ss - State-space model Use ss to create real-valued or complex-valued state-space models, or to convert dynamic system models to state-space model form. Creation Syntax sys = ss(A,B,C,D) sys = ss(A,B,C,D,ts) sys = ss(A,B,C,D,ltiSys) sys = ss(D) sys = ss(___,Name,Value) sys = ss(ltiSys) sys = ss(ltiSys,component) sys = ss(ssSys,'minimal') sys = ss(ssSys,'explicit') Input Arguments A - State matrix Nx-by-Nx matrix B - Input-to-state matrix Nx-by-Nu matrix C - State-to-output matrix Ny-by-Nx matrix D - Feedthrough matrix Ny-by-Nu matrix ts - Sample time scalar ltiSys - Dynamic system to convert to state-space form dynamic system model | model array component - Component of identified model 'measured' (default) | 'noise' | 'augmented' ssSys - Dynamic system model to convert to minimal realization or explicit form ss model object Output Arguments sys - Output system model ss model object | genss model object | uss model object Properties A - State matrix Nx-by-Nx matrix B - Input-to-state matrix Nx-by-Nu matrix C - State-to-output matrix Ny-by-Nx matrix D - Feedthrough matrix Ny-by-Nu matrix E - Matrix for implicit state-space models [] (default) | Nx-by-Nx matrix Offsets - Model offsets [] (default) | structure Scaled - Logical value indicating whether scaling is enabled or disabled 0 (default) | 1 StateName - State names ' ' (default) | character vector | cell array of character vectors StatePath - State path ' ' (default) | character vector | cell array of character vectors StateUnit - State units ' ' (default) | character vector | cell array of character vectors InternalDelay - Internal delays in the model vector InputDelay - Input delay 0 (default) | scalar | Nu-by-1 vector OutputDelay - Output delay 0 (default) | scalar | Ny-by-1 vector Ts - Sample time 0 (default) | positive scalar | -1 TimeUnit - Time variable units 'seconds' (default) | 'nanoseconds' | 'microseconds' | 'milliseconds' | 'minutes' | 'hours' | 'days' | 'weeks' | 'months' | 'years' | ... InputName - Input channel names '' (default) | character vector | cell array of character vectors InputUnit - Input channel units '' (default) | character vector | cell array of character vectors InputGroup - Input channel groups structure OutputName - Output channel names '' (default) | character vector | cell array of character vectors OutputUnit - Output channel units '' (default) | character vector | cell array of character vectors OutputGroup - Output channel groups structure Name - System name '' (default) | character vector Notes - User-specified text {} (default) | character vector | cell array of character vectors UserData - User-specified data [] (default) | any MATLAB data type SamplingGrid - Sampling grid for model arrays structure array Object Functions prescale - Optimal scaling of state-space models ssbal - Scale state/uncertainty while preserving uncertain input/output map of uncertain system estim - Form state estimator given estimator gain kalman - Design Kalman filter for state estimation kalmd - Design discrete Kalman estimator for continuous plant lqg - Linear-Quadratic-Gaussian (LQG) design lqgreg - Form linear-quadratic-Gaussian (LQG) regulator lqgtrack - Form Linear-Quadratic-Gaussian (LQG) servo controller lqi - Linear-Quadratic-Integral control lqr - Linear-Quadratic Regulator (LQR) design lqry - Form linear-quadratic (LQ) state-feedback regulator with output weighting reg - Form regulator given state-feedback and estimator gains totaldelay - Total combined I/O delays for LTI model balancmr - (Not recommended) Balanced model truncation via square root method bilin - Multivariable bilinear transform of frequency (s or z) dcgainmr - (Not recommended) Reduced order model hankelmr - (Not recommended) Hankel minimum degree approximation (MDA) without balancing hankelsv - (Not recommended) Compute Hankel singular values for stable/unstable or continuous/discrete system ltiarray2uss - Compute uncertain system bounding given LTI ss array schurmr - (Not recommended) Balanced model truncation via Schur method sectf - State-space sector bilinear transformation absorbDelay - Replace time delays by poles at z = 0 or phase shift bode - Bode frequency response of dynamic system bodemag - Magnitude-only Bode plot of frequency response c2d - Convert model from continuous to discrete time canon - (Not recommended) Canonical state-space realization chgTimeUnit - Change time units of dynamic system connect - Block diagram interconnections of dynamic systems d2c - Convert model from discrete to continuous time d2d - Resample discrete-time model dcgain - Low-frequency (DC) gain of LTI system delay2z - Replace delays of discrete-time TF, SS, or ZPK models by poles at z=0, or replace delays of FRD models by phase shift evalfr - Evaluate system response at specific frequency findop - Compute operating condition from specifications impulse - Impulse response plot of dynamic system; impulse response data interp - Interpolate FRD model iopzmap - Plot pole-zero map for input-output pairs of dynamic system using default options isct - Determine if dynamic system model is in continuous time isdt - Determine if dynamic system model is in discrete time isproper - Determine if dynamic system model is proper isstable - Determine if dynamic system model is stable lsim - Compute time response simulation data of dynamic system to arbitrary inputs nyquist - Nyquist response of dynamic system order - Query model order pole - Poles of dynamic system pzmap - Pole-zero map of dynamic system repsys - Replicate and tile models set - Set or modify model properties ssdata - Access state-space model data step - Step response of dynamic system tfdata - Access transfer function data zero - Zeros and gain of SISO dynamic system zpkdata - Access zero-pole-gain data allmargin - Gain margin, phase margin, delay margin, and crossover frequencies bandwidth - Frequency response bandwidth covar - Output and state covariance of system driven by white noise damp - Natural frequency and damping ratio getGainCrossover - Crossover frequencies for specified gain getPassiveIndex - Compute passivity index of linear system getPeakGain - Peak gain of dynamic system frequency response getSectorCrossover - Crossover frequencies for sector bound getSectorIndex - Compute conic-sector index of linear system initial - System response to initial states of state-space model isPassive - Check passivity of linear systems margin - Gain margin, phase margin, and crossover frequencies nichols - Nichols response of dynamic system norm - Norm of linear model passiveplot - Compute or plot passivity index as function of frequency sectorplot - Compute or plot sector index as function of frequency sigma - Singular values of frequency response of dynamic system stepinfo - Rise time, settling time, and other step-response characteristics tzero - Invariant zeros of linear system looptune - Tune fixed-structure feedback loops looptuneSetup - Convert tuning setup for looptune to tuning setup for systune loopview - Graphically analyze MIMO feedback loops pidtune - PID tuning algorithm for linear plant model rlocus - Root locus of dynamic system systune - Tune fixed-structure control systems modeled in MATLAB balreal - Balanced state-space realization dssdata - Extract descriptor state-space data fixInput - Fix value of some inputs and delete them freqsep - Slow-fast decomposition getDelayModel - State-space representation of internal delays mechssdata - Access second-order sparse state-space model data minreal - Minimal realization or pole-zero cancellation modalsep - Compute modal decomposition piddata - Access coefficients of parallel-form PID controller piddata2 - Access coefficients of parallel-form 2-DOF PID controller pidstddata - Access coefficients of standard-form PID controller pidstddata2 - Access coefficients of standard-form 2-DOF PID controller setDelayModel - Construct state-space model with internal delays sparssdata - Access first-order sparse state-space model data spectralfact - Spectral factorization of linear systems stabsep - Stable-unstable decomposition upsample - Upsample discrete-time models balred - (Not recommended) Model order reduction hsvd - (Not recommended) Hankel singular values of dynamic system hsvplot - (Not recommended) Plot Hankel singular values dksynperf - (Not recommended) Robust H_{∞} performance optimized by dksyn loopmargin - (Not recommended) Stability margin analysis of LTI and Simulink feedback loops mktito - Partition LTI system into two-input/two-output system modreal - (Not recommended) Modal form realization and projection ncfmr - (Not recommended) Model reduction from normalized coprime factorization wcmargin - (Not recommended) Worst-case disk stability margins of uncertain feedback loops wcsens - (Not recommended) Calculate worst-case sensitivity and complementary sensitivity functions of plant-controller feedback loop augw - Plant augmentation for weighted mixed-sensitivity H_{∞} and H_{2} loop-shaping design bstmr - (Not recommended) Balanced stochastic model truncation (BST) via Schur method diskmargin - Disk-based stability margins of feedback loops diskmarginplot - Visualize disk-based stability margins gapmetric - Gap metric and Vinnicombe (nu-gap) metric for distance between two systems h2hinfsyn - Mixed H_{2}/H_{∞} synthesis with regional pole placement constraints h2syn - Compute H_{2} optimal controller hinffc - Full-control H-infinity synthesis hinffi - Full-information H-infinity synthesis hinfnorm - H_{∞} norm of dynamic system hinfstruct - H_{∞} tuning of fixed-structure controllers hinfsyn - Compute H-infinity optimal controller lncf - Left normalized coprime factorization loopsens - Sensitivity functions of plant-controller feedback loop loopsyn - Loop-shaping controller design with tradeoff between performance and robustness ltrsyn - LQG loop transfer-function recovery (LTR) control synthesis mixsyn - Mixed-sensitivity H_{∞} synthesis method for robust control loop-shaping design ncfmargin - Calculate normalized coprime stability margin of plant-controller feedback loop ncfsyn - Loop shaping design using Glover-McFarlane method rncf - Right normalized coprime factorization robgain - Robust performance of uncertain system robstab - Robust stability of uncertain system robustperf - (Not recommended) Robust performance margin of uncertain multivariable system robuststab - (Not recommended) Calculate robust stability margins of uncertain multivariable system sdhinfnorm - Compute L_{2} norm of continuous-time system in feedback with discrete-time system sdhinfsyn - Compute H_{∞} controller for sampled-data system sdlsim - Time response of sampled-data feedback system slowfast - (Not recommended) Slow and fast modes decomposition ucover - Fit uncertain model to set of LTI responses wcdiskmargin - Worst-case disk-based stability margins of uncertain feedback loops wcdiskmarginplot - Visualize worst-case disk-based stability margins wcgain - Worst-case gain of uncertain system wcsigmaplot - Plot worst-case gain of uncertain system conj - Form model with complex conjugate coefficients ctranspose - Conjugate dynamic system model squeeze - Remove singleton dimensions for umat objects permute - Rearrange array dimensions in model arrays isfinite - Determine if model has finite coefficients isreal - Determine if model has real-valued coefficients hasInternalDelay - Determine if model has internal delays hasdelay - True for linear model with time delays isstatic - Determine if model is static or dynamic isempty - Determine whether dynamic system model is empty issiso - Determine if dynamic system model is single-input/single-output (SISO) ndims - Query number of dimensions of dynamic system model or model array nblocks - Number of control design blocks in generalized LTI model or generalized matrix nmodels - Number of models in model array isParametric - Determine if model has tunable parameters append - Group models by appending their inputs and outputs blkdiag - Block-diagonal concatenation of models get - Access model property values getValue - Current value of generalized model repmat - Replicate and tile array reshape - Change shape of model array size - Query output/input/array dimensions of input–output model and number of frequencies of FRD model stack - Build model array by stacking models or model arrays along array dimensions voidModel - Mark missing or irrelevant models in model array feedback - Feedback connection of multiple models getBlockValue - Get current value of Control Design Block in Generalized Model imp2exp - Convert implicit linear relationship to explicit input-output relation inv - Invert dynamic system models lft - Generalized feedback interconnection of two models (Redheffer star product) parallel - Parallel connection of two models replaceBlock - Replace or update control design blocks in generalized model rsampleBlock - Randomly sample Control Design blocks in generalized model sampleBlock - Sample Control Design blocks in generalized model series - Series connection of two models setBlockValue - Modify value of Control Design Block in Generalized Model showBlockValue - Display current values of Control Design Blocks in Generalized Model showTunable - Display current value of tunable Control Design Blocks in Generalized Model sminreal - Eliminates structurally disconnected states, delays, and blocks getNominal - Nominal value of uncertain model gridureal - Grid ureal parameters uniformly over their range lftdata - Decompose uncertain objects into fixed certain and normalized uncertain parts simplify - Simplify representation of uncertain object uscale - Scale uncertainty of block or system usubs - Substitute given values for uncertain elements of uncertain objects modred - (Not recommended) Eliminate states from state-space models xperm - Reorder states in state-space models augoffset - Map offset contribution to extra input channel augstate - Append state vector to output vector ctrb - Controllability of state-space model gram - Controllability and observability Gramians obsv - Observability of state-space model ss2ss - State coordinate transformation for state-space model ssequiv - Equivalence transformation for state-space models xelim - Eliminate states from state-space models Examples SISO State-Space Model Create Discrete-Time State-Space Model Continuous-Time MIMO State-Space Model Discrete-Time MIMO State-Space Model Specify State and Input Names for State-Space Model State-Space Model with Inherited Properties MIMO Static Gain State-Space Model Convert Transfer Function to State-Space Model Extract State-Space Models from Identified Model Explicit Realization of Descriptor State-Space Model Create State-Space Model with Both Fixed and Tunable Parameters State-Space Model with Input and Output Delay Stability Analysis of State-Space Systems Control Design Using State-Space Models Connect Specific Inputs and Outputs of State-Space Models in a Feedback Loop Create State-Space Model with Offsets See also dss, frd, get, set, ssdata, tf, zpk, genss, uss (Robust Control Toolbox), mechss, sparss, ltvss, lpvss Introduced in Control System Toolbox before R2006a Documentation for ss Other uses of ss

tf - Transfer function model Use tf to create real-valued or complex-valued transfer function models, or to convert dynamic system models to transfer function form. Creation Create Transfer Function Model sys = tf(numerator,denominator) sys = tf(numerator,denominator,ts) sys = tf(numerator,denominator,ltiSys) sys = tf(m) sys = tf(___,PropertyName=Value) Convert To Transfer Function Model sys = tf(ltiSys) sys = tf(ltiSys,Name=Value) sys = tf(ltiSys,component) Create Variable for Rational Expression s = tf('s') z = tf('z',ts) Input Arguments numerator - Numerator coefficients of the transfer function row vector | Ny-by-Nu cell array of row vectors denominator - Denominator coefficients of the transfer function row vector | Ny-by-Nu cell array of row vectors ts - Sample time scalar ltiSys - Dynamic system dynamic system model | model array m - Static gain scalar | matrix component - Component of identified model 'measured' (default) | 'noise' | 'augmented' Name-Value Arguments UseParallel - Use parallel computing 0 or false (default) | 1 or true RollOff - Roll-off slope 0 (default) | nonpositive scalar | matrix Focus - Frequency range of interest [0 Inf] (default) | vector MaxNumber - Maximum number of poles and zeros to compute 1000 (default) | positive integer Shift - Spectral shift 0 (default) | finite scalar Tolerance - Accuracy of computed poles and zeros 1e-12 (default) | positive finite scalar Display - Show or hide progress report "on" (default) | "off" Output Arguments sys - Output system model tf model object | genss model object | uss model object Properties Numerator - Numerator coefficients row vector | Ny-by-Nu cell array of row vectors Denominator - Denominator coefficients row vector | Ny-by-Nu cell array of row vectors Variable - Transfer function display variable 's' (default) | 'z' | 'p' | 'q' | 'z^-1' | 'q^-1' IODelay - Transport delay 0 (default) | scalar | Ny-by-Nu array InputDelay - Input delay 0 (default) | scalar | Nu-by-1 vector OutputDelay - Output delay 0 (default) | scalar | Ny-by-1 vector Ts - Sample time 0 (default) | positive scalar | -1 TimeUnit - Time variable units 'seconds' (default) | 'nanoseconds' | 'microseconds' | 'milliseconds' | 'minutes' | 'hours' | 'days' | 'weeks' | 'months' | 'years' | ... InputName - Input channel names '' (default) | character vector | cell array of character vectors InputUnit - Input channel units '' (default) | character vector | cell array of character vectors InputGroup - Input channel groups structure OutputName - Output channel names '' (default) | character vector | cell array of character vectors OutputUnit - Output channel units '' (default) | character vector | cell array of character vectors OutputGroup - Output channel groups structure Name - System name '' (default) | character vector Notes - User-specified text {} (default) | character vector | cell array of character vectors UserData - User-specified data [] (default) | any MATLAB data type SamplingGrid - Sampling grid for model arrays structure array Object Functions exp - Create pure continuous-time delays totaldelay - Total combined I/O delays for LTI model balancmr - (Not recommended) Balanced model truncation via square root method bilin - Multivariable bilinear transform of frequency (s or z) dcgainmr - (Not recommended) Reduced order model hankelmr - (Not recommended) Hankel minimum degree approximation (MDA) without balancing hankelsv - (Not recommended) Compute Hankel singular values for stable/unstable or continuous/discrete system ltiarray2uss - Compute uncertain system bounding given LTI ss array schurmr - (Not recommended) Balanced model truncation via Schur method sectf - State-space sector bilinear transformation absorbDelay - Replace time delays by poles at z = 0 or phase shift bode - Bode frequency response of dynamic system bodemag - Magnitude-only Bode plot of frequency response c2d - Convert model from continuous to discrete time canon - (Not recommended) Canonical state-space realization chgTimeUnit - Change time units of dynamic system connect - Block diagram interconnections of dynamic systems d2c - Convert model from discrete to continuous time d2d - Resample discrete-time model dcgain - Low-frequency (DC) gain of LTI system delay2z - Replace delays of discrete-time TF, SS, or ZPK models by poles at z=0, or replace delays of FRD models by phase shift evalfr - Evaluate system response at specific frequency findop - Compute operating condition from specifications impulse - Impulse response plot of dynamic system; impulse response data interp - Interpolate FRD model iopzmap - Plot pole-zero map for input-output pairs of dynamic system using default options isct - Determine if dynamic system model is in continuous time isdt - Determine if dynamic system model is in discrete time isproper - Determine if dynamic system model is proper isstable - Determine if dynamic system model is stable lsim - Compute time response simulation data of dynamic system to arbitrary inputs nyquist - Nyquist response of dynamic system order - Query model order pole - Poles of dynamic system pzmap - Pole-zero map of dynamic system repsys - Replicate and tile models set - Set or modify model properties ssdata - Access state-space model data step - Step response of dynamic system tfdata - Access transfer function data zero - Zeros and gain of SISO dynamic system zpkdata - Access zero-pole-gain data allmargin - Gain margin, phase margin, delay margin, and crossover frequencies bandwidth - Frequency response bandwidth covar - Output and state covariance of system driven by white noise damp - Natural frequency and damping ratio getGainCrossover - Crossover frequencies for specified gain getPassiveIndex - Compute passivity index of linear system getPeakGain - Peak gain of dynamic system frequency response getSectorCrossover - Crossover frequencies for sector bound getSectorIndex - Compute conic-sector index of linear system initial - System response to initial states of state-space model isPassive - Check passivity of linear systems margin - Gain margin, phase margin, and crossover frequencies nichols - Nichols response of dynamic system norm - Norm of linear model passiveplot - Compute or plot passivity index as function of frequency sectorplot - Compute or plot sector index as function of frequency sigma - Singular values of frequency response of dynamic system stepinfo - Rise time, settling time, and other step-response characteristics tzero - Invariant zeros of linear system looptune - Tune fixed-structure feedback loops looptuneSetup - Convert tuning setup for looptune to tuning setup for systune loopview - Graphically analyze MIMO feedback loops pidtune - PID tuning algorithm for linear plant model rlocus - Root locus of dynamic system systune - Tune fixed-structure control systems modeled in MATLAB balreal - Balanced state-space realization dssdata - Extract descriptor state-space data freqsep - Slow-fast decomposition mechssdata - Access second-order sparse state-space model data minreal - Minimal realization or pole-zero cancellation modalsep - Compute modal decomposition piddata - Access coefficients of parallel-form PID controller piddata2 - Access coefficients of parallel-form 2-DOF PID controller pidstddata - Access coefficients of standard-form PID controller pidstddata2 - Access coefficients of standard-form 2-DOF PID controller sparssdata - Access first-order sparse state-space model data spectralfact - Spectral factorization of linear systems stabsep - Stable-unstable decomposition upsample - Upsample discrete-time models balred - (Not recommended) Model order reduction hsvd - (Not recommended) Hankel singular values of dynamic system hsvplot - (Not recommended) Plot Hankel singular values dksynperf - (Not recommended) Robust H_{∞} performance optimized by dksyn loopmargin - (Not recommended) Stability margin analysis of LTI and Simulink feedback loops mktito - Partition LTI system into two-input/two-output system modreal - (Not recommended) Modal form realization and projection ncfmr - (Not recommended) Model reduction from normalized coprime factorization wcmargin - (Not recommended) Worst-case disk stability margins of uncertain feedback loops wcsens - (Not recommended) Calculate worst-case sensitivity and complementary sensitivity functions of plant-controller feedback loop augw - Plant augmentation for weighted mixed-sensitivity H_{∞} and H_{2} loop-shaping design bstmr - (Not recommended) Balanced stochastic model truncation (BST) via Schur method diskmargin - Disk-based stability margins of feedback loops diskmarginplot - Visualize disk-based stability margins gapmetric - Gap metric and Vinnicombe (nu-gap) metric for distance between two systems h2hinfsyn - Mixed H_{2}/H_{∞} synthesis with regional pole placement constraints h2syn - Compute H_{2} optimal controller hinffc - Full-control H-infinity synthesis hinffi - Full-information H-infinity synthesis hinfnorm - H_{∞} norm of dynamic system hinfstruct - H_{∞} tuning of fixed-structure controllers hinfsyn - Compute H-infinity optimal controller lncf - Left normalized coprime factorization loopsens - Sensitivity functions of plant-controller feedback loop loopsyn - Loop-shaping controller design with tradeoff between performance and robustness ltrsyn - LQG loop transfer-function recovery (LTR) control synthesis mixsyn - Mixed-sensitivity H_{∞} synthesis method for robust control loop-shaping design ncfmargin - Calculate normalized coprime stability margin of plant-controller feedback loop ncfsyn - Loop shaping design using Glover-McFarlane method rncf - Right normalized coprime factorization robgain - Robust performance of uncertain system robstab - Robust stability of uncertain system robustperf - (Not recommended) Robust performance margin of uncertain multivariable system robuststab - (Not recommended) Calculate robust stability margins of uncertain multivariable system sdhinfnorm - Compute L_{2} norm of continuous-time system in feedback with discrete-time system sdhinfsyn - Compute H_{∞} controller for sampled-data system sdlsim - Time response of sampled-data feedback system slowfast - (Not recommended) Slow and fast modes decomposition ucover - Fit uncertain model to set of LTI responses wcdiskmargin - Worst-case disk-based stability margins of uncertain feedback loops wcdiskmarginplot - Visualize worst-case disk-based stability margins wcgain - Worst-case gain of uncertain system wcsigmaplot - Plot worst-case gain of uncertain system conj - Form model with complex conjugate coefficients ctranspose - Conjugate dynamic system model squeeze - Remove singleton dimensions for umat objects permute - Rearrange array dimensions in model arrays isfinite - Determine if model has finite coefficients isreal - Determine if model has real-valued coefficients hasInternalDelay - Determine if model has internal delays hasdelay - True for linear model with time delays isstatic - Determine if model is static or dynamic isempty - Determine whether dynamic system model is empty issiso - Determine if dynamic system model is single-input/single-output (SISO) ndims - Query number of dimensions of dynamic system model or model array nblocks - Number of control design blocks in generalized LTI model or generalized matrix nmodels - Number of models in model array isParametric - Determine if model has tunable parameters append - Group models by appending their inputs and outputs blkdiag - Block-diagonal concatenation of models get - Access model property values getValue - Current value of generalized model repmat - Replicate and tile array reshape - Change shape of model array size - Query output/input/array dimensions of input–output model and number of frequencies of FRD model stack - Build model array by stacking models or model arrays along array dimensions voidModel - Mark missing or irrelevant models in model array feedback - Feedback connection of multiple models getBlockValue - Get current value of Control Design Block in Generalized Model imp2exp - Convert implicit linear relationship to explicit input-output relation inv - Invert dynamic system models lft - Generalized feedback interconnection of two models (Redheffer star product) parallel - Parallel connection of two models replaceBlock - Replace or update control design blocks in generalized model rsampleBlock - Randomly sample Control Design blocks in generalized model sampleBlock - Sample Control Design blocks in generalized model series - Series connection of two models setBlockValue - Modify value of Control Design Block in Generalized Model showBlockValue - Display current values of Control Design Blocks in Generalized Model showTunable - Display current value of tunable Control Design Blocks in Generalized Model sminreal - Eliminates structurally disconnected states, delays, and blocks getNominal - Nominal value of uncertain model gridureal - Grid ureal parameters uniformly over their range lftdata - Decompose uncertain objects into fixed certain and normalized uncertain parts simplify - Simplify representation of uncertain object uscale - Scale uncertainty of block or system usubs - Substitute given values for uncertain elements of uncertain objects Examples SISO Transfer Function Model Discrete-Time SISO Transfer Function Model Second-Order Transfer Function from Damping Ratio and Natural Frequency Discrete-Time MIMO Transfer Function Model Concatenate SISO Transfer Functions into MIMO Transfer Function Model Transfer Function Model Using Rational Expression Discrete-Time Transfer Function Model Using Rational Expression Transfer Function Model with Inherited Properties Array of Transfer Function Models Convert State-Space Model to Transfer Function Extract Transfer Functions from Identified Model Specify Input and Output Names for MIMO Transfer Function Model Specify Polynomial Ordering in Discrete-Time Transfer Function Tunable Low-Pass Filter Static Gain MIMO Transfer Function Model Compute Truncated Transfer Function Approximation of Sparse Model See also filt, frd, get, set, ss, tfdata, zpk, genss, realp, genmat, tunableTF Introduced in Control System Toolbox before R2006a Documentation for tf Other uses of tf

zpk - Zero-pole-gain model Use zpk to create zero-pole-gain models, or to convert dynamic system models to zero-pole-gain form. Creation Create ZPK Model sys = zpk(zeros,poles,gain) sys = zpk(zeros,poles,gain,ts) sys = zpk(zeros,poles,gain,ltiSys) sys = zpk(m) sys = zpk(___,PropertyName=Value) Convert To ZPK Model sys = zpk(ltiSys) sys = zpk(ltiSys,Name=Value) sys = zpk(ltiSys,component) Create Variable for Rational Expression s = zpk('s') z = zpk('z',ts) Input Arguments zeros - Zeros of the zero-pole-gain model row vector | Ny-by-Nu cell array of row vectors poles - Poles of the zero-pole-gain model row vector | Ny-by-Nu cell array of row vectors gain - Gain of the zero-pole-gain model scalar | Ny-by-Nu matrix ts - Sample time scalar ltiSys - Dynamic system dynamic system model | model array m - Static gain scalar | matrix component - Component of identified model 'measured' (default) | 'noise' | 'augmented' Name-Value Arguments UseParallel - Use parallel computing 0 or false (default) | 1 or true RollOff - Roll-off slope 0 (default) | nonpositive scalar | matrix Focus - Frequency range of interest [0 Inf] (default) | vector MaxNumber - Maximum number of poles and zeros to compute 1000 (default) | positive integer Shift - Spectral shift 0 (default) | finite scalar Tolerance - Accuracy of computed poles and zeros 1e-12 (default) | positive finite scalar Display - Show or hide progress report "on" (default) | "off" Output Arguments sys - Output system model zpk model object | genss model object | uss model object Properties Z - System zeros cell array | Ny-by-Nu cell array of row vectors P - System poles cell array | Ny-by-Nu cell array of row vectors K - System gains scalar | Ny-by-Nu matrix DisplayFormat - Specifies how the numerator and denominator polynomials are factorized for display 'roots' (default) | 'frequency' | 'time constant' Variable - Zero-pole-gain model display variable 's' (default) | 'z' | 'p' | 'q' | 'z^-1' | 'q^-1' IODelay - Transport delay 0 (default) | scalar | Ny-by-Nu array InputDelay - Input delay 0 (default) | scalar | Nu-by-1 vector OutputDelay - Output delay 0 (default) | scalar | Ny-by-1 vector Ts - Sample time 0 (default) | positive scalar | -1 TimeUnit - Time variable units 'seconds' (default) | 'nanoseconds' | 'microseconds' | 'milliseconds' | 'minutes' | 'hours' | 'days' | 'weeks' | 'months' | 'years' | ... InputName - Input channel names '' (default) | character vector | cell array of character vectors InputUnit - Input channel units '' (default) | character vector | cell array of character vectors InputGroup - Input channel groups structure OutputName - Output channel names '' (default) | character vector | cell array of character vectors OutputUnit - Output channel units '' (default) | character vector | cell array of character vectors OutputGroup - Output channel groups structure Name - System name '' (default) | character vector Notes - User-specified text {} (default) | character vector | cell array of character vectors UserData - User-specified data [] (default) | any MATLAB data type SamplingGrid - Sampling grid for model arrays structure array Object Functions exp - Create pure continuous-time delays totaldelay - Total combined I/O delays for LTI model balancmr - (Not recommended) Balanced model truncation via square root method bilin - Multivariable bilinear transform of frequency (s or z) dcgainmr - (Not recommended) Reduced order model hankelmr - (Not recommended) Hankel minimum degree approximation (MDA) without balancing hankelsv - (Not recommended) Compute Hankel singular values for stable/unstable or continuous/discrete system ltiarray2uss - Compute uncertain system bounding given LTI ss array schurmr - (Not recommended) Balanced model truncation via Schur method sectf - State-space sector bilinear transformation absorbDelay - Replace time delays by poles at z = 0 or phase shift bode - Bode frequency response of dynamic system bodemag - Magnitude-only Bode plot of frequency response c2d - Convert model from continuous to discrete time canon - (Not recommended) Canonical state-space realization chgTimeUnit - Change time units of dynamic system connect - Block diagram interconnections of dynamic systems d2c - Convert model from discrete to continuous time d2d - Resample discrete-time model dcgain - Low-frequency (DC) gain of LTI system delay2z - Replace delays of discrete-time TF, SS, or ZPK models by poles at z=0, or replace delays of FRD models by phase shift evalfr - Evaluate system response at specific frequency findop - Compute operating condition from specifications impulse - Impulse response plot of dynamic system; impulse response data interp - Interpolate FRD model iopzmap - Plot pole-zero map for input-output pairs of dynamic system using default options isct - Determine if dynamic system model is in continuous time isdt - Determine if dynamic system model is in discrete time isproper - Determine if dynamic system model is proper isstable - Determine if dynamic system model is stable lsim - Compute time response simulation data of dynamic system to arbitrary inputs nyquist - Nyquist response of dynamic system order - Query model order pole - Poles of dynamic system pzmap - Pole-zero map of dynamic system repsys - Replicate and tile models set - Set or modify model properties ssdata - Access state-space model data step - Step response of dynamic system tfdata - Access transfer function data zero - Zeros and gain of SISO dynamic system zpkdata - Access zero-pole-gain data allmargin - Gain margin, phase margin, delay margin, and crossover frequencies bandwidth - Frequency response bandwidth covar - Output and state covariance of system driven by white noise damp - Natural frequency and damping ratio getGainCrossover - Crossover frequencies for specified gain getPassiveIndex - Compute passivity index of linear system getPeakGain - Peak gain of dynamic system frequency response getSectorCrossover - Crossover frequencies for sector bound getSectorIndex - Compute conic-sector index of linear system initial - System response to initial states of state-space model isPassive - Check passivity of linear systems margin - Gain margin, phase margin, and crossover frequencies nichols - Nichols response of dynamic system norm - Norm of linear model passiveplot - Compute or plot passivity index as function of frequency sectorplot - Compute or plot sector index as function of frequency sigma - Singular values of frequency response of dynamic system stepinfo - Rise time, settling time, and other step-response characteristics tzero - Invariant zeros of linear system looptune - Tune fixed-structure feedback loops looptuneSetup - Convert tuning setup for looptune to tuning setup for systune loopview - Graphically analyze MIMO feedback loops pidtune - PID tuning algorithm for linear plant model rlocus - Root locus of dynamic system systune - Tune fixed-structure control systems modeled in MATLAB balreal - Balanced state-space realization dssdata - Extract descriptor state-space data freqsep - Slow-fast decomposition mechssdata - Access second-order sparse state-space model data minreal - Minimal realization or pole-zero cancellation modalsep - Compute modal decomposition piddata - Access coefficients of parallel-form PID controller piddata2 - Access coefficients of parallel-form 2-DOF PID controller pidstddata - Access coefficients of standard-form PID controller pidstddata2 - Access coefficients of standard-form 2-DOF PID controller sparssdata - Access first-order sparse state-space model data spectralfact - Spectral factorization of linear systems stabsep - Stable-unstable decomposition upsample - Upsample discrete-time models balred - (Not recommended) Model order reduction hsvd - (Not recommended) Hankel singular values of dynamic system hsvplot - (Not recommended) Plot Hankel singular values dksynperf - (Not recommended) Robust H_{∞} performance optimized by dksyn loopmargin - (Not recommended) Stability margin analysis of LTI and Simulink feedback loops mktito - Partition LTI system into two-input/two-output system modreal - (Not recommended) Modal form realization and projection ncfmr - (Not recommended) Model reduction from normalized coprime factorization wcmargin - (Not recommended) Worst-case disk stability margins of uncertain feedback loops wcsens - (Not recommended) Calculate worst-case sensitivity and complementary sensitivity functions of plant-controller feedback loop augw - Plant augmentation for weighted mixed-sensitivity H_{∞} and H_{2} loop-shaping design bstmr - (Not recommended) Balanced stochastic model truncation (BST) via Schur method diskmargin - Disk-based stability margins of feedback loops diskmarginplot - Visualize disk-based stability margins gapmetric - Gap metric and Vinnicombe (nu-gap) metric for distance between two systems h2hinfsyn - Mixed H_{2}/H_{∞} synthesis with regional pole placement constraints h2syn - Compute H_{2} optimal controller hinffc - Full-control H-infinity synthesis hinffi - Full-information H-infinity synthesis hinfnorm - H_{∞} norm of dynamic system hinfstruct - H_{∞} tuning of fixed-structure controllers hinfsyn - Compute H-infinity optimal controller lncf - Left normalized coprime factorization loopsens - Sensitivity functions of plant-controller feedback loop loopsyn - Loop-shaping controller design with tradeoff between performance and robustness ltrsyn - LQG loop transfer-function recovery (LTR) control synthesis mixsyn - Mixed-sensitivity H_{∞} synthesis method for robust control loop-shaping design ncfmargin - Calculate normalized coprime stability margin of plant-controller feedback loop ncfsyn - Loop shaping design using Glover-McFarlane method rncf - Right normalized coprime factorization robgain - Robust performance of uncertain system robstab - Robust stability of uncertain system robustperf - (Not recommended) Robust performance margin of uncertain multivariable system robuststab - (Not recommended) Calculate robust stability margins of uncertain multivariable system sdhinfnorm - Compute L_{2} norm of continuous-time system in feedback with discrete-time system sdhinfsyn - Compute H_{∞} controller for sampled-data system sdlsim - Time response of sampled-data feedback system slowfast - (Not recommended) Slow and fast modes decomposition ucover - Fit uncertain model to set of LTI responses wcdiskmargin - Worst-case disk-based stability margins of uncertain feedback loops wcdiskmarginplot - Visualize worst-case disk-based stability margins wcgain - Worst-case gain of uncertain system wcsigmaplot - Plot worst-case gain of uncertain system conj - Form model with complex conjugate coefficients ctranspose - Conjugate dynamic system model squeeze - Remove singleton dimensions for umat objects permute - Rearrange array dimensions in model arrays isfinite - Determine if model has finite coefficients isreal - Determine if model has real-valued coefficients hasInternalDelay - Determine if model has internal delays hasdelay - True for linear model with time delays isstatic - Determine if model is static or dynamic isempty - Determine whether dynamic system model is empty issiso - Determine if dynamic system model is single-input/single-output (SISO) ndims - Query number of dimensions of dynamic system model or model array nblocks - Number of control design blocks in generalized LTI model or generalized matrix nmodels - Number of models in model array isParametric - Determine if model has tunable parameters append - Group models by appending their inputs and outputs blkdiag - Block-diagonal concatenation of models get - Access model property values getValue - Current value of generalized model repmat - Replicate and tile array reshape - Change shape of model array size - Query output/input/array dimensions of input–output model and number of frequencies of FRD model stack - Build model array by stacking models or model arrays along array dimensions voidModel - Mark missing or irrelevant models in model array feedback - Feedback connection of multiple models getBlockValue - Get current value of Control Design Block in Generalized Model imp2exp - Convert implicit linear relationship to explicit input-output relation inv - Invert dynamic system models lft - Generalized feedback interconnection of two models (Redheffer star product) parallel - Parallel connection of two models replaceBlock - Replace or update control design blocks in generalized model rsampleBlock - Randomly sample Control Design blocks in generalized model sampleBlock - Sample Control Design blocks in generalized model series - Series connection of two models setBlockValue - Modify value of Control Design Block in Generalized Model showBlockValue - Display current values of Control Design Blocks in Generalized Model showTunable - Display current value of tunable Control Design Blocks in Generalized Model sminreal - Eliminates structurally disconnected states, delays, and blocks getNominal - Nominal value of uncertain model gridureal - Grid ureal parameters uniformly over their range lftdata - Decompose uncertain objects into fixed certain and normalized uncertain parts simplify - Simplify representation of uncertain object uscale - Scale uncertainty of block or system usubs - Substitute given values for uncertain elements of uncertain objects Examples Continuous-Time SISO Zero-Pole-Gain Model Discrete-Time SISO Zero-Pole-Gain Model Concatenate SISO Zero-Pole-Gain Models into a MIMO Zero-Pole-Gain Model Discrete-Time MIMO Zero-Pole-Gain Model Specify Input Names for Zero-Pole-Gain Model Continuous-Time Zero-Pole-Gain Model Using Rational Expression Discrete-Time Zero-Pole-Gain Model Using Rational Expression Zero-Pole-Gain Model with Inherited Properties Static Gain MIMO Zero-Pole-Gain Model Convert State-Space Model to Zero-Pole-Gain Model Array of Zero-Pole-Gain Models Extract Zero-Pole-Gain Models from Identified Model Zero-Pole-Gain Model with Input and Output Delay Control Design Using Zero-Pole-Gain Models Compute Truncated ZPK Approximation of Sparse Model See also filt, frd, get, set, ss, zpkdata, genss, tf, realp, genmat Introduced in Control System Toolbox before R2006a Documentation for zpk Other uses of zpk

--- help for DynamicSystem/pole --- DynamicSystem/pole - Poles of dynamic system This MATLAB function returns the poles of the SISO or MIMO dynamic system model sys. Syntax P = pole(sys) P = pole(sys,J1,...,JN) P = pole(___,Name=Value) Input Arguments sys - Dynamic system dynamic system model | model array J1,...,JN - Indices of models in array whose poles you want to extract positive integer Name-Value Arguments Focus - Frequency range of interest [0 Inf] (default) | vector MaxNumber - Maximum number of poles and zeros to compute 1000 (default) | positive integer Shift - Spectral shift 0 (default) | finite scalar Tolerance - Accuracy of computed poles 1e-12 (default) | positive finite scalar Display - Show or hide progress report "on" (default) | "off" Output Arguments P - Poles of the dynamic system column vector | array Examples Poles of Discrete-Time Transfer Function Poles of Transfer Function Poles of Models in an Array Compute Poles of Sparse Models See also damp, esort, dsort, pzmap, zero, step, impulse, pzplot Introduced in Control System Toolbox before R2006a Documentation for DynamicSystem/pole

feedback - Feedback connection of multiple models This MATLAB function returns a model object sys for the negative feedback interconnection of model objects sys1,sys2. Syntax sys = feedback(sys1,sys2) sys = feedback(sys1,sys2,feedin,feedout) sys = feedback(sys1,sys2,'name') sys = feedback(___,sign) Input Arguments sys1,sys2 - Systems to connect in a feedback loop dynamic system models feedin - Subset of inputs to be used vector feedout - Subset of outputs to be used vector sign - Type of feedback -1 (default) | +1 Output Arguments sys - Closed-loop system dynamic system model Examples Plant and Controller with Unit Feedback Plant with Controller in Negative Feedback Path Positive Feedback Loop with Plant and Controller Negative Feedback Loop with MIMO Systems Feedback Loop Based on I/O Names Specify Input and Output Connections in a Feedback Loop See also append, connect, series, lft, sumblk, parallel Introduced in Control System Toolbox before R2006a Documentation for feedback Other uses of feedback

series - Series connection of two models This MATLAB function forms the series interconnection of two dynamic system models by connecting the outputs of sys1 to the inputs of sys2, as shown in the diagram. Syntax sys = series(sys1,sys2) sys = series(sys1,sys2,out1,in2) sys = series(sys1,sys2,"name") Input Arguments sys1 - Systems to connect dynamic system model | control design block | model array sys2 - Systems to connect dynamic system model | control design block | model array out1 - Outputs to connect vector in2 - Inputs to connect vector Output Arguments sys - Interconnected system dynamic system model Examples Connect SISO Systems in Series Connect MIMO Systems in Series Connect Subset of Outputs and Inputs Using Signal Names Connect Subset of Outputs and Inputs Using Indices See also feedback, connect, parallel, append Introduced in Control System Toolbox before R2006a Documentation for series Other uses of series

Parallel Computing Toolbox Version 25.1 (R2025a) 21-Nov-2024 Toolbox Functions For Clusters batch - Run MATLAB function or script on worker fetchCloudClusters - Returns an array of your cloud clusters gcp - Returns the current parallel pool parcluster - Build a Cluster from a cluster profile parfeval - Execute function on worker in parallel pool parfor - Execute for loop in parallel on workers in parallel pool parpool - Create a parallel pool of workers on a cluster and return a pool object spmd - Single Program Multiple Data Other Toolbox Contents parallel.cluster - Use a cluster parallel.gpu - Use your computer's graphics processing unit (GPU) parallel/lang - Parallel computing programming language constructs parallel/mpi - Using the Message Passing Interface array/distributed - Parallel Algorithms See also demo toolbox parallel. parallel is both a directory and a function. parallel - Create parallel sum filter structure This MATLAB function returns an object PF of type dsp.ParallelFilter. Syntax PF = parallel(branch1,branch2,...,branchn) Input Arguments branch - Filter branch filter System object | scalar Output Arguments PF - Parallel filter dsp.ParallelFilter System object Examples Construct Parallel Filter See also dsp.ParallelFilter Introduced in DSP System Toolbox before R2006a Documentation for parallel Parallel Computing Toolbox Documentation Other uses of parallel Folders named parallel

connect - Block diagram interconnections of dynamic systems This MATLAB function connects the models and block diagram elements sys1,...sysN based on signal names. Syntax sysc = connect(sys1,...sysN,inputs,outputs) sysc = connect(sys1,...sysN,inputs,outputs,APs) sysc = connect(blksys,connections,inputs,outputs) sysc = connect(___,opts) Input Arguments sys - Dynamic system model or other element to interconnect LTI model object | control design block | AnalysisPoint block | identified model | sparse model object | time-varying model object | parameter-varying model object | time-varying model object inputs - Inputs of combined model character vector | cell array of character vectors | string | string vector | positive integer | vector of positive integers outputs - Outputs of combined model character vector | cell array of character vectors | string | string vector | positive integer | vector of positive integers APs - Locations of interest in the combined model character vector | cell array of character vectors | string | string vector blksys - Unconnected aggregate model dynamic system model connections - Connections and summing junctions matrix opts - Additional options for interconnection connectOptions options set Output Arguments sysc - Interconnected system state-space model | frequency-response data model Examples SISO Control System MIMO Control System Feedback Loop with Analysis Point Index-Based Interconnection Combine Two Interconnected Models with Analysis Points See also sumblk, AnalysisPoint, append, feedback, parallel, series, lft, connectOptions Introduced in Control System Toolbox before R2006a Documentation for connect Other uses of connect

initial - System response to initial states of state-space model For state-space and sparse state-space models, initial computes the unforced system response y to initial states xinit. Syntax [y,tOut] = initial(sys,IC) [y,tOut] = initial(sys,IC,t) [y,tOut] = initial(sys,IC,t,p) [y,tOut,x] = initial(___) [y,tOut,x,pOut] = initial(sys,IC,t,p) initial(___) Input Arguments sys - Dynamic state-space model dynamic state-space model | model array t - Time steps positive scalar | two-element vector | vector | [] IC - Initial condition vector | RespConfig object | operating condition created with findop p - LPV model parameter trajectory matrix | function handle Output Arguments y - Response data array tOut - Times at which response is computed vector x - State trajectories array pOut - Parameter trajectories array Examples Initial Conditions Response Plot Response of State-Space Model to Initial Condition Initial Condition Response Plot of MIMO System Initial Conditions Response Plot at Specified Time Initial Condition Responses of Multiple Systems Initial Condition Response Data of State-Space Model Initial Condition Response Data with Specified Time Initial Response of Linear-Parameter Varying State-Space Model Compute Initial-Condition Response of Model with Complex Coefficients See also initialplot, impulse, lsim, Linear System Analyzer, step Introduced in Control System Toolbox before R2006a Documentation for initial Other uses of initial