This paper proposes a design methodology of triangular controllers for full MIMO stable plants. The procedure is based on an optimal design for a triangular model of the plant. Stability and appropriate performance can be

achieved by adjusting a set of design parameters involved in a weighting function. The resulting controller provides integration and can be computed analytically

Oyarzún, Diego A.

Salgado, Mario E.

English

MURAL - Maynooth University Research Archive Library

SUBOPTIMAL TRIANGULAR CONTROLLERDESIGN METHODOLOGY FOR FULL MIMOSTABLE SYSTEMSDiego R. Oyarzu´n ∗,1 Mario E. Salgado ∗,2∗Department of Electronic EngineeringUniversidad Te´cnica Federico Santa Mar´ıaValpara´ıso, ChileAbstract: This paper proposes a design methodology of triangular controllersfor full MIMO stable plants. The procedure is based on an optimal design fora triangular model of the plant. Stability and appropriate performance can beachieved by adjusting a set of design parameters involved in a weighting function.The resulting controller provides integration and can be computed analytically.Keywords: Multivariable control, optimal control, sparse systems.1. INTRODUCTIONClassical multivariable (MIMO) control sys-tem design strategies (Goodwin et al., 2001; Sko-gestad and Postlethwaite, 1996) do not take intoaccount structural restrictions on the controller.This means that these controllers generate thecontrol inputs based on information from everymeasurement of the process, i. e. these techniquesdeliver full MIMO controllers. However, differentpractical reasons (ease of tuning, maintenance,cabling, etc.) may make full MIMO controllersunsuitable for certain applications (Salgado andConley, 2004) and hence, the need of imposingstructural constraints on the controller appearsnaturally. The structural restrictions in which weare interested are those known as sparse struc-tural constraints (Rotkowitz and Lall, 2002). Thisclass of structural constraints refer to which pro-1 d.o@elo.utfsm.cl2 msb@elo.utfsm.clcess measurements are used to build each controlinput. Common control structures in this classare the decentralised (diagonal) and triangularstructures. The first stage in designing a structurerestricted controller is the controller structure se-lection (Salgado and Conley, 2004). This task isusually accomplished with aid of interaction mea-sures such as the RGA and the PM (see (Salgadoand Conley, 2004) and the references therein). Inspite that there are many studies regarding thistopic, the lack of results on restricted structurecontroller design methodologies is evident. Themain difficulty behind this is that an optimalcontrol approach yields a nonconvex optimisationproblems (Qi et al., 2004).In the case of triangular controller structure,some interesting results are reported in (Qi etal., 2004; Claveau and Chevrel, 2005), where nu-merical methodologies are developed to build tri-angular design procedures. However, those meth-ods rely on the assumption that the plant is alsotriangular and do not make any considerations onhow to use the proposed techniques in the fullMIMO case.This paper proposes a novel methodologyfor the design of linear discrete time triangularcontrollers for full MIMO stable systems. Theproposed design procedure is based on computingan optimal triangular controller for a triangularmodel of the plant. This triangular controller pro-vides integration and can be computed explicitly.The stability and performance on the loop builtaround the full MIMO plant can be achieved byan appropriate adjustment of a weighting functioninvolved in the optimisation problem.The text is organised as follows: Section2 defines the main notation and mathematicalpreliminaries to be used on the remaining of thepaper. In Section 3 the main problem is defined.Sections 4 and 5 describe the design methodologyproposed in this paper. Finally, Sections 6 and 7expose some illustrative examples and concludingremarks, respectively.2. NOTATION AND PRELIMINARIESMost of the notation used throughout thispaper is standard. The (i, j)th element and the ithcolumn of a given matrixA are denoted as Aij (or[A]ij for clarity) and A∗i respectively. The n× nnull matrix is denoted as 0n and the ith elementof the canonical base of Rn is denoted as eni . Thecomplex function spaces (R)L2, (R)H2, (R)H⊥2 )and (R)H∞ are defined as usual (Silva and Sal-gado, 2005). The norm in L2 is called 2-norm anddenoted as ||·||2. GivenA(z) ∈ Cn×n, we define its`th submatrix as A`(z) ∈ C(n−`+1)×(n−`+1) suchthat[A`(z)]ij = A(i+`−1)(j+`−1)(z),∀i ≤ n− `+ 1∀j ≤ n− `+ 1(1)Note that according to this definition, A1(z) =A(z) and An(z) = Ann(z). Given a stable trans-fer matrix A(z), we define its generalised leftunitary interactor (GLUI), ξA(z), as a minimumphase matrix such that the product ξA(z)A(z) isbiproper, minimum phase and stable. Note thatsince ξA(z) is unitary, then ||ξA(z)A(z)||22 =||A(z)||22. An algorithm to build such interac-tor matrices can be found in (Silva and Sal-gado, 2005). In this paper we assume that allsystems are linear, time invariant and discretetime.3. PROBLEM DEFINITIONOptimal controllers must be thought interms of the minimisation of a cost functionalthat is meaningful for the control objectives. Acommon choice (Goodwin et al., 2001; Skoges-tad and Postlethwaite, 1996) is to consider theweighted loop sensitivity function, i. e. minimiseJ (G,W ) =∣∣∣∣∣∣∣∣So(z)W (z)z − 1∣∣∣∣∣∣∣∣22, (2)where So(z) is the loop sensitivity function andthe matrix W (z) is biproper, stable, minimumphase and such thatW (z) = diag {W1(z),W2(z), . . . ,Wn(z)} . (3)The weighting matrix W (z) is usually chosen todefine the bandwidth of the different channels ofthe control loop. This can be done with a diagonalW (z), so that (3) does not pose any serious limi-tation to the generality of the control problemsencompassed by minimisation of (2). Moreover,the weighting factor (z − 1)−1 in (2) makes thefunctional J (G,W ) well defined if and only ifSo(1) = 0, which is equivalent to consider onlycontrollers that provide integration and hence,allow perfect steady state tracking of constantreference signals.If the plant G(z) is stable, the Youlaparametrisation of all stabilising and proper con-trollers (Goodwin et al., 2001), leads to the loopsensitivity functionSo(z) = I −G(z)Q(z), (4)where Q(z) ∈ RH∞. A necessary and sufficientcondition for the controller to have integral actionis to force Q(1) = G−1(1). Thus an additionalassumption is necessary.Assumption 1. The plant has non singular DCgain, i. e. detG(1) 6= 0.Additionally, we are interested in triangular con-trollers belonging to the class St defined as followsSt ={X(z) ∈ Cn×n : Xij(z) ≡ 0∀j > i}. (5)Without loss of generality, we assume that thetransfer matrices in St are lower triangular.If the controller C(z) is constrained to belower triangular. This is equivalent to ensureQ(z) (I −G(z)Q(z))−1 ∈ St. (6)Using (2),(4) and (6), the optimal control prob-lem of interest can be stated as that of findingQt opt(z) such thatQt opt(z) = arg minQ∈RH∞∩Ω∣∣∣∣∣∣∣∣I −G(z)Q(z)z − 1 W (z)∣∣∣∣∣∣∣∣22,(7)where Ω is the set of all transfer matricesQ(z) such that (6) holds. The set Ω is noncon-vex (Rotkowitz and Lall, 2002; Qi et al., 2004;Rotkowitz, 2005). This poses a serious difficultyto the computation of an explicit solution of theproblem, so that the use of numerical search algo-rithms seems inescapable in the general formula-tion given by (7) (Qi et al., 2004).Previous research work (Silva et al., 2005)show that for the stable case, if the plant is trian-gular itself and non singular a. e., then problem(7) is convex and can be restated asQt opt(z) = arg minQ∈RH∞∩St∣∣∣∣∣∣∣∣I −G(z)Q(z)z − 1 W (z)∣∣∣∣∣∣∣∣22(8)This formulation is restricted only to those plantsbelonging to St, but its convexity allows to obtainan explicit solution to the optimisation problem.4. OPTIMAL DESIGN FOR TRIANGULARMODELNext lemma gives a preliminary result whichis necessary for the derivation of the main theoremof this section.Lemma 2. Let G(z) ∈ RH2 be a transfer matrixwithout zeros on the region |z| = 1 and let W (z)be a scalar, biproper, stable and minimum phasetransfer function. Consider the cost functionalJ (G,W ) defined in (2), thenQopt(z) = arg minQ(z)∈RH∞J (G,W ) (9)=(G˜(z)W (z))−1 {[ξG(z)W (z)]⊥|z=1+ [ξG(z)W (z)]2} , (10)where G˜(z) = ξG(z)G(z), ξG(z) is the GLUI ofG(z) and [·]2 ([·]⊥) denotes the portion in RH2(RH⊥2 ) of the argument.Proof: According to (2) and introducing theGLUI of G(z), the cost functional can be writtenasJ (G,W ) =∣∣∣∣∣∣∣∣∣∣ξG(z)W (z)− G˜(z)W (z)Q(z)z − 1∣∣∣∣∣∣∣∣∣∣22,(11)where it has been used the fact thatW (z) is scalarand G˜(z) = ξG(z)G(z). An orthogonal decom-position (Goodwin et al., 2001) of the productξG(z)W (z) can be made asξG(z)W (z) = [ξG(z)W (z)]⊥ + [ξG(z)W (z)]2 .(12)Substituting (12) in (11) and using the orthogonalseparation of the 2-norm, it follows thatJ (G,W ) = ||A(z)||22 + ||B(z)||22 , (13)where the matrixA(z) is independent ofQ(z) andB(z) =∣∣∣∣∣∣∣∣ [ξG(z)W (z)]⊥|z=1 + [ξG(z)W (z)]2z − 1−G˜(z)W (z)Q(z)z − 1∣∣∣∣∣∣∣∣∣∣22. (14)Then, the optimal Youla parameter that minimi-ses (13) is such that sets B(z) to zero, i. e.Qopt(z) =(G˜(z)W (z))−1 {[ξG(z)W (z)]⊥|z=1+ [ξG(z)W (z)]2} ∈ RH∞, (15)and the result follows. 2Lemma 2 gives an explicit solution to theproblem of minimising the cost functional in (2)with a scalar weighting function and withoutstructural restrictions on the controller.Suppose that a triangular model Gt(z) ∈ Stof the full MIMO plant G(z) is available. ThenLemma 2 allows to solve the optimisation problem(8) for Gt(z) and hence, to obtain the optimaltriangular controller for the triangular model. Thefollowing theorem is the key idea leading to themain result of this paper.Theorem 3. Let Gt(z) ∈ RH2 ∩ St be a trans-fer matrix without zeros in the region |z| = 1and W (z) = diag {W1(z),W2(z), . . . ,Wn(z)}a biproper, stable and minimum phase transfermatrix such that Wi(z) is scalar ∀i = 1, 2, . . . , n.Define Qi opt(z) as the unrestricted optimal so-lution of the optimisation problem in Lemma 2using the functional J (Gi,Wi), where Gi(z) isthe ith-submatrix of Gt(z), i. e.Qi opt(z) = arg minQ(z)∈RH∞J (Gi,Wi) . (16)ThenQt opt(z) = arg minQ(z)∈RH∞∩StJ (Gt,W ) (17)=n∑i=1diag {0i−1,Qi opt(z)}1ii, (18)where the special matrix 1ii ∈ Cn×n has zeros inall of its elements, except in the (i, i)th, in whichhas a 1.Proof: Using elementary properties of the 2-norm and taking advantage of the facts thatW (z)is diagonal and G(z) ∈ St, the cost functionalJ (G,W ) can be vectorised (Goodwin et al., 2001)and expressed asJ (G,W ) =∣∣∣∣∣∣∣∣Λ−GA(z)Qvec(z)z − 1∣∣∣∣∣∣∣∣22, (19)whereΛ(z) =[Λ1(z) Λ2(z) · · · Λn(z)]T, (20)Qvec(z) =[Q∗1T (z) Q22(z) Q32(z) · · ·Qn2(z) · · · Qnn(z)]T, (21)GA(z) = diag {G1(z)W1(z),G2(z)W2(z), . . . ,Gn(z)Wn(z)} , (22)and Λi(z) = Wi(z)en−i+11 . Expression (19)allows to separate the optimisation problem (17)in n independent problems of lower dimensions.In fact, using properties of the 2-norm, (19) canbe rewritten asJ (G,W ) =n∑i=1∣∣∣∣∣∣∣∣Λi(z)−Gi(z)Wi(z)Qivec(z)z − 1∣∣∣∣∣∣∣∣22(23)where Qivec(z) ∈ Cn−i+1 is a column vector con-taining every nonzero element of the ith columnof Q(z), i. e.Qivec(z) =[Qii(z) Q(i+1),i(z) · · ·Qni(z)]. (24)Since Λi(z) ∈ RH2, ∀i = 1, 2, . . . , n, a proceduresimilar to the one in Lemma 2 can be donein each term of (23). As a matter of fact, byintroducing the GLUI ξGi(z) of each submatrixGi(z) and making the appropriate orthogonaldecompositions on each term it followsJ (G,W ) =n∑i=1{||Mi(z)||22 + ||Ni(z)||22}, (25)where Mi(z) is independent of Qivec(z) andNi(z) =[ξGi(z)Λi(z)]⊥|z=1 + [ξGi(z)Λi(z)]2z − 1− G˜i(z)Wi(z)Qivec(z)z − 1 , (26)with G˜i(z) = ξGi(z)Gi(z). Since the productG˜i(z)Wi(z) is biproper and minimum phase, thenthe matrix(G˜i(z)Wi(z))−1∈ RH∞ and theoptimum Qivec opt(z) is such that sets Ni(z) tozero, i. e.Qivec opt(z) =(G˜i(z)Wi(z))−1{[ξGi(z)Λi(z)]2+ [ξGi(z)Λi(z)]⊥|z=1} ∈ RH∞.(27)Substituting the definition of Λi(z) in the last ex-pression, the result of Lemma 2 can be recognisedasQivec opt(z) =(G˜i(z)Wi(z))−1{[ξGi(z)Wi(z)]2+ [ξGi(z)Wi(z)]⊥|z=1}en−i+11(28)= Qi opt(z)en−i+11 , (29)whereQi opt(z) = arg minQ(z)∈RH∞J (Gi,Wi) . (30)Finally, the optimal triangular Youla parameter,Qt opt(z), can be reconstructed with an appropri-ate arrangement of the vectors Qivec opt(z) asQt opt =n∑i=1diag {0i−1,Qi opt(z)}1ii ∈ RH∞,(31)and the result follows. 2The form of (18) puts on evidence that theoptimal triangular controller is built upon the fullMIMO optimal controllers for every submatrixGi(z) of Gt(z) considering the scalar weightingfunction Wi(z) on each cost functional. Moreover,(18) shows that[Qt opt(z)]nn = arg minQ(z)∈RH∞J(Gnn,Wn), (32)which means that the (n, n)th element of theoptimal triangular Youla parameter must be builtbased on an optimal SISO loop around Gnn(z).This makes sense by noting that in a triangularsystem, the nth input is the only input that affectsa single output.5. TRIANGULAR CONTROL OF FULLMIMO SYSTEMSThis section deals with the problem of usingthe triangular controller defined by Qt opt(z) inTheorem 3 to build a closed loop around thefull MIMO plant G(z). Since the optimisationprocedure is done considering only the triangularportion of G(z), neither performance nor sta-bility are guaranteed when using Qt opt(z) tocontrol G(z). However, it should be noticed thatQt opt(1) = Gt−1(1), which implies that thetriangular controller of the previous section gua-rantees integration in the loop around G(z) (theproof is omitted due to lack of space).Assume that the procedure given in the pre-vious section delivers a Youla parameterQt opt(z)that does not stabilise the plant G(z). It is knownthat since G(z) is stable, this problem can besolved by reducing the bandwidth of Qt opt(z)(Goodwin et al., 2001). From (18) we have thatevery zero of W (z) is also pole of Qt opt(z). Thismeans that the bandwidth of Qt opt(z) can bearbitrarily diminished by choosing the zeros ofW (z) sufficiently close to z = 1. This analysisimplies that a convenient generic form for eachelement of W (z) isWi(z) =z − aiz(1− ai) , (33)where ai ∈ [0, 1 [ and i = 1, 2, . . . , n. The stabilityof the closed loop can then be attained by anappropriate election of the set of design parame-ters {ai}, i. e. {ai} must be chosen sufficientlyclose to z = 1 such that, at least, stabilityis achieved. When this is accomplished, a finetuning of the design parameters may be done inorder to obtain certain loop performance. Thisprocedure requires that the chosen interactionmeasure suggests that a triangular plant modelis a sensible choice.6. ILLUSTRATIVE EXAMPLESExample 4. Consider a plant modeled by thetransfer matrixG(z) =[z−1.5z2(z−0.9)−0.25(z−5)z2z−1.7z(z−0.9)1z−0.9]. (34)The Participation Matrix (PM) (Salgado andConley, 2004) associated to G(z) is0 10 20 30 40 50 60 7000.51Channel 1 output0 10 20 30 40 50 60 7000.20.40.60.81SamplesChannel 2 outputa1=a2=0.7a1=a2=0.8a1=a2=0.9Fig. 1. Step responses for different triangulardesigns in Example 4.Φ =[0.23 0.050.32 0.4]. (35)The small value of Φ12 suggests that a the use oflower triangular plant model for controller designis a sensible choice. Hence, we consider Gt(z)as the lower triangular portion of G(z). A firstattempt to the triangular design may considera1 = a2 = 0 which is equivalent to chooseW (z) = I. It can be verified that the controllerdefined by Qt opt(z) on a closed loop around G(z)does not stabilise the system. The bandwidthof Qt opt(z) can be reduced by choosing a1 =a2 = {0.7, 0.8, 0.9}. The closed loop is stable forthe three choices of the parameters and the stepresponses of each design are shown in Fig. 1.These results verify the bandwidth reduction asthe design parameters approach z = 1.Example 5. Consider the simplified model of ahigh purity distillation column of eq. (12.17) in(Skogestad and Postlethwaite, 1996) and its zeroorder hold discrete time equivalent given byG(z) =1z − 0.9934[0.58339 −0.574080.71893 −0.72824], (36)where the chosen sampling time is 0.5[min]. ThePM associated to G(z) isΦ =[0.198 0.1920.29 0.31], (37)so that using a triangular plant model is a rea-sonable choice. As in the previous example, thetriangular model Gt(z) is the lower triangularportion of G(z) and, on first instance, we canuse a1 = a2 = 0 in Theorem 3 to computeQt opt(z). It can be verified that, in this case, the0 10 20 30 40 50 6000.20.40.60.81Channel 1 output0 10 20 30 40 50 6000.20.40.60.81Time [min]Channel 2 outputa1=a2=0a1=0,a2=0.6a1=0.6,a2=0.8Fig. 2. Step responses for different triangulardesigns in Example 5.controller defined by Qt opt(z) on a loop aroundG(z) stabilises the plant. If it is needed, we canperform a fine tune on a1 and a2 to achieve adesired performance. Different choices for a1 anda2 are made and the step responses for each oneare shown in Fig. 2. The results are satisfactorysince we are able to modify the bandwidth of thedesign and hence, tune the performance to satisfysome additional transient requirements that couldarise depending on the application.7. CONCLUSIONSThis paper proposes a new methodology forthe design of triangular discrete time controllersfor full MIMO stable plants. The method is in-herently suboptimal since the optimisation proce-dure is formulated considering only the triangularportion of the plant. The full MIMO nature ofthe plant is taken into account by adjusting theloop bandwidth through the weighting functionin the cost index. The proposed method yields anexplicit solution for the controller, which providesintegral action.ACKNOWLEDGEMENTSThe authors gratefully acknowledge thesupport received from UTFSM and from CON-ICYT through grants FONDECYT 1040313 and1060437.REFERENCESClaveau, F. and Ph. Chevrel (2005). A sequentialdesign methodology for large-scale LBT sys-tems. In: Proceedings of the American ControlConference.Goodwin, Graham C., Stefan Graebe andMario E. Salgado (2001). Control System De-sign. Prentice Hall. New Jersey.Qi, Xin, Murti Salapaka, Petros Voulgaris andMustafa Khammash (2004). Structured opti-mal and robust control with multiple criteria:A convex solution. IEEE Transactions on Au-tomatic Control 49(10), 1623 – 1640.Rotkowitz, Michael (2005). Tractable Problems inOptimal Decentralized Control. PhD thesis.Stanford Univesity.Rotkowitz, Michael and Sanjay Lall (2002). De-centralized control information structurespreserved under feedback. In: 41st IEEE Con-ference on Decision and Control. Las Vegas,Nevada, USA.Salgado, Mario and Arthur Conley (2004). MIMOinteraction measure and controller structureselection. International Journal of Control77(4), 367–383.Silva, E. and M. Salgado (2005). Performancebounds for feedback control of nonminimum-phase MIMO systems with arbitrary delaystructure. IEE Proceedings - Control Theoryand Applications 152(2), 211–219.Silva, E., D. Rojas Oyarzu´n and M. Salgado(2005). On structural properties of a classof H2 optimal control problems. Submittedto IEE Proceedings - Control Theory andApplications.Skogestad, S. and I. Postlethwaite (1996). Multi-variable Feedback Control: Analysis and De-sign. Wiley. New York.

Suboptimal triangular controller design methodology for full MIMO stable systems

http://mural.maynoothuniversity.ie/1784/1/HamiltonSuboptimal.pdf

Suboptimal triangular controller design methodology for full MIMO stable systems

Abstract

Similar works

Full text

Available Versions

MURAL - Maynooth University Research Archive Library