Continuous-time systems 2 March 3, 2015 Continuous-time systems 2 - PowerPoint PPT Presentation

Properties of state-space representation Transfer functions Continuous-time systems 2 March 3, 2015 Continuous-time systems 2

Properties of state-space representation Transfer functions Properties of state-space representation 1 Transfer functions 2 Impulse response and time constant Relationship between state space and transfer functions Continuous-time systems 2

Properties of state-space representation Transfer functions Outline Properties of state-space representation 1 Transfer functions 2 Impulse response and time constant Relationship between state space and transfer functions Continuous-time systems 2

Properties of state-space representation Transfer functions Observability A measure of how well a system’s internal states x can be inferred by knowledge of its outputs y . Continuous-time systems 2

Properties of state-space representation Transfer functions Observability A measure of how well a system’s internal states x can be inferred by knowledge of its outputs y . Formally, a system is said to be observable if, for any possible sequence of state and control vectors, the current state can be determined in finite time using only the outputs. Continuous-time systems 2

Properties of state-space representation Transfer functions Observability A measure of how well a system’s internal states x can be inferred by knowledge of its outputs y . Formally, a system is said to be observable if, for any possible sequence of state and control vectors, the current state can be determined in finite time using only the outputs. This holds for linear, time-invariant systems with n states if:   C CA   rank ( O ) = n , O =  , O : observability matrix  .  .   .  CA n − 1 Continuous-time systems 2

Properties of state-space representation Transfer functions Controllability A measure of the ability to move a system around in its entire configuration space using only certain admissible manipulations. Continuous-time systems 2

Properties of state-space representation Transfer functions Controllability A measure of the ability to move a system around in its entire configuration space using only certain admissible manipulations. A system is controllable if its state can be moved from any initial state x 0 to any final state x f via some finite sequence of inputs u 0 . . . u f . Continuous-time systems 2

Properties of state-space representation Transfer functions Controllability A measure of the ability to move a system around in its entire configuration space using only certain admissible manipulations. A system is controllable if its state can be moved from any initial state x 0 to any final state x f via some finite sequence of inputs u 0 . . . u f . A linear, time-invariant system with n states is controllable if: � A n − 1 B � rank ( C ) = n , C = . . . , B AB where C is called the controllability matrix . Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Outline Properties of state-space representation 1 Transfer functions 2 Impulse response and time constant Relationship between state space and transfer functions Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Transfer function The transfer function of input i to output j is defined as: H i , j ( s ) = Y j ( s ) U i ( s ) , U ( s ) = L{ u ( t ) } , Y ( s ) = L{ y ( t ) } . MIMO systems with n inputs and m outputs have n × m transfer functions, one for each input-output pair. Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Transfer function The transfer function of input i to output j is defined as: H i , j ( s ) = Y j ( s ) U i ( s ) , U ( s ) = L{ u ( t ) } , Y ( s ) = L{ y ( t ) } . MIMO systems with n inputs and m outputs have n × m transfer functions, one for each input-output pair. The complex Laplace variable can be rewritten: s = σ + j ω . Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Transfer function The transfer function of input i to output j is defined as: H i , j ( s ) = Y j ( s ) U i ( s ) , U ( s ) = L{ u ( t ) } , Y ( s ) = L{ y ( t ) } . MIMO systems with n inputs and m outputs have n × m transfer functions, one for each input-output pair. The complex Laplace variable can be rewritten: s = σ + j ω . The frequency response of a system can be analyzed via H ( j ω ): e σ + j ω = e σ (cos ω + j sin ω ) . Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Illustration of Euler’s formula Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Poles and zeros In general, the transfer function can be written as: H ( s ) = N ( s ) D ( s ) . Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Poles and zeros In general, the transfer function can be written as: H ( s ) = N ( s ) D ( s ) . The poles of H ( s ) are zeros of D ( s ), ie { s : D ( s ) = 0 } . | H ( s ) | = ∞ if s is a pole. Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Poles and zeros In general, the transfer function can be written as: H ( s ) = N ( s ) D ( s ) . The poles of H ( s ) are zeros of D ( s ), ie { s : D ( s ) = 0 } . | H ( s ) | = ∞ if s is a pole. The zeros of H ( s ) are zeros of N ( s ), ie { s : N ( s ) = 0 } . H ( s ) = 0 if s is a zero. Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Poles and zeros In general, the transfer function can be written as: H ( s ) = N ( s ) D ( s ) . The poles of H ( s ) are zeros of D ( s ), ie { s : D ( s ) = 0 } . | H ( s ) | = ∞ if s is a pole. The zeros of H ( s ) are zeros of N ( s ), ie { s : N ( s ) = 0 } . H ( s ) = 0 if s is a zero. Poles and zeros may cancel, ie. if D ( s ) = N ( s ) = 0 for some s . Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Steady-state response The output of a linear time-invariant system yields consists of: a steady-state output y ss ( t ), which similar periodicity to u ( t ) → y ss comprises the same frequencies as u ( t ) a transient output y tr ( t ) → if the system is stable, then lim t →∞ y tr ( t ) = 0 → y tr ( t ) depends on the initial state x 0 ( t ) of the system Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Steady-state response The output of a linear time-invariant system yields consists of: a steady-state output y ss ( t ), which similar periodicity to u ( t ) → y ss comprises the same frequencies as u ( t ) a transient output y tr ( t ) → if the system is stable, then lim t →∞ y tr ( t ) = 0 → y tr ( t ) depends on the initial state x 0 ( t ) of the system If we apply an input u ( t ) = cos ( α t + θ ), then: y ss ( t ) = | H ( j α ) | cos ( α t + θ + ∠ H ( j α )) Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Steady-state response The output of a linear time-invariant system yields consists of: a steady-state output y ss ( t ), which similar periodicity to u ( t ) → y ss comprises the same frequencies as u ( t ) a transient output y tr ( t ) → if the system is stable, then lim t →∞ y tr ( t ) = 0 → y tr ( t ) depends on the initial state x 0 ( t ) of the system If we apply an input u ( t ) = cos ( α t + θ ), then: y ss ( t ) = | H ( j α ) | cos ( α t + θ + ∠ H ( j α )) The steady-state output y ss ( t ) of a linear time invariant system: consists of signals of same frequencies as the input signal u ( t ) which may have been magnified and/or phase changed Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Outline Properties of state-space representation 1 Transfer functions 2 Impulse response and time constant Relationship between state space and transfer functions Continuous-time systems 2

Properties of state-space representation Impulse response and time constant Transfer functions Relationship between state space and transfer functions Impulse response The impulse response h ( t ) of input i to output j is the output y j ( t ) of a system when an impulse δ ( t ) is applied at input u i ( t ). Continuous-time systems 2