双倒立摆最优控制策略探究与对比分析
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"这篇报告由Alexander Bogdanov于2004年12月发表，研究了双倒立摆（DIPC）在小车上的最优控制算法。通过使用欧拉-拉格朗日方程来建立动力学模型，探讨了线性二次调节器（LQR）、状态依赖的里卡蒂方程（SDRE）和最优神经网络（NN）控制等方法，并分析了它们在解决非完全受控问题中的性能。"
本文主要关注的是对一个双倒立摆系统（DIPC）的最优控制策略。双倒立摆是一个具有两个自由度的复杂动态系统，通常安装在一辆小车上，其运动由动能和势能的差值——即拉格朗日函数——推导出的非线性二阶微分方程系统来描述。为了便于控制设计，这个系统被转换为六个一阶常微分方程（ODE）的形式。
控制双倒立摆是一个挑战，因为与机器人不同，它是一个欠驱动系统，只有一个小车上的一个控制力，但需要处理三个自由度的运动。因此，报告中提出的问题是寻找最小化二次成本函数的最优控制策略。作者测试了以下几种方法：
1. **线性二次调节器（LQR）**：这是一种经典的最优控制方法，基于系统的线性化模型，通过解一组代数 Riccati 方程来计算控制器增益。
2. **状态依赖的里卡蒂方程（SDRE）**：这种方法考虑了系统的非线性特性，通过解一个与系统状态相关的里卡蒂方程来实现更精确的控制。
3. **最优神经网络（NN）控制**：利用神经网络的非线性映射能力来补偿模型的不足，尤其是对于复杂或难以建模的系统行为。
4. **NN与LQR和SDRE的组合**：试图结合神经网络的适应性和其他方法的稳定性优势，以提高控制性能。
仿真结果表明，SDRE在性能上优于LQR，而神经网络可以补偿LQR中模型不准确带来的影响。然而，由于神经网络在广义参数范围内的函数逼近能力有限，它无法显著提升SDRE的表现，只在较大的摆角偏离时提供边际效益。
这篇报告为双倒立摆的最优控制提供了深入的理论和实践见解，对理解欠驱动系统控制以及比较不同控制策略的优劣具有重要意义。通过这些方法，可以优化系统的稳定性和效率，对于实际的物理系统控制，如机器人平衡、运动控制等领域有着广泛的应用价值。
Lagrange equations for the DIPC system can be writ-
ten in a more compact matrix form:
D(θ)
¨
θ + C(θ,
˙
θ)
˙
θ + G(θ) = Hu (2)
where
D(θ)=
d
1
d
2
cos θ
1
d
3
cos θ
2
d
2
cos θ
1
d
4
d
5
cos(θ
1
−θ
2
)
d
3
cos θ
2
d
5
cos(θ
1
−θ
2
) d
6
!
(3)
C(θ,
˙
θ)=
0 −d
2
sin(θ
1
)
˙
θ
1
−d
3
sin(θ
2
)
˙
θ
2
0 0 d
5
sin(θ
1
−θ
2
)
˙
θ
2
0 −d
5
sin(θ
1
−θ
2
)
˙
θ
1
0
(4)
G(θ) =
0
−f
1
sin θ
1
−f
2
sin θ
2
!
(5)
H = (1 0 0)
T
Assuming that centers of mass of the pendulums are in
the geometrical center of the links, which are solid rods,
we have: l
i
= L
i
/2, I
i
= m
i
L
2
i
/12. Then for the ele-
ments of matrices D(θ), C(θ,
˙
θ), and G(θ) we get:
d
1
= m
0
+ m
1
+ m
2
d
2
= m
1
l
1
+ m
2
L1 =
1
2
m
1
+ m
2
L
1
d
3
= m
2
l
2
=
1
2
m
2
L
2
d
4
= m
1
l
2
1
+ m
2
L
2
1
+ I
1
=
1
3
m
1
+ m
2
L
2
1
d
5
= m
2
L
1
l
2
=
1
2
m
2
L
1
L
2
d
6
= m
2
l
2
2
+ I
2
=
1
3
m
2
L
2
2
f
1
= (m
1
l
1
+ m
2
L
1
)g = (
1
2
m
1
+ m
2
)L
1
g
f
2
= m
2
l
2
g =
1
2
m
2
L
2
g
Note that matrix D(θ) is symmetric and nonsingular.
4 Control
To design a control law, Lagrange equations of motion
(2) are reformulated into a 6-th order system of ordinary
differential equations. To do this, a state vector x ∈ R
6
is introduced:
x = (θ
˙
θ)
T
Then dropping dependencies of the system matrices on
the generalized coordinates and their derivatives, the
system dynamic equations appear as:
˙x =
0 I
0 −D
−1
C
x+
0
−D
−1
G
+
0
D
−1
H
u (6)
In this report, optimal nonlinear stabilization control de-
sign is addressed: stabilize the DIPC minimizing an ac-
cumulative cost functional quadratic in states and con-
trols. The general problem of designing an optimal con-
trol law involves minimizing a cost function
J
t
=
t
f inal
X
k=t
L
k
(x
k
, u
k
), (7)
which represents an accumulated cost of the sequence of
states x
k
and controls u
k
from the current discrete time t
to the final time t
final
. For regulation problems t
final
=
∞. Optimization is done with respect to the control
sequence subject to constraints of the system dynamics
(6). In our case,
L
k
(x
k
, u
k
) = x
T
k
Qx
k
+ u
T
k
Ru
k
(8)
corresponds to the standard Linear Quadratic cost. For
linear systems, this leads to linear state-feedback control,
LQR, designed in the next subsection. For nonlinear
systems the optimal control problem generally requires
a numerical solution, which can be computationally pro-
hibitive. An analytical approximation to the nonlin-
ear optimal control solution is utilized in subsection on
SDRE control, which represents a nonlinear extension
to the LQR and yields superior results. Neural net-
work (NN) capabilities for function approximation are
employed to approximate the nonlinear control solution
in subsection on NN control. And combinations of the
NN with LQR and SDRE are investigated in the subsec-
tion following the NN control.
4.1 Linear Quadratic Regulator
The linear quadratic regulator yields an optimal solu-
tion to the control problem (7)–(8) when system dynam-
ics are linear. Since DIPC is nonlinear, as described by
(6), it can be linearized to derive an approximate linear
solution to the optimal control problem. Linearization
of (6) around x = 0 yields:
˙x = Ax + Bu (9)
where
A =
0 I
−D(0)
−1
∂G(0)
∂θ
0
(10)
B =
0
D(0)
−1
H
(11)
and the continuous LQR solution is obtained then by:
u = −R
−1
B
T
P
c
x ≡ −K
c
x (12)
where P
c
is a steady-state solution of the differential
Riccati equation. To implement computerized digital
control, dynamic equations (9) are approximately dis-
cretized as Φ ≈ e
A∆t
, Γ ≈ B∆t, and digital LQR con-
trol is then given by
u
k
= −R
−1
Γ
T
Px
k
≡ −Kx
k
(13)
where P is the steady state solution of the difference
Riccati equation, obtained by solving the discrete-time
algebraic Riccati equation
Φ
T
[P − PΓ(R + Γ
T
PΓ)
−1
Γ
T
P]Φ − P + Q = 0 (14)
where Q ∈ R
6×6
and R ∈ R are positive definite state
and control cost matrices. Since linearization (9)–(11)
accurately represents the DIPC system (6) in the equi-
librium, the LQR control (12) or (13) will be a locally
near-optimal stabilizing control.
3
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