Signals and Systems

Question: Discuss about theSignals and Systems. Answer: State Space Approach The state space model is the mathematical description of the physical model that can be used for understanding the various parameters related to the model. The mathematical equations that are generated contain the inputs, outputs and the state variables that decide the system responses. The conventional control system mainly focuses on the frequency domain analysis of the system which mainly stresses on the SISO (Single Input and Single Output) models and the analysis of the MIMO (Multiple Input and Multiple Output) systems is extremely complex (Ogata, 2013). For Sorting out the various computational constraints and difficulties, the state space analysis has been presented which is based on time contrary to the conventional control theory. The state model also helps in analysis of the non-linear and the time-variant system. Hence the State Space Analysis of the system is the set of minimum variables, the knowledge about which at the initial instant t=to combined with the knowledge of the inputs at an instant tto is capable of defining the behavior of the system (Kirk, 2012). The state space representation is done in the form of the following equation: x(t) is the state vector of the model; x(t)?Rn y(t) is the output vector of the model; y(t) ?Rq u(t) is the input vector also known as the control vector to the system; u(t) ?Rp A(t) is the state matrix which has a dimension of p*q B(t) is the input matrix, which has the dimension of n*p C(t) is the output matrix, which has the dimension of q*n D(t) is the feedforward matrix, which has the dimension of q*p Advantages of State Space Approach The advantages of the state space approach are many as it has been developed after suffering the hands of the classical control system. Hence to understand the advantages of the state space model, the disadvantages of the classical models should be understood (Friedland, 2012). The disadvantages of the classical model are: The classical model or the transfer function model has to be defined under zero initial condition. The classical model can only be applied to the linear and time invariant system. The transfer function model cannot be applied to the MIMO systems. The transfer function analysis is difficult to perform on computers. The advantages of the state space analysis model are (Sontag, 2013): The model is easier to work with. The state space model can be easily applied to the time-invariant systems The state space model can be applied to the nonlinear systems. The state space model can be easily applied to the MIMO systems. The state space model can be easily done on the computers. The state space model can be used for understanding the internal state of the system. It can be seen that the modern control system or the state space control system is more efficient compared to the previous classical model. Apart from the stated advantages the various system analysis like observability and controllability can be easily done on the system thus helping in understanding the response of the system (Ogata, 2013). The easy matrix representation also eases the system understanding. Disadvantages of State Space Approach The state space model is up gradation to the classical control theory which used the transfer functions to solve only specific sets of problems, but the state space model has been developed to counter all the drawbacks of the traditional model. Hence the disadvantages have been drastically reduced. The major disadvantages of the state space approach are (Kar Das Ghosh, 2014): Complex techniques. Large computations are required for understanding the characteristics. The complexity of the state space model is because the representation is done in the form of matrices. The construction of the state matrix is done with the help of series of calculation based on the equation of the system, which is not very easy to find for every system. The simpler systems can be combined thus creating a complex system which is very difficult to handle manually (Kirk, 2012). Apart from the complexity, the matrix form causes the amount of calculation which needs computational assistance for a solution, thus making the system not user-friendly. Application of State Space Approach In the initial sections, we have learned about the state space analysis followed by the sections in which the advantages and the disadvantages have been presented. It can be seen from the analysis that the state space model can be easily applied to the time variant, non-linear system but the method is very much complex and involves a lot of calculations (Lathi Green, 2014). Hence it can be easily understood that the simpler model which involve time invariance and linearity should be left to the classical methods as the transfer functions are easier to perform compared to the state space model. Hence the systems which are inherently complex and involves and contains multiple inputs and multiple outputs paired with time invariance and non-linearity should be solved with the help of the state space model. IT should also be understood that the computers are needed for such systems as it will help in the calculation of the various parameters of the system readily compared to human calculations which might take longer duration and is prone to errors (Walter, 2013). The state space models have been applied to trend-cycle decomposition, missing value treatments, time varying parameters, etc. since the model offers the freedom to specify the process without observing certain variables. Various applications that have been analyzed by the model are: H control, estimating the price of the oil and its dynamics, time series analysis (Wang Ding Ximei, 2014). IT should be understood that the implementation has various inputs based upon which the equations of decencies are created, and finally the state matrix is presented which characterizes the model (Durbin Koopman, 2012). Then the inputs are applied to the model and based upon the response of the system; the output is generated. The plethora of academic journal that is available over the internet shows the success of the model and the ease of operation with it, which will mark an era of better controlled and observed systems. References Durbin, J., Koopman, S. J. (2012).Time series analysis by state space methods(No. 38). Oxford University Press. Friedland, B. (2012).Control system design: an introduction to state-space methods. Courier Corporation. Kar, S., Das, S., Ghosh, P. K. (2014). Applications of neuro-fuzzy systems: A brief review and future outline. Applied Soft Computing,15, 243-259. Kirk, D. E. (2012).Optimal control theory: an introduction. Courier Corporation. Lathi, B. P., Green, R. A. (2014).Essentials of digital signal processing. Cambridge University Press. Ogata, K. (2013).Modern Control Engineering: Pearson New International Edition. Pearson Education Limited. Sontag, E. D. (2013).Mathematical control theory: deterministic finite-dimensional systems(Vol. 6). Springer Science Business Media. Walter, E. (2013).Identifiability of state space models: with applications to transformation systems(Vol. 46). Springer Science Business Media. Wang, D., Ding, F., Ximei, L. (2014). Least squares algorithm for a nonlinear input system with a dynamic subspace state space model. Nonlinear Dynamics,75(1-2), 49-61.