RCPSP
RCPSP (Resource Constraint Project Scheduling Problem, 資源制約プロジェクトスケジューリング問題)とは、先行制約(Precedence Constraints)と資源制約(Resource Constraints)のもとで、決定変数を各作業の開始日とし、目的関数を最小化する組合せ最適化問題で、次のように定式化（またはモデル化）表されます。
![\[\displaystyle{\boxed{\begin{array}{cl} {\bf min}&\ Objective\ Function\\ {start\ times}&\\ {\bf subject\ to}&1^\circ\ Precedence\ Constraints\\ &2^\circ\ Resource\ Constraints \end{array}}}\]](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-e06beb8547aee1ddb83a85eeb795ffb6_l3.png "Rendered by QuickLaTeX.com")
これを拡張した問題に対する商用ソルバーとして、OptSeqがあります。そこで使用可能なクラス、インスタンス、メソッドを以下に示します。
●クラス「モデル」
model=Model()
●クラス「作業」
act=model.addActivity(name=””,duedate=”inf”,backward=False,weight=1,autoselect=False,quadratic=False)
act.addModes(mode1,mode2,…)
●クラス「モード」
mode=Mode(name,duration=0)
mode.addResource(resource,requirement={(start,finish):req},rtype=None)
mode.addBreak(start=0,finish=0,maxtime=’inf’)
mode.addBreak(start=0,{(start,finish):req},’break’)
mode.addParallel(start=1,finish=1,maxparallel=’inf’)
mode.addState(state,fromValue=0,toValue=0)
●クラス「資源」
res=model.addResource(name,capacity,rhs=0,direction=’<=',weight='inf')
res.addCapacity(start,finish,amount)
res.addTerms(coeffs,vars,values)
●クラス「時間制約」
model.addTemporal(pred,succ,tempType=’CS’,delay=0,pred_mode=None,succ_mode=None)
●クラス「状態」
model.addState(name)
基本的例題
例1（プッシュ型とプル型） ある作業の納期と作業期間が与えられるとき、納期までどれくらい余裕があるのか(prob1a)、または納期にジャストインするためにはいつから作業に取りかかればよいか(prob1b)という問題を考えます。そのために作業をどう前詰めするか、納期に後詰めするかが問われ、それぞれプッシュ型とプル型に対応します。
#prob1a.py
from optseq import *
#=====リソース
prob1=Model()
res=prob1.addResource("worker",1)
#=====データセット
#i:[期間、後続、納期]
data={
1:[5,2,'inf'],\
2:[1,0,10]\
}
#=====アクティビティ
act={}
for i in data:
act[i]=prob1.addActivity("Act[{0}]".format(i))
#=====先行制約
for i in data:
if data[i][1]!=0:
prob1.addTemporal(act[i],act[data[i][1]])
#-----納期日
for i in data:
if data[i][1]==0:
prob1.addTemporal("source",act[i],"SS",delay= data[i][2]-1)
prob1.addTemporal(act[i],"source","SS",delay=-data[i][2]+1)
#====資源制約
mode={}
for i in data:
mode[i]=Mode("Mode[{0}]".format(i),duration=data[i][0])
mode[i].addResource(res,requirement=1)
act[i].addModes(mode[i])
#=====最適化
prob1.Params.Makespan=True
prob1.Params.TimeLimit=1
#prob1.Params.OutputFlag=True
prob1.optimize()
prob1.write("prob1a.txt")
prob1.writeExcel("prob1a.csv")
#=====prob1a.txt
# activity mode duration 1 2 3 4 5 6 7 8 910
#---------------------------------------------------
# Act[1] Mode[1] 5 ==========
# Act[2] Mode[2] 1 ==
#---------------------------------------------------
# resource usage/capacity
#---------------------------------------------------
# worker 1 1 1 1 1 0 0 0 0 1
# 1 1 1 1 1 1 1 1 1 1
#---------------------------------------------------
OptSeqは指定がなければ前詰めが実行されます。納入作業は10日目に行うと考えて、開始のタイミングを９日目終了に合わせています。
#prob1b.py
from optseq import *
#=====リソース
prob1=Model()
res=prob1.addResource("worker",1)
#=====データセット
#i:[期間、後続、納期]
data={
1:[5,2,10],\
2:[1,0,10]\
}
#=====アクティビティ
act={}
for i in data:
act[i]=prob1.addActivity("Act[{0}]".format(i),duedate=data[i][2],backward=True)
#=====先行制約
for i in data:
if data[i][1]!=0:
prob1.addTemporal(act[i],act[data[i][1]])
#-----納期日
for i in data:
if data[i][1]==0:
prob1.addTemporal("source",act[i],"SS",delay= data[i][2]-1)
prob1.addTemporal(act[i],"source","SS",delay=-data[i][2]+1)
#====資源制約
mode={}
for i in data:
mode[i]=Mode("Mode[{0}]".format(i),duration=data[i][0])
mode[i].addResource(res,requirement=1)
act[i].addModes(mode[i])
#=====最適化
prob1.Params.Makespan=True
prob1.Params.TimeLimit=1
#prob1.Params.OutputFlag=True
prob1.optimize()
prob1.write("prob1b.txt")
prob1.writeExcel("prob1b.csv")
#=====prob1a.txt
# activity mode duration 1 2 3 4 5 6 7 8 910
#---------------------------------------------------
# Act[1] Mode[1] 5 ==========
# Act[2] Mode[2] 1 ==
#---------------------------------------------------
# resource usage/capacity
#---------------------------------------------------
# worker 0 0 0 0 1 1 1 1 1 1
# 1 1 1 1 1 1 1 1 1 1
#---------------------------------------------------
作業の定義において、納期の指定（duedate）と後詰めの指定（backward=True）を行っています。そのためにデータセットで主作業（Act[1]）にも納期を指定する必要があることに注意してください（後続作業として納品作業があるので本来は不要であるように思われますが）。
例2（その場待機と置き場待機） 種々の制約があると待機期間が生じます。リソース（作業員、作業機械、作業場所）によっては待機中も確保が必要になります。たとえば、リソースが作業場所の場合、その作業場所で待機をするその場待機と、どこか待機場所を確保する置き場待機が考えられます。
#prob2a.py
from optseq import *
#=====リソース
prob2=Model()
res=prob2.addResource("place",1)
#=====データセット
#i:[期間、後続、納期]
data={
1:[5,2,'inf'],\
2:[1,0,10]\
}
#=====アクティビティ
act={}
for i in data:
act[i]=prob2.addActivity("Act[{0}]".format(i))
#=====先行制約
for i in data:
if data[i][1]!=0:
prob2.addTemporal(act[i],act[data[i][1]],tempType="CS",delay=0)
prob2.addTemporal(act[data[i][1]],act[i],tempType="SC",delay=0)
#-----納期日
for i in data:
if data[i][1]==0:
prob2.addTemporal("source",act[i],"SS",delay= data[i][2]-1)
prob2.addTemporal(act[i],"source","SS",delay=-data[i][2]+1)
#====資源制約
mode={}
for i in data:
mode[i]=Mode("Mode[{0}]".format(i),duration=data[i][0])
mode[i].addResource(res,requirement=1)
mode[i].addBreak(0,'inf')
#mode[i].addBreak(0,'inf',maxtime=2)
#mode[i].addBreak(0,0)
mode[i].addResource(res,{(0,'inf'):1},'break')
act[i].addModes(mode[i])
#=====最適化
prob2.Params.Makespan=True
prob2.Params.TimeLimit=1
#prob2.Params.OutputFlag=True
prob2.optimize()
prob2.write("prob2a.txt")
prob2.writeExcel("prob2a.csv")
#=====prob2a.txt: mode[i].addBreak(0,'inf')
# activity mode duration 1 2 3 4 5 6 7 8 910
#---------------------------------------------------
# Act[1] Mode[1] 5 ==========........
# Act[2] Mode[2] 1 ==
#---------------------------------------------------
# resource usage/capacity
#---------------------------------------------------
# place 1 1 1 1 1 1 1 1 1 1
# 1 1 1 1 1 1 1 1 1 1
#---------------------------------------------------
#=====prob2a.txt: mode[i].addBreak(0,'inf',maxtime=2)
# activity mode duration 1 2 3 4 5 6 7 8 910
#---------------------------------------------------
# Act[1] Mode[1] 5 ========....==....
# Act[2] Mode[2] 1 ==
#---------------------------------------------------
# resource usage/capacity
#---------------------------------------------------
# place 1 1 1 1 1 1 1 1 1 1
# 1 1 1 1 1 1 1 1 1 1
#---------------------------------------------------
#=====prob2a.txt: mode[i].addBreak(0,0)
# activity mode duration 1 2 3 4 5 6 7 8 910
#---------------------------------------------------
# Act[1] Mode[1] 5 ........==========
# Act[2] Mode[2] 1 ==
#---------------------------------------------------
# resource usage/capacity
#---------------------------------------------------
# place 1 1 1 1 1 1 1 1 1 1
# 1 1 1 1 1 1 1 1 1 1
#---------------------------------------------------
ここでは、待機期間の取り方を３種類考えて、その場待機を実現しています。
#prob2b.py
from optseq import *
#=====リソース
prob2=Model()
res0=prob2.addResource("place",1)
res1=prob2.addResource("stock",1)
#=====データセット
#i:[期間、後続、納期]
data={
1:[5,-2,"inf"],\
2:[1,0,10]\
}
#=====アクティビティ
act={}
for i in data:
act[i]=prob2.addActivity("Act[{0}]".format(i))
#=====先行制約
for i in data:
if data[i][1]!=0:
prob2.addTemporal(act[i],act[data[i][1]],tempType="CS",delay=0)
#-----納期日
for i in data:
if data[i][1]==0:
prob2.addTemporal("source",act[i],"SS",delay= data[i][2]-1)
prob2.addTemporal(act[i],"source","SS",delay=-data[i][2]+1)
#====資源制約
mode={}
for i in data:
mode[i]=Mode("Mode[{0}]".format(i),duration=data[i][0])
mode[i].addResource(res0,1)
act[i].addModes(mode[i])
#=====仮想作業
d_act={}
for i in data:
if data[i][1]<0:
d_act[i]=prob2.addActivity("Wait[{0}]".format(i))
prob2.addTemporal(act[i],d_act[i],tempType="CS")
prob2.addTemporal(d_act[i],act[i],tempType="SC")
prob2.addTemporal(d_act[i],act[abs(data[i][1])],tempType="CS")
prob2.addTemporal(act[abs(data[i][1])],d_act[i],tempType="SC")
d_mode={}
for i in data:
if data[i][1]<0:
d_mode[i]=Mode("d_Mode[{0}]".format(i))
d_mode[i].addBreak(0,0)
d_mode[i].addResource(res1,{(0,"inf"):1},"break")
d_act[i].addModes(d_mode[i])
#=====最適化
prob2.Params.Makespan=True
prob2.Params.TimeLimit=1
#prob2.Params.OutputFlag=True
prob2.optimize()
prob2.write("prob2b.txt")
prob2.writeExcel("prob2b.csv")
#=====prob2b.txt
# activity mode duration 1 2 3 4 5 6 7 8 910
#---------------------------------------------------
# Act[1] Mode[1] 5 ==========
# Act[2] Mode[2] 1 ==
# Wait[1] d_Mode[1] 0 ........
#---------------------------------------------------
# resource usage/capacity
#---------------------------------------------------
# place 1 1 1 1 1 0 0 0 0 1
# 1 1 1 1 1 1 1 1 1 1
#---------------------------------------------------
# stock 0 0 0 0 0 1 1 1 1 0
# 1 1 1 1 1 1 1 1 1 1
#---------------------------------------------------
ここでは、待機場所を別に確保して、待機中はそこを使うようにしています。この置き場待機を実施するかどうかは、データセットにおける後続作業の番号が負であるかどうかで判断しています。
![\displaystyle{(4)\quad %\underbrace{ \left[\begin{array}{c} \dot{x}(t) \\ \dot{x}_I(t) \end{array}\right] %}_{\dot{x}_E(t)} = %\underbrace{ \left[\begin{array}{cc} A & 0 \\ C & 0 \end{array}\right] %}_{A_E} %\underbrace{ \left[\begin{array}{c} x(t) \\ x_I(t) \end{array}\right] %}_{x_E(t)} + %\underbrace{ \left[\begin{array}{c} B \\ 0 \end{array}\right] %}_{B_E} u(t) + %\underbrace{ \left[\begin{array}{c} w \\ -r \end{array}\right] %}_{w_E} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-8eae1302f0f9a0f92ca6ad1023069780_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(5)\quad \left[\begin{array}{c} 0 \\ 0 \end{array}\right] = \left[\begin{array}{cc} A & 0 \\ C & 0 \end{array}\right] \left[\begin{array}{c} x_\infty \\ x_{I\infty} \end{array}\right] + \left[\begin{array}{c} B \\ 0 \end{array}\right] u_\infty + \left[\begin{array}{c} w \\ -r \end{array}\right] }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-1f79558c23720658ef469eaed967aa38_l3.png "Rendered by QuickLaTeX.com")
,
,
は定数ベクトル)。![\displaystyle{(10)\quad \underbrace{ \left[\begin{array}{c} {\dot x}(t) \\ y(t)-r \end{array}\right] }_{x_{E2}(t)} = \underbrace{ \left[\begin{array}{cc} A & B \\ C & 0 \end{array}\right] }_{S_E} \underbrace{ \left[\begin{array}{c} x(t)-x_\infty \\ u(t)-u_\infty \end{array}\right] }_{x_{E3}(t)} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-d19845bc6a9ba5e6b6ffcc474aa170e5_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(15)\quad \begin{array}{l} \boxed{ \underbrace{ \left[\begin{array}{c} \dot x_1(t)\\ \dot x_2(t) \end{array}\right] }_{\dot{x}_{E3}(t)} = \underbrace{ \left[\begin{array}{cc} A & B \\ 0 & 0 \end{array}\right] }_{A_{E3}} \underbrace{ \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right] }_{x_{E3}(t)} + \underbrace{ \left[\begin{array}{c} 0\\ I_m \end{array}\right] }_{B_{E3}} {\dot u}(t)}\\ (x_1(t)=x(t)-x_\infty, x_2(t)=u(t)-u_\infty) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-2d3658577af35dd369549e995357ea2d_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(16)\quad s(t)= \underbrace{ \left[\begin{array}{cc} S_1C & S_2 \\ \end{array}\right] }_{S} \underbrace{ \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right] }_{x_{E3}(t)} = \underbrace{S_2 \left[\begin{array}{cc} M & I \\ \end{array}\right] }_{S} \underbrace{ \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right] }_{x_{E3}(t)} \ (M=S_2^{-1}S_1C) }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-69cd20c627c9c293f7697843f1f90dc6_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(17)\quad \underbrace{ \left[\begin{array}{c} x_1(t)\\ s(t) \end{array}\right] }_{x'_{E3}(t)} = \underbrace{ \left[\begin{array}{cc} I & 0 \\ S_1C & S_2 \\ \end{array}\right] }_{T_s} \underbrace{ \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right] }_{x_{E3}(t)}\\ \Leftrightarrow \underbrace{ \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right] }_{x_{E3}(t)} = \underbrace{ \left[\begin{array}{cc} I & 0 \\ -S_2^{-1}S_1C & S_2^{-1} \\ \end{array}\right] }_{T_s^{-1}} \underbrace{ \left[\begin{array}{c} x_1(t)\\ s(t) \end{array}\right] }_{x'_{E3}(t)} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-7e934ac1b10cbc9a5aedfe8a2c7344ed_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(18)\quad \underbrace{ \left[\begin{array}{c} \dot x_1(t)\\ \dot s(t) \end{array}\right] }_{\dot{x}'_{E3}(t)} = \underbrace{ \left[\begin{array}{cc} \bar{A}_{11} & \bar{A}_{12} \\ \bar{A}_{21} & \bar{A}_{22} \\ \end{array}\right] }_{T_sA_{E3}T_s^{-1}} \underbrace{ \left[\begin{array}{c} x_1(t)\\ s(t) \end{array}\right] }_{x'_{E3}(t)} + \underbrace{ \left[\begin{array}{cc} 0\\ S_2 \end{array}\right] }_{T_sB_{E3}} {\dot u}(t) }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-5ab2f4f17de6fee9605aa2556d7f248c_l3.png "Rendered by QuickLaTeX.com")
が安定行列となるようにスイッチング関数が選ばれていると仮定します。
![\displaystyle{(21)\quad \begin{array}{lll} V(\bar{x})= \underbrace{ \left[\begin{array}{c} x_1(t)\\ s(t) \end{array}\right]^T }_{\bar{x}^T(t)} \underbrace{ \left[\begin{array}{cc} P_1 & 0\\ 0 & P_2 \end{array}\right] }_{P} \underbrace{ \left[\begin{array}{c} x_1(t)\\ s(t) \end{array}\right] }_{\bar{x}(t)}\\ \Rightarrow \dot{V}(\bar{x})\le - \underbrace{ \left[\begin{array}{c} x_1(t)\\ s(t) \end{array}\right]^T }_{\bar{x}^T(t)} \underbrace{ \left[\begin{array}{cc} Q_1 & 0\\ 0 & I \end{array}\right] }_{Q} \underbrace{ \left[\begin{array}{c} x_1(t)\\ s(t) \end{array}\right] }_{\bar{x}(t)} \end{array}} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-7c77aa3ff56d9793c30ad999001a2185_l3.png "Rendered by QuickLaTeX.com")
, 
, 
）。![\displaystyle{(22)\quad \begin{array}{l} s(t)=0\Rightarrow\dot{s}(t)=0 \Rightarrow 0=\bar{A}_{21}x_1(t)+\bar{A}_{22}s(t)+S_2{\dot u}(t)\\ \Rightarrow {\dot u}_{eq}(t)=-\underbrace{S_2^{-1}}_{(SB_{E3})^{-1}} \underbrace{\left[\begin{array}{cc} \bar{A}_{21} & \bar{A}_{22} \\ \end{array}\right]x'_{E3}(t)}_{SA_{E3}x_{E3}(t)}} \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-3ad4a24162e39c9dd7341a0db806bc67_l3.png "Rendered by QuickLaTeX.com")
は、この等価制御をベースして![\displaystyle{(23)\quad \begin{array}{l} {\dot u}_\ell(t)=-\underbrace{S_2^{-1}}_{(SB_{E3})^{-1}} \underbrace{(\left[\begin{array}{cc} \bar{A}_{21} & \bar{A}_{22} \\ \end{array}\right]x'_{E3}(t)-\Phi s(t))}_{(SA_{E3}-\Phi S)x_{E3}(t)}} \end{array}} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-4da62d9774c6809e24338ce1ff05a030_l3.png "Rendered by QuickLaTeX.com")
は安定行列）。このとき閉ループ系は次式で与えられます。![\displaystyle{(24)\quad \begin{array}{l} \left[\begin{array}{c} \dot{x}_1(t)\\ \dot{s}(t) \end{array}\right] = \left[\begin{array}{cc} \bar{A}_{11} & \bar{A}_{12} \\ 0 & \Phi \\ \end{array}\right] \left[\begin{array}{c} x_1(t)\\ s(t) \end{array}\right] + \left[\begin{array}{c} 0\\ S_2{\dot u}_n(t) \end{array}\right] \end{array }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-099b020da252117a5d07864d8fd6a593_l3.png "Rendered by QuickLaTeX.com")
は定数）。このとき次式が成り立ちます。
![\displaystyle{(35)\quad {\dot u}_\ell(t)=- \underbrace{ (SB_{E3})^{-1}(SA_{E3}-\Phi S)S_E^{-1} }_{\left[\begin{array}{cc} F & F_I \end{array}\right]} \underbrace{ \left[\begin{array}{c} {\dot x}(t) \\ y(t)-r \end{array}\right] }_{x_{E2}(t)} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-edb851ab679593bf698b480644ba82ce_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(37)\quad {\dot u}_n(t) =-S_2^{-1}\rho\, {\rm sgn}( \underbrace{ P_2SS_E^{-1} }_{\left[\begin{array}{cc} G & G_I \end{array}\right]} \underbrace{ \left[\begin{array}{c} {\dot x}(t) \\ y(t)-r \end{array}\right] }_{x_{E2}(t)}) }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-913747aecf41d0c7b9c2a72e8ab5d520_l3.png "Rendered by QuickLaTeX.com")
は
が安定行列となるように選ばれているとします。
と、ある
に対して
は次式で与えます。
は適当な安定行列
![\displaystyle{(22)\quad \underbrace{ \left[\begin{array}{c} \dot x_1(t)\\ \dot x_2(t) \end{array}\right] }_{\dot{x}(t)} = \underbrace{ \left[\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{array}\right] }_{A} \underbrace{ \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right] }_{x(t)} + \underbrace{ \left[\begin{array}{c} 0\\ B_2 \end{array}\right] }_{B} (u(t)+\xi(t,x,u)) }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-afe711daafabb0f12e447bb07a3ace73_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(23)\quad s(t)= \underbrace{ \left[\begin{array}{cc} S_1 & S_2 \\ \end{array}\right] }_{S} \underbrace{ \left[\begin{array}{c} e_{z1}(t)\\ e_{z2}(t) \end{array}\right] }_{x(t)} = \underbrace{S_2 \left[\begin{array}{cc} M & I_m \\ \end{array}\right] }_{S} \underbrace{ \left[\begin{array}{c} e_{z1}(t)\\ e_{z2}(t) \end{array}\right] }_{x(t)} \ (M=S_2^{-1}S_1) }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-d0776a59ef9ed32eda82a08310faf1f3_l3.png "Rendered by QuickLaTeX.com")
と
に注意して
![\displaystyle{(25)\quad \begin{array}{l} \underbrace{\left[\begin{array}{c} e_{z1}(t)\\ s(t) \end{array}\right]}_{\bar{e}_z(t)}= \underbrace{\left[\begin{array}{cc} I_{n-m} & 0_{n\times m} \\ S_1 & S_2 \\ \end{array}\right]}_{\bar T} \underbrace{\left[\begin{array}{c} e_{z1}(t)\\ e_{z2}(t) \end{array}\right]}_{e_z(t)}\\ \Leftrightarrow \underbrace{\left[\begin{array}{c} e_{z1}(t)\\ e_{z2}(t) \end{array}\right]}_{e_z(t)}= \underbrace{\left[\begin{array}{cc} I_{n} & 0_{n\times m} \\ -M & S_2^{-1} \\ \end{array}\right]}_{\bar{T}^{-1}} \underbrace{\left[\begin{array}{c} e_{z1}(t)\\ s(t) \end{array}\right]}_{\bar{e}_z(t)}\quad(M=S_2^{-1}S_1) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-348e24095f272550ca7b0d85c1be93cf_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(26)\quad \begin{array}{l} \underbrace{ \left[\begin{array}{cc} I & 0 \\ -M & S_2^{-1} \\ \end{array}\right] }_{\bar{T}^{-1}} \underbrace{ \left[\begin{array}{c} \dot e_{z1}(t)\\ \dot s(t) \end{array}\right] }_{\dot{\bar{e}}_z(t)}\\ = (\left[\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{array}\right] - \left[\begin{array}{c} 0\\ B_2 \end{array}\right] (\left[\begin{array}{cc} S_1 & S_2 \\ \end{array}\right] \left[\begin{array}{c} 0\\ B_2 \end{array}\right])^{-1}\\ \times(\left[\begin{array}{cc} S_1 & S_2 \\ \end{array}\right] \left[\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{array}\right] -\Phi \left[\begin{array}{cc} S_1 & S_2 \\ \end{array}\right]) ) \underbrace{ \left[\begin{array}{cc} I & 0 \\ -M & S_2^{-1} \\ \end{array}\right] }_{\bar{T}^{-1}} \underbrace{ \left[\begin{array}{c} e_{z1}(t)\\ s(t) \end{array}\right] }_{\bar{e}_z(t)}\\ -G Ce(t)+\left[\begin{array}{c} 0\\ B_2 \end{array}\right](\nu_c+\xi(t,x,u)+\nu_o) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-8b57169deafca1bae478ad161a8da6bc_l3.png "Rendered by QuickLaTeX.com")
をかけて![\displaystyle{(27)\quad \begin{array}{l} \underbrace{ \left[\begin{array}{c} \dot e_{z1}(t)\\ \dot s(t) \end{array}\right] }_{\dot{\bar{e}}_z(t)} = \underbrace{ \left[\begin{array}{cc} I & 0 \\ S_1 & S_2 \\ \end{array}\right] }_{\bar{T}} (\left[\begin{array}{cc} A_{11}-A_{12}M & A_{12}S_2^{-1} \\ A_{21}-A_{22}M & A_{22}S_2^{-1} \\ \end{array}\right] -\left[\begin{array}{c} 0\\ B_2 \end{array}\right](S_2B_2)^{-1}\\ \times(\left[\begin{array}{cc} S_1 & S_2 \end{array}\right] \left[\begin{array}{cc} A_{11}-A_{12}M & A_{12}S_2^{-1} \\ A_{21}-A_{22}M & A_{22}S_2^{-1} \\ \end{array}\right] -\Phi \left[\begin{array}{cc} 0 & I_m \\ \end{array}\right]) ) \underbrace{ \left[\begin{array}{c} e_{z1}(t)\\ s(t) \end{array}\right] }_{\bar{e}_z(t)}\\ -\underbrace{\bar{T}G}_{\bar{G}} Ce(t) +\underbrace{\left[\begin{array}{cc} I & 0 \\ S_1 & S_2 \\ \end{array}\right]}_{\bar{T}} \left[\begin{array}{c} 0\\ B_2 \end{array}\right](\nu_c+\xi(t,x,u)+\nu_o)\\ = (\left[\begin{array}{cc} \bar{A}_{11} & \bar{A}_{12} \\ \bar{A}_{21} & \bar{A}_{22}\\ \end{array}\right] -\left[\begin{array}{c} 0\\ S_2B_2 \end{array}\right](S_2B_2)^{-1}\\ \times(\left[\begin{array}{cc} \bar{A}_{21} & \bar{A}_{22}\\ \end{array}\right] -\Phi \left[\begin{array}{cc} 0 & I_m \\ \end{array}\right]) ) \underbrace{ \left[\begin{array}{c} e_{z1}(t)\\ s(t) \end{array}\right] }_{\bar{e}_z(t)}\\ -\bar{G} Ce(t)+\bar{B}(\nu_c+\xi(t,x,u)+\nu_o)\\ =\underbrace{ \left[\begin{array}{cc} \bar{A}_{11} & \bar{A}_{12} \\ 0 & \Phi\\ \end{array}\right]}_{A_c} \underbrace{ \left[\begin{array}{c} e_{z1}(t)\\ s(t) \end{array}\right] }_{\bar{e}_z(t)} -\bar{G} Ce(t) +\underbrace{\left[\begin{array}{c} 0\\ S_2B_2 \end{array}\right]}_{\bar{B}}(\nu_c+\xi(t,x,u)+\nu_o) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-2b1c5781518b0452b1e2a88534ac95c5_l3.png "Rendered by QuickLaTeX.com")
が選ばれているとします。![\displaystyle{(28)\quad \begin{array}{l} \left[\begin{array}{c} \dot{e}(t)\\ \dot{\bar{e}}_z(t) \end{array}\right]= \underbrace{\left[\begin{array}{cc} A_o & 0\\ -\bar{G}C & A_c \end{array}\right]}_{A_G} \left[\begin{array}{c} e(t)\\ \bar{e}_z(t) \end{array}\right]\\ + \left[\begin{array}{c} 0\\ \bar{B} \end{array}\right]{\nu}_c + \left[\begin{array}{c} B\\ \bar{B} \end{array}\right]\nu_o + \left[\begin{array}{c} -B\\ \bar{B} \end{array}\right]\xi(t,x,\hat{u}) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-4780b23f37e18adf37d5716a3c232cfb_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(1.4)\quad \begin{array}{l} A_m=A+BL_x,\ B_m=BL_r\\ A_o=A-GC\\ A_c=\bar{T}(A_m-BL)\bar{T}^{-1}\\ L=(SB)^{-1}(SA_m-\Phi S)\\ \bar{B}=\left[\begin{array}{c} 0\\ S_2B_2 \end{array}\right],\ \bar{G}=\bar{T}G \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-5af9937c0ebb7b0466617bc2ea71afbe_l3.png "Rendered by QuickLaTeX.com")
、
は、それぞれ安定行列
、
のリャプノフ行列とし、次のリャプノフ方程式の解とします。
![\displaystyle{(4)\quad \underbrace{\left[\begin{array}{cc} \bar{P}_1 & 0\\ 0 & \bar{P}_2 \end{array}\right]}_{\bar P} \underbrace{\left[\begin{array}{cc} \bar{A}_{11} & \bar{A}_{12}\\ 0 & \Phi \end{array}\right]}_{A_c} +\underbrace{\left[\begin{array}{cc} \bar{A}_{11} & \bar{A}_{12}\\ 0 & \Phi \end{array}\right]^T}_{A_c^T} \underbrace{\left[\begin{array}{cc} \bar{P}_1 & 0\\ 0 & \bar{P}_2 \end{array}\right]}_{\bar P}<0 }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-9f1e612deced36f7c7db9b14fc59dc21_l3.png "Rendered by QuickLaTeX.com")
、
は、それぞれ安定行列
は、次のリャプノフ不等式の解を構成するとします。![\displaystyle{(7)\quad \underbrace{\left[\begin{array}{cc} P & 0\\ 0 & \bar{P} \end{array}\right]}_{P_G} \underbrace{\left[\begin{array}{cc} A_o & 0\\ -\bar{G}C & A_c \end{array}\right]}_{A_G} +\underbrace{\left[\begin{array}{cc} A_o & 0\\ -\bar{G}C & A_c \end{array}\right]^T}_{A_G^T} \underbrace{\left[\begin{array}{cc} P & 0\\ 0 & \bar{P} \end{array}\right]}_{P_G}<0 }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-dceeb6208f924a3f6efeaa77bb099a32_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(8)\quad \begin{array}{l} \underbrace{\left[\begin{array}{cc} PA_o & 0\\ -\bar{P}\bar{G}C & \bar{P}A_c \end{array}\right]}_{P_GA_G}+ \underbrace{\left[\begin{array}{cc} PA_o & 0\\ -\bar{P}\bar{G}C & \bar{P}A_c \end{array}\right]^T}_{(P_GA_G)^T} =\left[\begin{array}{cc} -Q & -C^T\bar{G}^T\bar{P}\\ -\bar{P}\bar{G}C & \bar{P}A_c+A_c^T\bar{P} \end{array}\right]<0\\ \Leftrightarrow \bar{P}A_c+A_c^T\bar{P}+\bar{P}\bar{G}CQ^{-1}C^T\bar{G}^T\bar{P}<0\\ \Leftrightarrow A_c\bar{P}^{-1}+\bar{P}^{-1}A_c^T+\bar{G}CQ^{-1}C^T\bar{G}^T<0\\ \Leftrightarrow A_c\bar{P}^{-1}+\bar{P}^{-1}A_c^T+\bar{G}CQ^{-1}C^T\bar{G}^T=-\bar{Q}\quad(\bar{Q}>0) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-c62b77d490c2fb89c0065a176246cf86_l3.png "Rendered by QuickLaTeX.com")
を導入し、次の公式を用いています。![\displaystyle{ \begin{array}{lll} && \left[\begin{array}{cc} P & M \\ M^T & Q \end{array}\right]<0\\ &\Leftrightarrow& P-MQ^{-1}M^T<0,\ Q<0\\ &\Leftrightarrow& P<0,\ Q-M^TP^{-1}M<0 \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-eee622434bc9e2fc505eb899dbb107cb_l3.png "Rendered by QuickLaTeX.com")
と
を与えて、
![\displaystyle{(11)\quad \begin{array}{l} \underbrace{\left[\begin{array}{cc} \bar{A}_{11} & \bar{A}_{12}\\ 0 & \Phi \end{array}\right]}_{A_c} \underbrace{\left[\begin{array}{cc} \bar{P}_1^{-1} & 0\\ 0 & \bar{P}_2^{-1} \end{array}\right]}_{\bar{P}^{-1}} + \underbrace{\left[\begin{array}{cc} \bar{P}_1^{-1} & 0\\ 0 & \bar{P}_2^{-1} \end{array}\right]}_{\bar{P}^{-1}} \underbrace{\left[\begin{array}{cc} \bar{A}_{11} & \bar{A}_{12}\\ 0 & \Phi \end{array}\right]^T}_{A_c^T}\\ +\underbrace{\left[\begin{array}{c} \bar{G}_1\\ \bar{G}_2 \end{array}\right]}_{\bar{G}} \underbrace{CQ^{-1}C^T}_{Q_{22}} \underbrace{\left[\begin{array}{c} \bar{G}_1\\ \bar{G}_2 \end{array}\right]^T}_{\bar{G}^T}\\ = \underbrace{\left[\begin{array}{cc} \bar{A}_{11}\bar{P}_1^{-1} & \bar{A}_{12}\bar{P}_2^{-1}\\ 0 & \Phi \bar{P}_2^{-1} \end{array}\right]}_{A_c\bar{P}^{-1}} + \underbrace{\left[\begin{array}{cc} \bar{A}_{11}\bar{P}_1^{-1} & \bar{A}_{12}\bar{P}_2^{-1}\\ 0 & \Phi \bar{P}_2^{-1} \end{array}\right]^T}_{(A_c\bar{P}^{-1})^T}\\ +\underbrace{\left[\begin{array}{cc} \bar{G}_1Q_{22}\bar{G}_1^T &\bar{G}_1Q_{22}\bar{G}_2^T\\ \bar{G}_2Q_{22}\bar{G}_1^T &\bar{G}_2Q_{22}\bar{G}_2^T \end{array}\right]}_{\bar{G}CQ^{-1}C^T\bar{G}^T}\\ = \left[\begin{array}{cc} \bar{A}_{11}\bar{P}_1^{-1}+\bar{P}_1^{-1}\bar{A}_{11}^T+\bar{G}_1Q_{22}\bar{G}_1^T & \bar{A}_{12}\bar{P}_2^{-1}+\bar{G}_1Q_{22}\bar{G}_2^T\\ \bar{P}_2^{-1}\bar{A}_{12}^T+\bar{G}_2Q_{22}\bar{G}_1^T & \Phi\bar{P}_2^{-1}+\bar{P}_2^{-1}\Phi^T+\bar{G}_2Q_{22}\bar{G}_2^T \end{array}\right]\\ = \left[\begin{array}{cc} -\hat{Q}_1+\bar{G}_1Q_{22}\bar{G}_1^T & \bar{A}_{12}\bar{P}_2^{-1}+\bar{G}_1Q_{22}\bar{G}_2^T\\ \bar{P}_2^{-1}\bar{A}_{12}^T+\bar{G}_2Q_{22}\bar{G}_1^T & -\hat{Q}_2+\bar{G}_2Q_{22}\bar{G}_2^T \end{array}\right]<0\\ \Leftrightarrow \left\{\begin{array}{l} \bar{G}_1Q_{22}\bar{G}_1^T +(\bar{A}_{12}\bar{P}_2^{-1}+\bar{G}_1Q_{22}\bar{G}_2^T)\\ \times(\bar{G}_2Q_{22}\bar{G}_2^T -\hat{Q}_2)^{-1}(\bar{P}_2^{-1}\bar{A}_{12}^T+\bar{G}_2Q_{22}\bar{G}_1^T) <\hat{Q}_1\\ \bar{G}_2Q_{22}\bar{G}_2^T <\hat{Q}_2 \end{array}\right. \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-d2a4be3ea4336e3bad8a74302a1a87d8_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(18)\quad \begin{array}{l} 2\bar{e}_z^T(t)\bar{P}\bar{B}(\nu_c+\nu_o+\xi(t,x,u))\\ =2\underbrace{\left[\begin{array}{c} e_{z1}(t)\\ s(t) \end{array}\right]^T}_{\zeta^T(t)} \underbrace{\left[\begin{array}{cc} \bar{P}_1 & 0\\ 0 & \bar{P}_2 \end{array}\right]}_{\bar P} \underbrace{\left[\begin{array}{c} 0\\ S_2B_2 \end{array}\right]}_{\bar{B}}(\nu_c+\nu_o+\xi(t,x,u))\\ =-2s^T(t)\bar{P}_2(S_2B_2)\rho_o(u_\ell,y)\frac{FCe(t)}{||FCe(t)||}\\ -2s^T(t)\bar{P}_2(S_2B_2)(S_2B_2)^{-1}\rho_c(u_\ell,y) \frac{\bar{P}_2s(t)}{||\bar{P}_2s(t)||}+2s^T(t)\bar{P}_2(S_2B_2)\xi(t,x,u)\\ =-2s^T(t)\bar{P}_2(S_2B_2)\rho_o(u_\ell,y)\frac{B^TPe(t)}{||B^TPe(t)||}\\ -2s^T(t)\bar{P}_2\rho_c(u_\ell,y)\frac{\bar{P}_2s(t)}{||\bar{P}_2s(t)||}+2s^T(t)\bar{P}_2(S_2B_2)\xi(t,x,u)\\ \le 2||\bar{P}_2s(t)||||S_2B_2||\rho_o(u_\ell,y)-2\rho_c(u_\ell,y)||\bar{P}_2s(t)||\\ +2||\bar{P}_2s(t)||||S_2B_2||||\xi(t,x,u)||\\ \le 2||\bar{P}_2s(t)||(\rho_o(u_\ell,y)||S_2B_2||-\rho_c(u_\ell,y))\\ +2||\bar{P}_2s(t)||||S_2B_2||(k_1||u||+\alpha(t,y))\\ \le 2||\bar{P}_2s(t)||(\rho_c(u_\ell,y)-\gamma_c-\rho_c(u_\ell,y))\\ +2||\bar{P}_2s(t)||||S_2B_2||(\rho_o(u_\ell,y)-\gamma_o)\\ =-2\underbrace{(\gamma_c-||S_2B_2||(\rho_o(u_\ell,y)-\gamma_o))}_{\gamma_c'}||\bar{P}_2s(t)|| \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-b653afff2b1cd408b6383ca8513746bc_l3.png "Rendered by QuickLaTeX.com")
の代わりに
を用いた制御則
を
で表します。![\displaystyle{(4)\quad \begin{array}{l} \hat{u}(t)=\underbrace{-L\left[\begin{array}{c} x_r(t)\\ z(t) \end{array}\right]-L_rr(t)+L_{\dot r}\dot{r}(t)}_{\hat {u}_L}+\hat{\nu}_c\\ \displaystyle{\hat{\nu}_c=L_n\frac{P_2(S\left[\begin{array}{c} x_r(t)\\ z(t) \end{array}\right]-S_rr(t))}{||P_2(S\left[\begin{array}{c} x_r(t)\\ z(t) \end{array}\right]-S_rr(t))||}}\\ L=(S\tilde{B})^{-1}(S\tilde{A}-\Phi S)\\ L_r=(S\tilde{B})^{-1}(\Phi S_r+S_2MB_r)\\ L_{\dot r}=(S\tilde{B})^{-1}S_r\\ L_n=(S\tilde{B})^{-1}\rho_c(\hat{u}_L,y) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-024f6914b3128301f6c6ae196567ba16_l3.png "Rendered by QuickLaTeX.com")
![\left[\begin{array}{c} x_r(t)\\ z(t) \end{array}\right]\in{\rm\bf R}^{m+n}](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-862ce405f31ebaf652230ab6f5133858_l3.png "Rendered by QuickLaTeX.com")
を![\left[\begin{array}{c} \tilde z_1(t)\\ \tilde z_2(t) \end{array}\right]\in{\rm\bf R}^{n+m}](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-5d06152b9334198e142b8c554e3d25a9_l3.png "Rendered by QuickLaTeX.com")
と分割して、次式で表されるとしています。![\displaystyle{(5)\quad s(t)= \underbrace{ \left[\begin{array}{cc} S_1 & S_2 \\ \end{array}\right] }_{S} %\underbrace{ \left[\begin{array}{c} \tilde z_1(t)\\ \tilde z_2(t) \end{array}\right] %}_{x(t)} = \underbrace{S_2 \left[\begin{array}{cc} M & I_m \\ \end{array}\right] }_{S} %\underbrace{ \left[\begin{array}{c} \tilde z_1(t)\\ \tilde z_2(t) \end{array}\right] %}_{x(t)} \ (M=S_2^{-1}S_1) }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-027f6a001ff2fd6efdf5b3456395f8b3_l3.png "Rendered by QuickLaTeX.com")
を用いて、(2.1)は
![\displaystyle{(9)\quad \boxed{\begin{array}{l} \left[\begin{array}{c} \dot x_r(t)\\ \dot z(t) \end{array}\right] = \left[\begin{array}{cc} 0_{m\times m} & -C \\ 0_{n\times m} & A \\ \end{array}\right] \left[\begin{array}{c} x_r(t)\\ z(t) \end{array}\right] + \left[\begin{array}{c} I_p\\ -G \end{array}\right] e_y(t)\\ + \left[\begin{array}{c} r(t)\\ B(\hat{u}(t)+\nu_o) \end{array}\right] \end{array}} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-65eb681dec39ab633fe7a5bbc2b85d2c_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(10)\quad \begin{array}{l} \left[\begin{array}{c} \tilde z_1(t)\\ s(t) \end{array}\right]= \underbrace{\left[\begin{array}{cc} I_{n} & 0_{n\times m} \\ S_1 & S_2 \\ \end{array}\right]}_{\bar T} \left[\begin{array}{c} \tilde z_1(t)\\ \tilde z_2(t) \end{array}\right]\\ \Leftrightarrow \left[\begin{array}{c} \tilde z_1(t)\\ \tilde z_2(t) \end{array}\right]= \underbrace{\left[\begin{array}{cc} I_{n} & 0_{n\times m} \\ -M & S_2^{-1} \\ \end{array}\right]}_{\bar{T}^{-1}} \left[\begin{array}{c} \tilde z_1(t)\\ s(t) \end{array}\right]\quad(M=S_2^{-1}S_1) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-4df32cc888cc01697c0d0a7a4a8de076_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(11)\quad \begin{array}{l} \bar{T}^{-1} \left[\begin{array}{c} \dot{\tilde{z}}_1(t)\\ \dot{s}(t) \end{array}\right] = \underbrace{\left[\begin{array}{cc|c} 0_{m\times m} & -C_1 & -C_2 \\ 0_{n-m\times m} & A_{11} & A_{12} \\\hline 0_{m\times m} & A_{21} & A_{22} \end{array}\right]}_{\left[\begin{array}{cc} \tilde{A}_{11} & \tilde{A}_{12} \\ \tilde{A}_{21} & \tilde{A}_{22} \end{array}\right]} \left[\begin{array}{cc} I_{n} & 0_{n\times m} \\ -M & S_2^{-1} \\ \end{array}\right] \left[\begin{array}{c} \tilde z_1(t)\\ s(t) \end{array}\right]\\ + \left[\begin{array}{c} I_p\\ -G_1\\\hline -G_2 \end{array}\right] e_y(t) + \left[\begin{array}{c} r(t)\\ 0\\\hline B_2(\hat{u}(t)+\nu_o) \end{array}\right]\\ = \underbrace{\left[\begin{array}{cc} \tilde{A}_{11}-\tilde{A}_{12}M & \tilde{A}_{12}S_2^{-1}\\ \tilde{A}_{21}-\tilde{A}_{22}M & \tilde{A}_{22}S_2^{-1} \end{array}\right]}_{\tilde{A}\bar{T}^{-1}} \left[\begin{array}{c} \tilde z_1(t)\\ s(t) \end{array}\right] - \left[\begin{array}{c} \bar{G}_1\\ G_2 \end{array}\right] e_y(t)\\ + \left[\begin{array}{c} B_rr(t)\\ B_2(\hat{u}(t)+\nu_o) \end{array}\right]\\ (\left[\begin{array}{c} G_1\\ G_2 \end{array}\right] =\left[\begin{array}{c} A_{12}C_2^{-1} \\ A_{22}C_2^{-1}-C_2^{-1}A_{22}^S \end{array}\right], \quad\bar{G}_1= \left[\begin{array}{c} -I_p\\ G_1 \end{array}\right]) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-da08553d36abc2e12fd0c39618cc6763_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(12)\quad \begin{array}{l} \left[\begin{array}{c} \dot{\tilde{z}}_1(t)\\ \dot{s}(t) \end{array}\right] = \underbrace{\left[\begin{array}{cc} I_{n} & 0_{n\times m} \\ S_1 & S_2 \\ \end{array}\right] \left[\begin{array}{cc} \tilde{A}_{11}-\tilde{A}_{12}M & \tilde{A}_{12}S_2^{-1}\\ \tilde{A}_{21}-\tilde{A}_{22}M & \tilde{A}_{22}S_2^{-1} \end{array}\right]}_{\bar{T}\tilde{A}\bar{T}^{-1}} \left[\begin{array}{c} \tilde z_1(t)\\ s(t) \end{array}\right]\\ - \left[\begin{array}{cc} I_{n} & 0_{n\times m} \\ S_1 & S_2 \\ \end{array}\right] \left[\begin{array}{c} \bar{G}_1\\ G_2 \end{array}\right] e_y(t) + \left[\begin{array}{cc} I_{n} & 0_{n\times m} \\ S_1 & S_2 \\ \end{array}\right] \left[\begin{array}{c} B_rr(t)\\ B_2(\hat{u}(t)+\nu_o) \end{array}\right]\\ \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-1da3c1791a8da84361163ffde267324f_l3.png "Rendered by QuickLaTeX.com")
とおいて![\displaystyle{(14)\quad \begin{array}{l} \dot{s}(t)= \underbrace{((S_1\bar{A}_{11}+S_2(\tilde{A}_{21}-\tilde{A}_{22}M))\tilde{z}_1(t)+(S_1\bar{A}_{12}+S_2\tilde{A}_{22}S_2^{-1})s(t)}_{S\tilde{A}\bar{T}^{-1}\left[\begin{array}{c} \tilde z_1(t)\\ s(t) \end{array}\right]=S\tilde{A}\left[\begin{array}{c} \tilde z_1(t)\\ \tilde z_2(t) \end{array}\right]}\\ -\underbrace{(S_1\bar{G}_1+S_2G_2)}_{\bar G_2}e_y(t)+S_1B_rr(t)\\ +\underbrace{S_2B_2}_{\Lambda}(L\left[\begin{array}{c} \tilde z_1(t)\\ \tilde z_2(t) \end{array}\right]+L_rr(t)+S_r\dot{r}(t)+\hat{\nu}_c+\nu_o)\\ =S\tilde{A}\left[\begin{array}{c} \tilde z_1(t)\\ \tilde z_2(t) \end{array}\right]-\bar{G}_2e_y(t)+S_1B_rr(t)\\ +\Lambda(-\Lambda^{-1}(S\tilde{A}-\Phi S)\left[\begin{array}{c} \tilde z_1(t)\\ \tilde z_2(t) \end{array}\right]-\Lambda^{-1}(\Phi S_r+S_2MB_r)r(t)+\Lambda^{-1}S_r\dot{r}(t))\\ +\Lambda(\hat{\nu}_c+\nu_o)\\ =S\tilde{A}\left[\begin{array}{c} \tilde z_1(t)\\ \tilde z_2(t) \end{array}\right]-\bar{G}_2e_y(t)+S_1B_rr(t)\\ -(S\tilde{A}-\Phi S)\left[\begin{array}{c} \tilde z_1(t)\\ \tilde z_2(t) \end{array}\right]-(\Phi S_r+S_2MB_r)r(t)+S_r\dot{r}(t)+\Lambda(\hat{\nu}_c+\nu_o)\\ =\Phi (\underbrace{S\left[\begin{array}{c} \tilde z_1(t)\\ \tilde z_2(t) \end{array}\right]}_{s(t)}-S_rr(t))+S_r\dot{r}(t)-\bar{G}_2e_y(t)+\Lambda(\hat{\nu}_c+\nu_o) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-ebec66163067a3490f53ebceab553d5e_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(16)\quad \begin{array}{l} \underbrace{\left[\begin{array}{c} \zeta_1(t)\\ \zeta_2(t) \end{array}\right]}_{\zeta(t)}= \left[\begin{array}{c} \tilde z_1(t)\\ s(t) \end{array}\right]+ \left[\begin{array}{c} \bar{A}_{11}^{-1}(\bar{A}_{12}S_r+B_r)r_c\\ -S_rr(t) \end{array}\right] \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-c676bab7a7f2b7c956feb633f175faa2_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(18)\quad \begin{array}{l} \underbrace{\left[\begin{array}{c} \dot \zeta_1(t)\\ \dot \zeta_2(t) \end{array}\right]}_{\dot{\zeta}(t)}= \underbrace{\left[\begin{array}{cc} \bar{A}_{11} & \bar{A}_{12}\\ 0 & \Phi \end{array}\right]}_{A_c} \underbrace{\left[\begin{array}{c} \zeta_1(t)\\ \zeta_2(t) \end{array}\right]}_{\zeta(t)}\\ - \underbrace{\left[\begin{array}{c} \bar{G}_1\\ \bar{G}_2 \end{array}\right]}_{\bar{G}} \underbrace{e_y(t)}_{Ce(t)} + \underbrace{\left[\begin{array}{c} B_r+\bar{A}_{12}S_r\\ 0 \end{array}\right]}_{\bar{G}_r}e_r(t) + \underbrace{\left[\begin{array}{c} 0\\ \Lambda \end{array}\right]}_{\bar\Lambda}(\hat{\nu}_c+\nu_o) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-e698246bb04c91be373318b20c960104_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(19)\quad \begin{array}{l} \left[\begin{array}{c} \dot{e}(t)\\ \dot \zeta(t) \end{array}\right]= \underbrace{\left[\begin{array}{cc} A_o & 0\\ -\bar{G}C & A_c \end{array}\right]}_{A_G} \left[\begin{array}{c} e(t)\\ \zeta(t) \end{array}\right]\\ + \left[\begin{array}{c} 0\\ \bar{G}_r \end{array}\right]e_r(t) + \left[\begin{array}{c} B\\ \bar\Lambda \end{array}\right]\nu_o + \left[\begin{array}{c} 0\\ \bar\Lambda \end{array}\right]\hat{\nu}_c - \left[\begin{array}{c} B\\ 0 \end{array}\right]\xi(t,x,\hat{u}) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-8e6c6626c17ebcf75a8e1b658d9e4738_l3.png "Rendered by QuickLaTeX.com")
は、それぞれ安定行列
のリャプノフ行列とし、次のリャプノフ方程式の解とします。
![\displaystyle{(18)\quad \begin{array}{l} 2\zeta^T(t)\bar{P}\bar{\Lambda}(\nu_o+\hat{\nu}_c)\\ =2\underbrace{\left[\begin{array}{c} \zeta_1(t)\\ \zeta_2(t) \end{array}\right]^T}_{\zeta^T(t)} \underbrace{\left[\begin{array}{cc} \bar{P}_1 & 0\\ 0 & \bar{P}_2 \end{array}\right]}_{\bar P} \underbrace{\left[\begin{array}{c} 0\\ \Lambda \end{array}\right]}_{\bar\Lambda}(\nu_o+\hat{\nu}_c)\\ =-2\zeta_2^T(t)\bar{P}_2\Lambda\rho_o(\hat{u}_L,y)\frac{FCe(t)}{||FCe(t)||}-2\zeta_2^T(t)\bar{P}_2\Lambda\rho_c(\hat{u}_L,y)\Lambda^{-1} \frac{\bar{P}_2\zeta_2(t)}{||\bar{P}_2\zeta_2(t)||}\\ \le 2||\bar{P}_2\zeta_2(t)||||\Lambda||\rho_o(\hat{u}_L,y)-2\rho_c(\hat{u}_L,y)||\bar{P}_2\zeta_2(t)||\\ =2||\bar{P}_2\zeta_2(t)||(\rho_o(\hat{u}_L,y)||\Lambda||-\rho_c(\hat{u}_L,y))\\ =2||\bar{P}_2\zeta_2(t)||(\rho_c(\hat{u}_L,y)-\gamma_c-\rho_c(\hat{u}_L,y))\\ =-2\gamma_c||\bar{P}_2\zeta_2(t)|| \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-79955beab6ef0705346e5157fe6ced17_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(1)\quad {\cal S}_c=\left\{\left[\begin{array}{c} e\\ \zeta_1\\ \zeta_2 \end{array}\right]\in{\bf R}^{2n+p}: \left\{\begin{array}{l} e_y(t)=Ce(t)=Cz(t)-y(t)=0\\ \zeta_2(t)=s(t)-S_rr(t)=0 \end{array}\right. \right\} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-198a3c5b246cf20144027bbb223bb2b6_l3.png "Rendered by QuickLaTeX.com")
より
を適用して
![\displaystyle{(9)\quad \Omega_\eta=\left\{\left[\begin{array}{c} e\\ \zeta_1\\ \zeta_2 \end{array}\right]\in{\bf R}^{2n+p}: \left\{\begin{array}{l} ||(CB)^{-1}CA_oe(t)||<\gamma_o-\eta\\ ||\bar{G}_2e_y(t)||<\gamma_c-\eta \end{array}\right. \right\} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-ba3afedc3c9f5d435d671a0460dfdda4_l3.png "Rendered by QuickLaTeX.com")
となり、超平面
に突入することが分かります（
の表現式はEdwards and Spurgeonによるものです）。
はモデル誤差、非線形要素、外乱などの影響を表し、次のマッチング条件を満たすとします。
は可制御対
（相対次数１）
の不変零点は安定（最小位相系）![\displaystyle{ \begin{array}{ll} (3.1) & \underbrace{ \left[\begin{array}{c} \dot x_1(t)\\ \dot x_2(t) \end{array}\right] }_{\dot{x}(t)} = \underbrace{ \left[\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{array}\right] }_{A} \underbrace{ \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right] }_{x(t)} + \underbrace{ \left[\begin{array}{c} 0\\ B_2 \end{array}\right] }_{B} u(t)\\ (3.2) & y(t) = \underbrace{ \left[\begin{array}{cc} C_1 & C_2 \end{array}\right] }_{C} \underbrace{ \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right] }_{x(t)} \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-6ec74dc0c4572a9c7fa91295a5ba283b_l3.png "Rendered by QuickLaTeX.com")
が安定行列となるように選ばれているとします。また、
を仮定します。座標変換![\displaystyle{(9)\quad T_c=\left[\begin{array}{cc} I_{n-p} & 0 \\ C_1 & C_2 \end{array}\right] \Leftrightarrow T_c^{-1}=\left[\begin{array}{cc} I_{n-p} & 0 \\ -C_2^{-1}C_1 & C_2^{-1} \end{array}\right] }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-3db2aaeb3c36b8bc6bf0ed0224760541_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(10)\quad \begin{array}{l} \underbrace{\left[\begin{array}{cc} {\cal A}_{11} & {\cal A}_{12}\\ {\cal A}_{21} & {\cal A}_{22} \end{array}\right]}_{{\cal A}}\\ =\underbrace{\left[\begin{array}{cc} {A}_{11}-{A}_{12}C_2^{-1}C_1 & A_{12}C_2^{-1} \\ C_1{\cal A}_{11}+C_2A_{21}-C_2A_{22}C_2^{-1}C_1 & C_1A_{12}C_2^{-1}+C_2A_{22}C_2^{-1} \end{array}\right] }_{T_cAT_c^{-1}}\\ {\cal B}=\underbrace{\left[\begin{array}{c} 0 \\ C_2B_2 \end{array}\right] }_{T_cB}\\ {\cal C}=\underbrace{\left[\begin{array}{cc} 0 & I_p \end{array}\right] }_{CT_c^{-1}}\\ {\cal P}= \underbrace{\left[\begin{array}{cc} I_{n-p} & 0 \\ -C_2^{-1}C_1 & C_2^{-1} \end{array}\right]^T \left[\begin{array}{cc} P_1 & 0\\ 0 & P_2 \end{array}\right] \left[\begin{array}{cc} I_{n-p} & 0 \\ -C_2^{-1}C_1 & C_2^{-1} \end{array}\right] }_{T_c^TPT_c}\\ =\left[\begin{array}{cc} I_{n-p} & -C_1^TC_2^{-T} \\ 0 & C_2^{-T} \end{array}\right]^T \left[\begin{array}{cc} P_1 & 0 \\ -P_2C_2^{-1}C_1 & P_2C_2^{-1} \end{array}\right]\\ =\left[\begin{array}{cc} P_1+C_1^TC_2^{-T}P_2C_2^{-1}C_1 & -C_1^TC_2^{-T}P_2C_2^{-1}\\ -C_2^{-T}P_2C_2^{-1}C_1 & C_2^{-T}P_2C_2^{-1} \end{array}\right] \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-23b25bc8ecbab40ee6e5976ebefccd82_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(11)\quad \begin{array}{l} G=\underbrace{\left[\begin{array}{cc} I_{n-p} & 0 \\ -C_2^{-1}C_1 & C_2^{-1} \end{array}\right] }_{T_c^{-1}} \underbrace{\left[\begin{array}{c} A_{12}C_2^{-1}\\ C_1A_{12}C_2^{-1}+C_2A_{22}C_2^{-1}-{\cal A}_{22}^s \end{array}\right] }_{\left[\begin{array}{c} {\cal A}_{12}\\ {\cal A}_{22}-{\cal A}_{22}^s \end{array}\right]}\\ =\left[\begin{array}{c} A_{12}C_2^{-1} \\ A_{22}C_2^{-1}-C_2^{-1}A_{22}^S \end{array}\right] \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-6f55e042f0e71a68f90ee8ddf5588d0c_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(12)\quad \begin{array}{l} \underbrace{\left[\begin{array}{cc} P_1+C_1^TC_2^{-T}P_2C_2^{-1}C_1 & -C_1^TC_2^{-T}P_2C_2^{-1}\\ -C_2^{-T}P_2C_2^{-1}C_1 & C_2^{-T}P_2C_2^{-1} \end{array}\right] \left[\begin{array}{c} 0 \\ C_2B_2 \end{array}\right] }_{{\cal P}{\cal B}} = \underbrace{\left[\begin{array}{cc} 0 & I_p \end{array}\right]^T }_{{\cal C}^T} F^T\\ \Rightarrow F=(C_2^{-T}P_2C_2^{-1}C_2B_2)^T=B_2^TP_2C_2^{-1} \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-d8371d4e684ace7595130cce38802819_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(13)\quad \boxed{\begin{array}{l} G= \left[\begin{array}{c} A_{12}C_2^{-1} \\ A_{22}C_2^{-1}-C_2^{-1}A_{22}^S \end{array}\right]\\ F=B_2^TP_2C_2^{-1} \end{array}} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-acc6f79294b0cb173b67e46276765b51_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(4)\quad \begin{array}{l} \left[\begin{array}{c} \dot x_r(t)\\ \dot x(t) \end{array}\right] = \left[\begin{array}{c|cc} 0 & -C_1 & -C_2\\\hline 0 & A_{11} & A_{12} \\ 0 & A_{21} & A_{22} \end{array}\right] \left[\begin{array}{c} x_r(t)\\ x(t) \end{array}\right] + \left[\begin{array}{c} 0\\\hline 0\\ B_2 \end{array}\right] u(t) + \left[\begin{array}{c} I_m \\\hline 0 \\ 0 \end{array}\right] r(t)\\ (x_r(t)\in{\rm\bf R}^m, x(t)\in{\rm\bf R}^n) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-1ab4b378422eddd1bff475a2408082e2_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(5a)\quad \begin{array}{l} \left[\begin{array}{c} \dot{x}_1(t)\\ \dot{x}_2(t) \end{array}\right] = \underbrace{ \left[\begin{array}{cc|c} 0 & -C_1 & -C_2\\ 0 & A_{11} & A_{12} \\\hline 0 & A_{21} & A_{22} \end{array}\right] }_{\tilde{A}=\left[\begin{array}{cc} \tilde{A}_{11} & \tilde{A}_{12} \\ \tilde{A}_{21} & \tilde{A}_{22} \\ \end{array}\right]} \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right] + \underbrace{ \left[\begin{array}{c} 0\\ 0\\\hline B_2 \end{array}\right] }_{\tilde{B}} u(t) + \left[\begin{array}{c} I_m \\ 0 \\\hline 0 \end{array}\right] r(t)\\ (x_1(t)\in{\rm\bf R}^n, x_2(t)\in{\rm\bf R}^m) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-a01e5393ef83d3a307441de8778d577b_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(5b)\quad %\begin{array}{l} \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right] = \left[\begin{array}{c} x_r(t)\\ x(t) \end{array}\right] %\end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-a936cb7479b75e7930c3bcfc0589d4fc_l3.png "Rendered by QuickLaTeX.com")
は定値となり、被積分項
の値は零となり、
は
へ漸近します。そこで、SM制御によって拡大系を安定化し、追従制御系を構成することを考えます。この制御系は特別な
の場合を含みますので、まずスイッチング関数として、次式を考えます。![\displaystyle{(6)\quad s(t)= \underbrace{ \left[\begin{array}{cc} S_1 & S_2 \\ \end{array}\right] }_{S} %\underbrace{ \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right] %}_{x(t)} = \underbrace{S_2 \left[\begin{array}{cc} M & I_m \\ \end{array}\right] }_{S} %\underbrace{ \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right] %}_{x(t)} \ (M=S_2^{-1}S_1) }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-01bb686061b94fbab2a0ff90e7d23f64_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(7)\quad \begin{array}{l} \left[\begin{array}{c} x_1(t)\\ s(t) \end{array}\right] = \underbrace{ \left[\begin{array}{cc} I_n & 0 \\ S_1 & S_2 \\ \end{array}\right] }_{T_s} \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right]\\ \Leftrightarrow \left[\begin{array}{c} x_1(t)\\ x_2(t) \end{array}\right] = \underbrace{ \left[\begin{array}{cc} I_n & 0 \\ -S_2^{-1}S_1 & S_2^{-1} \\ \end{array}\right] }_{T_s^{-1}} \left[\begin{array}{c} x_1(t)\\ s(t) \end{array}\right] \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-19e5643bcb296d8a517aba35bc4fe14e_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(8a)\quad \begin{array}{l} %\underbrace{ \left[\begin{array}{c} \dot x_1(t)\\ \dot s(t) \end{array}\right] %}_{\dot{x}'(t)} = \underbrace{ \left[\begin{array}{cc} \bar{A}_{11} & \bar{A}_{12} \\ S_2\bar{A}_{21} & S_2\bar{A}_{22}S_2^{-1} \\ \end{array}\right] }_{T_sA_ET_s^{-1}} %\underbrace{ \left[\begin{array}{c} x_1(t)\\ s(t) \end{array}\right] %}_{x'(t)} + \underbrace{ \left[\begin{array}{cc} 0\\ S_2B_2 \end{array}\right] }_{T_sB_E} u(t)\\ + \left[\begin{array}{cc} B_r \\ S_1B_r \end{array}\right] r(t) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-a23595fe93cd3fb99284c6e90f61bcf1_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(8b)\quad \left\{\begin{array}{l} \bar{A}_{11}=\underbrace{ \left[\begin{array}{cc} 0 & -C_1 \\ 0 & A_{11} \end{array}\right]}_{\tilde{A}_{11}} -\underbrace{\left[\begin{array}{c} -C_2\\ A_{12} \end{array}\right]}_{\tilde{A}_{12}}M\quad(M=S_2^{-1}S_1)\\ \bar{A}_{12}= \underbrace{\left[\begin{array}{c} -C_2\\ A_{12} \end{array}\right]}_{\tilde{A}_{12}}S_2^{-1}\\ \bar{A}_{21}=S_2(M\bar{A}_{11} + \left[\begin{array}{cc} 0 & A_{21} \end{array}\right] -A_{22}M)\\ \bar{A}_{22}=S_2(M \left[\begin{array}{c} -C_2\\ A_{12} \end{array}\right] +A_{22})S_2^{-1}\\ B_r=\left[\begin{array}{cc} I_m \\ 0 \end{array}\right] \end{array}\right. }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-a8cb0e83576eef13d13b8f669a414ae8_l3.png "Rendered by QuickLaTeX.com")
で表されますが、一般の
の場合のスライディングモードは
の選び方についてはあとで述べます。![\displaystyle{(10)\quad \begin{array}{l} u(t)=u_L(t)+\nu_c\\ u_L(t)=-\underbrace{(S\tilde{B})^{-1}(S\tilde{A}-\Phi S)}_{L=L_{eq}+L_\Phi}\left[\begin{array}{c} x_r(t)\\ x(t) \end{array}\right]\\ -\underbrace{(S\tilde{B})^{-1}(\Phi S_r+S_1B_r)}_{L_r} r(t) +\underbrace{(S\tilde{B})^{-1}S_r}_{L_{\dot r}} \dot{r}(t)\\ \nu_c=-\underbrace{(S\tilde{B})^{-1}\rho_c(u_L,y)}_{L_n}\frac{P_2(s(t)-S_rr(t))}{||P_2(s(t)-S_rr(t))||} \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-4fa5c5c72c2ee8614e09bcaa4631f9d9_l3.png "Rendered by QuickLaTeX.com")
）。![\displaystyle{(11)\quad \begin{array}{l} \hat{u}(t)=\underbrace{-L\left[\begin{array}{c} x_r(t)\\ z(t) \end{array}\right]-L_rr(t)+L_{\dot r}\dot{r}(t)}_{\hat{u}_L}+\hat{\nu}_c\\ \displaystyle{\hat{\nu}_c=-L_n\frac{P_2(S\left[\begin{array}{c} x_r(t)\\ z(t) \end{array}\right]-S_rr(t))}{||P_2(S\left[\begin{array}{c} x_r(t)\\ z(t) \end{array}\right]-S_rr(t))||}}\\ L=\Lambda^{-1}(S\tilde{A}-\Phi S)\\ L_r=\Lambda^{-1}(\Phi S_r+S_2MB_r)\\ L_{\dot r}=\Lambda^{-1}S_r\\ L_n=\Lambda^{-1}\rho_c(\hat{u}_L,y) \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-9b9975d5d229ccc8642e2fdd4c1c0561_l3.png "Rendered by QuickLaTeX.com")
とするためには
が正則であれば、
![\displaystyle{(15)\quad \begin{array}{l} \left[\begin{array}{cc} I & 0\\ B_r^T\tilde{A}_{11}^{-1} & -K_s \end{array}\right]= \left[\begin{array}{cc} \tilde{A}_{11} & \tilde{A}_{12}\\ B_r^T & 0 \end{array}\right] \left[\begin{array}{cc} \bar{A}_{11}^{-1} & -\bar{A}_{11}^{-1}\tilde{A}_{12}\\ 0 & I \end{array}\right]\\ = \left[\begin{array}{cc} \tilde{A}_{11} & \tilde{A}_{12}\\ B_r^T & 0 \end{array}\right] \left[\begin{array}{cc} I & 0\\ -M & I \end{array}\right] \left[\begin{array}{cc} \bar{A}_{11}^{-1} & -\bar{A}_{11}^{-1}\tilde{A}_{12}\\ 0 & I \end{array}\right] \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-359f0d62c20de545dd87152fb1c6a94b_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(16)\quad \begin{array}{l} {\rm det}\left[\begin{array}{cc} \tilde{A}_{11} & \tilde{A}_{12}\\ B_r^T & 0 \end{array}\right]\ne 0 \Leftrightarrow {\rm det}\left[\begin{array}{cc|c} 0 & -C_1 & -C_2\\ 0 & A_{11} & A_{12}\\\hline I & 0 & 0 \end{array}\right]\ne 0\\ \Leftrightarrow {\rm det}\left[\begin{array}{cc} -C_1 & -C_2\\ A_{11} & A_{12} \end{array}\right]\ne 0 \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-8d7129e0a72d941b228a1e4ecda1d741_l3.png "Rendered by QuickLaTeX.com")
）。
は
をフィードバックした上で、出力の推定誤差のスイッチング動作が加えられています。これにより、モデル誤差、非線形要素、外乱などの影響があるにも拘わらず、
の超平面に載せることが考えられます。
![\displaystyle{(11.1)\quad \begin{array}{l} \underbrace{ \left[\begin{array}{c} \dot x_1(t)\\ \dot y(t) \end{array}\right] }_{\dot{x}'(t)} = \underbrace{ \left[\begin{array}{cc} {\bar A}_{11} & {\bar A}_{12} \\ {\bar A}_{21} & {\bar A}_{22} \\ \end{array}\right] }_{{\bar A}=TAT^{-1}} \underbrace{ \left[\begin{array}{c} x_1(t)\\ y(t) \end{array}\right] }_{x'(t)} + \boxed{\underbrace{ \left[\begin{array}{c} 0_{n-p\times p}\\ {\bar B}_{2} \end{array}\right] }_{{\bar B}=TB}} (u(t)+\xi(t))\\ \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-df89f2c8c5027f65f3d1b001df56a333_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(11.2)\quad y(t) = \boxed{\underbrace{ \left[\begin{array}{cc} 0_{p\times n-p} & I_p \\ \end{array}\right] }_{{\bar C}=CT^{-1}}} \underbrace{ \left[\begin{array}{c} x_1(t)\\ y(t) \end{array}\right] }_{x'(t)} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-d15eb88c68556f6e054c3e64d14d9b4a_l3.png "Rendered by QuickLaTeX.com")
は安定行列です。![\displaystyle{(12)\quad \begin{array}{l} \dot{\hat x}(t)=A\hat{x}(t)+Bu(t)-G_\ell Ce(t)+G_n\nu\\ G_\ell=T_o^{-1}\left[\begin{array}{c} {\cal A}_{12}\\ {\cal A}_{22}-{\cal A}_{22}^{s} \end{array}\right]\\ G_n=||{\cal D}_{2}||T_o^{-1}\left[\begin{array}{c} 0_{n-p\times p}\\ I_p \end{array}\right]\\ \displaystyle{\nu=-\rho(t,y,u)\frac{P_2e_y(t)}{||P_2e_y(t)||} \quad(P_2{\cal A}_{22}^{s}+{\cal A}_{22}^{s}^TP_2<0)} \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-4b315d5f9deb1d92118c05143cc36e33_l3.png "Rendered by QuickLaTeX.com")
を、次のように定めることが提案されています。![\displaystyle{(13)\quad \boxed{G=(T_LT_bT_c)^{-1}\left[\begin{array}{c} {\cal A}_{12}\\ {\cal A}_{22}-{\cal A}_{22}^{s} \end{array}\right]} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-99565ec8e7483b355f59c24131882a38_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(1)\quad P=\left[\begin{array}{cc} {P}_{11} & {P}_{12} \\ {P}_{21} & {P}_{22} \end{array}\right]\quad({P}_{11}\in{\bf R}^{n-p\times n-p}) }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-636fb09f9302cb5c3d0752e5b2e18f9e_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(2)\quad T=\left[\begin{array}{cc} I_{n-p} & {P}_{11}^{-1}{P}_{12} \\ 0 & I_p \end{array}\right] }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-975d1217c9b83a1e84c83a6154c600bb_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(3)\quad \bar{C}=CT^{-1}=\left[\begin{array}{cc} 0 & I_p \end{array}\right] }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-82787eef12a2e88606fbe031fdc0378f_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(4)\quad P^{-1}=\left[\begin{array}{cc} {\tilde P}_{11} & {\tilde P}_{12} \\ {\tilde P}_{21} & {\tilde P}_{22} \end{array}\right]\quad({P}_{11}{\tilde P}_{12} + {P}_{12}{\tilde P}_{22}=0)](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-599bc13f9bd980496443996e87577ddd_l3.png "Rendered by QuickLaTeX.com")
に注意して![\displaystyle{(5)\quad \begin{array}{l} \bar{B}=TB= \left[\begin{array}{cc} I_{n-p} & {P}_{11}^{-1}{P}_{12} \\ 0 & I_p \end{array}\right] \left[\begin{array}{cc} {\tilde P}_{11} & {\tilde P}_{12} \\ {\tilde P}_{21} & {\tilde P}_{22} \end{array}\right] \left[\begin{array}{cc} 0 \\ I_p \end{array}\right] F^T\\ =\left[\begin{array}{cc} {\tilde P}_{12} + {P}_{11}^{-1}{P}_{12}{\tilde P}_{22} \\ {\tilde P}_{22} \end{array}\right] F^T= \underbrace{\left[\begin{array}{cc} 0 \\ {\tilde P}_{22}F^T \end{array}\right] }_{\left[\begin{array}{cc} 0 \\ {\bar B}_{2} \end{array}\right]} \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-ecf550e6e68e139dfd9b70a2c51821a0_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(6)\quad \begin{array}{l} \bar{P}=T^{-T}PT^{-1}\\ = \left[\begin{array}{cc} I_{n-p} & 0 \\ -{P}_{12}^T{P}_{11}^{-1} & I_p \end{array}\right] \left[\begin{array}{cc} {P}_{11} & {P}_{12} \\ {P}_{21} & {P}_{22} \end{array}\right] \left[\begin{array}{cc} I_{n-p} & -{P}_{11}^{-1}{P}_{12} \\ 0 & I_p \end{array}\right]\\ = \left[\begin{array}{cc} {P}_{11} & {P}_{12} \\ 0 & {P}_{22}-{P}_{12}^T{P}_{11}^{-1}{P}_{12} \end{array}\right] \left[\begin{array}{cc} I_{n-p} & -{P}_{11}^{-1}{P}_{12} \\ 0 & I_p \end{array}\right]\\ =\underbrace{\left[\begin{array}{cc} {P}_{11} & 0 \\ 0 & {P}_{22}-{P}_{21}{P}_{11}^{-1}{P}_{12} \end{array}\right] }_{\left[\begin{array}{cc} {\bar P}_{1} & 0 \\ 0 & {\bar P}_{2} \end{array}\right]} \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-d939c6b48cf57392a69499adfa73c15a_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(7)\quad \begin{array}{l} \bar{A}=TAT^{-1}\\ = \left[\begin{array}{cc} I_{n-p} & {P}_{11}^{-1}{P}_{12} \\ 0 & I_p \end{array}\right] \left[\begin{array}{cc} {A}_{11} & {A}_{12} \\ {A}_{21} & {A}_{22} \end{array}\right] \left[\begin{array}{cc} I_{n-p} & -{P}_{11}^{-1}{P}_{12} \\ 0 & I_p \end{array}\right]\\ = \left[\begin{array}{cc} {A}_{11}+{P}_{11}^{-1}{P}_{12}{A}_{21} & {A}_{12}+ {P}_{11}^{-1}{P}_{12}{A}_{22}\\ {A}_{21} & {A}_{22} \end{array}\right] \left[\begin{array}{cc} I_{n-p} & -{P}_{11}^{-1}{P}_{12} \\ 0 & I_p \end{array}\right]\\ =\underbrace{\left[\begin{array}{cc} {A}_{11}+{P}_{11}^{-1}{P}_{12}{A}_{21} & * \\ {A}_{21} & {A}_{22}-{A}_{21}{P}_{11}^{-1}{P}_{12} \end{array}\right] }_{\left[\begin{array}{cc} {\bar A}_{11} & {\bar A}_{12} \\ {\bar A}_{21} & {\bar A}_{22} \end{array}\right]} \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-818dd08ad7b025fcdf6ac9148b5e312b_l3.png "Rendered by QuickLaTeX.com")
![\displaystyle{(8)\quad \begin{array}{l} \bar{A}_o=\underbrace{TA_oT^{-1} }_{\left[\begin{array}{cc} {\bar A}_{o11} & {\bar A}_{o12} \\ {\bar A}_{o21} & {\bar A}_{o22} \end{array}\right]} =T(A-GC)T^{-1}\\ =\underbrace{TAT^{-1}}_{ \left[\begin{array}{cc} {\bar A}_{11} & {\bar A}_{12} \\ {\bar A}_{21} & {\bar A}_{22} \end{array}\right]} -\underbrace{TGCT^{-1}} _{\left[\begin{array}{cc} 0 & TG \end{array}\right]} = \left[\begin{array}{cc} {\bar A}_{11} & * \\ {\bar A}_{21} & * \end{array}\right] \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-ac09c08dae61878b5898912612961935_l3.png "Rendered by QuickLaTeX.com")
であることに注意します。
と
をかけて
![\displaystyle{(11)\quad \begin{array}{l} \left[\begin{array}{cc} {\bar P}_{1} & 0 \\ 0 & {\bar P}_{2} \end{array}\right] \left[\begin{array}{cc} {\bar A}_{o11} & {\bar A}_{o12} \\ {\bar A}_{o21} & {\bar A}_{o22} \ \end{array}\right] +\left[\begin{array}{cc} {\bar A}_{o11} & {\bar A}_{o12} \\ {\bar A}_{o21} & {\bar A}_{o22} \ \end{array}\right]^T \left[\begin{array}{cc} {\bar P}_{1} & 0 \\ 0 & {\bar P}_{2} \end{array}\right]\\ =- \left[\begin{array}{cc} \bar{Q}_{11} & \bar{Q}_{12} \\ \bar{Q}_{12}^T & \bar{Q}_{22} \end{array}\right] \end{array} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-f9c197610a2dc08d37d7807a77250d9d_l3.png "Rendered by QuickLaTeX.com")
は安定行列となります。![\displaystyle{(13)\quad \underbrace{\left[\begin{array}{cc} 0 & I_p \end{array}\right]}_{\bar{C}^T}F^T=\underbrace{\left[\begin{array}{cc} {\bar P}_{1} & 0 \\ 0 & {\bar P}_{2} \end{array}\right]}_{\bar P}\underbrace{\left[\begin{array}{cc} 0 \\ {\bar B}_{2} \end{array}\right]}_{\bar B} }](https://cacsd1.sakura.ne.jp/wp/wp-content/ql-cache/quicklatex.com-25481cf05987d97cf1d660f8462287a3_l3.png "Rendered by QuickLaTeX.com")
