概要

Sympy と Python-Control を用いて、Pythonによる直流モータの速度制御を解析する。Sympy は代数計算を行うPyrhonライブラリであり、複雑な代数方程式であっても解くことができる。Python-Control を用いることで Matlab と同じような解析が可能である。よって Sympy と Python-Control を組み合わせて使用することで、数理モデルの解析から、数値計算による評価まで Python だけで行うことができる。

背景

Pythonによる科学技術計算についてWeb上では情報が充実しており、これらの情報を用いて複雑な科学技術計算を行うことができる。科学技術計算で用いられるPythonライブラリとして Sympy と Python-control がある。 Sympy は代数計算ライブラリであり、 Mathematica や Maxima に近い解析が可能である。一方で Python-control は制御理論ライブラリであり、 matlab や Scilab , Octave に近い解析が可能である。 Sympy と Python-control の各々を説明するサイトは多いが、双方を組み合わせて使用する方法の情報があまり無かったので、ここにまとめる。

解析内容

モータ駆動装置と直流モータの制御システムを古典制御理論にて解析する。詳しい解説は下記のサイトを参照してほしい。

ここでは、Sympyを用いてモータ駆動装置＋直流モータとPID制御系を伝達関数で表し、システムの一巡伝達関数と閉ループ伝達関数を求める。得られた伝達関数に対して、control.matlabを用いて周波数応答解析と時間応答解析を行い、matplotlib.pyplotなどにより可視化する。

伝達関数の導出

sympyを用いてモータ駆動システムの一巡伝達関数と閉ループ伝達関数を求める。 sympyの基本的な使用方法は下記のサイトが詳しい。

import sympy
from IPython.display import display

IPython.displayはjupyter notebookでsympyのオブジェクトを表示するためであり、不要な場合はインポートしない。

V, I = sympy.symbols("V, I", cls=sympy.Function)
Tau, Omega = sympy.symbols("Tau, Omega", cls=sympy.Function)
s = sympy.Symbol("s")

ここでは、システムの変数をsympy.Symbolとして定義している。各変数の説明を下記に示す。

$V$ ラプラス変換後のモータ印加電圧
$I$ ラプラス変換後のモータ電流
$T$ ラプラス変換後のモータ負荷トルク
$\Omega$ ラプラス変換後のモータ速度
$s$ ラプラス変数

cls=sympy.Functionを引数にすることで、 $V$ , $I$ , $T$ , $\Omega$ を関数としている。

モータ回路の伝達関数

R, L, K_v = sympy.symbols("R, L, K_v")
K_t, J = sympy.symbols("K_t, J")

ここでは、システムの定数をsympy.Symbolとして定義している。各変数の説明を下記に示す。

$R$ 配線の抵抗
$L$ モータのインダクタ
$K_v$ 逆起電圧定数
$K_t$ トルク定数
$J$ モータのイナーシャ

モータの駆動回路の回路方程式のラプラス変換と、モータの運動方程式のラプラス変換は次式で表される。 Sympyで方程式はsympy.Eq(左辺, 右辺)で定義する。 cls=sympy.Functionを引数とした変数の後に( $s$ )を付けることで、 $s$ の関数であることを宣言する。

eq_e = sympy.Eq(R*I(s) + s*L*I(s) + K_v * Omega(s), V(s))
eq_m = sympy.Eq(Tau(s) + s*J*Omega(s), K_t*I(s))
display(eq_e)
display(eq_m)

$\displaystyle K_{v} \Omega{\left(s \right)} + L s I{\left(s \right)} + R I{\left(s \right)} = V{\left(s \right)}$

$\displaystyle J s \Omega{\left(s \right)} + \mathrm{T}{\left(s \right)} = K_{t} I{\left(s \right)}$

負荷トルク $T(s)$ を外乱としてシステムから除外すると、モータの運動方程式のラプラス変換は次式で書き直せる。

eq_m = sympy.Eq(s*J*Omega(s), K_t*I(s))
display(eq_m)

$\displaystyle J s \Omega{\left(s \right)} = K_{t} I{\left(s \right)}$

まず運動方程式のラプラス変換を電流 $I(s)$ について解く。方程式を解くときはsympy.solveメソッドを用いる。第二引数には解く変数を指定する。

I_solve= sympy.solve(eq_m, I(s))

これを回路方程式のラプラス変換に代入して次式が得られる。式への代入はsubsメソッドを使用する。またexpandメソッド、simplifyメソッドにより式を整理できる。

eq_e = eq_e.subs(I(s), I_solve[0])
eq_e = eq_e.expand().simplify()
display(eq_e)

$\displaystyle V{\left(s \right)} = \frac{\left(J L s^{2} + J R s + K_{t} K_{v}\right) \Omega{\left(s \right)}}{K_{t}}$

上式を $\Omega$ について解くことで、伝達特性が得られる。

Omega_solve= sympy.solve(eq_e, Omega(s))
display(Omega_solve[0])

$\displaystyle \frac{K_{t} V{\left(s \right)}}{J L s^{2} + J R s + K_{t} K_{v}}$

G_cir = Omega_solve[0] / V(s)
display(G_cir)

$\displaystyle \frac{K_{t}}{J L s^{2} + J R s + K_{t} K_{v}}$

ここで $G_{cir}$ がモータ駆動回路の入力電圧 $V$ からモータ速度 $\Omega$ までの伝達関数である。

制御系の伝達関数

PID制御系のPゲインを $C_{p}$ 、Iゲインを $C_{i}$ 、Dゲインを $C_{d}$ とする

C_p, C_i, C_d = sympy.symbols("C_p, C_i, C_d")

制御系の伝達関数 $G_{con}$ は次式で得られる。

G_con = C_p + C_i/s + C_d * s
display(G_con)

$\displaystyle C_{d} s + \frac{C_{i}}{s} + C_{p}$

システムの伝達関数

開ループ伝達関数 $G_{open}$ 、閉ループ伝達関数 $G_{close}$ は次式で得られる。

G_open = G_cir * G_con
display(G_open)
G_close = G_cir * G_con / (1 + G_cir * G_con)
display(G_close.expand().simplify())

$\displaystyle \frac{K_{t} \left(C_{d} s + \frac{C_{i}}{s} + C_{p}\right)}{J L s^{2} + J R s + K_{t} K_{v}}$

$\displaystyle \frac{K_{t} \left(C_{d} s^{2} + C_{i} + C_{p} s\right)}{C_{d} K_{t} s^{2} + C_{i} K_{t} + C_{p} K_{t} s + J L s^{3} + J R s^{2} + K_{t} K_{v} s}$

このように、Sympyを用いることで、代数計算をコンピュータで行うことができる。次に得られた伝達関数に具体的な値を代入して、図で表すことで可視化する。

定数の代入

例として、各定数の値を下記のように設定する。

const_params = {
    "R":100e-3,
    "L":1e-3, 
    "K_v":45e-3,
    "K_t":45.5, 
    "J":1.17e-3
}

control_params = {
    "C_p":1e-1, 
    "C_i":1e-1, 
    "C_d":1e-6
}

Sympyの変数のsubsメソッドを使用することで、辞書型で指定した複数の変数に代入することができる。

params = {}
params.update(const_params)
params.update(control_params)
G_open_num = G_open.subs(list(params.items()))
display(G_open_num.simplify())
G_close_num = G_close.subs(list(params.items()))
display(G_close_num.simplify())

$\displaystyle \frac{45.5 s \left(1.0 \cdot 10^{-6} s + 0.1\right) + 4.55}{s \left(1.17 \cdot 10^{-6} s^{2} + 0.000117 s + 2.0475\right)}$

$\displaystyle \frac{4.55 \cdot 10^{-5} s^{2} + 4.55 s + 4.55}{1.17 \cdot 10^{-6} s^{3} + 0.0001625 s^{2} + 6.5975 s + 4.55}$

上式は値を代入したときの $G_{open}$ と $G_{close}$ である。

システムの評価

伝達関数の解析にpython-controlのmatlabモジュールとnumpyを用いる。

from control import matlab
import numpy as np

最初に数値を代入した $G_{open}$ と $G_{close}$ の分母・分子を取り出す。与えられる式に対して、sympy.denomは分母を、sympy.numerは分子を返す。次に分母分子をそれぞれ $s$ の関数（sympy.poly型）として再定義する。

G_open_denom_poly = sympy.poly(sympy.denom(G_open_num.simplify()), s)
G_open_numer_poly = sympy.poly(sympy.numer(G_open_num.simplify()), s)
G_close_denom_poly = sympy.poly(sympy.denom(G_close_num.simplify()), s)
G_close_numer_poly = sympy.poly(sympy.numer(G_close_num.simplify()), s)

sympy.poly型のall_coeffsメソッドを用いることで、各関数の係数のリストを返す。リストの最初の項目は最大次数の係数、2番目の項目は2番目に大きい次数の係数…となる。係数がない次数は0で返る。リストの各係数をfloat型に変換してリストとする。

G_open_numer_poly

$\displaystyle \operatorname{Poly}{\left( 4.55 \cdot 10^{-5} s^{2} + 4.55 s + 4.55, s, domain=\mathbb{R} \right)}$

num = list(map(float, G_open_numer_poly.all_coeffs()))
den = list(map(float, G_open_denom_poly.all_coeffs()))

num = list(map(float, G_close_numer_poly.all_coeffs()))
den = list(map(float, G_close_denom_poly.all_coeffs()))

matlab.tf(num, den)のnumに分子係数リスト、denに分母係数リストを渡すことで、matlab.tf型の $G_{open}$ と $G_{close}$ を得る。

G_open_ml = matlab.tf(num, den)
G_open_ml

$\frac{4.55 \times 10^{-5} s^2 + 4.55 s + 4.55}{1.17 \times 10^{-6} s^3 + 0.0001625 s^2 + 6.598 s + 4.55}$

G_close_ml = matlab.tf(num, den)
G_close_ml

$\frac{4.55 \times 10^{-5} s^2 + 4.55 s + 4.55}{1.17 \times 10^{-6} s^3 + 0.0001625 s^2 + 6.598 s + 4.55}$

matlab.tf型の伝達関数が得られれば、python-control.matlabの各種解析メソッドが使用できる。

周波数応答解析

matlab.bode(matlab.tf)とすることで、ボード線図を描画できる。第二引数に周波数リストを指定すると、描画する周波数範囲を指定できる。

matlab.bode(G_open_ml, np.logspace(0, 8, 1000))

(array([8.02900970e-01, 8.00283658e-01, 7.97693529e-01, 7.95131853e-01, 7.92599839e-01, 7.90098627e-01, 7.87629292e-01, 7.85192837e-01, 7.82790201e-01, 7.80422250e-01, 7.78089781e-01, 7.75793522e-01, 7.73534132e-01, 7.71312200e-01, 7.69128250e-01, 7.66982736e-01, 7.64876049e-01, 7.62808515e-01, 7.60780395e-01, 7.58791894e-01, 7.56843153e-01, 7.54934258e-01, 7.53065241e-01, 7.51236080e-01, 7.49446700e-01, 7.47696981e-01, 7.45986756e-01, 7.44315812e-01, 7.42683898e-01, 7.41090720e-01, 7.39535950e-01, 7.38019226e-01, 7.36540152e-01, 7.35098303e-01, 7.33693228e-01, 7.32324449e-01, 7.30991465e-01, 7.29693757e-01, 7.28430783e-01, 7.27201987e-01, 7.26006796e-01, 7.24844627e-01, 7.23714883e-01, 7.22616958e-01, 7.21550239e-01, 7.20514105e-01, 7.19507931e-01, 7.18531088e-01, 7.17582946e-01, 7.16662872e-01, 7.15770234e-01, 7.14904402e-01, 7.14064747e-01, 7.13250643e-01, 7.12461469e-01, 7.11696608e-01, 7.10955449e-01, 7.10237387e-01, 7.09541824e-01, 7.08868170e-01, 7.08215840e-01, 7.07584262e-01, 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7.44512291e+07, 7.58367791e+07, 7.72481145e+07, 7.86857151e+07, 8.01500696e+07, 8.16416760e+07, 8.31610415e+07, 8.47086827e+07, 8.62851257e+07, 8.78909065e+07, 8.95265713e+07, 9.11926760e+07, 9.28897872e+07, 9.46184819e+07, 9.63793480e+07, 9.81729841e+07, 1.00000000e+08]))

png

上図は開ループ伝達関数のボード線図である。直流モータのシステムでは開ループ伝達関数と一巡伝達関数は同じである。上図では振幅が0dbとなるところで、位相が180deg回っていないので安定である。

matlab.bode(G_close_ml, np.logspace(0, 8, 1000))

png

上図は閉ループ伝達関数のボード線図である。上図では約 1k rad/sec で振幅が減少しており、ローパスフィルタのような特性となっている。

過渡解析

python-control.matlabは過渡解析もサポートしている。過渡解析の入力信号生成のためにscipy.signalを使用する。また過渡解析結果を描画するために、matplotlib.pyplotを使用する。

import matplotlib.pyplot as plt
from scipy import signal

matlab.impulse(matlab.tf)とすることで、伝達関数のインパルス応答が得られる。

yout, time = matlab.impulse(G_close_ml)

fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(time, yout)

[<matplotlib.lines.Line2D at 0x232cb752f90>]

png

後半部分はmatplotlib.pyplotの描画処理である。

matlab.step(matlab.tf)とすることで、伝達関数のステップ応答が得られる。

yout, time = matlab.step(G_close_ml)

fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(time, yout)

[<matplotlib.lines.Line2D at 0x232cb66abd0>]

png

matlab.lsim(matlab.tf, U, T)とすることで、入力信号Uに対する時間Tの間の伝達関数の過渡応答が得られる。前半部分はscipy.signalを用いて入力信号を作成している。後半部分は入力信号と出力信号の波形を描画している。

time = np.linspace(0, 0.05, int(5e5), endpoint=False)
f_mod, f_pwm = 1.6e2, 1e4
modulation = np.sin(2 * np.pi * f_mod * time)
u_pwm = signal.square(2 * np.pi * f_pwm * time, duty=(modulation + 1)/2)

yout, Time, x = matlab.lsim(G_close_ml, U=u_pwm, T=time)

fig = plt.figure()
ax1 = fig.add_subplot(211)
ax1.plot(time, u_pwm)

ax2 = fig.add_subplot(212)
ax2.plot(time, yout)

***.py:940: UserWarning: return_x specified for a transfer function system. Internal conversion to state space used; results may meaningless. warnings.warn(

[<matplotlib.lines.Line2D at 0x232cdaaa250>]

png

下記は時間0.04~0.05を拡大した波形である。

fig = plt.figure()
ax1 = fig.add_subplot(211)
ax1.plot(time, u_pwm)
ax1.set_xlim(0.04, 0.05)

ax2 = fig.add_subplot(212)
ax2.plot(time, yout)
ax2.set_xlim(0.04, 0.05)

(0.04, 0.05)

png

直流モータの司令値は 10k rad/sec で変化する、基本波 160 rad/sec のPWM信号としている。上図は、直流モータの司令値に極めて変化が速いPWM信号を入力しても、出力速度は入力信号よりも遅い基本波成分のみ出力されることを示している。これは閉ループ伝達関数 $G_{close}$ がローパスフィルタのような特性のためである。

おわりに

Sympy と Python-control の双方を組み合わせて、直流モータの速度制御システムの解析をすることができた。数式の変形から、得られた数式に代入して、数値解析できるので、計算が少し楽になった。

電気工学博士の雑記

電気工学の博士が徒然なるままに試した技術的なことをアウトプットする。

Python備忘録 SymPyとPython-Controlによる直流モータ動作解析

概要

背景