An Algebraic Geometric Approach to the Identification of a Class of Linear Hybrid Systems

1. An Algebraic Geometric Approach to the Identification of a Class of Linear Hybrid Systems

2. Modeling of a UAV, dynamic textures, human gaits 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

3. Given input/output data, identify Number of discrete states Model parameters of linear systems Hybrid state (continuous & discrete) Switching parameters (partition of state space) Challenging “chicken-and-egg” problem Given switching times, can estimate model parameters Given the model parameters, estimate hybrid state Given all above, estimate switching parameters Iterate Difficulties Very sensitive to initialization Needs a minimum dwell time Does not use all data Problem description and challenges

4. Prior work • Observability • Mixed-integer linear programming test for piece-wiseaffine systems (Bemporad et al. ’00) • Rank test for jump-linear systems (Vidal et al. ’02 and ‘03) • Design of Observers (known model parameters) • 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) • K-means Clustering + Regression+ Classification + Iterative Refinement: (Ferrari-Trecate et al. ’03) • Greedy/iterative approach: (Bemporad et al. ‘03)

5. Our approach to hybrid 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 PWARX models • Number of models =degree of a polynomial • Model parameters =roots (factors) of a polynomial • Hybrid decoupling constraint: independent of discrete state and of switching mechanism (no minimum dwell time) • There is a unique solution which is closed form iff nmodels ≤ 4 polynomial differentiation • The exact solution can be computed using linear algebra

6. Problem statement • Assume that each linear systems is in ARX form • input/output • discrete state • order of the ARX models • model parameters • Input/output data lives in a hyperplane • I/O data • Model params

7. Hybrid ID and hyperplane clustering • Given input/output data generated by ARX models with model parameters , identify • Number of models • Model parameters • Hybrid state • Problem is equivalent to clusteringhyperplanes • Number of hyperplanes • Hyperplane normals

8. Decoupling identification and filtering • The hybrid decoupling constraint • Independent of the value of the discrete state • Independent of the switching mechanism • Satisfied by all data points: no minimum dwell time • The hybrid model parameters • Veronese map Number of regressors Number of models

9. Solving for the hybrid model parameters Solving for the number of models Identifying the number of models Embedding Lifting Embedding

10. Identifying the model parameters I Theorem: Polynomial Factorization • Find roots of polynomial of degree n in one variable • Solve K-2 linear systems in n variables

11. Identifying the model parameters II Theorem: Polynomial Differentiation

12. Filtering the hybrid state • Discrete state: • Continuous state: linear system on initial state

13. Hybrid identification algorithm • Compute number of discrete states from rank of the data matrix • Compute hybrid parameters from nullspace of Ln • Compute the model parameters using polynomial differentiation (GPCA) • Compute the discrete state • Compute the continuous state (linear system)

14. Simulation results with noisy data • 1000 PWARX systems with n=3 discrete states Error in continuous state Error in discrete state

15. Example: switching circuit • Error in model parameters for different switching times and noise level Noise std

16. Conclusions • Identification of linear hybrid systems • Can decouple identification and filtering • There is an algebraic solution to hybrid identification • Number of discrete states: rank constraint • Model parameters: derivatives of a polynomial • Hybrid state: linear system • Ongoing work • Models of unknown/different orders: rank constraints • MIMO ARX models: multiple polynomials • Recursive hybrid identification • Least mean squares algorithm on hybrid model parameters

17. Ongoing work: modeling human gaits Walking Running Limping Data Synthesis