Decoupling State Feedback Control Design

A Quick Recap and Notation Update

Remember the general form of the state feedback law we introduced for decoupling: $u=Fx+Gv$. Today, we'll adopt a slightly different but equivalent notation that is common in literature, including in the resource we'll be using:

Here's how they map out:

• The state feedback matrix is now $-K$ (instead of $F$).
• The command input vector is $r$ (for reference, instead of $v$).
• The input gain matrix is $F$ (instead of $G$).

This is purely a change in symbols, but it's important to be aware of it. The goal remains the same: choose $K$ and $F$ so that the transfer function from the new command input $r$ to the system output $y$ is diagonal.

We will also use the term order difference ${\delta }_i$, which is identical to what we called the relative degree $d_i$ in the last lesson. It's the number of times we must differentiate output $y_i$ before an input $u_j$ appears.

The Decoupling Controller Design

The core of designing a decoupling controller involves finding the correct matrices $K$ and $F$. The method we'll use is a powerful technique developed by Falb and Wolovich, which allows for simultaneous decoupling and pole placement.

This is highly relevant to your work with the rotary inverted pendulum, where you'll want to control, for instance, the arm's angle and the pendulum's angle independently, while also ensuring the system is stable and responds quickly without excessive overshoot.

The following paper provides a concise and practical guide to this design method, using a 2-DOF helicopter as an example—an excellent mechanical analogue for your project.

Synthesizing the Falb-Wolovich Method

As you read, Theorem 1 presents a way to decouple the system into a series of pure integrators. While this achieves decoupling, it doesn't allow for pole placement, which is critical for performance. Theorem 2 provides the complete solution we need.

Let's break down the design procedure into clear steps.

Step 1: Choose Desired Pole Locations

For each of the $m$ input-output channels, you decide where you want to place the closed-loop poles to achieve the desired dynamic response (e.g., settling time, damping ratio).

• For output $y_i$, you can place ${\delta }_i$ poles.
• From your chosen poles $\{p_{i1},p_{i2},...,p_{i{\delta }_i}\}$, you form the desired characteristic polynomial for that channel: $$(s-p_{i1})(s-p_{i2})...(s-p_{i{\delta }_i})=s^{{\delta }_i}+q_{i,{\delta }_i-1}{s}^{{\delta }_i-1}+\cdots +q_{i1}s+q_{i0}$$ The coefficients $q_{i0},q_{i1},\dots ,q_{i,{\delta }_i-1}$ are now known values that define your desired dynamics.

Step 2: Construct the Necessary Matrices

You'll need three key matrices: $B^{\ast }$, $C^{\ast \ast }$, and a diagonal gain matrix for the feedforward term.

1. The Decoupling Matrix, $B^{\ast }$:
   You already know this from our last lesson. Its $i$-th row is $C_i{A}^{{\delta }_i-1}B$.

   $$B^{\ast }=\left[\begin{array}{c}{C}_1{A}^{{\delta }_1-1}B\\ C_2{A}^{{\delta }_2-1}B\\ ⋮\\ C_m{A}^{{\delta }_m-1}B\end{array}\right]$$

2. The Dynamic Matrix, $C^{\ast \ast }$:
   This matrix, defined in equation (17) of the paper, incorporates your desired dynamics through the $q_{iv}$ coefficients. Its $i$-th row is:

   $${\text{row}}_i(C^{\ast \ast })=C_i{A}^{{\delta }_i}+\sum _{v=0}^{{\delta }_i-1}{q}_{iv}{C}_i{A}^v$$

   Essentially, for each output, you're creating a new composite row vector that combines the system's natural dynamics ($C_i{A}^{{\delta }_i}$) with the desired ones (the weighted sum of $C_i{A}^v$).

3. The Input Gain Matrix, $F_q$:
   This is a simple diagonal matrix constructed from the constant terms of your desired characteristic polynomials.

   $$F_q=\text{diag}\{q_{10},q_{20},\dots ,q_{m0}\}=\left[\begin{array}{ccc}{q}_{10}& 0& \dots \\ 0& q_{20}& \dots \\ ⋮& & \ddots \end{array}\right]$$

Step 3: Calculate the Controller Matrices $K$ and $F$

With the preparatory work done, calculating the controller matrices is straightforward using the formulas from equation (18):

With these matrices, the control law $u=-Kx+Fr$ will decouple the system, and the transfer function from $r_i$ to $y_i$ will be:

Notice that the steady-state gain (at $s=0$) is $q_{i0}/q_{i0}=1$, ensuring that the output $y_i$ tracks the reference $r_i$ with no scaling error at steady state.

Worked Example: Designing the Controller

Let's use the same system from our previous lesson, for which we already confirmed decoupling is possible.
$A=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -1& -2& -3\end{array}\right],B=\left[\begin{array}{cc}0& 0\\ 1& 0\\ 0& 1\end{array}\right],C=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\end{array}\right]$
Last time, we found ${\delta }_1=2$, ${\delta }_2=1$, and $B^{\ast }=I$, which means $(B^{\ast }{)}^{-1}=I$. This simplifies the math, letting us focus on the process.

Step 1: Choose Desired Poles

• Channel 1 (${\delta }_1=2$): We can place two poles. Let's aim for a critically damped response with poles at $-5,-5$.
  The characteristic polynomial is $(s+5)(s+5)=s^2+10s+25$.
  So, $q_{11}=10$ and $q_{10}=25$.
• Channel 2 (${\delta }_2=1$): We can place one pole. Let's place it at $-8$.
  The characteristic polynomial is $s+8$.
  So, $q_{20}=8$.

Step 2: Construct $C^{\ast \ast }$

• Row 1 of $C^{\ast \ast }$: ${\text{row}}_1(C^{\ast \ast })=C_1{A}^2+q_{11}{C}_{1}A+q_{10}{C}_1$
  We need the components:

  • $C_1=[1,0,0]$
  • $C_{1}A=[0,1,0]$
  • $C_1{A}^2=(C_{1}A)A=[0,1,0]A=[0,0,1]$

  So, ${\text{row}}_1(C^{\ast \ast })=[0,0,1]+10[0,1,0]+25[1,0,0]=[25,10,1]$.

• Row 2 of $C^{\ast \ast }$: ${\text{row}}_2(C^{\ast \ast })=C_2{A}^1+q_{20}{C}_2$
  We need the components:

  • $C_2=[0,0,1]$
  • $C_{2}A=[-1,-2,-3]$

  So, ${\text{row}}_2(C^{\ast \ast })=[-1,-2,-3]+8[0,0,1]=[-1,-2,5]$.

Combining the rows gives:
$C^{\ast \ast }=\left[\begin{array}{ccc}25& 10& 1\\ -1& -2& 5\end{array}\right]$

Step 3: Calculate $K$ and $F$

• Calculate $K$:
  $K=(B^{\ast }{)}^{-1}{C}^{\ast \ast }=I\cdot C^{\ast \ast }=\left[\begin{array}{ccc}25& 10& 1\\ -1& -2& 5\end{array}\right]$
• Calculate $F$:
  First, build $F_q=\text{diag}(q_{10},q_{20})=\left[\begin{array}{cc}25& 0\\ 0& 8\end{array}\right]$.
  $F=(B^{\ast }{)}^{-1}{F}_q=I\cdot F_q=\left[\begin{array}{cc}25& 0\\ 0& 8\end{array}\right]$

We have successfully designed our decoupling controller.

Conclusion

Today we moved from theory to practice, learning a concrete method for designing a controller that achieves both decoupling and pole placement.

Key Takeaways:

• The Falb-Wolovich method provides a systematic way to design a decoupling state feedback controller of the form $u=-Kx+Fr$.
• The design process starts with choosing desired closed-loop poles for each independent channel, which defines the coefficients of your target characteristic polynomials ($q_{iv}$).
• These coefficients are used to construct the dynamic matrix $C^{\ast \ast }$.
• The final controller matrices are calculated with the formulas $K=(B^{\ast }{)}^{-1}{C}^{\ast \ast }$ and $F=(B^{\ast }{)}^{-1}\text{diag}\{q_{i0}\}$.
• This method guarantees not only decoupling and pole placement but also unity steady-state gain for reference tracking.

Preview of the Next Lesson:
We've now covered a significant portion of MIMO control design, from modeling to pole placement and decoupling. We've designed controllers that can theoretically place poles anywhere we want. However, making a system respond very quickly can require enormous control effort (e.g., massive motor torques or voltages). How do we balance performance with practicality?

In our next lesson, we will begin exploring optimal control with the Linear Quadratic Regulator (LQR). LQR provides a powerful framework to design a state feedback controller that optimally balances state regulation (performance) against control effort (cost).