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On Stabilizing with PID Controllers

Karim Saadaoui

, A. B¨ulent ¨

Ozg¨uler

§

LARA Automatique, ENIT BP 37, le Belv´ed`ere 1002, Tunis, Tunisie karimsa@postmaster.co.uk

§Department of Electrical and Electronics Engineering, Bilkent University, Bilkent, Ankara TR-06800, Turkey, Ozguler@ee.bilkent.edu.tr

Abstract— In this paper we give an algorithm that

de-termines the set of all stabilizing proportional-integral-derivative (PID) controllers that places the poles of the closed loop system in a desired stability regionS. The algorithm is applicable to linear, time invariant, input single-output plants. The solution is based on a generalization of the Hermite-Biehler theorem applicable to polynomials with complex coefficients and the the application of a stabilizing gain algorithm to three auxiliary plants.

I. INTRODUCTION

In many applications, stability of the closed loop sys-tem is not enough, and it is usually required that the poles of the closed loop system lie in more restrictive stability regions. It is known that time domain specifications for a closed loop system can be translated into desired closed loop pole locations in the frequency domain. These are specified in terms of the damping ratio and damped natural frequency of the closed loop poles. A desired stability region S in the complex plane is shown in Figure 1. −5 0 5 −6 −4 −2 0 2 4 6 Re[s] Im[s] −5 0 5 −5 0 5 Re[s] Im[s] −5 0 5 −5 0 5 Re[s] Im[s] S Sθ S−θ θ −θ

Fig. 1. Stability region S.

The region S is the intersection of two regions Sθ and

S−θ where

Sθ:= {s : s ∈ C, Re[se−jθ] < 0}.

S−θ:= {s : s ∈ C, Re[se] < 0}.

Sθand S−θare rotated Hurwitz stability regions. In [1], it

is stated that if all the poles of the closed-loop system lie in the region S, then the step response of the compensated system exhibits a maximum overshoot corresponding to

the angle θ. In [2], the region S is approximated by a cir-cular region and a design procedure that combines linear-quadratic optimal control with regional pole placement is given. Recently, a method for determining the set of all proportional controllers that places the closed-loop poles in the region S was given in [3]. In this paper, we give a method to determine the set of all PID controllers that places the poles of the closed-loop system in the region S.

The paper is organized as follows. In section 2, a gen-eralization of the Hermite-Biehler theorem applicable to polynomials with complex coefficients is stated. In section 3, the problem of stabilizing complex polynomials with proportional controllers is considered. In section 4, we give an algorithm that solves the problem of determining all stabilizing PID controllers that places the poles of the closed loop system in the stability region S.

II. A GENERALIZATION OF THEHERMITE-BIEHLER THEOREM

In this section, a generalization of the Hermite-Biehler theorem to polynomials with complex coefficients [4] is presented. Before proceeding any further, let us fix the notation used in this paper. Let R denote the set of real numbers and C denote the set of complex numbers and letC, C0, C+ denote the points in the open left half, jω-axis, and the open right half of the complex plane, respectively. Given a set of polynomials ψ1, ..., ψk∈ R[s]

not all zero and k > 1, their greatest common divisor (with highest coefficient 1) is unique and it is denoted by gcd{ψ1, ..., ψk}. If gcd {ψ1, ..., ψk} = 1, then we say 1, ..., ψk) is coprime. The derivative of ψ is denoted by

ψ. The set H of Hurwitz stable polynomials are H = {ψ(s) ∈ C[s] : ψ(s) = 0 ⇒ s ∈ C−}.

The signature σ(ψ) of a polynomial ψ ∈ C[s] is the difference between the number of its C roots and C+ roots. Given ψ∈ C[s], the real and imaginary parts (a, b) of ψ(s) are the unique polynomials a, b ∈ R[ω] such that ψ(jω) = a(ω) + jb(ω). Finally, let us define the signum functionS : R → {−1, 0, 1} by Sr =    −1 if r < 0 0 if r= 0 1 if r >0.

Theorem 1. [4] Let a non-zero polynomial ψ∈ C[s] of degree n have the real-imaginary parts (a, b). Suppose b≡ 0 and (a, b) is coprime. Let ω1< ω2< ... < ωk be

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the real, distinct finite roots of b with odd multiplicities. Also let ω0 = −∞, ωk+1 = ∞, and ξn be the leading

coefficient of ψ(s). Then σ(ψ) =                        1 2 

Sa(ω0)(−1)k+ 2ki=1Sa(ωi)(−1)k−i

− Sa(ωk+1)

 Sb(∞)

if n is even and ξn is purely real, or n is odd and ξn is purely imaginary.

1 2

 2k

i=1Sa(ωi)(−1)k−i

 Sb(∞) if n is even and ξn is not purely real, or n is odd and ξn is not purely imaginary.

(1) Proof. See [4], [5].

The following result transforms the problem of determining the number of real roots of a real polynomial to an equivalent problem of finding the signature of a complex polynomial.

Lemma 1. A non-zero polynomial ψ∈ R[u], has r real roots without counting the multiplicities if and only if the signature of the complex polynomial ¯ψ(jω) = ψ(w) + jψ(w) is −r.

Proof. See [6].

III. PROPORTIONALCONTROLLERS

We now describe an alternative method to the constant stabilizing gain method of [3] for complex polynomials. Our method avoids a search in an exponentially growing set. Given a plant g(s) = p(s)q(s), where p, q ∈ C[s] are coprime with m= deg p less than n = deg q, the set Ar(p, q) := {α ∈ R : σφ(s, α) = σ[q(s)+αp(s)] = r} is the set of all real α such that φ(s, α) has signature equal to r.

Let (h, g) and (f, e) be the real-imaginary parts of q and p, respectively, so that q(jω) = h(ω) + jg(ω), p(jω) = f(ω) + je(ω). Let d := gcd {f, e} so that f = d ¯f , e = d¯e, for coprime polynomials

¯

f ,¯e ∈ R[ω]. Then, the polynomial ¯p(s) such that ¯p(jω) := ¯f(ω) + j¯e(ω) is free of C0 roots. Let (H, G) be the real-imaginary parts of q(s)¯p∗(s) where ¯p∗(jω) :=

¯

f(ω) − j¯e(jω). Also let F (ω) := p(jω)¯p∗(jω). By a simple computation, it follows that

H(ω) = h(ω) ¯f(ω) + g(ω)¯e(ω), G(ω) = g(ω) ¯f(ω) − h(ω)¯e(ω), F(ω) = f(ω) ¯f(ω) + e(ω)¯e(ω).

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If G ≡ 0 and if they exist, let the real zeros with odd multiplicities of G(ω) be {ω1, ..., ωk} with the ordering

ω1 < ω2 <· · · < ωk, with ω0 := −∞ and ωk+1 := ∞

for notational convenience and let ξ be the leading coef-ficient of q(s)¯p∗(s). The following algorithm determines

whether Ar(p, q) is empty or not and outputs its elements

when it is not empty:

Algorithm 1. 1) Calculate αj=                        −H F(ωj), j = 1, . . . , k & F (ωj) = 0,

if n+ m is even and ξ is not purely real, or n+ m is odd and ξ is not purely imaginary.

−H

F(ωj), j = 0, . . . , k + 1 & F (ωj) = 0,

if n+ m is even and ξ is purely real, or n+ m is odd and ξ is purely imaginary. and sort the distinctαj’s in ascending order

¯α0<¯α1< . . . < ¯αk+2< ¯αk+3

where ¯α0= −∞ and ¯αk+3= ∞.

2) Identify all the sequences of signums

I =                    {i1, . . . , ik}

if n+ m is even and ξ is not purely real, or n+ m is odd and ξ is not purely imaginary. {i0, i1, . . . , ik+1}

if n+ m is even and ξ is purely real, or n+ m is odd and ξ is purely imaginary. where ij ∈ {−1, 1} for j = 0, 1, . . . , k + 1, that

correspond to the intervals (¯αj,¯αj+1) for j =

0, . . . , k + 2.

3) For each signum sequenceIj from step 2, if

r+σ(p∗) =                      {(−1)k−1i 0+ · · · + ik−2− ik−1 + ik}SG(∞)

if n + m is even and ξ is not purely real, orn + m is odd and ξ is not purely imaginary.

1

2{(−1)ki0+ · · · − 2ik−1+ 2ik − ik+1}SG(∞)

if n + m is even and ξ is purely real, orn + m is odd and ξ is purely imaginary.

holds, then (¯αj,¯αj+1) ∈ Ar(p, q)

Remark 1. By Step 3 of Algorithm 1, a necessary condition for the existence of an α ∈ Ar(p, q) is that the imaginary part of [q(s) + αp(s)]¯p∗(s) has at least ¯r = |r +σ(p∗)| real roots with odd multiplicities if n+m

is even and ξ is not purely real, or n+ m is odd and ξ is not purely imaginary, and ¯r = |r + σ(p∗) − 1| real roots with odd multiplicities if n+ m is even and ξ is purely real, or n+ m is odd and ξ is purely imaginary. 

IV. PID CONTROLLERS

Given a plant g(s) = p(s)q(s) and a PID controller c(s) = kds2+kps+ki

s , our objective is to find all values

of (kp, ki, kd) such that the closed loop characteristic

polynomial

φ(s, kp, ki, kd) = sq(s) + (kds2+ kps+ ki)p(s) has all its roots in the region S given in Figure 1. This is equivalent to solving two subproblems using the stability regions Sθ and S−θ and finding the intersection of the

solution sets.

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Let us first solve the problem for the stability region Sθ. Let s= s1ejθ, then

φ(s, kp, ki, kd) = sq(s) + (kds2+ kps+ ki)p(s),

= s1ejθq(s1ejθ) + (kds21ej2θ+ kps1ejθ

+ki)p(s1ejθ).

Since θ is constant, we have ejθ = c + jd, q(s

1ejθ) =

˜q(s1), and p(s1ejθ) = ˜p(s1) where ˜q(s1) and ˜p(s1) are

polynomials with complex coefficients. The new charac-teristic polynomial is given by

φ0θ(s1, kp, ki, kd) = (c + jd)s1˜q(s1) + [kd(c2− d2 +j2cd)s2 1+ kp(c + jd)s1+ ki]˜p(s1) = q0(s1) + kip0(s1). where q0(s1) = (c + jd)s1˜q(s1) + [kd(c2− d2+ j2cd)s21 +kps1(c + jd)]˜p(s1), p0(s1) = ˜p(s1). Roots of φ(s, kp, ki, kd) in stability

region Sθ is equivalent to roots of φ0θ(s1, kp, ki, kd) in

the open left half complex plane. Using the generalized Hermite-Biehler theorem applicable to complex polyno-mials and Lemma 1, we describe in what follows a method to compute all values of (kp, ki, kd) such that

φ0θ(s1, kp, ki, kd) is Hurwitz stable. Let

˜q(jω) = h(ω) + jg(ω), ˜p(jω) = f(ω) + je(ω), ¯p∗(jω) = ¯f(ω) − j¯e(ω),

recall that ¯f and¯e are coprime polynomials, then ˜q(jω)¯p∗(jω) = H(ω) + jG(ω),

˜p(jω)¯p∗(jω) = F (ω),

where H, G, and F are given by (2). Multiplying φ0θ(jω, kp, ki, kd) by ¯p∗(jω) we obtain ψ1θ(jω, kp, ki, kd) = φ0θ(jω, kp, ki, kd)¯p∗(jω) = [−ω(dH(ω) + cG(ω)) − kd(c2 −d22F(ω) − k pdωF(ω) + kiF(ω)] +j[H1(ω) + kpG1(ω) + kdF1(ω)] where H1(ω) = ω(cH(ω) − dG(ω)) G1(ω) = cωF (ω) F1(ω) = −2cdω2F(ω)

The reasoning behind the algorithm which determines the set of parameters kp, ki, kd of a stabilizing PID controller can be explained as follows. Suppose φ0θ(s) is Hurwitz stable for some kp, ki, kd∈ R. By Remark 1, it follows that the imaginary part H1(ω) + kpG1(ω) + kdF1(ω) of ψθ1(s) has at least r1= |n + 1 + σ(p∗)| real roots with odd multiplicities. Suppose the imaginary part of ψθ1(s) has r1 real roots with odd multiplicities. By Lemma 1, σ[φ1 θ(s1)] = −r1, where φ1θ(jω, kp, kd) = q1(jω) + kdp1(jω) (3) and q1(jω) = [H1(ω) + jH1(ω)] + kp[G1(ω) + jG1(ω)], p1(jω) = F1(ω) + jF1(ω).

Repeating the same process once more we get ψθ2(jω, kp, kd) = φθ1(jω, kp, kd)¯p∗1(jω) = [H2r(ω) + kpG2r(ω) + kdF2r(ω)] +j[H2(ω) + kpG2(ω)] where H2r(ω) = H1(ω)F1(ω) − H1(ω)F  1(ω) G2r(ω) = G1(ω)F1(ω) − G1(ω)F1(ω) F2r(ω) = F1(ω)F1(ω) − F1(ω)F1(ω) H2(ω) = H1(ω)F1(ω) − H1(ω)F1(ω) G2(ω) = G1(ω)F1(ω) − G1(ω)F1(ω) Once more, by Remark 1, it follows that the imaginary part H2(ω) + kpG2(ω) of ψ2θ(s) has at least r2 = |r1+ σ(p∗)| real roots with odd multiplicities. Now the

set of kp ∈ R which achieves r2 real roots with odd multiplicities in H2(ω) + kpG2(ω) can be determined by applying Algorithm 1 to q2(s) and p2(s) where

q2(jω) = H2(ω) + jH2(ω), p2(jω) = G2(ω) + jG2(ω).

The algorithm below traces the above steps backwards by repetition of the steps (i)-(iii) below:

(i) Pick a value ofkp such that the number of real roots with odd multiplicities of H2(ω) + kpF2(ω) is r2 or greater.

(ii) For every kp determined, find using Algorithm 1, all kd such that the imaginary part ofψ1θ(s) has at least r1 real roots.

(iii) For everykd determined, find using Algorithm 1, all

ki such thatφθ(s) is Hurwitz stable.

Algorithm 2.

1) Using Algorithm 1, partition the real axis into in-tervals (or union of inin-tervals) such that the number of real roots with odd multiplicities of H2(ω) + kpG2(ω) is constant in each interval.

2) Fix r1= |n + σ(p∗0) + 1|.

a) Find admissible range of kpfrom the intervals

found in the first step. (This corresponds to values ofkp such thatH2(ω) + kpG2(ω) has at leastr2= |r1+ σ(p∗1)| real roots with odd multiplicities).

i) Fix a kp in the admissible range.

ii) Apply Algorithm 1 to q1(s) and p1(s). (This calculates admissible values of kd

such thatH1(ω)+kpG1(ω)+kdF1(ω) has

r1 real roots).

A) Fix a kd in the admissible range. B) Apply Algorithm 1 to q0(s) and p0(s).

(This calculates admissible values ofki such thatφ0θ(s) is Hurwitz stable). C) Increment kd and go to step 2.a.ii.B.

iii) Increment kp and go to step 2.a.ii.

b) If r1 < deg(H1), then increment r1 by one and go to step 2.a.

For the stability region S−θ, it can be easily shown that the set of stabilizing PID controllers is exactly the same as the set of stabilizing PID controllers for

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. To see this, suppose that for (¯kp, ¯ki, ¯kd), s0 is a

root of φ(s, kp, ki, kd), then s0ejθq(s0ejθ) + (¯kds20ej2θ+

¯kps0ejθ+¯k

i)p(s0ejθ) = 0. As q(s) and p(s) are real

poly-nomials, it follows that s∗0e−jθq(s∗0e−jθ)+(¯kds∗20 e−j2θ+ ¯kps∗0e−jθ+ ¯ki)p(s∗0e−jθ) = 0 where s∗0 is the complex conjugate of s0. Since s∗0 and s0have the same real part, it follows that (¯kp, ¯ki, ¯kd) is stabilizing triplet for the stability region S−θ if and only if it is stabilizing triplet for the stability region Sθ.

Example 1. Consider a PID controller c(s) = kds2+ kps+ ki

s to stabilize the plant g(s) = p(s)

q(s) given in [5], where

q(s) = s5+ 8s4+ 32s3+ 46s2+ 46s + 17, p(s) = s3− 4s2+ s + 2.

The stability region S is given in Figure 1 and specified by the parameter θ=π4. For the rotated Hurwitz stability regions Sθ and S−θ, let s= s1ejπ4, then

˜q1(s1) = (−0.7071 + 0.7071j)s51− 8s41− (22.6274− 22.6274j)s3 1+ 46js21+ (32.5269+ 32.5269j)s1+ 17, ˜p1(s1) = (−0.7071 + 0.7071j)s31− 4js21 +(0.7071 + 0.7071j)s1+ 2.

Using the new polynomials ˜q1(s1), ˜p1(s1), and the

method described in this section, we obtain the stabilizing values of (kp, ki, kd) as shown in Figure 2. From these

−2 −1 0 1 −0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 kp kd ki

Fig. 2. Stabilizing values(kp, ki, kd).

results , for kp = −1 and ki = 0.4049 we obtain

(0.0255, 0.5612) as the stabilizing interval for kd. The

root-locus for the values of kd in this interval is shown in Figure 3. With kp = 0.5 and ki = 1.7, we obtain (0.9355, 1.0986) as the stabilizing interval for kd. The

root-locus for the values of kd in this interval is shown

in Figure 4.

V. CONCLUSION

In this paper, an algorithm is given for computing the set of all PID controllers that places the poles of a closed loop system in a desired stability region. The method is

Root Locus Real Axis Imag Axis −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 5 4 3 2 1 5 4 3 2 1 0.985 0.94 0.86 0.76 0.64 0.5 0.34 0.16 0.985 0.94 0.86 0.76 0.64 0.5 0.34 0.16

Fig. 3. Attainable roots with respect to regionsandS−θ.

Root Locus Real Axis Imag Axis −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 5 4 3 2 1 5 4 3 2 1 0.985 0.94 0.86 0.76 0.64 0.5 0.34 0.16 0.985 0.94 0.86 0.76 0.64 0.5 0.34 0.16

Fig. 4. Attainable roots with respect to regionS.

based on a generalization of the Hermite-Biehler theorem applicable to complex polynomials. We can also consider other stability regions. For example, by simply replacing s by s+ σ stabilizing with respect to shifted Hurwitz stability region is possible. We can combine both regions, shifted and rotated Hurwitz stability regions, and use the algorithm given in this paper to consider stabilizability with respect to other sectors of the left half-plane.

REFERENCES

[1] M. Vidyasagar, Control System Synthesis: a Factorization Approach, the MIT press, Cambridge, Massachussets, 1985.

[2] W. M. Haddad and D. S. Bernstein,“Controller Design With Re-gional Pole Constraints ”, IEEE Trans. Automat. Cont. , vol. 37, no.1, pp. 54-69, 1992.

[3] M. T. Ho, A. Datta, and S. P. Bhattacharyya, “ Constant Gain Stabilization with Specified Damping Ratio and Damped Natural Frequency ”, Proc. IFAC World Congress, Beijing, P.R.C., July 1999.

[4] M. T. Ho,,“Synthesis ofH∞ PID controllers: A Parametric Ap-proach ”, Automatica, vol. 39, pp.1069-1075, 2003.

[5] A. Datta, M. T. Ho, and S. P. Bhattacharyya, Structure and Synthesis

of PID controllers, New York: Springer-Verlag, 2000.

[6] K. Saadaoui and A. B. ¨Ozg¨uler,“ A new method for the computation of all stabilizing controllers of a given order ”, International Journal

of Control, vol. 78, no.1, pp. 14-28, 2005.

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Fig. 1. Stability region S.
Fig. 4. Attainable roots with respect to region S.

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