A Youla Parameter Approach to Robust Constrained Linear Model Predictive Control Qifeng Cheng, Basil Kouvaritakis∗ , Mark Cannon J. Anthony Rossiter University of Oxford, Department of Engineering Science, University of Shefﬁeld, Department of Automatic Control Oxford, OX1 3PJ, UK and Systems Eng., Shefﬁeld, S1 3JD, UK Abstract— Previous work (Kouvaritakis et al., 1992) proposed this control law for all initial states in the terminal set). the introduction of a Youla parameter into the receding As well as providing a means of systematically accounting horizon control strategy to enhance the degree of robustness for constraints on states and inputs, MPC provides a feedback of the closed loop system without affecting the optimality of nominal performance. The idea is clearly appealing and has mechanism through its receding horizon implementation been investigated further by a number of researchers. Yet which affords a certain degree of robustness to uncertainty to date the application of the idea has only been considered due to model mismatch. A convenient mathematical descrip- for the case of unconstrained systems, which is surprising tion of such mismatch has the form of polytopic parametric given that the presence of constraints is main justiﬁcation uncertainty. To retain the guarantee of closed loop stability for the use of MPC for linear systems. It is the purpose of the present paper to combine recent results on invariance, it is necessary to ensure that constraints are satisﬁed by the feasibility and an augmented and autonomous formulation of predicted behaviour over the entire prediction horizon (in the predicted dynamics to derive an effective robust constrained modes 1 and 2) and propagating the effects of uncertainty MPC algorithm that takes full advantage of the introduction over a future horizon poses some very signiﬁcant compu- of a suitable Youla parameter. The efﬁcacy of the proposed tational difﬁculties. To avoid these  removed mode 1 but algorithm is illustrated by means of a numerical example. allowed for the online adjustment of the terminal control law. Accordingly  ensured that the terminal set was invariant I. I NTRODUCTION and feasible over the entire uncertainty class and minimized Model Predictive Control (MPC) is a well established an upper bound (over the uncertainty class) on the cost- control strategy that uses a receding horizon strategy to to-go (in mode 2). Despite the elegance of the solution derive effective approximations to a closed loop constrained proposed, the restriction to a ﬁxed state feedback law over optimal control problem. Instead of solving a closed loop the entire prediction horizon implies that performance can be problem which is in essence an inﬁnite dimensional problem conservative while online computational load is quite high. that cannot be solved online, MPC uses the measured infor- Later work ,  produced less conservative results by mation at each instant of time to predict the future behaviour re-introducing mode 1 but at a considerable extra compu- of the controlled system for a given future trajectory of tational load. An alternative  which not only preserved control moves. This trajectory is then optimized in order mode 1 but also led to a signiﬁcant reduction in compu- to minimize a measure, J, of predicted behaviour; J here tational cost was based on an augmented autonomous state will be taken to be the usual linear-quadratic (LQ) cost. To space formulation of the predicted dynamics. According to reduce computational complexity, the future input trajectory this, the degrees of freedom (DOF) where reformulated as comprises only a ﬁnite number of free moves, say the ﬁrst perturbations on the unconstrained LQ optimal control law Nc , the remainder being those prescribed by a pre-speciﬁed and were introduced as additional states of an augmented state feedback control law, referred to as the terminal control state vector. In this framework it was possible to derive law, which is normally taken to be the unconstrained LQ the prediction dynamics as an autonomous system thereby optimal. Thus MPC effectively adopts a dual mode prediction converting the dual mode prediction problem into a single paradigm: mode 1 consisting of the ﬁrst Nc time instants, and mode problem and thus obviating the need to propagate the mode 2 of the remainder. MPC implements only the ﬁrst effects of uncertainty over the prediction horizon. element of the computed optimal future control sequence, In addition, given that the degrees of freedom are taken to and repeats the whole procedure at the next time instant. be perturbations on the unconstrained optimal law in , it To ensure a guarantee of closed loop stability , MPC can be shown that the optimization of nominal performance deploys: (i) a terminal penalty term in the cost index J, which is equivalent to the minimization of a weighted norm of the is normally taken to be a quadratic function of the terminal vector of degrees of freedom, f . This deﬁnes a cost which state (i.e. the state at the end of mode 1), and is equal to is quadratic in f , and, if this is combined with an ellipsoidal the cost-to-go, namely the cost incurred in mode 2; (ii) a terminal set which involves constraints that are quadratic in terminal inequality constraint that restricts the terminal state f , then the optimal solution can be obtained at a fraction of to lie in a terminal set which, under the terminal control law, the cost required by the usual MPC schemes, even those that is invariant and feasible (i.e. constraints are satisﬁed under use polytopic terminal sets enabling a solution through the use of quadratic programming (QP) with warm starting. linear-quadratic (LQ) problem with cost index Just as the worst case strategy adopted by  can be overly ∞ pessimistic (at least with respect to nominal behaviour), so J= 2 (yk + Ru2 ) k (5) the approach of  can be overly optimistic (at least with k=0 respect to worst case behaviour), especially so if the worst for the nominal model parameters, say (A(0) , B (0) , C). Then case and nominal performance differ signiﬁcantly in closed the prediction dynamics are given by the autonomous state loop behaviour. Reducing this difference provides an ideal space model remedy to this problem and an observation made in  is key to this. It noted in  that in the unconstrained case, the c0 x online optimization of MPC resulted in a ﬁxed term feedback zk+1 = Ψ(i) zk , zk = k , fk = . (6) . fk . control scheme which could be endowed with extra degrees cN −1 of freedom through the introduction of a Youla Parameter. The interesting observation made there was that this Youla A(i) −B (i) K B (i) eT 1 , M = 0N −1,1 IN −1 Ψ(i) = Parameter can be used to improve the robustness of the MPC 0 M 0 01,N −1 feedback scheme without affecting the optimality of nominal where e1 is the ﬁrst column of the N × N identity matrix performance. Thus the minimization of the RH∞ -norm of IN . A convenient way to ensure the feasibility of predictions an appropriate sensitivity transfer function would lead to is through the stability constraint: optimal robustness with respect to additive uncertainty to the T system transfer function; robustness to additive disturbances zk P zk ≤ 1 (7) could be optimized instead by minimizing the RH∞ -norm of where P is a positive deﬁnite matrix (denoted P 0) a different transfer function. This idea of achieving optimal satisfying the robust invariance and feasibility conditions: performance ﬁrst and subsequently optimizing performance gained favoured with many researchers in subsequent years Ψ(i) T P Ψ(i) P, i = 1, . . . p (8a) (e.g. , , , ). Surprisingly however, in all this ˆ ˆ ˆ KP −1 K T ≤ u2 , K = −K ¯ eT . (8b) 1 work, consideration was given only to the unconstrained case, which somehow seems to defeat the purpose of using It can be shown that the minimization of the cost (5) for the T MPC; after all, in the absence of constraints, MPC can be nominal model is equivalent to the minimization of fk fk , implemented as a ﬁxed term controller. It is the purpose of and on the basis of this an MPC algorithm with guaranteed the present paper to close this gap, through the combined robust stability/feasibility can be devised : use of the augmented autonomous formulation of , with Algorithm 1 (Robust efﬁcient MPC): At k = 0, 1, . . .: ∗ its attendant use of robust invariance and feasibility and the 1. Compute fk = arg minf f 2 subject to (7) 2 ∗ use of the Youla Parameter as recommended in . 2. implement uk = −Kxk + eT fk . 1 The online optimization in step 1 involves a quadratic cost II. ROBUST MPC BASED ON AUGMENTED AUTONOMOUS and is subject to quadratic constraints. This can be solved PREDICTION MODEL very efﬁciently (at a fraction of the computational cost of QP- Consider a SISO discrete time linear system with model: based MPC algorithms) by using a Newton-Raphson proce- dure with guaranteed quadratic convergence rate to determine xk+1 = Axk + Buk yk = Cxk (1) the unique negative root of a 2N th order polynomial. where x ∈ Rn , u, y ∈ R, subject to polytopic uncertainty III. T HE NEED FOR THE YOULA PARAMETER deﬁned by p Despite its signiﬁcant advantages in terms of computa- (A, B) = ηi (A(i) , B (i) ) (2) tional efﬁciency, Algorithm 1 optimizes the performance of i=1 the nominal system, which could differ considerably from that obtained for other models within the uncertain class and subject to the constraint of (2). The nominal performance objective of Algorithm 1 −¯ ≤ u ≤ u. u ¯ (3) would be better justiﬁed if it were possible to desensitize performance under model uncertainty. From the perspec- Although this constraint applies to the plant input, it is tive of frequency response desensitization, the Q parameter equivalent to a general constraint on both the states and the provides an effective solution and was introduced in the DOF if the closed loop paradigm is employed. According to context of MPC in the following way . In the absence of this, the DOF are not taken to be the future predicted inputs, constraints, the MPC control law can be implemented using but instead are perturbations on a state feedback control law: the ﬁxed term feedback scheme of Figure 1, where r denotes −Kxk+i + ci i = 0, 1, . . . N − 1 a reference set point signal and where the polynomials M (z), uk+i = (4) N (z) are solutions of −Kxk+i i = N, N + 1, . . . a(0) (z)M (z) + b(0) (z)N (z) = popt (z) (9a) with Nc denoting the prediction horizon, where K will be (0) (0) taken to be the optimal feedback gain for the unconstrained popt (z) = det[zI − A +B K]. (9b) Fig. 1. The MPC feedback loop Fig. 2. The MPC feedback loop with the Q parameter However (9a) is a Diophantine equation which admits a whole class of solutions: Since (11) is afﬁne in the parameter Q, it is amenable M (z) = Y (z) − b(0) (z)Q(z), N (z) = X(z) + a(0) (z)Q(z) to the usual RH∞ treatment (including that of  which is (10) based on the application of Lawson weighted least squares where for convenience X, Y , and Q can be chosen to algorithm ). Furthermore S is known to give a measure of be polynomials. Under these circumstances, the transfer the reduction of sensitivity to uncertainty in going from the functions from r to y and u are b(z)/popt (z), a(z)/popt (z), open to the closed loop system, but that reduction concerns respectively and as such are not affected by the introduction the frequency response sensitivity. Thus, choosing Q in of the free variable Q. The idea is therefore to use Q to this manner will optimize the worst case (over the entire optimize some other property without compromising the frequency range) frequency response sensitivity reduction. nominal optimality of the feedback system of Figure 1. For Clearly, if desired, S can be scaled by a suitable weighting example one could choose the Q that minimizes the RH∞ - factor that allows information about the frequency response norm of an appropriate transfer function in order to optimize of the open loop sensitivity to be taken into account. This robustness properties in terms of either uncertainty in the procedure will result in a Q which desensitizes the closed model, or additive disturbances. In the case of the former loop to additive uncertainty, but may well not be optimal in objective, the work of Caratheodory and Fejer  provides the context of desensitizing the cost of (5). Our simulation a justiﬁcation for the use of Lawson’s weighted least squares results show that the solutions for Q based on the well known algorithm  for the derivation of optimal polynomials Q RH∞ optimal solutions produced results which were very of any chosen order so as to maximize the tolerance of the close to optimal, even in the context of the cost of (5). system of Figure 1 to parametric uncertainty in the open loop The analysis above applies to the unconstrained case only system transfer function b(z)/a(z). This method of obtaining and will not in general hold true for all reference signals robustness over and above the optimality of MPC was ﬁrst r. However when constraints are present, as described in introduced in , and was subsequently explored further by Section 2, to avoid infeasibility it is necessary to perturb the many researchers (e.g. , , , ). Yet remarkably optimal control law, and this was achieved in (4) through the none of the results in this area address constraint handling, use of the closed loop paradigm. This paradigm can also be which is the main justiﬁcation for using MPC. applied to the system of Figure 1 through the introduction of the perturbation signal c depicted in Figure 2. Then, IV. R EDUCING SENSITIVITY THROUGH THE USE OF THE provided the MPC strategy is able to ensure the feasibility of YOULA PARAMETER predictions at all time instants, the feedback loop of Figure The current paper redresses this omission using the aug- 2 will always operate within its linear region and therefore mented autonomous MPC formulation described in Section the desensitization achieved through Q will remain valid, 2. We ﬁrst reformulate the DOF as perturbations on the throughout the receding horizon of the MPC strategy. The optimal feedback control law depicted in Figure 1. This leads following section shows how the strategy of Section 2 may to the feedback loop of Figure 2, where the perturbation be used to provide the required guarantee of feasibility. signal c is introduced in order to force the input u to respect the constraints of (3), but otherwise should be kept as small V. T HE USE OF THE YOULA PARAMETER IN ROBUST as possible in the interests of optimality. This strategy is CONSTRAINED MPC optimal for the nominal model only, but its approximate As explained above, the scheme of Figure 2 is the input- validity can be extended to a wider class of models (centred output version of the closed loop paradigm of (4). It therefore on the nominal) provided that the system of Figure 2 is follows that, if it were possible to derive a state space de- desensitized to the effects of model uncertainty. An indirect, scription for Figure 2, then it would be possible to apply the but convenient way to achieve this desensitization is to augmented autonomous formulation of Section 2 to derive minimize the RH∞ -norm of the sensitivity transfer function an MPC algorithm with the robust stability and feasibility X(z) + a(0) Q(z) b(0) −1 guarantees of Algorithm 1, but which would also beneﬁt from S(z) = 1 + the desensitization effects of Q. Y (z) − b(0) Q(z) a(0) (11) A simple procedure for the transformation of the input- (0) a Y (z) − a(0) b(0) Q(z) output setting of Figure 2 to a state space model is the = popt (z) following. First, in the interests of parsimony (i.e. in retaining a low model order), it is recommended that the optimal new augmented autonomous dynamics of (15)). As is the polynomial Q(z) is approximated by a transfer function, case for the predictions of (6), the minimization of the NQ (z)/DQ (z). Such an approximation is likely to result in nominal predicted cost of (5) can be likewise proved to T an increase in the norm of S, and hence a trade-off has to be equivalent to the minimization of fk fk . The procedure be struck between this effect and the desire to keep model differs however in one respect, since, unlike the model of complexity to a minimum. Substituting NQ (z)/DQ (z) in ˆ ˆˆ Section 2 (where uk = −Kzk ), here uk = C xk + Dvk , ˆ place of S yields: where C ˆ is subject to polytopic uncertainty with vertices ˆ C (i) = ( 0T CD θ(i) )T . This means that the conditions DQ (z)a(z) u(z) = v(z) ˆ to be satisﬁed by the parameter, P , deﬁning the augmented ˆ ˆ (z)a(z) + N (z)b(z) M (12) ellipsoid should be modiﬁed to the analogous invariance and DQ (z)b(z) y(z) = v(z) feasibility conditions: ˆ ˆ (z)a(z) + N (z)b(z) M ˆ ˆˆ Ψ(i) T P Ψ(i) ˆ ˆ ˆ ˆ P , K (i) P −1 K (i) T ≤ u2 , i = 1, . . . p ¯ where v(z) = r(z) + c(z) and (16) ˆ M (z) = Y (z)DQ (z) − b(0) (z)NQ (z) The overall MPC control strategy is summarized by the ˆ N (z) = X(z)DQ (z) − a(0) (z)NQ (z). following algorithm. Algorithm 2 (Robust MPC with Youla parameter): The observability canonical form for (12) can be written as: Ofﬂine: ˆ ˆx ˆ ˆ ˆ ˆˆ ˆ xk+1 = Aˆk + Bvk , yk = Cy xk , uk = C xk + Dvk ˆ ˆ a. Compute P = arg minP det(P ) subject to (16). ˆ 0 I 0 b. Compute the Q which minimizes the RH∞ -norm of the ˆ A= ˆ , B= ˆ , d = CM CN θ, D = 1 (13) −dT 1 ˆ ˆ (weighted) sensitivity function S of (11). T ˆT ˆ Cy = CD 0T θ, C T = 0T CD θ − d Online, at k = 0, 1, . . ., for xk = yk yk−1 · · · yk−ν+1 : ˆ ∗ ˆT ˆ ˆ 1. Compute fk = arg minf f 2 subject to zk P zk ≤ 1. 2 where θ is the column vector of coefﬁcients (in order of ˆ ˆ (0) ˆ 2. Implement uk = C xk + Dvk , ascending powers of z) of a(z), b(z) (with the coefﬁcients ˆ ˆ ˆ where C (0) = ( 0T CD θ(0) )T . of a(z) appearing ﬁrst), Cy , C are the row vectors of the ˆ Remark 1: Note that the ofﬂine optimization of P in step coefﬁcients of DQ a(z), DQ (z)b(z), respectively taken in ˆ ˆ ascending order, CM , CN , CD , CD are the convolution ma- a can be formulated as a convex LMI problem (e.g. see ˆ ˆ ˆ ˆ ). Furthermore the online optimization in step 1 can be trices that relate θ to the coefﬁcients of M (z)a(z), N (z)a(z), performed using the efﬁcient NRMPC optimization of . DQ (z)a(z), DQ (z)b(z), respectively. In this formulation, the states are given by: In common with Algorithm 1, Algorithm 2 enjoys the beneﬁt of an extremely efﬁcient online optimization and T xk = yk yk−1 · · · yk−ν+1 ˆ , ν = na +nb +nQ +1 (14) has the guarantee of closed loop stability and feasibility (provided that the online optimization is feasible at initial where na , nb , nQ denote the degrees of a(z), b(z), and time). However in addition to Algorithm 1, the desirability of DQ (z). The model (13) caters for a tracking problem where achieving optimality for the nominal predictions is enhanced r is assumed to be a constant setpoint. The more general on account of the use of the Q parameter. case requires knowledge of the dynamics that generate any particular reference signal r(t). With r a constant, it is VI. N UMERICAL E XAMPLE ˆ straightforward to compute the steady state value for x and This section illustrates the efﬁcacy of Algorithm 2 by re-write the model of (13) in terms of a regulation problem means of a numerical example. Consider a system with an by considering deviations away from the steady state. This is input-output description given by assumed to have already been done and correspondingly we take the input v to be c (the deviation of v from the steady a(z) b(z) = 1 z z 2 1 z θ (17) state). On account of the linear dependence on θ in (13), it is where θ lies in the convex hull of vertices θ(i) , i = 0, . . . 4: possible to generate the polytopic uncertainty model for the T θ(0) = 1 0.8 0.3 1 0.6 system of Figure 2 (with r = 0) given the vertices θ(i) T of a corresponding polytopic uncertainty class for the open (i) ± 0 0.16 0.06 0 0 , i = 1, 2 (18) θ = T loop input-output description. The problem then becomes ± 0 0 0 0.02 0.012 , i = 3, 4 equivalent to that deﬁned in Section 2, after the application of the closed loop paradigm law of (4). This leads to the For this example, R = 10−6 results in the optimal closed augmented autonomous model for the prediction dynamics: loop polynomial: ˆ x ˆ A(i) ˆ BeT popt (z) = 1 + 0.6z −1 (19) ˆ ˆ zk+1 = Ψ(i) zk , zk = k , Ψ(i) = ˆ ˆ 1 (15) fk 0 M and a pair of particular solutions to (9a) The perturbation ck can be optimized using an approach analogous to step 1 of Algorithm 1, but modiﬁed for the Y (z) = 1 + 0.6z −1 , X(z) = −0.8 − 0.3z −1 . (20) The 14th order polynomial Q that minimizes the sensi- Although it may seem paradoxical that the minimax costs tivity transfer function S of (11) for this system (computed given Algorithm by 3 are not the best, this can be explained using Lawson’s weighted least squares algorithm), and its by the fact that the costs displayed are closed loop costs and 2nd order transfer function approximation are given as: therefore do not bear a direct relation to the predicted costs that are optimized by any of the algorithms. Furthermore, in Q(z) = 0.1 3.514 − 0.651z −1 + 0.105z −2 − 0.176z −3 order to make the computation of Algorithm 3 tractable, an + 0.227z −4 − 0.175z −5 + 0.09z −6 − 0.027z −7 LTI assumption has been imposed. In summary, Algorithm 2 − 0.003z −8 + 0.009z −9 − 0.006z −10 is clearly the best, since it both produces the lowest nominal + 0.002z −11 − 0.001z −13 + 0.001z −14 cost (which, as expected, is the same as that of Algorithm (21) 1), while also resulting in smaller cost sensitivity. Moreover, Algorithm 2 achieves this with an online computational NQ (z) 0.3505 + 0.0599z −1 = (22) load which is comparable to that of Algorithm 1. Clearly DQ (z) 1 + 0.3570z −1 + 0.0229z −2 there is a certain amount of extra ofﬂine computation but The degree of optimality of Q is illustrated in Figures that involves two simple optimization problems which are 3-6 which give the gain of the frequency response of S known to converge and are not particularly computationally for the Q of (22) in transfer function form (solid line) demanding. and for Q = 0 (dashed line). The solid line plots would in fact be ﬂat for the optimal Q of (21), but deviate a VII. C ONCLUSIONS little from ﬂatness for the approximation of (22). Finally The use of the Youla parameter in the context of MPC the ofﬂine computation of Algorithm 2 is completed by the has been advocated in a paper as far back as 1992 and since ˆ determination of the parameter P of the augmented ellipsoid. then has been endorsed by various researchers. Yet, to date, For a control horizon of N = 10 and an input bound there has been no attempt to apply this in the presence of ¯ of u = 0.3, the maximum volume invariant and feasible constraints, which is very surprising when it is considered ˆ ˆ ellipsoid is obtained for P with determinant det(P ) = 0.093. that constraints offer the main justiﬁcation for the use of To illustrate the efﬁcacy of Algorithm 2, Table 1 lists MPC. This paper proposes a strategy that redresses this the closed loop costs obtained by Algorithms 1 and 2 for situation and demonstrates the efﬁcacy of the strategy by zero initial conditions and a reference setpoint r = 0.36. means of a numerical example. The costs are computed for ﬁxed values of θ, namely the There remains the issue of how to tune the Youla param- nominal value θ(0) and each of the vertices θ(i) , i = 1, . . . 4. eter. In this paper, the aim is to desensitize the closed loop To assess the desirability of the nominal objective employed system to additive disturbances. However this desensitization by these strategies, the table also includes the corresponding is performed for the sake of convenience (given the body of costs for a worst case algorithm, referred to as Algorithm 3. available work) in terms of the frequency response. As a A truly worst case algorithm would entail online computation result, there is no guarantee that the resulting Q parameter which is intractably computationally intensive, since it would is optimal for the purposes of the robust MPC framework have to assume an LTV model over the prediction horizon proposed in the paper. This particular point was tested at each time instant. To make the computational demand numerically using the example of section 6, and it was of Algorithm 3 a little more comparable with those of found that although some improvement to the results of the Algorithms 1 and 2, though in essence still a great deal proposed algorithm could be brought about by the further more intensive, the assumption is made at each instant that tuning of the Youla parameter, such improvements were very the dynamics are LTI over the prediction horizon, even modest indeed. It is nevertheless left as a topic for further though at each time instant, the algorithm implements the research to devise efﬁcient algorithms for the choice of Q ﬁrst element of the sequence that optimizes the predicted which address directly the needs of the robust constrained cost for whichever vertex gives the greatest cost. MPC algorithm proposed in the paper. As expected, the desensitization of the closed loop system to additive uncertainty has the signiﬁcant effect of reducing R EFERENCES (almost halving) the cost sensitivity (measured as a percent-  D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert. Con- age of nominal cost) with the maximum value for Algorithm strained model predictive control: Stability and optimality. Automatica, 2 being more than 4% smaller than that of Algorithm 1 but 36:789–814, 2000. also 3.6% smaller than that of Algorithm 3. In fact for this  M.V. Kothare, V. Balakrishnan, and M. Morari. 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Automatic Control, IEEE Transactions on, 45(8):1545– or Algorithm 2. 1549, 2000. θ(0) θ(1) θ(2) θ(3) θ(4) Max cost sensitivity Algorithm 1 0.1296 0.1402 0.1319 0.1445 0.1303 11.5% Algorithm 2 0.1296 0.1378 0.1280 0.1385 0.1271 6.9% Algorithm 3 0.1321 0.1357 0.1252 0.1453 0.1342 10.0% TABLE I C LOSED LOOP COST VALUES 0.165 0.3 0.16 0.25 0.155 | S [exp(jωT)] | 0.2 | S [exp(j ωT)] | 0.15 0.15 0.145 0.1 0.14 0.05 0 5 10 15 20 25 30 35 40 0.135 Frequency, ω 0 5 10 15 20 25 30 35 40 Frequency, ω Fig. 5. Sensitivity improvements due to Q – vertex 3. Fig. 3. Sensitivity improvements due to Q – vertex 1. 0.2 0.24 0.18 0.22 0.16 | S [exp(jωT)] | 0.2 | S [exp(jωT)] | 0.18 0.14 0.16 0.12 0.14 0.1 0.12 0.08 0 5 10 15 20 25 30 35 40 0.1 Frequency, ω 0 5 10 15 20 25 30 35 40 Frequency, ω Fig. 6. Sensitivity improvements due to Q – vertex 4. Fig. 4. Sensitivity improvements due to Q – vertex 2. e e ¨  C. Carath´ odory and L. Fej´ r. Uber den Zusammenhang der Extremen  B. Kouvaritakis, J.A. 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