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Fro the output to the control signals. This terminology was introduced by [24] who analyzed these systems systems carefully.
A very nice property of systems with two degrees of freedom is that the problem of set point response can be
separated from the problems of robustness and disturbance rejection. Referring to Figure 2 we will first design
a feedback by compromising between disturbance attenuation and robustness. When this is done we will then
design a model and a feedforward which gives the desired response to the setpoint.
There are many variations of systems with two degrees of freedom; the following quote from [24] is still valid
�Some structures have been presented as fundamentally different from the others. It has been suggested that they have virtues not possessed by others, and have been given special names�.all 2DOF configurations have basically the same properties and potentials�.�
Quantitative Feedback Theory (QFT)
Bode�s technique of dealing with gain variations were both elegant and effective. A limitation of Bode�s work was that it was limited to gain variations only. A very nice generalization of Bode�s work was done by Horowitz who extended it to arbitrary variations of a process transfer function. He characterized model uncertainty by sets of amplitudes and phase for each frequency called templates. Horowitz also developed graphical design techniques to design feedback systems that were robust to these types of disturbance. He used a system configuration with two degrees freedom to deal with set point responses. Horowitz design technique called quantitative feedback theory (QFT) is described in several books; see [24] and [26]. It has been applied successfully to a wide range of problems.
Summary
In this section we have reviewed classical control theory with a focus on model uncertainty and robustness. It is worthwhile to note that model uncertainty was a key motivation for introducing feedback and that classical control theory had very effective ways of dealing with uncertainty both qualitatively and quantitatively.
Process uncertainty could be described very easily as a variation in the process transfer function with the caveat that the disturbances do not change the number of right half plane poles of the system.
The theory has given important concepts and tools such as the transfer function, Nyquist�s curve, Bode diagrams, bode�s integrals and Bode�s integrals and Bode�s ideal loop transfer function. Robustness measures such as amplitude and phase margins and the maximum sensitivities were also introduced. Bode�s ideal loop transfer function is probably the first design method that addressed robustness explicitly. Horowitz quantitative feedback theory is a continuation of this idea.
2. State-Space Theory
The state-space theory represented a paradigm shift which led to many useful system concepts and new methods for analysis and design. The system was represented by differential equations instead of transfer function. For linear systems the standard model used was 
where is the input the output and is the state. The uncertainty is represented by the disturbances v and by variations in the elements of the matrices A,B and C. the disturbance e and u were typically described as stochastic processes, see [20] and [4]
The control problem was formulated as to minimize the criterion
Since the equations are linear with stochastic disturbances and the criterion is quadratic the problem was called the linear quadratic Gaussian control problem (LQG). The solution to the control problem is given by
This control law has a very nece interpretation as feedback from the error which is the difference between the ideal states and the estimated states . the estimated states are given by the kalmn filter. Controllability and absorbability are key conditions for solving the problem.
There are many other design methods based on the state-space formulation which gives controllers with the structure (22) for example pole placement. They differ from the LQG method in the sense that other techniques are used to obtain the matrices K and L.
In figure 8 we show a block diagram of the controller obtained from LQG theory in the figure we have also used a system configuration with two degrees of freedom. The system has a very attractive structure. The observer or the kalman filter delivers an estimate has a very attractive structure. The observer or the kalman filter delivers an estimate of the state based on a model of the system and the input and output signals of the system. Notice that the state may also have components that represent the disturbances. There is a feedback from the state deviations of the estimated state from its
Desired value set point following is obtained by the usual two degree of freedom
configuration.
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