Nonlinear Control Theory

Will be updated throughout the quarter.

Date (day) |
Topics |
Notes |

Jan 09 (Tues) | Welcome, course logistics, some math notations, modeling of nonlinear dynamical systems, systems theoretic terminology, groups encountered in control applications | Lecture 1 notes |

Jan 11 (Thu) | Modeling of control systems, open-loop and closed-loop, block diagrams, linear vs. nonlinear | Lecture 2 notes |

Jan 16 (Tues) | Planar autonomous systems: LTI systems and their phase portraits, hyperbolic and non-hyperbolic fixed points, Hartman-Grobman Theorem | Lecture 3 notes |

Jan 18 (Thu) | Planar autonomous systems (contd.): nonlinear systems and implication of Hartman-Grobman theorem for linearized analysis, when is a set positively invariant in time, Poincare-Bendixson criterion, two strategies to apply Poincare-Bendixson criterion in practice, Bendixson criterion, Bendixson-Dulac criterion, index theory | Lecture 4 notes |

Jan 23 (Tues) | Norms of vectors, matrices and functions; locally and globally Lipschitz functions, existence and uniqueness of solution for nonlinear systems | Lecture 5 notes |

Jan 25 (Thu) | Lyapunov stability theory for (a fixed point of) autonomous ODE: definitions for stability (S), asymptotic stability (AS), global asymptotic stability (GAS), exponential stability (ES); examples; Lyapunov's 1892 theorem (for S and AS); interpretation of level sets of a Lyapunov function; examples | Lecture 6 notes |

Jan 30 (Tues) | Lyapunov stability theory for autonomous ODE (contd.): Barbashin-Krasovskii theorem (for GAS), why radial unboundedness is necessary for GAS; LaSalle invariance theorem (for set), corollary of LaSalle invariance theorem (for fixed point), example proving origin is AS for pendulum with damping, example proving limit cycle is AS, Barbashin-Krasovskii version of LaSalle (GAS for set); Chetaev theorem (to prove fixed point is unstable), example |
Example of a positive definite function that is bounded in some radial direction |

Feb 01 (Thu) | Fixed point and stability for non-autonomous ODE, examples; different notions of stability in non-autonomous system: uniform stability (US), uniform asymptotic stability (UAS), global uniform asymptotic stability (GUAS), exponential stability (ES); Lyapunov-like theorems for US, UAS, GUAS, ES; examples; uniformly continuous functions, Barbalat's lemma and its corollary, application to $\dot{V}$, adaptive control example where $\displaystyle\lim_{t\rightarrow\infty}x_{i}(t)=0$ for some but not all $i$ | Lecture 8 notes |

Feb 06 (Tues) | Linear time varying systems: state transition matrix $\Phi(t,t_{0})$ and Peano-Baker series, stability (S, AS, UAS, GUAS $\Leftrightarrow$ ES) conditions in terms of $\Phi(t,t_{0})$, (Lyapunov stability for LTV and LTI systems in HW3); Using Lyapunov theory to estimate region-of-attraction for nonlinear systems, examples |
Counterexample on Hurwitz condition for LTV Constructing Lyapunov function using variable gradient method |

Feb 08 (Thu) | Techniques to construct Lyapunov functions: variable gradient method; Krasovskii's method: theorem, proof, example; sum-of-squares optimization: monomials and polynomials, positive definite and SOS polynomials: example and counterexample (Motzkin polynomial), Hilbert's theorem (1888), Searching SOS polynomial Lyapunov function as semi-definite programming (SDP) feasibility problem, SOSTOOLS | |

Feb 13 (Tues) | Input-to-state stability (ISS): motivation; when GAS of the unforced system + bounded input implies bounded state; case for LTI system; counterexample for nonlinear system; class $\mathcal{K},\mathcal{K}_{\infty},\mathcal{KL}$ functions; definition of ISS system; ISS Lyapunov function stability theorem; example; Input-Output stability, signal $\mathcal{L}_{p}$ norms | Lecture 11 notes |

Feb 15 (Thu) | Input-Output stability (contd.): $\mathcal{L}_{p}$ gain and worst-case $\mathcal{L}_{p}$ gain; static linear system case: induced $p$-norm of matrix; worst-case $\mathcal{L}_{2}$ gain of LTI system equals $\mathcal{H}_{\infty}$ norm of transfer matrix $G(j\omega)$; $\mathcal{L}_{2}$ gain theorem for control-affine nonlinear system without direct feedthrough: Hamilton-Jacobi partial differential inequality | Lecture 12 notes |

Feb 20 (Tues) | Compositional results for $\mathcal{L}_{p}$ stability: series, parallel and feedback connections of $\mathcal{L}_{p}$ stable sub-systems, small gain theorem; Passivity: energy of an input-output system, rate inequality, dissipativity inequality, dissipation equality; related notions: strictly passive (SP), input feedforward passive (IFP), input strictly passive (ISP), output feedback passive (OFP), output strcitly passive (OSP), zero state observability (ZSO); Six Passivity Theorems: (1) passivity implies stability of the unforced system, (2) OSP with parameter $\delta > 0$ implies finite gain $\mathcal{L}_{2}$ stability with gain $\leq \frac{1}{\delta}$, (3) either SP or {OSP+ZSO} implies (G.)A.S. for unforced system, (4) feedback preserves passivity, (5) feedback connection of two OSP subsystems with parameters $\delta_{1},\delta_{2}$ implies finitie gain $\mathcal{L}_{2}$ stability with gain $\leq \frac{1}{\min(\delta_{1},\delta_{2})}$, (6) when a sub-system in feedback connection is not necessarily passive but the overall system is finite gain $\mathcal{L}_{2}$ stable | Lecture 13 notes |

Feb 22 (Thu) | Designing feedback controllers: state and output feedback, static and dynamic feedback; Design Idea 1: passivity based control: if passive and ZSO, then origin can be globally stabilized by locally Lipschitz output feedback (theorem and proof); how to passivate: by choice of output (example), by designing feedback (feedback passivation); Design Idea 2: feedback stabilization: control affine system and control Lyapunov function (CLF), converse Lyapunov theorem, Sontag's formula for single input case (motivation, theorem, proof, example) |
Lecture 14 notes |

Feb 27 (Tues) | Design Idea 3: sliding mode control: motivation: robustness, sliding manifold, reach phase (design of feedback control to hit the sliding manifold in finite time) and sliding phase (design of sliding manifold such that trajectory will go to origin with control switched off), pro: sliding phase is independent of $f$ and $g$, con: chattering due to discretization or delay, remedy: approximate the discontinuous feedback; Design Idea 4: backstepping: integrator backstepping: theory |
Lecture 15 notes |

Mar 01 (Thu) | Design Idea 4: backstepping: integrator backestepping (contd.): theorem, example, general backstepping, recursive backstepping, block backstepping; Design Idea 5: feedback linearization and input-output linearization: motivation, definitions of (state) feedback linearization and input-output linearization, static and dynamic feedback linearization, example when "cancel the nonlinearity and make the closed loop stable LTI system" works -- pendulum in air, example when the same does not work but still feedback linearizable |
Lecture 16 notes |

Mar 06 (Tues) | Geometric ideas in modern nonlinear control: scalar and vector fields, manifold and tangent space, Lie derivative of a scalar field w.r.t. a vector field, Lie derivative of a vector field w.r.t. a vector field: Lie brackets and the ${\rm{ad}}_{\mathbf{f}}^{k}\mathbf{g}$ operator, example, properties of Lie bracket, space of vector fields as vector space: span and distribution $\Delta$, sum and intersection of distributions, the vector field stack matrix $\mathbf{F}$ and distribution $\Delta$ as its image, ${\rm{dim}}(\Delta)(\mathbf{x}) = {\rm{rank}}(\mathbf{F})(\mathbf{x})$, example, involutive distribution: definition and example, SISO relative degree: definition and examples | Lecture 17 notes |

Mar 08 (Thu) | Interpretation of SISO relative degree, examples, input-output linearization and normal form of SISO state + output equations, examples, feedback linearization of SISO state + output equations, example, what if output equation is not specified: necessary and sufficient conditions for feedback linearization, equivalent constructive conditions, step-by-step recipe for feedback linearization of nonlinear state equation | Lecture 18 notes |

Mar 13 (Tues) | Summary of input-output, partial and full-state feedback linearization for SISO nonlinear systems, "zeroing the output problem": zero dynamics for (partial) feedback linearized system, zero dynamics for LTI system, "reproduce the reference output trajectory problem": reference dynamics for (partial) feedback linearized system, zero dynamics as special case of reference dynamics, importance of zero dynamics: asymptotic feedback stabilizability can be checked from asymptotic stability (A.S.) of the zero dynamics, global asymptotic feedback stabilizability can be checked from ISS of the $\eta$ dynamics with $\xi$ thought of as input (GAS of zero dynamics is not enough); Controllability: definition and control-affine nonlinear systems | Lecture 19 notes |

Mar 15 (Thu) | Drift-free control system, involutive closure of $\Delta = {\rm{span}}\{g_{1}, g_{2}, ..., g_{m}\}$, filtration, degree-of-nonholonomy, holonomic and completely/partially non-holonomic control systems, Lie algebra, Rashevsky-Chow theorem, example: wheeled mobile robot/Dubin's car, interpretating Lie bracket as a measure of non-commutativity of flows, Lie bracket as infinitesimal composition of primitive flows, example: controllability of a realistic car and parking theorem, controllability of LTI and linearized system, Dubin's car counterexample showing the linearized system is not controllable but the nonlinear system is globally controllable | Lecture 20 notes |