《电子工程师手册》学习资料（英文版）Chapter 100 Control Systems.pdf_大学文库

100 Control Systems 100.1 Models University of Nevada, Las vegas Classes of Systems to Be Modeled. Two Major Approaches to Gordon K e lee Modeling. Forms of the Model. Nonuniqueness Approximation of Continuous Systems by Discrete Models 100.2 Dynamic Response Andrew P. Sage Computing the Dynamic System Response. Measures of the George Mason University Dynamic System Response 100.3 Frequency Response Methods: Bode Diagram Approach enjamin C. Kuo Frequency Response Analysis Using the Bode Diagram. Bode iversity of Illinois (Urbana Diagram Design-Series Equalizers. Composite Equalizers Charles L. Phillips 100.4 Root locus Auburn University Root Locus Properties. Root Loci of Digital Control Systems Design with Root Locus Royce D. Harbor nsation University of West Florida Control System Specifications. Design. Modern Control Raymond G. Jacquot Design. Other Modern Design Procedures 100.6 Digital Control Systems A Simple Example . Single-Loop Linear Control Laws John E. McInroy Proportional Control. PID Control Algorithm. The Closed Loop System. A Linear Control Example 100.7 Nonlinear Control System Derek P Atherton The Describing Function Method. The Sinusoidal Describing Function. Evaluation of the Describing FI Limit Cycles and Stability. Stability and Accuracy. Compensator John S Bay Design.Closed-Loop Frequency Response. The Phase Pla Virginia Polytechnic Institute and Method. Piecewise Linear Characteristics Discussion State University 100.8 Optimal Control and Estimation William. Baumann Linear Quadratic Optimal Estimation: The Kalman Virginia Polytechnic Institute and Filter. Linear-Quadratic-Gaussian(LQG)Control. H- Control· Example· Other Approaches 100.9 Neural Control Mo- Yuen Chow Brief Introduction to Artificial Neural Networks. Neural North Carolina State University Observer· Neural control· HVAC Ilustration· Conclusion 100.1 Models William L. Brogan A naive trial-and-error approach to the design of a control system might consist of constructing a controller, installing it into the system to be controlled, performing tests, and then modifying the controller until satisfactory performance is achieved. This approach could be dangerous and uneconomical, if not impossible. A more rational approach to control system design uses mathematical models. A model is a mathematical description of system havior, as influenced by input variables or initial conditions. The model is a stand-in for the actual system during the control system design stage. It is used to predict performance; to carry out stability, sensitivity, and trade-off c 2000 by CRC Press LLC

© 2000 by CRC Press LLC 100 Control Systems 100.1 Models Classes of Systems to Be Modeled • Two Major Approaches to Modeling • Forms of the Model • Nonuniqueness • Approximation of Continuous Systems by Discrete Models 100.2 Dynamic Response Computing the Dynamic System Response • Measures of the Dynamic System Response 100.3 Frequency Response Methods: Bode Diagram Approach Frequency Response Analysis Using the Bode Diagram • Bode Diagram Design-Series Equalizers • Composite Equalizers • Minor-Loop Design 100.4 Root Locus Root Locus Properties • Root Loci of Digital Control Systems • Design with Root Locus 100.5 Compensation Control System Specifications • Design • Modern Control Design • Other Modern Design Procedures 100.6 Digital Control Systems A Simple Example • Single-Loop Linear Control Laws • Proportional Control • PID Control Algorithm • The ClosedLoop System • A Linear Control Example 100.7 Nonlinear Control Systems The Describing Function Method • The Sinusoidal Describing Function • Evaluation of the Describing Function • Limit Cycles and Stability • Stability and Accuracy • Compensator Design • Closed-Loop Frequency Response • The Phase Plane Method • Piecewise Linear Characteristics • Discussion 100.8 Optimal Control and Estimation Linear Quadratic Regulators • Optimal Estimation: The Kalman Filter • Linear-Quadratic-Gaussian (LQG) Control • H∞ Control • Example • Other Approaches 100.9 Neural Control Brief Introduction to Artificial Neural Networks • Neural Observer • Neural Control • HVAC Illustration • Conclusion 100.1 Models William L. Brogan A naive trial-and-error approach to the design of a control system might consist of constructing a controller, installing it into the system to be controlled, performing tests, and then modifying the controller until satisfactory performance is achieved. This approach could be dangerous and uneconomical, if not impossible. A more rational approach to control system design uses mathematical models. A model is a mathematical description of system behavior, as influenced by input variables or initial conditions. The model is a stand-in for the actual system during the control system design stage. It is used to predict performance; to carry out stability, sensitivity, and trade-off William L. Brogan University of Nevada, Las Vegas Gordon K. F. Lee North Carolina State University Andrew P. Sage George Mason University Benjamin C. Kuo University of Illinois (UrbanaChampaign) Charles L. Phillips Auburn University Royce D. Harbor University of West Florida Raymond G. Jacquot University of Wyoming John E. McInroy University of Wyoming Derek P. Atherton University of Sussex John S. Bay Virginia Polytechnic Institute and State University William T. Baumann Virginia Polytechnic Institute and State University Mo-Yuen Chow North Carolina State University

arameter Stochastic Deterministic Discrete Nonlinear Linear arying coefficient FIGURE 100.1 Major classes of system equations. Source: W.L. Brogan, Modern Control Theory 3rd ed, Englewood Cliffs, N J. Prentice-Hall, 1991, P. 13. With permission.) studies; and answer various"what-if" questions in a safe and efficient manner. Of course, the validation of the model, and all conclusions derived from it, must ultimately be based upon test results with the physical hardwar The final form of the mathematical model depends upon the type of physical system, the method used to develop the model, and mathematical manipulations applied to it. These issues are discussed next Classes of Systems to Be Modeled Most control problems are multidisciplinary. The system may consist of electrical, mechanical, thermal, optical, fluidic, or other physical components, as well as economic, biological, or ecological systems. Analogies exist between these various disciplines, based upon the similarity of the equations that describe the phenomena. The discussion of models in this section will be given in mathematical terms and therefore will apply to several disciplines. Figure 100.1 [Brogan, 1991] shows the classes of systems that might be encountered in control systems modeling. Several branches of this tree diagram are terminated with a dashed line indicating that additional branches have been omitted, similar to those at the same level on other paths. Distributed parameter systems have variables that are functions of both space and time(such as the voltage along a transmission line or the deflection of a point on an elastic structure). They are described by partial differential equations. These are often approximately modeled as a set of lumped parameter systems(described by ordinary differential or difference equations) by using modal expansions, finite element methods, or other appro mations [Brogan, 1968]. The lumped parameter continuous-time and discrete-time families are stressed here. Two Major Approaches to Modeling In principle, models of a given physical system can be developed by two distinct approaches. Figure 100.2 shows the steps involved in analytical modeling. The real-world system is represented by an interconnection of idealized elements. Table 100.1[Dorf, 1989] shows model elements from several disciplines and their elemental equations. An electrical circuit diagram is a typical result of this physical modeling step(box 3 of Fig. 100.2). Application of the appropriate physical laws(Kirchhoff, Newton, etc. )to the idealized physical model (consisting of point masses,ideal springs, lumped resistors, etc. leads to a set of mathematical equations. For a circuit these will c 2000 by CRC Press LLC

© 2000 by CRC Press LLC studies; and answer various “what-if” questions in a safe and efficient manner. Of course, the validation of the model, and all conclusions derived from it, must ultimately be based upon test results with the physical hardware. The final form of the mathematical model depends upon the type of physical system, the method used to develop the model, and mathematical manipulations applied to it. These issues are discussed next. Classes of Systems to Be Modeled Most control problems are multidisciplinary. The system may consist of electrical, mechanical, thermal, optical, fluidic, or other physical components, as well as economic, biological, or ecological systems. Analogies exist between these various disciplines, based upon the similarity of the equations that describe the phenomena. The discussion of models in this section will be given in mathematical terms and therefore will apply to several disciplines. Figure 100.1 [Brogan, 1991] shows the classes of systems that might be encountered in control systems modeling. Several branches of this tree diagram are terminated with a dashed line indicating that additional branches have been omitted, similar to those at the same level on other paths. Distributed parameter systems have variables that are functions of both space and time (such as the voltage along a transmission line or the deflection of a point on an elastic structure). They are described by partial differential equations. These are often approximately modeled as a set of lumped parameter systems (described by ordinary differential or difference equations) by using modal expansions, finite element methods, or other approximations [Brogan, 1968]. The lumped parameter continuous-time and discrete-time families are stressed here. Two Major Approaches to Modeling In principle, models of a given physical system can be developed by two distinct approaches. Figure 100.2 shows the steps involved in analytical modeling. The real-world system is represented by an interconnection of idealized elements. Table 100.1 [Dorf, 1989] shows model elements from several disciplines and their elemental equations. An electrical circuit diagram is a typical result of this physical modeling step (box 3 of Fig. 100.2). Application of the appropriate physical laws (Kirchhoff, Newton, etc.) to the idealized physical model (consisting of point masses, ideal springs, lumped resistors, etc.) leads to a set of mathematical equations. For a circuit these will FIGURE 100.1 Major classes of system equations. (Source: W.L. Brogan, Modern Control Theory, 3rd ed., Englewood Cliffs, N.J.: Prentice-Hall, 1991, p. 13. With permission.)

© 2000 by CRC Press LLC In the MA model all ai = 0. This is alternatively called an all-zero model or a finite impulse response (FIR) model. In the AR model all bj terms are zero except b0. This is called an all-pole model or an infinite impulse response (IIR) model. The ARMA model has both poles and zeros and also is an IIR model [Makhoul, 1975]. Adaptive and learning control systems have an experimental modeling aspect. The data fitting is carried out on-line, in real time, as part of the system operation. The modeling described above is normally done off-line [Astrom and Wittenmark, 1989]. Forms of the Model Regardless of whether a model is developed from knowledge of the physics of the process or from empirical data fitting, it can be further manipulated into several different but equivalent forms. This manipulation is box 7 in Fig. 100.2. The class that is most widely used in control studies is the deterministic lumped-parameter continuous-time constant-coefficient system. A simple example has one input u and one output y. This might be a circuit composed of one ideal source and an interconnection of ideal resistors, capacitors, and inductors. TABLE 100.1 Summary of Describing Differential Equations for Ideal Elements Type of Physical Describing Energy E or Element Element Equation Power P Symbol Electrical inductance Inductive Translational storage spring Rotational spring Fluid inertia Electrical capacitance Translational mass Capacitive Rotational storage mass Fluid capacitance Thermal capacitance Electrical resistance Translational damper Energy Rotational dissipators damper Fluid resistance Thermal resistance v21 v2 v1 L i L= di dt v21 = dF dt 1 K 1 K 1 R w21 = dT dt P21 I= dQ dt i C = dv21 dt dP21 dt dv2 dt dw2 dt dt2 dt F M = T J = Q C = ƒ q C = t i = F v21 ƒv21 ƒw21 = T = Q P = 21 1 Rƒ q t = 21 1 Rt E Li 2 = E = F2 K 1 2 1 2 T 2 K 1 2 1 2 1 R E = E IQ2 1 2 Cv 2 = E = E = E = E = E Ct t = 2 = ƒv21 ƒw21 = = = P21 1 Rƒ t = 21 1 Rt 21 1 2 Mv 2 2 v 2 21 1 2 Jw2 2 1 2 CƒP 2 21 2 2 2 v2 v1 C i v2 v2 v1 v1 R i ƒ P2 F P1 I Q v2 v1 K F w2 w1 K T v2 v1 = M constant T w2 w1 = J constant constant Q P1 P2 Cƒ F q Ct 2 2 = P2 P1 Rƒ Q 2 1 Rt q w2 w1 ƒ T

The equations for this system might consist of a set of mesh or node equations. These could be reduced to a single nth-order linear ordinary differential equation by eliminating extraneous variables. d"y "y d t -1an-1+¨+a1+ay=bu+b (100.3) This nth-order equation can be replaced by an input-output transfer function Y(s) bs+b =H(s) (1004) The inverse Laplace transform L-H(s))= h(t) is the system impulse response function. Alternatively, by lecting a set of n internal state variables, Eq (100.3)can be written as a coupled set of first-order differential equations plus an algebraic equation relating the states to the original output y. These equations are called state equations, and one possible choice for this example is, assuming m=n, 100 0 b x(t)+ (t) 000 000 y(t)=[100….0]x(t)+bnu(t) (100.5 In matrix notation these are written more succinctly as 文=Ax+ Bu and y=Cx+Du (100.6) Any one of these six possible model forms, or others, might constitute the result of box 8 in Fig. 100.2.Discret time system models have similar choices of form, including an nth-order difference equation as given in Eq (100.1)or a z-transform input-output transfer function as given in Eq(100.2). A set of n first-order difference equations(state equations)analogous to Eq. (100.5)or(100.6)also can be written Extensions to systems with r inputs and m outputs lead to a set of m coupled equations similar to Eq (100.3), one for each output y. These higher-order equations can be reduced to n first-order state differential equations and m algebraic output equations as in Eq (100.5)or(100.6). The A matrix is again of dimension nX n, but B is now nXr C is mx n, and D is m xr. In all previous discussions, the number of state variables, n, is the order of the model In transfer function form, an mxr matrix H(s) of transfer functions will describe the input-output behavior Y(S)=HsU(s) (100.7) Other transfer function forms are also applicable, including the left and right forms of the matrix fraction description(MFD) of the transfer functions [ Kailath, 1980 e 2000 by CRC Press LLC

© 2000 by CRC Press LLC The equations for this system might consist of a set of mesh or node equations. These could be reduced to a single nth-order linear ordinary differential equation by eliminating extraneous variables. (100.3) This nth-order equation can be replaced by an input-output transfer function (100.4) The inverse Laplace transform L–1{H(s)} = h(t) is the system impulse response function. Alternatively, by selecting a set of n internal state variables, Eq.(100.3) can be written as a coupled set of first-order differential equations plus an algebraic equation relating the states to the original output y. These equations are called state equations, and one possible choice for this example is, assuming m = n, and y(t) = [100… 0]x(t) + bnu(t) (100.5) In matrix notation these are written more succinctly as · x = Ax + Bu and y = Cx + Du (100.6) Any one of these six possible model forms, or others, might constitute the result of box 8 in Fig. 100.2. Discretetime system models have similar choices of form, including an nth-order difference equation as given in Eq. (100.1) or a z-transform input-output transfer function as given in Eq. (100.2). A set of n first-order difference equations (state equations) analogous to Eq. (100.5) or (100.6) also can be written. Extensions to systems with r inputs and m outputs lead to a set of m coupled equations similar to Eq. (100.3), one for each output yi . These higher-order equations can be reduced to n first-order state differential equations and m algebraic output equations as in Eq. (100.5) or (100.6). The A matrix is again of dimension n ¥ n, but B is now n ¥ r, C is m ¥ n, and D is m ¥ r. In all previous discussions, the number of state variables, n, is the order of the model. In transfer function form, an m ¥ r matrix H(s) of transfer functions will describe the input-output behavior Y(s) = H(s)U(s) (100.7) Other transfer function forms are also applicable, including the left and right forms of the matrix fraction description (MFD) of the transfer functions [Kailath, 1980] d y dt a d y dt a dy dt a y b u b du dt b d u dt n n n n n m m m + - + + + = + + + - - 1 1 1 L L 1 0 0 1 Y s U s H s b s b s b s b s a s a s a m m m m n n n ( ) ( ) = = ( ) + + + + + + + + - - - - 1 1 1 0 1 1 1 0 L L ˙ ( ) – – – – ( ) – – – – x t x a a a a t b a b b a b b a b b a b n n n n n n n n n n = È Î Í Í Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ ˙ ˙ + È Î Í Í Í Í Í Í ˘ - - - - - - 1 2 1 0 1 1 2 2 1 1 0 0 100 0 011 0 000 1 000 0 L L M M M M M M L L M ˚ ˙ ˙ ˙ ˙ ˙ ˙ u( )t

H(s=P(s-N(s) or H(s)=N(SP(s-l (100.8) Both P and N are matrices whose elements are polynomials in s. Very similar model forms apply to continuous- time and discrete-time systems, with the major difference being whether Laplace transform or z-transfor transfer functions are involved When time-variable systems are encountered, the option of using high-order differential or difference equations versus sets of first-order state equations is still open. The system coefficients a( o), b( t) and/or the atrices A(O), B(n), C(o), and D(r) will now be time-varying. Transfer function approaches lose most of their utility in time-varying cases and are seldom used. with nonlinear systems all the options relating to the order and number of differential or difference equation still apply. he form of the nonlinear state equations is ⅸ=f(x,u,t) y=h(x,u, t) (100.9) where the nonlinear vector-valued functions f(x, u, t)and h(x, u, t)replace the right-hand sides of Eq (100.6) The transfer function forms are of no value in nonlinear cases Stochastic systems[ Maybeck, 1979]are modeled in similar forms, except the coefficients of the model and/or the inputs are described in probabilistic terms. There is not a unique correct model of a given system for several reasons. The selection of idealized elements to represent the system requires judgment based upon the intended purpose. For example, a satellite might be modeled as a point mass in a study of its gross motion through space. a detailed flexible structure model might be required if the goal is to control vibration of a crucial on-board sensor In empirical modeling, the assumed tarting form, Eq. (100.1), can vary There is a trade-off between the complexity of the model form and the fidelity with which it will match the data set. For example, a pth-degree polynomial can exactly fit to p+ 1 data points, but a straight line might be a better model of the underlying physics. Deviations from the line might be caused by extraneous measure- ment noise Issues such as these are addressed in Astrom [1980 The preceding paragraph addresses nonuniqueness in determining an input-output system description. In addition, state models developed from input-output descriptions are not unique. Suppose the transfer function of a single-input, single-output linear system is known exactly. The state variable model of this system is not unique for at least two reasons. An arbitrarily high-order state variable model can be found that will have this same transfer function. There is, however, a unique minimal or irreducible order nmin from among all state models that have the specified transfer function. A state model of this order will have the desirable properties of controllability and observability. It is interesting to point out that the minimal order may be less than the actual order of the physical syste The second aspect of the nonuniqueness issue relates not to order, i.e., the number of state variables, but to choice of internal variables(state variables). Mathematical and physical methods of selecting state variables are available [Brogan, 1991]. An infinite number of choices exist, and each leads to a different set (A, B, C, DI called a realization. Some state variable model forms are more convenient for revealing key system properties such as stability, controllability, observability, stabilizability, and detectability. Common forms include the controllable canonical form, the observable canonical form, the Jordan canonical form, and the Kalman canonical form The reverse process is unique in that every valid realization leads to the same model transfer function H(s=ClsI-A-IB+ D (100.10) e 2000 by CRC Press LLC

© 2000 by CRC Press LLC H(s) = P(s)–1N(s) or H(s) = N(s)P(s)–1 (100.8) Both P and N are matrices whose elements are polynomials in s. Very similar model forms apply to continuoustime and discrete-time systems, with the major difference being whether Laplace transform or z-transform transfer functions are involved. When time-variable systems are encountered, the option of using high-order differential or difference equations versus sets of first-order state equations is still open. The system coefficients ai (t), bj (t) and/or the matrices A(t), B(t), C(t), and D(t) will now be time-varying. Transfer function approaches lose most of their utility in time-varying cases and are seldom used. With nonlinear systems all the options relating to the order and number of differential or difference equation still apply. The form of the nonlinear state equations is · x = f (x, u, t) y = h(x, u, t) (100.9) where the nonlinear vector-valued functions f(x, u, t) and h(x, u, t) replace the right-hand sides of Eq. (100.6). The transfer function forms are of no value in nonlinear cases. Stochastic systems [Maybeck, 1979] are modeled in similar forms, except the coefficients of the model and/or the inputs are described in probabilistic terms. Nonuniqueness There is not a unique correct model of a given system for several reasons. The selection of idealized elements to represent the system requires judgment based upon the intended purpose. For example, a satellite might be modeled as a point mass in a study of its gross motion through space. A detailed flexible structure model might be required if the goal is to control vibration of a crucial on-board sensor. In empirical modeling, the assumed starting form, Eq. (100.1), can vary. There is a trade-off between the complexity of the model form and the fidelity with which it will match the data set. For example, a pth-degree polynomial can exactly fit to p + 1 data points, but a straight line might be a better model of the underlying physics. Deviations from the line might be caused by extraneous measurement noise. Issues such as these are addressed in Astrom [1980]. The preceding paragraph addresses nonuniqueness in determining an input-output system description. In addition, state models developed from input-output descriptions are not unique. Suppose the transfer function of a single-input, single-output linear system is known exactly. The state variable model of this system is not unique for at least two reasons. An arbitrarily high-order state variable model can be found that will have this same transfer function. There is, however, a unique minimal or irreducible order nmin from among all state models that have the specified transfer function. A state model of this order will have the desirable properties of controllability and observability. It is interesting to point out that the minimal order may be less than the actual order of the physical system. The second aspect of the nonuniqueness issue relates not to order, i.e., the number of state variables, but to choice of internal variables (state variables). Mathematical and physical methods of selecting state variables are available [Brogan, 1991]. An infinite number of choices exist, and each leads to a different set {A, B, C, D}, called a realization. Some state variable model forms are more convenient for revealing key system properties such as stability, controllability, observability, stabilizability, and detectability. Common forms include the controllable canonical form, the observable canonical form, the Jordan canonical form, and the Kalman canonical form. The reverse process is unique in that every valid realization leads to the same model transfer function H(s) = C{sI – A}–1B + D (100.10)

Digital controller Sampling Computer FIGURE 100.3 Digital output provided by modern sensor. Continuous-time model- Discrete-time mod differential equations sampling difference equation g,Eq.(93.3) Transfer functions Transfer functions eg,Fq(93.4) eg,Fq1(93.2) Select states Select states Continuous-time Discrete-time state equations: or sampling x=A(rx+ b(ju(r) t b(ju(k) y(=Cx(0)+ D(u(n) y()=C()x(k)+ D(kju(k) FIGURE 100.4 State variable modeling paradigm Approximation of Continuous Systems by Discrete Models Modern control systems often are implemented digitally, and many modern sensors provide digital output, as shown in Fig. 100.3. In designing or analyzing such systems discrete-time approximate models of continuous- me systems are frequently needed. There are several general ways of proceeding, as shown in Fig. 100.4. Many voices exist for each path on the figure. Alternative choices of states or of approximation methods, such as forward or backward differences lead to an infinite number of valid models. Defining Terms Controllability: A property that in the linear system case depends upon the A, B matrix pair which ensures the existence of some control input that will drive any arbitrary initial state to zero in finite time Detectability: A system is detectable if all its unstable modes are observable Observability: A property that in the linear system case depends upon the A, C matrix pair which ensures the ability to determine the initial values of all states by observing the system outputs for some finite time interval Stabilizable: A system is stabilizable if all its unstable modes are controllable. State variables: A set of variables that completely summarize the systems status in the following sense. If all states x, are known at time fo, then the values of all states and outputs can be determined uniquely for any time t >to, provided the inputs are known from to onward. State variables are components in the state vector. State space is a vector space containing the state vectors. e 2000 by CRC Press LLC

© 2000 by CRC Press LLC Approximation of Continuous Systems by Discrete Models Modern control systems often are implemented digitally, and many modern sensors provide digital output, as shown in Fig. 100.3. In designing or analyzing such systems discrete-time approximate models of continuoustime systems are frequently needed. There are several general ways of proceeding, as shown in Fig. 100.4. Many choices exist for each path on the figure. Alternative choices of states or of approximation methods, such as forward or backward differences, lead to an infinite number of valid models. Defining Terms Controllability: A property that in the linear system case depends upon the A,B matrix pair which ensures the existence of some control input that will drive any arbitrary initial state to zero in finite time. Detectability: A system is detectable if all its unstable modes are observable. Observability: A property that in the linear system case depends upon the A,C matrix pair which ensures the ability to determine the initial values of all states by observing the system outputs for some finite time interval. Stabilizable: A system is stabilizable if all its unstable modes are controllable. State variables: A set of variables that completely summarize the system’s status in the following sense. If all states xi are known at time t0, then the values of all states and outputs can be determined uniquely for any time t1 > t0, provided the inputs are known from t0 onward. State variables are components in the state vector. State space is a vector space containing the state vectors. FIGURE 100.3 Digital output provided by modern sensor. FIGURE 100.4 State variable modeling paradigm. Computer Zero order hold Continuous system uj (tk) y (tk y (t) ) Digital controller Sampling sensor Continuous-time model: differential equations [e.g., Eq. (93.3) or Transfer functions e.g., Eq. (93.4)] Approximation or sampling Approximation or sampling Select states Select states Discrete-time model: difference equations [e.g., Eq. (93.1) or Transfer functions e.g., Eq. (93.2)] Continuous-time state equations: x = A(t)x + B(t)u(t) y(t) = Cx(t) + D(t)u(t) Discrete-time state equations: y(k + 1) = A(k)x(k) + B(k)u(k) y(k) = C(k)x(k) + D(k)u(k)

Related Topic 6.1 Definitions and Properties References K J. Astrom,Maximum likelihood and prediction error methods, "Automatica, vol. 16, Pp. 551-574, 1980 .J. Astrom and B. Wittenmark, Adaptive Control, Reading, Mass. Addison-Wesley, 1989 W.L. Brogan,Optimal control theory applied to systems described by partial differential equations, "in Advances in Control Systems, vol 6, C. T Leondes(ed ) New York: Academic Press, 1968, chap 4. W L. Brogan, Modern Control Theory, 3rd ed, Englewood Cliffs, N J: Prentice-Hall, 1991 R C. Dorf, Modern Control Systems, 5th ed, Reading, Mass. Addison-Wesley, 1989. T Kailath, Linear Systems, Englewood Cliffs, N J. Prentice-Hall, 1980 J. Makhoul," Linear prediction: A tutorial review, Proc. IEEE, vol. 63, no. 4, PP. 561-580, 1975 P.S. Maybeck, Stochastic Models, Estimation and Control, vol 1, New York: Academic Press, 1979 J.L. Shearer, A.T. Murphy, and HH. Richardson, Introduction to Dynamic Systems, Reading, Mass. Addison Nesley, 1967 Further Information The monthly IEEE Control Systems Magazine frequently contains application articles involving models of interesting physical systems The monthly IEEE Transactions on Automatic Control is concerned with theoretical aspects of systems Model iscussed here are often the starting point for these investigations. Automatica is the source of many related articles. In particular an extended survey on system identification given by Astrom and Eykhoff in vol. 7, PP. 123-162, 1971 Early developments of the state variable approach are given by r. E. Kalman in"Mathematical description of linear dynamical systems, " SIAM J. Control Ser., vol. Al, no. 2, Pp. 152-192, 1963. 100.2 Dynamic response Gordon k. e lee Computing the Dynamic System Response Consider a linear time-invariant dynamic system represented by a differential equation form d"y(t) …+a (100.11) d…+b( d f(t) + bof(t) dt where yn) and f(r) represent the output and input, respectively, of the system Let p()(d/dr)() define the differential operator so that(100.11)becomes (p"+an1pm1+…+a1p+ao)y(t)=(bnpm+…+b1p+b)f(t)(100.12) The solution to (100.11)is given by y(t)=ys(1)+y1(t) (100.13) e 2000 by CRC Press LLC

© 2000 by CRC Press LLC Related Topic 6.1 Definitions and Properties References K.J. Astrom, “Maximum likelihood and prediction error methods,” Automatica, vol. 16, pp. 551–574, 1980. K.J. Astrom and B. Wittenmark, Adaptive Control, Reading, Mass.: Addison-Wesley, 1989. W.L. Brogan, “Optimal control theory applied to systems described by partial differential equations,” in Advances in Control Systems, vol. 6, C. T. Leondes (ed.), New York: Academic Press, 1968, chap. 4. W.L. Brogan, Modern Control Theory, 3rd ed., Englewood Cliffs, N.J.: Prentice-Hall, 1991. R.C. Dorf, Modern Control Systems, 5th ed., Reading, Mass.: Addison-Wesley, 1989. T. Kailath, Linear Systems, Englewood Cliffs, N.J.: Prentice-Hall, 1980. J. Makhoul, “Linear prediction: A tutorial review,” Proc. IEEE, vol. 63, no. 4, pp. 561–580, 1975. P.S. Maybeck, Stochastic Models, Estimation and Control, vol. 1, New York: Academic Press, 1979. J.L. Shearer, A.T. Murphy, and H.H. Richardson, Introduction to Dynamic Systems, Reading, Mass.: AddisonWesley, 1967. Further Information The monthly IEEE Control Systems Magazine frequently contains application articles involving models of interesting physical systems. The monthly IEEE Transactions on Automatic Control is concerned with theoretical aspects of systems. Models as discussed here are often the starting point for these investigations. Automatica is the source of many related articles. In particular an extended survey on system identification is given by Astrom and Eykhoff in vol. 7, pp. 123–162, 1971. Early developments of the state variable approach are given by R. E. Kalman in “Mathematical description of linear dynamical systems,” SIAM J. Control Ser., vol. A1, no. 2, pp. 152–192, 1963. 100.2 Dynamic Response Gordon K. F. Lee Computing the Dynamic System Response Consider a linear time-invariant dynamic system represented by a differential equation form (100.11) where y(t) and f(t) represent the output and input, respectively, of the system. Let pk (·)D =(dk /dtk )(·) define the differential operator so that (100.11) becomes (pn + an–1pn–1 + … + a1p + a0)y(t) = (bmpm + … + b1p + b0)f (t) (100.12) The solution to (100.11) is given by y(t) = yS(t) + yI(t) (100.13) dyt dt a d yt dt a dy t dt ayt b d ft dt b df t dt bft n n n n n m m m () () () () ( ) ( ) ( ) + ++ + = ++ + - - - 1 1 1 1 0 1 0 L L