Patricio Vela: Course Wiki

User Tools

Site Tools

Trace:
ece6554:project_ballbeam

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revisionPrevious revision
Next revision		Previous revision
ece6554:project_ballbeam [2019/04/20 20:14] – [Step 1: Baseline Adaptive Systems] pvelaece6554:project_ballbeam [2024/08/20 21:38] (current) – external edit 127.0.0.1
Line 90: Line 90:
 ===== Step 2: Full Linear System (An Almost Backstepping Approach) ===== ===== Step 2: Full Linear System (An Almost Backstepping Approach) =====
  
-=== Motor Control: Linear ===+==== Motor Control: Linear ====
  
 Add in the motor dynamics and have them be linearized around the steady-state. Work out what steady-state should be for a given non-zero target ball position $(x^*, 0)$ (it should be reasonably located, as in not too far out). Have your trajectories also be relative to this position so that the linearization remains valid.  Work out a good controller for this fifth order system. Remember that you are controlling $x$ via $\theta$ which is controlled through $I$ which has the control $V$.  Provide the $A$ and $B$ matrices. Add in the motor dynamics and have them be linearized around the steady-state. Work out what steady-state should be for a given non-zero target ball position $(x^*, 0)$ (it should be reasonably located, as in not too far out). Have your trajectories also be relative to this position so that the linearization remains valid.  Work out a good controller for this fifth order system. Remember that you are controlling $x$ via $\theta$ which is controlled through $I$ which has the control $V$.  Provide the $A$ and $B$ matrices.
Line 96: Line 96:
 This linear system is looking a lot like a backstepping control process, but where backstepping is usually not applied because the system should be controllable in the linearization. Since it is controllable in the linearization, all that is needed is to design a stabilizing controller for the fifth order system. Be careful about how quickly things stabilize as the main objective is to properly control the ball, then to control the beam. This linear system is looking a lot like a backstepping control process, but where backstepping is usually not applied because the system should be controllable in the linearization. Since it is controllable in the linearization, all that is needed is to design a stabilizing controller for the fifth order system. Be careful about how quickly things stabilize as the main objective is to properly control the ball, then to control the beam.
  
-=== Linear Adaptive Control ===+==== Linear Adaptive Control ====
  
 Incorporate adaptive control to correct for the unmatched uncertainty. Randomly perturb by about 15% the parameters.  Since the uncertainty is unmatched, you will have to work out a backstepping type of adaptive control law. The uncertainty hits the $\dot x$, $\dot \theta$, $\iota$ coordinates dynamics. Provide your derivation of the adaptive control law, the final set of equations, and simulations showing it in action. Incorporate adaptive control to correct for the unmatched uncertainty. Randomly perturb by about 15% the parameters.  Since the uncertainty is unmatched, you will have to work out a backstepping type of adaptive control law. The uncertainty hits the $\dot x$, $\dot \theta$, $\iota$ coordinates dynamics. Provide your derivation of the adaptive control law, the final set of equations, and simulations showing it in action.
 +
 +=== Tips: De-Coupled then Re-Coupled Adaptive Systems ===
  
 //Tip:// Designing a fifth-order backstepping method is going to be pretty nasty, especially because of the dependency structure of things.  Instead, decouple the system into two parts.  One is the subsystem from Step 1 and the second is the new motor dynamics subsystem.  Design a backstepping controller for the new motor dynamics subsystem independent of the ball subsystem.  You will have to treat the ball signal as an external input in the decoupled system.  Go ahead and do that when synthesizing the backstepping control policy for the motor subsystem.  Get it to work out, then add on the adaptive ball system from Step 1.  This is not 100% correct/ideal, but it is much simpler than attempting a full backstepping design that will end up with lord knows how many differentiated signals. //Tip:// Designing a fifth-order backstepping method is going to be pretty nasty, especially because of the dependency structure of things.  Instead, decouple the system into two parts.  One is the subsystem from Step 1 and the second is the new motor dynamics subsystem.  Design a backstepping controller for the new motor dynamics subsystem independent of the ball subsystem.  You will have to treat the ball signal as an external input in the decoupled system.  Go ahead and do that when synthesizing the backstepping control policy for the motor subsystem.  Get it to work out, then add on the adaptive ball system from Step 1.  This is not 100% correct/ideal, but it is much simpler than attempting a full backstepping design that will end up with lord knows how many differentiated signals.
 +
 +//Tip:// When designing the back-stepping type adaptive controller for the three new states, you should identify some candidate $x_1$ dynamics for the ball-beam part to be used in the motor controller. With $x_1$ specified in closed-form, the associated time derivatives can also be computed in closed form.  These should be used as exogenous signals in the linear, back-stepping type adaptive controller.  Later, when combining with the adaptive controller for the ball sub-system, you'll be trying to track the desired adaptive control law for the motor sub-system.
  
 --------------------------------------- ---------------------------------------
-==== Step 3: Nonlinear Adaptive Backstepping ====+===== Step 3: Nonlinear Adaptive Backstepping =====
  
 **This Step should not be done for ECE6554 Spring 2019. Only complete Steps 1 + 2.** **This Step should not be done for ECE6554 Spring 2019. Only complete Steps 1 + 2.**
Line 109: Line 113:
 Translate the linear adaptive backstepping strategy to the nonlinear system. Start with the vanilla nonlinear backstepping method, then make it adaptive. Translate the linear adaptive backstepping strategy to the nonlinear system. Start with the vanilla nonlinear backstepping method, then make it adaptive.
  
-=== Nonlinear Model Inversion with Backstepping ===+==== Nonlinear Model Inversion with Backstepping ====
  
 Consider the nonlinearities of the system and identify a backstepping approach to controlling the system. Get the baseline backstepping method working.  It should resemble that of the linear controller from Step 2 but with nonlinear terms. Consider the nonlinearities of the system and identify a backstepping approach to controlling the system. Get the baseline backstepping method working.  It should resemble that of the linear controller from Step 2 but with nonlinear terms.
  
  
-=== Nonlinear Model Inversion with Adaptive Backstepping ==+==== Nonlinear Model Inversion with Adaptive Backstepping ====
  
 Add in adaptation to the backstepping controller, using the same 15% perturbed model. Add in adaptation to the backstepping controller, using the same 15% perturbed model.
ece6554/project_ballbeam.1555805660.txt.gz · Last modified: 2024/08/20 21:38 (external edit)