Nonlinear Model Predictive
Control using
Automatic Differentiation
Yi Cao
Cranfield University, UK
4th April 2005
Colloquium on Predictive Control, Sheffield
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Outline
 Computation in MPC
 Dynamic Sensitivity using AD
 Nonlinear Least Square MPC
 Error Analysis and Control
 Evaporator Case Study
 Performance Comparison
 Conclusions
4th April 2005
Colloquium on Predictive Control, Sheffield
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Computation in Predictive Control
 Predictive control: at tk, calculate OC for tk· t·
tk+P, apply only u(tk), repeat at tk+1
 Prediction: online solving ODE
 Optimization: repeat prediction, sensitivity
required.
 Typically, over 80% time spend on solving
ODE + sensitivity
4th April 2005
Colloquium on Predictive Control, Sheffield
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Current Status
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Linear MPC successfully used in industry
Most systems are nonlinear. NMPC desired.
Computation: solving ODE and NLP online.
Difficult to get gradient for large ODE systems.
Finite difference: inefficient and inaccurate
Sensitivity equation: n×m ODE’s
Adjoint system: TPB problem
Other methods: sequential linearization and
orthogonal collocation
4th April 2005
Colloquium on Predictive Control, Sheffield
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Automatic Differentiation
 Limitation of finite and symbolic difference
 Function = sequence of fundamental OP
 Derivatives of fundamental OP are known
 Numerically apply chain rules
 Basic modes: forward and reverse
 Implementation:
 operating overloading
 Source translation
4th April 2005
Colloquium on Predictive Control, Sheffield
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ODE and Automatic Differentiation
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z(t)=f(x), x(t)=x0+x1t+x2t2+…+xdtd
z(t)=z0+z1t+z2t2+…+zdtd
AD forward: zk = zk(x0,x1,…,xk)
AD reverse: zk /xj = zk-j / x0 = Ak-j
f’(x)=A0+A1t++Adtd
ODE: dx/dt=f(x), dx/dt=z(t), xk+1=zk/(k+1).
x0=x(t0), x1=z0(x0), x2=z1(x0,x1), …
Sensitivity: Bk=dxk/dx0=1/kki=0 Ak-i-1Bi, B0=I
dB/dt=f’(x), B=B0+B1t++Bdtd
x(t0+1)=di=0 xi, dx(t0+1)/dx0=di=0Bi
4th April 2005
Colloquium on Predictive Control, Sheffield
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Non-autonomous Systems I
 For control systems: dx/dt=f(x,u)
 u(t)=u0+u1t+u2t2+…+udtd
 (x0, u(t)) → x(t), dx(t)/duk=?
 Method I: Augmented system:
 dv1/dt=v2, …, dvd+1/dt=0, vk+1(t0)=uk, u=v1
 X=[xT vT]T, dX/dt=F(X) (autonomous)
 High dimension system, n+md
 Not suitable for systems with large m
4th April 2005
Colloquium on Predictive Control, Sheffield
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Non-autonomous Systems II
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Method II: nonsquare AD
Let v=[u0T, u1T, …, udT]T
xk+1 = zk(x0,x1,…,xk,v)/(k+1)
Ak=[Akx | Akv] := [zk/x0 | zk/v]
Bk=[Bkx | Bkv] := [dxk/dx0 | dxk/dv]
Bk = Ak-1+k-1j=1A(k-j-1)xBj, B0=[I | 0]
x(t0+1)=dk=0xk,
dx(t0+1)/dv= dk=0Bkv , dx(t0+1)/dx(t0)= dk=0Bkx
4th April 2005
Colloquium on Predictive Control, Sheffield
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Nonlinear Least Square MPC
 Φ=½∑Pk=0 (x(tk)-rk)TWk(x(tk)-rk)
s.t. dx/dt=f(x,u,d), t[t0, tP], x(t0) given,
uj=u(tj)=u(t), t[tj, tj+1], uj=uM-1, j[M, P-1],
L≤u≤V, scale t=tj+1-tj=1.
 Nonlinear LS: minL≤U≤V Φ=½E(U)TE(U)
 Jacobian: J(U)=∂E/∂U
 Gradient: G(U)=JT(U)E(U)
 Hessian: H(U)=JT(U)J(U)+Q(U)≈JT(U)J(U)
4th April 2005
Colloquium on Predictive Control, Sheffield
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ODE and Jacobian using AD
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Efficient algorithm requires efficient J(U)
Difficult: E(U) is nonlinear dynamic
Ji,j=Wi½dx(ti)/duj-1 for i≥j, otherwise, Ji,j=0
Algorithm: for k=0:P-1, x0=x(tk)
Forward AD: xi, i=1,…,d, → x(tk+1)
Reverse AD: Aix, Aiu
Accumulate: Bix, Biu, → Bu(k)=Biu, Bx(k)=Bix
J(k+1)j=Wk+1½Bx(k)…Bx(j)Bu(j-1), j=1,…,k+1
K=k+1
4th April 2005
Colloquium on Predictive Control, Sheffield
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NLS MPC using AD
 Collect information: x, d, r, etc.
 Nonlinear LS to give a guess u
 Solve ODE and calculate J
 Update u and check convergence
 Implement the first move
4th April 2005
Colloquium on Predictive Control, Sheffield
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Error Analysis
 Taylor coefficients by AD is accurate.
 x(tk)= xk has truncation error, ek (local).
 ek will propagated to k+1, …, P (global).
 Local error controllable by order and step
 Global error depend on sensitivity dxk1/dxk
 Remainder: k≈C(h/r)k+1
 Convergence radius: r ≈ rk=|xk-1|/|xk|
 k-1=k(r/h)=k+|xk| → k=|xk|/(r/h-1)
4th April 2005
Colloquium on Predictive Control, Sheffield
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Error Control
 Tolerance  < d
 Increase order, d or decrease step, h?
 Decrease h by h/c (c>1): =d(1/c)d+1
 c=(d/)1/(d+1) , increase op by factor c
 Increase d to d+p (p>0): =d(h/r)p
 p=ln(/)/ln(h/r), increase op by (1+p/d)2
 c<(1+p/d)2 decrease h,
 otherwise increase d
4th April 2005
Colloquium on Predictive Control, Sheffield
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Case Study
 Evaporator process
 3 measurable states: L2, X2 and P2
 3 manipulates: 0≤F2≤4, 0≤P100,F200≤400
 Set point change: X2 from 25% to 15%
P2 from 50.5 kPa to 70 kPa
 Disturbance: F1, X1, T1 and T200 20%
 All disturbance unmeasured.
 T=1 min, M=5 min, P=10 min, W=[100,1,1]
4th April 2005
Colloquium on Predictive Control, Sheffield
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Simulation Results
(a)
(b)
30
X2, %
L2, m
1.2
1
20
0.8
10
(c)
(d)
4
F2, kg/min
P2, kPa
80
70
60
50
3
2
1
(e)
(f)
300
F200, kg/min
P100, kPa
300
200
200
100
100
0
(g)
(h)
6
X1, %
F1, kg/min
12
10
8
5
4
(i)
(j)
4th April 2005
30
T200, oC
T1, oC
50
40
30
0
20
25
20
Colloquium on Predictive Control,
Sheffield
40
60
time, min
80
100
0
20
40
60
time, min
80
100
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Performance Comparison
 CVODES, a state-of-the-art solver for
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dynamic sensitivity.
Simultaneously solves ODE and sensitivity
Two approaches: full & partial integration.
Three approaches programmed in C
Tested on Windows XP P-IV 2.5GHz
Solve evaporator ODE + sensitivity using
input generated by NMPC.
4th April 2005
Colloquium on Predictive Control, Sheffield
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Accuracy and Efficiency
Taylor AD
CVODES P CVODES F
Tol
d
time error
time
error time Error
1e-4
4
.008 7e-5
.022
4e-4 .812 4e-3
1e-6
6
.009 7e-8
.042
9e-5 1.64 8e-5
1e-8
8
.011 4e-11 .063
2e-6 2.36 2e-6
1e-11 11 .013 5e-13 .114
4th April 2005
6e-9 4.56 3e-11
Colloquium on Predictive Control, Sheffield
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Conclusions
 AD can play an important role to improve
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nonlinear model predictive control
Efficient algorithm to integrate ODE at the
same time to calculate sensitivity
Error analysis and control algorithm
Efficiency validated via comparison with
state-of-the-art software.
Satisfactory performance with Evaporator
study
4th April 2005
Colloquium on Predictive Control, Sheffield
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