36 Rosario Toscano / ISA Transactions 45, (2006) 35–44as the number of random trials can be evaluated. result which is less conservative than m + p Ͼ n.The robustness of the closed-loop system is im- Concerning this problem, further developments—proved by the minimization of a given cost func- including some necessary and sufﬁcienttion reﬂecting the performance of the controller conditions—can be found in Refs. ͓22,23,16͔.for a set of plants. The exact pole placement problem for output The paper is organized as follows. In Section 2, feedback is to ﬁnd such a K. However, it is com-the problem of static output feedback is formu- monly recognized that in practical applications,lated. Section 3 shows that the problem of regional the poles assigned are not required to be exactlypole placement ͑i.e., pole placement in a desirable the same as those speciﬁed. This is because thedomain͒ can be solved by an appropriate random closed-loop system with poles approximatelysearch algorithm. The robustness issue is dis- close to the desired one will possess similar de-cussed in Section 4, and Section 5 presents various sired behavior ͓24͔. In fact, from a practical pointsimulation results to demonstrate the effectiveness of view, it is sufﬁcient to consider the pole place-of this approach. Finally, Section 6 concludes this ment in a speciﬁed stable region D of the complexpaper. plane ͓16͔. As shown in Ref. ͓5͔, the problem of ﬁnding an2. Problem statement output feedback matrix K such that kij ഛ kij គ ഛk ¯ ij ∀ i , j, and such that A + BKC is a stable ma- Consider a multivariable linear dynamic system trix is NP-hard. In a more recent work, Fu ͓4͔ ͭ ͮdescribed by shows that the problem of pole placement via un- x͑t͒ = Ax͑t͒ + Bu͑t͒ ˙ constrained static output feedback is also NP-hard. , ͑1͒ This implies that no efﬁcient algorithm exists for y ͑t͒ = Cx͑t͒ solving such problems. In other words if a generalwhere x Rn, u Rm, and y R p represent the algorithm for solving the static output feedbackstate, input, and output vectors, respectively, A problem is derived, it is an exponential-time algo- Rnϫn, B Rnϫm, and C R pϫn are known con- rithm.stant matrices. As usual, it is assumed that An alternative approach to solve this kind ofrank͓B͔ = m and rank͓C͔ = p. By applying a con- problem is to use a nondeterministic algorithm.stant output feedback law, The drawback of this approach is that the prob- ability that the algorithm fails is not equal to zero u͑t͒ = r͑t͒ + Ky ͑t͒ , ͑2͒ for a ﬁnite number of iterations, but can be madeto Eq. ͑1͒, the closed-loop system is given as arbitrarily small as the number of iterations in- ͭ x͑t͒ = ͑A + BKC͒x͑t͒ + Br͑t͒ ˙ y ͑t͒ = Cx͑t͒ , ͮ ͑3͒ creases. In return for this compromise, one hopes that the algorithm runs in polynomial time. In the next section a random search algorithm iswhere r Rm is the reference input vector and K proposed in order to ﬁnd a constrained output Rmϫp is the output feedback gain matrix. feedback matrix K ͑i.e., such that kij ഛ kij គ It was established that under the condition of ഛk ¯ ij ∀ i , j͒ such that ͑A + BKC͒ ʚ D, where D͑A , B͒ controllable and ͑C , A͒ observable and that is a speciﬁed region of the complex plane deter-m + p Ͼ n ͑see Refs. ͓19,20͔͒ or mp Ͼ n ͑e.g., see mined in order to obtain a desired behavior. ThisRef. ͓21͔͒, there exists a feedback gain matrix K problem is also NP-hard.such that ͑A + BKC͒ = ⌳, where ⌳ is a given setof real and self-conjugate complex numbers, ⌳= ͕1 , 2 , . . . , n͖ are the desired poles of the 3. Random search algorithm approachclosed-loop system, and ͑M͒ is the spectrum ofthe square matrix M . In this section a possible approach to solve the More precisely, mp Ͼ n is a sufﬁcient condition problem of pole placement in a desirable domainfor the existence of a static output feedback to D ʚ C− is presented. Supposing the existence of asolve the problem of multivariable pole placement solution, the following theorem can be used for͑MVPP͒ for the generic system, i.e., for almost all ﬁnding a constrained output feedback matrix forsystems ͓21͔. The condition mp Ͼ n is a seminal the system ͑1͒.
Rosario Toscano / ISA Transactions 45, (2006) 35–44 37 Theorem 3.1. If there exists an output feedbackmatrix K such that kij ഛ kij ഛ¯ ij ∀ i , j , and such គ kthat ͑A + BKC͒ ʚ D, with D ʚ C−, then the algo-rithm 1. Generate a m ϫ p matrix K with random uniformly distributed elements kij on the in- tervals ͓kij ,¯ ij͔ ∀ i , j . គ k 2. If ͑A + BKC͒ D go to step 1, otherwise stop.converges certainly to a solution. Proof. Let K be the set of D-stabilizing outputfeedback matrices K deﬁned by K = ͕K Rmϫp:͑A + BKC͒ ʚ D,kij ഛ kij គ ഛ ¯ ij ∀ i, j͖ . k ͑4͒Let us consider n iterations of the algorithm, theprobability so that K K is given by the binomial Fig. 1. Surface of the region D പ D.probability distributionP͕K K͖ = ͫ n! r!͑n − r͒! r͑1 − ͒n−r ͬ r=0 = ͑1 − ͒ , n ഛ ␦ which gives n ജ ln͑␦͒ / ln͑1 − ͒. Suppose now that ͑Ac͒ ʚ C−, with Ac = A + BKC, the probabil- ity that K K is equal to the probability that ͑5͒ ͑Ac͒ ʚ D. Consider n random trials generating n independent identically distributed matrices K.where r is the number of successes ͑i.e., the num- If n goes to inﬁnity, the ratio between the num-ber of times that K K͒ and the probability of ber of successes ns and the number of trials n,elementary success. For Ͼ 0 it is clear that is equal, by deﬁnition, to the probabilitylimn→ϱ͑1 − ͒n = 0 the algorithm then certainly P͕͑Ac͒ ʚ D / ͑Ac͒ ʚ C−͖. This probability isconverges to a solution. bounded by the ratio between the surface Corollary 3.1. The average number of iterations A͑D പ D͒, where D is the speciﬁed region fornecessary to obtain a solution with a conﬁdence at pole placement and D is the half region ͓by as-least equal to 1 − ␦ is given by sumption ͑Ac͒ ʚ C−͔ generated by the maximum ln͑␦͒ over K of the spectral radius: nജ , with 0 Ͻ ln͑1 − ͒ ns P͕͑Ac͒ ʚ D/͑Ac͒ ʚ C−1͖ = lim 2A͑D പ D͒ P͕͑A + BKC͒ ʚ C−͖ n→ϱ n ഛ , ͑6͒ ͓max ͑A + BKC͔͒2 2A͑D പ D͒ K ഛ ͑7͒ max 2where C− is the left half plane ͑C is the set ofcomplex numbers͒, P͕͑A + BKC͒ ʚ C−͖ is the with max = maxK ͑Ac͒. The probability of el-probability that A + BKC is Hurwitz ͑i.e., a stable ementary success is given by matrix͒. ͑M͒ is the spectral radius of the matrix = P͕͓͑Ac͒ ʚ C−͔ പ ͓͑Ac͒ ʚ D͔͖, by the condi-M , that is ͑M͒ = max͉͑i͉͒, with i the eigenval- tional probability we have P͕͑Ac͒ues of M . The quantity A͑D പ D͒ is the surface ʚ D / ͑Ac͒ ʚ C−͖ = / P͕͑Ac͒ ʚ C−͖, which givesof the domain D പ D, where D is the speciﬁed ഛ 2A͑D പ D͒P͕͑Ac͒ ʚ C−͖ / ͑max͒. 2region for pole placement and D is the half re- Remark 3.1. The probability can be estimatedgion generated by the maximum over K of the ˆ as relative frequency N = Ns / N, where N is thespectral radius ͑see Fig. 1͒. total number of samples and Ns the number of Proof. From Eq. ͑5͒ we want to have ͑1 − ͒n samples such that ͑A + BKC͒ ʚ D. The problem
38 Rosario Toscano / ISA Transactions 45, (2006) 35–44is to determine the number of samples N in order Table 1to obtain a reliable probabilistic estimate. More Experiment results.precisely, given the accuracy ⑀ ͓0 , 1͔ and the n Matrix gain and closed-loop poles 2conﬁdence ␦ ͓0 , 1͔, the minimum of samples Nwhich guarantees that P͕͉ − N͉ ഛ ⑀͖ ജ 1 − ␦ isgiven by the Chernoff bound ͓25͔ N ˆ 688 K= ͓ 0.2327 7.8452 3.5994 −8.4385 −6.6849 5.4134 ͔ 12.5683ജ ln͑2 / ␦͒ / ͑2⑀2͒. Thus the probability can be es- ⌳ = ͕−2.3981± 1.3954j , −0.4670± 0.6628j͖timated using the following algorithm. 1. Choose a number of iterations N such that 1621 K= ͓ 7.7624 8.0917 4.0072 −8.8633 −6.9531 4.4514 ͔ 46.7701 N ജ ln͑2 / ␦͒ / ͑2⑀2͒. ⌳ = ͕−0.4981± 1.0898j , −2.4505± 0.3201j͖ 2. Generate a m ϫ p matrix K with random ͓ ͔ uniformly distributed elements kij on the in- 403 0.9900 2.1420 6.5898 8.0678 K= tervals ͓kij ,¯ ij͔ ∀ i , j . គ k −1.0529 −3.3637 2.7482 3. If ͑A + BKC͒ ʚ D then Ns = Ns + 1. ⌳ = ͕−2.4673, −0.3668± 0.9934j , −0.3957͖ 4. If the number of iterations is incomplete go to step 2, otherwise stop.The estimation of the probability is then given 175 K= ͓ 0.8021 3.6025 3.5335 −7.7666 −6.0396 3.0155 ͔ 19.8294 ⌳ = ͕−2.3630± 0.6832j , −0.3297± 0.8411j͖by Ns / N. One question arises, the feasibility prob- ͓ ͔lem. The feasibility of pole placement by con- 1501 −0.5261 0.2432 7.2889 8.6107strained output feedback is related to the spectral K= −3.4500 −2.6392 −0.9264radius of the closed-loop state matrix. Indeed, let l ⌳ = ͕−1.1096± 1.4612j , −0.3813± 0.6400j͖be the minimal distance between the origin of thecomplex plane and the domain D of the poleplacement ͑see Fig. 1͒. If maxK ͑A + BKC͒ Ͻ l,the problem is not feasible. Note that the probabil- ʈ⌬ʈ2 Ͻ min Re͑− i͒/2͑T͒ , ͑8͒ity P͕͓͑Ac͒ ʚ C−͔͖ as well as maxK ͑A + BKC͒ ican be estimated using the same approach as de-scribed in the above algorithm. where ʈ⌬ʈ2 is the 2-norm or spectral norm of ⌬, i ͑i = 1 , 2 . . . , n͒ are eigenvalues of A + BKC, 2͑T͒ is the spectral condition number of T, that is4. Robustness issue 2͑T͒ = ʈTʈ2ʈT−1ʈ2, and T is the eigenvector matrix In this section, our objective is to ﬁnd an output of A + BKC. From inequality ͑8͒ one can see that afeedback controller such that the closed-loop sys- smaller 2͑T͒ gives a largest bound of ʈ⌬ʈ2 andtem remains stable for a large variety of plants. thus increases the set of plants which can be sta-For this purpose, we consider the problem of mini- bilized. Hence the robustness of the closed-loopmal sensitivity ͑i.e., maximal robustness͒ of eigen- system can be improved by solving the followingvalues to unstructured perturbation in the system optimization problem:and controller parameters. An analytic solution tothe problem of minimal sensitivity in static outputfeedback design was ﬁrst given in Ref. ͓26͔. How- minimize J = ʈTʈ2ʈT−1ʈ2ever, as mentioned in the above paper, the mini-mum achievable condition number has a lowerbound ͑see also Ref. ͓27͔͒, the problem may not subject to K K. ͑9͒have a solution. Therefore the condition numberminimization approach is usually adopted. More A suboptimal solution of this optimization prob-precisely, if an additive uncertainty ⌬ exists in the lem can be found using Theorem 4.1. Let us startclosed-loop system matrix, according to Theorem with Lemma 4.1.6 in Ref. ͓27͔, the closed-loop state matrix A + ⌬ Lemma 4.1. There exists an optimal level of per-+ BKC is Hurwitz if formance ␥min Ͼ 1 such that
Rosario Toscano / ISA Transactions 45, (2006) 35–44 39Table 2 Table 3Optimization result. Experiment results. n Matrix gain and closed-loop poles 2 n Matrix gain and closed-loop poles 2 ͓ ͔ ͓ ͔75910 1.5474 7.7891 8.5192 5.2715 646 −2.8917 2.2234 0.0715 167.4183 K= −1.6813 −3.7358 −0.4161 K = −4.5221 0.1325 0.6331 ⌳ = ͕−2.4994, −0.3000± 1.4976j , −0.3004͖ −1.6834 2.6855 −2.6968 ⌳ = ͕−0.9769± 0.3738j , −0.1467± 0.4553j , −0.0537͖ ∃ K* K, J͑K*͒ = ␥min ഛ J͑K͒, ∀ K K. ͓ ͔ 6434 2.7175 −1.7948 −0.2750 677.7658 ͑10͒ K = −1.7312 4.5429 1.1175There exists a bound of performance level ␥max −0.8666 1.5499 −3.2881such that ⌳ = ͕−0.3659± 0.4066j , −0.0661, −0.6076, −1.0333͖ ∀ K K, J͑K͒ ഛ ␥max . ͑11͒ ͓ ͔ 3857 0.0123 0.9414 0.4097 18.8613For all levels of performance ␥min Ͻ ␥ Ͻ ␥max there K = 0.2420 0.0000 0.3112exists a nonempty set of solutions K␥ deﬁned by 0.6813 0.3795 0.8318 ⌳ = ͕−1.1476, −0.4345, ± 0.1868j , K␥ = ͕K K : J͑K͒ ഛ ␥͖ . ͑12͒ −0.0669, −0.0403͖ Theorem 4.1. For a given level of performance ␥with ␥min Ͻ ␥ Ͻ ␥max, the random optimization al- ͓ ͔ 10084 −2.8561 1.7753 0.2511 62.7497gorithm converges certainly to a solution K K␥: K = −4.2172 1.5954 1.5051 −3.3913 0.2017 −0.4808 1. Select an initial output feedback matrix K K, and a domain of exploration ͓−d , d͔, ⌳ = ͕−1.1092± 0.0535j , d Ͼ 0. −0.1583± 0.3618j , −0.0466͖ 2. Generate a m ϫ p matrix ⌬K with random ͓ ͔ −0.0125 0.1543 0.6328 uniformly distributed elements ⌬kij on the 5458 38.0574 interval ͓−d , d͔ ∀ i , j , such that K + ⌬K K = 0.2996 −0.3410 0.7724 K. −4.0131 −1.6788 0.0397 3. If J͑K + ⌬K͒ Ͻ J͑K͒ let K = K + ⌬K. ⌳ = ͕−1.1353, −0.1324+ 0.4316j , 4. If J͑K͒ Ͼ ␥, go to step 2, otherwise stop. −0.1272, −0.0513͖Proof. Consider an initial matrix K K for whichJ͑K͒ Ͼ ␥. By Lemma 4.1 there exists ⌬K, with More generally, an analog approach can be usedK + ⌬K K, such that J͑K + ⌬K͒ Ͻ J͑K͒. Consider to minimize a given cost function reﬂecting then iterations of the algorithm, the probability that performance of the controller for a given set ofJ͑K + ⌬K͒ Ͼ J͑K͒ is given by ͑1 − ͒n ͑see the plants ͑see Example 3 below͒.proof of Theorem 3.1͒, where Ͼ 0 is the prob-ability of success that is Pr͕J͑K + ⌬K͒ Ͻ J͑K͖͒. Itis clear that limn→ϱ͑1 − ͒n = 0, then repeatingsteps 2–4 we ﬁnally ﬁnd ⌬K such that J͑K+ ⌬K͒ Ͻ J͑K͒. If J͑K + ⌬K͒ ഛ ␥ then K + ⌬K 5. Simulation results K␥, if not, we consider K + ⌬K as a new initialmatrix and repeating the reasoning above we see In this section various numerical examples arethat the algorithm converges to an element of K␥ presented to illustrate the validity of the proposedwhich is a suboptimal solution. Obviously, the op- approach.timal solution is given by the smallest level of Example 1. Consider a four-state, two-input,performance ␥min which is unknown. three-output aircraft example ͓28͔ given by
40 Rosario Toscano / ISA Transactions 45, (2006) 35–44 ΄ ΅Table 4 0 0 − 0.0034 0 0Optimization result. 0 − 0.0410 0.0013 0 0 n Matrix gain and closed-loop poles 2 A= 0 0 − 1.1471 0 0 , ͓ ͔6288 −0.0135 0.0374 0.0870 6.9855 0 0 − 0.0036 0 0 K = 0.1409 −0.0761 0.3669 0 0.0940 0.0057 0 − 0.0510 −0.2131 −0.1487 1.1119 ΄ ΅ ⌳ = ͕−1.1435, −0.2865, −0.0422± 0.0474j , − 1.0000 0 0 −0.0414͖ 0 0 0 B= 0 0 0.9480 , ΄ ΅ 0.9160 − 1.0000 0 − 0.037 0.0123 0.00055 − 1.0 − 0.5980 0 0 0 0 1.0 0 A= , ΄ ΅ − 6.37 0 − 0.23 0.0618 0 0 0 0 1 1.25 0 0.016 − 0.0457 C= 1 0 0 0 0 ͑14͒ 0 0 0 1 0 ΄ ΅ 0.00084 0.000236 with its open-loop poles at 0, 0, −0.041, −0.051, ΄ ΅ 0 1 0 0 and −1.1471. We want to ﬁnd an output feedback 0 0 B= , C= 0 0 1 0 controller K, with ͉kij͉ ഛ 5 ∀ i , j , such that the 0.08 0.804 closed-loop poles are in the region deﬁned by D 0 0 0 1 − 0.0862 − 0.0665 = ͕␣ + j␤ : −1.2ഛ ␣ ഛ −0.04, −0.5ഛ ␤ ഛ 0.5͖. Us- ing the algorithm given in Remark 1, we obtain ͑13͒ = 5.13ϫ 10−4, and the average number of itera- tions necessary to ﬁnd a solution with a conﬁdencewith its open-loop poles at −0.0105, −0.2009, 1 − ␦ = 0.995 is 10 333. The upper bound given in−0.0507± 1.1168j. We want to ﬁnd an output Corollary 3.1 can be evaluated as follows. Usingfeedback controller K, with ͉kij͉ ഛ 10∀ i , j , the same principle given in Remark 3.1, the esti-such that the closed-loop poles are in the region mation of P͕͑Ac͒ ʚ C−͖ and maxK ͑Ac͒ aredeﬁned by D = ͕␣ + j ␤ : −2.5ഛ ␣ ഛ −0.3, −1.5ഛ ␤ given by 0.19 and 15.43, respectively. We haveഛ −1.5͖. Using the algorithm given in Remark 1, A͑D പ D͒ = ͑1.2− 0.04͒ ϫ 1 = 1.16. The upperwe obtain = 0.003, and the average number of bound of the probability is then ഛ 6.3ϫ 10−4.iterations necessary to ﬁnd a solution with a con- Table 3 summarizes various experimentations,ﬁdence 1 − ␦ = 0.995 is 1763. The upper bound where n is the number of iterations and 2 thegiven in Corollary 3.1 can be evaluated as follows. spectral condition number.Using the same principle given in Remark 3.1, the For the best result 2 = 18.86, using the randomestimation of P͕͑Ac͒ ʚ C−͖ and maxK ͑Ac͒ are optimization algorithm ͑with d = 0.025͒, we obtaingiven by 0.086 and 10.64, respectively. We have the matrix gain and spectral condition numberA͑D പ D͒ = ͑2.5− 0.3͒ ϫ 3 = 6.6. The upper shown in Table 4.bound of the probability is then ഛ 0.0032.Table 1 summarizes various experimentations, This spectral condition number is better thanwhere n is the number of iterations and 2 the that obtained in Ref. ͓29͔, which have 2 = 9.5.spectral condition number. Example 3. This example concerns the design of For the best result, using the random optimiza- an output dynamic feedback controller K͑s , p͒ fortion algorithm ͑with d = 0.025͒, we obtain the ma- the longitudinal axis of an aircraft modeled bytrix gain and spectral condition number shown in G͑s , ͒, where is the system parameters and pTable 2. the controller parameters. The closed-loop system Example 2. Consider a ﬁve-state, three-input, is shown in Fig. 2; for more details see Ref. ͓30͔.three-output pilot plant evaporator model ͓29͔ The problem is to minimize the weighted sensitiv-given by ity function over a set of uncertain plants, given
Rosario Toscano / ISA Transactions 45, (2006) 35–44 41 Table 5 Parameters of the aircraft model. Parameter Mean ͑0͒ Standard deviation ͑͒ Z␣ −0.9381 0.0736 Fig. 2. Block diagram of the closed-loop system. Zq 0.0424 0.0035 M␣ 1.6630 0.1385 Mq −0.8120 0.0676some constraints on the nominal system. The sys- Z ␦e −0.3765 0.0314tem G͑s , ͒ is given in the following state space M ␦e −10.8791 3.4695form: A= ͫ Z␣ 1 − Zq M␣ Mq , ͬ ͫ ͬB= Z ␦e M ␦e , C= ͫ ͬ 1 0 0 1 . ͫ K͑s,p͒ = − Ka − Kq 1 + s1 1 + s2 . ͬ ͑17͒ The controller parameters p = ͓KaKq12͔T have ͑15͒ uniform distributions in the rangesThe system parameters = ͓Z␣ZqM ␣M qZ␦eM ␦e͔T Ka ͓0,2͔, Kq ͓0,1͔, 1 ͓0.01,0.1͔,have Gaussian distribution with means and stan-dard deviations as in Table 5. 1 ͓0.01,0.1͔ . ͑18͒ The transfer function H͑s͒ models the different The objective is to ﬁnd the controller parametershardware components, such as the sensor, the ac- which solve the following problem:tuators, etc. It is given by minʈW͑I + GHK͒−1ʈϱ, H͑s͒ = 0.000 697s2 − 0.0397s + 1 0.000 867s2 + 0.0591s + 1 . ͑16͒ such that Ͱ 0.75KG0H 1 + 1.25KG0H Ͱ ϱ ഛ 1, ͑19͒The output dynamic feedback controllers have the where G0͑s͒ denote the nominal system, and W͑s͒following structure: is the weighting function given by ΄ ΅ 2.8 ϫ 6.28 ϫ 31.4 0 ͑s + 6.28͒͑s + 31.4͒ W͑s͒ = . ͑20͒ 2.8 ϫ 6.28 ϫ 3.14 0 ͑s + 6.28͒͑s + 31.4͒In order to solve this problem, consider the following cost function:Table 6Experiment results. ␥0 J͑0 , p͒ Controller parameters n0.9000 0.8013 Ka = 0.6124, Kq = 0.1122, 1 = 0.0499, 2 = 0.0520 20.8000 0.7329 Ka = 0.8874, Kq = 0.5925, 1 = 0.0673, 2 = 0.0579 50.7300 0.7204 Ka = 0.7223, Kq = 0.6570, 1 = 0.0599, 2 = 0.0450 160.7200 0.7170 Ka = 0.9046, Kq = 0.6471, 1 = 0.0835, 2 = 0.0718 460.7150 0.7100 Ka = 1.9004, Kq = 0.7735, 1 = 0.0417, 2 = 0.0107 629
42 Rosario Toscano / ISA Transactions 45, (2006) 35–44 Ͱ Ͱ Ά · 0.75KG0H 1, if the pair ͑G0,K͒ is unstable, or Ͼ1 1 + 1.25KG0H ϱ J͑0,p͒ = ͑21͒ ʈW͑I + G0HK͒−1ʈϱ , otherwise. 1 + ʈW͑I + G0HK͒−1ʈϱFor a given level of nominal performance ␥0 Ͻ 1, level of nominal performance and n the number ofa suboptimal controller can be found using the fol- iterations.lowing random search algorithm: For the best result J͑0 , p͒ = 0.7100, Fig. 3 shows the response of the system from the initial 1. Select a nominal level of performance ␥0 conditions y 1 = 1, y 2 = 1 for various ⌰. Ͻ 1. For this best result J͑0 , p͒ = 0.7100, the worst 2. Generate a controller parameter p with ran- case performance evaluated for 70 000 plants is dom uniformly distributed elements on the intervals deﬁned in Eq. ͑18͒. ˆ wc = 0.7549 and the average performance evalu- 3. If J͑0 , p͒ Ͼ ␥0 go to step 2, otherwise stop. ated for the same number of plants is ¯ 70000͑ , p0͒ = 0.7117. This result is better than JThe proof of convergence of this algorithm is that obtained in Ref. ͓17͔ which obtainssimilar to that of Theorem 3.1. Thus we ﬁnd con- ¯ 66 848͑ , p0͒ = 0.7149. In fact, for ⑀ = 0.005 and Jtroller parameters p0 such that J͑0 , p͒ ഛ ␥0. For ␦ = 0.005, only N = 1057 plants are needed tothis controller, it is crucial to verify that the worst evaluate the worst case performance and the aver-case performance is such that wc͑p0͒ age performance. This also is a good result com-= sup⌰J͑ , p0͒ Ͻ 1, where ⌰ is the set of more pared with N = 66 848 ͓17͔.representative system parameters. For instance, fora Gaussian distribution, one can choose ⌰ = ͓0− 3 , 0 + 3͔, where 0 is the mean and the 6. Conclusionstandard deviation. If there exists ⌰ such thatwc͑p0͒ = 1, the controller must be rejected because In this paper a simple but effective method towe want at least stability in the worst situation. ﬁnd a robust output feedback controller via a ran-The worst case performance wc͑p0͒ can be esti- dom search algorithm was presented. The outputmated using wc͑p0͒ = supiJ͑i , p0͒, where i ⌰ ˆ feedback controller can be static ͑see Examples 1 and 2͒ or dynamic ͑see Example 3͒. The robust-with i = 1 , . . . , N are N independent and identically ness of the closed-loop system is improved by thedistributed ͑i.i.d͒ samples generated according tothe probability measure P on the set ⌰. The num-ber of samples necessary to have P͕P͕wc Ͼ wc͖ ˆഛ ⑀͖ ജ 1 − ␦, for a given ⑀ ͓0 , 1͔ and ␦ ͓0 , 1͔,is such that N ജ ln͑1 / ␦͒ / ln͓1 / ͑1 − ⑀͔͒ ͑see Ref.͓14͔͒. In the same way, one can compute the aver-age performance N ¯ ͑,p ͒ = 1 ͚ J͑ ,p ͒ JN ͑22͒ 0 i 0 N i=1which reﬂects the performance of the controllermost often obtained for a given set of plants. Ob-viously, the optimal controller is obtained for thesmallest possible ¯ ͑ , p0͒ which is unknown but it Jcan be approached iteratively. Table 6 summarizessuccessive experiments where ␥0 is the speciﬁed Fig. 3. Simulation results for various plant parameters.
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44 Rosario Toscano / ISA Transactions 45, (2006) 35–44 Rosario Toscano was born in Catania, Italy, on 28 September 1962. He received his engineering degree in 1994, from the Conser- vatoire National des Arts et Métiers ͑CNAM͒, and Ph.D. de- gree in 2000 from the Ecole Cen- trale de Lyon. He is currently an assistant professor at the Ecole Nationale d’Ingénieurs de Saint- Etienne ͑ENISE͒. His research in- terests in the Laboratory of Tri- bology and Dynamical Systems ͑LTDS͒ include fault detection,robust control, and multimodel approach applied to diagnosis andcontrol.