Unity feedback

A unity feedback computer control system, has an open-loop pulse transfer function... 

A unity feedback continuous control system has a forward-path transfer function... 

Since H =, the system has unity feedback, and the closed-loop transfer function and step response is given by... 

In our examples, we will take Gm = Ga = 1, and use a servo system with L = 0 to highlight the basic ideas. The algebra tends to be more tractable in this simplified unity feedback system with only Gc and Gp (Fig. 5.6), and the closed-loop transfer function is... 

Example 5.6 Provide illustrative closed-loop time response simulations. Most texts have schematic plots to illustrate the general properties of a feedback system. This is something that we can do ourselves using MATLAB. Simulate the observations that we have made in previous examples. Use a unity feedback system. 

In these statements, we have used feedback () to generate the closed-loop function C/R. The unity feedback loop is indicated by the 1 in the function argument. Try first with Kc = 1, and Tj with values 10, 1, and 0.1. Next, select Xj = 0.1, and repeat with Kc = 0.1, 1, 5, and 10. In both... 

If you cannot follow the fancy generalization, think of a simple problem such as a unity feedback loop with a PD controller and a first order process. The closed-loop characteristic equation is... 

Example 8.14. Designing phase-lead and phase-lag compensators. Consider a simple unity feedback loop with characteristic equation 1 + GCGP = 0 and with a first order process... 

In the simplest scenario, we can think of the equation as a unity feedback system with only a proportional controller (i.e., k = Kc) and G(s) as the process function. We are interested in finding the roots for different values of the parameter k. We can either tabulate the results or we can plot the solutions s in the complex plane—the result is the root-locus plot. 

Here are some useful suggestions regarding root locus plots of control systems. In the following exercises, we consider only the simple unity feedback closed-loop characteristic equation ... 

With gain or phase margin, calculate proportional gain. Can also estimate the peak amplitude ratio, and assess the degree of oscillation. The peak amplitude ratio for a chosen proportional gain. Nichols chart is usually constructed for unity feedback loops only. 

A unity feedback control loop consists of a non-linear element N and a number of linear elements in series which together approximate to the transfer function ... 

Based on the linearized models around the equilibrium point, different local controllers can be implemented. In the discussion above a simple proportional controller was assumed (unity feedback and variable gain). To deal with multivariable systems two basic control strategies are considered centralized and decentralized control. In the second case, each manipulated variable is computed based on one controlled variable or a subset of them. The rest of manipulated variables are considered as disturbances and can be used in a feedforward strategy to compensate, at least in steady-state, their effects. For that purpose, it is t3q)ical to use PID controllers. The multi-loop decoupling is not always the best strategy as an extra control effort is required to decouple the loops. 

The analysis of feedback control systems can often be facilitated by conversion to an equivalent unity feedback system, i.e. a feedback loop in which the feedback path is represented by a steady-state gain of unity. There are two principal cases to consider. 

Conversion to Unity Feedback when the Transfer Function in the Feedback Part of the Loop is Represented by a Steady-State Gain K... 

Consider a simple feedback loop (Fig. 7.3a) in which the feedback path consists of elements which approximate to a steady-state gain K (Fig. 7.37). In this instance, the equivalent unity feedback loop is determined by placing 1 IK in the set point input to the main loop and compensating for this by adding an additional factor K in the forward part of the loop prior to the entry of the load disturbance, as in Fig. 7.38. It is easy to confirm that the standard closed loop transfer functions and are the same for the block diagrams in Figs 7.37 and 7.38. 

Fig. 7.38. Unity feedback system equivalent to the control loop shown in Fig. 7.37 when the dynamics of the feedback path are not significant...

In the unity feedback system shown in Fig. 7.59 (set-point following case), it is assumed that all the dead time is contained within the process and is represented by the transfer function G3(j), where ... 

Fig. 7.86. Equivalent unity feedback block diagram of control system illustrated in Fig. 7.42 including non-linear element...

The numerator and denominator polynomials of the closedloop servo transfer functions are formed using the [numcl,dencl] = cloop(numol,denol, - 1) command. This command converts an openloop transfer function into a closedloop transfer function, assuming negative unity feedback. A block diagram of a unity feedback system is given in Fig. 11.22. 

Calculate unity-feedback closedioop discrete transfer function [ncl,dclj-clo() Xnurud kc,dend. - i) printsys(ncl,dci z ... 

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