使用MATLAB实现多层神经网络的算例，包括扭摆系统、仿射非线性算例以及“质量-弹簧-阻尼”系统。
-
扭摆系统 (torsional pendulum system)
文献出处：
【1】Liu D , Wei Q . Policy Iteration Adaptive Dynamic Programming Algorithm for Discrete-Time Nonlinear Systems[J]. IEEE Trans Neural Netw Learn Syst, 2014, 25(3):621-634.
【2】Mu C , Wang D , He H . Novel iterative neural dynamic programming for data-based approximate optimal control design[J]. Automatica, 2017, 81:240-252.
Dynamic:
{
d
θ
d
t
=
ω
J
d
ω
d
t
=
u
−
M
g
l
sin
θ
−
f
d
d
θ
d
t
\left\{\begin{array}{l} \frac{d \theta}{d t}=\omega \\ J \frac{d \omega}{d t}=u-M g l \sin \theta-f_{d} \frac{d \theta}{d t} \end{array}\right.
{dtdθ=ωJdtdω=u−Mglsinθ−fddtdθ
where
M
=
1
/
3
k
g
M=1 / 3 \mathrm{kg}
M=1/3kg and
l
=
2
/
3
m
l=2 / 3 \mathrm{m}
l=2/3m are the mass and length of the pendulum bar, respectively. The system states are the current angle
θ
\theta
θ and the angular velocity
ω
.
\omega .
ω. Let
J
=
4
/
3
M
l
2
J=4 / 3 M l^{2}
J=4/3Ml2 and
f
d
=
0.2
f_{d}=0.2
fd=0.2 be the rotary inertia and frictional factor, respectively. Let
g
=
9.8
m
/
s
2
g=9.8 \mathrm{m} / \mathrm{s}^{2}
g=9.8m/s2 be the gravity. Discretization of the system function and performance index function using Euler and trapezoidal methods with the sampling interval
Δ
t
=
0.1
s
\Delta t=0.1 \mathrm{s}
Δt=0.1s leads to
[
x
1
(
k
+
1
)
x
2
(
k
+
1
)
]
=
[
0.1
x
2
k
+
x
1
k
−
0.49
×
sin
(
x
1
k
)
−
0.1
×
f
d
×
x
2
k
+
x
2
k
]
+
[
0
0.1
]
u
k
\begin{array}{r} {\left[\begin{array}{c} x_{1(k+1)} \\ x_{2(k+1)} \end{array}\right]=\left[\begin{array}{c} 0.1 x_{2 k}+x_{1 k} \\ -0.49 \times \sin \left(x_{1 k}\right)-0.1 \times f_{d} \times x_{2 k}+x_{2 k} \end{array}\right]} +\left[\begin{array}{c} 0 \\ 0.1 \end{array}\right] u_{k} \end{array}
[x1(k+1)x2(k+1)]=[0.1x2k+x1k−0.49×sin(x1k)−0.1×fd×x2k+x2k]+[00.1]uk
或者
x
t
+
1
=
[
x
1
t
+
0.1
x
2
t
0.2
(
−
0.49
sin
(
x
1
t
)
−
0.2
x
2
t
+
x
2
t
)
]
+
[
0
0.02
]
u
t
x_{t+1}=\left[\begin{array}{c} x_{1 t}+0.1 x_{2 t} \\ 0.2\left(-0.49 \sin \left(x_{1 t}\right)-0.2 x_{2 t}+x_{2 t}\right) \end{array}\right]+\left[\begin{array}{c} 0 \\ 0.02 \end{array}\right] u_{t}
xt+1=[x1t+0.1x2t0.2(−0.49sin(x1t)−0.2x2t+x2t)]+[00.02]ut
The initial state is $ x_{0}=[1,-1]^{T}$
仿真结果：ResultsCollation1.m
-
非线性算例
文献出处：
【1】Wang F Y , Jin N , Liu D , et al. Adaptive dynamic programming for finite-horizon optimal control of discrete-time nonlinear systems with ε-error bound.[J]. IEEE Trans Neural Netw, 2011, 22(1):24-36.
【2】Zhang H , Wei Q , Luo Y . A Novel Infinite-Time Optimal Tracking Control Scheme for a Class of Discrete-Time Nonlinear Systems via the Greedy HDP Iteration Algorithm[J]. IEEE Transactions on Systems Man & Cybernetics Part B, 2008, 38(4):937-942.
【3】Liu D , Wei Q . Policy Iteration Adaptive Dynamic Programming Algorithm for Discrete-Time Nonlinear Systems[J]. IEEE Trans Neural Netw Learn Syst, 2014, 25(3):621-634.
We consider the following nonlinear system:
KaTeX parse error: No such environment: align* at position 8: \begin{̲a̲l̲i̲g̲n̲*̲}̲ x_{k+1}=f\left…
variables, respectively. The system functions are given as
KaTeX parse error: No such environment: align* at position 8: \begin{̲a̲l̲i̲g̲n̲*̲}̲ f\left(x_{k}\r…
The initial state is $ x_{0}=[2,-1]^{T}$仿真结果：
ResultsCollation2.m
3. “质量-弹簧-阻尼”系统（Mass-Spring-Damper System）
文献出处：
Winston Alexander Baker. Observer incorporated neoclassical controller design: A discrete perspective[J]. Dissertations & Theses - Gradworks, 2010.
[
x
1
(
k
+
1
)
x
2
(
k
+
1
)
]
=
[
0.0099
x
2
k
+
0.9996
x
1
k
−
0.0887
x
1
k
+
0.97
x
2
k
]
+
[
0
0.0099
]
u
(
k
)
\left[\begin{array}{l} x_{1}(k+1) \\ x_{2}(k+1) \end{array}\right]=\left[\begin{array}{c} 0.0099 x_{2 k}+0.9996 x_{1 k} \\ -0.0887 x_{1 k}+0.97 x_{2 k} \end{array}\right]+\left[\begin{array}{c} 0 \\ 0.0099 \end{array}\right] u(k)
[x1(k+1)x2(k+1)]=[0.0099x2k+0.9996x1k−0.0887x1k+0.97x2k]+[00.0099]u(k)
The initial state vector is set as
x
0
=
[
−
1
,
1
]
T
x_{0}=[-1,1]^{T}
x0=[−1,1]T.
仿真结果：ResultsCollation3.m