登录建筑培训网优化网站标题是什么意思
一、文章摘要
状态空间模型（State-Space Model, SS）是一种广泛应用于控制系统、信号处理和系统建模的数学表示方式。MATLAB 提供的 ss 类用于描述线性时不变（LTI）系统的状态空间表示。本类实现了 LTI 系统的构造、属性设置、变换以及数学运算，并支持连续和离散系统的建模。该类允许用户创建模型、访问状态变量、执行系统转换（如最小实现或显式转换）等功能。
本文将详细介绍 ss 类的数学模型、关键公式，并对 MATLAB 代码进行逐行详细注释，以帮助读者理解 ss 的工作原理及其在优化和建模中的应用。
二、状态空间模型的数学定义
状态空间模型由以下微分方程（连续时间）或差分方程（离散时间）描述：
1. 连续时间状态空间方程
2. 离散时间状态空间方程
三、MATLAB ss 类的详细解析
MATLAB 提供了 ss 类来构造和操作状态空间模型，该类支持：
- 创建单个状态空间模型或模型数组；
- 转换其他模型格式（如传递函数、零极点-增益）为状态空间形式；
- 计算状态空间系统的最小实现；
- 访问和修改系统矩阵 A,B,C,DA, B, C, DA,B,C,D 以及其他相关属性。
四、MATLAB ss 代码详细注释
1. ss 类的定义
classdef (CaseInsensitiveProperties, TruncatedProperties, SupportExtensionMethods, ...InferiorClasses = {? matlab.graphics.axis.Axes, ? matlab.ui.control.UIAxes}) ...ss < numlti & StateSpaceModel
代码解析
-
ss类继承了numlti和StateSpaceModel，意味着ss继承了 LTI 系统的一般行为，并特化为状态空间模型。 -
CaseInsensitiveProperties：允许不区分大小写地访问类属性。 -
TruncatedProperties：支持自动补全属性名。 -
SupportExtensionMethods：支持扩展方法。 -
InferiorClasses：定义ss对 MATLAB 其他类（如Axes和UIAxes）的优先级较低，使得在绘图时Axes类的行为不会被ss覆盖。
2. ss 类的构造函数
function sys = ss(varargin)
代码解析
-
ss构造函数支持多种输入方式，包括：-
直接传入 A,B,C,DA, B, C, DA,B,C,D 矩阵；
-
仅传入 DDD（表示纯增益系统）；
-
通过参数对形式 (
Name-Value Pairs) 进行系统初始化。
-
3. ss 类的属性定义
properties (Access = public, Dependent)ABCDEScaledStateNameStatePathStateUnitInternalDelay
end
代码解析
-
A, B, C, D：状态空间方程中的四个核心矩阵。 -
E：描述符状态空间模型的附加矩阵（可选）。 -
Scaled：控制是否自动缩放系统的状态变量以提高数值稳定性。 -
StateName：存储各个状态变量的名称（用于可读性）。 -
InternalDelay：用于存储系统内部的延迟信息（如输入、输出或状态的延迟）。
4. get 与 set 方法
function Value = get.A(sys)Value = localGetABCDE(sys.Data_,'a',[0,0]);
endfunction sys = set.A(sys,Value)sys = localSetABCDE(sys,'a',Value);
end
代码解析
-
get.A方法：获取A矩阵的值，调用localGetABCDE处理数据。 -
set.A方法：修改A矩阵的值，并调用localSetABCDE进行验证和数据更新。
5. 状态空间转换
function sys = convert(X,varargin)if isnumeric(X) || isa(X,'StaticModel')sys = ss(double(X)); % 处理静态模型elseif isa(X,'DynamicSystem')sys = copyMetaData(X,ss_(X,varargin{:})); % 处理动态系统elseerror('Invalid conversion to ss')end
end
代码解析
-
convert方法用于将其他类型的系统（如传递函数tf或零极点-增益zpk）转换为ss。 -
当输入为数值或静态模型时，直接转换为
ss。 -
当输入为动态系统时，复制元数据并调用
ss_()进行转换。
6. balred 方法（平衡降阶）
function [sys,INFO] = reduceBT(sys,orders,BalInfo,MatchDC)Dss = sys.Data_;Dr = repmat(Dss,numel(orders),1);for ct=1:numel(orders)Dr(ct) = balred(Dss,orders(ct),BalInfo,MatchDC);endsys.Data_ = Dr;
end
代码解析
-
reduceBT实现了 基于平衡截断（Balanced Truncation, BT） 的模型降阶（Model Reduction）。 -
orders代表降阶的目标维度。 -
balred进行平衡变换并截断低能量的状态。
7. 状态反馈与状态观测
function [K,CL,Gamma] = lqr(sys,Q,R,N)[K,~,Gamma] = lqr(sys.A,sys.B,Q,R,N);CL = ss(sys.A-sys.B*K,sys.B,sys.C,sys.D,sys.Ts);
end
代码解析
-
lqr计算最优状态反馈增益 KKK，并返回闭环系统CL及性能指标Gamma。
五、总结
总体代码：
classdef (CaseInsensitiveProperties, TruncatedProperties, SupportExtensionMethods, ...InferiorClasses = {? matlab.graphics.axis.Axes, ? matlab.ui.control.UIAxes}) ...ss < numlti & StateSpaceModel%SS State-space models.%% Construction:% SYS = SS(A,B,C,D) creates an object SYS representing the continuous- % time state-space model% dx/dt = Ax(t) + Bu(t)% y(t) = Cx(t) + Du(t)% You can set D=0 to mean the zero matrix of appropriate size. SYS is % of type SS when A,B,C,D are dense numeric arrays, of type GENSS when % A,B,C,D depend on tunable parameters (see REALP and GENMAT), and % of type USS when A,B,C,D are uncertain matrices (requires Robust % Control Toolbox). Use SPARSS when A,B,C,D are sparse matrices.%% SYS = SS(A,B,C,D,Ts) creates a discrete-time state-space model with% sample time Ts (set Ts=-1 if the sample time is undetermined).%% SYS = SS(D) specifies a static gain matrix D.%% You can set additional model properties by using name/value pairs.% For example,% sys = ss(-1,2,1,0,'InputDelay',0.7,'StateName','position')% also sets the input delay and the state name. Type "properties(ss)" % for a complete list of model properties, and type % help ss.<PropertyName>% for help on a particular property. For example, "help ss.StateName" % provides information about the "StateName" property.%% Arrays of state-space models:% You can create arrays of state-space models by using ND arrays for% A,B,C,D. The first two dimensions of A,B,C,D define the number of % states, inputs, and outputs, while the remaining dimensions specify % the array sizes. For example,% sys = ss(rand(2,2,3,4),[2;1],[1 1],0)% creates a 3x4 array of SISO state-space models. You can also use% indexed assignment and STACK to build SS arrays:% sys = ss(zeros(1,1,2)) % create 2x1 array of SISO models% sys(:,:,1) = rss(2) % assign 1st model% sys(:,:,2) = ss(-1) % assign 2nd model% sys = stack(1,sys,rss(5)) % add 3rd model to array%% Conversion:% SYS = SS(SYS) converts any dynamic system SYS to the state-space % representation. The resulting model SYS is always of class SS.%% SYS = SS(SYS,'min') computes a minimal realization of SYS.%% SYS = SS(SYS,'explicit') computes an explicit realization (E=I) of SYS.% An error is thrown if SYS is improper.%% See also DSS, DELAYSS, RSS, DRSS, SPARSS, MECHSS, SSDATA, TF, ZPK, FRD, GENSS, USS, DYNAMICSYSTEM.% Author(s): A. Potvin, P. Gahinet
% Copyright 1986-2020 The MathWorks, Inc.% Add static method to be included for compiler%#function ltipack.utValidateTs%#function ss.loadobj%#function ss.make%#function ss.convert%#function ss.checkABCDE% Public properties with restricted valueproperties (Access = public, Dependent)% State matrix A.%% Set this property to a square matrix with as many rows as states, for % example, sys.a = [-1 3;0 -5] for a second-order model "sys".A% Input-to-state matrix B.%% Set this property to a matrix with as many rows as states and as many % columns as inputs, for example, sys.b = [0;1] for a single-input, % second-order system "sys".B% State-to-output matrix C.%% Set this property to a matrix with as many rows as outputs and as many % columns as states, for example, sys.c = [1 -1] for a single-output, % second-order system "sys".C% Feedthrough matrix D.%% Set this property to a matrix with as many rows as outputs and as many % columns as inputs, for example, sys.d = [1 0] for a single-output, % two-input system "sys".D% E matrix for implicit (descriptor) state-space models.%% By default E=[], meaning that the state equation is explicit. To % specify an implicit state equation E dx/dt = A x + B u, set this% property to a square matrix of the same size as A. Note that E% may be singular, for example, when modeling a pure derivative% element in state-space form. See DSS for more details on descriptor% state-space models.E% Enables/disables auto-scaling (logical, default = false).%% When Scaled=false, most numerical algorithms acting on this system% automatically rescale the state vector to improve numerical accuracy.% You can disable such auto-scaling by setting Scaled=true. See PRESCALE% for more details on scaling issues.Scaled% State names (cell array of char vectors, default = '' for all states).%% This property can be set to:% * A char vector for first-order models, for example, 'position'% * A cell array of char vectors for models with two or more states,% for example, {'position' ; 'velocity'}% Use the empty char array '' for unnamed states.StateName% State paths (cell array of char vectors, default = '' for all states).%% In the linearization of a Simulink model, each state originates from% a particular Simulink block, and this property gives the full pathname% of the block associated with each state.StatePath% State units (cell array of char vectors, default = '' for all states).%% Use this property to keep track of the units each state is expressed in.% It can be set to:% * A char vector for first-order models, for example, 'm/s'% * A cell array of char vectors for models with two or more states,% for example, {'m' ; 'm/s'}StateUnit% Internal delays (numeric vector, default = []).%% Internal delays arise, for example, when closing feedback loops with% delays or connecting delay systems in series or parallel. See the% documentation for details. For continuous-time systems, internal delays% are expressed in the time unit specified by the "TimeUnit" property.% For discrete-time systems, internal delays are expressed as integer% multiples of the sampling period "Ts", for example, InternalDelay=3% means a delay of three sampling periods.%% You can modify the values of internal delays but the number of entries% in sys.InternalDelay cannot change (structural property of the model).InternalDelayend% OBSOLETE PROPERTIESproperties (Access = public, Dependent, Hidden)% Obsolete property for state-space models.IODelayMatrixend% TYPE MANAGEMENT IN BINARY OPERATIONSmethods (Static, Hidden)function T = toClosed(~)T = 'ss';endfunction T = superiorTypes()T = {'ss'};endfunction A = getAttributes(A)% Override default attributesA.Varying = false;A.Structured = false;A.FRD = false;A.Sparse = false;endfunction T = toStructured()% Use virtual class to ensure op(ss,uss)=ussT = 'neutral_genss';endfunction T = toVarying()T = 'ltvss';endfunction T = toFRD()T = 'frd';endfunction T = toSparse()T = 'sparss';endendmethodsfunction sys = ss(varargin)% String support, revisitvarargin = controllib.internal.util.hString2Char(varargin);ni = nargin;% Handle conversion SS(SYS) where SYS is a class with constructor named "ss"% (no ss() converter can be defined for such class)if ni>0 && isa(varargin{1},'StateSpaceModel')sys0 = varargin{1};tryswitch nicase 1if isa(sys0,'ss') % Optimization for SYS of class @sssys = sys0;elsesys = copyMetaData(sys0,ss_(sys0)); % e.g., ltiblock.ssendcase 2optflag = ltipack.matchKey(varargin{2},{'minimal','explicit'});sys = copyMetaData(sys0,ss_(sys0,optflag));otherwiseerror(message('Control:ltiobject:construct1','ss'))endreturncatch MEthrow(ME)endend% Dissect input listDataInputs = 0;LtiInput = 0;PVStart = ni+1;for ct=1:ninextarg = varargin{ct};if isa(nextarg,'struct') || isa(nextarg,'lti')% LTI settings inherited from other modelLtiInput = ct; PVStart = ct+1; breakelseif ischar(nextarg)PVStart = ct; breakelseDataInputs = DataInputs+1;endend% Handle bad callsif PVStart==1if ni==1% Bad conversionerror(message('Control:ltiobject:construct3','ss'))elseif ni>0error(message('Control:general:InvalidSyntaxForCommand','ss','ss'))endelseif DataInputs>5error(message('Control:general:InvalidSyntaxForCommand','ss','ss'))end% Process numerical datatryswitch DataInputscase 0if nierror(message('Control:ltiobject:construct4','ss'))else% Empty modela = []; b = []; c = []; d = [];endcase 1% Gain matrixa = [];d = ss.checkABCDE(varargin{1},'D');[Ny,Nu,~] = size(d);b = zeros(0,Nu);c = zeros(Ny,0);% Optimization for fast pre-allocationCheckData = ~allfinite(d);% Handle ss(1,'explicit')if ni==2 && ~isempty(ltipack.matchKey(varargin{2},{'minimal','explicit'}))ni = ni-1; % ignore flagendcase {2,3}error(message('Control:general:InvalidSyntaxForCommand','ss','ss'))otherwise% A,B,C,D specified: validate dataa = ss.checkABCDE(varargin{1},'A');b = ss.checkABCDE(varargin{2},'B');c = ss.checkABCDE(varargin{3},'C');d = ss.checkABCDE(varargin{4},'D');CheckData = true;end% Sample timeif DataInputs==5% Discrete SSTs = ltipack.utValidateTs(varargin{5});elseTs = 0;endcatch MEthrow(ME)end% Determine I/O and array sizeif ni>0Ny = max(size(c,1),size(d,1));Nu = max(size(b,2),size(d,2));ArraySize = ltipack.getLTIArraySize(2,a,b,c,d);if isempty(ArraySize)error(message('Control:ltiobject:ss1'))endelseNy = 0; Nu = 0; ArraySize = [1 1];endNsys = prod(ArraySize);sys.IOSize_ = [Ny Nu];% Create @ssdata object array% RE: Inlined for optimal speedif Nsys==1Data = ltipack.ssdata(a,b,c,d,[],Ts);elseData = ltipack.ssdata.array(ArraySize);Delay = ltipack.utDelayStruct(Ny,Nu,true);for ct=1:NsysData(ct).a = a(:,:,min(ct,end));Data(ct).b = b(:,:,min(ct,end));Data(ct).c = c(:,:,min(ct,end));Data(ct).d = d(:,:,min(ct,end));Data(ct).e = [];Data(ct).Ts = Ts;Data(ct).Delay = Delay;endendsys.Data_ = Data;% Process additional settings and validate system% Note: Skip when just constructing empty instance for efficiencyif ni>0try% User-defined propertiesSettings = cell(1,0);if LtiInput% Properties inherited from other systemarg = varargin{LtiInput};if isa(arg,'lti')arg = getSettings(arg);end% @ss does not inherit internal delays (including IODelay)arg = rmfield(arg,intersect(fieldnames(arg),{'IODelay','FrequencyUnit'}));Settings = [Settings , lti.struct2pv(arg)];end[pvpairs,iodSettings] = LocalCheckDelaySettings(varargin(:,PVStart:ni));Settings = [Settings , pvpairs];% Apply settings except IODelayif ~isempty(Settings)sys = fastSet(sys,Settings{:});end% Consistency checkif CheckData || ~isempty(Settings)sys = checkConsistency(sys);end% I/O delay settings. Must be done after data checks to prevent errors in% setIODelay when A,B,C are not properly formatted (e.g., A=B=C=[] and D=1)if ~isempty(iodSettings)sys = fastSet(sys,iodSettings{:});endcatch MEthrow(ME)endendend%---------------- GET/SET ------------------------------------------function Value = get.A(sys)% GET method for a propertyValue = localGetABCDE(sys.Data_,'a',[0,0]);endfunction Value = get.B(sys)% GET method for b propertyValue = localGetABCDE(sys.Data_,'b',[0,sys.IOSize_(2)]);endfunction Value = get.C(sys)% GET method for c propertyValue = localGetABCDE(sys.Data_,'c',[sys.IOSize_(1),0]);endfunction Value = get.D(sys)% GET method for d propertyValue = localGetABCDE(sys.Data_,'d',sys.IOSize_);endfunction Value = get.E(sys)% GET method for e propertyValue = localGetABCDE(sys.Data_,'e',[0,0]);endfunction sys = set.A(sys,Value)% SET method for a propertysys = localSetABCDE(sys,'a',Value);endfunction sys = set.B(sys,Value)% SET method for b propertysys = localSetABCDE(sys,'b',Value);endfunction sys = set.C(sys,Value)% SET method for c propertysys = localSetABCDE(sys,'c',Value);endfunction sys = set.D(sys,Value)% SET method for d propertysys = localSetABCDE(sys,'d',Value);endfunction sys = set.E(sys,Value)% SET method for e property (cannot change state size)Value = ss.checkABCDE(Value,'E');Data = ltipack.utCheckAssignValueSize(sys.Data_,Value,2);for ct=1:numel(Data)Data(ct).e = Value(:,:,min(ct,end));if sys.CrossValidation_Data(ct) = checkData(Data(ct)); % Quick validationendendsys.Data_ = Data;endfunction Value = get.Scaled(sys)% GET method for Scaled property% True if all models are scaled, false otherwiseValue = true;Data = sys.Data_;for ct=1:numel(Data)if ~Data(ct).ScaledValue = false; breakendendendfunction sys = set.Scaled(sys,Value)% SET method for Scaled propertyif ~(isscalar(Value) && (islogical(Value) || isnumeric(Value)))error(message('Control:ltiobject:setSS4'))endValue = logical(Value);Data = sys.Data_;for ct=1:numel(Data)Data(ct).Scaled = Value;endsys.Data_ = Data;endfunction Value = get.StateName(sys)% GET method for StateName propertyValue = cellstr(ltipack.SystemArray.getStateInfo(sys.Data_,'StateName'));endfunction Value = get.StatePath(sys)% GET method for StatePath propertyValue = cellstr(ltipack.SystemArray.getStateInfo(sys.Data_,'StatePath'));endfunction Value = get.StateUnit(sys)% GET method for StateUnit propertyValue = cellstr(ltipack.SystemArray.getStateInfo(sys.Data_,'StateUnit'));endfunction sys = set.StateName(sys,Value)% SET method for StateName propertyData = sys.Data_;nsys = numel(Data);if nsys>0Value = ltipack.mustBeStringVector(Value,'StateName',false);nx = size(Data(1).a,1);for ct=1:nsysif size(Data(ct).a,1)~=nx% Not supported for varying state dimensionerror(message('Control:ltiobject:setSS2'))endendfor ct=1:nsysData(ct).StateName = Value;if sys.CrossValidation_Data(ct) = checkData(Data(ct));endendsys.Data_ = Data;endendfunction sys = set.StatePath(sys,Value)% SET method for StatePath propertyData = sys.Data_;nsys = numel(Data);if nsys>0Value = ltipack.mustBeStringVector(Value,'StatePath',false);% Replace carriage returns by blanksValue = regexprep(Value,'\n',' ');nx = size(Data(1).a,1);for ct=1:nsysif size(Data(ct).a,1)~=nx% Not supported for varying state dimensionerror(message('Control:ltiobject:setSS2'))endendfor ct=1:nsysData(ct).StatePath = Value;if sys.CrossValidation_Data(ct) = checkData(Data(ct));endendsys.Data_ = Data;endendfunction sys = set.StateUnit(sys,Value)% SET method for StateUnit propertyData = sys.Data_;nsys = numel(Data);if nsys>0Value = ltipack.mustBeStringVector(Value,'StateUnit',false);nx = size(Data(1).a,1);for ct=1:nsysif size(Data(ct).a,1)~=nx% Not supported for varying state dimensionerror(message('Control:ltiobject:setSS5'))endendfor ct=1:nsysData(ct).StateUnit = Value;if sys.CrossValidation_Data(ct) = checkData(Data(ct));endendsys.Data_ = Data;endendfunction Value = get.InternalDelay(sys)% GET method for InternalDelay propertyData = sys.Data_;Nsys = numel(Data);if Nsys==0Value = zeros(0,1);elseif Nsys==1Value = Data.Delay.Internal;elseRefValue = Data(1).Delay.Internal;ndf = length(RefValue);Value = zeros([ndf 1 size(Data)]);isUniform = true;for ct=1:NsysDf = Data(ct).Delay.Internal;isUniform = isUniform && isequal(Df,RefValue);if length(Df)==ndfValue(:,ct) = Df;elseerror(message('Control:ltiobject:get5'))endendif isUniformValue = Value(:,1);endendendfunction sys = set.InternalDelay(sys,Value)% SET method for InternalDelay propertyif ~(isnumeric(Value) && isreal(Value) && allfinite(Value) && all(Value(:)>=0))error(message('Control:ltiobject:setLTI1','InternalDelay'))elseValue = double(full(Value));endData = ltipack.utCheckAssignValueSize(sys.Data_,Value,2);for ct=1:numel(Data)NewValue = Value(:,:,min(ct,end));if numel(NewValue)~=numel(Data(ct).Delay.Internal)error(message('Control:ltiobject:setSS3'))elseif sys.CrossValidation_Data(ct).Delay.Internal = ...ltipack.util.checkInternalDelay(NewValue,Data(ct).Ts);elseData(ct).Delay.Internal = NewValue;endendsys.Data_ = Data;end function Value = get.IODelayMatrix(sys)% GET method for IODelay propertyValue = getIODelay(sys);endfunction sys = set.IODelayMatrix(sys,Value)% SET method for IODelay propertysys = setIODelay(sys,Value);endendmethods (Hidden)function sys = utSimplifyDelay(sys)% Replaces internal delays by input or output delays when possible% (used by SCD)sys.Data_ = simplifyDelay(sys.Data_);endfunction [sys,isProper] = ioderiv(sys,r)% Multiplies SYS by (s+r) or (z+r) (used by PID tool)[sys.Data_,isProper] = ioderiv(sys.Data_,r);endfunction sys = augmentMPCPlantModel(sys, b)% Add an "offset" signal to the plant model used by MPC object[ny, nu] = size(sys);data = sys.Data_;data = utGrowIO(data, ny, nu+1);data.b(:, nu+1) = b;sys.Data_ = data;endfunction [a,b1,b2,c1,c2,d11,d12,d21,d22,Ts] = titodata(P,nY,nU)% Extract data for two-input, two-output system% [y1;y2] = D * [u1;u2]% NY and NU are the sizes of y2 and u2.Dss = P.Data_;[a,b1,b2,c1,c2,d11,d12,d21,d22] = titodata(Dss,nY,nU);Ts = Dss.Ts;end% MUSYN supportfunction P = sector2gain(P,F,ny,nu)% Transforms the problem%% Find K such that [T1(jw);I]' Q(jw) [T1(jw);I] < 0, T1 = LFT(P1,K)%% into the HINFSYN problem%% Find K such that || T2 ||oo < 1 , T2 = LFT(P2,K) .%% T2 is related to T1 by%% T2 = Z1/Z2, [Z1;Z2] = F [T1;I]%% where%% Q(s) = F(s)' J F(s), J = diag(I,-I), F bi-stable%% is the J-spectral factorization of Q.P.Data_ = sector2gain(P.Data_,F.Data_,ny,nu);end% MOR APIfunction BalInfo = runBT(sys,Options)% Called by PROCESSBalInfo = hsvd(sys.Data_,Options);endfunction [sys,INFO] = reduceBT(sys,orders,BalInfo,MatchDC)% Called by GETROMnsys = numel(orders);Dss = sys.Data_;Dr = repmat(Dss,nsys,1);if nargout>1INFO = struct('PL',cell(nsys,1),'PR',[]);for ct=1:nsys[Dr(ct),INFO(ct)] = balred(Dss,orders(ct),BalInfo,MatchDC);endelsefor ct=1:nsysDr(ct) = balred(Dss,orders(ct),BalInfo,MatchDC);endendsys.Data_ = Dr;endfunction ModalInfo = runMT(sys,Options)% Called by PROCESSModalInfo = modsepMOR(sys.Data_,Options);endfunction [sys,INFO] = reduceMT(sys,select,ModalInfo,MatchDC)% Called by GETROMif isa(ModalInfo,'mor.util.ModalDecompositionCache')% Select modal componentsMC = ModalInfo.Component(select,:);d = ModalInfo.Feedthrough;if isempty(MC)Dr = createGain(sys.Data_,zeros(size(d)));elseDr = modsum(MC);endif MatchDCs = ModalInfo.DCFrequency; % complexX = ModalInfo.Component(~select);for ct=1:numel(X)d = d + evalfr(X(ct),s);endif ~allfinite(d)error(message('Control:transformation:MODALROM18'))endendDr.d = d;jsel = getSelectedCol(ModalInfo,select);INFO = struct('PL',ModalInfo.TL(:,jsel),'PR',ModalInfo.TR(:,jsel));else% Modal splitG = ModalInfo.SchurQZ;p = ModalInfo.Mode;e = p(select,:);% For real problems, selected modes should be self-conjugateif ModalInfo.REAL && ~isequal(sort(e(imag(e)>0)),sort(conj(e(imag(e)<0))))error(message('Control:transformation:MODALROM20'))end% Perform splitRegionFcn = @(p) 2-(ismember(p,e));[H,~,Info2] = modsepREG(G,2,RegionFcn,ltioptions.modalsep);Dr = H(1);% Adjust feedthroughif MatchDCDCcorr = evalfr(H(2),ModalInfo.DCFrequency);if ~allfinite(DCcorr)error(message('Control:transformation:MODALROM18'))endDr.d = G.d + DCcorr;elseDr.d = G.d;end% ProjectorsPL = ModalInfo.TL*Info2.TL';PR = ModalInfo.TR*Info2.TR;nsel = numel(e);INFO = struct('PL',PL(:,1:nsel),'PR',PR(:,1:nsel));endsys.Data_ = Dr;endendmethods (Access = protected)%% INPUTOUTPUTMODEL ABSTRACT INTERFACEfunction displaySize(sys,sizes)% Displays SIZE information in SIZE(SYS)ny = sizes(1);nu = sizes(2);nx = order(sys);if isempty(nx)nx = 0;endif length(sizes)==2disp(getString(message('Control:ltiobject:SizeSS1',ny,nu,nx)))elseArrayDims = sprintf('%dx',sizes(3:end));if all(nx(:)==nx(1))disp(getString(message('Control:ltiobject:SizeSS2',...ArrayDims(1:end-1),ny,nu,nx(1))))elsedisp(getString(message('Control:ltiobject:SizeSS3',...ArrayDims(1:end-1),ny,nu,min(nx(:)),max(nx(:)))))endendend%% DATA ABSTRACTION INTERFACEfunction sys = indexasgn_(sys,indices,rhs,ioSize,ArrayMask)% Data management in SYS(indices) = RHS.% ioSize is the new I/O size and ArrayMask tracks which% entries in the resulting system array have been reassigned.% Construct template initial value for new entries in system arrayD0 = ltipack.ssdata([],zeros(0,ioSize(2)),zeros(ioSize(1),0),...zeros(ioSize),[],sys.Ts);% Update datasys.Data_ = ltipack.reassignData(sys.Data_,indices,rhs.Data_,ioSize,ArrayMask,D0);% Update sampling gridif numel(indices)>2 && ~(isempty(sys.SamplingGrid_) && isempty(rhs.SamplingGrid_))sys.SamplingGrid_ = reassign(sys.SamplingGrid_,indices(3:end),rhs.SamplingGrid_,size(sys.Data_));endendfunction [rsys,Info] = balred_(sys,orders,Info,Options)% BALRED support (obsolete)R = reducespec(sys,'balanced');R.Options = mapOptions(R.Options,Options);if isempty(Info)R = process(R);elseR = setBalredInfo(R,Info);endrsys = getrom(R,Order=orders,Method=Options.StateProjection);Info = getBalredInfo(R);endfunction [sys,nx] = augstate_(sys)[sys,nx] = augstate_@ltipack.SystemArray(sys);sys = augmentOutput(sys,sys.StateName,sys.StateUnit,'states');endfunction [sys,offset,xkeep] = sminrealXC_(sys,offset)try[sys.Data_,offset,xkeep] = sminrealXC(sys.Data_,offset);catcherror(message('Control:transformation:sminreal2'))endendvsys = ssInterpolant_(asys,offsets,varargin)end%% STATIC METHODSmethods(Static, Hidden)sys = loadobj(s)function sys = make(D,IOSize)% Constructs SS model from ltipack.ssdata instancesys = ss;sys.Data_ = D;if nargin>1sys.IOSize_ = IOSize; % support for empty model arrayselsesys.IOSize_ = iosize(D(1));endendfunction sys = convert(X,varargin)% Safe conversion to SS.%% X = SS.CONVERT(X) safely converts the variable X to SS even when% X is a static model (in which case SS(X) returns a GENSS or USS % model rather than an SS model). This method is used in indexed % assignments and conversions from LTIBLOCK.SS to SS.% % X = SS.CONVERT(X,OPT) further specifies an option for the% DynamicSystem/ss converter.if isnumeric(X) || isa(X,'StaticModel')% Work around ss(GENMAT) yielding a GENSSsys = ss(double(X));elseif isa(X,'DynamicSystem')% DynamicSystem subclasssys = copyMetaData(X,ss_(X,varargin{:}));elseerror(message('Control:transformation:ss2',class(X)))endendfunction C = makeC(C,TU)% Applies ss.make to each entry of the cell array C.% Used by uncertainty analysis functionsfor ct=1:numel(C)V = C{ct};if isa(V,'ltipack.ssdata')C{ct} = setTimeUnit_(ss.make(V),TU);endendendfunction S = makeS(S,TU)% Applies ss.make to each field of a struct array S.% Used by uncertainty analysis functionsS = cell2struct(ss.makeC(struct2cell(S),TU),fieldnames(S),1);endfunction M = checkABCDE(M,MatrixName)% Checks A,B,C,D,E data is of proper typeif isnumeric(M)if ~(strcmp(MatrixName,'D') || allfinite(M))error(message('Control:ltiobject:ssProperties6',MatrixName))end% Convert to full doubleif issparse(M)warning(message('Control:ltiobject:SSSparse2Full',MatrixName))M = full(M);endM = double(M);elseif ~isa(M,'StaticModel')if isa(M,'InputOutputModel')error(message('Control:lftmodel:ss1',MatrixName))elseerror(message('Control:ltiobject:ssProperties6',MatrixName))endendendfunction [sys,offsets] = c2do(sys,offsets,Ts,options)% C2D discretization with offsets (assumes no delays)IMPULSE = strcmp(options.Method,'impulse');NOT_TUSTIN = ~strcmp(options.Method,'tustin');Data = sys.Data_;for ct=1:numel(Data)D = Data(ct);[nyz,nuw] = size(D.d);% Attach consolidated offsets[dx0,x0,u0,y0] = ltvpack.expandOffsets(offsets(ct).dx,offsets(ct).x,...offsets(ct).u,offsets(ct).y,size(D.a,1),nuw,nyz);if IMPULSE% Absorb input offset into dx,yD.f = dx0 - D.a * x0 - D.b * u0;D.g = y0 - D.c * x0 - D.d * u0;offsets(ct).u = [];else% Input offset unchanged for ZOH/TustinD.f = dx0 - D.a * x0;D.g = y0 - D.c * x0;end% ZOH/IMP require explicit model -> E must be invertible for% state consistencyif NOT_TUSTIND = des2exp(D,true);if ~isfinite(D)error(message('Control:transformation:c2d25'))endend% Discretize with offsetsD = c2d(D,Ts,options);% Detach and store discrete offsetsoffsets(ct).x = [];offsets(ct).dx = D.f;offsets(ct).y = D.g;D.f = []; D.g = [];Data(ct) = D;endsys.Data_ = Data;endfunction [sys,offsets] = d2co(sys,offsets,options)% D2C conversion with offsets (assumes no delays)sw = ctrlMsgUtils.SuspendWarnings; %#ok<NASGU>NOT_TUSTIN = ~strcmp(options.Method,'tustin');Data = sys.Data_;nsys = numel(Data);nx = zeros(nsys,1);for ct=1:nsysD = Data(ct);% Attach consolidated offsets (input offset unchanged)[dx0,x0,~,y0] = ltvpack.expandOffsets(offsets(ct).dx,offsets(ct).x,...[],offsets(ct).y,size(D.a,1),0,size(D.d,1));D.f = dx0 - D.a * x0;D.g = y0 - D.c * x0;% ZOH requires explicit model -> E must be invertible for% state consistencyif NOT_TUSTIND = des2exp(D,true);if ~isfinite(D)error(message('Control:transformation:c2d25'))endend% Convert with offsetsD = d2c(D,options);nx(ct) = size(D.a,1);% Detach and store continuous offsetsoffsets(ct).x = [];offsets(ct).dx = D.f;offsets(ct).y = D.g;D.f = []; D.g = [];Data(ct) = D;endsys.Data_ = Data;if NOT_TUSTIN && any(diff(nx(:)))error(message('Control:transformation:d2c13','D2C'))endendfunction [sys,offsets] = d2do(sys,offsets,Ts,options)% D2D resampling with offsets (assumes no delays)sw = ctrlMsgUtils.SuspendWarnings; %#ok<NASGU>NOT_TUSTIN = ~strcmp(options.Method,'tustin');Data = sys.Data_;nsys = numel(Data);nx = zeros(nsys,1);for ct=1:nsysD = Data(ct);% Attach consolidated offsets (input offset unchanged)[dx0,x0,~,y0] = ltvpack.expandOffsets(offsets(ct).dx,offsets(ct).x,...[],offsets(ct).y,size(D.a,1),0,size(D.d,1));D.f = dx0 - D.a * x0;D.g = y0 - D.c * x0;% ZOH requires explicit modelif NOT_TUSTIND = des2exp(D,true);if ~isfinite(D)error(message('Control:transformation:c2d25'))endend% Convert with offsetsD = d2d(D,Ts,options);nx(ct) = size(D.a,1);% Detach and store resampled offsetsoffsets(ct).x = [];offsets(ct).dx = D.f;offsets(ct).y = D.g;D.f = []; D.g = [];Data(ct) = D;endsys.Data_ = Data;if NOT_TUSTIN && any(diff(nx(:)))error(message('Control:transformation:d2c13','D2D'))endendendend%--------------------- Local Functions --------------------------------function [pvp,iodpvp] = LocalCheckDelaySettings(pvp)
% Pulls out IODelay settings, throws error when setting InternalDelays
if any(strncmpi(pvp(1:2:end),'int',3))error(message('Control:ltiobject:ss2'))
end
idx = find(strncmpi(pvp(1:2:end),'io',2));
iodpvp = cell(1,0);
for ct=length(idx):-1:1k = 2*idx(ct)-1;iodpvp = [iodpvp , pvp(k:min(k+1:end))]; %#ok<AGROW>pvp = [pvp(1:k-1) , pvp(k+2:end)];
end
endfunction Value = localGetABCDE(Data,ABCDE,DefaultSize)
% Return nominal values of A, B, C, D, or E
tmp = cell(1,6);
iselect = find(ABCDE=='abcd e');
ArraySize = size(Data);
if isempty(Data)% Empty arrayValue = zeros([DefaultSize,ArraySize]);
elseif isscalar(Data)[tmp{:}] = getABCDE(Data);Value = tmp{iselect};
elseValueArray = cell(ArraySize);for ct=1:numel(Data)[tmp{:}] = getABCDE(Data(ct));ValueArray{ct} = tmp{iselect};end% Replace E=[] by identity of proper size if some E's are non-emptyif strcmp(ABCDE,'e') && ~all(cellfun(@isempty,ValueArray(:)))for ct=1:numel(Data)if isempty(ValueArray{ct})ValueArray{ct} = eye(size(Data(ct).a));endendend% Turn into ND arraytryValue = cat(3,ValueArray{:});Value = reshape(Value,[size(Value,1) size(Value,2) ArraySize]);catch %#ok<CTCH>% A,B,C,E cannot be represented as ND arrayserror(message('Control:ltiobject:get4',upper(ABCDE)))end
end
end%%%%%%%%
function sys = localSetABCDE(sys,Property,Value)
% SET function for A,B,C,D
Value = ss.checkABCDE(Value,upper(Property));
% Check compatibility of RHS with model array sizes
Data = ltipack.utCheckAssignValueSize(sys.Data_,Value,2);
sv = size(Value);
SameSize = (~isempty(Data) && isequal(sv(1:2),size(Data(1).(Property))));
for ct=1:numel(Data)if isempty(Data(ct).Delay.Internal)Data(ct).(Property) = Value(:,:,min(ct,end));elseerror(message('Control:ltiobject:setSS1',Property))endif SameSize && sys.CrossValidation_Data(ct) = checkData(Data(ct)); % Quick validationend
end
sys.Data_ = Data;
if ~SameSize && sys.CrossValidation_% Note: Full validation needed because a single assignment can change I/O size,% e.g., sys = ss; sys.a = [1 2;3 4]% sys = ss(1); sys.d = [1 2;3 4]% sys = ss(1,[],[],[]); sys.b = 1;sys = checkConsistency(sys);
end
end
ss 类提供了一种高效的方式来创建和操作状态空间模型，它支持：
- 状态方程建模（适用于连续和离散系统）。
- 自动数值优化（如最小实现、显式变换）。
- LQR 反馈设计与模型降阶（如平衡截断）。