LMI Methods in Optimal and Robust Control

Transcription

1 LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 1: Introductions

2 Who Am I? Website: Research Interests: Computation, Optimization and Control Focus Areas: Expertise with LMI Methods: Control of Nuclear Fusion Immunology Thermostats, Renewable Energy, and Power Distribution My Background: B.Sc. University of Texas at Austin Ph.D. Stanford University Postdoc at INRIA Paris NSF CAREER Awardee Office: ERC 253; Lab: GWC 531 Optimization of Polynomials Parallel Computing for Control Control of Delayed Systems Control of PDE Systems Control of Nonlinear Systems M. Peet Lecture 1: Introduction 2 / 22

3 MAE 598: LMI Methods in Optimal and Robust Control References Required: LMIs in Control Systems by Duan and Yu LMIs in Systems and Control Theory by S. Boyd Link: Available Online Here Linear State-Space Control Systems by Williams and Lawrence Convex Optimization by S. Boyd Link: Available Online Here Link: Entire Course Online Here M. Peet Lecture 1: Introduction 3 / 22

4 MAE 598: LMI Methods in Optimal and Robust Control What are the challenges? Megatrends: Increased Complexity (Embedded Computation and Control) Increased Connectivity (Internet of Things) Robots, Drones and Self-Driving Cars Increased Demands (Higher Standards) Mobile Computing (Mobile Apps) M. Peet Lecture 1: Introduction 4 / 22

5 Challenges for Control in the 21st century Privatization of Space Travel Challenges Safety Complexity Uncertainty Links: Blue Origin Successful Landing Blue Origin Successful Landing: Flight 3 SpaceX Landing, Second Attempt Proton M launch Failure (FCS was for wrong rocket) Kepler Space Telescope M. Peet Lecture 1: Introduction 5 / 22

6 Challenges for Control in the 21st century UAVs and Drones Safe Interaction with Crowded Airspace Real-Time Obstacle Avoidance Precision Control with Delayed Feedback x (t) = Ax(t) + Bu(t τ ) Lossy Connections x (t) = Ax(t) + Bu(tk ) Links: X47 Drone Carrier Landing Raff s TED talk M. Peet Lecture 1: Introduction 6 / 22

7 Challenges for Control in the 21st century Self-Driving Vehicles Challenges: Safety (Provable) Uncertainty (in model, environment) Other Drivers (Multi-Agent) Obstacles Self-Driving Vehicles Google Über Tesla Toyota Links: Toyota s Research Expansion in Automation Uber s self-driving Taxis are in Pittsburg M. Peet Lecture 1: Introduction 7 / 22

8 Challenges for Control in the 21st century Interconnectivity mn Credits : layer1 : Esri, HERE, DeLorme, MapmyIndia, OpenStreetMap contributors, and the GIS user community; State Layers : ; layer0 : Esri, HERE, DeLorme, MapmyIndia, OpenStreetMap contributors, and the GIS user community Miles Surface Coal Mine Natural Gas Power Plant Wind Power Plant HGL Market Hub (z) Underground Coal Mine Biomass Power Plant Coal Power Plant Geothermal Power Plant Nuclear Power Plant Other Power Plant Petroleum Power Plant Pumped Storage Power Plant Petroleum Refinery Biodiesel Plant Ethanol Plant Natural Gas Processing Plant (z) Natural Gas Market Hub (z) Electricity Border Crossing Natural Gas Pipeline Border Crossing Hydroelectric Power Plant Solar Power Plant Ethylene Cracker M. Peet Lecture 1: Introduction 8 / 22

9 Challenges for Control in the 21st century Robotics HARD Robots Uncertain Terrain Interactions with the environment If x(t)>0: ẋ(t) = Ax(t) If x 1 (t)=0 AND x 2 (t)<0: Set x 2 (t) = x 2 (t) Link: Boston Dynamics, Atlas Mark 3 SOFT Robots Infinite Degrees of Freedom Material Dynamics Link: Robotic Worm M. Peet Lecture 1: Introduction 9 / 22

10 Challenges for Control in the 21st century Arduino and Raspberry Pi Trends: Rapid prototyping Internet of Things Control is Everywhere Challenges Noisy Sensors Data-Driven Modeling Dynamics with logical switching x = Ax + Bu(t) If Occupied=True : u(t) = K1 x(t) Else : M. Peet u(t) = K2 x(t) Lecture 1: Introduction 10 / 22

11 MAE 598: LMI Methods in Optimal and Robust Control This course is on RECENT Developments in Control Techniques Developed in the Last 20 years Computational Methods No Root Locus No Bode Plots No PID (Proportion-Integral-Differential) We focus on State-Space Methods In the time-domain We use large state-space matrices d dt x 1 (t) x 2 (t) x 3 (t) = x 4 (t) x 1 (t) x 2 (t) x 3 (t) x 4 (t) 0 0 We require Matlab Need robust control toolbox. Recommend using YALMIP. Link: Installs YALMIP and some other toolboxes [ ] u1 (t) u 2 (t) M. Peet Lecture 1: Introduction 11 / 22

12 So What is an Automatic Control System??? Well... What is a System? Definition 1. A System is anything with Inputs and Outputs There should ALWAYS be Inputs and Outputs! If No Inputs: You can t change anything. IF No Outputs: Then it doesn t matter anyway. M. Peet Lecture 1: Introduction 12 / 22

13 So What is an Automatic Control System??? put framework z regulated outputs y sensed outputs Plant exogenous inputs w actuator inputs u In Controls, we separate internal signals from external signals. Output Signals: puts u are those inputs to the system that can be mani z: Output to be controlled/minimized y: Output used by the controller Input Signals: inputs w are all other inputs. w: Disturbance, Tracking Signal, etc. u: Output from controller Input to actuator outputs z are every output signal from the model. M. Peet Lecture 1: Introduction 13 / 22

14 So What is an Automatic Control System??? State-Space System put framework z regulated outputs y sensed outputs Plant exogenous inputs w actuator inputs u A state-space system has the form puts u are those inputs ẋ = Ax to + the B 1 w system + B 2 u that can be mani z = C 1 x + D 11 w + D 12 u y = C 2 x + D 21 w + D 22 u inputs w are all other inputs. M. Peet Lecture 1: Introduction 14 / 22

15 K K So What is an Automatic Control System??? g interconnection is called the (lower) star-product of P and K, ort onal transformation (LFT). z w y P u K om w to z is given by The controller, K, determines how to use the signal y to get the signal u. S(P, K) =P 11 + P 12 K(I P 22 K) 1 P 21 Can be dynamic: u = F ˆx, ˆx = Aˆx + L(y C ˆx) Can be static: u = F y. Our job is to find the BEST K. M. Peet Lecture 1: Introduction 15 / 22

16 So What is an Automatic Control System??? TsConsider and stabilitythe Tracking Problem 200 ple: a tracking problem u n proc n sensor e r K + P + 0 z 1 = e z 2 = u P r = W err tracking input w 2 = n proc w 1 = rn = w W proc r = w 1 proc 2 e = tracking error w 3 = n sensor u = u n W act proc = process noise z 1 = e W sens n sensor = w y 3 1 = r n sensor = sensor noise z 2 = u y 2 = y p + P 0 + M. Peet Lecture 1: Introduction 16 / 22

17 r Tracking Control K + P + 0 P r = w 1 z 1 = e W err W proc n = w proc 2 z 2 = u W act W sens n sensor = w 3 + P 0 + y K u I P 0 0 P 0 P = I I P 0 I P 0 z 1 = r P 0 (n proc + u) z 2 = u y 1 = r y 2 = w 3 + P 0 (n proc + u) M. Peet Lecture 1: Introduction 17 / 22

18 What is Optimization? An Optimization Problem has 3 parts. min x F f(x) : subject to g i (x) 0 i = 1, K 1 h i (x) 0 i = 1, K 2 Variables: x F The things you must choose. F represents the set of possible choices for the variables. Can be vectors, matrices, functions, systems, locations, colors... However, computers prefer vectors or matrices. Objective: f(x) A function which assigns a scalar value to any choice of variables. e.g. [x1, x 2] x 1 x 2; red 4; et c. Constraints: g(x) 0; h(x) = 0 Defines what is a minimally acceptable choice of variables. Equality and Inequality constraints are common. x is OK if g(x) 0 and h(x) = 0. Constraints mean variables are not independent. (Constrained optimization is much harder). M. Peet Lecture 1: Optimization 18 / 22

19 How Hard is it to Solve Optimization Problems For Humans: Almost always IMPOSSIBLE (or at least tedious) For Computers: Easy if the Problem is CONVEX. (Polynomial Time) Otherwise IMPOSSIBLE. (NP-Hard) We will talk about this a bit more later! M. Peet Lecture 1: Optimization 19 / 22

20 Now What is an LMI? An LMI is a type of constraint Definition 2. A symmetric matrix (P = P T ) is Positive Definite (denoted P > 0) if all of its eigenvalues are positive. A Linear Matrix Inequality (LMI) is a constraint that looks like A i P B i + Q i > 0 where P is the variable and A i, B i, Q i are matrices. Question: Why do we have a whole controls course devoted to LMIs? LMI constraints are convex (Computers can solve them) Positive matrices can be used to study systems. This is because we are really optimizing Lyapunov functions. V (x) = x T P x 0 if P > 0. Almost ALL computational methods in Control are based on LMIs. Or at least be reformulated as an LMI. M. Peet Lecture 1: Optimization 20 / 22

21 Now What is an LMI? An Example The system ẋ = Ax is stable (eigenvalues have negative real part) if and only if there exists a P > 0 such that A T P + P A < 0 YALMIP Code for Stability Analysis: > A = [-1 2 0; ; 0 0-2]; > P = sdpvar(3,3); > F = [P >= eye(3)]; > F = [F, A *P+P*A <= 0]; > optimize(f); If Feasible, YALMIP Code to Retrieve the Solution: > Pfeasible = value(p); M. Peet Lecture 1: Optimization 21 / 22

22 Class Project In lieu of a final exam, we will have a class project. Work alone or in groups of at most two. Can be based on existing research. Some Project Ideas: Gain Scheduling for Missile Attitude Control (Switched Systems) Control of Robots over the internet (Sampled-Data Systems) Spacecraft Attitude Control with delayed communication (Delay Systems) Social Cognitive Therapy using Discrete Inputs (Mixed-Integer Control) Self-Driving Vehicles (Decentralized Control) Soft Robotics (Decentralized Control) Thermostat Programming (Dynamic Programming) Flow Control (PDEs) Controller/Estimator Design using Arduino and Simulink (Robust Control) System Identification using LMIs Mobile App for solving an optimization or control problem. For those who dislike Projects, we can arrange to take a Final Exam instead. M. Peet Lecture 1: Optimization 22 / 22