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Control of Quadruple-tank Using Passivity-Based Nonlinear MPC

This example shows how to design a passivity-based controller for a quadruple-tank using nonlinear model predictive control (MPC).

Overview

The dynamics for a quadruple tank system can be written as follows [1].

x˙=f(x)+g(x)u

where x denotes the heights of the four tanks and u denotes the flows of the two pumps. These dynamics are implemented in stateFcnQuadrupleTank.m.

The control objective is to select the flow of the pumps u such that (x,u) moves towards the equilibrium (xs,us).

To enforce closed-loop stability, the controller includes a passivity constraint.

To define the passivity constraint, first define the state error vector:

e=x-xs

Consider the storage function V=12(e32+e42) and take the derivative of V to obtain the relationship [1]

V˙upyp.

This means that, the system is passive from up=[u2;u1]-us to yp=[e3;e4]. The relationship for the passivity input up and the passivity output yp are described in the helper function getPassivityInputQuadrupleTank.m and getPassivityOutputQuadrupleTank.m .

To enforce closed-loop stability, define the passivity constraint as follows.

upyp-ρypyp with ρ>0.

For a nonlinear MPC controller, you define the passivity constraint by setting the Passivity property of the nonlinear MPC object.

Design Nonlinear MPC Controller

Create a nonlinear MPC object with four states, four outputs, and two inputs.

nlobj = nlmpc(4,4,2);
In standard cost function, zero weights are applied by default to one or more OVs because there are fewer MVs than OVs.

Specify the quadruple-tank dynamics function as the state function of the prediction model.

nlobj.Model.StateFcn = "stateFcnQuadrupleTank";

The default cost function of a nonlinear MPC problem is a standard quadratic cost function. For this example, keep a quadratic cost function and specify nonzero weights for the first two output variables [1].

nlobj.Weights.OutputVariables = [0.1 0.1 0 0];

Specify the passivity property fields of the nonlinear MPC object.

nlobj.Passivity.EnforceConstraint = true;
nlobj.Passivity.InputFcn = "getPassivityInputQuadrupleTank";
nlobj.Passivity.OutputFcn = "getPassivityOutputQuadrupleTank";
nlobj.Passivity.OutputPassivityIndex = 1;

Closed-Loop Simulation

Specify the initial conditions of the states.

x0 = [25;16;20;21];

Specify the equilibrium point for the quadruple-tank model.

xs = [28.1459,17.8230,18.3991,25.1192]';
us = [37,38]';

Open the Simulink model.

mdl = "quadrupleTankNLMPC";
open_system(mdl)

Run the model.

sim(mdl);

View the errors for the quadruple-tank states.

open_system(mdl + "/Quadruple-tank/state_error")

The errors go to zero and the closed-loop system is stable.

To view the performance of the nonlinear MPC controller without the passivity constraint, remove the passivity constraint from the controller.

nlobj.Passivity.EnforceConstraint = false;

Run the simulation.

sim(mdl);

Without the passivity constraint, the closed-loop system becomes unstable with the same controller design parameters.

References

[1] Raff, Tobias, Christian Ebenbauer, and Frank Allgöwer. "Nonlinear model predictive control: A passivity-based approach." In Assessment and Future Directions of Nonlinear Model Predictive Control, edited by Findeisen, Rolf, Frank Allgöwer, and Lorenz T. Biegler, 151-162; New York: Springer, 2007.

See Also

Functions

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