비선형 시스템의 부분공간-기반 예측제어

유기윤, 이현구

서울대학교 응용화학부·화학공정 신기술 연구소

Subspace-based Predictive Control of Nonlinear Systems

Kee-Youn Yoo and Hyun-Ku Rhee

School of Chemical Engineering and Institute of Chemical Processes,

Seoul National University

Preliminaries

Consider the following discrete-time nonlinear system

(1)

The standard nonlinear model predictive control setup, which uses the nonlinear model for prediction, is given as follows:

(2)

subject to the constraint

(3)

where N denotes the length of the control and prediction horizon in the model predictive control. Note that only for simplicity and without any loss of generality, it was assumed that the control and prediction horizons are of the same length. Such a setup generally involves non-convex nonlinear programs (NLP), whose solution may be problematic. In order to simplify the nonlinear model predictive control formulation and to make the implementation easier and more efficient, the idea explored here is to replace the nonlinear prediction model by a linear-fractional linear parameter-varying(LF-LPV) one, obtained by applying the subspace technique along some predefined nominal trajectories.  NLP can be reduced to a standard LP or QP (assuming that all the other, constraints are linear) by using an LF-LPV model originated from the LF-LPV subspace identification method [1] for prediction and choosing the performance objective to be linear or quadratic. This approach is explored in detail in the following sections. In this way, the problem setup of the standard nonlinear model predictive control approach is avoided, performance analysis may be simplified, and the computational efficiency is significantly improved. When compared with the standard local linearization-based model predictive control which uses linear time-invariant prediction models, the subspace-based predictive control using LF-LPV prediction models provides more accurate approximation to the true nonlinear system, resulting in the performance closer to that of the nonlinear model predictive control.

LF-LPV prediction model along the parameter trajectories

Assume that the parameter pre-trajectories, or the so-called nominal trajectories, over the prediction horizon N, given by

(4)

may be determined by some means. The actual trajectories can be measured at real time. The LF-LPV prediction model is given as follows:

(5)

where

(6)

For the design of predictive controller, the subspace identification problem of LF-LPV systems can be interpreted as follows: given the past inputs and outputs Wp and the future inputs Uf with the measurable scheduling variables[eqn: {hat delta}_k], find an optimal prediction of the future outputs Yfl. If we use an LF-LPV predictor

(7)

where the data matrices are defined in [1], the least squares prediction[eqn: {hat Y}_f] of [eqn: Y_f] can be found from the following least squares problem:

(8)

The solution to this problem is given by the orthogonal projection; i.e.,

(9)

Like the linear prediction models, the numerical implementation of the projection can be done in a very efficient way by making the following QR decomposition:

(10)

If we define

(11)

and

(12)

These two matrices will play a very important role in the next section where the link between subspace identification and model predictive control is made. Then the optimal prediction model can be represented as follows:

(13)

where

(14)

Subspace-based predictive control of LF-LPV systems

On the basis of the LF-LPV prediction model, the original nonlinear model predictive control optimization setup can be transformed into the following:

(15)

subject to the linear parameter-varying model dynamics

(16)

and constraints

(17)

If the performance index is chosen to be quadratic and the constraints are linear, then the above optimization is a standard QP. Here we choose the optimal control criterion as follows:

(18)

where

(19)

The control optimization and updating of nominal trajectories at each sampling time t are performed by the following procedure:

1. Find a nominal future parameter trajectory for time step t

2. Compute the control moves by solving the optimization problem of the predictive control setup.

3. Update the current parameter trajectory on the basis of the updated future process information[eqn: {haty}_k] and[eqn: {hat u}_k] within the prediction horizon. If the future parameter trajectory converges, go to step 4, otherwise go to step 2.

4. Implement the first control move[eqn: u_{t+1}].

5. Define a future nominal parameter trajectory for the next sampling time t' by using the converged parameter trajectory of the current step.

Conclusions

As an alternative to "full-fledged" nonlinear model predictive control, we introduced a different model predictive control algorithm that uses LF-LPV prediction models obtained by the LF-LPV subspace identification method. In this way, the problem setup of the standard nonlinear model predictive control approach is avoided and the computational efficiency is significantly improved. When compared with the standard local linearization-based model predictive control which uses linear time-invariant prediction models, subspace-based predictive control using LF-LPV prediction model provides performance closer to that of nonlinear model predictive control.

Acknowledgement

This work was supported by Korea Science and Engineering Foundation(KOSEF) through the Automation Research Center at Pohang University of Science and Technology.

References

[1] Kee-Youn Yoo, Subspace Identification and Control of Linear-Fractional LPV Systems, Ph.D. Thesis, Seoul National University, 2000.