Professor, Department of Mechanical Engineering

Transcription

1 State Space Approach in Modelling Dr Bishakh Bhattacharya Professor, Departent of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD

2 Answer of the Last Assignent Following Mason s law, there are two forward paths in the SFG: T 1 = G 1 G 2 G 3 and T 2 = G 4 There are four loops: L 1 = G 1 H 1 L 2 = G 3 H 2 L 3 = G 1 G 2 G 3 H 3 L 4 = G 4 H 3 Δ = 1 (L 1 + L 2 + L 3 + L 4 ) + L 1 L 2 Δ 1 = 1 Δ 2 = 1 Hence, the transfer function could be epressed as (T 1 + T 2 )/ Δ 2

3 The Lecture Contains State Space Modeling EOM of a SDOF syste in State Space For Response of a State Space Syste Eaples to Solve Joint Initiative of IITs and IISc - Funded by MHRD

4 State Space Modelling The state of a odel of a dynaic syste is a set of independent physical quantities, the specification of which (in the absence of ecitation) copletely deterines the future positions of the syste Dynaics describes how the state evolves The dynaics of a odel is an update rule for the syste state that describes how the state evolves, as a function on the current state and any eternal inputs 1 2 X n A X ( t) BU ( t)

5 When we talk tlkabout electro echanical lsystes odeled dldby differential equations, such as asses and springs, electric circuits or satellites (rigid bodies) rotating in space, we can attach soe additional intuition: the variables in the state should be adequate to specify the energy of the syste For eaple, take a ball free falling to earth: we can specify the position of the ball by specifying the height (h) above the ground, but we also need to include the velocity of the ball (dh/dt) to specify the total energy (E = 1/2**(dh/dt)^2 + gh) Therefore, the state of the ball is (h,dh/dt)

6 State t Space Modelling of a Single Degree of Freedo Syste Consider a SDOF syste (with ass M, stiffness K and Daping constant C) such that t the equation of otion corresponding to force ecitation ti is given by: M C K F ( t ) The following pair of states or their linear cobinations could be considered for the odelling:,,, 6

7 The EOM in State Space For p Consider for eaple, the position and velocity as the state coordinates The state vector could be written as: X Based on these states, the EOM could be rewritten as: f c k dt d ) / ( 1 0 / / 1 0 BU A X X f U B c k A /, 1, / /

8 Output of a State-space Syste Many a ties states of a syste are not directly easurable and hence are not of direct interest For eaple, if you consider, state space representation of a finite eleent odel pertaining to a Spacecraft The nuber of states could be as high as three to four thousand! However, one cannot have so any sensors to easure all the states In such cases, we fi a feasible nuber of outputs that are observable/easurable Suppose for a syste of n states sa there eare r outputs that are easurable ab e Then the output vector Y(t) of size r could be represented as a linear cobination of input to the syste and the states as follows: y1 ( t ) y2( t) Y ( t) C X ( t) DU ( t) y ( t) r Where C &D are constants for an LTIV syste For ajority of dynaic systes it is observed that D = 0, eaning outputs are not directly affected by the syste inputs 8

9 Tie Doain Solution for a Vector State t Equation X(t) = e At X A(t-τ) o + t o e BU(t) dt e At = I + At + (At) 2 /2! + (At) 3 /3! X(s) = (si-a) -1 B U(s) Find out the eigen values and eigen vectors of si A, Obtain the transforation atri and convert the state atri into diagonal for Solve using a Discrete Tie -Model

10 Special References for this Lecture Feedback Control of Dynaic Systes Franklin, Powell and Naeini, Pearson Education Asia Control Systes Engineering Noran S Nise, John Wiley & Sons Modern Control Engineering K Ogata, Prentice Hall Control Syste Design B Friedland, Dover 10

11 Find out the EOM for the following echanical syste in state space for :