CDS 101 Lecture 5 1 R M Murray Caltech CDS 101 Lecture 5 1 Controllability and State Space Feedback Richard M Murray 28 October 2002 Goals y Define controllability of a control system y Give tests for controllability of linear systems and apply to examples y Describe the design of state feedback controllers for linear systems Reading y Packard Poola and Horowitz Dynamic Systems and Feedback Section 25 y Optional Friedland Sections 5 1 5 4 controllability only y Optional A D Lewis A Mathematical Introduction to Feedback Control Chapter 2 available on web page Lecture Review4 1 from Linear Last Systems Week x Ax Bu y Cx Du x 0 0 u 1 1 0 1 0 1 0 5 10 0 1 0 y 1 0 0 5 5 10 5 10 5 10 0 5 10 2 0 5 0 2 0 0 2 0 2 0 5 10 t y t Ce At x 0 Ce A t Bu d Du t Properties of linear systems y Linearity with respect to initial condition and inputs y Stability characterized by eigenvalues y Many applications and tools available y Provide local description for nonlinear systems 0 28 Oct 02 10 28 2002 R M Murray Caltech CDS 2 1 CDS 101 Lecture 5 1 R M Murray Caltech Control Design Concepts System description single input single output nonlinear system MIMO also OK x f x u x n x 0 given y h x u u y Stability stabilize the system around an equilibrium point y Given equilibrium point xe n find control law u x such that lim x t xe for all x 0 n t Controllability steer the system between two points y Given x0 xf n find an input u t such that x0 xf x f x u t takes x t0 x0 x T x f Tracking track a given output trajectory y Given yd t find u x t such that lim y t yd t 0 for all x 0 n t 28 Oct 02 R M Murray Caltech CDS y t yd t t 3 Controllability of Linear Systems x Ax Bu y Cx Du x n x 0 given u y Defn A linear system is controllable if for any x0 xf n and any time T 0 there exists an input u 0 T such that the solution of the dynamics starting from x 0 x0 and applying input u t gives x T xf Remarks y In the definition x0 and xf do not have to be equilibrium points we don t necessarily stay at xf after time T y Controllability is defined in terms of states doesn t depend on output y Can characterize controllability by looking at the general solution of a linear system T x T e AT x0 e A T Bu d 0 If integral is full rank then we can find an input to achieve any desired final state 28 Oct 02 10 28 2002 R M Murray Caltech CDS 4 2 CDS 101 Lecture 5 1 R M Murray Caltech Tests for Controllability x Ax Bu x n x 0 given u y y Cx Du T x T e AT x0 e A T Bu d 0 Thm A linear system is controllable if and only if the n n controllability matrix B AB A2 B An 1B is full rank Remarks y Very simple test to apply In MATLAB use ctrb A B y If this test is satisfied we say the pair A B is controllable y Some insight into the proof can be seen by expanding the matrix exponential 1 1 e A T B I A T A2 T 2 An 1 T n 1 B 2 2 1 2 1 n 1 2 n 1 B AB T A B T A B T 2 2 28 Oct 02 R M Murray Caltech CDS 5 Example 1 Linearized pendulum on a cart m Approach look at the linearization around the upright position good approximation to the full dynamics if remains small x f 28 Oct 02 10 28 2002 Question can we locally control the position of the cart by proper choice of input M R M Murray Caltech CDS 6 3 CDS 101 Lecture 5 1 R M Murray Caltech Example 1 con t Linearized pendulum on a cart m x f M Controllability matrix B 28 Oct 02 AB A2B A3B Full rank as long as constants are such that columns 1 and 3 are not multiples of each other controllable as long as g M m 1 can steer linearization between points by proper choice of input R M Murray Caltech CDS 7 Constructive Controllability Given that system is controllable how do we find input to transfer between states Simple case chain of integrators x 1 x2 x 2 x3 x n 1 xn x n u Find curve x1 t such that x1 0 x 0 x0 1 n 1 x1 0 x1 T x T xf 1 n 1 x T 1 Choose input as u t x1 n t x0 xf Controllable canonical form y If controllable can show there exists a linear change of coordinates such that 1 0 0 0 0 z u z 1 0 an 1 a1 a2 28 Oct 02 10 28 2002 z1 z z 1 n 1 z1 R M Murray Caltech CDS u t z1 n t an z1 n 1 a1 z1 8 4 CDS 101 Lecture 5 1 R M Murray Caltech Control Design Concepts System description single input single output nonlinear system MIMO also OK x f x u y h x u x n x 0 given u y Stability stabilize the system around an equilibrium point y Given equilibrium point xe n find control law u x such that lim x t xe for all x 0 n t 9 Controllability steer the system between two points y Given x0 xf n find an input u t such that x0 xf x f x u t takes x t0 x0 x t f x f Tracking track a given output trajectory y Given yr t find u x t such that y t lim y t yd t 0 for all x 0 n yd t t 28 Oct 02 R M Murray Caltech CDS t 9 State space controller design for linear systems x Ax Bu y Cx Du x n x 0 given u y T x T e x0 AT e A T Bu d 0 Goal find a linear control law u Kx such that the closed loop system x Ax BKx A BK x is stable at xe 0 Remarks y Stability determined by eigenvalues use K to make eigenvalues of A BK stable y Can also link eigenvalues to performance eg initial condition response y Question when can we place the eigenvalues anyplace that we want Thm The eigenvalues of A BK can be set to arbitrary values if and only if the pair A B is controllable MATLAB K place A B eigs 28 Oct 02 10 …
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