controller synthesis free of analytical model: fixed-order controllers

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Controller synthesis free of analytical model: fixed-order controllersHossein Parastvand a & Mohammad–Javad Khosrowjerdi aa Department of Electrical Engineering , Sahand University of Technology , Sahand , Tabriz ,IranPublished online: 09 Jul 2013.

To cite this article: Hossein Parastvand & Mohammad–Javad Khosrowjerdi , International Journal of Systems Science(2013): Controller synthesis free of analytical model: fixed-order controllers, International Journal of Systems Science, DOI:10.1080/00207721.2013.815823

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International Journal of Systems Science, 2013http://dx.doi.org/10.1080/00207721.2013.815823

Controller synthesis free of analytical model: fixed-order controllers

Hossein Parastvand and Mohammad-Javad Khosrowjerdi∗

Department of Electrical Engineering, Sahand University of Technology, Sahand, Tabriz, Iran

(Received 24 February 2012; final version received 25 May 2013)

This paper extends the existing results on model-free approach for three-term controllers to fixed-order controllers. It isshown that knowing the frequency response of a plant is sufficient to calculate a subset of stabilising fixed-order controllersusing a set of linear inequalities. The main feature of the proposed approach is that the possible lowest order stabilisingcontrollers corresponding to any stable, unstable, minimum and non-minimum phase plants can be obtained. Also, it is shownthat the performance criterion can be transformed to simultaneously stabilising problem for a family of real and complexplants derived from the nominal plant. The usefulness of the proposed approach is illustrated by some examples.

Keywords: fixed-order controller; frequency response; model-free controller design; linear matrix inequality (LMI)

1. Introduction

There has been an increasing and considerable interest indesigning of low and fixed-order controllers, for exam-ple, see CSS-IEEE (2006) and references therein. This ismainly due to the fact that high-order controllers are rarelyimplemented in practical applications. However, there arefundamental difficulties inherent in the fixed-order con-troller design. Many researchers have attempted to tacklethese issues for three decades. For example, in Haddad,Huang, and Bernstein (1993), a solution has been pro-posed for the robust stability and performance based oncoupled algebraic Riccati equations (CAREs) using fixed-order dynamic controllers. The implicit small gain guaran-teed cost bound is used in Haddad, Chellaboina, Corrado,and Bernstein (1997) to address the problem of robust sta-bility andH2 performance via fixed-order controller design.Development of a mixed-norm H2/L1 controller synthe-sis framework via fixed-order dynamic compensation formulti-input/single-output (MISO) systems has been pro-posed in Haddad and Kapila (1999). In Doyle and Glover(1996), the Lagrange multiplier method has been proposedto design fixed-order controllers using CAREs in which theorder of controller should be smaller than or equal to theorder of plant. In Gahinet and Apkarian (1994) and Iwasakiand Skelton (1994), the fixed-order controller design prob-lem has been formulated in a state-space framework as alinear matrix inequality (LMI) minimisation problem sub-ject to an additional non-convex matrix rank constraint.In Henrion, Sebek, and Kucera (2003), an LMI formula-tion has been developed for fixed-order controller designin a polynomial framework based on polynomial positivityconditions. The method can assign the closed-loop poles

∗Corresponding author. Email: [email protected]

in a given region of the complex plane, solving the re-gional pole assignment problem (Chilali and Gahinet 1996).Non-convexity of the fixed-order controller design problemcan be resolved by choosing a particular tuning parame-ter, the so-called central polynomial. In Yang, Gani, andHenrion (2007), the problem of designing fixed-order ro-bust H∞ controllers has been considered for linear sys-tems affected by polytopic uncertainty. The design problemhas been formulated as an LMI constraint whose decisionvariables are controller parameters. In Khatibi, Karimi, andLongchamp (2008), convex parameterisation of fixed-orderrobust stabilising controllers for systems with polytopic un-certainty has been represented as an LMI using the Kalman–Yakubovich–Popov (KYP) lemma. In Malik, Darbha, andBhattacharyya (2008), a linear programming approach hasbeen proposed to the synthesis of fixed-order controllers.In Fujisaki, Oishi, and Tempo (2008), a general method-ology for designing fixed-order controllers for single-inputsingle-output (SISO) plants has been proposed using mixeddeterministic/randomised methods. In Jin and Kim (2008),for a two-parameter feedback configuration, the problemof finding a fixed or low-order controller to meet the de-sired time response specifications has been reduced to theleast square estimation (LSE) in the sense of partial modelmatching (PMM) that minimises a quadratic cost function.In Maruta, Kim, and Sugie (2009), it is shown how to ob-tain a fixed-order controller satisfying multiple H∞ speci-fications; but it is not guaranteed that the proposed methodalways results in a solution. For more references on fixed-order controller design see, for example, Huang and Huang(2005), Famularoa, Pugliese, and Sergeyev (2004), Allison,Breton, and Ridgfxy (1997) and Chung (2000).

C© 2013 Taylor & Francis

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http://dx.doi.org/10.1080/00207721.2013.815823

2 H. Parastvand and M.-J. Khosrowjerdi

It is important to note that all of the approachescited above are model-based. In recent years, severalimportant advances in model-free design using only thefrequency-domain data have been made, for example, seeLi, Qi, and Sheng (2011a), Karimi and Galdos (2010),Galdos, Karimi, and Longchamp (2010), Li, Qi, and Sheng(2011b), Keel and Bhattacharyya (2008), Parastvand(2010) and references therein. Frequency-domain datacan be easily obtained from input/output data usingFourier or spectral analysis (Pintelon and Schokens 2001;Wang, Lee, and Lin 2003). In Keel and Bhattacharyya(2008), it is shown interestingly that the set of all sta-bilising proportional-integral-derivative (PID) controllersachieving a desired gain and phase margin or H∞ normcan be obtained using only the frequency-domain data(Nyquist/Bode data) without constructing a state-spaceor transfer function model. As pointed out in Keel andBhattacharyya (2008), the open area of research understudy is the extension of this new model-free approach tofixed-order controllers that is the subject of this brief paper.

It is worth mentioning that in Karimi and Galdos (2010),the problem of robust fixed-order H∞ controller design inthe frequency domain for systems represented by nonpara-metric spectral models is also formulated as a convex fea-sibility problem with an infinite number of constraints. Anapproximate solution based on a set of linear or convexconstraints with respect to the parameters of a linearly pa-rameterised controller in the Nyquist diagram is proposedby choosing a finite number of frequencies ω ∈ {ω1, ω2,. . . , ωM}, where M should be sufficiently large. The overallapproximation and conservatism of the solution depends onthis frequency band. This can, however, be ended to a largeset of all possible linear inequalities with some numericalproblems.

Motivated by Keel and Bhattacharyya (2008) andKarimi and Galdos (2010), in this paper, the model-freeapproach proposed in Keel and Bhattacharyya (2008) forPID controllers has been extended to fixed-order controllersthat achieve some H∞ norms on sensitivity and comple-mentary sensitivity functions by linearly parameterisingcontrollers. It is shown that for fixed-order controllers, syn-thesis and design can also be carried out directly from onlythe frequency-domain data. The main contribution of theproposed model-free approach in comparison with Karimiand Galdos (2010) is that an exact low frequency band overwhich the plant data must be known with desired accuracyand beyond which the plant information may be rough orapproximate is sufficient for the controller synthesis. Thiscan be ended to a small set of all possible linear inequalitieswhose decision variables are controller parameters, allow-ing the use of standard convex optimisation softwares suchas Ghaoui, Nikoukhah, Delebecque, and Commeau (1998)and Gahinet, Nemirovski, Laub, and Chilali (1995).

This paper is organised as follows. In Section 2, somepreliminary results have been presented. The parameteri-

sation of stabilising fixed-order controllers is proposed inSection 3. It is shown that the design problem is formulatedas some linear inequalities, and a constructive algorithmis proposed. In Section 4, the effectiveness of the proposedapproach is illustrated by some examples. Some concludingremarks and future works are mentioned in Section 5.

2. Preliminaries

In this section, some mathematical preliminaries and a re-view on some results have been presented that are helpfulin this paper. The proof of lemmas and theorems of thissection can be found in Keel and Bhattacharyya (2008).

Consider a real rational transfer function

P (s) = A(s)

B(s),

where A(s) and B(s) are polynomials with real coefficientsand of degrees m and n, respectively. It is assumed thatA(s) and B(s) have no zero on jω axis. Let z+ , p+ (z−,p−) determine the number of open right half plane (RHP)(open left half plane (LHP)) zeros and poles of P(s). Alsolet �∠P(jω) denote the net change in phase of P(s) as ω

runs from 0 to +∞. Then, �∠P(jω) is given by

�∠P (jω) = π

2((z− − z+) − (p− − p+)) . (1)

The (Hurwitz) signature of P(s) is defined as

σ (P ) = z− − z+ − (p− − p+) = 2

π�∠P (jω) . (2)

Since P(s) has no pole and zero on jω axis, σ (P) is givenby

σ (P ) = −(n − m) − 2(z+ − p+), (3)

or

σ (P ) = −rp − 2(z+ − p+), (4)

where rp = n − m is the relative degree of plant P(s)and can be determined from high frequency slope of bodemagnitude. The frequency response of P(s) is given by

P (jω) = |P (jω)|ejφ(ω) = Pr (ω) + jPi(ω), (5)

where Pr(ω) and Pi(ω) denote the real and imaginary partsof P(jω), respectively. For stable plants, σ (P) and z+ can beobtained from Equations (2) and (4); but for unstable plantsthe frequency response cannot directly be obtained. In thiscase, there should be a stabilising controller C(s), then the

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International Journal of Systems Science 3

Figure 1. A stabilising controller C(s) for P(s).

frequency response of unstable plants can be obtained from

P (jω) = H (jω)

C(jω)(1 − C(jω)), (6)

where H(jω) is the frequency response of the closed-loopsystem as shown in Figure 1. The following theorem illus-trates how to calculate the number of RHP zeros and polesof unstable plant P(s).

Theorem 2.1:

z+ = 1

2[−rp − rc − 2z+

c − σ (H )] (7)

p+ = 1

2[σ (P ) − σ (H ) − rc] − 2z+

c , (8)

where z+c and rc are the number of RHP zeros and the

relative degree of C(s), respectively.

Since P(s) is real, ω0 = 0 is always a zero of Pi(ω).Assume that the real, distinct, finite zeros of Pi(ω) = 0 aredenoted by ω0, ω1,. . ., ωl − 1 such that

0 = ω0 < ω1 < · · · < ωl−1 < ωl = ∞,

and let

sgn[x] =⎧⎨⎩

−1 if x < 0,

0 if x = 0,

+1 if x > 0 .

(9)

Lemma 2.2: If rp is even, then

σ (P ) =(

sgn[Pr (ω0)] + 2l−1∑j=1

(−1)j sgn[Pr (ωj )]

+ (−1)lsgn[Pr (ωl)]

)(−1)l−1sgn[Pi(∞)],

and if rp is odd, then

σ (P ) =(

sgn[Pr (ω0)] + 2l−1∑j=1

(−1)j sgn[Pr (ωj ]

)

× (−1)l−1sgn[Pi(∞)]. (10)

Throughout this paper, Re(.) and Im(.) denote the realand imaginary parts of complex numbers, respectively.

3. Fixed-order controller synthesis

Consider the closed-loop system in Figure 1. Assume themathematical model of the plant P be unknown and the onlyavailable data is the frequency response P(jω). The mainobjective is to calculate a subset of fixed-order controllersthat stabilise P and achieve some performance criteria. It iswell-known (Doyle and Glover 1996) that the performancecriteria can be specified as

⎧⎨⎩

||W1(s)S(s)||∞ < γ,

||W2(s)T (s)||∞ < γ,

(11)

where γ > 0, S and T are sensitivity and complementarysensitivity functions, respectively and given by

S(s) = 1

1 + L(s), T (s) = L(s)

1 + L(s),

where L(s) := C(s)P(s) is the loop function and W1(s) andW2(s) are appropriate weighting functions. In this paper, thefollowing problems are considered.

Problem 3.1: Given the frequency response of the plantP, find the N-th order controller C that stabilises P inFigure 1.

Problem 3.2: Given the frequency response of the plant P,find the N-th order controller C that stabilises P in Figure 1and guarantees the performance criteria in Equation (11).

3.1. Stability attainment with fixed-ordercontrollers

In this subsection, an algorithm is proposed that gives asolution to Problem 3.1. Let the N-th order controller C inFigure 1 is parameterised as follows:

C(s) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

ρ1 + ρ2s +∑ N2 −1i=0 ρ(i+3)s

2(i+1)

sNi=2(s + Ti)

, N even

ρ1 + ρ2s

s, N = 1

ρ1 + ρ2s+∑ N−3

2i=0 ρ(i+3)s

2(i+1)

sNi=2(s + Ti)

, N ≥ 3 & odd

(12)

where ρ1, ρ2, . . ., {ρN+42

, ρN+32

} are design parameters andT1, T2, T3, . . . , TN are arbitrary constants selected by thedesigner. Without loss of generality, it is assumed that T1 =0 in the rest of the paper. Let

C(s) = Nc(s)

Dc(s),

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define

F (s) := Dc(s)(1 + L(s)).

It is clear that the zeros of F(s) are the closed-loop poles ifthere are no pole-zero cancellations in Re(s) ≥ 0 when theloop function L(s) is formed. The following lemma gives astability condition for the closed-loop system in Figure 1.

Lemma 3.3: The feedback control system in Figure 1 isinternally stable if and only if the following conditions aresatisfied:

(1) There are no pole-zero cancellations in Re(s) ≥ 0when the loop function L(s) is formed.

(2) σ (F̄ (s)) = rp+2z++N , where F̄ (s) = F (s)P (−s).

Proof: The feedback control system is internally stable ifand only if the following conditions are satisfied: (a) Allroots of the equation 1 + L(s) = 0 have strictly negativereal part and (b) There are no pole zero cancellations inRe(s) ≥ 0, see Doyle, Francis, and Tannenbaum (1990)and Goodwin, Graebe, and Salgado (2000). The part (a) isequivalent to the condition that all zeros of F(s) lie in theLHP. This in turn is equivalent to the condition

σ (F (s)) = n + N − (p− − p+). (13)

Note that

σ (F (s)) = σ (F (s)) + σ (P (−s)) .

Therefore, the stability condition becomes

σ (F (s)) = n + N − (p− − p+) + (z+ − z−) − (p+ − p−)

= n + N + z+ − z− = rp + 2z+ + N.

This completes the proof. �

Remark 3.4: The unstable pole-zero cancellation leads tointernal instability. Though this phenomenon occurs rarelyin practical systems, in Dehghani, Lecchini, Lanzon, andAnderson (2009), a novel test for internal stability utilisinga limited amount of experimental and possibly noisy fre-quency response data has been proposed. For the sake ofsimplicity in the rest of paper it is assumed that the unstablepole-zero cancellation is avoided. Moreover, the controllersthat satisfy the validation tests in Dehghani et al. (2009) cansatisfy some performance criteria such as gain margin andphase margin. Thus the approach can tolerate measurementerrors caused by possibly noisy data.

It can be written that

F (jω) = F r

(ω, ρ1, ρ3, ρ4, . . ., {ρN+4

2, ρN+3

2})

+ jωF i(ω, ρ2) ,

where F r (ω) and F i(ω) denote real and imaginary partsof F (jω), respectively. Let ω1 < ω2 < · · · < ωl − 1 denotethe distinct frequencies of odd multiplicities which are solu-tions of F i(ω, ρ2) = 0. After some algebra, it can be shownthat

ρ2 := g(ω) = −aPi(ω) + bPr (ω)

ω|P (jω)|2 , (14)

where a and b are obtained from real and imaginary partsof denominator of controller as

(jω)N∏

i=2

(jω + Ti) = a + jb.

Theorem 3.5: Let ω1 < ω2 < · · · < ωl − 1 denote the dis-tinct frequencies of odd multiplicities which are solutionsto Equation (14). Determine strings of integers {v0, v1, v2,. . . , vl} where v0, v1, . . . , vl belong to {−1, 1} such thatfor rp even

v0 − v1 + v2 + · · · + (−1)l−12vl−1 + (−1)lvl](−1)l−1J

= rp + 2z+ + N, (15)

and for rp odd

[v0 − v1 + v2 + · · · + (−1)l−12vl−1](−1)l−1J

= rp + 2z+ + N, (16)

where

vl = sgn(F r

(∞, ρ1, ρ3, ρ4, . . ., {ρN+42

, ρN+32

})), (17)

and

J = sgn[F i(∞, ρ2)]. (18)

Then, for every ρ2 ∈ [ρmin2 , ρmax

2 ], the values of(ρ1, ρ3, ρ4, . . ., {ρN+2

2, ρN+3

2}) in Equation (12) that solves

Problem 3.1 can be found such that the following frequencydomain inequality (FDI) is satisfied

f :=

⎛⎜⎜⎜⎜⎝

f 0r 0 . . . 0

0 f 1r . . .

...... . . .

. . ....

0 . . . . . . f l−1r

⎞⎟⎟⎟⎟⎠ < 0 , (19)

where

f kr := −F r

(ωk, ρ1, ρ3, ρ4, . . ., {ρN+4

2, ρN+3

2})vk , (20)

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for all k = 0, 1, . . . , l − 1 and vk’s are given by Equation(15) or (16) and ωk’s are solutions to Equation (14).

Proof: From Equation (13) the closed-loop stability isequal to

σ (F̄ (s)) = rp + 2z+ + N.

Using Lemma 2.1 to compute σ (F̄ (s)) leads to

σ (F̄ (s)) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

v0 − 2v1 + · · · + (−1)l−12vl−1

+ (−1)lvl](−1)l−1J if rp even

v0 − 2v1 + · · · + (−1)l−12vl−1](−1)l−1J

if rp odd.

The above equations with the stability criterion inEquation (13) can be combined and this results in Equa-tions (15) and (16). On the other hand, the closed-loopstability for the fixed ρ2 = ρ∗

2 is equivalent to

F r

(ωk, ρ1, ρ3, ρ4, . . ., {ρN+2

2, ρN+3

2})vk > 0,

and it is then equivalent to Equation (19), where vk’s aregiven by Equation (15) or (16) and ωk’s are solutions toEquation (14). It can be easily seen that Equation (20) forevery vk and ωk is a linear equation in terms of controllerparameters ρ1, ρ3, ρ4, . . ., {ρN+4

2, ρN+3

2} and ωk. Therefore,

the FDI in Equation (19) is also an LMI. Obviously, solvingthe feasibility problem of LMI (19) leads to stabilisingvalues of controller parameters. �

Remark 3.6: It is worth mentioning that almost in everyapplication the set of ωk

′s can be found in the low fre-quencies. This means that only an exact data of the lowfrequency band is sufficient for the controller synthesis andbeyond this band the plant information may be rough orapproximate. This is in contrast of the proposed approachby Karimi and Galdos (2010) that needs the whole accu-rate frequency spectrum and sweeps on a large number offrequencies. On the other hand, since here sweeping is onlyon the computed ωk

′s, the set of linear inequalities can besmaller than the resulting set using the approach proposedby Karimi and Galdos (2010).

Remark 3.7: Like other works, for example, see Karimiand Galdos (2010), the feasibility of the LMI in Equation(19) cannot be guaranteed. In fact, the conditions should berelaxed, for example, the different values for the parametersρ2 ∈ [ρmin

2 , ρmax2 ], N, T ′

i s and the weighting function W1

should be chosen that result in a different set of inequalities.

All controller parameters except ρ2 can be obtainedfrom Theorem 3.1. The following theorem shows how tocalculate the admissible range of ρ2.

Theorem 3.8: There exists an N-th order controller C(s)that solves Problem 3.1 if there exists ρ2 such that Equation(14) has at least R distinct roots of odd multiplicities suchthat

⎧⎪⎪⎪⎨⎪⎪⎪⎩

R ≥ rp + 2z+ + N

2− 1 if rp even,

R ≥ rp + 2z+ + N + 1

2− 1 if rp odd.

(21)

Proof: Equation (21) is the straight result of satisfyingEquation (15) or (16). �

In the above theorems, ρ2 belongs to the stabilising setof controller parameters where

ρmin2 < ρ2 < ρmax

2 ,

and

ρmin2 = min

ω∈g(ω), ρmax

2 = maxω∈

g(ω),

where is the range of frequencies that satisfyEquation (21).

Thanks to Theorem 3.5, Problem 3.1 can be solvedefficiently using the following algorithm.

Algorithm 3.9: Calculating the N-th order controllersRequired data: The frequency response of the plant

P(jω).

(1) Determine rp using high frequency slope of Bodemagnitude diagram.

(2) Determine z+ using Equation (4) or (7).(3) Set N = 1.(4) Determine the range of ρ2 using Equation (21).(5) If there is not any ρ2 that satisfies Equation (21)

then set N = N + 1 and go to Step 4, else go to thenext step.

(6) For every ρ2 = ρ∗2 ∈ [ρmin

2 , ρmax2 ] solve Equation

(14) and obtain roots with odd multiplicity as ω1 <

ω2 < · · · < ωl − 1.(7) Determine v0, v1, . . . , {vl − 1, vl} using Equation

(15) or (16).(8) For every ρ∗

2 , determine the values of(ρ1, ρ3, ρ4, . . ., {ρN+4

2, ρN+3

2}) using Equation

(19).(9) Construct the N-th order controller C(s) in Equation

(12).(10) Change the value of ρ2 by sweeping on its ad-

missible range and go to Step 6 to obtain anotherfixed-order controller.

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This algorithm is constructive and can be implementedusing standard scientific softwares such as Ghaoui et al.(1998) and Gahinet et al. (1995).

3.2. Performance attainment with fixed-ordercontrollers

In Section 3.1, the set of stabilising fixed-order controllersthat solve Problem 3.1 can be obtained by solving LMI inEquation (19). To obtain the set of possible fixed-order con-trollers that solve Problem 3.2, another constraints must beadded to Equation (19). In general, the criteria (11) cannotbe expressed as a linear constraint in terms of controllerparameters. Motivated by Keel and Bhattacharyya (2008)and Bhattacharyya, Chapellat, and Keel (1995), Pro-blem 3.2 can be transformed to simultaneously stabilisingof a family of real and complex plants. In fact, the followingcases can be considered:

(1) The H∞ norm attainment on sensitivity function inEquation (11) is equal to simultaneously stabilisingthe plant P(s) and the family of real and complexplants

P S(s) ={P (s)

(1

1 + 1γejθW1(s)

): θ ∈ [0, 2π ]

}.

(22)

(2) The H∞ norm attainment on complementary sen-sitivity function in Equation (11) is equal to simul-taneously stabilising the plant P(s) and the familyof real and complex plants

P T (s) ={P (s)

(1 + 1

γejθW2(s)

): θ ∈ [0, 2π ]

}.

(23)

The above criteria lead to the case of nominal plant sta-bilisation. In this way, Algorithm 3.9 should be appliedfor the family of plants PS(s) and PT(s) by sweeping on θ

and solving the resulted inequalities simultaneously. Thisleads to a solution to Problem 3.2. To satisfy stability andperformance criteria on sensitivity and complementary sen-sitivity functions in Equation (11), the following LMI mustbe feasible ⎛

⎝f 0 00 f S 00 0 f T

⎞⎠ < 0, (24)

where f is given by Equation (19) and f S and f T can beobtained by solving Equation (19) for PS(s) and PT(s),respectively.

Remark 3.10: Like the work by Keel and Bhattacharyya(2008), some values of θ such as {0, pi/3, 2pi/3, . . . } can

Figure 2. The Bode diagram of P(s) for Example 4.1.

be selected just for illustrating how the values of admissi-ble parameters change. This is a common problem in allnumerical methods that is widely used in literature, for ex-ample, see Bhattacharyya et al. (1995) and Barmish (1994).Here the intersection over all θ should be obtained and a setof parameters for the purpose of some smoothness at theboundary of the possibly high dimension geometry of LMI(24) should be obtained.

4. Simulation results

In this section, to show the effectiveness of the proposedapproach, Algorithm 3.9 is carried out on four differentexamples such as real and academic, minimum and non-minimum phase, stable and unstable plants.

Example 4.1: Suppose that the only required data is thebode diagram in Figure 2 generated by the following non-minimum phase plant

P (s) = 0.34662(s + 0.08655)(s2 − 60s + 1200)

(s + 6.251)(s + 0.1524)(s2 + 60s + 1200).

The required parameters can be determined from the Bodediagram. From Equations (2) and (3), σ (P) is given by

σ (P ) = 2

π.−5π

2= −5,

and z+ = 12 (−1 + 5) = 2. The main objective is to obtain

a second-order controller, i.e. N = 2 in Equation (12). Thedesired controller is then given by

C(s) = ρ1 + ρ2s + ρ3s2

s(s + T2),

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Figure 3. The diagram of g(ω) for Example 4.1.

where T2 = 1. From Equation (21), the inequality R ≥2+4+2

2 − 1 = 3 can be obtained. For N = 2, J is given by

J = sgn[F i(∞, ρ2)]

= sgn(ρ2|P (jω)|2 + Pr (jω) + ωPi(jω)

).

Using Equation (14), g(ω) is given by

g(ω) = −Pr − ωPi

|P (jω)|2 .

The diagram of g(ω) is illustrated in Figure 3. SinceR ≥ 3 the admissible range of ρ2 is corresponding to regionthat the diagram of g(ω) crosses 3 times over the horizontalline. From the maximised region in Figure 3, the admissiblerange of ρ2 can be obtained that is (−14, 393). Sweepingon this range leads us to the set of second-order controllersthat solves Problem 3.2. As shown in Figure 3, the exactfrequency information of plant in the small low frequencyspectrum, i.e. 0 ≤ ω ≤ 100, is sufficient for computationpurposes. Let ρ2 = 10. This leads us to J = 1 and l = 4.Solving Equation (16) results to the following values for

the string of integers

V = [v0, v1, v2, v3] = [1,−1, 1,−1].

The required three frequencies satisfying Equation (14) forρ2 = 10 are ω1 = 2.6, ω2 = 19.5 and ω3 = 80.5 as shownin Figure 3. Now from Equation (19), the following LMIcan be obtained

f =

⎛⎜⎜⎝

−ρ1 0 0 00 ρ1 − 6.76ρ3 − 214.6 0 00 0 −ρ1 + 380.2ρ3 − 22490 00 0 0 ρ1 − 6480ρ3 − 1509700

⎞⎟⎟⎠ < 0.

By solving the above LMI, for ρ2 = 10, the values ρ1 =106.0154 and ρ3 = 2.2084 × 10−15 are obtained. Theresulting second-order controller is given by

C(s) = 2.2084 × 10−15s2 + 10s + 106.0154

s(s + 1), (25)

and the corresponding step response of the closed-loopsystem in Figure 1 is shown in Figure 4(a). The over-shoot is 95% and the settling time is 50 sec. Obviously, thisstep response does not guarantee the desired performance.To improve the transient specifications, Algorithm 3.9 canbe applied for the family of plants (22) and (23) for θ ={0, π /3, 2π /3, . . . , 2π}. Let

W1(s) = s + 10

s + 0.1.

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Figure 4. The step response of the closed-loop system with controllers (a): (25); (b): (26) and (c): (27) for Example 4.1.

Simulation results for stabilising of the family of plants (22)for three values of θ = {0, π /3, 2π /3} lead to the followinginequalities with the corresponding ωk

′s

θ 0 π /3 2π /3

ωk {4.5, 22.3, 84.9} {3.6, 15.6, 56.8} {1.9, 12.3, 40.8}ρ1 − 20ρ3 − 307 < 0 ρ1 − 13ρ3 − 162 < 0 ρ1 − 3.6ρ3 − 77 < 0

ρ1 − 497ρ3 + 31939 > 0 ρ1 − 243ρ3 + 10535 > 0 ρ1 − 151ρ3 + 3751 > 0ρ1 − 7208ρ3 − 17∗105 < 0 ρ1 − 3226ρ3 − 4.3∗105 < 0 ρ1 − 1664ρ3 − 38882 < 0

Now the simultaneously stabilising of the family ofplants (22) for θ = {0, π /3, 2π /3, π , 4π /3, 5π /3, 2π} isequivalent to the solution of an LMI containing the inequal-ities shown in the above table together with the inequalitiesobtained from the rest of values of θ . For ρ2 = 10, the val-ues ρ1 = 23.0136 and ρ3 = 8.0182 × 10−15 are obtained.The resulting second-order controller is given by

C(s) = 8.0182 × 10−15s2 + 10s + 23.0136

s(1 + s). (26)

The step response is shown in Figure 4(b). It can be seenthat the transient response specifications such as settlingtime and overshoot have been improved in comparison withFigure 4(a). Choose

W2(s) = s + 0.1

s + 10.

For ρ2 = 10, the same approach can be applied for thefamily of plants (23) and obtain ρ1 = 13.8081 and ρ3 =1.9063 × 10−15. The resulting second-order controller is

given by

C(s) = 1.9063 × 10−15s2 + 10s + 13.8081

s(1 + s). (27)

The step response by the above controller is shown inFigure 4(c). As shown in this figure, there is no overshootand the transient response is acceptable. The sweeping onρ2 results in other possible controllers.

Example 4.2: This example is taken form Halim andMoheimani (2002) where the plant is a flexible beam con-sists of only one piezoelectric actuator-sensor pair. Here,the control input u is the voltage applied to the piezo-electric actuator and y is the induced voltage at the sen-sor that is proportional to the deflection of the beam at aspecific position. The purpose of the controller C(s) is toreduce the effect of disturbance w applied in the input ofthe plant on the output y. Globally, as shown in Halim andMoheimani (2002), a simplified model of the beam with alightly damped flexible mode and a low-pass filter is givenby

P (s) = K

(1

τs + 1

)(ω2

n

s2 + 2ζωns + ω2n

),

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Figure 5. The Bode diagram of P(s) for Example 4.2.

where K = 1, τ = 1, ωn = 3.16 and ζ = 0.16. The objec-tive is to find a solution to Problem 3.2 and the requireddata, i.e., the frequency spectrum has been illustrated inFigure 5. The values rp = 3, σ (p) = 3 and z+ = 0.5(−3 +3) = 0 can be determined. First a possible first-order con-troller is chosen. For N = 1, Equation (16) does not haveany solution. Then according to Algorithm 3.9 for N = 2,the inequality R ≥ 2 can be obtained. As shown in Figure 6,the admissible range of ρ2 is (−1.2, 3.1) and the requiredtwo frequencies to compute the parameters ρ1 and ρ3 forρ2 = 1 are ω1 = 1.35 and ω2 = 3.4. Then, it can be obtainedthat J = 1 and rp + 2z+ + N = (v0 − 2v1 + 2v2)(−1)2(1).Thus, the only possible string of integers is [1, −1, 1]. From

Equation (19), the following LMI can be obtained

f =⎛⎝−ρ1 0 0

0 ρ1 − 1.82ρ3 − 2.83 00 0 −ρ1 + 11.56ρ3 − 15.8

⎞⎠ < 0.

The resulting values are ρ1 = 1.47 and ρ3 =−2.65 × 10−12

and the corresponding second-order controller is given by

C(s) = −2.65 × 10−15s2 + s + 1.47

s(1 + s). (28)

A step disturbance signal was applied through the piezo-electric actuator. The simulation result is shown in Fig-ure 7 (a). To improve the time domain specifications, Alg-orithm 3.9 can be applied for the family of plants (22)and (23). Simulation results for stabilising of the family ofplants by gridding on θ lead to the following controller

C(s) = −7.9 × 10−13s2 + s + 0.6

s(1 + s)(29)

which satisfies the stability and performance criteria inEquation (11) and the corresponding simulation result isshown in Figure 7(b).

Example 4.3: In this example, the following unstabletransfer function is considered

P (s) = (s + 1)(s + 10)

(s + 2)(s + 4)(s − 1).


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Figure 7. The step response of the closed-loop system with con-trollers (a): (28) and (b): (29) for Example 4.2.

This process is the nominal form of the plant used by Karimiand Galdos (2010) and Djafari (1995). The correspondingBode diagram is shown in Figure 8. . For N = 1, there is notany feasible solution to Equation (16). Equation (21) resultsto R ≥ 1 for N = 2. To obtain the admissible range of ρ2 thefunction g(w) is plotted in Figure 9. This range is equal tothe values of ρ2 for which R ≥ 1. Thus, the feasible range ofρ2 is (0.82, + ∞). As shown in Figure 9, knowing the exactfrequency information of the plant below the frequency ω =20 is sufficient to compute the desired controller and beyondwhich the plant information may be rough or approximate.This is an important superiority of the proposed approachin comparison with the approach proposed by Karimi and

Figure 8. The Bode diagram of P(s) for Example 4.3

Galdos (2010) that needs a sufficiently large number offrequencies and as a result leads to a large number of linearinequalities. For N = 2 and ρ2 = 15, the value of J is givenby J = −1 and

(v0 − 2v1)(−1)1(−1) = 1 + 2 = 3 ⇒ {v0, v1} = {1,−1}.

As shown in Figure 9, the crossing frequency is ω1 = 5 andthe resulting linear inequalities(a). are

−ρ1 < 0, ρ1 − 25ρ3 + 19.4 < 0.


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Figure 10. The step response of the closed-loop system with controllers (a): (30) and; (b): (31) for Example 4.3.

One feasible solution to the above inequalities is ρ1 = 10.2and ρ3 = 2.5 and the corresponding second-order controlleris

C(s) = 10.2s2 + 15s + 2.5

s(s + 1). (30)

The corresponding step response with the above controlleris shown in Figure 10 By considering the performance cri-teria of Equations (22) and (23), the following second-ordercontroller is obtained

C(s) = 8.6s2 + 15s + 301.11

s(s + 1). (31)

The corresponding step response in Figure 10(b) demon-strates the improvement in time response specifications.

Example 4.4: There are cases that the model-free approachproposed by Keel and Bhattacharyya (2008) does not work.This means that some plants cannot be stabilised by firstor second-order controllers such as PID controllers. Thisrequires that the order of the controller should be increasedsufficiently. Consider the following plants

Pa(s) = 9673

(s2 + 142.6s + 7235.2)(s + 4)(s − 11),

Pb(s) = (s − 11)

(s − 20)(s + 3),

Pc(s) = 282710(s − 1)2

(s − .4858)(s + 2.38)(s + 12.205).

The main objective is to compute a second-order controllerfor the above plants. The corresponding functions of g(ω)have been shown in Figure 11. It can be shown that Ra ≥ 2,Rb ≥ 2 and Rc ≥ 3 for Pa(s), Pb(s) and Pc(s), respectively.According to these values, there is no admissible rangeof ρ2 for these plants. Thus, the design a second-ordercontroller for these plants by using the approach proposedin Keel and Bhattacharyya (2008) is impossible. However,there are other situations in which the approach proposedby Keel and Bhattacharyya (2008) does not work. One caseis when the resulting inequalities from Equation (19) or(24) do not have a feasible solution such as the followingexample. Let

Pd (s) = (s + 0.1)

(s + 17)(s + 45).

The values z+ = 0 and rp = 1 can be obtained. ForN = 2, the admissible value of ρ2 belongs to (−856, + ∞).Sweeping on this range of ρ2 does not lead to a stabilisingcontroller. For example, for ρ2 = 103, the value ρ1 = 7.1 ×104 and ρ3 = −104 can be obtained. This parameterleads to an unstable closed-loop transfer function. For N =3, there is not any feasible solution to Equation (16). Thus,the order of controller must be increased. For N = 4, Equa-tion (14) should be constructed by a = ω4 − 3ω2 and b =−3ω3 + ω. The corresponding function of g(ω) has beenshown in Figure 12. As shown in this figure, the admissi-ble range of ρ2 is [−6791, 1962]. For ρ2 = 103 and from

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Figure 11. The function g(ω) for (a): Pa(s); (b): Pb(s) and (c): Pc(s) for Example 4.4.

Figure 12. The function g(ω) for P(s) and N = 4 for Example 4.4.

Equation (20), the following inequalities can be obtained

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

ρ1 < 0,

−6.16 × 10−5 + 8.54 × 10−8ρ1 − 3.14 × 10−9ρ3

+ 1.36 × 10−10ρ4 > 0,

0.02 + 4.34 × 10−6ρ1 − 1.11 × 10−5ρ3

+ 2.84 × 10−5ρ4 < 0.

A feasible solution to the above inequalities for ρ2 = 103

is {ρ1, ρ3, ρ4} = {3.5 × 103, 10−7, 91}. The resultingforth-order controller is

C(s) = 91s4 + 10−7s2 + 103s + 3.5 × 103

s(s + 1)3. (32)

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Figure 13. The step response for the controller (32) for Example 4.4.

The step response is shown in Figure 13. As same as theprevious examples, the better transient specifications couldbe attained by satisfying Equation (24).

5. Conclusion

In this paper, a model-free approach for PID controller hasbeen extended to fixed-order controller synthesis. There isno need for transfer function or state space model of plant.Using only the frequency-domain data (Nyquist/Bode dia-grams), a subset of stabilising fixed-order controllers thatachieve some performance criteria can be obtained. First,the admissible range for one of the controller parame-ters is determined and then the stabilising controller canbe achieved from a set of linear inequalities in terms ofcontroller parameters. Some H∞ norms on sensitivity andcomplementary sensitivity functions are also transformedto the problem of simultaneously stabilising for a family ofplants. A constructive algorithm summarises our results onthe fixed-order controller design problem.

The open areas of research under study are (a) the designof such controllers for multivariable and nonlinear plants,and (b) robustness enhancements of the designs with respectto the measured data.

AcknowledgementsThe authors would like to thank the associate editor and the anony-mous reviewers for their valuable comments, criticisms and sug-gestions. They were very helpful for this study. This work hasbeen carried in Advanced Control Research Laboratory(ACRL)in the Sahand University of Technology (SUT).

Notes on contributorsHossein Parastvand received the BS andMS degrees in Instrument and Control En-gineering from Shiraz University of Tech-nology and Sahand University of Technol-ogy, Iran in 2008 and 2011, respectively. Hisresearch interests are model free and data-based control, fixed-order controller synthe-sis, control theory and applications.

Mohammad-Javad Khosrowjerdi was bornin Tehran, Iran, on May 1, 1970. He receivedthe BS, MS and PhD degrees in electricalengineering from K. N. Toosi University ofTechnology, Iran in 1993, 1996 and 2003,respectively. From 2001 to 2003, he workedas a visiting researcher at the French Na-tional Institute for Research in ComputerScience and Control (INRIA) in Rocquen-

court, France. At present, he is associate professor at the electricalengineering department of the Sahand University of Technology(SUT), Tabriz, Iran. His main research interests include Fault De-tection and Isolation (FDI), Fault Tolerant Control (FTC), Robotic,Multiobjective Control, Robust and Nonlinear Control.

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controller synthesis free of analytical model: fixed-order controllers

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