Topic #11 16.31 Feedback Control State-Space Systems State-space model features . Observability o Controllability e Minimal realizations Copyright[2001by JOnathan dHow.D
Topic #11 16.31 Feedback Control State-Space Systems • State-space model features • Observability • Controllability • Minimal Realizations Copyright 2001 by Jonathan How. 1
Fall 2001 16.3111-1 State-Space Model Features . There are some key characteristics of a state-space model that we need to identify Will see that these are very closely associated with the concepts of pole/zero cancellation in transfer functions Example: Consider a simple system 6 +2 for which we develop the state-space model Model#1 i==2c+2u . But now consider the new state space model t=[ a2T Model #2 a C 0-1 30x which is clearly different than the first model e But let's looks at the transfer function of the new model (s)=C(sI-A)B+D
Fall 2001 16.31 11—1 State-Space Model Features • There are some key characteristics of a state-space model that we need to identify. — Will see that these are very closely associated with the concepts of pole/zero cancellation in transfer functions. • Example: Consider a simple system G(s) = 6 s + 2 for which we develop the state-space model Model # 1 x˙ = −2x + 2u y = 3x • But now consider the new state space model ¯x = [x x2] T Model # 2 x¯˙ = −2 0 0 −1 x¯ + ∙ 2 1 ¸ u y = £ 3 0 ¤ x¯ which is clearly different than the first model. • But let’s looks at the transfer function of the new model: G¯(s) = C(sI − A) −1 B + D
Fall 2001 16.3111-2 This is a bit strange, because previously our figure of merit when comparing one state-space model to another(page 8-8)was whether they reproduced the same same transfer function Now we have two very different models that result in the same transfer function Note that i showed the second model as having 1 extra state but i could easily have done it with 99 extra states!! So what is going on The clue is that the dynamics associated with the second state of the model 2 were eliminated when we formed the product G(s)=[30] because the a is decoupled and there is a zero in the C matrix Which is exactly the same as saying that there is a pole-zero cancellation in the transfer function G(s) 6 6(s+1) s+2(s+2)(s+1 会G( Note that model #2 is one possible state-space model of G(s has 2 poles For this system we say that the dynamics associated with the second state are unobservable using this sensor( defines the C matrix There could be a lot "motion associated with o. but we would be unware of it using this sensor
Fall 2001 16.31 11—2 • This is a bit strange, because previously our figure of merit when comparing one state-space model to another (page 8-8) was whether they reproduced the same same transfer function — Now we have two very different models that result in the same transfer function — Note that I showed the second model as having 1 extra state, but I could easily have done it with 99 extra states!! • So what is going on? — The clue is that the dynamics associated with the second state of the model x2 were eliminated when we formed the product G¯(s) = £ 3 0 ¤ " 2 s+2 1 s+1 # because the A is decoupled and there is a zero in the C matrix — Which is exactly the same as saying that there is a pole-zero cancellation in the transfer function G˜(s) 6 s + 2 = 6(s + 1) (s + 2)(s + 1) , G˜(s) — Note that model #2 is one possible state-space model of G˜(s) (has 2 poles) • For this system we say that the dynamics associated with the second state are unobservable using this sensor (defines the C matrix). — There could be a lot “motion” associated with x2, but we would be unware of it using this sensor
Fall 2001 6.3111-3 There is an analogous problem on the input side as well. Consider Model #1 =2a + 2u with a=a a2 2 #3 0-1 32 which is also clearly different than model #1, and has a different form from the second model 20 [32]sr 0-1 2 6 0 s+2 Once again the dynamics associated with the pole at s=-1 are cancelled out of the transfer function But in this case it occurred because there is a o in the b matrix So in this case we can"see"the state a2 in the output C=3 2 but we cannot "influence that state with the input since b So we say that the dynamics associated with the second state are uncontrollable using this actuator(defines the b matrix)
Fall 2001 16.31 11—3 • There is an analogous problem on the input side as well. Consider: Model # 1 x˙ = −2x + 2u y = 3x with ¯x = [ x x2] T Model # 3 x¯˙ = −2 0 0 −1 x¯ + ∙ 2 0 ¸ u y = £ 3 2 ¤ x¯ which is also clearly different than model #1, and has a different form from the second model. Gˆ(s) = £ 3 2 ¤ sI − −2 0 0 −1 −1 ∙ 2 0 ¸ = £ 3 s+2 2 s+1 ¤ ∙ 2 0 ¸ = 6 s + 2 !! • Once again the dynamics associated with the pole at s = −1 are cancelled out of the transfer function. — But in this case it occurred because there is a 0 in the B matrix • So in this case we can “see” the state x2 in the output C = £ 3 2 ¤ , but we cannot “influence” that state with the input since B = ∙ 2 0 ¸ • So we say that the dynamics associated with the second state are uncontrollable using this actuator (defines the B matrix)
Fall 2001 6.3111-4 Of course it can get even worse because we could have C+ 0-1 ●5 o now we have 20 30 0-1 s+2s+1 +2 e Get same result for the transfer function, but now the dynamics associated with a are both unobservable and uncontrollable unnary Dynamics in the state-space model that are uncontrollable, un observable, or both do not show up in the transfer function Would like to develop models that only have dynamics that are both controllable and observable called a minimal realization It is has the lowest possible order for the given transfer function But first need to develop tests to determine if the models are ob- servable and or controllable
Fall 2001 16.31 11—4 • Of course it can get even worse because we could have x¯˙ = −2 0 0 −1 x¯ + ∙ 2 0 ¸ u y = £ 3 0 ¤ x¯ • So now we have G ](s) = £ 3 0 ¤ sI − −2 0 0 −1 −1 ∙ 2 0 ¸ = £ 3 s+2 0 s+1 ¤ ∙ 2 0 ¸ = 6 s + 2 !! • Get same result for the transfer function, but now the dynamics associated with x2 are both unobservable and uncontrollable. • Summary: Dynamics in the state-space model that are uncontrollable, unobservable, or both do not show up in the transfer function. • Would like to develop models that only have dynamics that are both controllable and observable V called a minimal realization — It is has the lowest possible order for the given transfer function. • But first need to develop tests to determine if the models are observable and/or controllable