按一下以編輯母片標題樣式,按一下以編輯母片,第二層,第三層,第四層,第五層,*,Chapter 7 Stability and Steady-State Error Analysis,7.1 Stability of Linear Feedback Systems,7.2 Routh-Hurwitz Stability Test,7.3 System Types and Steady-State Error,7.4 Time-Domain Performance Indices,Basic Concepts:,7.1 Stability of Linear Feedback Systems(1),(2)Stable System,System response can restore to initial equilibrium state under small,disturbance.,(3)Meaning of Stable System,Energy sense Stable system with minimum potential energy.,Signal sense Output amplitude decays or grows with different meaning.,Lyapunov sense Extension of signal and energy sense for state evolution,in state space.,(1)Equilibrium States,Plant Dynamics:,Regular pendulum(Linear),Inverted pendulum(Linear),Natural response,Natural response,7.1 Stability of Linear Feedback Systems(2),Natural behavior of a control system,r(t)=d(t)=0,Equilibrium state,Initial relaxation system,I.C.=0,No general algebraic solution for 5th-order and above polynomial equation,(Abel,Hamilton),Closed-loop System:,7.1 Stability of Linear Feedback Systems(3),G(s):Unstable plant,Closed-loop:Stable,Stabilization of unstable system,Destabilization Effect on stable system,G(s):stable plant,Closed-loop:Unstable,7.1 Stability of Linear Feedback Systems(4),Stability Problems:,Stability Definition,The impulse response of a system is absolutely integrable.,(1)Asymptotic stability,Stable system if the transient response decays to zero,(2)BIBO stability,Stable system if the response is bounded for bounded input signal,7.1 Stability of Linear Feedback Systems(5),(3)S-domain stability,System Transfer Function:T(s),Stable system if the poles of T(s)all lies in the left-half s-plane.,The definitions of(1),(2),and(3)are equivalent for LTI system.,7.1 Stability of Linear Feedback Systems(6),Hurwitz polynomial,All roots of D(s)have negative real parts.stable system,Hurwitzs necessary conditions:All coefficients(a,i,)are to be positive.,Define,7.2 Routh-Hurwitz Stability Test,(1865-1905),(1),Characteristic Polynomial of Closed-loop System,(1)The polynomial D(s)is a Hurwitz polynomial if are all positive,i.e.,are all positive.,(2)The number of sign changes in is equal to the,number of roots in the RH s-plane.,(3)If the first element in a row is zero,it is replaced by a small,and the sign changes when are counted after completing the,array.,(4)If all elements in a row are zero,the system has poles in the RH plane,or on the imaginary axis.,7.2 Routh-Hurwitz Stability Test,(1865-1905),(2),Routh-Hurwitz,Stability Criterion,Routh,Tabulation(array),For entire row is zero,Identify the auxiliary polynomial The row immediately above the zero row.,The original polynomial is with factor of auxiliary polynomial.,The roots of auxiliary polynomial are symmetric w.r.t.the origin:,7.2 Routh-Hurwitz Stability Test,(1865-1905),(3),7.2 Routh-Hurwitz Stability Test,(1865-1905),(4),Ex:For a closed-loop system with transfer function T(s),Ex:Find stability condition for a closed-loop system with,characteristic polynomial as,Sol:,7.2 Routh-Hurwitz Stability Test,(1865-1905),(5),Ex:For a colsed-loop system with characteristic polynomial,Ex:For ,determine if the system is stable,Sol:,Determine if the system is stable,Sol:,Characteristic equation,R-H Test on D(s),7.2 Routh-Hurwitz Stability Test,(1865-1905),(6),Absolute and Relative Stability,Absolute Stability,Relative Stability,Characteristic equation,R-H Test on D(p),7.3 System Types and Steady-State Error,(1),Steady-state error for unity feedback systems,For nonunity feedback systems,7.3 System Types and Steady-State Error,(2),Fundamental Regulation and Tracking Error,Regulation s.s.error,Tracking s.s.error,7.3 System Types and Steady-State Error,(3),Open-loop System Types,7.3 System Types and Steady-State Error,(4),Position Control of Mechanical Systems,(1)Command signal,(2)Error constants,Region 1 and 3:Constant acceleration and deceleration,Region 2:Constant speed,Region 4:Constant position,7.3 System Types and Steady-State Error,(5),(3)Systems control with finite steady-state position error,Constant position for Type 0 system,Constant velocity for Type 1 system,Constant acceleration for Type 2 system,7.3 System Types and Steady-State Error,(6),Output positioning in feedback control is driven by the dynamic positional error.,System nonlinearities such as friction,dead zone,quantization will introduce steady-state error in closed-loop position control.,Steady-state position errors for different types of system and input signal,7.3 System Types and Steady-State Error,(7),Ex:Find the value of K such that there is 10%error in the steady state,Sol:System G(s)is Type 1,s.s.error in ramp input,For velocity error constant,7.4 Time-Domain Performance Indices,(1),Stability,Transient Response,Steady-state Error,Performance of Control System,Performance Indices(PI),A scalar function for quantitative measure of the performance specifications of a control system.,error,command,state,output,Use P.I.To trade off transien