Question

A dynamical system with input signal $u$ is described by the following pair of differential equations $5\dot{x} + 0.5 \tan y - xy^2 = 0$, $(x^2 + 1)\ddot{y} + 8x\dot{x}y + 3x = u$. (i) Linearise this system around an equilibrium point that [14 marks] corresponds to $y = 0$. Hint: start by writing the system in its state space representation (it has 4 states) and use Prop. 3.4 to characterise its equilibrium points. Introduce deviation variables if necessary. (ii) Determine the poles of the transfer function of the [3 marks] linearised system. (iii) Show that a P-controller is not a suitable choice for this [4 points] system. Hint: you may use the BIBO instability Theorem.

          A dynamical system with input signal $u$ is described by the following pair of differential equations

$5\dot{x} + 0.5 \tan y - xy^2 = 0$,
$(x^2 + 1)\ddot{y} + 8x\dot{x}y + 3x = u$.

(i) Linearise this system around an equilibrium point that [14 marks]
corresponds to $y = 0$.

Hint: start by writing the system in its state space representation (it has 4 states) and use Prop. 3.4 to characterise its equilibrium points. Introduce deviation variables if necessary.

(ii) Determine the poles of the transfer function of the [3 marks] linearised system.

(iii) Show that a P-controller is not a suitable choice for this [4 points] system. Hint: you may use the BIBO instability Theorem.

A dynamical system with input signal u is described by the following pair of differential equations

5ẋ + 0.5 tan y - xy^2 = 0,
(x^2 + 1)ÿ + 8xẋy + 3x = u.

(i) Linearise this system around an equilibrium point that [14 marks]
corresponds to y = 0.

Hint: start by writing the system in its state space representation (it has 4 states) and use Prop. 3.4 to characterise its equilibrium points. Introduce deviation variables if necessary.

(ii) Determine the poles of the transfer function of the [3 marks] linearised system.

(iii) Show that a P-controller is not a suitable choice for this [4 points] system. Hint: you may use the BIBO instability Theorem.

Added by Mark S.

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