1 AUTOMATIC CONTROL Andrea M. Zanchettin, PhD Spring Semester, 2018 Linear systems (frequency domain)
2 Motivations Consider an LTI system Thanks to the Lagrange s formula we can compute the motion of the state (output) in closed form. However, this is usually unfeasible in practice and numerical methods (e.g. to compute integrals) should be adopted instead. This procedure is quite tedious if we want to quickly address and understand the properties of the system.
3 Motivations cont d In many physics and engineering disciplines, the frequency domain analysis a frequently adopted tool to describe signals as an alternative to time domain analysis. We want therefore characterize the property of a (linear) system in the frequency domain, as we can do with signals.
Laplace transform Pierre-Simon, marquis de Laplace (Beaumont-en-Auge 1749 - Paris 1827) 4 Given a function of time, defined for defined as follows its Laplace transform is For completeness, we also define the corresponding inverse (Laplace) transform as Since the integration is done on the vertical line, its value is selected in order to avoid singularities of.
5 Laplace transform cont d The Laplace transform has a significant number of properties, those useful for our purpose are: Linearity Translation in time domain Derivative in time domain Integration in time domain Initial and final values (under special conditions on F)
6 Laplace transform cont d Let s compute the Laplace transform of the most important signals: Step Ramp Exponential
7 Laplace transform cont d Time-scaled exponential Sine/cosine waves
8 Laplace transform cont d Time-scaled sine/cosine waves
9 Laplace transform & LTI systems Focus again on the LTI system Let s transform each quantity and apply the system equations
10 Laplace transform & LTI systems cont d Therefore, the Laplace transform of the state is as follows whilst for the output we obtain Definition: the quantity is called transfer function of the system and relates the Laplace transfer of the output to the Laplace transfer of the input (for null initial state).
11 Laplace transform & LTI systems cont d Recall the Lagrange s formula and compare it with the Laplace transform of the output Such quantities are related to the Laplace transform of the free and forced motions (as we introduced them within the Lagrange s formula)
12 Transfer function We just defined an operator relating the Laplace transform of the input to the Laplace transform of the output of an LTI system. How does it look like? For a system with p inputs and q outputs, the transfer function is a q-byp matrix of complex functions.
13 Transfer function cont d Example: consider RLC circuit u L C x 2 z 2 x 1 = z 1 R y
14 Transfer function cont d Focus on SISO (single input, single output system). Already seen, we called it characteristic polynomial In general, both the numerator and the denominator are polynomial functions in s (as we have seen from the example).
15 Transfer function cont d Consider a generic LTI system its transfer function a generic change of variables which leads to We wonder whether there is some relationship between and
16 Transfer function cont d Result: the transfer function (being a relationship between inputs and outputs) does not depend on the particular choice of the state variables. It is therefore invariant with respect to a change of variables. Proof: by sobstitution
17 Transfer function cont d We have seen, also through the example, that the transfer function of a SISO system appears as the ratio between two polynomials. This is general and we can focus on the following generic form Properties and facts: numerator and denominator are polynomials with real coefficients; the degree of the denominator n is always greater or equal to the degree of the numerator m (their difference is called relative degree); the root of those polynomials (real or complex conjugate) are called zeros (numerator) and poles (denominator)
18 Transfer function cont d Properties and facts cont d: when D = 0, the relative degree is at least one (the system is called strictly proper), otherwise it is zero (proper); poles are also eigenvalues of matrix A Example: compute the transfer function of the following system
19 Transfer function cont d The system is a second order one, we expect a second order transfer function (= order of the denominator) A simplification (cancellation) occurs and the resulting transfer function has reduced order. Eigenvalues -2 and -1 Poles -1 All poles are eigenvalues, but not necessarily vice versa (due to cancellation)
20 Transfer function cont d If we get back to the definition of the system We realize that the there is now way to modify the behaviour of the first state variable through the input u. We now wonder whether the system is reachable Indeed, the system is not reachable. This causes a reduced order transfer function which highlights the lower degree of the input/output relationship. A similar reasoning applies for non observable systems.
21 Transfer function cont d Properties and facts cont d: eigenvalues are also poles iff the system is both reachable and observable, poles are always eigenvalues; (asymptotic) stability can be inferred from poles, as long as the system is reachable and observable; an unstable system might have a stable transfer function (poles on the open left half-plane), in this case a cancellation occurs in the right half-plane) Examples:
22 Transfer function cont d So far, we introduced the following representation of a transfer function There are at least two other kinds of representation, definitely more useful for practical reasons. This, for example allows to highlights poles and zeros (as real or complex conjugate numbers).
23 Transfer function cont d Example: zeros: -1, -2 and -3; poles: 0 (twice) and +1. Another example: zero: 0; poles -2 and two complex & conjugate.
24 Transfer function cont d For complex and conjugate zeros or poles, it is usually better to adopt another form. As complex numbers can be written in terms of real/imaginary part and magnitude (natural frequency) and damping, this also applies to zeros and poles. Example:
25 Transfer function cont d Consider a complex and conjugate pair. Left half-plane root Right half-plane root
26 Transfer function cont d In general, one possible representation for the transfer function, considering real and complex conjugate roots separately, is therefore the following one
27 Transfer function cont d Another useful representation exists, in the case of real roots only. Where parameter is called type and represents the number of poles (when positive) or zeros (negative) in the origin; parameter is called (generalized) gain; parameters and are called zero/pole time constants; Finally we also have that and.
28 Transfer function cont d In case of complex and conjugate zeros or poles, the general form is the following one
29 Frequency response The transfer function we just introduced is a very interesting tool for studying the behaviour of linear systems subject to periodic inputs. Consider then an asymptotically stable LTI SISO system, for which we have computed the transfer function Consider then the input output look like?. How does the
30 Frequency response cont d Example: consider the following asymptotically stable LTI SISO system Compute its output when fed by the input. Apply the Lagrange s formula
31 Frequency response cont d Let s have a look at the output we computed: The first term is an exponential one (decreasing since the system is asymptotically stable). The second one is a pure sine wave at the same frequency of the input, however with different amplitude and phase. Notice that, since the system is asymptotically stable, we have
32 Frequency response cont d
33 Frequency response cont d This is true for the general case, in fact: Theorem (frequency response): given an asymptotically stable LTI SISO system having transfer function, given the input as the output is such that where
34 Frequency response cont d Example: let s verify our result on the last example The transfer function is while. which is what we obtained using the Lagrange s formula.
35 Representations From the thoerem of the frequency response, we have seen a first application of transfer function of an asymptotically stable LTI system to the case of periodic inputs. It could be simply applied by computing which is, in general, a complex number. We are now interested in understanding what happens to by varying the frequency.
36 Representations cont d As is, in general, a complex number, there are several ways two represent its dependency on. The two ways we are going to study are: Bode diagrams (two diagrams for magnitude and angle of ); Polar diagrams (real and imaginary parts).
Bode diagrams 37 Hendrik W. Bode (Madison 1905 - Cambridge 1982) Recall the generic form of a transfer function subsitute
38 Bode diagrams cont d We need now to recall some properties of complex numbers. Let be a pair of generic complex numbers. We than have and the following identities:
39 Bode diagrams cont d For the magnitude, we have shown that We can then proceed by computing the magnitude of each term and compute their product/ratio. However, there exists a trick which dramatically semplifies the computation!!
40 Bode diagrams cont d The trick is to compute rather than Thanks to the logarithmic identities, products (ratios) are transformed into sums (differences). The advantage is that we can study each contribution separately, and then sum up the corresponding results.
41 Bode diagrams cont d First contribution: gain This term does not depend on the frequency. Therefore, it can be represented by a constant function (horizontal line).
42 Bode diagrams cont d Before analysing other possible contributions, we introduce another trick which will ease the plot of each component: the logarithmic scale. 0 1 2 3 4 1 10 100 Linear scale (same difference between ticks) Logarithmic scale (same ratio between ticks) Space between to consecutive ticks is called decade
43 Bode diagrams cont d Second contribution: poles/zeros in the origin This time the term does depend on the frequency. Still a line, with slope depending on the type (notice the advantage of the logarithmic scale of the frequency axis). 40 20 0.01 0.1 1 10 100
44 Bode diagrams cont d Third contribution: real pole/zero 40 20 0.01/T 0.1/T 1/T 10/T 100/T
45 Bode diagrams cont d Comment: we are going to represent the asymptotic (approximate) behaviour only, however it is interesting to compute the approximation error (which occurs at frequency 1/T). 40 20 0.01/T 0.1/T 1/T 10/T 100/T
46 Bode diagrams cont d Fourth contribution: complex poles/zeros pairs As we should have understood that corresponding zeros and poles has similar behaviour (just mirrored with respect to the frequency axis), we are going to analyse the case of poles only.
47 Bode diagrams cont d The real diagram lies on top of the asymptotic approximation. This time, the quality of the approximation depends on. In particular: the higher the better approximation. We also have two different behaviours depending on. 0.01/ω n 0.1/ω n 1/ω n 10/ω n 100/ω n -40
48 Bode diagrams cont d All the contributions are then summed up to obtain the diagram. Example: consider the following transfer function 1 40 20 3 0.01 0.1-20 1 10 2-40 4
49 Bode diagrams cont d We then need to sum up all the terms (curves). 1 0.01 3 40 20 0.1 1 10-20 2-40 4 60 40 20 0.01 0.1-20 1 10-40
50 Bode diagrams cont d For the phase, we have As we have done for the magnitude, let s analyse each possible contribution.
51 Bode diagrams cont d First contribution: gain This term does not depend on the frequency. Therefore, it can be represented by a constant function (horizontal line). μ > 0-180 μ < 0
52 Bode diagrams cont d Second contribution: poles/zeros in the origin Still, this term does not depend on the frequency. 180 90 g = -2 g = -1 g = 0-90 g = 1
53 Bode diagrams cont d Third contribution: real pole/zero 90 zero, T > 0 or pole, T < 0 0.1/ T 1/ T 10/ T 100/ T -90 pole, T > 0 or zero, T < 0
54 Bode diagrams cont d Fourth contribution: complex poles/zeros pairs 180 zero, ξ > 0 or pole, ξ < 0 0.1 ω n ω n 10 ω n 100 ω n The smaller ξ the higher slope -180 zero, ξ < 0 or pole, ξ > 0
55 Bode diagrams cont d As for the magnitude, contributions are summed up. Example: consider again the transfer function 90 3 1 0.01 0.1 1 10 2-90 4-180
56 Bode diagrams cont d We then need to sum up all the terms (curves). 90 3 1 0.01 0.1 1 10 2-90 -180 4 0.1 1 0.01-90 -180 10
57 Bode diagrams cont d The procedure we have seen so far, is extremely time consuming. We wonder whether there exists a simpler way. Focus again on the example: 60 40 0.1 1 20 0.01-90 10 0.01 0.1 1 10-180 Magnitude and phase asymptotic diagrams are piecewise linear, changing only when singularities (zeros/poles) are encountered.
58 Bode diagrams cont d We realised the diagram only changes when running into singularities. Therefore we can state the following rules. Magnitude asymptotic diagram: at low frequency (before every singularities apart those in the origin) the diagram has slope and passes through when the frequency of a real pole (zero) is encountered the slope diminishes (increases) of one unit when the frequency of a pair of complex and conjugate poles (zeros) is encountered the slope diminishes (increases) of two units the final slope is equal to the opposite of the relative degree Notice: all slope changes are multiplied by the multiplicity of the corresponding poles/zeros.
59 Bode diagrams cont d Example: focus again on the example slope -1 (g=1) real zero, slope increases of one unit 60 40 20 real poles, slope decreases of two units 0.01 0.1 1 10 final slope -2 (#poles - #zeros = 3 1 = 2)
60 Bode diagrams cont d Phase asymptotic diagram: at low frequency the diagram takes the value when the frequency of a real pole with T>0 (zero with T<0) is encountered, the phase steps downward of 90 when the frequency of a real pole with T<0 (zero with T>0) is encountered, the phase steps upward of 90 when the frequency of a pair of complex and conjugate poles with ξ>0 (zeros with ξ<0) is encountered, the phase steps downward of 180 when the frequency of a pair of complex and conjugate poles with ξ<0 (zeros with ξ>0) is encountered, the phase steps upward of 180 Notice: all step changes are multiplied by the multiplicity of the corresponding poles/zeros.
61 Bode diagrams cont d Example: focus again on the example real zero, T>0 pair of real poles, T>0 0.1 1 gain is positive, pole in the origin 0.01-90 -180 10
62 Linear systems as filters So far, we analysed the behaviour of an LTI system subject to different input singlas (step, impulse, sine wave). There is another class of interesting signals which deserves some further analysis as it highlight some interesting properties of LTI systems. We focus of periodic signals with Fourier series, e.g. and we want to study how an LTI system with transfer function behaves when fed with such an input.
63 Linear systems as filters cont d As the system is linear, we can superpose (sum) the effect of each harmonic contribution. Moreover, assuming an asymptotically stable system we can simply derive the asymptotic behaviour of its output using the frequency respone. Therefore As we somehow expected, if we focus on the magnitude only, each harmonics has been multiplied by the corresponding value of the frequency response.
64 Linear systems as filters cont d For a periodic system, we can scompose the signal into harmonics through the Fourier series for each harmonics compute the frequency response (magnitude and phase) or evaluate them from Bode diagrams apply the frequency response (theorem) to evaluate the asymptotic behaviour sum up all contributions
65 Linear systems as filters cont d Example: consider a square wave with unitary amplitude and frequency as an input of the following LTI system u(t) t
66 Linear systems as filters cont d If we represent the spectrum of the input we obtain 1 3 5 7 which represents the amplitude of the (principal) harmonics of the square wave. Those numbers have to be multipled by the corresponding magnitude of the frequency response.
67 Linear systems as filters cont d Bode diagrams and frequency response (mangnitude)
68 Linear systems as filters cont d What we obtain on the output is as follows (spectrum of the asymptotic output) 1 3 5 7
69 Linear systems as filters cont d The output we obtain is the following one
70 Linear systems as filters cont d Going back to the Bode diagrams we notice that low frequency harmonics are preserved, harmonics around the complex poles are amplified, high frequency harmonics are attenuated.
71 Linear systems as filters cont d As we should have understood from the example, asymptotically stable LTI systems subject to periodic input signals are able to preserve, amplify or even damp on their output the corresponding harmonics of the input. Example: consider
72 Linear systems as filters cont d Based on their frequency response characteristics we can then classify asymptotically stable LTI systems in Low-pass filters (low frequancy harmonics are preserved on the output, higher frequency ones are damped) High-pass filter (high frequency harmonics are preserved on the output, lower frequency ones are damped) Band-pass filters (frequencies within a certain bandwidth are preserved, all the others are damped) Stop-band filters (frequencies within a certain bandwidth are damped, all the others are preserved) All-pass filters (all the frequencies are preserved)
73 Linear systems as filters cont d Examples: Low-pass High-pass Band-stop Band-pass All-pass
74 Systems with time delays We here introduce (and analyse) systems with time delays to complete our modelling capabilities. Observing very long distance space phenomenon is subject to delays (due to light travelling time). Imagine one switches on a bulb in another galaxy, it may conceptually take years for the light to reach the Earth. Neutrinos beam arrived after d/c seconds
75 Systems with time delays cont d Example: a conveyor is transfers material A to a mixer where it is composed with material B. Known the flows of A and B, what is the mass the mixer? A B v A+B
76 Systems with time delays cont d The equation describing the volume inside the mixer is where is the transport delay. Applying the Laplace s transform to the first equation we obtain while Therefore we get remember
77 Systems with time delays cont d In general, input and/or output of a system might be affected by (actuation/measurement) delays, e.g. we can define an additional input. Compute the corresponding transfer function (as usual) and then consider the effect of the delay
78 Systems with time delays cont d How does the frequency response of the delay look like? For any dalay, the frequency response has unitary magnitude and linearly decreasing (negative) phase. It is a particular case of all-pass filter.
79 Systems with time delays cont d Example: Bode diagrams of the delay are as follows:
80 Systems with time delays cont d Example: frequency response of the unit delay:
81 Systems with time delays cont d https://www.youtube.com/watch?v=_fnp37zfn9q
82 Minimum phase systems Given an LTI system and its transfer function the magnitude Bode diagram., we are able to plot We wonder whether we can infer how the phase diagram looks like from the magnitude diagram. This is usually impossible, as a slope change corresponds to either a zero or a pole, but we do not have any further information to decide whether this pole/zero has a positive or negative time constant (or damping). Definition: a transfer function (LTI system) is called minimum phase iff it has a positive gain and no delay; all singularities (zero and poles) are in the closed left-hand half-plane.
83 Minimum phase systems cont d Hence, for a minimum phase transfer function we can always infer the Bode phase diagram from the magnitude diagram, as we know that all singularities are in the left-hand half-plane. A generic transfer function, possibly containing right-hand side singularities and/or delays, can be always written as where is a minimum phase transfer function. Example: All-pass filter
Polar plots 84 Harry T. Nyquist (Nilsby 1889 - Harlingen 1976) We have been focused on Bode diagrams, as a graphic representation of the frequency response of an LTI system. Another way to represent the frequency response are the so called polar diagrams: real and imaginary parts of.
85 Polar plots cont d The basic idea behind the construnction of polar diagrams is to move on the imaginary axis (where lays) and compute the frequency response for each possible frequency. This procedure, quite tedious, can be simplified by means of graphical rules. We won t address this problem and we will try to draw polar diagrams from Bode ones.
86 Polar plots cont d Basic idea: Bode diagrams show magnitude and phase for varying frequency. big magnitude big positive phase small positive phase small magnitude negative phase
87 Polar plots cont d Example: consider the transfer function 20 db Magnitude is (monotonically) decreasing towards zero 1 10-90 Asymptotic value of the phase is -90
88 Polar plots cont d Example: consider the transfer function 20 db 40 db 0.1 1 10 90 10-90
89 Polar plots cont d The method, as seen so far, is not complete as we do not know what to do in case of purely imaginary poles (on imaginary axis). When a transfer function has a pole on the imaginary axis, the magnitude of the frequency response at the corresponding frequency tends to infinity, while the phase is discontinuous. Example: consider the transfer function
90 Polar plots cont d In case of imaginary pole(s), we modify the path on the imaginary axis by going around the pole along a half/quarter circle of infinitesimal radius on the right-hand half-plane. With this trick, the image, through, of this half/quarter anticlockwise infinitesimal circle is a half/quarter clockwise circle with infinite radius. This way the diagram is always connected.
91 Polar plots cont d Example: consider the following transfer function 0.1 1 20 db 10-180 -90 quarter circle with infinite radius Notice: the asymptote
92 Polar plots cont d Example: consider the following transfer function 1 y = mx + q -90 half circle with infinite radius Notice:
93 Stability of feedback system Open-loop control We may look for a controller that inverts the behaviour of the system but there is no way to modify the effect d on the output, moreover: G(s) might be not invertible G(s) might be non-minimum phase Closed-loop control We may look for a controller that makes the first transfer function as small as possible, while the second as closed as possible to be an all-pass filter. We are able to completely modify the behaviour of the system.
94 Stability of feedback system So far, we have been focusing on how to represent the frequency response of an LTI system. The two methods we study (Bode and polar plots) has tremendous implication on the stability assessment of feedback system. Let s first introduce the problem. Given a transfer function characterizing the system to be controlled consider the following diagram:
95 Stability of feedback system cont d There are two main problems concerning this kind of (feedback) systems: is the overall (closed- ANALYSIS: given the transfer function loop) system stable? SYNTHESIS: can we find a transfer function closed-loop system is stable? such that the We will first focus on the former problem.
96 Stability of feedback system cont d As is given (known) we can introduce the loop transfer function as follows. Given, what can we say about the stability of the closed-loop system? Are we already able to answer this question?
97 Stability of feedback system cont d Example: consider the loop transfer function which, as you may notice, is unstable. To study the stability of the overall system, let s compute its transfer function. The overall (closed-loop) system is asymptotically stable as the two poles are in the left open half-plane.
98 Stability of feedback system cont d Focus on the definition of the closed-loop transfer function and recall that, being a transfer function itself, is the ratio between two polynomials. Then we can write: Then, in general, we should focus on the root of the closed-loop characteristic polynomial.
99 Stability of feedback system cont d Notice that the polynomial has the same degree of since the loop transfer function cannot have more zeros than poles. For small order systems, we can explicitly compute the its roots (closed-loop poles), for higher order ones we can adopt the Routh criterion to address the stability of the closed-loop system. However, though being a complete method, it is not suited for synthesis problems. We need something else
100 Nyquist criterion Consider, once again, the closed-loop system We are looking at some methodology to address the stability of the closed-loop system directly from inpection of. Definition: we define the Nyquist diagram as the polar plot of for all. Notice that we define the polar plot for only. How can we draw the Nyquist plot from the corresponding polar diagram?
101 Nyquist criterion cont d Property: consider., then This property states that we can draw the Nyquist plot directly from the polar diagram, with no additional effort, by simply taking its mirrored version with respect to the real axis (e.g. by changing the sign of the imaginary part). Polar plot Nyquist diagram
102 Nyquist criterion cont d As we defined the Nyquist diagram we can state the main result. Result: the closed-loop system having asymptotically stable iff N = P, where P 0 is the number of poles of as loop transfer function is in the open right half-plane; N is the signed number of encirclements (positive when anticlockwise) around the point (-1,0) Provided that N is well defined (diagram does not pass through that point). Recall that, as we defined the polar diagram, the Nyquist diagram is always a closed curve, possibly closing at infinity.
103 Nyquist criterion cont d Examples: P = 1 (open-loop unstable) P = 0 (open-loop stable) N = 1, closed-loop asymptotically stable (as N = P) N = -2, closed-loop not asymptotically stable (as N P) Remark: feedback can stabilize (destabilize) open-loop unstable (stable) systems.
104 Nyquist criterion cont d Extensions: valid when P = 0 Small-gain theorem: if, then the closed-loop system is asymptotically stable Proof: all the Nyquist diagram stays inside the unit circle (N = 0) Small-phase theorem: if, then the closedloop system is asymptotically stable Proof: the Nyquist plot never crosses the real negative axis (N = 0) Im Im -1 Re -1 Re
105 Nyquist criterion cont d Consider an LTI system with time delay Result: the Nyquist result still holds provided that P is evaluated with respect to, whilst N is still computed from. This is the only result we have (so far) to address the stability of closedloop delayed systems! We will see how delays can affect the stability of a closed-loop system.
106 Nyquist criterion cont d Example: consider the following transfer function τ = 0, N = 0 τ = 0.1, N = 0 τ = 1, N = -2
107 Bode criterion We are about to introduce two additional useful criteria, which hold only in the case P = 0. As a consequence of the Nyquist criterion, when P = 0, N must be 0, as well in order to have closed-loop asymptotic stability. You shall not pass!
108 Bode criterion cont d Let s analyse what might happen around the point (-1,0). In particular how can we measure how far the diagram is from passing through that point? There are at least two major ways to quantify this distance.
109 Bode criterion cont d Gain margin criterion: assume has a positive gain, i.e. and that the corresponding polar plot intersects once the negative real semi-axis, then the closed loop system is asymptotically stable iff where P = 0 The gain margin is a measure of how far the diagram is from passing through (-1,0), the higher the more unlike to get unstable.
110 Bode criterion cont d This criterion can be easily checked from Bode diagrams.
111 Bode criterion cont d Phase margin criterion: assume has a positive gain, i.e. and that the corresponding polar crosses only once the unit circle from outside, the the closed-loop system is asymptotically stable iff where P = 0 The phase margin is a measure of how far the diagram is from passing through (-1,0), the higher the more unlike to get unstable.
112 Bode criterion cont d Also this criterion can be easily checked from Bode diagrams.
113 Bode criterion cont d The most important result we are going to use next is the following one. Bode criterion: given a loop transfer function P = 0 such that the magnitude Bode diagram crosses (from top) the 0 db axis exactly once then, the closed-loop system is asymptotically stable iff This criterion is extremely useful in synthesis problems, as we will see next.