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Rose-Hulman ECE 520 - ECE 520 Homework 5

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ECE-520: Linear Control Systems Homework 5 Due: Thursday January 12 at 5 PM Exam 1, Monday January 23 in Class 1) Consider the following state variable system 1z−DCGHpfG K()rk ()uk ()xk ()yk - + + + + + Here is the prefilter gainpfG, is the reference input()rk, and []12Kkk=is the feedback gain matrix. The state variable model for the plant is assumed to be (1) () (() () ())xkGxkHuyk Cxk Dukk+=+=+ From the diagram we have . () () ()pfuk G rk Kxk=− a) Determine an expression for the transfer function between the input ()Rz and the output . ()Yz b) Assuming and using the Final Value Theorem, show that for a single input single output system to have a zero steady state error for a unit step input we need to choose the prefilter to be 0D = 11()pfGCI G HK H−=−+ Hint: In order to have a zero steady state error for a unit step input, the final value of the output, ()ssyy∞= , should be 1.2) Consider the discrete-time state variable model (1) () ()xkGxkHuk+=+ with the initial state . Let (0) 0x = []01 1,,10,10 1GHCD⎡⎤ ⎡⎤===⎢⎥ ⎢⎥⎣⎦ ⎣⎦0= a) Determine the corresponding transfer function for the system. b) After 1 time step we have (1) (0) (0)xHu Mu== soMH=. After 2 time steps we have [](0)(2) (1) (1) (0) (1) (1)(1)uxGx Hu GHu Hu GH H Muu⎡⎤=+ = + = =⎢⎥⎣⎦ so []MGH H= and [](1) (0) (1)Tuuu=. Now assume we want [](2) 1 0Tx =. Can you find an input vector , and hence input values and , to make this happen? If you cannot go from the origin to any possible state in at most(1)u(0)u (1)u n steps, where , then the system is not controllable. Why at most n steps? See below….. ( )nxk∈\ c) The Cayley-Hamilton Theorem from Linear Algebra states that a matrix satisfies its own characteristic equation. The characteristic equation of a matrix G is found by setting the determinant of Wz equal to zero. Show that the characteristic equation for our system is given by and then verify that IG=−210z −=20GI−= . d) Now let’s look at the third time step 2(3) (2) (2) (0) (1) (2)xGx Hu G Hu GHu Hu=+ = + + Using the Cayley-Hamilton Theorem, we can write 2GI=. Show that we can then write [][](3) (2)(2) (1) (2) (0)TxGHHuuuuu==+ e) Show that we can write [][](4) (3)(3) (0) (2) (1) (3)TxGHHuuuuuu==+ + At this point, it should be clear that if we cannot find an input to go from the origin to a particular final state in n = 2 steps for a second order system we never will be able to get there, no matter how long we let the system run. If , then the controllability matrixnx∈ \ is defined to be 12...nnMGHGH GHH−−⎡=⎣⎤⎦. For a system to be controllable, this matrix must have rank n, or, equivalently, n linearly independent columns (or rows).f) Now assume we are using state variable feedback with a prefilter gain G, so . Here is the reference inputpf() () ()pfuk G rk Kxk=−()rk and []12Kkk=is the feedback gain matrix. With this form of state variable feedback, we have the system (1) () () ()[ ]()[ ](pf pf)xk Gxk H G rk Kxk G HKxk HG rk⎡⎤+= + − = − +⎣⎦ or (1) () ()xkGxkHr+= +k Note that now the system input is the reference input . Show that for the transfer matrix is given by ()rk0D = 11212(1)()() ( )() ( )( ) ( 1)( 1)pfGzYzFz CzI G HRz zkzk k k−+==− =++−− − g) Show that if and , the transfer function reduces to that found in part a. 1pfG =120kk== h) Is it possible to find and to place the poles of the closed loop system where ever we want? For example, can we make both poles be zero? 1k2k In summary, if the system is controllable • We can go from the origin to any final state in n steps ( the rank of the controllability matrix M is n) • We can place the poles of the closed loop system anywhere we want using state variable feedback3) Consider the discrete-time state variable model (1) () ()xkGxkHuk+=+ with the initial state . Let (0) 0x = []10 0,,01,11 1GHCD⎡⎤ ⎡⎤===⎢⎥ ⎢⎥⎣⎦ ⎣⎦0= a) Determine the corresponding transfer function for the system. b) Find the M matrix after two time steps. Now assume we want[](2) 1 1Tx =. Can you find an input vector , and hence input values and , to make this happen? (1)u(0)u (1)u c) Show that the characteristic equation for G is given by 221zz 0−+= and verify that . 22GG=−I d) Show that we can then write [][](3) (2)(2) 2 (0) (1) (2) (0)TxGHHuuuuuu==+ − e) Show that we can write [][](4) (3)(3) 3 (0) 2 (1) (2) 2 (0) (1) (3)TxGHHuuuuu uuu==++−−+ f) Now assume we are using state variable feedback with () () ()pfuk G rk Kxk=−. Show that for the transfer matrix is given by 0D = 2(1)()()() ( 1)( 1)pfGzYzFzRz z z k−==−+− g) Show that if and , the transfer function reduces to that found in part a. 1pfG =120kk== h) Is it possible to find and to place the poles of the closed loop system where ever we want? For example, can we make both poles be zero? 1k2k4) Consider the discrete-time state variable model (1) () ()xkGxkHuk+=+ with the initial state . Let (0) 0x = []01 0,,10,11 1GHCD⎡⎤ ⎡⎤===⎢⎥ ⎢⎥⎣⎦ ⎣⎦0= a) Determine the corresponding transfer function for the system. b) Find the M matrix after two time steps. Now assume we want[](2) 1 0Tx =. Can you find an input vector , and hence input values and , to make this happen? Now assume we want(1)u(0)u (1)u[](2) 0 1Tx =. Can you find an input vector to make this happen? (1)u c) Show that the characteristic equation for G is given by 210zz−−= and verify that . 2GG=+I d) Show that we can then write [][](3) (2)(2) (0) (1) (0) (2)TxGHHuuuuuu==+ + e) Show that we can write [][](4) (3)(3)2(0)(1)(2)(0)(1)(3)TxGHHuuuuuuuu==++ ++ f) Now assume we are using state variable feedback with () () ()pfuk G rk Kxk=−. Show that for the transfer matrix is given by 0D = 221()()() ( 1) ( 1)pfGYzFzRz z k z k==+−+− g) Show that if and , the transfer function reduces to that found in part a. 1pfG =120kk== h) Is it possible to find and to place the poles of the closed loop system where ever we want? For example, can we make both poles be zero? If we want the poles to be at 1k2k1pand 2p show that and . 111( )kp=− +2p2p211kp=+5) For the discrete-time state variable system given by []01 0(1) () (11 1() 1 0 ())xkxkyk xk⎡⎤ ⎡⎤+= +⎢⎥ ⎢⎥⎣⎦ ⎣⎦=uk a) Assuming state variable feedback, find a state variable


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