Topic #12 16.31 Feedback Control State-Space Systems e State-space model features o Controllability Copyright 2001 by Jonathan How
Topic #12 16.31 Feedback Control State-Space Systems • State-space model features • Controllability Copyright 2001 by Jonathan How. 1
Fall 2001 16.3112-1 Controllability Definition: An LTI system is controllable if, for every a*(t d every T>0, there exists an input function u(t),0<t<T, such that the system state goes from (0)=0 to a T)=I' Starting at 0 is not a special case- if we can get to any state in finite time from the origin, then we can get from any initial condition to that state in finite time as well Need only consider the forced solution to study controllability Bu(r)dr Change of variables T2=t-T, dr =-dr2 gives a(t)= eAry Bu(t-T2)dT2 0 This definition of observability is consistent with the notion we used before of being able to "influence"all the states in the system in the decoupled examples we looked at before ROT: For those decoupled examples, if part of the state cannot be"influenced"by ult), then it would be impossible to move that part of the state from 0 to *k
Fall 2001 16.31 12—1 Controllability • Definition: An LTI system is controllable if, for every x?(t) and every T > 0, there exists an input function u(t), 0 < t ≤ T, such that the system state goes from x(0) = 0 to x(T) = x?. — Starting at 0 is not a special case — if we can get to any state in finite time from the origin, then we can get from any initial condition to that state in finite time as well. — Need only consider the forced solution to study controllability. x(t) = Z t 0 eA(t−τ ) Bu(τ )dτ — Change of variables τ2 = t − τ , dτ = −dτ2 gives x(t) = Z t 0 eAτ2Bu(t − τ2)dτ2 • This definition of observability is consistent with the notion we used before of being able to “influence” all the states in the system in the decoupled examples we looked at before. — ROT: For those decoupled examples, if part of the state cannot be “influenced” by u(t), then it would be impossible to move that part of the state from 0 to x?
Fall 2001 16.31122 Definition: A state o*+0 is said to be uncontrollable if the forced state response a(t)is orthogonal to x*vt>0 and all input functions You cannot get there from here This is equivalent to saying that a* is an uncontrollable state if 2 Bu(t-T2)di 2B(t-2)d2=0 Since this identity must hold for all input functions u(t-r2), this can only be true if B≡0t>0
Fall 2001 16.31 12—2 • Definition: A state x? 6= 0 is said to be uncontrollable if the forced state response x(t) is orthogonal to x? ∀ t > 0 and all input functions. — “You cannot get there from here” • This is equivalent to saying that x? is an uncontrollable state if (x? ) T Z t 0 eAτ2Bu(t − τ2)dτ2 = Z t 0 (x? ) T eAτ2Bu(t − τ2)dτ2 = 0 • Since this identity must hold for all input functions u(t − τ2), this can only be true if (x? ) T eAtB ≡ 0 ∀ t ≥ 0
Fall 2001 16.3112-3 For the problem we were just looking at, consider Model #3 with x=[01]≠0,then 20 2 del #3 a C+ 0 32 (x)e4B=[01 01「 0e-t||0 2 0 t Sor*=[0 1 is an uncontrollable state for this system. But that is as expected, because we knew there was a problem with the state 2 from the previous analysis
˙ Fall 2001 16.31 12—3 • For the problem we were just looking at, consider Model #3 with x? = [ 0 1 ] T 6= 0, then − ∙ 2 0 Model # 3 x¯ = x¯ + 0 2 ¸ u 0 −1 £ 3 2 ¤ y = x¯ so ∙ e−2t 0 0 e−t ¸ ∙ 0 ¸ £ 2 0 1 ¤ e−t ¤ (x? ) T eAtB = ∙ 2 0 ¸ = 0 ∀ t £ = 0 So x? =[0 1 ] T is an uncontrollable state for this system. • But that is as expected, because we knew there was a problem with the state x2 from the previous analysis
Fall 2001 16.3112-4 Theorem: An Lti system is controllable iff it has no uncontrollable states We normally just say that the pair(A, B) is controllable Pseudo-Proof: The theorem essentially follows by the definition of an uncontrollable state If you had an uncontrollable state x*, then it is orthogonal to the forced response state a(t), which means that the system cannot reach it in finite time the system would be uncontrollable o Theorem The vector is an uncontrollable state iff (x)[BABA2B…A-B] See page 81 Simple test: Necessary and sufficient condition for controllability is that ankM会rank[BABA2B…A2-B]=n
Fall 2001 16.31 12—4 • Theorem: An LTI system is controllable iff it has no uncontrollable states. — We normally just say that the pair (A,B) is controllable. Pseudo-Proof: The theorem essentially follows by the definition of an uncontrollable state. — If you had an uncontrollable state x? , then it is orthogonal to the forced response state x(t), which means that the system cannot reach it in finite time ; the system would be uncontrollable. • Theorem: The vector x? is an uncontrollable state iff (x? ) T £ B AB A2B ··· An−1B ¤ = 0 — See page 81. • Simple test: Necessary and sufficient condition for controllability is that rank Mc , rank £ B AB A2B · · · An−1B ¤ = n