Model Predictive Control for Tracking Constrained …chemori/Temp/Predictive/MPC_course3/...An Educational Plant Based on The Quadruple-Tank Process 3 Model Predictive Control for

An Educational Plant Based on The Quadruple-Tank Process 1

Model Predictive Control for Tracking Constrained Linear Systems Page 1

Model Predictive Control for Tracking Constrained Linear

Systems

D. Limon, I. Alvarado, A. Ferramosca,T. Alamo, E.F. Camacho

Dept. Ingenieria de Sistemas y Automatica Universidad de Sevilla


Outline

o Motivation

o Theoretical results

MPC for tracking

Optimal MPC for tracking

Robust tube-based MPC for tracking

Output Feedback Robust MPC for Tracking

o Real applications

Linear motor

ACUREX

The quadruple-tank process

o Conclusions and future works



Motivation

Issues Constrained variables

Stability guarantee for every

operation point

Change in the dynamics

Traditional control scheme

Target optimizer

Plant

Adaptative strategy

Low-level Control Process

y

u

target

Advanced control

Processes with large changes in the operation point


Motivation

Proposed control scheme

Target optimizer

Plant

Predictive control

Low-level Control Process

y

u

target

Predictive Control Optimal performance

Constraint satisfaction

Stability guarantee

Robustness



Motivation Standard MPC: (Muske and Rawlings., 1993 ;Mayne et al., 2000 )

RTO: For a given target, a fixed steady state xs is selected.

MPC:

where:

A stabilizing design requires that:

K and P such that:

is an admissible invariant set around xs

If the target changes, the feasibility may be lost and the MPC must be redesigned.


Motivation Loss of feasibility



Outline o Motivation


MPC for tracking

• Problem Description

• Previous solutions

• Motivation example

• MPC for tracking

• Properties and an example


Robust tube-based MPC for tracking.


o Real applications



Problem Description: (Limon et al. Automatica 2008)

Consider the following discrete time LTI system

MPC for tracking

Objective: Given any target yt, design a control law such that:

y(k) tends to yt when k→

x(k) and u(k) are admissible for all k ≥ 0

Linear Process

u • x MPC for tracking

target yt

Target variable



MPC for tracking

Some existing solutions to this problem:

• Translation to the new setpoint (Muske and Rawlings., 1993)

• MPC with an infinite horizon (Constrained LQR)

• Reference Governors (Gilbert et al., 1994; Bemporad et al., 1997)

• CSGPC adds an artificial reference as a decision variable and a contraction constraint to ensure the convergence (Rossiter et al., 1996)

• Dual mode strategy for tracking (Chisci and Zappa., 2003)

• The change of reference considered as a disturbance to be rejected (Pannocchia and Kerrigan., 2005)

10

XN

Projx()


11

Invariant set for tracking

The set of states and targets such that the system starting from that state admissibly converges to the target


is an invariant set for tracking iff

Invariant set for tracking

The extended state xa is constrained to

Define the system:

Consider the stabilizing control law

Parametrization of the equilibrium point

Set of reachable targets



MPC for tracking Standard MPC vs MPC for tracking:


Theorem:

Consider that is such that is stable

Q>0, R>0, and P such that:

is and admissible invariant set for tracking for the system subject to the following constraints

Let the feasibility region of the optimization problem

MPC for tracking

Then, for any feasible initial state i.e., x N the system is steered asymptotically to any reachable target yt Yt satisfying the constraints



MPC for Tracking

Properties of the MPC for tracking with respect to the standard MPC:

1. Changing operation points. Since the constraints set doesn’t depend on the desired steady state and x(k) N for all k, the MPC is feasible for any admissible change of set point at any sample time

2. Larger domain of attraction. Since

3. Offset minimization. If the target is not reachable, then the system will converge to y*

s such that

7. Local optimality. For a parameter T big enough the cost function of the proposed MPC is arbitrarily close to the optimal one

8. Explicit solution. Due to only a QP have to be solved at each sample time, the explicit solution can be calculated


MPC for Tracking

Example: Consider the discrete time LTI system:

Subject to the following hard constraints:

Controller parameters:



MPC for tracking

N=3




MPC for tracking


• Drawback of the MPC for tracking

• Enhanced formulation

• Local optimality

• Example

Robust tube-based MPC for tracking.


o Real applications




Drawback of the MPC for tracking • Potentially loss of local optimality.

• It does not consider as target operating points, states and inputs not consistent with the prediction model.

• Solution: enhanced formulation with a general convex function as offset cost function.


Enhanced formulation

Properties: • Offset minimization. If the target is not reachable, then the

system will converge to y*s such that

• Local optimality. The cost function of the proposed MPC equals the optimal one



Local optimality

• The standard formulation of the MPC for regulation does not ensure the local optimality property, because the cost function is only minimized for a finite prediction horizon.

• Define:

• Then, 8 x 2 Υ(yt), the optimal value of the MPC for regulation equals the optimal one and the control laws are the same.

(Hu and Linnemann, 2002)


Local optimality (2)

• In the MPC for tracking, this property can be ensured thanks to the convex offset cost function.

• Property: There exists a ®*> 0 such that, for every ®1¸®*: – For all x 2 XNr(yt):

– If K is the one of the unconstrained LQR:

for all x 2 Υ(yt)



Determination of ® *

• Consider the following regulation problem, deriving from the tracking problem, with the offset cost function posed as an equality constraint.

• ® * is the maximum of the Lagrange multipliers of the equality constraint of the previous problem, in the set of parameters

xp=(x,yt)2 XN £ Ys


Determination of ® * (2)

• Consider the standard formulation of a mp-QP problem:

• The Karush-Kuhn-Tacker optimality conditions for this problem are:



Determination of ® * (3)

• ®* can be calculated by solving the following optimization problem:

• This problem is not easy to solve because the first constrained, the complementary slacknes, is not convex nor concave.


Optimal MPC for Tracking

Example: Consider the two cascade tanks system:






Example: State and time evolution



Example: Local optimality

The value of the Lagrange multiplier is 14.6611





MPC for tracking




• Preliminary Results

• Robust MPC for tracking


• Calculation of K


o Real applications



Robust MPC for Tracking

Consider the following discrete time LTI system with additive bounded uncertainties:

Objective: Given any admissible setpoint s, design a control law such that:

y(k) tends to the neighbourhood of yt when k→

x(k) and u(k) are admissible for all k ≥ 0 and all possible realizations of

Problem description

The system is constrained to: Linear

Process u x Robust MPC

for tracking

w

target yt



Robust MPC for tracking

Existing solutions to the Robust tracking problem:

• Dual mode strategy (Rossiter et al., 1996; Chisci and Zappa, 2003)

• In the context of uncertain systems, the change of reference considered as a disturbance to be rejected. (Pannocchia 2004; Pannocchia et al., 2005; Magni et al. 2001)

• Reference Governors (Gilbert et al., 1999; Bemporad et al., 1997)

Existing solutions to the Robust MPC:

• Min-Max

• Tube-Based robust MPC

• Stochastic approach

• LMI based solution

• It is based on nominal predictions

• All the computational load is made offline, suitable for fast systems

• Simple implementation, only requires the solution of a QP



Lemma (Langson 2004)



• Considering the tighter set of constraints for the nominal system

The tube: (Langson 2004 ; Bertsekas 1972)


(Mayne et al., 2005)


Robust MPC for Tracking MPC for tracking vs Robust MPC for tracking:




Then, for any feasible initial state i.e., x N and any reachable target, the uncertain system is steered asymptotically to the set for all possible realization of the disturbances, satisfying the constraints

Theorem: Consider that

is such that is stable


at is an admissible invariant set for tracking for the nominal system

subject to the following constraints K is such that (A+BK) is stable and are not empty sets

Let be the feasibility region of the optimization problem


Consider the uncertain LTI discrete time system described by the matrices:


The disturbance set is:

The controller parameters are:


Example:


Model Predictive Control for Tracking Constrained Linear Systems Page 37 Model Predictive Control for Tracking Constrained Linear Systems Page 38

Cancellation of the tracking error: In the case that the disturbance w tends to a constant value


Linear Process

u x q Robust MPC for tracking

w

Disturbance Estimator

w ^ Hs

N yt ytc




0 50 100 150 -10 -5

0 5

10 Output and Reference evolution

0 50 100 150 -0.5

0

0.5 Control action evolution

0 50 100 150 -1 -0.5

0 0.5

1 Disturbance evolution p.u.


Outline

o Motivation


MPC for tracking





• Preliminary Results

• Robust MPC for tracking


o Real applications




Problem Description Consider the following uncertain discrete time LTI system

Objective: Given any admissible setpoint s, design a control law such that:

y(k) tends to neighbourhood of yt when k→

x(k) and u(k) are admissible for all k ≥ 0 and possible realization of and

Output feedback MPC for Tracking

The system is constrained to:

Linear Process

u y Robust MPC for tracking

Estimator x ^

w u target

yt


Output feedback robust MPC for tracking

Existing solutions to the Robust tracking problem:

• Standard stable estimator (that provides a measure of the state) plus a

stable robust controller (Magni et al. 2001)

• Using a set-membership state estimator plus a robust controller that takes

into account the set

1. MPC for tracking for constrained linear systems (Bemporad and

Garrulli, 1997). This MPC uses a switching strategy, because of that this is suboptimal solution

2. Reference governor proposed by (Angeli, Casavola and Mosca, 2001). It doesn’t takes into account the performance




Let the observer system be

Let the state estimation error ee be defined by:

The state estimation error satisfies:

If AL is Hurtwitz and the disturbances are bounded, there exists an RPI

Thus if then

Preliminary results Estimation tube (Mayne et al., 2006)



Let the control error be

Then applying the control law:

The error satisfies:

If AK Hurwitz and the disturbances are bounded, there exist an RPI

Thus if then

Preliminary results

Control tube (Mayne et al., 2006)



Considering the tighter set of constraints for the

nominal system

The additional stabilizing constraint

If and then

applying

Output feedback MPC for Tracking Resulting tube

(Mayne et al., 2006)


Output feedback MPC for Tracking Robust MPC for tracking vs Output feedback Robust MPC for tracking:



Theorem: Consider that

is such that is stable


is an admissible invariant set for tracking for the nominal system subject to the following constraints T>0

K is such that (A+BK) is stable and are non-empty sets

The initial estimation error must be inside of the set ee

Let be the feasibility region of the optimization problem


Then, for any feasible initial estimated state i.e., and for any reachable target, the system is steered asymptotically to the set for all possible realizations of w and v satisfying the constraints


Output feedback MPC for Tracking Example Consider the LTI discrete time system:

Subject to the following hard constraints and the disturbance sets are:

The controller parameters:




50



Cancellation of the tracking error:

In the case that the disturbance w and v tends to a constant value


Linear Process

u y Robust MPC for tracking

Estimator

yt

x ^

w u

Disturbance Estimator

F

g ^

ytc



0 10 20 30 40 50 60 -50 -40 -30 -20 -10

0 10 Output and Reference evolution

0 10 20 30 40 50 60 -10

0

10 Control action evolution u u n

0 10 20 30 40 50 60 -1

0

1 Disturbance evolution p.u. w 1 w 2 v

yt

ys y

ytc



Outline

o Motivation


o Real applications

Linear Motor

• Linear Positioning system

• The linear motor controlled by MPC for tracking

• The linear motor controlled by robust MPC for tracking

ACUREX

The Quadruple-Tank process



Linear Motor

The linear positioning system:

The controlled plant is a positioning system driven by a linear motor Linear

Positioning

sensor

Rails

Primary

Secondary

• Linear Motor: Siemens 1FN3 050-2W00-0AA

• Absolute position sensor: LC181 of Heidenhein. Precision of 5m • dSpace card in a PC: dSpace card DS1103 PPC, based on PowerPC

processor 604e that works at 400 MHz. This processor is programmed in Simulink using Real-Time Interface



Linear Motor

The linear motor controlled by the MPC for tracking: From the response of the system to a PRBS, a linear discrete time model has been identified using least squares identification. The sample time is 10ms

The constraints:

The controller parameters are: Explicit solution:

The resulting MPC controller is defined by 514 regions. A binary search tree has been used with a depth of 13


Linear Motor

The linear motor controlled by the robust MPC for tracking: From the response of the system to a PRBS, a linear discrete time model has been identified using least squares identification. The sample time is 30ms

The constraints:

The controller parameters are: The disturbance set:





o Real applications

Linear Motor

ACUREX

• Acurex overview

• Identification

• Acurex controlled by RMPCT

• Acurex controlled by RMPCT with the output cancelation loop




ACUREX ACUREX is a solar plant that is located in the PSA of Almería in Taberna desert.

• The purpose of this plant is to produce hot oil that can be used to produce high pressure steam for an electrical turbine or for a desalination plant

• The control goal is keeping the oil’s temperature close to the reference actuating on the flow despite the disturbances.



ACUREX

The model that minimize the less square method error is a first order model without delay

Identification: Disturbances: • Solar radiation • Inlet oil temperature • Ambient temperature


ACUREX

Identification: To determine the set W the output of the model is compared with the real output as in the figure

Using the stored data of previous controllers W is obtained

The constraints sets are:



ACUREX

ACUREX controlled by robust MPC for tracking:

Pyrometer sensor error disturbance:

This experiment demonstrates the robustness of the proposed controller to an additional and unmodelled disturbance Controller parameters:


ACUREX

11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13 220 230 240 250 260

Local time

MPC#1 T out T ref Tref art

11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13 220 240 260 280

Local time

Control Action

Tff

11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13 -50 0

50 100

Local time

Disturbances

Pyrometer sensor error

Rad/10 Rad cor /10 w est × 10 T in /10

Clouds



ACUREX

ACUREX controlled by robust MPC for tracking with the output cancellation loop: Tin disturbance: A disturbance on the temperature at

the input of the collectors was introduced

The offset is removed due to the output cancellation loop



ACUREX



Outline

o Motivation


o Real applications

Linear Motor

ACUREX

The Quadruple-Tank Process

• Description of the plant

• Identification

• The Quadruple-Tank Process controlled by robust MPC for tracking




Interesting features:

1. The linearized model has multivariable zeros which can be located in either the right or left half-plane by simply changing a couple of valves.

2. All the states are measurable.

3. The outputs are strongly coupled.

4. The system is nonlinear.

5. The states and inputs are constrained.




Johansson, 2000



Linearizing the model:

The system is open loop stable with 2 multivariable zeros. The nature of these zeros is determined by parameters γa and γb as follows:

• If 0≤γa+γb<1 The system has Right Half Plane transmission zeros (RHPZ)

• If 1<γa+γb≤2 The system has Left Half Plane transmission Zeros (LHPZ)

The sign of the real part of the zeros does not depend on the operating point.



The Quadruple-Tank Process Identification:

The cross-section of the outlet holes can be adjusted, and the rest of the parameters are determined, so the only parameter to be identified is the set of the disturbances

Main source of disturbances:

• The linearization approximation error

• The discharge parameters are not constant

• The actuator dynamics




The gain K and the minimal RPI have been calculated using the procedure aforementioned

Plant parameters:








The Quadruple-Tank Process Using the offset cancellation loop


Conclusions & Future works

Extension to other robust MPC techniques (min-max) Address the problem of tracking arbitrary references Generalization to more complex systems

Piece-wise affine systems Nonlinear systems

Novel MPC for tracking Feasibility under any change of the target Single QP Larger domain of attraction

Robust MPC for tracking based on tubes Nominal predictions Output feedback Offset-free control

Real applications

Documents

Model Predictive Control for Tracking Constrained …chemori/Temp/Predictive/MPC_course3/...An Educational Plant Based on The Quadruple-Tank Process 3 Model Predictive Control for