A problem is to control an output $\bm{y}$ of following system using an input $\bm{u}$.

where $\bm{d}$ and $\bm{d}$ denote a plant model and disturbance. An input should be designed so that it make an output track a desired trajectory and a disturbance be suppressed. For this purpose, feedforward and feedback control should be introduced. Now, let us consider the nonideality of input and observation:

where $\bm{z}$, $\bm{v}$, and $\bm{w}$ stand for a sensor output, process noise, and sensor noise. A feedback controller is constructed using a sensor output.

1-DOF control systems enable the design of one characteristic of the control system. The variables $\bm{r}$, $\bm{y}$, $\bm{d}$, $\bm{v}$, and $\bm{w}$ are the reference, the output, the disturbance, the process noise, and the observation noise. The noises $\bm{v}$ and $\bm{w}$ are the Gaussian noises, which are seeds of the noises that superimpose into the system. The blocks $\bm{C}$, $\bm{P}$, $\bm{H}_1$, and $\bm{H}_2$ denote the feedback controller, the plant, and the noise-shaping filters for the process noise and the observation noise, respectively. These noise-shaping filters determine the colors and magnitudes of the noises. The term $\bm{\Delta}_P$ represents the fluctuation of the plant dynamics due to uncertainty or perturbation on the system. The multiplicative form $\bm{P}(\bm{I}+\bm{\Delta}_P)$ is useful in introducing a fluctuation rate into the controller design. For the simplicity of discussion, let the entire plant dynamics $\bm{P}(\bm{I}+\bm{\Delta}_P)$ be expressed as $\bm{P}_{\rm m}$. Herein, the output of this controller is generated as

where $\bm{I}$ is the unit matrix. The nominal output can be checked by replacing $\bm{P}_{\rm m}$ with $\bm{P}$. In the controller design, the uncertainty is once removed from the plant model, and the nominal model $\bm{P}$ is mainly used. Herein, the condition that the fluctuation of the plant system does not destabilize the control system is provided by the small gain theorem;

where $\|\cdot\|_{\infty}$ denotes the $H_{\infty}$ norm. In the equation above, the two coefficients are standing out;

where $\bm{S}$ is the sensitivity function, and $\bm{T}$ is the complementary-sensitivity function. As their names denote, these are in the complementary relationship, which is expressed as

The sensitivity means susceptibility of the transfer function from the command to the output. Here, let the transfer function from the command to the output be represented as

Herein, $\bm{T}_{ry}$ is described as the function of $\bm{P}$. When the plant system fluctuates from $\bm{P}$ to $\bm{P}_{\rm m}$, the rates of the changes of the plant system $\bm{R}_{P}$ and the transfer function $\bm{R}_{T}$ can be described as

With the simple algebraic operations, we obtain

and the following relationship can be found

Thus, the sensitivity function shows the robustness of the tracking performance against the fluctuation. On the other hand, the complementary-sensitivity function indicates the robustness of the stability against the fluctuation, as denoted in \eq{math:chap2_RobustStable}. These are called robust performance and robust stability. Besides, the index that represents the effect of the disturbance on the responses is called the quasi-sensitivity. The quasi-sensitivity function is defined as

Using these functions, the output of the nominal system can be rewritten as

This equation shows five tradeoffs between

The 1-DOF controller can handle only one characteristic due to the lack of DOFs. The first one is a well-known problem referred to as the pole-zero assignment problem. On the 1-DOF controller, the transfer function from the command to the response $\bm{T}_{ry}$ and that from the disturbance to the response $\bm{T}_{dy}$ have the same poles. Following the internal model principle \cite{IMC_Francis_1976}, most controllers have an integrator to remove a steady-state error, but this integrator puts an unexpected zero on $\bm{T}_{ry}$. Adding poles to compensate the disturbance increases the order of the system, and generates unexpected zeros. Therefore, a conservative design that sacrifices one side imposes on engineers. It should be noted that the added zeros and poles also influence the system noise. Furthermore, the controller should have a high gain so that both a numerator and a denominator of $\bm{T}_{ry}$ get almost the same value to improve the tracking performance. This approach causes to amplify the system noises and makes the response oscillatory. For those factors, it is hard to elicit the performance when using the 1-DOF controller.

The 1-DOF control architecture has been revised. Here, the blocks $\bm{K}$, $\bm{N}_{\rm n}$, and $\bm{D}_{\rm n}$ stand for the tracking controller and the irreducible numerator and denominator of the left coprime factorization of a nominal plant model$\bm{P}_{\rm n}$, i.e., $\bm{P}_{\rm n}=\bm{N}_{\rm n}\bm{D}^{-1}_{\rm n}$. This architecture enables the design of the tracking performance at will by introducing a model-based feedforward loop. Since the 1-DOF controller generates a control effort for tracking from trajectory error, a phase lag between the command and the response is inevitable. Furthermore, the tracking performance is affected by the controller since the control effort is the output of the controller. The solution which the architecture shows is to generate the control effort required for command tracking in a designed loop. This controller is referred to as the 2-DOF controller, which provides two available characteristics, which are tracking performance and disturbance suppression performance. In other words, the 2-DOF controller includes a feedback controller and a feedback controller. The feedforward controller determines the behavior of a plant system when a plant system is not disturbed from an external system. The feedback controller compensates tracking error which is caused by disturbance. Herein, the output is generated as

Because tracking performance relies on model-based feedforward control, the mismatch of a model causes an error. The additional factor is only the feedforward loop, and hence, the stability against plant fluctuation $\bm{T}$ is the same as that of the 1-DOF control system. Regarding to the sensitivity of the system response against plant fluctuation, the following conditions are found

Therefore, the sensitivity, the quasi-sensitivity, and the complementary-sensitivity function are the same as those of the 1-DOF controller. When the system model well describes the plant system or when the plant fluctuation is small, the output is rewritten as

In this equation, the coefficient of $\bm{r}$ does not have the characteristics of the controller $\bm{C}$. It means that the tradeoffs relating to the tracking performance, including the pole-zero assignment problem, are removed. Therefore, the remaining problems are the two tradeoffs between

It should be noted that the noise characteristics and the plant characteristics are not something that can be shaped by modifying a control architecture, and a design of the hardware is critical to reducing the constraints above. These should be handled by the integrated design of the control system, with the theory of control, power electronics, and computer science.