AUTHORS: MARCIO RIBEIRO DA SILVA GARCIA ADHEMAR DE BARROS FONTES

CARLOS EDUARDO TRABUCO DÓREA

An Iterative Algorithm For Constrained MPC With Stability Of Bilinear Systems

UNIVERSIDADE FEDERAL DA BAHIADEPARTAMENTO DE ENGENHARIA ELÉTRICA DA ESCOLA POLITÉCNICA

Table Of Contents

2

Purposes;

Predictive Control Concepts Model Prediction Constraints The Iterative Compensation Method Control Law Stability

Example

General Conclusions and Future Works;

References

Purposes3

1. Stablish a new control strategy with focus on reducing the prediction error in nonlinear systems and constraints satisfaction;

2. Formulation of an extension of the linear theory of invariant sets for the bilinear case;

3. Numerical examples regarding performance improvement and stability analysis

Predictive Control4

Features Application

Process model-based prediction;

Airplane system control;

Robustness regarding model –plant mismatch and process noise;

Robots control;

Easy tunning; Solar powerplant process control;

Easy threating of process constraints;

Several process of the chemical and petrochemical industry – FCC, Distillation columns, chemical reactors, furnace, etc;

Incorporation of the nonlinearities in the system model

Concepts5

r(k) - Reference signal;u(k) – Control signal;y(k) – System output; NY – Prediction horizon; NU – Control horizon;

Receiding Horizon: After the optimal sequence calculation, only the first element of the control vector is applied. In the next step, all calculation process is repeated.

FuturePast

Model6

Researches show that linear model-based controllers may present performance issues when applied to strongly nonlinear process;

Bilinear models:

Less complex than most nonlinear models;

More representative than linear models;

Linear in the parameters;

Bilinearity is presented in several process.

Model7

• Bilinear Model:• Linear Model:

Bilinear systems are Linear Parameter Varying (LPV) systems

Prediction8

a) Linear Prediction:

b) Bilinear Prediction:

Constraints9

All process are subjected to constraints;

The constraints are applied to the control signal and the process input / output variables;

Constraints violation may result in physical damage in the actuators or take the process output out of the specifications.

Constraints in predictive controllers are intrinsically related to prediction.

The Iterative Compensation Method10

Constraints satisfaction is easilly obtained in linear systems;

In nonlinear systems, future inputs sequences are unknown in the moment of prediction;

Prediction errors due to model linearization or uncertainties may take the state to violate the constraints.

The iterative compensation method consists in overcoming the prediction issue in bilinear systems so that the calculated control action do not violate the process constraints

11

Flowchart - The Iterative Compensation method

DataData

Calculation Of u(j=0)

u(k+i) = u(k-1), i=1,...,Hu

uj(k+i) = u(k-1)+u(j -1) (k+n),

i=1,...,Hu, n=1,...,i

CalculationOf u(j)

<orj > 20?

uoptimalu(j

)

uoptimalu(j

)

Yes

NoIter

ativ

e co

mpe

nsat

ion

j=j+1

Initi

al

Seq

uenc

e

Stop Criterion

The Iterative Compensation Method12

1-step ahead prediction for an unconstrined system:

Convergence is achieved if, and only if:

The convergence of the prediction depend on the system model, present state and tunning.

Control Law13

The minimum of J is calculated with Quadratic programming

Stability14

Bilinear systems are linear in the parameters;

An invariant set can be calculated in finite time considering that the resulting matrix A belongs to a convex set given by all possible combinations of the bounds on u(k);

The real maximum invariant set is unknow. The calculated set is an inner aproximation of the real maximal set;

The theory of stability, terminal constraints and invariant sets is well defined for linear systems. For bilinear systems, some assumptions may be done:

Stability15

Initial state lie inside the invariant set;

Contraction rate 0 < λ < 1, i.e, x(k) ∈ Ω ⇒x(k+1) ∈ λΩ;

The stronger constraint x(k+i) ∈ λiΩ must be imposed;

Zero prediction error and no model-plant mismatch. In practice this is not possible therefore, one must guarantee minimum erros.

Stability in bilinear systems can be guaranteed if the following conditions are fully satisfied:

Example16

System model:

Constraints:

Initial state = [3 2.8]T

Hp=10, =25:

-- blue line: Quasilinear controller- Black line: Iterative Compensation

State Trajectory

-4< x1(k) <4-4< x2(k) <4- red line: Invariant set for -1< u(k) <1-.- line: Invariant set for -0.5< u(k)

<0.5

Examples17

3 – Convergence of the prediction for a given iteration and effective cost improvements.

Convergence

-- Initial prediction -- Before convergence- After convergence

Cost Improvement

General Conclusion and Future Works

18

The iterative compensation method enables control without constraints violation;

The algorithm presents performance improvements over linear controllers and other technics for nonlinear control;

The calculation of invariant sets considering bilinear systems as LPV systems results in larger sets in comparison to existing strategies;

Future works should focus on the improvement of the developed strategy and a deeper analysis of the algorithm convergence and conservativness of the calculated invariant set.

References19

Bloemen, H. H. J., T. J. J. van den Boom e H. B. Verbruggen (2001). An optimization algorithm dedicated to a mpc problem for discrete time bilinear models, Proceedings of the american control conference. pp. 2376–2381.

Dórea, C.E.T., Hennet, J.C., (a,b)-invariant polyhedra of discrete-time systems. J. Optimiz. Theory Appl. 103(3), 521–542, 1999.

References20

Fontes, Desenvolvimento e Avaliação de Controladores Preditivos Baseados em Modelos Bilineares, PhD Thesis, Universidade Federal do Rio Grande do Norte, Natal-RN – Brasil, 2002.

Goodhart, Burnham, James, Bilinear, Self-tuning control of a high temperature heat treatment plant. IEEE Procedure – Control Theory Applications, vol. 141. N° 1, Jan, 1994.

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