Space Analysis Techniques Modern control engineering

Question: Describe about the Space Analysis Techniques for Modern control engineering. Answer: Investigation of state space analysis techniques Introduction The concept of state variables and state space equations is explained. The advantages, disadvantages of the state space analysis techniques are discussed. The applications of the state space analysis techniques are identified and explained. State Space Analysis The State Space Analysis techniques are applicable to Multiple Input Multiple Output (MIMO) systems, both linear and non-linear, time variant and time in-variant. The concepts of state, state variables, state vector, state space and state space equations are defined here. State The smallest set of variables, the knowledge of which at time along with the knowledge of input variables at , completely determines the behavior of the system at is called the state. State variables The variables that make up the smallest set of variables which determine the state are called the state variables. If n variables along with input are needed to predict the behavior of the system and future state can be determined, the n variables are called the state variables (Ogate, 2013). State Vector If the n state variables constitute the vector x, x is called the state vector. State Space The n-dimensional space with co-ordinate axes represented by axis, axis, etc., is called by the state space. Any state can be represented by a point in the state space. State Space Equations The three types of variables considered in the state-space analysis are the input variables, state variables and the output variables. The values of the input variables have to be stored in the memory devices for which integrators are used. The outputs of the integrators are considered as the state variables. The number of state variables is equal to the number of integrators used in the system. Let there are inputs represented by , outputs, and state variables represented by . Then the system can be described by, Equation 1 is called the state equation and equation 2 is called the output equation (Lyshevski, 2001; Choudhury, 2005) . If the state equation and the output equation are linearized about the operating state, - state matrix, - input matrix, output matrix, - Direct transmission matrix The block diagram of the state space representation of the linear, continuous time control system is shown in Figure 1. Figure 1 Block diagram of state space representation of the control system Relation between transfer function and state space equation Let the transfer function of the system is given by The system is represented in state space by the equations (6) and (7) are the state vector, input and output respectively. Taking laplace transforms for equations (6) and (7) If the initial conditions are assumed to be zero, Multiplying with on both the sides of equation (11) gives By substituting equation (12) in equation (9) Comparing equations (13) and (5) Thus the transfer function is expressed in terms of A, B, C and D. (State- Space System Representation of LTI Systems, Advantages of state space analysis over transfer function analysis Only linear time invariant systems can be analyzed using transfer function analysis while state space analysis can be used for non-linear and time variant systems. The transfer function approach can not be applied to multiple input - multiple output systems but the state space analysis techniques can be applied for MIMO systems. The internal state of the system can not be predicted by the transfer function analysis while the state space analysis gives clear idea about the internal state of the system. (Advantages of state space analysis, Electrical Engineering, Disadvantages of state space models Derivation of state space models for electrical circuits is difficult The state space model cannot be directly developed from the system diagram Applications of state space analysis State space techniques are applied in design and optimization of complex dynamic systems. They are used in Multiple Input Multiple Output systems such as air craft, space craft, servo mechanisms and robots. For example, in the electric drives, the output is the angular velocity of the shaft which is proportional to the angular velocity of the motor and it is the state variable. Similarly, in servo motors, the output linear position is proportional to the angular rotor displacement which is the state variable. One of the applications of state space analysis is tracking control problem for an unstable model of an aircraft. The different state variables of the system are forward velocity, angle of attack, pitch rate, pitch angle, side slip angle, roll rate, yaw rate, roll angle and yaw angle. The control inputs are the deflections of the right and left horizontal stabilizers, deflections of the right and left flaps and the canard and rudder deflections. Hence state space techniques are applied in design and optimization of complex dynamic systems both linear and non-linear, time variant and time invariant (Friedland, 2005). Conclusion The concept of state space representation, state variables and state equations are illustrated. The advantages and disadvantages over the transfer function analysis are detailed. The specific applications of the state space analysis techniques are discussed. References Ogata, K. (2013).Modern control engineering: Pearson new international edition. Pearson Education Limited. Lyshevski, S. E. (2001).Control systems theory with engineering applications. Boston: Birkhauser. Choudhury, D. R. (2005).Modern control engineering. New Delhi: Prentice Hall. Friedland, B. (2005).Control system design: An introduction to state-space methods. Mineola, NY: Dover. State- Space System Representation of LTI Systems. https://web.mit.edu/2.14/www/Handouts/StateSpace.pdf Advantages of state space analysis, Electrical Engineering. https://www.tutorsglobe.com/question/advantages-of-state-space-analysis-511506.aspx