In this section we present results that apply to the system described in equation (3.3), when one sensor and one actuator are used.
Since we have one sensor at the i-th location and the actuator at the j-th location, then the closed loop poles are given by
det hA˜1(s)
i
= det hA(s)˜
i
+àfi(s) ˜Aij(s) =p(s) +àq(s). (3.7) Hereq(s) =fi(s) ˜Aij(s), andp(s) = det
hA(s)˜ i
. The functionfi(s) is the product of the controller transfer functionτc(s) and the term exp[−sTi] due to the time delay.
Letg(s) =τc(s) ˜Aij(s), thenq(s) =g(s) exp[−sTi]. Since ˜Aij(s) is a polynomial, it is analytic, and if we assume that the controller transfer functionτc(s) is analytic in the right-half plane, theng(s) is analytic in the right-half plane. In most situations it is reasonable to expect thatp(s) goes to infinity faster thanq(s) does as|s| → ∞, so here we shall assume that
¯¯
¯¯p(s) q(s)
¯¯
¯¯→ ∞as|s| → ∞.
Assuming that the open loop poles have negative real parts, then Result 3.3 applies. Following the proof of the result, for large enough R, the bound on the gain for stability is given by
à∗< min
s∈ΩR
¯¯
¯¯p(s) q(s)
¯¯
¯¯= min
s∈ΓR
¯¯
¯¯p(s) q(s)
¯¯
¯¯.
The boundary ΓR is composed of a semicircleCεof radius Rand a segment of the imaginary axisIR of length 2R. The minimum must occur on the segmentIR, provided R is large enough and
¯¯
¯¯p(s) q(s)
¯¯
¯¯ → ∞ as |s| → ∞. Since
¯¯
¯¯p(s) q(s)
¯¯
¯¯ → ∞ as
|s| → ∞, then there exists an R∗ so that the minimum occurs for thisR∗, and for anyR∗< Rwe have min
s∈ΩR∗
¯¯
¯¯p(s) q(s)
¯¯
¯¯= min
s∈ΩR
¯¯
¯¯p(s) q(s)
¯¯
¯¯. ForR∗< R,CR∗ is in ΩR, by the minimum modulus principle, the minimum cannot occur in the interior of ΩR, thus
it occurs onIR∗. The above observations are used in the proof of the following result.
Result 3.4: When using one sensor and one actuator, there exists an interval [−à∗, à∗] in the gainà, withà∗>0, which isindependent of time delay, such that the closed loop poles have negative real parts, provided all the open loop poles have negative real parts, and
¯¯
¯¯p(s) q(s)
¯¯
¯¯→ ∞as |s| → ∞.
Proof: By the previous argument, for large enough R, the minimum min
s∈ΓR
¯¯
¯¯p(s) q(s)
¯¯
¯¯ will occur onIR. Say it occurs ats=iw∗∗ (i=√
−1) then for large enoughR, à∗< à∗∗= min
s∈ΩR
¯¯
¯¯p(s) q(s)
¯¯
¯¯= min
w∈IR
¯¯
¯¯ p(iw) g(iw) exp[−iwTi]
¯¯
¯¯= min
w∈IR
¯¯
¯¯p(iw) g(iw)
¯¯
¯¯=
¯¯
¯¯p(iw∗∗
g(iw∗∗)
¯¯
¯¯. (3.8)
This establishes the result. ¤
From the result, the boundà∗∗ can be obtained by considering the system with no time delay and finding the minimum of
¯¯
¯¯p(iw) q(iw
¯¯
¯¯forw∈IR. For some systems it may be possible thatà∗∗is actually the maximum gain for stability for the system with no time delay. This will be the case inExample 1shown after Result 3.6 as well as for the control systems explored in Section 4. That is, for zero time delay one finds the maximum gain for stability using positive and negative feedback, andà∗∗
is the minimum of the two maximum gains. In general, however, it is not necessarily true that the minimum of the maximum gains for stabilityà∗∗ is achieved for the system with zero time delay. It is possible for some systems that the maximum gain for stability of the system with zero time delay is larger than the gain that is independent of time delay. This case will be illustrated inExample 2.
Since the boundà∗∗occurs on the imaginary axis and it may not necessarily be the bound for the maximum gain for stability for the system with no time delay, and it is a bound on the gain for which the system is stable for all time delays, one might ask ifà∗∗ will actually be the maximum gain for stability at some time delay.
The following result answers this question.
Result 3.5: For systems with one sensor and one actuator, as described in Re- sult 3.4, suppose à∗∗ occurs at s = iw∗∗, then there exists a time delay T such that
p(iw∗∗) +à∗∗g(iw∗∗) exp(−iw∗∗T) = 0.
That is, the bound of the maximum gain for stability independent of time delay given in Result 3.4 is achieved at some time delayT. The value ofT will be given in the proof, and it is derived from the system properties without time delay.
Proof: Let the real and imaginary parts ofp(iw∗∗) andq(iw∗∗) from Result 3.4 be given by
p(iw∗∗) =pR+ipI, and g(iw∗∗) =gR+igI. (3.9) Consider
pR+ipI+à∗∗(gR+igI)(x−iy) = 0.
Then
x−iy=−pRgR+pIgI
à∗∗(gR2 +gI2)− pIgR−pRgI
à∗∗(gR2 +gI2). (3.10) Note thatà∗∗=
s
p2R+p2I
g2R+gI2. andx2+y2= 1. Therefore we can take cos(w∗∗T) =x
and sin(w∗∗T) =y, establishing the result. ¤
Result 3.5 indicates that there is a time delay at which the system will achieve the bound on the maximum gain for stability given in Result 3.4. The result also indicates if the bound of the maximum gain for stability that is independent of time delay is actually achieved or not for zero time delay. If x= 1, then for the system with zero time delay the maximum gain for stability will be reached with gain à =à∗∗. Similarly, if x=−1, then for the system with zero time delay the maximum gain for stability will be reached with gain à=−à∗∗. If |x| 6= 1, then the maximum gain for stability of the system with zero time delay is larger than the maximum gain for stability that is independent of time delay. The question of the uniqueness of the occurrence of poles at a gain ofà∗∗(for positive and negative feedback) is dealt with in the next result.
Result 3.6: For the case of a single actuator and a single sensor, suppose that a closed loop pole given by (3.7) is at s =iw∗, à =à∗ and with a time delay of T =T∗. Then there exist closed loop poles ats=iw∗, à=−à∗, with time delays T =T∗+(2n+ 1)π
w∗ , forn= 0,1,2, . . ., and at s=iw∗, à=à∗, with time delays T =T∗+2nπ
w∗ , forn= 1,2, . . . ,.
Proof: For a pole on the imaginary axis, we have p(iw) +àg(iw) exp[−iwT] =p(iw) +àg(iw)Ă
cos(wT)−isin(wT)¢
. (3.11) When the pole occurs atw=w∗, à=à∗ andT =T∗, then (3.11) is also satisfied withw=w∗,à=à∗, provided the time delayT satisfies the equations
cos(w∗T∗) = cos(w∗T) and sin(w∗T∗) = sin(w∗T). These equations are satisfied when T = T∗ +2nπ
w∗ , for n = 1,2, . . . ,. Similarly, when the pole occurs atw=w∗,à=à∗ andT =T∗, then (3.11) is also satisfied withw=w∗,à=−à∗, provided the time delayT satisfies the equations
cos(w∗T∗) =−cos(w∗T) and sin(w∗T∗) =−sin(w∗T). These equations are satisfied whenT=T∗+(2n+ 1)π
w∗ , forn= 0,1,2, . . . ,. ¤
One consequence of this result is that the maximum gain for stability which is independent of time delay (from Result 3.4) will be achieved for both positive and negative feedback, when an appropriate time delay is chosen. A time delayT∗ at which the minimum maximum gain for stability is achieved can be obtained using Result 3.6.
In the next example, the bound on the maximum gain for stability that is inde- pendent of time delay (given in Results 3.4) coincides with the maximum gain for stability of the system with zero time delay.
Example 1: As a simple illustration of Results 3.4, 3.5 and 3.6, consider a single degree of freedom oscillator with responsex(t), and with massm, damping c >0 and stiffness k >0. The oscillator is controlled by using a time-delayed negative velocity feedback −àx0(t−T), with time delayT and using a control gain à. The motion of the oscillator is described by the scalar differential equation
mx00(t) +cx0(t) +kx(t) =−àx0(t−T), x(0) = 0, x0(0) = 0. (3.12) The Laplace transform of (3.12) is
(ms2+cs+k+àsexp[−sT]) ˜x(s) = 0. (3.13) The closed loop poles of the system described by (3.12) are the zeros of the equation ms2+cs+k+àsexp[−sT] = 0. (3.14) From Result 3.4, we have that the maximum gain for stability which is inde- pendent of time delay occurs on the imaginary axis at somes=iw. Evaluation of (3.14) ats=iw gives
−mw2+icw+k+iàwĂ
cos(wT) +isin(wT)¢
= 0. (3.15)
The imaginary part of (3.15) is
cw+àwcos(wT) = 0 or cos(wT) = −c
à . (3.16)
Note thatw= 0 is not a solution, since it would violate (3.15). Equation (3.16) has a solution only if
¯¯
¯¯c à
¯¯
¯¯≤1, that is|à| ≥c. Thus for |à|< c, equation (3.15) has all poles in the left-half complex plane (i.e., all solutions have negative real parts).
For zero time delay (T = 0) the closed loop poles are given by the zeros of the equationms2+ (c+à)s+k= 0. From the Routh stability criterion, for negative feedback (à > 0) the system will be stable for all gains. On the other hand, for positive feedback (à <0), the system has a cross-over pole atà=−candw=
rk
m. Using Result 3.5 one can confirm this observation. From Result 3.5 we have
à∗∗ = min
w∈IR
¯¯
¯¯p(iw) g(iw)
¯¯
¯¯= min
w∈IR
¯¯
¯¯−mw2+k+icw iw
¯¯
¯¯=c ,
and the minimum occurs atw∗∗= rk
m.
Additionally from Result 3.5, for negative feedback we have that x = cos
Ãrk mT
!
= −1 and y = sin Ãrk
mT
!
= 0. The smallest time delay for