2.1 Double integrator | Contents | 2.3 Harvest system |
x(k+1=a·x(k)+b·u(k),
k=1,2,...,N.
f(x,u)=q·x(N+1)^2+sum(s·x(k)^2+r·u(k)^2),
k=1:N.
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1 | 45 | 100 | 1 | 1 | 1 | 1 | 1 | 16180.3399 |
2 | 45 | 100 | 10 | 1 | 1 | 1 | 1 | 109160.7978 |
3 | 45 | 100 | 1000 | 1 | 1 | 1 | 1 | 10009990.0200 |
4 | 45 | 100 | 1 | 10 | 1 | 1 | 1 | 37015.6212 |
5 | 45 | 100 | 1 | 1000 | 1 | 1 | 1 | 287569.3725 |
6 | 45 | 100 | 1 | 1 | 0 | 1 | 1 | 16180.3399 |
7 | 45 | 100 | 1 | 1 | 1000 | 1 | 1 | 16180.3399 |
8 | 45 | 100 | 1 | 1 | 1 | 0.01 | 1 | 10000.5000 |
9 | 45 | 100 | 1 | 1 | 1 | 1 | 0.01 | 431004.0987 |
10 | 45 | 100 | 1 | 1 | 1 | 1 | 100 | 10000.9999 |
Table 1: Parameter sets for linear-quadratic system
The ten test cases described by Michalewicz [Mic92] are all implemented in objlinq. The values of the parameter sets are shown in table 1. The solution obtained for the first test case is shown in figure 3.
Fig. 1: Optimal control vector for the linear-quadratic system set 1
The linear-quadratic system is identical to a single integrator with positive feedback. A continuous version of this problem using a Simulink model, an s-function and Control System Toolbox routines is implemented in objlinq2. The model used by the objective function is chosen through the first parameter of the function, the default is the Simulink model.
Fig. 2: Block diagram of linear-quadratic system
The Simulink method uses the model in figure 2 (simlinq1).
dx1/dt=x1+u;
y=x1.
G(s)=1/(s-1)
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