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Aerial Robotics

Control: LTI Systems

C. Papachristos

Robotic Workers (RoboWork) Lab

University of Nevada, Reno

CS-491/691

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LTI Systems Control

  • In general, robotic systems are Multiple-Input Multiple-Output (MIMO) systems and Nonlinear, and their control has to account for the possible existence of:
    • Actuation bandwidth limitations & absolute constraints
    • Other actuation non-linearities
    • Environment interaction dynamics:�Aerodynamics (Free-Flight / Ground-Effect / etc.)�Contact Stick-Slip Friction�Acoustics / Electromagnetics / etc.

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  • Often such nonlinearities can be disregarded in modeling for control as they do not affect most of the operational envelope�– and w.r.t. to the states we wish to control

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  • But couplings can still be prevalent in the linearized control models that we derive

CS491/691 C. Papachristos

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Pole Placement

C. Papachristos

 

 

 

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Optimal LTI Control

  • Optimal Control Principles

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  • We can leverage model knowledge to design optimal control behaviors

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  • The employed model must be simultaneously sufficiently accurate, but also simple enough to enable efficient control computation

  • Optimal control can support linear and nonlinear systems as well as systems subject to state, output and input constraints

� Further extensions (e.g. for hybrid systems) also exist

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  • Established method for unconstrained linear systems regulation:
    • Linear Quadratic Regulation (LQR)
    • Generalization:�Linear Quadratic Gaussian (LQG) control
  • Leads to applicable control practices for nonlinear systems:
    • Gauss-Newton LQR ( alternate between:�a) linearization around current trajectory�b) solving the associated LQR problem )

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CS491/691 C. Papachristos

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Optimal LTI Control

Error-State dynamics formulation

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  • Assume the following LTI system dynamics:

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  • Assume we want to track a set of State References:

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  • We define as the Error-State:

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    • so we have:

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We may represent the equivalent Error-State Dynamics:�

 

CS491/691 C. Papachristos

 

 

 

 

 

with Initial Condition:

 

where we can choose:

 

and we get:

 

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Linear Quadratic Regulator

 

 

 

 

CS491/691 C. Papachristos

 

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Linear Quadratic Regulator

 

 

Approach 1 (Computational):

CS491/691 C. Papachristos

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Linear Quadratic Regulator

 

 

Approach 1 (Computational):

C. Papachristos

Can arrive at the same via Lyapunov stability or via Dynamic Programming properties!

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Linear Quadratic Regulator

 

 

 

 

 

Approach 2 (Infinite Time-Horizon & Analytical via Lyapunov Stability):

CS491/691 C. Papachristos

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Linear Quadratic Regulator

 

 

 

 

C. Papachristos

Lyapunov Stability

xx

 

 

 

 

 

Approach 2 (Infinite Time-Horizon & Analytical via Lyapunov Stability):

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Linear Quadratic Regulator

 

 

CS491/691 C. Papachristos

Approach 2 (Infinite Time-Horizon & Analytical via Lyapunov Stability):

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Linear Quadratic Regulator

 

 

 

CS491/691 C. Papachristos

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Linear Quadratic Regulator

 

 

CS491/691 C. Papachristos

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Time for Questions !

CS-491/691

CS491/691 C. Papachristos