Enforcing Stability in Steady-State Optimization Radek Beˇ no * Daniel Pachner ** Vladim´ ır Havlena *,** * Department of Control Engineering, Czech Technical University in Prague. Czech Republic (e-mail: benorade@fel.cvut.cz, havlena@fel.cvut.cz) ** Honeywell Prague Laboratory, V Parku 2326/18, 148 00 Prague 4, Czech Republic (e-mail: daniel.pachner@honeywell.com, vladimir.havlena@honeywell.com) Abstract: The article deals with the stability constraint in nonlinear continuous-time dynamic model identification. The identification is formulated as a boundary value problem. Constraining the norm of the terminal sensitivity to the initial condition is used to drive the model state to a stable equilibrium. Solving such boundary value problem on an extending finite time horizon may be numerically more appealing than constraining the eigenvalues of the Jacobian matrix evaluated at the equilibrium point in the state space. Keywords: Steady-state stability, System identification, Nonlinear systems, Convex optimisation, Parameter identification, Continuous-time systems 1. INTRODUCTION This article treats of a nonlinear continuous-time dynamic model parameters identification based on the steady state data. Information in the steady state data is certainly not sufficient for model identification. Nonetheless, projection of that information to the estimates of model parameters is still a meaningful problem formulation. It will be shown how the steady state data matching a cost function can be constructed. However, it should be understood that the cost function, actually minimized during the parameter identification process, may include other terms as well. These terms may be related to transient data prediction errors or violations of some physical assumptions. There- fore, the algorithm presented should not be understood as a complete identification procedure, which may be more complex. A steady state fitting might be done simultaneously with the transient data fitting when minimizing the sum of two criteria. However, the identification process can be more efficient when using the steady state information first and adding the transient data prediction errors to the cost function later, see Stewart et al. [2010]. First, the extracted steady states can represent a significantly smaller amount of data compared to the transient data whereas being still representative of the process nonlinearity over its whole operating range. Because of the data set size, the steady state data fitting may be performed with much less effort. Second, it is often possible to guess which parameters have no impact on the steady state. Those parameters can be excluded from the optimization focused on the steady states. As an example, the fact that the heat accumulation rate is zero in the steady state may often result in the conclusion that the certain heat capacities cannot affect the steady state process values. Therefore, the resulting steady state parameter optimization problem may be {u 1 , y 1 } {u 2 , y 2 } Time Process values Process input Process output Averaged Fig. 1. Collection of the process steady state values during step testing. defined in a lower dimensional space which simplifies the optimization. Certainly , the parameters which have been left out must by identified in the next step of the modeling process when also matching the transient behavior. Another advantage of treating the steady state informa- tion separately is that the steady state model accuracy may be emphasized. This is especially useful when the model is used for the steady state process optimization. On the other hand this approach is not applicable to unstable, integrating or oscillating processes. Without a stability control, the numerical optimization of the steady states usually works only if the initial model is stable and the initial guess of the parameters is sufficiently accurate. With the stability control, the convergence can be significantly improved. It will be shown on a simple example that the model may lose stability during the numerical identification even if started from a stable 16th IFAC Symposium on System Identification The International Federation of Automatic Control Brussels, Belgium. July 11-13, 2012 978-3-902823-06-9/12/$20.00 © 2012 IFAC 1599 10.3182/20120711-3-BE-2027.00315