Optimal Design of PID Controllers with Filter on DSatisfying Gain Margin and Sensitivity Constraints

Simultaneously on a Set of Plants

Yaniv O. ∗

August 25, 2003

Abstract

This paper introduces a design methodology for PID-type controllers, includingthose augmented by a filter on the D element, satisfying required gain margin and anupper bound on the (complementary) sensitivity for a finite set of plants. Importantproperties of the methodology are: (i) the optimal controller for a practical definitionof optimality can easily be identified, (ii) the algorithm can be applied to a plant ofany order including non-minimum phase plants, plants with delay, plants characterizedby quasi-polynomials, unstable plants and plants described by measured data, (iii) thesensor used for the PI terms can be different than the one used with the D term (i.e.,they can have different transfer functions), (iv) the algorithm relies on explicit equa-tions, and thus is efficient and fast and can be used in near real-time to determine acontroller for on-line modification of the plant’s model including its uncertainty and/ormodification of the specifications, and (v) a single plot graphically highlights tradeoffsamong the gain margin, (complementary) sensitivity bound, low frequency sensitivityand high frequency sensor noise amplification.

Keywords: PID control, Dual loop control, Cascaded control, robustness, gain andphase margin, sensitivity.

1 Introduction

Methods for tuning PI and PID controllers have been widely reported and active research

continues due to the extensive use of such controllers in industry. The tuning methods can∗Faculty of Engineering, Department of Electrical Engineering Systems, Tel Aviv University, Tel Aviv

69978, Israel. E-mail: yaniv@eng.tau.ac.il

1

be divided into two main categories, those emphasizing gain and phase margin specifications

and those emphasizing sensitivity specifications. Design techniques based on gain and phase

margin specifications include those by Ho et al [1] and Ho et al [2]. They developed simple

analytical formulae to tune PI and PID controllers for commonly used first-order and second-

order plus dead time plant models to meet gain and phase margin specifications. Ho et al [3]

reported a method for designing PI controllers that meet gain and phase margin constraints

and optimized for setpoint response. Ho et al [4] presented tuning formulae for the design

of PID controllers that satisfy both robustness and performance requirements. Crowe and

Johnson [5] presented an automatic PI control design algorithm to satisfy gain and phase

margins based on a converging algorithm. Suchomoski [6] developed a tuning method for PI

and PID controllers that can shape the nominal stability, dynamic properties, and gain and

phase margins.

Gain and phase margin specifications may fail to guarantee a reasonable bound on the

sensitivity, which is a very important control design property. This point was considered by

several researchers. Ogawa [7] used the QFT-framework to propose a PI design technique

that satisfies a bound on the sensitivity for an uncertain plant. Poulin and Pomerleau [8]

developed a PI design methodology for integrating processes that bounds the maximum peak

resonance of the closed-loop transfer-function. The peak resonance constraint is equivalent to

bounding the complementary sensitivity, which can be converted to bounding the sensitivity.

In [9] a design method for bounding the sensitivity while achieving desired steady state

performance is given, the method can also be applied on measured data. However, plant

uncertainty is not considered, and the procedure fits a simple compensation structures.

Crowe and Johnson [10] reported a design approach to find a PI/PID controller that bounds

the sensitivity while satisfying a phase margin condition. Kristiansson and Lennartson [11]

emphasized the need to bound the sensitivity and complementary sensitivity. They suggested

the use of an optimization routine to design PI/PID controllers with low-pass filters on

the derivative gain to optimize for control efforts, disturbance rejection and bound on the

sensitivity. They also gave tuning rules for non-oscillatory stable plants or plants with a single

2

integrator. Astrom et al. [12] described a numerical method for designing PI controllers based

on optimization of load disturbance rejection with constraints on sensitivity and weighting

of set point response.

Other tuning methods have been proposed. Yeung et al [14] presented a non-trial and

error graphical design technique for controller design of the lead-lag structure that enables

simultaneous fulfillment of gain margin, phase margin and crossover frequency. Guillermo et

al [15] developed a theorem to calculate all stabilizing PID controllers for first-order delayed

plants. However, uncertainty, sensitivity and margins were not discussed.

These papers and others apply gain and phase margin constraints in finding PI and PID

controller designs. Some add limitations on the (complementary) sensitivity. However, there

are several differences between approaches reported in the literature and the idea proposed

here. First, the approach presented here bounds the sensitivity of the closed-loop transfer

function for all frequencies, not just at the crossover frequencies where the gain and phase

margins are satisfied. (It is possible that the gain and phase margin conditions are met with

a given PI design, but the sensitivity can be very high.) Second, the approach developed here

accounts for plant uncertainty, in that the controller design satisfies the specifications for a

set of plants. Third, the approach here provides explicit equations to determine the set of all

possible controllers. Fourth, with this method it is possible to extract the optimal solution

for many practical optimization criteria. Fifth, the algorithm presented can be applied to

any plant order including pure delay, unstable plants, and on plants given by measured data.

Sixth, it allows the PI terms (integration and proportional) to use different sensors then the

derivative term, D. Seventh, it is possible to extend the method to design dual loops and

other control structure and to sensitivity specifications depending on the frequency. Eighth,

the algorithm uses explicit equations, and, as such, it does not rely on optimization routines

and is very fast.

3

2 Problem Statement and Motivation

Consider the open-loop transfer function, L(s),

L(s) = a [P1(s) + bP2(s)] . (1)

If P1(s) = (1 + ki/s)P (s) and P2(s) = sP (s), (1) corresponds to the open-loop transfer

function of plant, P (s), with a PID controller

C(s) =aki

s+ a + abs.

It is possible to use a different sensor for the D term than for the PI terms. In this case, if

the transfer function of the sensor used by the D term is taken as H(s) (with H(s) being a

delay and/or a low-pass filter to decrease noise) and the transfer function of the sensor used

by the PI terms is 1(s), then P1(s) = (1 + ki/s)P (s), P2(s) = sH(s)P (s), and the controller

can be written as

C(s) =aki

s+ a + absH(s).

The gain and phase margin conditions, the typical measures of robustness, are replaced

by a condition on the closed-loop sensitivity inequality,∣

∣

∣

∣

∣

11 + kL(s)

∣

∣

∣

∣

∣

≤ M for s = jω, ∀ω ≥ 0, k ∈ [1, K], (2)

where the sensitivity bound M > 1 and the gain uncertainty of the plant, k, is in the

interval [1, K]. It can be shown [16] that when arg L(jω) = −π rad, then (2) requires

|L(jω)| ≤ (M − 1)/M for K = 1, and thus the gain margin for a given K is at least

GM = 20 log10 (K) + 20 log10

( MM − 1

)

. (3)

Similarly, when |L(jω)| = 1, (2) requires arg L(jω) > −π + 2 arcsin [(2M)−1], and thus the

phase margin is at least

PM = 2 arcsin( 1

2M

)

. (4)

4

Inequality (2) is a more encompassing measure of robustness than gain and phase margin. It

places a bound on the sensitivity at all frequencies, not just at the two frequencies associated

with the gain and phase margins.

Two design problems are considered here:

1. Determine all (a, b) pairs that satisfy (2) where the pair (P1, P2) is uncertain in the

sense that they belong to a finite set of pairs (P1m, P2m), m = 1, ..., n, and extract an

optimal pair (ao, bo) for a given optimality criterion.

2. Replacing L(s) in (2) by

L(s) = a[(

1 +ki

s

)

P1(s) + bP2(s)H(s)]

, (5)

find all a, b, ki and filters H(s) of a given structure that satisfy (2), and, in particular,

extract an optimal solution ao, bo, kio and Ho(s) for a given optimality criterion.

These two problems for the special case of P2 = P1s, without considering plant and gain

uncertainty, were solved in [12] and in [13] to determine a single controller for the case of

maximum kia or a, where their proposed solutions were not solved explicitly, but with the

Matlab 5 Optimization Toolbox.

3 Development of Methodology

In order to determine the (a, b) values for which the closed-loop system is stable and (2) is

satisfied, consider first the special case of no gain uncertainty, i.e., K = 1, and a single plant

pair (P1(s), P2(s)). Substituting (1) into (2) gives,

U + a(U1 + bU2) + a2(V1 + bV2 + b2V3) ≥ 0, ∀ω ≥ 0 . (6)

where

U = 1− 1/M2, U1 = 2 · Real(P1), U2 = 2 · Real(P2),

V1 = |P1|2, V2 = 2 · Real(P1P ∗2 ), V3 = |P2|2.

5

For an (a, b) pair which is on the boundary region of the allowed (a, b) values, there exists

ω such that (6) is an equality. Moreover, since at that particular ω, (6) is minimum, its

derivative (with respect to ω) at the same ω is zero (this observation was also used by [12]

to design a controller for maximum a). Thus,

U̇1 + bU̇2 + a(V̇1 + bV̇2 + b2V̇3) = 0. (7)

Solving (7) for a gives

a = − U̇1 + bU̇2

V̇1 + bV̇2 + b2V̇3. (8)

Substituting (8) into the equality of (6) gives the following fourth-order equation for b whose

coefficients are functions of ω.

x4b4 + x3b3 + x2b2 + x1b + x0 = 0 (9)

where1

x4 = UV̇ 23 + U̇2

2 V3 − U̇2U2V̇3

x3 = (−U2V̇3 + 2U̇2V3)U̇1 − U̇2U1V̇3 + 2UV̇2V̇3 + U̇22 V2 − U̇2U2V̇2

x2 = U̇21 V3 + (2U̇2V2 − U2V̇2 − U1V̇3)U̇1 + U̇2

2 V1 − U̇2U1V̇2 + UV̇ 22 + 2UV̇1V̇3 − U̇2U2V̇1

x1 = U̇21 V2 + (−U1V̇2 − U2V̇1 + 2U̇2V1)U̇1 + 2UV̇1V̇2 − U̇2U1V̇1

x0 = −U̇1U1V̇1 + UV̇ 21 + U̇2

1 V1.

The allowed (a, b) region for a given M value can be calculated as follows: For a given

ω solve (9) for b. Noting that b has four solutions (for a given ω), pick the positive real

solution(s) and use (8) to find their corresponding a. Select the (a, b) pairs for which the

resulting closed-loop system is stable and (2) is satisfied. Searching over a range of frequen-

cies, ω, gives two vectors of ω, (a(ω), b(ω)) which lie on the boundary of the allowed (a, b)

region. Note that for point (a, b) on the boundary, one of the following conditions can occur:

1The equations were derived using Matlab’s Symbolic Mathematics Toolbox from MathWorks, Inc., Nat-ick, MA.

6

(i) increasing a is inside the region, (ii) decreasing a is inside the region, or (iii) neither

increasing nor decreasing a is inside the region. Thus, for two points, (a1, b) and (a2, b), on

the boundary where a1 < a2, any a ∈ [a1, a2] and b is a pair within the region only if there

exist no (a, b) points on the boundary for any a ∈ (a1, a2) and the following is true

1. increasing a1 is within the region, and

2. decreasing a2 is within the region.

However it is enough to validate 1 or 2 because 1 implies 2 and vise versa. This validation can

be done easily according to the following algorithm. Let (a, b) be a boundary pair calculated

at frequency ω then (i) if |L(jω)| ≥√

1− 1/M2, only increasing a is within the boundary;

(ii) otherwise only decreasing a is within the boundary.

Since, as will be shown later, the optimal pair lies on the boundary of the (a, b) region,

internal points are not of interest.

Remark 3.1 The question arises as to how many frequencies should be chosen for calculat-

ing (a, b) pairs? If the frequencies are not dense enough the outcome can be that the (a, b)

boundary is not dense enough, and the optimal solution calculated from the given (a, b) list

is too far from the true optimum. The recommended density of the frequencies depends on

two factors: (i) how close a solution to the global optimum can be considered practically

optimal, and (ii) the particular plant, with more frequencies selected around under-damped

poles or zeros and less elsewhere.

Remark 3.2 If L(s) of (1) is replaced by the more general expression of (a, b)

L(s) = a (P1(s)f(b, s) + P2(s)g(b, s)) = aL0,

then the derivative condition which gave a of (8) will be replaced by (∗ denotes complex

conjugate)

a = − L̇0 + L̇∗0L̇0L∗0 + L̇∗0L0

. (10)

7

Therefore, if f and g are rational polynomial in b, then from (10) a is also a rational poly-

nomial in b. Substituting (10) in (7) gives a polynomial equation in b (replacing (9)), whose

coefficients are function of ω. The proposed design process will remain the same except that

the polynomial equation (9) for b will be replaced by a higher order equation. Examples:

Designing a lead element or two lead elements,

L(s) = aP (s)1 + bs1 + αbs

, or L(s) = aP (s)(

1 + bs1 + αbs

)2

where α is a chosen parameter (the chosen lead element depends on α, a parameter that will

be searched over as will be explained latter).

Remark 3.3 A PI controller exists if and only if there exists a frequency for which an (a, b)

pair solving (9) makes the resulting closed-loop system stable and (2) for K = 1 is satisfied.

3.1 Example 1

Consider an armature-controlled DC motor with the input being motor current and the

output being position. The motor transfer function is

P (s) =1s2 e−sT , T = 0.001 sec. (11)

It is required to find the region of the (a, b) pairs such that the complementary sensitivity

M ≤ 1.46, which allows for gain uncertainty k ∈ [1, K] and the pair for which a is maximum.

This is equivalent to a 40 deg phase margin or greater and a [10+20 log(K)] dB gain margin

or greater. Define

P1(s) =(

1 +ki

s

)

P (s), P2(s) = sP (s) (12)

Fig. 1 depicts the boundary of the allowed (a, b) pairs for ki = 80 and K = 1 (the (a, b)

values fall in both shaded regions). Fig. 1 can also be used to find the (a, b) values which

satisfy any gain uncertainty constraint k ∈ [1, K]. For example, if 6 dB gain uncertainty is

desired (that is, K = 2), then for any b, the allowed a values should be 6 dB less in order

to cope with the increase of a between 0 and 6 dB. The (a, b) region will therefore be the

8

lower shaded region depicted in Fig. 1 where the upper curve is shifted down by 6 dB. The

maximum a for K = 1 occurs at (a, b) = (94.2 dB, 0.063) and maximum a for K = 2 occurs

at (a, b) = (84.9 dB, 0.011), giving the controller designs corresponding to lowest sensitivity

at low frequencies. Qualitatively, this means that the price of protecting the system from a

possible gain uncertainty of 6 dB is increasing the low frequency sensitivity by 9.5 dB while

the high frequency noise decreases by 48 percent. The open-loop Nichols plot for maximum

a and K = 2 is shown in Fig. 2 for verification. It does not enter the ‘forbidden’ region and

confirms system stability.

0.004 0.008 0.012 0.016 0.0278

81

84

87

90

93

a (d

B)

b

0

6

K (dB)

Figure 1: Region of (a, b) values for M = 1.46, equivalent to 40 deg phase margin (PM)or greater and 10 dB gain margin or greater (GM for K = 1). Lower shaded region is forM = 1.46 with additional 6 dB plant gain uncertainty (K = 2), for a total of 16 dB orgreater.

9

−270 −240 −210 −180 −150 −120 −90−30

−20

−10

0

10

20

30

Am

plitu

de (d

B)

Phase (deg)

4040

5050

6060

70

808090

1001101201301401501701802002202402602803103403704004404805205706206807408108809601100110013001400150016001800190021002300250028003000

Figure 2: Nichols plot for M = 1.46 and K = 2, corresponding to 40 deg phase margin orgreater and 16 dB gain margin or greater. Frequencies are marked in rad/sec. The open-loop transfer function must not enter the shaded region in order to satisfy the M and Kconstraint.

3.2 Example 2

Let the plant be a loaded DC motor whose transfer function from current to position includes

two integrators, a real pole and a complex zero and pole

P (s) =1

s2(1 + s/p)1 + 2d1ω1/s + w2

1/s2

1 + 2d2ω2/s + w22/s2 .

It is assumed that a speed sensor measures the speed with a pure delay of T . Thus,

P1(s) = (1 + ki/s) P (s), P2(s) = sP (s)e−sT . (13)

The parameters for this example are ki = 10, p = 200, d1 = 0.03, ω1 = 120 rad/sec,

d2 = 0.05, ω2 = 150 rad/sec and T = 0.001 sec.

It is required to find the region of the (a, b) pairs such that the complementary sensitivity

M ≤ 1.46, which allows for gain uncertainty k ∈ [1, K] and the (a, b) pair for which a is

10

maximum. This is equivalent to a 40 deg phase margin or greater and a [10 + 20 log(K)] dB

gain margin or greater. Fig. 3 depicts the boundary of the allowed (a, b) pairs for K = 1.

(The (a, b) values fall in both shaded regions.) Fig. 3 can also be used to find the (a, b)

values which satisfy any gain uncertainty constraint k ∈ [1, K]. For example, if 6 dB gain

uncertainty is desired (that is K = 2), then for any b, the allowed a values should be 6 dB less

in order to cope with the increase of a between 0 and 6. The (a, b) region will therefore be the

lower shaded region depicted in Fig. 3 where the upper curve is shifted down by 6 dB. The

maximum a for K = 1 occurs at (a, b) = (62.1 dB, 0.037) and maximum a for K = 2 occurs

at (a, b) = (53.0 dB, 0.060), giving the controller designs corresponding to lowest sensitivity

at low frequencies. Qualitatively this means that the price of protecting the system from a

possible gain uncertainty of 6 dB is increasing the low frequency sensitivity by 9.1 dB while

the high frequency noise decreases by 43%. For validation, the open-loop Nichols plot for

maximum a and K = 2 is shown in Fig. 4

3.3 Extension to Complementary Sensitivity Requirements

It is possible to replace the sensitivity margin constraint (2) by the complementary sensitivity,∣

∣

∣

∣

∣

kL(jω)1 + kL(jω)

∣

∣

∣

∣

∣

≤ M, ∀ω ≥ 0, k ∈ [1, K]. (14)

It can be shown that L = L0 satisfies (2) if and only if L = M2

M2−1L0 satisfies (14). This leads

to the following lemma:

Lemma 3.1 The pair (a, b) solves problem 1 (problem 2) stated in Section 2 if and only if

the pair(

M2 − 1M2 a, b

)

solves problem 1 (problem 2) where (2) is replaced by (14).

3.4 Extension to Lower and Upper Gain Margin

The sensitivity margin constraint (2) has been written with k ∈ [1, K] under the assumption

of K ≥ 1, i.e., the system is protected against a plant gain increase. However, conditionally

11

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.235

40

45

50

55

60

65

a (d

B)

b

0

6

K (dB)

Figure 3: Region of (a, b) values for M = 1.46, equivalent to 40 deg phase margin (PM)or greater and 10 dB gain margin or greater (GM for K = 1). Lower shaded region is forM = 1.46 with additional 6 dB plant gain uncertainty (K = 2), for a total of 16 dB orgreater.

stable plants must be protected against a plant gain decrease, that is, the closed-loop transfer

function should satisfy the M constraints where k ∈ [1, K] and K < 1. It is possible to modify

(2) to satisfy∣

∣

∣

∣

∣

11 + kL(jω)

∣

∣

∣

∣

∣

≤ M, ∀ω ≥ 0, k ∈ [K1, K2] (15)

which is equivalent to∣

∣

∣

∣

∣

11 + kK1L(jω)

∣

∣

∣

∣

∣

≤ M, ∀ω ≥ 0, k ∈ [1, K2/K1]. (16)

The design process will then be: Replace the pair (P1, P2) by the pair (K1P1, K1P2) and the

gain uncertainty by k ∈ [1, K2/K1], and then calculate the (a, b) boundary. The boundary

(a/K1, b) will be the boundary for (15). Example: Fig. 5 is the (a, b) region for the plant of

Example 1 where the gain uncertainty is k ∈ [0.5, 2] and M = 1.46. Fig. 6 is the open-loop

Nichols plot for the (a, b) pair for which a is maximum.

12

−270 −240 −210 −180 −150 −120 −90 −60 −30 0−40

−30

−20

−10

0

10

20

30

Am

plitu

de (d

B)

Phase (deg)

6

8

10

12

15

2025

3040

60

80

100

110

115

120

125

130

135

140

145150155

160

200

300

400

600

800

Figure 4: Nichols plot for M = 1.46 and K = 2, corresponding to 40 deg phase margin orgreater and 16 dB gain margin or greater. Frequencies are marked in rad/sec. The open-loop transfer function must not enter the shaded region in order to satisfy the M and Kconstraint.

4 Optimization

The answer to the question “Which is the best (a, b) pair?” of course depends on the opti-

mization criterion. Seron and Goodwin [18] note that “In general, the process noise spectrum

is typically concentrated at low frequencies, while the measurement noise spectrum is typ-

ically more significant at high frequencies.” The conclusion is that an optimal controller is

a result of weighting the performance at low frequencies and of noise at high frequencies.

Since the high frequency noise is proportional to ab and low frequency performance to 1/a,

a practical optimal criterion can be

J = α · (1/a) + β · (ab)

whose solution must lie on the boundary of the (a, b) curve. When β is small enough or

zero (meaning the sensor noise is neglected), the optimal solution is the maximum possible

13

0.015 0.02 0.025 0.03 0.035 0.04 0.04570

72

74

76

78

80

82

a (d

B)

b

Figure 5: Region of (a, b) values for M = 1.46 and k ∈ [0.5, 2], equivalent to 40 deg phasemargin (PM) or greater, 16 dB lower gain margin or greater and 10.6 dB upper gain marginor greater. The chosen ki = 40

a. This is the criterion corresponding to the Nichols plots of examples 1 and 2.

Another practical optimal criterion is based on the observation that the noise should

be below a certain level. The optimal criterion will then be the (a, b) pair for which a is

maximum and ab is below a given level. This controller lies on the boundary of the (a, b)

region. Another required controller can be the one whose high frequency noise, ab, is the

lowest while the low frequency sensitivity, a, is better than a given figure. This controller

also lies on the boundary.

Any optimal criterion which is the absolute value of an analytic function of z = 1/a + jb

gets its maximum or minimum on the boundary of the (a, b) region. This is also true for

other definitions of z, for example, z = α/a + jβab, or z = f(a) + jg(ab) where f(a) and

g(ab) are monotonic, etc.

In many applications the criterion is to find a controller whose crossover frequency is

14

−270 −240 −210 −180 −150 −120 −90−30

−20

−10

0

10

20

30

Am

plitu

de (d

B)

Phase (deg)

30

4040

50

60

8090

110140

170200250300

360440

530650

780950

12001400

17002000

25003000

Figure 6: Nichols plot for M = 1.46 and k ∈ [0.5, 2]. Frequencies are marked in rad/sec. Theopen-loop transfer function must not enter the shaded region in order to satisfy the M andK constraint. Note that the minimum lower gain margin (dictated by the shaded region) is16 dB and the minimum upper gain margin is 10.6 dB

given, where the optimality criterion is a controller whose low frequency sensitivity, 1/a, is

minimum and/or high frequency noise is minimum. Under reasonable condition, given in

the following lemma, this optimal solution lies also on the boundary of the (a, b) region.

Lemma 4.1 Let (a, b) denote the set of all controllers solving the problem stated herein and

(aω, bω) denote the subset of (a, b) whose crossover frequency is ω, that is all (aω, bω) pairs

in (a, b) satisfying

|aω(P1(jω) + bωP2(jω))| = 1. (17)

Assuming all a’s and b’s in (a, b) are positive, all a’s are uniformly bounded from above and

a solution with cross-over frequency ω exist, then

1. If |P1(jω)+ bωP2(jω)| is an increasing function of bω, the maximum value of aω belong

to the boundary of (a, b);

15

2. The minimum value of aωbω belong to the boundary of (a, b).

Proof: We start with three observations: The first is that any pair a, b ∈ (a, b) which is

not on its boundary has an Euclidean neighborhood, inside (a, b). This is immediately from

(6) which defines the set (a, b). The second is that aω is a continuous function of bω and of

aωbω. The third is that there exist maximum aω and minimum aωbω, because the a values

are bounded from above and the ab from below and the set of solutions is closed.

Proof of 1: Let us assume that the pair(s) aω, bω ∈ (aω, bω) for which aω is maximum

is internal to (a, b), then there exist a > aω and b < bω in (a, b) satisfying (17), which is a

contradiction to the assumption.

Proof of 2: (17) can be written in the form

aω = −aωbω(real(P2/P1))±√

|1/P1|2 − (aωbω)2(imag(P2/P1))2,

hence if (aω, bω) is a solution of (17), there exists another solution, (a, b), in any small enough

neighborhood of (aω, bω) for which ab < aωbω. This fact will contradict the assumption that

minimum of aωbω satisfying (17) is internal to the (a, b) region. 2.

Example 1: C(s) = a(1 + bs). Condition 1 of lemma (4.1) is satisfied because

|P1(jω) + bωP2(jω)| =√

(1 + b2ω2)|P (jω)|

is an increasing function of b.

Example 2: C(s) = a(1 + bH(s)). Condition 1 of lemma (4.1) is satisfied if |1 +

bH(jω)||P (jω)| is an increasing function of b. This will be true if the phase of H(jω) is

in the interval [−π/2, π/2]. This condition can be relaxed to b|H(jω)| > −real(H(jω)).

Important H(s) filters with phase in the interval [−π/2, π/2] are: H = s/(1 + s/p), that is,

a low pass filter of order 1 on the derivative term, and (ii) H = s/(1 + 2ξs/ω + s2/ω2), that

is, a low pass filter of order 2 on the derivative.

4.1 Constraint Optimization

Other practical optimal criteria would include any optimal criterion subject to hard limita-

tion constraints on the control efforts. Two options are addressed here, one in which there

16

exist two independent actuators, which are the inputs to P1 and P2, and the second with a

single actuator.

The rest of this section is devoted to finding the boundary of the (a, b) region subjected

to the hard limitation constraint. The properties of the optimal criteria described above are

valid on these regions.

4.1.1 Two Actuator Case

It is required to find a controller whose input to P1, that is, ae(k) where e(k) is the input to

the controller at time k, is bounded, and the input to P2, that is, abe(k), is also bounded.

Given the constants umin1, umax1, umin2 and umax2, the controller should satisfy

umin1 ≤ ae(k) ≤ umax1

umin2 ≤ abe(k) ≤ umax2, (18)

which, geometrically, is all (a, ab) pairs in and on a rectangle in the a − ab plane. Assume

that the chosen (a, ab) pair is updated at any sampling time. Then the boundary of all

allowed (a, ab) pairs will be the boundary of the intersection of the original (a, ab) region

and the rectangular region dictated by (18).

4.1.2 Single Actuator Case

Consider the discrete version of the controller, that is, C(z) = a+ ab(1− z−1). It is required

to find a controller whose output is ae(k) + ab(e(k) − e(k − 1)) and is bounded where e(k)

is the input to the controller at time k. Given the constants umin and umax, the controller

should satisfy

umin ≤ ae(k) + ab(e(k)− e(k − 1)) ≤ umax, (19)

which, geometrically, is all a, ab pairs between and on two lines in the a − ab plane. Let

us now assume that the a, ab pair is updated at each sampling time, then the boundary of

the allowed (a, ab) pairs will be the intersection of the original (a, ab) region and the region

dictated by (19).

17

4.1.3 Extension to PID and PID with Filtered D

The discussion above applies to PD controllers. However, the extension to PID controllers

and PID controllers with a low-pass filter on the D element is trivial. For example, assume

the single-actuator case and a controller of the PID form whose D term is filtered by H(z),

that is

C(z) = a(

1 +kiz

z − 1

)

+ abH(z). (20)

its output for the input sequence e(k) is

u(k) = a(

1 +kiz

z − 1

)

e(k) + abH(z)e(k) = ae1(k) + abe2(k). (21)

where e1(k) is the output of (1 + kiz/(z − 1)) operating on the sequence {e(k)} and e2(k)

is the output of H(z) operating on the sequence {e(k)}. Constraints of the form umin ≤

u(k) ≤ umax are again, geometrically, all a, ab pairs between and on two lines in the a − ab

plane.

5 Extension to Uncertain Plants

Assume that the plant pair, (P1, P2), of the problem 1 stated in Section 2 is known to be

one of a finite set of plant pairs, (P1m, P2m), m = 1, .., n. It is required to find all (a, b) pairs

that solve the problem 1 where the pair (P1, P2) can be any member of the set. This (a, b)

region will be the intersection of all (a, b) regions of the members of the set. Note that if this

intersected region is empty, a PI feedback solution does not exist. As an example, consider

the plant

P1(s) =gain

s(1 + s/pole), for gain = [1, 3] and pole = [10, 12, 14, 16, 18, 20],

P2(s) = P1(s)s. (22)

Fig. 7 shows the intersection as the shaded region for M = 1.46 and K = 1. The (a, b) pair

corresponding to maximum a is (8.6 dB, 0.62).

The extension of problem 2 in Section 2 to the case of an uncertain plant is similar, since

its solution is based on the its reduction to problem 1.

18

0 1 2 3 4 5 6−40

−30

−20

−10

0

10

20

30

40

a (d

B)

b

Figure 7: Boundary curves of (a, b) regions for M = 1.46 and K = 1 for the plant set (22) .The intersection of all regions is the allowed region.

6 Discrete PI Controllers

The problem stated here can be expressed in its discrete form as follows: The plant pair is

(P1(z), P2(z)) and the open loop is

L(z) = a (P1(z) + bP2(z)) . (23)

Substituting z = esT into (23) or using the bilinear transformation translates the discrete

problem into the the original continuous problem.

7 Design Methodology for PID Controllers

In the previous sections, a design methodology of a PID controller whose three parameters

are (a, b, ki) was given, assuming the ki term is known (aki is the I term of the PID, see (5)).

The solution was presented by two vectors, a(ω) and b(ω) for a chosen vector of frequencies

19

ω. The (a(ω), b(ω)) vectors lie on the boundary of the region of the (a, b) pairs which solve

the problem. It was also claimed that the solution for practical optimization criteria lies on

these vectors. Here an extension to design the triple (a, b, ki) is proposed. The idea is to find

the boundary of the (aki, bki) regions for several ki values, and then apply an optimization

criterion on all the vectors (aki, bki) to extract the optimal one.

The answer to the question of how best to choose the ki values for the search is based on

the following observation: The PID controller is (if P2 6= P2 this is an approximation)

ao

(

1 +ki

s+ bs

)

= ao (1 + bs)(

1 +ki/s

1 + bs

)

. (24)

Let (ao, bo) denote the optimal solution for ki = 0 and ωo its lowest crossover frequency. Find

the range of ki values whose phase contribution to (24) at ωo is between −45 deg and about

−1 deg. That is, the ki values for which.

arg(

1 +ki/(jωo)1 + bjωo

)

= −45 deg, −1 deg. (25)

Use these two ki value as the largest and lowest value for the search on ki. Running on ki

values which differ by factor of 1.1, it is expected to have about 25 values.

Remark 7.1 The process of adding more terms, for example, notch filters, low-pass filters,

lead-lag elements, etc., can be extended naturally. The only limitation is the computing

effort.

7.1 Continuation of Example 1

This section evaluates the tradeoff between high frequency noise, that is the multiplication

a · b, and low frequency sensitivity, that is the multiplication a ·ki (this criterion was justified

in Section 4). For ki = 0 the pair (a, b) = (94.2dB, 0.0061) gives the optimal performance for

the maximum a criterion and the open-loop crossover frequency is 350 rad/sec. The phase

of (25) is −45 deg at ki = 620 and −1.2 deg at ki = 40. This suggests searching in the

range ki ∈ [40, 620]. Fig. 8 depicts the boundary of the allowed (aki, ab) pairs for different

ki’s. If it is assumed that the high frequency noise can be neglected, then the controller of

20

best performance has kia = 137.55 dB, ab = 360 and ki = 160. Its open-loop Nichols plot

is shown in Fig. 9. If it is assumed that the high frequency noise must be less than 180

deg, then the controller of best performance has kia = 121.9 dB, ab = 180 and ki = 60. Its

open-loop Nichols plot is presented in Fig. 10, showing that the solution touch the M, K

bound. The price for decreasing the high frequency noise by 50 percent is 15.7 dB increase

of low frequency sensitivity.

100 150 200 250 300 350 400 450110

115

120

125

130

135

140

40

60

80

100

120140160180

ki ⋅

a (d

B)

a ⋅ b

Figure 8: Example 1: Boundary curves of region of (aki, ab) values for M = 1.46 and K = 1,equivalent to 40 deg phase margin or greater and 10 dB gain margin or greater for severalki values marked on each curve (the (a, b) region for a given ki is on and below the curve).

7.2 Continuation of Example 2

This section evaluates the tradeoff between high frequency noise, that is the multiplication

a · b, and low frequency sensitivity, that is the multiplication a · ki. For ki = 0 the pair

(a, b) = (62.56dB, 0.0328) gives the optimal performance for the maximum a criterion and

the open-loop crossover frequency is 50 rad/sec. Since the phase of (25) is −45 deg at ki = 70

21

−270 −240 −210 −180 −150 −120 −90−30

−20

−10

0

10

20

30

Am

plitu

de (d

B)

Phase (deg)

70

80

100110

130

150180

210 240290

340390460

540630

730860

10001200

14001600

19002200

26003000

Figure 9: Example 1: Nichols plot for M = 1.46 and K = 1. Frequencies are marked inrad/sec. The open-loop transfer function must not enter the shaded region in order to satisfythe M and K constraint.

and −1.2 deg at ki = 4, the search is in the range ki ∈ [4, 70]. Fig. 11 depicts the boundary

of the allowed (aki, ab) pairs for different ki’s. If it is assumed that the high frequency noise

can be neglected, then the controller of best performance has kia = 87.36 dB, ab = 52.1

and ki = 24. Its open-loop Nichols plot is shown in Fig. 12. If it is assumed that the

high frequency noise must be less than 20 units, then the controller of best performance has

ki · a = 65.7 dB, ab = 20 and ki = 8, and its open-loop Nichols plot is shown in Fig. 13.

Here, the price for decreasing the high frequency noise from 52.1 to 20 is a 21.7 dB increase

of low frequency sensitivity.

22

−270 −240 −210 −180 −150 −120 −90−30

−20

−10

0

10

20

30

Am

plitu

de (d

B)

Phase (deg)

40

50

60

7080

100120

140170

200240

290350410

500590

710850

10001200

15001800

21002500

3000

Figure 10: Example 1: Nichols plot for M = 1.46 and K = 1. Frequencies are marked inrad/sec. The open-loop transfer function must not enter the shaded region in order to satisfythe M and K constraint and ab ≤ 180.

8 Design Methodology for PID Controllers with Low-Pass on D Term

In section 7 a design methodology of a PID controller (whose three parameters are a, b, ki)

was given. It was based on the design methodology for PD controllers search over ki. The

extension to include a filter, H, on the D parameter of the PID is again by searching over

both the ki and H(s). The idea is to choose the structure of the filter H(s), for example,

H(s) = sH0(s) where H0(s) = ps+p or H0(s) = p2

s2+ps+p2 , and search over the parameter p.

The question then is how best to choose the p values for the search. Since the reason for

introducing the filter H0(s) is to limit the sensor noise amplification of the D term and/or

reduce high-frequency resonances, it is recommended to perform an iterative search on p as

follows: Starting with very large p, measure the noise and if it is too large decrease p by

a factor of 1.5. When reaching a satisfactory noise level a refined search can be conducted

23

10 20 30 40 50 60 7050

55

60

65

70

75

80

85

90

ki ⋅

a (d

B)

a ⋅ b

4040

5050

6060

70

8080

90100

110120

130 140 150170180

200220240260280

310340370400

440480520

570620

680740

810880

9601100

11001300

14001500

16001800

19002100

23002500

28003000

4

6

8

12

1620 24 28

32

Figure 11: Example 2: Region of (aki, ab) values for M = 1.46 and K = 1, equivalent to 40deg phase margin or greater and 10 dB gain margin or greater.

around the satisfactory p.

A reasonable range for this search on p for first or second order filters can be calculated

as follows: If ω0 is the largest frequency where the open loop phase is −180 deg for a given

ki where H0(s) = 1, the search should not exceed about p = 10ω0 because at that value the

low-pass filter effect on frequencies less than ω0 is neglected (less than 5◦).

Remark 8.1 There exist control applications where the plant has high frequency resonance

that must be attenuated because (i) if it is not attenuated the achievable closed-loop perfor-

mance is restricted and (ii) the resonance might generate non-tolerable aliasing phenomena

in discrete systems. This attenuation can be done using a notch filter or a low-pass filter on

the D term or on both the D and P terms. The algorithm proposed here supports this kind

of filter where H(s) can be applied either on D or on the D and P terms.

24

−270 −240 −210 −180 −150 −120 −90 −60 −30 0−40

−30

−20

−10

0

10

20

30

Am

plitu

de (d

B)

Phase (deg)

0

6

K (dB)

10

12

15

20

2530

40

60

80

100

110

115

120

125

130

135

140

145150155

160

200

300

400

600

10

12

15

20

2530

40

60

80

100

110

115

120

125

130

135

140

145150155

160

200

300

400

600

Figure 12: Example 2: Nichols plot for M = 1.46 and K = 1. Frequencies are marked inrad/sec. The open-loop transfer function must not enter the shaded region in order to satisfythe M and K constraint.

8.1 Continuation of Example 1 for Designing H(s)

It is next of interest to evaluate the tradeoff between high frequency noise, that is a quantity

proportional to the multiplication a · b by the sensor’s noise of the D term filtered by H(s).

For simplicity, it is assumed that this noise is a · b · p and consider the tradeoff for ki = 100.

The frequency ω0 is 1500 rad/sec, and thus the search is in the range p ∈ [15000, 1500].

Fig. 14 depicts the boundary of the allowed (aki, abp) pairs for different p’s. Clearly, using

a low-pass filter with p = 5000 instead of p = 15000 decreases the noise by a factor of

three, while decreasing the performance by 2.1 dB. If the optimal criterion is that the high-

frequency noise must be less than 1.34 · 106, then p = 5000 gives the best performance. Its

controller parameters are ki · a = 30.9 dB, a · b · p = 1.34 · 106 and ki = 100. Its open-loop

Nichols plot is shown in Fig. 15.

25

−270 −240 −210 −180 −150 −120 −90 −60 −30 0−40

−30

−20

−10

0

10

20

30

Am

plitu

de (d

B)

Phase (deg)

5

7

1012

1520

253040

60

80

100

110

115

120

125

130

135

140

145150155

160

200

300

400

600

800

Figure 13: Example 2: Nichols plot for M = 1.46 and K = 1. Frequencies are marked inrad/sec. The open-loop transfer function must not enter the shaded region in order to satisfythe M and K constraint and ab ≤ 20.

9 Extension to Plants Given by Measured Data

9.1 First Method

If a plant identified at a list of frequencies is given and it is not possible to find a state-space

model or the accuracy of a chosen model is not sufficiently adequate, it is still possible to

design a controller. One option is to interpolate the plant pair(s) to correspond to a dense

list of frequencies (spline etc.) and replace the derivatives appearing in (7) by a numerical

derivative.

9.2 Second Method

Define the splitting of P1(jω) and P2(jω) at s = jω into real and imaginary parts,

P1(jω) = A1(ω) + jB1(ω)

P2(jω) = A2(ω) + jB2(ω).

26

0 1 2 3 4 5 6x 10

6

124

126

128

130

132

13415000

120009000

7000

5000

3000

2000

ki ⋅

a (d

B)

a ⋅ b ⋅ p

Figure 14: Region of (aki, abp) values for M = 1.46 and K = 1.

Any (a, b) on the boundary region must satisfy |1 + L(jω)| = 1/M , from which we make the

following observations: (i) The values of L(jω) for which |1 + L(jω)| = 1/M are

L(jω) = a [A1(ω) + bA2(ω) + j(B1(ω) + bB2(ω))] = x + jy, (26)

where (x, y) is any point on the circle (x − 1)2 + y2 = M−2 or equivalently any x = −1 +

cos θ/M and y = sin θ/M where θ ∈ [0, 2π], (ii) any such (x, y) satisfy |y/x| ≤√

1/(M2 − 1)

and x < 0 and (iii) solving (26) for b gives

b =B1x− A1yA2y −B2x

=B1 − A1y/xA2y/x−B2

. (27)

Based on the above three observations, the proposed design method is

1. Pick a dense list of y/x in the interval |y/x| ≤√

1/(M2 − 1).

2. Solve for b using (27). The sign of b dictates if it introduces a right or left half plane

zero and the sign of a if it is positive or negative feedback, therefore the sign of a

27

−270 −240 −210 −180 −150 −120 −90−30

−20

−10

0

10

20

30

Am

plitu

de (d

B)

Phase (deg)

50

60

70

80

100110

130160

180220

260300350420

490580

680800

9401100

13001500

18002100

2500

Figure 15: Nichols plot for M = 1.46 and K = 1. Frequencies are marked in rad/sec. Theopen-loop transfer function must not enter the shaded region in order to satisfy the M andK constraint.

and b are a-priori given in order to have a left-half-plane zero and a negative feedback

(assume both are positive). Therefore from all possible b values, pick only the positive

b’s for which the sign of x, that is, of a(A1(ω) + bA2(ω)) is negative.

3. Substitute b in (6) to get the following quadratic equality on a

a2[

(A1 + bA2)2 + (B1 + bB2)2]

+ a [A1 + bA2] + 1− 1/M2 = 0 (28)

and solve for a.

4. All the (a, b) pairs for which the closed-loop system is stable and (6) is satisfied lies on

the boundary of the (a, b) region.

5. Repeat the above steps for all measured data points to get a list of controllers. Apply

the optimal criterion on this list for finding the optimal solution.

28

The drawback of this technique compared to the one using a model is that the computation

time is expected to be much longer.

9.3 Example

Let the original plant pair be

P1(s) =1s2 e−sT , T = 0.001 sec.

P2(s) = sP1(s)

measured at 70 points equally spaced logarithmically in the range [3, 2500] rad/sec. It is

assumed that the amplitude and phase of each plant are corrupted by uniformly distributed

measurement noise of ±1 dB and ±10 deg. For this example 400 values are chosen for x+ jy

on the mentioned circle whose phases are equally spaced linearly. The five step algorithm

described above can now be executed. Fig. 16 depicts points on the boundary of the allowed

(a, b) pairs for K = 1. The maximum a occurs at (a, b) = (89.3723 dB, 0.0084), giving

the controller designs corresponding to lowest sensitivity at low frequencies. Its open-loop

Nichols plot is shown in Fig. 17.

10 Extension to Dual Loop Design

The transfer function of the dual loop system of Fig. 18 is

T (s) =C1P1C2P2

1 + (1 + C1P1)C2P2. (29)

Its stability and margin properties depend on the open-loop transfer function

L(s) = (1 + C1P1) C2P2, (30)

which can be written in the form

L(s) = [1 + kiP1(s)] a [P2(s) + bP2(s)H(s)] (31)

where the outer loop controller C1 = ki and the inner loop controller C2 = a(1 + bH(s)). A

design methodology for this structure is now straightforward.

29

0.006 0.009 0.012 0.015 0.018 0.021 0.02472

74

76

78

80

82

84

86

88

90

92

a (d

B)

b

Figure 16: Points on the boundary of the (a, b) allowed regions.

1. Design (a, b) assuming ki = 0 and some H(s) = H0(s) (if inner loop is PI, H0(s) = s).

2. Decide on the interval of ki to search over and redesign the triple (a, b, ki). That is,

for a given ki design (a, b) for (31) where P2 is replaced by P3 = P2 (1 + kiP1(s)). The

open-loop transfer function for designing (a, b) will then be

a (P3(s) + bP3(s)H0(s))

3. Decide on the filter structure to be applied on H0(s) (for example a low-pass, that is,

H(s) = H0(s) ps+p) and the interval of its parameter (p in the example) to search over

for a particular ki, then design a, b, ki, H(s). That is, for a given ki and H(s) design

(a, b) for (31) where P2 is replaced by P3 = P2 (1 + kiP1(s)). The open-loop transfer

function for designing (a, b) will then be

a (P3(s) + bP3(s)H(s))

30

−270 −240 −210 −180 −150 −120 −90−30

−20

−10

0

10

20

30

Am

plitu

de (d

B)

Phase (deg)

3842

46

51

56 626875

8291 100

110122134148

163180198219

241266 293

323356392433477526579639

704776856943 10001100

1300140015001700

19002100

23002500

Figure 17: Nichols plot for M = 1.46 and K = 1. Frequencies are marked in rad/sec. Theopen-loop transfer function must not enter the shaded region in order to satisfy the M andK constraint.

11 Use for Smith Predictor

A Smith predictor can be converted to a conventional feedback scheme whose loop transmis-

sion is

L(s) = C(s)(

P (s)e−sT + Pe(s)(1− e−sT ))

where the plant is P (s)e−sT and the estimated non-delayed plant is Pe(s). It is required to

find all controllers, C(s), satisfying specifications of the M, K form where P (s) is uncertain.

This is a special case of the technique described herein.

31

2 2P P1yr

- -1 CC

Figure 18: Schematic plot of a dual loop feedback structure

12 Two More Application

12.1 IMC Control Structure

Assume that it is required to design a controller C(s) for the open-loop transfer function

L(s) =C(s)P (s)

1 + P0(s)C(s)

where the plant P (s) can be a member of a finite set of plants and P0(s) is known. The

problem is to design C(s) such that the closed loop is stable and∣

∣

∣

∣

∣

1 +C(s)P (s)

1 + P0(s)C(s)

∣

∣

∣

∣

∣

−1

≤ M ; for s = jω, ∀ω ≥ 0,

or equivalently

|1 + [P (s) + P0(s)] C(s)| ≥ |1 + P0(s)C(s)| /M ; for s = jω, ∀ω ≥ 0. (32)

Assuming C(s) = a (1 + bH(s)), (32) is

|1 + (P (s) + P0(s))(a + abH(s))| ≥ |1 + P0(s)(a + abH(s))| /M ; for s = jω, ∀ω ≥ 0,

which can be manipulated to the form

|1 + aP1(s) + abP2(s)| ≥ |1 + aP3(s) + abP4(s)| /M for s = jω, ∀ω ≥ 0 (33)

where

P1 = P (s) + P0(s), P2 = P1H(s), P3 = P0(s), P4 = P0H(s).

32

Using the notation (∗ denotes complex conjugate)

Pa = P1 + P ∗1 − (P3 + P ∗

3 )/M2

Pab = P2 + P ∗2 − (P4 + P ∗

4 )/M2

Paa = P1P ∗1 − P3P ∗

3 /M2

Paabb = P2P ∗2 − P4P ∗

4 /M2

(33) is

1− 1/M2 + aPa + abPab + a2bPaab + a2b2Paabb ≥ 0 for s = jω, ∀ω ≥ 0.

The arguments following (6) for designing the pairs (a, b) can be applied on (33). That is, for

an (a, b) pair which is on the boundary region of the allowed (a, b) values, there exists ω such

that (33) is an equality. Moreover, since at that particular ω, (33) is minimum, its derivative

(with respect to ω) at the same ω is zero, which gives a rational polynomial equation on b

for a. Substituting this a into (33) gives a polynomial equation on b of order four, for each

ω.

12.2 Third Plant and Frequency Dependent Sensitivity Specifica-tions

It is possible to replace (1) by adding P3(s) to (1)

L(s) = a (P1(s) + bP2(s)) + P3(s). (34)

Equation (2) for k = 1 will then be∣

∣

∣

∣

∣

11 + a (P1 + bP2) + P3

∣

∣

∣

∣

∣

≤ M for s = jω, ∀ω ≥ 0,

which can be written in the form

U + a(U1 + bU2) + a2(V1 + bV2 + b2V3) = 0 for s = jω, ∀ω. (35)

33

where

U = |1 + P3|2 − 1/M2,

U1 = (1 + P3)P ∗1 + (1 + P ∗

3 )P1, U2 = (1 + P3)P ∗2 + (1 + P ∗

3 )P2

V1 = |P1|2, V2 = P1P ∗2 + P2P ∗

1 , V3 = |P2|2.

Taking the derivative of (35) and equating to zero gives

U̇ + a(U̇1 + bU̇2) + a2(V̇1 + bV̇2 + b2V̇3) = 0 for s = jω, ∀ω ≥ 0. (36)

Solving equations (35) and (36) for a gives

a =−U̇ U1 + UU̇1 +

(

−U̇ U2 + UU̇2

)

b

U̇ V1 − UV̇1 +(

U̇ V2 − UV̇2

)

b +(

U̇ V3 − UV̇3

)

b2. (37)

Substituting (37) into (35) gives a fourth order polynomial equation for b whose coefficients

are functions of ω. The design process to find all (a, b) that solve problem 1 (problem 2) for

the open loop (34) is similar as for the open loop (1).

Remark 12.1 The development herein allows the specification parameter M , that is, sen-

sitivity and disturbance rejection specs, being a function of ω.

Remark 12.2 Let us assume that the loop transmission is of the simplified load sharing

structure

L(s) = a (P1(s) + bP2(s)) + cP3(s),

where (a, b, c) are constants to be designed to satisfy given M, K constraints. By fixing c it is

possible to use the above algorithm to find all (a, b) pairs. Running over a dense enough c’s

will give the relevant (a, b, c) from which an optimal solution can be extracted. Clearly this

algorithm can be extended to more than three plant and/or filters on the sensors. However,

will be limited by computational efforts.

34

Remark 12.3 If L(s) is replaced by the more general expression of (a, b)

L(s) = a (P1(s)f(b, s) + P2(s)g(b, s)) + P3(s),

and if f and g are rational polynomial in b then it can be shown that a can be written

explicitly as a rational polynomial in b. The proposed design process will remain the same

except that the polynomial equation (9) for b will be replaced by a higher order polynomial

equation.

13 Extension to QFT

The Quantitative Feedback Theory (QFT) [16] is a feedback design tool based on sequential

loop shaping. In each step a one or two degree-of-freedom (DOF) single-input single-output

(SISO) feedback problem is solved, i.e., a feedback controller is designed around an uncertain

plant to satisfy frequency-dependent sensitivity specifications. The latter feedback problem

is solved here to find a PID with filtered D (or a more complicated controller structure).

Therefore, the design approach presented here can be extended for automated QFT design,

and could include SISO, cascaded SISO, multi-input multi-output (MIMO), cascaded MIMO

and many other control structures.

14 Conclusions

In this paper explicit equations are provided for determining controllers of the classical form,

i.e., PI, PID, and PID with D filtered, that simultaneously stabilize a given set of plants

and satisfy gain margin constraints and a bound on the (complementary) sensitivity. The

algorithm fits any plant dimension including pure delay, unstable plants, continuous plants,

discrete plants, and plants given by measured data.

The outcome of applying the algorithm is a list of controllers from which an optimal

controller can be extracted for many practical optimization criteria. Moreover, tradeoffs

among high frequency sensor noise, low frequency sensitivity, the parameters of the PID and

the filter on D are directly presented by design graphs (a single plot suffices for a single

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filter). Since the algorithm uses explicit equations, it can be executed very fast, and as such

the controller design can be updated in near real-time to reflect changes in plant uncertainty

and/or closed-loop specifications.

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