1. Observability and Identification of Linear Hybrid Systems
René Vidal
Dept. of EECS UC Berkeley

2. Vision based landing of a UAV
Landing on the ground
Tracking two meter waves

3. Probabilistic pursuit-evasion games
Hierarchical control architecture
High-level: map building and pursuit policy
Mid-level: trajectory planning and obstacle avoidance
Low-level: regulation and control

4. Formation control of nonholonomic robots
Examples of formations
Flocks of birds
School of fish
Satellite clustering
Automatic highway systems

5. Structure from motion & 3D reconstruction
Input: Corresponding points in multiple images
Output: camera motion, scene structure, calibration
Theory
Multiview geometry: Multiple view matrix
Multiple view normalized epipolar constraint
Linear self-calibration
Algorithms
Multiple view matrix factorization algorithm
Multiple view factorization for planar motions

6. Multibody structure from motion
Rotation:
Translation:
Epipolar constraint
Multiple motions
Write Sym(F_1,…., F_n).
Multibody epipolar constraint

7. Generalized Principal Component Analysis
Algebraic-geometric technique for simultaneous data segmentation and model estimation
Piecewise constant data
Piecewise linear data
Piecewise bilinear data

8. Some possible applications of GPCA
Segmentation
Image segmentation
Intensity
Color
Motion
Texture
Recognition
Faces (Eigenfaces)
Man - Woman
Human activities
Running, walking
Identification of hybrid systems
Data compression
UCLA database
CMU database

9. Texture-based image segmentation
Polysegment
Human

10. Texture-based image segmentation
Polysegment
Human

11. Affine Motion Segmentation Results
The affine motion model

12. 3D Motion Segmentation Results

13. Observability and Identification of Linear Hybrid Systems

14. What are hybrid systems?
Dynamical systems whose evolution is determined by
Continuous dynamics: Differential Equations
Discrete dynamics: Finite State Machine
Examples of hybrid systems
Flight envelope protection: Air Traffic Management
Platoons of cars: Automated Highway System
Hierarchical controllers of Unmanned Air Vehicle

15. Motivation Previous work on hybrid systems
Modeling, analysis, stability
Control: reachability analysis, optimal control
Verification: safety
In applications, one also needs to worry about observability and identifiability
Linear modeling of human motion
Hybrid model of a UAV

16. Observability & Identification of Linear Systems
Linear dynamical model
Observability: Given measurements of the output yt when can the state xt be uniquely estimated?
Filtering: Given measurements of the output yt how to estimate the state xt?
Kalman filter
Identification: Given measurements of yt how to estimate the state xt and the model parameters (A,C)
Subspace Identification: Van Overschee and De Moor [’95]
Popov-Belevic-Hautus rank test on the observability matrix

17. Linear Hybrid Systems: Models
Dynamic model: collection of linear systems
Continuous dynamics: linear in each mode
Discrete dynamics:
JLS: Piecewise constant input
JMLS: Markov chain
LHS: Guards and invariants
JLS input is more general for identification
In this paper we consider a special class of linear hybrid systems known as jump-linear systems.
The continuous dynamics are linear in each mode, and the discrete dynamics are determined by a piece-wise constant input.
While in control one usually assumes that transitions are determined by the continuous state, this could be too restrictive for identification purposes. So our model is more general since it assumes no constraint on the mechanism that generates the transitions.

18. Observability & Identification of Hybrid Systems
Observability Linear Hybrid Systems
Rank test for jump-linear systems with periodic and known transitions (Ezzine-Haddad ’89)
Mixed-integer linear programming test for piece-wise affine systems (Bemporad et.al ’00)
Design of Observers for Linear Hybrid Systems
Luenberger observers (Alessandri and Colleta ’01)
Location + Luenberger observers (Balluchi et al. ’02)
Moving horizon estimator via mixed-integer quadratic programming (Ferrari-Trecate et al. ’01)
Identification of Linear Hybrid Systems
Mixed-integer programming: (Bemporad et al. ’01)
Clustering + Regression + Classification + Iterative Refinement: (Ferrari-Trecate et al. ’03)
For linear systems, it is well known that observability reduces to checking the rank of the observability matrix, filtering can be done using the Kalman filter, and identification can be done using subspace techniques
Previous work on the observability of hybrid systems includes the case of piece-wise affine systems, which was studied by Bemporad. The authors propose a notion of observability which requires mixed-integer linear programming techniques in order to be checked. Rank constraints for jump linear systems where proposed by Ezzine, but under the assumptions that the discrete transitions are known and periodic.
As for identifiability, there has been a lot of work in the learning community for the past 30 years. Most of them are approximate and iterative algorithms. None of them enforces observability or guarantees a unique solution.

19. Outline In this talk, we concentrate on Observability Identification
Observability: of the state (continuous and discrete)
Identifiability: of the model parameters
Observability
Generalization of rank constraint to Jump-Linear Systems
Sufficient conditions for Linear Hybrid System
Gaining observability through switching
Identification
Algebraic geometric solution to identification
Closed form when Nmodels ≤ 4, formula for Nmodels

20. Linear Hybrid Systems: Models
Dynamic model: collection of linear systems
Continuous dynamics: linear in each mode
Discrete dynamics:
JLS: Piecewise constant input
JMLS: Markov chain
LHS: Guards and invariants
JLS input is more general for identification
In this paper we consider a special class of linear hybrid systems known as jump-linear systems.
The continuous dynamics are linear in each mode, and the discrete dynamics are determined by a piece-wise constant input.
While in control one usually assumes that transitions are determined by the continuous state, this could be too restrictive for identification purposes. So our model is more general since it assumes no constraint on the mechanism that generates the transitions.

21. Observability of JLS: assumptions
Given the model under what conditions is it possible to estimate the state (continuous and discrete) uniquely?
Assumption: there is a minimum separation between switching times
Continuous time:
Discrete time: Observability index of a Jump-Linear System

22. Observability of JLS: the initial state
Lemma 1: Observability of the initial continuous state and of the discrete sequence
: observability index of a jump-linear system
Each linear system is observable
Intersection of the observability subspaces must be trivial
This comes from indistinguishability condition
It turns out that for this class of linear hybrid systems, one can obtain a set of simple rank tests that determine its observability.
A first condition is that the rank of every pair of EXTENDED observability matrices has to be 2n. This implies that every linear system has to be observable, and that the observability subspaces of two different linear systems must not intersect. The observability index of a jump-linear system can also be naturally defined.
A second set of conditions has to do with the switching times, which we assume that are “separated enough”. Minimum separation is TWO times the observability index. The first contition guarantees the “detection” of a switch, i.e. it guarantees that a switch is “reflected” in the output, either immediately (right when it happens) or post-mortem. The second condition guarantees that if a switch is detected, then one can recover it UNIQUELY.
These conditions are necessary and sufficient provided the minimum separation
These conditions are SUFFICIENT for linear systems where the evolution of the discrete state is determined by the continuous state,
as long as the dynamics is such that the switching times are separated enough.

23. Observability of JLS: the switching times
Lemma 2: Observability of the switching times
This comes from
Discrete-time
Detection of a switch
Observability of a switch
It turns out that for this class of linear hybrid systems, one can obtain a set of simple rank tests that determine its observability.
A first condition is that the rank of every pair of EXTENDED observability matrices has to be 2n. This implies that every linear system has to be observable, and that the observability subspaces of two different linear systems must not intersect. The observability index of a jump-linear system can also be naturally defined.
A second set of conditions has to do with the switching times, which we assume that are “separated enough”. Minimum separation is TWO times the observability index. The first contition guarantees the “detection” of a switch, i.e. it guarantees that a switch is “reflected” in the output, either immediately (right when it happens) or post-mortem. The second condition guarantees that if a switch is detected, then one can recover it UNIQUELY.
These conditions are necessary and sufficient provided the minimum separation
These conditions are SUFFICIENT for linear systems where the evolution of the discrete state is determined by the continuous state,
as long as the dynamics is such that the switching times are separated enough.

24. Observability of JLS
Theorem 1: Observability of JLS

25. Observability of Linear Hybrid Systems
Sufficient condition for observability of LHS
What if linear systems are unobservable, but there is at least one switch in ?
One can reconstruct the hybrid state uniquely!!!

26. Observability of Linear Hybrid Systems
Theorem 3: Observability of LHS (sufficient)
Observability of the switching times
Observability of the discrete state
Observability of the continuous initial state

27. Identification of Linear Hybrid Systems
Given input/output data, estimate
The number of discrete states N
The model parameters for each linear system
The hybrid state (continuous and discrete)
The switching parameters (partition of state space)
Challenging “chicken and egg” problem
Use clustering to initialize model parameters
Given the model parameters, estimate discrete state
Given all above, estimate switching parameters
Iterate

28. Our approach to identification
Towards an analytic solution to identification of LHS
Can we estimate ALL models simultaneously using ALL data?
When can we do so analytically? In closed form?
Is there a formula for the number of models?
An algebraic geometric approach to identification of LHS
Number of models = degree of a polynomial
Model parameters = roots (factors) of a polynomial
=> polynomial factorization
In the case of PWARX models
There are algebraic constraints that are independent on the discrete state
There is a unique solution which is closed form iff Nmodels ≤ 4
The exact solution can be computed using linear algebra

29. Identification of LHS: modeling
Assume linear systems in ARX form
Define the vectors
Input/output data
Model parameters
Data lives in the subspace
Given input/output data generated by models with parameters
Number of models, model parameters
Hybrid state, switching mechanism

30. Identification of LHS and GPCA
Identify N k-dimensional subspaces of

31. Identification of LHS: model parameters
Can decouple identification from filtering
Solving for the hybrid model parameters
Number of models

32. Identification of LHS: model parameters
Theorem: Generalized PCA [Vidal et al. ‘03]
Find roots of polynomial of degree N in one variable
Solve K-2 linear systems in N variables
Discrete state:

33. Identification algorithm algorithm
Compute number of discrete states from rank of the data matrix
Compute hybrid parameters from nullspace of PN
Compute the model parameters using GPCA
Roots poly
Linear system
Compute the discrete state
Compute the continuous state (Linear system)
Objective function on p_n(x)

34. Multiple-input multiple-output
Multiple inputs
The same identification procedure can be applied
Multiple outputs
More challenging: multiple homogeneous polynomials representing the hybrid model
Can be reduced to single-input single-output case by suitable projection

35. Simulations 1000 PWARX systems with N=3 discrete states
Error in model parameters
Evolution of the continuous state
Error in discrete state
Evolution of the discrete state

36. Conclusions Observability of Linear Hybrid Systems
Observability of the hybrid state can be characterized in terms of rank constraints on the model parameters
Observability conditions for continuous-time systems are different from those in discrete time
Identification of Linear Hybrid Systems
One can decouple identification from filtering
There is a formula for the number of discrete states
Identification is equivalent to polynomial factorization
Future directions
Effect of the input on the observability conditions
Observability & Identification of Stochastic LHS
Do filtering and identification causally

37. Ongoing work and future directions
Statistical Geometry = Geometry + Statistics
Robust GPCA
Relations to other methods: Kernel PCA, etc.
Model selection
Estimating manifolds from sample data points
Applications of GPCA in Computer Vision
Optical flow estimation and segmentation
Motion segmentation: multiple views
Shape recognition: faces
A geometric/statistical theory of segmentation?

38. Dynamic GPCA: Recognition/Synthesis of Human Motion Recognition/Synthesis of Dynamic Textures
Given image data
Estimate a mixture of linear dynamical models (linear hybrid systems)
Use the models recognize
Human activity
Dynamic texture
Use the models to synthesize
human motion
dynamic textures
UCLA database

39. Thanks Computer Vision Pursuit-Evasion Games Vision Based Landing
Stefano Soatto, UCLA
Yi Ma, UIUC
Jana Kosecka, GMU
John Oliensis, NEC
Control/Hybrid Systems
John Lygeros, Cambridge
Shawn Schaffert
Research Advisor
Shankar Sastry
Pursuit-Evasion Games
Jin Kim
David Shim
Vision Based Landing
Omid Shakernia
Cory Sharp
Formation Control
Noah Cowan