46 Design of a Controller for Systems with Simultaneously Variable Parameters

The contribution of this paper is in suggesting an analysis and design of a control system with variab le parameters. By applying the recommended by the author method of the Advanced Dpartitioning the system’s stability can be analyzed in details. The method defines regions of stability in the space of the system’s parameters. The designed controller is enforcing desired system performa nce. The suggested technique for analysis and design is essential and beneficial for the further development of control theory in this area. Keyword: System with variable parameters, Stability regions, System performance

The contribution of this paper is in suggesting an analysis and design of a control system with variable recommended by the partitioning the system's stability can be analyzed in details. The method defines regions of stability in the space of the system's parameters. The designed controller is nce. The suggested technique for analysis and design is essential and beneficial for the further development of control

System with variable parameters, Stability
Control systems performance must be insensitive to parameter variations. In the process of design such a control system, it is important to determine the regions stability, related to the variation of the system parameters. The suggested by the author method, dealing with the effects of parameters variations on the system's stability, is classified as Advanced Dpartitioning [1], [2], [3]. It is an efficient tool for system stability analysis in case of variation of any of rther upgraded in this paper and can be used for simultaneously varying This research is also suggesting a method for design of a controller, by applying forward-series compensation. It can suppress the influence of any the control system. Innovation is demonstrated in the unique property of the designed controller that can operate effectively for variations of any one of the system's parameters within prescribed limits. The design of the controller criterion [4], [5]. For higher order systems, a pair of dominant poles represents the system dynamics. The relative damping ratio ζ of the system is taken as a performance objective for the optimization design.

II.
System with Simultaneously Variable Parameters The Advanced D-Partitioning analysis in case of two simultaneously variable parameters [2], [4], [6] can be demonstrated for a control system of the controlled dc motor and a type The gain and one of the time and variable. The open-loop transfer function of the system can be presented as: Since the gain may have only real values, the imaginary term of equation (3) is set to zero. Then: The result of (4) is substituted into the real part of equation (

System with Simultaneously Variable
Partitioning analysis in case of two simultaneously variable parameters [2], [4], [6] can be control system of the armaturecontrolled dc motor and a type-driving mechanism. The gain and one of the time-constants are uncertain loop transfer function of the equation (2) is modified to: Since the gain may have only real values, the imaginary term of equation (3) is set to zero. Then: The result of (4) is substituted into the real part of T Partitioning in terms of two (T) defines the border between the region of stability D (0) and instability D(1) for case of simultaneous variation of the two system parameters. Each point of the D-Partitioning curve represents the marginal values of the two simultaneously variable parameters. This is a unique advancement and an innovation in the theory of control systems stability analysis. The demonstration of the system performance in case of done at gain set to T > 1.5 sec the system is stable. But it becomes unstable in the range 0.25 sec < T < 1.5 sec. It is also obvious that the system performance and stability depends on the interaction between the two simultaneously varying parameters. If K < 8.3417, the system any value of the T. Higher values of 14), enlarge the range of T at which the system will fall into instability.

III. Design of a Controller for Systems with Simultaneously Variable Parameters
The open-loop transfer function of the plant modified and now presented in equation (6) considering the two variable parameters that are the system's gain K and time-constant suggested that the gain is set to variable [5], [7], [8].
The robust controller consists of a series stage and a forward stage G F0 (s). An integrating stage is also included in the controller as seen from 2.

Fig.2. Robust controller incorporated system
Initially, the plant transfer function standalone block, is involved in a unity feedback system having a closed-loop transfer function presented as:  (7) is used as a base in the design strategy for constructing the series stage of the robust controller. It has the task to place its two zeros near the desired dominant closedthe condition ζ = 0. system is stable. But it becomes unstable in the range < 1.5 sec. It is also obvious that the system performance and stability depends on the interaction between the two simultaneously varying < 8.3417, the system is stable for Higher values of K (K = 12, K = at which the system will Design of a Controller for Systems with Simultaneously Variable Parameters function of the plant G P0 (s) is modified and now presented in equation (6) the two variable parameters that are the constant T. Initially, it is suggested that the gain is set to K = 10, while T is The robust controller consists of a series stage G S0 (s) (s). An integrating stage G I0 (s) is also included in the controller as seen from Figure Equation (7) is used as a base in the design strategy for constructing the series stage of the robust controller. It has the task to place its two zeros near -loop poles, that satisfy = 0.707 [9], [10]. These zeros will become the dominant poles of the unity negative feedback system, involving the cascade connection of By substituting the values T = 27.06 sec and equation (7), the transfer function of the closed system becomes: The assessment of the system proves that the damping ratio becomes ζ = 0.707, when constant is T = 27.06 sec, resulting in system's desired closed-loop poles −0.466 ± j0.466. These outcomes are determined from the code: If initially the gain is set to K = ue of the time-constant T, corresponding to the relative damping ratio ζ = 0,707 loop system, is determined by the code:
= 27.06 sec and K = 10 in equation (7), the transfer function of the closed-loop 11 + The assessment of the system proves that the relative when the time-= 27.06 sec, resulting in system's desired j0.466. These outcomes The series robust controller zeros can be placed at the approximated values −0.5 ± transfer function of the series robust controller is: An integrating stage G I0 (s) is added to eliminate the steady-state error of the system. It is connected in cascade with the series controller. Then, the transfer function of the compensated open will be as follows:

IV.
Performance of the Compensated System The system is tested for insensitivity to variations of its gain K and its time-constant performance is done before and after applying the The series robust controller zeros can be placed at the ± j0.5. Therefore, the transfer function of the series robust controller G S0 (s) It is seen from the equation (11) that the closed-loop zeros will attempt to cancel the closed loop poles of the system, being in their area. This problem can be avoided if a forward controller G F0 (s) is added to the loop system, as shown in Figure 2. The poles (s) are designed to cancel the zeros of the loop transfer function G CL (s), as shown in (12) Finally, the transfer function of the total compensated system is derived considering the diagram in Figure 2.

Compensated System
The system is tested for insensitivity to variations of constant T. Comparison of its performance is done before and after applying the International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 @ IJTSRD | Available Online @ www.ijtsrd.com robust compensation. Initially at system gain is set to K = 10 and three different values of the time are set successively to T = 0.1 sec, T = 0.8 sec and 2 sec and are substituted in equation (6).
The case of T = 0.8 sec, corresponds to the region D(2) and definitely to an unstable control system. The cases of T = 0.1 sec and T = 2 sec, reflect regions D1(0) and D2(0) accordingly and are related to a stable control system.
The transient responses of the system before applying the robust compensation are illustrated in Figure 4 and Figure 5 and are achieved by the following codes: >> Gp001= tf ([0 10],[0.04 0.53 1.4  = 0.8 sec, corresponds to the region D(2) and definitely to an unstable control system. The = 2 sec, reflect regions D1(0) and D2(0) accordingly and are related to a
Since the system is with two variable parameters, now the gain will be changed, applying: 20, while Step responses of the system with a robust (T = 0. 1 sec, T = 0. 8 sec, T = 2 sec at K = , due to the applied robust controller, the control system becomes quite insensitive to variation of the time-constant T. The Step response of the original control system It is obvious that the cases of correspond to an unstable original control system. Next, the variable gain K = 5, applying to the robust compensated system, keeping the system's time-constant at values are substituted in equation following outcomes are delivered: It is obvious that the cases of K = 10 and K = 20, correspond to an unstable original control system. = 5, K = 10, K = 20 will be applying to the robust compensated system, keeping constant at T = 0.8 sec. These values are substituted in equation (13). As a result, the following outcomes are delivered: 5 To compare the system robustness before and after the robust compensation, the step responses for the three different cases, representing the gain variation of the plotted in Figure 9 with the aid of @ IJTSRD | Available Online @ www.ijtsrd.com will differ insignificantly from the other cases of the discussed variable K and T. The performance evaluation is achieved by following code: >> GT10=tf ([10], [0.32 1.44 22.1 21 10 -1.76e+000 + 7.88e+000i 2.18e-001  8.07e+000  -1.76e+000 -7.88e+000i 2.18e-001 8.07e+000 It is seen that the relative damping ratio enforced by the system's dominant poles is ζ = 0.709, being very close to the objective value of ζ = 0.707. insignificant difference is due to the rounding of the desired system's poles to −0.5 ± j0.5, during the design of the series robust controller stage.

V. Conclusions
The D-Partitioning analysis is further advanced for systems with multivariable parameters [8], [9], [10]. The Advanced D-partitioning in case two variable parameters is demonstrating the strong interaction between the variable parameters. Each point of the Partitioning curve represents the marginal values of the two simultaneously variable parameters, being a unique advancement and an innovation in the theory of control systems stability analysis.
The design strategy of a robust controller for linear control systems proves that by implementing desired dominant system poles, the controller enforces the required relative damping ratio and system performance. For systems Type 0, an additional integrating stage ensures a steady-state error equal to zero.
The designed robust controller brings the system to a state of insensitivity to the variation of within specific limits of the parameter variations. The experiments with variation of different parameters show only insignificant difference in performance for the different system conditions [10], [11], [12].
For the discussed case, the system becomes quite insensitive to variation of the time constant within the limits 0.1T < T < 10T. The system is quite insensitive to variations of the gain K within the limits 0.5 < 5K. Insignificant step response difference is observed also if the experiment is repeated with partitioning in case two variable parameters is demonstrating the strong interaction between the variable parameters. Each point of the D-Partitioning curve represents the marginal values of the two simultaneously variable parameters, being a unique advancement and an innovation in the theory The design strategy of a robust controller for linear ntrol systems proves that by implementing desired dominant system poles, the controller enforces the required relative damping ratio and system performance. For systems Type 0, an additional state error equal to The designed robust controller brings the system to a its parameters within specific limits of the parameter variations. The experiments with variation of different parameters performance for the different system conditions [10], [11], [12].
For the discussed case, the system becomes quite insensitive to variation of the time constant within the . The system is quite insensitive within the limits 0.5K < K . Insignificant step response difference is observed also if the experiment is repeated with different variation of the gain and different variation time-constants values.
Since the design of the robust controller is base the desired system performance in terms of relative damping, its contribution and its unique property is that it can operate effectively for any of the system's parameter variations or simultaneous variation of a number of parameters. This property is by the comparison of the system's performance before and after the application of the robust controller. demonstrate that the system performance in terms of damping, stability and time response and insensitive in case of any simultaneous variations of the gain and the time-constant within specific limits. The suggested analysis and design for further advancement of control theory in this Since the design of the robust controller is based on the desired system performance in terms of relative damping, its contribution and its unique property is that it can operate effectively for any of the system's parameter variations or simultaneous variation of a number of parameters. This property is demonstrated by the comparison of the system's performance before and after the application of the robust controller. Tests demonstrate that the system performance in terms of stability and time response remains robust any simultaneous variations constant within specific The suggested analysis and design is beneficial for further advancement of control theory in this field.