Topic =+8 16.31 Feedback Control State-Space Systems What are state-space models? Why should we use them? How are they related to the transfer functions used in classical control design and how do we develop a state- space model? What are the basic properties of a state-space model, and how do we analyze these? Copyright 2001 by Jonathan How
Topic #8 16.31 Feedback Control State-Space Systems • What are state-space models? • Why should we use them? • How are they related to the transfer functions used in classical control design and how do we develop a statespace model? • What are the basic properties of a state-space model, and how do we analyze these? Copyright 2001 by Jonathan How
Fall 2001 16.318-1 TFs to State-Space Models The goal is to develop a state-space model given a transfer function for a system G(s There are many, many ways to do this But there are three primary cases to consider 1. Simple numerator y=G(s)= s3+a152+a28+a 2. Numerator order less than denominator order y=G(s) b1s2+b2+b3 3+a152+02S+a3 3. Numerator equal to denominator order bos+618+b28+b s3+a1s2+a28+a3 These 3 cover all cases of interest
Fall 2001 16.31 8–1 TF’s to State-Space Models • The goal is to develop a state-space model given a transfer function for a system G(s). – There are many, many ways to do this. • But there are three primary cases to consider: 1. Simple numerator y u = G(s) = 1 s3 + a1s2 + a2s + a3 2. Numerator order less than denominator order y u = G(s) = b1s2 + b2s + b3 s3 + a1s2 + a2s + a3 = N(s) D(s) 3. Numerator equal to denominator order y u = G(s) = b0s3 + b1s2 + b2s + b3 s3 + a1s2 + a2s + a3 • These 3 cover all cases of interest
Fall 2001 16.318-2 Consider case 1 (specific example of third order, but the extension to nth follows easily) + a1s+ a2s+ a can be rewritten as the differential equation 98)+a1i+023+a33=u choose the output y and its derivatives as the state vector 0.9y then the state equations are 1 100 +0 010 =[0011y+ This is typically called the controller form for reasons that will become obvious later on There are four classic (called canonical) forms-oberver, con troller, controllability, and observability. They are all useful in their own way
Fall 2001 16.31 8–2 • Consider case 1 (specific example of third order, but the extension to nth follows easily) y u = G(s) = 1 s3 + a1s2 + a2s + a3 can be rewritten as the differential equation y(3) + a1y¨ + a2y˙ + a3y = u choose the output y and its derivatives as the state vector x = y¨ y˙ y then the state equations are x˙ = y(3) y¨ y˙ = −a1 −a2 −a3 100 010 y¨ y˙ y + 1 0 0 u y = 001 y¨ y˙ y + [0]u • This is typically called the controller form for reasons that will become obvious later on. – There are four classic (called canonical) forms – oberver, controller, controllability, and observability. They are all useful in their own way
Fall 2001 16.318-3 ● Consider case2 y=Gs=+a182+028+0a3D(s 618+b2s+b3 yy u 0 u where y/v=Ns) and v/u=1/D(s) Then the representation of o/=1/D(s) is the same as case 1 a10+a20+a use the state vector U to get A C+ Bu where a1-a2-a3 100andB2=0 010 Then consider y/v=N(s), which implies that b1i+620+b30 [b1 b2 b3 U C2x+[0
Fall 2001 16.31 8–3 • Consider case 2 y u = G(s) = b1s2 + b2s + b3 s3 + a1s2 + a2s + a3 = N(s) D(s) • Let y u = y v · v u where y/v = N(s) and v/u = 1/D(s) • Then the representation of v/u = 1/D(s) is the same as case 1 v(3) + a1v¨ + a2v˙ + a3v = u use the state vector x = v¨ v˙ v to get x˙ = A2x + B2u where A2 = −a1 −a2 −a3 100 010 and B2 = 1 0 0 • Then consider y/v = N(s), which implies that y = b1v¨ + b2v˙ + b3v = b1 b2 b3 v¨ v˙ v = C2x + [0]u
Fall 2001 16.318-4 ● Consider case3with y bos+0182+628+b3 a152+a28+a3 1s2+2s+/3 +D C15 Gi()+D where (s3+a1s2+a2s+a3) +(+/1s2+B2s+3) bos+0182+b2S +63 so that, given the bi, we can easily find the Bi D 61=01-Dar Given the Bi, can find G1(s) Can make a state-space model for G1(s)as described in case 2 Then we just add the"feed-through"term Du to the output equa tion from the model for Gi(s) e Will see that there is a lot of freedom in making a state-space model because we are free to pick the x as we want
Fall 2001 16.31 8–4 • Consider case 3 with y u = G(s) = b0s3 + b1s2 + b2s + b3 s3 + a1s2 + a2s + a3 = β1s2 + β2s + β3 s3 + a1s2 + a2s + a3 + D = G1(s) + D where D( s3 +a1s2 +a2s +a3 ) +( +β1s2 +β2s +β3 ) = b0s3 +b1s2 +b2s +b3 so that, given the bi, we can easily find the βi D = b0 β1 = b1 − Da1 . . . • Given the βi, can find G1(s) – Can make a state-space model for G1(s) as described in case 2 • Then we just add the “feed-through” term Du to the output equation from the model for G1(s) • Will see that there is a lot of freedom in making a state-space model because we are free to pick the x as we want