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 .  

Electrical power industry trends in Australia and the problems associated with it

Electrical energy is the main energy type we use in our almost every aspect of our lives. It is impossible to imagine a world without it. Although is a very basic need in modern society most people take it for  granted but generating electricity power and distributing it with a good efficiency to the places where it’s been used is a very complex and carefully designed process. I recently read a report by climate council of Australia (“Australia’s electricity sector: ageing inefficient and unprepared”) and as the title implies it focuses on the matters related to Australia electricity industry and propose the solutions for them .So I thought to share some key points I read in that report and share with you. I personally believe that it’s a responsibility of engineering community to educated general public about how science and technology is important for a better future and for a sustainable development. All the facts and data I go thorough in this article were extracted from that report and all the credit should go to the writer and the climate council of Australia.

Globally the energy sector accounts for the largest proportion of greenhouse gas emissions which is the main cause for climate change around the world as scientists speculate. In order to avoid the effects of climate changes countries have agreed to keep the temperature rise below 2 degrees and in order to achieve this goal every country has to reduce their greenhouse gas emissions it needs large scale changes to how countries produce their electricity and other energy methods.

Figure 1-Global energy related CO2 emission

The above graph shows how the energy sector is responsible for the emission of CO₂ one of the main component of greenhouse gases has increased over the time. As the graph shows it is an exponential growth. Coal is the main reason for this and as everybody knows coal is one of the main source for conventional electricity production. Australia places as 9th country in the top ten countries that emit greenhouse gases in electricity production. In 2011/2012 about 91% of Australia’s electricity was generated using fossil fuels.75% of the portion was from coal and 25% from natural gas. Australia’s per capita CO₂ emission is the world highest in 2011.So it is obvious that electricity generation in Australia is not ecofriendly as people might think. Australia produces 60% of emissions per MWh than USA. Even Australia is well behind China when reducing the emission of GHG (greenhouse gases) .Over the decade to 2012, China has reduced the GHG emission by electricity up to 16% but Australia was able to reduce that by less than 4%.As a result electricity sector responsible 33% of GHG emissions in Austalia.

Table 1-Global Emitters of CO2

The reason behind this is the way electricity is generated in Australia. As mentioned above coal and natural gas are the main sources used to generate electricity .Coal and gas power plants operates in Australia are getting old. Currently the average age of coal power plants in Australia is over 30 years where it is less than 2 years in the global scale and by 2030 over 65% of coal power plants will be over 40 years old. The age of the plant is an important parameter when it comes to it’s efficiency since an old plant operates on technology which is established over decades ago. The technology related with electricity generation is developing rapidly since these plants operates on low efficiency it becomes very expensive to produce electricity it’s and waste of natural resource. Some might see this is not a problem since we have time but building a megawatt scale power station will take at least a decade to plan, finance and build and operate in the required manner. Various much efficient technologies were introduced during the past decades in order to reduce the Carbon emission from plants but the old power plants are unable to adopt to these relatively new technologies since it’s not economically beneficial and those plants are constructed on old technology hence they are locked down to use that instead of moving to more efficient new technologies. One parameter to measure the emissions from a plant is cycle efficiency. It is determined by the steam temperature and the pressure conditions at the limits of a cycle. These efficiency are divided in to 3 categories, Subcritical, Super critical and ultra-critical. Most Australian coal power stations are subcritical hence they have become one of the least inefficient power stations in world.

Figure 2-Fuel usage for electricity generation in Australia 2012-13

Figure 3-Electricity production in Australia

As people realise the negative effects of using fossil fuel based electricity generation the world is slowly moving towards replacing fossil fuel with renewable energy sources like solar, wind, hydro and fuel cell (H₂) technologies. Hydro power is the leading renewable source in the world at the moment. AS it can be seen in above graphs it is the same trend in Australia. In Australia hydro power has supplied 8186MW (16.1% of total capacity) .New South Wales got the biggest portion of hydro power stations (55%). In global scale hydro power has supplied around 15% of electrical energy needs.

Table 2-Hydro power generation distribution worldwide

Wind power is also another emerging renewable source since it’s large reductions in costs associates with installation and equipment .Australia got one of the best wind fields in the world and Australia was able to generate over 6000 GWh in 2012/13 at an average capacity factor of 34%.(Capacity factor measures the ability to extract and store energy from wind by a turbine over a period of time).South Australia is the leading state is wind energy production. It is expected that there would be more 8900 MW of wind energy will be available by 2020 but still Australia is well behind most developed countries.

Solar power is another wildly used renewable energy source. It’s becoming more popular than wind because of it’s domestic use. While coal and other fossil fuels are expected to become expensive in future it’s obvious that solar energy will be less expensive than coal in near future because of the rising demand for solar PV. The advantage of solar is it can act as a distributed power source in houses and commercial premises or it can be scaled up to MW size and feed to the grid.

Over 1 157 000 Australian households had installed 3039 MW of solar PV by 2013. South Australia has the largest proportion of power generated by Solar. Although there is a decline in installing new solar PV in houses due to reductions in Feed in tariff rates it is expected that by 2020 the power generated by solar PV could be doubled. Although household solar PV is a popular alternative to grid power, Australia has only few field scale solar power stations. A 10 MW station at Greenough River in WA and 1MW project at Uterne in Northern territory. Besides these existing projects there are several new projects that are under construction. A 20 MW project at Royalla in ACT. A 102 MW project at Nyngan in NSW. A 3 MW project at Brokenhill.

The popularity of PV has caused them to reduce it’s price and more affordable to households. Furthermore many solar markets around the world getting closer to achieving ‘Socket parity’ point, where it is more economical to produce electricity using solar rather than buying it from the grid. Nowadays it’s more economically beneficial if people use electricity generated by solar for their needs rather than feeding into the grid. Because in that way they will reduce the amount of electricity they consume from grid. One of the major drawbacks of solar PV is it is unable to provide power when it is needed most. In generally the maximum solar insolation is received during noon time but the electrical peak arrives during evening times and most of the time solar power is unable to supply the demand. Therefore households have to use more grid generated power during this period. Therefore solution is to use battery system to store excess energy produced during day time and consume it during peak time. The price of the battery systems are relatively high but it’s becoming cheaper because of it’s popularity and with new battery designs like Tesla Power Wall it has become a worthy investment.    

There is no doubt that sooner or later the whole world has to change their way of producing energy therefore earlier is better. It could be cheaper to use fossil fuels at present but in about 20 years it won’t so nations should start planning and implementing renewable energy plants from now. It is a fact that Australia’s existing fossil fuelled plants can’t compete with renewable energy methods in long term because it’s obvious that price of fossil fuel is becoming higher while the cost of renewable energy solution becoming cheaper. According to the studies wind is the most suitable energy source for Australia along with field scale solar PV.

Table 3-Cost reduction potential for onshore wind systems

Table 4-Cost reduction potential for PV modules and systems

Although Solar, Wind, Hydro and other renewable energy sources are very promising technologies one problem associating with them is they are very sensitive to climate changes. It is very difficult to predict what would be the climate conditions in a region by 10 or 20 years. If the conditions changed dramatically in future the expected output will not be achieved by those plants and the investments done will be wasted. Therefore these aspects has to be taken into account when planning a large scale renewable plants .On the other hand when more renewable sources have been used , governments and other corresponding authorities will be forced to take environment protection more seriously. It will be helpful for a more sustainable and greener development. In conclusion, the fate of future generations in Australia and the rest of the world depends on how we address the energy crisis and take necessary steps to switch to renewable energy. 

References:

Australia’s Electricity Sector: Ageing, Inefficient and Unprepared        

esaa:data and statistics-energy in Australia