Ayrık zamanlı sistemlerde belli zaman özelliklerini sağlayan optimum PID kontrolör tasarımı

Abstract

Kapalı çevrim kontrol sistemlerinin tasarımında kutup sıfır yerleşiminin nasıl yapılacağı ve kutup sıfır yerleşiminin saman ve frekans domeninde, sistem performansı üzerinde etkisinin ne olacağı, karşılaşılan en önemli sorundur. Bu sorunun çözümünde bir yaklaşım, yüksek mertebeden sistemlerin, tasarım ve kontrolü için ikinci derece davranışı gösteren kutup-sıfır yerleşimini yapılması gerekmektedir. PID kontrolörlerin tasarımında çeşitli yöntemler uygulanmaktadır. Uygulanan yöntemlerden biri, parametre optimizasyonuna dayanan tasarım teknikleridir. Bu çalışmada da; parametrelerin, sistemin çıkış hatalarını minimum yapacak şekilde belirlenmesi sağlayacak bir yöntem üzerinde durulmuştur. Çıkış hatalarının miniminasyonu işleminde, amaç ölçütü olarak değişik fonksiyonlar düşünülebilir. Bu çalışmada, analitik olarak hesaplanması kolay olduğundan çıkış hatalarının karelerinden oluşan bir amaç ölçütü ku1lanılmıştır. Tanıtılacak olan yöntemle, digital kapalı çevrim kontrol siteminde çıkış hatalarının minimum olmasını sağlayan PID kontrol parametrelerinin belirlenmesi işlemi, eşitlik, eşitsizlik koşulları olan bir optimiazasyon problemi haline dönüştürülmüştür. Bu çalışmada, ikinci bölümde birinci, ikinci ve yüksek mertebeden sistemler ve bunların davranış özellikleri üzerinde durulmuştur. üçüncü bölümde, dijital sistemin çıkış hatalarını karelerinin minimum yapılması işleminde kullanılan amaç ölçütünün elde edilmesine yardımcı olan Parseval Teoremi tanıtılmıştır. Dördüncü bölümde, ele alman dört değişik sistem için, Parseval Teoremine göre amaç ölçütleri ve eşitlik koşullarını içeren matematik modeller oluşturulmuştur. Son bölümde ise, oluşturulan lineer olmayan program lama problemi için bir algoritma tanıtılmıştır. Bu algoritmada kullanılan değerler ve elde edilen sonuçlar da, Ekler kısmında verilmiştir.

In this work a procedure of synthesis of some digital control systems and their PID parameters to satisfy certain time domain specification have been examined. The procedure employs the "integral square error criterion" in the determination of the poles-seros of the closed- loop transfer functions. There has been considerable work in the literature on the synthesis for the feed-back control systems which can briefly outlined as follows: 1. Synthesis in the frequency domain 2. The root- locus technique 3. Synthesis in the s-plane 4. Some time domain synthesis techniques 5. Techniques for optimising the feedback control systems. The difficulty still remaining in the synthesis of the feedback control systems can be outlined as follows: what poles-aeros configurations or what frequency response gives the best systems performance in the time or frequency domain? The main characteristics for evaluating the system performance are: Maximum overshoot, time to reach the first maximum, settling time. The signals at the input assumed to be output in evaluating the system performance are usually assumed to be unit step functions. Efor the conventional control system there are several proposed criteria to obtain optimum systems performance. One of this criteria are Integral-square-error. In this work, PID controller parameters in the closed- loop digital system, can be determined for the minimum integral-square- error under the some time domain condition. In this work, higher order transfer functions are obtained as closed loop control system transfer functions. Let general term of the n th-order system be define by its closed- loop transfer function as, ciz) k n <*+*i> R{z) n-Z (z2+2Çoaz+tt£) U (z+p.f) Higher order system must have second order system response for more easy investigation. For this purpose, appropriate poles-zeros configurations should be done. Two system closed- loop poles are controlled the system. Other poles and aeros are replaced appropriate place for the system have second order system response. In the before equation, the terra z2+2%G>az+ti>a, related with the system controlled poles. Other poles and zeros to not effect the system controlled poles and systems response, are replaced appropriate placed. This places must be defined. Thus, system has second order system response and certain time domain specifications by means of two control poles. Therefore, with this procedure, in order to apply in any nonlinear algorithm, inequality constrained for poles and zeros of the system are obtained. In this work, integral-square-errors are obtained with Parseval's theorem, we know that this can be expressed as yr ez[ktf = -^ £ E(z)E(z-1)z-1 dz *-*-o 27tJ J* In this equation, J^Z_Q e2 [kTl is the integral-square- error and E(z) is the error function for the digital closed- loop system and x is the unit circle. E(s), error function for the digital closed- loop system is obtained easily. Therefore, integral-square-error can be obtained. In the right side term is carried out Residu Theorem for the each stabile poles to obtain integral-square-error. During applying Residu theorem, poles in s = 1 are accepted as nonstable. Therefore, objective function is obtained. Also equality constrained are obtained from that system transfer function which is equal to its general transfer function equation. As a result, this procedure is formed a nonlinear programming problem with equality and inequality constrained. This nonlinear programming problem can be solved by any nonlinear programming algorithm. In this work, Flexible Tolerance Algorithm are applied for this purpose. The general nonlinear problem is to find an extremum of an objective function subject to equality and/or inequality constraints. The constraints may be linear or nonlinear. The nonlinear programming problem can be formally stated as Minimise : J(x) x ? Ea (1) subject to m linear and/or nonlinear equality constraints and ( p - m ) linear and/or nonlinear inequality vi constraints hî{x) = 0 i b 1,....,m gi(x) h 0 i - (iîî+1),,p Each equality constraint absorbs one degree of freedom in the process model and results in one dependent variable being generated. It is usually assumed that the analyst prepare process-model statement carefully enough so that equality independent. The true residual number of degrees of freedom should correspond to the number of independent variables in the nonlinear programming problem. The number of residual degrees of freedom is an important concept in any type optimisation subject to equality constraints, because if the number variables equals the number of the independent equality constraints, no optimisation take place. Constrained nonlinear programming problems are much harder to solve than unconstrained problems with a comparable number of independent. variables and degree of nonlinearity because of additional requirement that the solutions must satisfy the constraints. The majority of constrained nonlinear optimisation procedures that have been proposed in the literature to date are centered around one of three bas ic concepts ; 1. Extension of linear methodology to nonlinear programming problems by means of repeated linear approximations 2. Transformation of the constrained nonlinear problem into a sequence of unconstrained problem through the use of penalty function 3. Use of flexible tolerances to accommodate both feasible and non feasible x vectors. In many nonlinear programming methods a considerable portion of the computation time spent on satisfying rather rigorous feasibility requirements. The flexible tolerance algorithm, on the other hands, improves the value of the objective function by using information provided by feasible points, as well as certain nonfeasible points termed near- feasible points. The near- feasibility limits are gradually made more restrictive as the search proceeds toward the solution of the programming problem, until in the limit only feasible x vectors are accepted. As a result of this basic strategy problem at the beginning, can replaced by a simpler problem, having the same solution: Minimise : J(x) x ? Ea Subject to : *(Jc) - T(x) h 0 (2) where $<="" q="" although="" polyhedron="" nelder="" mead="" to="" implement="" unconstrained="" searches="" method,="" particular="" minimization="" technique="" independent="" strategy.="" thus="" any="" other="" effective="" minimisation="" algorithm="" could="" replace="" method="" mead.="" *«*»="roin" {., -^yr1 S xj (&> XZ+2 *<0} =2 On+1) t where t = sise of initial polyhedron m = number of equality constraint xi m - ith vertex of polyhedron in E a r = (n-ro) number of degrees of freedom of J(x) xr*2 - vertex corresponding to centroid as define below k = 0,...is an index referring to number of completed stages of search 4>Uc-t) _ va]_ue 0f tolerance criterion on (k-1) st stages of search xn+2.j V^n+1.(» i' be any vector in En. The x(fc) vector is said to be viii 1. Feasible, if T{x<®) = 0 2. Near- feasible, if öi T(xtk)) * . *<*> Thus the region of near-feasibility is define as *<*> - T(x) h 0 The general strategy of the flexible tolerance algorithm is to reduce 0(Jc) as the search progress, thus tightening the region of neai'- feasibility, and to segregate the minimization of J(x) from the step taken the satisfy the constraint. For a given value of 0, in which case x(k+i> is either a feasible or near- feasible point will be accepted as a permitted move, or (2) T(x{k*1]) > . One way of getting an x(k+D closer to the feasible region is to minimise the value of T(x<*+i>) until T(x(wl) d.. <*». The starting point any minimisation of T(x) is always x. is the nonfeasible point on the kth stage of the search for minimum of J(x). To initiate the search to reduce the value of TCx) using the unconstrained method of Nelder and Mead, (n+1) initial vertices are required. The (n+1) vertices should be chosen in such a way that any subset of n vectors is linearly independent. For. all practical purposes it is most convenient to built a regular polyhedron using x(0) as the base point. The (n+1) vertices of the initial polyhedron in E" are found from X (0) + D i' i = 1,.n+1 where Di elements below. is a column vector in which the components are the of the ith column of n x (n+1) matrix D, define D = 0 dl d2 d2 0 d2 dl d2 0 d2 d2. d2 0 62 d2 '!!!!..! 1... dl IX where n s[2 (yn+l +J2-1 ) n 1/2 To initiate search for the minimisation of J(x) using the flexible tolerance algorithm, one needs to know an initial xi°), the sise of the initial polyhedron t, the value of <&<0), and r. In order to start the minimisation of J(x) with the appropriate polyhedron sise, t should be selected as a function of the expected range of variation of the variables x. Usually upper and lower bounds on x are known, in which case the following equation can be used as a reasonable estimate of t. *<« = min { [-9^- J^ (Ui- L£) ], (rç. -Lt) (Ua -Ln)) where (Ui - Li ) is the upper and lower bounds on the vari able xi. If upper and lower bounds of x are not known, a reasonable guess for t will have to suffice. The algorithm terminates under two circumstances; 1. When <6(jc) i e, in which case the search is considered completed and successful. 2. When a feasible or near-feasible point cannot be obtain by the procedure. In this case, user chose new starting point, «,P,Y»«.