State-space models nonlinear

Dynamic models derived from physical principles typically consist of one or more ordinary differential equations (ODEs). In this section, we consider a general class of ODE models referred to as state-space models, that provide a compact and useful representation of dynamic systems. Although we limit our discussion to linear state-space models, nonlinear state-space models are also very useful and provide the theoretical basis for the analysis of nonhnear processes (Henson and Se-borg, 1997 Khalil, 2002). 

A key featui-e of MPC is that a dynamic model of the pi ocess is used to pi-edict futui e values of the contmlled outputs. Thei-e is considei--able flexibihty concei-ning the choice of the dynamic model. Fof example, a physical model based on fifst principles (e.g., mass and energy balances) or an empirical model coiild be selected. Also, the empirical model could be a linear model (e.g., transfer function, step response model, or state space model) or a nonhnear model (e.g., neural net model). However, most industrial applications of MPC have relied on linear empirical models, which may include simple nonlinear transformations of process variables. 

As mentioned above, the backbone of the controller is the identified LTI part of Wiener model and the inverse of static nonlinear part just plays the role of converting the original output and reference of process to their linear counterpart. By doing so, the designed controller will try to make the linear counterpart of output follow that of reference. What should be advanced is, therefore, to obtain the linear input/output data-based prediction model, which is obtained by subspace identification. Let us consider the following state space model that can describe a general linear time invariant system ... 

The problem in obtaining a state space model for the dynamics of the CSD from this physical model is that the population balance is a (nonlinear) first-order partial differential equation. Consequently, to obtain a state space model the population balance must be transformed into a set of ordinary differential equations. After this transformation, the state space model is easily obtained by substitution of the algebraic relations and linearization of the ordinary differential equations. 

To determine the state space model with system Identification, responses of the nonlinear model to positive and negative steps on the Inputs as depicted in Figure 4 were used. Amplitudes were 20 kW for P,, . 4 1/s for and. 035 1/s for Q. The sample interval for the discrete-time model was chosen to be 18 minutes. The software described In ( 2 ) was used for the estimation of the ARX model, the singular value analysis and the estimation of the approximate... 

Early applications of MPC took place in the 1970s, mainly in industrial contexts, but only later MPC became a research topic. One of the first solid theoretic formulations of MPC is due to Richalet et al. [53], who proposed the so-called Model Predictive Heuristic Control (MPHC). MPHC uses a linear model, based on the impulse response and, in the presence of constraints, computes the process input via a heuristic iterative algorithm. In [23], the Dynamic Matrix Control (DMC) was introduced, which had a wide success in chemical process control both impulse and step models are used in DMC, while the process is described via a matrix of constant coefficients. In later formulations of DMC, constraints have been included in the optimization problem. Starting from the late 1980s, MPC algorithms using state-space models have been developed [38, 43], In parallel, Clarke et al. used transfer functions to formulate the so-called Generalized Predictive Control (GPC) [19-21] that turned out to be very popular in chemical process control. In the last two decades, a number of nonlinear MPC techniques has been developed [34,46, 57],... 

In order to suppress this oscillatory behaviour, the use of the automatic feedback control has been considered [4]. State space model and nonlinear full-state feedback have been used for stabilization of the system [5]. But, some of these state variables are not measurable, therefore, concept of state estimation from well-head measurements has been considered. A nonlinear observer is used for state estimation [6] which has shown satisfactory result in experiment [7]. As noted in [7], estimation is affected by noise. The standard Kalman filter has been used for state estimation and down-hole soft-... 

In this section, classical state-space models are discussed first. They provide a versatile modeling framework that can be linear or nonlinear, continuous- or discrete-time, to describe a wide variety of processes. State variables can be defined based on physical variables, mathematical solution convenience or ordered importance of describing the process. Subspace models are discussed in the second part of this section. They order state variables according to the magnitude of their contributions in explaining the variation in data. State-space models also provide the structure for... 

Various extensions of linear state-space approach have been proposed for developing nonlinear models [227, 274]. An extension of linear CVA for finding nonlinear state-space models was proposed by Larimore [160] where use of alternating conditional expectation (ACE) algorithm [24] was suggested as the nonlinear CVA method. Their examples used linear CVA to model a system by augmenting the linear system with pol3momials of past outputs. 

Vajda, S., Godfrey, K. R., and Rabitz, H. (1989). Similarity transformation approach to identifiability analysis of nonlinear compartmental models. Math. Biosci 93,217-248. Walter, E. (1982). Identifiability of State Space Models, Lect. Notes Biomath. No. 46. Springer-Verlag, New York... 

Zolghadri, A. and Cazaurang, F. Adaptive nonlinear state-space modelling for the prediction of daily mean PMio concentrations. Environmental Modelling and Software 21(6) (2006), 885—894. 

The notion and properties of, and the transformation to minimal models is well developed and understood in the area of linear and nonlinear system theory (Kailath, 1980 and Isidori, 1995). Moreover, a wide class of lumped process models can also be transformed into the form of nonlinear state-space models. Therefore, the case of nonlinear state-space models is used as a basic case for the notion and construction of minimal models. This is then extended to the more complicated case of general lumped process models. 

Minimality for nonlinear state-space models Lumped process models with differential index 1 can be described by input-affine state-space models, where the notion of minimality is related to the number of state variables necessary and sufficient to produce a given input-output behaviour. This means, that one naturally uses the single quality index = dim x = n for both linear and nonlinear state-space models and orders the functionally equivalent models accordingly. 

Model reduction of nonlinear state-space models Non-minimal input-affine state-space models can be transformed to a minimal realization form by applying a suitable nonlinear state transformation followed by state elimination. Such a model reduction is based on finding quantities which are constant along any trajectory in the state-space. These can be termed hidden conserved quantities (Szederkdnyi et al. 2002). 

Dynamic nonlinear analysis techniques (Isidori 1995) are not directly applicable to DAE models but they should be transformed into nonlinear input-affine state-space model form by possibly substimting the algebraic equations into the differential ones. There are two possible approaches for nonlinear stability analysis Lyapunov s direct method (using an appropriate Lyapunov-function candidate) or local asymptotic stability analysis using the linearized system model. 

Note that the state-space model for Example 6.6 has d = 0, because disturbance variables were not included in (6-77). By contrast, suppose that the feed composition and feed temperature are considered to be disturbance variables in the original nonlinear CSTR model in Eqs. 2-66 and 2-68. Then the linearized model would include two additional deviation variables cXi and 7y, which would also be included in (6-77). As a result, (6-78) would be modified to include two disturbance variables, diAcXianddiATi... 

State-space models provide a convenient representation of dynamic models that can be expressed as a set of first-order, ordinary differential equations. State-space models can be derived from first principles models (for example, material and energy balances) and used to describe both linear and nonlinear dynamic systems. 

A fired-tube furnace is one of the case studies in the Process Control Modules (PCM) in Appendix E. The PCM furnace model is a nonlinear state-space model that consists of 26 nonlinear ordinary differential equations based on conservation equations and reaction rate expressions for combustion (Doyle et al., 1998). The key process variables for the furnace model are listed in Table 13.1. 

The MFC predictions are made using a dynamic model, typically a linear empirical model such as a multivariable version of the step response or difference equation models that were introduced in Chapter 7. Alternatively, transfer function or state-space models (Section 6.5) can be employed. For very nonhnear processes, it can be advantageous to predict future output values using a nonlinear dynamic model. Both physical models and empirical models, such as neural networks (Section 7.3), have been used in nonlinear MFC (Badgwell... 

Most of the current MFC research is based on state-space models, because they provide an important theoretical advantage, namely, a unified framework for both linear and nonlinear control problems. State-space models are also more convenient for theoretical analysis and facilitate a wider range of output feedback strategies (Rawlings, 2000, Maciejowski, 2002 Qin and Badgwell, 2003). 

The EKF has by far been the most extensively used identification algorithm, for the case of nonlinear systems, over the past 30 years, and has been applied for a number of civil engineering applications, such as structural damage identification, parameter identification of inelastic structures, and so forth. It is based on the propagation of a Gaussian random variable (GRV) through the first-order linearization of the state-space model of the system. Despite... 

Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. Journal of Computational and Graphical Statistics, 5(1). 

To facilitate the design and application of the nonlinear robust control law, let us rewrite the pol3unerization reactor modeling equations (42) in the state space ... 

Since the orthogonal collocation or OCFE procedure reduces the original model to a first-order nonlinear ordinary differential equation system, linearization techniques can then be applied to obtain the linear form (72). Once the dynamic equations have been transformed to the standard state-space form and the model parameters estimated, various procedures can be used to design one or more multivariable control schemes. 

Note that in the following analyses, we will drop the prime symbol. It should still be clear that deviation variables are being used. Then this linear representation can easily be separated into the standard state-space form of Eq. (72) for any particular control configuration. Numerical simulation of the behavior of the reactor using this linearized model is significantly simpler than using the full nonlinear model. The first step in the solution is to solve the full, nonlinear model for the steady-state profiles. The steady-state profiles are then used to calculate the matrices A and W. Due to the linearity of the system, an analytical solution of the differential equations is possible ... 

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