2. Process models and uncertainties

Every detailed aspect of the real process cannot be adequately contemplated by mathematical models. Simplifying assumptions have to be made, especially due to the control purposes, where models with simple structures and sufficiently small size have to be used regarding to the available control techniques and real-time considerations. Therefore, mathematical control models can only describe the dynamics of the process in an approximate way.

Majority of modern control techniques need a control model of the plant with fixed structure and parameters, which is used throughout the design stage. For an exact description of the plant (neglecting external disturbances), processes could be controlled by an open-loop controller. However, feedback is necessary for process control because of the external perturbations and model inaccuracies in all real processes.

The objective of robust control is to design controllers which preserve stability and performance in spite of the modelling inaccuracies or uncertainties. Although the use of feedback contemplates the inaccuracies of the model simplicity, the term of robust control is used in [4, 5] to describe control systems that explicitly consider the discrepancies between the model and the real processes.

The combustion efficiency of pulverised fuel depends on several parameters. The commonly applied, low-emission techniques use recirculation vortexes that lengthen the paths of the coal grains passing through the flame to minimise generation of thermal oxides of nitrogen (NOx). To make co-combustion of pulverised coal more efficient and environment-friendly, it is nec-

The information taken at the output is delayed and averaged. Although there are several combustion diagnostic direct techniques, the most of them are expensive or impossible to utilise under industrial conditions. The radiation emitted by the flame reflects the combustion process occurring in chemical reactions and physical processes. The fast and minimally invasive optical methods allow to use image processing-based information in process control system. Such approach gives non-delayed and spatially selective additional information about the ongoing combustion process. The still and apparent position of flame is the result of dynamic equilibrium between the local flame propagation speed and the speed of the incoming fuel mixture. It allows assuming that the shape of a flame can be an indicator of the

As a result, the relationship between the parameters describes the variation of the flame and the temperature of the exhaust gas in the chamber or the amount of air flow in the secondary factor. Thus, if the temperature is slowly varying value, having an inert nature, the reasonable approach is including a single or a set of the image parameters that would provide fast information to the synthesis of the controller. Due to the incomplete knowledge about the control plant or various changes in its performance, the control system with fixed parameters is insufficient. Then, it is recommended to use the adaptive control approach. The required knowledge of the complex nonlinear object may be achieved using different methods but due to the process, they ought to be robust and secure. It seems to be a very interesting application

Every detailed aspect of the real process cannot be adequately contemplated by mathematical models. Simplifying assumptions have to be made, especially due to the control purposes, where models with simple structures and sufficiently small size have to be used regarding to the available control techniques and real-time considerations. Therefore, mathematical control

Majority of modern control techniques need a control model of the plant with fixed structure and parameters, which is used throughout the design stage. For an exact description of the plant (neglecting external disturbances), processes could be controlled by an open-loop controller. However, feedback is necessary for process control because of the external perturba-

The objective of robust control is to design controllers which preserve stability and performance in spite of the modelling inaccuracies or uncertainties. Although the use of feedback contemplates the

models can only describe the dynamics of the process in an approximate way.

essary to measure its key parameters.

236 Adaptive Robust Control Systems

for robust adaptive control algorithms.

2. Process models and uncertainties

tions and model inaccuracies in all real processes.

combustion process, occurring under certain conditions.

Depending on the technique used to design the controllers, there are different approaches in modelling uncertainties. The most extended techniques are frequency response uncertainties and transfer function parametric uncertainties. Most of the cases assume that the plant can be exactly described by one of the models belonging to a family. That is, if the family of models is composed of linear models, the plant is also linear. In case of model predictive control (MPC) approach, the uncertainties can be defined about the prediction capability of the model.

Frequency uncertainties are usually described by a band around nominal frequency response. The plant frequency response is presumed to be included in the band. In case of parametric uncertainties, each coefficient of the transfer function is presumed to be bounded by uncertainties limit. The plant is then presumed to have a transfer function with parameters within the uncertainty set. There is an assumption that the plant is linear with a frequency response within the uncertainty band for the first case and the plant is linear and of the same order as that of the family of models for the case of parametric uncertainties.

The control models in MPC are used to predict what is going to happen: future trajectories. The appropriate way to describe uncertainties in this context seems to be the model (or a set of models) that instead of generating a future trajectory may also generate the band of trajectories in which the process of trajectory will be included when the same input is applied, in spite of uncertainties. In case of availability of good process model, this band is narrow, and the uncertainty level is low.

The most general way of posing problem in MPC considers a process whose behaviour is dictated by the equation:

$$y(t+1) = f\left(y(t), \ldots, y\left(t - n\_y\right), \ u(t), \ldots, u(t - n\_u), \ z(t), \ldots, z(t - n\_z), \psi\right) \tag{1}$$

where y(t)∈Y and u(t)∈ U are n and m vectors of outputs and inputs, ψ∈ Ψ is a vector of parameters, possibly unknown, and z(t) ∈Z is a vector of possibly random variables.

Consider the model or family of models, for the process described by:

$$\hat{y}(t+1) = \hat{f}(y(t), \dots, y(t-n\_{\text{na}}), \ u(t), \dots, u(t-n\_{\text{nb}}), \ \Theta) \tag{2}$$

where <sup>b</sup>y tð Þ <sup>þ</sup> <sup>1</sup> is the prediction of output vector for instant <sup>t</sup> + 1 generated by the model <sup>b</sup><sup>f</sup> is a vector function, usually simplification of f, nna and nnb are the number of past outputs and inputs considered by the model and θ∈ Θ is a vector of uncertainties about the plant. Variables that are although influencing the plant dynamics are not considered in the model due to the necessary simplifications or for the other reasons are represented by the z(t).

The dynamics of the plant in (1) are completely described by the family of models (2) if for any y(t), ⋯, y(t � ny)∈Y, u(t), ⋯, u(t � nu) ∈ U, z(t), ⋯, z(t � nz)∈Z and ψ ∈ Ψ, there is a vector of parameters θi∈ Θ such that:

$$\begin{aligned} f\left(y(t),\ldots,y(t-n\_y),\ u(t),\ldots,u(t-n\_u),\ z(t),\ldots,z(t-n\_z),\psi\right) \\ = \widehat{f}\left(y(t),\ldots,y(t-n\_{\text{m}}),\ u(t),\ldots,u(t-n\_{\text{mb}}),\ \theta\right) \end{aligned} \tag{3}$$

In case quadratic cost function of θ for each value of u, used Hessian matrix (see [6]) can be assured to be positive definite. This implies that the function is convex and there are no local optimal solutions different from the global optimal solution. One of the main problems of nonlinear programming algorithms, the presence of local minima, is avoided. Such an approach can be prohibitive for real-time applications with long costing and control horizons. Of course, the problem gets more complex when the uncertainties on the input and output

Adaptive Robust Control of Biomass Fuel Co-Combustion Process

http://dx.doi.org/10.5772/intechopen.71576

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Campo and Morari have proved that the ∞-∞ norm reduces min-max problem; therefore, it requires fewer computation and can be solved using standard algorithms. Although ∞-∞ norm seems to be appropriate in terms of robustness, it is only concerned with maximum deviation and the rest of the behaviour is not taken explicitly into account. Other types of norms are more adequate for measuring the performance. Alwright [8] has shown that this method can

To guarantee robustness in MPC is imposing the stability conditions for all possible realisations of uncertainties [9]. The key ingredients of the stabilising MPC are a terminal set and a terminal cost. The terminal state (i.e. the state at the end of the prediction horizon) is forced to reach a terminal set that contains the steady state. An associated terminal cost is

The robust MPC consists of finding a vector of future control moves such that it minimises an objective function (including a terminal cost satisfying the stability conditions [9]) and forces the final state to reach the terminal region for all possible values of uncertainties, that is:

where the terminal set Ω<sup>T</sup> is usually defined by a polytope ΩT≜ {x : RTx ≤ rT}. The inequality Ru ≤ r + Vx(t) contains the operating constraints. If there are operating constraints on the

In general, industrial processes are nonlinear, but most of MPC applications are based on the

• Linear models provide good results when the plant is operating in the neighbourhood of the operating point. In the MPC appliances, the objective is to keep the process around the stationary state rather than perform frequent changes from one operating point to another, and therefore, a precise linear model is enough. The use of linear model together with a quadratic objective function gives rise to a convex problem whose solution is well studied and implemented in many commercial products. The existence of algorithms that can guarantee a convergent solution in a time shorter than sampling time is crucial in

Ru ≤ r þ Vx tð Þ x tð Þ þ N ∈ Ω<sup>T</sup>

(5)

<sup>u</sup><sup>∈</sup> <sup>U</sup> Jxt ð Þ ð Þ; <sup>u</sup> subject to∀θ<sup>∈</sup> <sup>Θ</sup>

process output and/or state, vector r is an affine function of the uncertainties θ.

• The identification of a linear model based on process data is relatively easy.

parameters are considered.

be extended to the 1-norm.

added to the cost function.

2.1. Robustness by imposing constraints

min

use of linear models. There are two reasons for this:

processes with the great number of variables.

The way in which uncertainties parameter θ and its domain Θ are defined mainly depends on the structures of f and bf and on the degree of certainty about the model. The following are the most popular structures in MPC approaches [5]:


The objective of prediction control is to compute the future control sequence u(t), u(t + 1), ⋯, u(t + Nu) in such way that the optimal j step ahead predictions y(t + j| t) are driven close to w(t + j) for the prediction horizon. The way in which system approach the desired trajectories is indicated by the function J which depends on the present and future control signals and uncertainties. Usually, for the stochastic type of the uncertainty, the function J minimization for the most expected situation, supposing that the future trajectories are going to be the future expected trajectories. In case bounded uncertainties are considered explicitly, bounds on the predictive trajectories can be calculated and more robust control would be obtained when controller tried to minimise the objective function for the worst situation, by solving:

$$\min\_{\mu \in \mathcal{U}} \max\_{\theta \in \Theta} J(\mu, \theta) \tag{4}$$

The function to be minimised is the maximum of the norm that measures how well the process output follows the reference trajectories.

Different types of norms can be used for this purpose, e.g. quadratic cost function [6], ∞-∞ norm [7] or 1-norm [8].

In case quadratic cost function of θ for each value of u, used Hessian matrix (see [6]) can be assured to be positive definite. This implies that the function is convex and there are no local optimal solutions different from the global optimal solution. One of the main problems of nonlinear programming algorithms, the presence of local minima, is avoided. Such an approach can be prohibitive for real-time applications with long costing and control horizons. Of course, the problem gets more complex when the uncertainties on the input and output parameters are considered.

Campo and Morari have proved that the ∞-∞ norm reduces min-max problem; therefore, it requires fewer computation and can be solved using standard algorithms. Although ∞-∞ norm seems to be appropriate in terms of robustness, it is only concerned with maximum deviation and the rest of the behaviour is not taken explicitly into account. Other types of norms are more adequate for measuring the performance. Alwright [8] has shown that this method can be extended to the 1-norm.
