Closed-loop state equations

We now finally launch into the material on controllers. State space representation is more abstract and it helps to understand controllers in the classical sense first. We will come back to state space controller design later. Our introduction stays with the basics. Our primary focus is to learn how to design and tune a classical PID controller. Before that, we first need to know how to set up a problem and derive the closed-loop characteristic equation. 

We can now state the problem in more general terms. Let us consider a closed-loop characteristic equation 1 + KCG0 = 0, where KCG0 is referred to as the "open-loop" transfer function, G0l- The proportional gain is Kc, and G0 is "everything" else. If we only have a proportional controller, then G0 = GmGaGp. If we have other controllers, then G0 would contain... 

Example 7.7 Consider installing a PI controller in a system with a first order process such that we have no offset. The process function has a steady state gain of 0.5 and a time constant of 2 min. Take Ga = Gm = 1. The system has the simple closed-loop characteristic equation ... 

The two nonlinear ordinary differential equations can be linearized around the steady-state values of the reactor compositions zA and zs. Laplace transforming gives the characteristic equation of the system. It is important to remember that we are looking at the closed-loop system with control structure CS2 in place. Therefore Eq. (2.13) is the closed-loop characteristic equation of the process ... 

This chapter has provided a brief overview of the application of optimal control theory to the control of molecular processes. It has addressed only the theoretical aspects and approaches to the topic and has not covered the many successful experimental applications [33, 37, 164-183], arising especially from the closed-loop approach of Rabitz [32]. The basic formulae have been presented and carefully derived in Section II and Appendix A, respectively. The theory required for application to photodissociation and unimolecular dissociation processes is also discussed in Section II, while the new equations needed in this connection are derived in Appendix B. An exciting related area of coherent control which has not been treated in this review is that of the control of bimolecular chemical reactions, in which both initial and final states are continuum scattering states [7, 14, 27-29, 184-188]. 

This equation, of course, contains information regarding stability, and as it is written, implies that one may match properties on the LHS with the point (-1,0) on the complex plane. The form in (7-2a) also imphes that in the process of analyzing the closed-loop stability property, the calculation procedures (or computer programs) only require the open-loop transfer functions. For complex problems, this fact eliminates unnecessary algebra. We just state the Nyquist stability criterion here.1... 

The loci typically drawn out by these equations as 0 varies are shown in Fig. 4.8(a). For y closed loop near the origin. Inside this loop, the stationary state is an unstable node. The larger outer loop separates stable focal character (inside curve) from stable nodal states (outside curve). As y increases beyond i the small inner loop shrinks to zero size the outer loop still exists. Stable focal character exists over some values of the parameters n and k for any value of y. 

Once the system s interactions have been removed, it can be placed under closed loop control as indicated in Figure 1. For illustrative purposes, a simplified form of equation 7 was selected. For the reference steady-state to be optimal, the first three terms are zero. The coefficient e was further assumed negligible compared to the coefficient 6. Actual plant economics will determine the coefficients of equation 7. 

Then one can state the following criterion for the stability of a closed-loop system A feedback control system is stable if all the roots of its characteristic equation have negative real parts (i.e., are to the left of the imaginary axis). If any root of the characteristic equation has a real positive part (i.e., is on or to the right of the imaginary axis), the feedback system is unstable. 

The stability criterion stated above secures stable response of a feedback system independently if the input changes are in the set point or the load. The reason is that the roots of the characteristic equation are the common poles of the two transfer functions, GSp and Gioad, which determine the stability of the closed loop with respect to changes in the set point and the load, respectively. 

The behavior of the fugacity shown in Figure 10.1 is representative, but it is not the only way that volumetric equations of state can produce changes in f x with changes in state. For example, at 10 bar, but at temperatures well below 275 K, the liquid branch of extends over all Xp and it is the vapor branch that can form a closed loop. At still other states, both the liquid and the vapor branches extend over all Xp and no loop (open or closed) occurs at all. These possibilities are discussed briefly in 8.4.2 and in more detail elsewhere [1]. The lesson here is that even simple cubic equations of state can provide relatively complicated forms for the fugacity, forms sufficiently complicated to satisfy the phi-phi equations (10.1.3) for phase equilibria. 

The closed-loop (controlled) steady-state equation is given by ssQ/ + - (Qss = VkCA +... 

The solution of Eqs (16)-(21) has physical significance if Da exceeds a critical value. This is the turning point of the Da - za,2 map, which represents a fold bifurcation of the mass balance equations (Fig. 3b). Here two steady states solutions are bom. The upper state (high za,2 and low-conversion) is closed-loop unstable. The instability, which can be proven on steady state considerations only, is independent of the d5mamics. Therefore, the fold point represents a feasibility and stability boundary. When zp =0 the coordinates of the fold point are given by ... 

A number of authors have established balance equations for these flows, using simplifying hypotheses. For examples, see Barral et al. [BAR 03] and Gascon, Dudeck and Barral [GAS 03] in the steady-state case, or Boeuf and Garrigues [BOE 98] for an initial approach to the low-frequency oscillations in plasma thrusters and, more recently, Barral and Miedzik [BAR 11] for a more elaborate model of these phenomena in the context of a closed-loop study of Hall-effect accelerators. Also see Dabiri et al. [DAB 13]. 

If ysp = Ku[) at the final steady state and the gain K is known, the appropriate preload expression is = yspIK. Then the closed-loop equation becomes... 

How do we find the best (t) that minimizes (5.144) We describe first a direct approach for an open-loop problem in which we compute the entire optimal trajectory for a specific initial state. Then, we outline an alternative dynamic programming approach that turns the integral equation (5.144) into a corresponding time-dependent partial differential equation, and generates a closed-loop optimal feedback control law. 

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