Robust Pole Placement for Plants with Semialgebraic ...

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Robust pole placement for plants with semialgebraic parametric uncertainty V. Cerone , , D. Piga § , D. Regruto Abstract— In this paper we address the problem of robust pole placement for linear-time-invariant systems whose uncer- tain parameters are assumed to belong to a semialgebraic region. A dynamic controller is designed in order to constrain the coefcients of the closed-loop characteristic polynomial within prescribed intervals. Two main topics arising from the problem of robust pole placement are tackled by means of polynomial optimization. First, necessary conditions on the plant parameters for the existence of a robust controller are given. Then, the set of all admissible robust controllers is sought. Convex relaxation techniques based on sum-of-square decomposition of positive polynomials are used to efciently solve the formulated optimization problems through semide- nite programming techniques. Index Terms— Robust coprimeness, Robust pole placement, Semialgebraic uncertainty, Sum-of-square decomposition. I. I NTRODUCTION Pole placement is a widely adopted technique to design feedback controllers for linear-time-invariant (LTI) systems (see, e.g., [1], [2]). Indeed, one of the most interesting properties of such a control design technique is the possibility to arbitrarily assign the roots of the closed-loop characteristic polynomial. In the case of plants described by proper transfer function, the well known Diophantine equation has to be solved (see, e.g., [3]). In the case of plants with parametric uncertainty, different techniques can be found in the literature to design robust controllers through the solution of uncertain Diophantine equation. More precisely, in [4] the set of admissible controllers is evaluated and, among them, the controller closest in some norm to a given nominal controller is chosen. In [5], the authors exploit linear programming (LP) optimization techniques for designing xed-order controllers with guaranteed and robust performance. The designed con- troller is robust in itself, in the sense that the closed-loop system guarantees stability and desired performance also for perturbations on the controller parameters. Similar results are obtained in [6] by solving interval Diophantine equations through interval analysis techniques, and in [7], where a two- degree-of-freedom methodology for designing a controller with robust stability and performance is presented. All the papers mentioned above consider the case of plants with interval uncertainty on the parameters. The case of plants V. Cerone and D. Regruto are with the Dipartimento di Au- tomatica e Informatica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy; e-mail: [email protected], [email protected]; Tel: +39-011-090 7064; Fax: +39-011- 090 7198 § D. Piga is with the Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, Delft, 2628 CD, The Netherlands; email: [email protected] Corresponding author with parameters belonging to a polytope is considered in [8] and [9], where a robust controller is designed in order to assign a closed-loop characteristic polynomial over a stability domain specied as a polytopic region on the space of characteristic polynomial coefcients. The same type of parameter uncertainty is assumed in [10] and [11], where semidenite programming (SDP) optimization techniques are employed to construct a convex inner approximation of the stability domain in the characteristic polynomial coefcients space. Indeed, the assumption of either interval uncertainty or polytopic uncertainty on the plant parameters might in- troduce conservativeness in robust controller design. As a matter of fact, in the case of interval uncertainties, each plant parameter is constrained to vary within an uncertainty range independently on the other parameters, while, in the case of polytopic uncertainty description, only a linear dependance among plant parameters can be considered. Motivated by this fact, robust pole placement techniques are proposed in [12], [13], [14], [15] to deal with uncertain plants with parameters belonging to a convex ellipsoidal region. In this way, a quadratic dependence of parameter variations can be modeled. Besides, this kind of uncertainty description often arises from parameter identication procedures, where the identied parameters are guaranteed to belong to ellipsoidal regions with a certain level of probability [16]. In this paper, we address the problem of robust pole place- ment when the plant parameters are assumed to lie into nonconvex semialgebraic regions, which include, indeed, interval, polytopic and ellipsoidal uncertainty sets as special convex cases. To the author’s best knowledge, no contribu- tion can be found in the literature addressing this problem. It is worth remarking that a semialgebraic description of the parameter uncertainty description allows us to consider any kind of polynomial dependence among the plant parameters. For instance, as recently shown in [17], [18], [19], this type of parameter uncertainty description arises from bounded error-in-variable identication of LTI systems, that is identi- cation of linear dynamical systems when both the input and the output measurements are corrupted by additive bounded noise. The paper is organized as follows. In Section II the problem of robust control synthesis is formulated in terms of solution of a robust Diophantine equation, in order to guarantee that the coefcients of the closed-loop characteristic polynomial lie in desired intervals for all possible values of the plant parameters belonging to the considered semialgebraic un- certainty set. It is known that, for zero-uncertainty plants, a necessary condition so that there exists a solution to the Dio- brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by IMT Institutional Repository

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Robust pole placement for plantswith semialgebraic parametric uncertainty

V. Cerone∗,�, D. Piga§, D. Regruto∗

Abstract— In this paper we address the problem of robustpole placement for linear-time-invariant systems whose uncer-tain parameters are assumed to belong to a semialgebraicregion. A dynamic controller is designed in order to constrainthe coefficients of the closed-loop characteristic polynomialwithin prescribed intervals. Two main topics arising from theproblem of robust pole placement are tackled by means ofpolynomial optimization. First, necessary conditions on theplant parameters for the existence of a robust controller aregiven. Then, the set of all admissible robust controllers issought. Convex relaxation techniques based on sum-of-squaredecomposition of positive polynomials are used to efficientlysolve the formulated optimization problems through semidefi-nite programming techniques.

Index Terms— Robust coprimeness, Robust pole placement,Semialgebraic uncertainty, Sum-of-square decomposition.

I. INTRODUCTION

Pole placement is a widely adopted technique to designfeedback controllers for linear-time-invariant (LTI) systems(see, e.g., [1], [2]). Indeed, one of the most interestingproperties of such a control design technique is the possibilityto arbitrarily assign the roots of the closed-loop characteristicpolynomial. In the case of plants described by proper transferfunction, the well known Diophantine equation has to besolved (see, e.g., [3]). In the case of plants with parametricuncertainty, different techniques can be found in the literatureto design robust controllers through the solution of uncertainDiophantine equation. More precisely, in [4] the set ofadmissible controllers is evaluated and, among them, thecontroller closest in some norm to a given nominal controlleris chosen. In [5], the authors exploit linear programming (LP)optimization techniques for designing fixed-order controllerswith guaranteed and robust performance. The designed con-troller is robust in itself, in the sense that the closed-loopsystem guarantees stability and desired performance also forperturbations on the controller parameters. Similar resultsare obtained in [6] by solving interval Diophantine equationsthrough interval analysis techniques, and in [7], where a two-degree-of-freedom methodology for designing a controllerwith robust stability and performance is presented. All thepapers mentioned above consider the case of plants withinterval uncertainty on the parameters. The case of plants

∗ V. Cerone and D. Regruto are with the Dipartimento di Au-tomatica e Informatica, Politecnico di Torino, corso Duca degliAbruzzi 24, 10129 Torino, Italy; e-mail: [email protected],[email protected]; Tel: +39-011-090 7064; Fax: +39-011-090 7198

§ D. Piga is with the Delft Center for Systems and Control, DelftUniversity of Technology, Mekelweg 2, Delft, 2628 CD, The Netherlands;email: [email protected]

� Corresponding author

with parameters belonging to a polytope is considered in[8] and [9], where a robust controller is designed in orderto assign a closed-loop characteristic polynomial over astability domain specified as a polytopic region on the spaceof characteristic polynomial coefficients. The same type ofparameter uncertainty is assumed in [10] and [11], wheresemidefinite programming (SDP) optimization techniques areemployed to construct a convex inner approximation of thestability domain in the characteristic polynomial coefficientsspace. Indeed, the assumption of either interval uncertaintyor polytopic uncertainty on the plant parameters might in-troduce conservativeness in robust controller design. As amatter of fact, in the case of interval uncertainties, each plantparameter is constrained to vary within an uncertainty rangeindependently on the other parameters, while, in the case ofpolytopic uncertainty description, only a linear dependanceamong plant parameters can be considered. Motivated bythis fact, robust pole placement techniques are proposedin [12], [13], [14], [15] to deal with uncertain plants withparameters belonging to a convex ellipsoidal region. In thisway, a quadratic dependence of parameter variations can bemodeled. Besides, this kind of uncertainty description oftenarises from parameter identification procedures, where theidentified parameters are guaranteed to belong to ellipsoidalregions with a certain level of probability [16].In this paper, we address the problem of robust pole place-ment when the plant parameters are assumed to lie intononconvex semialgebraic regions, which include, indeed,interval, polytopic and ellipsoidal uncertainty sets as specialconvex cases. To the author’s best knowledge, no contribu-tion can be found in the literature addressing this problem.It is worth remarking that a semialgebraic description of theparameter uncertainty description allows us to consider anykind of polynomial dependence among the plant parameters.For instance, as recently shown in [17], [18], [19], this typeof parameter uncertainty description arises from boundederror-in-variable identification of LTI systems, that is identi-fication of linear dynamical systems when both the input andthe output measurements are corrupted by additive boundednoise.The paper is organized as follows. In Section II the problemof robust control synthesis is formulated in terms of solutionof a robust Diophantine equation, in order to guarantee thatthe coefficients of the closed-loop characteristic polynomiallie in desired intervals for all possible values of the plantparameters belonging to the considered semialgebraic un-certainty set. It is known that, for zero-uncertainty plants, anecessary condition so that there exists a solution to the Dio-

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provided by IMT Institutional Repository

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phantine equation for any arbitrary characteristic polynomialis that numerator and denominator of the transfer functiondescribing the plant are coprime [3]. By the same token,in the case of uncertain plants, numerator and denominatorhave to be coprime for all possible values of the parametersin the uncertainty region. A novel approach to evaluaterobust coprimess of uncertain polynomials with coefficientsbelonging to a semialgebraic set is presented in Section III.In Section IV, convex relaxation techniques based on sum-of-square (SOS) decomposition of positive polynomials areused to efficiently solve the robust Diophantine equationformulated in Section II. The reported simulated examplesshow the effectiveness of the proposed approach.

II. PROBLEM FORMULATION

Let us consider the feedback control system depicted inFig. 1. The plant G to be controlled is a linear time invariant(LTI) system of order n described by the strictly properrational transfer function

G(s,p) =Np(s,p)Dp(s,p)

, (1)

where Np(s,p) and Dp(s,p) are polynomials in the Laplacevariable s, i.e.

Np(s,p) = bn−1(p)sn−1+bn−2(p)sn−2+ · · ·+b0(p), (2)

Dp(s,p) = an(p)sn + an−1(p)sn−1 + · · · + a0(p). (3)

Coefficients a(p) = [an(p), an−1(p), · · · , a0(p)]and b(p) = [bn−1(p), bn−2(p), · · · , b0(p)] dependpolynomially on a set of parameters p = [p1, p2, · · · , pnp],which are assumed to belong to a bounded semialgebraicregion P ⊆ R

np and an(p) is such that an(p) �= 0 ∀p ∈ P .

Remark 1: Since parameters p belong to abounded semialgebraic set and coefficients a andb depend polynomially on p, then also a and bbelong to a bounded semialgebraic set C defined asC : {(a,b) : gj(a,b,p) ≥ 0, j = 1, . . . , ng}, where gj ,with j = 1, . . . , ng, are appropriate polynomials taking intoaccount the constraints describing P and the dependance ofcoefficients a and b on parameters p. �

In this paper we address the problem of robust-poleplacement, whose formulation is provided below.

� �+ � ��−� �C(s, k) G(s, p)

Fig. 1. Feedback control system

Problem 1: Robust pole placementDesign the parameters k = [k1, k2, · · · , k2r+2] of an r-orderdynamic controller C(s):

C(s,k) =Nc(s,k)Dc(s,k)

=k1s

r + k2sr−1 + · · · + kr

kr+1sr + kr+2sr−1 + · · · + k2r+2,

(4)in order to constrain each coefficient fj , with j = 0, . . . , r+n, of the uncertain closed-loop characteristic polynomialF (s,p,k):

F (s,p,k) = Dc(s,k)Dp(s,p) + Nc(s,k)Np(s,p) =

= fr+nsn+r + fr+n−1sn+r−1 + · · · + f0

(5)

to lie in a given interval [fj; f j ] for all uncertain plants

G(s,p) with p ∈ P . �

In the case of plant models G without uncertainty, a solutionof the Diophantine equation (5) exists for any arbitrarycharacteristic polynomial F (s) if and only if r ≥ n − 1and the plant G has no common zero-pole pairs. Therefore,a necessary condition which guarantees the existence ofa solution to the robust pole placement problem is thatpolynomials Np(s,p) and Dp(s,p) are coprime for allvalues of parameters p ∈ P , or equivalently, for all valuesof the coefficients a and b in C. The following Lemmafrom commutative algebra provides a way to check robustcoprimeness of two polynomials with uncertain coefficients.

Lemma 1: Let Sylvester matrix S(a,b,p) associatedwith Np(s,p) and Dp(s,p) be a (2n−1)× (2n−1) squarematrix defined as

S(a,b,p) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 · · · 0 an 0 · · · 0

bn−1

. . . 0 an−1 an

. . . 0...

. . . 0... an−1

. . . 0...

. . ....

.... . .

. . .. . .

b0

. . . bn−1 a0

. . .. . . an−1

0. . . bn−2 0 a0

. . . an−2

.... . .

......

. . .. . .

...0 · · · b0 0 · · · · · · a0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(6)It is worth noting that the entries a0, . . . , an and b0, . . . , bn−1

of matrix S(a,b,p) are functions of parameter p. In thesequel, in order to simplify the notation, we will denote as S̃the Sylvester matrix S(a,b,p). The uncertain polynomialsNp(s,p) and Dp(s,p) are robustly coprime if and onlyif the Sylvester matrix S̃ associated with Np(s,p) andDp(s,p) is robustly nonsingular, that is, it is nonsingularfor all values of the coefficients a and b in C. �

In the following section a SOS-based method to checkrobust nonsingularity of Sylvester matrixes with uncertaincoefficients is presented. Then, in Section IV, we show howto design a controller to solve the robust pole placementformulated in Problem 1.

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III. ON THE ROBUST NONSINGULARITY OF UNCERTAIN

SYLVESTER MATRIXES

Checking robust nonsingularity of uncertain matrixes is anNP-hard problem (see, e.g. [20], [21]) and it has been exten-sively treated in the literature in the case of interval matrixes(see [22], [23], [24]). Unfortunately, the approaches proposedin [22], [23], [24] to evaluate if an interval Sylvester matrixis robustly nonsingular provide conservative nonsingularityconditions since each occurrence of an interval coefficient inthe Sylvester matrix has to be handled as if it was a differentinterval coefficient (see [6]). In this section we present anew approach to evaluate robust nonsingularity of uncertainSylvester matrixes. The advantages of the presented methodwith respect to the approaches in [22], [23], [24] are mainlythe following:

• uncertain coefficients appearing in the Sylvester matrixmay belong to a semialgebraic set;

• the dependence among same coefficients in theSylvester matrix is not lost. Therefore, in the case ofSylvester matrixes with interval polynomial coefficients,the approach proposed in the paper provides less con-servative nonsingularity conditions.

The following theorem provides necessary and sufficientconditions to check robust nonsingularity of uncertainSylvester matrixes.

Theorem 1: Let S(a, b) be the set of uncertain Sylvestermatrixes associated with polynomials Np(s,p) and Dp(s,p)with coefficients a, b ∈ C, i.e.

S(a,b) ={

S̃ : a,b ∈ C}

. (7)

The Sylvester matrix S̃ in (6) is robustly nonsingular onC if and only if the solution of the following optimizationproblem is bounded:

f∗ = maxx∈R2n−1

‖x‖22

s.t.

S̃x = 0; S̃ ∈ S(a,b).

(8)

or equivalently if and only if the set

F ={

x ∈ R2n−1 : S̃x = 0; S̃ ∈ S(a,b); ‖x‖2

2 ≥ ε}(9)

is empty for any ε ∈ R : ε > 0.

Proof: We first prove the “only if ” part. If matrixS̃ is robustly nonsingular, then, for all S̃ ∈ S(a,b), onlythe trivial solution x = 0 satisfies the constraint S̃x = 0.Therefore, the solution of problem (8) is equal to zero, thusbounded. The “if” part is proven by contradiction. Assumethat S̃ is not robustly nonsingular. This means that thereexists a value of a,b in C such that the solution of the linearsystem S̃x is not the trivial one, i.e. S̃x∗ = 0 for some x∗ �=0. Therefore, for every α ∈ R, also x = αx∗ satisfies theconstraint S̃x = 0 and thus it belongs to the set of feasibilityof problem (8). Then, solution to problem (8) is unbounded,

contradicting the hypothesis. The second part of the Theoremfollows from the fact that, when S̃ is robustly nonsingular,x = 0 is the only solution to equalities S̃x = 0 for allS̃ ∈ S(a,b).

On the basis of Theorem 1, checking robust nonsingularityof the Sylvester matrix on C is recast into the nonconvexoptimization problem (8). More precisely, (8) is asemialgebraic optimization problem since: (i) the objectivefunction is quadratic; (ii) the set of constraints S̃x = 0involves the product between the variable x and theuncertain coefficients a and b; (iii) coefficients a and b liein the semialgebraic set C.LMI-relaxation techniques based on Putinar’sPositivstellensatz [25] have been proposed in order torelax nonconvex polynomial optimization problems into ahierarchy of convex SDP problems, whose solutions areproven to monotonically converge to the optimal value ofthe original nonconvex problems (see, e.g., [26], [27] andthe survey paper [28] for an extensive review on this topic).Application of such LMI-based techniques to problem (8)leads to Theorem 2, whose statement is reported below.For a given integer δ such that 2δ ≥max{2, deg(g1), . . . , deg(gng)}, let us consider theconvex SDP problem

fsosδ = min

γ∈R

γ

s.t.

γ − ‖x‖22 = σ0 +

ng∑j=1

σjgj(a,b,p) +m∑

i=1

λiS̃ix,

for some σ0, σj ∈ Σ[x,a,b,p], λi ∈ R[x,a,b,p]

deg(σ0), deg(σjgj), deg(λiS̃ix) ≤ 2δ

(10)

where S̃i is the i-th row of Sylvester matrix S̃; R[x,a,b,p]is the ring of polynomials in the variables (x,a,b,p) andΣ[x,a,b,p] is the set of SOS polynomials in the samevariables.

Theorem 2: If fsosδ is bounded for some δ s.t. 2δ ≥

max{2, deg(g1), . . . , deg(gng)}, then the matrix S̃ is non-singular on C.

Proof: Let us rewrite the polynomial optimizationproblem (8) as

f∗ = minγ∈R

γ

s.t.

γ − ‖x‖22 ≥ 0, ∀x, S̃ s.t. S̃x = 0; S̃ ∈ S(a,b).

(11)

Indeed, γ −‖x‖22 = σ0 +

∑ngj=1 σjgj(a,b,p)+

∑mi=1 λiS̃ix

is nonnegative ∀x, S̃ s.t. S̃x = 0; S̃ ∈ S(a,b). Infact, when S̃x = 0 and S̃ ∈ S(a,b), γ − ‖x‖2

2 = σ0 +∑ngj=1 σjgj(a,b,p) +

∑mi=1 λiS̃ix ≥ 0 since S̃ix = 0;

gj(a,b,p) ≥ 0 and σ0, σj , with j = 1, . . . , ng, are sum-of-square polynomials, hence nonnegative. Then, fsos

δ is an

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upper bound of f∗, i.e. fsosδ ≥ f∗. Therefore, when fsos

δ

is bounded, f∗ is bounded as well, and from Theorem 1,uncertain Sylvester matrix S̃ is robustly nonsingular on C.

Remark 2: Alternatively, from the second part of The-orem 1, robust nonsingularity of matrix S̃ on C can bechecked by evaluating emptiness of the set F . On the basisof Positivstellensatz, emptiness of a semialgebraic set canbe efficiently evaluated through semidefinite programming.More precisely, F is empty if there exists a solution to thefollowing SDP feasibility problem for some integer δ:

σ0+ng∑

j=1

σjgj(a,b,p)+σng+1

(‖x‖22 − ε

)+

m∑i=1

λiS̃ix=0

for some σ0, σ1, . . . , σng+1 ∈ Σ[x,a,b,p];λi ∈ R[x,a,b,p]

deg(σ0), deg(σjgj), deg(σng+1‖x‖22), deg(λiS̃ix) ≤ 2δ.

(12)

Example 1: Let us consider the interval Sylvestermatrix associated with the uncertain polynomialsNp(s) = b2s

2 + b1s+ b0 and Dp(s) = s3 + a2s2 + a1s+ a0

with interval coefficients b2 ∈ [3 − Δ; 3 + Δ]; b1 ∈[8 − Δ; 8 + Δ]; b0 ∈ [−5 − Δ;−5 + Δ]; a2 ∈[4−Δ; 4+Δ]; a1 ∈ [3−Δ; 3+Δ]; a0 ∈ [−6−Δ;−6+Δ],where Δ ≥ 0 denotes the radius of the coefficient uncertaintyintervals. In this example, different values Δ are considered.In particular, the largest value of Δ such that polynomialsNp(s) and Dp(s) are guaranteed to be robustly coprime isequal to 0.208 when the algorithm in [24] is used to evaluaterobust nonsingularity of the associated interval Sylvestermatrix, while it is equal to 0.201 when the method in [22] isused. On the other hand, by using results in Theorem 2, weget that polynomials Np(s) and Dp(s) are guaranteed to berobustly coprime for Δ ≤ 0.420. It is worth remarking that,although Theorem 2 only provides a sufficient conditionfor checking robust nonsingularity of Sylvester matrixes,Δ = 0.420 is a good approximation of the exact radius ofrobust coprimeness of the considered polynomials, in fact,for instance, polynomials Np(s) and Dp(s) with coefficientsb2 = 3.4114, b1 = 7.6088 b0 = −4.6509, a2 = 4.4206,a1 = 2.5728 and a0 = −5.5760, belonging to uncertaintyintervals with Δ = 0.4273, are not coprime since they havea common root equal to −2.7298.

IV. DESIGN OF ROBUST CONTROLLER

In this section we show how to design the controllerparameters k in order to solve the robust pole placementformulated in Problem 1. Let us rewrite the Diophantineequation in (5) in the matrix form:

S̃k = f , (13)

where f = [fn+r, fn+r−1, . . . , f0] is the vector ofthe coefficients of the closed-loop characteristic polynomialF(s,p,k). The robust pole placement problem can be refor-mulated in terms of the following robust feasibility problem.

Problem 2: Find controller parameters k ∈ R2r+2 such

that {S̃ik − f

i> 0

f i − S̃ik > 0∀p ∈ P; i = 0, . . . , n + r, (14)

or equivalently{S̃ik − f

i> 0

f i − S̃ik > 0∀a,b ∈ C; i = 0, . . . , n + r. (15)

Note that (15) is a robust polynomial feasibility problembecause of the product between the controller parameters kand the uncertain plant coefficients a and b, and becausea and b belong to the semialgebraic region C. By usingPositivstellensatz, Problem 2 can be solved through SDPoptimization, as shown instated by the following theorem.

Theorem 3: For a given integer δ such that 2δ ≥max{2, deg(g1), . . . , deg(gng)}, let us consider the follow-ing SDP feasibility problem where k ∈ R

2r+2 is the designparameter:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

S̃ik−fi=σ0,i(a,b,p)+

ng∑j=1

σj,i(a,b,p)gj(a,b,p)+ε

f i−S̃ik=χ0,i(a,b,p)+ng∑

j=1

χj,i(a,b,p)gj(a,b,p)+ε

for some σ0,i, σ1,i, . . . , σng,i ∈ Σ[k,a,b,p];for some χ0,h, χ1,h, . . . , χng,h ∈ Σ[k,a,b,p];for some ε ∈ R : ε > 0;deg(σ0,i), deg(σ1,igj), . . . , deg(σng,igng) ≤ 2δdeg(χ0,i), deg(χ1,igj), . . . , deg(χng,igng) ≤ 2δi = 0, . . . , r + n.

(16)Let us denote with k∗

δ a feasible solution for the convexSDP problem (16); then k∗

δ belongs to set of admissiblecontroller parameters given by (15).

Proof: Theorem 3 can be proven on the basis ofsimilar reasonings used in the proof of Theorem 2. In fact,if polynomials S̃ik − f

iand f i − S̃ik can be written as⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

S̃ik−fi=σ0,i(a,b,p)+

ng∑j=1

σj,i(a,b,p)gj(a,b,p)+ε

f i−S̃ik=χ0,i(a,b,p)+ng∑

j=1

χj,i(a,b,p)gj(a,b,p)+ε

(17)for some SOS polynomials σj,i and χj,i, with j = 0, . . . , ng,and positive real number ε, then S̃ik− f

iand f i − S̃ik are

positive for all values of a and b belonging to C. Therefore,if parameters k∗

δ satisfy problem (16), then k∗δ is feasible for

problem (15).

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Remark 3: Since C is a bounded set, on the basis ofPutinar’s Positivstellensatz, if polynomials S̃ik − f

iand

f i − S̃ik are positive on C, then they can be always writtenas⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

S̃ik−fi=σ0,i(a,b,p)+

ng∑j=1

σj,i(a,b,p)gj(a,b,p)+ε

f i−S̃ik=χ0,i(a,b,p)+ng∑

j=1

χj,i(a,b,p)gj(a,b,p)+ε

for some σ0,i, σ1,i, . . . , σng,i ∈ Σ[k,a,b,p];for some χ0,i, χ1,i, . . . , χng,i ∈ Σ[k,a,b,p];for some ε ∈ R : ε > 0.

(18)This means that, whenever a solution of problem (15)exists, then the convex SDP problem (16) is guaranteed tobe feasible for δ large enough. Nevertheless, in practice,whenever problem (15) is feasible, then the SDP problem(16) is feasible also for small values of δ. �

Example 2: Let us consider the second-order plant withtransfer function

G(s, p) =Np(s,p)Dp(s,p)

=p1s + p2

s2 + p3p4s + p4, (19)

where the uncertain parameters p1 and p2 belong to thesemialgebraic region P12:

P12 ={(p1, p2) ∈ R

2 : (p1 − 1.8)2 + (p2 − 10)2 ≤ 0.25;

p2 [1 + 4 (p1 − 1.8)] ≤ 34− 39

4[1 + 4 (p1 − 1.8)] ,

p2 [1 + 4 (p1 − 1.8)] ≥ −34− 39

4[1 + 4 (p1 − 1.8)]

}.

(20)

while parameters p3 and p4 lie in the semialgebraic set P34

defined as:

P34 ={(p3, p4) ∈ R

2 : p3 ≥ −9.8; p4 ≤ 0.4;

p4 ≥ 45

(p3 + 9.3)3 + 0.3}

.(21)

Uncertainty sets P12 and P34 are shown, respectively, inFig. 2 and in Fig. 3. Note that all systems belonging to theconsidered family of uncertain plants are not stable, sincethe coefficient p3p4 of the second term of the denominatorDp(s,p) is negative for all pair of parameters (p3, p4) ∈ P34.First, robust coprimeness of the uncertain polynomialsNp(s,p) and Dp(s,p) describing the plant is checkedthrough the algorithm presented in Section III. Then, afeedback controller is designed in order to obtain the closed-loop characteristic polynomial:

F(s) = s3 + f2s2 + f1s + f0, (22)

with coefficients in the open intervals: f2 ∈ (6; 12), f1 ∈(19; 35) and f0 ∈ (25; 29). The spectral set of the familyof desired characteristic polynomials is shown in Fig. 4.The set of admissible controllers is obtained by solving the

1.2 1.4 1.6 1.8 2 2.2 2.49.4

9.6

9.8

10

10.2

10.4

10.6

parameter p1

para

met

er p

2

Fig. 2. Uncertainty set P12 (region inside black line) on parameters p1

and p2.

−9.8 −9.6 −9.4 −9.2 −9 −8.80.15

0.2

0.25

0.3

0.35

0.4

0.45

parameter p3

para

met

er p

4

Fig. 3. Uncertainty set P34 (region inside black line) on parameters p3

and p4.

convex SDP problem (16) for δ = 3. For instance, a feasiblecontroller C(s) satisfying the assigned specifications is givenby:

C(s) =3.684s + 2.544

s + 5.186. (23)

V. CONCLUDING REMARKS

Synthesis of feedback robust controllers for linear-time-invariant systems with uncertain parameters belonging tosemialgebraic regions is addressed in the paper. UncertainDiophantine equation is formulated in order to constrainthe coefficients of the closed-loop characteristic polynomialto lie within given intervals. Two main topics related toDiophantine equation are discussed. First, a new approachto evaluate robust nonsingularity of the uncertain Sylvestermatrix associated with the numerator and denominator of

Page 6: Robust Pole Placement for Plants with Semialgebraic ...

−12 −10 −8 −6 −4 −2 0−5

0

5

Real axis

Imag

inar

y ax

is

Fig. 4. Spectral set of the family of desired characteristic polynomials.

the transfer function describing the plant is presented. Theproblem of robust nonsingularity of the Silvester matrixis formulated as a nonconvex polynomial optimizationproblem, whose approximate solution is efficiently computedby means of sum-of-square convex-relaxation techniques.The discussed approach is able to deal with matrixes withuncertain entries belonging to a semialgebraic region and,in the case of interval uncertainty in the coefficients ofthe Sylvester matrix, the presented method provides lessconservative conditions on the nonsingularity of Sylvestermatrix with respect to previously published results. Finally,the design of a robust controller is reduced to a robustpolynomial feasibility problem, which is efficiently solvedthrough sum-of-square decomposition techniques similar tothe ones exploited to check robust coprimeness of uncertainSylvester matrixes. The reported simulated examples showthe effectiveness of the presented results.

VI. ACKNOWLEDGMENTS

This research was developed while Dr. D. Piga was Ph.Dstudent at the Politecnico di Torino.

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