LINEAR CONTROLTHEORY:Structures, Robustness, andDesignL. H. KeelTennessee State UniversityS. P. BhattacharyyaTexas A&M UniversityPART ILINEAR SYSTEMTHEORY2 ContentsChapter 1SYSTEMREPRESENTATION USINGSTATE VARIABLESThis chapter introduces the state space representation of linear time-invariantdynamic systems. The state spacce model is the foundation of modern optimalcontrol and filtering theory and has various applications in signal processing,circuit and control theory. in this course.1.1 INTRODUCTIONA dynamic system is a system which processes input signals and produces thecorresponding outputs. Thus, dynamic systems include not only engineeringsystems, but a much wider class of systems such as population, weather forecastsystems, social and economic systems etc. If all equations describing a systemare defined for all time, the system is called a continuous system. In discrete-time systems, some equations and variables are defined or used only at discretepoints in time. Continuous-time systems are described by differential equationswhile discrete-time systems are described by difference equations. Furthermore,if the input-output relationship of a dynamic system obeys the superpositionprinciple, such a system is called linear dynamic system. In other words, let yibe the corresponding output of the linear dynamic system to the input xi. Thena dynamic system is linear if and only if the output of the system to the inputPaixiisPaiyifor arbitrary scalar values of ai. It is important to note thatthe mathematical analysis of linear dynamic systems always results in lineardifferential equations.A system is called memoryless if its output at time t∗, y (t∗), depends onlyon the input applied at time t∗, u (t∗). An example of a memoryless system isan electrical circuit consisting of resistors only. Most systems possess memory.The instantaneous output y (t∗) of a system with memory depends on its presentinputs, u (t∗), as well as its past and/or future inputs. If the output y (t∗)depends only on past and present inputs, that is, u(t), t ≤ t∗, the system is34 SYSTEM REPRESENTATION USING STATE VARIABLES Ch. 1called causal. State variables of a system summarize the internal status of thesystem. Specifically, state x(t0) is the information at time t0that determinesthe entire output y(t) for all t > t0if the inputs are known.1.2 STATE SPACE REPRESENTATION OF LTI SYSTEMSLet y(t) be the output of a system with initial state x(t0) to the input u(t),t ≥ t0. Suppose that the initial state x(t0) is shifted to time t(t0+ T ) and thesame input is applied from time t0+ T instead of t0. If the new output is theexact t0+T shifted version of y(t), then such a system is called a time-invariantsystem. Simply speaking, if the initial state and the input are the same, theoutput is always the same regardless of the time when the input is applied.Typically, the input-output relationship of a linear time-invariant (LTI) sys-tem is written as differential equations, in most cases, of high order. The statespace equation is a first order vector differential equation which is equivalent tothe high order differential equation. We show how to obtain state space equa-tions from the differential equations considering, first, the case when there isno derivative terms in the input. Let us consider the following time-invariantsystem where u(t) is input and y(t) is output.dny(t)dtn+ an−1dn−1y(t)dtn−1+ · · · + a1dy(t)dt+ a0= u(t). (1.1)The define a set of new variables as follows.x1(t) := y(t)x2(t) := ˙x1(t) = ˙y(t)...xn(t) := ˙xn−1(t) = y(n−1).We now rewrite eq. (1.1) by using these variables.˙xn(t) + an−1xn(t) + an−2xn−1(t) + · · · + a1x2(t) + a0x1(t) = u(t) (1.2)ordotxn(t) = −an−1xn(t) − an−2xn−1(t) − · · · − a1x2(t) − a0x1(t) + u(t). (1.3)This expression can be put into matrix form.˙x1(t)˙x2(t)˙x3(t)...˙xn(t)| {z }˙x(t)=0 1 0 · · · 00 0 1 0......0 0 0 · · · 1−a0−a1−a2· · · −an−1| {z }A˙x1(t)˙x2(t)˙x3(t)...˙xn(t)| {z }x(t)+00...01| {z }Bu(t)Sec. 1.2. STATE SPACE REPRESENTATION OF LTI SYSTEMS 5(1.4)y(t) =£1 0 0 · · · 0¤| {z }C˙x1(t)˙x2(t)˙x3(t)...˙xn(t)| {z }x(t).The above is called a state space representation of the system and x(t) is calledstate vector. The state of a system at t∗is the amount of information at t∗that,together with u[t∗,∞), that determines uniquely the behaviour of the system forall t ≥ t∗.The general state space model of an LTI system is of the form˙x(t) = Ax(t) + Bu(t)y = Cx(t) + Du(t).where A, B.C.D are matrices of compatible order with A being nxn and n isthe dimension of the state vector or the order of the system. To illustate, wegive some examples.Example 1.1. Consider an electrical circuit shown in Figure 1.1.y+−x1x2+ −x3LC2C1Ru(t)+−iFigure 1.1. An electrical circuit.Choosing the inductor current and capacitor voltages as states and usingKirchoff’s laws, we haveu(t) = Ri(t) + x1(t), i(t) = C1˙x1(t) + C2˙x2(t), x3(t) = C2˙x2(t).Thus,u(t) = R (C1˙x1(t) + x3(t)) + x1(t)or˙x1(t) = −x1(t)RC1−x3(t)C1+u(t)RC1.and˙x2(t) =x3(t)C2, ˙x3(t) =x1(t)L−x2L.6 SYSTEM REPRESENTATION USING STATE VARIABLES Ch. 1We write this in matrix form.˙x1(t)˙x2(t)˙x3(t)| {z }˙x=−1R1C10 −1C10 01C21L−1L0| {z }Ax1(t)x2(t)x3(t)| {z }x+1RC100| {z }Bu(t).Since y = L ˙x3(t), we writey = L ˙x3(t) = x1(t) − x2(t)= [1 − 1 0]| {z }Cx1(t)x2(t)x3(t).This is the case of a single-input single-output (SISO) system.Example 1.2. Let us consider the following electromechanical system shownin Figure 1.2.+−uLaRaθ = yBTL+−ebiaFigure 1.2. An electromechanical system.It represents a separately excited dc motor driving a load producing a loadtorque TL. We denoteRaarmature-winding resistance, ΩLaarmature-winding inductance, Hiaarmature-winding current, Au applied armature voltage, Vebback emf, Vθ angular displacement of motor shaft, radianB equivalent viscous-friction coefficient of the motor and loadreferred to the motor shaft, ib-ft/radian/secJ equivalent moment of inertia of the motor and load referred tothe motor shaft, slug-ft2Kbback emf constnatKimotor-torque constantSec. 1.2. STATE SPACE REPRESENTATION OF LTI SYSTEMS 7and