Represent this system in terms of a block diagram consisting only of. g From Newton s law: 2 : θ sin θ 9 θ ` T

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1 Exercise (Block diagram decomposition). Consider a system P that maps each input to the solutions of 9 4 ` 3 9 Represent this system in terms of a block diagram consisting only of integrator systems, represented by the symbol solution p q P R of 9 ;, that map their input p q P R to the summation blocks, represented by the symbol output pq ; and, that map their input p q P R to the gain memoryless systems, represented by the symbol, that map their input p q P R to the output pq pq for some P R. m θ l g From Newton s law: Figure. Inverted pendulum : θ sin θ 9 θ ` T where T denotes a torque applied at the base, and is the gravitational acceleration. Exercise (Local linearization around equilibria). Consider the inverted pendulum in Figure and assume the input and output to the system are the signals and dened as T satpq θ where sat denotes the unit-slope saturation function that truncates at ` and. (a) Linearize this system around the equilibrium point for which θ. (b) Linearize this system around the equilibrium point for which θ π (assume that the pendulum is free to rotate all the way to this conguration without hitting the table). Linearize this system around the equilibrium point for which θ π 4. Does such an equilibrium point always exist? (d) Assume that { and {4. Compute the torque T pq needed for the pendulum to fall from θpq with constant velocity 9 Linearize the system around this trajectory. Exercise 3 (Local linearization around a trajectory). A single-wheel cart (unicycle) moving on the plane with linear velocity and angular velocity ω can be modeled by the nonlinear system 9 cos θ 9 sin θ 9 θ ω () where p q denote the Cartesian coordinates of the wheel and θ its orientation. Regard this as a system with input ω P R.

2 (a) Construct a state-space model for this system with state fl cos θ `p q sin θ sin θ `p q cos θfl 3 θ and output P R. (b) Compute a local linearization for this system around the equilibrium point eq, eq. Show that ωpq pq, pq sin, pq cos, is a solution to the system. (d) Show that a local linearization of the system around this trajectory results in an LTI system. Exercise 4 (Feedback linearization controller). Consider the inverted pendulum in Figure. (a) Assume that you can directly control the system in torque, i.e., that the control input is T. Design a feedback linearization controller to drive the pendulum to the upright position. Use the following values for the parameters: m, kg, Nm s, and 98 ms. Verify the performance of your system in the presence of measurement noise using Simulink. Attention! Writing the system in the carefully chosen coordinates 3 is crucial to getting an LTI linearization. If one tried to linearize this system in the original coordinates θ with dynamics given by (), one would get an LTV system. (b) Assume now that the pendulum is mounted on a cart and that you can control the cart s jerk, which is the derivative of its acceleration. In this case, T cos θ 9 Design a feedback linearization controller for the new system. What happens around θ π{? Note that, unfortunately, the pendulum needs to pass by one of these points for a swing-up, i.e., the motion from θ π (pendulum down) to θ (pendulum upright).

3 Exercise 5 (Observable canonical form). Given a transfer function Ĝpq, let pā B C Dq be a realization for its transpose Ḡpq Ĝpq. Show that pa B C Dq, where A Ā, B C, C B, and D D is a realization for Ĝpq. Note that if the realization pā B C Dq for Ḡpq is in controllable canonical form, then the realization pa B C Dq for Ĝpq so obtained is in observable canonical form. Exercise 6 (SISO realizations). This exercise aims at proving the theorem in Section Use the construction outlined in Section 4.3. to arrive at results consistent with the theorem in Section (a) Compute the controllable canonical form realization for the transfer function ˆpq ` α ` α ` `α ` α (b) For the realization in (a), compute the transfer function from the input to the new output, where is the th element of the state. Hint: You can compute pi Aq using the technique used in class for MIMO systems, or you may simply invert pi Aq using the adjoint formula for matrix inversion: M det M padj Mq adj M rcof Ms where cof M denotes the th cofactor of M. In this problem you actually need only to compute a single entry of pi Aq. Compute the controllable canonical form realization for the transfer function ˆpq β ` β ` `β ` β ` α ` α ` `α ` α () (d) Compute the observable canonical form realization for the transfer function in equation (). Hint: See Exercise 5. Exercise 7 (Equivalent realizations). Consider the following two systems: 9 fl ` fl 9 fl ` fl (a) Are these systems zero-state equivalent? (b) Are they algebraically equivalent?

4 Exercise 8 (State transition matrix). Consider the system 9 ` P R P R (a) Compute its state transition matrix (b) Compute the system output to the constant input for an arbitrary initial condition pq pq pq. Exercise 9 (Matrix powers and exponential). Compute A and A for the following matrices A fl A fl A 3 3 3fl (3) 3 Exercise (Jordan normal forms). Compute the Jordan normal form of the A matrix for the system represented by the following block diagram: u s ` ω y s s s ` ω y + y + y 3 Figure. Block interconnection for Exercise.

5 Exercise (Stability margin). Consider the continuous-time LTI system 9 A P R and suppose that there exists a positive constant µ and positive-denite matrices PQ P R for the Lyapunov equation A P ` PA ` µp Q (4) Show that all eigenvalues of A have real parts less than µ. A matrix A with this property is said to be asymptotically stable with stability margin µ. Hint: Start by showing that all eigenvalues of A have real parts less than µ if and only if all eigenvalues of A ` µi have real parts less than (i.e., A ` µi is a stability matrix). Exercise (Stability of nonlinear systems). Investigate whether or not the solutions to the following nonlinear systems converge to the given equilibrium point when they start sufciently close to it. (a) The state-space system with equilibrium point. (b) The second-order system 9 ` p ` q 9 ` p ` q : ` pq 9 ` with equilibrium point 9. Determine for which values of pq we can guarantee convergence to the origin based on the local linearization. This equation is called the Lienard equation and can be used to model several mechanical systems, depending on the choice of the function p q. Exercise 3. Consider the system 9 fl ` fl ` (a) Compute the system s transfer function. (b) Is the matrix A asymptotically stable, marginally stable, or unstable? Is this system BIBO stable?

6 Exercise 4 (A-invariance and controllability). Consider the LTI systems 9 A ` B { ` A ` B P R P R (AB-LTI) Prove the following two statements: (a) The controllable subspace C of the system (AB-LTI) is A-invariant. (b) The controllable subspace C of the system (AB-LTI) contains Im B. Exercise 5 (Satellite). The equations of motion of a satellite, linearized around a steady-state solution, are given by 9 A ` B, where and denote the perturbations in the radius and the radial velocity, respectively, 3 and 4 denote the perturbations in the angle and the angular velocity, and 3ω ω A ω fl B The input vector consists of a radial thruster and a tangential thruster. (a) Show that the system is controllable from. (b) Can the system still be controlled if the radial thruster fails? What if the tangential thruster fails? Exercise 6 (Controllable canonical form). Consider a system in controllable canonical form α I ˆ α I ˆ α I ˆ α I ˆ I ˆ ˆ ˆ ˆ A ˆ I ˆ ˆ ˆ fl ˆ ˆ I ˆ ˆ ˆ I ˆ ˆ B. ˆ fl ˆ ˆ C N N N N ˆ Show that such a system is always controllable. fl

7 Exercise 7 (Eigenvalue assignment). Consider the SISO LTI system in controllable canonical form 9 A ` B P R P R (AB-DLTI) where α α α α A fl ˆ B. fl ˆ (a) Compute the characteristic polynomial of the closed-loop system for K K Hint: Compute the determinant of pi A ` BKq by doing a Laplacian expansion along the rst line of this matrix. (b) Suppose you are given complex numbers λ, λ,...,λ as desired locations for the closed-loop eigenvalues. Which characteristic polynomial for the closed-loop system would lead to these eigenvalues? Based on the answers to parts (a) and (b), propose a procedure to select K that would result in the desired values for the closed-loop eigenvalues. (d) Suppose that A 3 fl B fl Find a matrix K for which the closed-loop eigenvalues are t u. Exercise 8 (Transformation to controllable canonical form). Consider the following third-order SISO LTI system 9 A ` B P R 3 P R (AB-CLTI) Assume that the characteristic polynomial of A is given by detpi Aq 3 ` α ` α ` α 3 and consider the 3 ˆ 3 matrix T α α α fl (5) where is the system s controllability matrix. (a) Show that the following equality holds: B T fl

8 (b) Show that the following equality holds: α α α 3 AT T fl Hint: Compute separately the left- and right-hand side of the equation above and then show that the two matrices are equal with the help of the Cayley-Hamilton theorem. Show that if the system (AB-CLTI) is controllable, then T is a nonsingular matrix. (d) Combining parts (a), you showed that, if the system (AB-CLTI) is controllable, then the matrix T given by equation (5) can be viewed as a similarity transformation that transforms the system into the controllable canonical form α T α α 3 AT fl T B Use this to nd the similarity transformation that transforms the following pair into the controllable canonical form A fl B fl 4 3 Hint: You may use the MATLAB R functions poly(a) to compute the characteristic polynomial of A and ctrb(a,b) to compute the controllability matrix of the pair (A,B). fl

9 Exercise 9 (Diagonal Systems). Consider the following system 9 fl 3 where,, and 3 are unknown scalars. (a) Provide an example of values for,, and 3 for which the system is not observable. (b) Provide an example of values for,, and 3 for which the system is observable. Provide a necessary and sufcient condition on the so that the system is observable. Hint: Use the eigenvector test. Make sure that you provide a condition that when true the system is guaranteed to be observable, but when false the system is guaranteed to not be observable. (d) Generalize the previous result for an arbitrary system with a single output and diagonal matrix A. Exercise (Diagonal Systems). Consider the system 9 fl 3 where,, and 3 are unknown scalars. (a) Provide a necessary and sufcient condition on the so that the system is detectable. (b) Generalize the previous result for an arbitrary system with a single output and diagonal matrix A. Exercise (Repeated eigenvalues). Consider the SISO LTI system 9{` A ` B C ` D P R P R (a) Assume that A is a diagonal matrix and B, C are column/row vectors with entries and, respectively. Write the controllability and observability matrices for this system. (b) Show that if A is a diagonal matrix with repeated eigenvalues, then the pair pa Bq cannot be controllable and the pair pa Cq cannot be observable. Given a SISO transfer function T pq, can you nd a minimal realization for T pq for which the matrix A is diagonalizable with repeated eigenvalues? Justify your answer. (d) Given a SISO transfer function T pq, can you nd a minimal realization for T pq for which the matrix A is not diagonalizable with repeated eigenvalues? Justify your answer. Hint: An example sufces to justify the answer yes in or (d).