Examples - Control Co-Design

Example 2.1: derive and analyze a state-space model¶

Consider a mass–spring–damper system with $m=2$ kg, $c=3$ N s/m, and $k=8$ N/m. The control input is force $u$ and the measured output is displacement $y=x$ .

Step 1: Write the dynamic equation.

2\ddot{x}+3\dot{x}+8x=u.

(1)

Step 2: Define the states.

x_1=x, \qquad x_2=\dot{x}.

(2)

Step 3: Write the first-order model.

\dot{x}_1=x_2, \qquad \dot{x}_2=-4x_1-1.5x_2+0.5u,

(3)

\dot{\mathbf{x}}= \begin{bmatrix}0&1\\-4&-1.5\end{bmatrix}\mathbf{x} +\begin{bmatrix}0\\0.5\end{bmatrix}u, \qquad y=\begin{bmatrix}1&0\end{bmatrix}\mathbf{x}.

(4)

Step 4: Analyze stability. The eigenvalues are the roots of

\lambda^2+1.5\lambda+4=0,

(5)

which are

\lambda=-0.75\pm j1.854.

(6)

Their real parts are negative, so the open-loop equilibrium is asymptotically stable.

Step 5: Interpret the result. The system oscillates, but the oscillations decay. If the designer raises stiffness to $k=18$ without changing damping, the oscillation frequency increases and the best feedback tuning may change.

Example 2.2: effect of PD feedback¶

Suppose Example 2.1 uses

u=-6x-4\dot{x}.

(7)

The closed-loop equation is

2\ddot{x}+(3+4)\dot{x}+(8+6)x=0,

(8)

with characteristic equation

2\lambda^2+7\lambda+14=0.

(9)

The closed-loop poles have a more negative real part than the open-loop poles, indicating increased damping and faster decay. Feedback improves the response by altering the effective dynamics.

Example 2.3: identify signals in cruise control¶

Consider a simple vehicle moving along a straight road:

m\dot{v}=F_t-c_rv-mg\sin\theta,

(10)

where $v$ is vehicle speed, $F_t$ is traction force, $c_r$ is a drag or rolling-resistance coefficient, and $\theta$ is road slope.

A sensible description is:

state: vehicle speed $v$ ;
control input: traction force $F_t$ or throttle command;
output: vehicle speed $v$ ; and
disturbance: road slope $\theta$ .

This first-order model reminds us that a useful state set need not contain many variables.

Common modeling mistakes¶

Treating outputs as states without justification. Outputs may equal states, but they do not have to.
Ignoring disturbances. This usually creates unrealistic conclusions about controller performance.
Using too many states. More states are not always better; the state set should be minimal and meaningful.
Forgetting units. Dynamic equations must be dimensionally consistent.
Assuming plant design changes only constraints or cost. In CCD, plant design often changes the differential equations themselves.
Confusing stability with performance. A stable system is not necessarily fast, efficient, comfortable, or optimal.