分数阶PIλ控制

Abstract —This article provides a graphical parameter tuningmethod of λPI controllers for fractional-order time-delay systems. First, the complete stabilizing region of λPI controllers in proportional- integral plane, for a fixed λ, is determined in terms of a graphical stability criterion applicable to time-delay systems. Then, the stabilizing region is maximized analytically with respect to parameter λ to expect the most various behaviors of the closed-loop systems. Finally, by defining appropriate functions relative to the requirements of gain and phase margins, the curves in the maximized stabilizing region satisfying the pre-specified gain and phase margins are drawn, which releases a flexible parameter tuning procedure. Numerical examples are given to illustrate the design steps.Keywords —Fractional order systems, λPI controllers, Time-delay systems, Graphical stability criterionI.INTRODUCTIONHE PID controller is the most commonly used control tool in industrial processes and is a very important topic in both academic research and control engineering applications [1-3]. The reason of this wide range of interest is its relatively simple structure that can be easily understood and implemented and its control ability in many practical processes. The conventional research efforts towards PID controllers was focuses mainly on the tuning rules of PID parameters, e.g., the well known Ziegler- Nicholes rules [4] for the processes with the S-shape reaction curve. In recent years, considerable attention has been paid to the so-called fractional-order systems whose models (the plants and/or the controllers) are described by fractional-order differential equations, i.e., equations involving noninteger-order derivatives [5]. This is due to the fact that many real control systems are well characterized by such equations. The frequency response and the transient response of the noninteger integral and its application to control systems were first introduced in [6]. Frequency analysis was also applied to fractional-order PID-like controllers, i.e., in the TID (tilt-integral-derivative) scheme proposed in [7], where the proportional compensating unit of a classical PID algorithm was replaced by an element referred to as a “tilt ” compensator. The synthesis approach to the CRONE control introduced in [8-9] pursues the “fractal robustness ” on the basis of a desired frequency template [10]. The conventional PID controller was extended to the fractional-order form of μλD PI both in the time domain [11] and in the frequency domain [12,5]. For time-delay systems, the design methods forλPI andμλD PI controllers have attracted more interests in recentThis work is supported by the National Nature Science Foundation of PR China under Grant 60874028.research [13-16]. In [13], taking advantage of the additional freedom λ provided by λPI controller, one more specification than in the case of conventional PI controller has achieved via a nonlinear function minimization subject to some given nonlinear constraints, improving the performance of the system and making it more robust to plant uncertainties (gain and time-constant changes). The robustness of λPI controller offered by fractional-order λ was further studied and confirmed in [14] by solving a nonlinear function minimization problem also. The results of D-decomposition method applied to the parameter space design of fixed structure controllers for the integer-order systems were generalized to the case of fractional-order μλD PI controllers [15-16]. The stability regions were derived analytically in different parameter space selected from five parameters of μλD PI controllers.In this article, a graphical stability criterion applicable to complex functions with delay factor is employed to investigate the stabilizing regions in the parameter space, i.e., in ),(i p k k -plane, of λPI controller for a fixed value of λ. By an analytical optimization procedure, the optimal value of λ corresponding to the maximum area of stabilizing region is computed. In this maximum stabilizing region, the specifications of phase and gain margins of fractional-order systems with time-delay is considered, which gives a quite flexible tuning method for λPI controllers.II. M AIN R ESULTSA. Stability CriterionConsider SISO unity feedback system shown in Fig.1, where )(s G is the transfer function of the plants with time-delayse s D s N s G θ-=)()()( (1) where D(s)andN(s)are integer order or fractional-orderpolynomials of complex variable s , 0>θis the time-delay, C(s)is the λPI controllerFig. 1.Unity feedback systemλsk k s C ip +=)( (2)A Graphical Tuning of λPI Controllers for Time-Delay SystemsDe-Jin Wang, and Jiang-Hui ZhangSchool of Electronic Information and Automation, Tianjin University of Science and Technology, Tianjin, 300222, PR China email: wdejin56@Twhere p k and i k are the proportional-gain and integral-gain, respectively,20<<λdenotes the fractional-order of the integrator. Taking 1=λ, the classical PI controller is recovered. The closed-loop fractional-order characteristic polynomial (quasi-polynomial, see [5]) is given by si p es N k s k s D s s θλλδ-++=)()()()(Multiplying both sides of the above equation by se θyields )()()()(*s N k s k e s D s s i p s++=λθλδ (3) The objective of this subsection is to determine thestabilizing region, based on a graphical stability criterion, in ),(i p k k -plane for a fixed )2,0(∈λ.To this end, substituting jw s = into (3), i.e. along the imaginary axis in s-plane, we get)(])([)()()(*jw N k jw k e jw D jw jw i p w j ++=λθλδ (4) Noting that2sin2cosλπλπλj j += (5) )()()(w jD w D jw D i r += (6) )()()(w jN w N jw N i r += (7) and partitioning )(*jw δ into its real and imaginarycomponents yields )()()(*w j w jw i r δδδ+=where)()2cos()()()2cos()()()(2222ωλπβωωωλπαθωωωωωδλλr i i r p i r r N k N N k D D +++++++= (8) )()2sin()()()2sin()()()(2222ωλπβωωωλπαθωωωωωδλλi i i r p i r i N k N N k D D +++++++= (9) where αdenotes the phase of the complex function (6), β the phase of (7), respectively.From (8) and (9), it is clear that both r δand i δdepend on parameters ),,,(ωλi p k k . Suppose that we have found, in one way or another, a point ),,,(0ωλi p k k on the imaginary axis such that⎪⎩⎪⎨⎧====0),,,(0),,,(0000ωλδδωλδδi p i i i p r r k k k k (10)i.e. there is a root on the imaginary axis in s-plane. Then, according to the Implicit Function Theorem, if the Jacobi Matrix),,,(00ωλδδδδi p k k i i p i i r pr k k k kJ ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡∂∂∂∂∂∂∂∂= (11)is nonsingular, the equations (10) has a unique local solution curve ))(),((ωωi p k k . Moreover, we have the following proposition [17].Proposition 1 The critical roots are in the right-half plane in the parameter space to the left of the curve ))(),((ωωi p k k when we follow this curve in the direction of increasing ω, whenever det 0<J and to the right when det 0>J . Here J is the Jacobi Matrix defined in (11).Solving equations (10) for p k and i k in terms of λ andω yields2sin)()()2sin()()(2222λπωωλπβαθωωωi r i r p N N D D k ++-++-= (12)2sin)()()sin()()(2222λπωωβαθωωωωλi r i r i N N D D k +-++= (13) On the other hand, it follows, from (8), (9) and (11), that20,0,0))()((2sindet 22<<>∀<+-=λωωωλπωλi r N N J (14)As the coefficients of the characteristic quasi-polynomial (4)and θare real, we have that if ωj is a root of (4), then, so toothe complex conjugate of it. Therefore, it is sufficient toconsider ),0[+∞∈ω, which corresponds to two cases:0=ωand ),0(+∞∈ω.When ),0(+∞∈ω, (12) and (13) determines a part of thestabilizing boundary line in ),(i p k k -plane, and fromProposition 1, we can identify which side of the line belongsto the stabilizing region as shown in the following examples. On the other hand, if the numerator )(s N of the transfer function )(s G in (1) contains a constant term, when 0=ω,0)(*=ωδ leads to0=i k (15) which gives another part of the stabilizing boundary line. Example 1 Integer- order plant controlled by fractional- order controller. Consider the following integer first-order plant with time-delay s e s s G 4.01105)(-+=For this plant, we determine the stabilizing parameters of thefractional-orderλPI controller (2). For fixed ,2.0=λ1=λ and 6.1=λ, respectively, from (12) and (13), the stabilizing boundary curves in ),(i p k k -plane can be drawnfor sufficiently large ω as shown in Fig.2. In the frequency range ),(10ωω, 00=ω, according to the sign of detJgiven by (14) and Proposition 1, the right side of each curve, as ω increases, belongs to stabilizing regions. Thus, the initial pieces of these curves in the frequency range ),(10ωω, together with the straight line (15), constitute the stabilizing regions. From Fig.2, it is clear that different values of λcorrespond to different shapes and areas of the stabilizing regions.kp k iFig. 2 Stabilizing boundary curves for different λExample 2 Fractional-order plant controlled by fractional- order controllers. Consider the following fractional-order plant with time-delays e a s s G 1.0221)(10)(-+=where a is a real number. Fixing 2.0=λ, we discuss theshapes of the stabilizing regions for different values of a . Similar to the procedure of Example 1, the stabilizing region in ),(i p k k -plane are plotted in Fig.3 for 5.0=a , 0=a , and 1-=a , respectively. It is observed that as a decreases, the stabilizing region shrinks. B. Optimization of λIt is expected that the bigger stabilizing region canλ=0.2kpk iFig. 3 Stabilizing region for different values of aprovides more various behaviors of control system, and achieves better response performances. From the above two examples, it is observed that the stabilizing regions have “sector” shapes which enables us to find the maximum stabilizing region by selecting an appropriate value of λ as discussed in the following.From the discussion of the above subsection, the parameter λof λPI controller affects the shape and area of the stabilizing region.We first note that when 0=i k , the stabilizing boundary curve intersects with p k axis, and the correspondingintersection frequency is1ω see Fig. 2. By letting0)(=ωi k in (13), we obtain that 1ωis the solution of theequation0)sin(=-+βαθωwhich is independent of λ.This fact leads us to utilize the Leibnitz Sector Formula to calculate the area of the stabilizing region,ωωωωωλωωd k k k k P i p i p ⎰'-'=10)]()()()([21)( (16)and the optimum value of λ corresponding to the biggeststabilizing region is determined by the solution of the equation0)(=λλd dP (17) Example 3 Recall Example 1. In this case, we have 00=ωand =1ω 3.99. By using the approximate integration, the relation curve between )(λP and λ is plotted in Fig. 4, from which it is seen that we have two optimum values of λ, i.e., 0→λ and 6.1=λapproximately. Choosing 2.0=λ and 6.1=λ, the stabilizing regions are shown in Fig. 2, which are bigger than the conventional case, 1=λ.λP (λ)Fig.4 Relation curve between )(λP and λIII.PERFORMANCE DESIGNOnce the stabilizing regions are known, we can further consider the synthesis of λPI controllers with given gain margin (GM) and phase margin (PM), i.e., the objective is to make the open-loop transfer function,)()(s G s C , meet the followingmj p p p i p p h e j D j N j k j k p 1)()()()(-=⋅+-θωλλωωωω (18) for the case of GM, andm gj j g g g i g p e e j D j N j k j k φθωλλωωωω-=⋅+-)()()()( (19) for the case of PM, respectively, where m h and m φ are the desired GM and PM, respectively, p ω and g ωare the phase and gain crossover frequencies of the open-loop system,respectively. For satisfactory response performance, the gain margin should be greater than 2, and the phase margin should be between30to60. In the sequel, let us only consider the case of phase margin since it is closely related to overshoot. (19) Leads us to define the following complex function of ω0)()()()()(=+⋅+=-m j j i p e e j D j N j k j k F φθωλλφωωωωω (20) from which, it follows that0)(])([)()()(=+++ωωωωλφθωλj N k j k e j D j i p j m (21)By comparing (4) with (21), it is exhibited that thedifference between (4) and (21) is the factor m φ in (21). Hence, for a given PM=m φ, the stabilizing analysis strategy in the previous section can be extended to the case of (21). First, from (21), along the same line as solving the solution (12) and (13), we have2sin)()()2sin()()(2222λπωωλπβαφθωωωi r m i r p N N D D k ++-+++-= (22)2sin)()()sin()()(2222λπωωβαφθωωωωλi r m i r iN N D D k +-+++= (23) which determines the achievable PM curve in the stabilizing region.Then, based on the above discussion, the following algorithm is developed on the achievement of the given PM for λPI controllers. Algorithm 1 For achieving the given PM of λPI controllers Step 1 Select an optimum value of λ in the range 20<<λ, using the formula (16) as shown in Fig.4. Step 2 For this λ, plot the curve ))(),((ωωi p k k using (12) and (13) for big enough ω, and identify the stabilizing region according to the sign of 0det <J given by (14) based on Proposition 1.Step 3 Given a desired m φ, draw the achievable PM curve given by (22) and (23) for big enough ω.Step 4 Choose a gain crossover frequency g ω along this curve in the stabilizing region according to the desired band width of closed-loop system, and compute the corresponding point ))(),((g i g p k k ωω.In the case of achieving the given GM of λPI controller, the design procedure is similar to Algorithm 1.If one wants to achieve the desired PM and GM simultaneously, the two curves for achieving the PM and GM can be drawn on the same graph, and the intersection of the two curves is the solution. Note that this intersection of the two curves may not exist in stabilizing region for bigger GM and/or PM.Example 4 Recall Example 1 again. First, selecting 2.0=λ, which corresponds to a relatively bigger stabilizing region (see Fig.2), and setting the desired PM to45, we draw the corresponding achievable PM curve in the stabilizing region using Algorithm 1, as shown in Fig.5 (a) (the inner solid line) . Next, for comparisons, choosing 1=λ, corresponding to the conventional PI controller, and PM=45, the achievable PM curve is plotted in the same figure (the inner dashed line). Clearly, the two PM curves have an intersection point A. The coordinates of point A reads 82.2=p k and 14.1=i k approximately, and the corresponding gain crossover frequencies are 89.1=g ω for2.0=λ and 46.1=g ω for 1=λ, respectively. Underthis set of parameters of λPI controller, the step responses are shown in Fig.5 (b) for the two cases. Although the given PM are the same (45), the overshoot for 2.0=λis lessthan that for 1=λ, and the response for 2.0=λ has a faster rise time than that for 1=λ, because the former has a bigger gain crossover frequency. Finally, it is observed, from Fig. 4, that the stabilizing region has another local maximum point at 6.1=λ. We compare the case of 2.0=λ with the case of 6.1=λ. Setting PM=45, the two PM curves have an intersection point B as shown in Fig. 6 (a) and the parameters at point B read 72.2=p k and 22.1=i k ,88.1=g ω for 2.0=λ, 86,0=g ω for 6.1=λ. Thestep responses are plotted in Fig. 6 (b), which shows that although 6.1=λ is a local maximum point of stabilizing region, the step response is worse than those for 2.0=λ and 1=λ.To summarize, a smaller value of λ can give a bigger stabilizing region and a better system performance.kpk i(a) Stabilizing regions and PM curvesStep ResponseTime (sec)A m p l i t u d e(b) Step responsesFig. 5 Comparisons between 2.0=λ and 1=λ.IV.C ONCLUSIONSIn this article, a graphical tuning method of λPI controllers for time-delay systems has been proposed. The maximum stabilizing region with respect to λ has been obtained analytically. In this region, the specifications ofachievable gain and phase margins have been discussed and the comparisons for different value of λ have been made bothin parameter space and in time domain.kpk i(a) Stabilizing region and PM curvesStep ResponseTime (sec)A m p l i t u d e(b) Step responsesFig. 6 Comparisons between 2.0=λ and 6.1=λ.R EFERENCES[1] K. Astrom, and T. Hagglund, PID controllers: Theory, Design, and Tuning. Research Triangle Park, NC: Instrument Society of American, 1995.[2] G.J. Silva, A. Datta, and S.P. Bhattacharyya, PID Controllers for Time- Delay Systems. Berlin, Germany: Birkhauser, 2004.[3] K. Astrom, and T.Hagglund, Advanced PID Control. Research T. Park, NC:ISA, 2005[4] J. G. Ziegler, and N. B. Nichols, “ Optimum setting for automatic controller,” Trans. ASME, Vol.64, pp. 759-768, 1942.[5] I. Podlubny, Fractional Differential Equations. London. UK, Academic Press, 1999[6] S. Manable, “The non- integer integral and its application to control systems ,” Electrotechnical Journal of Japan, Vol. 6, No. 3/4, pp. 83-87, 1961[7] B. J. Lurie, Three- parameter tunable tilt- integral- derivative (TID) controller. Us patent Us5371670, 1994[8] A. Oustaloup, B. Mathieu, and P. Lanusse, “The CRONE control of resonant plants: application to a flexible transmission, ” Euro. J. Control, Vol. 1, , pp. 113-121, Feb.., 1995[9] A. Oustaloup, X. Moreau, and M. Nouiliant, “ The CRONE suspension ,” Control Eng. Prac., Vol. 4, pp. 1101-1108, Aug., 1996 [10] A. Oustaloup, J. Sabatier, and P. Lanusse, “From fractal robustness toCRONE control ,” Fract. Calc. Appl. Anal., Vol. 2, pp. 1-30, Jan., 1999 [11] I. Petras, “The fractional-order controllers: methods for their synthesis and application ,” J. Electr. Eng., Vol. 50, No. 9-10, pp. 284-288, 1999 [12] I. Podlubny, “Fractional-order systems and μλD PI controllers,” IEEE Trans. Automat. Control, Vol. 44, pp. 208-214, Jan., 1999[13] C. A. Monje, A .J. Calderon, B .M. Vinagre, Y. Chen, and V. Feliu, “On fractional λPI controllers: Some tuning rules for robustness to plant uncertainties ,” Nonlinear Dynamics, Vol. 30, pp.369-381, 2004 [14] G. Maione, and P. Lino, “New tuning rules for fractional λPI controllers ,” Nonlinear Dynamics, Vol. 49, pp. 251-257, 2007[15] S. E. Hamamci, “Stabilization using fractional- order PI and PID controllers ,” Nonlinear Dynamics, Vol. 51, pp. 329-343, 2008[16] S. E. Hamamci, “An algorithm for stabilization of fractional-order time delay systems using fractional- order PID controllers ,” IEEE Trans. Automat. Control, Vol. 52, pp. 1964-1969, Oct., 2007[17] O. Diekmann, S. A. van Gils, S. M. verduyn Lunel, and H.-O. Walther, Delay Equations: Functional-, Complex- and Nonlinear Analysis. Applied Mathematical Sciences. Springer-Verlag, 1995.时滞系统λPI 控制器参数整定的图解法王德进 张江辉天津科技大学电子信息与自动化学院 天津 300222摘要：本文给出一种分数阶时滞系统λPI 控制器参数整定的图解法。