The worldwide market for control and monitoring products is around 188 ... indicating the theoretical best achievable MPC performance on the same system and ...
MPC Performance Monitoring and Evaluation Principles Luo Ji
James B. Rawlings
Department of Chemical and Biological Engineering University of Wisconsin-Madison
AlChE (October 2011)
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
1 / 22
Outline
1
Motivation
2
Method
3
Results KPI of Unconstrained, Linear Systems KPI of Constrained, Nonlinear Systems Case study I: Deterministic Disturbances Case study II: Nonlinear Polymerization System
4
Future work
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
2 / 22
Process Control Performance Process control is an integral part of industrial process engineering. The worldwide market for control and monitoring products is around 188 billion euros 1 . However, only one third of controllers actually provide an acceptable level of performance 2 .
Output
Output
Output setpoints Input change?
Input Input setpoint Time
1 “Impact of Information and Communication Technology on Monitoring and Control: Today’s Market, its Evolution till 2020.” (http://www.decision.eu/smart2007.htm).
2 D. B. Ender (1993) Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
3 / 22
Performance Benchmarks Minimum Variance Benchmark The most famous benchmark is Minimum Variance (MV) Benchmarka : Derived for linear time-invariant systems with no constraints Use the output variance to represent the performance Not designed for MPC performance evaluation a T. J. Harris (1989)
Other Benchmarks Historical benchmarka PID benchmarkb LQG benchmarkc Other covariance-based benchmarkd a R. S. Patwardhan, S. Shah, G. Emoto, H. Fujii (1998) b B. S. Ko, T.F. Edgar (1998) c B. Huang, S.L. Shah (1999) d S. J. Qin, J. Yu (2007) Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
4 / 22
MPC Introduction MPC Model Predictive Control (MPC) has been widely applied and studieda because it: handles constraints directly. handles large multi-variable systems. a S. J. Qin, T. A. Badgwell (2003)
MPC Performance Issues in MPC performancea : How well is the current MPC system performing? What is the rank ordering of many MPC loops? Has the performance changed over time? Do some systems require maintenance?
Controller No. 1 2 3 4 5 ...
Status Good Bad Good Terrible Normal ...
a R. Patwardhan, S. Shah (2002), M. Jelali (2006) Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
5 / 22
MPC Performance However, there is no established, systematic control theory or technology that enables MPC practitioners to routinely assess, monitor and diagnose performances of their MPC systems.
Factors Affecting MPC Performance Large number of interacting loops Plant model mismatch Unmodeled deterministic disturbances Controller is nonlinear Disturbance and state estimation Many vendor implementations Nonlinear nature of industrial systems
MPC Performance Benchmark Define an MPC performance benchmark, which takes advantage of the process model, MPC control law, and estimated disturbances. Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
6 / 22
Basic Idea 1
Define a Key Performance Index (KPI). Pk −1 KPI ≡ k1 j=0 |yk − ysp |2Q + |uk − usp |2R + |uk − uk −1 |2S
2
Plant KPI is calculated from raw data and represents the performance of the studied plant.
3
Simulated KPI is defined as the expectation of the same objective, indicating the theoretical best achievable MPC performance on the same system and under the same modeled condition. sKPI = E(|yk − ysp |2Q + |uk − usp |2R + |uk − uk −1 |2S )
4
Compare Plant KPI with Simulated KPI. KPI Definition Apply to operating data
Plant KPI
Calculated by simulation
Comparison
Simulated KPI
How to ensure the simulation and the plant have similar disturbance effects? Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
7 / 22
Outline
1
Motivation
2
Method
3
Results KPI of Unconstrained, Linear Systems KPI of Constrained, Nonlinear Systems Case study I: Deterministic Disturbances Case study II: Nonlinear Polymerization System
4
Future work
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
8 / 22
KPI of Unconstrained, Linear Systems Linear Time-Invariant (LTI) system xk +1 = Axk + Buk + Gwk yk = Cxk + vk
Zero-mean, Normally Distributed Disturbances w is the process disturbance: w ∼ N(0, Qw ) v is the measurement noise: v ∼ N(0, Rv ) Qw and Rv are estimated by ALS a from operating data. a B. J. Odelson, M. R. Rajamani, J. B. Rawlings(2006)
Estimation and Regulation xˆk +1 = Axˆk + AL(yk − C xˆk ) + Buk uk = K xˆk in which L is the Kalman Filter gain and K is the LQ regulator gain. Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
9 / 22
Estimate Disturbance Covariance from Data The state noise wk propagates but the measurement noise vk does not: k ≡ yk − C xˆk
yk = Cxk yk +1 = CAxk yk +2 = CA2 xk
Condenser
XD X1
y1
X2
v1 Xf , Ff
X N −1
v2
XB Reboiler
w
y2
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
10 / 22
Estimate Disturbance Covariance from Data The state noise wk propagates but the measurement noise vk does not: k ≡ yk − C xˆk
yk = Cxk yk +1 = CAxk + CGwk yk +2 = CA2 xk + CAGwk + CGwk +1
Condenser
XD X1
y1
X2
v1 Xf , Ff
X N −1
v2
XB Reboiler
w
y2
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
10 / 22
Estimate Disturbance Covariance from Data The state noise wk propagates but the measurement noise vk does not: k ≡ yk − C xˆk
yk = Cxk + vk yk +1 = CAxk + CGwk + vk +1 yk +2 = CA2 xk + CAGwk + CGwk +1 + vk +2
Condenser
XD X1
y1
X2
v1 Xf , Ff
X N −1
v2
XB Reboiler
w
y2
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
10 / 22
Estimate Disturbance Covariance from Data The state noise wk propagates but the measurement noise vk does not: yk = Cxk + vk
k ≡ yk − C xˆk
yk +1 = CAxk + CGwk + vk +1
E[k Tk ] = Rv
yk +2 = CA2 xk + CAGwk + CGwk +1 + vk +2
Condenser
XD X1
y1
X2
v1 Xf , Ff
X N −1
v2
XB Reboiler
w
y2
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
10 / 22
Estimate Disturbance Covariance from Data The state noise wk propagates but the measurement noise vk does not: yk = Cxk + vk
k ≡ yk − C xˆk
yk +1 = CAxk + CGwk + vk +1
E[k Tk ] = Rv
yk +2 = CA2 xk + CAGwk + CGwk +1 + vk +2
E[k +2 Tk+1 ] = CAGQw GT C T
Condenser
XD X1
y1
X2
v1 Xf , Ff
X N −1
v2
XB Reboiler
w
y2
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
10 / 22
Estimate Disturbance Covariance from Data The state noise wk propagates but the measurement noise vk does not: yk = Cxk + vk
k ≡ yk − C xˆk
yk +1 = CAxk + CGwk + vk +1
E[k Tk ] = Rv
yk +2 = CA2 xk + CAGwk + CGwk +1 + vk +2
E[k +2 Tk+1 ] = CAGQw GT C T
Autocovariance Least Squares (ALS)
Condenser
XD X1
y1
X2
φ = min |AN Qw ,Rv
(Qw )s ˆ2 − b| (Rv )s
v1 Xf , Ff
X N −1
v2
XB Reboiler
w
y2
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
10 / 22
Estimate Disturbance Covariance from Data The state noise wk propagates but the measurement noise vk does not: yk = Cxk + vk
k ≡ yk − C xˆk
yk +1 = CAxk + CGwk + vk +1
E[k Tk ] = Rv
yk +2 = CA2 xk + CAGwk + CGwk +1 + vk +2
E[k +2 Tk+1 ] = CAGQw GT C T
Autocovariance Least Squares (ALS)
Condenser
XD X1
φ = min |AN
y1
Qw ,Rv
X2
v1
1
(Qw )s ˆ2 − b| (Rv )s
Form AN from known system matrices
Xf , Ff
X N −1
v2
XB Reboiler
w
y2
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
10 / 22
Estimate Disturbance Covariance from Data The state noise wk propagates but the measurement noise vk does not: yk = Cxk + vk
k ≡ yk − C xˆk
yk +1 = CAxk + CGwk + vk +1
E[k Tk ] = Rv
yk +2 = CA2 xk + CAGwk + CGwk +1 + vk +2
E[k +2 Tk+1 ] = CAGQw GT C T
Autocovariance Least Squares (ALS)
Condenser
XD X1
φ = min |AN
y1
Qw ,Rv
X2
v1
1
Xf , Ff
2 X N −1
v2
XB Reboiler
w
y2
Luo Ji, James Rawlings (UW-Madison)
(Qw )s ˆ2 − b| (Rv )s
Form AN from known system matrices ˆ is a vector containing the estimated correlations b from data yk ykT T X .. ˆ= 1 b . T T k =1 y y k +N−1 k s MPC Performance Monitoring
AlChE 2011
10 / 22
Simulated KPI of Unconstrained, Linear Systems Combine the States and their Estimates
xˆ w ˜ +G k ˆ x − x vk k +1 k xˆ xˆ ˜ ˜ yk = C + vk , uk = K x − xˆ k x − xˆ k ALC ˜ = 0 ˜ = A + BK A , G 0 A − ALC G ˜ ˜ C= C C , K = K 0
xˆ x − xˆ
˜ =A
AL −AL
Expectations and Covariances
xˆ x − xˆ
∼ N(mk , Pk ),
˜ k, mk +1 = Am
k
˜ Qw ˜ kA ˜T + G Pk +1 = AP 0
0 ˜T G Rv
Analytical Expression of Simulated KPI Assume the whole system is stable and has zero offset:
Simulated KPI = f (ysp , usp , Q, R, S, A, B, C, G, Qw , Rv , K , L) ˜ T Q CP ˜ k ) + tr (QRv ) + tr (K ˜ T RK ˜ Pk ) + tr ((A ˜ − I)T K ˜ T SK ˜ (A ˜ − I)Pk ) = tr (C Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
11 / 22
Results of Plant KPI and Simulated KPI Imperfect model:
Perfect model: 7
7
Plant KPI Simulated KPI
(a)
Plant KPI Simulated KPI Simulated KPI (mis)
(b) 6
5
5
4
4 KPI
KPI
6
3
3
2
2
1
1
0 0
100
200
300
400
500
Time
0 0
100
200
300
400
500
Time
With perfect model, Simulated KPI converges to Plant KPI quickly.
With plant model mismatch, Plant KPI is larger than Simulated KPI.
The transient behavior is not important.
Plant KPI converges to another value.
About 250 steps are enough to get a reasonable Simulated KPI result.
Plant model mismatch could be detected assuming there is no other faults.
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
12 / 22
Outline
1
Motivation
2
Method
3
Results KPI of Unconstrained, Linear Systems KPI of Constrained, Nonlinear Systems Case study I: Deterministic Disturbances Case study II: Nonlinear Polymerization System
4
Future work
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
13 / 22
Constrained or Nonlinear Simulated KPI Constraints of Inputs or Outputs umin < u < umax ymin < y < ymax
Nonlinear System xk +1 = F (x, u) + Gwk yk = h(x) + vk
Simulated KPI of Constrained or Nonlinear MPC Inputs, outputs, and states are now nonlinear functions of disturbances. They are not normally distributed. Analytical expression of Simulated KPI is unavailable. One solution is to calculate them from Monte Carlo simulations. Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
14 / 22
Outline
1
Motivation
2
Method
3
Results KPI of Unconstrained, Linear Systems KPI of Constrained, Nonlinear Systems Case study I: Deterministic Disturbances Case study II: Nonlinear Polymerization System
4
Future work
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
15 / 22
Case study I: Deterministic Disturbances Performance evaluation for a linear system dominated by a significant deterministic disturbance (supported by Air Products):
pk is estimated by: ˆk = yk − C xˆk p 0.4
unmodelled disturbance
0.3
Model the disturbance on the output:
disturbance
0.2
xk +1 = Axk + Buk + Gwk
0.1 0 -0.1 -0.2 -0.3
yk = Cxk + pk + vk
-0.4
0
500
1000
1500 Time
2000
2500
3000
Disturbance
We can estimate the disturbance within a limited time window.
Estimated Disturbance Real Disturbance Disturbance Forcast
Sample Window
To calculate the new Simulated KPI, just assume the disturbance is periodic... Time 0
T
2T
Luo Ji, James Rawlings (UW-Madison)
3T
MPC Performance Monitoring
AlChE 2011
16 / 22
Simulated KPI with Deterministic Disturbance 30
Plant KPI Simulated KPI Simulated KPI (dis)
25
KPI
20 15 10 5 0 0
500
1000
1500
2000
2500
3000
Time
ALS is not designed to detect the deterministic disturbance. The nominal Simulated KPI would over-estimate the plant performance. Plant KPI will converge to Simulated KPI (dis). Simulated KPI (dis) provides a more realistic performance benchmark for this problem. Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
17 / 22
Outline
1
Motivation
2
Method
3
Results KPI of Unconstrained, Linear Systems KPI of Constrained, Nonlinear Systems Case study I: Deterministic Disturbances Case study II: Nonlinear Polymerization System
4
Future work
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
18 / 22
Case study II: Nonlinear Polymerization System Performance evaluation for a nonlinear polymerization system (supported by ExxonMobil Chemical):
The plant is regulated by PI controllers but a nonlinear state space model is also available.
SP
CV
PI MV
CW
SP
Catalyst
MV
PI
Plant
CV CV
CV
Product SP
PI
SP
PI
...
PI Simulated KPI indicates the best achievable theoretical performance of PI control. MPC Simulated KPI indicates the best achievable theoretical performance of MPC.
MV
MV
Fresh Feed
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
19 / 22
KPIs of Nonlinear Systems With some chosen weighting parameters in the performance objective: KPI Source Plant Simulated Simulated Simulated Simulated
Controller Type PI PI LTI MPC LTV MPC NMPC
KPI 9.44 5.91 2.56 2.52 2.23
Further Considerations Decisions should be made based on the performance work: Is it worthwhile to replace the PI controllers by MPC controllers? Linear or nonlinear MPC? Estimation strategy?
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
20 / 22
Future Work
Examine the diagnosis scenario with real case studies Future study on constrained and nonlinear systems Systems with deterministic disturbances corrupting the states Diagnosis nonlinear state and disturbance estimation problems
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
21 / 22
Acknowledgment
Dr. George Shen from Air Products Dr. Tyler Soderstrom from ExxonMobil Chemical Company Dr. Fernando Lima from University of Minnesota Industrial members of TWCCC
Luo Ji, James Rawlings (UW-Madison)
MPC Performance Monitoring
AlChE 2011
22 / 22
