Objectives of the lecture Chapter 5: Time response analysis 5.4 Steady state errors and error constants Dr. Soumita Ghosh Steady state errors Steady state error – an important aspect of system performance. It is a measure of the system accuracy. They arise from the nature of inputs, type of system and from nonlinearities of system components such as static friction, backlash and more. Aggravated by aging, wear and tear. Steady state performance of a stable control system is judged by its steady state error to step, ramp and parabolic inputs. Steady state errors The deviation of the output of control system from desired response during steady state is known as steady state error. E(s) is the Laplace transform of the error signal, e(t) Steady State Errors for Unity Feedback Systems Consider the following block diagram of closed loop control system, which is having unity negative feedback. R(s) is the Laplace transform of the reference Input signal r(t) C(s) is the Laplace transform of the output signal c(t) Steady State Errors for Unity Feedback Systems The output of the summing point is – Substitute C(s) value in the above equation. Steady State Errors for Unity Feedback Systems Substitute E(s) value in the steady state error formula Steady State Errors for Unity Feedback Systems The following table shows the steady state errors and the error constants for standard input signals like unit step, unit ramp and unit parabolic signals. Steady State Errors for Unity Feedback Systems Where, Kp, Kv and Ka are position error constant, velocity error constant and acceleration error constant respectively. Note − If any of the above input signals has the amplitude other than unity, then we should multiply corresponding steady state error with that amplitude. Note − We can’t define the steady state error for the unit impulse signal because, it exists only at origin. So, we can’t compare the impulse response with the unit impulse input as t denotes infinity. Example Let us find the steady state error for an input signal of unity negative feedback control system with Example The given input signal is a combination of three signals step, ramp and parabolic. The following table shows the error constants and steady state error values for these three signals. Example Steady State Errors for Non-Unity Feedback Systems Consider the following block diagram of closed loop control system, which is having nonunity negative feedback. Steady State Errors for Non-Unity Feedback Systems We can find the steady state errors only for the unity feedback systems. We have to convert the non-unity feedback system into unity feedback system. For this, include one unity positive feedback path and one unity negative feedback path in the above block diagram. The new block diagram looks like as shown below. Steady State Errors for Non-Unity Feedback Systems Simplify the block diagram by keeping the unity negative feedback as it is. This block diagram resembles the block diagram of the unity negative feedback closed loop control system. We can now calculate the steady state errors by using steady state error formula given for the unity negative feedback systems. Steady State Errors for Non-Unity Feedback Systems It is meaningless to find the steady state errors for unstable closed loop systems. So, we have to calculate the steady state errors only for closed loop stable systems. This means we need to check whether the control system is stable or not before finding the steady state errors. Objectives of the lecture Analysis of control systems for stability Dr. Soumita Ghosh Concept of Stability A system is said to be stable if it does not exhibit large changes in its output for a small change in its input, initial conditions or its system parameters. In a stable system, the output is predictable and finite for a given input. https://electronics-club.com/concept-of-stability-control-system/ Concept of Stability Stable System: Maintains its equilibrium position (original position) when a small disturbance is applied to it. ◦ When the ball is disturbed, the equilibrium position will not change. ◦ Hence, the system is called asymptotically stable. Unstable System: Attains a new equilibrium position that does not resemble the original equilibrium position when a small disturbance is applied to it. ◦ If the ball is slightly disturbed, it will upset the equilibrium condition and the ball will roll down from the hill and it will never return back to the equilibrium state. ◦ Hence, the system is called unstable. Concept of Stability Marginally Stable System: Attains a new equilibrium position that resembles the original equilibrium position when a small disturbance is applied to it. ◦ When an impulsive force is applied, the ball will move for a fixed distance and then stop due to friction. ◦ Although the position is changed, the ball will come to a new equilibrium position. ◦ Hence, the system is called marginally or non-asymptotically stable. Oscillatory System: ◦ When we apply a disturbing force, the ball will move in both directions (oscillatory motion). ◦ Eventually, these oscillations will cease due to the presence of friction and the ball will return back to its stable equilibrium state. Transfer function The transfer function of a single-input, single-output system can be written as The denominator of the transfer function is called characteristic polynomial and its roots are the poles of the transfer function. The characteristic equation is formed by equating the characteristic polynomial to zero. Thus, for the above case, the characteristic equation is S-plane In mathematics and engineering, the s-plane is the complex plane on which Laplace transforms are graphed. Transfer function The stability of the closed-loop system can be determined by examining the poles of the closed-loop system, that is, by the roots of the characteristic equation. The nature of the time response of a system is related to the location of the roots of the characteristic equation in s-plane. Transfer function A system will be stable, unstable, or oscillatory depending upon the positions of the roots of the characteristic equation. Stability of a system The necessary conditions for stability are as follows. The coefficients of characteristic equation should be necessarily positive. This is the condition for stability of the systems of the first and second order. For third and higher-order systems, positive coefficients of the characteristic equation do not ensure the negative real parts of complex roots. Therefore, it is a necessary but not sufficient condition for the systems of a third and higher-order. If all the coefficients are positive, the possibility of stability of the system exists. Methods for Determining the Stability of the System When the characteristic equation of the system has known parameters, it is very easy to determine the stability of the system by finding the roots of characteristic equation. But when there are some unknown parameters in the equation, the stability of the system can be determined by using the methods given here. Routh–Hurwitz Criterion: Routh–Hurwitz criterion was developed independently by E. J. Routh (1892) in the United States of America and A. Hurwitz (1895) in Germany, which is useful to determine the stability of the system without solving the characteristic equation. This is an algebraic method that provides stability information of the system that has the characteristic equation with constant coefficients. It indicates the number of roots of the characteristic equation, which lie on the imaginary axis and in the left half and right half of the s-plane without solving it. Methods for Determining the Stability of the System Root Locus Technique: The technique which was first introduced by W. R. Evans in 1946 for determining the trajectories of the roots of the characteristic equation is known as the root-locus technique. Root locus technique is also used for analyzing the stability of the system with a variation of gain of the system. Bode Plot: This is a plot of the magnitude of the transfer function in decibel versus ω, and the phase angle versus ω , as ω is varied from zero to infinity. By observing the behaviour of the plots, the stability of the system is determined. Methods for Determining the Stability of the System Polar Plot: It provides the magnitude and phase relationship between the input and output response of the system, i.e., a plot of magnitude versus phase angle in the polar coordinates. The magnitude and phase angle of the output response is obtained from the steadystate response of a system on a complex plane when it is subjected to sinusoidal input. Nyquist Criterion: This is a semi-graphical method that gives the difference between the number of poles and zeros of the closed-loop transfer function of the system, which are in the right half of the s-plane. Objectives of the lecture Stability analysis of control systems Dr. Soumita Ghosh Stability Stable System: If all the roots of the characteristic equation lie on the left half of the ‘s' plane then the system is said to be a stable system. Marginally Stable System: If all the roots of the system lie on the imaginary axis of the ‘s' plane then the system is said to be marginally stable. Unstable System: If all the roots of the system lie on the right half of the ‘s' plane then the system is said to be an unstable system. Routh Hurwitz criterion Routh-Hurwitz stability criterion is having one necessary condition, and one sufficient condition for stability. If any control system doesn’t satisfy the necessary condition, then we can say that the control system is unstable. However if the control system satisfies the necessary condition, then it may or may not be stable. Therefore the sufficient condition is helpful for knowing whether the control system is stable or not. Necessary Condition for Routh-Hurwitz Stability The necessary condition is that the coefficients of the characteristic polynomial should be positive. • This implies that all the roots of the characteristic equation should have negative real parts. Consider a system with characteristic equation: There should be no missing term. • This means that the nth order characteristic equation should not have any coefficient that is of zero value. Sufficient Condition for Routh-Hurwitz Stability The sufficient condition is that all the elements of the first column of the Routh array should have the same sign. This means that all the elements of the first column of the Routh array should be either positive or negative. Advantages of Routh- Hurwitz Criterion We can find the stability of the system without solving the equation. We can easily determine the relative stability of the system. By this method, we can determine the range of ‘K’ for stability. Limitations of Routh- Hurwitz Criterion This criterion is applicable only for a linear system. It does not provide the exact location of poles on the right and left half of the s plane. In case of the characteristic equation, it is valid only for real coefficients. Routh Array Method We have to find the roots of the characteristic equation to know whether the control system is stable or unstable. • But, it is difficult to find the roots of the characteristic equation as order increases. Routh array method • In this method, there is no need to calculate the roots of the characteristic equation. • First we formulate the Routh table, and find the number of the sign changes in the first column of the Routh table. • The number of sign changes in the first column of the Routh table gives the number of roots of characteristic equation that exist in the right half of the ‘s’ plane and the control system is unstable. Routh Array Method The procedure for forming the Routh table. • Fill the first two rows of the Routh array with the coefficients of the characteristic polynomial. • Start with the coefficient of 𝑠 𝑛 and continue up to the coefficient of 𝑠 0 . • Fill the remaining rows of the Routh array. • Continue this process till you get the first column element of row 𝑠 0 is 𝑎𝑛 . • Here, 𝑎𝑛 is the coefficient of 𝑠 0 in the characteristic polynomial. Note − If any row elements of the Routh table have some common factor, then we can divide the row elements with that factor for the simplification will be easy. Routh Array Method The following table shows the Routh array of the nth order characteristic polynomial. 𝑠𝑛 𝑎0 𝑠 𝑛−1 𝑎2 𝑎1 𝑠 𝑛−2 𝑎3 𝑠 𝑛−3 ⋮ 𝑠1 𝑠0 𝑎4 𝑎5 ⋮ ⋮ ⋮ an ⋮ ⋮ ⋮ 𝑎6 ... 𝑎7 ... ... ... Example Let us find the stability of the control system having characteristic equation, Step 1 − Verify the necessary condition for the Routh-Hurwitz stability. All the coefficients of the characteristic polynomial are positive. So, the control system satisfies the necessary condition. Example Step 2 − Form the Routh array for the given characteristic polynomial. Example Step 3 − Verify the sufficient condition for the Routh-Hurwitz stability. All the elements of the first column of the Routh array are positive. There is no sign change in the first column of the Routh array. So, the control system is stable. https://www.tutorialspoint.com/control_systems/control_systems_stability_analysis.htm Routh-Hurwitz Stability Criterion The two special cases are − The first element of any row of the Routh array is zero. All the elements of any row of the Routh array are zero.
