and the composite system output is y(t) = y 1 (t). We thus obtain (24.146) (24.147) Parallel Connection The system interconnection shown in Fig. 24.11 is known as a parallel connection. To obtain the desired state space model we observe that the input is u(t) = u 1 (t) = u 2 (t) and the output for the whole system is y(t) = y 1 (t) + y 2 (t). We obtain (24.148) (24.149) Feedback Connection The system interconnection shown in Fig. 24.12 is known as feedback connection (with unit negative feedback), and it corresponds to the basic structure of a control loop, where S 1 is the plant and S 2 is the controller. To build the composite state space model we observe that the overall system input satisfies the equation u(t) = u 2 (t) + y 1 (t), and the overall system output is y(t) = y 1 (t). Furthermore, we assume FIGURE 24.11 Parallel connection. FIGURE 24.12 Feedback connection. x ˙ 1 t() x ˙ 2 t() A 1 B 1 C 2 0 A 2 x 1 t() x 2 t() B 1 D 2 B 2 u t()+= y t() C 1 D 1 C 2 [] x 1 t() x 2 t() D 1 D 2 [] u t()+= x ˙ 1 t() x ˙ 2 t() A 1 0 0A 2 x 1 t() x 2 t() B 1 B 2 u t()+= y t() C 1 C 2 [] x 1 t() x 2 t() D 1 D+ 2 [] u t()+= x (t) 1 y (t) 1 y (t) 2 u (t) 2 u (t) 1 y(t)u(t) 2 x (t) + + x (t) 12 x (t) y (t) 1 y(t)u(t) u (t) 2 y (t) 2 1 u (t) + − 0066_Frame_C24 Page 25 Thursday, January 10, 2002 3:45 PM ©2002 CRC Press LLC and the composite system output is y(t) = y 1 (t). We thus obtain (24.146) (24.147) Parallel Connection The system interconnection shown in Fig. 24.11 is known as a parallel connection. To obtain the desired state space model we observe that the input is u(t) = u 1 (t) = u 2 (t) and the output for the whole system is y(t) = y 1 (t) + y 2 (t). We obtain (24.148) (24.149) Feedback Connection The system interconnection shown in Fig. 24.12 is known as feedback connection (with unit negative feedback), and it corresponds to the basic structure of a control loop, where S 1 is the plant and S 2 is the controller. To build the composite state space model we observe that the overall system input satisfies the equation u(t) = u 2 (t) + y 1 (t), and the overall system output is y(t) = y 1 (t). Furthermore, we assume FIGURE 24.11 Parallel connection. FIGURE 24.12 Feedback connection. x ˙ 1 t() x ˙ 2 t() A 1 B 1 C 2 0 A 2 x 1 t() x 2 t() B 1 D 2 B 2 u t()+= y t() C 1 D 1 C 2 [] x 1 t() x 2 t() D 1 D 2 [] u t()+= x ˙ 1 t() x ˙ 2 t() A 1 0 0A 2 x 1 t() x 2 t() B 1 B 2 u t()+= y t() C 1 C 2 [] x 1 t() x 2 t() D 1 D+ 2 [] u t()+= x (t) 1 y (t) 1 y (t) 2 u (t) 2 u (t) 1 y(t)u(t) 2 x (t) + + x (t) 12 x (t) y (t) 1 y(t)u(t) u (t) 2 y (t) 2 1 u (t) + − 0066_Frame_C24 Page 25 Thursday, January 10, 2002 3:45 PM ©2002 CRC Press LLC 25 Response of Dynamic Systems 25.1 System and Signal Analysis Continuous Time Systems • Discrete Time Systems • Laplace and z -Transform • Transfer Function Models 25.2 Dynamic Response Pulse and Step Response • Sinusoid and Frequency Response 25.3 Performance Indicators for Dynamic Systems Step Response Parameters • Frequency Domain Parameters 25.1 System and Signal Analysis In dynamic system design and analysis it is important to predict and understand the dynamic behavior of the system. Examining the dynamic behavior can be done by using a mathematical model that describes the relevant dynamic behavior of the system in which we are interested. Typically, a model is formulated to describe either continuous or discrete time behavior of a system. The corresponding equations that govern the model are used to predict and understand the dynamic behavior of the system. A rigorous analysis can be done for relatively simple models of a dynamic system by actually computing solutions to the equations of the model. Usually, this analysis is limited to linear first and second order models. Although limited to small order models, the solutions tend to give insight in the typical responses of a dynamic system. For more complicated, higher order and possibly nonlinear models, numerical simulation tools provide an alternative for the dynamic system analysis. In the following we review the analysis of linear models of discrete and continuous time dynamic systems. The equations that describe and relate continuous and discrete time behavior are presented. For the analysis of continuous time systems extensive use is made of the Laplace transform that converts linear differential equations into algebraic expressions. For similar purposes, a z -transform is used for discrete time systems. Continuous Time Systems Models that describe the linear continuous time dynamical behavior of a system are usually given in the form of differential equations that relate an input signal u ( t ) to an output signal y ( t ). The differential equation of a time invariant linear continuous time model has the general format (25.1)a j d j dt j yt() j=0 n a ∑ b j d k dt j ut() j=0 n b ∑ = Raymond de Callafon University of California 0066_Frame_C25 Page 1 Wednesday, January 9, 2002 7:05 PM ©2002 CRC Press LLC 26 The Root Locus Method 26.1 Introduction 26.2 Desired Pole Locations 26.3 Root Locus Construction Root Locus Rules • Root Locus Construction • Design Examples 26.4 Complementary Root Locus 26.5 Root Locus for Systems with Time Delays Stability of Delay Systems • Dominant Roots of a Quasi- Polynomial • Root Locus Using Padé Approximations 26.6 Notes and References 26.1 Introduction The root locus technique is a graphical tool used in feedback control system analysis and design. It has been formally introduced to the engineering community by W. R. Evans [3,4], who received the Richard E. Bellman Control Heritage Award from the American Automatic Control Council in 1988 for this major contribution. In order to discuss the root locus method, we must first review the basic definition of bounded input bounded output (BIBO) stability of the standard linear time invariant feedback system shown in Fig. 26.1, where the plant, and the controller, are represented by their transfer functions P ( s ) and C ( s ), respectively. 1 The plant, P ( s ), includes the physical process to be controlled, as well as the actuator and the sensor dynamics. The feedback system is said to be stable if none of the closed-loop transfer functions, from external inputs r and v to internal signals e and u , have any poles in the closed right half plane, . A necessary condition for feedback system stability is that the closed right half plane zeros of P ( s ) (respectively C ( s )) are distinct from the poles of C ( s ) (respectively P ( s )). When this condition holds, we say that there is no unstable pole–zero cancellation in taking the product P ( s ) C ( s ) =: G ( s ), and then checking feedback system stability becomes equivalent to checking whether all the roots of (26.1) are in the open left half plane, . The roots of (26.1) are the closed-loop system poles. We would like to understand how the closed-loop system pole locations vary as functions of a real parameter of G ( s ). More precisely, assume that G ( s ) contains a parameter K , so that we use the notation 1 Here we consider the continuous time case; there is essentially no difference between the continuous time case and the discrete time case, as far as the root locus construction is concerned. In the discrete time case the desired closed-loop pole locations are defined relative to the unit circle, whereas in the continuous time case desired pole locations are defined relative to the imaginary axis. ރ + := s ރŒ : Re s() 0≥{} 1 Gs() 0=+ ރ - := s ރ : Re(s)∈ 0<{ } Hitay Özbay The Ohio State University ©2002 CRC Press LLC 27 Frequency Response Methods 27.1 Introduction 27.2 Bode Plots 27.3 Polar Plots 27.4 Log-Magnitude Versus Phase plots 27.5 Experimental Determination of Transfer Functions 27.6 The Nyquist Stability Criterion 27.7 Relative Stability 27.1 Introduction The analysis and design of industrial control systems are often accomplished utilizing frequency response methods. By the term frequency response, we mean the steady-state response of a linear constant coefficient system to a sinusoidal input test signal. We will see that the response of the system to a sinusoidal input signal is also a sinusoidal output signal at the same frequency as the input. However, the magnitude and phase of the output signal differ from those of the input signal, and the amount of difference is a function of the input frequency. Thus, we will be investigating the relationship between the transfer function and the frequency response of linear stable systems. Consider a stable linear constant coefficient system shown in Fig. 27.1. Using Euler’s formula, e j ω t = cos ω t + j sin ω t , let us assume that the input sinusoidal signal is given by (27.1) Taking the Laplace transform of u ( t ) gives (27.2) The first term in Eq. (27.2) is the Laplace transform of U 0 cos ω t , while the second term, without the imaginary number j , is the Laplace transform of U 0 sin ω t . Suppose that the transfer function G ( s ) can be written as (27.3) ut() U 0 e jwt U 0 wtcos jU 0 wtsin+== Us() U 0 sjw– U 0 sjw+ s 2 w 2 + U 0 s s 2 w 2 + j U 0 w s 2 w 2 + +== = Gs() ns() ds() ns() sp 1 +()sp 2 +() … sp n +() == Jyh-Jong Sheen National Taiwan Ocean University 0066_frame_C27 Page 1 Wednesday, January 9, 2002 7:10 PM ©2002 CRC Press LLC . understand the dynamic behavior of the system. Examining the dynamic behavior can be done by using a mathematical model that describes the relevant dynamic behavior of the system in which we are. s ), and then checking feedback system stability becomes equivalent to checking whether all the roots of (26.1) are in the open left half plane, . The roots of (26.1) are the closed-loop system poles We will see that the response of the system to a sinusoidal input signal is also a sinusoidal output signal at the same frequency as the input. However, the magnitude and phase of the output signal