Robust controller for nonlinear MIMO systems
View Patent ↗The robust controller for nonlinear MIMO systems uses a radial basis function (RBF) neural network to generate optimal control signals abiding by constraints, if any, on the control signal or on the system output. The weights of the neural network are trained in the negative direction of the gradient of output squared error. Nonlinearities in the system, as well as variations in system parameters, are handled by the robust controller. Simulation results are included in the end to assess the performance of the proposed controller.
1. A robust controller for nonlinear MIMO systems, comprising:
a radial basis function (RBF) neural network accepting reference trajectory inputs for a nonlinear process, the RBF neural network having an unconstrained output responsive to the nonlinear process control inputs;
means for transforming the RBF output into constrained control outputs acceptable by the nonlinear process, the nonlinear process having inputs accepting the constrained control outputs from the transforming means;
a summer accepting high fidelity and noisy outputs of the nonlinear process, the summer providing a computed sum of the high fidelity and noisy outputs of the nonlinear process;
a difference calculator performing a difference calculation between the reference trajectory control inputs and the computed sum provided by the summer, a result of the difference calculation being provided by an output of the difference calculator, the difference calculator output representing an error of the nonlinear process output, the nonlinear process output error being characterized by an equation:
e ( k )=[ e 1 ( k ) . . . e m ( k )] T ;
an adaptation algorithm processor accepting the output of the difference calculator;
a linear estimator outputting a linear estimation signal to the adaptation algorithm processor; and
a weight updater accepting an output of the adaptation algorithm processor, the weight updater having a weight updater output connected to the RBF network for adjusting weights of the RBF network, the weight updater output being characterized by an equation,
w
j
(
k
+
1
)
=
w
j
(
k
)
+
2
η
∑
l
=
1
m
e
l
(
k
)
(
ψ
lj
α
2
β
ϕ
(
k
-
1
)
ⅇ
β
w
j
T
ϕ
(
k
-
1
)
(
ⅇ
β
w
j
T
ϕ
(
k
-
1
)
+
1
)
2
+
d
lj
α
2
β
ϕ
(
k
)
ⅇ
β
w
j
T
ϕ
(
k
)
(
ⅇ
β
w
j
T
ϕ
(
k
)
+
1
)
2
)
.
,
where e l (k) corresponds to error at the l th output, ψ lj is the element at the l th row and j th column of the matrix Ψ, η is the learning rate of the RBF neural network, w j is the vector for the weights of j th RBF output, m is the number of outputs of the process, and φ(k) is the basis function vector, α is an upper and lower limit of a control signal constraint, and β is a slope adjustment parameter of a linear part of the transforming means.
2. The robust controller for nonlinear MIMO systems according to claim 1 , wherein said linear estimator includes an input accepting an output signal from the nonlinear process.
3. The robust controller for nonlinear MIMO systems according to claim 1 , further comprising a stabilizing controller disposed in operable communication between said difference calculator and said adaptation algorithm processor;
wherein said means for transforming includes a summing calculator disposed in operable communication between said stabilizing controller and the nonlinear process, the summing calculator accepting as operands an output of said stabilizing controller and the unconstrained output of said RBF neural network, a result computed by said summing calculator forming the constrained control output values fed to the nonlinear process.
4. The robust controller for nonlinear MIMO systems according to claim 1 , wherein the nonlinear process is an automatic generation control process of a power distribution system.
5. The robust controller for nonlinear MIMO systems according to claim 1 , further comprising means for training the weights of said RBF network in a negative direction of a derivative of a cost function I characterized by the equation: I=e T (k)e(k).
6. The robust controller for nonlinear MIMO systems according to claim 1 , wherein said stabilizing controller is a proportional controller having a fixed gain.
7. The robust controller for nonlinear MIMO systems according to claim 4 , further comprising means for computing an objective function, the objective function minimizing a change in frequency of the power distribution system under load disturbance conditions.
8. The robust controller for nonlinear MIMO systems according to claim 7 , further comprising means for minimizing error in a minimum time using minimum effort in the presence of the load disturbance conditions.
9. The robust controller for nonlinear MIMO systems according to claim 8 , wherein a generation rate constraint of the power distribution system is approximately 0.6 p.u MW min −1 =0.01 p.u MW sec −1 .
10. The robust controller for nonlinear MIMO systems according to claim 1 , wherein said means for transforming includes an activation function generator connected to the unconstrained output of said RBF neural network, the activation function generator providing the constrained control outputs responsive to said RBF neural network unconstrained output.
11. The robust controller for nonlinear MIMO systems according to claim 10 , wherein said activation function generator is a tangent-sigmoid activation function generator, thereby forming a constrained control signal at said activation function generator's control output.