A simple method to find a robust output feedback controller by random search approach
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A random search algorithm is proposed to find a robust output feedback controller for uncertain linear systems. The algorithm generates random feedback gain matrices and evaluates the closed-loop pole locations. If the poles lie in a specified stable region, the gain matrix is a solution. The probability of finding a solution and the number of trials needed can be estimated. Simulation results demonstrate the effectiveness of using this random approach. The method provides a simple way to design output feedback controllers for systems where traditional techniques are intractable, such as problems with nonconvex constraints.
A simple method to find a robust output feedback controller by random search approach
- 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 2005͒ Abstract This paper presents a simple but effective method for finding a robust output feedback controller via a random search algorithm. The convergence of this algorithm can be guaranteed. Moreover, the probability to find a solution as well as the number of random trials can be estimated. The robustness of the closed-loop system is improved by the minimization of a given cost function reflecting the performance of the controller for a set of plants. Simulation studies are used to demonstrate the effectiveness of the proposed method. © 2006 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Output feedback; Pole placement; Random search algorithm; Robustness; Condition number; Cost function; NP-hard problem 1. 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 the back so that the closed-loop system has some de- time but most of the time; in return, this kind of sirable performances. It is well known that the algorithm runs in polynomial time ͓8͔. The idea of performances of feedback control systems are using a random algorithm to solve a complex mainly determined by the locality of their closed- problem is not new, and was first proposed, in the loop 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, and of 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 objective feedback is more complex. In fact, the static out- of this paper is to present a new random approach put feedback problem in the case where the feed- to find a static output feedback for uncertain linear back 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: [email protected] method, the probability to find a solution as well 0019-0578/2006/$ - see front matter © 2006 ISA—The Instrumentation, Systems, and Automation Society.
- 2. 36 Rosario Toscano / ISA Transactions 45, (2006) 35–44 as 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 sufficient tion 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 exactly pole placement ͑i.e., pole placement in a desirable the same as those specified. This is because the domain͒ can be solved by an appropriate random closed-loop system with poles approximately search 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 point simulation 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 complex paper. plane ͓16͔. As shown in Ref. ͓5͔, the problem of finding an 2. 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 general where x Rn, u Rm, and y R p represent the algorithm for solving the static output feedback state, 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 of rank͓B͔ = 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 made to 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 is where 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 D ͑A , 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. This Ref. ͓21͔͒, there exists a feedback gain matrix K problem is also NP-hard. such that ͑A + BKC͒ = ⌳, where ⌳ is a given set of real and self-conjugate complex numbers, ⌳ = ͕1 , 2 , . . . , n͖ are the desired poles of the 3. Random search algorithm approach closed-loop system, and ͑M͒ is the spectrum of the 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 domain for the existence of a static output feedback to D ʚ C− is presented. Supposing the existence of a solve the problem of multivariable pole placement solution, the following theorem can be used for ͑MVPP͒ for the generic system, i.e., for almost all finding a constrained output feedback matrix for systems ͓21͔. The condition mp Ͼ n is a seminal the system ͑1͒.
- 3. Rosario Toscano / ISA Transactions 45, (2006) 35–44 37 Theorem 3.1. If there exists an output feedback matrix K such that kij ഛ kij ഛ¯ ij ∀ i , j , and such គ k that ͑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 output feedback matrices K defined by K = ͕K Rmϫp:͑A + BKC͒ ʚ D,kij ഛ kij គ ഛ ¯ ij ∀ i, j͖ . k ͑4͒ Let us consider n iterations of the algorithm, the probability so that K K is given by the binomial Fig. 1. Surface of the region D പ D. probability distribution P͕K 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 probability limn→ϱ͑1 − ͒n = 0 the algorithm then certainly P͕͑Ac͒ ʚ D / ͑Ac͒ ʚ C−͖. This probability is converges 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 for necessary 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 2 where C− is the left half plane ͑C is the set of complex 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 P͕͑Ac͒ ues of M . The quantity A͑D പ D͒ is the surface ʚ D / ͑Ac͒ ʚ C−͖ = / P͕͑Ac͒ ʚ C−͖, which gives of the domain D പ D, where D is the specified ഛ 2A͑D പ D͒P͕͑Ac͒ ʚ C−͖ / ͑max͒. 2 region for pole placement and D is the half re- Remark 3.1. The probability can be estimated gion generated by the maximum over K of the ˆ as relative frequency N = Ns / N, where N is the spectral 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. 38 Rosario Toscano / ISA Transactions 45, (2006) 35–44 is to determine the number of samples N in order Table 1 to obtain a reliable probabilistic estimate. More Experiment results. precisely, given the accuracy ⑀ ͓0 , 1͔ and the n Matrix gain and closed-loop poles 2 confidence ␦ ͓0 , 1͔, the minimum of samples N which guarantees that P͕͉ − N͉ ഛ ⑀͖ ജ 1 − ␦ is given by the Chernoff bound ͓25͔ N ˆ 688 K= ͓ 0.2327 7.8452 3.5994 −8.4385 −6.6849 5.4134 ͔ 12.5683 ജ ln͑2 / ␦͒ / ͑2⑀2͒. Thus the probability can be es- ⌳ = ͕−2.3981± 1.3954j , −0.4670± 0.6628j͖ timated 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± 0.8411j͖ by Ns / N. One question arises, the feasibility prob- ͓ ͔ lem. The feasibility of pole placement by con- 1501 −0.5261 0.2432 7.2889 8.6107 strained output feedback is related to the spectral K= −3.4500 −2.6392 −0.9264 radius of the closed-loop state matrix. Indeed, let l ⌳ = ͕−1.1096± 1.4612j , −0.3813± 0.6400j͖ be the minimal distance between the origin of the complex plane and the domain D of the pole placement ͑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 , ͑8͒ ity P͕͓͑Ac͒ ʚ C−͔͖ as well as maxK ͑A + BKC͒ i can 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 is 4. 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 a feedback controller such that the closed-loop sys- smaller 2͑T͒ gives a largest bound of ʈ⌬ʈ2 and tem 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-loop mal sensitivity ͑i.e., maximal robustness͒ of eigen- system can be improved by solving the following values to unstructured perturbation in the system optimization problem: and controller parameters. An analytic solution to the problem of minimal sensitivity in static output feedback design was first given in Ref. ͓26͔. How- minimize J = ʈTʈ2ʈT−1ʈ2 ever, as mentioned in the above paper, the mini- mum achievable condition number has a lower bound ͑see also Ref. ͓27͔͒, the problem may not subject to K K. ͑9͒ have a solution. Therefore the condition number minimization 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 start closed-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. Rosario Toscano / ISA Transactions 45, (2006) 35–44 39 Table 2 Table 3 Optimization 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 1.1175 There exists a bound of performance level ␥max −0.8666 1.5499 −3.2881 such that ⌳ = ͕−0.3659± 0.4066j , −0.0661, −0.6076, −1.0333͖ ∀ K K, J͑K͒ ഛ ␥max . ͑11͒ ͓ ͔ 3857 0.0123 0.9414 0.4097 18.8613 For all levels of performance ␥min Ͻ ␥ Ͻ ␥max there K = 0.2420 0.0000 0.3112 exists 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 62.7497 gorithm 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, −0.0513͖ Proof. Consider an initial matrix K K for which J͑K͒ Ͼ ␥. By Lemma 4.1 there exists ⌬K, with More generally, an analog approach can be used K + ⌬K K, such that J͑K + ⌬K͒ Ͻ J͑K͒. Consider to minimize a given cost function reflecting the n iterations of the algorithm, the probability that performance of the controller for a given set of J͑K + ⌬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. It is clear that limn→ϱ͑1 − ͒n = 0, then repeating steps 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 initial matrix and repeating the reasoning above we see In this section various numerical examples are that the algorithm converges to an element of K␥ presented to illustrate the validity of the proposed which 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. 40 Rosario Toscano / ISA Transactions 45, (2006) 35–44 ΄ ΅ Table 4 0 0 − 0.0034 0 0 Optimization 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 confidence with 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. Using feedback 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 are defined by D = ͕␣ + j  : −2.5ഛ ␣ ഛ −0.3, −1.5ഛ  given by 0.19 and 15.43, respectively. We have ഛ −1.5͖. Using the algorithm given in Remark 1, A͑D പ D͒ = ͑1.2− 0.04͒ ϫ 1 = 1.16. The upper we 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 the given 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 random estimation of P͕͑Ac͒ ʚ C−͖ and maxK ͑Ac͒ are optimization algorithm ͑with d = 0.025͒, we obtain given by 0.086 and 10.64, respectively. We have the matrix gain and spectral condition number A͑D 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 than where 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 for tion algorithm ͑with d = 0.025͒, we obtain the ma- the longitudinal axis of an aircraft modeled by trix gain and spectral condition number shown in G͑s , ͒, where is the system parameters and p Table 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. 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 0.0676 some constraints on the nominal system. The sys- Z ␦e −0.3765 0.0314 tem G͑s , ͒ is given in the following state space M ␦e −10.8791 3.4695 form: 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 ranges The 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 parameters hardware 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, ͑19͒ The output dynamic feedback controllers have the where G0͑s͒ denote the nominal system, and W͑s͒ following 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 + 31.4͒ In order to solve this problem, consider the following cost function: Table 6 Experiment results. ␥0 J͑0 , p͒ Controller parameters n 0.9000 0.8013 Ka = 0.6124, Kq = 0.1122, 1 = 0.0499, 2 = 0.0520 2 0.8000 0.7329 Ka = 0.8874, Kq = 0.5925, 1 = 0.0673, 2 = 0.0579 5 0.7300 0.7204 Ka = 0.7223, Kq = 0.6570, 1 = 0.0599, 2 = 0.0450 16 0.7200 0.7170 Ka = 0.9046, Kq = 0.6471, 1 = 0.0835, 2 = 0.0718 46 0.7150 0.7100 Ka = 1.9004, Kq = 0.7735, 1 = 0.0417, 2 = 0.0107 629
- 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 J͑0,p͒ = ͑21͒ ʈW͑I + G0HK͒−1ʈϱ , otherwise. 1 + ʈW͑I + G0HK͒−1ʈϱ For a given level of nominal performance ␥0 Ͻ 1, level of nominal performance and n the number of a 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 J The proof of convergence of this algorithm is that obtained in Ref. ͓17͔ which obtains similar to that of Theorem 3.1. Thus we find con- ¯ 66 848͑ , p0͒ = 0.7149. In fact, for ⑀ = 0.005 and J troller parameters p0 such that J͑0 , p͒ ഛ ␥0. For ␦ = 0.005, only N = 1057 plants are needed to this 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, for a Gaussian distribution, one can choose ⌰ = ͓0 − 3 , 0 + 3͔, where 0 is the mean and the 6. Conclusion standard deviation. If there exists ⌰ such that wc͑p0͒ = 1, the controller must be rejected because In this paper a simple but effective method to we 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 output mated 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 the distributed ͑i.i.d͒ samples generated according to the probability measure P on the set ⌰. The num- ber of samples necessary to have P͕P͕wc Ͼ wc͖ ˆ ഛ ⑀͖ 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=1 which reflects the performance of the controller most often obtained for a given set of plants. Ob- viously, the optimal controller is obtained for the smallest possible ¯ ͑ , p0͒ which is unknown but it J can be approached iteratively. Table 6 summarizes successive experiments where ␥0 is the specified Fig. 3. Simulation results for various plant parameters.
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- 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 and control.