29. Reachability and Controllability#

29.1. Time-continuous linear systems#

The solution of the linear system

\[\begin{split}\begin{aligned} \dot{\mathbf{x}} & = \mathbf{A} \mathbf{x} + \mathbf{B} \mathbf{u} \\ \mathbf{y} & = \mathbf{C} \mathbf{x} + \mathbf{D} \mathbf{u} \\ \end{aligned}\end{split}\]

with initial conditions \(\mathbf{x}(0) = \mathbf{x}_0\) reads

\[\mathbf{x}(t) = \boldsymbol\Phi(t,t_0) \mathbf{x}_0 + \int_{\tau=0}^{t} \boldsymbol\Phi(t,\tau) \mathbf{B}(\tau) \mathbf{u}(\tau) d \tau \ ,\]

29.1.1. Reachability#

A state \(\mathbf{x}_1\) is reachable if there’s a control input \(\mathbf{u}(t)\) that can move the system from the initial state \(\mathbf{x}_0 = \mathbf{0}\) to the final state \(\mathbf{x}_1\) in finite time.

29.1.2. Controllability#

A state \(\mathbf{x}_0\) is reachable if there’s a control input \(\mathbf{u}(t)\) that can move the system from the initial state \(\mathbf{x}_0\) to the final state \(\mathbf{x}_1 = \mathbf{0}\) in finite time.

Optimal input and controllability Gramian

Let’s solve the optimal input defined as the one with the minimal cost \(\frac{1}{2}\int_{\tau=0}^{t_1} \left| \mathbf{u}(\tau) \right|^2\), subject to the equations of motion as constraints. todo Here the final time \(t_1\) is assumed to be known, but in general it’s not: include \(t_1\) in the optimization, or run several optimizations to find optimal \(t_1^*\).

The cost function of the constrained problem reads

\[\widetilde{J}(\mathbf{u};\boldsymbol\lambda) = \frac{1}{2} \int_{\tau=0}^{t_1} \mathbf{u}^T(\tau) \mathbf{u}(\tau) d \tau + \boldsymbol\lambda^T \left( \mathbf{x}_1 - \int_{\tau=0}^{t_1} \boldsymbol\Phi(t_1,\tau) \mathbf{B}(\tau) \mathbf{u}(\tau) d \tau \right) \ .\]

Variation w.r.t. \(\boldsymbol\lambda\) gives the solution of the equation, i.e. the constraints. Variation w.r.t. the control \(\mathbf{u}(\tau)\) gives the optimal control as a function of \(\boldsymbol\lambda\)

\[\mathbf{u}(\tau) = \mathbf{B}^T(\tau) \boldsymbol\Phi^T(t_1,\tau) \boldsymbol\lambda \ .\]

Replacing this expression in the solution of the dynamical system, an expression of \(\boldsymbol\lambda\) in terms of the final state \(\mathbf{x}_1\) follows,

\[\mathbf{x}_1 = \int_{\tau=0}^{t_1} \boldsymbol\Phi(t_1,\tau) \mathbf{B}(\tau) \mathbf{B}^T(\tau)\boldsymbol\Phi^T(t_1,\tau) d \tau \boldsymbol\lambda = \mathbf{W}_c(t_1) \boldsymbol\lambda \ ,\]

having introduced the definition of the controllability Gramian, \(\mathbf{W}_c(t_1)\).

  • If \(\mathbf{x}_1 \in R(\mathbf{W}_c(t_1))\) is in the range of the controllability Gramian, at least a value of the Lagrange multiplier \(\boldsymbol\lambda\) that satisfies the relation - and thus a control driving the system from \(\mathbf{0}\) to \(\mathbf{x}_1\) - exists.

  • If the controllability Gramian is full-rank, it’s invertible, the solution of the problem is unique and reads

    \[\boldsymbol\lambda = \mathbf{W}_c^{-1}(t_1) \mathbf{x}_1 \ ,\]

    and the cost of the control is

    \[J = \frac{1}{2} \mathbf{x}_1^T \mathbf{W}_c^{-1} \int_{\tau=0}^{t_1} \boldsymbol\Phi(t_1,\tau) \mathbf{B}(\tau) \mathbf{B}^T(\tau)\boldsymbol\Phi^T(t_1,\tau) d \tau \mathbf{W}_c^{-1} \mathbf{x}_1 = \frac{1}{2} \mathbf{x}_1^T \mathbf{W}_c^{-1} \mathbf{x}_1 \ .\]

  • In general, using singular value decomposition of the Gramian \(\mathbf{W}_c = \mathbf{U} \boldsymbol\Sigma \mathbf{U}^*\) (as the Gramian is symmetric, \(\mathbf{V} = \mathbf{U}\))

    \[\mathbf{u}_{\mathbf{x}_1} = \boldsymbol\Sigma \mathbf{u}_{\boldsymbol\lambda} \ ,\]

    with \(\mathbf{u}_{\mathbf{v}} := \mathbf{U}^* \mathbf{v}\), the projection of the vector \(\mathbf{v}\) onto the unit orthogonal base of \(\mathbb{R}^n\). A solution \(\mathbf{u}_{\boldsymbol\lambda}\) exists if zero singular values \(\sigma_i = 0\) correspond to null projection of the vector \(\mathbf{x}_1\) onto the \(i^{th}\) vector of the base \(\mathbf{u}_i^* \mathbf{x}_1 = \left\{ \mathbf{u}_{\mathbf{x}_1} \right\}_i = 0\). Otherwise a solution of the equation does not exist: in this situation, there’s no control law driving the system into \(\mathbf{x}_1\), an error in the final state exists, and the problem can be formulated as a optimization problem weighting both the control and the error (resulting in a least square solution of a linear system with no solution?).

Controllability Gramian satisfies the Lyapunov equation

\[ \dot{\mathbf{W}}_c(t) = \mathbf{B}(t) \mathbf{B}^T(t) + \mathbf{A}(t) \mathbf{W}_c(t) + \mathbf{W}_c(t) \mathbf{A}^T(t) \ , \]

with initial conditions \(\mathbf{W}_c(0) = \mathbf{0}\), as the integral vanishes with the same values of the lower and upper extremes of integration.

Lyapunov equation for the controllability Graminan
\[\begin{split}\begin{aligned} \frac{d}{dt} \mathbf{W}_c(t) & = \frac{d}{dt} \int_{\tau=0}^{t} \boldsymbol\Phi(t,\tau) \mathbf{B}(\tau) \mathbf{B}^T(\tau)\boldsymbol\Phi^T(t,\tau) d \tau = \\ & = \mathbf{B}(t) \mathbf{B}^T(t) + \int_{\tau=0}^{t} \partial_t \boldsymbol\Phi(t,\tau) \mathbf{B}(\tau) \mathbf{B}^T(\tau)\boldsymbol\Phi^T(t,\tau) d \tau + \int_{\tau=0}^{t} \boldsymbol\Phi(t,\tau) \mathbf{B}(\tau) \mathbf{B}^T(\tau) \partial_t \boldsymbol\Phi^T(t,\tau) d \tau = \\ & = \mathbf{B}(t) \mathbf{B}^T(t) + \mathbf{A}(t) \mathbf{W}_c(t) + \mathbf{W}_c(t) \mathbf{A}^T(t) \ , \end{aligned}\end{split}\]

as \(\boldsymbol\Phi(t,t) = \mathbf{I}\), and \(\partial_t \boldsymbol\Phi(t,\tau) = \mathbf{A}(t) \boldsymbol\Phi(t,\tau)\).

Controllability for LTI systems

For LTI systems \(\boldsymbol\Phi(t,t_0) = e^{\mathbf{A}(t-t_0)}\). The controllability Gramian becomes

\[\mathbf{W}_c(t) = \int_{\tau=0}^{t} e^{\mathbf{A} (t-\tau)} \mathbf{B} \mathbf{B}^T e^{\mathbf{A}^T (t-\tau)} d \tau\]

A vector \(\mathbf{v}\) belongs to the null space of \(\left( e^{\mathbf{A}t} \mathbf{B} \right)^T\) if

\[0 = \mathbf{v}^T e^{\mathbf{A}t} \mathbf{B} \ .\]

Using the definition of the exponential matrix \(e^{\mathbf{A}t} := \sum_{n=0}^{+\infty} \frac{\mathbf{A}^n t^n}{n!}\), it follows

\[0 = \mathbf{v}^T \sum_{n=0}^{+\infty} \frac{\mathbf{A}^n t^n}{n!} \mathbf{B}\]

Thus…, a vector \(\mathbf{v}\) is in the null-space of \(\mathbf{W}_c\) (symmetric! Be precise and use orthogonality between range and kernel of the adjoint if it’s orthogonal to the columns of the matrix

\[\mathcal{C} = \begin{bmatrix} \mathbf{B} & \mathbf{A} \mathbf{B} & \dots & \mathbf{A}^{n-1} \mathbf{B} \end{bmatrix} \ ,\]

i.e. \(\mathbf{v}^T \mathcal{C} = \mathbf{0}\). If the column space of matrix \(\mathcal{C}\) spans \(\mathbb{R}^n\), then the system is fully controllable.

Obs. If \(\mathbf{v}^T \mathbf{A}^n \mathbf{B}\) for \(n = 0:N-1\), thus it holds for all the integer exponent \(n \ge N\), as a consequence of the Cayley-Hamilton theorem, i.e. \(\mathbf{A}^n\) can be written as a linear combination of the powers \(\mathbf{A}^k\), \(k = 0:n-1\).

29.2. Time-discrete linear systems#