Digital Control Systems Part 1 Fundamentals and sampling process Z-transform and inverse z-transform Analysis Methods Design Techniques Dr. Kalyana Veluvolu

WEEK 2: FUNDAMENTAL AND SAMPLING PROCESS Main topics: Basic concepts Linear time-invariant systems Sampling and signal reconstruction Z-transform Dr. Kalyana Veluvolu

BASIC CONCEPTS Signal Form Classification: Continuous-time signals Discrete-time signals Dr. Kalyana Veluvolu

Dr. Kalyana Veluvolu

Sampler and Holder: Dr. Kalyana Veluvolu

Ideally sampled signal: Dr. Kalyana Veluvolu

The ideal sampler represents the unit impulse train of (c). In discrete-time control systems, digital computers are used to perform the control logic and algorithm computations. A data-hold device and the simplified representation of the sampling and hold devices are shown in (a) and (b). The ideal sampler represents the unit impulse train of (c). The output of the sampler is the amplitude-varying impulse train 𝑒 ∗ 𝑡 𝑜𝑓 𝑑 . The utilization of the hold device simplifies the mathematical analysis of the sampled signal. This in turn makes the analysis of the sampled-data control system model easier. The type of continuous-time systems where the controller contains sampler and holders of (a) is called sampled-data control systems. Dr. Kalyana Veluvolu

Four types of signal characteristics: Discrete amplitude-discrete time (D-D) Discrete amplitude-continuous time (D-C) Continuous amplitude-discrete time (C-D) Continuous amplitude-continuous time (C-C) Dr. Kalyana Veluvolu

Dr. Kalyana Veluvolu

Advantages of digital control systems: Permits the use of sensitive control elements with relatively low-energy signals. Digital transducer has the relative immunity of its digital signals to distortion by noise and nonlinearities and its relatively high accuracy and resolution as compared to analog transducers. Digital manipulation enables design and development of complex and sophisticated control systems because the ability of a digital control system to store discrete information for long time intervals, to process complicated algorithms, and to transmit them with high accuracy. Dr. Kalyana Veluvolu

Advantages of digital control systems: Requires only one communication channel to transmit discrete control signals for more than one control system(multiplexing). The digital processing is accomplished by small digital systems. These computers can perform sophisticated algorithms such as multi-dimensional state-variable manipulations, Kalman filtering, stochastic system control, and adaptive mode control. The utilization of the computer as controller has provided flexibility and versatility in the design of the control systems Dr. Kalyana Veluvolu

Disadvantages of digital control systems: System design. The mathematical analysis and design of a sampled-data control system is sometimes more complex and more tedious as compared to continuous- data control systems System stability. Converting a given continuous-data control system into a sampled-data system not changing any system parameter, the addition of the required hold device, however, degrades the system stability margin. Signal information. The hold device is to reconstruct the continuous signal e(t) from the discrete signal 𝑒 ∗ 𝑡 . The best is the reconstructed signal m(t) that approximates e(t). Thus there is a loss of signal information. Dr. Kalyana Veluvolu

Disadvantages of digital control systems (more): Software errors. The complexity of the control process is in the software – implemented algorithms which may contain errors; the software is not correct. Controller dynamic update. Because the A/D, D/A, and digital computer in reality delay the signal, performance objectives may be difficult to achieve since theoretically it assumes no delay Dr. Kalyana Veluvolu

LINEAR TIME-INVARIANT SYSTEMS Linear systems: if and only if superposition theorem holds. For either a continuous – or a discrete-time systems to be linear, the relationship 𝑐=𝑓 𝑎 1 𝑟 1 + 𝑎 2 𝑟 2 = 𝑎 1 𝑓 𝑟 1 + 𝑎 2 𝑓[ 𝑟 2 ] Where 𝑎 1 and 𝑎 2 are constant real number and 𝑟 1 and 𝑟 2 are any two input vectors, must be satisfied. Otherwise, the system is said to be nonlinear. Dr. Kalyana Veluvolu

Where t is the sampling time of r(t) and k = 0,1,2 Example An approximation for numerical differentiation is given by the discrete-time equation 𝑐 𝑘𝑇 = 1 𝑇 𝑟 𝑘𝑇 −𝑟[ 𝑘−1 𝑇] Where t is the sampling time of r(t) and k = 0,1,2 𝑐 1 (𝑘𝑇)=1/𝑇{ 𝑟 1 (𝑘𝑇)− 𝑟 1 [(𝑘−1)𝑇]} 𝑐 2 (𝑘𝑇)=1/𝑇{ 𝑟 2 (𝑘𝑇)− 𝑟 2 [(𝑘−1)𝑇]} Dr. Kalyana Veluvolu

Take 𝑟 𝑘𝑇 = 𝑎 1 𝑟 1 kT + 𝑎 2 𝑟 2 kT 𝑐 𝑘𝑇 = 𝑎 1 𝑇 𝑟 1 𝑘𝑇 − 𝑟 1 [ 𝑘−1 𝑇] + 𝑎 2 𝑇 𝑟 2 𝑘𝑇 − 𝑟 2 [ 𝑘−1 𝑇] For 𝑐 𝑘𝑇 = 𝑎 1 𝑐 1 𝑘𝑇 + 𝑎 2 𝑐 2 𝑘𝑇 As both are identical, it is linear. Dr. Kalyana Veluvolu

Example 1.2: Determine if 𝑐 𝑘𝑇 = 𝑟 2 𝑘𝑇 is a linear equation. Take 𝑐 1 𝑘𝑇 = 𝑟 1 2 (𝑘𝑇) and 𝑐 2 𝑘𝑇 = 𝑟 2 2 (𝑘𝑇) For 𝑟 𝑘𝑇 = 𝑎 1 𝑟 1 (𝑘𝑇) + 𝑎 2 𝑟 2 (𝑘𝑇), 𝑐 𝑘𝑇 = 𝑎 1 2 𝑟 1 2 𝑘𝑇 +2 𝑎 1 𝑎 2 𝑟 1 𝑘𝑇 𝑟 2 𝑘,𝑇 + 𝑎 2 2 𝑟 2 2 𝑘𝑇 If the given equation were linear, we would have 𝑐 𝑘𝑇 = 𝑎 1 𝑐 1 𝑘𝑇 + 𝑎 2 𝑐 2 𝑘𝑇 = 𝑎 1 𝑟 1 2 𝑘𝑇 + 𝑎 2 𝑟 2 2 𝑘𝑇 As the two are not identical, the given equation is nonlinear. Dr. Kalyana Veluvolu

Time-invariant systems: The coefficients of the differential or difference equation do not depend on time Dr. Kalyana Veluvolu

Time-invariance concept: With a time delay of 𝜏 seconds, if the output quantities of path 1 are equal to the corresponding outputs of path2, the system is said to be time-invariant, i.e., 𝑐 𝑡−𝑟 =𝑓[𝑟(𝑡−𝑟)] for a continuous-time systems 𝑐 𝑘−𝑝 𝑇 =𝑓{𝑟[ 𝑘−𝑝 ]} for a discrete-time systems Dr. Kalyana Veluvolu

Example 1.3: Determine if the system is time-invariant. 𝑐 ′ 𝑡 +𝑐 𝑡 =𝑡𝑟 𝑡 For path 1: 𝑐 1 ′ 𝑡 + 𝑐 1 𝑡 =𝑡𝑟 → 𝑐 1 ′ 𝑡−𝜏 + 𝑐 1 (𝑡−𝜏) = 𝑡−𝜏 𝑟 𝑡−𝜏 =𝑡𝑟 𝑡−𝜏 −𝜏𝑟 𝑡−𝜏 For path 2: 𝑐 2 ′ 𝑡−𝜏 + 𝑐 2 𝑡−𝜏 =𝑡𝑟 𝑡−𝜏 =𝑡𝑟 𝑡−𝜏 −𝜏𝑟 𝑡−𝜏 +𝜏𝑟 𝑡−𝜏 As 𝑐 1 𝑡−𝜏 ≠ 𝑐 2 𝑡−𝜏 , the system is not time-invariant Dr. Kalyana Veluvolu

Example: The discrete-time system 𝑐 𝑘𝑇 =𝑘𝑇𝑟(𝑘𝑇) For path 1: 𝑐 1 𝑘𝑇 =𝑘𝑇𝑟 𝑘𝑇 𝑐 1 𝑘−𝑝 𝑇 = 𝑘−𝑝 𝑇 𝑟 𝑘−𝑝 𝑇 For path 2: 𝑐 2 𝑘−𝑝 𝑇 =𝑘𝑇𝑟 𝑘−𝑝 𝑇 = 𝑘−𝑝 𝑇 𝑟 𝑘−𝑝 𝑇 +𝑝𝑇𝑟 𝑘−𝑝 𝑇 As the two are not identical, 𝑐 1 𝑘−𝑝 𝑇 ≠ 𝑐 2 𝑘−𝑝 𝑇 the system is not time-invariant. Dr. Kalyana Veluvolu

NUMERICAL APPROXIMATION Given the set of discrete values 𝑐 𝑁𝑇 , 𝑐 𝑁−1 𝑇 ,…,𝑐 0 , where 𝑁= 1,2,3,.., one defines the 𝑓𝑖𝑟𝑠𝑡−𝑏𝑎𝑐𝑘𝑤𝑎𝑟𝑑 Difference 𝛻𝑐 𝑁𝑇 =𝑐 𝑁𝑇 −𝑐 𝑁−1 𝑇 This permits its solution on a digital computer. Dr. Kalyana Veluvolu

𝑐 𝑘+1 𝑇 −𝑐 𝑘𝑇 = 𝑒 𝑘+1 𝑇 +𝑒(𝑘𝑇) 2 𝑇 Integral approximation of 𝑐 𝑡 = 0 𝑡 𝑒 𝑓 𝑑𝑓 First forward difference (Euler’s Technique) 𝑐 𝑘+1 𝑇 −𝑐 𝑘𝑇 =𝑇𝑒 𝑘𝑇 Trapezoidal: 𝑐 𝑘+1 𝑇 −𝑐 𝑘𝑇 = 𝑒 𝑘+1 𝑇 +𝑒(𝑘𝑇) 2 𝑇 Dr. Kalyana Veluvolu

Example: Evaluate 𝑐 𝑡 = 0 𝑡 𝑒 𝑓 𝑑𝑓 by a numerical integration technique 𝑐 𝑁𝑇 = 0 𝑁𝑇 𝑒 𝑓 𝑑𝑓 ≈ 𝑘=0 𝑁−1 𝑒 𝑘𝑇 ∆𝑡= 𝑘=0 𝑁−1 𝑇𝑒(𝑘𝑇) Dr. Kalyana Veluvolu

Notice that due to finite word length, Decreasing T increases N and hence the computation time required and increases the roundoff and truncation errors by use of recursive equation. If it is not small enough, aliasing or frequency folding problems will exist. Thus the selection of T is based on trade-off when all factors are considered Dr. Kalyana Veluvolu