Zusammenhänge
x(t+1)=A*x(t)+B*u(t)
y(t)=C*x(t)
Bezeichnungen
r(t) in R^l .. Die Werte die erreicht werden sollen (Sollwert)
x(t) in R^n .. Umgebungs-Zustand
u(t) in R^m .. Stellgröße
y(t) in R^l .. Output/Messwert (Istwert)
A Matrix .. wie sich der vorherige Umgebungszustand auf den Nächsten auswirkt
B Matrix .. wie sich die Stellgröße auf den nächsten Umgebungszustand auswirkt
C Matrix .. wie aus der Umgebung gemessen wird
%hide
%latex
"Stellgröße" $u_1$ von der Folge $u:=\arg \min\limits_{u} ( \sum\limits_{t=1}^T{(w*(r_t-y_t)^2+(u_t-u_{t-1})^2)})$
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def component_pow(x,p): return vector([a**p for a in x])
def component_abs(x): return vector([abs(a) for a in x])
class DMC:
def __init__(self,A,B,C,x_0,T):
self.x_0=x_0; self.x=x_0; self.T=T;
self.A=A; self.B=B; self.C=C; self.dim=len(x_0)
self.lasty=vector([0.0 for i in range(self.dim)])
def plant_reset(self): self.x=self.x_0
def plant_step(self,u):
self.x=self.A*self.x+self.B*u
self.lasty=self.C*self.x
return self.lasty
def x(self,t): return self.x_0 if t==0 else self.A*x(t-1)+self.B*u(t-1)
def J(self,uarr): #qualität (todo wieder auf nxn verallgemeinern)
def u(t): return vector([uarr[t]]) if t>0 else self.lasty #vector([uarr[t],uarr[t+1]])
val=vector([0.0 for i in range(self.dim)])
self.plant_reset(); z=0.0
for t in range(1,self.T+1):
y=self.plant_step(u(t))
#val+=component_pow(u(t)-u(t-1),2.0)+self.w(t)*component_pow(y-self.r[t],2) #TODO downhill-simplex
Du=component_abs(u(t)-u(t-1))
val+=Du+component_abs(y-self.r[t]) #linear
if max(Du)>2: z+=1000#constraints rudimentär inzwischen
return val.norm(1)+z
def doStep(self,r,start,w):
self.w=w;
self.r=r
solution=minimize(self.J,start,maxfun=100000)
print "sol1",solution[1]
self.plant_reset()
self.plant_step(vector([solution[1]]))
##self.plant_step(vector([solution[0],solution[1]]))
self.x_0=self.x #new base state of plant
return solution
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Anwendung:
A=matrix([[1.0]]); B=matrix([[2.0]]); C=matrix([[3.0]])
T=8; x_0=vector([55.0]); dim=len(x_0)
mpc=DMC(A,B,C,vector([55.0]),T)
lastsolution=[0.0 for i in range(T+1)]
lastpoint=(0,0); lastsoll=(0,0)
pic=point(lastpoint)
for k in range(7):
val=90.0
if k>3: val=120
lastsolution=mpc.doStep([vector([val])for i in range(T+1)],lastsolution,lambda x:0.002)
print "y",mpc.lasty,k
P=(k,val); S=(k,mpc.lasty[0])
if k>0:
pic+=line([lastpoint,P],color="red")
pic+=line([lastsoll,S])
lastpoint=P; lastsoll=S
show(pic)
Optimization terminated successfully.
Current function value: 1527.045670
Iterations: 1200
Function evaluations: 1986
sol1 -3.60485954731
y (143.370842716168) 0
Optimization terminated successfully.
Current function value: 1159.069120
Iterations: 707
Function evaluations: 1088
sol1 -4.8446141587
y (114.303157763943) 1
Optimization terminated successfully.
Current function value: 1118.002876
Iterations: 410
Function evaluations: 656
sol1 -3.0252707202
y (96.1515334427210) 2
Optimization terminated successfully.
Current function value: 1105.576060
Iterations: 503
Function evaluations: 799
sol1 -1.02495513765
y (90.0018026168245) 3
Optimization terminated successfully.
Current function value: 1144.418209
Iterations: 1198
Function evaluations: 1924
sol1 2.50958118815
y (105.059289745714) 4
Optimization terminated successfully.
Current function value: 1121.637500
Iterations: 641
Function evaluations: 951
sol1 2.22221549594
y (118.392582721354) 5
Optimization terminated successfully.
Current function value: 1120.996438
Iterations: 479
Function evaluations: 749
sol1 0.117018980809
y (119.094696606205) 6
Optimization terminated successfully.
Current function value: 1527.045670
Iterations: 1200
Function evaluations: 1986
sol1 -3.60485954731
y (143.370842716168) 0
Optimization terminated successfully.
Current function value: 1159.069120
Iterations: 707
Function evaluations: 1088
sol1 -4.8446141587
y (114.303157763943) 1
Optimization terminated successfully.
Current function value: 1118.002876
Iterations: 410
Function evaluations: 656
sol1 -3.0252707202
y (96.1515334427210) 2
Optimization terminated successfully.
Current function value: 1105.576060
Iterations: 503
Function evaluations: 799
sol1 -1.02495513765
y (90.0018026168245) 3
Optimization terminated successfully.
Current function value: 1144.418209
Iterations: 1198
Function evaluations: 1924
sol1 2.50958118815
y (105.059289745714) 4
Optimization terminated successfully.
Current function value: 1121.637500
Iterations: 641
Function evaluations: 951
sol1 2.22221549594
y (118.392582721354) 5
Optimization terminated successfully.
Current function value: 1120.996438
Iterations: 479
Function evaluations: 749
sol1 0.117018980809
y (119.094696606205) 6
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