VALLIAMMAI ENGINEERING COLLEGE

VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 6 DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING QESTION BANK ME-Power Systems Engineering I st Year SEMESTER I IN55- SYSTEM THEORY Regulation 7 Academic Year 7 8 Prepared by DrRArivalahan, Associate Professor/EEE,

IN55 SYSTEM THEORY L T P C 4 NIT I STATE VARIABLE REPRESENTATION 9 Introduction-Concept of State-State equations for Dynamic Systems -Time invariance and linearity- Non uniqueness of state model- Physical Systems and State Assignment - free and forced responses- State Diagrams NIT II SOLTION OF STATE EQATIONS 9 Eistence and uniqueness of solutions to Continuous-time state equations - Solution of Nonlinear and Linear Time Varying State equations - State transition matri and its properties Evaluation of matri eponential- System modes- Role of Eigen values and Eigen vectors NIT III STABILITY ANALYSIS OF LINEAR SYSTEMS 9 Controllability and Observability definitions and Kalman rank conditions -Stabilizability and Detectability-Test for Continuous time Systems- Time varying and Time invariant case- Output Controllability-Reducibility- System Realizations NIT IV STATE FEEDBACK CONTROL AND STATE ESTIMATOR 9 Introduction-Controllable and Observable Companion Forms-SISO and MIMO Systems- The Effect of State Feedback on Controllability and Observability-Pole Placement by State Feedback for both SISO and MIMO Systems-Full Order and Reduced Order Observers NIT V LYAPNOV STABILTY ANALYSIS 9 Introduction-Equilibrium Points- BIBO Stability-Stability of LTI Systems- Stability in the sense of Lyapunov - Equilibrium Stability of Nonlinear Continuous-Time Autonomous Systems-The Direct Method of Lyapunov and the Linear Continuous-Time Autonomous Systems-Finding Lyapunov Functions for Nonlinear Continuous-Time Autonomous Systems Krasovskil s and Variable-Gradiant Method TOTAL : 45+ = 75 PERIODS TEXT BOOKS: M Gopal, Modern Control System Theory, New Age International, 5 K Ogatta, Modern Control Engineering, PHI, John S Bay, Fundamentals of Linear State Space Systems, McGraw-Hill, 999 4 D Roy Choudhury, Modern Control Systems, New Age International, 5 5 John J D Azzo, C H Houpis and S N Sheldon, Linear Control System Analysis and Design with MATLAB, Taylor Francis, 6 Z Bubnicki, Modern Control Theory, Springer, 5 7 CT Chen, Linear Systems Theory and Design Oford niversity Press, rd Edition, 999 8 M Vidyasagar, Nonlinear Systems Analysis, nd edition, Prentice Hall, Englewood Cliffs, New Jersey

NIT I STATE VARIABLE REPRESENTATION Introduction-Concept of State-State equations for Dynamic Systems -Time invariance and linearity- Non uniqueness of state model- Physical Systems and State Assignment - free and forced responses- State Diagrams PART A QNo Questions BTL Level Eamine the general form of the state space model for continuous system BTL And also write the state diagram Domain Define the following terms such as (i) State (ii) State Variable (iii) State BTL Vector (iv) State Space Model Give any two approach to convert the transfer function approach to the state BTL space model 4 List the drawbacks in transfer function model analysis? BTL 5 Define the terms (i) Linearity (ii) Time invariance BTL 6 Define the term (i) Non-uniquness of the state space model BTL 7 Epress the state space model for a simple (i) Mass Spring Damper System BTL (ii) Mechanical Rotational System 8 Epress the Formula in which the general form of state space model into BTL transfer functional approach 9 How the Diagonal canonical form is distinguished with Jordon Canonical BTL form Distinguish the difference between (i) Physical variable model (ii) Phase BTL Variable model

Obtain the state space model for the given differential equation BTL d Y dt d Y 6 dt dy dt 6 Y (t) Solve and obtain the transfer function model Consider a system whose transfer function is given by Y(S)/(S) = BTL (S+)/S +6s +5s+ Calculate state model for this system A discrete time system is described by the difference equation BTL Y(K+)+5Y(K+)+6Y(K) =(K)Solve and find the transfer function of the system 4 Compare the merits and demerits of realizing a given system in state variable and transfer function form 5 Derive and eplain the transfer function model of a LTI system whose state equation is given by 6 Eplain the applications of state space model for the different system 7 Summarize the draw backs of transfer function model compare with state space model 8 Judge any -methods for the conversion of transfer functional model into 9 state space model Formulate state space model with state diagram for observable canonical form Develop state space model with state diagram for controllable canonical form

PART B Evaluate the state space model for the mechanical system as shown in Fig Where u(t) is input and y(t) is output Also derive the transfer function from the state space equations (6) Design eplain (i) Armature control of DC Moto r (ii) Field Control of DC Motor And also draw the (i) Block diagram (ii) State diagram and state space model for the system (6) Analyze the state model of the following electrical system (6) 4 Calculate the state space model for (i) Series RLC Circuit (6) BTL 5 Illustrate the epression for the state space model for the continuous system BTL and also draw the state diagram for it (6) 6 Formulate the epression for the (i) Controllable canonical form (ii) Observable Canonical form (6) 7 Solve the state space model for the given system(i) BTL Y(S)/(S)=/S +4S +S+ by the method of (i) Laplace Transform (ii) Signal Flow Graph Method (6)

8 Evaluate the state space model for the given differential equation d Y dt d Y 6 dt dy dt 6 Y (t) by Canonical form or companion form method and also draw the state diagram for it (6) 9 A discrete time system is represented by the differential equation BTL y( n ) 6y( n ) 8y( n) u( n) in which the initial condition y()=y()= with T=Second (i)estimate the controllable Canonical discrete state space model (6) Define the terms (i) Linearity (ii) Non uniqueness (iii) Time invariance for BTL state space model (6) Illustrate the epression for the state space model for continuous system (6) BTL Evaluate the method for converting the transfer function into state space model (i) Bush Form (or) Companion form (ii)signal Flow Graph Method (iii) Canonical Form Method (6) Solve the state space model for the given system(i) BTL Y(S)/(S)=8/(S+)(S+)(S+) by the method of (i) Laplace Transform (ii) Signal Flow Graph Method (6) 4 Create the state space model by using signal flow graph for the given problem (i) Y(S)/(S)=/(S +5S +4S+) (6) NIT II SOLTION OF STATE EQATIONS Eistence and uniqueness of solutions to Continuous-time state equations - Solution of Nonlinear and Linear Time Varying State equations - State transition matri and its properties Evaluation of matri eponential- System modes- Role of Eigen values and Eigen vectors PART A QNo Questions BTL Level Domain

What is the state transition matri? List any two methods for findingbtl state transition matri Quote the formula for the solution of the state equation in time domain? BTL 4 What is eigen values and eigen vectors? Eamine how the eigen values BTL can be calculated? What is state transition matri and identify how it is related to state of a BTL system? 5 Describe the formula for Matri eponential method BTL 6 Quote the different methods available for computing e At? BTL 7 Summarize the term Jordan canonical form BTL 8 Predict the transformation used to diagonalize a system matri? BTL 9 Estimate the transformed canonical state model of a system? BTL Epress the term Model Matri and eplain with suitable formulae BTL Demonstrate how the modal matri can be determined? BTL Illustrate Cayley-Hamilton theorem BTL Eamine How the state transition matri e At is computed by canonical BTL transformation 4 Analyze how the state transition matri e At is computed using Cayley- Hamilton theorem? 5 Point out how the state transition matri and how it is related to state of a system? 6 Eplain the solution of homogeneous state equations 7 Write the solution of non-homogeneous state equations 8 Judge the term resolvant matri

9 Formulate the state transition matri by Laplace Transform method 5 A Formulate the state transition matri by Matri Eponential method A PART B Illustrate the epression by (i) Matri Eponential Method (ii) Laplace Transform Method for state transistion of matri (6) BTL Obtain the state space Model 5 4 ; y=[ ] Convert the state space model into canonical form state space model And also calculate the value of state transition matri (6) BTL State Cayley-Hamilton s Theorem Derive and eplain the epression state transition matri using Cayley- Hamilton s theorem for continuous system (6) 4 Evaluate the state transition matri for the given discrete system matri 7 A By i Z-transform technique iicayley-hamilton s theorem (6) 5 The given state space model A ; Calculate the (i) Eigen values and Eigen vectors (ii) Rank of the matri (6) BTL 6 Evaluate the value of e At by (i) Trial and Error Method (ii) Cayley Hamilton s Theorem A (6)

7 Analyze the value of state transition matri or e At by using (a) Laplace Transform Method (b) Cayley Hamilton s Theorem (c)a in which 7 A (6) 8 The given matri A ; Calculate the state transistion matri by using Laplace transform method (6) BTL 9 The given matri 4 A ; Estimate the value of state transistion matri by using Cayley Hamilton s Theorem (6) BTL Convert the transfer function for the state space model and calculate ; y=[ ] 5 5 ; y=[ ] 4 ; y=[8 ] (6) BTL Solve and Calculate the value of state transition matri or e At by using (a) Laplace Transform Method (b) Cayley Hamilton s Theorem(c) A in which 7 A (6) BTL Find and point out the value of e At by (i) Trial and Error Method (ii) Cayley Hamilton s Theorem A (6)

Consider a system whose transfer function is given by Y(S)/(S) = (S+)/S +6S +5S+ Evaluate the state model for the system (i) by Block diagram reduction (ii) Signal flow graph Method (6) 4 Create the epression for the following Methods for State Transition Matri as follows (i) Trial and Error Method (iii) Laplace Transform Method (iv) Canonical Form (6) NIT III STABILITY ANALYSIS OF LINEAR SYSTEMS Controllability and Observability definitions and Kalman rank conditions-stabilizability and Detectability- Test for Continuous time Systems-Time varying and Time invariant case- Output Controllability- Reducibility- System Realizations PART A QNo Questions BTL Level Domain Quote what is meant by the rank of the matri? BTL Define the duality of the system between controllability and BTL observability concept? The given state space model BTL 4 5 ; y=[ ] Tell whether the given is controllable 4 Eamine the need for observability test? BTL 5 Describe the condition for observability by Gilbert s method BTL 6 When the Controllability test is normally applicable? BTL 7 Summarize the condition for controllability by Gilbert s method BTL 8 Describe the condition for controllability by Kalman s method BTL 9 What is meant by minimal realization? Give the epression for it BTL

Discuss the effect of pole zero cancellation in transfer function BTL approach Illustrate the concept of stabilizability BTL Illustrate the concept of detectability BTL Discover the advantage and disadvantage in Kalman s test for BTL observability? 4 The given state space model BTL ; y=[ ] 4 5 Solve whether the given is controllable 5 Analyze the condition for O/P controllability by Kalman;s method 6 Analyze the condition for observability by Kalman s method 7 Given two equivalent mathematical epressions which state that a given pair of matrices(a, B) is controllable Evaluate the epression 8 Write the Ackerman s formula for state feed back gain and eplain it 9 Create the formula for Observability of the system What is meant by duality of the system Develop the epression by Kalman s Method PART B What is meant by rank of the matri? Tell Whether the rank of the matri depends on controllability or not eplain it (6) BTL Define the concept of Controllability and observability of the systemwrite the epression for the controllability and observability in (i) Kalman s Method (ii) Gillbert s Method (6) BTL

Derive and Eamine the epression for the (i) Controllable canonical form (ii) Observable Canonical Form (6) BTL 4 The given state space model ; y=[ ] Check and Tell whether the given is controllable and observable or not by Kalman s approach and Gilbert s method (6) BTL 5 The given state space model 6 5 ; y=[ ] Check and discuss whether the given is controllable and observable or notand also check the duality by Kalman s approach and Gilbert s method (6) BTL 6 The transfer function of the system Y(S)/(S)=/S +6S +S+6 Check and epress whether the given system is controllable as well as observable And also check the duality by Kalman s approach and Gilbert s method (6) BTL 7 Estimate the controllable canonical realization of the following systems Hence, obtain the state space model in controllable canonical form (i) H(S)=(S+)/(S+5) (ii) H(S)=(S+)/(S +S+5) (iii) H(S)=(S+9)/(S +8S +S+) (iv) H(S)=(S +S+)/(S 4 +S +S +9S+) (6) BTL 8 Illustrate the epression for the Controllability and Observability in (i) Kalman s Method (ii)gilbert s Method (6) BTL

9 Eplain and demonstrate the (i) Reduciability (ii) System Realization Eplain with an eample for each for solving a problem (6) BTL Eplain with an eample eplain (i) Output Controllability and Observability (ii) Control lable and Observability concept applicable for time varying and invariant system (6) Consider a system with state space model is given below 4 ; y=[ -4 ] Point out that the system is observable and controllable (6) The given state space model 6 6 ;y=[ ] Check whether the given is controllable and observable or not And also Point out the duality by Kalman s approach and Gilbert s method (6) With the case study Summarize (i) Armature control of DC Motor (ii) Field Control of DC Motor And also draw the (i) Block diagram (ii) State diagram and state space model for the system (6) 4 (i) Consider a system whose transfer function is given by Y(S)/(S) = (S+)/S +6s +5s+ Solve and eplain state model for this system (ii) The given state space model 5 4 5 ; y=[ ] Point out whether the given is controllable or observable or not And also check the duality principle (6)

NIT IV STATE FEEDBACK CONTROL AND STATE ESTIMATOR Introduction-Controllable and Observable Companion Forms-SISO and MIMO Systems- The Effect of State Feedback on Controllability and Observability-Pole Placement by State Feedback for both SISO and MIMO Systems-Full Order and Reduced Order Observers PART A QNo Questions BTL Level Domain What is the state observer? Draw the diagram for State Observer and point out main features Analyze the need for state observer for the system? Summarize the following terms (i) F ull-order observer (ii) BTL Reduced-order observer (iii) Minimum-order state observer? 4 BTL What is the necessary condition to be satisfied for the design of state observer? Also Write the Ackermann s formula to identify the state observer gain matri, G 5 Define the term Pole Placement of controller BTL 6 7 Formulate the Ackermann s formula to find the state feedback gain matri, K BTL Illustrate the general form of observable phase variable form of state model 8 Summarize the pole placement controller by state feedback? 9 How will you Evaluate the transformation matri, Po to the state model to observable phase variable form? How control system design is carried in state space and discuss BTL with an suitable eample BTL Quote the necessary condition to be satisfied for design using state feedback?

Illustrate the block diagram of a system with state feedback BTL concept for controller Epress the general form of controllable phase variable form of BTL state model approach 4 What is meant by Control law? And also write the gain formulae and analyze the Ackermann s Method 5 Illustrate how will you find the transformation matri, Pc to BTL transform the state model to controllable phase variable form using the characteristic equation? 6 BTL How will you eamine the transformation matri, Pc to transform the state model to controllable phase variable form using the characteristic equation? 7 A system ehibits critically damped response for a step input and BTL has a natural frequency of oscillation of rad/sec Quote the equivalent pole locations 8 Discuss obsevability of the system and eplain with an diagram BTL 9 Formulate the state space model with state diagram for controllable canonical form Formulate the state space model with state diagram for observable canonical form PART B Eamine the design of pole placement concept for SISO and MIMO System with suitable diagram and epression (6) BTL Consider the state space model described by X ( t ) Y ( t ) AX ( t ) CX ( t ) BTL A ; C=[ ] Design and eamine a full-order state observer The desired Eigen values for the observer matri 5; 5 (6)

Eplain the controllable canonical form and observable canonical form for an eample (6) 4 Obtain and analyze the epression for (i) Full order observer (ii) Reduced Order Observer (iii) Pole Placement of Controller (6) 5 Describe the effect of feedback on the concept of Controllability BTL and Observability of the system (6) 6 Describe in detail the concept of state space model for full order BTL observer and reduced order observer (6) 7 Discuss briefly about the controllable and observable forms of SISO BTL systems (6) 8 (i) Illustrate the Controllable Canonical Form Canonical forms for MIMO system BTL and Observable (ii) Illustrate the effect of state feedback on Controllability and Observability (6) 9 What is meant by state observer? Draw and analyze the state diagram and eplain with an eample for state space with feed back (i) Full Order (ii) Reduced Order Observer, (6) The given state space model as follows ; y=[ ] Convert the state model into observable phase variable format and evaluate it (6) Illustrate the effect of state feedback gain by pole placement(i) Open loop state space without feedback gain (ii) Closed loop state feedback gain with control law for obtaining gain K by any one of the method with necessary condition (6) BTL

Consider the state space model described by X ( t ) Y ( t ) AX ( t ) CX ( t ) A ; C=[ ] Design and epress a full-order state observer The desired Eigen values for the observer matri 5; 5 (6) What is meant by observer? How the observer coincept related with Observability Eamine the following types of observer (i) BTL Full Order Observer (ii) Reduced Order Observer (6) 4 Estimate the following types of Canonical form with epression BTL for (i) Controllable Canonical Form (ii) Observable Canonical Form (6) NIT V LYAPNOV STABILTY ANALYSIS Introduction-Equilibrium Points- BIBO Stability-Stability of LTI Systems- Stability in the sense of Lyapunov - Equilibrium Stability of Nonlinear Continuous-Time Autonomous Systems-The Direct Method of Lyapunov and the Linear Continuous-Time Autonomous Systems-Finding Lyapunov Functions for Nonlinear Continuous-Time Autonomous Systems Krasovskil s and Variable- Gradiant Method PART A QNo Questions BTL Level Domain Define (i) Conditionally Stable (ii) Limited ly Stable (iii) Marginally Stable (iv) nstable Eplain BIBO stability BTL BTL Define positive definiteness of scalar functions Give an eample? BTL 4 Point out Lyapunov s asymptotic stability 5 Define the term stability BTL 6 Evaluate the concept of equilibrium points? 7 Eamine what is meant by autonomous system? BTL 8 Summarize the negative definiteness of scalar functions Give an eample? BTL

9 Illustrate the Lyapunov s instability theorem BTL Define positive semi definiteness of scalar functions Give an eample? BTL Draw and quote graphical representation of stable, asymptotic stable and unstable equilibrium states with their trajectory BTL Show that the following quadratic form is + ve definite BTL V(X)= +4 + + - -4 Determine whether the following quadratic form is ve definite BTL V (X) = - - - + -4-4 In routh array Analyze the conclusions you can make when there is arrow of all zeros? 5 Analyze the concept of limitedly stable system? 6 Invent the necessary and sufficient condition for stability? 7 A system has repeated Eigen values on imaginary ais What can you create about the asymptotic stability of the system? 8 List out various methods for stability analysis of non linear system BTL 9 Define Lyapunov s sufficient condition for asymptotic stability BTL Mention the advantages of Lyapunov s stability criteria BTL PART B Describe the modeling energy system in terms of quadratic function BTL Eplain the Lyapunov s stability criteria with diagrammatic representation (i) Asymptotically stable (ii) Stable (iii) nstable Eamine the Lyapunov s stability analysis for (i) Linear time BTL invariant system (ii) Nonlinear Continuous system 4 Eplain the Lyapunov s criterion stability analysis for (i) Continuous system (ii) Discrete time systems 5 BTL Summarize Krasovskii method and how it can be applicable for stability analysis eplain with an eample for it

6 Summarize direct method of Lyapunov s function how it can be applicable for nonlinear continuous time system 7 Eamine Lyapunov s direct method of Lyapunov for Continuous time autonomous system BTL BTL 8 Eamine the following terminology (i) Stability in the sense of BTL Lyapunov (ii) BIBO Stability for Linear Time Invarient System 9 Design and determine if the following matri is positive definite V(X)= +4 + + - -4 Estimate the direct method of Lyapunov s function how it can be BTL applicable for nonlinear continuous time system Illustrate the following methods for stability analysis (i) Krasovskii Method (ii) Variable-Gradiant Method with suitable eample Describe Lyapunov s Method Stability analysis for (i) Linear System (ii) Non-Linear System with suitable eample What is meant by autonomous of the system? Analyze how the Lyapunov s Method applicable for Linear and Nonlinear autonomous system 4 Eplain following stability concepts (I) Lyapunov s Method stability at orgin (ii) Lyapunov s Method stability in stable boundary (iii) Lyapunov s Method for unstable condition BTL BTL