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 wayFall 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, con￾troller, controllability, and observability. They are all useful in their own way