State space form of differential equation equal to constant

Question

How to represent a differential equation in state space form when differential equation is equal to constant.

I am trying to represent following two equations

$$\dfrac{d\,x_1}{dt} = u(t) - c_{1},$$

and

$$\dfrac{d\,x_2}{dt} = c_{2}.$$

From first equation, I removed constant term using the deviation variables, i.e, subtracting the steady state from the input and multiply by appropriate partial derivative, etc.

I don't know what to do with second equation. I want to represent this as system of equations of the form

$$ \dot{x}(t) = A\,x(t) + B\,u(t). $$

Any help is appreciated. Thanks!!

Answer 1

Define new variables

$$z_1(t) = x_1(t) + c_1 t$$ $$z_2(t) = x_2(t) - c_2$$

and differentiate with respect to time $t$ to obtain

$$\dot{z}_1 = \dot{x}_1+c_1 = u(t)-c_1+c_1=u(t)$$ $$\dot{z}_2 = \dot{x}_2-c_2 = c_2-c_2=0.$$

Finally, we obtain:

$$\dot{\boldsymbol{z}}=\begin{bmatrix}0 &0\\ 0 &0 \end{bmatrix}\boldsymbol{z}+\begin{bmatrix}1\\0 \end{bmatrix}u.$$

Note, that the system is not controllable because $u(t)$ is only influencing the first state $z_1$ (which could be seen from the first form of the equation as well).

If you have discretized the state equation you can also discretize the substitution as

$$z_1(k\Delta t)=x_1(k\Delta t)+c_1k\Delta t$$ $$z_2(k\Delta t) = x_2(k\Delta t)-c_2.$$