State-space models linear

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. 

The PBL reactor considered in the present study is a typical batch process and the open-loop test is inadequate to identify the process. We employed a closed-loop subspace identification method. This method identifies the linear state-space model using high order ARX model. To apply the linear system identification method to the PBL reactor, we first divide a single batch into several sections according to the injection time of initiators, changes of the reactant temperature and changes of the setpoint profile, etc. Each section is assumed to be linear. The initial state values for each section should be computed in advance. The linear state models obtained for each section were evaluated through numerical simulations. 

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 ... 

From the last example, we may see why the primary mathematical tools in modem control are based on linear system theories and time domain analysis. Part of the confusion in learning these more advanced techniques is that the umbilical cord to Laplace transform is not entirely severed, and we need to appreciate the link between the two approaches. On the bright side, if we can convert a state space model to transfer function form, we can still make use of classical control techniques. A couple of examples in Chapter 9 will illustrate how classical and state space techniques can work together. 

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. 

Even after linearization, the state-space model often contains too many dependent variables for controller design or for implementation as part of the actual control system. Low-order models are thus required for on-line implementation of multivariable control strategies. In this section, we study the reduction in size, or order, of the linearized model. 

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 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. 

There have been relatively few studies of multi-variable controllers for continuous crystallizers. Most studies of MIMO control algorithms are based on linear state-space models of the form... 

The procedure for deriving the linear state-space model and the input-output transfer function model in Eq. (21.1) involves the following steps ... 

Assume that the dynamic behaviour of a process is within a neighbourhood of an operating point and can be described sufficiently accurate by a linear time-invariant state space model. Then sensor and actuator faults, e.g. leakage from a tank, are additional external input signals to the process. They are commonly taken into account as additive terms in the state space equations and are classified as additive faults [7, 8]. 

Let A, B, C, D, E, F, G, K be constant coefficient matrices of appropriate dimensions and let x denote the state vector, u the vector of known inputs, y the vector of measured outputs, fit) additive faults and d (t) disturbances. The dynamic behaviour of a process subject to additive faults can then be described by the linear state space model... 

Consideration of the equations of a faulty LTI system and of a Luenberger state variable observer reveals that any faults affecting the system have an affect on the observer output error which, after transients have settled, can be used as a fault indicator ([34], Sect. 5.2.2). Assume that thedynamics of a system may be represented by the linear time-invariant state space model... 

Suppose that the dynamic behaviour of a real system can be described by means of a linear time-invariant (LTI) system given by the state space model... 

In this section we briefly describe a linear state space model that serves as a building block for the main models of collaborative forecasting processes we present in this chapter. We then present a well-known forecasting technique associated with this model namely, the Kalman filter. Let Xt be a finite, n-dimensional vector process called the state of the system. In the context of inventory management, this vector may consist of early indicators of future demand in the channel, actual demand realizations at various points of the channel, and so forth. Suppose that the state vector evolves according to ... 

Suppose that the demand evolves according to the linear state space model (10.1)-(10.3). Aviv (2002b) presents a dynamic programming model for this case by extending the state-space as follows. Let... 

An adaptive replenishment policy for the linear state-space model... 

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