Theory of Linear Systems Exercises. Luigi Palopoli and Daniele Fontanelli
|
|
- Dana Griffin
- 5 years ago
- Views:
Transcription
1 Theory of Linear Systems Exercises Luigi Palopoli and Daniele Fontanelli Dipartimento di Ingegneria e Scienza dell Informazione Università di Trento
2
3 Contents Chapter. Exercises on the Laplace Transform Chapter. Exercises on the Z-Transform 9 Chapter 3. Exercises on the State Space Representation 9 v
4
5 CHAPTER Exercises on the Laplace Transform Exercise Consider the following system expressed with the differential equation determine if it is BIBO stable. ÿ(t) = ẏ(t) 3y(t) u(t)+4 u(t), In order to determine the BIBO stability, we first compute the Laplace transform of the impulsive response imposing null initial conditions, i.e., that yields to s Y(s) = sy(s) 3Y(s) U(s)+4sU(s), G(s) = 4s s +s+3. Since the roots of the denominator are p = ±j, it follows that the system is BIBO stable. This fact can be proved by assuming u(t) = (t) and computing the forced response, i.e., 4s Y(s) = s(s++j )(s+ j ) = A s + A s++j + A 3 s+ j, where A = 3, A = +j3, A 3 = j3, where A 3 = A (the complex and conjugated). It then follows ( ) L (Y(s)) = 3 + +j3 e ( j )t + j3 e ( +j )t (t). By rewriting +j3 one has L (Y(s)) = ( 4+ 3 = Me jφ, with M =, φ = arctan 3 ), ( ) ( 3 +Me t e j( t φ) +Me t e j( t φ) (t) = 3 +Me t cos( ) t φ) (t), which is of course bounded.
6 . EXERCISES ON THE LAPLACE TRANSFORM Exercise Consider the following system expressed with the differential equation ÿ(t) = ẏ(t) 3y(t) u(t)+4 u(t), determine its unforced response given ẏ(0) = and y(0) =. To compute the unforced we need first to compute the Laplace transform of the system, that is s Y(s) sy(0) ẏ(0) = sy(s)+y(0) 3Y(s) U(s)+4sU(s) 4u(0), that yields to Y(s) = 4s s +s+3 U(s) 4 (s+)y(0)+ẏ(0) s u(0)+ +s+3 s. +s+3 Since we are interested in the unforced response, we have Y(s) = (s+)y(0)+ẏ(0) s +s+3 where = (s+)+ s +s+3 = A s++j + A s+ j, A =, A =, where again A = A. It then follows that L (Y(s)) = ( ) e ( j )t +e ( +j )t (t). Hence L (Y(s)) = e t( e j t +e j t ) (t) = e t cos( t)(t).
7 . EXERCISES ON THE LAPLACE TRANSFORM 3 Exercise 3 Consider the following system expressed with the differential equation ÿ(t) = 5 3ẏ(t) 4 y(t) 3 u(t)+ 4 3 u(t), determine its response to the input u(t) = k(t) given ẏ(0) = 0, y(0) = and u(0) =. To compute the response we need first to compute the Laplace transform of the system, that is 3s Y(s) 3sy(0) 3ẏ(0) = 5sY(s)+5y(0) 3 4 Y(s) U(s)+4sU(s) 4u(0), that yields to 4s 4 Y(s) = 3s +5s+ 3 U(s) 4 3s +5s+ 3 4 that turns to Y(s) = k(4s ) s(3s +5s+ 3 4 ) 4 3s +5s+ 3 4 u(0)+ (3s+5)y(0)+3ẏ(0) 3s +5s s+5 + 3s +5s+ 3, 4 i.e., the superposition of three different components. Therefore: For the first, we have where Y (s) = It then follows k(4s ) s(3s +5s+ 3 4 ) = A s + A s+ 6 A = k 4 3, A = k 5, A 3 = k 7 6. L (Y (s)) = k + A 3 s+ 3, ( e 6 t 76 e 3 t ) (t). For the second term, related to the initial condition of the input, we have 4 Y (s) = 3s +5s+ 3 = A 4 4 s+ + A 5 6 s+ 3, where It then follows L (Y (s)) = A 4 =, A 5 =. ( e 6 t +e 3 t) (t).
8 4. EXERCISES ON THE LAPLACE TRANSFORM For the third term, related to the initial condition of the output, we have 3s+5 Y 3 (s) = 3s +5s+ 3 = A 6 4 s+ + A 7 6 s+ 3, where It then follows L (Y 3 (s)) = A 6 = 9 8, A 7 = 8. ( 9 8 e 6 t ) 8 e 3 t (t). Therefore, we finally have by the superposition principle: [ y(t) = 4k 0k + + e 6 t 8k e 3 ](t). t 3 8 4
9 . EXERCISES ON THE LAPLACE TRANSFORM 5 Exercise 4 Consider the following system G(s) = 3s+7 s s 3 and determine if it is BIBO unstable. Since the coefficients of the polynomial at the denominator changes in sign, the system is BIBO unstable since it has one root with positive real part.
10 6. EXERCISES ON THE LAPLACE TRANSFORM Exercise 5 Consider the following Laplace transform of two transfer functions G (s) = 3s 7 s +s+3 s+ G (s) = s(s +s+3) determine if they are BIBO stable. Both of them are not unstable. However, the first one has two roots with negative real part (persistency of the sings of the polynomial), while the second has the same two roots plus a pole in the origin. Therefore, only the first one is BIBO stable. To verify this fact, it is sufficient to apply a unitary step as input and then compute the inverse Laplace transform of the output.
11 . EXERCISES ON THE LAPLACE TRANSFORM 7 Exercise 6 Given a system whose transfer function is s 5 G(s) = (s+) (s+) determine the time evolution of the output when the input is u(t) = e 3t + (t ). One way to takle this problem is to first compute the Laplace transform of the input, which is given by U(s) = s+3 + e s s, and then apply the inverse Laplace transform to the output Y(s) = s 5 (s+) (s+)(s+3) + (s 5)e s s(s+) (s+). Another, quite similar approach, would be to compute the inverse Laplace transform of the two output functions as they are applied at time t = 0, i.e., ( ) y (t) = L s 5 (s+), (s+)(s+3) ( ) y (t) = L s 5 s(s+), (s+) and then define y(t) = y (t)+y (t ).
12
13 CHAPTER Exercises on the Z-Transform Exercise Given the following Z-Transform of an input sequence U(z) = +3z z 5 +z 7, find the corresponding signal representation Straightforwardly, we apply the inverse Z-Transform of the Kronecker delta function to find: u(t) = δ(t)+3δ(t ) δ(t 5)+δ(t 7). 9
14 0. EXERCISES ON THE Z-TRANSFORM Exercise Given two sequences find y(t) = h(t) u(t). u(t) = (t) (t 4) and h(t) = (t) (t 5), We first recall that Then, we noticed that that yields Similarly y(t) = h(t) u(t) = Z (H(z)U(z)). u(t) = δ(t)+δ(t )+δ(t )+δ(t 3) U(z) = +z +z +z 3. h(t) = δ(t)+δ(t )+δ(t )+δ(t 3)+δ(t 4) H(z) = +z +z +z 3 +z 4. As a consequence: and then Y(z) = +z +3z +4z 3 +4z 4 +3z 5 +z 6 +z 7, y(t) = δ(t)+δ(t )+3δ(t )+4δ(t 3)+4δ(t 4)+3δ(t 5)+δ(t 6)+δ(t 7).
15 . EXERCISES ON THE Z-TRANSFORM Exercise 3 FindtheinverseZ-TransformwithROC z > forthefollowingtransfer function G(z) = +z +z 3 z + z. Furthermore, determine if the system is BIBO stable. We first rewrite G(z) in the more suitable form G(z) = z +z + z 3 z +. Being the roots of the denominator equal to z = and z =, we have where It then follows that and finally G(z) z = A z + A z + A 3 z, A =, A = 9, A 3 = 8. Z (G(z)) = Z () 9Z ( Z (G(z)) = δ(t)+ [ z z 8 9 ) ( ) z +8Z z ( ) ] t (t). Obviously, the system is BIBO unstable since z =. To further prove it, we compute the output of the system when the input u(t) is a unitary step, i.e., which yields to with Y(z) = G(z)U(z) = z +z + z 3 z + Y(z) z Z (Y(z)) = 9Z ( and then = A z z z y(t) = [ ) z z, + A 3 z + A 4 (z ), ( ( ) z 8Z )+8Z z z (z ) 8t 8+9 ( ) ] t (t).
16 . EXERCISES ON THE Z-TRANSFORM Exercise 4 Consider the following difference equation representing the output y(t) of a system subjected to the input u(t) y(t+) = 3u(t+) u(t+)+u(t)+ 5 6 y(t+) 6 y(t) compute the impulse response of the system for null initial conditions. Since the initial conditions are null, we have that y(0) = y() = u(0) = u() = 0. Therefore the Z-Transform is given by which yields to z Y(z) = 3z U(z) zu(z)+u(z)+ 5 6 zy(z) 6 Y(z), Y(z) = 3z z + z 5 6 z + U(z) = 3z z + ( )( 6 z z )U(z). 3 The impulse response is then given by Hence, where It then follows that and finally H(z) = 3z z + ( )( z z ). 3 ( ) H(z) Z = A z z + A z A = 6, A = 9, A 3 =. Z (H(z)) = Z (6)+9Z ( Z (H(z)) = 6δ(t)+ [ 9 z z + A 3 z, 3 ) Z ( z z 3 ( ) t ( ) ] t (t). 3 Comparison Inordertoprovethecorrectnessoftheanalysis, wecancomputethetime evolution of the output y(t) when the input is a Kronecker delta u(t) = δ(t) using the difference equation and then compare it with the time evolution Z (H(z)) computed for every t, as reported in Table. As shown in the Table, the correctness of the analysis carried out is further verified. )
17 . EXERCISES ON THE Z-TRANSFORM 3 t y(t) Z (H(z)) t 0 3δ(t) = 3 6δ(t)+[9 ] = 3 δ(t)+ 5 6 y(0) = [ 9 ] 3 = δ(t)+ 5 6 y() 9 6y(0) = 4 9 = y() y() = = Table. Comparison of the two impulse responses.
18 4. EXERCISES ON THE Z-TRANSFORM Exercise 5 Consider the following difference equation representing the output y(t) of a system subjected to the input u(t) y(t) = 3u(t) u(t )+u(t )+ 5 6 y(t ) 6 y(t ) compute the forced response of the system for ( ) t u(t) = (t) 4 The forced response is computed for null initial conditions, i.e., y( ) = y( ) = 0. Therefore the Z-Transform is given by Y(z) = 3U(z) z U(z)+z U(z)+ 5 6 z Y(z) 6 z Y(z), which yields to Hence, Y(z) = 3z z + z 5 6 z + U(z) = 6 ( ) Y(z) Z = A z z ( z z(3z z +) )( z 3 + A z 3 + A 3 z, 4 )( z 4 where A = 8, A = 48, A 3 = 33. It then follows that ( ( ( ) Z (Y(z)) = 8Z z ) 48Z z )+33Z z z z 3 z 4 and finally Z (Y(z)) = [ 8 ( ) t 48 ( ) t ). ( ) ] t (t). 4 Comparison In order to prove the correctness of the analysis, we can compute the time evolution of the output y(t) when the input is u(t) using the difference equation and then compare it with the time evolution Z (Y(z)) computed for every t, as reported in Table. As shown in the Table, the correctness of the analysis carried out is further verified.
19 . EXERCISES ON THE Z-TRANSFORM 5 t y(t) Z (H(z)) t 0 3 [ ] = y(0) = 5 [ ] 4 = y() 6 y(0) = = Table. Comparison of the two vanishing step responses.
20 6. EXERCISES ON THE Z-TRANSFORM Exercise 6 Consider the following difference equation representing the output y(t) of a system subjected to the input u(t) y(t+) = 3u(t+) u(t+)+u(t)+ 5 6 y(t+) 6 y(t) compute the unforced response of the system for y() = 0 and y(0) =. The Z-Transform is given by z Y(z) z y(0) zy() = 3z U(z) z U(z)+U(z)+ 5 6 zy(z) 5 6 zy(0) 6 z Y(z), which yields to Y(z) = (3z z +) ( )( z z )U(z)+ z(z 5 ( 6 )y(0)+zy() )( ) 3 z z. 3 Hence (U(z) = 0), Y(z) = z(z 5 ( )( 6 ) ) z z 3 and, therefore where It then follows that and finally ( ) Y(z) Z = A z z Z (Y(z)) = Z ( Z (Y(z)) = A =, A = 3. [ z z + A z, 3 ) 3Z ( z z 3 ( ) t ( ) ] t 3 (t). 3 Comparison In order to prove the correctness of the analysis, we can compute the time evolution of the output y(t) when the input is u(t) using the difference equation and then compare it with the time evolution Z (Y(z)) computed for every t, as reported in Table 3. As shown in the Table, the correctness of the analysis carried out is further verified. )
21 . EXERCISES ON THE Z-TRANSFORM 7 t y(t) Z (H(z)) t 0 [ 3] = 0 [ ] = y() 6 y(0) = y() 6 y() = = = Table 3. Comparison of the responses for the selected initial conditions.
22
23 CHAPTER 3 Exercises on the State Space Representation Exercise Given a continuous time linear system [ ] [ 0 ẋ(t) = x(t)+ u(t) = Ax(t)+Bu(t), 0 0 ] compute the unforced response for a generic x(0) and verify if the system is BIBO stable. The unforced response is given by x u (t) = e At x(0) = Te Jt T x(0), where T is the transformation matrix that transforms the dynamic matrix A in the Jordan form J. Therefore, T is a base of generalised eigenvectors of the matrix A. However, it is evident that the matrix A has two eigenvalues (i.e., it is upper triangular), that are λ = and λ = 0. It is then clear that [ ] 0 J =. 0 0 The transformation matrix T = [v,v ], where v i = [v i,,v i, ] T, is then obtained by solving the eigenvector problem, i.e., { v +v = v Av i = λ i v i For the first eigenvector v, we have while for the second v, we have v = v = 0 = v { v +v = 0 0 = 0 [ α 0], [ β β], 9
24 0 3. EXERCISES ON THE STATE SPACE REPRESENTATION where α R and β R, and α 0 and β 0. Therefore, a possible choice for T is [ ] [ ] T = T 0 =. 0 It then follows that for a generic x(0) = [x (0),x (0)] T, we have [ ][ ][ ][ ] [ x u (t) = Te Jt T e t 0 x (0) x (0)+e x(0) = = t ] (x (0) x (0)) x (0) x (0) For what concerns the BIBO stability, it is immediate to notice that the system has an eigenvalue in zero, so it is marginally stable. Therefore, it has a real pole in the origin, that corresponds to an integrator. It finally yields that the system is not BIBO stable.
25 3. EXERCISES ON THE STATE SPACE REPRESENTATION Exercise Given a continuous time linear system in Jordan form whose unforced response for the initial condition x(0) = [,,0,3] T is x u (t) = e Jt x(0) = cos(t)+sin(t) cos(t) + sin(t) 3te t 3e t determine the explicit expression of the matrix J., We first recall that a matrix in Jordan form is block diagonal. We then recall that in order to have sinusoidal functions in the expression of x u (t), at least two complex and conjugated eigenvalues are needed. Since the state has dimension 4 and the sinusoidal functions comes without the presence of the time t, we conclude that we have only two complex and conjugated eigenvalues that has geometric and algebraic multiplicity equals to one (i.e., eigenvalues λ = σ + jω and λ = σ jω). Hence, the submatrix associated to such eigenvalues is diagonalisable. On the contrary, the other two eigenvalues are coincident, with geometric multiplicity equals to one (since we have the explicit presence of the time t that multiplies the exponential function). To summarise, we have two Jordan blocks, one associated to a pair of complex and conjugated eigenvalues, the other with only one mini-block (hence associated to a chain of generalised eigenvalues whose length is two). It then follows that e Jt = [ ] e J t 0 0 e J t = e σt cos(ωt) e σt sin(ωt) 0 0 e σt sin(ωt) e σt cos(ωt) e λ 3t te λ 3t e λ 3t Therefore, by recalling the initial condition x(0) and the expression of x u (t), it follows that σ = 0, ω = and λ 3 = and, then [ ] J 0 J = = J
26 3. EXERCISES ON THE STATE SPACE REPRESENTATION Exercise 3 Given the following continuous time linear invariant system [ ] [ ẋ(t) = x(t)+ u(t) = Ax(t)+Bu(t), 4 3 ] compute the response of the system assuming an initial condition x(0) = [0,] T and an input u(t) = α(t t 0 ), where t 0 > 0. To solve this problem we have to recall the explicit form of the response of a linear continuous time system, i.e., x(t) = e At x(0)+ t 0 e A(t τ) Bu(τ)dτ. Due to the particular form of the input, we have x(t) = e At x(0)+ t t 0 e A(t τ) Bu(τ)dτ = e At x(0)+α t t 0 e A(t τ) dτb, where the second equation is obtained by noticing that both B and u(t) are constant for t t 0. We then recall that the exponential of the matrix can be readily obtained using the Jordan form, i.e., x(t) = Te Jt T x(0)+α = Te Jt T x(0)+αt t Te J(t τ) T dτb = t 0. t t 0 e J(t τ) dτt B It is then evident that it is necessary to compute the Jordan form J of the matrix A to solve the described problem. To this end, let us first compute the eigenvalues, which are det(λi A) = 0 λ +λ+ = 0 λ = λ =, so an eigenvalue with algebraic multiplicity equals to. Let us check the geometric multiplicity by tackling the eigenvector problem { vi v i = v i Av i = λv i 4v i 3v i = v i whichyieldstoonlyoneconstraintv i = v i, hencetheequationissolvedby eigenvectors linearly dependent with v = [,] T. Therefore, the geometric multiplicity is one, so the matrix is defective. It follows that the second generalised eigenvector will solve (A λi)v () = v () = v Av () = λv () +v () v () v () = v () + 4v () 3v () = v () +
27 3. EXERCISES ON THE STATE SPACE REPRESENTATION 3 which yields to v () = v (), therefore a solution like v () = [,] T is linear independent to v () = [,] T. We finally have that T = [ ] T = [ with [ ] [ J = T AT = e 0 Jt e t te = t ] 0 e t, that is one Jordan block with an associated mini block. Finally, the unforced response is given by [ x u (t) = Te Jt T te x(0) = t ] e t te t, while the forced response is t [ t x f (t) = αt e J(t τ) dτt B = αt t 0 e τ t t ] dτ t 0 (t τ)e τ t dτ t t 0 0 t 0 e τ t T B = dτ [ e t 0 t (t = αt 0 )e t0 t t(e t0 t ] ) t+ 0 e t 0 t T B = [ α(t+e t 0 = t +t(e t0 t ) e t0 t ] (t 0 ) ) α(t+e t0 t +t(e t0 t ) e t0 t (t 0 ) 3) Finally, we have x(t) = x u (t)+x f (t). ],
28 4 3. EXERCISES ON THE STATE SPACE REPRESENTATION Exercise 4 Given a discrete time linear system [ ] [ x(k+) = 0 0 x(k)+ u(k) = Ax(k)+Bu(k), 3 ] compute the unforced response for a generic x(0) and the steady state value (if any). The unforced response is given by x u (k) = A k x(0) = TJ k T x(0), where T is the transformation matrix that transforms the dynamic matrix A in the Jordan form J. Therefore, T is a base of generalised eigenvectors of the matrix A. However, it is evident that the matrix A has two eigenvalues (i.e., it is lower triangular), that are λ = and λ =. It is then clear that [ J = 0 0 The transformation matrix T = [v,v ], where v i = [v i,,v i, ] T, is then obtained by solving the eigenvector problem, i.e., v = v Av i = λ i v i For the first eigenvector v, we have while for the second v, we have ]. 3v +v = v v = v 3v +v = v v = v = [ ] α, α [ 0 β], where α R and β R, and α 0 and β 0. Therefore, a possible choice for T is [ ] [ ] 0 T = T 0 =. It then follows that for a generic x(0) = [x (0),x (0)] T, we have [ ][ x u (k) = TJ k T 0 x(0) = k ][ ][ ] [ ] 0 0 x (0) = k x (0) 0 x (0) ( k. )x (0)+x (0)
29 3. EXERCISES ON THE STATE SPACE REPRESENTATION 5 For the steady state value, we have just to compute [ ] lim x 0 u(k) =. k + x (0)+x (0)
30 6 3. EXERCISES ON THE STATE SPACE REPRESENTATION Exercise 5 Given a discrete time linear system in Jordan form whose unforced response for the initial condition x(0) = [,,0,3] T is k x u (k) = J k x(0) = k cos(k π ) k sin(k π ), 3 k 4 determine the explicit expression of the matrix J. We first recall that a matrix in Jordan form is block diagonal. We then recall that in order to have sinusoidal functions in the expression of x u (k), at least two complex and conjugated eigenvalues are needed. Since the state has dimension 4 and the sinusoidal functions comes without a multiplication by the time k, we conclude that we have only two complex and conjugated eigenvalues that has geometric and algebraic multiplicity equals to one (i.e., eigenvalues λ = σ+jω and λ 3 = σ jω). Hence, the sub-matrix associated to such eigenvalues is diagonalisable. By the particular form of the other two entries, we conclude that there are two additional and distinct real eigenvalues (λ and λ 4 ). To summarise, we have three Jordan blocks, one associated to a pair of complex and conjugated eigenvalues, and the other two associated with two distinct mini-blocks (in other words, the matrix is diagonalisable). It then follows that J k J k 0 0 = 0 J k 0 = 0 0 J3 k λ k ρ k cos(kθ) ρ k sin(kθ) 0 0 ρ k sin(kθ) ρ k cos(kθ) λ k 4 where ρ = λ = σ +ω and θ = arctan( Imag(λ ) Real(λ ) ) = arctan(ω σ ). Therefore, by recalling the initial condition x(0) and the expression of x u (k), it follows that σ = 0, ω =, λ = and λ 4 = 4, then J = J J 0 = 0 cos( π ) sin(π ) sin( 0 0 J π ) cos(π ) 0 = ,
31 3. EXERCISES ON THE STATE SPACE REPRESENTATION 7 Exercise 6 Given the following discrete time linear invariant system 3 x(k+) = 4 4 x(k)+ 0 0 u(k) = Ax(k)+Bu(k), 0 0 compute the response of the system assuming an initial condition x(0) = [,, ] T and an input u(k) = α(k k 0 ), with k 0 Z and k 0 > 0. To solve this problem we have to recall the explicit form of the response of a linear discrete time system, i.e., k x(k) = A k x(0)+ A k i Bu(i). i=0 Due to the particular form of the input, we have k x(k) = A k x(0)+ A k i Bu(i) = A k x(0)+α for k > k 0 and i=0 k k x(k) = A k x(0)+ A k i Bu(i) = A k x(0), i=0 i=k 0 A k i B, for k k 0. We then recall that the power of the matrix can be readily obtained using the Jordan form, i.e., k x(k) = TJ k T x(0)+α TJ k i T B = i=k 0. k = Te Jt T x(0)+αt T B i=k 0 J k i It is then evident that it is necessary to compute the Jordan form J of the matrix A to solve the described problem. To this end, let us first compute the eigenvalues, which are det(λi A) = 0 λ 3 3λ +3λ = 0 λ = λ = λ 3 = λ =, so an eigenvalue with algebraic multiplicity equal to 3. In other words, the Jordan form will have only one Jordan block. Let us check the geometric multiplicity by tackling the eigenvector problem 3v i v i v i3 = v i Av i = λv i 4v i v i 4v i3 = v i v i3 = v i3
32 8 3. EXERCISES ON THE STATE SPACE REPRESENTATION which yields to only one constraint v i = v i v i3, hence the equation is solved by linearly independent eigenvectors, for example (3.) v = η and v = β 0, 0 β or the more straightforward (3.) v = η η and v = 0 β. 0 β η In both cases, η,β R and η 0 and β 0. It is then obvious that the geometric multiplicity is, which is less than 3, hence we have two Jordan mini-blocks of dimensions and, respectively, associated to the eigenvalue λ. We can then readily define the Jordan form J of the matrix A as J = T AT = In order to fully characterise the Jordan form we need also to define the transformation matrix T, which is formed by a basis of generalised eigenvectors. To this end, let us first define v () = v and v () = v, making the choice reported in (3.). Then, (A λi)v () = v () v () v() v() 3 = η 4v () v() 4v() 3 = η 0 = 0 which yields to v () = v() v() 3 η. Following the same steps of the eigenvector, in this particular case we have two straightforward solutions v = γ γ η and v = 0 δ, 0 δ or v = γ 0 γ and v = δ η, 0 δ where γ,δ R and γ 0 and δ 0. For the first choice, v is linearly dependent to v (), hence v() = v. For the second choice, v is linearly dependent to v (), hence v() = v. Notice that if one tries to continue the chain by computing v (3), it will obtain from the first choice η = 0, which violates the hypothesis, and from the second choice δ = 0, which again violates the hypothesis. So the chain comprises only v () and v () (se the mini-block of dimension ).
33 3. EXERCISES ON THE STATE SPACE REPRESENTATION 9 Finally, by computing v () starting from v (), one immediately has that β = 0, which violates the hypothesis. It then follows that the matrix T = [v (),v(),v() ] has two possible values T = η γ 0 η γ η β and T = η 0 0 η δ η β, 0 0 β 0 δ β that, by simply choosing η = β = γ = δ =, yields to T = 0 and T =, and T = and T = Whatever is the choice made (i.e., T = T or T = T ), we have J = T AT = J k = k 0 0 0, that is one Jordan block with two mini blocks. Notice that the same results would be obtained if the choice of (3.) would be selected, instead (prove it on your own). Finally, the unforced response is given by x u (k) = TJ k T x(0) = +k +4k, while the forced response is k x f (k) = αt T B = = αt = αk αk. α J k i i=k 0 k k 0 Finally, we have λ k i k k 0 (k i )λ k i 0 0 k k 0 λ k i x(t) = x u (t)+x f (t). k k 0 λ k i T B =
34
Identification Methods for Structural Systems
Prof. Dr. Eleni Chatzi System Stability Fundamentals Overview System Stability Assume given a dynamic system with input u(t) and output x(t). The stability property of a dynamic system can be defined from
More informationSome solutions of the written exam of January 27th, 2014
TEORIA DEI SISTEMI Systems Theory) Prof. C. Manes, Prof. A. Germani Some solutions of the written exam of January 7th, 0 Problem. Consider a feedback control system with unit feedback gain, with the following
More informationIdentification Methods for Structural Systems. Prof. Dr. Eleni Chatzi System Stability - 26 March, 2014
Prof. Dr. Eleni Chatzi System Stability - 26 March, 24 Fundamentals Overview System Stability Assume given a dynamic system with input u(t) and output x(t). The stability property of a dynamic system can
More informationState will have dimension 5. One possible choice is given by y and its derivatives up to y (4)
A Exercise State will have dimension 5. One possible choice is given by y and its derivatives up to y (4 x T (t [ y(t y ( (t y (2 (t y (3 (t y (4 (t ] T With this choice we obtain A B C [ ] D 2 3 4 To
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 2 Laplace Transform I 1/52
1/52 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 2 Laplace Transform I Linear Time Invariant Systems A general LTI system may be described by the linear constant coefficient differential equation: a n d n
More informationLTI Systems (Continuous & Discrete) - Basics
LTI Systems (Continuous & Discrete) - Basics 1. A system with an input x(t) and output y(t) is described by the relation: y(t) = t. x(t). This system is (a) linear and time-invariant (b) linear and time-varying
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.6 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS
More information21 Linear State-Space Representations
ME 132, Spring 25, UC Berkeley, A Packard 187 21 Linear State-Space Representations First, let s describe the most general type of dynamic system that we will consider/encounter in this class Systems may
More informatione st f (t) dt = e st tf(t) dt = L {t f(t)} s
Additional operational properties How to find the Laplace transform of a function f (t) that is multiplied by a monomial t n, the transform of a special type of integral, and the transform of a periodic
More informationGATE EE Topic wise Questions SIGNALS & SYSTEMS
www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)
More informationSYSTEMTEORI - ÖVNING 1. In this exercise, we will learn how to solve the following linear differential equation:
SYSTEMTEORI - ÖVNING 1 GIANANTONIO BORTOLIN AND RYOZO NAGAMUNE In this exercise, we will learn how to solve the following linear differential equation: 01 ẋt Atxt, xt 0 x 0, xt R n, At R n n The equation
More informationAutomatic Control Systems theory overview (discrete time systems)
Automatic Control Systems theory overview (discrete time systems) Prof. Luca Bascetta (luca.bascetta@polimi.it) Politecnico di Milano Dipartimento di Elettronica, Informazione e Bioingegneria Motivations
More informationEE Homework 12 - Solutions. 1. The transfer function of the system is given to be H(s) = s j j
EE3054 - Homework 2 - Solutions. The transfer function of the system is given to be H(s) = s 2 +3s+3. Decomposing into partial fractions, H(s) = 0.5774j s +.5 0.866j + 0.5774j s +.5 + 0.866j. () (a) The
More informationChapter 7. Canonical Forms. 7.1 Eigenvalues and Eigenvectors
Chapter 7 Canonical Forms 7.1 Eigenvalues and Eigenvectors Definition 7.1.1. Let V be a vector space over the field F and let T be a linear operator on V. An eigenvalue of T is a scalar λ F such that there
More informationControl Systems I. Lecture 5: Transfer Functions. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 5: Transfer Functions Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 20, 2017 E. Frazzoli (ETH) Lecture 5: Control Systems I 20/10/2017
More information19 Jacobian Linearizations, equilibrium points
169 19 Jacobian Linearizations, equilibrium points In modeling systems, we see that nearly all systems are nonlinear, in that the differential equations governing the evolution of the system s variables
More informationSystems and Control Theory Lecture Notes. Laura Giarré
Systems and Control Theory Lecture Notes Laura Giarré L. Giarré 2017-2018 Lesson 7: Response of LTI systems in the transform domain Laplace Transform Transform-domain response (CT) Transfer function Zeta
More informationControl Systems I. Lecture 6: Poles and Zeros. Readings: Emilio Frazzoli. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 6: Poles and Zeros Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 27, 2017 E. Frazzoli (ETH) Lecture 6: Control Systems I 27/10/2017
More informationAutomatique. A. Hably 1. Commande d un robot mobile. Automatique. A.Hably. Digital implementation
A. Hably 1 1 Gipsa-lab, Grenoble-INP ahmad.hably@grenoble-inp.fr Commande d un robot mobile (Gipsa-lab (DA)) ASI 1 / 25 Outline 1 2 (Gipsa-lab (DA)) ASI 2 / 25 of controllers Signals must be sampled and
More informationLaplace Transform Part 1: Introduction (I&N Chap 13)
Laplace Transform Part 1: Introduction (I&N Chap 13) Definition of the L.T. L.T. of Singularity Functions L.T. Pairs Properties of the L.T. Inverse L.T. Convolution IVT(initial value theorem) & FVT (final
More informationMath Matrix Algebra
Math 44 - Matrix Algebra Review notes - 4 (Alberto Bressan, Spring 27) Review of complex numbers In this chapter we shall need to work with complex numbers z C These can be written in the form z = a+ib,
More informationDefinition of the Laplace transform. 0 x(t)e st dt
Definition of the Laplace transform Bilateral Laplace Transform: X(s) = x(t)e st dt Unilateral (or one-sided) Laplace Transform: X(s) = 0 x(t)e st dt ECE352 1 Definition of the Laplace transform (cont.)
More informationCONTROL OF DIGITAL SYSTEMS
AUTOMATIC CONTROL AND SYSTEM THEORY CONTROL OF DIGITAL SYSTEMS Gianluca Palli Dipartimento di Ingegneria dell Energia Elettrica e dell Informazione (DEI) Università di Bologna Email: gianluca.palli@unibo.it
More informationDiscrete and continuous dynamic systems
Discrete and continuous dynamic systems Bounded input bounded output (BIBO) and asymptotic stability Continuous and discrete time linear time-invariant systems Katalin Hangos University of Pannonia Faculty
More informationLinear algebra II Tutorial solutions #1 A = x 1
Linear algebra II Tutorial solutions #. Find the eigenvalues and the eigenvectors of the matrix [ ] 5 2 A =. 4 3 Since tra = 8 and deta = 5 8 = 7, the characteristic polynomial is f(λ) = λ 2 (tra)λ+deta
More informationTransform Solutions to LTI Systems Part 3
Transform Solutions to LTI Systems Part 3 Example of second order system solution: Same example with increased damping: k=5 N/m, b=6 Ns/m, F=2 N, m=1 Kg Given x(0) = 0, x (0) = 0, find x(t). The revised
More informationRaktim Bhattacharya. . AERO 422: Active Controls for Aerospace Vehicles. Dynamic Response
.. AERO 422: Active Controls for Aerospace Vehicles Dynamic Response Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. . Previous Class...........
More informationRaktim Bhattacharya. . AERO 632: Design of Advance Flight Control System. Preliminaries
. AERO 632: of Advance Flight Control System. Preliminaries Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. Preliminaries Signals & Systems Laplace
More informationMA2327, Problem set #3 (Practice problems with solutions)
MA2327, Problem set #3 (Practice problems with solutions) 5 2 Compute the matrix exponential e ta in the case that A = 2 5 2 Compute the matrix exponential e ta in the case that A = 5 3 Find the unique
More informationControl Systems I. Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback. Readings: Emilio Frazzoli
Control Systems I Lecture 4: Diagonalization, Modal Analysis, Intro to Feedback Readings: Emilio Frazzoli Institute for Dynamic Systems and Control D-MAVT ETH Zürich October 13, 2017 E. Frazzoli (ETH)
More informationGeneralized Eigenvectors and Jordan Form
Generalized Eigenvectors and Jordan Form We have seen that an n n matrix A is diagonalizable precisely when the dimensions of its eigenspaces sum to n. So if A is not diagonalizable, there is at least
More informationTime Response of Systems
Chapter 0 Time Response of Systems 0. Some Standard Time Responses Let us try to get some impulse time responses just by inspection: Poles F (s) f(t) s-plane Time response p =0 s p =0,p 2 =0 s 2 t p =
More informationMath Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More informationControl Systems Design
ELEC4410 Control Systems Design Lecture 13: Stability Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 13: Stability p.1/20 Outline Input-Output
More informationTime Response Analysis (Part II)
Time Response Analysis (Part II). A critically damped, continuous-time, second order system, when sampled, will have (in Z domain) (a) A simple pole (b) Double pole on real axis (c) Double pole on imaginary
More informationEE Control Systems LECTURE 9
Updated: Sunday, February, 999 EE - Control Systems LECTURE 9 Copyright FL Lewis 998 All rights reserved STABILITY OF LINEAR SYSTEMS We discuss the stability of input/output systems and of state-space
More informationLaplace Transforms and use in Automatic Control
Laplace Transforms and use in Automatic Control P.S. Gandhi Mechanical Engineering IIT Bombay Acknowledgements: P.Santosh Krishna, SYSCON Recap Fourier series Fourier transform: aperiodic Convolution integral
More informationExplanations and Discussion of Some Laplace Methods: Transfer Functions and Frequency Response. Y(s) = b 1
Engs 22 p. 1 Explanations Discussion of Some Laplace Methods: Transfer Functions Frequency Response I. Anatomy of Differential Equations in the S-Domain Parts of the s-domain solution. We will consider
More informationSTABILITY. Have looked at modeling dynamic systems using differential equations. and used the Laplace transform to help find step and impulse
SIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 4. Dr David Corrigan 1. Electronic and Electrical Engineering Dept. corrigad@tcd.ie www.sigmedia.tv STABILITY Have looked at modeling dynamic systems using differential
More informationLaplace Transforms Chapter 3
Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important
More informationOrdinary Differential Equations
Ordinary Differential Equations for Engineers and Scientists Gregg Waterman Oregon Institute of Technology c 2017 Gregg Waterman This work is licensed under the Creative Commons Attribution 4.0 International
More informationChapter 6: The Laplace Transform. Chih-Wei Liu
Chapter 6: The Laplace Transform Chih-Wei Liu Outline Introduction The Laplace Transform The Unilateral Laplace Transform Properties of the Unilateral Laplace Transform Inversion of the Unilateral Laplace
More informationMath 215/255 Final Exam (Dec 2005)
Exam (Dec 2005) Last Student #: First name: Signature: Circle your section #: Burggraf=0, Peterson=02, Khadra=03, Burghelea=04, Li=05 I have read and understood the instructions below: Please sign: Instructions:.
More informationNotes for ECE-320. Winter by R. Throne
Notes for ECE-3 Winter 4-5 by R. Throne Contents Table of Laplace Transforms 5 Laplace Transform Review 6. Poles and Zeros.................................... 6. Proper and Strictly Proper Transfer Functions...................
More informationENGIN 211, Engineering Math. Laplace Transforms
ENGIN 211, Engineering Math Laplace Transforms 1 Why Laplace Transform? Laplace transform converts a function in the time domain to its frequency domain. It is a powerful, systematic method in solving
More informationDynamic Response. Assoc. Prof. Enver Tatlicioglu. Department of Electrical & Electronics Engineering Izmir Institute of Technology.
Dynamic Response Assoc. Prof. Enver Tatlicioglu Department of Electrical & Electronics Engineering Izmir Institute of Technology Chapter 3 Assoc. Prof. Enver Tatlicioglu (EEE@IYTE) EE362 Feedback Control
More information22.2. Applications of Eigenvalues and Eigenvectors. Introduction. Prerequisites. Learning Outcomes
Applications of Eigenvalues and Eigenvectors 22.2 Introduction Many applications of matrices in both engineering and science utilize eigenvalues and, sometimes, eigenvectors. Control theory, vibration
More informationReview of Linear Time-Invariant Network Analysis
D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x
More informationCHEE 319 Tutorial 3 Solutions. 1. Using partial fraction expansions, find the causal function f whose Laplace transform. F (s) F (s) = C 1 s + C 2
CHEE 39 Tutorial 3 Solutions. Using partial fraction expansions, find the causal function f whose Laplace transform is given by: F (s) 0 f(t)e st dt (.) F (s) = s(s+) ; Solution: Note that the polynomial
More informationMethods for analysis and control of. Lecture 6: Introduction to digital control
Methods for analysis and of Lecture 6: to digital O. Sename 1 1 Gipsa-lab, CNRS-INPG, FRANCE Olivier.Sename@gipsa-lab.inpg.fr www.lag.ensieg.inpg.fr/sename 6th May 2009 Outline Some interesting books:
More information4. Linear transformations as a vector space 17
4 Linear transformations as a vector space 17 d) 1 2 0 0 1 2 0 0 1 0 0 0 1 2 3 4 32 Let a linear transformation in R 2 be the reflection in the line = x 2 Find its matrix 33 For each linear transformation
More informationThe Laplace Transform
The Laplace Transform Introduction There are two common approaches to the developing and understanding the Laplace transform It can be viewed as a generalization of the CTFT to include some signals with
More informationControl System. Contents
Contents Chapter Topic Page Chapter- Chapter- Chapter-3 Chapter-4 Introduction Transfer Function, Block Diagrams and Signal Flow Graphs Mathematical Modeling Control System 35 Time Response Analysis of
More informationCDS Solutions to Final Exam
CDS 22 - Solutions to Final Exam Instructor: Danielle C Tarraf Fall 27 Problem (a) We will compute the H 2 norm of G using state-space methods (see Section 26 in DFT) We begin by finding a minimal state-space
More informationLinear Systems. Chapter Basic Definitions
Chapter 5 Linear Systems Few physical elements display truly linear characteristics. For example the relation between force on a spring and displacement of the spring is always nonlinear to some degree.
More informationOctober 4, 2017 EIGENVALUES AND EIGENVECTORS. APPLICATIONS
October 4, 207 EIGENVALUES AND EIGENVECTORS. APPLICATIONS RODICA D. COSTIN Contents 4. Eigenvalues and Eigenvectors 3 4.. Motivation 3 4.2. Diagonal matrices 3 4.3. Example: solving linear differential
More informationLinear Differential Equations. Problems
Chapter 1 Linear Differential Equations. Problems 1.1 Introduction 1.1.1 Show that the function ϕ : R R, given by the expression ϕ(t) = 2e 3t for all t R, is a solution of the Initial Value Problem x =
More informationHomogeneous Constant Matrix Systems, Part II
4 Homogeneous Constant Matrix Systems, Part II Let us now expand our discussions begun in the previous chapter, and consider homogeneous constant matrix systems whose matrices either have complex eigenvalues
More informationE2.5 Signals & Linear Systems. Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & 2)
E.5 Signals & Linear Systems Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & ) 1. Sketch each of the following continuous-time signals, specify if the signal is periodic/non-periodic,
More informationModule 03 Linear Systems Theory: Necessary Background
Module 03 Linear Systems Theory: Necessary Background Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html September
More informationA system that is both linear and time-invariant is called linear time-invariant (LTI).
The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped
More informationAdvanced Control Theory
State Space Solution and Realization chibum@seoultech.ac.kr Outline State space solution 2 Solution of state-space equations x t = Ax t + Bu t First, recall results for scalar equation: x t = a x t + b
More informationSeptember 26, 2017 EIGENVALUES AND EIGENVECTORS. APPLICATIONS
September 26, 207 EIGENVALUES AND EIGENVECTORS. APPLICATIONS RODICA D. COSTIN Contents 4. Eigenvalues and Eigenvectors 3 4.. Motivation 3 4.2. Diagonal matrices 3 4.3. Example: solving linear differential
More informationEE102 Homework 2, 3, and 4 Solutions
EE12 Prof. S. Boyd EE12 Homework 2, 3, and 4 Solutions 7. Some convolution systems. Consider a convolution system, y(t) = + u(t τ)h(τ) dτ, where h is a function called the kernel or impulse response of
More informationStability. X(s) Y(s) = (s + 2) 2 (s 2) System has 2 poles: points where Y(s) -> at s = +2 and s = -2. Y(s) 8X(s) G 1 G 2
Stability 8X(s) X(s) Y(s) = (s 2) 2 (s 2) System has 2 poles: points where Y(s) -> at s = 2 and s = -2 If all poles are in region where s < 0, system is stable in Fourier language s = jω G 0 - x3 x7 Y(s)
More informationLecture 7: Laplace Transform and Its Applications Dr.-Ing. Sudchai Boonto
Dr-Ing Sudchai Boonto Department of Control System and Instrumentation Engineering King Mongkut s Unniversity of Technology Thonburi Thailand Outline Motivation The Laplace Transform The Laplace Transform
More information2.3 Oscillation. The harmonic oscillator equation is the differential equation. d 2 y dt 2 r y (r > 0). Its solutions have the form
2. Oscillation So far, we have used differential equations to describe functions that grow or decay over time. The next most common behavior for a function is to oscillate, meaning that it increases and
More informationEcon Lecture 14. Outline
Econ 204 2010 Lecture 14 Outline 1. Differential Equations and Solutions 2. Existence and Uniqueness of Solutions 3. Autonomous Differential Equations 4. Complex Exponentials 5. Linear Differential Equations
More informationSolutions to Math 53 Math 53 Practice Final
Solutions to Math 5 Math 5 Practice Final 20 points Consider the initial value problem y t 4yt = te t with y 0 = and y0 = 0 a 8 points Find the Laplace transform of the solution of this IVP b 8 points
More informationA proof of the Jordan normal form theorem
A proof of the Jordan normal form theorem Jordan normal form theorem states that any matrix is similar to a blockdiagonal matrix with Jordan blocks on the diagonal. To prove it, we first reformulate it
More informationDiscrete Systems. Step response and pole locations. Mark Cannon. Hilary Term Lecture
Discrete Systems Mark Cannon Hilary Term 22 - Lecture 4 Step response and pole locations 4 - Review Definition of -transform: U() = Z{u k } = u k k k= Discrete transfer function: Y () U() = G() = Z{g k},
More informationReachability and Controllability
Capitolo. INTRODUCTION 4. Reachability and Controllability Reachability. The reachability problem is to find the set of all the final states x(t ) reachable starting from a given initial state x(t ) :
More informationControl Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:
More informationMA Ordinary Differential Equations and Control Semester 1 Lecture Notes on ODEs and Laplace Transform (Parts I and II)
MA222 - Ordinary Differential Equations and Control Semester Lecture Notes on ODEs and Laplace Transform (Parts I and II Dr J D Evans October 22 Special thanks (in alphabetical order are due to Kenneth
More informationMath 308 Exam II Practice Problems
Math 38 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..
More informationModule 4. Related web links and videos. 1. FT and ZT
Module 4 Laplace transforms, ROC, rational systems, Z transform, properties of LT and ZT, rational functions, system properties from ROC, inverse transforms Related web links and videos Sl no Web link
More informationGEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM. COURSE: ECE 3084A (Prof. Michaels)
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 09-Dec-13 COURSE: ECE 3084A (Prof. Michaels) NAME: STUDENT #: LAST, FIRST Write your name on the front page
More informationControl Systems. Frequency domain analysis. L. Lanari
Control Systems m i l e r p r a in r e v y n is o Frequency domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic
More informationDiscrete-time linear systems
Automatic Control Discrete-time linear systems Prof. Alberto Bemporad University of Trento Academic year 2-2 Prof. Alberto Bemporad (University of Trento) Automatic Control Academic year 2-2 / 34 Introduction
More informationModule 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control
Module 09 From s-domain to time-domain From ODEs, TFs to State-Space Modern Control Ahmad F. Taha EE 3413: Analysis and Desgin of Control Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/
More information7.2 Relationship between Z Transforms and Laplace Transforms
Chapter 7 Z Transforms 7.1 Introduction In continuous time, the linear systems we try to analyse and design have output responses y(t) that satisfy differential equations. In general, it is hard to solve
More informationNotes on the matrix exponential
Notes on the matrix exponential Erik Wahlén erik.wahlen@math.lu.se February 14, 212 1 Introduction The purpose of these notes is to describe how one can compute the matrix exponential e A when A is not
More informationLTI system response. Daniele Carnevale. Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome Tor Vergata
LTI system response Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome Tor Vergata Fondamenti di Automatica e Controlli Automatici A.A. 2014-2015 1 / 15 Laplace
More informationSignals and Systems I Have Known and Loved. (Lecture notes for CSE 3451) Andrew W. Eckford
Signals and Systems I Have Known and Loved (Lecture notes for CSE 3451) Andrew W. Eckford Department of Electrical Engineering and Computer Science York University, Toronto, Ontario, Canada Version: December
More information2.004 Dynamics and Control II Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8 I * * Massachusetts
More informationProfessor Fearing EECS120/Problem Set 2 v 1.01 Fall 2016 Due at 4 pm, Fri. Sep. 9 in HW box under stairs (1st floor Cory) Reading: O&W Ch 1, Ch2.
Professor Fearing EECS120/Problem Set 2 v 1.01 Fall 20 Due at 4 pm, Fri. Sep. 9 in HW box under stairs (1st floor Cory) Reading: O&W Ch 1, Ch2. Note: Π(t) = u(t + 1) u(t 1 ), and r(t) = tu(t) where u(t)
More informationEE263: Introduction to Linear Dynamical Systems Review Session 6
EE263: Introduction to Linear Dynamical Systems Review Session 6 Outline diagonalizability eigen decomposition theorem applications (modal forms, asymptotic growth rate) EE263 RS6 1 Diagonalizability consider
More informationUnit 2: Modeling in the Frequency Domain Part 2: The Laplace Transform. The Laplace Transform. The need for Laplace
Unit : Modeling in the Frequency Domain Part : Engineering 81: Control Systems I Faculty of Engineering & Applied Science Memorial University of Newfoundland January 1, 010 1 Pair Table Unit, Part : Unit,
More informationModels for Sampled Data Systems
Chapter 12 Models for Sampled Data Systems The Shift Operator Forward shift operator q(f[k]) f[k +1] In terms of this operator, the model given earlier becomes: q n y[k]+a n 1 q n 1 y[k]+ + a 0 y[k] =b
More information7 Planar systems of linear ODE
7 Planar systems of linear ODE Here I restrict my attention to a very special class of autonomous ODE: linear ODE with constant coefficients This is arguably the only class of ODE for which explicit solution
More informationFrequency domain analysis
Automatic Control 2 Frequency domain analysis Prof. Alberto Bemporad University of Trento Academic year 2010-2011 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011
More informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
More informationControl Systems. Dynamic response in the time domain. L. Lanari
Control Systems Dynamic response in the time domain L. Lanari outline A diagonalizable - real eigenvalues (aperiodic natural modes) - complex conjugate eigenvalues (pseudoperiodic natural modes) - phase
More informationProblem Set 3: Solution Due on Mon. 7 th Oct. in class. Fall 2013
EE 56: Digital Control Systems Problem Set 3: Solution Due on Mon 7 th Oct in class Fall 23 Problem For the causal LTI system described by the difference equation y k + 2 y k = x k, () (a) By first finding
More informationFINAL EXAM SOLUTIONS, MATH 123
FINAL EXAM SOLUTIONS, MATH 23. Find the eigenvalues of the matrix ( 9 4 3 ) So λ = or 6. = λ 9 4 3 λ = ( λ)( 3 λ) + 36 = λ 2 7λ + 6 = (λ 6)(λ ) 2. Compute the matrix inverse: ( ) 3 3 = 3 4 ( 4/3 ) 3. Let
More informationSpectral Theorem for Self-adjoint Linear Operators
Notes for the undergraduate lecture by David Adams. (These are the notes I would write if I was teaching a course on this topic. I have included more material than I will cover in the 45 minute lecture;
More informationECE 388 Automatic Control
Controllability and State Feedback Control Associate Prof. Dr. of Mechatronics Engineeering Çankaya University Compulsory Course in Electronic and Communication Engineering Credits (2/2/3) Course Webpage:
More informationThe Laplace Transform
The Laplace Transform Generalizing the Fourier Transform The CTFT expresses a time-domain signal as a linear combination of complex sinusoids of the form e jωt. In the generalization of the CTFT to the
More information1. The Transition Matrix (Hint: Recall that the solution to the linear equation ẋ = Ax + Bu is
ECE 55, Fall 2007 Problem Set #4 Solution The Transition Matrix (Hint: Recall that the solution to the linear equation ẋ Ax + Bu is x(t) e A(t ) x( ) + e A(t τ) Bu(τ)dτ () This formula is extremely important
More informationLABORATORY OF AUTOMATION SYSTEMS Analytical design of digital controllers
LABORATORY OF AUTOMATION SYSTEMS Analytical design of digital controllers Claudio Melchiorri Dipartimento di Ingegneria dell Energia Elettrica e dell Informazione (DEI) Università di Bologna email: claudio.melchiorri@unibo.it
More information