introduction for mpc

1B Erik Ydstie, CMU

Course Objectives: 1. The McNamara Program for MPC2. The Feldbaum Program for MPC3. From Optimal Control to MPC to SYSID for Control and Back4.Towards Tractable Optimization Formulations5. State of Art (Challenging – Problem of Integration and Software)

System Identification for MPC Conflict of Conflux?B. Erik Ydstie, Carnegie Mellon University

System Identification (SYSID) Review

Mass and Energy Balance Constraints (nonlinear)

dzidt

=pi(z) +nMV +nDVX

j=1

fi(uj , z), i = 1, ..., n

yk =hk(z), k = 1, ..., nPV

Linear (output) error model

e(t) = y(t)�Gp(q�1)u(t)

LT

Feed

Product

Cooling water return

FT

TT

Interface Layer (SCADA)

Measured Outputs y

Control Inputs u

CT FT

FT

Distributed Control System (DCS) Setpoints y*

Model Predictive Controller

• Capture Flowsheet structure• Energy and material balances

- Collinearity- Uncollinearity

Used for very large systems 50 + MV/DVs 100+ CVs

B. Erik Ydstie, ILS Inc. 3

Data Flow

MPC Control

ABBHoneywellAspenEmerson Process

Prior InformationStep-responseState SpaceLaguerre,…Tuning Parameters,…

.XML

.TXT

B.ErikYdstie

Model estimated using output error identification (global optimality)

MPC –SYSID • Boiler Master –Turbine Master Controls (Emerson/Ovation)• Turbine Controls for SiemensSYSID Data from

B.ErikYdstie

Problem: Define the Operator G that “best” matches the prior information and process data.

Bayes Estimation Problem with Constraints

• Prior structure of G• Digraph (edges in the process network)• Parametric representation for each Gij (nodes)’• Information of collinearity structure

• Process Data• Semi-closed loop• Experiment Design

G25

G38

0 1000 2000 3000 4000 5000 6000 7000-50

0

50

100

150

200

250

300

350

400

Prior Information: System Strucure

10 MV/DVs12 CVs

I/O Data

DigraphNetwork

Collinearity:SVDRGAAngles

N(n,m) =

(n� 1)n(m� 1)m

4

= 2970, 7! 0.12 deg separation

Bilinear constraints

Prior Information: Model Structures


Polynomials used NameB FIR, SRA,B ARX, Equation error,

Instrumental variables,..A,B,C ARMAX, AMLB,F OE (Output Error,

Markov-Laguerre, Kautz,…..)

State space representations have become popular for multivariable systems after the introduction of sub-space identification.

Halfway Conclusion: • The components are in place for systematic SYSID• Software is lacking• Quite difficult to do due to non-stationary disturbances• Theory not that easy to understand completely• Comprehensive Software solutions not available yet.

8

n

CA

CACAC

n

=

úúúúúú

û

ù

êêêêêê

ë

é

×-1

2rank

[ ] n BABAABB n =× -12 ||||rank

)()()()( tuqAqBty =

• A(q) and B(q) no common factors = Observable+Controllable (Co-prime)• A(q) and B(q) no common unstable factors = Detectable+Stabilizable

Reachability: Any state can be reached in a finite amount of time

Observability: Any state can be determined in a finite amount of time

Detectable: Any unstable state is observableStabilizable: Any unstable state is reachable

The Admissibility Problem

The FIR / Markov-Laguerre Models are automatically stabilizable


MISO Identification

Data is persistently excited from a SISO case.

0 500 1000 1500 2000 2500-4

-2

0

2

4

6

8

10

12

14

16

MV7 is most excited

MV2 is least excited

Cond(F) = O(1)

1

N

NX

k=1

u(k)u(k � i)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-3

-2

-1

0

1

2

3

4

5

6

CV 3Prior (blue)Update (red)Data (yellow)

ExcitationMV 1 DMV2 AMV3 AMV 4 AMV 5 AMV 6 A

We get (Ljung, Wahlberg, Forsell) Bias and Variance:

Bias

Variance


Systemu1

u2

y1y2

Generating Multivariable Input Signals

Same results hold as long as PE and independent noise and disturbance sequences. Results based law of large numbers, difficult to achieve using PRBS type excitation.

1

N

NX

k=1

u(k)u(k � i) = V (N)TV (N � i) = R(N) =

⇢R > 0, i = 0

0, i 6= 0

Input sequences must be independent in time wrt to the network.

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

0 1000 2000 3000 4000 5000 6000 700028

30

32

34

36

38

40

42

44

0.2210 �0.2206

�0.2206 0.3301

�

[0.2019]

108

64

200

24

68

1

-0.5

0

0.5

10

Orthogonal inputs:

Mass balance constraints in the process


y(t) = Gc0(q

�1)u(t) +Hc0(q

�1)v(t)

Gc0(q

�1) = Sc0(q

�1)G0(q�1)

Hc0(q

�1) = Sc0(q

�1)H0(q�1)

Sc0(q

�1) =1

1 +G0(q�1)K(q�1)

Closed Loop System:

Issues for closed loop identification:

• Model parameterization• Algorithm and mathematical approach• Filters to shape bias and variance• Excitation (complete theory for SISO, Lacking for MIMO, some progress for Networks• Extension to multivariable case (treated very superficially in most books and papers)

Methods that may fail:• Regression type models (equation error, instrumental variables)• Subspace methods• Compensation methods (direct and indirect)• Correlation/spectral methods

Closed Loop Identification

Use output error methods for identification (open and closed loop)

Excitation

Process

MPC

12

Integration of SYSID with MPC:The Decision Problem

1. Defining clear business objectives (Stable/Robust Performance)2. Developing plans to achieve the objectives (Predictive Control)3. Systematically monitoring progress against the plan (Feedback, Filter)4. Adapt objectives/plans as new needs and opportunities arise (Identification)

Control/Plan Process MeasureRobert McNamara,1960, (CEO Ford, US Secretary of State)

Repeated Identification -> Iterative Learning -> Adaptive Control

Model and Desired Performance Objectives

13

The McNamara Program for MPC

MPC Process Measure EvaluateCritic

Model and Desired Performance Objectives MPC Design/Identify/Adapt

1. Measure, evaluate and critique (Gap analysis)2. Control strategies (Optimal Control/Model Predictive Control/Hinfinity)3. Identification, Learning, Adaptation

a) Adapt Controllers (directly or indirectly)b) Adapt Performance Objectives (closed loop, Q,R/move suppression)

Performance Objectives

Predictive Model

• The Decision problem is driven by Uncertainty (more than accurate models)• Numerous Practical and Theoretical Challenges Remain• MPC provides a fruitful Paradigm to Study these Challenges

Current Practice

B Erik Ydstie, CMU 14

The Feldbaum (1961) Program

Each field well advanced, but poorly integrated(Especially on the software side)

minu2U,y2Y

1X

i=1

(y(t+ i+ 1)� y(t+ i+ 1)⇤)2 + ru(t+ i)2

• Optimal (Certainty Equivalent, LQ Optimal Control, 1980 to MPC)• Caution (Robust Control, 1980 ) • Probing (System ID / Adaptation, 1980 )

H1

min

u(t+i)

TX

i=0

x(t+ i)

TQx(t+ i) + u(t+ i)

TRu(t+ i)

| {z }Finite Horizon Cost

+ x(t+ i)

TPx(t+ i)| {z }

Terminal Cost

Subject to: x(t+ 1) = Ax(t) +Bu(t)

x(t+ i)min x(t+ i) x(t+ i)max

u(t+ i)min u(t+ i) u(t+ i)max

*

*


From LQ to MPC and Back Again

Step 2: Split Objective in Two and use Predictions from Model

Step 1: Formulate a (linear) model

Step 3: Ignore last part Step 4: Solve QP and use first control.Step 3: Repeat Step 4 (and hope for the best)

Theory for robust stability and performance

x(t+ 1) = Ax(t) +Buf (t) +Ke(t)

y(t) = ✓

Tx(t) +Duf (t) + e(t)

x(t+ T )TPTx(t+ T )

minu2U,y2Y

T�1X

i=0

(y(t+ i+ 1)� y(t+ i+ 1)⇤)2 + ru(t+ i)2| {z }Model Predictive Control

+1X

i=T

(y(t+ i+ 1)� y(t+ i+ 1)⇤)2 + ru(t+ i)2| {z }LQ Control


MPC and SYSID:Learn from the Past and Control Into the Future

Step 2B: Split Objective in Three and use Past Information

Model Identified from Past Data

Control Into the Future

min✓2⇥

NX

i=0

(y(t� i)� y(t� i)⇤)2 + (✓ � ✓0

)TF0

(✓ � ✓0

)| {z }SYSID�Bayes

minu2U,y2Y

1X

i=1

(y(t+ i+ 1)� y(t+ i+ 1)⇤)2 + ru(t+ i)2| {z }Robust MPC

• Adaptive Control• Iterative Control• Closed Loop Identification• Identification for Control+++

Basic Idea: Controller works while data is collected

✓(0) 7! ✓(t1) 7! ✓(t2), ....

SYSID and MPC - Conflict or Conflux?Adapted from Polderman (1986)

Exampe: LTI System:

Linear feedback:

System

y(t) = ay(t� 1) + bu(t� 1)

e(t) = y(t)� y(t)

Model+

-

Question: Will system satisfy performance specifications when the same control is applied to both systems?

(The question of (Roust Lagrange) stability for closed loop identification and control was addressed by 1995)

uy

u(t) = K(a, b)y(t)

Definition: An Identification Based Control is said to be Self-Tuning if SYSID gives the “correct control”

Set of Identified Models : GSet of Parameters with correct controls : HControl and Estimation are Self Tuningif : H ✓ G


G = {a, b : ay(t� 1)� bK(✓)x(t)| {z }y(t)

= ay(t� 1)� bK(✓)x(t)| {z }y(t)

}

Analysis: Assume model output matches plant output

An infinite number of solutions. These depend on K.

Example 1: One step ahead predictive control

Solve for u(t) : y(t+ 1)

⇤= ay(t) + bu(t)

G =

⇢a, b :

a

b=

a

b

�u(t) = � a

by(t)

Get correct control even if parameter estimates are off.

Thm: Any identifier that minimizes prediction error is self tuning when used with minimum variance control.

Admissibility Problem (close to singularity gives large, oscillatory controls)(Problem of “direction”)

a-1 -0.5 0 0.5 1

b

-1

-0.5

0

0.5

1

The admissible set


Example 2: Pole placement control (Vogel and Edgar,

Find gains so that : y(t) = a0y(t� 1), 0 < a0 < 1

u(t) = � a� a0

by(t)

H =

⇢a, b :

a

by(t)) =

a

by(t)

�The set that gives correct controls H ✓ G

H =

⇢a, b : �a� a0

by(t)) =

�a� a0

by(t)

�

H ✓ G

Thm: Any identifier that minimizes prediction error is self tuning when used with pole-placement.

In this case Admissibility is more Complex as we require:

Observability and Controllability

Can be expressed as Bilinear Constraints in SYSID problem.

It is going well so far!!

a-1 -0.5 0 0.5 1

b

-3

-2

-1

0

1

2

3

4

Admissible set


Example 3: Model Predictive Control

H =

⇢a, b : �a� a0

by(t)) =

�a� a0

by(t)

�

Thm: MPC does NOT satisfy the self-tuning principle.

minu

(y(t+ 1)2 + ru(t)2) + py(t+ 1)2

u(t) = � ba

r + b2G =

(a, b : a� a = (b� b)

ba

r + b2

)

H 6✓ Gunless :

r = 0 and/or

a = a, ˆb = b

-1 -0.5 0 0.5 1-5

0

5

H

H

G


Problem: Information in the Past is Not Connected to Future Information

Additional means are needed to get optimal controls for MPC.• Persistent Excitation to Converge Controls• More Complex Controls to Align sets G and H?• “Intelligent” Excitation (SYSID for Control, Dual Control)


From Feldbaum to MPC and Back

minu2U,y2Y

TX

i=1

(y(t+ i)� y(t+ i)⇤)2 + ru(t+ i)2

| {z }Model Predictive Control

+1X

i=T+1

(y(t+ i)� y(t+ i)⇤)2 + ru(t+ i)2

| {z }LQ Control

y(t+ i)�Maximum Likelihood Estimate

F (t+ i)� Fisher Information Matrix

minu2U,y2Y

TX

i=1

(y(t+ i)� y(t+ i)⇤)2 + ru(t+ i)2| {z }Robust CE AMPC

+x(t+ i)TF (t+ i)�1

x(t+ i)| {z }Information Gathering

+x(t)PTx(t)

Challenges (identification using past data is the easiest): • Solve a robust control problem on line (structured and unstructured uncertainty)• “Back out control signals” from forward propagation of Fisher matrix• What to do with the arrival cost

Special Case (TA Heirung/ J Morinelly)

• Fix the transition matrix A (step-response/Kautz model)• Solve CE (rather than robust Hinfinity) control problem (caution

related to parameter uncertainty only)• Ignore arrival cost

Computationally Expensive and Untested

24

So What are the Issues?Data Rich – But Information Poor Systems

(Nature is not a kind adversary)

• MPC and SYSID - Conflict or Conflux?• How to represent/parametrize the system• How to excite the system• How to manage changing models• Directionality• Complexity• Software

25

Conclusions• MPC is not self tuning• There is a “strong" inter-action between

control and identification• Different ways to “solve the problem”

– More complex “control”– External Excitation (setpoints/inputs)– Identification for control

• Need to Retune Controller when model changes

• Collinearity issue is not well understood• Very Large scale Applications are challenging• MPC Maintenance is still challenging

4756 linesof assemblycode (1983)

15 linesof MATLABcode (2014)

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introduction for mpc

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