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

§ 7.1 Stability of Linear Feedback Systems (1) Basic Concepts: (1) Equilibrium States, (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.

§ 7.1 Stability of Linear Feedback Systems (2) Plant Dynamics: Regular pendulum (Linear) Inverted pendulum (Linear) Natural response Natural response

§ 7.1 Stability of Linear Feedback Systems (3) Closed-loop System: 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)

§ 7.1 Stability of Linear Feedback Systems (4) Stability Problems: Stabilization of unstable system Destabilization Effect on stable system G(s) : Unstable plant Closed-loop : Stable G(s) : stable plant Closed-loop : Unstable

§ 7.1 Stability of Linear Feedback Systems (5) Stability Definition (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 The impulse response of a system is absolutely integrable.

§ 7.1 Stability of Linear Feedback Systems (6) (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.

§ 7.2 Routh-Hurwitz Stability Test (1865-1905) (1) Characteristic Polynomial of Closed-loop System Hurwitz polynomial All roots of D(s) have negative real parts. stable system Hurwitz’s necessary conditions: All coefficients (ai) are to be positive. Define Note: Any zero root has been removed in D(s).

§ 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 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) (3) 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) (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 Determine if the system is stable Sol: Ex: For , determine if the system is stable Sol:

§ 7.2 Routh-Hurwitz Stability Test (1865-1905) (6) Absolute and Relative Stability Absolute Stability Relative Stability Characteristic equation R-H Test on D(s) 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 Region 1 and 3: Constant acceleration and deceleration Region 2: Constant speed Region 4: Constant position (2) Error constants

§ 7.3 System Types and Steady-State Error (5) (3) Systems control with non-zero 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) Steady-state position errors for different types of system and input signal 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.

§ 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) Performance of Control System Stability Transient Response Steady-state Error 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 transient response and steady-state error with sufficient stability margin.

§ 7.4 Time-Domain Performance Indices (2) Systems Control (1) Classical control Plant: Input-Output Model Controller: PID Control P-Proportional control: I-Integral control: D-Differential control: (2) Modern control Plant: State-space Model P. I.: Usually Quadratic functional Controller: States feedback control

§ 7.4 Time-Domain Performance Indices (3) Optimal Control Given: Plant model Control configuration (Usually feedback) Controller structure (Usually linear) Design constraints Objective: Minimize P. I. (P. I. to be selected) Find: Optimal parameters in controller Ex: Design optimal proportional control system Find optimal K to minimize P. I.