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Controller Architecture

Problem statement

A problem is to control an output y\bm{y} of following system using an input u\bm{u}.
y=P(u+d),\begin{align} \bm{y} &= \bm{P}(\bm{u} + \bm{d}), \end{align}
where P\bm{P} and d\bm{d} denote a plant model and disturbance. An input should be designed so that it make an output track a desired trajectory r\bm{r} 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:
u=uref+vz=y+w,\begin{align} \bm{u} &= \bm{u}^{\rm ref} + \bm{v} \\ \bm{z} &= \bm{y} + \bm{w}, \end{align}
where z\bm{z}, v\bm{v}, and w\bm{w} stand for a sensor output, process noise, and sensor noise. A feedback controller is constructed using a sensor output.

Simple feedback control

Let us consider a feedback controller
uref=C(r−z).\begin{align} \bm{u}^{\rm ref} &= \bm{C}(\bm{r} - \bm{z}). \end{align}
When this controller is applied to a system, an output is expressed as
y=(I+PC)−1PCr+(I+PC)−1Pd+(I+PC)−1Pv−(I+PC)−1PCw=T(r−w)+SP(d+v),\begin{align} \bm{y}=&(\bm{I}+\bm{P}\bm{C})^{-1}\bm{P}\bm{C}\bm{r}+(\bm{I}+\bm{P}\bm{C})^{-1}\bm{P}\bm{d}+(\bm{I}+\bm{P}\bm{C})^{-1}\bm{P}\bm{v}-(\bm{I}+\bm{P}\bm{C})^{-1}\bm{P}\bm{C}\bm{w} \\ =&\bm{T}(\bm{r} - \bm{w})+\bm{S}\bm{P} (\bm{d}+\bm{v}), \end{align}
where S\bm{S} and T\bm{T} are complementary parameters that are expressed as
S≡(I+PC)−1T≡(I+PC)−1PC\begin{align} \bm{S}\equiv&(\bm{I}+\bm{P}\bm{C})^{-1}\\ \bm{T}\equiv&(\bm{I}+\bm{P}\bm{C})^{-1}\bm{P}\bm{C} \end{align}
Using a high gain controller that make S→0\bm{S} \rightarrow \bm{0} and T→I\bm{T} \rightarrow \bm{I} apparently seems like good. However, it can be achieved due to the existence of noise. Furtherover, a controller cannnot suppress tracking error or disturbance unless satisfy the inner-model principle. An important point is that the pole-zero placements of a transfer functions related to tracking and diturbance suppression are the same. This problem makes it difficult to be compliant with inner-model principle. This controller architecture provides a degree-of-freedom (DoF) for either one, and it is called 1-DoF controller.

Adding feedforward controller

Feedforward control using a plant model ensures a tracking performance and solves a pole-zero placement problem that is associated with a 1-DoF controller. We introduce a plant model Pn=NnDn−1\bm{P}_{\rm n}=\bm{N}_{\rm n}\bm{D}^{-1}_{\rm n}, where Nn\bm{N}_{\rm n} and D\bm{D} stand for a irreducible numerator and denominator of the left coprime factorization of a nominal plant model. Let us consider a feedback controller
uref=DnKr+C(NnKr−z),\begin{align} \bm{u}^{\rm ref} &= \bm{D}_{\rm n}\bm{K}\bm{r} + \bm{C}(\bm{N}_{\rm n}\bm{K}\bm{r} - \bm{z}), \end{align}
where K\bm{K} is a smoother that makes a controller be a proper system. An output is expressed as
y=(I+PC)−1(PPn−1+PC)NnKr+(I+PC)−1Pd+(I+PC)−1Pv−(I+PC)−1PCw=(I+PC)−1(PPn−1+PC)NnKr−Tw+SP(d+v)≈NnKr−Tw+SP(d+v).\begin{align} \bm{y}=&(\bm{I}+\bm{P}\bm{C})^{-1}(\bm{P}\bm{P}^{-1}_{\rm n} + \bm{P}\bm{C})\bm{N}_{\rm n}\bm{K}\bm{r}+(\bm{I}+\bm{P}\bm{C})^{-1}\bm{P}\bm{d}+(\bm{I}+\bm{P}\bm{C})^{-1}\bm{P}\bm{v}-(\bm{I}+\bm{P}\bm{C})^{-1}\bm{P}\bm{C}\bm{w} \\ =&(\bm{I}+\bm{P}\bm{C})^{-1}(\bm{P}\bm{P}^{-1}_{\rm n} + \bm{P}\bm{C})\bm{N}_{\rm n}\bm{K}\bm{r} - \bm{T}\bm{w}+\bm{S}\bm{P} (\bm{d}+\bm{v})\\ \approx& \bm{N}_{\rm n}\bm{K}\bm{r} - \bm{T}\bm{w}+\bm{S}\bm{P} (\bm{d}+\bm{v}). \end{align}
This architecture enables the design of the tracking performance at will. This controller is referred to as the 2-DOF controller because it provides two available characteristics. 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. 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, a disturbance suppression performance and stability against plant fluctuation are the same as that of the 1-DOF controller.

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