MIMO State-Space Equations

The MIMO State-Space Framework

You are already familiar with the standard state-space representation for a linear time-invariant (LTI) system:

The great strength of this framework is its scalability. To move from a SISO to a MIMO system, we don't change the equations; we simply expand the dimensions of the input and output vectors, and consequently, the dimensions of the B, C, and D matrices.

The following resource provides a formal definition of the state and output equations for a general MIMO system.

As the text explains, for a system with $n$ state variables, $m$ inputs, and $p$ outputs:

• The state vector $\mathbf{x}$ is of size $n\times 1$.
• The input vector $\mathbf{u}$ is of size $m\times 1$.
• The output vector $\mathbf{y}$ is of size $p\times 1$.

This requires the system matrices to have the following dimensions:

• State matrix $A$: $n\times n$ (always square)
• Input matrix $B$: $n\times m$
• Output matrix $C$: $p\times n$
• Feedforward matrix $D$: $p\times m$

Notice that $B$, $C$, and $D$ are now generally rectangular, reflecting the mapping between different numbers of inputs, states, and outputs.

From Coupled Equations to a MIMO State-Space Model

The process of deriving a MIMO state-space model usually begins by writing down the physical laws governing the system. This often results in a set of coupled, higher-order differential equations. Let's see this in action with a mechanical example.

Example 1: A Quarter-Car Suspension Model

A common way to model a car's suspension is the "quarter-car model," which includes the mass of the car body and the mass of the wheel. This creates a two-mass system where the motions are coupled.

Let's break down the key steps from the resource:

1. System Dynamics: The system is described by two coupled second-order differential equations, one for the car's mass ($m_c$) and one for the wheel's mass ($m_w$). $$m_c{\ddot{x}}_1=-K_1(x_1-x_2)-c({\dot{x}}_1-{\dot{x}}_2)$$ $$m_w{\ddot{x}}_2=K_1(x_1-x_2)-K_2(x_2-u)+c({\dot{x}}_1-{\dot{x}}_2)$$
2. State Variable Selection: Since we have two second-order equations, we need $2+2=4$ state variables. A natural choice, based on the energy storage elements (the kinetic energy of the masses and potential energy in the springs), is the position and velocity of each mass.
   • $z_1=x_1$ (car position)
   • $z_2={\dot{x}}_1$ (car velocity)
   • $z_3=x_2$ (wheel position)
   • $z_4={\dot{x}}_2$ (wheel velocity)
3. Formulating First-Order Equations: We create four first-order differential equations:
   • ${\dot{z}}_1=z_2$ (by definition)
   • ${\dot{z}}_3=z_4$ (by definition)
   • ${\dot{z}}_2={\ddot{x}}_1=\frac{1}{m_c}[-K_1(z_1-z_3)-c(z_2-z_4)]$ (from the first dynamic equation)
   • ${\dot{z}}_4={\ddot{x}}_2=\frac{1}{m_w}[K_1(z_1-z_3)-K_2(z_3-u)+c(z_2-z_4)]$ (from the second dynamic equation)
4. Assembling the Matrices: By rearranging these equations and grouping terms by $z_1,z_2,z_3,z_4$, and $u$, we can directly construct the $A$ and $B$ matrices as shown on slide 29 of the notes. If our outputs are the positions of the car and wheel, $y_1=x_1=z_1$ and $y_2=x_2=z_3$, the output equation becomes: $$\mathbf{y}=\left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\end{array}\right]\left[\begin{array}{c}{z}_1\\ z_2\\ z_3\\ z_4\end{array}\right]$$

This gives us a $2\times 4$ C matrix, as expected for a 4-state, 2-output system.

From a Block Diagram to a MIMO State-Space Model

Another common starting point for modeling is a block diagram, especially in fields like signal processing and controls. Your EE background in analyzing such diagrams will be useful here.

Let's look at a 2-input, 2-output example.

In this example, the state variables ($x_1,x_2,x_3$) are the outputs of the integrators. By tracing the signals through the summing junctions and gains, we can write down the equations directly:

• State Equations:
  • ${\dot{x}}_1=2x_1-x_2-u_1$
  • ${\dot{x}}_2=x_1+x_2+2x_3+u_1$
  • ${\dot{x}}_3=-x_1+x_3+2u_2$
• Output Equations:
  • $c_1=x_1+x_2$
  • $c_2=x_1+x_3$

From these, assembling the matrices is a matter of organization:

Here, you can clearly see the rectangular B ($3\times 2$) and C ($2\times 3$) matrices characteristic of a MIMO system.

Test your understanding!

Consider the MIMO system represented by the block diagram below. The outputs of the two integrators are the state variables, $x_1$ and $x_2$. Derive the state-space representation ($A,B,C,D$ matrices) for this system.

Show answer

First, we write the equations for the derivatives of the states by looking at the inputs to the integrators:

• ${\dot{x}}_1=-2x_1+6x_2+2u_1$
• ${\dot{x}}_2=-5x_2+3u_2$

Next, we write the equations for the outputs:

• $y_1=x_1$
• $y_2=4x_1+7x_2$

Now, we can assemble these equations into the standard matrix form: $\dot{\mathbf{x}}=A\mathbf{x}+B\mathbf{u}$ and $\mathbf{y}=C\mathbf{x}+D\mathbf{u}$.

$$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\underset{A}{\underbrace{\left[\begin{array}{cc}-2& 6\\ 0& -5\end{array}\right]}}\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\underset{B}{\underbrace{\left[\begin{array}{cc}2& 0\\ 0& 3\end{array}\right]}}\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]$$ $$\left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]=\underset{C}{\underbrace{\left[\begin{array}{cc}1& 0\\ 4& 7\end{array}\right]}}\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\underset{D}{\underbrace{\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]}}\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right]$$

Conclusion

In this lesson, we have successfully extended our state-space modeling skills from SISO systems to the more general and powerful MIMO case. You have learned how the matrix dimensions adapt and have practiced deriving complete MIMO state-space models from both physical principles and block diagrams.

Key Takeaways:

• The standard state-space equations apply directly to MIMO systems; the input $\mathbf{u}$ and output $\mathbf{y}$ are treated as vectors.
• The dimensions of the matrices $B$, $C$, and $D$ become rectangular ($n\times m$, $p\times n$, and $p\times m$, respectively) to accommodate multiple inputs and outputs.
• The general procedure for modeling a physical MIMO system involves deriving its coupled differential equations, selecting an appropriate set of state variables, and arranging the resulting first-order equations into matrix form.

Preview of the Next Lesson:

Now that you have a general method for modeling MIMO systems, it's time to apply it to your specific problem of interest. In the next lesson, we will derive the state-space model for a rotary inverted pendulum. This will be a more challenging and rewarding application of the techniques you learned today, bringing you one step closer to designing a controller for it.