ASPECTS OF POLE PLACEMENT TECHNIQUE IN SYMMETRICAL OPTIMUM METHOD FOR PID CONTROLLER DESIGN

Transcription

1 ASES OF OLE LAEMEN EHNIQUE IN SYMMERIAL OIMUM MEHOD FOR ID ONROLLER DESIGN Viorel Nicolau *, onstantin Miholca *, Dorel Aiordachioaie *, Emil eanga ** * Deartment of Electronics and elecommunications, Dunarea de Jos University of Galati, 47 Domneasca Street, Galati, 88, Galati, ROMANIA Viorel.Nicolau@ugal.ro, onstantin.miholca@ugal.ro, Dorel.Aiordachioaie@ugal.ro ** Research enter in Advanced rocess ontrol Systems, Dunarea de Jos University of Galati, 47 Domneasca Street, Galati, 88, Galati, ROMANIA Emil.eanga@ugal.ro Abstract: he aer resents an analytical aroach to the design of ID controllers by combining ole lacement with symmetrical otimum method, for the integration lus first-order lant model. he desired closed-loo transfer function (c.l.t.f.) contains a second-order oscillating system and a lead-delay comensator. It is shown that the ero value of c.l.t.f. deends on the real-ole value of c.l.t.f. and in addition, there is only one ole value, which satisfies the assumtions of symmetrical otimum method. In these conditions, the analytical eressions of the controller arameters can be simlified. he method is alied to design a ID autoilot for heading control of a shi with firstorder Nomoto model. oyright 5 IFA Keywords: ID control, ole lacement, symmetrical otimum method, autoilot. INRODUION Due to widesread industrial use of ID controllers, it is clear that even a small ercentage imrovement in ID design could have a major imact worldwide (Silva, et al., ). uning of ID controllers is a difficult task, as a three-arameter model should be defined and it must be accurate at higher frequencies (Astrom and Hagglund, 995). Although, analytical methods are more convenient than grahical methods based on frequency diagrams, in industry, most controllers are tuned using frequency resonse methods (ang and Ortega, 993). Analytical methods rely on low-order models characteried by a small number of arameters. he most emloyed models are the integration lus firstorder model, which is used for thermal and electromechanical rocesses, and the first-order lus dead-time model, which is used for chemical rocesses (Datta, et al., ). In this aer, the integration lus first-order model tye is used for shi dynamics modelling. Also, analytical design of ID autoilot for heading control is considered. If ole lacement method (M) is used to synthesie the ID controller, the first ste is to secify some erformance conditions of the closedloo system, which lead to the eression of the closed-loo transfer function (c.l.t.f.) (Yu and Salgado, 3). In this aer, the desired c.l.t.f. contains a second-order oscillating system and a ole-ero air, with real and negative values. Alying M, the ero value deends on the ole value, and the controller arameters deend on the arameters of c.l.t.f. he resulting oen-loo transfer function (o.l.t.f.) contains a double-integrator element and a ole-ero air, with real and negative values. Hence, the symmetrical otimum method (SOM) can be used (Kessler, 958).

2 Imosing symmetrical characteristics of the oenloo transfer function, the analytical eressions of the controller arameters can be simlified. he goal of this aer is to find the ole-ero values of c.l.t.f. and the simlified analytical eressions of ID controller arameters, which satisfy two simultaneous conditions: the desired close-loo transfer function and symmetrical characteristics of the oen-loo transfer function. he aer is organied as follows. Section rovides mathematical models used in simulations. In section 3, the analytical eressions of ID controller arameters are obtained using M. In section 4, the eressions of the controller arameters are simlified, imosing symmetrical characteristics of the o.l.t.f. Section 5 describes the simulation results, using a ID autoilot for heading control of a shi. onclusions are resented in section 6.. MAHEMAIAL MODELS onsider the classical structure of the control loo without disturbances, as shown in Fig.. he lant model contains an integrator and the controller is of ID tye. r + - ε ontroller (ID) Fig.. lassical structure of the control loo he erformance conditions of the closed-loo system can be secified imosing the eression of system transfer function. In general, a second order reference model is chosen to aroimate the behaviour of the closed-loo system: H (, () s + ζ s + where > is the natural frequency and ζ > is the daming coefficient. Because the lant model contains an integrator and another one is included into the ID controller, the oen-loo transfer function contains a doubleintegrator, which can not be obtained with c.l.t.f. given in (). herefore, the reference model must be comleted with a lead-delay comensator, which contains a ole-ero air, with real and negative values (eanga, et al., ): ( s + ) H s () ( s + ζ s + ) ( s + ) where > and >. u lant H( (with integrator) y he lant model contains an integrator, and it is characteried by a dominant time constant ( ) and a gain coefficient (k ). he eression of the model deends on the rocess tye. a) If the rocess is fast, then the small time constants can not be neglected and the model contains an equivalent small time constant ( Σ ), corresonding to the sum of arasitic time constants: H ( s ( s ( s +, (3) ) where Σ <. In this case, the ID controller is of the form: H (, (4) where < < Σ. ( s ( s he oen-loo transfer function is: ( ( (5) s ( s ( s Using ole cancellation, the non-ero dominant ole of the lant model is cancelled by choosing: (6) hus, only two controller arameters must be determined: k and. It can be observed that, if the rocess does not have any non-ero dominant ole (model without time constant ), then the controller is of I tye (without time constant ) and the same arameters must be determined (k and ). b) If the rocess is slow, the equivalent small time constant ( Σ ) can be neglected and the lant model is: H ( (7) s ( s he ID controller contains a sulementary degree of freedom and it is of the form: + H ( (, (8) s + where Σ < < <. he oen-loo transfer function is: + H ( ( (9) s + s ( s Again, using ole cancellation, the non-ero dominant ole of the lant model is cancelled, resulting equation (6):. In this case, three controller arameters must be determined: k, and. his is a more general case because the time constant is not imosed by the rocess and it can be chosen. So, in this aer, the lant model given in (7) is used for comutations, but discussions are made also for model given in (3).

3 In both cases, the oen-loo transfer functions, given in (5) and (9), have similar eressions: ( ( s H ( H ( () s With this oen-loo transfer function, the symmetrical otimum method (SOM) can be used. he time constant has different meanings: in the first case, it reresents the equivalent small time constant ( Σ ) imosed by the rocess, while in the second case, it is a controller arameter ( ). 3. ID ONROLLER DESIGN USING OLE LAEMEN MEHOD onsider the control system illustrated in Fig. with the desired closed-loo transfer function given in (). roosition : For every ole value (s -) of the desired closed-loo transfer function given in (), there is only one ero value (s -) for which the oen-loo transfer function has a double-ole in origin, and in addition, the ero frequency is smaller than the ole frequency: <. he roosition demonstration includes the net lemma results. Lemma : he necessary and sufficient condition, for the eistence of a double-ole in origin for the oen-loo transfer function of a control system illustrated in Fig., starting from a desired c.l.t.f. of the form given in (), is:,,, > ζ ζ () roof L: he transfer function of the oen-loo system can be comuted starting from the desired closed-loo transfer function (Nicolau, 4): H ( H ( H ( ζ s ζ s + s From () it can be observed that a double-ole in origin is obtained if the equation below is satisfied: ζ, (3) which is equivalent with: ζ + (4) ( s + ) () From (4) it results the necessary and sufficient condition indicated in () (q.e.d.). Imlicitly, the unique ero value results, whose eression deends on the selected ole value: s,,, ζ > (5) ζ For every frequency () of the ole, the corresonding frequency () of the ero is smaller: > < (6) ζ So, for the lead-delay comensator introduced into the desired c.l.t.f. given in (), the hase-lead effect is dominantly. In this case, the real values of ole-ero air and conjugate comle oles, of the desired c.l.t.f. given in (), are illustrated in Fig.. s o s ζ s s ζ Im Fig.. oles and ero of the desired c.l.t.f. Re Using () and (3) in (), the eression of the oen-loo transfer function is obtained: ( ζ ) s + ζ + H s (7) s s + + ζ [ ] Denote by and, resectively, the ero and ole frequencies of the oen-loo transfer function: ζ + (8) ζ he oen-loo transfer function can be rewritten: (ζ ) ( s + ) H ( (9) s [ s + ] utting into evidence the time constants, the oenloo transfer function can be rewritten, like in (): ζ s + H ( () + s (ζ ) + s ζ + From (), using (5) or (9) corresonding to the lant model indicated in (3) or (7), resectively, it results: + ( s s + s s + ζ s +, () s (ζ + ) s + ζ + where Σ or.

4 Equation () can be reduced to an equality of two olynomials of 3 rd order in s variable: (ζ + ) ζ + ( s ( s s + ζ + + s s s +, () where no ole cancellation was considered. he equality must be true for every frequency, resulting a four equation system: ( ζ + ) (3.) ζ (3.) ζ + + ( + ) ζ + ζ + + ( + ) (3.3) ζ (3.4) ζ + he solutions of the equation system are the ID controller arameters (Nicolau, 4): ζ k (4.) k ζ + ζ (4.) (4.3) (4.4) ζ + It can be observed that the solution (4.3) reresents the ole cancellation condition, considered in (6), and it does not deend on the ole value. If the rocess is slow and the equivalent small time constant ( Σ ) is ignored, the time constant reresents a controller arameter given in (4.4). he time constants must satisfy the inequalities: Σ < < <, (5) which can be transosed into frequency domain: < < < (6) herefore, the reference model must be chosen so that the arameters of c.l.t.f. (, ζ and ) to satisfy the system of inequalities: ζ + < (7.) ζ + > (7.) ζ + > (7.3) If the time constant Σ is imosed by the rocess and the lant model given in (3) is considered, then the solution (4.4) becomes:, (8) ζ + which reresents a sulementary condition for arameters of c.l.t.f. (, ζ and ). In addition, the time constants must satisfy the inequalities: < Σ <, (9) which can be transosed into frequency domain: < < (3) In this case, the reference model must be chosen so that the arameters of c.l.t.f. (, ζ and ) to satisfy the following conditions: ζ + (3.) ζ + > (3.) ζ > (3.3) It can be observed that the first condition in (3) is more restrictive than the corresonding one in (7), while the last two conditions are the same. 4. SYMMERIAL HARAERISIS OF OEN-LOO RANSFER FUNION he ID controller arameters deend on the arameters of c.l.t.f. (, ζ and ). In general, and ζ characterie the desired system behaviour and they have fied values, while the ole value can be chosen. Secific ole values can be imosed by using sulementary conditions. In this aer, the conditions for choosing the ole value refer to the symmetrical otimum method, which simlify the eressions of ID arameters. he goal is to find that ole value of the c.l.t.f., which satisfies the assumtions of symmetrical otimum method around natural frequency, for the transfer function of oen-loo system given in (9). Using this value, the eressions of ID arameters in (4) are simlified. roosition : here is only one admissible value for the ole (s - ) of c.l.t.f. given in (), so that the corresonding o.l.t.f. given in (7) to have symmetrical characteristics around :,, ζ (3), > roof: For the secified oen-loo transfer function, the symmetry of magnitude-frequency characteristic around natural frequency imlies the symmetry of hase-frequency characteristic. herefore, only the symmetry of former characteristic must be imosed.

5 he general form of the symmetrical otimum method imoses two conditions for magnitudefrequency characteristic: a) the central frequency must be equally laced between ero and ole frequencies on the -base logarithmic scale: (33) b) for central frequency, the magnitude-frequency characteristic of o.l.t.f. must have db: H ( j ) (34) Using (8) in (33), the first condition becomes: (ζ + ) (35) ζ From (35), it results:,,, ζ > (36) So, the condition (33) is satisfied if. For the second condition in (34), the magnitude of oen-loo transfer function in frequency is comuted from (9): (ζ ) + H ( j ) (37) + he frequencies and are relaced with their eressions from (8), resulting: + 4ζ + 4ζ H ( j ) (38) + 4ζ + 4ζ Using (38) in (34), the same solution in (36) is obtained: oncluding, there is only one admissible value for the ole of c.l.t.f., so that the corresonding o.l.t.f. to have symmetrical characteristics around (q.e.d.). From (), it results: (39) ζ + he eression of c.l.t.f. becomes: (ζ s + ζ + H s (4) ( s + ζ s + ) ( s + ) Also, from (8), the ero and ole frequencies of o.l.t.f. are obtained:, ( ζ (4) ζ + he oen-loo transfer function can be rewritten: (ζ s + ζ + H s (4) s s + (ζ [ ] he real values of ole-ero air and conjugate comle oles, of the c.l.t.f. given in (4), are illustrated in Fig. 3. s s s ζ o s ζ + Im ζ Re Fig. 3. oles and ero of the c.l.t.f. with he osition of the ero s deend on ζ + the arameter ζ: - if ζ,, then < < ζ and the ero is laced between the two oints: s and s ζ, resectively; - if ζ, then ζ. his is the articular case of the Kessler s symmetrical otimum method; - if ζ >, then > > ζ and the ero is laced to the right of the oint s ζ. his is the case illustrated in Fig. 3. Knowing the ole value of c.l.t.f. ( ), the ID controller arameters result from (4): (43.) ζ + (43.) (43.3) (43.4) (ζ he arameters in (43) corresond to the lant model given in (7) and ID controller given in (8). In this case, the conditions from (7) deend on the arameters and ζ. Hence, the reference model must be chosen so that the arameters of c.l.t.f. ( and ζ) to satisfy the system of inequalities: ζ + < (44.) Σ ζ + > (44.) ζ + > (44.3)

6 roosition 3: In the case of symmetrical characteristics of the o.l.t.f. given in (4) around the natural frequency, the hase margin and the distance between the frequency oints on the -base logarithmic scale deend only on the arameter ζ. roof: he distance between the frequency oints on the -base logarithmic scale can be easily obtained, using (4) in (33): ζ + (45) he hase margin is: ϕm π + arg( j) ) (46) Using the o.l.t.f. given in (4), results: ϕ m arctg(ζ arctg (47) ζ + It can be observed that, for articular case of the Kessler s symmetrical otimum method ( ζ. 5 ), the distance between frequency oints is equal with an octave and the hase margin is ϕ [deg]. 5. SIMULAION RESULS For simulations, the heading control roblem of a shi is considered, using a ID autoilot. he shi model is linear, being identified for a shi seed of knots (Nicolau, 4). It is a first order Nomoto model of the form given in (7): ψ ( H (, (48) δ ( s ( s where ψ( and δ( reresent the Lalace transforms of yaw angle and rudder angle, resectively. he shi model arameters are: k.834 [s - ], [s] (49) he autoilot model is given in (8) and the desired c.l.t.f. is given in (). he arameters and ζ are chosen from erformance conditions (Fossen, 994): ζ.9,. [rad/s] (5) Starting from the desired c.l.t.f. and imosing symmetrical characteristics of the o.l.t.f., the eressions in (4) and (4) are obtained. he ste resonse of the c.l.t.f. is illustrated in Fig. 4. From (43), the autoilot arameters are obtained: m k., 8 [s], [s], [s] Fig. 4. Ste resonse of the closed-loo system onsidering [s], the conditions in (44) are satisfied. he symmetrical characteristics of the o.l.t.f. are illustrated in Fig. 5. It can be observed that the hase margin is ϕ [deg]. Σ m Fig. 5. Symmetrical characteristics of the o.l.t.f. 6. ONLUSIONS here is only one ossible air for the ole-ero values of c.l.t.f. so that the corresonding arameters of ID controller to satisfy two simultaneous conditions: the desired behaviour of close-loo system and symmetrical characteristics of the oenloo transfer function. REFERENES Astrom, K. and. Hagglund (995). ID ontrollers: heory, Design, and uning. Research riangle ark, N: Instrument Society of America. eanga, E,. Nichita, L. rotin and N. A. utululis (). heorie de la ommande des Systemes. Ed. ehnica, Bucharest, Romania. Datta, A., M.. Ho and S.. Bhattacharyya (). Structure and Synthesis of ID ontrollers. Londin, UK: Sringer-Verlag. Fossen,. I. (994). Guidance and ontrol of Ocean Vehicles. John Wiley and Sons Ltd, NY. Kessler,. (958). Das Symmetrische Otimum. Regelungstechnik, 6, and Nicolau, V. (4). ontributions in Advanced Automatic ontrol of Naval Systems. h.d.hesis (in romanian), University of Galati, Romania. Silva, G. J., A. Datta and S.. Bhattacharyya (). New Results on the Synthesis of ID ontrollers. IEEE ransactions on Automatic ontrol, 47, no., ang, Y. and R. Ortega (993). Adative uning to Frequency Resonse Secification. Automatica, 9, Yu, J. I. and M. E. Salgado (3). From lassical to State-Feedback-Based ontrollers. IEEE ontrol Systems Magaine, 3, no. 4,