Stability of a control system

Stability is something we all need in our lives .You may have a different perspectives about what exactly meant by being in a stable state but we all have some common aspects about it. In control systems engineering it’s essential to have a stable system otherwise the system will be impractical to setup in real world no matter how elegant is your design. When we are talking about the stability of a control system it corresponds the ability of a system to produce bounded output signals to bounded input signals. When we are talking about bounded signal I would think it as a signal with a finite energy (magnitude).When I first learn about BIBO systems (Bounded Input Bounded Output) I always had a confusion what is actually meant by bounded in real case, especially with the impulse signal .The definition tells us that it has an infinite magnitude at t=0 (when the duration is close to zero) so I wondered how could be the impulse signal is bounded signal but if we relate it to real world cases we can see this concept can model lots of forces like impulse force which acts on a body for a very short time and has a very high magnitude compared to a conventional force . Impulse signals always have a finite energy therefore I consider a bounded signal means the signal has a finite amount of energy, It can’t be infinite. So if a system is a BIBO system it will not produce an unbounded signal to a bounded signal. we only consider about BIBO systems. As well as that, all the systems we consider are LTI system (Linear Time invariant signals) which means that there is only single output to a single input(one to one relation) for all t and if a system has an output value other than zero at t=0 the system is not considered as a linear in control system. Time invariant means the system will produce the same output regardless of the time difference if the inputs and the initial conditions are identical in two instances. I mentioned those concepts here although they are not relevant here since we talk about LTI, BIBO system only.

Before I proceed to talk about the stability of a system it may be good to explain few things about a typical control system. Most of the time all the analysis of control systems are done using transfer functions of real system. Laplace transformation converts all the time varying functions to frequency varying functions in complex domain(s domain) and this transformation make the analysis very easy to carry out. The following diagram shows a typical block diagram of a control system.

Plant: Plant is the system we need to control described as a transfer function(s domain). It would be any physical system and several very different physical systems can be modelled using the same transfer function.

Sensor: The sensor component measures the value of an output specified by the design and sends it as a feedback to the controller.  

Controller:  Controller is a computing device (A computer) that compares the sensor output and reference provided as the input to the system and generates control signals to Actuator using PLCs.

Actuator: A device or a mechanism that controls the plant in a desired manner.

Now back to stability, 

having a stable system is a good and essential thing in control engineering but finding the perfect stable state for a given system will be impossible since we can’t predict how the system will be behaving in future. Any physical system change it’s qualities with time and these changes might cause the system to behave in an unpredictable manner. For example, a motor will not output it desired torque after some years due to mechanical ware and a motor control system might behave unexpectedly if this doesn’t take into account. Therefore rather than using just one stable state engineers design the system with stability margins. So system will be able to handle small parameter changes to input.  This helps to tackle any unpredictability in system and makes it more robust.

How do we determine the stability of a system? The method is easy to understand but very hard to do it without using a computer  most of the time. It’s not that difficult to determine the stability nowadays with the help of advanced computer softwares like MATLAB. But before the arrival of computers people had to find methods to determine the stability of a system by hand. Few methods are listed below. I’m hoping to discuss these topics in future. Each of this method has it’s own pros and cons and becomes handy according the problem you are dealing with.  Although using computer programs are the wildly used method in control engineering, an engineer should have an in depth knowledge about those methods in order to understand and interpret the results.

• ·         Root locus method.
• ·         Bode Plots.
• ·         Nyquist plots.
• ·         Routh-Hurwitz criterion.

Determining the stability

As I mentioned above in order to assess the stability of a system all we have to know is whether the poles of the system (closed loop) is located at the right or left side of s plane.                                          If all the poles are located in left half plane then the system will be stable.
If a single pole is located on right half s plane then the system will be unstable.
Furthermore if there is at least single multipole on the imaginary axis the system will be unstable.
If there is at least single pole on the imaginary axis then the system is said to be critically stable. It’s on the verge of being unstable.

Let’s take a simple example .Let G(s) = N(s)/P(s) = 1/(s+2)(s+3) . G(s) is the open loop transfer function . So the poles of the system are -2 and -3. And these values(poles ) are negative and therefore they are located on left half plane of s plane. Therefore the system will be stable. If we convert into time domain this result will be obvious.

By taking partial fractions of G(s),             G(s) = 1/(s+2)- 1/(s+3)

And By taking Laplace inverse of G(s) ,   g(t)= exp(-2t)-exp(-3t). 

So it is obvious when t goes to infinity g(t) will decay in an exponential manner. Therefore this system will be a stable system. But instead of the system poles were +2 and +3 then g(t) = exp(2t)-exp(3t) and when t goes to infinity t this system will produce unbounded (infinite)results, which cause malfunctions in a realistic system.

If above system (g(t)) was connected to a unity feedback system , the system’s new transfer function will be H(s)=G(s)/1+G(s) and all we have to do is to solve the denominator for roots and the roots will be new poles of the system. This 1+G(s) =0 Is called the characteristic equation of the system since it will determine the stability of the system.

In conclusion it is very important to know whether a system will be able to stabilize itself and fortunately there are many tools we can use to verify the stability and the stability margins of the systems. It is an essential aspect in Control engineering .