Introduction to Signals and Systems General Informations. Guillaume Drion Academic year

Transcription

1 Introduction to Signals and Systems General Informations Guillaume Drion Academic year

2 SYST General informations Website: Contacts: Guillaume Drion - gdrion@ulg.ac.be Marie Wehenkel (teaching assistant) - m.wehenkel@ulg.ac.be Organization: 11 or 12 main lessons - Wednesdays 13:30 or 13:30? 10 tutorial sessions split in 7 groups (B5b, see website) Theory and exercises follow the textbooks provided on the website (in French). The textbooks are the same as last year!

3 Schedule of the year Tutorials will start on Wednesday, October 4!

4 Goals of the course and evaluation Goals of the course: Lessons: intuition! The main goal of this course is to provide a general (and simple) framework for the analysis of possibly complex systems. Tutorials: develop your technical skills, methods. SYST0002 Level of complexity Past Future Time

5 Goals of the course and evaluation Goals of the course: Lessons: intuition! The main goal of this course is to provide a general (and simple) framework for the analysis of possibly complex systems. Tutorials: develop your technical skills, methods. Evaluation: 2 short assignments (matlab). Exam: 2-3 questions to test your technical skills (tutorial style). 1-2 questions to test your basic knowledge and intuition.

6 What changes this year? The course absorbs the basic signal processing course: Addition of the concept of Fourier series, increased focus on Fourier transforms from a signal processing viewpoint. Addition of 1-2 courses on signal processing (sampling, windowing). On the other hand, some concepts regarding the analytical solutions of state-space equations will be removed: This part has already been taught in the calculus course (linear differential equations with constant coefficients).

7 Introduction to Signals and Systems Lecture #1 - Introduction to Systems Theory Guillaume Drion Academic year

8 Introduction to Signals and Systems Neville Hogan (MIT): The Paradox of Human Performance

9 Introduction to Signals and Systems

10 Introduction to Signals and Systems Neville Hogan (MIT): The Paradox of Human Performance As engineers, your task will not only be to use existing tools to design complex systems, but also to develop novel tools to shape tomorrow s technology.

11 Introduction to Signals and Systems Example: can you accurately describe the motion of this simple pendulum

12 Introduction to Signals and Systems Example: can you accurately describe the motion of this simple pendulum without using Newton s laws of motion?

13 Introduction to Signals and Systems What was mechanics like before Newton? How much did Newton s laws changed our understanding of mechanics?

14 Mathematical modeling can turn any problem into an engineering problem Real life problems

15 Mathematical modeling can turn any problem into an engineering problem Real life problems Engineering problems

16 Mathematical modeling can turn any problem into an engineering problem Real life problems Engineering problems SYSTEMS MODELING Analysis Design Implementation Applied mathematics

17 Systems modeling is a key method for the development of novel engineering tools Example: drones flying in formation. ithout using Newton s laws of motion?

18 Systems modeling is a key method for the development of novel engineering tools Example: drones flying in formation. Source of inspiration? ithout using Newton s laws of motion?

19 Systems modeling is a key method for the development of novel engineering tools Example: neuroscience and deep learning. Synaptic plasticity Neuromodulation

20 Systems modeling is a key method for the development of novel engineering tools Contemporary examples: analysis and design of next generation materials (graphene). engineering in life sciences (cardiovascular physiology, neuroscience).

21 Systems modeling in three courses SYST0002: Introduction to signals and systems: open loop. Observing and analyzing the environment Input SYSTEM Output SYST0003: Linear control systems: closed loop. Interacting with the environment Input SYSTEM CONTROLLER Output SYST0017: Advanced topics in systems and control: goes further. (nonlinear systems, chaos, etc.)

22 Systems modeling in three courses SYST0002: Introduction to signals and systems: open loop. Observing and analyzing the environment Input SYSTEM Output SYST0003: Linear control systems: closed loop. Interacting with the environment Input SYSTEM CONTROLLER Output SYST0017: Advanced topics in systems and control: goes further. (nonlinear systems, chaos, etc.)

23 What is the value of a mathematical model? What was mechanics like before Newton? How much did Newton s laws changed our understanding of mechanics? And what about after Einstein? v 1 v 2 Galilean relativity v = v 1 + v 2 v

24 What is the value of a mathematical model? What was mechanics like before Newton? How much did Newton s laws changed our understanding of mechanics? And what about after Einstein? v 1 v 2 Galilean relativity v = v 1 + v 2 Which one is correct, which one is wrong? v Special relativity v = v 1 + v 2 1+ v 1v 2 c 2

25 What is the value of a mathematical model? What was mechanics like before Newton? How much did Newton s laws changed our understanding of mechanics? And what about after Einstein? v 1 v 2 Galilean relativity v = v 1 + v 2 v Special relativity v = v 1 + v 2 1+ v 1v 2 c 2 Which one is correct, which one is wrong? Which one is useful?

26 What is the value of a mathematical model? What was mechanics like before Newton? How much did Newton s laws changed our understanding of mechanics? And what about after Einstein? All models are wrong, some are useful. George E. P. Box

27 What is the value of a mathematical model? Contemporary examples: Graphene Theory showed that 2D crystals are unstable ( L. D. Landau, and E. M. Lifshitz, 1980)

28 What is the value of a mathematical model? Contemporary examples: Graphene Theory showed that 2D crystals are unstable ( L. D. Landau, and E. M. Lifshitz, 1980) Andre Geim and Konstantin Novoselov did it anyway (2004) The micromechanical cleavage technique ( Scotch tape method) for producing graphene

29 What is the value of a mathematical model? Contemporary examples: Graphene Theory showed that 2D crystals are unstable ( L. D. Landau, and E. M. Lifshitz, 1980) Andre Geim and Konstantin Novoselov did it anyway (2004)

30 What is the value of a mathematical model? Contemporary examples: Standard model in physics ( theory of everything ). Correctly deducts the existence of the Higgs Boson.

31 What is the value of a mathematical model? Contemporary examples: Standard model in physics ( theory of everything ). Correctly deducts the existence of the Higgs Boson. Fails to explain the origin of gravitational forces (so far). Adding a graviton?

32 Why do we need a theory to analyse and design dynamical systems? Dynamical systems can have counter-intuitive properties. Example: Briggs-Rauscher reaction (color shows iodine concentration)

33 Why do we need a theory to analyse and design dynamical systems? Example: mathematical modelling in ecology In 1838, Pierre-François Verhulst proposed a dynamical model for the growth of a population (N) depending on the intrinsic growth rate (r) and the maximum number of individuals the environment can support (K). This equation is called the logistic equation. Simple behavior: If r >> and N << K: the population grows fast. If N = K: the population does not grow anymore.

34 The logistic equation Simulation of the logistic equation for different growth rates and K=1. 1 Logistic equation, r=2 0.8 N Logistic equation,r=4 0.8 N time

35 A discrete equivalent of the logistic equation: the logistic map In 1976, Robert May proposed a discrete equivalent vs As opposed to the continuous system, the dynamics of the discrete system is extremely rich, and can be chaotic for certain values of α.

36 Dynamical behavior of the logistic map 1 Logistic map, a= Logistic map, a= Logistic map, a=

37 Dynamical behavior of the logistic map 1 Logistic map, a= Logistic map, a= Logistic map, a=

38 Dynamical behavior of the logistic map: chaos. 1 Logistic map, a= Logistic map, a= Logistic map, a=

39 Dynamical behavior of the logistic map: chaos. In 1976, Robert May proposed a discrete equivalent vs As opposed to the continuous system, the dynamics of the discrete system are extremely rich, and can be chaotic for certain values of α. This dynamical richness comes from the nonlinearity of the system. But it highlights the fact that continuous and discrete systems are not always equivalent.

40 Why do we need a theory to analyse and design dynamical systems? Because every system is dynamical in nature

41 Why do we need a theory to analyse and design dynamical systems? Because every system is dynamical in nature Systems theory studies systems dynamical behavior: Stability Oscillations Response speed Overshoots Resonance etc.

42 Where is systems theory useful? High levels of automation. Example: SpaceX automatic landing.

43 Where is systems theory useful? Engineering in life sciences. Example: cochlear implants.

44 Where is systems theory useful? Engineering in life sciences. Example: deep brain stimulation in Parkinson s disease.

45 Where is systems theory useful? Engineering in life sciences. Example: deep brain stimulation in Parkinson s disease.

46 Where is systems theory useful? Civil engineering. How can we study the effect of a fire on a beam in a building? Construct a whole building in a laboratory? Possible solution: submit a real beam to a fire in a laboratory, measure the forces and displacements and feed them to a numerical model of the building. Aerospace engineering. Will my satellite survive the launch to space? Identification of resonance peaks and nonlinearities using system identification.

47 What does systems theory consist of? In this course, we will mainly focus on methods that have been developed in the case of linear, time-invariant (LTI) systems. The course will introduce two main approaches: The state-space approach (exhaustive description of the system) The input-output approach (the system under study is a black box ) The course will introduce novel mathematical methods that will look complex at first sight but, when used wisely, can drastically simplify your engineering work.

48 What does systems theory consist of? Example: design of an electrical circuit in the frequency domain. The order of the set of differential equations describing the typical negative feedback amplifier used in telephony is likely to be very much greater. As a matter of idle curiosity, I once counted to find out what the order of the set of equations in an amplifier I had just designed would have been, if I had worked with the differential equations directly. It turned out to be 55. Hendrik Bode, 1960 The use of frequency domain methods has made the design of complex systems possible. But it first requires to master the concepts of Fourier transforms, Laplace transforms, etc.

49 A unified theory to study systems is possible because systems share a lot of properties Illustration: what is the common point between a simple suspension, an electrical circuit and the mammalian cardiovascular system?

50 A unified theory to study systems is possible because systems share a lot of properties Illustration: what is the common point between a simple suspension, an electrical circuit and the mammalian cardiovascular system? Answer: they all have very similar dynamical and input/output behaviors.

51 A unified theory to study systems is possible because systems share a lot of properties Illustration: what is the common point between a simple suspension, an electrical circuit and the mammalian cardiovascular system? Answer: they all have very similar dynamical and input/output behaviors. = m B = B K = RC = L R m = mass B = damping K = stiffness R = resistance C = capacitance L = inductance Stimulation ON Stimulation OFF

52 A unified theory to study systems is possible because systems share a lot of properties Illustration: what is the common point between a simple suspension, an electrical circuit and the mammalian cardiovascular system? Answer: they all have very similar dynamical and input/output behaviors.? = m B = B K m = mass B = damping K = stiffness = RC = L R R = resistance C = capacitance L = inductance Stimulation ON Stimulation OFF

53 Open loop systems modeling: analyzing the environment Case study: cardiovascular physiology. Our system: heart + vessels + blood. Question: how can the blood flow be continuous knowing that the heart generates pulses? vs

54 Modeling the cardiovascular system Measurements: pressure in the left ventricle LV (input) and in the Aorta Ao(output).

55 Modeling the cardiovascular system Measurements: pressure in the left ventricle LV (input) and in the Aorta Ao(output). LV: large variations. Ao: stays highly positive (between 80 and 120 mmhg). Input Output LV and Ao pressure variations over time are signals.

56 Modeling the cardiovascular system Pathology: some patients have higher systolic pressure with lower diastolic pressure. Why? (It happens mostly in older patients). Answering this question is very important because these patients are prone to heart failures. What can we do to fix the problem?

57 Modeling the cardiovascular system: Otto Frank. In 1899, german physiologist Otto Frank came up with a first mathematical representation of the LV-Ao system: the Windkessel Model. We will use this example to introduce the different ways to model a system 1. Find an equivalent representation of the system under study (Ch2, Ch3) 2. Put system into equations (Ordinary Differential Equations or Difference Equations) State-space representation (Ch2-3-4) 3. Extract system input/output properties (Laplace/Fourier or z-transform) (Ch 5-6) Transfer function (Ch7) System analysis (effects of changes in parameters?) (Ch8-9-10)

58 Modeling scheme 1. Find an equivalent representation of the system under study 2. Put system into equations (Ordinary Differential Equations or Difference Equations) State-space representation 3. Extract system input/output properties (Laplace/Fourier transform or z-transform) Transfer function System analysis (effects of changes in parameters?)

59 Find an equivalent representation of the system under study Mathematical analysis Ordinary differential equations Series Fourier transform Convolution Linear algebra Physics Laws of mechanics Laws of electricity and electromagnetism Chemistry Chemical reactions Organic chemistry Thermodynamics Matrix algebra Difference equations Informatics Algorithms Programming Numerical analysis Numerical methods Optimization

60 Equivalent representation of the left ventricle-aorta (LV-Ao) system Otto Frank took advantage of the water circuit analogy to electric circuit. Left ventricle Aorta Aortic valve (r) Arterial compliance (C a ) Periphery vessels (R 1, R 2,..., R n ) u(t)

61 Equivalent representation of the left ventricle-aorta (LV-Ao) system Otto Frank took advantage of the water circuit analogy to electric circuit.

62 The 3-Element Windkessel model - Circuit diagram 1. Equivalent circuit of the LV-Ao system Left ventricle Aorta Aortic valve (r) Arterial compliance (C a ) Periphery vessels (R 1, R 2,..., R n ) u(t) r P(t) P r (t) P Ca (t) C a R

63 The 3-Element Windkessel model - Circuit diagram 1. Equivalent circuit of the LV-Ao system Left ventricle Aorta Aortic valve (r) Arterial compliance (C a ) Periphery vessels (R 1, R 2,..., R n ) u(t) r P(t) P r (t) P Ca (t) C a R

64 The 3-Element Windkessel model - Circuit diagram 1. Equivalent circuit of the LV-Ao system Left ventricle Aorta Aortic valve (r) Arterial compliance (C a ) Periphery vessels (R 1, R 2,..., R n ) u(t) r P(t) P r (t) P Ca (t) C a R

65 The 3-Element Windkessel model - Circuit diagram 1. Equivalent circuit of the LV-Ao system Left ventricle Aorta Aortic valve (r) Arterial compliance (C a ) Periphery vessels (R 1, R 2,..., R n ) u(t) r P(t) P r (t) P Ca (t) C a R

66 The 3-Element Windkessel model - Circuit diagram 1. Equivalent circuit of the LV-Ao system u(t) r P(t) P r (t) P Ca (t) C a R

67 Modeling scheme 1. Find an equivalent representation of the system under study 2. Put system into equations (Ordinary Differential Equations or Difference Equations) State-space representation 3. Extract system input/output properties (Laplace/Fourier transform or z-transform) Transfer function System analysis (effects of changes in parameters?)

68 The 3-Element Windkessel model - Circuit diagram 2. Mathematical description of the dynamical system: ordinary differential equations u(t) r P(t) P r (t) P Ca (t) C a R

69 The 3-Element Windkessel model - ODE s 2. Mathematical description of the dynamical system: ordinary differential equations u(t) r P(t) P r (t) P Ca (t) C a R Kirchhoff s voltage law: P(t) =P r (t)+p Ca (t) =ru(t)+p Ca (t)

70 The 3-Element Windkessel model - ODE s 2. Mathematical description of the dynamical system: ordinary differential equations u(t) r P(t) P r (t) P Ca (t) C a R Kirchhoff s voltage law: P(t) =P r (t)+p Ca (t) =ru(t)+p Ca (t) Kirchhoff s current law: u(t) =i Ca (t)+i r (t) =C a dp Ca (t) dt + P C a (t) R

71 The 3-Element Windkessel model - ODE s 2. Mathematical description of the dynamical system: ordinary differential equations u(t) r P(t) P r (t) P Ca (t) C a R C a dp Ca (t) dt + P C a (t) R = u(t) P(t) =ru(t)+p Ca (t)

72 The 3-Element Windkessel model - ODE s P(t) 2. Mathematical description of the dynamical system: ordinary differential equations u(t) r Elements that vary over time are called P r (t) variables. Ex: P Ca (t) P Ca (t) C a R dp Ca (t) C a + P C a (t) = u(t) dt R P(t) =ru(t)+p Ca (t) Elements that are fixed are called parameters. Ex: r, R, C a u(t) is the input (commonly used), P(t) is the output.

73 The 3-Element Windkessel model - Simulation P(t) 2A. Validation of the model: simulation (ex: matlab) u(t) r P r (t) P Ca (t) C a R u(t) dp Ca (t) C a + P C a (t) = u(t) dt R P(t) =ru(t)+p Ca (t) P(t)

74 The 3-Element Windkessel model - State-space 2B. State-space canonical representation Linear, time-invariant (LTI) dynamical systems can be represented in the form ẋ = Ax + Bu y = Cx + Du where A is the dynamics matrix, B is the input matrix, C the output matrix, D the feedthrough matrix. Linear: the output is a linear function of the input (not y = x 2 for instance) Time-invariant: parameters do not change over time. (A, B, C and D does not depend on time). Here: the values of r, R and C a are fixed.

75 The 3-Element Windkessel model - State-space 2B. State-space canonical representation Linear, time-invariant (LTI) dynamical systems can be represented in the form ẋ = Ax + Bu dp Ca (t) = 1 P Ca (t)+ 1 u(t) dt RC a C a y = Cx + Du where A is the dynamics matrix, B is the input matrix, C the output matrix, D the feedthrough matrix. P(t) =P Ca (t)+ru(t) Linear: the output is a linear function of the input (not y = x 2 for instance) Time-invariant: parameters do not change over time. (A, B, C and D does not depend on time). Here: the values of r, R and C a are fixed.

76 The 3-Element Windkessel model - State-space 2B. State-space canonical representation Linear, time-invariant (LTI) dynamical systems can be represented in the form ẋ = Ax + Bu dp Ca (t) = 1 P Ca (t)+ 1 u(t) dt RC a C a y = Cx + Du where A is the dynamics matrix, B is the input matrix, C the output matrix, D the feedthrough matrix. P(t) =P Ca (t)+ru(t) x(t) =P Ca (t), y(t) =P(t) A = 1 RC a, B = 1 C a, C =1,D = r Linear: the output is a linear function of the input (not y = x 2 for instance) Time-invariant: parameters do not change over time. (A, B, C and D does not depend on time). Here: the values of r, R and C a are fixed.

77 Why is the state-space representation important? ẋ = Ax + Bu y = Cx + Du General representation! At this stage, we use the same tools, whether the system is a car suspension, an electrical circuit, a chemical reaction, the cardiovascular system, etc. Four matrices summarize the behavior of any LTI system, regardless of its complexity. We can use this representation to analyze the key features of the system: stability, reachability, observability, etc. Very important for system realization: still contains the real system parameters.

78 Back to our case study ẋ = Ax + Bu y = Cx + Du x(t) =P Ca (t), y(t) =P(t) A = 1 RC a, B = 1 C a, C =1,D = r Questions: How can the blood flow be continuous knowing that the heart generates pulses? Some patients have higher systolic pressure with lower diastolic pressure. Why?

79 Modeling scheme 1. Find an equivalent representation of the system under study 2. Put system into equations (Ordinary Differential Equations or Difference Equations) State-space representation 3. Extract system input/output properties (Laplace/Fourier transform or z-transform) Transfer function System analysis (effects of changes in parameters?)

80 Frequency domain: introduction (Some of) you have seen the Fourier transform in calculus. In this course, we will use the Fourier transform, and others such as the Laplace transform (continuous time) and z-transform (discrete time) to move from the time domain to the frequency domain. where ω is an angular frequency (rad/s).

81 Frequency domain: introduction (Some of) you have seen the Fourier transform in calculus. In this course, we will use the Fourier transform, and others such as the Laplace transform (continuous time) and z-transform (discrete time) to move from the time domain to the frequency domain. where ω is an angular frequency (rad/s).

82 Frequency domain: introduction The idea is to decompose a signal into the frequencies that compose it and analyze how a system transmit/transform these frequencies.

83 Frequency domain: introduction The idea is to decompose a signal into the frequencies that compose it and analyze how a system transmit/transform these frequencies.

84 Frequency domain: introduction The idea is to decompose a signal into the frequencies that compose it and analyze how a system transmit/transform these frequencies.

85 Frequency domain: introduction The idea is to decompose a signal into the frequencies that compose it and analyze how a system transmit/transform these frequencies. Low frequency High frequency

86 Frequency domain: Fourier transform vs Laplace transform Fourier transform where ω is an angular frequency (rad/s). Laplace transform where s is the complex frequency s = σ + jω.

87 Why working in the frequency domain? Many advantages, here is one of them: Time domain Frequency domain

88 The 3-Element Windkessel model - Transfer function 3. Input/output properties: transfer function (frequency domain via Laplace transform) Idea: describe the system through a simple function that characterizes the way it affects an input U(s) U(s) H(s) Y(s) and s is the complex number frequency (s = σ+jω). If σ=0: Fourier transform!

89 The 3-Element Windkessel model - Transfer function 3. Input/output properties: transfer function (frequency domain via Laplace transform) Idea: describe the system through a simple function that characterizes the way it affects an input U(s) U(s) H(s) Y(s) and s is the complex number frequency (s = σ+jω). If σ=0: Fourier transform! There are different ways to compute the transfer function of a system. However, it is convenient to start from the canonical state-space representation (if available) ẋ = Ax + Bu y = Cx + Du which gives H(s) = Y (s) U(s) = C(sI A) 1 B + D (see next slide)

90 The 3-Element Windkessel model - Transfer function Transfer function from state-space representation: which gives (1) (1) (2) and therefore (1) (2)

91 The 3-Element Windkessel model - Transfer function 3. Input/output properties: transfer function (frequency domain via Laplace transform) Transfer function of the 3-Element Windkessel model ( ) H(s) = Y (s) U(s) = C(sI A) 1 B + D A = 1, B = 1, C =1,D = r RC a C a

92 The 3-Element Windkessel model - Transfer function 3. Input/output properties: transfer function (frequency domain via Laplace transform) Transfer function of the 3-Element Windkessel model ( A = 1, B = 1, C =1,D = r ) RC a C a H(s) = Y (s) U(s) = C(sI A) 1 B + D =1(s + 1 RC a ) 1 1 C a + r

93 The 3-Element Windkessel model - Transfer function 3. Input/output properties: transfer function (frequency domain via Laplace transform) Transfer function of the 3-Element Windkessel model ( A = 1, B = 1, C =1,D = r ) RC a C a H(s) = Y (s) U(s) = C(sI A) 1 B + D =1(s + 1 RC a ) 1 1 C a + r = R RC a s +1 + r

94 The 3-Element Windkessel model - Transfer function 3. Input/output properties: transfer function (frequency domain via Laplace transform) Transfer function of the 3-Element Windkessel model ( A = 1, B = 1, C =1,D = r ) RC a C a H(s) = Y (s) U(s) = C(sI A) 1 B + D =1(s + 1 RC a ) 1 1 C a + r = R RC a s +1 + r K τs +1 + r => Low pass filter! (r<< physiologically) K=R: gain τ=rc a : time constant ω c =1/τ: cutoff frequency

95 The 3-Element Windkessel model - Transfer function Low-pass filter K τs +1 + r

96 The transfer function of the Windkessel model helps making predictions on the potential effects of physiological and/or pathological conditions on blood pressure Low pass filter R H(s) = RC a s +1 + r τ = RC a Low frequency High frequency

97 The transfer function of the Windkessel model helps making predictions on the potential effects of physiological and/or pathological conditions on blood pressure Pathology: some patients have higher systolic pressure with lower diastolic pressure. Why? (It happens mostly in older patients).

98 The transfer function of the Windkessel model helps making predictions on the potential effects of physiological and/or pathological conditions on blood pressure Pathology: some patients have higher systolic pressure with lower diastolic pressure. Why? (It happens mostly in older patients). Loss of low-pass filtering properties

99 The transfer function of the Windkessel model helps making predictions on the potential effects of physiological and/or pathological conditions on blood pressure Pathology: some patients have higher systolic pressure with lower diastolic pressure. Why? (It happens mostly in older patients). Low pass filter R H(s) = RC a s +1 + r τ = RC a Loss of low-pass filtering properties

100 The transfer function of the Windkessel model helps making predictions on the potential effects of physiological and/or pathological conditions on blood pressure Atherosclerosis: loss of arterial compliance => C a decreases => τ=rc a decreases Low pass filter R H(s) = RC a s +1 + r τ = RC a

101 Modeling the cardiovascular system: conclusion The vascular system acts as a low-pass filter, following slow heart movements but filtering fast heart movements. This allows to maintain a rather constant blood flow in the system. Input Output

102 Modeling scheme 1. Find an equivalent representation of the system under study 2. Put system into equations (Ordinary Differential Equations or Difference Equations) State-space representation 3. Extract system input/output properties (Laplace/Fourier transform or z-transform) Transfer function System analysis (effects of changes in parameters?)

103 Systems theory: state-space vs input-output approaches State-space approach Potentially complex and high-dimensional Non unique Closely relates to physics/biology Input-output approach Low dimensional, simple to interpret Unique More abstract dp Ca (t) C a + P C a (t) = u(t) dt R P(t) =ru(t)+p Ca (t) H(s) = R RC a s +1 + r