DEPARTMENT OF MECHANICAL TECHNOLOGY VI -SEMESTER AUTOMATIC CONTROL 1 CHAPTER NO.6 State space representation of Continuous Time systems 1 Teaching Innovation - Entrepreneurial - Global The Centre for Technology enabled Teaching & Learning, M G I, India DTEL DTEL (Department for Technology Enhanced Learning)

DTEL 2 DEPARTMENT OF MECHANICAL TECHNOLOGY VI -SEMESTER CONTROL SYSTEMS ENGINEERING 1 CHAPTER NO.6 State space representation of Continuous Time systems

CHAPTER 6:- SYLLABUSDTEL State space representation for Discrete time systems. State equations, transfer function from state variable representation – solutions of the state equations. Concepts of Controllability and Observability.. 4 Introduction to control system design lag lead compensation.

CHAPTER-6 SPECIFIC OBJECTIVE / COURSE OUTCOMEDTEL Stability analysis using analytical and graphical techniques, The student will be able to: To understand the concepts of time domain and frequency domain analysis of control system.

DTEL 5 LECTURE 1 5 Linear Continuous-Time State Space Models A continuous-time linear time-invariant state space model takes the for where x  n is the state vector, u  m is the control signal, y  p is the output, x 0  n is the state vector at time t = t 0 and A, B, C, and D are matrices of appropriate dimensions. X(t) = AX(t) +B u(t) x(t o ) = x o Y(t) =C x(t) +Du(t) State equations

DTEL 6 LECTURE 2 6 Transfer Functions vs. State-Space Models Transfer functions provide only input and output behavior – No knowledge of the inner workings of the system – System is essentially a “black box” that performs some functions State-space models also represent the internal behavior of the system H(s) X(s)Y(s) State equations Fig 6.1 Transfer Function

DTEL 7 LECTURE 2 7 Linear State-Space Equations system matrix input matrix output matrix matrix representing direct coupling from system inputs to system outputs If A, B, C, D are constant over time, then the system is also time invariant → Linear Time Invariant (LTI) system. State-Space Models

DTEL 8 LECTURE 2 8 Transfer Functions vs. State-Space Models Transfer Functions Defined as G(s) = Y(s)/U(s) Represents a normalized model of a process, i.e., can be used with any input. Y(s) and U(s) are both written in deviation variable form. The form of the transfer function indicates the dynamic behavior of the process. Transfer Functions

9 LECTURE 9 LECTURE 3 Controllability 제 14 강 nnnn nmnm Controll able nnnn DTEL A system is completely controllable if there exists an unconstrained control u(t) that can transfer any initial state x(t 0 ) to any other desired location x(t) in a finite time t 0  t  T. Controllability Fig 6.2 Controllability

DTEL 10 LECTURE 4 제 14 강 Controllable 1n1n n1n1 nnnn ObservabilityDTEL 10 Observability A system is completely observable if and only if three exists a finite time T such that the national state x(0) can be demined from the observation history y(t) given the control u(t) Fig 6.3 Observability

DTEL 11 LECTURE 5 1. Control system design and compensation Design need to design the whole controller to satisfy the system requirement. Compensation only need to design part of the controller with known structure. 2. Three elements for compensation Original part of the system Performance requirement Compensation device Stability criterion

DTEL 12 LECTURE 6 1.Frequency Response Based Method Main idea ： By inserting the compensator, the Bode diagram of the original system is altered to achieve performance requirements. 2. Root Locus Based Method Main idea: Inserting the compensator introduces new open-loop zeros and poles to change the closed-loop root locus to satisfy. Method of compensator design Original open-loop Bode diagram Bode diagram of compensator alteration of gain open-loop Bode diagram with compensation Method of compensator design

DTEL 13 LECTURE 7DTEL 13 Transfer function: Fig 6.4 Passive phase lag network Phase Lag Compensation

DTEL 14 LECTURE 8 Fig 6.5 Maximum phase lag DTEL 14 The compensator has no filtering effect on the low frequency signal, but filters high frequency noise. The smaller β is ， the lower the noise frequency where the noise can pass. Maximum phase lag Stability criterion

DTEL 15 LECTURE 8DTEL 15 Comments on phase lag compensation: 1.Phase lag compensator is a low-pass filter. It changes the low-frequency part to reduce gain crossover frequency. The phase is of no consequence around the gain crossover frequency. 2.Be able to amplify the magnitude of low-frequency part, and thus reduce the steady-sate error. 3.The slope around gain crossover frequency is -20dB/ dec. Resonance peak is reduced, and the system is more stable. 4.Reduce the gain crossover frequency, and then reduce the bandwidth. The rising time is increased. The system response slows down. phase lag compensation

DTEL 16 THANK YOU

DTEL References Books: 1.Automatic control system by Farid Golnaraghi. 2.Modern control System Engineering by Katsuhiko Ogata 3.Feedback Control System by R. A. Barapate 4.Automatic control system by Benjamin 17