Data-driven subspace predictive control: Stability and horizon tuning

doi:10.1016/j.jfranklin.2018.07.032

Journal of the Franklin Institute

Volume 355, Issue 15, October 2018, Pages 7509-7547

https://doi.org/10.1016/j.jfranklin.2018.07.032 Get rights and content

Abstract

Data-driven Subspace Predictive Control (SPC) is an advanced model-free process control strategy in the presence of system constraints. Efficient implementation of SPC requires appropriate tuning of the controller horizons, which are called Prediction Horizon and Control Horizon. This tuning is a critical step to guarantee the SPC closed-loop stability and to enhance the closed-loop performance and robustness. In this paper we propose an optimal tuning method for unconstrained SPC, which can guarantee stability, computational efficiency and optimality of the unconstrained SPC closed-loop system and is applicable to non-minimum phase open-loop stable or marginally stable systems. Derivation of general form of closed-loop transfer function for unconstrained SPC, and providing a necessary and sufficient condition of the closed-loop stability is the primary contribution of this work. In addition, the stability analysis enabled us to propose an algorithm to determine the shortest-feasible-prediction-horizon and the feasible range of prediction horizon. Consequently, these results are used in proposing a new algorithm to determine the SPC horizons in optimal manner. Simulation results illustrate effectiveness and importance of our proposed stability analysis and horizons tuning algorithm for unconstrained SPC.

Introduction

Data-driven Subspace Predictive Control (SPC) is one of the most popular predictive control strategies in industry over the past decade [1], [2], [3], [4]. SPC was first introduced in [5], and it is based on the combination of subspace predictor and Model Predictive Control (MPC) algorithm. In SPC the subspace predictor matrices are obtained directly from the experimental input-output (I/O) data by using the subspace matrices, which eliminates the intermediate parametric model identification step. Therefore, SPC is called a model-free or data-driven approach. Some features of SPC, such as no pre-assumptions about the system model and calculation of prediction matrices without iteration and solving Diophantine equation are advantages of SPC in practical applications [2], [6].

MPC and SPC have same cost function and tuning parameters that includes prediction horizon, which shows number of sample times requires to estimate the future output, control horizon that is the number of sample times to calculate the optimal control signal sequence, and weighting matrices to penalty the tracking error and the control signal. Appropriate choice of these parameters can significantly influence the closed-loop stability, performance and robustness. Poor tuning of these parameters makes the closed-loop system more sensitive to changes in system parameters, noise and disturbances. There are extensive studies in the literature that provide several tuning strategies for MPC [7], [8], [9], [10], and a survey of tuning methods was provided in [11]. However, existing a complex interaction between the system and controller parameters, and desired performance and stability criteria, makes the MPC tuning procedure a tricky problem, specially in the presence of active constraints [12]. On the other hand, the proposed tuning methods are based on model-based techniques of MPC and require dynamic model of the system or its parameters to obtain some a prior information, such as open-loop settling time, rise time, delay time or system matrices [11], [13], [14], [15]. Moreover, closed-loop stability of the predictive control and effect of tuning parameters on stability are still open issues [16]. Stability of MPC is addressed in several papers [17], [18], [19], [20], [21], and [22] provided an exhaustive review of stability solutions for MPC. However, all of these MPC stability strategies were developed with the assumption that the system model is available and all state variables can be measured. Since, SPC is a model-free approach the existing model-based techniques for tuning and stability are not applicable for SPC. In SPC approach the controller is designed based on subspace predictor coefficient matrices that is constructed directly from I/O data with no need to system dynamic model or its parameters [2], [5].

Consequently, in this paper we focus on stability analysis of unconstrained SPC and tuning the SPC parameters based on model-free algorithms in data-driven manner. This feature makes the proposed SPC tuning algorithm suitable for on-line applications, where the tuning parameters and the controller can be updated by collecting new I/O data and updating the subspace predictor matrices. Although this paper focuses on unconstrained SPC, however, it provides a unique insight into the structure of the unconstrained SPC closed-loop characteristic equation. Specifically, it proves the role of SPC gain and DC-gain of a process on the stability of the closed-loop system, and shows how these gains dictate the required Shortest-Feasible-Prediction-horizon value to achieve a stable closed-loop system. The main contribution of this paper is deriving a necessary and sufficient condition for closed-loop stability of unconstrained SPC for non-minimum phase systems with stable or marginally stable poles. For this purpose, general form of closed-loop transfer function for unconstrained SPC is obtained for the first time. The necessary and sufficient condition for the closed-loop stability is derived based on small-gain stability analysis of the SPC closed-loop transfer function in inspired by the stability analysis of self-tuning controllers in [23] and [24]. The second contribution is in providing a model-free algorithm to determine the Feasible Range of Prediction Horizon that guarantees closed-loop stability of the unconstrained SPC. For this purpose, the SPC stability graphs are introduced to ease the SPC stability analysis. In addition, we provide a model-free methodology to tune and determine the control horizon and prediction horizon in efficient way, which makes the proposed method suitable for on-line applications and on-line updating. For this purpose, a criteria is introduced to determine the Efficient-Control-horizon by efficiently minimizing the dimension of subspace predictor matrix based on the Shortest-Feasible-Prediction-horizon. The Efficient-Prediction-horizon is then provided by optimizing the unconstrained SPC cost function. Another advantage of our proposed SPC horizons tuning method is decreasing computational cost of the unconstrained SPC design. This feature is provided by showing that performance of the SPC closed-loop system does not improve significantly by increasing the control horizon from the Efficient-Control-horizon, which avoids of selecting longer control horizons. The feature also makes the proposed method more appropriate for industrial applications. Therefore, our proposed SPC tuning algorithm is founded based on closed-loop stability, computational efficiency and optimality of the unconstrained SPC design. This appropriate selection of SPC tuning parameters increases stability and robustness of the unconstrained SPC closed-loop system and decreases its sensitivity to disturbance and variation of system parameters. As a result, our proposed method decreases the requirement of applying all-time persistently excitation (PE) signals to update the SPC controller [3], [25]. The simulation results show advantages and effectiveness of the proposed algorithms.

The paper is organized as follows: Section 2 is mathematical formulation of SPC. Section 3 is the problem statement and motivation. Stability analysis of unconstrained SPC is discussed in Section 4. Section 5 is about the role of SPC tuning parameters. Simulation results are provided in Section 6; and finally, Section 7 is conclusion.

Section snippets

SPC mathematical formulation

Consider the following stable discrete linear system in innovation form as below [5], $\begin{matrix} x (k + 1) & = & A x (k) + B u (k) + K ζ (k) \end{matrix}$ $\begin{matrix} y (k) & = & C x (k) + D u (k) + ζ (k) \end{matrix}$ where k is the k-th sampling time, $x (k) \in R^{n}$ is the state of the system, $y (k) \in R^{l}$ is the output, $u (k) \in R^{m}$ is the input, and $ζ (k) \in R^{l}$ is an unknown innovation sequence with $E [ζ (k) ζ^{T} (k)] = S \in R^{l \times l}$ as the innovation covariance matrix. The matrices $A \in R^{n \times n},$ $B \in R^{n \times m},$ $C \in R^{l \times n},$ $D \in R^{l \times m}$ and $K \in R^{n \times l}$ are the state-space matrices that describe the system [5].

Problem statement and motivation

In SPC implementation, value of the SPC gain matrices and control signal in Eqs. (21), (22) and (23) are functions of the SPC tuning parameters. Therefore, appropriate choice of the SPC tuning parameters have a significant impact on stability, robustness and performance of the SPC closed-loop system. Moreover, the SPC gains are calculated directly by using I/O data, and computational complexity of the optimization algorithm depends on dimension of the system, which is also affected by the

Unconstrained SPC stability analysis

This section provides closed-loop small-gain stability condition for unconstrained SPC based on the SPC gains and cost function tuning parameters.

SPC tuning parameters

The SPC cost function in Eq. (11) has four tuning parameters: Weighting matrices Q and R, Prediction horizon N_p and Control horizon N_c. Appropriate selection of them has significant effect on stability, robustness and performance of the SPC controller. Weighting matrices have similar roles and characteristics as in Linear Quadratic (LQ) controllers [11]. However, prediction horizon N_p and control horizon N_c are specific parameters for SPC, as LQ controllers use infinite horizon in optimization

Simulation

In this section, three examples are presented to demonstrate the proposed unconstrained SPC stability condition by applying Algorithm 1. Simulation results show effectiveness of the proposed SPC horizon tuning technique in Algorithm 2.

Conclusion

In this work an optimal SPC horizons tuning method was proposed, which guarantees the closed-loop stability and performance of unconstrained SPC. The necessary and sufficient condition to assure the unconstrained SPC closed-loop stability was provided for non-minimum phase open-loop stable (or marginally stable) systems. Based on the derived unconstrained SPC stability condition, Algorithm 1 was provided to determine the Shortest-Feasible-Prediction-horizon (N_SFP) and the Feasibility Range (FR)

References (34)

WuX. et al.
Data-driven modeling and predictive control for boiler-turbine unit using fuzzy clustering and subspace method’s
ISA Trans.
(2014)
A.S. Yamashita et al.
Tuning the model predictive control of a crude distillation unit
ISA Trans.
(2016)
J.O. Trierweiler et al.
RPN tuning strategy for model predictive control
J. Process Control
(2003)
S. Kanev et al.
Robust asymptotically stable finite-horizon MPC
Automatica
(2006)
R. Scattolini et al.
On the choice of the horizon in long-range predictive control- some simple criteria
Automatica
(1990)
D.G. Mayne et al.
Constrained model predictive control: stability and optimality
Automatica
(2000)
P.R. Belanger
On type 1 systems and the clarke-gawthrop regulator
Automatica
(1983)
R. Hallouzi et al.
Persistency of excitation in subspace predictive control
Proceedings of the 17th World Congress IFAC
(2008)
R.K. Wood et al.
Terminal composition control of a binary distillation column
Chem. Eng. Sci.
(1973)
TianX. et al.
A data-based approach for multivariate model predictive control performance monitoring
Neurocomputing
(2011)

M. Verhaegen

Identification of the deterministic part of MIMO state space models given in innovations form from input-output data

Automatica

(1994)

R. Hallouzi et al.

Fault-tolerant subspace predictive control applied to a boeing 747 model

Journal of Guid. Control Dyn.

(2008)

R. Kadali et al.

A data driven subspace approach to predictive controller design

Control Eng. Pract.

(2003)

N.A. Mardi et al.

Subspace-based model predictive control of time-varying systems

Proceedings of Joint 48th IEEE Conference on Decision and Control and 28th Chinese Conference

(2009)

W. Favoreel et al.

SPC: Subspace predictive control

Proceeding of the 14th IFAC World Congress, H

(1999)

S.T. Navalkar et al.

Wind tunnel testing of subspace predictive repetitive control for variable pitch wind turbines

IEEE Trans. Control Syst. Technol.

(2015)

J.L. Garriga et al.

Model predictive controller tuning via eigenvalue placement

Proceeding of American Control Conference

(2008)

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View full text

Data-driven subspace predictive control: Stability and horizon tuning

Abstract

Introduction

Section snippets

SPC mathematical formulation

Problem statement and motivation

Unconstrained SPC stability analysis

SPC tuning parameters

Simulation

Conclusion

ISA Trans.

ISA Trans.

J. Process Control

Automatica

Automatica

Automatica

Automatica

Chem. Eng. Sci.

Neurocomputing

Automatica

Fault-tolerant subspace predictive control applied to a boeing 747 model

Journal of Guid. Control Dyn.

A data driven subspace approach to predictive controller design

Control Eng. Pract.

Subspace-based model predictive control of time-varying systems

Proceedings of Joint 48th IEEE Conference on Decision and Control and 28th Chinese Conference

SPC: Subspace predictive control

Proceeding of the 14th IFAC World Congress, H

Wind tunnel testing of subspace predictive repetitive control for variable pitch wind turbines

IEEE Trans. Control Syst. Technol.

Model predictive controller tuning via eigenvalue placement

Proceeding of American Control Conference