State Space Full State Feedback Design for a Mass$-$Spring$-$Damper Double Pendulum$1.$ IntroductionThis project aims to assess the GA$4$: INVESTIGATIONS, EXPERIMENTS AND DATA ANALYSIS. This will require the student to investigate and plan experiments and conduct data analysis within an appropriate discipline. The student is assessed through a report submission where he$/$she critically engages with the impact in the context of control system modelling, controller design and performance analysis. The assessment will cover seven competency indicators for GA$4$ as stated below:$($i$)$ Investigations and experiments are planned and conducted within an appropriate discipline$($ii$)$ Available literature is searched, and material is critically evaluated for suitability to the investigation.$($iii$)$ Analysis is performed as necessary to the investigation$($iv$)$ Equipment or software is selected and used as appropriate in the investigations$($v$)$ Information is analyzed, interpreted and derived from available data$($vi$)$ Conclusions are drawn from an analysis of all available evidence$($vii$)$ The purpose, process and outcomes of the investigation are recorded in a technical report The abovementioned indicators will be evaluated using the rubric listed in Table $1.$ After the consideration of the assessment, the internal examiner will be required to complete scores in the assessment rubric to indicate whether the candidate has demonstrated competence for GA$4.$Satisfactory performance for GA$4$ will be a sub$-$minimum score of $3$ for all competency indicators to be competent in GA$4.$The unsatisfactory performance will be leaded to$,$ if a student obtains a score of less than $3$ for any competency indicator. The student will have two additional opportunities to demonstrate satisfaction with this GA after the submission of the final assignment report. The student will rectify the report and re$-$submit for re$-$evaluation. If$,$ after the $3$rd submission, the student still does not obtain a minimum mark of $3$ for all competency indicators, he$/$she will fail the subject.The assessment will contribute $1\%$ to the final mark. $2.$ TasksTo fulfil the required skills and knowledges defined in GA$4,$ the following tasks are requested$\mathrm{.}2\mathrm{.}1$ A double pendulum as shown in Fig$\mathrm{.}1$ is requested to be implemented in Simulink Simscape,$2\mathrm{.}2$ The system transient response will be analyzed,$2\mathrm{.}3$ Controller design specifications will be given, then a state space full state feedback controller will be designed$\mathrm{.}2\mathrm{.}4$ On the designed controller the system performance will be analyzed via experiment results and validated against the given specifications$\mathrm{.}2\mathrm{.}5$ Comment on the transient responses using the key indicators$\mathrm{.}2\mathrm{.}6$ Write a report to reflect your ideas on system behavior investigation, modelling from the first principal method, controller design for given specifications, experiments on the designed controller, data analysis on the resulting system, and comment on your design$\mathrm{.}0\mathrm{.}10\mathrm{.}10\mathrm{.}1$ Fig. $1.$ A double pendulum made of springs, dampers and mass blocksFig$\mathrm{.}1$ shows the structure of the double pendulum, while the right part gives the parameters. M$1$ and M$2$ are the mass of the two blocks. K$1,$ K$2$ are the coefficients of the two springs. fv$1,$ fv$2,$ and fv$3$ are the damping factor of the dampers. y$=$x$2$ is the output. $3.$ Design specificationsFor the system given in Fig. $1,$ the rest state $($i$.$e$.$ both mass blocks are static, if no external forces applied$)$ is the initial state. The force f$($t$)$ is applied to M$2,$ which is the controller output, to quickly control the mass blocks to a desired output, that is $.$The requested peak time $($Tp$)$ is $2$s$,$ and the settling time is $5$s$.4.$ Requirements on report$4\mathrm{.}1$ Cover page$4\mathrm{.}2$ Title$4\mathrm{.}3$ Content list$4\mathrm{.}4$ Figure list$4\mathrm{.}5$ Table list$4\mathrm{.}6$ Introduction of tasks and the system to be implemented$4\mathrm{.}7$ Model the system using the first principal method$4\mathrm{.}8$ Analyse the system transient responses $($overshot$,$ rising time, peak time, settling time$)4\mathrm{.}9$ The response indicators obtained in $3\mathrm{.}8,$ should be shown in the plotted system response$\mathrm{.}4\mathrm{.}10$ For the given specification, design a full state feedback controller$4\mathrm{.}11$ Conclusion and discussion$\mathrm{.}4\mathrm{.}12$ Summary of Matlab$/$Simulink$/$Simscape$/$Multibody functions$/$components used in the assignment. $($list the functions$/$components$,$ explain how they work, and demonstrate how did you use them$)4\mathrm{.}13$ References