Question
- Find the steady state values of the current ‘i’ and voltage v, which are needed to balance the ball at a desired position, y = r > 0.

- Compute the linearized model at this equilibrium point and show that the equilibrium point obtained by taking u = v(steady state value of the voltage) is unstable.

- Write a Matlab function that returns the equilibrium current, voltage and state space matrices as a function of r.

- Using linearization, design state feedback control laws to stabilize the ball at y = 1cm, y = 5cm and y = 8cm. You may use the commands place and 1qr. In each case, carry out closed loop performance analysis (e.g time responses to step commands).

- Assume that the permissible range of y is 0 to 10 cm and the permissible range of the voltage is 0 to 50 V. Starting with the ball at equilibrium, move it a small distance up (and the down) and let it go. Using simulation, determine the largest range of initial disturbance for which the ball will return to the equilibrium position without violating the constraints of y and v. To account for the constraints on v, include a limiter in your simulation.

- The non-linear closed loop system responses will be computed with Matlab/Simulink. The effect of 20% parameter variations from nominal values for all parameters will be investigated as well as the effect of the current protection circuits (‘i’ will be set to zero if it goes over a certain value ‘imax’ = 5A.

Values used in Matlab for disturbance testing (proper values can be acquired by running maglevlinearizationandstep.m):
r=0.01
usim=1.9809;
x3sim=1.1885;

r=0.05
usim=9.904;
x3sim=1.9809;

r=0.08
usim=15.847;
x3sim=2.5752;
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%Clearing memory and screen, closing all open windows
clear;
clc;
close all;

%Defining symbolic variables
syms x1 x2 x3 m g k L0 a R L1 u r x1e x2e x3e ue

%Defining differential equations
x1dot=x2;
x2dot=g-(k*x2)/m-(L0*a*x3^2)/(2*m*(a+x1)^2);
x3dot=(u-R*x3+(L0*a*x2*x3)/(a+x1)^2)/(L1+L0/(x1/a+1));

%Jacobian linearization
A=[diff(x1dot,x1) diff(x1dot,x2) diff(x1dot,x3);diff(x2dot,x1) diff(x2dot,x2) diff(x2dot,x3);diff(x3dot,x1) diff(x3dot,x2) diff(x3dot,x3)]
B=[diff(x1dot,u);diff(x2dot,u);diff(x3dot,u)]

%Defining matrix C
C=[1 0 0];

%Finding and defining equilibrium states
x2e=solve(x1dot)
x2=x2e;
x1e=r
x1=x1e;
x2dot=eval(x2dot)
x3e=solve(x2dot,x3)
x3=x3e(1,1)
x3dot=eval(x3dot)
ue=solve(x3dot,u)
u=ue;

Alin=eval(A)
Blin=eval(B)

%Defining parameters
m=0.01; %mass in kg
k=0.001; %in N/m/s;
g=9.81; %m/s^2
a=0.05; % in meters
L0=0.01;
L1=0.02;
R=10;    % in Ohms

Alin=eval(Alin)
Blin=eval(Blin)
x2sim=eval(x2e);
s=tf('s');
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      • Illuminate\Cache\Events\CacheHit (187μs)
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      • Illuminate\Cache\Events\CacheHit (205μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (244μs)
      • Illuminate\Cache\Events\CacheHit (154μs)
      • Illuminate\Cache\Events\CacheHit (172μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (196μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (159μs)
      • Illuminate\Cache\Events\CacheHit (235μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (196μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (205μs)
      • Illuminate\Cache\Events\CacheHit (156μs)
      • Illuminate\Cache\Events\CacheHit (177μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (193μs)
      • Illuminate\Cache\Events\CacheHit (213μs)
      • Illuminate\Cache\Events\CacheHit (218μs)
      • Illuminate\Cache\Events\CacheHit (173μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheMissed (284μs)
      • Illuminate\Database\Events\StatementPrepared (848μs)
      • Illuminate\Database\Events\QueryExecuted (1.28ms)
      • eloquent.retrieved: App\Models\SubjectCat (85μs)
      • Illuminate\Cache\Events\KeyWritten (1.61ms)
      • Illuminate\Cache\Events\CacheHit (704μs)
      • Illuminate\Cache\Events\CacheHit (248μs)
      • Illuminate\Cache\Events\CacheHit (149μs)
      • Illuminate\Cache\Events\CacheHit (145μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (124μs)
      • Illuminate\Cache\Events\CacheHit (106μs)
      • Illuminate\Cache\Events\CacheHit (156μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (145μs)
      • Illuminate\Cache\Events\CacheHit (111μs)
      • Illuminate\Cache\Events\CacheHit (120μs)
      • Illuminate\Cache\Events\CacheHit (102μs)
      • Illuminate\Cache\Events\CacheHit (114μs)
      • Illuminate\Cache\Events\CacheHit (107μs)
      • Illuminate\Cache\Events\CacheHit (312μs)
      • Illuminate\Cache\Events\CacheHit (131μs)
      • creating: site.layouts.app (464μs)
      • composing: site.layouts.app (18μs)
      • creating: components.canonical (297μs)
      • composing: components.canonical (79μs)
      • creating: components.open-graph (130μs)
      • composing: components.open-graph (50μs)
      • creating: site.headers.header (178μs)
      • composing: site.headers.header (49μs)
      • Illuminate\Cache\Events\CacheHit (1.96ms)
      • creating: components.footer (82μs)
      • composing: components.footer (75μs)
      • Illuminate\Cache\Events\CacheHit (683μs)
      • Illuminate\Cache\Events\CacheMissed (266μs)
      • Illuminate\Database\Events\StatementPrepared (648μs)
      • Illuminate\Database\Events\QueryExecuted (912μs)
      • eloquent.retrieved: App\Models\SubjectCat (87μs)
      • Illuminate\Cache\Events\KeyWritten (317μs)
      • Illuminate\Cache\Events\CacheHit (265μs)
      • Illuminate\Cache\Events\CacheHit (138μs)
      • Illuminate\Cache\Events\CacheHit (201μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (148μs)
      • Illuminate\Cache\Events\CacheHit (246μs)
      • Illuminate\Cache\Events\CacheHit (449μs)
      • Illuminate\Cache\Events\CacheHit (160μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (119μs)
      • Illuminate\Cache\Events\CacheHit (126μs)
      • Illuminate\Cache\Events\CacheHit (123μs)
      • creating: components.forms.contact-us (248μs)
      • composing: components.forms.contact-us (141μs)
      • creating: components.forms.get-started (155μs)
      • composing: components.forms.get-started (50μs)
      • creating: components.forms.free-tool-download (90μs)
      • composing: components.forms.free-tool-download (41μs)
      • creating: components.forms.claim-free-worksheet (83μs)
      • composing: components.forms.claim-free-worksheet (40μs)
      • creating: components.forms.tutor-subscription-waitlist (80μs)
      • composing: components.forms.tutor-subscription-waitlist (38μs)
      • creating: components.forms.tutor-subscription-join (99μs)
      • composing: components.forms.tutor-subscription-join (40μs)
      • creating: components.forms.tutor-support (84μs)
      • composing: components.forms.tutor-support (40μs)
      • 311 x Illuminate\Cache\Events\CacheHit (16.23%)
        79.16ms
        20 x Illuminate\Database\Events\QueryExecuted (7.46%)
        36.36ms
        20 x Illuminate\Database\Events\StatementPrepared (7.2%)
        35.11ms
        11 x Illuminate\Cache\Events\CacheMissed (2.24%)
        10.91ms
        11 x Illuminate\Cache\Events\KeyWritten (1.63%)
        7.94ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (1.05%)
        5.14ms
        1 x Illuminate\Routing\Events\Routing (0.29%)
        1.42ms
        12 x eloquent.retrieved: App\Models\SubjectCat (0.25%)
        1.24ms
        1 x creating: homework.show (0.2%)
        971μs
        1 x Illuminate\Database\Events\ConnectionEstablished (0.19%)
        916μs
        1 x Illuminate\Routing\Events\RouteMatched (0.12%)
        572μs
        1 x composing: components.breadcrumbs (0.1%)
        467μs
        1 x creating: site.layouts.app (0.1%)
        464μs
        1 x creating: components.breadcrumbs (0.08%)
        392μs
        1 x eloquent.booting: App\Models\SubjectCat (0.08%)
        383μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        312μs
        1 x creating: components.canonical (0.06%)
        297μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        274μs
        1 x creating: components.forms.contact-us (0.05%)
        248μs
        2 x eloquent.retrieved: App\Models\Subject (0.05%)
        230μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.05%)
        228μs
        1 x composing: homework.show (0.04%)
        200μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        187μs
        1 x creating: site.headers.header (0.04%)
        178μs
        1 x creating: components.forms.get-started (0.03%)
        155μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.03%)
        151μs
        1 x composing: components.forms.contact-us (0.03%)
        141μs
        1 x creating: components.open-graph (0.03%)
        130μs
        1 x creating: components.forms.tutor-subscription-join (0.02%)
        99μs
        1 x creating: components.forms.free-tool-download (0.02%)
        90μs
        1 x creating: components.forms.tutor-support (0.02%)
        84μs
        1 x creating: components.forms.claim-free-worksheet (0.02%)
        83μs
        1 x creating: components.footer (0.02%)
        82μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.02%)
        80μs
        1 x composing: components.canonical (0.02%)
        79μs
        1 x composing: components.footer (0.02%)
        75μs
        1 x eloquent.booting: App\Models\Subject (0.01%)
        66μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        61μs
        1 x composing: components.open-graph (0.01%)
        50μs
        1 x composing: components.forms.get-started (0.01%)
        50μs
        1 x composing: site.headers.header (0.01%)
        49μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        48μs
        1 x composing: components.forms.free-tool-download (0.01%)
        41μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        40μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        40μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        40μs
        1 x composing: components.forms.tutor-support (0.01%)
        40μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        38μs
        1 x composing: site.layouts.app (0%)
        18μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      20 statements were executed, 5 of which were duplicates, 15 unique. Show only duplicated52.62ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '7424' limit 1
        14.07mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 7424
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (395)
        1.23mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (7424)
        9.74mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        1.56mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 10
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 7424 and `subject` = 395 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        3.94mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 7424
        • 1: 395
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (395)
        1.21mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 7424 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1.75mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 7424
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 7424 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        1.27mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 7424
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 7424 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        2.25mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 7424
        • 1: solution
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        1.36mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 10
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.2mstwenty4_siteSubject.php#100
        Bindings
        • 0: 1
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.64mstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        1.02mstwenty4_siteSubject.php#100
        Bindings
        • 0: 4
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        2.35mstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        1.09mstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        1.12mstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        1.19mstwenty4_siteSubject.php#100
        Bindings
        • 0: 5
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        2.25mstwenty4_siteSubject.php#100
        Bindings
        • 0: 34
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        1.32mstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.06mstwenty4_siteSubject.php#100
        Bindings
        • 0: 36
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\SubjectCat
      12SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      7HomeworkLibraryFile.php
      App\Models\Subject
      2Subject.php
          _token
          pVMQTMZPnyE10pdIZspmTcoUejN4LwwCt81Xe5xy
          utm_source
          direct
          redirectUrl
          /college-homework-library/Engineering/MATLAB-for-Engineering/7424
          _previous
          array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Engineering/MAT...
          _flash
          array:2 [ "old" => [] "new" => [] ]
          PHPDEBUGBAR_STACK_DATA
          []
          path_info
          /college-homework-library/Engineering/MATLAB-for-Engineering/7424
          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
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