## Question:

## Answer:

### Introduction:

The state space analysis of the various control systems is lying on the basis of modern theories. It has been applicable to all kind of systems. This includes the single and multiple inputs and output systems, the linear and non-linear systems and so on.

The following report explains the state space analysis along with various advantages and disadvantages. It also includes the instances of applications, reviewing of methods and the results. Lastly, it provides examples of the state space assessment of the accelerometer system.

### Explanation of the state space analysis system:

The state space analysis has been an effective approach to design and examines the control systems. The old and conventional approaches to design and examine the control systems have been the transfer function method. However, the transfer function method to analyze and design has possessed various drawbacks.

There have been few basic terms that are needed to be known related to the state space analysis. The first one is the state of the systems (Azar & Vaidyanathan, 2015). This has been describing the future type of the system through using the current and the past. The former one indicates the input variables and the later one denotes the state variables. Then there is the state variable. It is the set of variables that the along with the input has been defining the system’s behaviors. These behaviors have been representing the future values.

The number of the state variables in the system is intended to be minimum, same as the system’s order. Moreover, the output of the system and the derivatives could be regarded as the “state variables”. Lastly, the state space defines the set of the possible values. These are assumed by the state variables.

The advantages of the state space analysis lie in the fact that it could be applied to the non-linear systems. Moreover, it could be applied to the tile invariant systems, multiple inputs and output systems and the idea regarding the internal state of the system. Further, it has been possible to examine the time-invariant and time-varying, single or multiple and linear or non-linear input-output systems (Isidori, 2013). Then there is the possibility of confirming the parameters of the system’s state apart from the input-output relations. Further there lies the scope of the optimizing the systems. It is also helpful for the optimal designs. Lastly, it has been possible to incorporate the initial situations.

Further, the analysis of state variables could be done on any kind of systems. It has been very simple to do the tasks of state variable analysis over the computers. One of the most interesting facts of the analysis is that the chosen state variables to describe the system never require being the physical quantities that are related to the system. The variables not related to the physical quantities and associated with that system could also be chosen as the state variables. Moreover, the variables that are unobservable and immeasurable could be chosen as the state variables.

Despite all these, one of the disadvantages of the analysis is that the transfer function has been defined within the initial conditions which are zero (Yin, Luo & Ding, 2014). Moreover, the approach is applicable only to the systems that are linear time-invariant. It never provides the idea about what the internal state of the systems have been. It is never applicable to the multiple input and output systems. It has been difficult to do the analysis of transfer function on the computers in a comparative manner.

### Examples of applications, review of methods and outcomes:

### The classes of problems:

Every problem defines a searched space comprising of objects known as the solution candidates. Let the search space be S. Particularly the expected solution is represented in terms of S. There have been primarily two kinds of problems. The first one is the constraint satisfaction problems. Here the constraints are needed to be fulfilled and should be determined with the least search effort. The next one is the optimization problems. Here all the constraints are needed to be fulfilled. It has been standing out as every other candidate related to the special property.

The computing of the optimum has been generally infeasible. The semi-optimization has been the relaxation of the optimality necessities. For example, let two kinds of optimality relaxation is provided. The first one is the near optimization. Here the threshold of the deviation of the maximum cost has been provided (Kumar & Varaiya, 2015). Then there is the approximate optimization. This has been even more relaxed in nature. Here the deviation threshold or the near optimization has been required to be adopted with as particular probability only.

### Review of the methods:

For example, let the 8-queen’s problem be taken as the optimization problem. Here the users have been capable of controlling the trade-offs happening between the efficiency and the deviation from the optimization solution. This has been ideally with some of the hyper-parameters (Wang, Blaabjerg & Wu, 2014).

Alternative methods could be applied to solve this type of problems as proposed by McCarthy. The first one is the well-defined problems. Here there is an entire knowledge of the beginning state, goal state, and the operators. Then there are the ill-defined problems. Here the knowledge regarding the necessary operators or the goal states has been incomplete.

Other methods to overcome the barriers were suggested by Doner. The first barrier is the interpolation barrier (Box et al., 2015). There has been a total knowledge of the operators, goal states, and the initial states. Then there is the synthesis barrier. Here the knowledge of the required operators has been incomplete. Then there is the dialectic barrier. Here the solutions should be examined regarding the external or internal conflicts.

### Outcomes of the solutions:

The solution candidates or the objects in S are needed to be distinguished along with their encodings. This encoding is understood as the reference to the object or the set of objects in S. This encoding must take benefits of the structural properties of the problem domain. Generally, the structural properties have been captured by the type of partial or incomplete representation of the object. This has been against the unique or complete representation (Stevens, Lewis & Johnson, 2015). The encoding of the second one has been corresponding to the solution candidate. This indicates the one-to-one to the object. For instance, in the 8 queen’s problem, the encoding denotes to the board configuration with the exact number of 8 queens. Further, the partial representation could be extended or elaborated (Nikiforov et al., 2017). Hence this has been corresponding to the set of objects or solution base. The encoding of this type could be referred to either the solution candidate or the solution base.

### The examples of the state space analysis of the accelerometer system:

The measuring of the acceleration, along with the core element of the inertial guidance systems is applicable to a broad range of commercial problems. The various transducer mechanisms are utilized in the accelerometer that is micro-fabricated. For example, the tunneling has been a highly sensitive approach to measure the position. This has been inspired by the previous works on the microscope of the scanning tunnels. Here the electrostatic actuator is utilized. This is more effective than the piezo-resistive, piezoelectric and capacitive displacement transducers (Dzunic et al., 2017). Moreover, the tunneling current is never dependent on the temperature. The realizing of the accelerometer, controller systems, and the tunneling detector is included in the state space model.

The analysis of the state space done here provides the internal view of the stochastic behavior of the tunneling current. This also includes the way in which the noise has been propagating through the system. The controller that is utilized has been an easy integrating controller set up. The results verify the conclusion of Gabrielson (Aoki, 2013). This states that the thermal noise provides the main contribution. Further, the bandwidth could be raised through utilizing the more advanced controller. This kind of controller possesses the larger bandwidth and the more noise of resistor that is meant to be fed in the system (Walter, 2013). The impacts of the more complex controllers have been currently explored. This provides the tool with the larger bandwidth.

## Conclusion and discussion:

The state equations could be yield as a great deal of data regarding the system. This occurs when they are unable to be solved explicitly. Further, the state equations of the system could be penned down directly from the system structure’s knowledge from the system equations. Thus the analysis procedure has been consisting of the calculation of the state equations at the very first step. Then the output equations are solved. It is seen that the description of the state space has been able to determine all the possible system variables or outputs. This happens from the knowledge of the initial state and the various inputs of the system. Through the usage of that kind of transformation, it is clearly seen what systems are manageable and which one is observable.

## References:

Andriluka, M., Pishchulin, L., Gehler, P., & Schiele, B. (2014). 2d human pose estimation: New benchmark and state of the art analysis. In Proceedings of the IEEE Conference on computer Vision and Pattern Recognition (pp. 3686-3693).

Aoki, M. (2013). State space modeling of time series. Springer Science & Business Media.

Azar, A. T., & Vaidyanathan, S. (Eds.). (2015). Chaos modeling and control systems design. Germany: Springer.

Box, G. E., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015). Time series analysis: forecasting and control. John Wiley & Sons.

Dzunic, Z., Chen, J. G., Mobahi, H., B?y?k?zt?rk, O., & Fisher, J. W. (2017). A Bayesian state-space approach for damage detection and classification. Mechanical Systems and Signal Processing, 96, 239-259.

Isidori, A. (2013). Nonlinear control systems. Springer Science & Business Media.

Kumar, P. R., & Varaiya, P. (2015). Stochastic systems: Estimation, identification, and adaptive control. Society for industrial and applied mathematics.

Nikiforov, V. M., Gusev, A. A., Zolotukhin, S. S., Zhukova, T. A., & Nizhegorodov, A. A. (2017, May). Identification of pendulous accelerometer mathematical model taking into account parametric uncertainty. In Integrated Navigation Systems (ICINS), 2017 24th Saint Petersburg International Conference on (pp. 1-4). IEEE.

Stevens, B. L., Lewis, F. L., & Johnson, E. N. (2015). Aircraft control and simulation: dynamics, controls design, and autonomous systems. John Wiley & Sons.

Walter, E. (2013). Identifiability of state space models: with applications to transformation systems (Vol. 46). Springer Science & Business Media.

Wang, X., Blaabjerg, F., & Wu, W. (2014). Modeling and analysis of harmonic stability in an AC power-electronics-based power system. IEEE Transactions on Power Electronics, 29(12), 6421-6432.

Yin, S., Luo, H., & Ding, S. X. (2014). Real-time implementation of fault-tolerant control systems with performance optimization. IEEE Transactions on Industrial Electronics, 61(5), 2402-2411.