Synthesis of restricted complexity controllers for l2 parametric ...

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Synthesis of restricted complexity controllers for l2 parametric stability margin maximization Gianni Bianchini† , Paola Falugi‡ , Alberto Tesi‡ , and Antonio Vicino† †

Dipartimento di Ingegneria dell’Informazione, Universit`a di Siena, Via Roma 56, 53100 Siena, Italy. Tel. +39 0577 234644 - Fax. +39 0577 233609 Email: [email protected], [email protected] ‡ Dipartimento di Sistemi e Informatica, Universit`a di Firenze, Via di S. Marta 3, 50139 Firenze, Italy. Tel. +39 055 4796263 - Fax. +39 055 4796363 Email: [email protected], [email protected]

Abstract In this paper the problem of restricted complexity stability margin maximization (RCSMM) for single-input single-output (SISO) plants affected by rank one real perturbations is considered. This problem amounts to maximizing the l 2 parametric stability margin over an assigned class of restricted complexity controllers, which are described by rational transfer functions of fixed order with coefficients depending affinely on some free parameters. It is shown that the RCSMM problem, which in general admits local maxima, can be approached by means of convex optimization methods. Specifically, a lower bound of the stability margin, whose maximization can be accomplished via Linear Matrix Inequality (LMI) techniques, is developed and employed to devise an iterative procedure. Several examples are presented to illustrate the main features of the proposed approach.

I. I NTRODUCTION The parametric stability margin is a typical measure of the robustness of feedback control systems subject to parametric plant uncertainty. In particular, the stability margin maximization (SMM) problem, i.e., the design of controllers maximizing the parametric stability margin, has received considerable attention in the literature (see [1] for a comprehensive reference on this problem). The most general result is given in [2] and it concerns the class of uncertain systems with rank one real perturbations, which includes single-input single-output (SISO) systems with transfer function coefficients depending affinely on the uncertain parameters. In that paper, Rantzer and Megretski showed that the optimization problem can be made convex provided that a suitable controller parameterization is employed. Despite this remarkable result, some issues still deserve to be investigated in order to devise satisfactory design techniques. In particular, since the optimization problem in [2] is infinite dimensional, suboptimal solutions must be looked for through finite dimensional convex programming [3],[4]. Hence, a suitable structure for the approximating solution must be found in order to tackle the computational burden and the complexity of the resulting controller [2],[5]. It is also important to recall that in the majority of practical applications it is often mandatory to employ controllers with a prescribed structure, such as PID or lead-lag compensators [6]. In this respect, it is worth to remark that a renewed interest in the synthesis of low-order controllers, mainly based on robust parametric control techniques, has been observed recently [7],[8],[9],[10]. In particular, the set of stabilizing PID controllers for a given plant is characterized in [7], while [8] considers an additional H∞ constraint and [9] proposes a more general approach, based on simple geometrical considerations, to cover also the case of lead-lag compensators. An LMI-based method for the design of fixed order controllers based on an inner approximation of the stability domain of polynomials is proposed in [10]. The above considerations naturally lead to the introduction of the restricted complexity stability margin maximization (RCSMM) problem, which amounts to maximizing the parametric stability margin of a family of SISO uncertain plants with rank one real perturbations over a given class of controllers. Such a class of restricted complexity controllers is characterized by fixed order rational transfer functions with coefficients depending affinely on some free design parameters. Many widely used controller structures indeed lie in this class. Despite its simple formulation, the RCSMM problem displays several difficulties. In particular, the solution of this problem implicitly calls for an efficient characterization of the restricted complexity controllers which stabilize the nominal plant of the uncertain family. This is a very difficult task and few results have been obtained so far (see [7],[11] for the PID case). It is also to remark that in general the RCSMM problem cannot be solved directly by means of convex optimization techniques because of the restricted complexity condition imposed on the controller class. In this paper, a comprehensive analysis of the RCSMM problem, which extends in several respects the preliminary ones reported in [12],[13],[14], is provided in order to relax the above difficulties. A new characterization of the l2 parametric stability margin pertaining to a given stabilizing controller of the restricted complexity class is given. More specifically, on the basis of previous

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results on the design of filters ensuring robust strict positive realness for affine families of polynomials [15], a strictly positive real (SPR) rational function is associated to each stabilizing controller, thus providing an implicit parameterization of the stabilizing controllers. Moreover, such a characterization makes it possible to compute a suitable lower bound of the stability margin pertaining to each stabilizing controller. In particular, the maximization of this lower bound, which provides a locally optimal approximation of the stability margin in some neighborhood of the given stabilizing controller, amounts to solving a Generalized Eigenvalue Problem (GEVP) [16]. can be accomplished via standard LMI techniques. Finally, it is shown how an LMI-based iterative procedure, which requires at each step the computation of a SPR rational function via a polynomial factorization and to solving an LMI feasibility problem with respect to the free controller parameters, can be devised for the RCSMM problem. The paper is organized as follows. The RCSMM problem is introduced in Section 2. Section 3 collects the main results of the paper, namely, the new characterization of the l2 parametric stability margin, the LMI-based lower bound and the iterative algorithm for the RCSMM problem. Some numerical examples are developed in Section 4. Section 5 contains some concluding remarks. Notation Rn : real n-space; v ∈ Rn : vector of Rn ; v 0 : transpose of v; kvk2 : l2 norm of v; A ∈ Rn×m : real n × m matrix; C: the complex plane; s ∈ C: complex number; Re[s], Im[s]: real and imaginary parts of s; π(s): a polynomial in the complex variable s; ∂π: ¯ (s): rational function vector/matrix. degree of a polynomial π(s); P (s): rational function of s; M II. P ROBLEM FORMULATION AND PRELIMINARIES A. Problem formulation In this paper we consider the following class of uncertain SISO plants  ¯ B0 (s) + δ 0 B(s)   ; δ ∈ Rn P (s; δ) = ¯ A0 (s) + δ 0 A(s) P=   ¯ ¯ B(s) = [B1 (s) . . . Bn (s)]0 ; A(s) = [A1 (s) . . . An (s)]0

  

(1)

 

where δ = [δ1 . . . δn ]0 is the uncertain parameter vector and B0 (s), B1 (s), . . . , Bn (s), A0 (s), A1 (s), . . . , An (s) are given polynomials. In the sequel the following simplifying assumption on the degree of the polynomials involved in (1) will be enforced. Assumption 1: (Plant degree constraint) ∂B0 < ∂A0 ; ∂Ai < ∂A0 , ∂Bi < ∂B0 , i = 1, . . . , n. (2) The above assumption ensures that P (s; δ) is strictly proper and its order is invariant with respect to δ. Throughout the paper we denote by Pδ ∈ P the plant with transfer function P (s; δ). In particular, P0 will be referred to as the nominal plant of the family. We denote by S0 the class of controllers which stabilize P0 . Consider the feedback interconnection of the uncertain plant Pδ and a controller C ∈ S0 with transfer function C(s) =

N (s) D(s)

(3)

where N (s) and D(s) are given polynomials. A standard representation of the resulting feedback control loop can be obtained via a linear fractional transformation (LFT) in the form reported in Fig. 1, where PSfrag replacements C(s) ¯ (s) M u y

Fig. 1.

w∈R

δ

0

z ∈ Rn

¯ G(s)

Closed loop LF T representation.

¯ G(s) =−

1 ¯ + N (s)B(s)]. ¯ [D(s)A(s) D(s)A0 (s) + N (s)B0 (s)

(4)

Since the perturbation δ is a vector (i.e., w is a scalar), this representation is usually said to be rank one. A well-known measure of the robustness of this feedback control system is the l 2 parametric stability margin, whose definition is recalled next [1].

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Definition 1: Given the uncertain plant class P and a controller C ∈ S0 , the l2 parametric stability margin ρC pertaining to such controller is defined as the maximal ρ such that the closed loop is stable for all kδk2 < ρ. We have the following well-known expression for ρC . Lemma 1: Let P and C ∈ S0 be given. Then, ρC = sup ρ s.t. (5) ¯ 1 − δ 0 G(jω) 6= 0, ∀ω ≥ 0 ∀δ : kδk2 < ρ. One of the most investigated robust control problems is the parametric stability margin maximization (SMM) which amounts to finding the controller which maximizes ρC over the class of stabilizing controllers S0 [1],[2]. Despite the fundamental results given in [2], where the SMM problem is shown to be convex with respect to a suitable parameterization of robustly stabilizing controllers, the actual computation of the optimal controller shows practical difficulties, especially when constraints on the controller structure are imposed, a very common situation in practical applications. Motivated by the above consideration, in this paper we consider the SMM problem when the controllers are assumed to have a given structure. In particular, we introduce the following class of restricted complexity controllers:   ¯ (s) N (s; ϑ) N0 (s) + ϑ0 N     C(s; ϑ) = = : ϑ ∈ Θ 0 D(s) ¯ D(s; ϑ) D (s) + ϑ 0 (6) C=    ¯ 0 0  ¯ N (s) = [N1 (s) . . . Nm (s)] ; D(s) = [D1 (s) . . . Dm (s)] where • N0 (s), N1 (s), . . . , Nm (s), D0 (s), D1 (s) . . . Dm (s) are given polynomials, m 0 • ϑ = [ϑ1 . . . ϑm ] ∈ R is a free controller parameter vector, m • Θ⊆R is of the form Θ = {ϑ ∈ Rm : Qj (ϑ) > 0 ; j = 1, . . . , k}

(7)

being Qj (ϑ), j = 1, . . . , k symmetric matrices whose entries depend affinely on ϑ. Some remarks pertaining to C will be reported in Subsection II-B. Here, we only note that the choice of the structure of the admissible parameter set Θ is motivated by the fact that such structures enter naturally into LMI problems. Throughout the paper we refer to Cϑ as the controller with transfer function C(s; ϑ). In particular, without loss of generality, C0 is assumed to be a stabilizing controller as stated next. Assumption 2: C0 stabilizes the nominal plant P0 , i.e., C0 ∈ S0 . Also, to avoid unnecessary technicalities, the following condition on the degrees of the polynomials involved in (6) is enforced. Assumption 3: (Controller degree constraint) ∂N0 ≤ ∂D0 , ∂Ni ≤ ∂N0 , ∂Di < ∂D0 , i = 1, . . . , m. (8) The above assumption ensures that C(s; ϑ) is proper and that its order is invariant with respect to ϑ. The problem of stability margin maximization over the class of restricted complexity controllers C can be stated as follows, where ρ(ϑ) denotes the parametric stability margin achieved by Cϑ according to Definition 1, i.e., ρ(ϑ) = ρCϑ . RCSMM problem. Given the uncertain plant family P and the class of restricted complexity controllers C, find a parameter vector ϑ∗ ∈ Θ such that the controller Cϑ∗ ∈ C achieves the maximum of the closed loop stability margin over the class C, i.e., (9) ρCϑ∗ = sup ρ(ϑ). ϑ∈Θ

B. Remarks on the class of restricted complexity controllers The choice of the controller class C in (6) is motivated mainly by practical reasons, since many widely used structures can be seen as members of such a class. For example, consider the case of PI compensators, i.e., ½ ¾ KI C(s) = KP + , KP ∈ R, KI ∈ R . (10) s Suppose that there exist constants KP0 and KI0 such that the related PI controller KI0 s stabilizes the nominal plant. Then, (10) can be rewritten in the form (6) once ¯ (s) = [1 s]0 ; D(s) ¯ = [0 0] N0 (s) = KI0 + KP0 s ; D0 (s) = s ; N ϑ = [KI − KI0 KP − KP0 ]0 ; Θ = R2 . C0 (s) = KP0 +

(11)

(12)

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Note that in (10) the gains KP and KI are supposed to have any real value. Assume instead that KP and KI are restricted to be positive. It is easily checked that in this case it is enough to replace Θ in (12) with ½ · ¸ ¾ ϑ1 + KI0 0 Θ = ϑ ∈ R2 : Q1 (ϑ) = > 0 . (13) 0 ϑ2 + KP0 An interesting property of the class C concerns the strict connection with the suboptimal finite dimensional solutions to the SMM problem derived in [2]. Exploiting a Ritz series expansion technique, the optimal controller is approximated via rational transfer functions of the form PN PN X(s)[1 + i=1 θi Ri (s; p)] − A(s) i=0 θN +1+i Ri (s; p) C(s; θ; N, p) = (14) PN PN Y (s)[1 + i=1 θi Ri (s; p)] + B(s) i=0 θN +1+i Ri (s; p)

where X(s), Y (s), A(s), and B(s) are given rational functions depending on the plant class, θ ∈ R 2N +1 is the tunable parameter vector, and Ri (s; p) is the i−th Ritz term given by ¶i µ s−p Ri (s; p) = , p > 0. s+p Indeed, it is straightforward to verify that controllers of the form (14) can be seen as members of the class C for a suitable ¯ (s), D(s). ¯ choice of the polynomials N0 (s), D0 (s), N III. M AIN RESULTS

Several difficulties arise in solving the RCSMM problem. In particular, in view of Definition 1, it is to observe that ρ(ϑ) is well-defined only for ϑ belonging to the set Θstab ∩ Θ where Θstab = {ϑ ∈ Rm : Cϑ ∈ S0 } .

(15)

Therefore, solving (9) implicitly calls for an efficient characterization of Θstab . Unfortunately, this is a very difficult task and few results have been presented so far in the literature (see, e.g., [7],[11]). It is also to observe that the RCSMM problem cannot be solved in general via convex techniques because of the restricted complexity condition imposed on the controllers. Indeed, it is not difficult to single out examples which display local maxima. In this paper, a new approach to the RCSMM problem is developed in order to relax the above difficulties. First, in Subsection III-A a new characterization of the parametric stability margin ρ(ϑ) pertaining to C ϑ , ϑ ∈ Θstab is developed. More specifically, exploiting earlier results on Strict Positive Realness (SPR) [15], a suitable SPR rational function is associated to each C ϑ , ϑ ∈ Θstab , thus providing an implicit parameterization of the set Θstab . Subsequently, on the basis of such characterization, a lower bound of ρ(ϑ) is developed in Subection III-B. In particular, this bound provides a locally optimal approximation of ρ(ϑ) in some neighborhood of a given controller parameter vector, and its maximization can be accomplished via LMI techniques. Finally, in Subsection III-C, it is shown how such lower bound can be employed to devise an LMI-based algorithm for the RCSMM problem. A. Characterization of the l2 parametric stability margin ¯ The stability margin ρ(ϑ) pertaining to Cϑ , ϑ ∈ Θstab , can be computed in accordance with Lemma 1 once G(s) is replaced with 1 ¯ + N (s; ϑ)B(s)]. ¯ ¯ ϑ) = − [D(s; ϑ)A(s) (16) G(s; D(s; ϑ)A0 (s) + N (s; ϑ)B0 (s) However, we look for a special characterization of ρ(ϑ) which is obtained by associating to C ϑ a suitable SPR rational function. ¯ ϑ) as To proceed, let us rewrite G(s; · ¸0 π1 (s; ϑ) πn (s; ϑ) ¯ G(s; ϑ) = − ,..., (17) π0 (s; ϑ) π0 (s; ϑ) where, according to the expressions in (6), the polynomials πi (s; ϑ) are given by ¯ ¯ πi (s; ϑ) = D0 (s)Ai (s) + N0 (s)Bi (s) + ϑ0 [D(s)A i (s) + N (s)Bi (s)] ; i = 0, . . . , n.

(18)

Note that, since Cϑ is assumed to stabilize the nominal plant, the polynomial π0 (s; ϑ) is Hurwitz. From (17) it follows that the characteristic polynomial of the closed loop system has the expression π(s; δ; ϑ) = π0 (s; ϑ) +

n X i=1

δi πi (s; ϑ).

(19)

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It is not difficult to check that Assumption 1 and Assumption 3 ensure that the degree of the characteristic polynomial is invariant with respect to δ and ϑ, specifically ∂π = ∂D0 + ∂A0 . Let us now introduce the two functions ¯ ¯ ¯ ϑ) = Im[G(jω; ¯ R(ω; ϑ) = Re[G(jω; ϑ)] ; I(ω; ϑ)],

(20)

© ª ¯ ϑ) = 0 , Ωϑ = ω ≥ 0 : I(ω;

(21)

the related set and the polynomial Π(s; ϑ) =

n X i=1

π0 (s; ϑ)πi (−s; ϑ) [π0 (−s; ϑ)πi (s; ϑ) − π0 (s; ϑ)πi (−s; ϑ)] .

(22)

It can be easily verified that for each ϑ, any nonzero frequency ω ˆ ∈ Ωϑ must be a common root of n polynomials in ω, which implies that the existence of such ω ˆ is not generic especially for large n (see also Remark 3). Therefore, the assumption Ωϑ = {0} will be enforced in the sequel. Under this assumption, it can be easily shown that the following factorization holds for Π(s; ϑ): Π(s; ϑ) = αϑ sqϑ Π1 (s; ϑ)Π2 (−s; ϑ) (23) where αϑ is a real constant, qϑ ≥ 1 is an integer and Π1 (s; ϑ), Π2 (s; ϑ) are uniquely determined monic Hurwitz polynomials. The following result provides the sought characterization of ρ(ϑ). Theorem 1: Let ϑ ∈ Θstab and suppose Ωϑ = {0}. Introduce the rational function  Π1 (s; ϑ)   (1 + τ s)∂Π2 −∂Π1 for even qϑ    Π2 (s; ϑ)   (24) Φ(s; ϑ; ε, τ ) = Π1 (s; ϑ)  ξ1  · (s + ε)   for odd qϑ Π2 (s; ϑ)    ·(1 + τ s)ξ2 where Π1 (s; ϑ), Π2 (s; ϑ), αϑ , qϑ are as in (23), ε and τ are positive scalars, and ξ1 = sgn[αϑ (−1)(qϑ −1)/2 ] ξ2 = ∂Π2 − ∂Π1 − sgn[αϑ (−1)(qϑ −1)/2 ]. Then, for all ρ such that 0 < ρ < ρ(ϑ), there exist ε > 0 and τ > 0 such that 1) Φ(s; ϑ; ε, τ ) is SPR 2) The following inequality holds ρ < ρΦ (ϑ; ε, τ ) ≤ ρ(ϑ)

(25)

where ρΦ (ϑ; ε, τ ) = inf rΦ (ω; ϑ; ε, τ ) ω≥0

being rΦ (ω; ϑ; ε, τ ) = and

1 ¯ ¯ ϑ)k2 kR(ω; ϑ) − γΦ (ω; ϑ; ε, τ )I(ω;

(26)

(27)

Im[Φ(jω; ϑ; ε, τ )] . (28) Re[Φ(jω; ϑ; ε, τ )] Proof: Condition 1. follows from [15] where it is shown (see proof of Theorem 2) that Φ(s; ϑ; ε, τ ) is SPR for sufficiently small ε and τ . More specifically, it is shown that Φ(s; ϑ; 0, 0) is positive real and ε and τ are introduced to ensure that Φ(s; ϑ; ε, τ ) is SPR. In particular, ε takes into account the presence of a singularity at s = 0 of Π(s; ϑ) in (23) when q ϑ is odd, while τ is for making Φ(s; ϑ; ε, τ ) biproper. Let us now consider condition 2. For any ϑ ∈ Θstab , it can be shown (see Lemma 5, Lemma 6, and eq. (23) in [15]) that ¯ 0 (ω; ϑ)I(ω; ¯ ϑ) R γΦ (ω; ϑ; 0, 0) = ∀ω ∈ / Ωϑ . (29) ¯ kI(ω; ϑ)k2 γΦ (ω; ϑ; ε, τ ) =

2

Hence (see Lemma 2 in [15] and (27)), under the assumption Ωϑ = {0}, and taking into account that the degree of the closed loop characteristic polynomial is invariant with respect to δ, the parametric stability margin ρ(ϑ) is given by ρ(ϑ) = min{ρ0 (ϑ), ρˆ(ϑ)}

(30)

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where ρ0 (ϑ) = and

1 ¯ ϑ)k2 kR(0;

(31)

ρˆ(ϑ) = inf rΦ (ω; ϑ; 0, 0).

(32)

ω>0

¯ ¯ ϑ)k2 for γ ∈ R. It reduces to kR(0; ¯ ϑ)k2 for ω = 0, while for any ω > 0 its Let us consider the expression kR(ω; ϑ) − γ I(ω; minimum with respect to γ is achieved for γ = γΦ (ω; ϑ; 0, 0). According to (27), this implies rΦ (0; ϑ; ε, τ ) = ρ0 (ϑ)

∀ε ≥ 0, τ ≥ 0

(33)

∀ω > 0, ε > 0, τ > 0

(34)

and rΦ (ω; ϑ; ε, τ ) ≤ rΦ (ω; ϑ; 0, 0)

and hence the second inequality in (25) is proven. To complete the proof, we need to show that for any ρ < ρ(ϑ) there exist ε, τ > 0 such that ρ < inf rΦ (ω; ϑ; ε, τ )

(35)

ω≥0

which, taking (33) into account, reduces to ρ < rΦ (ω; ϑ; ε, τ )

∀ω > 0.

(36)

Note that there exist εˆ > 0 and τˆ > 0 such that Φ(s; ϑ; ε, τ ) is a SPR rational function with relative degree zero for all 0 < ε ≤ εˆ, 0 < τ ≤ τˆ and hence γΦ (ω; ϑ; ε, τ ) satisfies lim γΦ (ω; ϑ; ε, τ ) = 0

ω→0

lim γΦ (ω; ϑ; ε, τ ) = 0

ω→+∞

¯ ϑ) is strictly proper and hence for all 0 < ε ≤ εˆ, 0 < τ ≤ τˆ. Moreover, G(s; ¯ ¯ ϑ) = 0. lim R(ω; ϑ) = 0, lim I(ω; ω→+∞

ω→+∞

Therefore, for any given ρ < ρ(ϑ) there exist 0 < ω < ω and arbitrarily small ε¯ > 0, τ¯ > 0 such that 1 ρ< ¯ ¯ ϑ)k2 = rΦ (ω; ϑ; ε¯, τ¯) ∀ω ≥ ω kR(ω; ϑ) − γΦ (ω; ϑ; ε¯, τ¯)I(ω; and 1 ρ< ¯ ¯ ϑ)k2 = rΦ (ω; ϑ; ε¯, τ¯) ∀ 0 < ω ≤ ω, kR(ω; ϑ) − γΦ (ω; ϑ; ε¯, τ¯)I(ω; where the latter also follows from the fact that ¯ ¯ ϑ)k2 = 1 ≤ 1 . lim kR(ω; ϑ)k2 = kR(0; ω→0 ρ0 (ϑ) ρ

(37)

(38)

Now, it is easily shown that γΦ (ω; ϑ; ε, τ ) converges uniformly to γΦ (ω; ϑ; 0, 0) for (ε, τ ) → (0, 0) in any compact subset of ω > 0. Then, for any ρ < ρ(ϑ) there exist ε˜, τ˜ > 0 such that ρ≤

inf

ω∈[ω,ω]

rΦ (ω; ϑ; ε, τ ) ≤

inf

ω∈[ω,ω]

rΦ (ω; ϑ; 0, 0)

∀0 ≤ ε ≤ ε˜, 0 ≤ τ ≤ τ˜.

(39)

Finally, from the observation that ε¯ and τ¯ are arbitrarily small, it follows that ε < min{¯ ε, εˆ, ε˜} and τ < min{¯ τ , τˆ, τ˜} exist such that (36) holds, thus concluding the proof. The above Theorem provides a new characterization of the l2 parametric stability margin ρ(ϑ) pertaining to a controller Cϑ , ϑ ∈ Θstab . The key element is the SPR rational function Φ(s; ϑ; ε, τ ), which is computed via the polynomial factorization (23) and the appropriate selection of the parameters ε and τ . In this respect, some remarks are in order. Remark 1: The proof of Theorem 1 makes it clear that, for sufficiently small ε, τ , Φ(s; ϑ; ε, τ ) is SPR and ρΦ (ϑ; ε, τ ) is a quite good approximation of ρ(ϑ). By taking into account the role of ε (Π(s; ϑ) has a singularity at s = 0) and τ (Φ(s; ϑ; ε, 0) may not be biproper), it turns out that ε (resp. 1/τ ) should be chosen at least one decade smaller (resp. larger) than all the singularities of Π1 (s; ϑ) and Π2 (s; ϑ). Remark 2: It is not difficult to show that the order of Φ(s; ϑ; ε, τ ) is related to that of the plant class P in (1) and the controller class C in (6). Indeed, exploiting (18), (19), and (23), it turns out that the order of Φ(s; ϑ; ε, τ ) is no larger than 2(∂A0 + ∂D0 ) (see [15]). Remark 3: An expression for Φ(s; ϑ; ε, τ ) having the properties stated in Theorem 1 can be derived also in the non-generic case of Ωϑ containing frequencies other than ω = 0 (see [15]). It is well known that in this case the function ρˆ(ϑ) in (30) may be discontinuous (see, e.g., [1],[17]). On the other hand, exploiting the link between ρˆ(ϑ) and the structured singular value function µ, it turns out that such a discontinuity does not affect the RCSMM problem (9) [18],[17].

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B. An LMI lower bound to the l2 parametric stability margin The characterization in Subsection III-A provides an implicit parameterization of Θ stab . However, the maximization of the approximating quantity ρΦ (ϑ; ε, τ ) with respect to ϑ ∈ Θstab ∩ Θ cannot be solved by convex optimization methods directly. In the sequel, a suitable lower bound of ρ(ϑ), whose maximization can be carried out by means of a standard GEVP, is developed. To proceed, let θ ∈ Θstab be given, and consider the corresponding closed loop nominal characteristic polynomial π 0 (s; θ) in (19). Let Φ(s; θ; ε, τ ) be computed as in Theorem 1 for some positive ε, τ and consider the rational function Ψ(s; ϑ; θ; ε, τ ) = Φ(s; θ; ε, τ )

π0 (s; ϑ) π0 (s; θ)

(40)

which depends affinely on ϑ and obviously satisfies Ψ(s; θ; θ; ε, τ ) = Φ(s; θ; ε, τ ). Moreover, since Φ(s; θ; ε, τ ) is SPR, it follows that also Ψ(s; ϑ; θ; ε, τ ) is SPR for all ϑ in some neighborhood of θ. In this respect, we have the following result. Lemma 2: Let Θθ = {ϑ ∈ Θ : Ψ(s; ϑ; θ; ε, τ ) is SPR }. Then, Θθ ⊆ Θstab . Proof: We first observe that, since Ψ(s; ϑ; θ; ε, τ ) depends affinely on ϑ, Θθ is a convex set. Since Φ(s; θ; ε, τ ) is SPR, from (40) we have that for any ϑ ∈ Θθ the following inequality holds |arg[π0 (jω; ϑ)] − arg[π0 (jω; θ)]| = |arg[Ψ(jω; ϑ; θ; ε, τ )] − arg[Φ(jω; θ; ε, τ )]| < π,

∀ ω ≥ 0.

(41)

Recalling that the degree of π0 (s; θ)is invariant with respect to ϑ, from the Bounded Phase Lemma [1] we have that all the polynomials of the following one-parameter family {(1 − λ)π0 (s; θ) + λπ0 (s; ϑ), λ ∈ [0, 1]} have the same stability properties. Hence, the proof follows for Hurwitzness of π0 (s; θ) and convexity of Θθ . The above lemma ensures that, for any ϑ ∈ Θθ , the rational function γΨ (ω; ϑ; θ; ε, τ ) =

Im[Ψ(jω; ϑ; θ; ε, τ )] Re[Ψ(jω; ϑ; θ; ε, τ )]

(42)

is bounded for all ω ≥ 0. As a consequence, we can define the function of ϑ ρΨ (ϑ; θ; ε, τ ) = inf rΨ (ω; ϑ; θ; ε, τ )

(43)

ω≥0

where rΨ (ω; ϑ; θ; ε, τ ) =

1 ¯ ¯ ϑ)k2 . kR(ω; ϑ) − γΨ (ω; ϑ; θ; ε, τ )I(ω;

(44)

Note that, according to (40), the dependence of ρΨ (ϑ; θ; ε, τ ) on ε and τ is inherited by the computation of Φ(s; θ; ε, τ ) as in Theorem 1 (see also Remark 1). The result that follows relates ρΨ (ϑ; θ; ε, τ ) with the stability margin ρ(ϑ) and its approximating quantity ρΦ (ϑ; ε, τ ). Lemma 3: Let θ ∈ Θstab . Then, the following properties hold for the function ρΨ (ϑ; θ; ε, τ ). 1) ρΨ (ϑ; θ; ε, τ ) is a lower bound of ρ(ϑ), i.e., ρΨ (ϑ; θ; ε, τ ) ≤ ρ(ϑ)

∀ϑ ∈ Θθ .

(45)

2) ρΨ (ϑ; θ; ε, τ ) is equal to ρΦ (ϑ; ε, τ ) for ϑ = θ, i.e., ρΨ (θ; θ; ε, τ ) = ρΦ (θ; ε, τ ). (46) Proof: Lemma 2 ensures that both ρΨ (ϑ; θ; ε, τ ) and ρ(ϑ) are well defined for any ϑ ∈ Θθ . Moreover, observe that, for fixed θ, ε, τ , both γΨ (ω; ϑ; θ; ε, τ ) and rΨ (ω; ϑ; θ; ε, τ ) are continuous functions of ω for all ω ≥ 0, and it holds that rΨ (0; ϑ; θ; ε, τ ) = Since

1 , ρ0 (ϑ). ¯ kR(0; ϑ)k2

¯ ¯ ϑ)k2 γΦ (ω; ϑ; 0, 0) = arg min kR(ω; ϑ) − γ I(ω; γ

(47)

∀ω > 0

(48)

we get rΨ (ω; ϑ; θ; ε, τ ) ≤ rΦ (ω; ϑ; 0, 0)

∀ω > 0.

(49)

From (47), (49) and continuity of rΨ (ω; ϑ; θ; ε, τ ) with respect to ω for all ω ≥ 0, we get that (45) holds. Finally, condition 2. directly follows from (40). Condition 2 of Lemma 3 ensures that ρΨ (ϑ; θ; ε, τ ) provides a quite good approximation of ρ(ϑ) in some neighborhood of θ for sufficiently small values of ε and τ , as depicted in Fig. 2. Moreover, it turns out that the maximization of ρΨ (ϑ; θ; ε, τ ) amounts to solving a GEVP. Consider the rational matrix ¸ · ¯ ϑ) In×n ρG(s; ¯ . (50) T (s; ϑ; θ; ρ; ε, τ ) = Ψ(s; ϑ; θ; ε, τ ) ¯ 0 (s; ϑ) ρG 1

8

It turns out that T¯(s; ϑ; θ; ρ; ε, τ ) depends affinely on ϑ for fixed ρ. In particular, the poles ofT¯(s; ϑ; θ; ρ; ε, τ ) do not depend on ϑ. This observation suggests that a canonical controllable state space realization of T¯(s; ϑ; θ; ρ; ε, τ ) has the form [A(θ; ε, τ ), B, C(ϑ; θ; ρ; ε, τ ), D(ϑ; θ; ρ; ε, τ )]

(51)

where C(ϑ; θ; ρ; ε, τ ) and D(ϑ; θ; ρ; ε, τ ) are of the form C(ϑ; θ; ρ; ε, τ ) = C0 (ϑ; θ; ε, τ ) + ρCρ (ϑ; θ; ε, τ ) D(ϑ; θ; ρ; ε, τ ) = D0 (ϑ; θ; ε, τ ) + ρDρ (ϑ; θ; ε, τ ).

(52)

Obviously, once θ is given, the matrices A(θ; ε, τ ) and B and the affine functions of ϑ C0 (ϑ; θ; ε, τ ), Cρ (ϑ; θ; ε, τ ), D0 (ϑ; θ; ε, τ ), Dρ (ϑ; θ; ε, τ ) can be computed explicitly. The following result provides the sought maximization of ρΨ (ϑ; θ; ε, τ ) with respect to ϑ ∈ Θθ ∩ Θ by means of the solution of a GEVP. Theorem 2: Let θ ∈ Θstab ∩ Θ. Then, max ρΨ (ϑ; θ; ε, τ ) = min µ (53) ϑ∈Θθ ∩Θ

µ,X,ϑ

s.t. B(X, ϑ; θ; ε, τ ) > 0 µB(X, ϑ; θ; ε, τ ) − A(X, ϑ; θ; ε, τ ) > 0 X = X0 > 0

(54)

and Qj (ϑ) > 0 ; j = 1, . . . , k where B(X, ϑ; θ; ε, τ ) = −

·

A0 (θ; ε, τ )X + XA(θ; ε, τ ) B0 X − C0 (ϑ; θ; ε, τ )

and A(X, ϑ; θ; ε, τ ) = Proof: We first prove that

·

0 −Cρ (ϑ; θ; ε, τ )

(55)

XB − C00 (ϑ; θ; ε, τ ) −D0 (ϑ; θ; ε, τ ) − D00 (ϑ; θ; ε, τ )

−C0ρ (ϑ; θ; ε, τ ) −Dρ (ϑ; θ; ε, τ ) − D0ρ (ϑ; θ; ε, τ )

¸

max ρΨ (ϑ; θ; ε, τ ) = max ρ

ϑ∈Θθ

¸

(56)

(57) (58)

ρ,ϑ

s.t. T¯(s; ϑ; θ; ρ; ε, τ ) in (50) is SPR. Indeed, for any ρ > 0 and any ϑ ∈ Θθ , by (43)-(44) it turns out that ρ < ρΨ (ϑ; θ; ε, τ ) if and only if ¯ ¯ ϑ)k2 < kR(ω; ϑ) − γΨ (ω; ϑ; θ; ε, τ )I(ω;

1 ρ

∀ω ≥ 0

which by strict positive realness of Ψ(s; ϑ; θ; ε, τ ) is equivalent to ¯ Re[Ψ(jω; ϑ; θ; ε, τ )] − kRe[ρG(jω; ϑ)Ψ(jω; ϑ; θ; ε, τ )]k2 > 0

∀ω ≥ 0.

(59)

In turn, by a standard Schur complement argument on the matrix function Re[ T¯(jω; ϑ; θ; ρ; ε, τ )] it follows that (59) holds if and only if T¯(s; ϑ; θ; ρ; ε, τ ) is SPR. By the Kalman-Yacubovich-Popov Lemma, strict positive realness of T¯(s; ϑ; θ; ρ; ε, τ ) for given ρ > 0 is equivalent to the feasibility of the LMI · 0 ¸ A (θ; ε, τ )X + XA(θ; ε, τ ) XB − C0 (ϑ; θ; ρ; ε, τ ) 0 In turn, by (52),(56),(57), and taking µ = ρ−1 , the latter condition can be rewritten as µB(X, ϑ; θ; ε, τ ) − A(X, ϑ; θ; ε, τ ) > 0 X = X0 > 0 i.e., the second and third of (54). Moreover, the condition B(X, ϑ; θ; ε, τ ) > 0, i.e., the first of (54), is equivalent to strict positive realness of T¯(s; ϑ; θ; ρ; ε, τ ) for ρ = 0, i.e., strict positive realness of Ψ(s; ϑ; θ; ε, τ ) (see (50)), and hence it holds for all ϑ ∈ Θθ . Finally, the set of additional constraints (55) force ϑ to belong to Θ. Therefore, the maximization of ρ Ψ (ϑ; θ; ε, τ ) over Θθ ∩ Θ amounts to solving the GEVP (53),(54),(55).

9

Remark 4: From the proof of the above theorem, it is readily checked that for given ρ > 0, the solution of the LMI feasibility problem (60) with the additional condition (55) is equivalent to the existence of ϑ ∈ Θ θ ∩ Θ such that ρΨ (ϑ; θ; ε, τ ) > ρ. Remark 5: Condition 2 of Lemma 3 implies that for any θ ∈ Θstab ∩ Θ the value of ρΦ (θ; ε, τ ) can be computed by maximizing ρ such that the LMI (60) with ϑ = θ holds for some X = X0 > 0. According to Theorem 1, for sufficiently small ε and τ , the solution of this LMI problem also provides an approximation from below of the parametric stability margin ρ(θ). Remark 6: An estimate of the size of the LMI (60) can be provided. Indeed, from (40) and the fact that π 0 (s; θ) is a factor of the numerator of Φ(s; θ), it follows that the Mac-Millan degree of T¯(s; ϑ; θ; ρ; ε, τ ) in (50) is less or equal to 2(∂A0 + ∂D0 ). Hence, the dimension of matrix X in (60) turns out to be less or equal to 2(∂A0 + ∂D0 ), while the dimension of the first inequality matrix in (60) is no larger than 2(∂A0 + ∂D0 ) + n + 1. Summing up, the LMI (60) involves no more than (∂A0 + ∂D0 )[2(∂A0 + ∂D0 ) + 1] + m parameters and a symmetric matrix of dimension no larger than 2(∂A 0 + ∂D0 ) + n + 1 in addition to the matrices Qj (ϑ), i = 1, . . . , k. C. LMI-based algorithm for the RCSMM problem Several iterative algorithms can be devised to maximize ρ(ϑ) on the basis of the fact that ρ Ψ (ϑ; θ; ε, τ ), which can be maximized according to Theorem 2, provides a quite good approximation of ρ(ϑ) at ϑ ≈ θ, for sufficiently small ε and τ . In the sequel we describe one possible algorithm, which is based on the following idea. Let θ ∈ Θ stab ∩ Θ. Pick a value ρΦ (θ; ε, τ ) ρΨ (ϑ; θ; ε, τ ) ρ(ϑ) PSfrag replacements

ϑ

θ Fig. 2.

LMI lower bound for the parametric stability margin.

ρ > ρΦ (θ; ε, τ ). Then, according to Theorem 2 and Remark 4, find, if possible, some value ϑf ∈ Θ of ϑ ensuring that ρΨ (ϑf ; θ; ε, τ ) > ρ. If such ϑf exists, then by Lemma 3 we have that ϑf ∈ Θstab ∩ Θ and ρ is a lower bound for ρΦ (ϑf ; ε, τ ) (see Fig. 3) . If possible, repeat the procedure with θ equal to the value ϑf previously found and a larger value of ρ, otherwise try with a smaller ρ, until some optimality criterion is met. As a result, the solution of the RCSMM problem can be addressed by means of a procedure involving, at each step, the

ρ(ϑ)

PSfrag replacements ρΦ (ϑf ; ε, τ ) ρ ρΦ (θ; ε, τ )

ρ(ϑf ) ρΨ (ϑ; θ; ε, τ ) ρ(θ) ρΦ (ϑ; ε, τ ) Fig. 3.

θ

ϑf

ρΨ (ϑ; ϑf ; ε, τ )

ϑ

Optimization procedure based on the LMI lower bound

computation of a rational function Φ(s; θ; ε, τ ) as in Theorem 1 and the solution of an LMI feasibility problem of the form

10

(60) with additional constraints (55) for given ρ. This procedure can be formalized according to Algorithm 1 below, where a bisection strategy on ρ provides a sequence of feasible ϑf ∈ Θstab ∩ Θ such that the corresponding sequence of lower bounds is nondecreasing. Algorithm 1: Given the uncertain plant family P, the class of restricted complexity controllers C, a tolerance value σ > 0 and the maximum number of iterations kmax . 1) Set θ = 0; ¯ 0) (see (16)) and the transfer function Φ(s; 0; ε, τ ) as in Theorem 11 ; Compute the vector G(s; Compute ρΦ (0; ε, τ ) according to Theorem 2 and Remark 5; 2) Set k = 0, ϑf = θ = 0, ρf = ρΦ (0; ε, τ ), Φ(s; θ; ε, τ ) = Φ(s; 0; ε, τ ); Set ρc = 2ρΦ (0; ε, τ ); 3) Repeat a) Compute the coefficients of the state space realization of T¯(s; ϑ; θ; ε, τ ) in (50); For ρ = ρc , solve the LMI feasibility problem (60),(55) according to Theorem 2 and Remark 4; b) If the problem has a feasible solution ϑf , then i) set θ = ϑf ; ii) set ρf = ρc ; iii) set ρc = 2ρc ; iv) compute a new Φ(s; θ; ε, τ ) as in Theorem 11 ; else i) if |ρc − ρf | < σ then exit: θ, ρf is the solution. ii) set ρc = (ρc + ρf )/2; c) set k = k + 1; until k ≥ kmax ; 4) Maximum number of iterations reached. IV. E XAMPLES Some numerical examples are reported to illustrate the main features of the proposed approach and the behavior of Algorithm 1 in Section III-C. In each example, the values of ε and τ are chosen according to Remark 1 and the tolerance value σ is set to 10−4 . The MatlabTM LMI Toolbox solver has been employed. Example 1: Consider the class of uncertain plants P defined by P (s; δ) =

1 + δ1 . s + 1 + δ2

According to (1), we have ¯ ¯ B0 (s) = 1; A0 (s) = s + 1; B(s) = [1 0]0 ; A(s) = [0 1]0 ; δ = [δ1 δ2 ]0 , and hence Assumption 1 is satisfied. Consider the class of proportional controllers C(s; ϑ) = −0.5 + ϑ ; ϑ ∈ R which is of the form (6) where ¯ (s) = 1; D(s) ¯ N0 (s) = −0.5; D0 (s) = 1; N = 0; Θ = R. Note that Assumptions 2 and 3 are satisfied. In particular, we have Θstab = {ϑ ∈ R : ϑ > −0.5} . The parametric stability margin ρ(ϑ) can be computed analytically as ρ(ϑ) = p

0.5 + ϑ (−0.5 + ϑ)2 + 1

.

In Fig. 4, ρ(ϑ) is depicted showing that only one maximum is present. The vector function G(s; ϑ) in (17) is given by ¯ ϑ) = − G(s; 1 We

1 [−0.5 + ϑ 1]0 . s + 0.5 + ϑ

recall that the computation of Φ(s; ϑ; ε, τ ) requires the choice of the positive scalars ε and τ (see also Remark 1).

11

ρ(ϑ) 1.4

1.2

1

0.8

0.6

0.4

0.2

PSfrag replacements 0

Fig. 4.

0

1

2

3

4

5

6

7

8

9

10

ϑ

Example 1: Parametric stability margin ρ(ϑ).

and, according to Theorem 1, the rational function Φ(s; ϑ; ε, τ ) is then computed as Φ(s; ϑ; ε, τ ) =

s + 0.5 + ϑ . s+ε

From (40) we get

s + 0.5 + ϑ . s+ε We note that in this case Ψ(s; ϑ; θ; ε, τ ) does not depend on θ, i.e., ρΨ (ϑ; θ; ε, τ ) = ρΦ (ϑ; ε, τ ) for any θ. In turn, this implies that the maximization of ρΦ (ϑ; ε, τ ), which approximates the true parametric stability margin ρ(ϑ), can be actually carried out via the solution of one GEVP as in Theorem 2. This also implies that Algorithm 1 is expected to converge to the unique optimum of the RCSMM problem. The progress of Algorithm 1, initialized at θ = 0, is illustrated in Fig. 5, where the sequence of lower bounds ρf is reported. Example 2: Consider the uncertain plant 1 + δ1 P (s; δ) = (61) s − 4 + δ2 and the class of controllers © ª s + 4 + ϑ1 ; Θ = ϑ ∈ R2 : ϑ1 > −4, ϑ2 > −10 . (62) C(s; ϑ) = 20 s + 10 + ϑ2 Note that Θ ensures that the pole and the zero of the controller are negative. Clearly, (61) is in the form (1) and satisfies Assumption 1, while (62) can be put in the form (6) by introducing the constraint ¸ · 4 + ϑ1 0 > 0. Q(ϑ) = 0 10 + ϑ2 Ψ(s; ϑ; θ; ε, τ ) =

Moreover, Assumption 2 holds since

s+4 s + 10 is a stabilizing controller for the nominal (unstable) plant. In this respect, it is easily seen that the set of controller parameters yielding a stable nominal control system is given by C(s; [0 0]) = 20

Θstab ∩ Θ = {(ϑ1 , ϑ2 ) : ϑ2 < 5ϑ1 + 10, ϑ2 > −10} .

(63)

Moreover, an expression of the l2 parametric stability margin as a function of the controller parameters ϑ1 , ϑ2 can be derived analytically as ) ( |26 + ϑ2 | 4|10 + 5ϑ1 − ϑ2 | . (64) , √ ρ(ϑ) = min p 401 400(4 + ϑ1 )2 + (10 + ϑ2 )2

12

ρf 1.6

1.4

1.2

1

0.8

0.6

PSfrag replacements

Fig. 5.

0.4

0

10

20

30

40

50

60

70

80

90

100

k

Example 1: Sequence of lower bounds ρf

Note that ρ(ϑ) turns out to be non-smooth at some ϑ as clearly depicted in Figure 6. The supremum value of ρ(ϑ) is equal to 1 and can be attained only asymptotically for ϑ1 → +∞ on the curve on which ρ(ϑ) is non-smooth.

ρ(ϑ) 1

0.8

0.6

0.4

0.2

PSfrag replacements

0 20

20 15

15 10

ϑ2

10

5

5

0 0

−5 −10

Fig. 6.

ϑ1

−5

Example 2: Parametric stablity margin ρ(ϑ).

Algorithm 1 initialized at θ1 = 0, θ2 = 0 yields the behavior depicted in Figs. 7,8. Note that after kmax = 100 iterations, the value of θ gets close to the region where the stability margin is non-smooth and hence the LMI solver experiences problems in finding new feasible values of θ. The result is that the optimization is unable to proceed. Since s + 14 C(s; [10 10]) = 20 s + 20 is also a stabilizing controller, Algorithm 1 can be initialized at θ1 = 10, θ2 = 10. In this case, after kmax = 500 iterations, it yields θ1 = 23.97 ; θ2 = −6.47 and correspondingly ρf = 0.97.

13

ρf

margine di stabilita‘

0.85

0.8

0.75

0.7

0.65

0.6

0.55

PSfrag replacements

0.5

0.45 0

Fig. 7.

20

40

60 passi k

80

100

120

Example 2: Sequence of feasible ρf . 0 −1 −2

θ1 −3 −4 gamma1 gamma2

−5 −6 −7 −8

PSfrag replacements

θ2

−9 −10 0

10

20

30

40

50 passi

60

70

80

90

100

k

Fig. 8.

Example 2: Sequence of feasible θ1 (solid) and θ2 (dashed).

The algorithm progress is illustrated in Figs. 9,10,11. Example 3: (PI control of a third-order plant with uncertain gain and damping factor). Consider the uncertain plant 10 + δ1 P (s; δ) = 2 (s + 1)(s + (2 + 0.1δ2 )s + 10) and the PI controller

(65)

0.5 + ϑ2 ; ϑ ∈ R2 . (66) s Again, it is easily verified that (65) and (66) are in the form (1) and (6), respectively, and Assumptions 1-3 hold. The set Θstab is given by ½ ¾ 55 4 10 8 1 Θstab = (ϑ1 , ϑ2 ) : ϑ2 < − ϑ1 − ϑ21 , ϑ1 < , ϑ2 > − 18 9 9 5 2 C(s; ϑ) = 1 + ϑ1 +

14

ρf

margine di stabilita‘

1

0.95

0.9

0.85

0.8

0.75

PSfrag replacements

0.7

0.65

0

100

200

300

400

500

600

kpassi Fig. 9.

Example 2: Sequence of feasible ρf .

θ1

gamma1

25

20

15

10

5

PSfrag replacements 0

0

100

200

300

400

500

600

kpassi Fig. 10.

Example 2: Sequence of feasible θ1 .

and is depicted in Fig. 12. The lower bound ρΦ (ϑ; ε, τ ) of the stability margin ρ(ϑ) is depicted in Fig. 13. The plot is obtained numerically by gridding the parameter space and performing a sequence of LMI optimizations according to Remark 5. It is worth noting that the parametric stability margin drops from non-zero values to zero when crossing the line ϑ 2 = − 21 . The behavior of Algorithm 1, initialized at θ1 = 0, θ2 = 0, is summarized in Figs. 14 (ρf ), 15 (θ1 ), 16 (θ2 ).

After kmax = 100 iterations, Algorithm 1 yields the parameter values θ1 = −0.54 and θ2 = −0.45, which correspond to the PI controller 0.05 C(s; [−0.54 − 0.45]0 ) = 0.46 + . s It can be verified by inspection of Fig. 13 that such controller achieves the maximum of ρΦ (ϑ; ε, τ ). V. C ONCLUSION In the present paper the problem of maximizing the l2 parametric stability margin via the design of restricted complexity controllers (RCSMM problem) has been addressed. The class of SISO uncertain plants with rank one parametric perturbations

15

θ2

gamma2

10

8

6

4

2

0

−2

−4

PSfrag replacements

−6

−8

0

100

200

300 passi

400

500

600

k Fig. 11.

Example 2: Sequence of feasible θ2 .

ϑ2 3.5 3 2.5 2 1.5

Θstab 1 0.5 0

PSfrag replacements

−0.5 −1 −2

Fig. 12.

−1.5

−1

−0.5

0

0.5

1

1.5

ϑ1

Example 3: Region Θstab .

and a controller family characterized by transfer functions of fixed degree, which depend affinely on a tunable parameter vector, have been considered. The main contribution of the paper is the development of a lower bound of the parametric stability margin, which provides a locally optimal approximation in some neighborhood of any given stabilizing controller, and whose maximization is performed via the solution of a Generalized Eigenvalue Problem (GEVP). It is believed that the lower bound can be fruitfully exploited to devise LMI-based algorithms, as the one proposed in the present paper, to solve the RCSMM problem. The development of efficient optimization procedures based on the proposed characterization is the subject of current research. R EFERENCES [1] S. Bhattacharrya, H. Chapellat, and L. Keel, Robust control, the parametric approach. Englewood Cliffs, USA: Prentice Hall, 1995. [2] A. Rantzer and A. Megretski, “A convex parameterization of robustly stabilizing controllers,” IEEE Transactions on Automatic Control, vol. 39, no. 9, pp. 1802–1808, 1994. [3] J. Langer and I. Landau, “Combined pole placement/sensitivity function shaping method using convex optimization criteria,” Automatica, vol. 35, pp. 1111–1120, 1999. [4] D. Henrion, D. Arzelier, and D. Peaucelle, “Positive polynomial matrices and improved LMI robustness conditions,” in Proceedings of the 15th IFAC World Congress, Barcelona, Spain, 2002. [5] A. Gulchak and A. Rantzer, “Robust controller design via linear programming,” in Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, Arizona, USA, 1999, pp. 1815–1820. [6] K. J. Astrom and T. Hagglund, PID controllers: theory, design, and tuning. Research Triangle Park, NC: Instrum. Soc. Amer., 1995.

16

ρΦ (ϑ; ε, τ ) 10 9 8 7 6 5 4 3 2 1 0 2 1.5 1 0.5 0 3.5 3

−0.5

ϑ1

2.5 2

−1

replacements

1.5 1

−1.5

0.5 −2

Fig. 13.

0

ϑ2

−0.5

Example 3: Lower bound of the parametric stability margin.

ρf

margine di stabilita‘ 10 9.8 9.6 9.4

ro

9.2 9 8.8 8.6 8.4

PSfrag replacements 8.2 0

20

40

60

80

100

120

kpassi Fig. 14.

Example 3: Sequence of feasible ρf .

[7] A. Datta, M. T. Ho, and S. Bhattacharrya, Structure and Synthesis of PID Controllers. London, U.K.: Springer-Verlag, 2000. [8] M. T. Ho, “Synthesis of H∞ PID controllers: a parametric approach,” Automatica, vol. 39, no. 6, pp. 1069–1075, 2003. [9] F. Blanchini, A. Lepschy, S. Miani, and U. Viaro, “Characterization of PID and lag/lead compensators satisfying given H ∞ specifications,” IEEE Transactions on Automatic Control, vol. 49, no. 5, pp. 736–740, 2004. [10] D. Henrion, M. Sebek, and V. Kucera, “Positive polynomials and robust stabilization with fixed-order controllers,” IEEE Transactions on Automatic Control, vol. 48, no. 7, pp. 1178–1186, 2003. [11] M. T. Soylemez, N. Munro, and H. Baki, “Fast calculation of stabilizing PID controllers,” Automatica, vol. 39, no. 1, pp. 121–126, 2003. [12] A. Tesi, A. Torrini, and A. Vicino, “Low order suboptimal robust controller design for systems with structured uncertainty,” in Proceedings of the 2nd

17

θ1

gamma1 0

−0.1

−0.2

−0.3

−0.4

−0.5

PSfrag replacements −0.6

0

20

40

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80

100

120

80

100

120

kpassi Fig. 15.

Example 3: Sequence of feasible θ1 .

θ2

gamma2 0

−0.05 −0.1 −0.15 −0.2 −0.25 −0.3 −0.35 −0.4

PSfrag replacements −0.45 0

Fig. 16.

20

40

60 passi k

Example 3: Sequence of feasible θ2 .

IFAC Symposium on Robust Control Design, Budapest, Hungary, 1997, pp. 65–70. [13] G. Bianchini, P. Falugi, A. Tesi, and A. Vicino, “Restricted complexity robust controllers for uncertain systems with rank one real perturbations,” in Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, Florida, USA, 1998, pp. 1213–1218. [14] ——, “On the synthesis of restricted complexity controllers for uncertain plants with ellipsoidal perturbations,” in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida, USA, 2001, pp. 1562–1567. [15] G. Bianchini, A. Tesi, and A. Vicino, “Synthesis of robust strictly positive real systems with l 2 parametric uncertainty,” IEEE Transactions on Circuits and Systems-I, vol. 48, no. 4, pp. 438–450, 2001. [16] S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakhrishnan, Linear matrix inequalities in system and control theory. SIAM, 1994. [17] A. Vicino and A. Tesi, “Regularity conditions for robust stability problems with parametric perturbations,” SIAM Journal of Optimization and Control, vol. 36, pp. 1000–1016, 1995. [18] A. Packard and P. Pandey, “Continuity properties of the real/complex structured singular value,” IEEE Transactions on Automatic Control, vol. 38, no. 3, pp. 415–428, 1993.