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2.1 Double integrator Contents2.3 Harvest system

2 Optimization of dynamic systems


2.2 Linear-quadratic system

The linear-quadratic system [Mic92] is one-dimensional:
x(k+1=a·x(k)+b·u(k), k=1,2,...,N.
The objective function for minimization is defined as:
f(x,u)=q·x(N+1)^2+sum(s·x(k)^2+r·u(k)^2), k=1:N.

set
N
x(0)
s
r
q
a
b
exact solution
145100111 1116180.3399
2451001011 11109160.7978
345100100011 1110009990.0200
4451001101 1137015.6212
545100110001 11287569.3725
645100110 1116180.3399
745100111000 1116180.3399
845100111 0.01110000.5000
945100111 10.01431004.0987
1045100111 110010000.9999

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

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

Block diagram of linear-quadratic system

The Simulink method uses the model in figure 2 (simlinq1).

The s-function solves the linear equations (simlinq2):
dx1/dt=x1+u; y=x1.

and the third method (Control System Toolbox) uses the transfer function:
G(s)=1/(s-1).

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