On Stabilizing Sets of PI Controllers for Multiple Time Delayed Process

Joumal of Donghua University (Eng. Ed.) Vol.26, No.1 (2009)On Stabilizing Sets of PI Controllers for Multiple Time DelayedProcessZHENG Da(郑达), REN Zheng-yun(任正云)，FANG Jian-an(方建安)"College of Information Science and Technology. Donghua University, Shanghai 201620, ChinaAbstract: This paper considers the problem of stabilizingscts of stabilizing PID parameters. With the complcte setsmultiple time delayed processes using proportional integralof stabilizing PID controller parameters being available for(PI) controller. The presented approach is based on findinga given proccss, it can avoid the time consuming stabilityall possible values of control parameters which will result incheck for each set of PID controller parameters in thepure imaginary roots of closed loop characteristic equationsearch process and thereby save the controller tuning timc.under all process parameters fixed. The ergodic search ofThe boundaries of the stabilizing PID controller parameterthree PI control parameters are converted from the range ofrcgion can be determined by the technique of Dinfinity to finite range by introducing trigonometric tangentparitionE5s] . Recently, a mcthod based on using a versionfunction. After all possible stability boundaries arof Hermitc Bichler theorem applicable to quasi-polynomialsobtained, the Nyquist stability method is used to determinehas been used to determine the completc set of stabilizingthe actual stability region of the controller parameters. ThisPID contoller gain parameters for first order plus timemethod also permits design for simultaneous minimum gaindelay systems ' 10.。The mcthod involves finding the zerosand phase margin requirement. An ilustrative example caseof a transcendental equation to determine the range ofis also presented.stabilizing gains.Key words: proportiomal-integral controller ; muliple timeMost industrial processes can be modeled as processesdelay; robust stability; stabilily regionwith single time delay. However, in order to model theCLC number: TP 273Document code: Apractical processes more accuratcly, models with multipleArticle ID: 1672 - 5220(2009)01 - 0036 - 04time delays are requircd. And in parallel cascadecontro1ll-8], the primary loop PID is actually designed fora multiple time delayed process.IntroductionUnfortunately, most methods determining theboundaries of the stabilizing PI controller parameter regionDespite continual advances in control theory anddeal with only single dclay process, which motivated thedevelopment of advanced control stratcgics, thpresent paper.proportional, intcgral, and derivative ( PID) controlThe novel method to stabilize the PI control systemalgorithm still finds wide applications in industrial processpresented here is based on determining all possible values ofcontrol systems. It has been reportedI] that 98% in of thecontrol parameters which will result in pure imaginary rootscontrol loops ihe pulp and paper[1] industries areof characteristic equation. Since the system only carry outcontrolled by proportional integral ( PI)controllers.stability shift on these points, it means some of these valuesMoreover, more than 95% of the controllers used inof control paramcters constitute the stability boundaries.process control applications are of the PID typet2]. Thepopularity among industrial practitioners stems from theThen the Nyquist stability method is uscd to determine theactual stability region.facts that the PID control structure is simple and itprinciple is easy to understand and that the PID controllersare deemed to be satisfactory and robust for a vast majority1 Main Resultsof processes. The primary problem associated with the use ofThe control structure is shown in Fig. 1.PID controllers is tuning, that is, the determination of PIDThe multiple time delayed processes controlled by a PIcontroller parameters for satisfactory control performance.Since the primary requirement of the candidate PIDcontroller with the form ofcontroller parameters is that of making the closcd-loopC(s)=k.+k(1)systcm stable, it is often desircd to construct the complete中国煤化工Reccived date; 2007- 10- 29YHCNMHGFoundation item: National Natural Science Foundation of China (No.60674088)* Corrcspondence should be addresscd l0 FANG Jjan-an, professor, E-mail: jafang@dhu. edu. cnJourmal of Donghua University (Eng. Ed.) Vol. 26, No.1 (2009)37It usually has the following form:τ,=二[tan -(Tw) + kx]CE(s, K)= Po(s, K) + PNu(s, K, r)+0≤tan -'(Tw)≤π, k =0, 1, 2, .(4)ZP.s, K,n, ..心仙ev)erSubstituting (3) into (2) result in a rational2)polynomial; CE(s， T, K) = 0, further transforming itwhere K = (kp, k:) is the vector of control paramctersinto a simpler form by multiplying it with IT (1 + Ts)" :andτ=(π, . tw) is the vector of time delays. Po(s)s an n" dcgree polynomial in s, Pi (s) arc quasi-CE(s, T, K)=CE(s, r, K) (1+ T;s)9vpolynomials in s and all the delays except t. ni is thehighcst order of commensuracy of dclay τi in the dynamics之be(T, K)s*=0(5)(n≤n). PN+I is another quasi-polynomial whichcontains all the remaining terms with lower commensuracySubstituting s= wi for (5) andlevels than n. In short, P((s) are factors multiplying therepresentative exponential expressions of the highcstπ=⊥tan (号)=ucommensuracy of ti.vi=τw∈[0, 2n], i=1,2, . N6)yicldC(s)G(s)ControllerPlant .P(u. K,w)=ce(, Juv.x)|..文0.("，k)awi)*=0 (7)Fig.1 Feedback control systcmReplace the transcendentalities in (1) with RckasiusDividing P(u ,K ,w) into real part and imaginary parttransformation[4], we gctRe(P(u, K, w))= 2 fe(u, K)w*=0 (8a)'e=i+Ts'T,∈R, i=1,..N (3)Im(P(u, K, w))= 2gl(u, K)w'=0 (8b)Eq. (3) is an exact mapping and not an approximationThe ncessary condition for (8a) and (8b) to have afor s= wi, w∈R, with the obvious mapping conditioncommon root (real or complex) is that Sylvester's RcsultantofMatrix[5s] which is defined as:(fu(K, u) fu-.(K, u) fu-z(K, u)0fL(K, u) fL-n(K, u) fL-z(K, u)fz(K, u) f(K, u) f。(K, u)M=gL(K，u) gL-1(K, u) 8L-z(K, u) gL-3(K, u)gL(K, u) 8L-i(K, u) gL-2(K, u) gL-s(K, u)g3(K, u) gr(K, u) g1(K, u) go(K, u)) 2Lx21.(9)It should be singular, which results in terms of k and ukp =tan(P,), q,∈[一气.部](or )det(M)=F(K.u)=F(K, tan号)=0 (I0)k.an(4),.E[告到](11)Such a mapping converts K∈(-∞，+∞) of infiniteIf we regard τ as an uncertain paramcter, then Eq(10) defincs all possible (K，r) on the boundaries ofrange to4∈| -分. 只| of finite rangc, which will makcstability region.the lat中国煤化Iier.Now the bounds of Vi arc known. but the bounds of, by scanning thekp，k, are (一∞，+∞) respectivcly, thus we introducetrajeteMHCNM H! hallalus of(k，the fllowing mapping:r) which will result in pure imaginary roots of (2). But8Joumal of Donghua University (Eng. Ed.) Vol. 26, No. 1 (2009)only valucs of K along the stability boundaries are neededsince τ is constant vector. Notice that if there is certain2 Example Case Study(K"，v" ) which satisfies (10)， there might be acorresponding w° which satisfies ( 8a) and ( 8b )Consider the fllowing combined integrating system:concurrently. Moreover, if the prospective (K*, w*) isalong the stability boundaries, the following condition mustP(s)=-(1 -e9)e中15)be satisfied;This is a combined integrating process companyingtun(“)=tan(器), vi∈[0, 2*],i=1,2.-.Nwith time delays, the whole character of which is stableinstead of being unstable or integrating. Such kind of(12)process exists extensively in steel, petrochemical, grainTherefore, when scanning the trajoctory defined by (10),processing, tobacco, and mineral mining industry but few(12) can be used to chock whether the prospoctive K is on thepeople pay much attention to it. Combined integratingstability boundaries. Now the numerical procedure is as follows;systems are open-loop stable in essence and can be changed(1) Find the prospective trajetories of (K， v)into first order system by first order Pade approximating.from (10).Now most of combined integrating systems are taken as(2) Evaluate the corresponding w∈R satisfying (8a)first order plus time delay process and controlled by routineand (8b) concurrently for each point on these trajectories.control strategy such as PI contoller. But such kind ofIf no such root, w∈R, disqualify the prospective pointapproximation will lead to incorrect result when determine(K，v)。 If such a root exists, w∈R, check whether K isthe stable region of controller parameters.on the boundary of stability region by (12). If all划, i=Here the PI controller is cascaded with gain phase1, 2, . N stisfying (12), save the prospective (K, w).tester and the controller can be defined asOtherwise, disqualifty the prospective (K, w).(3) Repeat step (1) and step (2) until all trajectoriesGc(s)=k, +k(16)are scanned.The closed loop transfer function is:In order to complete stability analysis sufficiently,there are some special and critical cases in our procedure4()=- K(k,s+k)(1-e "e(17)which have to be taken into account.s2+ K(kps+k;)(1 -e"I1)e"?2'We discuss the standing root at s=0 as follows. OneThe characteristic equation of the closed loop transferor more roots at s=0 may exist when K is fixeud to certainfunction is;values, and this case should be treated separately.CE(s, Kp, K,)=s2 +(Kps+ K,)(1 -e")e"72"Understandably there can be many scenarios of such cases,(18)such as double root at s= 0, an inflection point along theroot locus at s= 0. We take two typical cases where therewhere Kp= Kkp, K;= Kk;,exists only one root s= 0 and double root.Following the procedure shown in Section 1, weCE(s=0, K)=f(K)=0(13)replaces with wi in (19) and transform into P(u， Kp,Ki, w) by Eq. (7).Solving (13) for K, the prospective K is also on thestability boundaries.P(u，Kp, K, w)=Re(P(u, Kp, K, w))+If there exists double root at the origin, (1) and thei Im(P(u, Kp, K, w)) (19)first derivative of (1) should be equal to zero concurrently.whereHence we should solve the following equation for K.Re(P(u, Kp, K, w))=(u1u:-1)w +(CE(s=0, K)=02Kpu1w+2K;u;Ux (20a)aCE(s, K)=0(14)Im(P(u, Kp, K, w))=(-u,- u2)w2 +as2Kpu1u:w+2K;ul (20b)And the prospective K is also on the stability boundaries.If s=0 is a multiple root ( more than double) inFrom (20a) and (20b) we form the Sylvcster Recsultantgeneral, one has to check higher order derivatives of (2)Matrix (4X4) as follows;with respect to s in order to construct the stability analysis_”1u20similar to the double root case.中国煤化工u2K,u1uzAfter all possible stubility boundaries are obtained, theTHCNMHG"1Nyquist stibility method is used to deternine the :nctual-u1一u2 2KpU1u: 2K.,u1stability region. in which no right hialf plane root exists.(21)Journal of Donghua University (Eng. Ed.) Vol. 26, No.1 (2009)39The corresponding Sylvcster Resultant Matrix is trivialNC, USA, 1995.to obtain from (21)[3] Cheng s L，Hwang C. On Stablization of Time-dclayUnstablc Systcms Using PID Controllers [J]. Joumal of theF(K, u)= dct(M)=Chinese Institute of Chemical Eniners. 199, 30(2);4K;u{(2Kξu{ +4K:ul -123 -140.2K})u} + [4(K; - K})u2 -4(K:+[4] Hwang C, HuangJ H. On Siabilization of First-order PlusK})ur]u:+ K;(ut -u经+1)=0 (22Dead-time Unstable Proccess Using PID Contrllers[J]. IEEProcedings on Control Theory and Application, 2004 151Obviously, s= 0 is a standing root of (18) independent(1): 89-94.of the valuc of Kp, Ki. The first derivative of (18) is[5] Ncimark Ju. I. Ddccomposition of the space ofaCE(s, Kp. K.)quasipolynonials (on the Suability of Linearizod Distributive=ηKr=0≈k;=0 (23)Systcms)[C]. Amcrian Mathermutical Socicty Tanslations,Series 2. Vo. 102: 'Ten Papurs in Andyis. AmericanThe sccond derivative of (18) isMauthemstical Society, Providence R.I.. 1973; 95 - 131.aCE(s, K, K) |=2 +[2t;Kp- r(τ +2r:)K;][6] Porter B. Bradshaw A. Efet of Integral Action on theas2Stabilizability of Continuous-time Lincar Dynamical[K;=0{k,=0.Systcms with Retarded Control[J]. International Journalof Systems Science, 1974,5(9); 807 - 815.Kp=-二kp=-一kT1[7] SilvaG J, Datta A. Bhatacharyya S P. Stabilizaion of(24)Time Delay Systems[C]. Amcrican Control Confcrence,2000: 963 - 970.Thus the stability region has a singular boundary k.=0.[ 8] Silva GJ, Dalta A, Bhattacharyya S P. PI Stabilization ofNowset K=1,t1=3,Tz= 1.2, the stability region isFirst Order Systems with Time Delay [J]. Aulomatica,shown in Fig.2.2001, 37(12); 2025 - 2031.[9] Silva G J, Datta A. Bhatacharyya S P. Stabilization of0.35First-order Systcms with Time Delay Using the PID0.Sstability boundaryContoller[C]. 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