TP model transformation in control theory

Key takeaways

TP model transformation in control theory belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect TP model transformation in control theory to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of TP model transformation in control theory from memory before moving on to harder problems.

Reference excerpt

Baranyi and Yam proposed the TP model transformation as a new concept in quasi-LPV (qLPV) based control, which plays a central role in the highly desirable bridging between identification and polytopic systems theories. It is also used as a TS (Takagi-Sugeno) fuzzy model transformation. It is uniquely effective in manipulating the convex hull of polytopic forms (or TS fuzzy models), and, hence, has revealed and proved the fact that convex hull manipulation is a necessary and crucial step in achieving optimal solutions and decreasing conservativeness in modern linear matrix inequality based control theory. Thus, although it is a transformation in a mathematical sense, it has established a conceptually new direction in control theory and has laid the ground for further new approaches towards optimality. For details please visit: TP model transformation.

TP-tool MATLAB toolbox

A free MATLAB implementation of the TP model transformation can be downloaded at [1] or an old version of the toolbox is available at MATLAB Central [2]. Be careful, in the MATLAB toolbox the assignments of the dimensions of the core tensor is in the opposite way in contrast to the notation used in the related literature. In some variants of the ToolBox, the first two dimension of the core tensor is assigned to the vertex systems. In the TP model literature the last two. A simple example is given below.

clear M1=20; % Grid density M2=20; omega1=[-1,1]; %Interval omega2=[-1,1]; domain=[omega1; omega2];

for m1=1:M1 for m2=1:M2 p1=omega1(1)+(omega1(2)-omega1(1))/M1*(m1-1); %sampling grid p2=omega2(1)+(omega2(2)-omega2(1))/M2*(m2-1); SD(m1,m2,1,:)=[1 0]; % SD is the discretized system matrix SD(m1,m2,2,:)=[(-1-0.67*p1*p1) (1.726*p2*p2)]; end end

[S,U, sv]=hosvd(SD,[1,1,0,0],1e-12); % Finding the TP structure UA{1}=U{1}; % This is the HOSVD based canonical form UA{2}=U{2}; ns1 = input('Results of SNNN TS fuzzy model'); UC=genhull(UA,'snnn'); % snnn weightinf functions UCP{1}=pinv(UC{1}); UCP{2}=pinv(UC{2}); SC=tprods(SD,UCP); %This is to find the core tensor H(:,:)=SC(1,1,:,:) %This is to show the vertices of the TP model H(:,:)=SC(1,2,:,:) H(:,:)=SC(2,1,:,:) H(:,:)=SC(2,2,:,:) figure(1) hold all plothull(U{1}, omega1) %Draw the waiting functions of p1 title('Weighting functions for p_{1}'); xlabel('p_{1}') ylabel('Weighting functions')

grid on box on

figure(2) hold all plothull(UC{2}, omega2) %Show the waiting functions of p2 title('Weighting functions for p_{2}'); xlabel('p_{2}') ylabel('Weighting functions')

grid on box on

ns2 = input('Results of CNO TS fuzzy model'); UC=genhull(UA,'cno'); %Create CNO type waiting functions UCP{1}=pinv(UC{1}); UCP{2}=pinv(UC{2}); SC=tprods(SD,UCP); %Find the cortensor H(:,:)=SC(1,1,:,:) %Show the vertices of the TP model H(:,:)=SC(1,2,:,:) H(:,:)=SC(2,1,:,:) H(:,:)=SC(2,2,:,:) figure(1) hold all plothull(U{1}, omega1) %Show the waiting functions of p1 title('Weighting functions for p_{1}'); xlabel('p_{1}') ylabel('Weighting functions')

grid on box on figure(2) hold all plothull(UC{2}, omega2) %Show the waiting functions of p2 title('Weighting functions for p_{2}'); xlabel('p_{2}') ylabel('Weighting functions')

Once you have the feedback vertexes derived to each vertexes of the TP model then you may want to calculate the controller over the same polytope (see PDC design by Tanaka) W = queryw1(UC,domain,p); % computing the weighting values over the parameter vector F = tprods(K,W); % calculating the parameter dependent feedback F(p) F = shiftdim(F) U=-F*x % calculate the control value.

Key features for control analysis and design The TP model transformation transforms a given qLPV model into a (tensor product type) polytopic form, irrespective of whether the model is given in the form of analytical equations resulting from physical considerations, or as an outcome of soft computing based identification techniques (such as neural networks or fuzzy logic based methods, or as a result of a black-box identification). Further the TP model transformation is capable of manipulating the convex hull defined by the polytopic form that is a necessary step in polytopic qLPV model-based control analysis and design theories.

Related definitions Linear Parameter-Varying (LPV) state-space model

( x ˙ ( t ) y ( t ) ) = S ( p ( t ) ) ( x ( t ) u ( t ) ) , {\displaystyle {\begin{pmatrix}{\mathbf {\dot {x}} }(t)\\{\mathbf {y} }(t)\end{pmatrix}}={\mathbf {S} }({\mathbf {p} }(t)){\begin{pmatrix}{\mathbf {x} }(t)\\{\mathbf {u} }(t)\end{pmatrix}},}

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Worked examples

Example 1 — a first encounter with TP model transformation in control theory

Start with the simplest possible case. Write down what TP model transformation in control theory claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to TP model transformation in control theory before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about TP model transformation in control theory ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of TP model transformation in control theory

In research: TP model transformation in control theory appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses TP model transformation in control theory in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: TP model transformation in control theory is common in secondary-school and first-year university syllabi. It links to neighbouring topics Control theory, so understanding it makes those chapters shorter.
In everyday life: Look for TP model transformation in control theory outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study TP model transformation in control theory in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what TP model transformation in control theory means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain TP model transformation in control theory out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is TP model transformation in control theory in simple terms?

Baranyi and Yam proposed the TP model transformation as a new concept in quasi-LPV (qLPV) based control, which plays a central role in the highly desirable bridging between identification and polytopic systems theories. It is also used as a TS (Takagi-Sugeno) fuzzy model transformation.

Why does TP model transformation in control theory matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study TP model transformation in control theory?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on TP model transformation in control theory.