More on the controllability of linear time-invariant systems

Control of a VTOL Aircraft: Motion Planning and Trajectory Tracking

“…Using the transformation suggested in [12], let z 1 = x 1 − ε sin θ, z 2 = x 2 − εω cos θ, w 1 = y 1 + ε(cos θ − 1), w 2 = y 2 − εω sin θ, ξ 1 = θ, and ξ 2 = ω, (2) reduces tȯ [2] and [3]. Let the system state vector be x = [z 1 ,z 2 ,w 1 ,w 2 ,ξ 1 ,ξ 2 ] T .…”

“…Using the approach in [2] and [3] the polynomials z 1 (t) = p(t) and w 1 (t) = q(t) will determine the entire open-loop trajectory connecting the initial state with the desired final target in the state-space.…”

“…However as far as the underactuated VTOL aircraft is concerned, we present here a novel application of the polynomial control approach for synthesizing controllers. Essentially we adopt tools from the earlier works [2], [3], and [1], which show that in linear time-invariant systems controllability is equivalent to polynomial controllability and it is always possible to transfer a controllable linear system from a given state to a desired one along a polynomial trajectory, by means of a polynomial input. The polynomial coefficients are computed simply by solving a linear algebraic equation.…”

Applications of polynomial controllability to optimal control in linear and nonlinear systems: the vehicle kinematic model case

“…Furthermore, from [4] the procedure to determine a polynomial pair {x(·), u(·)} that satisfies the differential equation (1) and the end-point conditions is obtained as follows. Let…”

“…In particular, a sequence of papers have shown that a controllable linear system is also polynomial controllable ( [3], [4], and [1]). The implication of this result is that a polynomial input ensures the transfer of the system along a polynomial trajectory and the control function can be computed by solving a set of linear algebraic equations (rather than by the controllability Grammian), the solution of which yields the polynomial coefficients of the desired input.…”