Robust PID Controller Design - CiteSeerX

Robust PID Controller Design Yuzo Ohta∗ , Jing Li∗∗ , Kiyoharu Tagawa∗ and Hiromasa Haneda∗ ∗ Kobe University, Kobe 657, Japan, [email protected] ∗∗ Sin-sei Denki Co. Ltd., Kobe 657, Japan

Abstract— In this paper, a method to compute an almost correct region of PID parameters which guarantees robust stability and several robust performances for systems with real uncertain parameters is proposed. The proposed method computes the region in a short computing time by adopting the idea of computational geometry approach and using the Nonconvex Polygon Interval Arithmetic. I. INTRODUCTION PID controllers have been commonly used in the industry applications because of the functional simplicity and the ability of the final tuning of control parameters according to the actual plant. In the classical design methods of PID controllers, the characteristic for nominal plant is mainly concerned, and the robustness for stability and performances was not clear for general cases. On the other hand, in recent years, several design methods of robust PID controllers have been proposed under several problem setting [1] – [8]. The interval system framework and/or the unstructured uncertainty setting easily gives over bounding of uncertainty, and, hence, the obtained results might be conservative. In this sense, it is desired to treat the structural uncertainty or real physical parameters uncertainty. In the theoretical point of view, the approach adopted in [7], [8] can treat the latter type of uncertainty and general robust performances problem as long as the functionals corresponding to performances considered are multi polynomials of uncertain parameters. However this approach requires the huge computing time and storage to execute the algorithm. In this paper, we treat the uncertainty of real physical parameters, that is, we assume that the transfer function of a given plant is given by G(s, q), where q is the vector of uncertain parameters and q belongs a given box Q in Rn . We consider to compute almost correct sets of regions of parameters of the PID controller given by C(s, a) = aP (1 +

s aI + ), s 1/aD + s · τ

(1)

which guarantees robust stability and robust performances. We assume an initial box A = AP × AI × AD of the parameter vector a of the PID controller. As performances, we consider the disturbance suppres-

sion characteristic, the closed-loop band width characteristic, and the closed-loop gain peak assign characteristic. To solve this problem, we adopt the idea of Saeki’s method [3], which determine the feasible region of PID parameters by using the frequency response of the plant having no uncertainty. To treat the uncertainty of real physical parameters, we extend the idea by introducing the concept of the value set of the plant defined by V (G; s, Q) = {G(s, q)| q ∈ Q}

(2)

The issues here are 1) how to compute and maintain feasible regions of PID parameters, and 2) how to compute the value set or its ”good” estimate. For the first issue, we adopt a kind of ”splitting” method: we split the given box A into slender bars. According to the requirement corresponding to robust performances, we cut down these bars and remove the infeasible parts. Therefore the number of bars is crucial for computing time. To avoid the unnecessary small splitting of boxes, we change the way of splitting box according to the frequency we are concerned. Moreover, we adopt the computational geometry approach to reduce the time for executing the algorithm. For the second issue, we use the Non-convex Polygon Interval Arithmetic (NPIA) [9]. By using NPIA, we compute an estimate N(G; s, Q) of the value set V (G; s, Q). If the function G(s, q) has a totally decomposable expression, that is, in the expression each qi appears only once, then the obtained estimate N(G; s, Q) is ”good” in the sense that outbd[V (G; s, Q)]

⊆ N(G; s, Q) ⊆ N ( outbd[V (G; s, Q); ε]), (3)

where outbd[V ] denotes the region which is surrounded by the outer boundary of V and N (W ; ε) is an ε neighborhood of W . We note here that we compute ”good” estimates of the value sets of the given transfer function for many frequencies by using NPIA in the reasonable computing time in contrast to the rumor that it is very time consuming. When G has no totally decomposable expression, we compute estimate of outbd[V (G; s, Q)] by grid-ding intervals of qj ’s, which appear more than one in the expression of G, and by applying NPIA for the resultant ”totally decomposable expression” (since grid-ed parameters are

just constants). We say that the expression has a nice structure if the number of parameters appearing more than once in G is not so many. The PID controller has a very nice structure since it has the expression given by (1). One of the crucial point of the proposed approach is that it can keep the in-dependency of PID parameters and real physical parameters in the process of solving the problem. If the expression of the transfer function of the plant has a nice structure, then we can effectively use it in estimating the value set V (G; s, Q). If we transformed the problem to the non-negativity of functionals corresponding to performances, then it is hard to keep the nice structure of uncertain parameters, and this is one of the reasons why the approach in [7], [8] requires the huge computing time and storage even for the case when the expression of transfer function has a nice structure. Notation: In this paper, R and C denote the real line and the complex plane, respectively. For integers i and j such that i < = j, [i..j] denotes the set of integers {i, i + 1, · · · , j}. B[z; r] denotes the circle in the complex plane whose center and radius are z and r, respectively. V(f ; s, Q) denotes the value set of f (s, q), where q ∈ Q. N(f ; s, Q) denotes the estimate of value set of f (s, q) computed by using NPIA. For a set V ⊆ C, outbd(V ) denotes the region surrounded by the outer boundary of V , and N (V ; ε) denotes the ε neighborhood of V . II. PROBLEM STATEMENT Let us consider the system given by y = G(s, q)u, u = C(s, a)e + Wd (s, q)d, e = r − y,

5) |S(jω, q, a)| < = |σ(jω)| ∀ ω ∈ R, ∀ q ∈ Q, ∀ a ∈ A, where N (s, q, a), T (s, q, a) and S(s, q, a) are given by N (s, q, a) =

Wd (s, q)G(s, q) , 1 + G(s, q)C(s, a)

T (s, q, a) =

G(s, q)C(s, a) , 1 + G(s, q)C(s, a)

S(s, q, a) = 1 − T (s, q, a) and α(s), β(s) and γ(s) are given functions, and β satisfied |β(jω)| < 1. Given jω and let AN (ω) = {a ∈ A| |N (jω, q, a)| < = |α(jω)| ∀ q ∈ Q}, AB (ω) = {a ∈ A| |T (jω, q, a)| > = |β(jω)| ∀ q ∈ Q}, AT (ω) = {a ∈ A| |T (jω, q, a)| < = |γ(jω)| ∀ q ∈ Q}. AS (ω) = {a ∈ A| |S(jω, q, a)| < = |σ(jω)| ∀ q ∈ Q}. Then, we need to determine a region A such that A ⊆ A N ∩ AB ∩ AT ∩ AS , AN =

\

AN (ω), AB =

ω

AT =

\ ω

\

AB (ω),

ω

AT (ω), AS =

\

AS (ω),

ω

and the closed system is stable for all q ∈ Q and a ∈ A. We note that we can not take ∩ω in the above equations in the exact sense, we just take ∩ωk ∈Ω , where

where y, u, e, d and r denote the output, the control input, the disturbance, the tracking error and the reference input, respectively; G(s, q) is the transfer function of the plant, q = [q1 , q2 , · · · , qNq ]T , q ∈ Q, denotes the vector of physical system parameters; C(s, a) is the transfer function of the PID controller, a = [aP , aI , aD ]T , aP , aD and aI are tuning parameters; and Wd (s, q) is the model of disturbance. C(s, a) is given by (1). We assume that

is the set of frequencies we actually compute. This occurs usually when we use frequency response and it does not cause any serious trouble in the practical point of view. We also note that AN ∩ AB ∩ AT ∩ AS may consist of several connected components. Suppose that AN ∩ AB ∩ AT ∩ AS is obtained. Since |σ(jω)| < ∞, we have

Q = Q1 × Q2 × · · · × QNq , qi ∈ Qi ∀ i ∈ [1..Nq ],

0 ∈| {1 + G(jω, q)C(jω, a)| q ∈ Q, a ∈ AN }.

aP ∈ A P , a I ∈ A I , a D ∈ A D , A = A P × A I × A D , and Qi , AP , AI , and AD are given closed intervals in R. The problem we will consider is the following: Compute a feasible region A ⊆ AP ×AI ×AD satisfying 1) The closed system is stable for all q ∈ Q and a ∈ A; 2) |N (jω, q, a)| < = |α(jω)| ∀ ω ∈ R, ∀ q ∈ Q, ∀ a ∈ A, 3) |T (jω, q, a)| > = |β(jω)| ∀ ω ∈ R, ∀ q ∈ Q, ∀ a ∈ A, 4) |T (jω, q, a)| < = |γ(jω)| ∀ ω ∈ R, ∀ q ∈ Q, ∀ a ∈ A, and

Ω = {ωk , k ∈ [1..Nω ]}

(4)

Therefore, as shown in [3], if there is no unstable pole and zero cancellation in G(s, q)C(s, a) then we can conclude that for each connected component Ai of AN ∩ AB ∩ AT ∩ AS and for each a ∈ Ai and q ∈ Q closed system is stable if and only if there exists a0 ∈ Ai and q 0 ∈ Q such that closed system is stable, and the latter condition is easily examined. Thus, our main task is compute AN ∩ AB ∩ AT ∩ AS . At this point we note the following result which is automatic from the definition of T [3]:

Proposition 1 For given jω and q ∈ Q, the condition |T (jω, q, a)| < = |γ(jω)| is satisfied if and only if 1) C(jω, a) is in the inside of the circle B[zT (ω, q); rT (ω, q)] when |γ(jω)| < 1, where zT (ω, q) =

{BH (`, m)} of Fig. 1 (b). Step 6. For each `, m, and ω ∈ ΩH compute AT D (ω, AP` , AIm ) = {aD ∈ AD | |T (jω, q, a)| < = γ(jω),

|zT (ω, q)| 1 γ2 , rT (ω, q) = ; G(jω, q) 1 − γ 2 |γ(ω)|

2) C(jω, a) is at the outside of the circle B[zT (ω, q); rT (ω, q)] when |γ(jω)| > 1; and 3) C(jω, a) is in the inside of the half plane defined by

∀ q ∈ Q, ∀ aP ∈ AP` , ∀ aI ∈ AIm } and renewal BH (`, m) BH (`, m) := BH (`, m) ∩ AT D (ω, AP` , AIm )

when |γ(jω)| = 1. Because of the limitation of space, we do not show the results for AN , AB and AS , but they require that C(jω, a) is at the outside of appropriate circles. And we confine ourselves to illustrate the method to compute AT in the following.

AD

AD

III. OUTLINE OF THE SCHEME TO COMPUTE AT

AI (a)

(5)

and let V(CI ; s, AIm ) and V(CD ; s, ADn ) be value sets of CI and CD , where AIm and ADn are sub intervals of AI and AD , respectively. Let us divide Ω of (4) into the low frequency range ΩL and the high frequency range ΩH . We note that V(CI ; s, AIm ) is not so large in ΩH even if AIm is not so small, and that V(CD ; s, ADn ) is not so large in ΩL even if ADn is not so small. These facts are useful to reduce the number of splitting and the outline of the scheme to compute AT is the following: Algorithm get AT Step 1. Divide AP and AD into NP and ND numbers of subintervals {AP` } and {ADn }, respectively. Step 2. For each ` and n, let BL (`, n) := AI . Step 3. For each ` and n and ω ∈ ΩL , compute AT I (ω, AP` , ADn ) = {aI ∈ AI | |T (jω, q, a)| < = γ(jω),

Because of the limitation of the space, we will illustrate the method to compute AT D (ω, AP` , AIm ) in Step 6. Computation of AT I (ω, AP` , ADn ) can be done in a similar way. IV. COMPUTATION OF AT D (ω, AP` , AIm ) In this section, we fix ω and we omit the symbol ω as long as it is clear from the context. Let VG−1 = V(G−1 ; jω, Q) be the value set of G−1 and let PG−1 (jω) be a polygon which includes outbd[VG−1 ] and is included an ε neighborhood of outbd[VG−1 ]. We define PG in a similar way. Moreover, let |γ|2 z |zT (z)| ] , D(z) = B[zT (z); 2 1 − |γ| |γ| [ \ DU = D(z), DI = D(z) z∈PG−1 z∈PG−1

zT (z) =

N CI m = −

∀ q ∈ Q, ∀ aP ∈ AP` , ∀ aD ∈ ADn } (6) and renewal BL (`, n) as follows. BL (`, n) := BL (`, n) ∩ AT I (ω, AP` , ADn ).

AI (b)

Fig. 1 Reconstruction of AI × AD . (a) {BL (`, n)}, (b) {BH (`, m)}.

Let s · aD aI , CD (s, aD ) = , s 1 + s · τ · aD

(9)

Step 7. Let Amin and Amax be the minimum and D D maximum values of {BH (`, m)}. Divide the interval max [Amin D , AD ] into ND number of subintervals {ADn } and round BH (`, m) according to {ADn }.

C(jω, a) · G(jω, q) + |G(jω, q)|2 /2 > = 0,

CI (s, aI ) =

(8)

(7)

Step 4. Let Amin and Amax be the minimum and I I maximum values of {BL (`, n)}. Divide the interval ] into NI number of subintervals {AIm }. , Amax [Amin I I Round BL (`, n) according to {AIm } Step 5. Reconstruct {BL (`, n)} of Fig. 1 (a) into

(10) (11)

jω j AIm , CD (aD ) = (12) ω 1/aD + jω · τ

NC`,m (aD ) = AP` (1 ⊕ CIm ⊕ CD (aD )),

(13)

where ⊕ and denote the addition and the multiplication in NPIA [9]. From Proposition 1, ADn ∈ AT D (ω, AP` , AIm ) if 1) NC`,m (aD ) is in the inside of DI when |γ| < 1; 2) NC`,m (aD ) is at the outside of DU when |γ| > 1; 3) for all zG ∈ PG , z ∈ NC`,m (aD ) satisfies 2 Re[z · zG ] + 1 > =0 when |γ| = 1.

(14)

The conditions above are sufficient conditions but they are close to necessary conditions since PG−1 , PG and NC`,m (aD ) are almost equal to corresponding value sets. In the following, we assume that nodes of a polygon are sorted anti-clockwise, and we denote the set of all nodes of a polygon, say PG , by node(PG ). A. Case 1: (|γ| < 1) In this case, we compute intervals of aD such that NC`,m (aD ) is in the inside of disk D(zk ) = B[zT ; rT ] for each zk ∈ node(PG−1 ). Let A a1 , a ˆ2 ] and P` = [ˆ node(1 ⊕ NCIm ) = {CI1 , CI2 }, then a necessary condition for NC`,m (aD ) to be in the inside of B[zT ; rT ] is ˆ T,a CD (jω, aD ) is included in 4 circles D(z ˆk , CIj ) (k, ˆ j = 1, 2), where D(zT , a ˆk , CIj ) are defined by ˆ T,a D(z ˆk , CIj ) = B[zT · a ˆk − CIj ; rT · a ˆk ].

(15)

Now we note that the set {CD (aD ) | aD ∈ AD } is a a part of circle Bτ = B[rτ + j0; rτ ]. By using this fact, we can compute intervals of aD such that CD (jω, aD ) is in the inside of the above 4 circles. The above process means that we are examining the necessary condition for NC`,m (aD ) to be in the inside of DI , but it is very close to the sufficiency if adjoining nodes of PG−1 , A P` and NCIm are very close. B. Case 2: (|γ| > 1) The intervals of aD such that NC`,m (aD ) is at the outside of D(z) = B[zT ; rT ] is computed in a similar way that we illustrated in the above. Moreover, we have the following: Lemma 1 Let zk and zk+1 be nodes of an edge Ek of PG−1 . Suppose that we get the outer common tangents L1 and L2 to circles D(zk ) and D(zk+1 ), and let v1` , v1r v2` , and v2r be points of tangency of L1 and L2 . Moreover, let Pc (k) be the quad-angle obtained by connecting these points of tangency. Then, for each λ ∈ [0, 1], the corresponding point z = λzk + (1−λ)zk+1 on the edge Ek , the circle D(z) = B[zT (z); r(z)], r(z) = |zT (z)|/|γ|, is included in Pc (k) ∪ D(zk ) ∪ D(zk+1 ) By Lemma 1, if NC`,m (aD ) is at the outside of {D(zk )} ∪ {Pc (k)}, then NC`,m (aD ) is at the outside of DU . Intervals of aD such that NC`,m (aD ) is at the outside of disk D(zk ) = B[zT ; rT ] for all zk ∈ node(PG−1 ) can be computed in a similar way that we illustrated in A.. Moreover, NC`,m (aD ) is at the outside of the quadangle Pc (k) is equivalent with that CD (jω, aD ) is at the outside of the 8-gon A P` Pc (k) ⊕ (−1 (1 ⊕ NCIm )), and we can get intervals of aD to be at the outside of the 8-gon by assigning the part of the circle Bτ which is at the outside of the 8-gon.

C. Case 3: (|γ| = 1) We note that if 2 Re[z · zk ] + 1 > = 0, 2 Re[z · zk+1 ] + 1 > =0 then, for each λ ∈ [0, 1], we have 2 Re[z · (λzk + (1 − λ)zk+1 )] + 1 > = 0. Therefore, it suffices to compute the interval of aD such that z ∈ NC`,m (aD ) satisfies (14) for all zG ∈ PG . V. CONCLUSION In this paper, we proposed a method to compute a feasible set of parameters of PID controller satisfying the disturbance suppression characteristic, the closedloop band width characteristic and the bound of the peak of the closed system. Reference [1] B. Barmish, New Tools for Robustness of Linear Systems, Macmillan, (1994). [2] R. Kondo, S. Hara and T. Kaneko, ”Parameter space design for H∞ control, ” Trans. SICE, Vol. 27, No. 6, pp. 714–716 (1991) (in Japanease). [3] M. Saeki, ”A design method for the optimal PID controller for a two disk type mixed sensitivity problem, ” Trans. of ISCIE, 7, 12, pp.520-527 (1994) (in Japan-ease). [4] M. Saeki, ”An optimal design method of the three-term (PID) controller for multi-disk type robust control problem,” Preprint of 24th SICE Symp. on Control Theory, pp.87-90 (1995). [5] Y. Ohta and M. Saeki, ”Computation of feasible region of PID controller which satisfies several performances, ” Preprint of 25th SICE Symp. on Control Theory, pp. 435–438 (1996) (in Japanease). [6] M. Saeki and D. Hirayama, ”Parameter space design method of PID controller for robust sensitivity minimization problem, ” Preprint of 25th SICE Symp. on Control Theory, p. 429–434 (1996) (in Japan-ease). [7] S. Malan, M. Milanse and M. Taragna: Robust tuning for PID controllers with multiple performance specifications; Proc. the 33rd CDC, pp. 2684–2689 (1994) [8] S. Malan, M. Milanse and M. Taragna, ”Robust analysis and design of control systems using interval arithmetic,” Proc. the 13th IFAC World Congress, 25/30 (1996). [9] Y. Ohta, H. Kakiuch, M. Watabiki, K. Tagawa and H. Haneda: NPIA for fast computation of almost correct estimate of value sets of transfer functions; Proc. of the 35th CDC, pp. 2878–2883 (1996)