Input-output feedback linearization

Since, 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. 

Zames, G. (1966) On the Input-Output Stability of Time-Varying Non-Linear Feedback Systems. Parts I and II. IEEE Trans, on Automat. Contr., AC-11(2 3), pp. 228-238, 465-476. 

In order to briefly introduce the basic principle of the feedback linearizing control, consider the following Single Input Single Output (SISO) nonlinear model ... 

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... 

This is a well known result from linear systems theory. The function Sp is called the sensitivity function as it gives the relative change in the input-output sensitivity due the presence of feedback. The index p is here used to denote that it is the sensitivity function of the process itself For the case with multivariable recycle, we get identical results, except that the sensitivity function in this case is a matrix Sp = I — G22 s)Gr s)). ... 

The transfer matrices Ta and Tb are computed from the transfer matrices of the plant and the base functions in the series expansion of the Youla parameter. Eq. (17) is a complete and convex description of all possible input-output behaviours of stable linear closed loops with the given plant. x,tj) depends on the choice of the external inputs Mf) and the external outputs z(0 as well as on the signals used for feedback and the available control inputs. 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

State estimators are basically just mathematical models of the system that are solved on-line. These models usually assume linear DDEs, but nonlinear equations can be incorporated. The actual measured inputs to the process (manipulated variables) are fed into the model equations, and the model equations are integrated. Then the available measured output variables are compared with the predictions of the model. The differences between the actual measured output variables and the predictions of the model for these same variables are used to change the model estimates through some sort of feedback. As these differences between the predicted and measured variables are driven to zero, the model predictions of all the state variables are changed. 

Non-linearity with Memory. The AR-MNL model is clearly somewhat restrictive in that most distortion mechanisms will involve memory. For example an amplifier with a non-linear output stage will probably have feedback so that the memoryless non-linearity will be included within a feedback loop and the overall system could not be modelled as a memory less non-linearity. The general NARMA model incorporates memory but its use imposes a number of analytical problems. A special case of the NARMA model is the NAR (Non-linear AutoRegressive) model in which the current output x[w] is a non-linear function of only past values of output and the present input s [n ]. Under these conditions equation 4.27 becomes ... 

The controllability tools presented in here are based on the theory of linear systems, which is valid for relatively small disturbances around the stationary state. A non-linear approach, more suited for investigating the effect of large variations, will be developed in Chapter 13. The chapter starts with a brief introduction in process dynamics, followed by the properties of linear systems. The controllability analysis begins with SISO (single input/single output) systems and reviews the major concepts in feedback control. Then, the analysis is extended to MIMO (multi input/multi output) systems, with emphasis on decentralised control systems (multi SISO control loops), which is the most encountered in plantwide applications. 

Here we adopt the following definition valid for linear systems a system is stable if bounded input variations produce bounded output variations as t —>oo otherwise the system is unstable (Ogunnaike Ray, 1994). One of the main issues in designing feedback controllers is stability. Let consider the response of a closed-loop system under proportional control, as deviation in outputy vs. time (Fig. 12.5). If the controller gain is moderate then y goes to zero after some oscillations. By increasing gain... 

In this work, the above mentioned simple scheme to implement the oscillator is considered. The non-linear feedback element is connected to the biological filter to implement the oscillator. In this way, only the input and output of the biological filter are manipulated to perform the test allowing a low intrusion in the structure. 

Rule 1 When the op-amp output is in linear range (for example, when there is negative feedback between output and negative input terminal), the two input terminals are at the same voltage. 

The feedback loop sees Tt as its input and W, as its output. But within the loop, Ti is subtracted from, and Wp multiplied by, the controller output. Subtraction is a linear operation, so gain is not changed therein but multiplication is nonlinear, causing feedback gain to vary directly with flow Correct loop-gain adaptation cannot be achieved if the feedback is introduced in any other place. If the output of the feedback controller were to set K, then feedback gain would vary both with Wp and with — Ti. But process gain does not vary with... 

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