System Stability Using MATLAB Software 2

In this chapter we explored the concepts of system stability from both the classical and the state-space viewpoints. We found that for linear systems, stability is based on a natural response that decays to zero as time approaches infinity. On the other hand, if the natural response increases without bound, the forced response is overpowered by the natural response, and we lose control. This condition is known as instability. A third possibility exists: the natural response may neither decay nor grow without bound but oscillate. In this case the system is said to be marginally stable.

We also used an alternative definition of stability when the natural response is not explicitly available. This definition is based on the total response and says that a system is stable if every bounded input yields a bounded output (BIBO) and unstable if any bounded input yields an unbounded output.

Mathematically, stability for linear, time-invariant systems can be determined from the location of the closed-loop poles:

• If the poles are only in the left half-plane, the system is stable.
• If the poles are only in the right half-plane, the system is unstable.
• If the poles are on the jω-axis and in the left half-plane, the system is marginally stable as long as the
poles on the jω-axis are of unit multiplicity; it is unstable if there are any multiple jω-axis.

In this post we will look for the stability in aeroplane by looking to the control system block diagram below


The block diagram represent the system for determining the angle (theta) of the aerofoil in aeroplane system in order to produce the output from the desire input.

After getting the equation from the diagram, with matlab we can find the stability of the system by determining the value K (Gain) from Routh-Hurwitz Criterion

From the program and result above, we can use the MATLAB Simulation program to look either the gain that we get from the calculation can be use through the system. the value K (gain) is inserted through the simulated block diagram and the graph will show the final value which consist the transient response and the stready state error of the system

System Stability Using MATLAB Software 1

Stability is the most important system specification. If a system is unstable, transient response and steady-state errors are moot points. An unstable system cannot be designed for a specific transient response or steady-state error requirement. There are many definitions for stability, depending upon the kind of system or the point of view.

• A linear, time-invariant system is stable if the natural response approaches zero as time approaches infinity.
• A linear, time-invariant system is unstable if the natural response grows without bound as time approaches infinity.
• A linear, time-invariant system is marginally stable if the natural response neither decays nor grows but remains constant or oscillates as time approaches infinity.

Thus, the definition of stability implies that only the forced response remains as the natural response approaches zero. a stable system needed because in order to build certain system such as an aerofoil in aeroplane system a stable system play most important part in building in because the unstable system will make the whole part of the object that we will build is dangerous to human.