Q / √ τ K ... √ τ l K l , G = R / √ τ G ... √ τ l G l , E = GG T ,C = h √ τ C . . . √ τ l C l i for given parameters τ > , τ > ,..., τ l > . Theparameters τ j for j = 1 , · · · , l are selected such that theygive a minimum value of a bound on the following costfunction F = Z ∞ [ x ( t ) T Qxt ) + v ( t ) T Rv ( t )] dt, (5)where, Q = Q T > and R = R T > are the state andcontrol weighting matrices respectively and the solution X τ of the Riccati equation (4) should be symmetric and positivedefinite. The bound on the cost function is given asmin [ x T (0) X τ x (0) + l X j =1 τ j x T (0) d j x (0)] , (6)where the initial condition x (0) = 0 ∈ R n is assumed to beknown. Alternatively, the initial condition can be assumed tobe a zero mean unity variance random vector, in which casethe trace of the matrix in (6) would be minimized. Theorem 1:
Consider the uncertain linear system (1) withcost function (5). Then for any τ > , τ > ,..., τ l > suchthat Riccati equation (4) has a positive definite solution, thecontroller (3) is a guaranteed cost controller for this uncertainsystem with any initial condition x (0) ∈ R n . Furthermore thecorresponding value of the cost function (23) is bounded bythe quantity (6) for all admissible uncertainties and moreover,the closed loop system is absolutely stable. Proof:
See [12], [13]. (cid:4)
III. T HE A NTIWINDUP P ROBLEM
In real control problem, the input u ( t ) is subject to satu-ration and thus the control law (3) may not give satisfactoryperformance. Indeed, each component u i of the input vector u is subject to a saturation nonlinearity of the form shownin Fig. 2. The antiwindup problem here is to design asuitable antiwindup compensator which guarantees adequateperformance of the closed loop system in the presence ofsaturation nonlinearity. We represent saturation nonlinearityig. 1: Closed loop system structure with antiwindup aug-mentation.as a deadzone type sector bounded uncertainty and definescorresponding deadzone function φ i ( u i ) as follows: φ i ( u i ) = u i − u i sat = ( , | u i | < | u i max | sgn ( u i )( u i − | u i max | ) , | u i | > | u i max | (7)where, u sat is the input vector which is subject to saturation.Also, we augment the controller in (3) with an antiwindupaugmentation (see Fig. 1) as follows: u ( t ) = − G τ x ( t ) + v, (8)where v is the signal from the antiwindup compensator ofthe form ˙ x aw = A aw x aw + B aw ¯ ζ,v = C aw x aw , (9)where A aw , B aw and C aw are the matrices of suitabledimension and ¯ ζ is the input to the compensator. Theprocedure to obtain these matrices will be given in sequel. A. Antiwindup controller design
The design of an antiwindup augmentation system forthe uncertain linear system is designed by defining a newequivalent uncertain model, considering the saturation non-linearities as sector bounded uncertainties (deadzone type)and then froming an equivalent closed loop system using (8).Firstly, we define a domain of attraction D c by restricting u i ∈ [ u i , ¯ u i ] for i = 1 , · · · , m where, ¯ u i is the maximumallowed value of u i and u i is the minimum allowed valueof u i . The corresponding sector bound can be selected byappropriately choosing < ǫ i ≤ (see Fig. 2 and Fig. 3)and by using the following equation. ¯ u i − u i max = ǫ i ¯ u i , ¯ u i = u i max − ǫ i (10) Fig. 2: Control input saturation.Fig. 3: Deadzone uncertainty representation with domain ofattraction.Also, note that u i min = − u i max . The open loop systemcan be written using (1) considering all input saturations asfollows: ˙ x ( t ) = Ax ( t ) + Bu sat + l X j =1 C j ζ j ;= Ax ( t ) + B ( u − φ ( . )) + l X j =1 C j ζ j ;= Ax ( t ) + Bu − Bφ ( . ) + l X j =1 C j ζ j ; (11)where, φ ( · ) = [ φ i ( u i ) , · · · , φ m ( u m )] T .The sector bound on each φ i ( u i ) for i = 1 , · · · , m can bewritten as follows: ≤ φ i ( u i ) u ≤ ǫ i ≤ φ ( u i ) u i ≤ ǫ i u i . (12)We then define a new uncertainty input as ˆ w i ( u i ) = φ i ( u i ) − ǫ i u i ⇒ ˆ w ( u i ) + ǫ i u i φ i ( u i ) , (13)ig. 4: Uncertainty representation with new sector.and write the uncertainty corresponding to this uncertaintyinput in a new sector as follows: ≤ ˆ w i ( u i ) u i + ǫ i u i φ ( u i ) u i ≤ ǫ i u i ⇒ − ǫ i u i ≤ ˆ w ( u i ) u i ≤ ǫ i u i . (14)Also, the sector bound (14) on the saturation uncertainty canbe written in the following form (See Fig. 4). k ˆ w i k ≤ ǫ i | u i | | ¯ z | , (15)where, ¯ z = ǫ i u i . Remark 1:
The formulation presented here is applicableto the case where | u i min | = | u i max | ; i.e. the symmetricsaturation case (see Fig. 2). However, it is straightforwardto extend this formulation to the asymmetric saturation casewhere | u i min | 6 = | u i max | .We can satisfy the bound (11) using the new uncertaintyin (15) as follows: ˙ x ( t ) = Ax ( t ) + Bu + l X j =1 C j ζ j − B ( ˆ w ( · ) + E u x ( t ) = Ax ( t ) + B (1 − E u + l X j =1 C j ζ j − B ˆ w (16)where, ˆ w ( · ) = [ ˆ w ( u ) , · · · , ˆ w m ( u m )] T , E = diag [ ǫ i , · · · , ǫ m ] , ¯ B = B (1 − E ) , ¯ G is a scaling matrixcorresponding to the bound in (15) as given below: ¯ G = ǫ i / · · · . . .
00 0 ǫ m / . (17)Since, we have considered additional uncertainties cor-responding to the saturation uncertainties in (16), the un-certainty outputs of the original system (1) along with theuncertainty output ¯ z corresponding to ˆ w can be written asfollows: ˜ z = ˜ Kx + ˜ Gu, (18) where ˜ z = z z ... z l ¯ z ; ˜ K = K K ... K l ; ˜ G = G G ... G l ¯ G . Also, we can write the complete uncertain linear modelof the system in the presence of actuator saturation as givenbelow: ˙ x ( t ) = Ax ( t ) + ¯ Bu + l X j =1 C j ζ j − B ˆ w ( · ) , ˜ z = ˜ Kx + ˜ Gu. (19)The minimax optimal controller in (8) can be used toobtain closed loop system which guarantees stability forthe uncertain system (1) without saturation uncertainty. Theclosed loop system incorporating the antiwindup signal v in(19) can be written as follows: ˙ x ( t ) = Ax ( t ) + ¯ B ( Gx + v ) + l X j =1 C j ζ j − B ˆ w ( · ) , = ( A − ¯ BG ) x ( t ) + ¯ Bv + l X j =1 C j ζ j − B ˆ w ( · ) . (20)In a similar way using (8) in (19), the uncertainty output canbe written as follows: ˜ z = ˜ Kx + ˜ G ( Gx + v ) , = ( ˜ K + ˜ GG ) x + ˜ Gv. (21)Finally, we can write the closed loop system consideringsaturation uncertainty with antiwindup signal v as follows: ˙ x = ¯ Ax ( t ) + B v + ˜ B ζ, ˜ z = ˜ C x + ˜ D v, ˜ y = ˜ C x + ˜ D ζ, (22)where, ¯ A = ( A − ¯ BG ) , ˜ C = ˜ K + ˜ GG , ˜ D = [ ] , B = ¯ B , ˜ C = , ˜ B = (cid:2) C C · · · C l − B (cid:3) ,ζ = (cid:2) ζ ζ · · · ζ l ˆ w ( · ) (cid:3) T . B. Minimax LQG controller synthesis for antiwindup aug-mentation
We now design an antiwindup controller of the form (9)using minimax LQG design procedure. The design procedurefor the standard minimax LQG is given in [12]. Here, wepresent a summary of the method and then present ourapproach to designing a minimax LQG antwindup controller.The controller (9) can be designed after stabilizing the system(1) using (3) and writing the closed loop system in theform (22). The minimax LQG control problem [12] involvesig. 5: Scaled H ∞ control problem.finding a controller which minimizes the maximum value ofthe following cost function: J = lim T →∞ T E Z T ( x ( t ) T Qx ( t ) + v ( t ) T Rv ( t )) dt, (23)where R > and Q > . The maximum value of the costis taken over all uncertainties satisfying the IQC (2). If wedefine a variable ψ = (cid:20) Q / xR / v (cid:21) , (24)the cost function (23) can be written as follows: J = lim T →∞ ( 12 T ) E Z T k ψ k dt. (25)The minimax optimal controller problem can now besolved by solving a scaled risk-sensitive control problem [12]which corresponds to a scaled H ∞ control problem; e.g. see[14]. The scaled risk-sensitive control problem consideredhere (see Fig. 5) allows for a tractable solution in terms ofthe following pair of H ∞ type algebraic Riccati equationsfor C = 0 .: ¯ AY ∞ + Y ∞ ¯ A T + Y ∞ ( τ − R τ ) Y ∞ + ˜ B ( I − ˜ D T Γ − ˜ D ) ˜ B T = 0 , (26)and X ∞ ( ¯ A − B G − τ Υ Tτ ) + ( ¯ A − B G − τ Υ Tτ ) T X ∞ − X ∞ ( B G − τ B T − τ − ˜ B ˜ B T ) X ∞ + ( R τ − Υ τ G − τ Υ Tτ ) = 0 , (27)where, R τ , Q + τ ˜ C T ˜ C , G τ , R + τ ˜ D T ˜ D , Γ τ , τ ˜ C T ˜ D . In order to obtain solutions to both of the algebraic Riccatiequations Y ∞ > , X ∞ > , the system (22) is required tosatisify the following assumption: Assumption 1:
1. The matrix ˜ C and ˜ D satisfy the condition ˜ C T ˜ D = 0 .2. The matrix ˜ D satisfies the condition ˜ D T ˜ D > .3. ¯ A is Hurwitz.4. The pair ( ¯ A, ˜ B ) is stabilizable and ˜ B = 0
5. The pair ( ¯ A, ˜ B ) is stabilizable and y ( t ) is measurable.6. The matrix ˜ B and ˜ D satisfy the condition ˜ B ˜ D T = 0 . 7. R τ − Υ τ G − τ Υ Tτ ≥
8. The pair ( ¯ A − ˜ B G − τ Υ Tτ , R τ − Υ τ G − τ Υ Tτ ) is de-tectable.9. The pair ( ¯ A, ˜ B ( I − ˜ D T Γ − ˜ D ) ) is stabilizable.If the solutions of the Ricatti equations satisfy I − τ − Y ∞ X ∞ > and the parameter τ > is chosen suchthat it minimizes the cost bound ( W τ ) defined by W τ , tr [( ˜ B ˜ D T )( D D T ) − × ( ˜ D ˜ B T ) X ( I − Y X ) − + τ Y R τ ] , (28)then the antiwindup controller matrices in (9) can be obtainedas follows: C aw = − G − τ ( ˜ B T X ∞ + Υ Tτ ); B aw = ( I − τ − Y ∞ X ∞ ) − ( ˜ B ˜ D T )Γ − ; A aw = ˜¯ A + ˜ B C aw + τ − ( ˜ B − B c ˜ D ) ˜ B T X ∞ . Theorem 2:
Consider the uncertain linear system (22)with the cost function (23) and suppose assumption 1 issatisfied. Then the controller (9) minimzes the bound on thecost function (23) such that W τ ( · ) = inf v ( · ) sup J ( · ) andguarantees the stability of the system (within certain domainof attraction) in the presence of saturation nonlinearity if thefollowing conditions are hold for an arbitrary τ > :1. The algebraic Ricatti equation (26) admits a minimalpositive-definite solution Y ∞ .2. The algebraic Ricatti equation (27) admits a minimalnonnegative-definite solution X ∞ .3. The matrix I − τ Y ∞ X ∞ has only positive eigenvalues;that is the spectral radius of the matrix Y ∞ X ∞ satisfiesthe condition ρ ( Y ∞ X ∞ ) < τ . Proof:
See [12]. (cid:4)
IV. E
XAMPLE
In this section, we apply our proposed antiwindup syn-thesis approach to design a velocity and attitude trackingcontroller with antiwindup augmentation for an air-breathinghypersonic flight vehicle (AHFV). This design examplehas been taken from our previous work [15], [16] and itis observed that the AHFV system is subject to actuatorsaturation. Here, we use the uncertain linearized modelof AHFV which was obtained using the robust feedbacklinearization method in [16]. The linearized model is 2-input and 2-output system which is subject to uncertaintyparameters p , p , · · · , p . For the ease of reference thecorresponding linearized model of the form (1) is shownbelow: ˙ χ ( t ) = Aχ ( t ) + B ¯ v ( t ) + X j =1 C j ζ j ; (29) z i ( t ) = K i χ ( t ) + G i ¯ v ( t ); i = 1 , (30) y ( t ) = ¯ C χ ( t ) + D ¯ v ( t ) (31)here, A = , B = ; C = · · · · · · · · · · · · · · · ... ... . . . ... ... · · · , C = · · · · · · · · · · · · ... ... . . . ... ... · · · · · · ; K = ∆ ˜ w p ) ∆ ˜ w p ) ... ∆ ˜ w p ) ∆ ˜ w p ) , K = ∆ ˜ w p ) ∆ ˜ w p ) ... ∆ ˜ w p ) ∆ ˜ w p ) ; G = ∆ ˇ w p ) ∆ ˇ w p ) ... ∆ ˇ w p ) ∆ ˇ w p ) , G = ∆ ˇ w p ) ∆ ˇ w p ) ... ∆ ˇ w p ) ∆ ˇ w p ) , and χ ( t ) ∈ R is the state vector, and ¯ v ( t ) = [¯ v ¯ v ] T ∈ R is the control input vector. Note that the size of matrix Cj is × . Also, ζ ∈ R , and ζ ∈ R are uncertainty inputs, z ( t ) ∈ R , and z ( t ) ∈ R are the uncertainty outputs, ∆ ˇ w ( · ) and ∆ ˜ w ( · ) represent the magnitude of the uncertaintiesin the system.As a first step, a controller of the from (3) is designedwhich gives a stable close loop system in the absence ofactuator saturations as discussed in the Section II. In thesecond step, we augment the controller with antiwindupcompensation (8) so that the performance degradation isminimized and system remains stable in the selected domainof attraction as discussed in Section III. The antiwindupcompensator of the form (9) is obtained by selecting ap-propriate state and control weighting matrices Q and R . Forthis example, the following parameters are selected to obtaina good antiwindup controller: Q = 1000 × diag (cid:2) , , , , , , , , (cid:3) , (32) R = (cid:20) (cid:21) , τ = 35 . (33)The parameter τ has been selected which gives the minimumcost bound (28) as shown in Fig 6. Remark 2:
The selection of state and control weightingmatrices plays a significant role in the synthesis of the τ Fig. 6: Bound on the cost function with varying τ .minimax LQG antiwindup controller. The selection should bemade such that the dynamic gain of the closed loop transferfunction should not be too high or too low. The antiwindupsignal v should be smaller than the actual input u . A. Simulation Results
Simulation results using the antiwindup augmentation pro-cedure discussed above are shown in Fig. 7- Fig. 8. Thesolid (blue) line shows the response of the nominal systemwithout actuator saturation, the dashed (red) line shows theresponse with the actuator saturation and the dashed-dot(black) line shows the response with the antiwindup aug-mentation compensator. The results show that in the presenceof actuator saturation, the antiwindup augmentation removesthe actuator saturation degradation in a very effective way.The tracking errors remain small and bounded for both thecases of velocity and altitude reference input commands.V. C
ONCLUSION
In this paper, a minimax linear quadratic Gaussian (LQG)antiwindup augmentation compensator has been proposed foran uncertain linear plant subject to input saturation. The de-sign employs a two-stage process in which a robust controlleris designed using minimax linear quadratic regulator (LQR)without considering actuator saturation as the first step.Then antiwindup augmentation is provided in the secondstep. The proposed approach has been applied to a trackingcontrol problem for an air-breathing hypersonic flight vehicle(AHFV) system. Simulation results show that the proposedapproach is very effective in dealing with actuator saturation.It is observed that the proposed antiwindup augmentation,reduces the degradation in performance. Antiwindup designsfor nonlinear uncertain systems using feedback linearizationare areas for future research. ft/ s e c Velocity error0 1 2 3 4 5−40−2002040 ft Altitude errorTime (sec) NominalWith saturation With antiwindup
Fig. 7: Velocity and altitude tracking error responses. Thesolid line shows the response with the nominal model, ‘--’ shows the response with input saturation, ‘-.’ shows theresponse with antiwindup augmentation.VI. ACKNOWLEDGMENTSThis research was supported by the Australian ResearchCouncils and Australian Space Research Program.R
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