PI/PID control via State Space Formulation

Example I

Consider the first example of
$$G(s) =\frac{k}{2s+1} = \frac{b}{s+a}$$

We shall try to control this system with a PI control.

$$G(t) = K_I \int e dt + K_C e(t) = [K_I, K_C] [\int e dt, e(t)]^T$$

Let $x_1 = \int e dt$, this means that $\dot x_1 = x_2 = e(t)$

$$G(t) = [K_I, K_C][\int e dt, e(t)]^T = [K_I, K_C] [x_1, x_2]^T$$

As per the State Space formulation,

$$\dot X = AX + BU$$

Note,
$\dot x_2 = \dot e$
$e = r -y = \frac{-bu}{s+a}$
$\dot e +ea = -bu$
$$\dot x_2 = -x_2a -bu$$
and from the definition given above,
$$\dot x_1 = x_2$$

Thus we can form both A and B matrices.
$A = \begin{array}{cc} 0 & 1 \\ 0 & -a \\ \end{array}$

$B = \begin{array}{cc} 0 \\ -b \\ \end{array}$

Also $U = -kX$,

$$\dot X = AX + BU$$
$$sX = AX + B(-kX)$$
$$X(sI -A+kB) = 0$$

Thus the eigenvalues for the closed loop system will be the solutions to the equation,
$$|sI -A+kB| = 0$$

Desired closed loop Transfer Function is,
$D(s) = \frac{1}{\lambda s +1}$
But as we have a second order system, we restate the desired closed loop T.F for comparison purposes,
$D(s) = \frac{(z s+1)}{(\lambda s +1)(z s+1)}$

Thus to find the k value we can now compare the desired to actual closed loop pole equations.

Example II

Given system T.F $G(s) = \frac{1}{s}$,
we need to find the Proportional control for this system.

$$u(t) = K_c e(t)$$

we define $x_1 = e(t)$
from the diagram we can see,
$e = r - y$
$e = \frac{-u}{s}$
$\dot e = -u$

which means $\dot x_1 = -u$

Thus,
$A_{1 \times 1} = 0$
$B_{1 \times 1} = -1$

The desired closed loop T.F is,
$$G_{CL}(s) = \frac{1}{\lambda s +1}$$

Similarly as the first example find $k$ by solving,
$$|sI -A+kB| = \lambda s +1$$

Example III

This time consider a 2nd order system,

$$G(s) = \frac{k}{(z_1s+1)(z_2s+1)}$$

While the desired closed loop system is,

$$G_{CL}(s) = \frac{1}{\lambda s +1}$$

We restate the specifications because the system is a second order system.

$$G_{CL}(s) = \frac{(z_1s+1)(z_2s+1)}{(z_1s+1)(z_2s+1)(\lambda s +1)}$$

We will now control this system using PID,

$$u = K_I \int e dt + K_C e(t) +K_D \frac{de}{dt}$$
$$u =[K_I, K_C, K_D] [\int e dt, e(t), \frac{de}{dt}]^T$$

Lets define the following terms,
$$x_1 = \int e dt ; \dot x_1 = x_2 = e(t) ; \dot x_2 = x_3 = \frac{de}{dt}$$

So we can rewrite $u$ as,
$$u =[K_I, K_C, K_D] [x_1, x_2, x_3]^T$$

As $e = r - y = \frac{-k u}{(z_1s+1)(z_2s+1)}$
double differentiating,
$$\ddot e = f(x_1,x_2,x_3) + g(u)$$

From these we can find $A_{3 \times 3}$ and $B_{3 \times 1}$

Using these we can find $k$ by solving,
$$|sI -A+kB| = (\lambda s +1) (z_1s+1)(z_2s+1)$$

Inference

• So we have used PI/PID control via State Space formulation for the above system specifications.
• IMC based tuning or synthesis may also be used for these problems
• The tuning procedure is straightforward and relatively simple. The solution of all systems take the same form.
• We prefer PI/PID control via State Space formulation when the desired closed loop system specifications are performance based.