Ordinary Differential Equation

For homogeneous ($c = 0$) linear error dynamics:

Using $p$ coupled first order ODE’s defining $x = (x_1, \dots , x_p)$ where:

The ODE can be written as:

Then $\dot{x}(t) = A\;x(t)$ where:

The vector differential equation $\dot{x}(t) = A\;x(t)$ has the solution: $x(t) = e^{At}\;x(0)$

Error Response

Unit error response:

• response $\theta_e(t), t \gt 0$ of the controlled system
• for the initial conditions:
  • $\theta_e(t) = 1$ and
  • $\dot{\theta}_e(0) = \ddot{\theta}_e(0) = \cdots = 0$

Error response can be described by:

• Steady-state response
  • steady-state error: $e_{ss}$: asymptotic error $\theta_e(t)$ as $t \rightarrow \infty$
• Transient response
  • overshoot (if error response initially overshoots final steady state error): $\left|\frac{\theta_{e,min}-e_{ss}}{\theta(0)-e_{ss}}\right| \times 100 \%$
  • (2%) settling time: the first time $T$ such that $\left| \theta_e(t) - e_{ss} \right| \leq 0.02 (\theta_e(0)-e_{ss})$ for all $t \ge T$

Stability

$\dot{x}(t) = A\;x(t)$ converges to $x = 0$ if:

$A$ is negative definite. i.e. all eigenvalues of $A$ (which can be complex) have negative real components.

Eigenvalues

The eigenvalues of $A$ are given by the roots of the characteristic polynomial. i.e. complex values $s$ satisfying:

$det(sI - A) = s^p + a'_{p-1}s^{p-1}+ \cdots +a'_ss^2+a'_1s + a'_0 = 0$

Necessary condition for each root to have a negative real component: all coefficients $a'_0, \cdots, a'_{p-1}$ must be positive.

Stability

Error dynamics are:

• stable: if each root has negative real component
• unstable: if any root has positive real component

First-Order Error Dynamics

or

where:

• $\tau = b/k$: time constant

solution:

Second-Order Error Dynamics

Standard Second-Order Form

with:

• $\omega_n$: natural frequency
• $\zeta$: damping ratio

For mass-spring-damper analogy:

• $\omega_n = \sqrt{k/m}$
• $\zeta = b/(2\sqrt{k\,m})$

Characteristic polynomial:

Roots:

Stable if and only if $\zeta\omega_n >0$ and $\omega_n^2 >0$.

Damping

The system behaves different depending on the roots $s_{1,2}$

Overdamped

• $\zeta > 1$
• Roots $s_{1,2}$ are real and distinct ($s_1 \neq s_2$)
• Solution to ODE: $\theta_e(t) = c_1\,e^{s_1t} + c_2\,e^{s_2t}$
  • $c_1$ and $c_2$ can be calculated from initial condition

Critically Damped

• $\zeta = 1$
• Roots $s_{1,2} = -\omega_n$ are real and equal
• Solution to ODE: $\theta_e(t) = ( c_1 + c_2\,t) e^{-\omega_n t}$

Underdamped

• $\zeta < 1$
• Roots $s_{1,2}$ are complex conjugates
  • $s_{1,2} = -\zeta\omega_n \pm j\,\omega_d$
  • where $\omega_d = \omega_n \sqrt{1-\zeta^2}$: damped natural frequency
• Solution to ODE: $\theta_e(t) = (c_1\cos{\omega_d t} + c_2\sin{\omega_d t}) e^{-\zeta\omega_n t}$

Controller

P Controller

Simplest feedback controller:

with:

• actual position: $\theta(t)$
• desired position: $\theta_d(t)$
• error: $\theta_e(t) = \theta_d(t) - \theta(t)$

Error Dynamics

With constant desired (reference) velocity: $\dot{\theta}_d(t) = c$:

Solution to first-order homogeneous linear ODE:

This converges to $c/K_p$ as time goes to infinity.

PI Controllers

Error Dynamics

With constant desired velocity: $\dot{\theta}_d(t) = c$

Taking derivative:

Stable if $K_i > 0$ and $K_p > 0$.

The roots of the characteristic equation:

Feedforward Plus Feedback PI Controller

Add a term with the desired velocity $\dot{\theta}_d(t)$:

Multi-joint Robot

Same as for scalars but with vectors and matrices:

• actual position: $\theta(t) \in \mathbb{R}^n$
• desired (reference) position: $\theta_d(t) \in \mathbb{R}^n$
• P gain: $K_p = k_p\,I\in \mathbb{R}^{n \times n}$
• I gain: $K_i = k_i\,I\in \mathbb{R}^{n \times n}$

with:

• $k_p \in \mathbb{R}^{+}$ and $k_i \in \mathbb{R}^{+}$
• $I$: Identity matrix