Reference Input Tracking

• Adding a Reference Input
• Intuition Behind Feedforward Control

Adding a Reference Input

A block diagram of a system with state feedback and a reference input $\mathbf{r}$

In my previous post, I discussed how to determine if a system is controllable. If it was controllable, I showed this meant its poles could be arbitrarily chosen allowing the dynamics and stability characteristics of the system to be manipulated as desired. However, in many cases we want the system to not only stabilise but stabilise to a desired reference or perhaps even track a desired reference input.

A block diagram of a system with state feedback

To achieve this, we introduce a reference input signal $\mathbf{r}$ which represents the desired output of the system. Naively, we might try adding this reference input directly to our feedback control signal so that $\mathbf{u = -Kx + r}$. Therefore we have,

\[\begin{align*} \mathbf{\dot{x}} &= \mathbf{Ax + Bu} \\ \implies \mathbf{\dot{x}} &= \mathbf{Ax + B(-Kx + r)} \\ \implies \mathbf{\dot{x}} &= \mathbf{(A - BK)x + Br} \end{align*}\]

If we assume the system has been stabilised appropriately with state feedback then at steady state $\mathbf{\dot{x}} = 0$ by definition. Also noting that $\mathbf{y = Cx + Du},

\[\begin{align*} \mathbf{x}_{ss} &= \mathbf{(A - BK)^{-1} Br} \\ \mathbf{y}_{ss} &= \mathbf{Cx_{ss} + D(-Kx_{ss} + r)} \\ \implies \mathbf{y}_{ss} &= \mathbf{((C - DK)(A - BK)^{-1} B + D)r} \end{align*}\]

where \(\mathbf{x}_{ss}\) and \(\mathbf{y}_{ss}\) are the steady state values for the state and output, respectively. Clearly, \(\mathbf{y}_{ss} \neq \mathbf{r}\) in general. But what if we add a matrix gain $\mathbf{\bar{N}}$ to the reference input so that $\mathbf{u = -Kx + \bar{N}r}$. Then the above equation becomes

\[\mathbf{y}_{ss} = \mathbf{((C - DK)(A - BK)^{-1} B + D) \bar{N}r}\]

Now if we let $\mathbf{y}_{ss} = \mathbf{r}$,

\[\begin{align*} \mathbf{I} &= \mathbf{((C - DK)(A - BK)^{-1} B + D) \bar{N}} \\ \implies \mathbf{\bar{N}} &= \mathbf{((C - DK)(A - BK)^{-1} B + D)^{-1}} \end{align*}\]

Or if $\mathbf{D = 0}$ which is fairly common,

\[\mathbf{\bar{N}} = \mathbf{(C(A - BK)^{-1} B)^{-1}}\]

This gives us a system block diagram like this.

A block diagram of a system with state feedback and a reference input $\mathbf{r}$

Intuition Behind Feedforward Control

As we just saw, the reference input signal is simply solving for the required steady state input to achieve a desired steady state output. Therefore, it can be thought of as a predictive feedforward control since it blindly (without any knowledge of the actual current state) computes what would be the required input to keep the system at a desired steady state.

This contrasts with the state feedback control which reacts to the current state to choose an appropriate response that will stabilise the system. As such, the feedforward control from the reference input is not only beneficial since it allows an arbitrary reference input but also because, similarly to derivative control in PID, it generally decreases the response time of the system by predicting the required input.