The Complete Set of PID Controllers with Guaranteed Gain and Phase Margins

Proceedings of the th IEEE onference on Decision and ontrol, and the European ontrol onference 5 Seville, Spain, December -5, 5 ThA7.5 The omplete Set of PID ontrollers with Guaranteed Gain and Phase Margins Keunsik Kim and Young hol Kim Abstract The problem of determining the entire set of PID controllers that stabilize a linear time invariant(lti) plant has been recently solved in [], []. In this paper, we extend these results to the problem of obtaining the complete set of PID parameters that attains prescribed gain and phase margins. An example is given for illustration. I. INTRODUTION The basic approach in modern control theories is first to obtain the entire set of stabilizing controllers as in the Youla parameterization, and then to search over the set to get a controller that meets given performance criteria. Using this approach, Datta et al. [] have recently reported very important results about computation of all stabilizing P, PI and PID controllers for a LTI system of arbitrary order. This complete solution is based on the generalized Hermite- Biehler theorem. They have shown that this problem is equivalent to the problem of linear programming. In [], the result has been extended to an arbitrary order LTI plant with time delay. In this paper, we consider a problem of computing PID controllers, (s) = ( + k p s + s )/s, along with an arbitrary LTI plant, P (s) = N(s)/D(s)e T ds, in the unit feedback configuration, that meet the stability margin specifications of gain and/or phase margins. We will show that this problem can be solved by extending the results from [], []. Here let A s denote the complete set of PID controllers which stabilizes a given LTI plant. If we wish to search a PID set over A s which attains merely a phase margin requirement, it is straightforwardly solved by using the method in []. However, it will be shown that more conditions should be considered for the case the gain margin specification is imposed. For this problem, a different computation method based on the characteristic polynomial with frequency sweeping has been proposed in []. Applying this method to PID controllers, we need to search three kinds of -D sets. Those shall be computed on (k p, ), (k p, ) and (, ) planes respectively. The final set is determined by taking the intersections of those sets. As a similar result, a problem of determining the complete set of stabilizing first order controllers that guarantees prespecified gain and phase margins has been solved in []. This research was supported in part by grant R---78- from the Basic Research Program of the Korea Science & Engineering Foundation and in part by grant N-55 from the KOTEF. Keunsik Kim is with the School of Automotive Eng., Ajou ollege, Boryong, 55-769, Korea kskim@motor.ac.kr Young hol Kim is with the School of Elec.&omputer Eng., hungbuk National University, heongju, 6-76, Korea yckim@cbu.ac.kr In the present paper, we start by finding the set of all stabilizing PID controllers, by means of the methods in [], []. We also use the important property that at a fixed k p, a subset of A s,kp on the (, ) plane appears only in the form of a polygon. Due to the fact that there must exist at least one characteristic root on the imaginary axis if one chooses the PID gains at the boundary of the stabilizing region, the magnitude and phase conditions of the Nyquist plot derived at the point +j are used to map the A s,kp onto the set A R,kp in the (, ) plane. The A R,kp represents all (, ) parameters that satisfy the prescribed stability margins at a fixed k p A s. Then the complete set of PID controllers attaining the given gain and/or phase margin will be determined by sweeping over k p. All these sets can be displayed using and -D graphics. An example is given for illustration. II. PROBLEM FORMULATION AND PRELIMINARIES In this section, we present several definitions and preliminaries regarding the relationship between the Nyquist plot and the boundary of the stability region in controller parameter space. onsider the unit feedback system given by a LTI plant and PID controller: P (s) := N(s) D(s) e T ds, (s) := ( + k p s + s ) s The open loop transfer function is written as G (s) =(s)p (s) =( + k p s + s ) G s (s), () G s (s) := N(s) sd(s) e T ds. We first clarify the definitions of gain and phase margins. Let us replace the plant P (s) by KP(s) for gain margin. If the loop transfer function G (s) has N unstable poles in the right half plane(rhp) of the s-plane, the Nyquist plot has to encircle the point +j as much by N times in order for the overall system to be stable. Thus, there may exist two intervals of gain K for stability, which are [K, ] and [,K + ]. K and K + are referred to as lower gain margin and upper gain margin, respectively. For simplicity, the prespecified gain margin is denoted by [K,K + ]. Similarly, let us replace the plant P (s) by e jθ P (s) for the phase margin. In the case that G (s) has zeros in RHP(or the plant is of nonminimum phase or time delay), the phase margin -78-9568-9/5/$. 5 IEEE 65

is also characterized by lower phase margin θ and upper phase margin θ +, which are written simply as [θ,θ + ]. Let A s be the set of parameters of all PID controllers that stabilizes the given plant P (s). A s can be computed using the results from [], []. It is important to note that the stabilizing region in the (, ) plane for a fixed value k p is a convex polygon as shown in Fig.. Let A s,kp denote a subset of A s with k p fixed at some value. A s is then the boundary of A s. Fig.. Stabilizing PID set: A s,kp, A K,kp, A θ,kp forafixedk p. For any value of A s, the Nyquist curve of the corresponding G (s) should cross (,j) for some ω. At this point, we have G (jω)=. This, in turn, is equivalent to the following two conditions for (k p,, ) A s : arg[( ω + jk p ω)g s (jω)] = π, () ( ω + jk p ω)g s (jω) =. () Here arg( ) [, π). Obviously, every PID controller with (k p,, ) A s gives rise to the gain/phase margin of [db] and o. On the other hand, any values inside A s provide a larger stability margin than that of the stability boundary. Therefore it is natural to assume that there exists a subset of A s attaining the given gain margin [K,K + ] and/or phase margin [θ,θ + ]. The problem of interest is to find the complete set of (k p,, ) from A s. The approach developed here to solve this problem proceeds as follows: (i) ompute all stabilizing PID gains using the result of [], []. (ii) ompute the set of PID parameters attaining the given gain margins [K,K + ]. The set is denoted by A K.Todo this, replace the plant P (s) in () by KP(s). ompute the set A c K : A c K := {(k p,, ) (k p,, ) A s and L(jω):=K( ω + jk p ω)g s (jω) =, for K [K,K + ],ω Ω K }, () Ω K will be defined in the next section. By the definition above, A c K is the subset of A s which makes KG (s) have a minimal destabilizing K. Then A K is determined by excluding A c K from A s. That is, A K = A s \A c K. This is a PID design with guaranteed gain margin. (iii) ompute the set of PID parameters attaining the given phase margins [θ,θ + ]. The set is denoted by A θ. To do this, replace the plant P (s) in () by e jθ P (s). ompute the set A c θ := {(k p,, ) (k p,, ) A s and G(jω):=e jθ ( ω + jk p ω)g s (jω) =, for θ [θ,θ + ],ω Ω θ }, (5) Ω θ will be defined in section IV. From the definition above, A c θ is the subset of A s which makes e jθ G (s) have a minimal destabilizing θ. Then A θ is determined by excluding A c θ from A s. That is, A θ = A s \A c θ. This is a PID design with guaranteed phase margin. (iv) If A K A θ Ø, it is clear that the set of PID controllers simultaneously satisfying both guaranteed gain and phase margins shall be obtained by A R = A s A K A θ. To proceed, we first investigate another equivalent condition for () as follows. Proposition The phase condition () yields the following identity. ω k p ω = tan{arg[g s (jω)]}, (6) with (a) <arg[g s (jω)] <π if k p >, (b) arg[g s (jω)] = or π if k p =, (7) (c) π < arg[g s (jω)] < π if k p <. Proof: The following equation necessarily implies the phase condition (). arg[( ω + jk p ω]= arg[g s (jω)] + iπ, (i =, ). (8) Expressing the left hand side of (8) in terms of arctangent, tan k p ω ( ω ) ± lπ = arg[g s(jω)] + iπ, (l =, ). (9) Using tan{θ +(i ± l)π} = tan(θ) and tan( θ) = tan(θ), (9) becomes k p ω ω = tan{ arg[g s (jω)]+(i ± l)π} = tan{arg[g s (jω)]}. () Equation () results in (6). This means that (6) is identical to two cases of (); arg[g (jω)] = and arg[g (jω)] = π. Thus, we need to impose other conditions on (6) to establish (). Rewriting (8) with i =, arg[g s (jω)] = π arg[ ω + jk p ω]. () learly, <arg[ ω + jk p ω] <πif k p > and π < arg[ ω + jk p ω] < π if k p <. Furthermore, if k p =, arg[ ω ]=or π. Therefore, the conditions (a), (b) and (c) must hold. 65

"! ( ' & % III. ALL PID SET WITH GUARANTEED GAIN MARGIN Since we have assumed that A s is known, solving problem (i) described in the previous section is equivalent to the problem of finding A c K in (). onsider a gain margin specification of [,K + ] for a stable plant and [K,K + ] for unstable plant. According to the definition of stability margins, the Nyquist plot of G (jω) should cross over the negative real axis at least [ K ]g L and [K + ]g U from (,j) in order for the PID controllers to guarantee the gain margin. This is shown in Fig..! " # ω + i := {ω K(ω) =K + }, for a stable or unstable P (s) ω i := {ω K(ω) =} for a stable P (s), := {ω K(ω) =K } for an unstable P (s). We see that it is sufficient to search over ω Ω K when we ) * + ' (, % & (a) G (s) stable (a) G (s) stable (b) G (s) unstable Fig.. Nyquist plots attaining the given gain margin The Nyquist curve corresponding to the boundary A s shown in Fig. must go through the point (,j) for some ω. For example, when the upper gain margin K + and the lower gain margin K are given, a PID controller that meets the stability margins should make the Nyquist curve of L(jω) cross over the negative real axis outside the interval [g L,g U ] in Fig.. From these, we know that computing such all PID set can be achieved by finding all regions in A s which provide gain margins [K, ] and [,K + ] reversely. The subset has been defined by A c K. In the sequel, the solution to problem (i) will be obtained by extruding A c K from A s. Therefore it is sufficient to compute A c K for the guaranteed gain margin problem. Let us define (6) again as follows: For afixedkp A s, kp ω M(ω) := tan{arg [G s (jω)]} =( ω ). () From the magnitude condition KG (jω) =,wehave K(ω) := = ( ω ) +(kp ω) G s (jω). () M(ω) +(kp ω) G s (jω) The frequency plot of K(ω) for ω [, ) as shown in Fig.. It is worth comparing the regions [,K + ] and [K,K + ] in Fig. with those in Fig.. Here we define a set of ω as follows: Ω K := {ω ω {[ω,ω+ ] [ω m,ω + m]}} () Fig.. #! " $ (b) G (s) unstable K(ω) vs. ω forafixedk p find the set A c K from A s with a fixed k p. Note that we need not consider the case of =because the corresponding point does not exist in the Nyquist plane []. However, we need to deal with two cases, k p and k p =, separately. A. k p Since K(ω) is the function for which the phase is π (see Proposition ), () can be represented by a set of straight lines in the (, ) plane. That is, for a fixed kp, using (), we have A c K,k p A c K,k p := {(, ) (, ) A s and ωj = M(ω j ), for ω j Ω K }. (5) denotes the subset of A c K with a fixed k p. Once we determine the set A c K,k p at a kp, we can compute A K,kp = A s,kp \A c K,k p. B. k p = If k p =, then, according to condition (b) in Proposition, (6) can not be occupied in (5). The plant P (s) can be written in terms of its even and odd parts. N(s) = N e (s )+sn o (s ), D(s) = D e (s )+sd o (s ). ondition (b) is equivalent to the condition that the imaginary part of G s (jω) equals zero. This is derived from the 655

definition of G s (s) and substituting s = jω as follows: (N e( ω ) D e( ω )+ω N o( ω ) D o( ω ))cosωt d + ω(n o( ω )D e( ω ) N e( ω )D o( ω ))sinωt d = (6) We solve the identity (6) for both cases of arg[g s (jω)] = and π. Let the solution be ω i for i =,,. Let s define Ω := { ω i arg[g s (j ω i )] = }, Ω π := { ω i arg[g s (j ω i )] = π} for i =,,. At these points, the magnitude condition () yields ω k = ± K G s ( ω k ), for ω k [Ω Ω π ]. (7) In the sequel, we can compute A c K,k p with k p =as follows: A c K,k p := {(, ) (, ) A s and ω j = K G s ( ω j ), for ω j Ω, ω k = K G s ( ω k ), for ω k Ω π }, (8) K [K,K + ]. Finally, by sweeping over k p, we will have the entire set of PID gains that guarantees the given gain margin: A K = k p A K,kp. (9) IV. ALL PID SET WITH GUARANTEED PHASE MARGIN To implement problem (ii) formulated in section II, we first replace plant P (s) by e jθ P (s). Then, G(s) :=e jθ G (s) =e jθ ( +k p s+ s ) G s (s). () Similar to the guaranteed gain margin, we consider a phase margin specification of θ + for a stable plant and [θ,θ + ] for a nonminimum phase plant. According to the definition of stability margins, the Nyquist plot of G(jω) should cross over the unit circle at least θ + or [θ,θ + ] from (,j) in order for the PID controllers to guarantee the phase margin. This is shown in Fig.. margin either θ + or [θ,θ + ]. The gain and phase conditions for G(jω) can be written as arg[( ω + jk p ω)g s (jω)] θ = π, () G(jω) = ( ω + jk p ω)g s (jω) =. () For a fixed k p, () becomes ω = ± Π(ω), () Π(ω) := G s (jω) (k p ω). () Note that Π(ω) should be greater than or equal to zero since ω R. Furthermore, rewriting (), we have θ(ω) =arg[(± Π(ω)+jω k p) G s (jω)] π. (5) Fig. 5 shows a typical plot of θ(ω). From these figures, we define Ω θ := {ω ω {[ω,ω+ ] [ω t,ω + t ]}} (6) ω + i := {ω θ(ω) =θ + }, for any G (s) ω i := {ω θ(ω) =}, for stable G (s), := {ω θ(ω) =θ }, for other G (s). (a) G (s) stable (a) G (s) stable (b) G (s) nonminimum phase Fig.. Nyquist plots attaining the given phase margin It is clear that if one chooses PID gains on the boundary A s, there exists at least one ω such that the Nyquist curve of G(jω) goes through the point (,j). That is, those PID gains give a phase margin of. Fig. shows that A c θ in (5) is nothing but the subset of A s that makes the phase (b) G (s) nonminimum phase or time delay Fig. 5. θ(ω) vs. ω forafixedkp Therefore it is sufficient to search over only ω Ω θ for the sake of finding A c θ from A s. ombining this set and (), we can compute A c θ with fixed k p. Let s define A c θ,k p := {(, ) (, ) A s and ωj = ± Π(ω)(ω j ), for ω j Ω θ }. (7) 656

Here A c θ,k p denotes the subset of A c θ for the fixed k p, which is a set of straight lines. After determining A c θ,k p at a kp,we compute A θ,kp = A s,kp \A c θ,k p. Finally sweeping over k p, we will have the complete set of PID gains that guarantees the given phase margin. That is, A θ = k p A θ,kp. (8) K(ω) 8 6 8 arg[g(jω)]=π arg[g(jω)]= This is the solution to problem (ii) in section II. Remark PI and PD controllers are special cases of a PID controller with either =or =. Thus the above algorithm can be extended to these cases as well. V. ILLUSTRATIVE EXAMPLES In this section, we will give two examples; a time delayed plant and the same plant with delay free. Example onsider the following nonminimum phase plant in []: s s + s + P (s) = s 5 +8s +s +6s +6s +7 e T ds (9) with time delay T d =[sec]. We wish to find the entire set of PID controllers that guarantees the following gain and phase margins: Gain margin : K + = (about 6 [db]), Phase margin : [θ,θ + ]=[ o, 6 o ]. To begin with, we use the method proposed in [], [] to obtain the complete set of PID controllers A s that stabilize the closed loop system in the plant above. Fig. 6 shows the set. 6 K + 6 8 6 8 ω Fig. 7. K(ω) vs. ω for Example From Fig. 6 k p =.88 has been chosen. Fig. 7 shows the K(ω) plot over ω [, ][rad/sec]. Solving () with K(ω) =K + =, we obtain Ω K = {[.59,.56] [.86,.] [.567,.67] [.8,.] [6.597, 6.67] [9.65, 9.75] [.9,.87] [.97,.98] [7.958, 7.9565]}. Next we compute a set of straight lines on the (, ) plane, which is obtained by (5) at every ω = {ω + i,ω i }, as shown in Fig. 8. Then has been determined, and appear in the same figure. A K,kp 5 A K, Kp A K, Kp K p 6 8 Kp=.56 Kp=.89 Kp=. Kp=.55 Kp=.88 Kp=. Kp=.6 Kp=. Kp=.79 Kp=.6 Kp=. Kp=.8 Kp=.8 Kp= 5.5 Kp= 5.8 Kp= 6.9 6 Kp= 7.6 Kp= 7.8 Fig. 6. All stabilizing PID controllers in the Example A. Stabilizing Region for Guaranteed Gain Margin Based on the discussion in section III, the following steps can be used for computing A K : ) hoose a k p from the results shown in Fig. 6. ) ompute Ω K using () after setting K(ω) =K +. ) ompute A c K,k p using (5) if k p or (8) if k p =. ) Find A K,kp by extruding A c K,k p from A s,kp..5.5.5 k.5.5 5 i Fig. 8. The region for guaranteed gain margin at k p =.88 B. Stabilizing Region for Guaranteed Phase Margin The procedures computing A θ from A s are as follows: ) hoose a k p from the result shown in Fig. 6. ) ompute Ω θ using (6) after setting θ(ω) = θ and θ +. ) ompute A c θ,k p using (7). ) Find A θ,kp by extruding A c θ,k p from A s,kp. Similar to the previous step, we choose k p =.88 from Fig. 6. The θ(ω) plot over ω [, ] [rad/sec] is shown in Fig. 9. By solving (6) with θ + = 6 o and θ = o, we obtain Ω θ = {[.96,.57] [.77,.655] [5.875, 6.7] [.56,.77] [6.97, 8.] [.985,.899] [.656,.6] [8.78, 9.7] [.95, 5.8] [9.976,.]}. 657

By means of (7), we constitute a set of straight lines at 5 every ω = {ω +,ω }. Fig. shows A θ,kp, A c θ,k p and A s,kp respectively. Region A θ,kp indicates the complete set of all PID parameters that guarantees the phase margin requirement when k p =.88. Fig. shows A K,kp, A θ,kp, A s,kp and their intersection, A R,kp. Thus any point (, ) A R,kp constitutes a controller that guarantees the previously specified gain and phase margins. To verify this, let s pick four points out on the Fig.. The gain and phase margins corresponding to these PID gains are given in Table I. # A R, Kp # # A K, Kp # θ(ω) 5 +Γ Π ΓΠ A θ, Kp.5.5.5.5.5 5 θ + 5 Fig.. The region for guaranteed gain and phase margins at k p =.88 θ 5 Kp=. Kp=.55 Kp=.88 5 Kp=. 6 8 6 8 ω 5 A θ,kp Fig. 9. θ(ω) vs. ω at k p =.88 A θ,kp A S,KP A S,KP.5.5.5.5.5 5 Fig.. The region for guaranteed phase margin at k p =.88 TABLE I GAIN AND PHASE MARGINS OF THE PID GAINS SELETED IN FIG. No. k p,, Gm[dB] Pm[deg] Remark #.88,.5,.98 6. 85. A K,kp #.88,.6,. 8. 6. A θ,kp #.88,.7,. 6. 6. A K,kp A θ,kp #.88,.9,.. 69.7 inside A R,kp If we repeat this procedure over the whole range of k p A s, we obtain the complete set A R of PID controllers which guarantees both gain and phase margins simultaneously, as shown in Fig.. k p Kp=. Kp=.6 Kp=.79 Kp=. Kp=.6 Fig.. The complete set of PID gains with guaranteed gain and phase margins in Example VI. ONLUDING REMARKS In this paper, we have determined for a given LTI plant, the complete set of stabilizing PID controllers that attains the given gain and phase margins. This computing method extends the results of [], [] in all stabilizing PID controllers for a LTI system were found and also for a LTI plant with time delay. The procedures are simple and constitutional. -D and -D graphics of the PID set will be very useful for computer-aided design because they allow for much better designs than those for obtaining a single controller, as mentioned in []. REFERENES [] A. Datta, M.T. Ho, and S.P. Bhattacharyya, Structure and Synthesis of PID ontrollers, Springer-Verlag, London, UK.,. [] H. Xu, A. Datta and S.P. Bhattacharyya, PID Stabilization of LTI Plants with Time-Delay, Proc. of the nd IEEE conf. on Decision and ontrol, Maui, Hawaii,, pp 8-. [] M. Tan, I. Kaya and D.P. Atherton, omputation of Stabilizing PI and PID ontrollers, Proc. of the IEEE conf. on ontrol Applications,, pp 876-88. [] R.N. Tantaris, L.H. Keel and S.P. Bhattacharyya, Gain/Phase Margin Design with First Order ontrollers, Proc. of the American ontrol onf., Denver, olorado,, pp 97-9. 658