2.1. Robustness by imposing constraints

f ytð Þ; …; y t � ny

238 Adaptive Robust Control Systems

most popular structures in MPC approaches [5]:

� �; u tð Þ;…; u tð Þ � nu ; z tð Þ;…; z tð Þ � nz ;<sup>ψ</sup> � �

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

• Truncated impulse response uncertainties—Suitable when the plant model is nonlinear and linear (obtained at different operating regimes), so the plant is described by a linear combination q known stable linear time-invariant plants with unknown weighting θj. • Matrix fraction description uncertainties—Frequently, the state space description is used and each of the entries of the transfer matrix is characterised by its static gain, time

obtained on the gain and time constants. However, uncertainties about the dead time are difficult to handle. If the uncertainty band about the dead time is smaller, the pure delay of the discrete-time model does not have to be changed. The fractional delay time can be modelled by the Pade expansion and the uncertainty bound of these coefficients can be calculated from the uncertainties of the dead time. It is imperfect for real-time applications due to min-max problem solving. If the uncertainties only affect polynomial matrix B, the prediction equation is an affine function of the uncertainty parameter and the resulting

• Global uncertainties—Based on assumption that all modelling errors are globalised in a vector of parameters, the process can be approximated by a linear model in the sense that all trajectories will be included in bands that depend on θ(t). If the process variables are

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:

max

The function to be minimised is the maximum of the norm that measures how well the process

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

min u ∈ U

constant and dead time. Bounds on the coefficients of matrices A(z

min-max problem is less computationally expensive.

bounded, the global uncertainties are also be bounded.

output follows the reference trajectories.

norm [7] or 1-norm [8].

<sup>¼</sup> <sup>b</sup>f ytð Þ;…; y tð Þ � nna ð Þ ; u tð Þ;…; u tð Þ � nnb ; <sup>θ</sup> (3)

�1

<sup>θ</sup><sup>∈</sup> <sup>Θ</sup> J uð Þ ; <sup>θ</sup> (4)

) and B(z

�1

) can be

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 added to the cost function.

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:

$$\min\_{u \in \mathcal{U}} J(\mathbf{x}(t), u) \text{ subject to} \\ \forall \theta \in \Theta \begin{cases} \mathbf{R} \mathbf{u} \le \mathbf{r} + \mathbf{V} \mathbf{x}(t) \\ \mathbf{x}(t+N) \in \Omega\_{\Gamma} \end{cases} \tag{5}$$

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

In general, industrial processes are nonlinear, but most of MPC applications are based on the use of linear models. There are two reasons for this:


However, the dynamic response of the resulting linear controllers is unacceptable when applied to processes that are nonlinear to varying degrees of severity. Despite the fact that in many situations the process will be operating in the neighbourhood of a steady state, and therefore a linear representation will be adequate, there are some very important situations where it does not occur. There are processes for which the nonlinearities are so severe and so crucial to the closed loop stability that a linear model is not sufficient.

the same in the nonadaptive case and (2) when one of these controllers becomes nonimplemen-

The proposed approach is significant because it constructs globally stabilising controllers for a wider class of plants than multi-input feedback linearizable systems and parametric-pure-feedback

where x∈ R<sup>n</sup>and u∈ R<sup>m</sup>denote the state and the control input vectors, F, G, ϑi, i = 1, 2 are constant unknown matrices and l0, l<sup>1</sup> are continuous matrix functions, non-singular for all x. The existing adaptive designs guarantee [10–12] closed-loop stability only if the constant

<sup>x</sup>\_ <sup>1</sup> <sup>¼</sup> xiþ<sup>1</sup> <sup>þ</sup> <sup>θ</sup>Tf <sup>i</sup> xi ð Þ ;…; xiþ<sup>1</sup> <sup>1</sup> <sup>≤</sup> <sup>i</sup> <sup>≤</sup> <sup>n</sup> � <sup>1</sup>

where θ is a vector of unknown constant parameters, x denotes the state vector of the system and fi, ɡni are continuous functions. The global stability procedures, presented in [13, 15, 20],

bances and f, g, gw are C<sup>1</sup> vector-fields of appropriate dimensions. We assume that the disturbance vector w is bounded. The control objective is to find the control input u as a function of x such that all closed-loop signals are bounded and x ! 0 as t! ∞. Since w(t) 6¼ 0,

The system (8) is robustly asymptotically stabilizable (RAS) when there exists a control law u = k (x), where k is appropriate feedback, such that the closed-loop solutions are robustly globally uniformly asymptotically stabilizable (RGUAS), according to definitions given in [16, 24].

Other approaches involve artificial intelligence methods to guarantee robust adaptive control for MIMO nonlinear systems. Fuzzy logic controllers have proven to have great potential in applications to complex or poorly modelled systems. Wang and Mendel in [25, 26] have started studies regarding fuzzy control of uncertain nonlinear systems. According to Wang [26], it is possible to find control law to achieve a stable control loop system. Chiu [27] proposed a universal fuzzy approximator for feedback cancellation, and the stability is guaranteed by Lyapunov's method. While the system is composed of tunable fuzzy sets, the approach is called Mamdani fuzzy approximation (MFA) control. Such a MFA controller is often extended

For the problem formulation, the nonlinear system of the following form is considered:

t ≥ 0 and is assumed to be any general unknown bounded continuous time function.

<sup>1</sup>l0ð Þþ <sup>x</sup> <sup>ϑ</sup><sup>τ</sup>

<sup>2</sup>l1ð Þ<sup>x</sup> u, (6)

Adaptive Robust Control of Biomass Fuel Co-Combustion Process

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

241

ɡn2(x) + ɡn1(x) is independent θ

<sup>x</sup>\_<sup>n</sup> <sup>¼</sup> <sup>θ</sup>Tf <sup>n</sup>ð Þþ <sup>x</sup> <sup>θ</sup><sup>T</sup>ɡn2ð Þþ <sup>x</sup> <sup>ɡ</sup>n1ð Þ<sup>x</sup> <sup>u</sup><sup>0</sup> (7)

x\_ ¼ f xð Þþ ɡð Þx u þ ɡwð Þx w, (8)

, u∈Rmand w ∈R<sup>k</sup> denote vectors of system states, control inputs and distur-

<sup>x</sup>\_ <sup>¼</sup> Fx <sup>þ</sup> <sup>G</sup> ϑτ

In case of the parametric-pure-feedback system, denoted by:

guarantee the global stability only if the input vector field θ<sup>T</sup>

and the functions fi are independent of xi + 1.

table, the other one is implementable.

systems and can be expressed as:

matrices G, ϑ<sup>2</sup> are known.

where x∈R<sup>n</sup>
