Assessment and Future Directions of Nonlinear Model Predictive ...

142 M. Srinivasarao, S.C. Patwardhan, and R.D. GudiWeiner structure. The linear dynamic component of these models is parameterizedusing generalized orthonormal basis filters (GOBF) [3, 4]. We then proceedto show how the identified models can be used for inter-sample inferential estimationof the slowly sampled variable and also for model predictive control ofsuch irregularly sampled, multi-rate systems. The efficacy of the proposed modelingand control scheme is demonstrated by conducting simulation studies ona benchmark CSTR system [5] which exhibits input multiplicity and change inthe sign of steady state gain in the operating region.2 Development of Multi-rate NOE+NARMA ModelIn this work, we make following assumptions: (a) Sampling rates for all measurementsare integer multiples of some time period called ‘shortest time unit’ (T ).(b)All actuators are to be manipulated at a frequency corresponding to the ‘shortesttime unit’ (T ).and (c) The effect of unmeasured disturbances on the outputscan be adequately captured by an additive nonlinear noise model. Thus, the manipulatedinputs are changed at {t k = kT : k =0, 1, 2, ...} while the i th outputmeasurements are assumed to be available only at sampling instants given bythe sub-sequence {k i0 ,k i1 ,k i2, ...} such that the difference k il − k il−1 = q il (> 1)where q il is an integer.We propose to use nonlinear output error (NOE) structure to model the deterministiccomponent in the data. This choice is motivated by the fact thatthe internal model in an NMPC formulation is required to have good predictionability with respect to the manipulated inputs. It can be argued that the NOEmodels, which are driven only by the manipulated inputs, have good long rangeprediction ability. The choice of NOE structure implies that the deterministiccomponent of a r × m MIMO system can be modeled as r MISO NOE models.The irregularly sampled model residuals generated while identifying the deterministiccomponent are further modeled as r SISO nonlinear stochastic models.Thus, in order to simplify the notation, without loss of generality, we carry outthe model development for a MISO system. Therefore in the sequel we dropthe subscript ’i’ that we used above to indicate variables associated with i’ thoutput . Given input sequence {u(k) :k =0, 1, 2, .....N} and the correspondingirregularly sampled output data {y(k l ):k l = k 0 ,k 1 ,k 2 , ...} collected from a plantwhere k l represents sampling instants. We propose a two step approach to thedevelopment of deterministic and stochastic models for a general multivariate,nonlinear system with fading memory. In the first step, we develop a MISO fastrate Weiner type NOE model of the form,X u (k +1) = Φ u X u (k)+Γ u u(k) (1)ŷ u (k) =Ω u [ X u (k)] (2)for k l ≤ k