Chapter 2

# 2.4 Determining Stable Range for Proportional Controller Operations

Consider a closed loop system under Proportional Control in Figure 2‑2:

The closed loop transfer function and the system Characteristic Equation are:

 $G_{cl}(s) = \frac{K_{prop}G(s)}{1 +K_{prop}G(s)\cdot H(s)}$ $1+ K_{prop}G(s)\cdot H(s) = 0$ Equation 2‑5

The variable parameter – the Proportional Controller Gain – is now a part of the system Characteristic Equation and will feature in the Routh Array. This allows us to define the required condition for a stable, safe, range of Controller operations.

### Example

Consider the following control system:

• Is this system open-loop stable?
• Determine a range of gains K required for a stable operation of this closed loop system.

Solution:

#### Part 1: Open Loop

Open loop characteristic equation is:

$Q(s) = s^{3}+ 11s^{2}+ 38s+ 4 = 0$

The system is open-loop stable, based on the Routh-Hurwitz Stability Criterion. We can also find numerically (MATLAB), what the poles of the open system are: -5.45 +j2.68, -5.45 – j2.68, -0.11.

The system Open Loop Transfer Function components are:

$G(s)=\frac{4K_{prop}}{s^{3} +11s^{2}+ 38s+ 4}, H(s) = 1$

### Part 2: Closed Loop

The system Closed Loop Transfer Function is:

$G_{cl} = \frac{G(s)}{1 +G(s)H(s)}=\frac{\frac{4K_{prop}}{s^{3}+ 11s^{2}+ 38s +4}}{1 +\frac{4k_{prop}}{s^{3} +11s^{2}+ 38s+ 4}} = \frac{4K_{prop}}{s^{3}+ 11s^{2} +38s +(4K_{prop} +4)}$

Apply the Routh-Hurwitz criterion to the closed loop characteristic equation:

$s^{3} +11s^{2}+ 38s +(4K_{prop}+ 4) = 0$

The Necessary Condition is:

$4 +4K_{prop} > 0, K_{prop} >-1$

The Sufficient Conditions from Routh Array:

The resulting conditions are:

$\frac{11\cdot 38 - (4+ 4K_{prop})}{11} > 0, 4K_{prop} < 414, K_{prop} < 103.5$

The total condition is: $-1 < K_{prop} < 103.5$

Note that the practical range of stable controller operations, as opposed to the previous purely mathematical condition, is $0Critical Gain values. Again, of practical interest is only the upper, positive Critical gain value. What happens when the Controller gain reaches that Critical gain value? When [latex]K_{prop} = 103.5$ :

$Q_{aux}(s) = 11s^{2} +418$

$Q_{aux}(s) = 0$

$s^{2} +38 = 0$

$s_{1} = j\sqrt{38} = j6.16$

$s_{2} = -j\sqrt{38} = -j6.16$

As the pole-zero map illustrates, the location of marginally stable poles on the Imaginary axis corresponds to the frequency of sustained oscillations, \$\omega_{crit} = 6.16 rad/sec$. This corresponds to the period of oscillations equal to 1.02 seconds. Step response of the system when $K_{prop}=103.5$ is also shown.