NONLINEAR CONTROLLER COMPARISON ON A BENCHMARK ...

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To apply this algorithm to the exible beam system, we made use of a Matlab implementation of this algorithm contained in the le hjb.m (see Appendix B). It is a straightforward numerical implementation of the preceding algorithm, and it only required three items: the set , the basis elements f jg N j=1, and the initial stabilizing control u (0) . In the hardware implementation we used the following initializing parameters: =[;:2:2] [;2:5 2:5] [; 2 2 ] [;20 20] f jg = fx 2 1x 2 2x 2 3x 2 4x 1x 2x 1x 3x 1x 4x 2x 3x 2x 4x 3x 4g u (0) (x) =41x 1 ; 1:5x 2 ; 2:6x 3 ; :19x 4: We added higher order terms in simulation, though they only slightly improved the performance: =[;:05:05] [;5 5] [; 2 2 ] [;10 10] f jg = fx 2 1x 2 2x 2 3x 2 4x 1x 2x 1x 3x 1x 4x 2x 3x 2x 4x 3x 4x 3 1x 2x 1x 3 2x 3 1x 3x 3 2x 3x 1x 3 3 x 2x 3 3x 3 1x 4x 3 2x 4x 3 3x 4x 1x 3 4x 2x 3 4x 3x 3 4g u (0) (x) = 120x 1 ; 25x 2 ; 4:5x 3 ; :6x 4: In other words, we made the initial stabilizing control the linearized LQR control developed previously. was constructed so that the control would be stabilizing for displacements of up to 20 cm in either direction and for rotations of up to 90 degrees by the proof mass, in the hardware. The basis functions were simply a set of second order polynomials and their corresponding cross terms, and in the simulation fourth order terms were added as basis functions. This control has several bene cial qualities. First, it uses all of the available outputs to generate a truly non-linear control signal { u(x) can depend on the square of the states, and the g T (x) term renders even a second order SGA control signal nonlinear. Second, the SGA algorithm is approximating an optimal solution: the designer can feel assured that the control approaches optimality with respect to the desired cost functional. Third, the SGA algorithm is easy to tune 26
ecause one can quickly adjust the Q and R matrices to change the state penalty weightings. Finally, this design technique can be interpreted as improving upon the initial stabilizing control (developed through some sub-optimal approach, such as a backstepping or PBC strategy) at each iteration of the algorithm. 3.5.2 The H1 Problem The successive Galerkin approximation technique for the Hamilton-Jacobi- Isaacs (HJI) equation is described in [5]. The basic idea is similar to the previous section except that the successive iteration step requires two nested loops. The nonlinear H1 problem data is given by _x = f(x)+g(x)u + k(x)w Z T 0 l(x)+kuk 2 R dt x(0) = 0 8 T 0 2 Z T 0 kwk 2 P dt where it is also desirable to compute the smallest possible > 0. In other words, the objective is to minimize the L 2 gain from an exogenous disturbance signal, w, to an output de ned by R l(x)+kuk 2 R dt. The solution to this minimization problem is the HJI equations, given by @V T @x f + hT h + 1 4 @V @x 1 ( 2 2 kP ;1 k T ; gR ;1 g T T @V ) @x =0: (3.28) In a manner directly analogous to the previous section we can write the HJI equation in a way that decouples u, w, and V , and then we can solve for the optimal u iteratively: we start, as before, with an initial stabilizing control u (0) , and then iterate between the disturbance and V untilitistheworst disturbance for the given control. Then we update the control u (1) and iteratively compute the worst disturbance for 27
Page 1 and 2: NONLINEAR CONTROLLER COMPARISON ON
Page 3 and 4: BRIGHAM YOUNG UNIVERSITY GRADUATE C
Page 5 and 6: ABSTRACT NONLINEAR CONTROLLER COMPA
Page 7 and 8: Contents Acknowledgments vi List of
Page 9: List of Tables 4.1 Tabular Comparis
Page 12 and 13: 4.19 SGA: Nonlinear H1 vs. Linear O
Page 14 and 15: Figure 1.1: TORA System One of the
Page 16 and 17: 1.3 Literature Review In surveying
Page 19 and 20: Chapter 2 Plant Speci cations and M
Page 21 and 22: Figure 2.2: Mechanical Model of Fle
Page 23 and 24: Figure 2.4: Translational Oscillati
Page 25 and 26: 2.3 Software Set-up All of the cont
Page 27 and 28: Chapter 3 Overview of Control Strat
Page 29 and 30: The optimal gain matrix was given b
Page 31 and 32: 3.3 Passivity-Based Control Certain
Page 33 and 34: limitations, k 1 and k 2 are tuning
Page 35 and 36: Now we regard as the control variab
Page 37: where V satis es the well known Ham
Page 41 and 42: Chapter 4 Simulation Results 4.1 Th
Page 43 and 44: disturbances, whereas the hardware
Page 45 and 46: Response in cm 4 3 2 1 0 −1 −2
Page 47 and 48: Response in cm Response in cm 4 3 2
Page 51 and 52: Response in cm 4 3 2 1 0 −1 −2
Page 57 and 58: It is seen in Figure 4.23 that the
Page 59: all of the physical parameters exce
Page 62 and 63: Response in m 0.05 0.04 0.03 0.02 0
Page 64 and 65: Response in m 0.05 0.04 0.03 0.02 0
Page 66 and 67: Response in m Response in m 0.05 0.
Page 72 and 73: control was that it requires only t
Page 74 and 75: to the Galerkin approximation techn
Page 77 and 78: Bibliography [1] P. Kokotovic M. Ja

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