Linear Controls Notes

\[ \left| \begin{array}{cc} x_{11} & x_{12} \\ x_{21} & x_{22} \end{array}\right| = x_{11}x_{22} - x_{12}x_{21} \]

\[ {\rm null}({\bf A})\ \equiv\ \{ {\bf x}\ \arrowvert\ {\bf A}: m\times n \ \land\ {\bf x} \in \Re^n\ \land\ {\bf A} {\bf x} = 0 \} \]

\[ {\rm span}({\bf v_1}, {\bf v_2}, \dots, {\bf v_n})\ \equiv\ % \{ {\bf x}\ \arrowvert\ % {\bf x} = c_1{\bf v_1} + c_2{\bf v_2} + \dots + c_n {\bf v_n}\ % \forall\ % \{c_1, c_2, \dots, c_n\} \in \Re \} \]

\[ {\rm rank}({\bf A})\ \equiv\ {\rm number\ of\ linearly\ independent\ colums\ of\ }{\bf A} \] \[ {\rm rank}(A) = {\rm rank}(A^T) \]

\[ \mathcal{R}({\bf A})\ \equiv\ {\rm span}({\rm columns}({\bf A})) \]

\[ {\rm row\_space}({\bf A})\ \equiv\ {\mathcal R}({\bf A}^T)\]

Let \(\lambda\) be the eigenvalues of matrix \(\bf A\).

\[ {\bf A}{\bf x} = \lambda {\bf x} \]

Characteristic Equation:

\[ \lvert {\bf A} - \lambda {\bf I} \rvert = 0 \] \[ ({\bf A} - \lambda {\bf I}) {\bf x} = 0 \]

\[ \lvert {\bf A} - {\bf A} {\bf I} \rvert = 0 \]

Thus, you can substitute \(\bf A\) in it's own characteristic equation. This can be used to reduce the order of a matrix polynomial by using the characteristic equation to find the value of a high order matrix power as a linear combination of lower order powers of that matrix.

\[ {\bf P}^{-1} {\bf A} {\bf P} = {\bf B}\ \iff\ {\bf P} {\bf B} {\bf P}^{-1}= {\bf A} \]

Similarity Transform: \({\bf A} \to {\bf P} {\bf B} {\bf P}^{-1}\)

\[ {\bf z} \in {\bf x}^\perp \implies {\bf z}^T{\bf x} = 0 \]

\[ e^{\bf A} = \sum_{k=0}^\infty \frac{{\bf A}^k}{k!}\] \[e^{{\bf A} t} = \mathscr{L}^{-1}\left\{(s {\bf I} - {\bf A} )^{-1}\right\} \]

\[ {\bf A} {\bf A}^{-1} = {\bf I} \]

For matrix \({\bf A} \in \Re^{2\times 2}\): \begin{eqnarray} {\bf A} = \left[\begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] \nonumber \\ {\bf A}^{-1} = \frac{1}{\left|{\bf A}\right|} \left[\begin{array}{cc} a_{22} & -a_{12} \\ -a_{21} & a_{11} \end{array}\right] \nonumber \end{eqnarray}

Higher order functions – the derivative, integral, and Laplace transform – operate on matrices elementwise.

\begin{eqnarray} \frac{d {\bf A}}{dt} = \left[\begin{array}{cccc} \frac{ d a_{11}}{dt} & \frac{d a_{12}}{dt} & \cdots & \frac{d a_{1n}}{dt} \\ \frac{ d a_{22}}{dt} & \frac{d a_{22}}{dt} & \cdots & \frac{d a_{2n}}{dt} \\ \vdots & \vdots & \ddots & \frac{d a_{1n}}{dt} \\ \frac{ d a_{n1}}{dt} & \frac{d a_{n2}}{dt} & \cdots & \frac{d a_{nn}}{dt} \\ \end{array}\right] \end{eqnarray}

\[ {\bf A} = \left[ \begin{array}{cccc} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & 0 \\ 0 & 0 & \cdots &a_n \end{array} \right] \implies e^{\bf A} = \left[ \begin{array}{cccc} e^{a_1} & 0 & \cdots & 0 \\ 0 & e^{a_2} & \cdots & 0 \\ \vdots & \vdots & \ddots & 0 \\ 0 & 0 & \cdots & e^{a_n} \end{array} \right] \] \[ {\bf A} = {\bf P}{\bf D}{\bf P}^{-1} \implies e^{\bf A} = {\bf P}e^{\bf D}{\bf P}^{-1} \]

If matrix \({\bf A}\) is nilpotent, then

\[{\bf A}^q = 0 \implies e^{\bf A} = \sum_{k=0}^{q-1} \frac{{\bf A}^k}{k!}\]

stable

\(\forall\ \epsilon>0,\quad \exists\, \delta(\epsilon,t_0)\ \arrowvert\ x(t_0) = \delta \land x(t) < \epsilon \forall t\)

asymptotically stable

stable and \(\exists\, \delta'(t_0)\ \arrowvert\ x(t_0) = \delta' \land \lim_{t \to \infty } x(t) = 0\)

global asymptotically stable

\(\bf x \to \bf 0\ {\rm as}\ t\to\infty \ \forall\ \bf x(t_0) = \bf x_0\)

Employs a Lyapunov function, \(V({\bf x}): \Re^n \mapsto \Re\). Analogous to physical energy, but must satisfy certain mathematical considerations.

positive definite

for some region \(\Omega\) about \({\bf x} = {\bf 0}\), \[ \left(V({\bf 0}) = 0 \right)\ \land\ \left( V({\bf x}) > 0\ \forall\ \{{\bf x} \arrowvert {\bf x} \neq {\bf 0} \land {\bf x} \in \Omega \} \right)\]

positive semidefinite

for some region \(\Omega\) about \({\bf x} = {\bf 0}\), \[ \left(V({\bf 0}) = 0 \right)\ \land\ \left( V({\bf x}) \geq 0\ \forall\ \{{\bf x} \arrowvert {\bf x} \neq {\bf 0} \land {\bf x} \in \Omega \} \right)\]

negative definite

for some region \(\Omega\) about \({\bf x} = {\bf 0}\), \[ \left(V({\bf 0}) = 0 \right)\ \land\ \left( V({\bf x}) < 0\ \forall\ \{{\bf x} \arrowvert {\bf x} \neq {\bf 0} \land {\bf x} \in \Omega \} \right)\]

negative semidefinite

for some region \(\Omega\) about \({\bf x} = {\bf 0}\), \[ \left(V({\bf 0}) = 0 \right)\ \land\ \left( V({\bf x}) \leq 0\ \forall\ \{{\bf x} \arrowvert {\bf x} \neq {\bf 0} \land {\bf x} \in \Omega \} \right)\]

Stable

positive definite \(V({\bf x})\) and negative semidefinite \(\dot{V}({\bf x})\), system is stable about origin

Asymptotically Stable

positive definite \(V({\bf x})\) and negative definite \(\dot{V}({\bf x})\), system is stable about origin

GAS

\[V({\bf x}) > 0\,\forall\,{\bf x}\neq 0 \quad\land\quad V(\bf 0)=0 \quad\land\quad \dot{V}(\bf x)<0\,\forall\,x\neq 0 \quad\land\quad V(\bf x) \to \infty\ {\rm as}\ \lVert\bf x\rVert \to \infty\]

\[ W(t_0,t_1) = \int_{t_0}^{t_1} \Phi(t_1,s) B(s) B^T(s) \Phi^T(t_1,s) ds \] \[ x(t_0) = x_0 \to x(t_1) = x_1 \iff x_t - \Phi(t_1,t_0)x_0 \in {\mathcal R}(W(t_0,t_1)) \]

\[ {\bf \Gamma} = \left[ {\bf A}^0 {\bf B}\ {\bf A}^1 {\bf B}\ \dots\ {\bf A}^{n-1} {\bf B} \right] \] \[ {\rm reachable}({\bf x}) \iff {\bf x} \in \mathcal{R}({\bf \Gamma}) \] \[ {\rm controllable}({\bf x}) \iff e^{{\bf A}t}{\bf x} \in \mathcal{R}({\bf \Gamma}) \] \[{\rm rank}({\bf \Gamma}) = n \implies {\rm completely\ controllable} \]

\[ \bf \Omega = \left[ \begin{array}{c} \bf C \\ \bf C \bf A \\ \bf C \bf A^2 \\ \vdots \\ \bf C \bf A^{n-1} \end{array} \right] \] \[ {\rm rank}(\bf \Gamma) = n \implies {\rm completely\ observable} \]

Transformation by \[ \bf T = \left[\begin{array}{cccc} {\bf v}_{co} & {\bf v}_{c\bar{o}} & {\bf v}_{\bar{c}o} & {\bf v}_{\bar{c}\bar{o}} \end{array}\right] \] where: \begin{eqnarray} {\bf v}_{co} \equiv \left\{{\bf x} \arrowvert {\bf x} \in \mathcal{R}({\bf \Gamma}) \cap \mathcal{N}({\bf \Omega})^\bot\right\} \nonumber \\ {\bf v}_{c\bar{o}} \equiv \left\{{\bf x} \arrowvert {\bf x} \in \mathcal{R}({\bf \Gamma}) \cap \mathcal{N}({\bf \Omega})\right\} \nonumber \\ {\bf v}_{\bar{c}o} \equiv \left\{{\bf x} \arrowvert {\bf x} \in \mathcal(R)({\bf \Gamma})^\bot \cap \mathcal{N}({\bf \Omega})^\bot\right\} \nonumber \\ {\bf v}_{\bar{c}\bar{o}} \equiv \left\{{\bf x} \arrowvert {\bf x} \in \mathcal{R}({\bf \Gamma})^\bot \cap \mathcal{N}({\bf \Omega})\right\} \nonumber \end{eqnarray} with: \begin{eqnarray} \tilde{\bf x} = \bf T^{-1} \bf x \nonumber \\ \dot{\tilde{\bf x}} = \bf T^{-1} \bf A \bf T \tilde{\bf x} + \bf T^{-1} \bf B \bf u \nonumber \\ \bf y = \bf C \bf T \tilde{\bf x} \end{eqnarray} Noting that \begin{eqnarray} \bf C \bf T = \left[\begin{array}{cccc} \bf c_1 & 0 & \bf c_3 & 0 \end{array}\right] \nonumber \\ \bf T^{-1} \bf B = \left[ \begin{array}{c} \bf b_1 \\ \bf b_2 \\ 0 \\ 0 \end{array}\right] \nonumber \\ \bf T^{-1} \bf A \bf T = \left[ \begin{array}{cccc} A_{11}& 0 &A_{12} & 0 \\ A_{21}& A_{22}&A_{23} &A_{24} \\ 0 & 0 &A_{33} & 0 \\ 0 & 0 &A_{43} & A_{44} \end{array}\right] \nonumber \end{eqnarray} Since:

• Controllable states cannot affect uncontrollable states
• Uncontrollable states may affect controllable states
• Observable states may affect unobservable states
• Unobservable states cannot affect observable states

The idea is to pick eigenvalues, \(\lambda\), giving the desired behavior, then determine a gain matrix \(\bf K\) that will give the controlled system those eigenvalues. This is done by solving the characteristic equation for \(\bf K\). This gives you one scalar equation for each \(\lambda\).

\[ \left| (\bf A - \bf B \bf K) - \lambda \bf I \right| = 0 \]

An identical but computationally easier approach is to solve this equation by matching the proper coefficients:

\[ \left| (\bf A - \bf B \bf K) - \lambda \bf I \right| = \prod_{j=1}^n(\lambda - \lambda_j) \]

We define some metric that our controller must minimize. For the LQ problem, we minimize:

\[ J = \int_{t_0}^{t_1} ({\bf x}^T{\bf Q}{\bf x} + {\bf U}^T {\bf R} {\bf u})dt + {\bf x}^T(t_1) {\bf S} {\bf x}(t_1) \]

This gives:

\[ {\bf u}(t) = -{\bf R}^{-1}{\bf B}^T{\bf P}(t){\bf x}(t) \]

Where \({\bf P}(t)\) solves the Riccati Equation:

\begin{eqnarray} \dot{\bf P} = -{\bf A}^T {\bf P} - {\bf P}{\bf A} + {\bf P}{\bf B} {\bf R}^{-1}{\bf B}^T{\bf P} - {\bf Q} \nonumber \\ {\bf P}(t_1) = {\bf S}\nonumber \end{eqnarray}

With our system

\[ \dot{{\bf x}} = {\bf A}{\bf x} + {\bf B} {\bf u}\quad {\rm and}\quad {\bf y} = {\bf C}{\bf x} \]

We have an observed state, \(\hat{\bf x}\):

\begin{eqnarray} \hat{ \bf y } = {\bf C} \hat{\bf x}\\ \dot{ \hat{\bf x}} = {\bf A} \hat{\bf x} + {\bf B} {\bf u} + {\bf L} \left( {\bf y} - \hat{\bf y} \right) \\ \end{eqnarray}

Where \({\bf L}\) is the gain matrix for the Luenberger observer.

\begin{eqnarray} {\bf e} = {\bf x} - \hat{\bf x} \\ \dot{{\bf e}} = \dot{{\bf x}} - \dot{\hat{\bf x}} \nonumber \\ \dot{{\bf e}} = ({\bf A} - {\bf L} {\bf C}) {\bf e} \end{eqnarray}

Where \({\bf e}\) is the error. The derivation is \(\dot{\bf e}\) is simple algebra. Thus, the problem of observation reduces to the problem of control. Note that the poles for \({\bf A} - {\bf L} {\bf C}\) can essentially be arbitrarily fast as the observer system only exists inside a computer. In practice, observer poles should be about 10 times faster than controller poles.

• \({\bf x} \in \Re^n\) – state vector
• \({\bf u} \in \Re^{m}\) – input vector
• \({\bf z} \in \Re^s\) – measurement vector
• \({\bf A} \in \Re^{n \times n}\) – process matrix
• \({\bf B} \in \Re^{n \times m}\) – input gain matrix
• \({\bf H} \in \Re^{s \times n}\) – state to measurement mapping matrix (${\bf z} = {\bf H} {\bf x} $)
• \({\bf K} \in \Re^{n \times s}\) – Kalman gain
• \({\bf P} \in \Re^{n \times n}\) – Error Covariance
• \({\bf Q} \in \Re^{n \times n}\) – Process Noise Covariance
• \({\bf R} \in \Re^{s \times s}\) – Measurement Noise Covariance

The separation principle states that one may design a feedback controller assuming real \(\bf x\) and an observer to find \(\hat{\bf x}\) independently, then feed \(\hat{\bf x}\) to the controller and have everything still work. This is explained by:

\begin{eqnarray} \dot{\bf x} = \bf A \bf x - \bf B \bf K \hat{\bf x} = (\bf A - \bf B \bf K)\bf x + \bf B \bf K \bf e \nonumber\\ \dot{\bf e} = (\bf A - \bf L \bf C)\bf e \nonumber \end{eqnarray}

These equations can then be combined into the following system:

\[ \left[ \begin{array}{c} \dot{\bf x}\\ \dot{\bf e} \end{array}\right] = \left[ \begin{array}{cc} \bf A - \bf B \bf K & \bf B \bf K \\ 0 & \bf A - \bf L \bf C \end{array}\right] \]

Because this is an upper triangular block matrix, only the diagonals determine its stability and the controller and observer design already proved those two terms are stable.