Second Order Transfer Function Discrete Equations

Transcription

1 Second Order Transfer Function Discrete Equations J. Riggs 23 Aug 2017

2 Transfer Function Equations pg 1 1 Introduction The objective of this paper is to develop the equations for a discrete implementation of a generic second order transfer function for use in digital simulation. The general form for the second order transfer function is given by where 2 Cs is the function output Rs is the function input. is the oscillation frequency when is zero is the damping ratio. As increases from zero, oscillations dampen exponentially. When 1 oscillation does not occur. The equation represents a system with the following differential equation: This second order system has two states. Let 2 Where is the function input. State might represent position, and therefore would be the first derivative of with respect to time, which is velocity. The original differential equation can now be written in terms of two linear equations; 2 The time history of the transfer function is obtained by solving the initial value problem represented by these two simultaneous differential equations.

3 Transfer Function Equations pg 2 2 Discrete Implementation The transfer function can be represented in state space, canonical form which can be solved (approximately) using numerical techniques. Take the general form for the linear function: The companion matrix, P, represents the system dynamics and input matrix, B, represents how the inputs couple into the system. Together, they define the change in system states. Propagating these changes over time step gives the value for the system states at time. The details of P and B are obtained from the system differential equations. Using the two linear differential equations, and the definitions of the system states, above, we can determine P and B for our generic second order system; companion matrix Input matrix Euler Integration The derivative can be approximated numerically (using the forward difference method) by Using this representation for the derivative in the state space equation gives This equation can be solved for the future value of, which is : (or) The above Equation gives the formula for the value of the state vector, x, for the next sample in the time series (updated at sample time)

4 Transfer Function Equations pg 3 Applying these substitutions and performing the algebraic expansion gives the following equations for the future value of the states; Δ Δ Runge Kutta 2 nd Order Integration For increased accuracy (compared to Euler integration), the transfer function equations can also be solved using a higher order method to approximate the function derivative, such as the second order Runge Kutta method. Since this is a multi pass method, the solution approach is a little different than the previous derivation. The implementation of 2 nd order Runge Kutta integration is performed as follows; Given; initial values for the two state variables x and x function input r(t), Equations for the state derivatives, x, x (i.e. x fx,x,t, x fx,x,t Apply 2 nd order Runge-Kutta integration to propagate the value of x and x forward in time. The algorithm is as follows; 1. Compute 1 x f x,x,r x f x,x,r 2. Compute x x Δt x x x Δt x tt Δt 3. Compute x f x,x,r x f x,x,r 4. Compute x x x x x x Estimated state values at time t Δt 1 Note that the time dependency of the function is contained within the function input, r. For example, r = k*t (ramp), r = sin(omega*t) (sine wave), etc.

5 Transfer Function Equations pg 4 Using the second order transfer function equations, define f x f ω x r 2ζω x The function equations can be expanded algebraically, in accordance with the 4 step solution algorithm, above, to compute the future value for the state variables. Step 1: x x x ω x r 2ζω x Step 2: x x Δt x x x Δt ω x r 2ζω x tt Δt Step 3: x x x ω x r 2ζω x x x Δt ω x r 2ζω x x ω x Δt x r 2ζω x Δt ω x r 2ζω x Step 4: x x Δt x x x Δt x Applying substitution and some algebraic manipulation, the updated states can be expressed in terms of the previous state values. These two equations give the future value at time of the two system states based on the current system state values at time t and the function input, r, using a second order Runge Kutta approximation to propagate the state derivative. x x Δt x ω x r 2 ζω x x x Δt 2 ω 2 Δt ζ ω 1x r x r 2 2Δt ζ ω 2Δt ζω Δt 2 ω x Note that in the above equation; r is the commanded value of the function at time t, which is the time of validity for x and x. r is the commanded value of the function at time Δ.

6 Transfer Function Equations pg 5 3 Performance Characterization This section illustrates the performance of the two numerical solutions to the second order transfer function using a side by side comparison of the two sets of equations. Figure 2 illustrates the discrete implementation of the transfer function equation for the forward difference (Euler) implementation, and Figure 2 shows the implementation using the second order Runge Kutta integration technique. The key difference between these two is that the second order method requires a memory state from the previous time step. For initialization, the two function states, x 1 and x 2 are assigned initial values. The function input, r(t), is the desired (i.e. commanded) value of the x 1 state, and the function outputs are the current state (x 1 ) and its first derivative (x 2 ). For the performance evaluation and comparison, we choose to use a nominal 2 Hz period (i.e. 4) and will propagate the solution of the transfer function for a unit step response, therefore x 1 initial value = 0 (initial position), and r(t) = 1 (unit step, from 0). We also set the initial value of x 2 (initial velocity) to zero. The performance of the two numerical solutions are compared by varying the damping ratio,, and the calculation time step, Δt. Figure 1 Transfer Function Discrete Implementation Euler Integration Figure 2 Transfer Function Discrete Implementation RK2 Integration

7 Transfer Function Equations pg Critically Damped Function The first comparison is for a function with critical damping ( 1.0) and the two function implementations will be compared at multiple calculation time steps. = 2 Hz (4 ) = 1.0 t = 0.05 (20 Hz) Figure 3 Unit Step response Comparison; Omega = 2 Hz, Zeta = 1, dt =.05 = 2 Hz (4 ) = 1.0 t = (40 Hz) Figure 4 Unit Step response Comparison; Omega = 2 Hz, Zeta = 1, dt =.025

8 Transfer Function Equations pg 7 = 2 Hz (4 ) = 1.0 t = 0.01 (100 Hz) Figure 5 Unit Step response Comparison; Omega = 2 Hz, Zeta = 1, dt =.01 = 2 Hz (4 ) = 1.0 t = (200 Hz) Figure 6 Unit Step response Comparison; Omega = 2 Hz, Zeta = 1, dt =.005

9 Transfer Function Equations pg 8 The preceding sequence of four charts illustrates the effect of increasing the calculation frequency (i.e. reducing the calculation time step). The first figure, above, shows the effect of a 20 Hz calculation frequency. Since the fundamental frequency of the function represented is 2 Hz, this represents a calculation rate of only 10 samples per period. There is a significant difference in the peak velocity between the two methods. The true peak is at a value of (at T= ), indicated by the short dashed line. The Euler method over estimates the peak velocity by more than 70%, while the Runge Kutta method under estimates the peak velocity by 27.7%. Comparing the first and second figures, above, shows the effect of doubling the calculation frequency from 20 Hz o 40 Hz. Notice that the first order method shows much greater change in the peak velocity as the time step is reduced, indicating that it is less accurate at the larger time step. The Euler method is now 20% high, and the Runge Kutta method is 5% low. Reducing the time step further improves the accuracy of both calculations. Table 1 Effect of Calculation Rate on Peak Velocity Error ( = 2Hz, = 1) Calculation Peak Velocity Error Freq (Hz) t Euler RK % 27.7% % 5.0% % 0.61% % 0.14% % 0.09% % 0.035% % 0.023% 1, % % 2, % % 4, % % 5, % % 6, % % 10, % % 20, % % Table 1 shows the improvement in the calculation as the calculation rate is increased. Clearly the second order Runge Kutta function improves at a much faster rate compared to the Euler integration implementation. For example, from increasing the calculation rate from 100 Hz to 10,000 Hz (two orders of magnitude) The Euler function improves by two orders of magnitude, from 6.8% to.063%), where the RK2 function improves by four orders of magnitude, from 0.6% to %. Another metric of comparison is to observe the calculation frequency required for a particular level of accuracy, say 0.1% or better. The RK2 method provides this level of accuracy at a calculation rate of 250 Hz, where the Euler method requires a rate of 6250 Hz, (25 times higher).

10 Transfer Function Equations pg Lightly damped Function Next we look at the two functions when the damping ratio is small. This is an area where the two functions will show some of the greatest differences. It must be borne in mind that the Euler implementation is a first order approximation to the function, therefore, it will eventually break down when trying to follow a second order function. In contrast, the 2 nd order Runge Kutta implementation will have a significant advantage in reproducing the second order transfer function. This is observed in the case of very low damping. To begin, we use a damping ratio of zero, resulting in a function that is in undamped oscillation at 2 Hz. Figure 7 shows the comparison of the two function implementations. Note that even at a calculation rate of 100 Hz, the first order Euler implementation is unstable, and shows a negatively damped characteristic. But the second order Runge Kutta implementation exhibits pure oscillation. By increasing the damping slightly to 0.063, the first order function becomes stable as seen in Figure 8. Note that in this figure, the first order function exhibits no damping under these conditions, while the second order function shows positive damping, more faithfully reproducing the expected result. Thus, a small increase in the damping ratio provides an improvement in the numerical stability of the Euler integration method = 2 Hz (4 ) = 0.0 t = 0.01 (100 Hz) Vel (EU) Vel (RK2) Pos (EU) Pos (RK2) Figure 7 Unit Step response Comparison; Omega = 2 Hz, Zeta = 0, dt =.001

11 Transfer Function Equations pg = 2 Hz (4 ) = t = 0.01 (100 Hz) Vel (EU) Vel (RK2) Pos (EU) Pos (RK2) Figure 8 Unit Step response Comparison; Omega = 2 Hz, Zeta = 0.063, dt =.001 Increasing the time step for the zero damping case will eventually make the second order implementation go unstable, as is illustrated in the following two figures. = 2 Hz (4 ) = 0.0 t = 0.02 (50 Hz) Figure 9 Unit Step response RK2 Implementation; Omega = 2 Hz, Zeta = 0.0, dt =.02

12 Transfer Function Equations pg 11 = 2 Hz (4 ) = 0.0 t = 0.04 (25Hz) Figure 10 Unit Step response RK2 Implementation; Omega = 2 Hz, Zeta = 0.0, dt =.04 Figure 9 shows that the second order Runge Kutta method is slightly unstable at a calculation rate of 50 Hz, as is evident by the slightly increasing amplitude over time in the velocity curve. Comparing Figure 10 to Figure 9 shows that reducing the calculation rate from 50 Hz to 25 Hz further increases the instability. Conclusions For the zero damping case, a calculation rate of about 100 Hz is necessary when using the second order Runge Kutta integrator, but the Euler integration method requires a significantly faster calculation rate (several orders of magnitude faster) in order to maintain numerical stability. Increasing the damping in the transfer function helps improve the stability of the Euler method tremendously.

13 Transfer Function Equations pg 12 4 Applying a Rate Limit It is herein assumed that the transfer function state x 1 represents a position and the user desires to apply a rate limit to the function output. The limit process is applied as follows; let be the rate limit. Compute an incremental limit for x 1 The position state, x, is expressed in terms of the previous value plus an increment; A limit of ± is placed on x x x x A limit of ± is placed on the value of x 4.1 Euler Method with Rate Limit Δ Δ 2 x Δ x,, nd Order Runge Kutta Method with Rate Limit x x Δt x ω x r 2 ζω x x x Δt 2 ω 2 Δt ζ ω 1x r x r 2Δt ζ ω 2Δt ζω Δt 2 ω x x Δt x ω x r 2 ζω x x,,

14 Transfer Function Equations pg 13 5 Observations and Conclusions Two numerical implementations of the generic second order transfer function have been developed and compared, one using a first order forward difference (Euler) method to propagate the state derivatives, and the second using a second order Runge Kutta algorithm. The RK2 based algorithm provides significant improvement in accuracy compared to the forward difference Euler method. Both methods break down in their ability to accurately represent the function under certain conditions, which depend on the calculation rate, the function natural frequency and damping ratio. Break down occurs at very small, and very large values of the damping ratio. Both numerical methods become unstable if the calculation time step is too large. The allowable time step magnitude is a function of the frequency and damping. For a damping ratio of 1, the largest allowable time step is approximately 1/(7 x ) ( in Hz), i.e. Δ 1 7 One interpretation of this is that we require at least 7 samples per cycle to maintain numeric stability in the calculation (with a damping ratio of 1). Reducing the damping ratio to less than 1 will require a smaller time step, with the Euler method requiring an even greater reduction in the time step compared to the Runge Kutta method. The Euler method is completely unsuitable in order to represent a function with zero damping. Because both methods have limitations, users should explore the function behavior under the conditions of their intended application in order to verify that the method will provide accurate results, especially if very high frequencies or low damping is desired.