MATLAB and its Control ToolboxOutlineMATLABSlide 4MATLAB ToolboxesMATLAB and ControlSlide 7Slide 8Slide 9Control System ToolboxSlide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18State-Space ModelsSlide 20Slide 21Conversion between different modelsModel DynamicsTime Response of SystemsSlide 25Slide 26Slide 27Frequency Domain Analysis and DesignSlide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 4401/14/19 MATLAB 1MATLAB MATLAB and ToolboxesMATLAB and ControlControl System ToolboxSimulink01/14/19 MATLAB Control Toolbox 2Aerospace and DefenseAutomotiveBiotech, Medical, and PharmaceuticalChemical and PetroleumCommunicationsComputers and Office EquipmentEducationElectronics and SemiconductorFinancial ServicesIndustrial Equipment and MachineryInstrumentationUtilities and Energy01/14/19 MATLAB Control Toolbox 301/14/19 MATLAB Control Toolbox 4The MathWorks Product Family01/14/19 MATLAB Control Toolbox 5MATLABMath and optimizationToolboxesOptimization Symbolic Math Partial Diff. Eq.…Signal Processingand communicationsToolboxesSignal Processing Communications Filter Design Filter Design HDL Coder …SimulinkProduct FamilySimulink® Simulink Accelerator Simulink Report Generator ……..Control System Design and AnalysisToolboxesSimulink Control Design Simulink ResponseSimulink Parameter …01/14/19 MATLAB Control Toolbox 6MATLAB-Toolboxes for ControlLinear ControlControl System ToolboxSimulink®Mu ToolboxNonlinear ControlNonlinear Control ToolboxFuzzy ToolboxSimulink®Identification Identification ToolboxFrequency-Domain ID ToolboxSimulink®Control Design Process01/14/19 MATLAB Control Toolbox 7Modeling Tools01/14/19 MATLAB Control Toolbox 8Design and Analysis01/14/19 MATLAB Control Toolbox 9Core FeaturesTools to manipulate LTI modelsClassical analysis and designBode, Nyquist, Nichols diagramsStep and impulse responseGain/phase marginsRoot locus designModern state-space techniquesPole placementLQG regulation01/14/19 MATLAB Control Toolbox 10LTI Objects (Linear Time Invariant)4 basic types of LTI modelsTransfer Function (TF)Zero-pole-gain model (ZPK)State-Space models (SS)Frequency response data model (FRD)Conversion between modelsModel properties (dynamics)01/14/19 MATLAB Control Toolbox 1111 2 111 1 11 2 11 1 1...( )..., ... numerator coefficients, ... denominator coefficientsn nnm mmnmp s p s pH sq s q s qwherep p pq q q-+-++++ + +=+ + +01/14/19 MATLAB Control Toolbox 12Transfer FunctionConsider a linear time invariant (LTI) single-input/single-output systemApplying Laplace Transform to both sides with zero initial conditions'' 6 ' 5 4 ' 3y y y u u+ + = +2( ) 4 3( )( )6 5Y s sG sU ss s+= =+ +01/14/19 MATLAB Control Toolbox 13Transfer Function01/14/19 MATLAB Control Toolbox 14>> num = [4 3];>> den = [1 6 5];>> sys = tf(num,den)Transfer function: 4 s + 3-----------------s^2 + 6 s + 5Transfer Function>> [num,den] = tfdata(sys,'v') num = 0 4 3 den = 1 6 501/14/19 MATLAB Control Toolbox 15Zero-pole-gain model (ZPK)1 21 21 2 11 1 1( )( ) ... ( )( )( )( ) ... ( ), ... the zeros of H(s), ... the poles of H(s)nmnms p s p s pH s Ks q s q s qwherep p pq q q++- - + + -=- - + + -'' 6 ' 5 4 ' 3y y y u u+ + = +01/14/19 MATLAB Control Toolbox 16Consider a Linear time invariant (LTI) single-input/single-output systemApplying Laplace Transform to both sides with zero initial conditions2( ) 4 3 4( 0.75)( )( ) ( 1)( 5)6 5Y s s sG sU s s ss s+ += = =+ ++ +Zero-pole-gain model (ZPK)01/14/19 MATLAB Control Toolbox 17Zero-pole-gain model (ZPK)>> sys1 = zpk(-0.75,[-1 -5],4) Zero/pole/gain:4 (s+0.75)-----------(s+1) (s+5)>> [ze,po,k] = zpkdata(sys1,'v')ze = -0.7500po = -1 -5k = 4. state vector input and output vectors, , state-space matricesx A x B uy C x D uwherexu and yA B C and D= += +01/14/19 MATLAB Control Toolbox 18State-Space Model (SS)Consider a Linear time invariant (LTI) single-input/single-output systemState-space model for this system is'' 6 ' 5 4 '' 3y y y u u+ + = +1 12 2'0 1 0'5 6 1x xux x� � � �� � ��= +� � � �� � ��- -� � ��� � � �01/14/19 MATLAB Control Toolbox 1912(0)0(0)0xx� ���=� ������ �[ ]123 4xyx� �=� �� �Control System Toolbox>> sys = ss([0 1; -5 -6],[0;1],[3,4],0) a = x1 x2 x1 0 1 x2 -5 -6 b = u1 x1 0 x2 101/14/19 MATLAB Control Toolbox 20c = x1 x2 y1 3 4 d = u1 y1 0 State-Space ModelsState Space Modelsrss, drss - Random stable state-space models.ss2ss - State coordinate transformation.canon - State-space canonical forms.ctrb - Controllability matrix.obsv - Observability matrix.gram - Controllability and observability gramians.ssbal - Diagonal balancing of state-space realizations.balreal - Gramian-based input/output balancing.modred - Model state reduction.minreal - Minimal realization and pole/zero cancellation.sminreal - Structurally minimal realization.01/14/19 MATLAB Control Toolbox 2101/14/19 MATLAB Control Toolbox 22Transfer functionState SpaceZero-pole-gaintf2ssss2tftf2zpzp2tfss2zpzp2sspzmap: Pole-zero map of LTI models.pole, eig - System poleszero - System (transmission) zeros.dcgain: DC gain of LTI models.bandwidth - System bandwidth.iopzmap - Input/Output Pole-zero map.damp - Natural frequency and damping of systemesort - Sort continuous poles by real part.dsort - Sort discrete poles by magnitude.covar - Covariance of response to white noise.01/14/19 MATLAB Control Toolbox 23Impulse Response (impulse)Step Response (step) General Time Response (lsim)Polynomial multiplication (conv)Polynomial division (deconv)Partial Fraction Expansion (residue)gensig - Generate input signal for lsim.01/14/19 MATLAB Control Toolbox 24Control System Toolbox01/14/19 MATLAB Control Toolbox 25The impulse response of a system is its output when the input is a unit impulse.The step response of a system is its output when the input is a unit step.The general response of a system to any input can be computed using the lsim command.Control System Toolbox1s5s4s22s3)s(G23)t5.0sin()t(u 01/14/19 MATLAB Control Toolbox 26ProblemProblem Given the LTI systemGiven the LTI