EC Control Engineering Quiz II IIT Madras

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1 EC34 - Control Engineering Quiz II IIT Madras Linear algebra Find the eigenvalues and eigenvectors of A, A, A and A + 4I Find the eigenvalues and eigenvectors of the following matrices: (a) cos θ sin θ sin θ cos θ (b) (c) B C Find the eigenvalues for the matrix Show that for matrix A, the algebraic multiplicity and geometric multiplicity of its eigenvalue is not the same 4 Find the eigenvalues of matrices A, B and A + B 3 B 3 Are the eigenvalues of A + B equal to sum of eigenvalues of A and eigenvalues of B? 5 Find the eigenvalues of matrices A, B, AB and BA B (a) Are the eigenvalues of AB equal to eigenvalues of A times eigenvalues of B? (b) Are the eigenvalues of AB equal to the eigenvalues of BA?

2 6 The eigenvalues of A equal the eigenvalues of A T This is because det(a λi) equals det(a T λi) That is true because Show by an example that the eigenvectors of A and A T are not the same 7 Let X and X be the eigenvectors corresponding to distinct eigenvalues of the symmetric matrix A ( A T ) Show that X is orthogonal to X (X T X X T X ) 8 Show that inverse A exists if and only if none of the eigenvalues λ, λ λ n is zero, and then A has the eigenvalues /λ, /λ /λ n 9 Let λ, λ λ n be the eigenvalues of matrix A Show that the eigenvalues of e A are given by e λ, e λ e λn Find the eigenvalues and eigenvectors of matrix 6 6 Find the eigenvectors X, X and X 3 corresponding to the distinct eigenvalues λ, λ and λ 3 of matrix A Create the 3 3 matrix P X X X 3, and show that λ AP P λ λ 3 Diagonalize the following matrices and find the eigenvector of à P AP () 5 55 (3) () 7 3 (4) 8 5 Given the following state space model x + u x 3 6 and y Transform the above state model using x P z 3 Find the matrix P such that following state space equation is transformed into diagonal canonical form: a a x + u 3 a 3 4 Obtain a state transition matrix Φ(t) of the following system, x 3 x x x 3 + If u(t) is a step input function, find the state response of the system u

3 5 Find the transfer function of the following state space model, x x + u and y x 6 For the following set of differential equations, write down the canonical state space model: (i) (ii) 7 Find f(a)a, where d3 y(t) 3 d 3 y(t) d y(t) + 5 d y(t) + 5 dy + y(t) u(t), + 3 dy + 3 t y(τ)dτ u(t) Stability 8 Stability of time delayed dynamical system as covered in class Chapter 6 9 Write down the definition of reachability Draw the block diagram of a system in reachable canonical form Åström and Murray, Example 63 Consider a simple two-dimensional system of the form, dx α ω x + u ω α Find the transformation that converts the system into reachable canonical form Åström and Murray, Exercise 64 (Integral feedback for rejecting constant disturbances) Consider a linear system of the form, dx Ax + Bu + F d, and y Cx, where u is a scalar and d is a disturbance that enters the system through a disturbance vector F R n Assume that the matrix A is invertible and the zero frequency gain CA B is nonzero Show that integral feedback can be used to compensate for a constant disturbance by giving zero steady-state output error even when d 3

4 3 (Exercise 67)(Reachability matrix for reachable canonical form) Consider a system in reach- able canonical form Show that the inverse of the reachability matrix is given by a a a n a a n Wr 4 (Exercise 68)(Non-maintainable equilibria)consider the normalized model of a pendulum on a cart d x u, d θ θ + u where x is cart position and θ is pendulum angle Can the angle θ θ for θ be maintained? 5 Åström and Murray, Exercise 6 (Cayley-Hamilton theorem) Let A R n n be a matrix with characteristic polynomial, λ(s) det(si A) s n + a s n + + a n s + a n Assume that the matrix A can be diagonalized and show that it satisfies, λ(a) A n + a An + + a n A + a n I, Use the result to show that A k, k n, can be rewritten in terms of powers of A of order less than n Chapter 7 6 Write down the definition of observability 7 Draw the block diagram of a system in observable canonical form 8 Show that the inverse of the observability matrix has a form given by a Wo a a a n a n a n 3 9 (Example 7)Consider the two-compartment model in which drug with concentration c is injected in compartment at a volume flow rate of u and that the concentration in compartment is the output Let c and c be the concentrations of the drug in the compartments, V and V be the volumes of the compartments and q be the outflow rate Introducing the variables k q /V, k q/v, k q/v and b c /V, the system can described by the linear system and its model is, dc k k k C + k k b u, and, y C 3 (Example 7)Consider the compartment model characterized by the matrices 4

5 k k k b, B k k, C (a) Find the condition for observability of the system (b) Let the desired characteristic polynomial for the observer be s + p s + p Obtain the observer gain (L) so that the desired characteristic polynomial is achieved 3 (Exercise 7)Consider a system under a coordinate transformation z T x, where T R n n is an invertible matrix Show that the observability matrix for the transformed system is given by W W T and hence observability is independent of the choice of coordinates 3 (Exercise 73) Show that if a system is observable, then there exists a change of coordinates z T x that puts the transformed system into observable canonical form 33 (Exercise 74)(Bicycle dynamics) The linearized model for a bicycle is given in equation (35), which has the form J d φ Dv dδ b mghφ + mv h δ b where φ is the tilt of the bicycle and δ is the steering angle Give conditions under which the system is observable and explain any special situations where it loses observability 5