Video 6.1 Vijay Kumar and Ani Hsieh

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1 Video 6.1 Vijay Kumar and Ani Hsieh Robo3x-1.6 1

2 In General Disturbance Input + - Input Controller + + System Output Robo3x-1.6 2

3 Learning Objectives for this Week State Space Notation Modeling in the time domain Solutions in the time domain From Frequency Domain to Time Domain and Back Design in the Time Domain Linearization Robo3x-1.6 3

4 State-Space Representation Converts N-th order differential equation into N simultaneous FIRST-ORDER differential equations Allows for multiple inputs and/or outputs Versatility our initial conditions DO NOT have to be 0 Robo3x-1.6 4

5 State Variables Smallest set of linearly independent system variables s.t. state_variables(t_0) + known input (or forcing) functions completely determines the system. # of state variables = Dimension of the State Space # of state variables = order of the original diff eqn Robo3x-1.6 5

6 Example Robo3x-1.6 6

7 Another Example (1) y 1 x 1 y 0 q 2 P y 2 Q O q 1 x 0 x 2 Robo3x-1.6 7

8 Another Example (2) Robo3x-1.6 8

9 Video 6.2 Vijay Kumar and Ani Hsieh Robo3x-1.6 9

10 Transfer Function State Space (1) Given Robo3x

11 Transfer Function State Space (2) Given Robo3x

12 Transfer Function State Space (3) Given Robo3x

13 State Space Transfer Function (1) Given Robo3x

14 State Space Transfer Function (2) Given Robo3x

15 Solutions in the Time Domain Given w/ Robo3x

16 Time Domain Solution (2) Robo3x

17 Output of the System Natural Response Particular Response Thus, output of the system is given by Robo3x

18 Video 6.3 Vijay Kumar and Ani Hsieh Robo3x

19 The Matrix Exponential Properties of If then Robo3x

20 Characterizing System Response Since, for constant, then w/ constant and Then, Robo3x

21 Steady-State Performance Since System is stable if and only if Robo3x

22 Transient Performance What about complex eigenvalues? Recall with results in, then Complex eigenvalues come in pairs, terms will cancel out. Robo3x

23 Example (1) Given w/ Eigenvalues and eigenvectors of and are Robo3x

24 Example (2) Robo3x

25 Example (3) Robo3x

26 State Space Design Given w/ Linear State Feedback Control Law Closed-loop system Choose K such that CL response is stable. Robo3x

27 Advantages of State Space Design Given w/ with 1. Not restricted to 2 nd order approximations 2. Access to a larger range of closed-loop poles 3. Allows for full state feedback Robo3x

28 Video 6.4 Vijay Kumar and Ani Hsieh Robo3x

29 A Caveat Controllability A linear system is controllable if for each and, there exists a that can get the system from to at time. Such a linear system is controllable if and only if Controllability Matrix Robo3x

30 Feedback Law and Controllability Let s.t. are real coefficients. Then there exists such that if and only if is controllable. State feedback enables any controllable linear system to have arbitrary closed-loop poles! Robo3x

31 Linear Quadratic (LQ) Control Find optimal feedback control strategy that Constraints Cost Function Ø Constrained Optimization Problem Ø Optimal Control Robo3x

32 Optimal LQ Controller w/ symmetric, positive definite matrix Q and R where w/ Robo3x

33 A Few More Words The Algebraic Riccati Equation P is n x n matrix P is unique P is symmetric and positive definite w/ is optimal w.r.t. the cost function J Robo3x

34 Caveat for the Caveat Observability Recall: If is controllable, then K can be chosen s.t. achieves arbitrary CL poles. Assumption: Robo3x

35 Example Robot Joint Control (1) Robo3x

36 Example Robot Joint Control (2) With Note: 1. Only θ m is can be measured 2. requires θ l Robo3x

37 Observers State estimators Use system model and measured output to estimate the full state Also a dynamical system Robo3x

38 Obtaining Let A, B, C are known Solve for Use Pick L s.t. from any initial condition in feedback law as Error between plant & estimate output Robo3x

39 Video 6.5 Vijay Kumar and Ani Hsieh Robo3x

40 Performance of the Estimator Let be the estimation error Then, 1. Dynamics determined by 2. Pick L s. t. as Eigenvalues of can be arbitrarily set if and only if system is observable. Robo3x

41 Observability A linear system is observable if every can be exactly determined from in a finite time interval. and The pair is observable if and only if Observability Matrix Robo3x

42 Example Robot Joint Control (3) With Separation Principle: Allows us to separately design the feedback control and the state estimator Robo3x

43 For nonlinear systems In general, given where f is a nonlinear function in x, possibly u What if 1. We want to analyze system behavior around? 2. We want to control system behavior around? Robo3x

44 Linearization Given and, Let and, substitute into f(x,u) Apply Taylor series expansion about Then Robo3x

45 Linearization Let and Proceed Robo3x

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