A simple method to find a robust output feedback controller by random search approach

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A simple method to find a robust output feedback controller
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A simple method to find a robust output feedback controller by random search approach

  1. 1. ISA Transactions® Volume 45, Number 1, January 2006, pages 35–44 A simple method to find a robust output feedback controller by random search approach R. Toscano* Laboratoire de Tribologie et de Dynamique des Systèmes CNRS UMR5513, ECL/ENISE, 58 rue Jean Parot, 42023 Saint-Etienne Cedex 2, France ͑Received 19 October 2004; accepted 25 May 2005͒Abstract This paper presents a simple but effective method for finding a robust output feedback controller via a random searchalgorithm. The convergence of this algorithm can be guaranteed. Moreover, the probability to find a solution as well asthe number of random trials can be estimated. The robustness of the closed-loop system is improved by theminimization of a given cost function reflecting the performance of the controller for a set of plants. Simulation studiesare used to demonstrate the effectiveness of the proposed method. © 2006 ISA—The Instrumentation, Systems, andAutomation Society.Keywords: Output feedback; Pole placement; Random search algorithm; Robustness; Condition number; Cost function; NP-hard problem1. Introduction Starting from this negative result numerous progress has been made which modifies our notion The static output feedback is an important issue of solving a given problem. In particular, random-not yet entirely solved ͑e.g., see Refs. ͓1–4͔͒. The ized algorithms have recently received more atten-problem can be stated as follows: given a linear tion in the literature. Indeed, for a randomized al-time-invariant system, find a static output feed- gorithm it is not required that it works all of theback so that the closed-loop system has some de- time but most of the time; in return, this kind ofsirable performances. It is well known that the algorithm runs in polynomial time ͓8͔. The idea ofperformances of feedback control systems are using a random algorithm to solve a complexmainly determined by the locality of their closed- problem is not new, and was first proposed, in theloop poles; it follows that a natural design ap- domain of automatic control, by Matyas ͓9͔; fur-proach to find a static output feedback is by means ther developments can be found in Ref. ͓10͔, andof pole placement. Compared to pole placement references therein, and Refs. ͓11–18͔.via state feedback, the same problem via output In the same way as Ref. ͓16͔, the main objectivefeedback is more complex. In fact, the static out- of this paper is to present a new random approachput feedback problem in the case where the feed- to find a static output feedback for uncertain linearback gains are constrained to lie in some intervals systems, which is simple and easy to use. Com-is NP-hard ͓5–7͔. pared with the work of Khaki-Sedigh and Bavafa- Toosi ͓16͔, the main contribution of the present paper is a mathematical justification in the use of *Tel.: ϩ33 477 43 84 84; fax: ϩ33 477 43 84 99. E-mail the random search approach. In the proposed address: toscano@enise.fr method, the probability to find a solution as well0019-0578/2006/$ - see front matter © 2006 ISA—The Instrumentation, Systems, and Automation Society.
  2. 2. 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.
  3. 3. 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 P͕␭͑Ac͒ 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 D␳͒, 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 P͕␭͑Ac͒ ʚ D/␭͑Ac͒ ʚ C−1͖ = lim 2A͑D പ D␳͒ P͕␭͑A + BKC͒ ʚ C−͖ n→ϱ n ഛ , ͑6͒ ␲͓max ␳͑A + BKC͔͒2 2A͑D പ D␳͒ K ഛ ͑7͒ ␲␳max 2where C− is the left half plane ͑C is the set ofcomplex numbers͒, P͕␭͑A + 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 = P͕͓␭͑Ac͒ ʚ C−͔ പ ͓␭͑Ac͒ ʚ D͔͖, by the condi-M , that is ␳͑M͒ = max͉͑␭i͉͒, with ␭i the eigenval- tional probability we have P͕␭͑Ac͒ues of M . The quantity A͑D പ D␳͒ is the surface ʚ D / ␭͑Ac͒ ʚ C−͖ = ␰ / P͕␭͑Ac͒ ʚ C−͖, which givesof the domain D പ D␳, where D is the specified ␰ ഛ 2A͑D പ D␳͒P͕␭͑Ac͒ 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
  4. 4. 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 P͕͓␭͑Ac͒ 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
  5. 5. 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
  6. 6. 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 P͕␭͑Ac͒ 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 D␳͒ = ͑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 P͕␭͑Ac͒ 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 D␳͒ = ͑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
  7. 7. 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 + s␶1 1 + s␶2 . ͬ ͑17͒ The controller parameters p = ͓KaKq␶1␶2͔T 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 J͑␪0 , 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
  8. 8. 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 J͑␪0,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 J͑␪0 , 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 J͑␪0 , 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 J͑␪0 , 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 J͑␪0 , 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 + 3␴͔, 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 = sup␪iJ͑␪i , 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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  10. 10. 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.

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