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Advanced Control of Continuous-Time LTI-SISO Systems

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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. 1.1.1 Ordinary differential equations An ordinary differential equation describes the dynamic relation between the input u(t) and output y(t) of a LTI-SISO system. It takes on the general form (n) (n−1) (q) y (t) + an−1y (t) + ··· + a1y˙(t) + a0y(t) = bqu (t) + ··· + b1u˙(t) + b0u(t) , (1.1) 11 i i i i i “MAIN” — 2015/11/19 — 9:36 — page 12 — #12 i i i 12 CHAPTER 1. REVIEW OF LINEAR SYSTEMS THEORY with the initial conditions (n−1) y (0) = y0n, ··· , y(0) = y00, (1.2) (q) u (0) = u0q, ··· , u(0) = u00, (1.3) and with the convention dny(t) = y(n)(t). dtn The value y(n)(0) is not required in (1.2), since it can be calculated by (1.1) when (1.2) and (1.3) are given. Causality. In order to describe a technically realizable system, n ≥ q must hold. If this is not the case, and n < q, then y(t) depends on future values of u(t), which is physically impossible. This can be made more clear considering the example where n = 0 and q = 1. A certain choice of constants could then lead to the differential equation y(t) =u ˙(t). The problem with this equation is that in order to determine the output y(t0) at a certain time t0, a pure differentiation of the input signal at time t0 is necessary (see Fig 1.2). The physical realization of a pure differentiation requires knowledge of the value of u(t) for t > t0, which is physically impossible. u(t) u_(t0) =? t0 t Figure 1.2: To evaluate the output y(t0) of the system y(t) =u ˙(t), an instantaneous evaluation of u˙(t0) is needed. This evaluation is not possible without any information about u(t) for t > t0. Characteristic polynomial. The characteristic polynomial p(λ) of an ODE of the form (1.1), is given by n n−1 p(λ) = λ + an−1λ + ··· + a1λ + a0 = 0 . (1.4) (k) Notice that the characteristic polynomial only depends on the coefficients ak of the derivatives y (t) in (1.1) and not of the coefficients bk. The characteristic polynomial can be derived by using the homogeneous solution approach y(t) = ceλt (for uq(0) = 0, ··· , u(0) = 0 and c a constant) in the λt ODE. Thus, the solutions λi of 1.4 yield together with the approach y(t) = ce a solution for the homogenous ODE. The λi are called also called the roots of p(λ). These roots play an essential role in the analysis of the system dynamics, since they directly determine the natural response and transient response of the system. These partial solutions of the ODE are discussed next. i i i i i “MAIN” — 2015/11/19 — 9:36 — page 13 — #13 i i i 1.1. DESCRIPTIONS AND PROPERTIES OF LINEAR SYSTEMS 13 Basic ODE solution structure. The solution y(t) of an ODE is the sum of three partial solutions: the natural response, the transient response and the steady state response. This can be seen by considering inputs of the type X X i µmt u(t) = u¯i,mt e , (1.5) i m where µm and u¯i,m are real numbers or complex conjugate pairs (this ensures that u(t) is a real signal and has no imaginary part). Nearly all interesting input signals can be described by this class of inputs. In the following, some examples of inputs of this form are shown (see also Fig. 1.3). The convention is that u(t) = 0 for t < 0: 0t Step: i = 0, µm = 0 u(t) =u ¯1e =u ¯1σ(t) 0t Ramp: i = 1, µm = 0 u(t) =u ¯1te =u ¯1t jωt −jωt Sinusoids: i = 0, µm = ±jω u(t) = e + e = 2 cos(ωt) (−a+jω)t (−a−jω)t −at Damped Sinusoids: i = 0, µm = −a ± jω u(t) = e + e = 2e cos(ωt) u(t) u(t) u(t) u(t) t t t t (a) Step input (b) Ramp input (c) Sinusoidal input (d) Damped sinusoid P P i µmt Figure 1.3: Examples of inputs of the type u(t) = i j u¯i,mt e . For every input u(t) of the type above, the output y(t) can be computed analytically. It can be shown that the output consists of a natural response ynat(t) that appears due to the initial value of the system, and of a forced response yforced(t) that is caused by the input signal. The forced response itself consists of the transient response ytrans(t) and the steady state response yss(t). y(t) = ynat(t) + yforced(t) (1.6) | {z } | {z } natural response forced response (depending on initial conditions (depending on input but not on system input) but not on initial conditions) = ynat(t) + ytrans(t) + yss(t) (1.7) X λit X λit X X i µmt = y¯nati e + y¯transi e + y¯ssi,m t e . (1.8) i i i m It can be shown that the coefficients y¯nati , y¯transi and y¯ssi,m do not depend on time. Hence, the λit time behavior is only determined by the terms e (with λi are the solutions of the characteristic polynomial) and the terms tieµmt of the input. An important consequence is that if the real part of all λi is negative, then the natural response and the transient response will go to zero for t → ∞. The system output for t → ∞ is then solely represented by the steady state response. However, if at i i i i i “MAIN” — 2015/11/19 — 9:36 — page 14 — #14 i i i 14 CHAPTER 1. REVIEW OF LINEAR SYSTEMS THEORY least one λi has a positive real part, the natural and transient response will not die out and will grow unbounded. In addition, it is interesting to note that the coefficients y¯nati , y¯transi and y¯ssi,m can be interpreted as the intensity or the "amplitude" of the time-dependent exponential terms. It turns out that y¯transi depends on the input parameters u¯i,m and µm but not on the initial conditions (1.2) and (1.3). Hence, the transient behavior is triggered by the input, but evolves similarly to the natural response. The example of a (linearised) pendulum system can be used to illustrate the different components of the output response y(t). In this example the output is defined as the angle between the pendulum and the vertical axis, as shown in Fig. 1.4a. Suppose now that the pendulum has an initial non-zero displacement from its equilibrium position. Apart from gravity, no external forces are present. When released from its initial position, the pendulum will start an oscillating motion, that will fade out if any friction is present, until equilibrium is reached. This oscillating motion is described by the natural response ynat(t) (see Fig 1.4b) and is solely a function of the initial displacement and of the roots λi of the characteristic polynomial. ynat(t) y(t) t (a) Non-zero initial state (b) Natural response Figure 1.4: Natural response of a pendulum system. Suppose now that the initial displacement of the pendulum is zero (hence the natural response is zero), but that it is placed in a homogeneous wind field, that exerts an external force u(t) on the pendulum, as shown in Fig. 1.5a. The pendulum will be displaced from its equilibrium, following an oscillating motion, until the new equilibrium point is reached. This motion is described by the forced response yforced(t), shown in Fig. 1.5b. The transient response ytrans(t) describes the oscillating motion around the new equilibrium point, and the steady state response y(t) describes precisely this equilibrium point. Linearization. The state equations of a physical systems are in general nonlinear.
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