PID Controller Design With Guaranteed Stability Margin
Proceedings of the World Congress on Engineering and Computer Science 2007WCECS 2007, October 24-26, 2007, San Francisco, USAPID Controller Design with Guaranteed Stability Marginfor MIMO SystemsT. S. Chang and A. N. GündeşAbstract— Closed-loop stabilization with guaranteed stabil-The goal of this paper is to study closed-loop stabiliza-ity margin using Proportional Integral Derivative (PID) con-tion with guaranteed stability margins using PID-controllers.trollers is investigated for a class of linear multi-input multi-A sufficient condition is presented for existence of PID-output plants. A sufficient condition for existence of suchPID-controllers is derived. A systematic synthesis procedurecontrollers that stabilize linear, time-invariant, MIMO stableto obtain such PID-controllers is presented with numericalplants, where the closed-loop poles are guaranteed to haveexamples.real-parts less than a pre-specified h. A systematic designKeywords– Simultaneous stabilization and tracking, PIDcontrol, integral action, stability margin.I. INTRODUCTIONProportional Integral Derivative (PID) controllers are thesimplest integral-action controllers that achieve asymptotictracking of step-input references [1]. Although the simplicityof PID-controllers is desirable due to easy implementationand from a tuning point-of-view, it also presents a majorrestriction that only certain classes of plants can be controlledby using PID-controllers. Rigorous PID synthesis methodsbased on modern control theory are explored recently ine.g., [2], [3], [4], [5], [6]. Sufficient conditions for PIDstabilizability of multi-input multi-output (MIMO) plantswere given in [6] and several plant classes that admit PIDcontrollers were identified.procedure is proposed and illustrated with several numericalexamples. The choice of the free parameters can be optimized with a chosen cost function. Although stability margincan be considered as an important performance measure,there are other factors effecting the performance of thesystem and hence, “good” choice for the design parameters for overall performance is case-specific and cannot begeneralized.The paper is organized as follows: Section II shows themain result, where a sufficient condition for stabilizabilityusing a PID-controller with guaranteed stability margin isgiven. Section III presents a systematic procedure to synthesize PID controllers and gives several illustrative examples.Section IV gives a short discussion, concluding remarks andsome future directions.The systematic controller design method given in [6]II. MAIN RESULTSallows freedom in several of the design parameters. Althoughthese parameters may be chosen appropriately to achievevarious performance goals, these issues were not explored.The authors are with the Department of Electrical and Com-Notation: Let CI , IR, IR denote complex, real, positivereal numbers. The extended closed right-half complex planeis U {s CI Re(s) 0} { }; Rp denotes realputer Engineering, University of California, Davis, CA 95616. (emails:proper rational functions of s; S Rp is the stable subsetchang@ece.ucdavis.edu, angundes@ucdavis.edu)with no poles in U; M(S) is the set of matrices with entriesISBN:978-988-98671-6-4WCECS 2007
Proceedings of the World Congress on Engineering and Computer Science 2007WCECS 2007, October 24-26, 2007, San Francisco, USAin S ; In is the n n identity matrix. The H -norm ofplants. Furthermore, even when it is achievable, it may beM (s) M(S) is M : sup σ̄(M (s)), where σ̄ is thepossible to place the closed-loop poles to the left of a shifted-maximum singular value and U is the boundary of U. Weaxis that goes through h only for certain h IR . We startdrop (s) in transfer-matrices such as G(s) wherever thisour investigation of plant classes for which we can achievecauses no confusion. We use coprime factorizations over S ;our goal by considering stable plants. The class of plantsi.e., for G Rp ny nu , G Y 1 X denotes a left-coprime-under consideration, denoted by Gh , is described as follows:s Ufactorization (LCF), where X, Y M(S), det Y ( ) 0.Let G Gh Sm m , i.e., let the given plant be stable.Consider the linear time-invariant (LTI) MIMO unity-Furthermore, let G have no poles with real parts in [ h, 0].feedback system Sys(G, C) shown in Fig. 1, where G Assume that G(s) has no transmission-zeros (or blocking-m mzeros) at s 0, i.e., G(0) is invertible (note that thisis the controller’s transfer-function. Assume that Sys(G, C)condition is necessary for existence of PID-controllers withis well-posed, G and C have no unstable hidden-modes,nonzero integral-constant Ki [6]). The plant G may haveRpm mis the plant’s transfer-function and C Rpand G Rpm mis full (normal) rank. We considerthe realizable form of proper PID-controllers given by (1),where Kp , Ki , Kd IRm m are the proportional, integral,transmission-zeros (or blocking-zeros) elsewhere in U butnot at s 0.Now definederivative constants, respectively, and τ IR [7]:CpidKd sKi . Kp sτs 1(1)action in Cpid is present when Ki 0. The subsets of(2)Ĝ(ŝ) : G(ŝ h) ;(3)andFor implementation, a (typically fast) pole is added to thederivative term so that Cpid in (1) is proper. The integral-ŝ : s h, or s : ŝ hthen Ĝ(ŝ) has no poles in the closed right ŝ-plane. Similarly,define Ĉpid asPID-controllers obtained by setting one or two of the threeĈpid (ŝ) : Kp constants equal to zero are denoted as follows: (1) becomes aKd (ŝ h)Ki .ŝ h τ (ŝ h) 1(4)PI-controller Cpi when Kd 0, an ID-controller Cid whenLet Sh (G) denote the set of all PID-controllers thatKp 0, a PD-controller Cpd when Ki 0, a P-controllerstabilize G Gh , with real parts of the closed-loop poles ofCp when Kd Ki 0, an I-controller Ci when Kp the system Sys(G, Cpid ) less or equal to h; i.e.,Kd 0, a D-controller Cd when Kp Ki 0.Sh (G) : { Cpid Ĉpid stabilizes Ĝ(ŝ) } .Definition 2.1: a) Sys(G, C) is said to be stable iff thetransfer-function from (r, v) to (y, w) is stable. b) C is saidto stabilize G iff C is proper and Sys(G, C) is stable.The problem addressed here is the following: Supposethat h IR is a given constant. Can we find a PIDcontroller Cpid that stabilizes the system Sys(G, Cpid ) witha guaranteed stability margin, i.e., with real parts of theclosed-loop poles of the system Sys(G, Cpid ) less or equalto h? It is clear that this goal is not achievable for someISBN:978-988-98671-6-4(5)Proposition 2.1: (A sufficient condition):Let h IR and G Gh be given. If for some K̂p IRm m , K̂d IRm m and τ 1/h, the given h IR satisfiesh 1γ(h, K̂p , K̂d ),2(6)where γ γ(h, K̂p , K̂d ) is defined asγ(h, K̂p , K̂d )WCECS 2007
Proceedings of the World Congress on Engineering and Computer Science 2007WCECS 2007, October 24-26, 2007, San Francisco, USA: Ĝ(ŝ)(K̂p Ĝ(ŝ)G(0) 1 I 1K̂d (ŝ h)) , (7)τ (ŝ h) 1ŝ hthen there exists a PID-controller Cpid of the form in (1) thatwhere Kp (α h)K̂p , Kd (α h)K̂d , Ki (α h)G(0) 1 (α h)Ĝ(h) 1 andW : stabilizes G Gh , with real parts of the closed-loop poles ofthe system Sys(G, Cpid ) less or equal to h. Furthermore,a PID-controller Cpid Sh (G) is given by(α h)G(0) 1 (α h)K̂d s , (8)sτs 1Cpid (α h)K̂p Note thatIR satisfiesh α γ(h, K̂p , K̂d ) h .Kd (ŝ h)(α h)KiI Ĝ(Kp )ŝ hŝ h τ (ŝ h) 1 (α h)[Ĝ(K̂p where K̂p , K̂d IRm m are arbitrary, τ 1/h, and α (α h)I ĜĈpidŝ hĜ(ŝ)G(0) 1 IK̂d (ŝ h)) ].τ (ŝ h) 1ŝ h(17)Ĝ(ŝ)G(0) 1 Iŝ h Ĝ(ŝ)Ĝ(h) 1 Iŝ h M(S). If (6) and(9) hold, then h α and α h γ(h, K̂p , K̂d ) imply(9) (ŝ h)α hW W 1ŝ αγ(h, K̂p , K̂d )Remark:and hence, M̂ in (16) is unimodular by the “small-gainCondition (6) is obviously satisfied if h 0, i.e., there existstheorem” [8]. Therefore, Ĉpid stabilizes Ĝ and hence, Cpid a PID-controller Cpid of the form in (1) that stabilizes aSh (G).given stable plant G, where the closed-loop poles of theIII. PID CONTROLLER SYNTHESISsystem Sys(G, Cpid ) may be anywhere in the open left-halfFrom the sufficient condition in Proposition 2.1, the fol-complex plane [6].lowing systematic procedure to synthesize a PID controllerProof of Proposition 2.1:is obtained: Given h IR and G Gh , defineWrite G and Cpid given by (8) asG I 1 G ,Cpid (ssCpid )(I) 1 .s αs αΔ(10)(18)(11)Then Cpid stabilizes G if and only ifM : ssI G(Cpid )s αs α(12)(4) and write Ĝ(ŝ), Ĉpid (ŝ) asĈpid (ŝ hŝ h 1Ĉpid )(I) .ŝ αŝ αthen h β. Choose any K̂p and K̂d and computeγ(h, K̂p , K̂d ) given by (7). If γ(h, K̂p , K̂d ) 2h as inis unimodular. Similarly, substitute ŝ s h as in (2), (3),Ĝ I 1 Ĝ ,β max{x p x jy, where p is a pole of G(s)};condition (6), then it is possible to find α IR satisfying(9). The PID-controller Cpid Sh (G) is then given by (8). If(6) is not satisfied, the process can be repeated for a smallerh value.(13)(14)The following examples illustrate the PID-controller synthesis procedure and some of its properties.Example 3.1: Consider the plant transfer-functionThen Ĉpid stabilizes Ĝ if and only ifŝ hŝ hI Ĝ(Ĉpid )M̂ ŝ αŝ α(15)is unimodular. Write M̂ asM̂ I ŝ hŝ hα hI (ĜĈpid ) : I W, (16)ŝ αŝ αŝ αISBN:978-988-98671-6-4G(s) (s 5)(s2 8s 32)(s 2)(s 8)(s2 12s 40)(19)By (18), β 2. Suppose that h 1. Fig. 2 showsthe constant contour of γ(K̂p , K̂d ), where the solid linerepresents γ 2h as the upper-bound for condition (6).WCECS 2007
Proceedings of the World Congress on Engineering and Computer Science 2007WCECS 2007, October 24-26, 2007, San Francisco, USAEach contour is evaluated in one point denoted by , whichthat are manipulating two pumps. The transfer-matrix of theis given in Table 1.linearized model at some operating point is given by Table 1: Evaluated points for contours in Example 3.1G 3.7b162s 13.7(1 b2 )(23s 1)(62s 1)4.7(1 b1 )(30s 1)(90s 1)4.7b290s 1 S2 2 .(21)xyγxyγxyγ-2.5-3.50.40-1-20.700-11.41One of the two transmission-zeros of the linearized system102.0920.17.802.503.83dynamics can be moved between the positive and negativereal-axis by changing a valve. The adjustable transmission-Note that any (K̂p , K̂d ) inside the solid boundary can bezeros depends on parameters γ1 and γ2 (the proportions ofchosen. Suppose that we choose (K̂p 2.5, K̂d 0.2) andwater flow into the tanks adjusted by two valves). For theτ 0.05. We compute γ 4.7 2h 2, and set α 0.5γ.values of b1 , b2 chosen as b1 0.43 and b2 0.34, theThe closed-loop poles are 1.79, 2.66, 4.93 j2.53i,plant G has transmission-zeros at z1 0.0229 0 and 6.87, 42.58, which all have real-parts less than h 1.z2 0.0997.For a given (K̂p , K̂d ), there may exist a maximum valuehmax such that condition (6) is violated, as indicated by theBy (18) β 1/90 0.0111. Suppose that h 0.004,and chooseintersection point about hmax 1.81 in Fig. 3. The solid line represents the γ curve in terms h for the selected (K̂p , K̂d ),K̂p and the dash-dotted line represents the straight line 2h. 22.6137.6172.14 43.96 Example 3.2: Consider the same transfer-function as inK̂d ,(22) 5.286.216.537.84 ,(23)(19), except the real zero is now in the right-half complexand τ 0.05. We can compute γ 0.0099 2h 0.008,plane, i.e.,(s 5)(s2 8s 32).G(s) (s 2)(s 8)(s2 12s 40)and set α 0.5γ. The maximum of the real-parts of the(20)closed poles can now be computed as 0.0059, which isLet h 1 as in Example 3.1. Fig. 4 shows the constantless than h 0.004. Thus the requirement is fulfilled. Incontour of γ(K̂p , K̂d ). Clearly, the feasible region in thisthis example, hmax is very small as shown in Fig. 5, due tocase is very different from the previous one in Example 3.1.the fact that β is very close to the imaginary-axis.Suppose that we choose (K̂p 3, K̂d 0.2) and τ 0.05. We compute γ 3.22 2h 2, and set α 0.5γ.The closed-loop poles are 1.32, 2.66 j3.31, 7.62, 4.73 j12.69, which all have real-parts less than h 1.The maximum value hmax can be similarly obtained, whichis about 1.3 and is lower than that in Example 3.1.Example 3.4: The PID-synthesis procedure based onProposition 2.1 involves free parameter choices. Considerthe same transfer-function as in (19) of Example 3.1. Leth 1, choose τ 0.05, and set α 0.5γ as before. Ifwe choose (K̂p 2.5, K̂d 0.2), then the the dash line inFig. 6 shows the closed-loop step response. However, if weExample 3.3: Consider the quadruple-tank apparatus inchoose (K̂p 2, K̂d 0.1), then we obtain a completely[9], which consists of four interconnected water tanks anddifferent step response as shown with the dash-dotted line intwo pumps. The output variables are the water levels of theFig. 6. It is natural to ask then if the free parameters can betwo lower tanks, and they are controlled by the currentschosen optimally in some sense.ISBN:978-988-98671-6-4WCECS 2007
Proceedings of the World Congress on Engineering and Computer Science 2007WCECS 2007, October 24-26, 2007, San Francisco, USAConsider a prototype second order model plant, with ζ 0.7 and ωn 6; i.e.,R EFERENCES[1] K. J. Aström and T. Hagglund, PID Controllers: Theory, Design,Tmodelωn2 2s 2ζω ωn2(24)We want the closed-loop step response sm (t) using the modelplant Tmodel to be as close as possible to the actual stepresponse so (t). The step response using Tmodel is shownwith the solid line in Fig. 6. Let us consider the cost function 1 3error (sm (t) so (t))2 dt,(25)3 0where so (t) is the step response for any choice of (K̂p , K̂d ).By plotting the contour of the error in terms of (K̂p , K̂d ) inFig. 7, we find the global minimum of the error to occur at(K̂p 1.47, K̂d 0.15). The step response correspondingto this choice of (K̂p , K̂d ) is shown with the solid line witha circle in Fig. 6, which is closer to the model step responseand Tuning, Second Edition, Research Triangle Park, NC: InstrumentSociety of America, 1995.[2] M. Morari, “Robust Stability of Systems with Integral Control,” IEEETrans. Autom. Control, 47: 6, pp. 574-577, 1985.[3] M.-T. Ho, A. Datta, S. P. Bhattacharyya, “An extension of the generalized Hermite-Biehler theorem: relaxation of earlier assumptions,”Proc. 1998 American Contr. Conf., pp. 3206-3209, 1998.[4] G. J. Silva, A. Datta, S. P. Bhattacharyya, PID Controllers for TimeDelay Systems, Birkhäuser, Boston, 2005.[5] C.-A. Lin, A. N. Gündeş, “Multi-input multi-output PI controllerdesign,” Proc. 39th IEEE Conf. Decision & Control, pp. 3702-3707,2000.[6] A. N. Gündeş, A. B. Özgüler, “PID stabilization of MIMO plants,”IEEE Transactions on Automatic Control, to appear.[7] G. C. Goodwin, S. F. Graebe, M. E. Salgado, Control System Design,Prentice Hall, New Jersey, 2001.[8] M. Vidyasagar, Control System Synthesis: A Factorization Approach,MIT Press, 1985.than the other two.[9] K. H. Johansson, “The quadruple-tank process: A multivariable labo-Table 2: Evaluated points for contours in Example 3.4ratory process with an adjustable zero,” IEEE Trans. Control SystemsTechnology, 8, (3), pp. 456-465, .132.50.214.15IV. CONCLUSIONSFor stable plants whose poles have negative real-parts lessthan a pre-specified h, we obtained a sufficient conditionfor existence of PID-controllers that achieve integral-actionvre- h - C 6w- ?h- Gy-and closed-loop poles with real-parts less than h. We proposed a systematic design procedure, which allows freedomFig. 1.Unity-Feedback System Sys(G , C).in the choice of parameters. We showed in an example howthis freedom can be used to improve a single-input singleoutput system’s performance. Extending the optimal parameter selection to MIMO systems would be a challenging goal.These results are limited to stable plants. Future directions of this study will involve extension to certain classesof unstable MIMO plants. In addition, optimal parameterselections for the MIMO case will be explored.ISBN:978-988-98671-6-4WCECS 2007
Proceedings of the World Congress on Engineering and Computer Science 2007WCECS 2007, October 24-26, 2007, San Francisco, USAFig. 2.Contour of γ(K̂p , K̂d ) for Example 3.1Fig. 5.Finding hmax for Example 3.3Finding hmax for Example 3.1Fig. 6.Step responses for Example 3.4Fig. 3.Fig. 4.Contour of γ(K̂p , K̂d ) for Example 3.2ISBN:978-988-98671-6-4Fig. 7.Contour of error(K̂p , K̂d ) for Example 3.4WCECS 2007
PID-controllers is derived. A systematic synthesis procedure to obtain such PID-controllers is presented with numerical . and from a tuning point-of-view, it also presents a major restriction that only certain classes of plants can be controlled by using PID-con
PID-controller Today most of the PID controllers are microprocessor based DAMATROL MC100: digital single-loop unit controller which is used, for example, as PID controller, ratio controller or manual control station. Often PID controllers are integrated directly into actuators (e.g valves, servos)File Size: 1MBPage Count: 79Explore furtherWhen not to use PID-controllers - Control Systems .www.eng-tips.comPID Controller-Working and Tuning Methodswww.electronicshub.org(PDF) DC MOTOR SPEED CONTROL USING PID CONTROLLERwww.researchgate.netTuning for PID Controllers - Mercer Universityfaculty.mercer.eduLecture 9 – Implementing PID Controllerscourses.cs.washington.eduRecommended to you b
4.1 Simulink Block of PID Controller 58 4.2 Detailed Simulink Block of the System 60 4.3 Output of DC Motor without PID Controller 60 4.4 Detail Simulink Block of the System with PID Controller 61 4.5 Output of DC Motor without PID Controller 62 4.6 Sim
Fig.4.1 Step response using P controller Fig.4.2 Fig.4.3 Fig.4.4 Fig.4.5 Step response using PID controller Fig.4.6 Comparison of bode plot for smith predictor and PI . Step response of IMC PDF control structure Step response of PDF controller Step response using PI controller Step response using PID controller Step response using PI controller
fuzzy controller are 69.9 % and 67.9 % less than PID controller. 6. CONCLUSION Theses. Paper This paper presents the control of the level in a single tank using different two type controllers PID and fuzzy. From program simulation that built it was indicates that the fuzzy controller has more Advantages to the system than the PID controller.
SPEED CONTROL WITH PID CONTROLLER A proportional-integral-derivative controller (PID controller) is widely used in industrial control systems. It is a generic control loop feedback mechanism and used as feedback controller. PID working principle is that it calculates an error
Plot System Responses . (time-domain response) or Bode plots (frequency-domain response). For 1-DOF PID controller types such as PI, PIDF, and PDF, PID Tuner computes system responses based upon the following single-loop control architecture: For 2-DOF PID controller types such as PI2, PIDF2, and I-PD, PID Tuner computes responses based upon .
Currently, PID controller has been used to operate in electric furnace temperature control system because its structure is simpler compared to others. However, the issue of tuning and designing PID controller adap-tively and efficiently is still open. This paper presents an improved PID controller effic
4. Design of PID Controller . Industrial PID controllers are usually available as a packaged, and it's performing well with the industrial process problems. The PID controller requires optimal tuning. Figure 3shows the diagram of a simple closed loop - cont
