Presentation on theme: "Chapter 7 Stability and Steady-State Error Analysis"— Presentation transcript:
1 Chapter 7 Stability and Steady-State Error Analysis § Stability of Linear Feedback Systems§ Routh-Hurwitz Stability Test§ System Types and Steady-State Error§ Time-Domain Performance Indices
2 § 7.1 Stability of Linear Feedback Systems (1) Basic Concepts:(1) Equilibrium States,(2) Stable SystemSystem response can restore to initial equilibrium state under smalldisturbance.(3) Meaning of Stable SystemEnergy 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 evolutionin state space.
3 § 7.1 Stability of Linear Feedback Systems (2) Plant Dynamics:Regular pendulum (Linear)Inverted pendulum (Linear)Natural responseNatural response
4 § 7.1 Stability of Linear Feedback Systems (3) Closed-loop System:Natural behavior of a control system, r(t)=d(t)=0Equilibrium stateInitial relaxation system, I.C.=0No general algebraic solution for 5th-order and above polynomial equation(Abel , Hamilton)
5 § 7.1 Stability of Linear Feedback Systems (4) Stability Problems:Stabilization of unstable systemDestabilization Effect on stable systemG(s) : Unstable plantClosed-loop : StableG(s) : stable plantClosed-loop : Unstable
6 § 7.1 Stability of Linear Feedback Systems (5) Stability Definition(1) Asymptotic stabilityStable system if the transient response decays to zero(2) BIBO stabilityStable system if the response is bounded for bounded input signalThe impulse response of a system is absolutely integrable.
7 § 7.1 Stability of Linear Feedback Systems (6) (3) S-domain stabilitySystem 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.
8 § 7.2 Routh-Hurwitz Stability Test (1865-1905) (1) Characteristic Polynomial of Closed-loop SystemHurwitz polynomialAll roots of D(s) have negative real parts stable systemHurwitz’s necessary conditions: All coefficients (ai) are to be positive.DefineNote: Any zero root has been removed in D(s).
9 § 7.2 Routh-Hurwitz Stability Test (1865-1905) (2) Routh Tabulation (array)Routh-Hurwitz Stability Criterion(1) The polynomial D(s) is a stable polynomial if are all positive, i.e.are all positive.(2) The number of sign changes in is equal to thenumber of roots in the RH s-plane.(3) If the first element in a row is zero, it is replaced by a smalland the sign changes when are counted after completing thearray.(4) If all elements in a row are zero, the system has poles in the RH planeor on the imaginary axis.
10 § 7.2 Routh-Hurwitz Stability Test (1865-1905) (3) For entire row is zeroIdentify 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:
11 § 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 withcharacteristic polynomial asSol:
12 § 7.2 Routh-Hurwitz Stability Test (1865-1905) (5) Ex: For a colsed-loop system with characteristic polynomialDetermine if the system is stableSol:Ex: For , determine if the system is stableSol:
13 § 7.2 Routh-Hurwitz Stability Test (1865-1905) (6) Absolute and Relative StabilityAbsolute StabilityRelative StabilityCharacteristic equationR-H Test on D(s)Characteristic equationR-H Test on D ’(p)
14 § 7.3 System Types and Steady-State Error (1) Steady-state error for unity feedback systemsFor nonunity feedback systems
15 § 7.3 System Types and Steady-State Error (2) Fundamental Regulation and Tracking ErrorRegulation s.s. errorTracking s.s. error
16 § 7.3 System Types and Steady-State Error (3) Open-loop System Types
17 § 7.3 System Types and Steady-State Error (4) Position Control of Mechanical Systems(1) Command signalRegion 1 and 3: Constant acceleration and decelerationRegion 2: Constant speedRegion 4: Constant position(2) Error constants
18 § 7.3 System Types and Steady-State Error (5) (3) Systems control with non-zero steady-state position errorConstant position for Type 0 systemConstant velocity for Type 1 systemConstant acceleration for Type 2 system
19 § 7.3 System Types and Steady-State Error (6) Steady-state position errors for different types of system and input signalOutput 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.
20 § 7.3 System Types and Steady-State Error (7) Ex: Find the value of K such that there is 10% error in the steady stateSol: System G(s) is Type 1s.s. error in ramp inputFor velocity error constant
21 § 7.4 Time-Domain Performance Indices (1) Performance of Control SystemStabilityTransient ResponseSteady-state ErrorPerformance Indices (PI)A scalar function for quantitative measure of the performance specifications of a control system.errorcommandstateoutputUse P.I. To trade off transient response and steady-state error with sufficient stability margin.
22 § 7.4 Time-Domain Performance Indices (2) Systems Control(1) Classical controlPlant: Input-Output ModelController: PID ControlP-Proportional control:I-Integral control:D-Differential control:(2) Modern controlPlant: State-space ModelP. I.: Usually Quadratic functionalController: States feedback control
23 § 7.4 Time-Domain Performance Indices (3) Optimal ControlGiven: Plant modelControl configuration (Usually feedback)Controller structure (Usually linear)Design constraintsObjective: Minimize P. I. (P. I. to be selected)Find: Optimal parameters in controllerEx: Design optimal proportional control systemFind optimal K to minimize P. I.