3. Robust control of biomass fuel co-combustion

The design of stabilising controllers for nonlinear controllers with known and unknown constant parameters has significant improvement within the last decades. It involves design techniques such as adaptive feedback linearization [10–12], adaptive backstepping [13–15], robust Lyapunov functions (CLFs and RCLFs) [16–19], nonlinear damping and swapping [14, 20] as well as switching adaptive control [21, 22]. They are applicable for globally stabilising controllers for single input feedback linearizable systems [10, 11, 22] and parametric-strict-feedback systems [13–15]. Despite this, the problem of adaptive control of a big class of nonlinear systems still remains unexplored.

The procedure presented in Ref. [23] for designing robust adaptive controllers for a large class of multi-input nonlinear systems with exogenous bounded input disturbances results in approach that combines the theory of control Lyapunov functions and the switching adaptive controller to overcome the problem of computing the control law in the case where estimation model becomes uncontrollable.

It is important that the control law depends on estimates of the Lie derivative LgV, which depends both on the system vector-fields and robust control Lyapunov function (RCLF) V. The class of systems for which the proposed approach is applicable can be characterised by the following assumption: LgV depends linearly on unknown constant parameters, where g denotes the input vector field and V is CLF (RCLF) for the system.

Contrary to the classical adaptive approach where the control law depends on estimates of the system vector-fields, in the presented case, it depends on estimates of the RCLF term [23]. LgV depends on both system vector-fields and RCLF function V.

On the one hand, the main advantage of such approach is that Lyapunov inequalities relating to the parameter estimation errors and the time derivative of the RCLF are easy to handle. But on the other hand, the designed controllers depend critically on the knowledge of LgV. In case of adaptive versions of such controllers, there is the risk of failure regarding to the fact that the estimate of LgV may have a different sign at certain times than the actual LgV. Similarly, when the estimate of LgV is close to zero, the actual LgV is not. Such divergences imply uncontrollability of the estimation model, even if the actual model is not.

To overcome these problems, the switching control law is used, which is modified version of control law presented in Ref. [22]. Such control law approximately switches between two adaptive controllers, which have the following properties: (1) both controllers behave approximately the same in the nonadaptive case and (2) when one of these controllers becomes nonimplementable, the other one is implementable.

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 systems and can be expressed as:

$$\dot{\mathbf{x}} = F\mathbf{x} + G\left[\mathfrak{S}\_1^\tau l\_0(\mathbf{x}) + \mathfrak{S}\_2^\tau l\_1(\mathbf{x})\right] \mathbf{u},\tag{6}$$

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 matrices G, ϑ<sup>2</sup> are known.

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

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

The design of stabilising controllers for nonlinear controllers with known and unknown constant parameters has significant improvement within the last decades. It involves design techniques such as adaptive feedback linearization [10–12], adaptive backstepping [13–15], robust Lyapunov functions (CLFs and RCLFs) [16–19], nonlinear damping and swapping [14, 20] as well as switching adaptive control [21, 22]. They are applicable for globally stabilising controllers for single input feedback linearizable systems [10, 11, 22] and parametric-strict-feedback systems [13–15]. Despite this, the problem of adaptive control of a big class of nonlinear

The procedure presented in Ref. [23] for designing robust adaptive controllers for a large class of multi-input nonlinear systems with exogenous bounded input disturbances results in approach that combines the theory of control Lyapunov functions and the switching adaptive controller to overcome the problem of computing the control law in the case where estimation

It is important that the control law depends on estimates of the Lie derivative LgV, which depends both on the system vector-fields and robust control Lyapunov function (RCLF) V. The class of systems for which the proposed approach is applicable can be characterised by the following assumption: LgV depends linearly on unknown constant parameters, where g denotes

Contrary to the classical adaptive approach where the control law depends on estimates of the system vector-fields, in the presented case, it depends on estimates of the RCLF term [23]. LgV

On the one hand, the main advantage of such approach is that Lyapunov inequalities relating to the parameter estimation errors and the time derivative of the RCLF are easy to handle. But on the other hand, the designed controllers depend critically on the knowledge of LgV. In case of adaptive versions of such controllers, there is the risk of failure regarding to the fact that the estimate of LgV may have a different sign at certain times than the actual LgV. Similarly, when the estimate of LgV is close to zero, the actual LgV is not. Such divergences imply uncontrolla-

To overcome these problems, the switching control law is used, which is modified version of control law presented in Ref. [22]. Such control law approximately switches between two adaptive controllers, which have the following properties: (1) both controllers behave approximately

crucial to the closed loop stability that a linear model is not sufficient.

3. Robust control of biomass fuel co-combustion

the input vector field and V is CLF (RCLF) for the system.

depends on both system vector-fields and RCLF function V.

bility of the estimation model, even if the actual model is not.

systems still remains unexplored.

240 Adaptive Robust Control Systems

model becomes uncontrollable.

$$\begin{aligned} \dot{\mathbf{x}}\_1 &= \mathbf{x}\_{i+1} + \boldsymbol{\theta}^T f\_i(\mathbf{x}\_i, \dots, \mathbf{x}\_{i+1}) \ 1 \le i \le n-1 \\ \dot{\mathbf{x}}\_n &= \boldsymbol{\theta}^T f\_n(\mathbf{x}) + \left[ \boldsymbol{\theta}^T g\_{n2}(\mathbf{x}) + g\_{n1}(\mathbf{x}) \right] \boldsymbol{u}' \end{aligned} \tag{7}$$

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], guarantee the global stability only if the input vector field θ<sup>T</sup> ɡn2(x) + ɡn1(x) is independent θ and the functions fi are independent of xi + 1.

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

$$
\dot{\mathbf{x}} = f(\mathbf{x}) + g(\mathbf{x})\boldsymbol{\mu} + g\_w(\mathbf{x})\mathbf{w},\tag{8}
$$

where x∈R<sup>n</sup> , u∈Rmand w ∈R<sup>k</sup> denote vectors of system states, control inputs and disturbances 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, t ≥ 0 and is assumed to be any general unknown bounded continuous time function.

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 to robust adaptive controllers due to [28, 29], but this requires a large number of fuzzy rules to achieve satisfactory approximator. To cope this problem, in [30–33], Takagi-Sugeno fuzzy approximator (TSFA) is involved. The invertible fuzzy approximated input matrix needs to be imposed in case of MIMO systems [34–36]. Furthermore, some examples of combining fizzy adaptive and sliding mode control can be found in [37, 38]. The examples of robust fuzzy adaptive control schemes with guaranteed H∞ control performance for a specific class of MIMO nonlinear systems can be found in [39–41].
