Input-Output Linearization

in process control, input-output linearization techniques are usually preferred to state-space approaches, mostly due to the higher complexity of the latter, in the following, only input-output feedback linearization basic concepts are briefly reviewed. 

The same definition applies for Lgh, where the function g plays the same role as / in (5.8). Of course, higher-order Lie derivatives can be defined in a recursive way, e.g., 

If the relative order y, of y with respect to u is l,1 then Lgh 0, and the procedure stops here. Otherwise, the procedure is iterated until LgL xh 0, i.e., until the derivative of order y, is computed  

---

M.A. Henson and D.E. Seborg. Input-output linearization of general non-linear process. A.I.Ch.E. Journal, 36 1753-, 1990. 

Proposition 4. Since the AD model (2) is a minimum-phase system and has a well-defined relative degree r under NOC. Then, the following input-output linearizing controller will make converge exponentially the total concentration of organic substrate St to a desired value for all t > 0... 

Proof. By implementing the input-output linearizing controller (6) to the AD model in the normal form (5), it is obtained that... 

Remark 1. From Proposition 4, the existence of an input-output linearizing control law capable to regulate exponentially the total concentration of organic substrate St in a desired value S was demonstrated. However, in order to implement this controller in practice, a perfect knowledge of the process dynamics is required. In other words, this implies that either the influent composition St,in or the process kinetics k, /j,. ) must be perfectly known. Nevertheless, this condition is difficult to satisfy in practice limiting its application. But what about if the uncertain terms can be estimated from available measurements and a control scheme with a similar structure to that of the input-output linearizing controller (6) is used. In the next section, a robust approach is proposed based in this fact. 

The GMC control response can be designed via the tuning parameters K and K2 based on the tuning curve given by [24], It should be noted that the GMC approach is a special case of the global input output linearizing control technique in which a transformed control action is chosen properly with the external PI controller. The use of Eq. (16) forces y toward its set point, ysp, with zero offset. If Eq. (15) is differentiated, and the Eq. (16) is substituted into the resulting equation, the GMC control law is... 

Several process control design methods, such as the Generic Model Control (GMC) [41], the Globally Linearizing Control (GLC) [37], the Internal Decoupling Control (IDC) [7], the reference system synthesis [8], and the Nonlinear Internal Model Control (NIMC) [29], are based on input-output linearization. 

The Generic Model Control (GMC) is a model-based control strategy developed by Lee and Sullivan in 1988 [41], It can be shown that GMC is an input-output linearization technique for processes with unitary relative order [31],... 

C. Kravaris and C.B. Chung. Nonlinear state feedback synthesis by global input-output linearization. AIChE Journal, 33 592-603,1987. 

We used the model of the fast dynamics of the system in Equation (4.36) to design a nonlinear input-output linearizing output feedback controller with integral action (Daoutidis and Kravaris 1992) for x. The controller was designed to produce the critically damped second-order response... 

We then used the model (5.51) as the basis for synthesizing an input-output linearizing controller with integral action (Daoutidis and Kravaris 1992) for the product purity xB = 1 — Cs - C6, using the condenser vapor holdup setpoint... 

The non-stiff, low-dimensional model (6.63) is ideally suited for control purposes. We used it to design an input-output linearizing controller with integral action that manipulates QH and enforces a first-order response in the Tr dynamics, namely... 

The term Cb + 71,1 dCB/dt is statically equivalent to Cb, and corresponds to requesting a first-order response in Cb when using a standard input-output linearizing controller. However, such a controller would lead to closed-loop instability, and the output requires a statically equivalent addition that would allow one to overcome this limitation. 

With the outputs y and M, using the reduced-order model (7.38), a multi-variable input-output linearizing controller with integral action (Daoutidis and Kravaris 1994) was designed for the product purity and reactor holdup, requesting a decoupled first-order response ... 

Listing C.l. Symbolic derivation of reduced-order model of the slow dynamics, and of the input-output linearizing temperature controller for the reactor-feed effluent heat exchanger system in Section 6.6... 

Kloppenburg E. and Gilles E.D., Automatic control of the simulated moving bed process for Cg aromatics separation using asymptotically exact input/output-linearization. J. of Process Control 9 (2000) pp. 41-50. 

A general input-output linear model for a single-output system with input u and output y can be written ... 

R. Marino, S. Peresada, and P. VaUgi. Adaptive input-output linearizing control of induction motors. IEEE Transaction on Autemiatic Control, 3S(2) 20 22l, 1993. 

---