A Framework for Distributed Model Predictive Control Leonardo Giovanini Industrial Control Centre...

A Framework for Distributed Model Predictive Control

Leonardo Giovanini

Industrial Control CentreUniversity of Strathclyde

Glasgow

April 2005 IEEE Colloquium on MPC University of Sheffield

2

Outline

• Background and motivation

• Problem Formulation

• Communication based MPC

Properties (Stability, Convergence, Performance)

• Cooperative based MPC

Properties (Stability, Convergence, Performance)

• Simulations

• Conclusions and Future directions


3

Background

• Decentralized control and the traditional design mindset.

Several small units rather than a single monolithic unit

• Significant literature in the late seventies and early nineties focused on improved decentralized control.

• Limited focus on handling constraints, optimality stability of control methods.


4

Background

• Since nineties linear model predictive control became a dominant advanced control technology.

• Properties of single linear MPC are well established (stability, performance, feasibility).

• Potential benefits of integrating several MPCs (system resilience, reconfigurability).


5

BackgroundExample: Heat-exchanger networks

I3

I2 I1

X 1

c2 c1

h1

s 1

S1

S2

s 2

s 3

S3 h2

X 2

X 3

inCT

2 in

CT1

inHT

1

inHT

2 out

HT2

outHT

1

outCT

2

outCT

1

w S1

w S3

w S2

The objective are:

• Control the output temperatures of streams C1, C2, H1 and H2 in the whole operating space

• Minimize the use of services S1, S2 and S3

Let consider a heat-exchanger network.


6

BackgroundExample: Heat-exchanger networks

• The control inputs are constrained

• The network needs to employ the extra services (S2 and S3 ) to achieve some output targets and reject some disturbances.

• Therefore, a control strategy capable of make the sub systems work cooperatively and optimize the system behavior.

0 1 1,2,3

0 1l

l

X l

s


7

BackgroundWhy not centralized MPC?

• The main reasons are organizational and computational.

• Different sections of the networked systems may be owned by different organizations (power systems, integrated chemical plants, communication networks, manufacturing supply chain).

• Large centralized controllers are inflexible, hard to maintain and modify.

• Need of control methods for effective collaboration between the different subsystems.


8

The Problem Formulation

Complex high dimensional systems need efficient control architectures and algorithms.

Conventional approach is based on the decomposition of the system (to reduce the computational burden) and coordination of the subsystems (to tackle with interactions).

Syst2

Controller 1 Controller 2 Controller i Controller m

Communication Network

Subsystem 1 Subsystem 2 • • • Subsystem i • • • Subsystem m

System

Control System

The development of distributed control systems brings new requirements and potential benefits to control field.


9

The Problem Formulation

The requirements for a game approach to decentralized MPC are

• ModelingOptimal partitioning of the systemModels for the decentralized and interaction models

• Assumptions All cost functions are positive definite Linear models Convex inequalities Synchronous communication

• Levels of collaboration


10

The global optimization problem has been decomposed into a number of small coupled local optimization problem.

Each optimization problems only consider the local part of objective function.

The optimization problems are solved using only the local decision variables and communicating the result to others problems at each iteration.

Communication based MPC


11

An iterative algorithm for solving this problem is

1. Exchange state and inputs trajectories information between controllers,

2. Solve each sub problem using the trajectories provided by other agents and exchange information till all

trajectories converge,

3. Once the problems converge, apply the first element of the inputs trajectories of each subsystem.

Communication based MPCThe basic algorithm


12

Communication based MPCThe basic algorithm

Definition: A group of control decision is called to be Nash optimal solution if the following relation is held

1 2( )p p p pMU k u u u

11 1 1 1 1,2, ,p p p p p p p p

i i m i i i i mJ u u u J u u u u u i m

If the Nash solution is achieved, each

agent (i) does not change its control

decision (ui) because it has achieved

the local optimum, otherwise the local

performance index Ji will degrade.

1

u 1

u 2

Paretto set (P )

Equilibrium point (N )

Iterates

J 2 J 1


13

The control actions are given by

where

The convergence of control action only depends on

Therefore the algorithm will converge if

Communication based MPC Convergence

11 0( ) ( ) ( ) ( ) 1,2,p pU k D R k x k D U k p

11 12 11 1

11 122 21 22 2

0 1

1 22 2

00

0; ;

00

m

T Tmii ii i ii i ii i

mmmm m m

D A D AD

D A D AD D D H Q H R H Q

DD A D A

10( ) ( ) 1,2,p pU k D U k p

0 1D


14

To model the effects of communication failure, the matrices Tr and Tc are introduced into the control law.

Their elements only assume the values 1 or 0. Then, the control actions are given by

Communication based MPCConvergence

0 1r cT D T

11 0( ) ( ) ( ) ( ) 1,2,p p

r cU k D R k x k T D T U k p

Therefore the algorithm will converge if


15

The system can be written as

where

Then, the closed-loop system is given by

which is stable if

Communication based MPCStability

( 1) ( ) ( )X k AX k B U k

1

0 1 1A B I D D

1 1

0 1 0 1( 1) ( ) ( )X k A B I D D X k B I D D R k

1

10 00

11 0; 1, , .

011 1

i

m

SS i m

S


16

Communication based MPCPerformance

The solutions are Nash equilibrium, they are generally not Paretto optimal when the number of interacting agents is finite (Ramachadran et al., 1992; Dubbey and Rogawski, 2002).

1

u 1

u 2

P

N

2 1 22

( , )J u uu

1 1 21

( , )J u uu

For example, lets assume

two subsystems, two inputs,

one output each and the

prediction horizon is 1.


17

Communication based MPCRecovering Performance

Regulatory constraints: The presence of constraints can change the number and location of the equilibrium points.

This can used to improve the solution,

approaching N to the Paretto set P.

u 1

u 2

Feasible region

N Regulatory constraints

P

1

N u 1

u 2

P

Deference: If one of the agents allows the other to use both decision variables, under the condition that new decision are not worse than the equilibrium point, the agent can drag the attractor inside of the Paretto set.


18

Communication based MPCProperties

Trajectories generated by this scheme at each iteration p

• Feasible

• Guarantee the closed-loop stability

The cooperation based scheme can be terminated in any intermediate iteration.

Then


19

The global optimization problem has been decomposed into a number of small coupled optimization problem.

Each optimization problems consider the global objective function.

Each optimization problem is solved using only local decision variables and communicating the result to others problems at each iteration.

Cooperative based MPC


20

Cooperative based MPCThe basic algorithm

An iterative algorithm for solving this problem is

1. Exchange state and inputs trajectories information between controllers,

2. Solve each subproblem using the trajectories provided by other agents and exchange information till all

trajectories converge,

3. Once the problems converge, apply the first element of the inputs trajectories of each subsystem.


21

Cooperative based MPCProperties: feasibility and convergence

• All iterates are system wide feasible.

• The sequence of cost functions is a non-increasing function of the iteration number p

( ), ( ), ( )i j iJ u k u k x k

1 1( ), ( ), ( ) ( ), ( ), ( )p p p pi j i i j iJ u k u k x k J u k u k x k

• The sequence of iterates converges to an optimal limit point (centralized MPC).


22

Cooperative based MPCPerformance

1

u 1

u 2

P

N

2 1 22

( , )J u uu

1 1 21

( , )J u uu

1 2

1 1

1J J

u u

1 2

2 2

1J J

u u

0.5

Returning to the initial

example, two subsystems

with two inputs, one

output each and the

prediction horizon is 1.


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Cooperative based MPCProperties: closed-loop stability

Feasibility at initial time implies feasibility at all futures times.

Additionally

• (Aii, Bii) i=1,2, … ,m stabilizable ( Decentralized model )

• (Aij, Bij) i=1,2, … ,m stable ( Interaction model )

• Stabilizing constraints ( ui(k+j) = 0 j > N )

The origin is an asymptotically stable equilibrium point for the closed loop system for all initial states x(k) and all iteration numbers p.

Then


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Cooperative based MPCProperties

Trajectories generated by this scheme at each iteration p

• Feasible

• Guarantee the closed-loop stability

The cooperation based scheme can be terminated in any intermediate iteration.

Then


25

Simulations and ResultsThe heat-exchanger network

Let consider the following heat-exchanger network

The objective are:

• Guarantee the controllability of the system for any operational condition,

• Provide an optimal response for steady-state,

• Provide an optimal performance for any change.

I3

I2 I1

X 1

c2 c1

h1

s 1

S1

S2

s 2

s 3

S3 h2

X 2

X 3

inCT

2 in

CT1

inHT

1

inHT

2 out

HT2

outHT

1

outCT

2

outCT

1

w S1

w S3

w S2


26

Simulations and ResultsThe heat-exchanger network

Table 1: Stream conditions

Stream T in T out w c (Kw/C)

H 1 90 40 50 H 2 130 100 20 C 1 30 80 40 C 2 20 40 40 S 1 15 --- 35 S 2 30 --- 30 S 3 200 --- 10

The parameters of the streams are


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Simulations and ResultsSet point changes

The simulations consist in a sequence of set point and load disturbances changes that drive the network outside of the original operational space.

The set point changes are:

• Firstly, TC1 goes from 80 °C to 70 °C at 5 min.

• then, TC2 goes from 40 °C to 45 °C at 15 min.

• finally, TH2 goes from 100 C to 90 C at 25 min.

Simulations and ResultsThe system response

68

72

76

80

B C B B

TC

1

0 5 10 15 20 25 30 3538

40

42

44

46

Centralized MPC Cooperative based MPC Communication based MPC

Time [ min ]

TC2

TH1

TC

2 and

TH

1

84

88

92

96

100

T H2

Simulations and ResultsThe steady state results

Steady-Sate Change 1 Change 2 Change 3

300

400

500

600

700

800

2400

2500

2600

2700

2800

Se

rvic

e E

ne

rgy

Reference Changes


Re

cove

ry E

ne

rgy


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Simulations and ResultsLoad disturbances changes

The simulations consist in a sequence of load disturbances changes that drive the network outside of the original operational space.

The load changes are:

• Firstly, T inH1 goes from 90 °C to 80 °C at 5 min.

• then, T inH2 goes from 130 °C to 140 °C at 15 min.

• finally, T inC1 goes from 30 C to 40 C at 25 min.

Simulations and ResultsThe system response

99

100

101

102

103

104


TH

2

80

81

82

83

84

TC

1

0 5 10 15 20 25 30 3539

40

41

TC

2 and

TH

1

Time [ min ]

Simulations and ResultsThe steady state results

2400

2500

2600

2700

2800

Steady-Sate Change 1 Change 2 Change 3

200

400

600

800


Re

cove

ry E

ne

rgy

Se

rvic

e E

ne

rgy

Disturbance Changes


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Conclusions

A general framework for decentralized model predictive control has been presented.

The proposed framework is able to take advantage of system flexibility to handle conflicting situations while maintain operation in the optimal conditions.

The framework allows to explicit handle the trade off between optimality and performance with a minimum computational effort.

The proposed framework has a modular design, that can be easily update and allow to take advantage of decentralize systems.


34

Future Directions

Extend these ideas to uncertain and nonlinear systems.

Develop procedures and algorithms for the optimal partition of the controlled system.

Explore the inclusion of adaptation and learning capabilities into the decentralized scheme.

A Framework for Distributed Model Predictive Control

Leonardo Giovanini

Industrial Control CentreUniversity of Strathclyde

Glasgow

A Framework for Distributed Model Predictive Control Leonardo Giovanini Industrial Control Centre...

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