An Overview of Linear Systems The content from this course was hosted on TechOnline.com from 999-4. TechOnline.com is now targeting commercial clients, so the content, (without animation and voice) is now being hosted here. Description This course provides an introduction to linear systems with a view towards modeling, simulation, filtering, and control system design. The material introduces linear, time-invariant systems that can be modeled with ordinary, constant coefficient, differential equations. The Laplace transform and transfer functions are used to simply the analysis. Bode and Nyquist plots are used to present the system frequency response. Module List: ) Modeling of continuous time invariant linear systems ) Analysis of linear systems, state space representation, numerical simulation 3) Analysis of linear systems, control system design and synthesis 4) Implementation of control systems, discretization, z-transforms. Author: Duane Mattern Background: Duane Mattern is an independent contractor specializing in modeling, simulation, control system design and implementation. He is experienced with rapid prototyping software tools like the Mathwork s Matlab/Simulink/Controls/RTW and Integrated System s Xmath/MatrixX/ SystemBuild/Autocode. As a mechanical engineer specializing in instrumentation and controls with more than years of experience, he has a broad range of practical knowledge, including automatic testing machines, turbofan engine control, integrated flight and propulsion control, servo-systems including voice coil and electromagnetic actuation, diagnostics, and neural networks. His current interests are in modeling, real-time simulation, control system design and embedded system programming for control system implementation, Prerequisites: Familiarity with the following concepts: (i) phasor notation and the fundamentals of complex variables; (ii) integration and differentiation; (iii) superposition, the Laplace transform and transfer functions; (iv) frequency response using Bode and Nyquist plots, (v) basic linear algebra. Intended Audience: () Engineer or practitioner who would like to renew their knowledge of linear systems; () Engineer or practitioners who would like a fast introduction to linear systems; (3) College student who desires an alternative presentation to linear systems, separate from what they receive in their normal courses. Estimated Total Learning Time: hours 999 DLMattern
HTML text - Module 4a introduction Module : Sampled data systems and z-transforms Purpose - To introduce sampled data systems, aliasing and z-transforms. - To introduce process of converting a continuous system to a discrete system. Objective: - Understand sampled data systems, aliasing, and z-domain transfer functions. Contents: 9 pages test questions Learning Time: minutes This module introduces sampled data systems, aliasing, discrete time systems and z-transforms as an introductory step prior to discussing the implementation of a controller in a digital system. Upon completion, you will be able to accomplish the objective listed here. Click the Forward arrow when you re ready to continue. 999 DLMattern
Content Slide : Continuous closed-loop control Position Command err unity gain feedback b s + b s + b Current Cmd r(s) e(s) s + a s u(s) Continuous Controller, K(s) Continuous controller differential equations Linear Voice Coil Actuator Plant G(s) Position mm Velocity m/s Current, amps Volts, volts y(s) Block diagram of continuous controller 3 4 x& a x e x& = x + x u = [ b ba b] b e x + [ ] e(t) + b b / s / s x x b + u(t) -a = integration with respect to time [E high light row ] In a previous module we designed a continuous time, linear controller for a linear plant comprised of a voltage controlled current source and a voice coil motor. The output from the controller is a command signal, u(t), for the desired current. The input to the controller is an error signal, e(t), which is the difference between the desired position, r(t), and measured position, y(t), of the voice coil. [E high light row ] Using the inverse Laplace transform we can convert the controller transfer function back to the time domain to obtain a set of differential equation that relate the error to the commanded current. A state space representation of this set of differential equations is shown above in control canonical form. It is possible to implement this controller in continuous time using analog components with a couple of quad op-amp IC s and the appropriate resistors and capacitors. However, we want to implement this control system in a digital device so any required modifications are made in software. We could implement the continuous version of the controller in software, but this requires online numerical integration and adds to the computational overhead. Instead, we will convert the continuous controller to a discrete representation resulting in a system of difference equations suitable for digital implementation. Before we can consider the conversion of the continuous controller to a discrete version, we need to review sampling theory and aliasing and all the components necessary to implement the discrete controller. 999 DLMattern 3
Content Slide : Discrete closed-loop control Digital System Analog to Digital T sampler Continuous System Sensor Position Command err Clock Discrete Controller K(z) Digital to Analog zero order hold Current Cmd Linear Voice Coil Actuator Plant G(s) Position mm Force, Newtons Current, amps Volts, volts 3 4 [E3] The figure shows the components necessary to implement a discrete version of our controller. Lets walk around the signal path to examine each of the components, starting with the voice coil position measurement. [E3 highlight sensor block] The optical sensor that we are using in this control system has an analog interface and provides a voltage signal that is proportional to the voice coil position in millimeters. [E3b high light sampler block] The position signal will be sampled ever T seconds, T being the sampling period. The sampler latches the signal to give the analog to digital converert time to convert the voltage signal. [E3c highlight ADC block] The voltage signal is digitized to an integer count by the analog to digital converter. If we assume - bit conversion devices over a ± volt range, we obtain 496 counts over volts, or about count per 5 millivolts of the input. [E3d highlight Discrete Controller K(z) block] This integer count is then acted upon by the software control program. The control program generates an output signal that is proportional to the current command in amps. Ever T seconds, the discrete controller presents a new current command in the form of an integer count to the digital to analog converter. [E3e highlight DAC block] The digital to analog converter then updates the controller output signal, which is held constant by the zero order hold until the next time period. There are quantization errors caused by convert from analog to digital and from digital to analog. For a -bit ADC and DAC this process only has a resolution of in 496. We will assume that the scale factors within the position sensor and the voltage controlled current source are such that -bit resolution is sufficient for this application. [E3f highlight clock block] Our main concern is with the selection of the clock sampling period T and how this selection affects the measured signal and the controller output signal. 999 DLMattern 4
Content Slide 3: Sampled data systems A 6 Hertz signal sampled at,, and 5 Hertz to show the effects of aliasing. Sampled 6 Hz Signal Magnitude of FFT of Sampled Signal - - Sampled at Hz Sampled at Hz Aliased from 6 to 4 Hz T=ms T=.83ms 5 Hz fold freq. 6 Hz fold freq. Sampled at 75 Hz 5 Hz fold freq. - T=.673ms.5. 4 6 8 Time (seconds) Signal Frequency (Hertz) [E4] The sampling theory of Nyquist and Shannon states that in order for a sampled signal to accurately represent the original signal, that the sampling frequency (one over the sampling period) must be at least twice the frequency of the highest frequency component within the signal. If a signal contains frequency components greater than one half the sampling frequency, then aliasing occurs. Aliasing introduces lower frequency components into the sampled signal that do not exist in the original signal. The three figures on the left shown a 6 Hertz analog signal as a function of time. The sampled data points, shown by linearly connected discrete points, corresponding to sampling frequencies of,, and 5 Hertz. On the right are the corresponding plots of the magnitude of the discrete Fourier transform of the sampled signal. The magnitude of the discrete Fourier transform is plotted versus frequency. According to the sampling theory, the 6 Hz signal must be sampled at least two times 6, or at Hertz to prevent aliasing. The top figure on the right shows what happens to the signal when it is not sampled fast enough. Instead of seeing a 6 Hz signal in the discrete Fourier transform, the data sampled at Hz appears as a 4 Hz signal! Any signal with frequency content above one half the sampling frequency is folded back unto the lower frequencies as a mirror image. So when sampled at Hertz, a 6 Hertz signal looks like 4 Hertz and a 9 Hertz signal would looks like a Hertz. This frequency folding effect occurs over frequency bands such that and 6 Hertz signals sampled at Hertz would appear as and 4 Hertz signals respectively. 999 DLMattern 5
Content Slide 4: Anti-aliasing filters Digital System Analog to Digital T sampler Analog Anti- Aliasing Filter Continuous System Sensor Position Command err Clock Discrete Controller K(z) Digital to Analog zero order hold Current Cmd Linear Voice Coil Actuator Plant G(s) Position mm Force, Newtons Current, amps Volts, volts 3 4 [E5 highlight analog anti-aliasing filter] If aliasing occurs, it can not be corrected in the sampled data. To avoid aliasing, an analog anti-aliasing filter can be used to attenuate signals above one half the sampling frequency. Anti-aliasing filters are typically high-order, low-pass filters and can have significant phase lag. If the anti-aliasing filter design frequency is not significantly larger than the controller bandwidth, the phase contribution of the antialiasing filter may affect the controller performance. Note that occasionally, the sensing device itself provides sufficient low pass filtering to avoid the requirement for a separate anti-aliasing filter. Let s look at a pair of anti-aliasing filters for our case study problem to see how much phase lag they contribute. 999 DLMattern 6
Content Slide 5: Sampling rate Frequency response of anti-aliasing filters phase (degs) gain (db) - - 6th order low-pass filters @ 3kHz and 6kHz phase = -3, -46 degs - -4 design bandwidth -6 3 6Hz 4 5 6 Frequency (radians/sec) Control design bandwidth = 6 Hz Phase lag of anti-aliasing filter at 6 Hz 3 Hertz -46 degrees 6 Hertz -3 degrees [E6] Consider two 6 th order filter with a pass-band of 3kHz and 6 khz. These filter attenuate the high frequency components, but both filters also contribute to the phase lag. The 3kHz filter contributes (- 46) degrees at 6 Hz, while the 6kHz filter contributes (-3) degrees of phase at 6 Hz. Recall that the phase margin of the control design was 55 degrees at 6 Hz. If the phase contribution of the antialiasing filter is significant, then a model of the filter can be lumped into the model of the original plant and a control system can be designed that includes the effect of the anti-aliasing filter. This will increase the phase lead required in the control design. If we decide that our problem requires an anti-aliasing filter and that the above 6kHz filter with (-3) degrees of phase lag at 6 Hz is acceptable, then what sampling frequency should we select? To be effective, the anti-aliasing filter should provide at least db of attenuation at one-half the sampling frequency. For the 6kHz filter, db of attenuation occurs at 85 Hz. This implies that when using the above 6kHz filter, the minimum sampling frequency is * 85 = 7 khz. Reference [] suggestions the when approximating a continuous design with a discrete representation, a sampling rate of -3 times the controller bandwidth will provide a reasonable approximation of the continuous design. A sampling rate of 7kHz for a 6 Hz bandwidth control falls within the region denoted by this rule of thumb. We will discuss the conversion of the continuous controller to a discrete representation next, but this approximation is improved with a higher sampling rate. There is a cost associated with the higher sampling rate, in the form of new hardware or in a reduction of the total number of calculations that can be performed each time period. It may be possible to achieve a lower sampling rate by performing a direct digital control design. This would require a discrete approximation of the plant, but we will not consider the topic of digital control system design here. Instead, we ll focus on the conversion of the continuous design to a discrete design. Reference: [] Feedback Control of Dynamics Systems, G.F. Franklin, J.D. Powell, A. Emami- Naeini, 3 rd edition, Addison-Wesley Publishing Co. 994, ISBN --5747-, p67. 999 DLMattern 7
Content Slide 6: Conversion from continuous to discrete us () Transfer function for a integral controller es ( ) = s Rearranging su( s) = e( s) du(t) Taking the inverse Laplace transform = et () dt Integrating both sides with respect to time ut () = etdt () Breaking off the last time period. (k-)t kt ukt ( ) = ektdt ( ) + ektdt ( ) = u(( k ) T) + e( kt) dt (k-)t t kt (k-)t [E7 high light row] Consider the transfer function of a pure integral controller with an error signal as input, e, and output u. Using the inverse Laplace transform, we can convert this system to a continuous time, differential equation where the time derivative of u(t) is equal to e(t). We can integrate this differential equation as a function of time to obtain the controller output, u(t) as a function of the integral of the error, with respect to time. [E7 high light row] Now let s consider a sampled version of this equation. We can substitute kt for the time, where T is the fixed sampling period and k is equal to the integer time index. We can break this integral up into two parts represent all previous values, and just the current time interval. How should we approximate this integral over the interval? We only have the end point data at e(kt) and e((k-)t) to work with. Let s consider trapozoidal integration. 999 DLMattern 8
Content Slide 7: Numerical integration (k-)t kt ukt ( ) = ektdt ( ) + ektdt ( ) = u(( k ) T) + e( kt) dt (k-)t Trapezoidal integration e(t) area = T*(e(kT)+e((k-)T) )/ e(kt) e((k-)t) (k-)t kt (k+)t time kt (k-)t Trapezoidal integration kt u(( k ) T) + e( kt) dt ukt ( ) = u(( k ) T) + T Introduce the z-transform uz ( ) = z uz ( ) + ( ez ( ) + z ez ( ) ) (k-)t [ ( ) + (( ) )] TekT e k T T ( ) ( ) ( ) ( z u z = + z e z) uz ( ) T z ez ( ) = + z Tustin s Transformation (bilinear) T + z z s s z T z + or z T + z [E8 highlight row ] The figure shows the discrete sampling of this error signal. If we assume that the function behaves linearly between the sampling points, we can calculate the integral for the current time step by summing the trapezoidal area under the curve. Trapezoidal integration only requires data at the individual sampling times and does not require information between time steps. Thus, we can write the integration as the sum of the integral from the previous time step and the incremental addition between the current time step. This equation is in the form of a discrete time, difference equation where (kt) denotes the current time and ((k-)t) denotes the previous sample. [E8 high light row3] For notational purposes we can introduce the z-transform and the unit time delay operator z -. The z- transform of u(k) is u(z) and the z-transform of u(k-) is z - u(z). Using this transformation, we can write the difference equation as a discrete time transfer function, parameterized as a function of z. [E8 high light row4] We started with a transfer function for an continuous time, s-domain, integral and we end up with a transfer function for a discrete time, z-domain, integral. If we equate these two transfer functions, we define the Tustin s or bilinear transformation. We can use this transformation to approximate any continuous time transfer function with a discrete time transfer function by substituting the above function of z for the Laplace variable, s. We ll discuss the z-transform a bit more in the next module. For now, let s convert the original continuous controller to a discrete approximation. 999 DLMattern 9
Content Slide 8 Continuous to discrete Continuous to discrete approximation using Tustin s transformation (bilinear) z us () bs + bs+ b s Ks ( ) = = T z + uz ( ) β z + β z + β K( z) = = es ( ) s + as ez ( ) z + α z + α Discrete transfer function coefficients Discrete transfer function β = (4b + b T + b T ) / (4 + a T) β = (b T - 8b ) / (4 + a T) uz ( ) z z β = (4b - b T + b T ) / (4 + a T) ez ( ) = β + β + β + α z + α z α = (-8) / (4 + a T) α = (4-a T) / (4 + a T) Rearrange discrete transfer function ( + α z + α z ) u( z) = ( β + β z + β z ) e( z ) Solve for the controller output, u(z) uz ( ) = ( β + β z + β z ) ez ( ) ( α z + α z ) u(z) Inverse z-transform to get to a difference equation ukt ( ) = β ekt ( ) + β e(( k ) T) + β e(( k ) T) α u( ( k) T) α u( ( k ) T) [E9 high light row ] Now we can consider the steps necessary to go from a continuous, s-domain transfer function to a discrete, z-domain transfer function. We will start with the continuous control system, K(s) and the bilinear transformation. [E9 high light row ] By substituting the function of z for all occurrences of the Laplace variable s in the transfer function, K(s), we arrive at a transfer function for K(z). This substitution process is automated in most computer aided control design software packages. The coefficients of the discrete time transfer function were calculated by hand for this simple transfer functions. Note that the coefficients of the discrete time transfer function are parameterized by the sampling period, T. [E9 high light row 3] We would like to implement this discrete transfer function, so we have to convert it back to a difference equation. We can rearrange the equation to solve for u(z) as a function of e(z) and other terms that are multiplied by powers of the time delay operator z -. By taking the inverse z-transform, we can convert this equation back to a difference equation. The updated control output, u(kt) requires the current error signal, e(kt) and the two prior error signals, e((k-)t, e((k-)t), and the two prior controller outputs, u((k-)t), u((k-)t). We can easily implement this difference equation in a digital computer. 999 DLMattern
Content Slide 9: Summary err Implemented in continuous time using differential equations e(s) b s + b s + b s + a s Continuous Controller K(s) Current Cmd u(s) z s T z + err e(z) Implemented in discrete time using difference equations β z + β z + β z + α z+α Discrete Controller K(z) Current Cmd u(z) [E] We have introduced the process of converting a continuous control system design to a difference equation suitable for implementation on a digital computer. First we explored some new components: sampler, analog to digital and digital to analog converts. Then we reviewed sampling theory and considered aliasing. We examined anti-aliasing filters and how they contribute to the phase lag of the measured signal. Finally we looked at a numerical approximation technique to approximate the continuous controller as a discrete controller. This method is called Tustin s transform or the bilinear transformation. In the next module, we ll look at some of the mathematics behind the z-transform and we ll discuss the implementation of difference equations on a digital computer to complete the implementation of this control design. 999 DLMattern
Question ) Given a signal with frequency content from to Hertz, what is the minimum sampling frequency that can be used to log this data in a digital computer? a) Hertz b) Hertz c) 3 Hertz d) 4 Hertz ) Given a signal with frequency content from to Hertz, what is the minimum sampling frequency that can be used to log this data in a digital computer? The correct answer is (d). According to sampling theory, you can accurately represent signals with frequency content up to one half the sampling frequency. Thus you need to sample twice as fast as the highest frequency of the signal you which to obtain. 999 DLMattern
Question ) What is the bilinear or Tustin transformation? z a) s = T + z b) z = e Ts c) The Laplace Transform of a sampled data system. d) A numerical technique for approximating continuous systems using digital devices. Both a and d are correct. The bilinear or Tustin transformation is: A numerical technique for approximating continuous systems using digital devices. This approximation is equal to : z s = T + z By substituting the above for s in a continuous, s-domain transfer function, we can obtain a discrete, z-domain transfer function. 999 DLMattern 3