Introductory Control: First Order System Response - Assignment Example

INTRODUCTORY CONTROL By + FIRST ORDER SYSTEM RESPONSE Comparison of the system response for different value of α Taking into the consideration the system equation of G(s) = 1/ (s+ α). The student ID is 972 Where; α = a, b, c, a, b and c = 9,7 and 2 Step response Taking the step response to be designated by R(s) and the Laplace transformation for R(s) and the system response is represented as R(s) =1/s, Y(s) = R(s) x G(s) = 1/s (s+ α) = 1/ α [1/s-1/ (s+ α)] Employing inverse Laplace transformation, the system response in the time domain will be y (t) =1/ α( 1-e-at).. Thus, the step response has one constant term and exponent term. The exponential term is subtracted from corresponding constant term. Swift decay of the exponent terms depicts relatively faster response. Moreover, depending on the value of α, the underlying exponential decay will be either quicker or slower. Relatively higher of α depict faster decay of the exponential that ensure faster response. For α=9, the system response faster than corresponding values of α=, 2 and 7. Table 1‑1: Rise time for different alpha value Alpha Value 9 7 2 Rise time 0.2984 0.4206 0.7686 Thus, as the value of alpha escalates the system responses relatively quickly as depicted by the underlying lowest rise on the table above. The underlying step response depicts that the output of the system commence to reach the final value succeeding the first order system response, which essentially exponential increase of the output. The speed of the system depends on the value of the alpha that it α it dictates whether a system will be either faster or slower. Moreover, relatively higher value of α normally results to faster response. Evaluation of steady state value yss of system output for the different values of α used Based on the final value theorem, the steady state values is represented by Thus, the steady state response is , which can be correspondingly computed through substitution of t=∞ within the equation results Steady State Value Tabulation Table 1‑2: Yss value for different values of alpha Alpha Value 9 7 2 Yss 1/9=0.111 1/7=0.143 1/2=0.500 This can be verified by the three graphs below that the steady state value of every response. For relatively value of alpha, the underlying steady values translate to the lower as the prevailing alpha appears at the denominator. Thus, the alpha value of eight possesses steady state value of 0.125, which is the lowest. Values of α for which the system stable Transfer function is significant for the analysis of the system stability. A system is stable in case e poles of the underlying transfer function possess negative real parts. This implies that the quantities of s at the existing denominator become zero. When the poles are on the complex s-plane that is all the poles are within the left half of the plane then it is under stable operation. The system marginally stable when two poles on the imaginary axis and the corresponding system have oscillation. The system is stable since the pole’s real part is negative. Moreover, the system’s stability is attained from the state-space model. Transfer function poles are the system matrix A’s eigenvalues. To make the system stable, the alpha value ought to be that the underlying poles stay left side of the S-plane, which depicts that the α>0. When the value of α=0, the system is stable and called marginal stable Table 1 3: Pole location and stability for different values of alpha Alpha Value 9 7 2 Pole location -9 -7 -2 Stability Stable Stable Stable Because the alpha (α) value is 8,7 and 6 and all are positive, the prevailing poles will always remain at the left side of the S-plane. Thus, the system is stable for all the prevailing values of α When α is negative, the prevailing poles become positive and remain at the right side of the S-plane. Consequently, the system becomes unstable. Figure 1 7: Step response for alpha = -0.5 Figure 1 8: Step response for alpha = -1 Because the prevailing negative value of the alpha makes the underlying exponential term positive, the corresponding output escalates exponentially. For any relatively higher value of alpha, that is when all the existing values are negatives; the exponential term normally takes relatively longer time to go infinitely high than compared to the lower value. For instance, -0.5 input takes a period of 120 seconds to reach the infinite value. Conversely, -1.0 values take a period of 6o seconds to attain the infinite value. Ramp response Taking the ramp response designated by R(s) and its corresponding Laplace transformation being R(s)= 1/s2, Y(s)= R(s)* G(s) = 1/s2(s+ α) = 1/ α [1/s2-1/ αs-1/ α(s+ α)] Applying Laplace transformation, the system response in time domain becomes Y (t) =1/ α [t-1/ α-e-at] The step response has three terms namely time, increase of time and the step response scaled by alpha. Scaled step response is normally subtracted from the linearly the corresponding escalated time value. Moreover, there is also scaled version for the entire outcome. Because the step response is normally scaled by alpha depicts that the output is linearly increase with the underlying ramp input. It implies that the output is a ramped signal scaled by the alpha as depicted by the graphs. Impulse response Taking the impulse response designated by R(s) and its corresponding Laplace transformation being R(s)= 1, Y(s)= R(s)* G(s) = 1/(s+ α) Applying Laplace transformation, the system response in time domain becomes Y (t) = (e-at] The step response has a single exponential tern, which contains alpha value. Consequently, depending on the prevailing value of α, the corresponding exponential decay will be either swifter or slower. Relatively higher value of α designates swifter decay of the exponential. Thus, α=9 depicts that it decays relatively faster than corresponding α=2, 7. SECOND ORDER SYSTEM RESPONSE Transfer function for the mass, spring and damper system is G(s) =1/ (s2+ αs+1) The poles are computed from the denominator Denominator: s2+αs+1 Poles: Pole-zero Plot Figure 2-1: Pole values for alpha = 9 Figure 2-2: Pole values for alpha = 7 Figure 2-3: Pole values for alpha = 2 Through comparison of the prevailing equation with the corresponding standard second order equation, Damping Factor (zeta),, parameter can be computed as Where, K = Gain τ = Natural Period of Oscillation ζ = Damping Factor (zeta) Thus, the damping factor is Based on the value, the systems can be of three types of damping namely over Damped Systems, Critically Damped Systems and Undamped Systems. Over Damped Systems occur when >1 and both of its poles are real, negative thus making the system to be stable, and lacks oscillation. A system is critically damped when =1, both poles are real and possesses identical values, Sp=-. Moreover, critically damped system normally extends to stable condition extremely fast devoid of any oscillation. A system is considered to be undamped when =0, and the poles are imaginary thus making the system to be marginally stable and normally oscillates for infinite time. Table 2‑3: Damping Factor for different alpha value Alpha Value 9 7 2 Damping Factor 4.5 3.5 1 System types Over Damped Systems Over Damped Systems Over Damped Systems Pole-zero Plot Figure 2-4: Pole values for alpha =9 Figure 2-5: Pole values for alpha =7 Figure 2-6: Pole values for alpha =2 As depicted by the pole-zero plot of the system, the entire poles are at the left hand side of the prevailing S-plane for every value of the alpha. Thus, the system is steady for the entire value of the alpha. Impulse, step and ramp response of the system Figure 2‑1: Impulse, step and ramp response of the system for alpha=9 Figure 2‑2: Impulse, step and ramp response of the system for alpha=2 Figure 2‑3: Impulse, step and ramp response of the system for alpha=7 Impulse response: Taking the impulse response designated by R(s) and its corresponding Laplace transformation being R(s) =1 Where, and are two roots of the system. Applying inverse Laplace transformation, the system response in time domain will be The step response has two exponentials terms, which also has two roots. Consequently, based on the value of the alpha, the prevailing peak value varies. The peak value is highest occur when there is the lowest value of the alpha. Since the roots are relatively smaller for the low value of alpha, it takes more duration to decay. Step response: Taking the impulse response designated by R(s) and its corresponding Laplace transformation being Where, and are two roots of the system. Applying inverse Laplace transformation, the system response in time domain The step response has tow two terms namely constant and exponential term. The exponential terms are normally added to the constant term. Thus, swift decay of exponent terms depicts relatively slower response since the extra terms does not contribute to escalate the value. Consequently, relying on the prevailing value of α, the exponential decay will be either quicker or slower. Relatively higher value of α depicts quicker decay of the exponential thus ensuring there is slower response. Hence, for α=9, the system will normally response slower as compared to those of α=2, 7. Table 1‑4: Rise time for different alpha value Αlpha Value 9 7 2 Rise time 19.4 15.1 4.3 Thus, as the value of alpha escalates the system responses more swiftly that is verified by the corresponding lowest increase time as depicted in table 1-1. Ramp response: Taking the ramp response designated by R(s) and its corresponding Laplace transformation being Applying inverse Laplace transformation, the system response in time domain will be The step response has four terms thus subsequent to grouping, one term will be time and will increase and the other term is step response scale by.Scaled step response is then added to the prevailing linearly escalated time value . Moreover, there exist another scaled version for the entire result. Because the step response is normally scaled by it can then be assumed depicting that the underlying output is linearly escalates with the ramp input. Thus, the output is also taken as the ramp for the entire values of the alpha Steady state term, transient term and the steady state error: a.) For impulse response: The impulse of the underlying system is represented as Transient is normally the term that has time parameter. Thus, for the impulse response system has solely time parameter based on term. Thus, the entire response is the transient. Moreover, it has no steady state term Subsequent to application of the t=∞, the corresponding steady state error is computed, which is zero. a.) For Step response: The impulse of the underlying system is represented by The term having the time parameter is the transient. Thus, for the impulse response system there is solely time parameter based on the prevailing time. Thus, the transient portion is Moreover, any prevailing constant parameter is normally taken as a steady state term and exists after the time transient has disappeared. The steady state term will be Subsequent to the application of the t=∞, the final value of the system is computed thus allowing estimation of the steady state error. The final value and input value is 1. Thus, the error is zero. According to the final value theorem, the prevailing steady state value will be Because the input is 1 and corresponding final input value is 1, the steady state error is zero. b.) For Ramp response: The impulse of the system is represented by Transient is contained in the time parameter. For the impulse response system the time parameter is based time. Thus, the transient portion is Steady state time is considered at any constant parameter and exists subsequent to the disappearance of the time transient, thus, the steady state term is Subsequent to the application of the t=∞, the final value of the system is computed thus allowing the estimation of the steady state error. The final value is t and the corresponding e input is 1. Thus, the underlying error escalates within time then become infinite. With the final value theorem, the steady state value is Because the input and final value is 1, the steady state error is infinite. Relationship with damping parameter zeta Taking into the consideration the second order system to be designated by Thus, the general step response of the system when 0 Read More