# (264g) On the Selection of Input-Output Pairs to Achieve Global Stability of Nonlinear Process Systems

- Conference: AIChE Annual Meeting
- Year: 2009
- Proceeding: 2009 AIChE Annual Meeting
- Group: Computing and Systems Technology Division
- Session:
- Time:
Tuesday, November 10, 2009 - 2:30pm-2:50pm

We highlight the significance of input output pair selection in the design of inventory controllers for nonlinear process systems. A process system is a high order system, potentially infinite dimensional. Macroscopic dynamics are constrained by conservation laws, (like mass and energy balance equations), and the existence of a convex extension which is continuous and homogeneous degree one (commonly referred to as the Second Law of Thermodynamics). These properties lead to a very elegant and natural definition of control Lypunov functions, which can be used for nonlinear control system design using passivity theory or input-output linearization.

The key idea of inventory control is to reformulate control problems stated for intensive variables in terms of conjugate inventories. For example, temperature is conjugate to internal energy, composition (chemical potential) is conjugate to total molar hold-up, pressure (liquid level) is conjugate to mass or molar hold-up and surface tension is conjugate to area. Flow variables are then used to control inventories by adjusting flows using a combination of feedback and feedforward control.

The main advantage of the approach is that it leads to linearization and an effective means to choose synthetic input output pairs for very complex process control problems. The reason why the approach is effective that the operator mapping flows to inventories is passive. The passivity theorem states that any input strictly passive feedback stabilizes the inventory around time-invariant or time-varying trajectories. The approach can be applied to networks and distributed processes. However, in order to achieve internal stability it is also necessary that the so-called zero dynamics are stable, or equivalently, that the feedback control system is strictly state passive.

The main purpose of this paper is to strengthen connections amongst the classical theories of nonlinear control and process thermodynamics by using a thermodynamics based storage (Lyapunov) function to and analyze the zero dynamics of closed loop inventory control systems. We show that in some systems it is possible to apply control to some inventories so that global stability is guaranteed whereas other inventories cannot be stabilized using the approach due to lack of internal stability. These properties are related to classical minimum phase (passivity) conditions which have been studied extensively in the area of nonlinear feedback control.

The contribution of the paper is two-fold. First we review new connections between thermodynamic stability theory and nonlinear systems theory. Second we develop novel case studies which highlight the application of the theory. These applications concern continuous stirred tank reactors operating under a variety of inventory control strategies. The first example shows input multiplicity and has dynamics determined by mass balances only. We demonstrate that the choice of inventories is critical to achieve closed loop stability. The second example shows the output multiplicity, which is typical of systems with three steady states, one unstable bracketed by two. In such systems the mass and energy balance equations are integrated and again we find that the choice of input output pairing provides the key to achieve global stability of the zero-dynamics of the system. We finally develop a multi-variable control structure for the inventory control which provides guaranteed stability of the mass and energy balance systems.

The stability theory is based on the use of a universal Lyapunov function derived from the entropy. This Lyapunov function, called the available work, was in fact proposed by Gibbs for investigation of phase stability and later generalized by Keenan for the analysis of thermodynamic efficiency of coal fired power plants. Michelsen used the same concept to study fluid phase equilibria and Ydstie and Alonso used it for control system design and stability analysis of infinite dimensional process systems. We believe this is the first instance where the approach is applied for stability analysis of finite dimensional systems under inventory control.