Networks and Linear Systems Unit 4 Lecture PDF

NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Networks and Linear Systems Unit – IV Concepts of state and state variables – state space modeling for simple electrical and mechanical systems – s...

NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Networks and Linear Systems Unit – IV Concepts of state and state variables – state space modeling for simple electrical and mechanical systems – state transition matrix - solution of state equations Dr. Karthik Thirumala Email: [email protected] Reference textbooks: 1) D. Roy Choudhury, ‘Networks and Systems’, New Age International Publications, 1st Edition, 2013. 2) I. J. Nagrath, M. Gopal, ‘Control Systems Engineering’, New Age International Publisher, 4th edition, 2008. 3) A. Nagoor kani, ‘Control Sytems’, RBA Publications, 3rd Edition, 2017. 4) M. Gopal, ‘Control Systems Principles and Design’, 3rd edition, 2009. 1 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Lecture - 1 Introduction In many practical applications, circuits consist of numerous energy storage elements. Differential equations describing such circuits are then generally of high order. A nth order differential equation is not generally suitable for computer simulation; it is best to obtain a set of n-differential equation from the given nth order differential equation, using a set of auxiliary variables called state variables. The transfer function is the classical approach, using frequency domain technique deals with input and output only and is unable to give any information about the internal state of the system. On the other hand, modern control theory based on the state variable approach gives all the internal states of the system. In order to have a perfect design of the feedback control system, all the states may be required to be fed back with proper weights. The state variable approach is a time-domain technique. The state of a dynamic system is the minimal amount of information required together with the initial conditions at t=t0 and input excitation, to completely specify the future behavior of the system for any time t> t0. 2 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 3 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 4 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 5 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Lecture – 2 Example: Write the state space model of the Series RL network shown below. Solution: Write the set of equations When an inductor is present in a system, current through the inductor is commonly chosen as a state variable. In this system, source voltage (input) and current through inductor (state variable) are sufficient to determine this system's future voltage across the resistor (output). Let the state variable be x(t) = i(t), input is u(t) = v(t) and the output is y(t) = VR(t) State equation: Output equation: Example: Write the state space model of the Series RLC network shown below. Solution: There are two storage elements (inductor and capacitor) in this circuit. So, the number of the state variables is equal to two and these state variables are the current flowing through the inductor, i(t) and the voltage across capacitor, vc(t). Let the state variable be x1(t) = i(t), x2(t) = vc(t), input is u(t) = vi(t) and the output is y(t) = vo(t)= vc(t). The loop equations are and We can arrange the differential equations and output equation into the standard form of state space model as, Example: Write the state space model of the Series RLC network shown below with three outputs. Solution: Consider the voltage across all the three passive elements as three outputs. The state vector is 6 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Input vector is Output vector is The state space model is Example: Write the state equation of the parallel RLC network shown below. Solution: KCL equation at the top node: We retain the differentiated variables on the left hand side and move all others to the right hand side to get the following state model Example: For the network shown below, obtain the state space model of the system. Solution: At node A, KCL equation Using KVL equation, 7 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Example: For the network shown below, obtain the state space model of the system. Solutions: Let the state variables are v1 and v2. Now, write the nodal equations Rearrange the equations in form of state model The state space model is Example: Write the state equation of the network shown in below. Let R1 = R2 = 1Ω, L = 1H, C1 = C2= 1F. Solution: Choose v1, i2 and v3 as state variables, where v1 and v3 are the voltages across the capacitors C1 and C2. I2 is the current through the inductance. Write the independent node and loop equations at node n 1, n2 and loop l1. and Rearranging the equations, we get 8 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 The state equation is Example: For the network in previous problem, choose charges and fluxes as the state variables and write the state space model. Example: Write the state space model of the Spring mass damper system 9 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Example: Translational system Example: Write the state and output equation of the given mechanical system Solution: Input: fa(t); Outputs: Tensile force in spring-2 and total momentum of the masses Number of States = no. of energy storing elements = 4 (2 springs & 2 masses) One possible choice of states: x1, x2, v1, v2 Elongation of spring 1= x1; Elongation of spring 2= x2-x1 Velocity of mass 1 =v1; Velocity of mass 2 =v2; Example: Obtain state space model of an armature-controlled DC motor The loop equation for the armature circuit is 10 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 11 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Lecture - 3 Derivation of Transfer function from State variable model The two most powerful and common ways to represent systems are the transfer function form and the state space form. We shall derive the transfer function of a SISO system from the Laplace transformed version of the state and output equations. Consider the state model of a SISO system as Setting the denominator of the transfer function, we get the characteristic equation |sI-A| = 0 The roots of this equation are the poles of the transfer function. An important observation that needs to be made here is that while the state model is nonunique, the transfer function of a system is unique i.e., the transfer function must be same irrespective of which particular state model is used to describe the system. For a multiple-input multiple-output system, the X(s) could be written in terms of input vector U(s) as The output equation is 12 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 The matrix transfer function H(s) relating the output vector Y(s) to the input vector U(s) is For a system with r inputs U1(s),... , Ur(s) and m outputs Y1(s),... , Ym(s), H(s) is a m×r matrix whose elements are individual scalar transfer functions relating a given component of the output Y(s) to a component of the input U(s). Expansion of H(s) generates a set of equations: where the ith component of the output vector Y(s) is: The elemental transfer function Hij(s) is the scalar transfer function between the ith output component and the jth input component. All of the Hij(s) transfer functions in H(s) have the same denominator factor det[sI-A], giving the important result that all input-output differential equations for a system have the same characteristic polynomial. Eigen values of the matrix A The state model for some cases is not convenient for investigation of system properties and evaluation of time response. The canonical state model wherein matrix A is in diagonal form is most suitable for this purpose. These techniques are called as diagonalization techniques. The eigen values are helpful in determining the controllability and observability of a system. 13 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 14 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Example: Obtain the transfer function from the given state space model Lecture - 4 Solution of State Equation 15 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Solution of Homogeneous State Equation 16 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 17 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Example: Example: Determine the State Transition matrix for the network given here. 18 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Lecture - 5 Solution of Non-Homogenous State equation 19 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 20 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Example: Example: Obtain the time-response of the following system 21 NLS, UNIT-IV, KARTHIK THIRUMALA JULY 2022 Example: Obtain the output time-response at t=0.1 sec of the system given in the previous example with initial condition x(0)=[1 0]T when subjected to a unit step input. 22