Presentation on theme: "David Rosen Goals Overview of some of the big ideas in autonomous systems Theme: Dynamical and stochastic systems lie at the intersection of mathematics."— Presentation transcript:
Sensing and Estimation How do we know what the state is at a given time? Generally, we have some sensors: ◦ Laser rangefinders ◦ GPS ◦ Vision systems ◦ etc…
Great! Well, not quite… ◦ In general, can’t measure all state variables directly. Instead, an observation function H : M → O maps the current state x to some manifold O of outputs that can be directly measured ◦ Usually, dim O < dim M ◦ Given some observation z = H(x), can’t determine x !
Maybe we can use the system dynamics (f ) together with multiple observations? Observability: Is it possible to determine the state of the system given a finite-time sequence of observations? ◦ “Virtual” sensors! Detectability (weaker): Are all of the unobservable modes of the system stable?
What about noise? In general, uncorrected/unmodeled error accumulates over time. Stochastic processes: nondeterministic dynamical systems that evolve according to probability distributions.
New model: for randomly distributed variables w t and v t. We assume that x t conditionally depends only upon x t-1 and the control u t (completeness): Stochastic processes that satisfy this condition are called Markov chains.
Similarly, we assume that the measurement z t conditionally depends only upon the current state x t :
Thus, we get a sequence of states and observations like this: This is called the hidden Markov model (HMM).
How can we estimate the state of a HMM at a given time? Any ideas?
Bayes’ Rule Punchline: If we regard probabilities in the Bayesian sense, then Bayes’ Rule provides a way to optimally update beliefs in response to new data. This is called Bayesian inference. It also leads to recursive Bayesian estimation.
Define Then by conditional independence in the Markov chain: and by Bayes’ rule:
Recursive Bayesian Estimation: The Bayes Filter This shows how to compute given only and the control input. Recursive filter!
Initialize the filter with initial belief Recursion step: ◦ Propagate: ◦ Update:
Benefits of recursion: ◦ Don’t need to remember observations ◦ Online implementation ◦ Efficient! Applications: ◦ Guidance ◦ Aerospace tracking ◦ Autonomous mapping (e.g., SLAM) ◦ System identification ◦ etc…
This clip was reportedly sampled from an Air Force training video on missile guidance, circa 1955. It is factually correct. See also: ◦ Turboencabulator ◦ Unobtainium Rudolf Kalman
How do we identify trajectories of the system with desirable properties?
Controllability: given two arbitrary specified states p and q, does there exist a finite-time admissible control u that can drive the system from p to q ? Reachability: Given an initial state p, what other states can be reached from p along system trajectories in a given length of time? Stabilizability: Given an arbitrary state p, does there exist an admissible control u that can stabilize the system at p ?
Several methods for generating trajectories ◦ Rote playback ◦ Online synthesis from libraries of moves ◦ etc… Optimal control: Minimize a cost functional amongst all controls whose trajectories have prescribed initial and final states x 0 and x 1.
Provides a set of necessary conditions satisfied by any optimal trajectory. Can often be used to identify optimal controls of a system. Lev Pontryagin
We have a desired trajectory that we would like to follow, called the reference. At each time t, we can estimate the actual state of the system. In general there is some nonzero error at each time t.
What to do? Maybe we can find some rule for setting the control input u (t ) at each time t as a function of the error e (t ) such that the system is stabilized? In that case, we have a feedback control law:
Many varieties of feedback controllers: Proportional-integral-derivative (PID) control Fuzzy logic control Machine learning Model adaptive control Robust control H ∞ control etc…
We started with what (at least conceptually) were very basic problems from engineering e.g., make do this this
and ended up investigating all of this: Dynamical systems Stochastic processes Markov chains The hidden Markov model Bayesian inference Recursive Bayesian estimation The Pontryagin Maximum Principle Feedback stabilization and this is just the introduction!