control system
Say there is an system with open-loop transfer function $$G(s)=\frac{1}{s-1}$$
The system is definitely unstable.
If I put an any nonlinear system in cascade with \$G(s)\$ & from the output, a negative unity feedback. Is it possible to make the system close-loop transfer function is stable? It is assumed that the higher order harmonics are filtered in the close loop path.
The stability depends on both the poles and the zeros of the plant and controller.
To see this consider the asymptotes of the diagram of the loop gain/phase. Imagine moving the zeros changes the phase at crossover and therefore affects the stability.
Looking at it more formally, the dynamic characteristics and stability are determined by the poles of the closed loop transfer function.
Consider a control system as shown below.
Let the plant be represented as a transfer function of poles and zeros $$G=B/A$$ and let the Feedback (or controller) be represented as a transfer function of poles and zeros $$H=S/R$$
The closed loop transfer function is therefore: $$\dfrac{B/A}{1+\dfrac{BS}{AR}}$$ Where the demoniator is the characteristic equation which determines stability. Simplifying: $$\dfrac{BR}{AR+BS}$$
So the stability is determined by the characteristic equation (poles): $$AR+BS=0$$
Therefore the stability depends on the poles and the zeros.
For a minimum-phase system, the gain and phase margins must be positive for the system to be stable.
This system has two poles at the origin, and hence is not minimum-phase. Thus it can be stable even if the gain or phase margins are negative.
Best Answer
If you connect a simple linear gain, K, in series with the 1/(s-1) block, giving a transfer function: K/(s-1), and then form a negative feedback closed loop system around this new block, the closed loop transfer function will be K/[s+(K-1)], which is stable if K>1.
Another simple block that is unstable in the open loop is an integrator, which has transfer function: 1/s. It's easy to see that this is unstable by considering a unit step input. This will result in a unit ramp output (integral of a constant, A, is ramp, At), and this goes to infinity for large t. However, closing the loop around the integrator (with or without a series gain, K) will give a stable closed loop with transfer function: K/(s+K), or 1/(s+1). As well as being stable, the closed loop system has unity steady state gain (or 'DC gain'), which is characteristic of closed loop systems that have a pure integrator in the forward path transfer function.