Semi-Mechanistic Modeling for Control
During last years, a number of nonlinear control system design techniques, including differential geometric-based control, reference system synthesis and nonlinear model predictive control, have been developed to control chemical processes. Most of nonlinear control methods are based on the first-principle model of the controlled process. The disadvantage of nonlinear model-based control techniques is that they require a detailed first-principle model which is not always available. Sometimes, there are situations when some parts of a white-box model are in fact not known. For such situations, this chapter proposes the application of semi-mechanistic modeling approach where the difficult part of the white-box model is represented by a feedforward Artificial Neural Network. The proposed approach will be illustrated through an exam-ple of the temperature control of a continuous stirred tank reactor, where the controller is based on Globally Linearizing Control scheme.
This chapter is organized as follows. Firstly, Sect. 5.1 surveys Globally Lin-earizing Control, then the identification and application of semi-mechanistic neural network model is presented in Sect. 5.2, and finally Sect. 5.3 gives the application example.
5.1 Globally Linearizing Control
5.1.1 Introduction to Globally Linearizing Control
In recent years, differential geometry has been used as an effective tool for the analysis and design of nonlinear control systems. An overview of geometrical methods for process control is given by Kravaris and Kantor [30], Bequette [70], Isidori [71], Nijmeijer and Schaft [72]. These approaches are based on the feedback linearization of nonlinear systems. Basically, two approaches can be distinguished:
• The input-output relation is linear (I-O feedback linearization)
• The input-state relation is linear (I-S feedback linearization)
These approaches can be applied for multivariable processes [73, 74] or sys-tems with input constraints [75, 76]. Furthermore, the robustness of feedback linearization control has been also discussed [77–80].
ysp Linear controller
State feedback compensator
v Process Output
map u
x
y +
-Fig. 5.1. Scheme of Globally Linearizing Control
Globally Linearizing Control (GLC) is an input-output feedback lineariza-tion control strategy where the input-output feedback linearizalineariza-tion generates a static feedback control law which linearizes the input-output behavior of the controlled system, and a linear controller controls the linearized system, see Fig. 5.1. GLC incorporates a nonlinear state-space model directly within the control algorithm. It means that the nonlinear, multivariable and time-dependent model comprise the dominant structure of the controller. For a practical implementation of GLC, two problems have to be addressed:
• One problem is the stability of the internal dynamics when the relative degree is less than the system order.
• The other problem is how to acquire a model of the plant that leads to a reasonably useful result.
This chapter concentrates on the second problem, and suggests the application of semi-mechanistic modeling approach for GLC. First let us see how the control law of GLC controller can be derived.
5.1.2 GLC for Affine Systems
In case of continuous-time systems, there is a vast amount of theory for de-veloping feedback linearization of affine systems, where the u input of the system appears in linear form in the state space equation. Such affine systems are given by
˙
x =f(x) +g(x)u
y =h(x), (5.1)
where x = [x1, . . . , xn]T is the state vector, u is the manipulated input and y is the controlled output of the system. In general, f, g and h are smooth vector fields.
5.1 Globally Linearizing Control 71 The objective of input-output linearization is to obtain a nonlinear control law in the form
u=p(x) +q(x)v, (5.2)
in such a way that the obtained system, shown in Fig. 5.1, be linear. So the input-output linearization results in a linear transfer between v and y
y(s)
v(s) = 1
βrsr+· · ·+β1s+β0, (5.3) where r is the relative degree of the nonlinear system. This relative degree at the point x0 is defined by the integer r which satisfies
LgLif−1h(x) = 0 ∀ i < r and x near x0
LgLr−1f h(x) = 0 ∀ x near x0, (5.4) where Lg andLf are Lie derivatives defined as
Lfh(x) = n i=1
fi(x)∂h
∂xi
(x) (5.5)
and
LgLfh(x) = n i=1
gi(x)∂Lfh
∂xi (x), (5.6)
while the higher order Lie derivatives can be written as Lαfh(x) =Lf
Lαf−1h
. (5.7)
The time derivatives of the output of the system can be expressed as algebraic functions of these Lie derivatives
dy
dt = Lfh(x) ...
dr−1y
dtr−1 = Lrf−1h(x) dry
dtr = Lrfh(x) +LgLrf−1h(x)u. (5.8) Equation (5.8) shows that the relative degree (or relative order) represents how many times the output must be differentiated with respect to the time to recover explicitly the input u.
Globally Linearizing Control (GLC) is an input-output linearization tech-nique for processes of arbitrary relative degree. From the above equations, the feedback control law can be expressed as
u=
v− r
k=0
βkLkfh(x)
βrLgLr−1f h(x) . (5.9) Figure 5.1 illustrates that the scheme of GLC contains a linear feedback con-troller which controls the linearized system. In most process control applica-tions, the objective is to maintain the output at a non-zero setpoint despite model error and unmeasured disturbances. Consequently, the linear controller is usually a PI or a PID controller [81]. (It should be noted that, in theory, if the first-principle model was ’perfect’ and it was possible to measure every variable and disturbance accurately, then this linear controller would not be absolutely necessary, instead the input of feedback state compensator would be the setpoint:v=ysp andβ0 = 1.) If, for example, a PI controller is applied
where the gain Kc and the time constant Ti are additional controller tuning parameters, then the complete GLC control law yields the following closed-loop transfer function for setpoint-changes:
y(s) Certainly, one can use other linear controller, especially for systems that have high relative degree. If the relative order of the system is 1 or 2, the PID controller is a good choice; but for higher relative orders, it may be more useful to design the linear controller directly to the linearized system (e.g. by direct synthesis method).