q - Rosario Toscano - Free

More documents

Recommendations

Info

q where Tw ( s, q) denotes the closed-loop transfer matrix from wi to zi, q ∈ R is the vector of izi decision variables regrouping the entries of the matrices Kp, Ki, Kd, and the time constant τ, λi(q) denotes the i th pole of the closed-loop system and s is the Laplace variable. In the formulation (34) the constraint g1(q) is required to ensure the stability of the closed-loop system. To do so, the parameter λmin must be set to a negative value. Note that the formulations (34) and (35) are quite general and can be used to specify many control objectives. For instance, the formulation (34) includes the PID loop-shaping design problem whereas (35) includes the single or mixed sensitivity PID control problem. In the numerical experiments we will see some applications belonging to theses two kind of control problems. The constrained optimization problem (34) or (35) can be transformed into an unconstrained one, by introducing a new objective function which includes penalty functions (see relation (2) and (3)). Remark concerning the feasibility issue. In many engineering problems the bounds of a feasible search domain are often known a priori because they are linked to purely material, physical considerations. This is not so clear in control problem for which we have to impose a priori an hyperbox search domain containing stabilizing controllers (i.e. potential solutions of the optimal HHHH∞ problem). Finding a priori such a hyperbox is not trivial at all. However, for a given hyperbox search domain it is possible, using HKA, to say whether or not the problem is feasible. More precisely, the feasibility problem can be stated as follows. Given the hyperbox search domain DDDD = { q i ∈ R : q ≤ q i i ≤ qi , i = 1, L, nq } is there a stabilizing controller? This important issue can be treated via HKA by solving the following optimization problem: Minimize ( q) = arg max{ Re( λ ( q)), ∀i} Subject to: J λ i λi ( q) q ≤ q ≤ q , i = 1, L, n i i i q where λi(q) represents the i th pole f the closed-loop system. Let q* the solution found by HKA to the problem (36). If Jλ(q*) < 0, then the problem is feasible within D. In the opposite case, because of the stochastic nature of HKA, this does not necessarily mean that the problem is not feasible (in this case we will say that the problem is probably not feasible). 1.2. Numerical experiments Mixed sensitivity approach. In this example, for comparison purpose, the same optimization 8 problem as the one presented in Kim, Maruta & Sugie, (2008), is considered: Minimize { } [ ] T Subject to: J ( q) = arg max Re( λ ( q)), ∀i , q = q λ 1 2 λi ( q) g ( q) = W ( s) S( s, q) g ( q) = W ( s) T ( s, q) i ∞ ∞ −1 ≤ 0 −1 ≤ 0 8 Note that the optimization problem (37) is of the form (35). S T 1 q 2 q 3 q 4 n (36) (37)
where [ ] T q = q1 q2 q3 q is the vector of decision variables, s is the Laplace variable, S(s,q) is 4 the sensitivity function defined as S(s,q) = 1/(1+L(s,q)), T(s,q) is the closed-loop system defined as T(s,q) = L(s,q)/(1+L(s,q)), L(s,q) is the open-loop transfer function defined as L(s,q) = G(s)K(s,q) where G(s) is the transfer function of the system to be controlled and K(s,q) is the transfer function of the PID controller which depends upon the decision variable as follows: q3 q ⎛ 1 10 s ⎞ 1 ( s, q) = 10 ⎜ ⎜1+ + ⎟ (38) q2 ( q3 − 4 ) ⎝ 10 s 1+ 10 s ⎠ K q Note that the relationship between the decision variables of the optimization problem and the parameters of the PID controller are defined as: K = 10 = (39) p q1 q2 q3 q4 , Ti = 10 , Td = 10 , N 10 This is done to ensure a broader parameter space of (Kp, Ti, Td, N). The frequency-dependent weighting functions WS(s) and WT(s) are set in order to meet the performance specifications of the closed-loop system. The optimization problem (37), has been solved for the magnetic levitation system described in Sugi, Simizu & Imura, (1993). The process model is defined as: 7. 147 G ( s) = (40) ( s − 22. 55)( s + 20. 9)( s + 13. 99) The frequency-dependent weighting functions WS(s) and WT(s) are respectively given as: 5 43. 867( s + 0. 066)( s + 31. 4)( s + 88) S ( s) = , W ( s) = (41) 4 2 s + 0. 1 ( s + 10 ) W T The search space is: 2 ≤ q 1 ≤ 4, −1 ≤ q2 ≤1, −1 ≤ q3 ≤ 1, 1 ≤ q4 ≤ 3 . In this test, we performed the minimization 30 times and we compared our results with those obtained via ALPSO (Augmented Lagrangian Particle Swarm Optimization see Kim, Maruta & Sugie, 2008). The following parameters have been used: N = 50, Nξ = 5 and α = 0.4. The best solutions obtained via ALPSO and HKA are listed in Table 13 and the statistical results are shown in Table 14 (the mark "-" means that the corresponding result is not available). Table 13. Comparison of the best solutions found via ALPSO and HKA. ALPSO HKA q1 3.2548 3.2542 q2 -0.8424 -0.8634 q3 -0.7501 -0.7493 q4 2.3137 2.3139 g1(q) 6.1e-3 -9.6e-4 g2(q) -4.0e-4 -1.2e-3 J(q) -1.7197 -1.7106
Page 1 and 2: Stochastic Stochastic Methods Metho
Page 3 and 4: The problem is then how to tune the
Page 5 and 6: where k T > 0 is a normalizing cons
Page 7 and 8: Number of repetitions of the Metrop
Page 9 and 10: where q1 and q2 are coded with 12 b
Page 11 and 12: i i i i Inversely, the decoding of
Page 13 and 14: Each individual of the population o
Page 15 and 16: 3. Particle Swarm Optimization (PSO
Page 17 and 18: The inertia weight. The inertia coe
Page 19 and 20: 3.3. Advantages and disadvantages o
Page 21 and 22: Our ignorance about the optimum can
Page 23 and 24: Note that all the matrices used in
Page 25 and 26: 5.1. Comparison of HKA with SA, GA
Page 27 and 28: 5.2. Comparison of HKA with other m
Page 29 and 30: and: c ( q) = 1 0. 10471, 7 G = 1.
Page 31: and K(s) is, for instance, the tran
Page 35 and 36: where q is the decision vector whic
Page 37 and 38: We want the residual r insensitive
Page 39 and 40: For comparison, Figure 14 shows the
Page 41 and 42: Constrained test problem (F8) (3 va
Page 43 and 44: Coello, C.A.C. (2002). Theoretical
Page 45: Siarry, P., Berthiau, G., Durbin, F

q - Rosario Toscano - Free

Create successful ePaper yourself

Delete template?

Save as template?