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

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

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

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

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

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

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

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

Texture-based image segmentation Polysegment Human

Texture-based image segmentation Polysegment Human

Affine Motion Segmentation Results The affine motion model

3D Motion Segmentation Results

Observability and Identification of Linear Hybrid Systems

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

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

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

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.

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.

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

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.

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

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.

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.

Observability of JLS Theorem 1: Observability of JLS

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!!!

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

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

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

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

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

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

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:

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)

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

Simulations 1000 PWARX systems with N=3 discrete states Error in model parameters Evolution of the continuous state Error in discrete state 1 30 60 10 Evolution of the discrete state

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

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?

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

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