Page 1 NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 State Space Approach in Modelling State Space Approach in Modelling D Bi h kh Bh tt h Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD Page 2 NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 State Space Approach in Modelling State Space Approach in Modelling D Bi h kh Bh tt h Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 Answer of the Last Assignment g 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 3 G 1 G 2 G 3 H 3 L 4 = -G 4 H 3 ?= 1 –(L 1 + L 2 + L 3 + L 4 ) + L 1 L 2 ? 1 = 1 ? 2 = 1 h ffi ldb d ( )/ 2 Hence, the transfer function could be expressed as (T 1 + T 2 )/ ? Joint Initiative of IITs and IISc -Funded by MHRD Page 3 NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 State Space Approach in Modelling State Space Approach in Modelling D Bi h kh Bh tt h Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 Answer of the Last Assignment g 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 3 G 1 G 2 G 3 H 3 L 4 = -G 4 H 3 ?= 1 –(L 1 + L 2 + L 3 + L 4 ) + L 1 L 2 ? 1 = 1 ? 2 = 1 h ffi ldb d ( )/ 2 Hence, the transfer function could be expressed as (T 1 + T 2 )/ ? Joint Initiative of IITs and IISc -Funded by MHRD NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 The Lecture Contains ? State Space Modeling ?EOM of a SDOF system in State Space Form ?Response of a State Space System ?Examples to Solve Joint Initiative of IITs and IISc - Funded by MHRD Page 4 NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 State Space Approach in Modelling State Space Approach in Modelling D Bi h kh Bh tt h Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 Answer of the Last Assignment g 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 3 G 1 G 2 G 3 H 3 L 4 = -G 4 H 3 ?= 1 –(L 1 + L 2 + L 3 + L 4 ) + L 1 L 2 ? 1 = 1 ? 2 = 1 h ffi ldb d ( )/ 2 Hence, the transfer function could be expressed as (T 1 + T 2 )/ ? Joint Initiative of IITs and IISc -Funded by MHRD NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 The Lecture Contains ? State Space Modeling ?EOM of a SDOF system in State Space Form ?Response of a State Space System ?Examples to Solve Joint Initiative of IITs and IISc - Funded by MHRD NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 State-Space Modelling The stateof a model of a dynamic system is a set of independent physical quantities, the specification of which (in the absence of excitation) completely determines the future positions of the system Dynamics describes how the state evolves. The dynamicsof a model is an update rule for the system state that describes howthestateevolves asafunctiononthecurrentstateand how the state evolves, as a function on the current state and any external inputs . . 1 x ? ? ? ? ? ? ) ( ) ( . . 2 . t U B t X A x X ? ? ? ? ? ? ? ? ? ? ? ? ? Joint Initiative of IITs and IISc -Funded by MHRD . x n ? ? ? ? ? ? Page 5 NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 State Space Approach in Modelling State Space Approach in Modelling D Bi h kh Bh tt h Dr. Bishakh Bhattacharya Professor, Department of Mechanical Engineering IIT Kanpur Joint Initiative of IITs and IISc - Funded by MHRD NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 Answer of the Last Assignment g 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 3 G 1 G 2 G 3 H 3 L 4 = -G 4 H 3 ?= 1 –(L 1 + L 2 + L 3 + L 4 ) + L 1 L 2 ? 1 = 1 ? 2 = 1 h ffi ldb d ( )/ 2 Hence, the transfer function could be expressed as (T 1 + T 2 )/ ? Joint Initiative of IITs and IISc -Funded by MHRD NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 The Lecture Contains ? State Space Modeling ?EOM of a SDOF system in State Space Form ?Response of a State Space System ?Examples to Solve Joint Initiative of IITs and IISc - Funded by MHRD NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 State-Space Modelling The stateof a model of a dynamic system is a set of independent physical quantities, the specification of which (in the absence of excitation) completely determines the future positions of the system Dynamics describes how the state evolves. The dynamicsof a model is an update rule for the system state that describes howthestateevolves asafunctiononthecurrentstateand how the state evolves, as a function on the current state and any external inputs . . 1 x ? ? ? ? ? ? ) ( ) ( . . 2 . t U B t X A x X ? ? ? ? ? ? ? ? ? ? ? ? ? Joint Initiative of IITs and IISc -Funded by MHRD . x n ? ? ? ? ? ? NPTEL >> Mechanical Engineering >> Modeling and Control of Dynamic electro-Mechanical System Module 1- Lecture 4 Wh tlkbt lt hi l t dldb diff ti l When we talk about electro-mechanical systems modeled by differential equations, such as masses and springs, electric circuits or satellites (rigid bodies) rotating in space, we can attach some additional intuition: the variables inthestate shouldbeadequate tospecifytheenergyofthe variables in the state should be adequate to specify the energyof the system. For example, 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*m*(dh/dt)^2 + mgh). Therefore, the state of the ball is (h,dh/dt). Joint Initiative of IITs and IISc -Funded by MHRDRead More