SIGNALS AND SYSTEMS-VII PROPERTIES OF SYSTEMS STABILITY

EDITOR: B. SOMANATHAN NAIR

1. INTRODUCTION

A system is said to be stable, if at time t = nT = ¥, the system reaches its steady-state condition. In the steady-state condition, a system does not vibrate or oscillate and its amplitude remains within safe limits. It may be noted that if the amplitude exceeds these safe limits, the system will fail or get destroyed. Stability of systems can best be illustrated by the considering the following examples.

 Consider a chair with three legs of equal lengths. Let us assume that it is standing on these legs over a flat ground. We find that the chair is in a very stable state in this position and that we can sit on it without fear of falling down. If we try to swing the chair by applying a swinging force, it will only move slightly, and will immediately return to the stable state again. The chair resting in this position is said to be asymptotically stable.

            Now, consider the situation, wherein we remove one leg of the chair.  We can still balance it into a position that appears to be stable. However, we know that it is not really stable in this position, and that a small force (applied horizontally) can easily topple it from this position to ground. In this condition, we say that the chair is in an unstable state.

            Consider a third situation, wherein we have the chair modified such that it is converted into a swinging chair. Such a swinging chair is said to be marginally stable, since the chair is not really occupying a very stable position. In a marginally stable condition, a system will be in an unstable state; but the instability will be such that the amplitudes of vibration involved in the process will be well within safe limits.

            We can illustrate stability of electronic devices (and hence, systems) by considering the case of a bipolar junction transistor. Assume that the transistor under consideration is capable of handling an average value of 1 A of collector current. Let its maximum current rating be 2 A. If this transistor is operated at 1 A, then the heat developed in its collector region will be well within safe limits.

Now, consider the case of a transistor being used as a power amplifier. Suppose we are using a heat sink on top of the transistor. Then, we can increase the collector current beyond its rated capacity to some extent, i.e., we are operating the transistor above its rated current by using a heat dissipating device, which under normal circumstances will ensure its safe operation. However, in such situations, the transistor will also be getting heated above its normal temperature-withstanding capacity. But the heat sink will dissipate away the excess heat from its collector surface into the surrounding atmosphere. This keeps the transistor heat well within safe limits. If, however, by any chance, the current exceeds this safe value, then we find that the transistor gets destroyed. We therefore state that our transistor is in a marginally stable condition.

Now, suppose we are trying to operate the transistor at a current much higher than that required for marginally stable operation. Then, definitely the transistor will get destroyed as the heat generated in the collector region will be well above the safe limits. We call this state as the unstable condition.

Finally, one word about oscillations produced by oscillators: we can construct several sinusoidal and non-sinusoidal oscillators using electronic circuits. Such oscillators produce waveforms whose amplitudes vary at every instant; yet we find that they are stable, as their amplitudes are well within safe limits. These oscillators, therefore, belong to the class of marginally stable systems.

            As a final example of instability, consider a place where an extremely cold atmosphere exists. In such cold conditions, the human body may start vibrating. We know that every physical object or material having a mass has a natural frequency of vibration. When these vibrations, created by an external force like the cold atmosphere, coincide with the natural frequency of vibration of the human body, the amplitude of vibrations goes excessively high. This results in the sudden collapse of the owner of the body. So, we see that instability is an extremely dangerous condition in many cases.

 2. BOUNDED-INPUT, BOUNDED-OUTPUT CONDITION FOR STABILITY

We now derive the necessary and sufficient condition for testing whether a given system is stable or not. We have already seen that the relation between the input and output of a system given by the convolution equation:

  │ y(n) │  = │Σx(n)h(n‒k)│= Σ│x(n)││h(n‒k)│ (2)          

Equation (2) may also be written as:

                        │y(n)│= Σ│x(n ‒ k)││h(k)│ (3)    

                                                         

If the output is to be bounded (or finite) under steady-state, then we must have

                                   │y(n)│= K­₁           (4)                                                    

where K₁ = a constant. Using (3) and (4), we get

                      K₁ = Σ│x(n ‒ k)││h(k)│ (5)                                                                                       

Now, if the input is assumed to be bounded, then we have

           

                                 K₂ = │x(n ‒ k)│  (6)                                                      

where K₂ = another constant. Using this in (5), we find

                        K₁ = K₂ Σ│h(k)│  (7)

Equation (7) can be simplified to get the necessary condition for stability as

                              K =  Σ│h(k)│  (8)                                           

                                                          

where K = K₁/K₂ is a third constant. Equation (6) states that for a given system to be stable, its impulse response must be finite. Equation (8) can also be stated in another form. Thus, for a given system to be stable, we must have

                              Σ│h(k)│< K < ∞   (10)                                                 

Equation (10) is the necessary and sufficient condition for testing the stability of a given system. Since this is derived based on the bounded input/bounded output conditions, it is called the BIBO (Bounded-Input, Bounded-Output) condition for stability.

            We may approach the problem in a slightly different way. We can rewrite (4) by using (7) as:

                                 

                           │y(n)│= K₂ Σ│h(k)│  (11)

                                                        

Let us now assume that

                          │h(n)│= K (12)                                                    

where K  is a constant, as discussed above. Using (12) in (11) yields

                           │y(n)│= K₂ K = K₃  (13)                    

                                            

where K₃ is a new constant. Equation (13) says that to test for stability, we should see whether the output is bounded or not for a bounded input. The conditions based on this theory for testing BIBO stability are discussed in the following examples.

                                                                          

Example 1: Test the stability of the system given by the relation

y(n) = Ax­(n) (1)

Solution: For testing BIBO stability, the following rules are used:

·     First, assume that a bounded input is applied to the system to be tested. In some cases, we assume that the input is a constant. In other cases, we use the delta function, unit step function, etc., which are typical examples of bounded inputs. For example, the amplitude of the unit step function u(n) is always unity for any value of time n, and hence is always bounded at 1.

·  See whether the amplitude of the output of the system remains constant (i.e., bounded) at a fixed value, or tends to infinity, as n tends to infinity.

·       Conclude that the given system is stable, if the output remains less than or equal to a fixed value; otherwise it is unstable.

Using the first rule given above, let us rewrite (1) as

                                                y(n) = Au­(n)   (2)                                     

Since u(n) is bounded at 1 (or, unity), y(n) will be bounded at A. If A is a constant, then we conclude that the system is stable.

Example 2: Test the system given by y(n) = A cos (nω_o) u(n) for stability.

Solution: Following the procedure given above, we find that since u(n) is bounded at 1, y(n) will be bounded at A cos (nω_o). If A cos (nω_o) is constant, then we conclude that the system is stable.

Example 3: Test the stability of the system governed by y(n) = n cos (nω_o) u(n).

Solution: As n tends to infinity, we find that y(n) = n cos (nω_o) u(n) also tends to  infinity. This makes the function y(n) tend to infinity and hence the system is unstable.