EEE58 Exam, Fall 6 AA Rodriguez Rules: Closed notes/books, No calculators permitted, open minds GWC 35, 965-37 Problem (Dynamic Augmentation: State Space Representation) Consider a dynamical system consisting of several subsystems The describing equations are as follows: y x a, ẋ a y p y y p y p C p x p + D p u p, ẋ p A p x p + B p u p u p u, u p x a, ẋ a x a + u Sketch a block diagram and determine a state space representation for the above dynamical system Problem (Gaussian Elimination, Fundamental Spaces, Least Squares, Minimum Norm) Consider the following linear algebraic system of equations: A 5 b () (a) Determine the general solution (b) Determine a basis for the four fundamental subspaces: R(A),R(A T ),N(A),N(A T ) Now let b T and consider the vector norm z def z T z (c) Determine the sef all possible x that minimizes the error b Ax (d) Determine the projection P R(A) b of the vector b onto the range of A (e) Determine the minimum norm x which minimizes the error b Ax Problem 3 (Controllability, State Transfer, Pole Placement) p z p Consider the LTI system defined by the state space dual A, B where z p p and z p Sketch a block diagram for the system (a) Is the system controllable? Explain your answer Are there any pole-zero cancellations? (b) Does there exist a control law which will transfer the state of the system from x o to x()? If so, show how to determine one Also, determine a minimum energy state transferring control law (c) Does there exist a full state feedback control law u Gx such that the closed loop system has poles at s ± j? If so, determine a suitable control gain matrix G If not, explain why Problem 4 (Controllability, State Transfer, Pole Placement) p z p Consider the LTI system defined by the state space dual A, B where z p p and z p (a) Is the system controllable? Explain your answer Are there any pole-zero cancellations? (b) Assuming x o, determine an expression for x(t) using modal analysis concepts Determine a basis for the sef states that are reachable from x o (c) Does there exist a control law which will transfer the state of the system from x o to x()? If so, determine one Also, determine a minimum energy state transferring control law (d) Does there exist a control law which will transfer the state of the system from x o to x()? If so, determine one If not, determine what state closest to x() is reachable Then determine a control law which will achieve the transfer to this reachable state Is your control law unique? If not, show how to determine another state transferring control law and determine the minimum energy state transferring control law (e) Does there exist a full state feedback control law u Gx such that the closed loop system has poles at s p, (where p )? If so, determine a suitable control gain matrix G If not, explain why (f) Does there exist a full state feedback control law u Gx such that the closed loop system has poles at
s, 5 (both distinct from p )? If so, determine a suitable control gain matrix G If not, explain why Problem 5 (Observability, State Reconstruction, Pole Placement) p Consider the LTI system defined by the state space triple A p where z p and z p Sketch a block diagram for the system, B, C z p (a) Is the system observable? Explain your answer Are there any pole-zero cancellations? (b) Suppose that u and y is known on t, Determine an expression for the sef all possible initial conditions x o Can x o be determined uniquely? Explain Determine the minimum norm initial condition (c) Can one design an observer with closed loop poles at s,? If not, explain why If so, determine a suitable observer gain matrix H Moreover, determine the state estimation error x def x ˆx when u, the initial system state is x o, and the initial state estimate (used in your observer) is ˆx o Problem 6 (Observability, State Reconstruction, Pole Placement) p Consider the LTI system defined by the state space triple A, B, C z p p where z p and z p (a) Is the system observable? Explain your answer Are there any pole-zero cancellations? (b) Suppose that u and y is known on t, Determine an expression for the sef all possible initial conditions x o Can x o be determined uniquely? Explain Determine the minimum norm initial condition (c) Can one design an observer with closed loop poles at s p, (where p )? If so, determine a suitable observer gain matrix H If not, explain why (d) Can one design an observer with closed loop poles at s, 5 (both distinct from p )? If so, determine a suitable observer gain matrix H If not, explain why (e) Discuss how the answers to (a)-(d) change if z p and z p? Hint: Examine block diagram Problem 7 (Model Based Compensator Design) Consider the linear time invariant plant P (s) s 5 with state space quadruple A p 5,B p,c p,d p (a) Show how to design a model based compensator which satisfies the following design specifications: (i) zero steady state error to step reference commands, (ii) closed loop poles at s 4 ± j3, s ± j (b) Discuss how one might minimize the overshoot due to a step reference command (c) Summarize the design process if one desires to follow sinusoidal reference commands with frequency ω o
EEE58 Exam Solutions, Fall 6 AA Rodriguez Rules: Closed notes/books, No calculators permitted, open minds GWC 35, 965-37 Problem (Dynamic Augmentation: State Space Representation) Consider a dynamical system consisting of several subsystems The describing equations are as follows: y x a, ẋ a y p y y p y p C p x p + D p u p, ẋ p A p x p + B p u p u p u, u p x a, ẋ a x a + u Sketch a block diagram and determine a state space representation for the above dynamical system Problem -Partial Solution: u p u x a T () ẋ a y p (C p x p + D p u p ) (3) ẋ p A p x p + B p u p (4) ẋ a x a + u (5) y x a (6) y y p (C p x p + D p u p ) (7) B p B p B p (8) D p D p D p (9) ẋ a ẋ p ẋ a C p B p A p B p x a x p x a + B p B p u u () y y C p D p x a x p + D p u u () x a Problem (Gaussian Elimination, Fundamental Spaces, Least Squares, Minimum Norm) Consider the following linear algebraic system of equations: A b () 5 (a) Determine the general solution (b) Determine a basis for the four fundamental subspaces: R(A),R(A T ),N(A),N(A T ) (c) Determine the minimum norm solution Now let b T and consider the vector norm z def z T z (d) Determine the sef all possible x that minimizes the error b Ax (e) Determine the projection P R(A) b of the vector b onto the range of A (f) Determine the minimum norm x which minimizes the error b Ax Problem Solution: (a) Gaussian Elimination yields U ˆb b b b b 3 b (3)
From this, it follows that rank(a) This is also the number of basic variables The basic variables are x and x 4 From the last equation, we have x 4 b 3 b From this and the the first equation, it follows that x b x 4 b (b 3 b )5b b 3 The general solution is given by x 5b b 3 x + x x 3 + x 3 (4) x 4 b + b 3 b constrainn b; required for existence (5) Check, via substitution into Ax, that the above general solution does in fact satisfy Ax b b b b 3 T (b) Dimensions and bases for each of the 4 fundamnntal subspaces are as follows: From the above Gaussian elimination, it follows that dimr(a) rank(a) A basis for R(A) is:, (6) 5 Similarly, dimr(a H )rank(a) A basis for R(A H ) is as follows:, 5 (7) or, (8) From the above Gaussian elimination, it follows that dimn(a) n rank(a) 4 (number of free variables) A basis for N(A) is given by, (9) From the above Gaussian elimination, it follows that dimn(a H )m rank(a) 3 (number of constraints on b) A basis for N(A H ) is given by () This follows from the constraint equation (Equation (5)) that resulted from the above Gaussian elimination process (c) To determine the minimum norm solution (which is unique), we need to determine a solution that lies in the row space of A (or row space of U) Two approaches are used to address this componenf the problem Approach : Using U and ˆb It suffices to solve UU H z ˆb for any z and choose x U H z Doing so yields UU H z z 5 z ˆb b b b b 3 b ()
Performing Gaussian elimination on this yields 4 b z b b or 5 5b 3 b 4 z b b b 5b 3 b () From this, it follows that z is free, z and z 3 are basic, z 3 b +5b 3, z b 4z 3 b 4( b +5b 3 )b +48b b 3 5b b 3 The general z solution is then z 5b b 3 z + z (3) z 3 b +5b 3 From this, we obtain the minimum norm solution z 5b b 3 x U H z z + z (4) z 3 b +5b 3 5b b 3 5b b 3 (5) b 4b 3 b +5b 3 b + b 3 Approach : Using A and b Note that using the pair AA H w b, x A H w, yields the same result; but it generally involves more cumbersome arithmetic Lets show this: 5 5 b AA H w w 5 5 w b b (6) 5 9 b 3 5 Gaussian elimination on this yields 6 6 44 b w b b b 6 6 45 5b 3 or 44 6 w b b b b 5b 3 b (7) From this, we obtain w 3 5b 3 b and w b 44 6 w 3 w b 44 6 (5b 3 b ) w b b 3 +88b w 9b b 3 w From this, it follows that w 9b b 3 w + w (8) w 3 b +5b 3 This yields the minimum norm solution 9b b 3 4b +b 3 5b b 3 9b b 3 x A H w + w b +5b 3 5 58b 4b 3 6b +5b 3 b + b 3 (9)
It should be noted that the minimum norm solution is the unique projection of any solution onto the row space of A (or U) (d) To determine the sef all vectors x that minimize b Ax, it is necessary and sufficient to determine the sef all vectors x that satisfy the normal equations A H Ax A H b To do so, we proceed as follows: 6 4 A H Ax x x A H b 5 5 4 33 5 b b b 3 (3) This yields A H Ax 6 4 4 33 x A H b b + b +b 3 b +b +5b 3 (3) From this, it follows that x x 4 6 4 4 33 b + b +b 3 b +b +5b 3 33 4 b + b +b 3 4 6 b +b +5b 3 65 7 b + b +b 3 (3) 7 3 b +b +5b 3 65(b + b +b 3 ) 7(b +b +5b 3 ) (33) 7(b + b +b 3 ) + 3(b +b +5b 3 ) 5b +5b b 3 (34) b b + b 3 (35) The general solution is then given by x x x 3 x 4 5b +5b b 3 b b + b 3 + x + x 3 (36) (e) To determine the unique projection of b onto R(A), we need any solution to the normal equations A H Ax A H b and then P R(A) b Ax Using the result from (d) yields P R(A) b 5b +5b b 3 Ax + x + 5 b b + b 3 (5b +5b b 3 )+( b b + b 3 ) 5b +5b (5b +5b b 3 )+( b b + b 3 ) 5b +5b (5b +5b b 3 )+5( b b + b 3 ) b 3 x 3 (37) (38)
Recall that b R(A) if and only if b b In such a case, we note that P R(A) b b b b 3 as expected (f) To determine the minimum norm x that minimizes b Ax, we must solve AA H z P R(A) b for any z and then take x A H z Doing so yields AA H z z 5 5 5 5 5 5 w P R(A)b 9 5b +5b 5b +5b b 3 (39) Gaussian elimination yields 6 6 44 z 6 6 45 6b +6b 5b 3 or 44 6 z P R(A)b b +b 6b 6b +5b 3 (4) The latter equations yield z 3 6b 6b +5b 3 and z b +b 44 6 z 3 z b +b +44b + 44b b 3 z 45b +45b b 3 z From this, we obtain the general solution z z z 3 45b +45b b 3 6b 6b +5b 3 + From this, the minimum norm x that minimizes b Ax is given by x A H z z (4) 45b +45b b 3 + z 5b 5b +5b 3 5 (45b +45b b 3 )+( 6b 6b +5b 3 ) 5b +5b b 3 (4) (43) (45b +45b b 3 )+5( 6b 6b +5b 3 ) b b + b 3 Finally, it is worth noting that if b b then b R(A), P R(A) b b, and the above minimum norm x that minimizes Ax b takes the form x A H z 5b b 3 b + b 3 T which is in agreement with the answer obtained in (c) Problem 3 (Controllability, State Transfer, Pole Placement) p z p Consider the LTI system defined by the state space dual A, B where z p p and z p Sketch a block diagram for the system (a) Is the system controllable? Explain your answer Are there any pole-zero cancellations? (b) Does there exist a control law which will transfer the state of the system from x o to x()? If so, show how to determine one Also, determine a minimum energy state transferring control law (c) Does there exist a full state feedback control law u Gx such that the closed loop system has poles at s ± j? If so, determine a suitable control gain matrix G If not, explain why Problem 3-Partial Solution: (a) From the block diagram, one can identify two systems in series One system (the input system) has
transfer function z p s+p + s+z s+p This system feeds into a system (the output system) with transfer s+z function s+p The transfer function of the composite system is therefore s+p s+p From this, one sees that we have a pole-zero cancellation, the state x is uncontrollable, and p is an uncontrollable mode if and only if z p The controllability matrix is given by z p p C B AB (44) p From this, we see that det C p +p +p z p z Given this, the system is uncontrollable if and only if z p Since z p, it follows that the system is controllable and there are no pole-zero cancellations It should be noted that the condition z p is NOT needed for controllability (b) Since the system is controllable, there exists a control u that will transfer the state from any initial state x( )x o to any final state x(t f )x f in finite time Such a state transferring control law is provided by any u satisfying tf e A(to t f ) x f x o e A(to τ) Bu(τ)dτ (45) Such a u is given by ( tf ) u M(t) e A(to τ) BM(τ)dτ e A(to t f ) x f x o (46) where M(t) is any matrix -valued function of t such that the above inverse exists A particular control law that does the job is the minimum energy control law This control law is obtained when M(t) B H e AH ( t) In such a case, we have the following Gramian formula: u B H e AH ( t) G c (,t f ) e A(to t f ) x f x o (47) where G c (,t f ) tf For our problem, e At may be computed as follows ( s + p p z ) (si A) s + p e A(to τ) BB H e A H ( τ) dτ (48) ( ) s + p z p (49) (s + p )(s + p ) s + p ( z p ) s+p (s+p )(s+p ) (5) s+p From this, it follows that e At ( e p t z p p p e pt e pt ) e pt (5) (c) Since the system is controllable, there exists a full state feedback control gain matrix G which will place the eigenvalues of A BG anywhere in the complex plane - modulo complex conjugate constraints For our problem, p z p A BG p g g p g z p g g p g (5) To place closed loop poles at ± j, we require det(si A + BG) s + p + g det g + p z g s + p + g (53) s +(p + g + p + g )s +(p + g )(p + g )+g (z p g ) (54) s +(g + g + p + p )s + p p + g (p + z p )+g p (55) s +s + (56)
The above yields the algebraic system of equations: g p p z p g p p (57) As expected, this system possesses a solution if and only if z p ; ie if and only if the system is controllable In such a case, the solution is given by g p p p p p (58) g z p p p p z z p p p ( p p ) ( p p ) (59) p z z( p p ) + ( p p ) Problem 4 (Controllability, State Transfer, Pole Placement) p z p Consider the LTI system defined by the state space dual A, B where z p p and z p (a) Is the system controllable? Explain your answer Are there any pole-zero cancellations? (b) Assuming x o, determine an expression for x(t) using modal analysis concepts Determine a basis for the sef states that are reachable from x o (c) Does there exist a control law which will transfer the state of the system from x o to x()? If so, determine one Also, determine a minimum energy state transferring control law (d) Does there exist a control law which will transfer the state of the system from x o to x()? If so, determine one If not, determine what state closest to x() is reachable Then determine a control law which will achieve the transfer to this reachable state Is your control law unique? If not, show how to determine another state transferring control law and determine the minimum energy state transferring control law (e) Does there exist a full state feedback control law u Gx such that the closed loop system has poles at s p, (where p )? If so, determine a suitable control gain matrix G If not, explain why (f) Does there exist a full state feedback control law u Gx such that the closed loop system has poles at s, 5 (both distinct from p )? If so, determine a suitable control gain matrix G If not, explain why Problem 4 - Solution: (a) From the block diagram, one sees that we have a pole-zero cancellation, the state x is uncontrollable, and p is an uncontrollable mode if and only if z p The controllability matrix is given by z p p C B AB (6) p From this, we see that det C p + p + p z p z Given this, the system is uncontrollable if and only if z p Since z p, it follows that the system is uncontrollable and there is a pole-zero cancellation at s p It should be noted that the condition z p is NOT needed for controllability We now use PBH eigenvalue-eigenvector ideas to show that p is an uncontrollable mode Since A is diagonal, it follows that the system eigenvalues can be determined by inspection to be λ p and λ p We s + p p note that that (si A) p s + p For s p, we have (si A) p p It follows that v p p T is an eigenvector associated with p p p For s p, we have (si A) p p It follows that v T is an eigenvector associated with p
The matrix of right eigenvectors is given by V v v (6) Since the eigenvectors ar linearly independent, this matrix is invertible, and the system A matrix is said to be diagonalizable The corresponding matrix of left eigenvectors is then given by w H W w H V (6) Next, we note that w H B It thus follows from the PBH eigenvalue-eigenvector modal controllability test that p is an uncontrollable mode This, of course, was expected from the pole-zero cancellation within the system block diagram (b) To determine an expression for x(t), we use the above modal information Specifically, we note that e At B e λt v w H B + e λt v w H B e λt v w H B e pt It thus follows that x() x f A basis for the sef all reachable states from x o is then v e pt (63) e p( τ) u(τ)dτ (64) (c) There exists a control law which transfers the state from x o to satisfying In fact, any control law e p( τ) u(τ)dτ e p e pτ u(τ)dτ (65) will do the job The following is such a control law: u e p (66) epτ dτ where it is assumed that the integral in the denominator is non-zero This will be the case for any finite p One can also determine the minimum energy control law that will perform the desired state transfer To do so, we apply our minimum energy control law Gramian formula with x o,,x f,t f,a p, B This works because x f t f e A(t f τ) Bu(τ)dτ matches the above (desired) integral constraint for the selected values For the above selected values, we get the following from our minimum energy Gramian formula tf G c (,t f ) e A(to τ) BB H e A H ( τ) dτ e pτ dτ e p (67) p for p Forp, we obtain u B H e AH ( t) G c (,t f ) e A(to t f ) x f x o G c (,t f ) tf e pt p e p ep (68) e A(to τ) BB H e A H ( τ) dτ e pτ dτ (69) u B H e AH ( t) G c (,t f ) e A(to t f ) x f x o e pt e p (7)
(d) Given the analysis in (b), it follows that there does not exist a control law that will transfer the state from x o tox() x f because does not lie along the span of v In such a case, we can determine a control law that will minimize b Ax x where x e p( τ) u(τ)dτ To determine x, we solve the normal equations A H Ax A H b or x This yields x orx We thus need to determine a u such that e p( τ) u(τ)dτ One u that does the job is u e p ep τ dτ There are infinitely many control laws that will do the job Using x o,, x f, t f,a p, B, the minimum energy control law is given by G c (,t f ) tf e A(to τ) BB H e A H ( τ) dτ e pτ dτ p e p (7) u B H e AH ( t) G c (,t f ) e A(to t f ) x f x o e pt p e p e p (7) for p Forp, we obtain G c (,t f ) tf e A(to τ) BB H e A H ( τ) dτ e pτ dτ (73) u B H e AH ( t) G c (,t f ) e A(to t f ) x f x o e pt e p (74) (e) Since p is an uncontrollable mode, it cannot be moved via full state feedback (linear or otherwise) Given this, it is possible to find a control gain matrix G such that A BG has eigenvalues at p and where p That is, one pole stays fixed at p ; the other can be moved to any desired (real) location With z p, it follows that det(si A + BG) s + p + g det g + p p g s + p + g (75) s +(p + g + p + g )s +(p + g )(p + g )+g (p p g ) (76) s +(g + g + p + p )s + p (g + g + p ) (77) (s + p )(s + g + g + p ) (78) This corroborates the above; namely that p cannot be moved; the other closed loop eigenvalue (pole) can be placed at any desired real location g + g + p will, for example, place the other pole at (f) Since p is uncontrollable and hence cannot be moved, it is NOT possible to move both poles to arbitrary locations If and 5 are distinct from p, then no G exists If one of these are p, then from above, a G can be found Problem 5 (Observability, State Reconstruction, Pole Placement) p Consider the LTI system defined by the state space triple A p where z p and z p Sketch a block diagram for the system, B, C z p (a) Is the system observable? Explain your answer Are there any pole-zero cancellations? (b) Suppose that u and y is known on t, Determine an expression for the sef all possible initial conditions x o Can x o be determined uniquely? Explain Determine the minimum norm initial condition (c) Can one design an observer with closed loop poles at s,? If not, explain why If so, determine a suitable observer gain matrix H Moreover, determine the state estimation error x def x ˆx when u, the initial system state is x o, and the initial state estimate (used in your observer) is ˆx o
Problem 5-Partial Solution: (a) From the block diagram, one can identify two systems in series One system (the input system) has transfer function s+p This system feeds into a system (the output system) with transfer function z p s+p + s+z s+p s+z The transfer function of the composite system is therefore s+p s+p From this, one sees that we have a pole-zero cancellation, the state x is unobservable, and p is an unobservable mode if and only if z p Problem 6 (Observability, State Reconstruction, Pole Placement) p Consider the LTI system defined by the state space triple A, B, C z p p where z p and z p (a) Is the system observable? Explain your answer Are there any pole-zero cancellations? (b) Suppose that u and y is known on t, Determine an expression for the sef all possible initial conditions x o Can x o be determined uniquely? Explain Determine the minimum norm initial condition (c) Can one design an observer with closed loop poles at s p, (where p )? If so, determine a suitable observer gain matrix H If not, explain why (d) Can one design an observer with closed loop poles at s, 5 (both distinct from p )? If so, determine a suitable observer gain matrix H If not, explain why (e) Discuss how the answers to (a)-(d) change if z p and z p? Hint: Examine block diagram Problem 7 (Model Based Compensator Design) Consider the linear time invariant plant P (s) s 5 with state space quadruple A p 5,B p,c p,d p (a) Show how to design a model based compensator which satisfies the following design specifications: (i) zero steady state error to step reference commands, (ii) closed loop poles at s 4 ± j3, s ± j (b) Discuss how one might minimize the overshoot due to a step reference command (c) Summarize the design process if one desires to follow sinusoidal reference commands with frequency ω o