13. Controller Design for MIMO Linear Systems

1. Pole Placement (Pole Assignment) Method

Given a linear system with the state space equation:

\[ \begin{aligned} \dot x(t) &= Ax(t) + Bu(t) \\ \end{aligned} \]

Assume the state is measurable, given the control law:

\[ u(t) = -Kx(t) + \gamma(t) \]

The feedback control scheme is:

This state feedback control could change the system performance and become a new system, we can design a controller based on this new system:

Given the enlarged system description:

\[ \dot x_e(t) = Ax_e(t) + Bu(t) \]

• \(x_e = \begin{bmatrix} x \\ v \end{bmatrix}\)

We recall the MIMO system design in the previous part:

For the stabilizing regulator, there have:

\[ u = -Kx_e = - \begin{bmatrix} K_x & K_v \end{bmatrix} \begin{bmatrix} x \\ v \end{bmatrix} \]

By knowing the control law, we can rearrange the control scheme:

Replace \(u = -Kx(t) + \gamma(t)\) term into the state space equation, we can get:

\[ \begin{aligned} \dot x(t) &= Ax(t) - BKx(t) + B\gamma(t) \\ &= (A-BK)x(t) + B\gamma(t) \end{aligned} \]

The goal of the pole-placement control is:

• Choose a suitable \(K\) to set the eigenvalues of \(A-BK\) in arbitrarily selected positions.
• An N&S condition for such a \(K\) to exist is \((A,B)\) is reachable.

Info

Proof (necessity):

Assume that \((A,B)\) is not reachable

\[ \begin{aligned} \begin{bmatrix} \dot x_R \\ \dot x_{NR} \end{bmatrix} = \begin{bmatrix} A_R&A_X \\ 0&A_{NR} \end{bmatrix} \begin{bmatrix} x_R \\ x_{NR} \end{bmatrix} + \begin{bmatrix} B_R \\ 0 \end{bmatrix}u \end{aligned} \]

And we can design the controller \(u = -\begin{bmatrix} K_R & K_{NR} \end{bmatrix} \begin{bmatrix} x_R \\ x_{NR} \end{bmatrix}\) , replace it to the system dynamics:

\[ \begin{aligned} \begin{bmatrix} \dot x_R \\ \dot x_{NR} \end{bmatrix} = \begin{bmatrix} A_R-B_RK_R&A_X-B_RK_{NR} \\ 0&A_{NR} \end{bmatrix} \begin{bmatrix} x_R \\ x_{NR} \end{bmatrix} \end{aligned} \]

And the eigenvalues of \(A-BK\) are \(\text{eig}(A_R-B_RK_R)\cup \text{eig}(A_X-B_RK_{NR})\)

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Proof (Sufficiency):

Assume \((A,B)\) is reachable, \(m = 1\), \(\exists T_{n\times n}\), \(\det(T) \neq 0\), such that \(\tilde x = Tx\).

The controllable canonical form of the system have:

\[ \tilde A = TAT^{-1} = \left[\begin{array}{c:c} \mathbf{0} & \mathbf{I}_{n-1} \\ \hdashline -a_0 & \begin{matrix} -a_1&\cdots&-a_{n-1}\end{matrix} \end{array}\right] \]

The system characteristic polynomial is:

\[ p_A(\lambda) = p_{\tilde A}(\lambda) = \lambda^n + a_{n-1}\lambda^{n-1}+\dots+a_1\lambda + a_0 \]

And the controllable canonical form also gives:

\[ \begin{aligned} \tilde B &= TB = \begin{bmatrix} 0 \\ \vdots \\ 0 \\ 1\end{bmatrix} \end{aligned} \]

Given the state feedback control law:

\[ u(t) = -\tilde K \tilde x(t) + \gamma(t) \]

Thus we have:

\[ \tilde A - \tilde B \tilde K = \left[\begin{array}{c:c} \mathbf{0} & \mathbf{I}_{n-1} \\ \hdashline (-a_0-\tilde K_0) & \begin{matrix} (-a_1-\tilde K_1)&\cdots&(-a_{n-1}-\tilde K_{n-1})\end{matrix} \end{array}\right] \]

The characteristic polynomial of the closed loop system is:

\[ \begin{aligned} p_{\tilde A - \, \tilde B \tilde K}(\lambda) &= p_{A-BK}(\lambda) = \lambda^n + (a_{n-1}+\tilde K_{n-1})\lambda^{n-1}+\dots+(a_1 + \tilde K_1)\lambda + (a_0 + \tilde K_0) \\ \end{aligned} \]

And we want to design system into a desired characteristic polynomial function:

\[ p_n^*(\lambda) = \prod_{i=1}^n(\lambda - \lambda_i^*) = \lambda^n + \alpha_{n-1}\lambda^{n-1} + \dots + \alpha_0 \]

• \(\lambda_i^*\) is desired eigenvalues

We determine the \(\tilde K_i = \alpha_i - a_i\), \(i=0,\dots,n-1\), and \(K=\tilde K T\), in the MIMO case, where \(m > 1\), \((A, B)\) is reachable,

• if \((A, b_i)\) is reachable

  \[ \begin{aligned} u_j(t) &= 0, &\forall j \neq i \\ u_i(t) &= -K_ix(t) + \gamma(t) \\ K &= \begin{bmatrix} 0 & \cdots & K_i & \cdots & 0\end{bmatrix}^T & \end{aligned} \]

Steps (Single Input):

1. Compute \(p^*(\lambda)\) based on the desired set of \(n\) eigenvalues, \(\lambda_i^*\), \(i=1,\dots,n\)
2. Compute \(p_A(\lambda)\)
3. Build \(\tilde A\), \(\tilde B\) in controllable canonical form

4. Determine the \(M_R\) and \(\tilde M_R\) \(\Rightarrow\) \(T = \tilde M_R M_R^{-1}\), where:

   \[ \begin{aligned} M_R &= \begin{bmatrix} B&AB&\cdots&A_{n-1}B\end{bmatrix} \\ \tilde M_R &= \begin{bmatrix} \tilde B&\tilde A\tilde B&\cdots&\tilde A_{n-1}\tilde B\end{bmatrix} = TM_R\\ \end{aligned} \]

   Where \(\tilde A = TAT^{-1}\), and \(\tilde B = TB\)

5. \(\tilde K = \begin{bmatrix} (\alpha_0-a_0) & \cdots & (\alpha_{n-1}-a_{n-1})\end{bmatrix}\)

6. \(K=\tilde KT\)

1.1 Ackermann's Formula:

\[ K = \begin{bmatrix} 0&\cdots&0&1\end{bmatrix}M_R p^*(A) \]

where:

• \(p^*(\lambda) = \lambda^n + \alpha_{n-1}\lambda^{n-1} + \dots + \alpha_0\)
• \(p^*(A) = A^n + \alpha_{n-1}A^{n-1} + \dots + \alpha_0I\)

Example

\[ \begin{aligned} &A = \begin{bmatrix} -1&3\\0&2 \end{bmatrix} &B = \begin{bmatrix} 1\\2 \end{bmatrix} \\ &\lambda_1^* = -3 & \lambda_2^* = -4 \end{aligned} \]

Method 1:

• \(p^*(\lambda) = (\lambda + 3)(\lambda + 4) = \lambda^2 + 7\lambda + 12\)

• \(M_R = \begin{bmatrix} 1&5\\2&4 \end{bmatrix}\)

• \((A,B)\) is reachable,

\[ K = \begin{bmatrix} 0\\1 \end{bmatrix} \begin{bmatrix}-\frac46 &\frac56 \\ \frac26 &-\frac16 \end{bmatrix} (A^2+7A+12I)=\begin{bmatrix} 2&3 \end{bmatrix} \]

Method 2:

• \((A, B)\) is reachable

\[ \begin{aligned} A-BK &= \begin{bmatrix} -1-K_1 & 3-K_2 \\ -2K_1 & 2-2K_2 \end{bmatrix} \\ p_{A-BK}(\lambda) &= \det(\lambda I - (A-BK)) \\ &= \lambda^2 + (-1+K_1+2K_2) + 4K_1 + 2K_2 - 2 \end{aligned} \]

• \(p^*(\lambda) = \lambda^2 + 7\lambda + 12\)

\[ \left\{\begin{aligned} -1 + K_1 + 2K_2 = 7 \\ 4K_1 + 2K_2 - 2 = 12 \end{aligned}\right. \rightarrow \left\{\begin{aligned} K_1=2 \\ K_2=3 \end{aligned}\right. \]

1.2 Effect of Poles on the System Response

The poles have the effect on the transient in the step response:

For the complex poles, the system will oscillates, the maximum value in the transient state could be calculated:

\[ s = \frac{y_{\max} - \mu}{\mu} = \exp(\frac{\pi \xi}{\sqrt{1 - \xi^2}}) \]

In a conjugate complex poles system,

• The natural frequency \(\omega_n\) is the length between poles to the origin.
• The damping \(\xi\) is associated to the angle \(\alpha\) in the figure, \(\xi = \cos\alpha\)

In practice, to realize the pole-placement control, it could be done by executing the matlab function:

[k, precision] = place(A, B, P)

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