Robustness region method for PID controller design Miloš Schlegel, Jiří Mertl, Martin Čech Department of Cybernetics University of West Bohemia in Pilsen Plzeň, Czech Republic,
[email protected] Abstract: An approach for tuning PID-type controllers is developed for single input single output LTI systems based on an extension to the classical method of D-partition. This method permits to shape the sensitivity and complementary sensitivity functions of the closed loop. Analytical formulas for the boundaries of robustness regions in the parameter plane are given in the general case of stable and unstable as well as irrational systems with time delay. It is shown that the new method can be used to design a fixed robust controller for several process models. Key words: PID control, D-partition, parameter plane method, robust design
1 Introduction Several methods of tuning PID controllers for satisfactory behaviour are used in practice. Most of them are only semi-empirical method selected for specific problem requirements [Äström and Hägglund 1995]. There exist very few really systematic procedures applicable for more complex systems such as non-minimum-phase systems, unstable systems and systems with significant time delay. One exceptional method is based on the classical D-partition [Neimark 1948]. Different modifications of this method for the design of the PID controller are described in [Shafiei and Shenton 1994, 1997], [Ho, Datta and Bhattacharyya 1996, 1997], [Munro, Löylemer and Baki 1999], [Äström and Hägglund 2001]. The key concept in these methods is the stability or robustness region in the parameter plane. It turns out that much insight into PID control can be obtained by analyzing such regions. The design technique presented in this paper gives a further step in this direction. The main result obtained isolates the robustness region in parametric plane for PID controller when we require that the open loop transfer function does not have intersection with the given circle in the complex plane. Moreover, the solution is completely analytical. For plotting the regions, only the procedure for determining roots of polynomials is needed.
2 Robustness regions arising in PID control design Consider the PID feedback control system, in which P(s) represents the process and C (s) the PID controller with transfer function
k 1 C ( s) k 1 Td s k i k d s , (1) T s s i where k, Ti and Td are its engineering parameters and k i k / Ti , k d kTd are gains used in the following. For this control loop, the open loop transfer function L(s) , the sensitivity function S (s) and complementary sensitivity function T (s) are, respectively, defined by L( s ) C ( s ) P ( s ) ,
S (s)
1 , 1 L( s )
T ( s)
L( s ) . 1 L( s )
The maximum sensitivity M S and maximum complementary sensitivity M P are, respectively, defined by M S sup S ( j ) , M P sup S ( j ) .
The controller design, according to [Äström 1998], involves the determination of three controller gains (k p , k i , k d ) in such a way that the disturbance – rejection performance index J 1 / k i is minimized while the closed loop is stable and satisfies the specified sensitivity M S and/or complementary sensitivity M P . Both of these constraints can be interpreted as a limitation on the open loop L( j ) in the following form: Nyquist plot
L( j ), 0,) is outside a circle with center s c and radius r 0 . More formally,
L( j ) U (c, r ) for arbitary 0 , where U (c, r ) s C : s c r. The sensitivity M S corresponds with sensitivity M P with
c 1, r 1 / M S
and the complementary
c M P2 / (1 M P2 ), r M P2 / ( M P2 1)
To be able to use the D-partition method, we restrict the above problem to the case of two design parameters k and d for which it holds k i kd ,
kd k
f fk 2 , d ki
(2)
where d 1 / Ti and f is a fixed parameter with the meaning f Td / Ti . Classically it was quite common to postulate f 1 / 4 . Note, that the given point in k-ki plane determines unambiguously the gain k d according to (2).
3 Sensitivity and complementary sensitivity regions We want to find all points in the parameter plane k-ki for which the corresponding Nyquist plot does not have an intersection with the disc U (c, r ) , where c is real and r 0 . For this purpose, we consider the following notation
L( j ) u jv , C( j ) x jy , P( j ) a jb , f d xk , y k . d
where
(3)
Further denote
d L( j ) u1 jv1 , d
d P( j ) a1 jb1 . d
(4)
Clearly, the Nyquist plot L( j ) is tangential to the circle with the center c and radius r at the frequency iff it holds (u c) 2 v 2 r 2 , (u c)u1 vv1 0 . From (5), by substituting (3 – 4) into u and v , we obtain 2
(5) 2
f d f d c ak bk r 2 ak bk d d
and
(6)
d f d f d f k ak bk c * a1 b1 b 2 d d d f d f d f d k 2 a b * a1 a 2 b1 0 . d d d
(7)
Eq. (7) has explicit solution for indeterminate variable k . By substituting this solution into (6), we obtain a polynomial equation of order eight for indeterminate variable d . Let d l , l 1, 2, ..., m denote the real roots of this equation, then the pairs
(k , k i kdl ), l 1, 2, ..., m describe parametric curves with the parameter which create the boundaries of robustness regions in the parameter plane k-ki.
4 MATLAB program for PID design The results obtained in Section 3 are used in a program for PID controller design. Here, we can define up to ten processes in one of three forms of transfer functions. We can also define up to ten conditions for the open loop transfer functions. The region boundaries are then plotted in the parameter plane k-ki. By choosing an arbitrary point in parameter plane k-ki the open loop frequency responses, step and load responses of the closed loops and sensitivity functions are depicted immediately. The following example illustrates the above PID design method. e 0.5 s 1 Example 1: Consider the following two processes: F1 ( s) , F2 ( s) 0.5s 1 0.25 s 14 and a condition M S 2.0 .The Figure 1 shows the results of the design.
(a)
(b)
(c)
Figure 1 (a) Robustness regions for M S 2.0 and processes F1 (s) (full), F2 (s) (dashed). The chosen point (marked with “+”) is k p 0.98, k i 2.02, k d 0.12 ; (b) step and load disturbance responses for the chosen point in the parameter plane k-ki; (c) frequency responses for the point and the condition M S 2.0
5 Conclusion An approach for tuning PID-type controllers is developed for single input single output LTI systems based on an extension to the classical method of D-partition. . It is shown that the new method can be used to design a fixed robust controller for several process models. A user friendly Matlab-program developed on the basis of the new method demonstrates these features. It is suggested that the proposed technique may suit a large range of applications in industrial practice. This work was particularly supported by MŠMT ČR – project no. MSM 235200004 and GAČR – project no. 102/01/1347.
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