DOCSLIB.ORG

Stability of Digital Control Systems Outline Asymptotic Stability

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Stability of Digital Control Systems Outline Asymptotic Stability Outline • Asymptotic stability. Stability of Digital • Input-output stability. Control Systems • Stability conditions. • Routh-Hurwitz criterion. M. Sami Fadali •Jury test. Professor of Electrical Engineering UNR 1 2 Bounded-Input-Bounded-Output Asymptotic Stability (BIBO) Stability Response due to any initial conditions Response due to any bounded input decays to zero asymptotically in the steady remains bounded. state, i.e. the response due to the initial conditions satisfies u()k < bu ,k = 012 ,,,... Limy(k) = 0 0 <<∞bu k→∞ Marginal Stability: response due to any y(k )< by ,k = 012 , , ,... initial conditions remains bounded but does not decay to zero. 0 <<∞by 3 4 1 Stable Z-Domain Pole Time Sequence Locations n n z f (k) = A pk , k = 0,1,2,... ←⎯→Z F(z) = A ∑ i i ∑ i z − p Sampled exponential and its z-transform i=1 i=1 i with p real or complex ¾ Bounded sequence for poles in the closed unit z disc (i.e. on or inside the unit circle). pk , k = 0,1,2,... ←⎯→Z⎯ z − p ¾ Sequence decays exponentially for poles in the open unit disc (i.e. inside the unit circle). ⎧0, p <1 ¾ Unbounded sequence for repeated poles on the k ⎪ unit circle. Limp → ⎨1, p =1 k→∞ ¾ For real time sequences poles and partial ⎪ ⎩∞, p >1 fraction coefficients are either real or complex conjugate pairs. 5 6 Proof y(k + n) + a y(k + n −1) + ...+ a y(k +1) + a y(k) Theorem: Asymptotic Stability n−1 1 0 = bmu(k + m) + bmu(k + m −1) +...+ b1u(k +1) + b0u(k) k = 0,1,2,... In the absence of pole-zero cancellation, a Response of the system due to ICs y(0), y(1), …, y(n) LTI digital system is asymptotically stable N ()z Yz()= if its transfer function poles are in the open zazn ++++n −1 ... aza unit disc and marginally stable if the poles n −1 10 are in the closed unit disc with no repeated 1. Denominator of the output z-transform = denominator of the z-transfer function for no pole-zero cancellation. poles on the unit circle. 2. Poles of Y(z) = poles of transfer function. 3. Y(z) due to initial conditions is bounded for system poles in the closed unit disc unit (no repeated poles on the unit circle) & decays exponentially for system poles 7 in the open unit disc (inside the unit circle). 8 2 Example Solution Determine the asymptotic stability of the • Use Theorem 1 can (a) and (b) without pole- following systems: zero cancellation. • Ignore zeros, (do not affect response due ICs) 4()z − 2 4()z − 0.2 a) H (z) = b) H (z) = a) Pole outside the unit circle ⇒ unstable ()()z − 2 z − 0.1 ()()z − 0.2 z − 0.1 b) All poles inside the unit circle ⇒ asymptotically stable. 5()z − 0.3 8()z − 0.2 c) H (z) = d) H (z) = ()()z − 0.2 z − 0.1 ()()z − 0.1 z −1 c) All poles inside the unit circle ⇒ asymptotically stable. d) One pole on the unit circle ⇒ marginally stable. 9 10 BIBO Stability Theorem 4.2:BIBO Stability k yk()=−∑ hk ( iui )(), k =012 ,,,... i = 0 A discrete-time linear system is BIBO Is system BIBO stable if its impulse stable if and only if its impulse response response h(k) is bounded? NO sequence is absolutely summable i.e. Bounded & strictly positive impulse response ∞ 0 < b < h(k) < b < ∞, k = 0,1,2,... h1 h2 hi() <∞ k k ∑ y(k) = ∑ h(k − i)u(i) > bh1∑u(i), k = 0,1,2,... i = 0 i=0 i=0 u(i) = 1,i = 0,1,L ⇒ unbounded output 11 12 3 Proof of Necessity (Only if) Proof of Sufficiency (If) • Assume the system is BIBO stable but the • Assume an absolutely summable impulse impulse response is not absolutely summable. response and show that the system is BIBO stable. • Unbounded output with the bounded input (contradiction) • Use the input bound bu in the convolution summation ⎧10,()hi ≥ k uk()−= i ⎨ ⎩−<10,()hi y(k) ≤ ∑ h(i) u(k − i) i=0 k k yk()=→∞→∞ hi (), yk () as k ∑ < bu ∑ h(i) < ∞, ∀k ≥ 0 i=0 i=0 13 14 Theorem 4.3: BIBO Stability Proof of Necessity n n z A discrete-time linear system is BIBO h(k) = A pk , k = 0,1,2,... ←⎯→Z⎯ H (z) = A ∑ i i ∑ i z − p stable if and only if the poles of its transfer i=1 i=1 i function lie inside the unit circle. • Impulse response is bounded if the poles of the transfer function are in the closed unit disc and decays exponentially if the poles are in the open unit disc. • Systems with a bounded impulse response that does not decay exponentially are not BIBO stable. 15 16 4 Proof of Sufficiency Example 4.2 Assume exponentially decaying impulse Investigate the BIBO stability of systems response (i.e. poles inside the unit circle). with the impulse response Ar is the coefficient of largest magnitude |ps | < 1 is the largest pole magnitude. ⎧ Kkm, 0 ≤≤< ∞ hk()= ⎨ The impulse response is bounded by ⎩ 0, elsewhere n n k k k hk( )=≤∑∑ Apii Aii p ≤ nArs p , k =012 , , ,... i ==11i where K is a finite constant. ∞ ∞ i 1 ∑∑hi() ≤= nArs p n Ar <∞ i = 00i = 1− ps 17 18 Solution Example 4.3 BIBO stable since the impulse response Investigate BIBO stability for Example 4.1 satisfies ∞ m hi()==+<∞ hi () ( m1) K 4(z − 2) 4()z − 0.2 ∑∑ a) H (z) = b) H (z) = ii= 00= ()()z − 2 z − 0.1 ()()z − 0.2 z − 0.1 • Let K= upper bound for any impulse 5(z − 0.3) 8()z − 0.2 response of finite duration. c) H (z) = d) H (z) = ()()z − 0.2 z − 0.1 ()()z − 0.1 z −1 • Any FIR system is BIBO stable. 19 20 5 Solution Z-plane Stable Pole Locations After pole-zero cancellation Im[z] a) BIBO stable, all poles inside unit circle. Unit Circle b) BIBO stable, all poles inside unit circle. c) BIBO stable, all poles inside unit circle. STABLE d) Not BIBO stable, a pole on unit circle. Re[z] 21 22 MATLAB Stability Determination MATLAB: ddamp Obtain roots of polynomial: Gives the pole locations, ζ and ωn » roots(den) % denominator coeffts. den Eigenvalue Magnitude Equiv. Damping Equiv. Freq. (rad/sec) 0.2306 + 0.7428I 0.7778 0.1941 12.9441 » zpk(g) %g = transfer function 0.2306 - 0.7428I 0.7778 0.1941 12.9441 -0.6612 0.6612 0.1306 31.6871 Stable for roots inside the unit circle. Closed-loop transfer function >> H = feedback(gforward, gfeedback, ±1) 23 24 6 Routh-Hurwitz Criterion Advantages/Disadvantages 1. Transform the inside of the unit circle to • Easy stability test for low-order polynomials. the LHP (bilinear transformation). • Difficult for high order z-polynomials. 2. Use the Routh-Hurwitz criterion for the • For high order polynomials, use symbolic investigation of discrete-time system manipulation. stability. n n−1 Fz( )=+ azn an−1 z ++ ... a0 1 + w z − 1 z w 1+w n n−1 = ⇔= z= ⎛1+ w⎞ ⎛1+ w⎞ 1 − w z + 1 ⎯→⎯1⎯−w a ⎜ ⎟ + a ⎜ ⎟ ++... a n ⎝1− w⎠ n−1 ⎝1− w⎠ 0 25 26 Example 4.4 Solution: 1st order Find stability conditions for • Solve for the root. a) The first order polynomial • Stability conditions a1 z + a0 , a1 > 0 a 0 < 1 b) The second order polynomial a1 2 az2 ++ az10 a, a 2 >0 27 28 7 nd Bilinear Transformation Solution: 2 order 2 ⎛1+ w⎞ ⎛1+ w⎞ a2 ⎜ ⎟ + a10⎜ ⎟ + a 2 ⎝1− w⎠ ⎝1− w⎠ −±aaaa11 −4 02 z12, = 2 a2 2 (aaaw210−+) +2( aawaaa20 −) +( 210 ++) ¾ Stability determination by solving for roots is difficult. Routh-Hurwitz criterion: poles of 2nd order w-polynomial ¾ Monic polynomial remain in the LHP iff its coefficients are all positive. aaa− +>0 ¾ constant term = product of poles 210 ¾ For pole magnitudes < 1 aa20−>0 ¾ Necessary stability condition ¾ Sufficient for complex conjugate poles ) aaa210++ >0 a Adding the first and third conditions gives the condition 0 < 1 ⇔ −a < a & a < a obtained earlier 0 2 0 2 aa+ > 0 a2 29 20 30 Stable Parameter Range for 2nd Comments Order z-polynomial • Recall: The condition is sufficient for a0 a0=1 complex conjugate roots and only 1 necessary for real roots. stable a0= a1−1 a0= −a1−1 • For real roots, if the three conditions are a1 −1 1 substituted in the expression (4.14) we a2 =1 obtain roots between −1 and +1. −1 1− a1 + a0 > 0 1− a0 > 0 1+ a1 + a0 > 0 31 32 8 Jury Test Jury Table Row z0 z1 z2 … zn − k … zn − 1 zn 1 a0 a1 a2 … an − k … an − 1 an The roots of the polynomial are inside the unit 2 an an − 1 an − 2 … ak … a1 a0 3 b b b … b … b circle iff 0 1 2 n − k n − 1 4 bn − 1 bn − 2 bn − 3 … bk … b0 Fz()=+ azn a zn −1 +++= az a00 , a > n n −1 L 10 n 5 c0 c1 c2 … … cn − 2 6 c c c … … c (1) F(1) > 0 n − 2 n − 3 n − 4 0 n . … … (2) (−1) F(−1) > 0 . … … . … … (3) | a0 | < an 2 n − 5 s s s s (4) | b0 | > | bn − 1 | 0 1 2 3 2 n − 4 s3 s2 s1 s0 (5) | c0 | > | cn − 2 | 2 n − 3 r r r M 0 1 2 (n+1) | r | > | r | 0 2 33 34 Table Entries Remarks 1.
Load more

Recommended publications
  • Control Theory
    Control theory S. Simrock DESY, Hamburg, Germany Abstract In engineering and mathematics, control theory deals with the behaviour of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain ref- erence over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. Rapid advances in digital system technology have radically altered the control design options. It has become routinely practicable to design very complicated digital controllers and to carry out the extensive calculations required for their design. These advances in im- plementation and design capability can be obtained at low cost because of the widespread availability of inexpensive and powerful digital processing plat- forms and high-speed analog IO devices. 1 Introduction The emphasis of this tutorial on control theory is on the design of digital controls to achieve good dy- namic response and small errors while using signals that are sampled in time and quantized in amplitude. Both transform (classical control) and state-space (modern control) methods are described and applied to illustrative examples. The transform methods emphasized are the root-locus method of Evans and fre- quency response. The state-space methods developed are the technique of pole assignment augmented by an estimator (observer) and optimal quadratic-loss control. The optimal control problems use the steady-state constant gain solution. Other topics covered are system identification and non-linear control. System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data.
    [Show full text]
  • EECS 451 LECTURE NOTES BASIC SYSTEM PROPERTIES What: Input X[N] → |SYSTEM| → Y[N] Output
    EECS 451 LECTURE NOTES BASIC SYSTEM PROPERTIES What: Input x[n] → |SYSTEM| → y[n] output. Why: Design the system to filter input x[n]. DEF: A system is LINEAR if these two properties hold: 1. Scaling: If x[n] → |SYS| → y[n], then ax[n] → |SYS| → ay[n] for: any constant a. NOT true if a varies with time (i.e., a[n]). 2. Superposition: If x1[n] → |SYS| → y1[n] and x2[n] → |SYS| → y2[n], Then: (ax1[n]+ bx2[n]) → |SYS| → (ay1[n]+ by2[n]) for: any constants a, b. NOT true if a or b vary with time (i.e., a[n], b[n]). EX: y[n]=3x[n−2]; y[n]= x[n+1]−nx[n]+2x[n−1]; y[n] = sin(n)x[n]. NOT: y[n]= x2[n]; y[n] = sin(x[n]); y[n]= |x[n]|; y[n]= x[n]/x[n − 1]. NOT: y[n]= x[n] + 1 (try it). This is called an affine system. HOW: If any nonlinear function of x[n], not linear. Nonlinear of just n OK. DEF: A system is TIME-INVARIANT if this property holds: If x[n] → |SYS| → y[n], then x[n − N] → |SYS| → y[n − N] for: any integer time delay N. NOT true if N varies with time (e.g., N(n)). EX: y[n]=3x[n − 2]; y[n] = sin(x[n]); y[n]= x[n]/x[n − 1]. NOT: y[n]= nx[n]; y[n]= x[n2]; y[n]= x[2n]; y[n]= x[−n]. HOW: If n appears anywhere other than in x[n], not time-invariant.
    [Show full text]
  • Digital Signal Processing Lecture Outline Discrete-Time Systems
    Lecture Outline ESE 531: Digital Signal Processing ! Discrete Time Systems ! LTI Systems ! LTI System Properties Lec 3: January 17, 2017 ! Difference Equations Discrete Time Signals and Systems Penn ESE 531 Spring 2017 - Khanna Penn ESE 531 Spring 2017 - Khanna 2 Discrete Time Systems Discrete-Time Systems Penn ESE 531 Spring 2017 - Khanna 4 System Properties Examples ! Causality ! Causal? Linear? Time-invariant? Memoryless? " y[n] only depends on x[m] for m<=n BIBO Stable? ! Linearity ! Time Shift: " Scaled sum of arbitrary inputs results in output that is a scaled sum of corresponding outputs " x[n]y[n =] x[n-m]= x[n − m] " Ax [n]+Bx [n] # Ay [n]+By [n] 1 2 1 2 ! Accumulator: ! Memoryless " y[n] n " y[n] depends only on x[n] y[n] = x[k] ! Time Invariance ∑ k=−∞ " Shifted input results in shifted output " x[n-q] # y[n-q] ! Compressor (M>1): ! BIBO Stability y[n] x[Mn] " A bounded input results in a bounded output (ie. max signal value = exists for output if max ) Penn ESE 531 Spring 2017 - Khanna 5 Penn ESE 531 Spring 2017 - Khanna 6 1 Non-Linear System Example Spectrum of Speech ! Median Filter " y[n]=MED{x[n-k], …x[n+k]} Speech " Let k=1 " y[n]=MED{x[n-1], x[n], x[n+1]} Corrupted Speech Penn ESE 531 Spring 2017 - Khanna 7 Penn ESE 531 Spring 2017 - Khanna 8 Low Pass Filtering Low Pass Filtering Corrupted Speech LP-Filtered Speech Penn ESE 531 Spring 2017 - Khanna 9 Penn ESE 531 Spring 2017 - Khanna 10 Median Filtering LTI Systems Corrupted Speech Med-Filter Speech Penn ESE 531 Spring 2017 - Khanna 11 Penn ESE 531 Spring 2017 - Khanna
    [Show full text]
  • MTHE/MATH 332 Introduction to Control
    1 Queen’s University Mathematics and Engineering and Mathematics and Statistics MTHE/MATH 332 Introduction to Control (Preliminary) Supplemental Lecture Notes Serdar Yuksel¨ April 3, 2021 2 Contents 1 Introduction ....................................................................................1 1.1 Introduction................................................................................1 1.2 Linearization...............................................................................4 2 Systems ........................................................................................7 2.1 System Properties...........................................................................7 2.2 Linear Systems.............................................................................8 2.2.1 Representation of Discrete-Time Signals in terms of Unit Pulses..............................8 2.2.2 Linear Systems.......................................................................8 2.3 Linear and Time-Invariant (Convolution) Systems................................................9 2.4 Bounded-Input-Bounded-Output (BIBO) Stability of Convolution Systems........................... 11 2.5 The Frequency Response (or Transfer) Function of Linear Time-Invariant Systems..................... 11 2.6 Steady-State vs. Transient Solutions............................................................ 11 2.7 Bode Plots................................................................................. 12 2.8 Interconnections of Systems..................................................................
    [Show full text]
  • A Note on BIBO Stability
    1 A Note on BIBO Stability Michael Unser Abstract—The statements on the BIBO stability of continuous- imposes that the result of the convolution be continuous, anda time convolution systems found in engineering textbooks are second (Theorem 3) that characterizes the BIBO-stable filters often either too vague (because of lack of hypotheses) or in full generality. The mathematical derivations are presented mathematically incorrect. What is more troubling is that they usually exclude the identity operator. The purpose of this note in Section IV, where we also make the connection with known is to clarify the issue while presenting some fixes. In particular, results in harmonic analysis. we show that a linear shift-invariant system is BIBO-stable in We like to mention a similar clarification effort by Hans Fe- L the ∞-sense if and only if its impulse response is included in ichtinger, who proposes to limit the framework to convolution the space of bounded Radon measures, which is a superset of operators that are operating on C0(R) (a well-behaved subclass L1(R) (Lebesgue’s space of absolutely integrable functions). As we restrict the scope of this characterization to the convolution of bounded functions) in order to avoid pathologies [3]. This operators whose impulse response is a measurable function, we is another interesting point of view that is complementary to recover the classical statement. ours, as discussed in Section IV. II. BIBO STABILITY:THE CLASSICAL FORMULATION I. INTRODUCTION The convolution of two functions h,f : R → R is the A statement that is made in most courses on the theory of function usually specified by linear systems as well as in the English version of Wikipedia1 △ is that a convolution operator is stable in the BIBO sense t 7→ (h ∗ f)(t) = h(τ)f(t − τ)dτ (1) (bounded input and bounded output) if and only if its impulse ZR response is absolutely summable/integrable.
    [Show full text]
  • Controllability, Observability, Realizability, and Stability of Dynamic Linear Systems
    Electronic Journal of Differential Equations, Vol. 2009(2009), No. 37, pp. 1–32. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu CONTROLLABILITY, OBSERVABILITY, REALIZABILITY, AND STABILITY OF DYNAMIC LINEAR SYSTEMS JOHN M. DAVIS, IAN A. GRAVAGNE, BILLY J. JACKSON, ROBERT J. MARKS II Abstract. We develop a linear systems theory that coincides with the ex- isting theories for continuous and discrete dynamical systems, but that also extends to linear systems defined on nonuniform time scales. The approach here is based on generalized Laplace transform methods (e.g. shifts and con- volution) from the recent work [13]. We study controllability in terms of the controllability Gramian and various rank conditions (including Kalman’s) in both the time invariant and time varying settings and compare the results. We explore observability in terms of both Gramian and rank conditions and establish related realizability results. We conclude by applying this systems theory to connect exponential and BIBO stability problems in this general setting. Numerous examples are included to show the utility of these results. 1. Introduction In this paper, our goal is to develop the foundation for a comprehensive linear systems theory which not only coincides with the existing canonical systems theories in the continuous and discrete cases, but also to extend those theories to dynamical systems with nonuniform domains (e.g. the quantum time scale used in quantum calculus [9]). We quickly see that the standard arguments on R and Z do not go through when the graininess of the underlying time scale is not uniform, but we overcome this obstacle by taking an approach rooted in recent generalized Laplace transform methods [13, 19].
    [Show full text]
  • Chapter 3 Control Theory 1 Introduction 2 Transfer Functions
    Chapter 3 Control Theory 1 Introduction To control an object is to influence its behaviour so as to achieve some desired objective. The object to be controlled is usually some dynamic process (a process evolving in time), for example, the UK economy, illness/disease in a patient, air temperature in this lecture theatre. The controls are inputs to the process which can influence its evolution (for example, UK interest rates, medication in patient treatment, radiators in a heating system). The evolution of the process may be monitored by observations or outputs (for example, UK inflation, medical biopsies, temperature sensors in a heating system). A central concept in control theory is that of feedback: the use of information available from the observations or outputs to generate the controls or inputs so as to cause the process to evolve in some desirable manner. Input Dynamical process Output ? Much of control theory is focussed on the question ? . We start with a mathematical description of the dynamic process linking the output to the input. 2 Transfer functions We restrict attention to linear, autonomous, single-input, single-output systems of the form x˙(t)= Ax(t)+ bu(t), x(0) = ξ, A ∈ Rn×n, b ∈ Rn (1) y(t)= cT x(t)+ du(t), c ∈ Rn, d ∈ R , (2) where u is a scalar-valued input and y is a scalar-valued output. The variable x(t) is referred to as the state of the system. By the variation of parameters formula, the solution of (1) is t t 7→ x(t) = (exp At)ξ + (exp A(t − τ))bu(τ) dτ .
    [Show full text]
  • Advanced Control of Continuous-Time LTI-SISO Systems
    i “MAIN” — 2015/11/19 — 9:36 — page 9 — #9 i i i Part I Advanced Control of continuous-time LTI-SISO Systems 9 i i i i i “MAIN” — 2015/11/19 — 9:36 — page 10 — #10 i i i i i i i i “MAIN” — 2015/11/19 — 9:36 — page 11 — #11 i i i Chapter 1 Review of linear systems theory This chapter contains a short review of linear systems theory. The contents of this review roughly correspond with the materials of “Systems and Control 1”. Here and there however, new concepts are introduced that are relevant for the themes that will be treated further on in this course. 1.1 Descriptions and properties of linear systems The first step in feedback controller design is to develop a mathematical model of the process or plant to be controlled. This model should describe the relation between a given input signal u(t) and the system output signal y(t). In this review, we will discuss the modelling of linear, time-invariant (LTI), single-input-single-output (SISO) systems, as depicted in Fig 1.1. These models can be described by differing dynamic models. In the time domain, the physics of an LTI system can be modelled by a linear, ordinary differential equation (ODE) or by a state space model. In the frequency domain, the dynamics of the system can be described by a transfer function or by the frequency response (in the form of a Bode or a Nyquist diagram). u(t) y(t) System SISO linear time-invariant Figure 1.1: Basic setup of a LTI-SISO system.
    [Show full text]
  • Stability Analysis
    12 Stability Analysis 12.1 Introduction..................................................................... 12-1 12.2 Using the State of the System to Determine Stability ............. 12-2 12.3 Lyapunov Stability Theory.................................................. 12-3 12.4 Stability of Time-Invariant Linear Systems............................ 12-4 Stability Analysis with State-Space Notation * The Transfer Ferenc Szidarovszky Function Approach University of Arizona 12.5 BIBO Stability ................................................................. 12-11 A. Terry Bahill 12.6 Bifurcations..................................................................... 12-12 University of Arizona 12.7 Physical Examples ............................................................ 12-12 12.1 Introduction In this chapter, which is based on Szidarovszky and Bahill (1998), we discuss stability. We start by discussing interior stability, where the stability of the state trajectory or equilibrium state is examined and then we discuss exterior stability, in which we guarantee that a bounded input always evokes a bounded output. We present four techniques for examining interior stability: (1) Lyapunov functions, (2) checking the boundedness or limit of the fundamental matrix, (3) finding the location of the eigenvalues for state-space notation, and (4) finding the location of the poles in the complex frequency plane of the closed-loop transfer function. We present two techniques for examining exterior (or BIBO) stability (1) use of the weighting pattern of the system and (2) finding the location of the eigenvalues for state-space notation. Proving stability with Lyapunov functions is very general: it even works for nonlinear and time-varying systems. It is also good for doing proofs. However, proving the stability of a system with Lyapunov functions is difficult. And failure to find a Lyapunov function that proves a system is stable does not prove that the system is unstable.
    [Show full text]
  • ESE 531: Digital Signal Processing
    ESE 531: Digital Signal Processing Lec 3: January 24, 2019 Discrete Time Signals and Systems Penn ESE 531 Spring 2019 - Khanna Lecture Outline ! Discrete Time Systems ! LTI Systems ! LTI System Properties ! Difference Equations Penn ESE 531 Spring 2019 - Khanna 3 Discrete-Time Systems Discrete Time Systems ! Systems manipulate the information in signals ! Examples " Speech recognition system that converts acoustic waves into text " Radar system transforms radar pulse into position and velocity " fMRI system transform frequency into images of brain activity " Moving average system smooths out the day-to-day variability in a stock price Penn ESE 531 Spring 2019 - Khanna 5 System Properties ! Causality " y[n] only depends on x[m] for m<=n ! Linearity " Scaled sum of arbitrary inputs results in output that is a scaled sum of corresponding outputs " Ax1[n]+Bx2[n] # Ay1[n]+By2[n] ! Memoryless " y[n] depends only on x[n] ! Time Invariance " Shifted input results in shifted output " x[n-q] # y[n-q] ! BIBO Stability " A bounded input results in a bounded output (ie. max signal value exists for output if max ) Penn ESE 531 Spring 2019 - Khanna 6 Stability ! BIBO Stability " Bounded-input bounded-output Stability Penn ESE 531 Spring 2019 - Khanna 7 Examples ! Causal? Linear? Time-invariant? Memoryless? BIBO Stable? ! Time Shift: " x[n]y[n =] x[n-m]= x[n − m] ! Accumulator: " y[n] n y[n] = ∑ x[k] k=−∞ ! Compressor (M>1): y[n] = x[Mn] Penn ESE 531 Spring 2019 - Khanna 8 Non-Linear System Example ! Median Filter " y[n]=MED{x[n-k], …x[n+k]} " Let k=1
    [Show full text]
  • EE2S11 Signals and Systems, Part 2 Ch.10 the Z-Transform
    EE2S11 Signals and Systems, part 2 Ch.10 The z-transform Contents definition of the z-transform region of convergence convolution property, stability inverse z-transform Skip sections 10.5.3, 10.6, 10.7 1 12a z-transform January 7, 2020 The z-transform Sampled sequence Suppose that we have a sampled signal: xs (t) = x[n]δ(t − nTs ) , x[n] := x(nTs ) X The Laplace transform L{xs (t)} is −sTs n −n Xs (s) = x[n]L{δ(t − nTs )} = x[n]e = x[n]z , X X X where z := esTs . jΩTs jω For s = jΩ we obtain z = e = e , with ω = ΩTs . More generally: s = σ + jΩ becomes z = eσTs ejΩTs = rejω. 2 12a z-transform January 7, 2020 The z-transform Note that the mapping s → z = esTs is not one-to-one. For a given z = ejω we can take −π ≤ ω ≤ π, this corresponds to π π − ≤ Ω ≤ : the fundamental interval. Ts Ts Complex numbers s = jΩ with Ω outside this interval are mapped onto the same z. Left half-plane is mapped to the inside of the unit circle. s-vlak z-vlak (aliasing) π/Ts ω Ω π 0 −π −π/Ts (aliasing) 3 12a z-transform January 7, 2020 The z-transform From now on, we will work with z and apply this transform to time series, even if there is no connection to continuous-time signals. The z-transform The (two-sided) z-transform of a time series x[n] is defined as ∞ X(z) = Z(x[n]) := x[n]z −n , z ∈ ROC n=X−∞ We also need to indicate the region of convergence (ROC).
    [Show full text]
  • On BIBO Stability of Systems with Irrational Transfer Function
    On BIBO stability of systems with irrational transfer function Ansgar Tr¨achtler* * Heinz Nixdorf Institute, University of Paderborn, F¨urstenallee11, D-33098 Paderborn March 4, 2016 Abstract We consider the input/output-stability of linear time-invariant single-input/single- output systems in terms of singularities of the transfer function F(s) in Laplace domain. The approach is based on complex analysis. A fairly general class of transfer functions F(s) is considered which can roughly be characterized by two properties: (1) F(s) is meromorphic with a finite number of poles in the open right half plane and (2), on the imaginary axes, F(s) may have at most a finite number of poles and branch points. For this class of systems, a complete and thorough characterization of BIBO stability in terms of the singularities in the closed right half-plane is developed where no necessity arises to differentiate between commensurate and incommensurate orders of branch points as often done in literature. A necessary and sufficient condition for BIBO stability is derived which mainly relies on the asymptotic expansion of the impulse response f(t). The second main result is the generalization of the well-known Nyquist citerion for testing closed-loop stability in terms of the open-loop frequency response locus to the class of systems under consideration which is gained by applying Cauchy's argument principle. 1 Introduction arXiv:1603.01059v1 [math.DS] 3 Mar 2016 Non rational transfer functions arise in various context: spatially distributed systems de- scribed by partial differential equations [Sch81, Fra74, Deu12] or, more general, infinite- dimensional systems [CZ95], or systems with one or more dead-time elements, e.g.
    [Show full text]