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Stability Analysis of Continuous-Time Linear Time-Invariant Systems Kam Modjtahedzadeh Department of Physics, UC Santa Barbara

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Stability Analysis of Continuous-Time Linear Time-Invariant Systems Kam Modjtahedzadeh Department of Physics, UC Santa Barbara Stability Analysis of Continuous-Time Linear Time-Invariant Systems Kam Modjtahedzadeh Department of Physics, UC Santa Barbara Abstract This paper focuses on the mathematical approaches to the analysis of stability that is a crucial step in the design of dynamical systems. Three methods are presented, namely, absolutely integrable impulse response, Fourier integral, and Laplace transform. The superiority of Laplace transform over the other methods becomes clear for several reasons that include the following: 1) It allows for the analysis of the stable, as well as, the unstable systems. 2) It not only determines absolute stability (a yes/no answer), but also shines light on the relative stability (how stable/unstable the system is), allowing for a design with a good degree of stability. 3) Its algebraic and convolution properties significantly simplify the mathematical manipulations involved in the analysis, especially when tackling a complex system composed of several simpler ones. A brief relevant introduction to the subject of systems is presented for the unfamiliar reader. Additionally, appropriate physical concepts and examples are presented for better clarity. Introduction later work of Kalman. Current research includes 퐻∞ (H infinity) and Hardy space [6]. However, Stability is a vast subject in the field of signals coverage of such advanced methods is beyond the and systems, in general, and, controls, in scope of this paper. particular. All controlled dynamical systems, being biological, chemical, physical, nuclear, or System Classifications of any other domain, must have a good degree of stability. The Three Mile Island and Chernobyl Strictly speaking, there are no linear-time catastrophes that occurred due to the process’s invariant (LTI) systems in reality. Elements of the temperatures going out of control are still fresh in systems are functions of the dependent, our memories. independent (e.g., time), or both variables. However, such effects become minimal when the Seldom, dynamical systems are intentionally systems operate near their equilibrium points, and made unstable. That adds an extra layer of temperature variation and other environmental difficulty in the design of the systems for effects are kept within a range. Thus the LTI acceptable overall stability [5]. Examples are models of the systems results in ordinary Segway, essentially an inverted pendulum for fun differential equations (ODE) with constant and space efficiency, and jet fighter planes, for coefficients. high maneuverability. This paper only concerns the continuous-time LTI Considering the degree of complexity of today’s systems. Although digital controllers have been systems, stability issue is a huge liability and a replacing the analog ones, the concepts headache for the control engineer; hence, the developed for continuous domain are extendable need for relevant and powerful mathematical to the discrete domain in the framework of z- tools. transform [3], [4]. A lot has been developed by control theoreticians Another classification of the systems is causality [4] such as earlier works of Bode, Nichol, and (causal versus non-causal). Almost all real Nyquist, building upon Laplace transform, and systems are causal as the output depends on the 1 1 1 past and present inputs but not on the future ℎ(푡) = 푒−푎푡 푢(푡), 푎 = inputs. However, time advance (positive time 푅 푅퐶 1 1 shift), such as preview, and anticipation is 퐻(푠) = 픏{ℎ(푡)} = ( ) , 퐼퐶 = 0 actually employed in practice and is non-causal. 푅 푠+푎 푄(푠) 1 1 Also, 퐻(푠) = = ( ) , 퐼퐶 = 0 Impulse Response and Transfer 푉(푠) 푅 푠+푎 Function An alternative name for the transfer function is As the name implies, the impulse response, ℎ(푡), system function. Note that the impulse response of an LTI system is defined as the output when and the transfer function characterize the system the input is a unit-impulse function, 푢(푡). and are neither input nor IC dependent; hence the Practical impulse resembling inputs include, name system function. force and voltage gendered by a hammer stroke A transfer function is a rational function when and fast on-off switching respectively. both the numerator and the denominator are The transfer function, 퐻(푠), of an LTI system is polynomials. The roots of the denominator and defined as the Laplace transform of the impulse the numerator are called poles and zeros response, with all the initial conditions (IC) set to respectively. The denominator polynomial set to zero. Alternatively, it can be defined as the zero forms the characteristic equation of the Laplace transform of the output over the Laplace system (the roots, or poles, determine the types of transform of the input, when 퐼퐶 = 0. functions in the partial fraction expansion). Laplace transform is covered in a dedicated Summarizing the system of figure 1, it is a causal section later in this paper. However, some of its continuous-time LTI system with a rational basic operations are employed in this section, in transfer function. Throughout the rest of this order to accomplish rudimentary system concepts paper, the assumptions of continuous-time linear all at once in the beginning. time-invariant will remain. Consider the RC circuit of figure 1 when a voltage source, 푣(푡), is placed in the loop, as the input, and the capacitor charge, 푞(푡), as the output. The parameters are constants for the range of operation and with respect to time. The voltage Fig. 2, Block Diagram of a Basic System source provides a continuous signal. The system is causal as it is a real practical circuit. Systems are often depicted by their corresponding block diagrams, showing their transfer functions, impulse responses, or even Fourier integrals in the blocks. For instance, figure 2 represents a simple system such as that of figure 1. The solution for the output , 푦(푡), of figure 2 via Fig. 1, RC Circuit impulse response is through the convolution The corresponding ODE and Laplace transform integral of the input, 푥(푡), and the impulse equations, when 퐼퐶 = 0, are: response, ℎ(푡): 푑 1 ( ) ∞ ( ) ( ) 푅 푞(푡) + 푞(푡) = 푣(푡) 푦 푡 = ∫−∞ 푥 푡 − 휏 ℎ 휏 푑휏 푑푡 퐶 ∞ 1 = 푥(휏)ℎ(푡 − 휏)푑휏 푅푆푄(푠) + 푄(푠) = 푉(푠) ∫−∞ 퐶 The integration becomes cumbersome for The impulse response and the transfer function complicated functions. A much more efficient are obtained according to their definitions: and systematic solution is by the Laplace 2 ∞ transform method, applying its convolution 푦(푡) = ∫ ℎ(휏)푥(푡 − 휏)푑휏 property: −∞ ∞ 푦(0) = ∫ ℎ(휏)푠푔푛(ℎ(휏))푑휏 푌(푠) = 퐻(푠)푋(푠) −∞ ∞ | | Inverse Laplace transform can be performed = ∫−∞ ℎ(휏) 푑휏 = ∞ subsequently with ease, with the aid of partial fraction expansion and an inverse table. As an example, the exponentially decaying However, the concept behind convolution is impulse response of the RC circuit is absolutely insightful, and also is employed in derivations integrable and stable. However, had the exponent of the impulse response been positive, then the and proofs. system would be neither absolutely integrable nor Absolutely Integrable Impulse stable. A negative resistor, i.e., physically a Response voltage source in phase with the input voltage source, would put energy into the system rather Bounded input bounded output (BIBO) stability than damping out energy, so causing the definition has been vastly accepted as the exponentially growing output. criterion for the stability of systems [3], [4]. It states that: a system is stable, if for every bounded Note that the absolutely integrable method only input the corresponding output is bounded. In reveals the absolute stability, but not the relative other words, the system is stable if the output is stability. finite for all possible finite inputs. Fourier integral For the particular case of continuous-time LTI Spectrum, 푋(푗휔), of a continuous-time systems, it can be proven that a system is (BIBO) aperiodic signal, 푥(푡), is obtained by the Fourier stable, if and only if, the impulse response ℎ(푡) is integral, also known as the Fourier transform absolutely integrable. of 푥(푡), if the integral converges: ∞ |ℎ(푡)|푑푡 < ∞ ∞ ∫−∞ ( ) { ( )} ( ) −푗휔푡 푋 푗휔 = ℱ 푥 푡 = ∫−∞ 푥 푡 푒 푑푡 The sufficiency part of the proof is obtained by The transformation is valid, provided that 푥(푡) manipulating the absolute value of the can be reconstructed accurately by an integral convolution integral: called the synthesis equation [1]: ∞ |푦(푡)| = |∫ ℎ(휏)푥(푡 − 휏)푑휏| 1 ∞ −∞ 푥(푡) = ∫ 푋(푗휔)푒푗휔푡푑휔 2휋 −∞ ∞ ≤ ∫ |ℎ(휏)푥(푡 − 휏)푑휏| −∞ The synthesis and the spectrum equations form ∞ the Fourier transform pair. A sufficient (but not ≤ ∫ |ℎ(휏)||푥(푡 − 휏)|푑휏 −∞ necessary) condition for the validity of the ∞ transformation is, if 푥(푡) is square integrable ≤ 퐵 ∫ |ℎ(휏)|푑휏 ≤ 퐵 푥 −∞ 푦 [1]: Where 퐵 and 퐵 are the input and output bounds ∞ 푥 푦 ∫ |푥(푡)|2푑푡 < ∞ respectively and are positive numbers. −∞ For the necessity part of the proof, it can be Alternatively, 푥(푡) should be absolutely shown that if the integral is not bounded, then integrable [1] (as one of the 3 Dirichlet there exists at least a bounded input that will drive conditions, where the other 2 are normally the output out of bound. satisfied for practical signals and systems). ∞ 1 ℎ(−푡) > 0 ∫−∞ |푥(푡)|푑푡 < ∞ 푥(푡) = 푠푔푛(ℎ(−푡)) = { 0 ℎ(−푡) = 0 −1 ℎ(−푡) < 0 3 The alternative condition also implies that any working in frequency domain (i.e., Bode, Nichol, stable system possesses Fourier transform of its and Nyquist methods not covered here) [4]. impulse response. Bilateral Laplace Transform The RC circuit of Figure 1 has an absolutely integrable, as well as a square integrable, impulse Bilateral Laplace transform of a continuous-time response. Thus, its Fourier transform exists and signal, 푥(푡), is defined as: is: { ( )} ∞ −푠푡 푋(푠) = 픏 푥 푡 = ∫−∞ 푥(푡)푒 푑푡 1 1 1 퐻(푗휔) = ( ) , 푎 = > 0 푅 푗휔+푎 푅퐶 It converts 푥(푡) into the complex function 푋(푠) of complex variable 푠 = 휎 + 푗휔, provided that The Fourier transform would not exist for if the the integral exists.
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