Model Predictive Control of Cyclic Systems Using Linear Matrix Inequalities
Pradeep Y. Tiwari and Mayuresh V. Kothare, Department of Chemical Engineering, Lehigh University, 111 Research Drive, Bethlehem, PA 18017

Cyclic processes occupy a significant place in nature, technology and economics. The motion of a spacecraft and satellite, the change of seasons of the year, the corresponding cycles of farm work, biological rhythms, etc. provide examples of systems in this category. Cyclic processes also occur in a variety of manufacturing systems, e.g., Pressure Swing Adsorption (PSA), chromatography and robotic motions. Typically, a cyclic system is characterized by two distinct time scales, viz, the time scale within each repeating cycle and the cycle index. As a result, cyclic systems are also sometimes referred to as 2-dimensional (2D) systems. Conventional control formulations do not explicitly incorporate and/or exploit this 2D representation of cyclic systems.

In this paper, we propose a novel MPC formulation for cyclic processes in an optimization framework. The proposed approach explicitly incorporates the two time scales of cyclic systems as well as input and output constraints. The resulting optimization problem is recast as a convex problem involving LMIs.

Following [3], a 2D cyclic system can be defined using the following discrete time state space model equations:

\begin{displaymath}\begin{array}{lll} x_{k+1}(p+1) & = & Ax_{k+1}(p)\;+\;B u_{k+... ...& = & Cx_{k+1}(p)\;+\;D u_{k+1}(p) \;+\; D_0 y_k(p) \end{array}\end{displaymath} (1)

where, $ k \geq 0$ is the cycle index, $ 0 \leq p \leq \alpha $ is discrete time within the cycle, $ \alpha \geq 0$ is cycle length. $ x_k(p) \in \Re^n$ is the plant state vector and $ y_k(p) \in \Re^m$ is the plant output vector at discrete time $ p$ within cycle $ k$. $ u_k(p) \in \Re^l$ is plant input vector. $ A,\; B, \; B_0,\; C,\; D, \; D_0$ are state space parameters. The terms $ B_0$ and $ D_0$ characterize the effect of the previous cycle on the state evolution in the current cycle.

Any control strategy designed for a cyclic system must explicitly account for stability along the two distinct dimensions of the system viz. stability along the pass and asymptotic stability. Although there have been attempts to establish 2D stability for repetitive systems in the literature based on proportional and proportional integral action, none account for process constraints. Furthermore, no reported approach attempts to incorporate optimality within a stabilizing framework. Recently, [1] used a more robust LMI based approach for discrete linear repetitive processes using lyapunov stability condition. A combination of state feedback and output feedback was used to accomplish 2D control. This idea was later extended [4] to a PI based control action where integral information of output error was used to incorporate output tracking. However, no attempt was made to incorporate frequently occurring input/output constraints in the proposed framework. Also the static controller designed using this approach did not consider any performance based majors. This approach does not directly extend to optimal control.

In this paper, we approach the stability problem of 2D systems using a Lyapunov framework, which also allows the formulation of an optimal control problem. A 2D Lyapunov function for the system under consideration is defined as:

$\displaystyle V(X_{k+1}(p)) \; = \; X_{k+1}^T P X_{k+1}(p), P > 0$ (2)

where $ X$ is the augmented state vector and $ P$ is a symmetric positive definite matrix.

The stability in the sense of Lyapunov requires that the Lyapuonv function decrease with time. To establish the stability along the cycle we require (2) to satisfy:

$\displaystyle V(X_{k+1}(p+i+1\vert p)) - V(X_{k+1}(p+i\vert p)) \;\leq \; - X^T_{k+1}(p+i+1\vert p) Q_1 X_{k+1}(p+i+1\vert p)$ (3)

To ensure asymptotic stability (stability across the cycle) we require:

$\displaystyle V(X_{k+1}(\alpha\vert p)) \; \leq \beta V(X_{k}(\alpha\vert p))$ (4)

where $ 1\geq \beta \geq 0$. Since $ X_k(\alpha)$ is known, $ V(X_{k}(\alpha))=\bar{V}$ is also known and remains constant over the cycle $ k+1$. These two stablity requirement can be transformed to (5), (3) using steps that closely follow the development in [2]. We can also show that input and output constraints can be readily incorporated within the proposed optimal control strategy for 2D systems. We can summerize the main result as follows:

Theorem 1   A cyclic system given by (1) with input bounded by $ u^j_{max}$ is stable along the pass and stable asymptotically iff there exist matrix variables $ \gamma$, $ \beta$ $ Q$ and $ Y$ with $ Q=(\gamma+\beta\bar{V})P^{-1}$, $ Y=KQ$ that are the solution to the following optimization problem:

$\displaystyle \displaystyle \min \gamma $

subject to

$\displaystyle \left[ \begin{array}{cc} 1 & X_{k+1}^T(p\vert p) X_{k+1}(p\vert p) & Q \end{array}\right] \; \geq \;0$ (5)

$\displaystyle \left[ \begin{array}{ccc} Q & Q \Phi^T + Y^T R^T & Q\Phi^TQ_1^{\f... ...Q_1^{\frac{1}{2}}RY & 0 & (\gamma+\beta \bar{V}) I \end{array}\right] \geq \; 0$ (6)

$\displaystyle \left[ \begin{array}{cc} X & Y Y^T & Q \end{array}\right], \;\; X_{jj} \leq (u^j_{max})^2$ (7)

$\displaystyle Q \geq 0 \;\;\;(\gamma+\beta \bar{V}) \geq 0 $

which are LMIs in $ Q,\; Y,\; \gamma$ and $ \beta$.

We demonstrate the applicability of the proposed approach using a benchmark example and also a cyclic manufacturing example.

Bibliography

1
K. Galkowski, E. Rogers, S. Xu, J. Lam, and D. H. Owens. LMIs-A fundamental tool in analysis and controller design for discrete linear repetitive processes. IEEE Transactions on Circuits and Systems, 49(6):768-778, June 2002.

2
M. V. Kothare, V. Balakrishnan, and M. Morari. Robust constrained model predictive control using linear matrix inequalities. Automatica, 32(10):1361-1379, October 1996.

3
E. Rogers and D. H. Owens. Stability Analysis for Linear Repetitive processes. ser. Lecture Notes for Control and Information Sciences, Springer-Verlag, 175, 1992.

4
B. Sulikowski, K. Galkowski, E. Rogers, and D.H. Owens. PI control of discrete linear repititive processes. Automatica, 42:877-880, 2006.

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