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Exercise 5.5. Consider the linear system 2 as in (5.44) with A-4 0 C [1 0...
Problem 1 (25 pts): Consider the following non-linear autonomous system Where a>o,b0,c 0,d >0 and...
Problem 1 (25 pts): Consider the following non-linear autonomous system Where a>o,b0,c 0,d >0 and k >a. Consider the following Lyapunov Function: Where p >0. Answer the following questions: . Is V(x) a good candidate Lyapunov function? Explain 2. Is the origin at least stable? Explain (Hint: set p c) 3. Show that the system is Globally Asymptotically Stable.
Problem 1 (25 pts): Consider the following non-linear autonomous system Where a>o,b0,c 0,d >0 and k >a. Consider the following Lyapunov...
Problem 5 Consider the linear system [1 2 0 2 -4 7x(t) 1 -4 6 y(t)...
Problem 5 Consider the linear system [1 2 0 2 -4 7x(t) 1 -4 6 y(t) [1 -2 2] (t). (4) a(t = (a) Is the system (4) observable? (b) Give a basis for the unobservable subspace of the system (4). In the remainder of this problem, consider the linear system а — 3 8— 2а 0 1 2a u(t) (t) (5) x(t) = with a a real parameter. (c) Determine all values of a for which the system (5)...
1. Consider the system described by: *(t) - 6 m (0) + veu(t): y(t) = 01...
1. Consider the system described by: *(t) - 6 m (0) + veu(t): y(t) = 01 (1) 60 = {1, 1421 a) Find the state transition matrix and the impulse response matrix of the system. b) Determine whether the system is (i) completely state controllable, (ii) differentially control- lable, (iii) instantaneously controllable, (iv) stabilizable at time to = 0. c) Repeat part (b) for to = 1. d) Determine whether the system is (i) observable, (ii) differentially observable, (iii) instanta-...
Problem 1 (25 pts): Consider the following non-linear autonomous systerm Where a 0,b0.c O,d > 0 a...
Problem 1 (25 pts): Consider the following non-linear autonomous systerm Where a 0,b0.c O,d > 0 and k> a. Consider the following Lyapunov Function Where p >0. Answer the following questions: 1. 1s V (x) a good candidate Lyapunov function? Explairn 2. Is the origin at least stable? Explain (Hint: set p c) 3. Show that the system is Globally Asymptotically Stable
Problem 1 (25 pts): Consider the following non-linear autonomous systerm Where a 0,b0.c O,d > 0 and k>...
Consider the linear system given by the following differential equation y(4) + 3y(3) + 2y +...
Consider the linear system given by the following differential equation y(4) + 3y(3) + 2y + 3y + 2y = ů – u where u = r(t) is the input and y is the output. Do not use MATLAB! a) Find the transfer function of the system (assume zero initial conditions)? b) Is this system stable? Show your work to justify your claim. Note: y(4) is the fourth derivative of y. Hint: Use the Routh-Hurwitz stability criterion! c) Write the...
The objective of the controller design in Fig. 1 is to find the controller Gc(s) such...
The objective of the controller design in Fig. 1 is to find the
controller Gc(s) such that I.)The closed-loop system is stable
ii.)The output of the system above (y(t)) can track the reference
input r(t) = At (A>0 is any real number). Use the Nyquist plot
and Nyquist criterion to show that: a.)The portional controller
Gc(s)=K can achieve asymptotic tracking of the ramp input r(t)=At
but cannot meet the stability requirement for 0<K<+inf. b.)
Can the PI controller Gc(s)=K(1+10/s) be...
b(t) 1. Consider the system described by: 2. Consider the sy uuu It tet i(t) =...
b(t) 1. Consider the system described by: 2. Consider the sy uuu It tet i(t) = 0 -1 ] y(t) = (1 out) u(t) , 0, \t <1 (1, t > 1 a) Find the state transition matrix and the impulse response matrix of the system. 2D) Determine whether the system is (i) completely state controllable, (ii) differentially control lable, (iii) instantaneously controllable, (iv) stabilizable at time to = 0. (c) Repeat part (b) for to = 1. gd) Determine...
Problem 3. (15 points) Consider the feedback system in Figure 3, where G(s)1 (s -1)3 Ge(s)...
Problem 3. (15 points) Consider the feedback system in Figure 3, where G(s)1 (s -1)3 Ge(s) G(s) Figure 3: Problem 3 1. Let the compensator be given by a pure gain, ie, Ge(s)-K, K 0 (a) Draw the root locus of the compensated system (b) Is it possible to stabilize the systems by selecting K appropriately? If so, find the range for K such that the closed-loop system is BIBO stable. If not, explain. 2. This time, let the compensator...
Problem 2: Output-feedback stabilization Consider the following system 0 -8 3-3 4 [2-92]z y = a)...
Problem 2: Output-feedback stabilization Consider the following system 0 -8 3-3 4 [2-92]z y = a) Verify that the system is observable and controllable. Then, design an output-feedback controller (based on a full-order observer) by placing the poles of the closed loop system at -1 j, -3, 12 ±j2. and-30 (mention which desired poles you select for your observer design and why).
Assume a =500 4. Consider the following system [ 1.2 1 0 1 x (k +...
Assume a =500
4. Consider the following system [ 1.2 1 0 1 x (k + 1) = 0.6 0 1 x (k) + | –0.8 0 0 y (k) = [ 5 +a 0 0 ]x (k) 0 1 | 0.8 u(k) where a is the last three digits of your student ID number. (a) Obtain the transfer function of the system. Is the origin a stable equilibrium point? (b) Is the system controllable? Provide your reasoning. If your...
Problem 1 (25 pts): Consider the following non-linear autonomous system Where a>o,b0,c 0,d >0 and k >a. Consider the following Lyapunov Function: Where p >0. Answer the following questions: . Is V(x) a good candidate Lyapunov function? Explain 2. Is the origin at least stable? Explain (Hint: set p c) 3. Show that the system is Globally Asymptotically Stable.
Problem 1 (25 pts): Consider the following non-linear autonomous system Where a>o,b0,c 0,d >0 and k >a. Consider the following Lyapunov...
Problem 5 Consider the linear system [1 2 0 2 -4 7x(t) 1 -4 6 y(t) [1 -2 2] (t). (4) a(t = (a) Is the system (4) observable? (b) Give a basis for the unobservable subspace of the system (4). In the remainder of this problem, consider the linear system а — 3 8— 2а 0 1 2a u(t) (t) (5) x(t) = with a a real parameter. (c) Determine all values of a for which the system (5)...
1. Consider the system described by: *(t) - 6 m (0) + veu(t): y(t) = 01 (1) 60 = {1, 1421 a) Find the state transition matrix and the impulse response matrix of the system. b) Determine whether the system is (i) completely state controllable, (ii) differentially control- lable, (iii) instantaneously controllable, (iv) stabilizable at time to = 0. c) Repeat part (b) for to = 1. d) Determine whether the system is (i) observable, (ii) differentially observable, (iii) instanta-...
Problem 1 (25 pts): Consider the following non-linear autonomous systerm Where a 0,b0.c O,d > 0 and k> a. Consider the following Lyapunov Function Where p >0. Answer the following questions: 1. 1s V (x) a good candidate Lyapunov function? Explairn 2. Is the origin at least stable? Explain (Hint: set p c) 3. Show that the system is Globally Asymptotically Stable
Problem 1 (25 pts): Consider the following non-linear autonomous systerm Where a 0,b0.c O,d > 0 and k>...
Consider the linear system given by the following differential equation y(4) + 3y(3) + 2y + 3y + 2y = ů – u where u = r(t) is the input and y is the output. Do not use MATLAB! a) Find the transfer function of the system (assume zero initial conditions)? b) Is this system stable? Show your work to justify your claim. Note: y(4) is the fourth derivative of y. Hint: Use the Routh-Hurwitz stability criterion! c) Write the...
The objective of the controller design in Fig. 1 is to find the
controller Gc(s) such that I.)The closed-loop system is stable
ii.)The output of the system above (y(t)) can track the reference
input r(t) = At (A>0 is any real number). Use the Nyquist plot
and Nyquist criterion to show that: a.)The portional controller
Gc(s)=K can achieve asymptotic tracking of the ramp input r(t)=At
but cannot meet the stability requirement for 0<K<+inf. b.)
Can the PI controller Gc(s)=K(1+10/s) be...
b(t) 1. Consider the system described by: 2. Consider the sy uuu It tet i(t) = 0 -1 ] y(t) = (1 out) u(t) , 0, \t <1 (1, t > 1 a) Find the state transition matrix and the impulse response matrix of the system. 2D) Determine whether the system is (i) completely state controllable, (ii) differentially control lable, (iii) instantaneously controllable, (iv) stabilizable at time to = 0. (c) Repeat part (b) for to = 1. gd) Determine...
Problem 3. (15 points) Consider the feedback system in Figure 3, where G(s)1 (s -1)3 Ge(s) G(s) Figure 3: Problem 3 1. Let the compensator be given by a pure gain, ie, Ge(s)-K, K 0 (a) Draw the root locus of the compensated system (b) Is it possible to stabilize the systems by selecting K appropriately? If so, find the range for K such that the closed-loop system is BIBO stable. If not, explain. 2. This time, let the compensator...
Problem 2: Output-feedback stabilization Consider the following system 0 -8 3-3 4 [2-92]z y = a) Verify that the system is observable and controllable. Then, design an output-feedback controller (based on a full-order observer) by placing the poles of the closed loop system at -1 j, -3, 12 ±j2. and-30 (mention which desired poles you select for your observer design and why).
Assume a =500
4. Consider the following system [ 1.2 1 0 1 x (k + 1) = 0.6 0 1 x (k) + | –0.8 0 0 y (k) = [ 5 +a 0 0 ]x (k) 0 1 | 0.8 u(k) where a is the last three digits of your student ID number. (a) Obtain the transfer function of the system. Is the origin a stable equilibrium point? (b) Is the system controllable? Provide your reasoning. If your...
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