PID controller synthesis for a class of unstable MIMO plants with I/O delays

PID CONTROLLER SYNTHESIS FOR A CLASS OF UNSTABLE MIMO PLANTS WITH I/O DELAYS

A. N. G¨unde¸s∗ H. ¨Ozbay∗∗ A. B. ¨Ozg¨uler∗∗∗

∗_{Dept. Electrical & Computer Eng., Univ. of California,}

Davis, CA 95616, U.S.A., angundes@ucdavis.edu

∗∗_{Dept. Electrical & Electronics Eng., Bilkent Univ.,}

Ankara, 06800 Turkey, hitay@bilkent.edu.tr on leave from Dept. Electrical & Computer Eng., The Ohio State Univ.,Columbus, OH 43210, U.S.A.

∗∗∗_{Dept. Electrical & Electronics Eng., Bilkent Univ.,}

Ankara, 06800 Turkey, ozguler@ee.bilkent.edu.tr

Abstract: Conditions are presented for closed-loop stabilizability of linear time-invariant (LTI) multi-input, multi-output (MIMO) plants with I/O delays (time delays in the input and/or output channels) using PID (Proportional + Integral + Derivative) controllers. We show that systems with at most two unstable poles can be stabilized by PID controllers provided a small gain condition is satisfied. For systems with only one unstable pole, this condition is equivalent to having sufficiently small delay-unstable pole product. Our method of synthesis of such controllers identify some free parameters that can be used to satisfy further design criteria than stability. Copyright c
2006 IFAC

Keywords: PID Control, Time Delay, Unstable Systems, MIMO Systems

1. INTRODUCTION

While finite dimensional LTI systems are suffi-ciently accurate models for a wide range of dy-namical phenomena, there are many cases in which delay effects cannot be ignored and have to be in-cluded in the model, (Gu et al., 2003). An r input and r output LTI system with I/O delays (time delays in the input and/or output channels) can be represented by G_Λ(s) := Λo(s)G(s)Λi(s), where G

is the finite dimensional part (an r× r rational matrix), and Λ
(s) = diag

e−T1
s_,· · · , e−Tr
s is the delay matrix, where
stands for i (input delay case) or o (output delay case). This paper consid-ers closed-loop stabilization (see Fig. 1) of such

_{This work was supported in part by the European}

Commission (contract no. MIRG-CT-2004-006666) and by T ¨UB˙ITAK (grant no. EEEAG-105E065 and BAYG).

systems using proper PID-controllers (Goodwin et al., 2001): Cpid(s) = Kp+ Ki s + Kd s τds + 1 , (1)

where Kp, Ki, Kd are real matrices and τd > 0.

Stability of delay systems of retarded type, or even neutral type, is extensively investigated and many delay-independent and delay-dependent sta-bility results are available, (Gu et al., 2003), (Niculescu, 2001). Also, since delay element is an integral part of process control systems, most of the tuning and internal model control techniques used in process control systems apply to delay systems, (Astrom and Hagglund, 1995). The more special, but practically very relevant problem of existence of stabilizing PID-controllers is unfor-tunately not easy to solve even for the delay-free case. One way of gaining insight into the difficulty of the problem is to note that the

ex-istence of a stabilizing PID-controller for a plant of transfer matrix G(s) is equivalent to that of a constant stabilizing output feedback for a trans-formed MIMO plant (at this point an LMI ap-proach can be used, see e.g. (Lin et al., 2004) and the references therein). Alternatively, the prob-lem can be posed as determining conditions of existence of a stable and fixed-order controller for the extended plant G(s)s+1_s , which is again well-known to be a difficult problem, (Blondel et al., 1994; Vidyasagar, 1985). It should be men-tioned that there are some computational PID-stabilization methods, which consist of “efficient search” in the parameter space, recently developed for single-input single-output (SISO) delay-free systems (see (Saadaoui and Ozguler, 2005) and the references therein). Some of these techniques have been extended to cover scalar, single-delay systems,(Silva et al., 2005).

In this paper, making a novel use of the small gain theorem, we obtain two main results: First, for MIMO plants with input and/or output delays, we obtain some sufficient conditions on the exis-tence of stabilizing PID controllers, and second, we explicitly construct PID controllers for plants having only one unstable pole (under the condition that the product of the unstable pole with delay is sufficiently small). This construction is extended to the case of two unstable real or complex poles. As our goal is to establish existence of stabilizing PID controllers at this point, we do not consider performance issues but propose freedom in the design parameters that can be used to satisfy performance criteria.

Notation: As usual, R, C, C−, C+ denote real, complex, open left-half plane complex and open right-half plane complex numbers; U denotes the extended closed right-half plane, i.e., U = { s ∈ C | Re(s) ≥ 0 } ∪ {∞}; Rp denotes proper rational functions; S denotes stable proper real rational functions of s. The set of matrices whose entries are in S is denoted by M(S). The space H∞ is the set of all bounded analytic functions

in C₊. For h ∈ H_∞, the norm is defined as h∞= ess sups∈C₊|h(s)|, where ess sup denotes

the essential supremum. A matrix-valued function H is in M(H_∞) if all its entries are in H_∞, and in this case H_∞ = ess sups∈C+σ(H(s)), where ¯σ denotes the maximum singular value. From the induced L2gain point of view, a system with transfer matrix H is stable if and only if H ∈ M(H∞). Moreover, for square H∈ M(H∞), we say that H is unimodular if H−1 ∈ M(H∞). For simplicity, we drop (s) in transfer matrices such as G(s) where this causes no confusion. Also, since all norms we are interested in areH_∞ norms, we will drop the norm subscript, i.e. · _∞≡  ·  whenever this is clear from the context.

2. PROBLEM DESCRIPTION

Consider the standard unity-feedback system shown in Fig. 1, where G ∈ Rpr×r and C ∈ Rpr×r

denote the plant without the time delay term (non-delayed plant, for short) and the controller transfer-functions. It is assumed that the feedback system is well-posed and that the non-delayed plant and the controller have no unstable hidden-modes. It is also assumed that G∈ R_pr×r is full normal rank. The delay terms are in the form Λ
=

diag
e−sT1
_{, . . . , e}−sTr
, where, for 1≤ j ≤ r, we have T

j ∈ Θ
j = [0 , Tj,
max)⊂ R+ and
stands

for i (input channel delays) or o (output channel delays). We assume that the delay upper bound Tj,
max is known for all input and output

chan-nels j = 1, . . . , r. Define T
:= (T₁
, . . . , Tr
) and

Θ
:= (Θ
₁, . . . , Θ
r). As a shorthand notation we

will write (Ti

,To) =:T ∈ Θ := (Θi, Θo) to repre-sent all possibilities Tj
∈ Θ
j, 1 ≤ j ≤ r. We

de-note the delayed plant by G_Λ := Λo(s)G(s)Λi(s).

The closed-loop transfer matrix Hcl from (r, v) to

(u, y) is Hcl=  C(I + G_ΛC)−1 _{−C(I + G} ΛC)−1GΛ GΛC(I + GΛC)−1 (I + GΛC)−1GΛ  . (2)

We consider the proper form of PID-controllers in (1), where the real matrices Kp, Ki, Kd are called

the proportional constant, the integral constant, and the derivative constant, respectively. Due to implementation issues of the derivative action, a pole is typically added to the derivative term (with τd∈ R, τd> 0 when Kd= 0) so that the

transfer-function Cpid in (1) is proper. If one or more of

the three terms Kp, Ki, Kd is zero, then the

corresponding subscript is omitted from Cpid.

Definition 1. a) The feedback system Sys(G_Λ, C), shown in Fig. 1, is said to be stable iff the closed-loop map Hcl is in M(H∞). b) A delayed plant

GΛ, where G ∈ Rpr×r, is said to admit a PID-controller iff there exists a PID-PID-controller C =

Cpid as in (1) such that the system Sys(GΛ, C)

is stable. We say that G_Λ is stabilizable by a PID-controller, and Cpid is a stabilizing

PID-controller. 2

Let G = Y−1X be any left coprime factorization (LCF) of the plant, C = NcD−1c be any right

co-prime factorization (RCF) of the controller, where we use coprime factorizations over S ; i.e., for G ∈ R_pr×r, X, Y ∈ M(S) and det Y (∞) = 0, and similarly for C∈ R_pr×r, Nc, Dc∈ M(S) and

det Dc(∞) = 0. Let XΛ denote the “numerator”

matrix of GΛ, i.e., XΛ := Λo(s)X(s)Λi(s). Now

if the “denumerator” matrix Y of G = Y−1X is diagonal, then the delayed plant GΛ can be

- h - C - h? - _ΛoGΛi

-6 −

y_{ref e u} v _y

expressed as G_Λ = Y−1X_Λ. The controller C stabilizes G_Λif and only if M_Λ:= Y Dc+ X_ΛNc ∈

M(H∞) is unimodular, i.e., MΛ−1 ∈ M(H∞),

(Smith, 1989).

3. MAIN RESULTS

Throughout the paper we assume that Y−1 is diagonal, hence it commutes with Λo. Thus G_Λ=

Y−1X_Λin all cases studied here.

The result in Lemma 1 below will be used in designing PI or PID controllers from P or PD controllers, i.e., integral action will be added once P and D terms are designed. This result is a slight extension of Theorem 5.3.10 of (Vidyasagar, 1985) to systems with time delays.

Lemma 1. (Two-step controller synthesis): Let G∈ R_pr×r. Suppose that Cg is a controller that

stabilizes G_Λ, and Ch is a controller that stabilizes

the stable system H_Λ := G_Λ(I + CgG_Λ)−1 ∈

M(H∞). Then C = Cg+ Ch is also a controller

that stabilizes G_Λ. 2

Although it is obvious that stable plants admit PID-controllers, the freedom in the stabilizing controller parameters is still worth investigating. We propose a PID-controller synthesis for stable plants in Proposition 2 below, which will be fre-quently referred to in the sequel.

Proposition 2. (PID-controller synthesis for sta-ble plants): Let G ∈ Sr×r and assume (nor-mal) rankG(s) = r. i) PD-design: Choose any

ˆ

Kp Kˆd∈ Rr×r, τd> 0. Define Cpd:= Kˆp+

ˆ

K_ds τ_ds+1. Then, for any α satisfying0 < α <
GCpd
−1a

PD-controller that stabilizes G_Λ forT ∈ Θ is

Cpd(s) = αCpd= α ˆKp+

α ˆKds

τds + 1

. (3)

ii) PID-design: Let rankG(0) = r. Choose any

ˆ Kp, ˆKd∈ Rr×r, τd> 0. Define Cpid= Kˆp+G(0) −1 s + ˆ Kds

τ_ds+1. Then, for any γ satisfying

0 < γ < max{min

T ∈ΘΨ  −1_{, min}

T ∈ΘΨ

−1_{}, (4)}

where Ψ = sGΛ(s) ˆ_sCpid−I,Ψ = s ˆCpidG_sΛ(s)−I, a

PID-controller stabilizing G_Λ forT ∈ Θ is

Cpid(s) = γ Cpid. 2 (5)

Proposition 3 below gives general existence condi-tions for stabilizing PID controllers. If a stabilizing P, I, or D-controller exists, then it can be extended to a stabilizing PI, ID, PD, PID-controller: Proposition 3. (General existence conditions for stabilizing PID-controllers): Let G ∈ R_pr×r. Let (normal) rankG(s) = r. a) If G_Λ admits a PID-controller such that the integral constant Ki ∈

Rr×r_{is nonzero, then G has no transmission-zeros}

at s = 0 and rankKi= r . b) If G_Λadmits a

PID-controller such that any one of the three constants Kp, Kd, Ki is nonzero, then G_Λ admits a

PID-controller such that any two of the three constants is nonzero, and GΛ admits a PID-controller such

that all of the three constants is nonzero. c) If G_Λ admits a PID-controller such that two of the three constants Kp, Kd, Ki is nonzero, then G_Λ

admits a PID-controller such that all of the three constants is nonzero. In b) and c), the integral constant Ki = 0 only if G has no

transmission-zeros at s = 0. 2

Proposition 3 does not explicitly define which plant classes admit P, I, or D-controllers. We investigate specific classes of plants and propose stabilizing PID-controller design methods next in Section 3.1.

3.1 Delayed plants that admit PID-controllers Lemma 2. (Strong stabilizability is a necessary condition for PID stabilization): Let G∈ R_pr×r. Let rankG(s) = r . If G_Λ admits a PID-controller for any T ∈ Θ, then G is strongly stabilizable. 2 We now consider plants with a limited number of U-poles, including the origin. Such limitations on the number of U-poles are not surprising. Clearly, plants with an odd number of positive real-axis poles are not even strongly stabilizable if there are two or more positive real-axis zeros (including infinity). But even when the parity-interlacing-property is satisfied, plants that have more than two U-poles do not necessarily admit PID-controllers. For example, by using the Routh-Hurwitz test it can easily be shown that the plant (s− p)−3 does not admit a stabilizing PID controller for p≥ 0.

3.1.1. Plants with only one unstable real-axis pole We consider transfer matrices G in the form G = Y−1X = [ (s− p)

as + 1 I ]

−1 _[ (s− p)

as + 1 G ] , (6) where p ∈ R, p ≥ 0 and a ∈ R, a > 0, and rankX(p) = rank(s− p)G(s)|s=p = r .

Further-more, since G has no transmission-zeros at s = 0, rankX(0) = rank(s − p)G(s)|s₌₀ = r. In this

paper, by a slight abuse of notation, we say that G has only one unstable pole if Y (s), in (6), is identity times a scalar transfer function with a single zero in the closed right half plane.

Proposition 4. Let G ∈ Rpr×r, be as in (6), with X = (s−p)_as+1G ∈ M(S), rankX(p) = r. Let X(0) be nonsingular, G−1_{(0) = −p X(0)}−1_.

i) PD-design: Choose any Kˆd ∈ Rr×r, τd >

0. Define Cpd := X(0)−1 + ˆ Kds τ_ds+1 and ΦΛ := (s−p)GΛ(s)Cpd(s)−I s , ΦΛ :=  Cpd(s)(s−p)GΛ(s)−I s . If 0 ≤ p < max{min_{T ∈Θ}Φ_Λ−1, min_{T ∈Θ}Φ_Λ−1}, then for any positive α ∈ R satisfying (7), a PD-controller that stabilizes G_ΛforT ∈ Θ is given by (8); if Kˆd= 0, (8) is a P-controller: p < α + p < max{min T ∈ΘΦΛ −1_{, min} T ∈ΘΦΛ −1_},(7) Cpd(s) = ( α + p )Cpd(s). (8)

ii) PID-design: Let Cpd be as in (8). Let Hpd :=

(9), a PID-controller that stabilizes G_Λ forT ∈ Θ is given by (10) where Hpd(0)−1 = α X(0)−1; if ˆ Kd= 0, is a PI-controller: 0 < γ < max{min T ∈ΘΥ −1_{, min} T ∈ΘΥ −1_{}, (9)} where Υ = Hpd(s)Hpd(0)−1−I s , Υ = H_pd(0)−1H_pd(s)−I s Cpid(s) = Cpd(s) + γ α X(0)−1 s . 2 (10)

Example 1. Consider the delayed plant G_Λ(s) =

e−sT

s_−p, where p > 0. Then for a > 0, X :=

1/(as + 1), X(0) = 1. Choose any Kˆd ∈ R,

τd> 0. By Proposition 4, if p < minT∈ΘΦΛ−1= minT∈Θe −sT₋₁ s +e−sT ˆ Kd τds+1

−1_{, then for any α as}

in (7), Cpd(s) = (p+α)+(p+α)

ˆ Kds

τ_ds₊₁ is a stabilizing

PD-controller for G_Λ. Note that for SISO plants, Φ_Λ = ˜Φ_Λ. Now consider proportional controller design for a fixed T and p in this example. It is easy to show that a stabilizing P-controller exists if and only if pT < 1. Moreover, for any fixed pT < 1, there is a maximum allowable gain Kmax for the proportional controller; this is shown in Fig. 2 (a) as the exact bound. On the other hand, our approach uses the small gain argument and leads to Cp = (p + α) as the

controller gain. With Φ_Λ = T (e−sT_sT−1) = T , the condition p < Φ_Λ−1 is the same as pT < 1. From the bound given in (7), α < T−1 − p ; the largest controller gain we can use in our case is 1/T . This bound is also shown in Fig. 2 (a), which illustrates that the approach used here is not too conservative. Fig. 2 (a) also demonstrates the difficulty of controlling this plant using a proportional controller when the product of the unstable pole with delay is relatively large. Other fundamental performance limitations can also be quantified in terms of smallest achievable sensitivity level, (Stein, 1989), or mixed sensitivity H∞ cost, (Enns et al., 1992). It is also clear

that by using the derivative term we can improve the bound on largest allowable pT . The largest pole delay product for which we can find a PD-controller is 1.38 = 1/0.725, and that corresponds

to τd→ 0 and ˆKd/T = 0.31. 2

Example 2. Consider the transfer matrix G(s) of a distillation column, (Friedland, 1986), where G(s) = 1_sGoG1(s) with Go=  _{3.04 −}278.2 180 0.052 206.6₁₈₀  and G₁(s) = _{1 0} 0 _(s+6)(s+30)180  . An LCF of the plant is G(s) = Y (s)−1X(s), with X(s) = _as1₊₁GoG1(s),

Y (s) = _ass₊₁I, a > 0. Assume that the delays in the input channels are h₁ and h₂, and consider proportional control only. In this case we have



Cp = X(0)−1 = G−1o , Cp(s) = αX(0)−1 =

αG−1o , and ΦΛ(s) = (

Go G1(s)Λi(s) G−1o −I

s ). Fig. 3

shows Φ_Λ−1 versus h₁ and h₂, from which we see that the largest value 4.86 is obtained for

10−2 10−1 100

100 101 102

(a) Maximum allowable gain versus p*T

p*T

Kmax

/p

Conservative bound

Exact bound

Fig. 2. Maximum Kp versus pT .

h₁ = 0.18 and h₂ = 0. Note that 0.18 sec delay is needed in the first channel to equalize the phase lag in the input channels of G1Λi. In this case

0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 h 1 || Φ_Λ ||−1 versus h 1 and h2 h2 1 1.5 2 2.5 3 3.5 4 4.5

Fig. 3. Maximum Φ_Λ−1 versus h₁ and h₂. stability is guaranteed if α < ˜ΦΛ−1, where ˜ΦΛ = max{h1 , h2+ 0.2}. Clearly, the largest

gain allowable is αmax = 5, for h2 = 0 and 0 ≤ h1 < 0.2. This result is less conservative than the one obtained using the bound α < ΦΛ−1. Note that for h2 = 0, and h1 > 0.2

we have α_max = 1/h₁. But, when C(s) = αG−1o ,

the characteristic equation of this system is (1 +

α e−h1s s )(1 +

α_180e−h2s

s_(s+6)(s+30)) = 0. When h2 = 0,

actual largest gain we can use is αmax,act = min{αmax,1 , 36}, where αmax,1 = _2hπ

1, and for h1> 0.2 we have α_max,act= π 2h1 ≈ 1.57 h1 > αmax= 1 h1. (11)

The level of conservatism in this example is char-acterized by (11). Now consider the PD-controller

Cpd = α(I + ˜

K_ds

τ_ds+1)G−1o in (8), where ˆKd =:

˜

KdG−1o . The optimal derivative gain matrix ˆKd=

˜

KdG−1o is the one which minimizes ˜ΦΛ. Since

˜

Φ_Λ is diagonal, we restrict ˜Kd to be in the form

diag(Kd,1, Kd,2). Fig. 4 shows optimal Kd,1(resp.

Kd,2) versus h1 (resp. h2). 2

3.1.2. Plants with two unstable poles Let G(s)∈

R_pr×r have full (normal) rank. Let G have no transmission-zeros at s = 0. Define d := (a₁s + 1)(a₂s + 1) and n := (s − p₁)(s− p₂), where p₁, p₂∈ U, a₁, a₂ ∈ R, a₁, a₂ > 0, and let G have an LCF G = Y−1X of the form

G = Y−1X = [ n dI ]

−1 _[ n

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 h Kd K d1 versus h1 K d2 versus h2

Fig. 4. Optimal Kd,₁and Kd,₂.

where rankX(pj) = rank nG(s)|s=pj = r , j = 1, 2. Furthermore, since G has no transmission-zeros at s = 0, rankX(0) = rank nG(s)|s=0 = r . We

consider real and complex-conjugate pairs of poles as two separate cases:

Case a) The two unstable poles are real, i.e.,

pj ∈ R, pj ≥ 0, j = 1, 2. Proposition 5-(a)

shows that under certain assumptions, the delayed plant G_Λ admits PD and PID-controllers. Some plants in this class (for example, G =_(s−p 1

1)(s−p2), p₁≥ 0, p₂≥ 0) do not admit P, D, or I-controllers.

Case b) The two poles are a complex-conjugate

pair, i.e., p₁= ¯p₂, n = s2−(p₁+p₂)s+p₁p₂= s2− 2f s + g2, f ≥ 0, g > 0, f < g. In this case, X(0) = g2G(0) . Proposition 5-(b) shows that under certain assumptions, the delayed plant G_Λ admits D, PD, ID, PID-controllers. Some plants in this class (for example, G = _s2_+g1 2, g ≥ 0) do not admit P-controllers or I-controllers.

Proposition 5. Let G be as in (12), with X =

n

dG ∈ S r_×r

, rankX(pj) = r, j = 1, 2. Let

X(0) be nonsingular. Choose any τd > 0.

De-fine Φ_Λ:= s−1 n (τds+1)GΛ(s) X(0) −1_{− I, ˜Φ} Λ:= s−1 n (τds+1)X(0) −1_{GΛ(s)− I. a) Let pj ∈ R,} pj≥ 0, j = 1, 2. i) PD-design: If 0 ≤ p₁< Ω where

Ω := max{min_{T ∈Θ}Φ_Λ−1, min_{T ∈Θ}Φ_Λ−1}, then choose any α₁∈ R satisfying

p₁< α₁+ p₁< Ω. (13) Define W := (s − p2)GΛ(s)X(0)−1 and W = (s − p₂)X(0)−1G_Λ(s). Let Φ_2Λ:= α1(I+ (α1+p1) τds+1 W)−1W−I s , Φ_2Λ := α1(I+ (α1+p1) τds+1 W)−1W−I s . If 0 ≤ p2 < Ω2,

where Ω₂:= max{min_{T ∈Θ}
Φ_2Λ
−1, min_{T ∈Θ}
Φ_2Λ
−1},

then choose any α₂∈ R satisfying

p₂< α₂+ p₂< Ω₂. (14) Let Kp = (α1α2 − p1p2)X(0)−1, Kd = (α1+

p₁) (1 + τdp2)X(0)−1; then a PD-controller that

stabilizes G_ΛforT ∈ Θ is given by Cpd(s) = Kp+ K_ds

τds+1. ii) PID design: Let Cpd be as above. Then for any γ ∈ R satisfying (9) with Hpd(0)−1 =

α1α2X(0)−1, a PID-controller that stabilizes GΛ

forT ∈ Θ is given by (15): Cpid(s) = Cpd(s) + γ α1α2X(0)−1 s . (15) b) Let p₁ = ¯p₂ ∈ C, n = s2 − (p₁ + p₂)s + p1p2= s2− 2fs + g2, f ≥ 0, g > 0, f < g. i)

PD-design: If f + 2g < Ω, then choose any β₁, β₂∈ R, β₁, β₂≥ 0, satisfying

β₁+ β₂+ (f + 2g) < Ω. (16) Let Kp = [β₁β₂+ β₁(g− f) + β₂g− fg ]X(0)−1,

Kd = (β₁+ β₂+ f + 2g)X(0)−1− τdKp; then a

PD-controller that stabilizes G_Λ forT ∈ Θ is

Cpd(s) = Kp+ Kds τds + 1 = ϑ(s)G(0) −1 g2 (17) ϑ :=(β1+ β2+ f + 2g)s + β1(β2+ g− f) + β2g − fg τds + 1 . If 2(f + g) < Ω, let Kd= 2(f + g)X(0)−1; then a

D-controller that stabilizes G_Λ is

Cd(s) = Kds τds + 1 = 2 (f + g) ) G(0) −1_s g2(τds + 1) . (18)

ii) PID-design: Let Cpd be as in (17). Then

for any γ ∈ R satisfying (9) with Hpd(0)−1 =

(β1+ g)(β2+ g− f) X(0)−1, a PID-controller that stabilizes GΛ forT ∈ Θ is Cpid(s) = Cpd(s) + γ(β₁+ g)(β₂+ g− f) s G(0)−1 g2 . (19) Let Cd be as in (18). Then for any γ∈ R satisfying

(9) with Hd(0)−1 = g2X(0)−1 = G−1(0), an

ID-controller that stabilizes G_ΛforT ∈ Θ is

Cid(s) = Cd(s) +

γ G(0)−1

s . 2 (20)

4. CONCLUSIONS

We showed existence of stabilizing PID-controllers for a class of LTI, MIMO plants with delays in the input and/or output channels. Moreover, for plants with only one or two unstable poles (and finitely many C₋ poles) we gave explicit formu-lae for PID controller parameters. These results are obtained from a small gain based argument. Therefore, they are conservative. We were able to quantify the level of conservatism on the examples given.

In the light of inequality conditions (7) and (9) of Proposition 4, an interesting problem to study is the computation of optimal Kd which minimizes

Φ or Φ, and optimal α, Kd minimizing Υ

or Υ. Figure 4 answers this question partially for the specific example considered. The numerical values in this figure are computed from a brute-force search. An analytic solution is possible, see (Ozbay and Gundes, 2006) for further details.

APPENDIX: PROOFS

Proof of Lemma 1: Let G = Y−1X be an LCF; let

Cg = NgD−1g be an RCF. The controller Cg = NgDg−1

stabilizes GΛ = Y−1XΛ if and only if MΛ := Y Dg + X_ΛNg is unimodular. Since Cg stabilizes G, the

transfer-functions H_Λ = G_Λ(I + CgGΛ)−1 and I− CgHΛ = (I +

CgGΛ)−1 are stable. Now Ch stabilizes HΛ ∈ M(H∞)

if and only if Ch(I + HΛCh)−1 ∈ M(H∞), and (I + H_ΛCh)−1 ∈ M(H∞). Write C = Cg+ Ch= [Ng+ (I− CgHΛ)Ch(I + HΛCh)−1Dg] [(I + HΛCh)−1Dg]−1. Define Nc:= [Ng+ (I− CgHΛ)Ch(I + HΛCh)−1Dg]∈ M(H∞), Dc:= (I + HΛCh)−1Dg∈ M(H∞). Then Y Dc+ XΛNc= Y [(I +H_ΛCh)−1+ HΛCh(I + HΛCh)−1]Dg+ XΛNg= MΛ

is unimodular. Therefore, C = NcD−1c is a stabilizing

Proof of Proposition 2: i) Let Mpd:= I + GΛCpd= I + αG_Λ Cpd; then Mpdis unimodular since α
GΛCpd
= α
G Cpd
< 1. Therefore, Cpd stabilizes GΛ. Since

ˆ

Kp, ˆKd are arbitrary, they can be zero. ii) The controller Cpid stabilizes GΛ if and only if Mpid := _s+γs I + G_Λ s

s+γ Cpid is unimodular, and equivalently Mpid := s

s+γI +s+γs CpidGΛ is unimodular. Writing Mpid = I + γ s

s+γ (s GΛ(s) 

Cpid −I)

s , a sufficient condition for Mpdto be

unimodular is that γ satisfies the first upper bound in (4). Similarly, writing Mpid = I + s+γs (s



Cpid GΛ(s)−I)

s ,

a sufficient condition for ˜Mpid to be unimodular is that γ satisfies the second upper bound in (4). Since Mpid is

unimodular if and only if M˜pid is unimodular, the less

conservative one of these bounds suffices and hence, Cpid

in (5) stabilizes G_Λfor γ∈ R satisfying (4). 2

Proof of Proposition 3: a) Let G = Y−1X be an LCF of G. Let Cpid= Kp+K_si+_τKds

ds+1 be a PID-controller that

stabilizes G_Λ. For any positive a ∈ R, an RCF Cpid = NcDc−1 is
(Kp+_τKds ds+1) s s+a+sK+ai 
_s s+aIr −1 . Since

Cpid stabilizes GΛ, M_Λ = Y Dc+ X_ΛNc is unimodular,

which implies rankM_Λ(0) = r = rankX(0) Ki. Therefore,

rankX(0) = r, equivalently, G has no transmission-zeros at

s = 0, and rankKi= r . b) Suppose that GΛ is stabilized by Cp, equivalently Hp= GΛ(I + CpGΛ)−1∈ M(H∞); or by Cd, equivalently Hd= GΛ(I + CdGΛ)−1∈ M(H∞); or

by Ci, which implies Hi= GΛ(I + CiGΛ)−1 ∈ M(H∞). The (normal) ranks of Hp, Hd, Hiare equal to rankG = r .

By Proposition 2-(i), there exists a a P-controller for Hd,

for Hi, and for Hid; there exists a D-controller for Hp,

for Hi, and for Hpi. By Proposition 2-(ii), there exists an

I-controller for Hp, for Hd, and for Hpd. Consider Hp ∈ M(H∞) : If G has no transmission-zeros at s = 0, then rankHp(0) = rank(Y + XΛCp)−1(0)XΛ(0) = rankX(0) =

r. Let Cdh be a D-controller and Cih be an I-controller for Hp. By Lemma 1, the PD-controller Cpd= Cp+ Cdh and

the PI-controller Cpi = Cp+ Cih stabilize GΛ. Similarly, consider Hd ∈ M(H∞) : Since MdΛ := (Y + X_ΛCd) is

unimodular, rankMdΛ(0) = rankY (0) = r ; i.e., G has no poles at s = 0. If G has no transmission-zeros at s = 0, then rankHd(0) = rankMd−1Λ(0)XΛ(0) = rankX(0) = r . Let Cph be a P-controller and Cih be an I-controller for Hd. By Lemma 1, the PD-controller Cdp= Cd+ Cph and

the ID-controller Cdi = Cd+ Cih stabilize GΛ. Consider Hi ∈ M(H∞) : Let Cph be a P-controller and Cdh be

a D-controller for Hi. By Lemma 1, the PI-controller Cip = Ci+ Cph and the ID-controller Cid = Ci+ Cdh

stabilize G_Λ. c) Suppose that G_Λ is stabilized by Cpd,

equivalently Hpd= GΛ(I + CpdGΛ)−1∈ M(H∞); or by Cpi, which implies Hpi= GΛ(I + CpiGΛ)−1∈ M(H_∞); or by Cid, which implies Hid = GΛ(I + CidGΛ)−1 ∈

M(H∞). The (normal) ranks of Hpd, Hpi, Hid are equal

to rankG = r . Consider Hpd ∈ M(H∞) : If G has no

transmission-zeros at s = 0, then rankHpd(0) = rank(Y + X_ΛCpd)−1(0)XΛ(0) = rankX(0) = r . Let Cih be an

I-controller for Hpd. Let Cdh be a D-controller for Hpi. Let Cph be a P-controller for Hid. By Lemma 1, each of the

PID-controllers Cpdi = Cpd+ Cih, Cpid = Cpi+ Cdh,

and Cidp= Cid+ Cph stabilize GΛ. 2

Proof of Lemma 2: Let G = Y−1X be an LCF of

G. Let Cpid be a PID-controller that stabilizes GΛ. An RCF Cpid = NcDc−1 is given in Proposition 3. Then

det Dc(zi) = det _zzi

i+aIr > 0 for all zi > 0. If Cpid

stabilizes G_Λ, then M_Λ = Y Dc+ XΛNc is unimodular,

which implies det M_Λ(zi) = det Y (zi) det Dc(zi) has the

same sign for all zi∈ U such that X(zi) = 0; equivalently,

det Y (zi) has the same sign at all blocking-zeros of G.

Therefore, G has the parity-interlacing-property; hence, it is strongly stabilizable, (Vidyasagar, 1985). 2

Proof of Proposition 4: i) The controller Cpd stabilizes G_Λ if and only if Mpd := Y + XΛ Cpd = (s−p)_as+1[I + G_Λ Cpd] is unimodular. Now Mpd is unimodular if and

only if det(s−p)_as+1[I + G_ΛCpd] = det(s−p)_as+1 det[I + CpdGΛ] is a unit in H∞, and equivalently, Mpd := (s−p)as+1[I +

CpdGΛ] = Y + CpdXΛ is unimodular. Writing Mpd =

(s−p)

as+1 [I + (α + p) GΛ Cpd] = [ I + (α+p) ss+α ΦΛ](as+1)(s+α) ,

a sufficient condition for Mpd to be unimodular is that

(α + p) < min_{T ∈Θ}
Φ_Λ
−1. Similarly, writingMpd= [I +

(α+p)s

s+α ΦΛ](as+1)(s+α), a sufficient condition for Mpd to be

unimodular is that (α + p) < min_{T ∈Θ}
Φ_Λ
−1. Since

Mpd is unimodular if and only if Mpd is unimodular,

the less conservative one of these bounds suffices and hence, Cpd in (8) stabilizes GΛ for α satisfying (7). ii) Since Cpd stabilizes GΛ, Hpd := Mpd−1XΛ = GΛ(I + CpdGΛ)−1 ∈ M(H∞), where Hpd(0)−1 = G−1(0) + Kp = X(0)−1Y (0) + (α + p)X(0)−1 = α X(0)−1. Using

similar steps as in the proof of Proposition 2, the I-controller Ki/s = γHpd(0)−1/s stabilizes Hpd for any γ ∈ R satisfying (9). So, Cpid in (10) stabilizes GΛ. 2 Proof of Proposition 5: Omitted due to space restrictions.

See the full version of the paper submitted for publication in Automatica.

REFERENCES

Astr¨om, K. J., T. Hagglund, PID Controllers: Theory,

De-sign, and Tuning, Second Edition, Research Triangle

Park, NC: Instrument Society of America, 1995. Blondel, V., M. Gevers, R. Mortini, and R. Rupp,

“Simul-taneous stabilization of three or more plants: Condi-tions on the positive real axis do not suffice,” SIAM

Journal on Control and Optimization, vol. 32, no. 2,

pp. 572–590, 1994.

Enns, D., H. ¨Ozbay and A. Tannenbaum, “Abstract model and controller design for an unstable aircraft,” AIAA

Journal of Guidance Control and Dynamics,

March-April 1992, vol. 15, No. 2, pp. 498–508.

Friedland, B., Control System Design, An Introduction to

State Space Methods, McGraw-Hill, 1986.

Goodwin, G. C., S. F. Graebe, and M. E. Salgado, Control

System Design, Prentice Hall, 2001.

Gu, K., V. L. Kharitonov, and J. Chen, Stability of

Time-Delay Systems, Birkh¨auser, Boston, 2003.

Lin, C., Q.-G. Wang, T. H. Lee, “An improvement on multivariable PID controller design via iterative LMI approach,” Automatica, 40, pp. 519-525, 2004. Niculescu, S.-I., Delay Effects on Stability: A Robust

Con-trol Approach, Springer-Verlag: Heidelberg, LNCIS,

vol. 269, 2001. ¨

Ozbay, H., A. N. G¨unde¸s, “Resilient PI and PD controllers for a class of unstable MIMO plants with I/O delays,” to appear in the Proc. of 6th IFAC Workshop on Time

Delay Systems, L’Aquila, Italy, July 2006.

Saadaoui, K., A. B. ¨Ozg¨uler, “A new method for the computation of all stabilizing controllers of a given order,” Int. J. Control, vol. 78, no. 1, pp. 14–28, 2005. Silva, G. J., A. Datta, S. P. Bhattacharyya, PID

Con-trollers for Time-Delay Systems, Birkh¨auser, Boston, 2005

Smith, M. C., “On stabilization and existence of coprime factorizations,” IEEE Trans. Automatic Control, pp. 1005–1007, 1989.

Stein, G., “Respect the Unstable,” Bode Lecture IEEE

CDC Tampa, FL, December 1989, see also IEEE Control Systems Magazine, August 2003, pp. 12–25.

Vidyasagar, M., Control System Synthesis: A Factorization